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{"url":"http:\/\/mathhelpforum.com\/algebra\/69158-divide-1-polynomial.html","text":"# Math Help - Divide 1 by a polynomial\n\n1. ## Divide 1 by a polynomial\n\nI know how to use synthetic division to divide a polynomial by one of lower degree, but how does one divide a poly. by one of higher degree?\n\nIn particular, a text I am studying says\n1 divided by x-1 =1 + x + x^2 + x^3 + ...\niff -1 < x < 1,\nbut I don't know how to find this out myself.\n\nHow do you divide 1 by x-1 ?\n\n2. Originally Posted by mnova\nI know how to use synthetic division to divide a polynomial by one of lower degree, but how does one divide a poly. by one of higher degree?\n\nIn particular, a text I am studying says\n1 divided by x-1 =1 + x + x^2 + x^3 + ...\niff -1 < x < 1,\nbut I don't know how to find this out myself.\n\nHow do you divide 1 by x-1 ?\nThis is done using the geometric series formula:\n\n$\\displaystyle \\sum_{k=0}^{\\infty} ar^k = \\frac{a}{1-r}$\n\nIn your case r = x, a = 1\n\n$\\displaystyle \\frac{1}{1-x} = \\sum_{k=0}^{\\infty} x^k = x^0 + x^1 + x^2 + x^3 + x^4 ...$\n\nAs you can see, the result goes on to infinity. There is no method of doing this in the same way that you can divide using synthetic division\/long division.","date":"2014-10-01 17:39:31","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\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"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\": 0.6561744213104248, \"perplexity\": 347.39606124630376}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2014-41\/segments\/1412037663467.44\/warc\/CC-MAIN-20140930004103-00133-ip-10-234-18-248.ec2.internal.warc.gz\"}"}
| null | null |
Q: Is there a way to pass an argument to a class constructor when used as a type parameter? I'm trying to make my code as re-usable as possible, and ran into a problem when trying to pass a constuctor argument to a class used as a type parameter.
What I currently have is this:
public sealed class SubmitForm : DerivedClass
{
}
public void TestMethod_Simulated()
{
var foo = GetObject<SubmitForm>();
SubmitData(foo);
}
private void SubmitData(SubmitForm form)
{
// Do work
}
public T GetObject<T>()
where T : class
{
// Work
}
This was working because there was no explicit constructor and parameter for SubmitForm. However I want to extend the functionality and require an explicit constructor expecting an argument.
After adding the constructor, I am looking to call "GetObject" with the type parameter "SubmitForm" INCLUDING an argument for its explicit constructor.
This is clearly not possible in any straightforward way, but I don't want to believe there is not way of getting this to work.
The following shows what I'm trying to achieve, which is to use the class as a type parameter while instantiating it in order to access the constructor I've created:
public sealed class SubmitForm : DerivedClass
{
public SubmitForm(EnumType typeVar)
{
_derivedVar = typeVar;
}
}
public void TestMethod_Simulated()
{
var foo = GetObject<new SubmitForm(EnumType.FormA)>();
SubmitData(foo);
}
private void SubmitData(SubmitForm form)
{
// Do work
}
public T GetObject<T>()
where T : class
{
// Work
}
I hope this makes sense. I can't seem to find a way to first instantiate the class AND then use it as a type parameter.
A:
I hope this makes sense. I can't seem to find a way to first
instantiate the class AND then use it as a type parameter
Well, it is not possible to pass an instance of a class as generic type parameter.
A type parameter is just a placeholder for type of object. Compiler uses it to infer the type and substitute T with a real type. It has nothing to do with actual parameters.
If you want to use a specific instance of SubmitForm class in your function you could first create an instance:
var submitForm = new SubmitForm(EnumType.FormA);
And then change your function to accept an argument of type T:
public T GetObject<T>(T instance)
where T : class
{
// .. do something with instance
}
And now you can make a call to the new overload:
var someResult = GetObject(submitForm);
However from your code sample it's not clear why you need generic type parameters at all. It seems that you might just as well change the signature of GetObject function from T GetObject<T>() to SubmitForm GetObject(SubmitForm form)
We usually use type parameters when we have a generic function that needs to execute more or less same logics for different types and we don't want to duplicate the same code.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The rotational evolution of solar-type stars, from the pre-main sequence to nearly the age of the sun, is increasingly well characterized \citep[for a recent review seen][]{Bouvier2013-rotation-evol-review}. Stars early on the pre-main sequence (PMS) strongly interact with their disks, and this regulates their rotation rates, despite accretion and contraction. Stars eventually stop strongly interacting with their disks while still on the PMS, and by conservation of angular momentum spin up. Stars also lose angular momentum, by the interaction of stellar winds and magnetic fields. This is a slower process than spin up due to contraction, thus once stars reach the main sequence they begin spinning down.
The magnetic evolution of young solar-type stars is less well characterized. In particular direct observations of the large-scale magnetic field are needed. The strong rotational evolution likely affects the dynamo generated magnetic fields in these stars, thus there should be large changes in the stellar magnetic properties. Additionally, the spin-down of these stars is controlled by the magnetic field, thus to fully understand the angular momentum loss, we must have well characterized large-scale magnetic properties \citep[e.g.][]{Vidotto2011-MHD-wind-V374Peg, Matt2012-magnetic-breaking-formulation, Reville2015-MHD-wind-torque, Reville2016-3D-MHD-wind-Jdot}. Indeed, for studies of stellar winds and angular momentum loss, it is the large-scale component of the magnetic field that is important \citep{Jardine2017-wind-lowres-mag, See2017-spindown-ZDImaps}.
Studies of individual young solar-like stars, using spectropolarimetry to measure large-scale magnetic fields, have been performed for a number of stars. The earliest studies focused on very rapid rotators with strong magnetic fields \citep[e.g][]{Donati1999-ABDor-mag-geom, Donati2003-sempol-monitoring-cool-active}. More recently a number of slower rotators with weaker magnetic fields have been investigated \citep[e.g][]{Petit2008-sunlike-mag-geom, Marsden2011-hd141943-sempol-hd101412, Jeffers2014-epsEri-mag-var, Waite2015-2-young-solar-B, BoroSaikia2015-HNPeg, doNascimento2016-kappaCeti-mag-wind, Hackman2016-young-solarlike-small, Waite2017-young-solar-EKDra}. However many of these stars are field objects and have poorly determined ages, and they span a wide range of masses and spectral types, making an inhomogeneous sample.
Long term variability in the large-scale magnetic fields of G and K stars has been studied using spectropolarimetry for a number of stars \citep[e.g.][]{Donati2003-sempol-monitoring-cool-active, Jeffers2011-nonSolar-HD171488-3epochs, Jeffers2014-epsEri-mag-var, Mengel2016-tauBoo-mag-update, BoroSaikia2016-61CygA-solar-like, Scalia2017-B-linear-regression-cool-stars}. In some cases cyclical variability in the large-scale magnetic field has been found \citep[e.g.][]{Mengel2016-tauBoo-mag-update, BoroSaikia2016-61CygA-solar-like}, but in other cases no clear periodicity is yet known. This variability amounts to a factor of a couple in magnetic field strength, and often large changes in magnetic geometry. Despite this variability, trends in large-scale magnetic field strength and geometry with mass and rotation period have been found using large samples of stars \citep[e.g.][]{Donati2009-ARAA-magnetic-fields, Vidotto2014-magnetism-age-rot, Folsom2016-Toupies1}, and multi-epoch studies of stars have suggestions of further trends \citep[e.g.][]{See2016-mag-geom-cycles-PolTor-Rossby}.
A study compiling literature magnetic field results was performed by \citet{Vidotto2014-magnetism-age-rot}. They found a clear trend of decreasing magnetic field with age, although the somewhat heterogeneous sample produced significant scatter. They also found trends in decreasing magnetic field with rotation period and Rossby number. This is expected in the context of stellar spin-down, where the older stars rotate more slowly, and thus have weaker dynamos. \citet[][the first paper in this series]{Folsom2016-Toupies1} began a study focusing on a more well defined sample, with ages established from clusters or co-moving groups, and found broadly similar results to \citet{Vidotto2014-magnetism-age-rot}. \citet{Rosen2016-mag-young-solar-twins} performed a study on a small sample of stars, although with multiple epochs of observation for most stars in their sample. They also found similar results, although with much of the scatter in their sample apparently driven by intrinsic long-term variability of the large-scale magnetic fields.
We present here the second set of results from our ongoing study of magnetic fields in young solar-type stars. The first set of results from this study was published in \citet{Folsom2016-Toupies1} (henceforth Paper I). The whole study focuses on stars between 20 and 650 Myr old, and 0.7 to 1.2 $M_{\odot}$, while this paper focuses on a subset of older stars with a narrower range of mass. We use spectropolarimetry to directly detect Zeeman splitting in polarized spectra. With time a series of rotationally modulated spectra, we can use Zeeman Doppler Imaging (ZDI) to invert the polarized spectra and reconstruct the large-scale stellar magnetic field strength and geometry. We observe a relatively large number of stars in order to investigate trends in magnetic field with age, rotation rate, and Rossby number, and to overcome scatter due to long term variability in the magnetic fields. In this paper we specifically focus on completing the older part of our sample, from 250 to 650 Myr old, with three additional stars in AB Dor (120 Myr old).
This work is part of the `TOwards Understanding the sPIn Evolution of Stars' (TOUPIES) project\footnote{http://ipag.osug.fr/Anr\_Toupies/}. In particular, observations from the Canada France Hawaii Telescope are from the large program `History of the Magnetic Sun'.
\section{Observations}
\label{observations}
Observations for this study were obtained at the Canada France Hawaii Telescope (CFHT) using the ESPaDOnS instrument (\citealt{Donati2003-ESPaDOnS-descript}; see also \citealt{Silvester2012-data-paper}),
and at the T\'elescope Bernard Lyot (TBL) at the Observatoire du Pic du Midi, France, using the Narval instrument \citep{Auriere2003-Narval-early}. Both instruments are high resolution \'echelle spectropolarimeters, and Narval is a direct copy of ESPaDOnS. Both instruments have a Cassegrain mounted polarimeter module, connected by optical fiber to a bench mounted, cross dispersed \'echelle spectrograph. They have a resolution of R$\sim$65000 and cover the wavelength range from 3700 to 10500 \AA.
Observations were obtained using spectropolarimetric mode, which provides simultaneous Stokes $V$ (circularly polarized) and $I$ (total intensity) spectra. Observations were reduced using the Libre-ESpRIT package \citep{Donati1997-major}, as in Paper I.
A series of observations were obtained for each star, with a goal of 15 observations distributed evenly over a couple rotation cycles of the star.
Observations of a single star were typically obtained within a two week period, to limit the possibility of intrinsic variations in the magnetic field during the observations. This is the same general observing strategy as in Paper I. However, due to varying observing conditions, for some targets fewer observations were obtained, or the time span of the observations was longer.
A minimum target peak S/N in the reduced $V$ spectra of 100 (per spectral pixel) was adopted, although for stars with weaker magnetic fields a higher target S/N was used. This was achieved for almost all observations, except for a few cases of observations obtained in poor weather conditions. For a couple targets (BD-072388 and HD 6569) exposure times were increased during the observing run, to ensure we obtained consistent detections. A summary of the observations is presented in Table \ref{observations-table}.
\begin{table*}
\centering
\caption{Summary of observations obtained. Exposure times are for a full sequence of 4 sub-exposures,
and the S/N values are the peak in the $V$ spectrum (per 1.8 km\,s$^{-1}$\ spectral pixel, typically near 730 nm).
For three stars the exposure times were modified during the set of observations to ensure an adequate S/N (for BD-072388 and HD 6569), or to make efficient use of time when the target S/N was exceeded (for Mel 25-151). }
\begin{tabular}{lcc@{~ }cccccc}
\hline
Object & Assoc. & RA & Dec. & Dates of &Telescope &Integration& Num & S/N \\
& & & & Observations &Semester & Time (s) & Obs & Range \\
\hline
BD-072388 & AB Dor & 08 13 50.99 & -07 38 24.6 & 16 Jan - 25 Jan 2016 & CFHT 15B & 496, 992 & 20 & 126-224 \\
HIP 10272 & AB Dor & 02 12 15.41 & +23 57 29.5 & 16 Oct - 01 Nov 2014 & TBL 14B & 400 & 13 & 80-165 \\
HD 6569 & AB Dor & 01 06 26.15 & -14 17 47.1 & 18 Sep - 29 Sep 2015 & CFHT 15B & 576, 1152 & 12 & 77-170 \\
HH Leo & Her-Lyr & 11 04 41.47 & -04 13 15.9 & 06 Mar - 26 May 2015 & TBL 15A & 1520 & 14 & 225-352 \\
EP Eri & Her-Lyr & 02 52 32.13 & -12 46 11.0 & 24 Oct - 01 Nov 2014 & TBL 14B & 160 & 9 & 158-265 \\
EX Cet & Her-Lyr & 01 37 35.47 & -06 45 37.5 & 01 Sep - 27 Sep 2014 & TBL 14B & 800 & 13 & 158-264 \\
AV 2177 & Coma Ber & 12 33 42.13 & +25 56 34.1 & 09 Apr - 19 Jun 2014 & CFHT 14A & 3600 & 21 & 124-212 \\
AV 1693 & Coma Ber & 12 27 20.69 & +23 19 47.5 & 24 Mar - 09 Apr 2015 & CFHT 15A & 5608 & 15 & 170-327 \\
AV 1826 & Coma Ber & 12 28 56.43 & +26 32 57.4 & 09 Apr - 19 Jun 2014 & CFHT 14A & 3600 & 22 & 115-199 \\
TYC 1987-509-1 & Coma Ber & 11 48 37.71 & +28 16 30.6 & 24 Mar - 01 Apr 2015 & CFHT 15A & 6740 & 9 & 312-330 \\
AV 523 & Coma Ber & 12 12 53.24 & +26 15 01.5 & 16 Feb - 02 Mar 2016 & CFHT 16A & 3920 & 30 & 108-179 \\
Mel 25-151 & Hyades & 05 05 40.38 & +06 27 54.6 & 13 Jan - 28 Jan 2016 & CFHT 15B & 3080, 2584 & 15 & 280-207 \\
Mel 25-43 & Hyades & 04 23 22.85 & +19 39 31.2 & 17 Nov - 02 Dec 2015 & CFHT 15B & 1880 & 13 & 219-293 \\
Mel 25-21 & Hyades & 04 16 33.48 & +21 54 26.9 & 18 Sep - 01 Oct 2015 & CFHT 15B & 1420 & 14 & 136-223 \\
Mel 25-179 & Hyades & 04 27 47.04 & +14 25 03.9 & 17 Nov - 02 Dec 2015 & CFHT 15B & 2200 & 12 & 266-303 \\
Mel 25-5 & Hyades & 03 37 34.98 & +21 20 35.4 & 18 Sep - 01 Oct 2015 & CFHT 15B & 2120 & 14 & 179-263 \\
\hline
\end{tabular}
\label{observations-table}
\end{table*}
\subsection{Sample Selection}
The sample of stars in this paper followed the same selection criteria as in Paper I, however in here we focus on stars in the age range from 250 to 650 Myr, with three additional targets of interest in AB Dor (120 Myr). Targets were selected from lists of members in nearby stellar associations or clusters, and only stars with published rotation periods were used. In this paper we focused on a mass range from 0.8-0.95 $M_{\odot}$, and attempted to cover the full range of periods available in the associations.
Relatively bright targets were selected to ensure we could meet our S/N targets.
The targets in this study are from the AB Dor association \citep[120 Myr][]{Luhman2005-ABDor-age, Barenfeld2013-ABDor-memb-age}, the Her-Lyr association \citep[257 Myr][]{Lopez-Santiago2006-HerLyr-ABDor-assoc, Eisenbeiss2013-HerLyr-age}, the Coma Ber cluster \citep[584 Myr][]{CollierCameron2009-ComaBer-periods, Delorme2011-periods-Hyades-Praesepe}, and the Hyades \citep[625 Myr][]{Perryman1998-Hyades-age-dist}. They span a range of effective temperatures from 4700 K to 5400 K and masses from 0.8 to 0.95 $M_{\odot}$. This relatively narrow range in mass provides a sample with relatively consistent internal structure. Rotation periods range from 0.326 days to 10.5 days, although almost all stars in this study rotate slower than 6 days, while in Paper I most stars rotated faster than 6 days. Combined, these two studies provide a wide range of rotation rates and Rossby numbers.
Individual stars are discussed in Appendix \ref{Individual Targets}, and the physical parameters of the stars are summarized in \ref{fundimental-param-table}.
\section{Fundamental physical parameters}
\label{fund-param}
\subsection{Spectroscopic analysis}
\label{spectrum-fitting}
\subsubsection{Primary analysis}
The physical atmospheric parameters $T_{\rm eff}$, $\log\,{g}$, $v \sin i$, and microturbulence ($\xi$) were derived for all stars in the sample. This was done by directly fitting synthetic spectra to our observed Stokes $I$ spectra. The initial analysis was done using the {\sc Zeeman} spectrum synthesis program \citep{Landstreet1988-Zeeman1,Wade2001-zeeman2_etc}, using the same methodology as Paper I.
The observed spectra were first normalized to their continuum. A low order polynomial was fit through carefully chosen continuum points in the observations, then the observations were divided by that continuum polynomial, as in Paper I.
Stellar parameters were derived by fitting synthetic spectra to observed spectra, though $\chi^2$ minimization, using metallic lines. The synthetic spectra were produced with {\sc Zeeman} \citep{Landstreet1988-Zeeman1,Wade2001-zeeman2_etc}, and fit to observations using the Levenberg-Marquardt procedure of \citet{Folsom2012-HAeBe-abundances} \citep[see also][]{Folsom2013-PhD-thesis}. Atomic data were extracted from the Vienna Atomic Line Database \citep[VALD][]{Ryabchikova1997-VALD-early, Kupka1999-VALD, Ryabchikova2015-VALD3}, through `extract stellar' requests. Model atmospheres from the MARCS grid \citep{Gustafsson2008-MARCS-grid} were used. A comparison between fitting results with MARCS and {\sc atlas9} \citep{Kurucz1993-ATLAS9etc} models, in this parameter range, showed that these models produced results consistent to much less than the uncertainties.
Solar chemical abundances were assumed for this analysis, except for the Hyades targets. Since all stars in this study are relatively young and near the sun, they likely have very nearly solar abundances.
The Hyades is well established to have a mildly enhanced metallicity of [Fe/H] = 0.13, with a star to star dispersion of $\sim$0.05 \citep{Perryman1998-Hyades-age-dist, Paulson2003-Hyades-abun, Heiter2014-clusters-metallicity}. Thus in our primary spectroscopic analysis, we assume a metallicity of [Fe/H] = 0.13 for the spectrum synthesis. However the exact value of [Fe/H] has a relatively small impact on the derived stellar parameters, at the level of the quoted uncertainties or smaller.
Spectra were fit independently in five spectral windows (6000-6100, 6100-6276, 6314-6402, 6402-6500, and 6600-6700 \AA\ excluding telluric features and Balmer lines). The results of these five independent fits were averaged to produce the final best values, and the standard deviation of these results was taken as the final uncertainty. This allows for a robust inclusion of systematic errors, such as errors in atomic data or continuum placement. If we consider the differences in results for the different windows to be driven primarily by random errors, then a better uncertainty estimate would be the standard error on the mean (standard deviation divided by the square root of the number of windows). This would scale our formal uncertainties down by a factor of $\sim$0.45. However to be cautious and ensure we account for the range of possible systematic errors, we report standard deviation here.
\subsubsection{Secondary analysis and comparison}
A second independent analysis of the stellar parameters was performed, as in Paper I, which was crosschecked against the analysis with {\sc Zeeman}. This used spectrum synthesis from MARCS models of stellar atmospheres, and fit for lithium abundances ($A_{\rm Li}$) and metallicity in addition to $T_{\rm eff}$, $\log\,{g}$, $v \sin i$, and microturbulence values. This analysis also proceeded by directly fitting synthetic spectra to observations though $\chi^2$.
This analysis used the region around the 6707.8 \AA\ lithium line, with checks from regions around the Ca IR triplet and H$\beta$. It used synthetic spectra from the TurboSpectrum code \citep{Alvarez1998-TurboSpectrum-etc}, atomic data from VALD with some modifications, and the fitting procedure discussed in \citet{CantoMartins2011-Li-abun-M67}. Sample fits to the 6707.8 \AA\ lithium line are provided in Fig.~\ref{fig-Li-line}. While the underlying model atmospheres in both analyses are similar, the spectral regions and spectrum synthesis tools are entirely independent, thus the more important systematic uncertainties are independent.
\begin{figure*}
\centering
\includegraphics[width=6.5in]{Figures/fig-Li-line-fits-v4.eps}
\caption{The region around the 6707.8 \AA\ lithium line (indicated with a vertical tick) for the stars in the sample. Observations are points and the best fit synthetic spectra are over-plotted with smooth lines. }
\label{fig-Li-line}
\end{figure*}
In this analysis metallicity was included as a free parameter. For most stars we find metallicities consistent with zero, supporting the assumption used in the previous analysis, with an average (excluding Hyades) members of [Fe/H] = 0.036, relative to an uncertainty of 0.05. However, for several Hyades members we find enhanced metallicities by [Fe/H] of +0.1 to +0.15.
The agreement between the two analyses is generally good, with our values usually consistent within $1\sigma$. For $\log\,{g}$, and microturbulence our values always differ by less than $2\sigma$, and the majority agree within $1\sigma$. For $v \sin i$, three of the stars (HIP 10272, HD 6569, and Mel25-151) disagree by a little over $2\sigma$ (a little over 1 km\,s$^{-1}$), however the rest show better agreement with the majority (12 stars) better than $1\sigma$. Thus we don't consider this marginal disagreement serious.
$T_{\rm eff}$\ is the most sensitive parameter to systematic errors in metallicity or continuum normalization, relative to the formal uncertainties. However, all stars agree in $T_{\rm eff}$\ within $2\sigma$, and the majority agree within $1\sigma$.
\begin{table*}
\caption{Derived fundamental parameters for the stars in our sample. $P_{\rm rot}$ are the adopted rotation periods, containing a mix of literature and our spectropolarimetric periods, as discussed in Appendix~\ref{Individual Targets}. Radial velocities ($v_r$) are the averages and standard deviations of our observations. Lithium abundances are in the form $\log(N_{\rm Li}/N_{\rm H})+12$.
Distance references: $^1$ \citet{van_Leeuwen2007-Hipparcos_book}, $^2$ \citet{Torres2008-youngNearbyAssoc}, $^3$ \citet{Gaia2016-data-release1} }
\begin{sideways}
\begin{tabular}{lccccccccc}
\hline
Star & Assoc. & Age & $P_{\rm rot}$ & $T_{\rm eff}$ & $\log\,{g}$ & $v \sin i$ & $\xi$ & $v_r$ & $i$ \\
& & (Myr) & (days) & (K) & & (km/s) & (km/s) & (km/s) & ($^{\circ}$) \\
\hline
BD-072388 & AB Dor & $ 120 \pm 10 $ & $0.32595\pm 0.0005$ & $ 5121 \pm 137$ & $ 4.44 \pm 0.18$ & $126.1 \pm 1.7 $ & $ 1.26 \pm 0.40$ & $26.14 \pm 0.56 $ & $ 38^{+13}_{-13}$ \\
HIP10272 & AB Dor & $ 120 \pm 10 $ & $ 6.13 \pm 0.03 $ & $ 5281 \pm 79 $ & $ 4.61 \pm 0.15$ & $ 7.17 \pm 0.21 $ & $ 1.05 \pm 0.33$ & $ 0.490 \pm 0.028$ & $ 55^{+20}_{-20}$ \\
HD 6569 & AB Dor & $ 120 \pm 10 $ & $ 7.13 \pm 0.05 $ & $ 5118 \pm 95 $ & $ 4.56 \pm 0.08$ & $ 5.25 \pm 0.29 $ & $ 1.14 \pm 0.09$ & $ 7.940 \pm 0.029$ & $ 77^{+13}_{-15}$ \\
HH Leo & Her-Lyr & $ 257 \pm 46 $ & $ 5.915 \pm 0.017 $ & $ 5402 \pm 73 $ & $ 4.62 \pm 0.11$ & $ 6.70 \pm 0.27 $ & $ 1.31 \pm 0.31$ & $ 18.931 \pm 0.035$ & $ 67^{+19}_{-9 }$ \\
EP Eri & Her-Lyr & $ 257 \pm 46 $ & $ 6.76 \pm 0.20 $ & $ 5125 \pm 87 $ & $ 4.53 \pm 0.13$ & $ 5.37 \pm 0.38 $ & $ 1.25 \pm 0.21$ & $ 18.273 \pm 0.036$ & $ 85^{+5 }_{-30}$ \\
EX Cet & Her-Lyr & $ 257 \pm 46 $ & $ 7.15 \pm 0.10 $ & $ 5326 \pm 63 $ & $ 4.65 \pm 0.13$ & $ 2.83 \pm 0.48 $ & $ 1.12 \pm 0.28$ & $ 11.827 \pm 0.034$ & $ 28^{+8 }_{-6 }$ \\
AV 2177 & Coma Ber & $ 584 \pm 10 $ & $ 8.98 \pm 0.12 $ & $ 5316 \pm 61 $ & $ 4.69 \pm 0.12$ & $ 4.62 \pm 0.31 $ & $ 1.00 \pm 0.25$ & $-1.30 \pm 1.26$ (SB1) & $ 77^{+14}_{-22}$ \\
AV 1693 & Coma Ber & $ 584 \pm 10 $ & $ 9.05 \pm 0.10 $ & $ 5372 \pm 63 $ & $ 4.69 \pm 0.10$ & $ 4.77 \pm 0.31 $ & $ 1.06 \pm 0.26$ & $ 0.628 \pm 0.024$ & $ 75^{+15}_{-20}$ \\
AV 1826 & Coma Ber & $ 584 \pm 10 $ & $ 9.34 \pm 0.08 $ & $ 5098 \pm 76 $ & $ 4.67 \pm 0.35$ & $ 4.66 \pm 0.35 $ & $ 1.12 \pm 0.2 $ & $ 0.744 \pm 0.090$ (SB1) & $ 65^{+25}_{-18}$ \\
TYC 1987-509-1 & Coma Ber & $ 584 \pm 10 $ & $ 9.43 \pm 0.10 $ & $ 5379 \pm 68 $ & $ 4.68 \pm 0.11$ & $ 4.88 \pm 0.30 $ & $ 1.09 \pm 0.26$ & $ 0.663 \pm 0.020$ & $ 67^{+15}_{-15}$ \\
AV 523 & Coma Ber & $ 584 \pm 10 $ & $ 11.1 \pm 0.20 $ & $ 4769 \pm 74 $ & $ 4.61 \pm 0.14$ & $ 3.89 \pm 0.26 $ & $ 1.03 \pm 0.20$ & $ 0.509 \pm 0.032$ & $ 50^{+10}_{-10}$ \\
Mel25-151 & Hyades & $ 625 \pm 50 $ & $ 10.41 \pm 0.10 $ & $ 4920 \pm 73 $ & $ 4.43 \pm 0.10$ & $ 4.83 \pm 0.33 $ & $ 1.27 \pm 0.26$ & $37.98 \pm 0.24 $ (SB1) & $ 52^{+12}_{-12}$ \\
Mel25-43 & Hyades & $ 625 \pm 50 $ & $ 9.90 \pm 0.10 $ & $ 5121 \pm 71 $ & $ 4.51 \pm 0.17$ & $ 4.01 \pm 0.28 $ & $ 0.88 \pm 0.18$ & $37.78 \pm 0.16 $ (SB1) & $ 46^{+13}_{-9 }$ \\
Mel25-21 & Hyades & $ 625 \pm 50 $ & $ 9.73 \pm 0.20 $ & $ 5236 \pm 88 $ & $ 4.39 \pm 0.18$ & $ 3.65 \pm 0.38 $ & $ 1.03 \pm 0.20$ & $38.285 \pm 0.023$ & $ 53^{+18}_{-11}$ \\
Mel25-179 & Hyades & $ 625 \pm 50 $ & $ 9.70 \pm 0.10 $ & $ 5023 \pm 55 $ & $ 4.46 \pm 0.12$ & $ 4.04 \pm 0.28 $ & $ 1.15 \pm 0.19$ & $39.684 \pm 0.053$ (SB1) & $ 65^{+17}_{-17}$ \\
Mel25-5 & Hyades & $ 625 \pm 50 $ & $ 10.57 \pm 0.10 $ & $ 4916 \pm 97 $ & $ 4.35 \pm 0.22$ & $ 3.28 \pm 0.34 $ & $ 1.01 \pm 0.19$ & $31.618 \pm 0.024$ & $ 61^{+29}_{-14}$ \\
\hline
Star & Distance & $L$ & $R$ & $M$ & $\tau_{\rm conv}$ & Rossby & $A_{\rm Li}$ & ${\rm d}\Omega$ & ${\rm d}\Omega / \Omega_{\rm eq}$\\
& (pc) & ($L_\odot$) & ($R_\odot$) & ($M_\odot$) & (days) & number & (dex) & (rad/day) & \\
\hline
BD-072388 & $93.0 \pm 18.6$ $^2$ & $0.38 \pm 0.08$ & $0.78 \pm 0.09$ & $0.85^{+0.05}_{-0.04}$ & $22.5^{+5.5}_{-2.2}$ &$0.014^{+0.002}_{-0.003}$ & $2.7 \pm 0.1$ & $0.16^{+0.23}_{-0.23}$ n & $0.008^{+0.012}_{-0.012}$ \\
HIP10272 & $36.6 \pm 1.6 $ $^1$ & $0.45 \pm 0.10$ & $0.80 \pm 0.08$ & $0.90^{+0.04}_{-0.04}$ & $20.2^{+3.0}_{-1.0}$ & $0.30^{+0.02 }_{-0.04 }$ & $2.2 \pm 0.1$ & $0.20^{+0.10}_{-0.10}$ m & $0.20^{+0.10}_{-0.10}$ \\
HD 6569 & $45.8 \pm 0.5 $ $^3$ & $0.36 \pm 0.01$ & $0.76 \pm 0.03$ & $0.85^{+0.04}_{-0.04}$ & $23.3^{+1.2}_{-1.2}$ & $0.32^{+0.02 }_{-0.02 }$ & $2.0 \pm 0.1$ & $0.30^{+0.25}_{-0.50}$ n & $0.34^{+0.28}_{-0.57}$ \\
HH Leo & $25.9 \pm 0.3 $ $^3$ & $0.54 \pm 0.02$ & $0.84 \pm 0.03$ & $0.95^{+0.05}_{-0.05}$ & $18.1^{+2.6}_{-0.9}$ & $0.33^{+0.02 }_{-0.04 }$ & $2.5 \pm 0.1$ & $0.10^{+0.02}_{-0.02}$ D & $0.10^{+0.02}_{-0.02}$ \\
EP Eri & $10.35 \pm 0.04$ $^1$ & $0.30 \pm 0.06$ & $0.72 \pm 0.08$ & $0.85^{+0.04}_{-0.05}$ & $22.5^{+3.1}_{-1.1}$ & $0.30^{+0.03 }_{-0.04 }$ & $2.9 \pm 0.1$ & $< 0.2$ n & $< 0.2$ \\
EX Cet & $24.0 \pm 0.2 $ $^3$ & $0.46 \pm 0.02$ & $0.86 \pm 0.05$ & $0.90^{+0.04}_{-0.04}$ & $20.1^{+1.0}_{-1.0}$ & $0.36^{+0.02 }_{-0.02 }$ & $2.3 \pm 0.1$ & - & - \\
AV 2177 & $82.3 \pm 2.6 $ $^3$ & $0.43 \pm 0.03$ & $0.78 \pm 0.03$ & $0.90^{+0.04}_{-0.04}$ & $20.1^{+1.0}_{-1.0}$ & $0.45^{+0.03 }_{-0.03 }$ & $1.6 \pm 0.1$ & $0.05^{+0.05}_{-0.02}$ m & $0.07^{+0.07}_{-0.03}$ \\
AV 1693 & $84.7 \pm 2.0 $ $^3$ & $0.52 \pm 0.03$ & $0.83 \pm 0.03$ & $0.90^{+0.05}_{-0.05}$ & $20.1^{+1.0}_{-2.3}$ & $0.45^{+0.07 }_{-0.03 }$ & $1.4 \pm 0.1$ & $0.22^{+0.10}_{-0.08}$ D & $0.32^{+0.14}_{-0.12}$ \\
AV 1826 & $87.4 \pm 3.4 $ $^3$ & $0.39 \pm 0.03$ & $0.80 \pm 0.04$ & $0.85^{+0.04}_{-0.04}$ & $22.6^{+3.6}_{-1.1}$ & $0.41^{+0.03 }_{-0.06 }$ & $0.7 \pm 0.2$ & $0.09^{+0.04}_{-0.03}$ D & $0.13^{+0.06}_{-0.04}$ \\
TYC 1987-509-1 & $88.3 \pm 2.4 $ $^3$ & $0.52 \pm 0.03$ & $0.83 \pm 0.03$ & $0.90^{+0.05}_{-0.05}$ & $20.1^{+1.0}_{-2.4}$ & $0.47^{+0.07 }_{-0.03 }$ & $1.0 \pm 0.2$ & $0.07^{+0.20}_{-0.15}$ n & $0.11^{+0.30}_{-0.23}$ \\
AV 523 & $85.7 \pm 2.6 $ $^3$ & $0.24 \pm 0.02$ & $0.72 \pm 0.03$ & $0.80^{+0.04}_{-0.05}$ & $26.2^{+2.6}_{-1.3}$ & $0.42^{+0.03 }_{-0.04 }$ & $0.2 \pm 0.2$ & $0.00^{+0.20}_{-0.20}$ n & $0.00^{+0.35}_{-0.35}$ \\
Mel25-151 & $52.0 \pm 8.2 $ $^1$ & $0.35 \pm 0.11$ & $0.82 \pm 0.13$ & $0.85^{+0.05}_{-0.05}$ & $26.9^{+6.0}_{-1.8}$ & $0.39^{+0.03 }_{-0.07 }$ & $0.2 \pm 0.3$ & $0.03^{+0.06}_{-0.07}$ n & $0.05^{+0.10}_{-0.12}$ \\
Mel25-43 & $60.6 \pm 4.8 $ $^1$ & $0.38 \pm 0.08$ & $0.79 \pm 0.08$ & $0.85^{+0.05}_{-0.04}$ & $22.6^{+3.6}_{-1.1}$ & $0.44^{+0.03 }_{-0.06 }$ & $0.2 \pm 0.2$ & $0.18^{+0.16}_{-0.16}$ n & $0.28^{+0.25}_{-0.25}$ \\
Mel25-21 & $51.1 \pm 0.7 $ $^3$ & $0.56 \pm 0.02$ & $0.91 \pm 0.04$ & $0.90^{+0.05}_{-0.05}$ & $24.2^{+1.5}_{-3.7}$ & $0.40^{+0.08 }_{-0.03 }$ & $0.8 \pm 0.3$ & $0.20^{+0.15}_{-0.13}$ m & $0.31^{+0.23}_{-0.20}$ \\
Mel25-179 & $49.1 \pm 0.7 $ $^3$ & $0.40 \pm 0.02$ & $0.84 \pm 0.03$ & $0.85^{+0.04}_{-0.04}$ & $27.6^{+1.9}_{-5.3}$ & $0.35^{+0.09 }_{-0.03 }$ & $0.2 \pm 0.3$ & $0.10^{+0.08}_{-0.08}$ n & $0.15^{+0.12}_{-0.12}$ \\
Mel25-5 & $46.2 \pm 0.7 $ $^3$ & $0.43 \pm 0.02$ & $0.91 \pm 0.04$ & $0.85^{+0.05}_{-0.04}$ & $29.5^{+5.8}_{-3.0}$ & $0.36^{+0.04 }_{-0.06 }$ & $0.0 \pm 0.4$ & $-0.17^{+0.18}_{-0.18}$ n& $-0.29^{+0.30}_{-0.30}$ \\
\hline
\end{tabular}
\end{sideways}
\label{fundimental-param-table}
\end{table*}
\subsection{H-R diagram and evolutionary tracks}
\label{H-Rdiagram}
The stars in our sample were placed on a Hertzsprung-Russell (H-R) diagram, in order to derive masses. Luminosities were derived as in Paper I, based on J-band photometry from 2MASS \citep{Cutri2003-2MASS}. The bolometric correction of \citet{Pecaut2013-PMS-BC-withJ} was used, together with our $T_{\rm eff}$. Reddening was assumed to be negligible, since our targets are all near the sun ($<100$ pc).
To derive distances to the stars in the sample we used parallax measurements from the Gaia Data Release 1 \citep{Gaia2016-data-release1} when possible. For a few stars, Gaia parallaxes were not yet available, so we used Hipparcos parallax measurements \citep{van_Leeuwen2007-Hipparcos_book}. When both values were available for a target the parallaxes were consistent but the Gaia values were more precise.
The Hyades cluster has a known distance \citep[e.g. 46.3 pc, tidal radius $\sim$10 pc][]{Perryman1998-Hyades-age-dist}, however there are significant star to star differences in the measured parallax, thus we prefer the individual parallax measurements, which are relatively precise.
The Coma Ber cluster also has a known distance of 86.7 pc \citep{vanLeeuwen2009-cluster-distances}, and a radius of $\sim$9.1 pc (based on the $\sim 6^\circ$ radius of the cluster). However, these targets all have precise Gaia parallaxes, thus we use the more precise Gaia values, although they are all consistent with the cluster distance.
BD-072388 (in AB Dor) does not have a Hipparcos parallax and does not yet have a Gaia parallax, so we used the dynamical distance from \citet{Torres2008-youngNearbyAssoc} and arbitrarily assumed a 20\% uncertainty on the value. With these distances we derive absolute luminosities in Table \ref{fundimental-param-table}.
For three stars, BD-072388, HIP 10272, and Mel25-43, there are significant uncertainties in their luminosity due to binarity. These three stars all fall significantly above their association isochrones, as noted in Appendix \ref{Individual Targets}, using luminosities from photometry. Thus for these targets we estimate their luminosity by fixing them to their association isochrones, as this is likely more accurate.
Stellar radii were derived from the Stefan-Boltzmann law, using our $T_{\rm eff}$\ and luminosities.
Masses were derived by comparing with a grid of evolutionary tracks (c.f.\ Fig.~\ref{fig-hr-diagram}). The evolutionary tracks were computed with the STAREVOL V3.30 stellar evolution code, as discussed in \citet{Amard2016-eol-traks-rotating}, and are the same tracks described in Paper I.
These evolutionary tracks assumed initial solar abundances, since the associations have nearly solar abundances and the stars are too young to have undergone significant chemical evolution \citep{VianaAlmeida2009-SACY-abun-young-assoc,Biazzo2012-abun-3nearby-assoc}. A constant mixing length was used since neither the range of metallicities nor masses is large enough to significantly affect this approximation.
The stars Mel25-5, Mel25-21 and Mel25-179 fall above the their cluster isochrone (and the ZAMS). This may be due to binarity, indeed Mel25-179 is an SB1 in our observations, alternately a small overestimate in the distance could cause this, and using the Hyades distance of \citet{Perryman1998-Hyades-age-dist} is sufficient to bring Mel25-21 and Mel25-179 onto the ZAMS. These evolutionary tracks were also used to derive convective turnover times for the stars. The convective turnover time at one pressure scale height above the base of the convective envelope was used, as discussed in Paper I. These, combined with the rotation periods of the stars, were used to compute Rossby numbers ($R_{o} = P_{\rm rot}/\tau_{\rm conv}$).
\begin{figure}
\centering
\includegraphics[width=3.3in]{Figures/hr-diagram-paper2.eps}
\caption{H-R diagram of the stars in this study. Evolutionary tracks are from \citet{Amard2016-eol-traks-rotating}, plotted in 0.1$M_{\odot}$\ increments for the masses labeled on the right (in $M_{\odot}$). Isochrones are shown for 24 Myr, 42 Myr, and the ZAMS, as in Paper I. The stars are grouped by age and association, as indicated. }
\label{fig-hr-diagram}
\end{figure}
\section{Spectropolarimetric analysis}
\subsection{Least squares deconvolution}
\label{Least squares deconvolution}
The signature of the Zeeman effect in Stokes $V$ is typically quite weak for solar-like stars, and undetectable in individual lines for any practical S/N. Therefore, we used the multi-line technique Least Squares Deconvolution \citep[LSD;][]{Donati1997-major,Kochukhov2010-LSD} to produce a pseudo-average line profile with much higher S/N. The LSD procedure used here was identical to that from Paper I. The same line masks were used, based on data from the VALD using `extract stellar' requests, and rounded to the nearest 500 K in $T_{\rm eff}$. The same normalization parameters for LSD were also used, specifically a line depth of 0.39, Land\'e factor of 1.195, and a wavelength of 650~nm. Sample LSD profiles for each star in this study are presented in Fig.~\ref{fig-lsd-grid}.
\begin{figure*}
\centering
\includegraphics[width=6.8in]{Figures/sample-lsd-grid-n.eps}
\caption{Sample LSD Stokes $V$ profiles for the stars in this study. The associated diagnostic null profile for each observation is plotted in the background as a dashed line, with a second horizontal dashed line indicating zero. }
\label{fig-lsd-grid}
\end{figure*}
\subsection{Longitudinal magnetic field measurements}
\label{longitudinal-magnetic}
Longitudinal magnetic field ($B_l$) measurements were made, as in Paper I, for all observations. This quantity represents the disk averaged line of sight component of the magnetic field. These values were primarily used to investigate the rotation period of the star, since $B_l$ should vary smoothly as the star rotates. However, they can also provide an estimate of the strength and degree of axisymmetry of the global stellar magnetic field. This was measured with Eq.~2 from Paper I \citep[e.g.][]{Rees1979-magnetic-cog}. This requires a wavelength and Land\'e factor, and the normalizing values from our LSD analysis were used (Sect.~\ref{Least squares deconvolution}). The resulting $B_l$ measurements, phased with rotation period, are presented in Figs.~\ref{fig-bz} and \ref{fig-bz2}, and the maximum absolute value of $B_l$ and full amplitude of variability for each star is reported in Table \ref{table-mag-param}.
We find peak $B_l$ between 20 and 7 G for most stars in the sample. For BD-072388 we find a peak $B_l$ of 320 G, which is much stronger than the rest of the sample, but consistent with the stars much shorter rotation period. For EX Cet we find no magnetic field, and $B_l$ is consistent with zero with uncertainties between 1.5 and 2 G (usually 1.7 G). Thus $B_l$ must remain below 5 G, with a $3\sigma$ confidence. EX Cet clearly has the weakest $B_l$ of the sample, despite having a $T_{\rm eff}$\ and a literature rotation period in the middle of the sample's range. Unless the literature rotation period is incorrect (the star has the lowest $v \sin i$\ in the sample, hinting at a possible error) we have no explanation for the weakness of the magnetic field.
For every star in the sample we performed a period analysis using $B_l$. This was done as in Paper I, by fitting sinusoids with a grid of periods to the data by minimizing $\chi^2$, and constructing a periodogram in $\chi^2$. When an adequate fit to the data could not be achieved with a simple sine curve, due to more complex magnetic field topology, a higher order sinusoid was used (e.g. $\sin + \sin^2 ...$). This accounts for a quadrupole component (for $\sin^2$) and an octupole component (if extended to $\sin^3$), and is equivalent to a Lomb-Scargle periodogram for a simple sine. The results of this for individual stars are discussed in Appendix \ref{Individual Targets}. The rotation periods for all the stars are consistent with the best literature periods, and we do not find any stars lacking accurate literature periods as we did in Paper I. However, our periods are typically more uncertain than the literature values, due to the relatively short timespan of our observations. For AV 523, we are able to resolve a possible ambiguity in the literature rotation period. We need sinusoids beyond first order for AV 1693, AV 1862, TYC 1987-509-1, Mel 25-151, and Mel 25-179 to achieve an adequate fit to the data. For EX Cet, we do not detect a magnetic field, and thus cannot derive a rotation period from this method.
\subsection{Radial velocity}
\label{Radial velocity}
Radial velocities were measured for all observations by fitting a Gaussian to the Stokes $I$ line profiles by $\chi^2$ minimization, and taking the center of the Gaussian of the to be radial velocity ($v_r$). Uncertainties on individual $v_r$ measurements were taken from the covariance matrix of the $\chi^2$ fit. While this method is potentially influenced by spots on the stellar surface, the influence of spots is useful for our study as it provides another way to check the rotation period of the star, and this study does not require extremely high precision velocimetry. The $v_r$ value averaged over all observations of a target is reported in Table \ref{fundimental-param-table}, with the standard deviation of the $v_r$ values reported as an uncertainty.
For each star we checked for systematic variations in $v_r$ with Julian Date. In particular, we looked for trends on longer timespans than the rotation period of the star. This was done by plotting $v_r$ versus Julian Date and looking for cases were there were significant differences between the earlier and later $v_r$ measurements, based on the uncertainties for individual $v_r$ values. For the stars AV 2177, AV 1826, Mel25-151, and Mel25-43, there are clear systematic trends in the $v_r$ measurements, strongly suggesting that they are the primaries of SB1 systems. For Mel25-179 we find a similar but weaker trend, tentatively suggesting it may be an SB1. This is reflected in the larger standard deviations in $v_r$ for these stars. We do not have sufficient data to find an orbital solution, thus we do not subtract off their orbital motion in the reported $v_r$. However, since the $v_r$ for these stars are consistent with their cluster $v_r$, the amplitude of variability is likely small and this likely does not introduce a large error in our reported values.
We performed a period analysis on $v_r$, similar to the analysis used for $B_l$, as in Paper I. The apparent variability in $v_r$ used here is assumed to be due to surface features on the star distorting line profiles, not due to actual motion of the star. However, since most of these stars were less spotted than the stars in Paper I, this analysis was less useful. Significant unambiguous periods were only found for EP Eri and BD-072388, both of which were consistent with literature. For HIP 10277 and AV 1693 pairs of ambiguous minima were found that were also consistent with their literature periods. For EX Cet, for which we have no constraint on the period from magnetic data, the $v_r$ data are not able to strongly constrain the rotation period either.
\section{Magnetic mapping}
\label{ZDI}
Magnetic mapping was done in two stages, first a preliminary map was made, to check the quality of the Stokes $V$ data and to ensure the stellar parameters were correct. Then a more detailed search for an optimal rotation period and differential rotation value was made, around the rotation period from the literature and $B_l$. This further refined the rotation period, and where possible derived a differential rotation estimate. Then the final best magnetic map was made using these optimal parameters. The search for differential rotation was not performed in Paper I, however some of the stars in this paper with datasets spanning a longer time period (particularly AV 1826, AV 2177, and HH Leo) required non-zero differential rotation to achieve an acceptable fit to the observations.
For the ZDI analysis in this paper we developed a new code, which implements the same physical model and analysis principles as the code used in Paper I. This code has the practical advantages of being easier to use and easier to modify in the future, however the scientific output of the two codes is identical. The code used in Paper I was described in that paper and was based on the code of \citet{Donati2006-tauSco}, while the new code used here is described in Appendix \ref{ZDIpy}. Both codes use Gaussian model Stokes $I$ line profiles and the weak field approximation for Stokes $V$ profiles. They both use the spherical harmonics description of the magnetic field from \citet{Donati2006-tauSco}, and use the maximum entropy fitting routine from \citet{Skilling1984-max-entropy-regularisation} to find the regularized best fit solution.
The two ZDI codes were extensively tested to ensure they produced identical results for identical input parameters. Indeed, for every star in this sample we produced a ZDI map with both codes and compared them to ensure the results were identical. Thus, despite changing the underlying ZDI code, the results from this paper and Paper I are homogeneous, since the performance of the two codes is identical.
While ZDI has been used successfully for many years, concerns continue to be raised \citep[e.g.][]{Stift2012-ZDI-systematics-atmo}. Indeed, ZDI maps do not represent a complete picture of a stellar magnetic field, but only the components of the field that are constrained observationally. A wide range of studies have shown the general reliability of ZDI \citep[e.g.][]{Donati1997-ZDI-tests, Hussain2000-ZDI-code-comparison, Hussain2001-DOTS-descript, Kochukhov2002-MDI-intro2, Yadav2015-MHD-models-ZDI-recon}. However, there are some potential systematic trends that need to be considered. In particular, the resolution of the map is dependent on the $v \sin i$\ of the star (e.g. Morin 2010), and we provide a discussion of this in Appendix \ref{Trends in magnetic geometry and resolution}. There is the possibility of cross-talk between radial and azimuthal magnetic field, at least for some inclinations, when only Stokes $V$ is used. The map is also somewhat sensitive to the degree of regularization used, mostly for the amount of energy in higher degree harmonics, although this also has a small impact on the total magnetic field strength. While ZDI may contain some biases, we are using a consistent methodology across our sample, and the same basic methodology as the BCool \citep[][Petit et al. in prep.]{Marsden2014-Bcool-survey1}, MaPP \citep{Donati2008-BPTau-ZDI}, and MaTYSSE \citep{Donati2014-LkCa4-wTTs-mag-planet} samples. This crucially provides results that can be directly compared for a large number of stars at different evolutionary stages.
The input parameters for the ZDI model were the same as in Paper I. Specifically, the model line used the normalizing Land\'e factor and wavelength from LSD, a Gaussian line full width at half maximum of 7.8 km\,s$^{-1}$\ ($1\sigma$ width of 3.2 km\,s$^{-1}$) was used (see Paper I), and the line strength was set by fitting the central line depth of the $I$ LSD profile for each star. The stellar model again used a linear limb darkening law with a coefficient of 0.75. For computing disk integrated model lines, the stellar surface was modeled using 2000 surface elements.
The spherical harmonic expansion was carried out to 15th degree in $l$, although for most stars in the sample the higher degrees are unnecessary, since they are unresolved in the observations due to the low $v \sin i$.
We find very little information (with values close to zero) in the higher degree harmonics, confirming that we are not reconstructing spurious smaller scale magnetic field. A uniform maximum $l$ degree was used to provide a more uniform analysis of the sample.
As in Paper I, a uniform surface brightness was assumed. Since the Stokes $I$ line profile variability is very weak or undetectable in these stars (except for BD-072388, c.f.\ the standard deviation of radial velocities in Table \ref{fundimental-param-table}) the stars are not strongly spotted and this approximation should not affect the results.
Inclinations of the stellar rotation axis relative to the line of sight ($i$) were, when possible, derived from our measured $v \sin i$\ (Sect.\ \ref{spectrum-fitting}), radius (Sect.\ \ref{H-Rdiagram}), and rotation period. However, in cases where the radius has a large uncertainty, or $v \sin i$\ is very small (significantly below the instrumental resolution) this becomes unreliable. In these cases we used ZDI to derive an inclination angle. For this we generated ZDI maps for a grid of inclinations, and selected the map with the best maximum entropy. Then using this entropy as a target, we performed ZDI with a fixed target entropy and variable minimum $\chi^2$ \citep[as in][]{Petit2002-diff-rot-DI}, for the same grid of inclinations, and selected the model with a minimum $\chi^2$. Generally these two inclinations agreed, however the curve of $\chi^2$ as a function of inclination allows us to derive formal uncertainties on the inclination. Uncertainties were taken to be the variation in $i$ around the minimum needed to produce a $1\sigma$ difference according to $\chi^2$ statistics. This approach allowed for a sensible target entropy, and allowed us to check that the maximum in entropy for a target $\chi^2$, and minimum in $\chi^2$ for a target entropy, are consistent. Details of the derivation of $i$ are given in Appendix \ref{Individual Targets} for stars where the ZDI method was used, and our adopted values are given in Table \ref{fundimental-param-table}.
\subsection{Rotation period and differential rotation}
\label{Rotation period and differential rotation}
In order to verify and possibly refine the rotation periods of the stars, we performed a rotation period search using ZDI, initially assuming no differential rotation. This proceeded by assuming a grid of rotation periods, and performing ZDI for each assumed rotation period, similar to in Paper I. From this a periodogram in entropy and rotation period can be constructed, and the period that produces the maximum entropy can be selected. While the assumption of no differential rotation at this stage may be inaccurate, this allows us to efficiently explore a wide range of periods, since we only have one dimension of parameter space to search. Thus we can ensure we find a global maximum, not just a local maximum.
Then we repeat this analysis, but rather than maximizing entropy for a fixed target $\chi^2$ as done by \citet{Skilling1984-max-entropy-regularisation}, we can minimize $\chi^2$ for a fixed target entropy as done by \citet{Petit2002-diff-rot-DI}. The target entropy used is the previous global maximum, and this produces a curve of $\chi^2$ across the parameter space. From the change in $\chi^2$ around the minimum we can define a confidence region at $1\sigma$ \citep[e.g.][]{numerical-recipes-Fortran}, and we use the extent of that region as our formal uncertainty. First performing the search in entropy for fixed $\chi^2$ allows us to chose an appropriate target entropy, for the search in $\chi^2$ at fixed entropy. Thus we get a periodogram in $\chi^2$, with formal uncertainties on the period.
In order to further verify the rotation periods, and when possible to derive differential rotation estimates, we used a second search based on ZDI, simultaneously probing rotation period and differential rotation following the method of \citet{Petit2002-diff-rot-DI}. In this we assume a solar-like differential rotation law in the form
\begin{equation}
\Omega(\theta) = \Omega_{\rm eq} + {\rm d} \Omega \sin^2 \theta ,
\end{equation}
where $\Omega(\theta)$ is the angular frequency at latitude $\theta$, $\Omega_{\rm eq}$ is the angular frequency at the equator, and ${\rm d} \Omega$ is the difference in angular frequency between the equator and pole.
A ZDI fit is performed for each point in a grid of $\Omega_{\rm eq}$ and ${\rm d} \Omega$, using a range of periods around the global best period found in the previous analysis. This produces a map of maximum achievable entropy in the $\Omega_{\rm eq}$ - ${\rm d} \Omega$ parameter space, and from this we can select the pair of parameters that produce the global maximum entropy.
Similar to the simple period analysis, we repeat this analysis but minimizing $\chi^2$ for a fixed target entropy as done by \citet{Petit2002-diff-rot-DI}. Again, a $\chi^2$ contour around the minimum provides a confidence region at $1\sigma$, the extent of which defines our formal uncertainty.
This analysis only produced reliable differential rotation values for some stars in our sample. This approach requires observations at similar rotation phases but on different rotation cycles. Larger differences in time between observations provides more sensitivity, as long as there has not been significant intrinsic evolution of the magnetic field. This approach requires good S/N to detect changes in line profiles due to differential rotation, and it requires a reasonably large number of observations.
Thus, due to the limited time span of our observations, no reliable value of differential rotation could be found for EP Eri, TYC 1987-509-1, AV 523, Mel25-151, Mel25-43, Mel25-179, and Mel25-5, all of which have observations covering less than 1.5 rotation cycles. Limitations from the S/N do not allow us to detect differential rotation in BD-07 2388 and HD 6569. Marginally significant values of differential rotation were found for AV 2177 and HIP 10272, limited by S/N, and for Mel25-21, limited by phases with repeated observations. More reliable differential rotation values were found for HH Leo, AV 1693, and AV 1826, aided by the relatively long time span over which the observations were obtained.
A detailed discussion of the attempted differential rotation measurements is reported in Appendix \ref{Individual Targets}, and the values found are summarized in Table \ref{fundimental-param-table}.
\begin{figure}
\centering
\includegraphics[width=3.3in]{Figures/plot-p-dOmega-chi2-AV1826.eps}
\caption{Sample reduced $\chi^2$ map, as a function of rotation frequency and differential rotation, for AV 1826. Contours corresponding to $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels, calculated from the changes in $\chi^2$ from the minimum, are show. A well defined non-zero value of ${\rm d}\Omega$ is found. }
\label{fig-sample-diff-rot-search}
\end{figure}
\subsection{ZDI Results}
\begin{figure}
\centering
\includegraphics[width=3.in]{Figures/sample-zdi-fit-Mel25-151.eps}
\caption{Sample ZDI fit for Mel25-151. Solid lines are the observed Stokes $V$ LSD profiles, and dashed lines are the best fit synthetic ZDI line profiles. The line profiles are shifted vertically by their rotation phase, and labeled by rotation cycle. Error bars for the observations are given on the left. }
\label{fig-sample-zdi-fit}
\end{figure}
The final magnetic maps derived for the stars in this paper are presented in Figs.~\ref{fig-zdi-maps} and \ref{fig-zdi-maps2}, and a sample ZDI fit to $V$ LSD profiles is provided in Fig.~\ref{fig-sample-zdi-fit}. We find a wide range of magnetic field strengths and geometries.
In order to effectively compare this large number of stars, we parameterize the magnetic field in a number of ways, with those parameters given in Table \ref{table-mag-param}. For the global large-scale magnetic strength, we consider the unsigned (magnitude of the vector) field averaged over the surface of the star ($\langle B \rangle$). To describe the geometry we consider the square of the magnetic field in different components, which is proportional to the magnetic energy, averaged over the surface of the star. In the spherical harmonic description of \citet{Donati2006-tauSco} the $\alpha_{l,m}$ and $\beta_{l,m}$ terms are poloidal components, while the $\gamma_{l,m}$ terms are toroidal components, and we consider terms with $m = 0$ to be the axisymmetric components (about the rotation axis). For geometry independent of field strength, we consider ratios of these components, and refer to them as fractions of energy, since magnetic energy is proportional to $B^2$. We include the dipolar ($l=1$) quadrupolar ($l=2$) and octupolar ($l=3$) components of the poloidal field in Table \ref{table-mag-param}. Some energy is present in higher degree spherical harmonics for some maps, however those are more sensitive to the spacial resolution of the maps, and for most stars this is enough to capture most of the poloidal energy.
We also include the axisymmetry of the total magnetic field, just the poloidal part of the field, and just the toroidal part of the field.
BD-072388 has by far the strongest and most complex magnetic field in this paper, with an average surface field of 195 G. This is likely due to it having by far the shortest rotation period, driving a much stronger dynamo. However, BD-072388 has a similar strength and morphology to LO Peg in Paper I, which is a similarly fast rotator. The rest of the sample has somewhat more similar field strengths, with surface average value from 34 to 8.5 G.
The stars generally have significant toroidal components to their fields (e.g. HIP10272 at 68\% total energy and EP Eri at 77\% total energy, the weakest toroidal field being Mel25-21 at 20\% total energy), but it is never completely dominant. The toroidal magnetic field components are generally axisymmetric, while the poloidal field components are generally less than 50\% axisymmetric (except for HD 6569). In comparison to Paper I, the magnetic fields are on average weaker, due to these older stars rotating more slowly and hence having weaker dynamos.
\begin{table*}
\centering
\caption{Derived magnetic properties for the stars in our sample. The maximum disk integrated longitudinal magnetic field is in column 2, and the amplitude of variability in the longitudinal field is in column 3. The surface averaged large-scale magnetic field strength from the ZDI map is in column 4, and the maximum field value from the ZDI map is in column 5. The remaining columns present the percent of the magnetic energy in different components of the field (poloidal, toroidal, dipolar, quadrupolar, octupolar and axisymmetric), as percentages of the total, poloidal, or toroidal field. For EX Cet we do not detect a magnetic field, the limits on $B_{l}$ are $3\sigma$ upper limits. }
\begin{sideways}
\begin{tabular}{lcccccccccccccc}
\hline
Star & Assoc. &$B_{l, {\rm max}}$&$B_{l, {\rm range}}$&$\langle B \rangle$& $|B_{\rm peak}|$ & pol. & tor. & dip. & quad. & oct. & axisym. & axisym. & axisym. & axisym.\\
& & (G) & (G) & ZDI (G) & ZDI (G) & (\%tot) & (\%tot) & (\%pol) & (\%pol) & (\%pol) & (\%tot) & (\%pol) & (\%tor) & (\%dip) \\
\hline
BD-072388 & AB Dor & 320 & 440 & 195.5 & 1015.7 & 39.7 & 60.3 & 35.0 & 7.7 & 6.7 & 62.5 & 34.7 & 80.8 & 81.5 \\
HIP10272 & AB Dor & 18 & 28 & 21.2 & 40.2 & 32.0 & 68.0 & 83.0 & 10.9 & 2.3 & 74.6 & 29.8 & 95.7 & 34.0 \\
HD 6569 & AB Dor & 20 & 19 & 25.0 & 48.6 & 60.0 & 40.0 & 88.5 & 7.3 & 3.3 & 85.2 & 76.2 & 98.7 & 79.4 \\
HH Leo & Her-Lyr & 18 & 33 & 28.9 & 66.2 & 45.9 & 54.2 & 57.0 & 18.3 & 8.7 & 49.2 & 1.9 & 89.3 & 2.6 \\
EP Eri & Her-Lyr & 10 & 11 & 34.3 & 82.3 & 22.4 & 77.6 & 65.5 & 30.7 & 0.8 & 77.3 & 21.4 & 93.4 & 2.9 \\
AV 2177 & Coma Ber & 10 & 15 & 10.3 & 28.9 & 49.7 & 50.3 & 59.6 & 20.9 & 8.1 & 47.1 & 8.7 & 85.0 & 6.4 \\
AV 1693 & Coma Ber & 13 & 20 & 33.7 & 71.0 & 50.5 & 49.5 & 31.3 & 51.1 & 10.8 & 51.9 & 17.7 & 86.8 & 42.8 \\
AV 1826 & Coma Ber & 14 & 25 & 25.1 & 57.8 & 41.7 & 58.3 & 31.6 & 38.1 & 21.7 & 63.6 & 26.9 & 89.9 & 74.9 \\
TYC 1987-509-1 & Coma Ber & 11 & 16 & 25.0 & 62.8 & 55.7 & 44.3 & 35.4 & 27.2 & 9.8 & 59.4 & 38.1 & 86.3 & 74.0 \\
AV 523 & Coma Ber & 10 & 12 & 22.8 & 56.3 & 32.9 & 67.1 & 39.9 & 24.8 & 24.0 & 78.3 & 48.0 & 93.1 & 76.4 \\
Mel25-151 & Hyades & 15 & 23 & 23.7 & 74.5 & 42.2 & 57.8 & 49.5 & 22.1 & 8.7 & 63.4 & 34.5 & 84.5 & 45.3 \\
Mel25-43 & Hyades & 7 & 14 & 8.5 & 18.5 & 61.3 & 38.7 & 71.8 & 22.3 & 4.3 & 36.2 & 0.6 & 92.5 & 0.4 \\
Mel25-21 & Hyades & 14 & 18 & 12.7 & 43.9 & 80.0 & 20.0 & 71.8 & 14.0 & 6.0 & 31.0 & 22.4 & 65.5 & 29.5 \\
Mel25-179 & Hyades & 17 & 26 & 26.0 & 63.1 & 52.5 & 47.5 & 70.3 & 18.4 & 9.5 & 61.0 & 34.6 & 90.1 & 43.9 \\
Mel25-5 & Hyades & 11 & 15 & 13.0 & 32.8 & 35.8 & 64.2 & 73.4 & 16.4 & 7.2 & 69.6 & 19.9 & 97.4 & 24.1 \\
EX Cet & Her-Lyr & <5 & <10 & & & & & & & & & & & \\
\hline
\end{tabular}
\end{sideways}
\label{table-mag-param}
\end{table*}
\section{Discussion}
The expanded sample of stars with magnetic properties derived here strengthens many of the trends we found in Paper 1. We again find a clear decreasing trend in the average large-scale magnetic field strength (the unsigned magnetic field strength from our maps averaged over the surface of the star, $\langle B \rangle$) with age, shown in Fig.~\ref{fig-trend-age-B}. We also find a decreasing trend with rotation period, shown in Fig.~\ref{fig-trend-period-B}. Having older, slower rotating stars in our sample improves these correlations. There is also a decreasing trend in $\langle B \rangle$\ with Rossby number, shown in Fig.~\ref{fig-trend-rossby-B}, which provides a tighter correlation than simply rotation period.
The trend we find in the average large-scale magnetic field strength can be described by a power law:
$\langle B \rangle = (466 \pm 290) t ^{-0.49 \pm 0.12}$, for age $t$ in Myr, based on the Toupies sample (Fig.~\ref{fig-trend-age-B} left). Including T Tauri stars from the MaPP and MaTYSSE projects would produce an exponent of $-0.68 \pm 0.05$.
This is consistent within $1.5\sigma$ with the trend found by \citet{Vidotto2014-magnetism-age-rot}, who found an exponent of $-0.655 \pm 0.045$. The ages of the stars in our sample are much more accurate than in \citet{Vidotto2014-magnetism-age-rot}. However, due to the large range of magnetic fields found around an age of $\sim$120 Myr, the scatter in our relationship is similar.
\citet{Rosen2016-mag-young-solar-twins} studied six young solar analogues using ZDI and also found a decreasing trend in $\langle B \rangle$\ with age. Their results are consistent with our trend, although the trend is much clearer here due to the larger sample size.
In rotation period we find a power law trend in $\langle B \rangle$\ of:
$\langle B \rangle = (207 \pm 71) P_{\rm rot} ^{-1.05 \pm 0.19}$ with a saturation below periods of 1 or 2 days (Fig.\ \ref{fig-trend-period-B}).
This trend has an exponent slightly smaller than \citet{Vidotto2014-magnetism-age-rot}, who found an exponent of $-1.32 \pm 0.14$, but it is consistent within $1.5\sigma$.
In Rossby number we find a power law trend with $\langle B \rangle$\ of:
$\langle B \rangle = (8.4 \pm 1.8) R_o ^{-0.89 \pm 0.13}$.
This assumes saturation for values below 0.06 (Fig.\ \ref{fig-trend-rossby-B}), however the exact saturation value of Rossby number is not strongly constrained, and could be as high as 0.1. We also note that the convective turnover time depends on how deep in the convective envelope this is calculated. Using a different choice of depth will shift all Rossby numbers (as discussed in Paper I), and would lead to a somewhat different saturation value.
This trend is qualitatively consistent with the trend found by \citet{Vidotto2014-magnetism-age-rot}, however the exponent we find is smaller by roughly $2.5\sigma$ than their value of $-1.38 \pm 0.14$. This could partly be due to us including stars near the saturated regime, with Rossby numbers between 0.06 and 1.0. However repeating the power law fit restricting it to $R_o > 0.1$, we still find an exponent of -0.90, which is not enough for a good agreement.
Our two studies use different sources for convective turnover times, which could contribute to this discrepancy.
The scatter in our trend of $\langle B \rangle$\ with Rossby number is much smaller than the trend from \citet{Vidotto2014-magnetism-age-rot}, since our sample is much more homogeneous. Thus our power law fit may in fact be closer to the correct value.
Interestingly the exponents for both the trends in $R_o$ and $P_{\rm rot}$ are close to -1.0.
The very fast rotator BD-072388, together with LO Peg from Paper I, supports the hypothesis that we are seeing a saturation of the large-scale magnetic field strength due to increasing rotation period. This star has a magnetic field of similar strength to LO Peg in Paper I, with a qualitatively similar complex geometry. Both BD-072388 and LO Peg have magnetic field strengths similar to stars with rotation periods around 2 days and Rossby numbers around 0.1, despite having much shorter rotation periods and smaller Rossby numbers (0.3-0.4 days and $R_o$ 0.01-0.02). The star AB Dor, while slightly more massive than BD-072388 and LO Peg ($M \sim 1.0$ $M_{\odot}$, $P \sim 0.514$ d), has been studied using ZDI \citep{Donati1999-ABDor-mag-geom, Donati2003-sempol-monitoring-cool-active, Hussain2007-ABDor-mag-x-ray}, and was found to have a similar large-scale magnetic strength ($\langle B \rangle$\ $\sim 125$ G), and a similar complex geometry.
The star LQ Hya ($M \sim 0.8$ $M_{\odot}$, $P \sim 1.60$ d) is less confidently in the saturated regime by Rossby number, but also has a strong ($\langle B \rangle$\ $\sim 100$ G) complex magnetic field \citep{Donati2003-sempol-monitoring-cool-active}, which is comparable to BD-072388 and LO Peg.
These four stars are consistent with the saturation of the large-scale magnetic field due to rapid rotation.
Saturation of magnetic proxies, such as X-ray emission, at low Rossby number are well established \citep[e.g.][]{Noyes1984-CaHK-Rossby, Pizzolato2003-Xray-saturation-rossby, Wright2011-x-ray-rossby-relations}.
However, those proxies are only indirectly related to the large-scale magnetic field, by several physical processes (they depend on small-scale magnetic field, a filling factor, and magnetic reconnection or chromospheric heating), thus it is not clear that the large-scale magnetic field should behave similarly.
The behavior of the large-scale component of the field is perhaps the most direct observational constraint for dynamo simulations. Saturation of the large-scale magnetic field at low Rossby number, due to changing convective properties, has been observed in comparisons of mostly convective M-dwarfs to K-stars \citep[e.g.][]{Morin2008-Mdwarf-topo, Donati2009-ARAA-magnetic-fields, Vidotto2014-magnetism-age-rot}. However, this is due to the growth of the convective zone to dominate the star. Thus an independent constraint is the saturation of the large-scale magnetic field due to rapid rotation, for stars with approximately the same size of convective envelope.
To search for a trend in the mean large-scale magnetic field strength as a function of age, beyond the trend as a function of Rossby number, we calculated the difference between the power law fit in $R_o$ and the observed mean large-scale field values, excluding the two saturated regime stars (BD-072388 and LO Peg). This is the residuals to the power law fit in Rossby number. Plotting this residual against age shows a decreased scatter to older ages, illustrated in Fig.\ \ref{fig-residualB-age}.
Rotational evolution models predict the development of a steep gradient in the internal rotation profile of solar-type stars at the zero-age main sequence, with a rapidly rotating core and a slowly rotating outer convective envelope, which gradually becomes flatter as the star evolves on the early main sequence \citep[e.g.][]{Gallet2013-Bouvier-ang-mom-evol,Gallet2015-Bouvier-ang-mom-evol2}. The decreasing magnetic field scatter observed between 120 and 650 Myr is qualitatively consistent with these predictions, provided the dynamo process is indeed sensitive to the early rotational history of solar-type stars.
However, if we plot this residual as a fraction of the power law values, effectively the fractional residuals of the fit, the scatter appears to be constant as a function of age (Fig.\ \ref{fig-residualB-age}).
If this scatter is physical, then the process giving rise to it, such as cyclical magnetic variability, appears to operate as a fraction of the magnetic field value. However, this fractional process does not seem to be age dependent, with the precision currently allowed by our sample. The possible impact of long term magnetic variability is discussed further in Appendix \ref{Long term magnetic variability}.
\begin{figure*}
\centering
\includegraphics[width=3.4in]{Figures/plot-v3-age-B-zoom.eps}
\includegraphics[width=3.4in]{Figures/plot-v3-age-B.eps}
\caption{Mean large-scale magnetic field from ZDI as a function age for the stars in our study (left) and compared with some literature results (right). The dotted line is a power law fit. }
\label{fig-trend-age-B}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=3.3in]{Figures/plot-v3-prot-B-zoom.eps}
\caption{Mean large-scale magnetic field from ZDI as a function rotation period for the stars in our study. The dotted line is a power law fit. }
\label{fig-trend-period-B}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=3.4in]{Figures/plot-v3-rossby-B-zoom.eps}
\includegraphics[width=3.4in]{Figures/plot-v3-rossby-B-trends.eps}
\caption{Mean large-scale magnetic field from ZDI as a function Rossby number for the stars in our study (left) and compared with some literature results (right). In the left panel, a power law fit to the data is presented (solid line), an extrapolation of this to lower Rossby numbers (dashed line) and a hypothetical saturation level (dashed line) are also shown. }
\label{fig-trend-rossby-B}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=3.3in]{Figures/plot-v3-age-B-diff-model-v3.eps}
\caption{Residual values of the power law fit to $\langle B \rangle$\ as a function of Rossby number (top) and those residuals divided by the predicted $\langle B \rangle$\ (bottom). }
\label{fig-residualB-age}
\end{figure}
Trends in magnetic geometry are more challenging to find. Multi-epoch studies of stars with ZDI generally show large changes in the magnetic field geometry over a time span of years. This can be due to stellar magnetic cycles \citep[e.g.][]{Donati2003-sempol-monitoring-cool-active, Mengel2016-tauBoo-mag-update, BoroSaikia2016-61CygA-solar-like}, or apparently more chaotic long term magnetic variability \citep[e.g.][]{Jeffers2011-nonSolar-HD171488-3epochs, Jeffers2014-epsEri-mag-var, BoroSaikia2015-HNPeg}. This intrinsic variability complicates searches for trends in magnetic geometry. However, our results continue to support the observation from \citet{Petit2008-sunlike-mag-geom}, Paper I, and \citet{See2016-mag-geom-cycles-PolTor-Rossby}, that very slowly rotating stars have dominantly poloidal fields, while faster rotators have a wider range of poloidal/toroidal ratios. This transition appears to occur around a Rossby number of 1.0, or a rotation period of 15-20 days, and seems to occur at longer rotation periods than are available in our sample. However, the transition is not precisely defined, partly because a given star can exhibit a range of poloidal/toroidal ratios, due to its long term magnetic variability.
We also support the trend from \citet{See2015-toroidal-axisymm} and Paper I that dominantly toroidal magnetic fields are dominantly axisymmetric (i.e. symmetric about the stellar rotation axis), while dominantly poloidal magnetic fields have a wide range of axisymmetries.
In their study of six solar analogues between 100 and 600 Myr old, \citet{Rosen2016-mag-young-solar-twins} found that the magnetic energy in $l=3$ spherical harmonics was larger than the energy in the $l=2$ harmonics for their two oldest stars ($\sim$600 Myr). We do not find the same trend in our sample. None of our stars older than 200 Myr have an $l=3$ energy above 50\% of the $l=2$ energy. The only two stars with this ratio significantly above one are BD-072388 and HII 739 (from paper I). BD-072388 is the fastest rotator in our sample, while HII 739 is the hottest star in our sample ($T_{\rm eff}$\ $= 6066 \pm 89$ K). Our sample is largely cooler than the sample of \citet{Rosen2016-mag-young-solar-twins} (4400-5400 K, vs 5800 K). So if this effect is strongly dependent on temperature, that could explain the difference. Or this may simply be a coincidence due to their small sample size.
The stars TYC 6349-0200-1, HIP 76768, TYC 5164-567-1, HII 296, LO Peg all have ratios of $l=3$ to $l=2$ energy near 1, and these are all fast rotators with some of the smallest Rossby numbers in the sample. This is consistent with the general trend of stars with smaller Rossby number and faster rotation having more complex ZDI maps. However, it is still unclear how much of this is driven by changing resolution of the maps and how much of this is real changes in the magnetic field structure.
\begin{figure*}
\centering
\includegraphics[width=5.0in]{Figures/cg-hr-diagram-names.eps}
\includegraphics[width=5.0in]{Figures/cg-age-rossby-names.eps}
\caption{H-R diagram (top) and age-Rossby number plane (bottom), with stars from the Toupies project (thin black outlines), together with some classical T Tauri stars from the MaPP project (thick blue outlines), and some weak-line T Tauri stars from the MaTYSSE project (thick green outlines). Symbol size corresponds to magnetic field strength, symbol color is how poloidal or toroidal the magnetic field is, and shape is the axisymmetry of the poloidal component of the magnetic field. In the H-R diagram, evolutionary tracks (dashed lines) are shown for 1.2, 1.0, 0.8 and 0.6 $M_{\odot}$, isochrones (dotted lines) are shown for 10, 20, 50 and 100 Myr.
The development of a significant radiative core corresponds to the bend where the stars move from the Hayashi track (largely vertical) onto the Henyey track (more horizontal).
Evolutionary tracks and isochrones models are from \citet{Amard2016-eol-traks-rotating}. }
\label{fig-cg-hrd-age-ro}
\end{figure*}
Comparing the stars in our sample to younger T Tauri stars is helpful for investigating evolutionary changes in magnetic fields on the pre-main sequence. In particular, we consider the classical T Tauri stars from the MaPP project (BP Tau, \citealt{Donati2008-BPTau-ZDI}; AA Tau, \citealt{Donati2010MNRAS-AATau-ZDI}; TW Hya, \citealt{Donati2011-TWHya-ZDI}; V4046 Sgr A \& B, \citealt{Donati2011-V4046Sgr-ZDI}; GQ Lup, \citealt{Donati2012-GQLup-ZDI}; and DN Tau, \citealt{Donati2013-DNTau-ZDI}), and weak-line T Tauri stars from the MaTYSSE project (LkCa 4, \citealt{Donati2014-LkCa4-wTTs-mag-planet}; V819 Tau \& V830 Tau, \citealt{Donati2015-V819Tau-WTTs-mag}; and TAP 26, \citealt{Yu2017-TAP26-wTTs-mag-planet}), in roughly the same mass range as our sample. We considered this in Paper I, but we revisit it here with our expanded sample of stars, and with the expanded sample of weak-line T Tauri stars. The classical and weak-line T Tauri stars differ in that the classical stars are strongly accreting, while the weak-line stars are accreting at a much lower level.
The classical and weak-line T Tauri stars are shown on an H-R diagram, together with our stars, in Fig.~\ref{fig-cg-hrd-age-ro}. The classical T Tauri stars show stronger, more poloidal, and more axisymmetric magnetic fields than our sample (apart from the two most evolved T Tauri stars V4046 Sgr A \& B). This follows the proposal of \citet{Gregory2012-TTauri-B-structure}, that the different magnetic properties are driven by different convective properties, with the T Tauri stars being mostly convective. This is essentially the same result as in our Paper I, since the new stars we add here are clustered around the ZAMS. The weak-line T Tauri stars complicate the hypothesis somewhat. TAP 26 is partly convective and has a similar magnetic field strength and geometry to our later pre-main sequence stars, which is consistent with the magnetic field being driven by structure. The star LkCa 4 is mostly or fully convective and has a consistent strength and axisymmetry to the classical T Tauri stars, although it may be less poloidal. However, the two stars V819 Tau and V830 Tau seem to have intermediate magnetic properties between our sample and the classical T Tauri stars, with magnetic field strengths closer to our sample, and magnetic geometries that are mostly poloidal but more complex and non-axisymmetric, and seem to fall in the mostly or fully convective regime of the H-R diagram. Thus it remains unclear how well the weak-line T Tauri stars fall into this scenario, but observations of more stars are needed to further test this idea.
\subsection{Differential rotation}
\begin{figure*}
\centering
\includegraphics[width=3.3in]{Figures/plot-v3-dOmega-teff.eps}
\includegraphics[width=3.3in]{Figures/plot-v3-dOmega-age.eps}
\caption{Differential rotation rate (${\rm d} \Omega$) as a function of effective temperature (left) and age (right). Stars with significantly non-zero differential rotation are in blue. }
\label{fig-dOmega-trends}
\end{figure*}
We have searched for trends in latitudinal differential rotation using the values derived this paper, and the literature value for LO Peg from \citet{Barnes2005-LOPeg-DI}.
We see no clear trend in latitudinal differential rotation with rotation period, although there are a large uncertainties on our ${\rm d} \Omega$\ values, and most stars are non-detections. There is a trend towards decreasing values of the ratio ${\rm d}\Omega / \Omega_{\rm eq}$ for the faster rotators, although that is largely driven by BD-072388 and LO Peg. The large $\Omega_{\rm eq}$ of BD-072388 and LO Peg implies small values for the ratio, and smaller limits on the ratio provided by our uncertainties.
We find a weak trend towards increasing ${\rm d} \Omega$\ with $T_{\rm eff}$, illustrated in Fig.\ \ref{fig-dOmega-trends}. The range of $T_{\rm eff}$\ in our sample is small, thus the tend is not strong. However, the hotter stars have larger ${\rm d} \Omega$, while the cooler stars have a small value (LO Peg) or are non-detections typically with smaller limits. A similar trend, with a similar degree of confidence, appears in ${\rm d}\Omega / \Omega_{\rm eq}$.
If we consider convective turnover time rather than $T_{\rm eff}$, we get a similar quality trend, with ${\rm d} \Omega$\ decreasing with increasing convective turnover time. Indeed this correlation may be slightly better, but the larger uncertainties on convective turnover time makes this unclear. We can speculate that larger convective turnover times, and larger convective cells, redistribute angular momentum more efficiently, leading to less differential rotation.
This trend in ${\rm d} \Omega$\ with $T_{\rm eff}$\ is qualitatively similar to the trend reported by Barnes et al. (MNRAS, in press, doi:10.1093/mnras/stx1482),
who considered a range of literature ${\rm d} \Omega$\ values for stars from 3000 to 7000 K. A few other overviews of literature ${\rm d} \Omega$\ values have found similar trends \citep[e.g.][]{CollierCameron2007-diff-rot-review}.
We find no clear trend in ${\rm d} \Omega$\ with age, illustrated in Fig.\ \ref{fig-dOmega-trends}. There seems to be a comparable range of values in the sample around 120 Myr as there is in the sample around 600 Myr. However, the age sampling of our ${\rm d} \Omega$\ values is sparse, and does not extend to the youngest portion of our sample. One of the motivations for investigating ${\rm d} \Omega$\ is the large radial internal differential rotation predicted by rotational evolution models. If the internal radial differential rotation changes importantly between 120 and 600 Myr, it does not seem to be reflected in surface latitudinal differential rotation. However, if the surface latitudinal differential rotation is primarily controlled by the convective properties of the stellar envelope, this would not be surprising.
We find no trend in the mean magnetic field $\langle B \rangle$\ with ${\rm d} \Omega$. This is in strong contrast to the trends with rotation period and Rossby number. The ${\rm d} \Omega$\ values carry large uncertainties and we lack stars in the saturated regime, however the latitudinal surface differential rotation we measure does not seem to be important for the generation of large-scale magnetic fields. If differential rotation is important for the dynamo generation of magnetic fields in these stars, it must not be related to the latitudinal surface differential rotation ${\rm d} \Omega$.
In magnetic geometry, we find no clear trend in the fraction of toroidal magnetic field with ${\rm d} \Omega$.
This would seem to argue against the toroidal field being generated by latitudinal differential rotation shearing poloidal field. However, we caution that there are large uncertainties on ${\rm d} \Omega$, and it may be that ${\rm d} \Omega$\ is harder to to measure in strongly toroidal stars, thus no strong conclusions can be drawn.
There is no trend in axisymmetry of the magnetic field and ${\rm d} \Omega$. However, the uncertainties on ${\rm d} \Omega$\ are noticeably larger for strongly axisymmetric fields, particularly when the total axisymmetry reaches $\sim$70\%. This is because measuring ${\rm d} \Omega$\ requires detectable non-axisymmetric features in the magnetic field at different latitudes, thus when the non-axisymmetric features becomes weak, our ability to measure ${\rm d} \Omega$\ becomes weak.
\section{Conclusions}
We have derived detailed magnetic maps for 15 young solar-like stars and characterized their large-scale magnetic field strength and geometry. We also derived fundamental physical parameters for these stars. The stars were selected from members of four stellar associations with ages from 120 to 625 Myr. We find a narrow range of $T_{\rm eff}$\ for the stars, from 4769 to 5402 K, and most of the stars have rotation periods between 5.9 and 10.6 days, except for one very fast rotator at 0.326 days. This extends the sample from Paper I to older, slower rotating stars.
We find that the average large-scale magnetic field decreases with increasing age across our sample, although there is a large scatter around 120 Myr. The average large-scale magnetic field also decreases with rotation period and Rossby number within our sample, with Rossby number providing the tighter correlation. At very low Rossby number we see further tentative evidence for saturation of the large-scale magnetic field, with a second apparently saturated star BD-072388. This star has a similar rotation rate and similar magnetic properties to LO Peg, both of which are similar to the literature values for AB Dor. This helps further support the hypothesis of saturation of large-scale magnetic fields due to increasing rotation rate, rather increasing convective turnover time.
Among stars older than $\sim$20 Myr, the evolution of the large-scale magnetic field strength can be explained sufficiently well by changing Rossby number. Once the trend in Rossby number has been subtracted, there is no clear residual trend in age. However, comparing to T Tauri stars in the same mass range, there are clear differences that cannot be explained by Rossby number. The oldest T Tauri stars fall close to our proposed saturation value, however the younger objects have much stronger magnetic fields. This is likely a consequence of changing internal structure, as proposed by \citet{Gregory2012-TTauri-B-structure}, since the youngest T Tauri stars are largely convective. However the possible impact of accretion and star-disk interactions cannot be completely ruled out from the current initial studies of weak-line T Tauri stars.
This paper has largely strengthened the conclusions of our Paper I. In the next paper in this series, we will focus on younger stars, and further probe the saturation of the large-scale magnetic field at rapid rotation.
\section*{Acknowledgments}
This study was supported by the grant ANR 2011 Blanc SIMI5-6 020 01 ``Toupies: Towards understanding the spin evolution of stars'' ({http://ipag.osug.fr/Anr\_Toupies/}).
This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{http://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{http://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
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A sturdy new edition of a classic novelty board book for babies and toddlers.
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{
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| 5,924
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\section{Introduction}
Let $X$ be an abelian variety of positive dimension $g$ over a finite field $k=\F_q$ of characteristic $p$ (where $q$ is a power of $p$), $\Fr_X$ the
Frobenius endomorphism of $X$, and $\mathcal{P}_X[t]\in \Z[t]$ the characteristic polynomial of $\Fr_X$, which is a degree $2g$ monic polynomial with integer coefficients.
\cite{Mumford,Tate1}. Let $L=L_X$ be the splitting field of $\mathcal{P}_X[t]$ over the field $\Q$ of rational numbers and therefore is a number field. Since $\deg(\mathcal{P}_X)=2g$, the degree $[L_X:\Q]$ divides $(2g)!$. (In fact, one may prove that
$[L_X:\Q]$ divides $2^g g!$, see below). We write $R_X$ for
the set of eigenvalues of $\Fr_X$; clearly, $R_X$ coincides with the set of roots of $\mathcal{P}_X[t]\in \Z[t]$ and is viewed as a certain finite subset of $L_X^{*}$.
Clearly, $R_X$ consists of algebraic integers and $\#(R_X) \le 2g$. (The equality holds if and only if $\mathcal{P}_X[t]$ has no repeated roots.) By a classical theorem of A.Weil \cite{Mumford},
all algebraic numbers $\alpha\in R_X$ have the same archimedean value $\sqrt{q}$.
In addition, $\alpha \mapsto q/\alpha$ is a {\sl permutation} of $R_X$. If $\alpha$ is a root of $\mathcal{P}_X[t]$ (i.e., $\alpha\in R_X$) then we write $\mult_X(\alpha)$ for its multiplicity. It is well known that if $\alpha\in R_X$ then
\begin{equation}
\label{multR}
\mult_X(\alpha)=\mult_X(q/\alpha); \ \text{ if } \alpha=q/\alpha \ \text{ then } \mult_X(\alpha) \ \text{ is even}.
\end{equation}
In particular, the constant term $\prod_{\alpha\in R_X}\alpha^{\mult_X(\alpha)}$ of $\mathcal{P}_X[t]$ is $q^g$.
The Galois group $\Gal(L_X/\Q)$ of $L_X/\Q$ permutes elements of $R_X$ and
\begin{equation}
\label{sigmaRX}
\mult_X(\sigma(\alpha))=\mult_X(\alpha) \ \forall \sigma\in \Gal(L_X/\Q) , \alpha\in R_X.
\end{equation}
In this paper we continue our study of multiplicative relations between elements of $R_X$ that was started in \cite{ZarhinK3,LenstraZarhin,ZarhinEssen,ZarhinJPAA}.
(In \cite{ZarhinK3,LenstraZarhin} we concentrated on abelian varieties with rather special type of Newton polygons; in \cite{ZarhinEssen,ZarhinJPAA} we studied
abelian varieties of small dimension).
In order to state results of the present paper, we need the following definitions.
\begin{defn}
An integer-valued function $e \colon R_X \to \Z$ is called
\begin{itemize}
\item[(i)]
{\bf admissible} if there exists an integer $d$ such that
\begin{equation}
\label{relationM}
\prod_{\alpha\in R_X}\alpha^{e(\alpha)}=q^d;
\end{equation}
Such a $d$ is called the {\bf degree} of $e$ and is denoted $\deg(e)$.
The nonnegative integer
$\sum_{\alpha\in R_X}|e(\alpha)|$ is called the {\bf weight} of $e$ and denoted $\wt(e)$.
\item[(ii)]
{\bf trivial} if
\begin{equation}
\label{relationT}
e(\alpha)=e(q/\alpha) \ \forall \alpha\in R_X; \ e(\beta)\in 2\Z \ \forall \beta\in R_X \text{ with } \beta^2=q.
\end{equation}
\end{itemize}
\end{defn}
\begin{defn}
An {\sl admissible} integer-valued function $e \colon R_X \to \Z$ is called
{\bf reduced} if it enjoys the following properties:
\begin{itemize}
\item[(i)]
$\deg(e) \ge 1$ and all $e(\alpha)\ge 0$.
\item[(ii)]
If $\alpha \in R_X$ and $\alpha \ne q/\alpha$ then either $e(\alpha)=0$ or $e(q/\alpha)=0$.
\item[(iii)]
$e(\beta)=0$ or $1$ $\forall \beta\in R_X$ with $\beta^2=q$.
\end{itemize}
\end{defn}
\begin{rems}
\begin{itemize}
\item[]
\item[(i)] It follows from \eqref{multR} that $\alpha \mapsto \mult_X(\alpha)$ is a trivial admissible function of degree $g$ and weight $2g$.
\item[(ii)] Every trivial function is admissible.
\item[(iii)] If $e \colon R_X \to \Z$ is admissible then it follows from Weil's theorem that
\begin{equation}
\label{degreeE}
2\deg(e)=\sum_{\alpha\in R_X}e(\alpha).
\end{equation}
\item[(iv)] If $e \colon R_X \to \Z$ is reduced admissible then it follows from \eqref{degreeE} that
\begin{equation}
\label{weightE}
\wt(e)=2\deg(e).
\end{equation}
\end{itemize}
\end{rems}
Our first main result is the following assertion.
\begin{thm}
\label{mainRelation}
Let $g$ be a positive integer.
There exists a positive integer $N=N(g)$ that depends only on $g$ and enjoys the following property.
Let $X$ be a $g$-dimensional abelian variety over a finite field $k$ such that there exists a nontrivial
admissible function $R_X \to \Z$. Then there exists a reduced admissible function of degree $\le N(g)$.
\end{thm}
Our main tool in the proof of Theorem \ref{mainRelation} is the multiplicative (sub)group $\Gamma(X,k)\subset L_X^{*}$ generated by $R_X$, which was first introduced in \cite{ZarhinIzv79,ZarhinInv79} (see also \cite{ZarhinK3,LenstraZarhin,ZarhinEssen,ZarhinJPAA}).
\begin{defn}
\begin{itemize}
\item[]
\item[(i)]
We say that $k=\F_q$ is {\bf small} with respect to $X$ if there exist {\sl distinct } $\alpha_1, \alpha_2 \in R_X$ such that
$\alpha_1/\alpha_2$ is a {\sl root of unity}.
\item[(ii)]
We say that $k$ is {\bf sufficiently large} with respect to $X$ if $\Gamma(X,k)$ does {\sl not} contain roots of unity except $1$ (see \cite{ZarhinEssen,ZarhinJPAA}).
\end{itemize}
\end{defn}
\begin{rems}
\label{smallN}
\begin{itemize}
\item[]
\item[(i)] If $k$ is {\sl not} small with respect to $X$ then there is at most one $\beta\in R_X$ with $\beta^2=q$.
\item[(ii)] If $k$ is sufficiently large with respect to $X$ then it is {\sl not} small.
\end{itemize}
\end{rems}
The role of $\Gamma(X,k)$ is explained by the following statement.
\begin{lem}
\label{rankGamma}
Suppose that $k$ is not small with respect to $X$. Then the following three conditions are equivalent.
\begin{itemize}
\item[(i)] There exists a nontrivial admissible function $R_X \to \Z$.
\item[(ii)] There exists a reduced admissible function $R_X \to \Z$.
\item[(iii)] The rank of $\Gamma(X,k)$ does not exceed $\lfloor\#(R_X)/2\rfloor$.
\end{itemize}
\end{lem}
Our second main result deals with Tate classes on abelian varieties (see \cite{Tate0,Tate1,Tate94,ZarhinTokyo,ZarhinEssen} and Section \ref{etaleT} below for the definition of these classes
and their basic properties).
Recall that a Tate class is called {\sl exotic} if it {\sl cannot} be presented as a linear combination of products of divisor classes.
\begin{thm}
\label{mainTate}
Let $g$ be a positive integer and let $N=N(g)$ be as in Theorem \ref{mainRelation}.
Let $X$ be a $g$-dimensional abelian variety over a finite field $k$ of characteristic $p$. Assume that
there exist a positive integer $n$ and a prime $\ell\ne p$ such that
the self-product $X^n$ of $X$ carries an exotic $\ell$-adic Tate cohomology class.
Then the self-product $X^{2N}$ of $X$ carries an exotic $l$-adic cohomology Tate class for all primes $l\ne p$.
\end{thm}
\begin{rem} Theorem \ref{mainTate} gives a positive answer to a question of Kiran Kedlaya, who pointed out
that this result is related to the algorithmic problem of deciding whether or not a given
abelian variety (specified by its Weil polynomial) is {\sl neat} in a sense of \cite[Sect. 3]{ZarhinEssen}, \cite{ZarhinJPAA}).
\end{rem}
Is it possible to get all Tate classes on all self-products of $X$, using only Tate classses
of bounded dimension? In order to answer this question, we need the following result about nonnegative admissible functions.
\begin{thm}
\label{semigroupADM}
Let $g$ be a positive integer. Then there exists a positive even integer $H=H(g)$ that enjoys the following property.
Let $X$ be a $g$-dimensional abelian variety over a finite field $k$. Then there exist a positive integer $d$
and $d$ nonnegative admissible functions $e_i \colon R_X \to \Z_{+}$ such that:
\begin{itemize}
\item[(i)]
the weight of each $e_i$ does not exceed $H(g)$;
\item[(ii)]
each nonnegative
admissible function $e \colon R_X \to \Z_{+}$ may be presented as a linear combination of $e_1, \dots, e_d$ with nonnegative integer coefficients.
\end{itemize}
\end{thm}
Theorem \ref{semigroupADM} implies the following assertion.
\begin{thm}
\label{semigroupTate}
Let $g$ be a positive integer. Let $H=H(g)$ be as in Theorem \ref{semigroupADM}.
Let $X$ be a $g$-dimensional abelian variety over a finite field $k$. Assume that $k$ is sufficiently large w.r.t $X$.
Let $\ell$ be a prime different from $\fchar(k)$ and
$n$ be a positive integer. Then every $\ell$-adic Tate cohomology class on $X^n$ may be presented as a linear conbination of products of
$\ell$-adic Tate cohomology classes of dimension $ \le H(g)$.
\end{thm}
The paper is organized as follows. In Section \ref{neat} we discuss basic useful results about $R_X$ and related objects,
including the Newton polygons. In addition, we discuss roots of unity in $\Gamma(X,k)$ (Lemma \ref{rootsG})
and the structure and degree of $L_X$ (Lemma \ref{basicL}).
In Section \ref{MultRel} we study multiplicative relations between Weil numbers (i.e., admissible functions) and their weights; in particular, we prove Lemma \ref{rankGamma} (see Lemma \ref{rank}). In Sections \ref{RelationProof} and \ref{GordanL} we prove Theorems \ref{mainRelation} and \ref{semigroupADM} respectively.
Section \ref{linAlgebra} contains certain constructions from multilinear algebra that we use
in Section \ref{TateProof} in order to prove Theorems \ref{mainTate} and \ref{semigroupTate}.
As usual, $\ell$ and $l$ are primes different from $p$, and $\NN, \Z,\Z_{\ell},\Q,\RR, \C,\Q_{\ell}$ stand for the set of positive integers, the rings of integers and $\ell$-adic integers, the fields of rational, real, complex, and $\ell$-adic numbers respectively. We write $\Z_{+}$ and $\RR_{+}$ for the additive semigroups of {\bf nonnegative} integers and of {\bf nonnegative} real numbers respectively.
If $z$ is a complex number then we write $\bar{z}$ for its complex-conjugate. Similarly, if $\phi \colon E \hookrightarrow \C$ is a field embedding then we write $\bar{\phi}$ for the corresponding complex-conjugate field embedding
$$\bar{\phi} \colon E \hookrightarrow \C, \ x \mapsto \overline{\phi(x)}.$$
If $M$ is a positive integer and $v$ and $w$ are two vectors in $\RR^M$ then we write $v\cdot w$ for their scalar product.
If $A$ is a finite set then we write $\#(A)$ for the number of its elements. We write $\rk(\Delta)$ for the rank of a finitely generated commutative group $\Delta$.
{\bf Acknowledgements}. I am deeply grateful to Kiran Kedlaya for interesting stimulating questions and to the referee, whose comments helped to improve the exposition.
Part of this work was done during my stay at Centre \'Emile Borel (Institut Henri Poincar\'e, Paris) in June-July 2019, whose hospitality and support are gratefully acknowledged.
\section{Preliminaries}
\label{neat}
In this section we discuss basic properties of $L=L_X, R_X, \Gamma(X,k)$. Let us start with the formal definition of $\mathcal{P}_X(t)$.
Throughout this paper $k$ is a finite field of characteristic $p$ that consists of $q$ elements, $\bar{k}$ an algebraic closure of $k$ and $\Gal(k)=\Gal(\bar{k}/k)$ the absolute Galois group of $k$. It is well known that the profinite group $\Gal(k)$ is procyclic and the {\sl Frobenius automorphism}
$$\sigma_k \colon \bar{k} \to \bar{k}, \ x \mapsto x^q$$
is a topological generator of $\Gal(k)$. If $\ell \ne p$ is a prime then we write
$$\chi_{\ell} \colon \Gal(k) \to \Z_{\ell}^{*}$$
for the $\ell$-adic {\sl cyclotomic character} that defines the Galois action on all $\ell$-power roots of unity in $\bar{k}$. By definition,
$$\chi_{\ell}(\sigma_k)=q \in \Z_{\ell}^{*}.$$
Let $X$ be an abelian variety of positive dimension over $k$. We write $\End(X)$ for the ring of its $k$-endomorphisms and $\End^0(X)$ for the corresponding (finite-dimensional semisimple) $\Q$-algebra $\End(X)\otimes\Q$. We write $\Fr_X=\Fr_{X,k}$ for the Frobenius endomorphism of $X$. We have
$$\Fr_X \in \End(X)\subset \End^0(X).$$
It is well known that
\begin{equation}
\label{sigmaFr}
\sigma_k(x)=\Fr_X(x) \ \forall x \in X(\bar{k}).
\end{equation}
By a theorem of Tate \cite[Sect. 3, Th. 2 on p, 140]{Tate1}, the $\Q$-subalgebra $\Q[\Fr_X]$ of $\End^0(X)$ generated by $\Fr_X$ coincides with the center of $\End^0(X)$. In particular, if $\End^0(X)$ is a field then $\End^0(X)=\Q[\Fr_X]$.
If $\ell$ is a prime different from $p$ then we write $T_{\ell}(X)$ for the $\Z_{\ell}$-Tate module of $X$ and $V_{\ell}(X)$ for the corresponding $\Q_{\ell}$-vector space
$$V_{\ell}(X)=T_{\ell}(X)\otimes_{\Z_{\ell}}\Q_{\ell}.$$
It is well known \cite[Sect. 18]{Mumford} that $T_{\ell}(X)$ is a free $\Z_{\ell}$-module of rank $2\dim(X)$ that may be viewed as a $\Z_{\ell}$-lattice in the $\Q_{\ell}$-vector space $V_{\ell}(X)$ of dimension $2\dim(X)$. The Galois action on $X(\bar{k})$ induces the continuous group homomorphism \cite{SerreAbelian,SerreRibet}
$$\rho_{\ell}=\rho_{\ell,X} \colon \Gal(K) \to \Aut_{\Z_{\ell}}(T_{\ell}(X))\subset \Aut_{\Q_{\ell}}(V_{\ell}(X)).$$
In addition, there is a canonical isomorphism of $\Gal(k)$-modules
$X[\ell] \cong T_{\ell}(X)/\ell$ where $X[\ell]$ is the kernel of multiplication by $\ell$ in $X(\bar{k})$.
By functoriality, $\End(X)$ and $\Fr_X$ acts
on ($T_{\ell}(X)$ and) $V_{\ell}(X)$; it is well known that the action of $\Fr_X$ coincides with the action of $\rho_{\ell}(\sigma_k)$. By a theorem of A. Weil \cite[Sect. 19 and Sect. 21]{Mumford}, $\Fr_X$ acts on $V_{\ell}(X)$ as a semisimple linear operator, its characteristic polynomial
$$\P_X(t)=\P_{X,k}(t)=\det (t \I -\Fr_X, V_{\ell}(X)) \in \Z_{\ell}[t]$$
lies in $\Z[t]$ and does not depend on a choice of $\ell$. In addition, all eigenvalues of $\Fr_X$ (which are algebraic integers) have archimedean absolute value equal to $q^{1/2}$,
and if an eigenvalue of $\Fr_X$ is a square root of $q$ then its multiplicity is even (see \cite[p. 267]{ZarhinK3}).
This implies that the constant term of $\P_{X,k}(t)$ is $q^{\dim(X)}$. In particular, $\Fr_X$ acts as an {\sl automorphism} of the free $\Z_{\ell}$-module $T_{\ell}(X)$.
This means that if
$$L=L_X \subset \C$$
is the splitting field of $\P_X(t)$ and $$R_X=R_{X,k} \subset L$$ is the set of roots of $P(t)$ then $L$ is a finite Galois extension of $\Q$ such that for every field embedding $L \hookrightarrow \C$ we have $\mid \alpha \mid =q^{1/2}$ for all $\alpha \in R_X$. Let $\Gal(L/\Q)$ be the Galois group of $L/\Q$.
Clearly, $R_X$ is a $\Gal(L/\Q)$-invariant (finite) subset of $L^{*}$. It follows easily that if $\alpha \in R_X$ then $q/\alpha \in R_X$. Indeed,
$q/\alpha$ is the {\sl complex-conjugate} $\bar{\alpha}$ of $\alpha$. We have
$$q^{-1}\alpha^2=\frac{\alpha}{q/\alpha}.$$
\begin{defn}
Let $\ell$ be a prime and $n$ a positive integer. We write $\mathbf{e}_{\ell}(n)$
for the largest order of elements of the general linear group $\GL(n,\F_{\ell})$.
We write $\exp_{\ell}(n)$ for the {\sl exponent} of $\GL(n,\F_{\ell})$.
\end{defn}
Recall that $\Gamma(X,k)$ is the multiplicative subgroup of $L$ generated by $R_X$.
\begin{lem}
\label{rootsG}
If $\gamma \in \Gamma(X,k)$ is a root of unity then there is a positive integer
$m \le \max(2 \mathbf{e}_2(2g), \mathbf{e}_3(2g))$ such that $\gamma^m=1$. In addition,
$\gamma^{D(g)}=1$ where
$$D(g):=\mathrm{LCM}(2\exp_{2}(2g),\exp_{3}(2g)), \ \text{ and } \ m | D(g).$$
\end{lem}
\begin{proof}
In what follows we choose a prime $\ell \ne p$ and view $\Fr_X$ as the automorphism of free $\Z_{\ell}$-module $T_{\ell}(X)$ of rank $2g$. Then
$\Fr_X$ induces the automorphism $\Fr_X \bmod \ell$ of the $2g$-dimensional $\F_{\ell}$-vector space
$$T_{\ell}(X)/\ell=X[\ell].$$
Let $r$ be the order of
$$\Fr_X \bmod \ell \in \Aut_{\F_{\ell}}(X[\ell]) \cong \GL(2g, \F_{\ell}).$$ Clearly,
$$r \le \mathbf{e}_{\ell}(2g) \ \text{ and } r \ | \exp_{\ell}(2g).$$
In addition,
$$\Fr_X^r \in \mathrm{Id}+\ell \End_{\Z_{\ell}}(T_{\ell}(X)).$$
Let $\Delta$ be the multiplicative group generated by all the eigenvalues of $\Fr_X^r$.
Clearly, $\delta=\gamma^r\in \Delta$.
Applying a variant of Minkowski's Lemma \cite[Lemma 2.4]{SZcomp},
we obtain that $\delta=1$ if $\ell>2$ and $\delta^2=1$ if $\ell=2$.
This implies that $\gamma^r=1$ if $\ell>2$ and $\gamma^{2r}=1$ if $\ell=2$.
Now let us put $\ell=2$ if $p\ne 2$ and $\ell=3$ if $p=2$.
The rest is clear.
\end{proof}
Let $\Oc_L$ be the ring of integers in $L$. Clearly, $R_X \subset \Oc_L$.
By a classical theorem of A. Weil (Riemann's hypothesis) \cite{Mumford}, if $j \colon L_X=L \hookrightarrow \C$ is a field embedding then $j(\alpha)\overline{j(\alpha)}=q$.
This implies that if $\B$ is a maximal ideal in $\Oc_L$ such that $\fchar(\Oc_L/\B) \ne p$ then all elements of $R_X$ are $\B$-adic units.
The $p$-adic behaviour of $R_X$ is described in terms of the set $\Sl_X$ of {\sl slopes of the Newton polygon} of $X$ \cite{OortG} (see also \cite[Sect. 4]{ZarhinJPAA}). Recall that
$\Sl_X$ is a finite nonempty set of rational numbers that enjoys the following properties.
\begin{itemize}
\item[(i)] $0 \le c \le 1$ for all $c\in \Sl_X$.
\item[(ii)] $c\in \Sl_X$ if and only if $1-c\in \Sl_X$.
\item[(iii)] If $c\in \Sl_X$ then either $c=1/2$ or there is a positive integer $h \le g=\dim(X)$ such that $c\in \frac{1}{h}\Z$.
\item[(iv)] Let $\mathfrak{P}$
be any maximal ideal in $\Oc_L$ such that $\fchar(\Oc_L/\mathfrak{P})= p$ and let
$$\ord_{\PP} \colon L^{*} \to \Q$$
be the discrete valuation map attached to $\PP$ that is normalized by the condition
$$\ord_{\PP}(q)=1.$$
Then
$$\ord_{\PP}(R_X)=\Sl_X.$$
\item[(v)] If $\alpha\in R_X$ then
$$\ord_{\PP}(q/\alpha)=1-\ord_{\PP}(\alpha).$$
\item[(vi)] Let $\mu_{L}$ the multiplicative group of all roots of
unity in $L$. Then its image $\ord_{\PP}(\mu_{L})=\{0\}$.
\end{itemize}
Properties (i)-(vi) imply readily the following assertion.
\begin{lem}
\label{SLP}
Let $g$ be a positive integer.
Let us consider the set $\mathrm{Slp}(g)$ of all rational numbers $c$ that enjoy the following properties.
\begin{enumerate}
\item
$0 \le c \le 1$.
\item
Either $c=1/2$ or there exists a positive integer $h \le g$ such that $c\in \frac{1}{h}\Z$.
\end{enumerate}
Then $\mathrm{Slp}(g)$ is a {\sl finite nonempty} set that enjoys the following property.
{\sl If $X$ is a $g$-dimensional abelian variety over a finite field then $\Sl_X$ lies in} $\mathrm{Slp}(g)$.
\end{lem}
\begin{rems}
\label{ordP}
Let $S(p)$ be the set of all maximal ideals in $\Oc_L$ such that $\fchar(\Oc_L/\mathfrak{P})= p$. Since $L/\Q$ is Galois, $\#(S(p))$ divides $[L:\Q]$;
in particular, $\#(S(p))$ divides $g! 2^g$ (see Lemma \ref{basicL}(ii) below). Let us define a group homomorphism
$$w_X \colon \Gamma(X,K) \to \Q^{S(p)}, \gamma \mapsto \{\ord_{\PP}(\gamma)\}_{\PP\in S(p)}.$$
\begin{itemize}
\item[(a)]
Clearly,
$w_X(q)$ is the vector ${\bf 1}\in \Q^{S(p)}$, all whose coordinates equal $1$, hence
$$w_X\left(\frac{q}{\alpha}\right)={\bf 1}-w_X(\alpha) \ \forall \alpha \in R_X.$$
\item[(b)]
By Property (iv) and Lemma \ref{SLP},
$$w_X(R_X) \subset \Sl_X^{S(p)} \subset \mathrm{Slp}(g)^{S(p)} \subset \Q^{S(p)}.$$
\item[(c)]
It follows from Property (v) that a vector $\tilde{c} \in \Q^{S(p)}$ lies in $w_X(R_X)$ if and only if
${\bf 1}-\tilde{c} \in w_X(R_X)$.
\item[(d)]
In light of Property (vi), $w_X(\gamma)=0$ if $\gamma$ is a root of unity. The converse is also true:
it is proven in \cite[Prop. 2.1 on p. 249]{ZarhinTokyo} (see also \cite[Prop. 3.1.5]{ZarhinInv79}) that $\ker(w_X)$ {\sl consists of roots of unity}.
\item[(e)]
It follows readily from (d) that:
\begin{enumerate}
\item[(1)] none of elements in $R_X$ lies in $\ker(w_X)$;
\item[(2)]
if $k$ is {\sl not small} w.r.t $X$ and $\alpha_1, \alpha_2$ are distinct elements of $R_X$ then
$$w_X(\alpha_1) \ne w_X(\alpha_2).$$
\end{enumerate}
\end{itemize}
\end{rems}
\begin{lem}
\label{basicL}
\begin{itemize}
\item[]
\item[(i)]
The field $L_X$ is either $\Q$ or $\Q(\sqrt{p})$ or a CM field.
\item[(ii)]
The field $L_X$ is a finite Galois extension of $\Q$ and its degree $[L_X:\Q]$ divides $g! 2^g$.
\item[(iii)] $\#(S(p))$ divides $g! 2^g$.
\end{itemize}
\end{lem}
\begin{proof}
Let us prove (i,ii).
By definition of the splitting field, $L_X/\Q$ is Galois.
Suppose that $X$ is simple. According to \cite{Tate1,Tate2},
$\mathcal{P}_X(t)$ is a power $\mathcal{P}_{\mathrm{irr}}(t)^{a}$ of a $\Q$-irreducible monic
polynomial $\mathcal{P}_{\mathrm{irr}}(t)$ where $a$ is a positive integer dividing $2g$ and
$$\deg(\mathcal{P}_{\mathrm{irr}})=\frac{2g}{a}.$$
Clearly, $L_X$ is the normal closure of the degree $2g/a$ number field $E_X:=\Q[t]/\mathcal{P}_{\mathrm{irr}}(t)$.
According to \cite[Exemples]{Tate2}, $E_X$ is either $\Q$ or $\Q(\sqrt{p})$ or a CM field.
In the first two cases $L_X=\Q$ or $\Q(\sqrt{p})$; in particular, it is a totally real number field, whose
degree divides $2g=2\dim(X)$.
In the third case let $E_X^{+}$ be the maximal totally real subfield of $E_X$;
$[E_X^{+}:\Q]=g/a$ and $E_X$ is a purely imaginary quadratic extension $E_X^{+}(\sqrt{-\delta})$ of $E_X^{+}$.
Here $\delta$ is a totally positive element of $E_X^{+}$.
Let $L_X^{+}$ be the normal closure of $E_X^{+}$. Since $E_X^{+}$ is totally real, $L_X^{+}$ is totally real as well, and its degree $[L_X^{+}:\Q]$ divides
$[E_X^{+}:\Q]!=(g/a)!$. Since $-\delta \in E_X^{+}\subset L_X^{+}$, its Galois orbit in $L_X^{+}$ consists at most
of $[E_X^{+}:\Q]=g/a$ elements. This implies that $[L_X: L_X^{+}]$ divides $2^{g/a}$, since $L_X$ is obtained from
$L_X^{+}$ by adjoining square roots of all the (totally negative) Galois conjugates $-\delta$. This implies that
$L_X$ is a CM field and $[L_X:\Q]$ divides $(g/a)!\cdot 2^{g/a}$, which in turn, divides $g! \cdot 2^g$.
Now let us consider the general case when $X$ is isogenous to a product $\prod_{i=1}^m X_i$
of $m$ nonzero simple abelian varieties $X_i$.
It is well known that if we put
$$g_i:=\dim(X_i), \ \text{ and } L_i:=L_{X_i}$$
then
$$g=\dim(X)=\sum_{i=1}^m g_i, \ \mathcal{P}_{X}(t)=\prod_{i=1}^m \mathcal{P}_{X_i}(t).$$
Let $\bar{\Q}$ be an algebraic closure of $\Q$. We may and will view all $L_i$ as subfields of $\bar{\Q}$. Then
$L_X$ is the {\bf compositum} of $m$ number fields $L_{X_i}=L_i$ in $\bar{\Q}$. Applying (i,ii) to simple $X_i$'s,
we obtain that $L_X$ is either $\Q$ or $\Q(\sqrt{p})$ or a CM field, which proves (i).
In order to prove (ii), recall that all $L_{i}/\Q$ and $L_X/\Q$ are finite Galois extensions.
Let $\Gal(L_{i}/\Q)$ and $\Gal(L_{X}/\Q)$ be the corresponding (finite) Galois groups.
Clearly, each $L_{i}$ is a $\Gal(L_{X}/\Q)$-invariant subfield of $L_X$, and the corresponding restriction map
$$\Gal(L_{X}/\Q) \to \Gal(L_{i}/\Q), \ s \mapsto s_i$$
is a surjective group homomorphism. On the other hand, since all the $L_i$'s generate $L_X$ as a field, the product-map
$$\Gal(L_X/\Q) \to \prod_{i=1}^m \Gal(L_{i}/\Q), \ s \mapsto \{s_i\}_{i=1}^m$$
is a {\sl group embedding}. By Lagrange's theorem, $\#(\Gal(L_X/\Q) )$ divides $\prod_{i=1}^m \#(\Gal(L_{i}/\Q))$. In other words,
$[L_X:\Q]$ divides
$\prod_{i=1}^m [L_i:\Q]$, which, in turn, divides
$$\prod_{i=1}^m g_i! 2^{g_i}=2^g \prod_{i=1}^m g_i!.$$
Since $\sum_{i=1}^m g_i=g$, the product $\prod_{i=1}^m g_i!$ divides $g!$. This implies that
$[L_X:\Q]$ divides $2^g g!$, which ends the proof of (ii).
Let us prove (iii). Since $L_X/\Q$ is Galois, $\#(S(p))$ divides $[L_X:\Q]$. Now (iii) follows readily from (ii).
\end{proof}
\section{Multiplicative relations between Weil numbers}
\label{MultRel}
This section contains auxiliary results that will be used in Section \ref{RelationProof} in the proof of Theorem \ref{mainRelation}.
\begin{sect}
\label{involution}
{\bf The involution}.
Recall that there is an involution map
\begin{equation}
\label{iotaR}
\iota: R_X \to R_X, \ \alpha \mapsto \frac{q}{\alpha}=\bar{\alpha}.
\end{equation}
Let $R_X^{\iota}$ be the subset of fixed points of $\iota$. Its elements
(if there are any) are square roots of $q$; hence,
\begin{equation}
\label{iota2}
\#(R_X^{\iota})\le 2.
\end{equation}
In addition, if $k$ is {\sl not} small with respect to $X$ then at most {\sl one square root} of $q$ lies in $R_X$, hence,
\begin{equation}
\label{iota1}
\#(R_X^{\iota})\le 1.
\end{equation}
\begin{rem}
\label{ssF}
Suppose that $k$ is {\sl not} small w.r.t $X$.
If $\beta$ is an element of $R_X$ such that $\beta^2/q$ is a root of unity then
$q/\beta \in R_X$ and the ratio
$$\frac{\beta}{q/\beta}=\frac{q}{\beta^2}$$
is a root of unity. This implies that $\beta=q/\beta$, i.e., $\beta \in R_X^{\iota}$.
\end{rem}
Let us consider the free abelian group $\Z^{R_X}$ of functions $e: R_X \to \Z$. The involution $\iota$
induces an automorphism (also an involution)
$$\iota^{*} \colon \Z^{R_X} \to \Z^{R_X}, \ \iota^{*}e(\alpha):=e(\iota \alpha)=e(q/\alpha).$$
Let us consider the group homomorphism
$$\Pi \colon \Z^{R_X} \to \Gamma(X,k)\subset L_X^{*}, \ e \mapsto \prod_{\alpha\in R_X}\alpha^{e(\alpha)}.$$
We have
$$\Pi(\iota^{*}e)=\overline{\Pi(e)}.$$
\begin{rems}
\label{supersing}
\begin{itemize}
\item[]
\item[(i)]
If $\iota^{*}e=e$ then $\Pi(e)=\overline{\Pi(e)}$ is (totally) real; it follows from Weil's theorem that
$\Pi(e)^2$ is an integral power of $q$. In particular, if $\sum_{\alpha\in R_X}e(\alpha)$ is even then it follows
from Weil's theorem that $\Pi(e)$ is $\pm$ integral power of $q$.
\item[(ii)]
If $f$ is a function $R_X \to \Z$ then the function $e:=f+\iota^{*}f$ is obviously trivial.
Conversely, one may easily check that a function $e \colon R_X \to \Z$ is trivial if and only if there exists $f \colon R_X \to \Z$ such that
$e=f+\iota^{*}f$.
\item[(iii)]
Clearly, $e$ is admissible if and only if $\Pi(e)$ lies in the cyclic multiplicative subgroup $q^{\Z}$ generated by $q$.
\end{itemize}
\end{rems}
\end{sect}
\begin{sect}
\label{Ror}
{\bf Ranks and Orbits}.
The complement $R_X\setminus R_X^{\iota}$ splits (if it is not empty) into a disjoint union of
$2$-element orbits of $\iota$ say $\{\alpha,q/\alpha\}$. Let $r_X$ be the number of such orbits, which is a nonnegative integer that vanishes
if and only if $R_X = R_X^{\iota}$. We have
\begin{equation}
\label{rXdef}
\#(R_X\setminus R_X^{\iota})=2 r_X; \ r_X \le \frac{\#(R_X)}{2} \le \frac{2g}{2}=g.
\end{equation}
If $r_X\ge 1$ (i.e., $R_X \ne R_X^{\iota}$) then we have $r_X$ $2$-elements $\iota$-orbits $O_1, \dots, O_{r_X}$ in $R_X\setminus R_X^{\iota}$.
By choosing arbitrarily an element $\alpha_i \in O_i$ for all $i=1, \dots, r_X$, we get
\begin{equation}
\label{OR}
O_i=\{\alpha_i,q/\alpha_i\} \ \forall i=1, \dots, r_X; \ R_X \setminus R_X^{\iota}=\{\alpha_1, q/\alpha_1, \dots, \alpha_{r_X},q/\alpha_{r_X}\}.
\end{equation}
Recall \cite{ZarhinEssen}, that $\Gamma(X,k)$ always contains $q$. Since $\beta^2=q$ for all $\beta \in R_X^{\iota}$,
the subgroup $\Gamma_1(X,k)$ of $\Gamma(X,k)$ generated by $q$ and all elements of $R_X\setminus R_X^{\iota}$ has finite index
in $\Gamma(X,k)$. In particular,
\begin{equation}
\label{rkGamma1}
\rk(\Gamma_1(X,k))= \rk(\Gamma(X,k)).
\end{equation}
Clearly, if $r_X=0$ then $\Gamma_1(X,k)=q^\Z$ has rank $1$, hence $\rk(\Gamma(X,k))=1$.
It follows from \eqref{OR} that if $r_X \ge 1$ then $\Gamma_1(X,k)$ is generated by $\{\alpha_1, \dots, \alpha_{r_X};q\}.$
In particular,
\begin{equation}
\label{rankGamma1rX}
\rk(\Gamma_1(X,k)) \le r_X+1.
\end{equation}
\begin{rem}
\label{rankSmall}
Suppose that $k$ is {\sl not small} w.r.t $X$. Then $\#(R_X^{\iota})=0$ or $1$ and therefore
$\#(R_X)=2 r_X$ or $2 r_X+1$ respectively. In both cases
\begin{equation}
\label{rXR}
r_X=\lfloor\frac{\#(R_X)}{2}\rfloor.
\end{equation}
Combining \eqref{rXR} with \eqref{rankGamma1rX} and \eqref{rkGamma1}, we obtain that
\begin{equation}
\label{smallrXr}
\rk(\Gamma(X,k))= \rk(\Gamma_1(X,k))\le r_X+1=\lfloor\frac{\#(R_X)}{2}\rfloor+1.
\end{equation}
\end{rem}
\end{sect}
\begin{sect}{\bf Nontrivial and reduced admissible functions}.
\label{nontrivial}
The existence of a nontrivial admissible function implies certain restrictions on $R_X$.
\begin{lem}
\label{multiply}
Suppose that there exists a nontrivial admissible function $e \colon R_X \to \Z$ of degree, say, $d$.
Then the following conditions hold.
\begin{itemize}
\item[(i)] $R_X \ne R_X^{\iota}$, i.e.,
$R_X \setminus R_X^{\iota}$ is a nonempty subset of $R_X$.
\item[(ii)]
For each nonzero integer $m$ the function
$$m \cdot e \colon R_X \to \Z, \ \alpha \mapsto m\cdot e(\alpha)$$
is also nontrivial admissible.
\item[(iii)]
Let us consider the function
$e_0 \colon R_X \to \Z$ that vanishes identically on $R_X^{\iota}$ (if this subset is nonempty)
and coincides with $e$ on $R_X \setminus R_X^{\iota}$. Then $e_0$ is nontrivial.
In addition,
for each nonzero even integer $m$
the function
$$m \cdot e_0: R_X \to \Z, \ \alpha \mapsto m\cdot e_0(\alpha)$$
is nontrivial admissible, and its weight
\begin{equation}
\label{wte0}
\wt(m\cdot e_0)=|m|\wt(e_0) \le |m|\wt(e).
\end{equation}
\end{itemize}
\end{lem}
\begin{proof}
If $R_X^{\iota}=\emptyset$ then all three assertions of Lemma are obviously true. So, let us assume that
$R_X^{\iota}\ne\emptyset$.
(i) Suppose that $R_X=R_X^{\iota}$. Then
$$\prod_{\alpha\in R_X^{\iota}}\alpha^{e(\alpha)} =q^d \text{ and } \sum _{\alpha\in R_X^{\iota}}e(\alpha)=2d$$
is an even integer. Since $ R_X^{\iota}$ consists of one or two elements and $e$ is {\sl nontrivial}, there is $\beta \in R_X^{\iota}$
such the $e(\beta)$ is odd. This implies that $ R_X^{\iota}$ consists of two elements, say, $\beta$ and $-\beta$,
$$e(\beta)+e(-\beta)=2d$$
and both integers
$e(\beta)$ and $e(-\beta)$ are {\sl odd}. This implies that (recall that $\beta^2=q$)
$$q^d =\beta^{e(\beta)} \cdot (-\beta)^{e(-\beta)}=\beta^{e(\beta)} \cdot (-1)\cdot \beta^{e(-\beta)}=
-\beta^{e(\beta)+e(-\beta)}=-\beta^{2d}=-q^d.$$
So, $q^d=-q^d$, which is absurd. The obtained contradiction proves (i).
(ii).
The admissibility of $m\cdot e$ is obvious. The nontriviality is also clear if there exists $\alpha \in R_X\setminus R_X^{\iota}$ with
$$e(\alpha) \ne e(q/\alpha).$$
So, we may assume that $m\cdot e$ is trivial (we are going to arrive to a contradiction),
and
\begin{equation}
\label{push}
e(\alpha)=e(q/\alpha) \ \forall \alpha\in R_X\setminus R_X^{\iota}.
\end{equation}
This implies that
there is an integer $n$ such that
$$\prod_{\alpha\in R_X^{\iota}}\alpha^{e(\alpha)}=q^n.$$
It follows that
the sum
\begin{equation}
\label{TwoN}
\sum_{\alpha\in R_X^{\iota}}e(\alpha)=2n\in 2\Z
\end{equation}
is an {\sl even} integer. On the other hand,
the nontriviality of $e$ combined with \eqref{push} implies that there is $\beta \in R_X^{\iota}$ with {\sl odd} $e(\beta)$.
Since $\#(R_X^{\iota})\le 2$, it follows from \eqref{TwoN} that integer $e(\alpha)$ is odd for all $\alpha \in R_X^{\iota}$.
It follows that
$\prod_{\alpha\in R_X^{\iota}}\alpha =q^d$ for some integer $d$. Therefore $\#(R_X^{\iota})=2d$ is a positive even integer,
i.e., $R_X^{\iota}$ consists of two elements $\beta,-\beta$ with $\beta^2=q$; in addition, both integers
$e(\beta)$ and $e(-\beta)$ are odd. The same computations as in the proof of (i) give us that
$$q^d=\beta^{e(\beta)} (-\beta)^{e(-\beta)}=-q^d,$$
hence, $q^d=-q^d$. The obtained contradiction proves the nontriviality of $m \cdot e$.
(iii) Suppose that $e_0$ is trivial, i.e.,
$$e(\alpha)=e(q/\alpha) \ \forall \alpha \in R_X\setminus R_X^{\iota}.$$
Then it is admissible and therefore there is an integer $h$ such that
$$\prod_{\alpha \in R_X\setminus R_X^{\iota}}\alpha^{e_0(\alpha)}=\prod_{\alpha \in R_X\setminus R_X^{\iota}}\alpha^{e(\alpha)}=q^h.$$
Since $e$ is admissible of degree $d$,
$$ \prod_{\beta \in R_X^{\iota}}\beta^{e(\beta)}=q^{d-h}.$$
The nontriviality of $e$ implies that there is $\beta \in R_X^{\iota}$ such that integer $e(\beta)$ is {\sl odd}.
Now the same computations as in the proof of (i) give us that $R_X^{\iota}$ consists of two elements $\beta$ and $-\beta$,
both integers $e(\beta)$ and $e(-\beta)$ are odd and eventually, $q^{d-h}=-q^{d-h}$. The obtained contradiction proves that $e$ is {\sl nontrivial},
which is the first assertion of (iii).
Let us prove the second asssertion of (iii). Since $m$ is even, there is an integer $n$ such that $m=2n$.We have
$$q^{md}=\left(\prod_{\alpha \in R_X}\alpha^{e(\alpha)}\right)^m=\left(\prod_{\alpha \in R_X\setminus R_X^{\iota}}\alpha^{e(\alpha)}\right)^m
\times \left(\prod_{\beta \in R_X^{\iota}}\beta^{ e(\beta)}\right)^m=$$
$$\left(\prod_{\alpha \in R_X}\alpha^{m\cdot e_0(\alpha)}\right)
\times \left(\prod_{\beta \in R_X^{\iota}}\beta^{2n\cdot e(\beta)}\right)=$$
$$\left(\prod_{\alpha \in R_X}\alpha^{m \cdot e_0(\alpha)}\right)
\times \left(\prod_{\beta \in R_X^{\iota}} q^{n\cdot e(\beta)}\right).$$
This implies that $\prod_{\alpha \in R_X}\alpha^{m \cdot e_0(\alpha)}$ is an integral power of $q$,
i.e., $m \cdot e_0$ is {\sl admissible}. The nontriviality of $m \cdot e_0$ follows from the nontriviality of $e_0$,
because $e_0$ vanishes identically on $R_X^{\iota}$. This ends the proof of (iii).
The last assertion of (iii) about weights follows readily from obvious inequality $\wt(e_0) \le \wt(e).$
\end{proof}
It turns out that one may easily construct a reduced admissible function when $k$ is {\sl small} w.r.t $X$.
\begin{lem}
\label{rootsR}
Assume that there are distinct $\alpha_1, \alpha_2 \in R_X$ such that $\gamma:=\alpha_2/\alpha_1$
is a root of unity. Then there is a reduced admissible function $e \colon R_X \to \Z$,
whose weight $w$ enjoys the following properties.
$$w \le 4 \mathbf{e}_2(2g) \ \text{ if } p \ne 2; \ w \le 2 \mathbf{e}_3(2g)) \text{ if } p = 2.$$
\end{lem}
\begin{proof}
Clearly, $\gamma \in \Gamma(X,k)$.
By Lemma \ref{rootsG}, there is a positive integer $m$ such that
$$\gamma^m=1; \ m \le 2 \mathbf{e}_2(2g) \ \text{ if } p \ne 2; \ m \le \mathbf{e}_3(2g)) \text{ if } p = 2.$$
Hence, it suffices to produce a reduced multiplicative relation of weight $2m$.
To this end, notice that $q/\alpha_1\in R_X$ and
$$\alpha_2^m (q/\alpha_1)^m=q^m.$$
If $\alpha_2 \ne q/\alpha_1$ then we may define
$$e \colon R_X \to \Z, \ e(\alpha_2):=m, e(q/\alpha_1):=m; \ e(\alpha):=0 \ \text{ for all other } \alpha.$$
Clearly, $e$ is a {\sl reduced admissible} function of weight $2m$.
Suppose that $\alpha_2 = q/\alpha_1$. Since $\alpha_1 \ne \alpha_2,$
$$\alpha_1 \ne q/\alpha_1, \ \alpha_1^2 \ne q.$$
Then we have
$$q^m=(\alpha_1 \alpha_2)^m=\alpha_1^{2m}, \
\text{ i.e., }
\alpha_1^{2m}=q^m.$$
Now let us consider
$$e \colon R_X \to \Z, \ e(\alpha_1):=2m; \ e(\alpha):=0 \ \text{ for all other } \alpha.$$
Clearly, $e$ is a {\sl reduced admissible} function of weight
$2m$.
\end{proof}
The next Lemma asserts that the existence of a nontrivial admissible function implies the existence of a reduced admissible function,
whose weight we can control.
\begin{lem}
\label{nonTrivRed}
Let $w$ be a positive integer.
Suppose that $k$ is not small w.r.t $X$ and there exists a nontrivial admissible function of weight $\le w$.
Then there exist a nonempty subset $A_1\subset R_X$, an integer-valued function
$\tilde{e} \colon A_1 \to \Z$, and a positive integer $s\le w$ that enjoy the following properties.
\begin{enumerate}
\item[(1)]
$\forall \alpha \in A_1$ we have $\frac{q}{\alpha}\not\in A_1, \ \tilde{e}(\alpha)>0$.
\item[(2)]
\begin{equation}
\label{A1R}
\prod_{\alpha\in A_1}\alpha^{\tilde{e}(\alpha)}=q^s.
\end{equation}
In particular, if we define
$$f \colon R_X \to \Z, \ f(\alpha):=\tilde{e}(\alpha) \ \forall \alpha\in A_1; \ f(\alpha):=0 \ \forall \alpha\not\in A_1$$
then $f$ is a reduced admissible function of weight $2s\le 2w$ that vanishes identically on $R_X^{\iota}$.
\end{enumerate}
\end{lem}
\begin{proof}
By Lemma \ref{multiply}, $R_X\setminus R_X^{\iota}$ is {\sl not} empty.
Let $e \colon R_X \to \Z$ be a {\sl nontrivial} admissible function $e$ of weight $\le w$.
Let us consider (in the notation of Lemma \ref{multiply}) the function
$$h_2=2\cdot e_0 \colon R_X \to \Z.$$
It follows from Lemma \ref{multiply} that $h_2$ is nontrivial admissible,
it vanishes identically on $R_X^{\iota}$ and its weight does not exceed $2w$. This implies that
\begin{equation}
\label{needReduced}
\prod_{\alpha\in R_X\setminus R_X^{\iota}} \alpha^{h_2(\alpha)}=q^d, \ 2w \ge \sum_{\alpha\in R_X}|h_2(\alpha)|
\end{equation}
where $d$ is an integer such that
$$|d| \le \wt(h_2) \le 2w.$$
The nontriviality and vanishing everywhere at $R_X^{\iota}$ of $h_2$ imply that the subset $A$ of $R_X$ defined by
$$A: =\{\alpha \in R_X\mid h_2(\alpha) \ne h_2(q/\alpha)\}$$
is {\sl nonempty}. It follows from the very definition that $A$ is $\iota$-invariant and does {\sl not} meet $R_X^{\iota}$.
Let us define the subset $A_1$ of $A$ by
$$A_1:=\{\alpha\in A\mid h_2(\alpha)>h_2(q/\alpha)\} \subset A\subset R_X.$$
Clearly, if $\alpha \in A$ then $\alpha\in A_1$ if and only if $\bar{\alpha}=q/\alpha\not\in A_1$. This implies that
$A_1$ is {\sl nonempty} and $A$ is the {\sl disjoint union} of $A_1$ and $\iota(A_1)$. In particular,
$\#(A)=2\#(A_1)$.
On the other hand, if
$$B:=\{\beta\in R_X\setminus R_X^{\iota}\mid h_2(\beta)=h_2(q/\beta)\}\subset R_X\setminus R_X^{\iota}\}$$
then $B$ is $\iota$-invariant, $R_X\setminus R_X^{\iota}$ is a disjoint union of $A$ and $B$, and
$$\prod_{\beta\in B}\beta^{h_2(\beta)}=q^n$$
for some integer $n$ with
$$|n| \le \wt(h_2) \le 2w.$$
Since $R_X\setminus R_X^{\iota}$ is a disjoint union of $A$ and $B$, it follows from
\eqref{needReduced} that
$$\prod_{\alpha\in A}\alpha^{h_2(\alpha)}=\frac{q^d}{q^n}=q^{d-n}.$$
Since $A$ is a disjoint union of $A_1$ and $\iota(A_1)$, we get
$$q^{d-n}=\left(\prod_{\alpha\in A_1}\alpha^{h_2(\alpha)}\right)\times \left(\prod_{\alpha\in A_1}\iota(\alpha)^{h_2(\iota \alpha)}\right)=$$
$$\left(\prod_{\alpha\in A_1}\alpha^{h_2(\alpha)}\right)\times \left(\prod_{\alpha\in A_1}(q/\alpha)^{h_2(q/ \alpha)}\right)=$$
$$\left(\prod_{\alpha\in A_1}\alpha^{h_2(\alpha)-h_2(q/\alpha))}\right)\times q^m$$
where $m:=\sum_{\alpha\in A_1}h_2(q/\alpha)\in \Z$.
If we define the function
$$\tilde{e} \colon A_1 \to \Z, \ \alpha\mapsto h_2(\alpha)-h_2(q/\alpha)$$
then $\tilde{e}(\alpha)>0 \ \forall \alpha \in A_1$,
$$\sum_{\alpha \in A_1}\tilde{e}(\alpha)\le \sum_{\alpha\in A_1}\left(|h_2(\alpha)|+|h_2(q/\alpha)|\right)=\sum_{\alpha\in A}|h_2(\alpha)|\le \wt(h_2)\le 2w,$$
and
$$q^{d-n}=\left(\prod_{\alpha\in A_1}\alpha^{\tilde{e}(\alpha)}\right)\times q^m,$$
i.e.,
$$\prod_{\alpha\in A_1}\alpha^{\tilde{e}(\alpha)}=q^{d-n-m}.$$
It remains to put $s:=d-n-m$.
This ends the proof.
\end{proof}
\end{sect}
The following assertion contains Lemma \ref{rankGamma}. (Recall that $\Gamma_1(X,k)$ is defined in Subsection \ref{Ror}.)
\begin{lem}
\label{rank}
Suppose that $k$ is not small w.r.t $X$.
Then the following conditions are equivalent.
\begin{itemize}
\item[(1a)]
There is a nontrivial admissible function on $R_X$.
\item[(1b)]
There is a nontrivial admissible function on $R_X$ that vanishes at $R_X^{\iota}$.
\item[(2a)]
There is a reduced admissible function on $R_X$.
\item[(2b)]
There is a reduced admissible function on $R_X$ that vanishes at $R_X^{\iota}$.
\item[(3a)]
$\rk(\Gamma(X,k)) \le \lfloor\#(R_X)/2\rfloor$.
\item[(3b)]
$\rk(\Gamma_1(X,k)) \le \lfloor\#(R_X)/2\rfloor$.
\end{itemize}
\end{lem}
\begin{proof}
Obviously, (1b) implies (1a), (2b) implies (2a), (2a) implies (1a), and (2b) implies (1b). By Lemma \ref{nonTrivRed}, (1a) implies (2b).
This implies that (1a), (1b), (2a), (2b) are equivalent.
In light of \eqref{rkGamma1}, conditions (3a) and (3b) are equivalent.
In order to handle conditions (3), let us discuss the parity of $\#(R_X)$, using the observations and notation of Subsection \ref{involution}.
In order to check the equivalence of (1) and (3), let us start with the ``degenerate'' case $r_X=0$, i.e., $R_X=R_X^{\iota}=\{\beta\}$.
Then $\Gamma(X,k)$ is an infinite cyclic group generated by $\beta$ containing the index $2$ subgroup generated by $\beta^2=q$.
Therefore $\rk(\Gamma(X,k)) =1 > 0$, i.e., (3a) does {\sl not} hold. On the other hand, we have already seen (Lemma \ref{multiply})
that if $R_X=\{\beta\}=R_X^{\iota}$ then (1a) does {\sl not} hold.
So, we may assume that $R_X \ne R_X^{\iota}$. Then the positive integer
$r_X=\lfloor\#(R_X)/2\rfloor$ is the number of all $\iota$-orbits $O_1, \dots, O_{r_X}$ in $R_X \setminus R_X^{\iota}$, see Subsection \ref{Ror}.
If we choose any element $\alpha_i$ of $O_i$ for all $i$ then the $2r_X$-element set
$$R_X \setminus R_X^{\iota}=\{\alpha_1, q/\alpha_1, \dots, \alpha_{r_X},q/\alpha_{r_X}\}$$
and $\Gamma_1(X,k)$ is generated by $q$ and $\{\alpha_1,\dots, \alpha_{r_X}\}$, see Subsection \ref{Ror}.
Suppose that (3b) holds. This means that $\rk(\Gamma_1(X,k))\le r_X$.
Hence, there are $(r_X+1)$ integers $f_1, \dots, f_{r_X}; d$ {\sl not all} zeros, such that
\begin{equation}
\label{Delta}
\prod_{i=1}^{r_X}\alpha_i^{f_i}=q^d.
\end{equation}
Clearly, {\sl not all} $f_1, \dots, f_{r_X}$ are zeros. Let us define the function
\begin{equation}
\label{Deltaf}
e \colon R_X \to \Z, \ e(\alpha_i)=f_i \ \forall i=1, \dots, r_X; \ f(\alpha)=0 \ \text{ for all other } \alpha.
\end{equation}
In light of \eqref{Delta} and \eqref{Deltaf}, $e$ is a nontrivial admissible function. Hence,
(1a) holds.
Now assume that (1a) holds. Then (2b) holds, i.e., there are a {\sl nonempty} subset $A_1\subset R_X$, a function
$\tilde{e} \colon A_1 \to \Z$ and a positive integer $s$ that enjoy the following properties.
\begin{itemize}
\item[(i)] $A_1$ and $\iota(A_1)$ do not meet each other;
\item[(ii)] $ \tilde{e}(\alpha)>0 \ \forall \alpha\in A_1$;
\item[(iii)] $\prod_{\alpha \in A_1}\alpha^{\tilde{e}(\alpha)}=q^s$.
\end{itemize}
Let us put $n:=\#(A_1)$ and let $A_1=\{\alpha_1, \dots \alpha_n\}$. Then all
$O_i=\{\alpha_i,q/\alpha_i\}$ are disjoint $2$-element orbits in $R_X\setminus R_X^{\iota}$. In particular, $n\le r_X$.
If $n=r_X$ then
$\{\alpha_1, \dots, \alpha_{r_X}; q\}$ generate $\Gamma_1(X,k)$. The property (iii)
implies that the rank of this group does not exceed $r_X$, i.e., (3b) holds.
Now assume that $n<r_X$. Then there are precisely $(r_X-n)$ {\sl other} two-element $\iota$-orbits $O_j$
in $R_X$ ($j=n+1, \dots r_X$). If we pick for all $j$ an element $\delta_j \in O_j$ then $O_j=\{\delta_j, q/\delta_j\}$
($n+1 \le j \le r_X$). Then $\{\alpha_1, \dots, \alpha_n; \delta_{n+1}, \dots, \delta_{r_X};q\}$ generate a
subgroup of finite index in $\Gamma_1(X,k)$. The property (iii)
implies that the rank of this group does not exceed $r_X$, i.e., (3b) holds. This ends the proof.
\end{proof}
\section{Frames and Skeletons of Abelian Varieties over Finite Fields}
\label{RelationProof}
In the course of our proof of Theorem \ref{mainRelation} we will need the following notion.
\begin{defn}
Let $g$ be a positive integer. A $g$-frame is a triple $(M,r,U)$ that consists
of positive integers $M$ and $r$, and a finite subset
$$U \subset \Q^M$$ of {\sl nonzero} vectors
that enjoy the following properties.
\begin{itemize}
\item[(i)]
$M$ divides $2^g g!$, $r\le g$, and $\#(U)=2r$.
\item[(ii)]
$U \subset \Sl(g)^M\subset \Q^M$
(see Lemma \ref{SLP} for the definition of the finite subset $\Sl(g)\subset \Q$).
\item[(iii)] A vector $u\in \Q^M$ lies in $U$ if and only if
$\mathbf{1}-u$ lies in $U$. Here
$\mathbf{1}=(1, \dots,1)\in \Q^M$ is the vector, all whose coordinates are $1$.
\item[(iv)]
Let $\Delta(U)$ be the additive subgroup of $\Q^M$ generated by $\mathbf{1}$ and all elements of $U$.
Then the rank of $\Delta(U)$ does not exceed $r$.
\end{itemize}
\end{defn}
\begin{rem}
\label{finiteF}
The {\sl finiteness} of $S(g)$ implies that the set of all frames (for a given $g$) is finite.
\end{rem}
\begin{sect}
\label{invF}
The map
\begin{equation}
\label{iotaF}
\iota_F \colon \Q^M \to \Q^M, \ u \mapsto \mathbf{1}-u
\end{equation}
is an involution, whose only fixed point is
$$\frac{1}{2}\cdot \mathbf{1}=(1/2, \dots,1/2).$$
Notice that
$$\iota_F(U)=U.$$
Since $\#(U)$ is {\bf even}, $U$ does {\sl not} contain the {\sl fixed point} $\frac{1}{2}\cdot \mathbf{1}$
and therefore splits
into a disjoint union of $2$-element $\iota_F$-orbits $O_1, \dots, O_r$.
If we choose in each $O_i$ a vector $u_i \in O_i$ then
\begin{equation}
\label{UrOrbits}
O_i=\{u_i,\mathbf{1}-u_i\} \ \forall i=1, \dots, r; \ U=\{u_1, ,\mathbf{1}-u_1, \dots, u_i, \mathbf{1}-u_i, \dots, \mathbf{1}-u_r\}
\end{equation}
Property (iv) combined with \eqref{UrOrbits} implies that there exist integers $a_1, \dots, a_r$ not all zeros and an integer $d$
such that
\begin{equation}
\label{relationF}
\sum_{i=1}^r a_i u_i=d \cdot \mathbf{1}=(d,\dots, d).
\end{equation}
\end{sect}
\begin{lem}
\label{boundU}
Let $g$ be a positive integer. Then there is a positive integer $C(g)$ that depends only on $g$ and enjoys the following property.
Let $(M,r,U)$ be a $g$-frame. Then there are exist $r$ integers $a_1, \dots, a_r$ not all zeros, an integer $d$,
and $r$ distinct vectors $u_1, \dots ,u_r$ in $U$ such that:
\begin{itemize}
\item[(i)]
the $2r$-element set
$U=\{u_1, \mathbf{1}-u_1, \dots, u_i,\mathbf{1}-u_r\}$;
\item[(ii)] $\sum_{i=1}^r a_i u_i=d \cdot \mathbf{1}=(d,\dots, d)$;
\item[(ii)] $\sum_{i=1}^r |a_i| \le C(g)$.
\end{itemize}
\end{lem}
\begin{proof}
The assertions follow readily from the construction of Subsection \ref{invF} combined with Remark \ref{finiteF}.
\end{proof}
\begin{sect}
\label{skeleton}
Let $X$ be a $g$-dimensional abelian variety over a finite field $k$ of characteristic $p$.
Suppose that $k$ is {\sl not small} with respect to $X$ and there exists a nontrivial admissible function $R_X \to \Z$. The aim of this subsection is to assign to $X$
a certain $g$-frame that we call the {\sl skeleton} of $X$.
First, let us put $r:=r_X$ and
$M:=M_X:=\#(S(p))$ where $S(p)$ is the set of maximal ideals in $\Oc_{L_X}$ that lie above $p$ (see Remark \ref{ordP}).
It follows from Lemma \ref{basicL} that $M$ divides $2^g\cdot g!$.
By Lemma \ref{multiply}, the existence of a nontrivial admissible function implies that $R_X \ne R_X^{\iota}$ and $r=r_X$ is a positive integer.
In addition (see \eqref{rXdef}),
$$r \le g, \ 2r=\#(R_X\setminus R_X^{\iota}).$$
Let us choose an order on the $M$-element set $S(p)$. This allows us to identify $S(p)$ with $\{1, \dots, M\}$ and $\Q^{S(p)}$ with $\Q^M$.
Let us put
$$U=U_X:=w_X(R_X\setminus R_X^{\iota})\subset \Q^{S(p)}=\Q^M$$
(where homomorphism $w_X$ is defined in Remark \ref{ordP}).
It follows from Remark \ref{ordP}(d) that the map
\begin{equation}
\label{inRX}
R_X\setminus R_X^{\iota} \to U_X, \ \alpha \mapsto w_X(\alpha)
\end{equation}
is {\sl injective}; in particular,
$$2r=2 r_X=\#(R_X\setminus R_X^{\iota})=\#(U_X).$$
Since $\ker(w_X)$ consists of roots of unity (see Remark \ref{ordP}(d)), the rank of $\Delta(U_X)$ coincides
with the rank of multiplicative $\Gamma_1(X,k)$ generated by $R_X\setminus R_X^{\iota}$. The existence of a nontrivial
admissible function implies (thanks to Theorem \ref{rank}) that
\begin{equation}
\label{rankU}
\rk(\Delta(U_X))=\rk(\Gamma_1(X,k)) \le r_X.
\end{equation}
I claim that $(M_X,r_X,U_X)$ is a $g$-frame. Indeed, it follows from Remarks \ref{ordP} that
\begin{equation}
\label{ordPrevisited}
w_X(q)=\mathbf{1}; w_X(\alpha)\ne 0, \ w_X(q/\alpha)=\mathbf{1}-w_x(\alpha) \ \forall \alpha \in R_X\setminus R_X^{\iota}.
\end{equation}
This implies that $(M_X,r_X,U_X)$ enjoys the properties (i)-(iii).
As for (iv), its validity follows from \eqref{rankU}.
\end{sect}
\begin{proof}[Proof of Theorem \ref{mainRelation}]
Let $g$ be a positive integer.
In light of Lemma \ref{rootsR}, we may and will assume that $k$ is {\sl not small} w.r.t. $X$.
In light of Lemma \ref{nonTrivRed}, it suffices to prove the following assertion.
{\bf Claim}. {\sl There exists a positive integer $E(g)$ that depends only on $g$ and enjoys the following property.
Suppose that $X$ is a $g$-dimensional abelian variety over a finite field $k$ such that $k$ is not small w.r.t. $X$ and there exists a nontrivial admissible function $R_X \to \Z$.
Then there exists a nontrivial admissible function $R_X \to \Z$
of weight $\le E(g)$.}
\begin{proof}[Proof of Claim]
Let $X$ be an $g$-dimensional abelian variety over a finite field $k$ such that $k$ is not small w.r.t. $X$ and there exists a nontrivial admissible function $R_X \to \Z$.
Let us consider the corresponding $g$-frame $(M_X,r_X,U_X)$. It follows from the injectiveness of the map \eqref{inRX} combined with Lemma \ref{boundU}
that there exist $r_X$ distinct elements $\alpha_1, \dots, \alpha_{r_X} \in R_X\setminus R_X^{\iota}$, $r_X$ integers $a_1, \dots, a_{r_X}$, and an integer $d$
that enjoys the following properties.
\begin{enumerate}
\item[(1)]
$R_X\setminus R_X^{\iota}=\{\alpha_1,q/\alpha_1 \dots, \alpha_{r_X},q/\alpha_{r_X}\}$.
\item[(2)]
Not all $a_1, \dots a_{r_X}$ are zero.
\item[(3)]
$\sum_{i=1}^{r_X} a_i w_X(\alpha_i)=d \cdot \mathbf{1}=(d,\dots, d)$.
\item[(4)] $\sum_{i=1}^r |a_i| \le C(g)$.
(Here $C(g)$ is as in Lemma \ref{boundU}.)
\end{enumerate}
It follows from Remark \ref{ordP}(d) that there exists
a root of unity $\gamma \in \Gamma(X,k)$ such that
$$\prod_{i=1}^{r_X}\alpha_i^{a_i}=q^d \gamma.$$
According to Lemma \ref{rootsG}, there exists a positive integer
$m \le D(g)$ such that $\gamma^m=1$. (See Lemma \ref{rootsG}
for the explicit formula of $D(g)$.) This implies that
$$\prod_{i=1}^{r_X}\alpha_i^{ma_i}=q^{md}.$$
This implies that the function
$$e \colon R_X \to \Z, \ e(\alpha_i)=m\cdot a_i \ \forall \alpha_i, \ e(\alpha)=0 \ \text{ forall other } \alpha$$
is admissible. On the other hand, it follows from properties (1) and (2) that $e$ is {\sl nontrivial}.
In order to finish the proof of Claim, one has only to notice
that
$$\wt(e)=\sum_{i=1}^{r_X}|a_i|=m \sum_{i=1}^{r_X}|a_i| \le D(g)\cdot C(g)=:E(g).$$
\end{proof}
This ends the proof of Theorem \ref{mainRelation}.
\end{proof}
\section{Applications of Gordan's Lemma}
\label{GordanL}
In order to prove Theorem \ref{semigroupADM}, we need the following
variant of a classical result of P. Gordan.
\begin{lem}
\label{GordanVar}
Let $m$ and $s$ be positive integers and $v_1, \dots v_s$ be elements of $\Q^m$. Let us consider the addditive semigroup
$$W=\{u \in \Z_{+}^m \mid u \cdot v_j=0 \ \forall j=1, \dots, s\}\subset \Z_{+}^m.$$
Then $W$ is a finitely generated semigroup of $\Z_{+}^m$.
\end{lem}
\begin{proof}
Replacing all $v_j$ by $N v_j$, where $N$ is a sufficiently divisible positive integer, we may and will assume that $v_j \in \Z^m$ for all $j=1, \dots, s$.
Let us consider the rational polyhedral cone
$\sigma\subset \RR^m$ that is generated by the standard basis of $\RR^m$ and all the vectors $\{v_1, \dots v_s\}$. Then
the dual cone is
$$\sigma^{\vee}=\{u\in \RR_{+}^m\mid u \cdot v_j\ge 0 \ \forall j=1, \dots, s\}.$$
By Gordan's Lemma \cite[Ch. 1, Prop. 1.2.17]{Cox}, $\sigma^{\vee}\cap \Z^m$ is a finitely generated additive semigroup. Let $G$ be a finite subset of $\sigma^{\vee}\cap \Z^m$
that contains $0$
and generates $\sigma^{\vee}\cap \Z^m$.
Then the intersection $G\cap W$ is a finite subset of $W$ that contains $0$. I claim that $G\cap W$ generates $W$ as a semigroup. Indeed, if $w \in W$ then $w \in \sigma^{\vee}\cap \Z^m$ and therefore there exists a positive integer $r$ and (not necessarily distinct) $r$ elements $g_1, \dots g_r \in G$ such that
$w=\sum_{i=1}^r g_i$. We have for all $j=1, \dots, s$
$$0=w \cdot v_j=\sum_{i=1}^r g_i\cdot v_j, \ g_i\cdot v_j \ge 0 \ \forall i=1, \dots, s.$$
This implies that all $g_i\cdot v_j=0$ and therefore all $g_i \in W$, i.e., $g_i \in G\cap W$.
It follows that $G\cap W$ generates $W$ as a semigroup.
\end{proof}
We also need the following elementary observation.
\begin{lem}
\label{F2q}
Suppose that $X$ is an abelian variety of positive dimension $g$ over a finite field $k$ with $q$ elements.
Suppose that $k$ is sufficiently large w.r.t. $X$.
Then a nonnegative integer-valued function $e \colon R_X \to \Z_{+}$ of even weight is admissible if and only if
\begin{equation}
\label{normalizedF}
w_X\left(\prod_{\alpha\in R_X}(\alpha^2/q)^{e(\alpha)}\right)=0 \in \Q^{S(p)}.
\end{equation}
\end{lem}
\begin{rem}
\label{AdmEven}
Let $e \colon R_X \to \Z_{+}$ be an admissible nonnegative integer-valued function.
Then its weight is twice its degree and therefore is {\sl even}.
\end{rem}
\begin{proof}[Proof of Lemma \ref{F2q}]
Since $e$ is nonnegative, its weight coincides with
$$\sum_{\alpha\in R_X} e(\alpha)=:n.$$
Since this weight is even,
there is a nonnegative integer $d$ such that $n=2d$.
Now notice that in light of Remark \ref{ordP}(d),
\eqref{normalizedF}
holds if and only if $\prod_{\alpha\in R_X}(\alpha^2/q)^{e(\alpha)}$ is a root of unity.
This means that $\prod_{\alpha\in R_X}(\alpha^2/q)^{e(\alpha)}=1$, because $k$ is sufficiently large w.r.t. $X$. Hence,
\eqref{normalizedF} means that
$$\left(\prod_{\alpha\in R_X}\alpha^{e(\alpha)}\right)^2=q^n \ \text{ with } n=\sum_{\alpha\in R_X} e(\alpha)=2d.$$
This means that
\begin{equation}
\label{pmqd}
\prod_{\alpha\in R_X}\alpha^{e(\alpha)}=\pm q^d.
\end{equation}
Since torsion-free $\Gamma(X,k)$ does not contain $-1$, \eqref{pmqd} is equivalent to
$$\prod_{\alpha\in R_X}\alpha^{e(\alpha)}= q^d,$$
i.e, $e$ is admissible.
\end{proof}
\begin{proof}[Proof of Theorem \ref{semigroupADM}]
Let $X$ be an abelian variety of positive dimension $g$ over a finite field $k$ of characteristic $p$. Suppose that $k$ is sufficiently large w.r.t. $X$.
Let us put $s:=\#(S(p))$. By Lemma \ref{basicL}, $s$ divides $2^g \cdot g!$. Let us choose an order in $S(p)$. This allows us to identify $S(p)$ with $\{1, \dots,s\}$
and $ \Q^{S(p)}$ with $\Q^s$.
Let us choose an order on $R_X$: it allows us to list elements of $R_X$ as $\{\alpha_1, \dots, \alpha_m\}$ with $m=\#(R_X)$. We have $m \le 2g$. Let us consider an additive group homomorpism
$$\tilde{w}_X \colon \Z^m \to \Q^{S(p)}=\Q^s, \ u=(a_1, \dots a_m)\mapsto w_X\left(\prod_{i=1}^m (\alpha_i^2/q)^{a_i}\right)=$$
$$2\sum_{i=1}^m a_i w_X(\alpha_i)-\left(\sum_{i=1}^m a_i\right)\cdot \mathbf{1}.$$
Clearly, there is a unique collection of $s$ vectors $v_1, \dots v_s \in \Q^m$ such that
$$\tilde{w}_X(u)=(u\cdot v_1, \dots , u\cdot v_s) \ \forall u \in \Z^m.$$
It is also clear that all the coordinates of all $v_j$'s lie in the same finite set
$$2\cdot S(g)-1:=\{2c-1\mid c \in S(g)\} \subset \Q$$
that depends only on $g$. This implies that all the $v_j$'s lie in the same finite subset
$$\left(2\cdot S(g)-1\right)^m \subset \Q^m$$
of $\Q^m$ that depends only on $g$ and $m$. Combining this assertion with Lemma \ref{GordanVar}, we obtain
that for each positive integers $m \le 2g$ and $s$ dividing $2^g \cdot g!$ there is a {\sl finite subset}
$F_0(g,m,s) \in \Z_+^{m}$ that depends only on $g$, $m$, $s$ and enjoys the following property.
{\sl If $\#(R_X)=m$ and $\#(S(p))=s$ then the additive semigroup $\ker(\tilde{w}_X)\cap \Z_{+}^m$ of $\Z_{+}^m$ is generated by a certain subset of $F_0(g,m,s)$.}
Now let as define the weight $\wt(u)$ of any $u=(a_1, \dots, a_m) \in \Z_{+}^m$ as $\sum_{i=1}^m a_i$. It follows from Remark \ref{AdmEven} combined with Lemma \ref{F2q}
that an integer-valued nonnegative function
$$\mathbf{b}_u \colon R_X \to \Z_{+}, \ \alpha_i \mapsto a_i$$
is {\sl admissible} if and only if $u \in \ker(\tilde{w}_X)\cap \Z_{+}^m$ and $\wt(u)$ is {\sl even}. (It is also clear that each admissible nonnegative function $e: R_X \to \Z_{+}$
coincides with $\mathbf{b}_u$ for exactly one vector $u\in \Z_{+}^m$.) Then such $u$ may be presented as a sum of (not necessarily distinct) elements of $F_0(g,m,s)$. It may happen that
some elements of $F_0(g,m,s)$ in this sum have odd weight. Since the weight of $u$ is even, the number of such summands is even. By grouping them in pairs, we obtain
that $u$ is a finite sum of some even weight elements from $F_0(g,m,s)$ and even weight elements from $F_0(g,m,s)+F_0(g,m,s)\subset \Z_{+}^m$. Now let $F(g,m,s) \subset \Z_{+}^m$ be
the (finite) set of all even weight vectors from $F_0(g,m,s)$ and from
$F_0(g,m,s)+F_0(g,m,s)$. Clearly, each $\mathbf{u} \in F(g,m,s)$ gives rise to nonnegative admissible
$\mathbf{b}_{\mathbf{u}} \colon R_X \to \Z_{+}$ and each nonnegative admissible $e \colon R_X \to \Z_{+}$ may be presented as a linear combination of such $\mathbf{b}_{\mathbf{u}}$'s with nonnegative integer coefficients.
Now one only has to choose as $H(g)$ the largest of the weights of $\mathbf{b}$ among all $\mathbf{u}$ (with {\sl even weight})
in the union of all
$F(g,m,s)$ where $1 \le m \le 2g$ and $s\mid 2^g \cdot g!$.
\end{proof}
Theorem \ref{semigroupADM} implies readily the following assertion.
\begin{cor}
\label{contractionE}
Let $g$ be a positive integer and $H(g)$ be as in Theorem \ref{semigroupADM}.
Let $X$ an abelian variety of positive dimension $g$ over a finite field $k$. Let $e: R_X \to \Z_{+}$
be a nonnegative admissible function. If $\wt(e)>H(g)$ then $e$ may be presented as a sum
$e=f_1+f_2$ of two nonnegative admissible functions
$$f_1 \colon R_X \to \Z_{+}, \ f_2 \colon R_X \to \Z_{+}$$
such that
$2 \le \wt(f_2)\le H(g)$.
\end{cor}
\section{Linear Algebra}
\label{linAlgebra}
Throughout this section, $V$ is a nonzero vector space of finite dimension $n$ over a field $K$ of characteristic $0$, and $E$ is an overfield of $K$. We write
$V_E$ for the $E$-vector space $V\otimes_K E$ of the $E$-dimension $n$. Let us put
$$V^{*}=\Hom_K(V,K), \ V_E^{*}=\Hom_E(V_E,E).$$
Let $A \colon V \to V$ be a $K$-linear operator and
$$A^{*} \colon V^{*} \to V^{*}, \ \phi \mapsto \phi\circ A \ \forall \phi\in V^{*}.$$
As usual, let us define
$$A_E \in \End( V_E), \ A_E (v\otimes e)= Av\otimes e \ \forall v \in V, e \in E.$$
Clearly,
$$(A_E)^{*}=(A^{*})_E \colon V_E^{*} \to V_E^{*}.$$
\begin{rem}
\label{eigenKE}
Let $a \in K\subset E$ and $V(a)$ (resp. $V_E(a)$) be the eigenspace of $A$ (resp. of $A_E$) attached to eigenvalue $a$.
It is well known that the natural $E$-linear map
$$V(a)\otimes_K E \to V_E(a)$$ is an isomorphism of $E$-vector spaces; in particular,
$$\dim_K(V(a))=\dim_E( V_E(a)) \ \forall a \in K\subset E.$$
\end{rem}
There are well known natural isomorphisms \cite[Ch. III, Sect. 7, Prop. 7]{Bourbaki} of graded $K$-algebras
$$\wedge(V^*)=\oplus_{j=0}^{n}\wedge_K^{j} (V^*)=\oplus_{j=0}^{n}\Hom_K(\wedge_K^j (V),K)$$
and of graded $E$-algebras \cite[Ch. III, Sect. 7, Prop. 8]{Bourbaki}
$$\wedge(V_E^*)=\oplus_{j=0}^{n} \wedge_E^j (V_E^*)=\oplus_{j=0}^{n}\Hom_E(\wedge_E^j (V_E),E)=\oplus_{j=0}^{n}\Hom_K(\wedge_K^j (V),K)\otimes_K E,$$
which give rise to the natural isomorphisms of $E$-vector spaces
\begin{equation}
\label{baseEKchange}
\wedge_K^{j} (V^*)_E \cong \wedge_E^j (V_E^*).
\end{equation}
\begin{sect}
\label{wedgeImage}
Let $i$ and $j$ be nonnegative integers.
The multiplication in $\wedge(V^*)$ (resp. in $\wedge(V_E^*)$) gives rise to the surjective $K$-linear map
\begin{equation}
\label{wedgeKsurGeneral}
\Lambda_{i,j,K} \colon \wedge_K^{i} (V^*)\otimes_K \wedge_K^{j} (V^*)\twoheadrightarrow \wedge_K^{i+j} (V^*), \ \psi_i\otimes \psi_j \mapsto \psi_i\wedge \psi_j
\end{equation}
and to the surjective $E$-linear map
\begin{equation}
\label{wedgeEsurGeneral}
\Lambda_{i,j,E} \colon \wedge_E^{i} (V_E^*)\otimes_E \wedge_E^{j} (V_E^*)\twoheadrightarrow \wedge_E^{i+j} (V_E^*), \ \psi_i\otimes \psi_j \mapsto \psi_i\wedge \psi_j.
\end{equation}
Let $U$ be a $K$-vector subspace in $\wedge_K^{i} (V^*)$ and $W$ be a $K$-vector subspace in $\wedge_K^{j} (V^*)$. Then obviously the images
$\Lambda_{i,j,K}(U\otimes_K W)\subset \wedge_K^{i+j} (V^*)$ and $\Lambda_{i,j,E}(U_E\otimes_E W_E)\subset \wedge_E^{i+j} (V_E^*)$ are related by
\begin{equation}
\label{wedgeUVKE}
\Lambda_{i,j,E}(U_E\otimes_E W_E)=\Lambda_{i,j,K}(U\otimes_K W)_E.
\end{equation}
Here we identify $U_E$ (resp. $W_E$) with its isomorphic image in $\wedge_K^{i} (V^*)_E=\wedge_E^{i} (V_E^*)$
(resp. in $\wedge_K^{j} (V^*)_E=\wedge_E^{j} (V_E^*)$).
The equality \eqref{wedgeUVKE} implies readily its own generalization. Namely, let $n$ be a positive integer and suppose that for each positive integer $r\le n$
we are given $K$-vector subspaces
$$U_r \subset \wedge_K^{i} (V^*), \ W_r \subset \wedge_K^{j} (V^*).$$
Then
\begin{equation}
\label{wedgeUWrKE}
\sum_{r=1}^n\Lambda_{i,j,E}(U_{r,E}\otimes_E W_{r,E}) =\left(\sum_{r=1}^n \Lambda_{i,j,K}(U_r\otimes_K W_r)\right)_E.
\end{equation}
Here
$$U_{r,E}=U_r\otimes_K E, \ W_{r,E}=W_r\otimes_K E.$$
\end{sect}
\begin{sect}
The operators $A^{*}$ and $A_E^{*}$ give rise to the graded $K$-algebra and graded $E$-algebra {\sl endomorphisms}
$$\wedge(A^*) \colon \wedge(V^*) \to \wedge(V^*), \ \wedge(A_E^*): \wedge(V_E^*) \to \wedge(V_E^*)$$
\cite[Ch. III, Sect. 7, Prop. 2]{Bourbaki}, whose homogeneous components are
$K$-linear and $E$-linear operators
$$\wedge^j(A^{*})\colon \wedge_K^{j} (V^*) \to \wedge_K^{j} (V^*), \ \wedge^j_E(A_E^{*}):\wedge_E^{j} (V_E^*) \to \wedge_E^{j} (V_E^*)$$
respectively,
such that
\begin{equation}
\label{WEK}
\wedge^j(A_E^{*})=\wedge^j (A^{*})_E.
\end{equation}
Since $\wedge(A)$ and $\wedge(A_E^*)$ respect the multiplication in $\wedge(V^*)$ and $ \wedge(V_E^*) $ respectively,
\begin{equation}
\label{multipliWedge}
\wedge^i(A^{*})(\psi_i)\wedge \ \wedge^j(A^{*})(\psi_j)=\wedge^{i+j}(A^{*})(\psi_i\wedge \psi_j)\in \wedge^{i+j} (V^{*}) \ \forall \psi_i \in \wedge^i (V^{*}), \psi_j \in \wedge^j (V^{*});
\end{equation}
$$\wedge^i(A_E^{*})(\psi_{i,E})\wedge \ \wedge^j(A_E^{*})(\psi_{j,E})=\wedge^{i+j}(A_E^{*})(\psi_{i,E}\wedge \psi_{j,E}) \in \wedge^{i+j} (V_E^{*})$$
$$ \forall \psi_{i,E} \in \wedge^i (V_E^{*}),
\psi_{j,E} \in \wedge^j (V_E^{*}).$$
The following assertion is an immediate corollary of \eqref{multipliWedge} and \eqref{WEK}.
\end{sect}
\begin{lem}
\label{product EigenSpaces}
Let $j_1, j_2$ be positive integers such that $j_1+j_2 \le \dim(V)$. Let $\lambda_1,\lambda_2$ be elements of $K$. Let
$\wedge^{j_r}_K (V^{*})(\lambda_r)\subset \wedge^{j_r}_K (V^{*})$ be the eigenspace of $\wedge^j_r(A^{*})$ attached to $\lambda_r$ ($r=1,2$).
Then the image of the $K$-linear map
$$\wedge^{j_1}_K (V^{*})(\lambda_1)\otimes_K \wedge_K^{j_2} (V^{*})(\lambda_2) \to \wedge_K^{j_1+j_2} (V^{*}), \ \psi_{j_1}\otimes\psi_{j_2} \mapsto \psi_{j_1}\wedge\psi_{j_2}$$
lies in the eigenspace $\wedge^{j_1+j_2}_K (V^{*})(\lambda_1\lambda_2)$ of $\wedge^{j_1+j_2}(A^{*})$ attached to $\lambda_1 \lambda_2$.
\end{lem}
\begin{rem}
\label{restrictionLambdaK}
The $K$-linear map in Lemma \ref{product EigenSpaces}
is the restriction of $\Lambda_{j_1,j_2,K}$ defined in Subsection \ref{wedgeImage}.
\end{rem}
\begin{rem}
\label{eigenKEwedge}
Applying Remark \ref{eigenKE} to $\wedge^j (A^{*}) \colon \wedge_K^{j} (V^*) \to \wedge_K^{j} (V^*)$ (instead of $A \colon V \to V$), we obtain that if $\lambda \in K\subset E$
and $\wedge_K^{j} (V^*)(\lambda)$ (resp. $\wedge^j(V_E^{*})(\lambda)$) is the attached to $\lambda$ eigenspace of $\wedge^j (V^{*})$ (resp. of $\wedge^j(V_E^{*})$)
then the natural $E$-linear map
$$\wedge_K^{j} (V^*)(\lambda)\otimes_K E \to \wedge^j(V_E^{*})(\lambda)$$
induced by \eqref{baseEKchange} is an isomorphism. Combining this assertion with Lemma \ref{product EigenSpaces} applied twice (over $K$ and over $E$),
we get immediately the following assertion.
\end{rem}
\begin{lem}
\label{WedgeEigen} Let $j_1, j_2$ be positive integers such that $j_1+j_2 \le \dim(V)$.
Let $\lambda_1,\lambda_2 \in K \subset E$.
We keep the notation and assumptions of Lemma \ref{product EigenSpaces}.
The $E$-linear map
$$\wedge^{j_1}_E (V_E^{*})(\lambda_1)\otimes_E \wedge^{j_2}_E (V_E^{*})(\lambda_2) \to \wedge_E^{j_1+j_2} (V^{*})(\lambda_1\lambda_2), \ \psi_1\otimes\psi_1 \mapsto \psi_1 \wedge\psi_2$$
is not surjective if and only if the $K$-linear map
$$\wedge^{j_1}_K(V^{*})(\lambda_1)\otimes_K \wedge^{j_2}_K (V^{*})(\lambda_2) \to \wedge^{j_1+j_2}_K (V^{*})(\lambda_1\lambda_2), \ \psi_1\otimes\psi_1 \mapsto \psi_1 \wedge\psi_2$$
is not surjective. Here $\wedge^{j}_E (V_E^{*})(\lambda)\subset \wedge^{j}_E (V_E^{*})$ is the eigensubspace of $\wedge^j(A_E^{*})$ attached to $\lambda$.
\end{lem}
\begin{sect}{\bf Main construction.}
\label{multEigen}
We keep the notation of Remark \ref{eigenKEwedge}. Suppose that $A_E \colon \to V_E \to V_E$ is {\sl diagonalizable},
$\mathrm{spec}(A)\subset E$ is the set of its {\sl eigenvalues}, and $\mult_A \colon \mathrm{spec}(A) \to \Z_{+}$ is the integer-valued function
that assigns to each eigenvalue of $A_E$ its multiplicity.
Let $\lambda \in K$ and $j \le \dim(V)$ be a positive integer.
Let us consider an integer-valued
function $e \colon \mathrm{spec}(A) \to \Z_{+}$ that enjoys the following properties.
\begin{itemize}
\item[(i)] $e(\alpha)\le \mult_A(\alpha) \ \forall \alpha \in \mathrm{spec}(A)$;
\item[(ii)] $\sum_{\alpha\in \mathrm{spec}(A)} e(\alpha)=j$;
\item[(iii)] $\prod_{\alpha\in \mathrm{spec}(A)}\alpha^{ e(\alpha)}=\lambda$.
\end{itemize}
Let us choose an {\sl eigenbasis} $B$ of $E$-vector space $V_E$ w.r.t. $A_E$ and let
$$\pi \colon B \twoheadrightarrow \mathrm{spec}(A)$$ be
the surjective map that assigns to each eigenvector $x \in B$ the corresponding eigenvalue of $A_E$. Clearly,
for every eigenvalue $\alpha\in \mathrm{spec}(A)$ the preimage $\pi^{-1}(\alpha)$ consists of $\mult_A(\alpha)$ elements of $B$.
Let $$B^{*}=\{x^{*}\mid x\in B\}$$ be the basis of $V_E^{*}$ that is dual to $B$.
Let us choose an order on $B$ and define for each $j$-element subset $C\subset B$ an element
$$y_C:=\wedge_{x \in C} x^{*}\in \wedge^j(V_E^{*}).$$
Clearly, all $y_C$'s constitute an {\sl eigenbasis} of $\wedge^j(V_E^{*})$ w.r.t. $\wedge^j (A^{*})$. Actually,
$$\wedge^j (A^{*}) (y_C)= \left(\prod_{x\in C}\pi(x)\right) y_C.$$
Let us assign to $C$ the integer-valued function
\begin{equation}
\label{eCdef}
e_C \colon \mathrm{spec}(A) \to \Z_{+}, \ \alpha \mapsto \#(\{x \in C \mid \pi(x)=\alpha\}).
\end{equation}
Clearly, $y_C$ is an eigenvector of $\wedge^j (A^{*})$ with eigenvalue $\lambda$ if and only if $e_C$ enjoys the properties (i)-(iii).
This implies that the set of $y_C$'s such that $e_C$ satisfies (i)-(iii) is a $E$-basis of the eigenspace $\wedge^j(V_E^{*})(\lambda)$.
Conversely, suppose that $e \colon \mathrm{spec}(A) \to \Z_{+}$ is an integer-valued function that enjoys the properties (i)-(iii). I claim that
there exists a $j$-element subset $C\subset B$ such that $e=e_C$. Indeed, let us choose a $e(\alpha)$-element subset
$C_{\alpha}\subset \pi^{-1}(\alpha)\subset B$ for all $\alpha \in \mathrm{spec}(A)$ with $e(\alpha)>0$. The property (i) guarantees that such a choice is possible
(but not necessarily unique). Now define $C$ as the (disjoint) union of all these $C_{\alpha}$'s. Property (ii) implies that $B$ is a $j$-element subset of $B$.
It follows from (iii) that $y_C \in \wedge^j(V_E^{*})(\lambda)$.
\end{sect}
The following assertion will be used in the proof of Theorem \ref{mainTate} (with $K=\Q_{\ell}, V=V_{\ell}(X^n), A=\Fr_{X^n}$).
\begin{prop}
\label{exoticKE}
We keep the notation and assumptions of Subsection \ref{multEigen}, Remark \ref{eigenKE} and Lemma \ref{WedgeEigen}. In particular, $A_E \colon V_E \to V_E$
is diagonalizable.
Assume additionally that $A \colon V \to V$ is invertible, $j_1=j-2$ and $j_2=2$.
Suppose that $\lambda_1$ and $\lambda_2$ are nonzero elements of $K$ and $j>2$, i.e., $j_1\ge 1$.
Then the following conditions are equivalent.
\begin{itemize}
\item[(a)] The $K$-linear map
\begin{equation}
\label{imageWDGK}
\wedge_K^{j-2}(V^{*})(\lambda_1) \otimes_K \wedge_K^2(V^{*})(\lambda_2) \to \wedge_K^j(V^{*})(\lambda_1\lambda_2), \psi\otimes \phi \mapsto \psi\wedge \phi
\end{equation}
is not surjective.
\item[(b)]
There exists a function $e \colon \mathrm{spec}(A) \to \Z_{+}$ that enjoys the following properties.
\begin{itemize}
\item[(i)] $e(\alpha)\le \mult_A(\alpha) \ \forall \alpha \in \mathrm{spec}(A)$;
\item[(ii)] $\sum_{\alpha\in \mathrm{spec}(A)} e(\alpha)=j$;
\item[(iii)] $\prod_{\alpha\in \mathrm{spec}(A)}\alpha^{ e(\alpha)}=\lambda_1\lambda_2$.
\item[(iv)]
If $\alpha \in \mathrm{spec}(A)$ and
$e(\alpha) \ne 0$ then
$e(\alpha) \ge 1$ and one of the following conditions holds.
\begin{enumerate}
\item[(1)] $\lambda_2/\alpha \not\in \mathrm{spec}(A)$;
\item[(2)] $\lambda_2/\alpha \in \mathrm{spec}(A)$ but $e(\lambda_2/\alpha)=0$.
\item[(3)] $\alpha=\lambda_2/\alpha$ (i.e., $\alpha^2=\lambda_2$) and $e(\alpha)=1$.
\end{enumerate}
\end{itemize}
\end{itemize}
\end{prop}
\begin{rem}
The invertibility of $A$ means that $0\not\in \mathrm{spec}(A)$.
\end{rem}
\begin{rem}
\label{KESUR}
In light of Lemma \ref{WedgeEigen}, it suffices to check that condition (b) is equivalent (in the obvious notation) to the {\sl non-surjectiveness} of the $E$-linear map
\begin{equation}
\label{imageWDG}
\wedge_E^{j-2}(V_E^{*})(\lambda_1) \otimes_E \wedge_E^2(V_E^{*})(\lambda_2) \to \wedge^j(V^{*})(\lambda_1\lambda_2), \psi\otimes \phi \mapsto \psi\wedge \phi.
\end{equation}
\end{rem}
\begin{proof}[Proof of Proposition \ref{exoticKE}]
We start with the following lemma that describes the image of map \eqref{imageWDG}.
\begin{lem}
\label{imageWedgePr}
The image of map \eqref{imageWDG} is generated by all $y_C$'s where $C$ is any $j$-element subset of $B$ that enjoys the following properties.
The set $C$ is a disjoint union of a $(j-2)$-element subset $S$ and a $2$-element subset $T$ such that the corresponding functions
$$e_S \colon \mathrm{spec}(A) \to \Z_{+}, \ e_T \colon \mathrm{spec}(A) \to \Z_{+}$$
defined by\eqref{eCdef} enjoy the following properties.
\begin{equation}
\label{STC}
\prod_{\alpha\in \mathrm{spec}(A)}\alpha^{ e_S(\alpha)}=\lambda_1, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{ e_T(\alpha)}=\lambda_2.
\end{equation}
\end{lem}
\begin{proof}[Proof of Lemma \ref{imageWedgePr}]
It follows from arguments of Subsection \ref{multEigen} that all the $y_S$'s (resp. all the $y_T$'s) where $S$ is any $(j-2)$-element subset of $B$
(resp. where $T$ is any $2$-element subset of $B$) that satisfies \eqref{STC} constitute a basis of $\wedge_E^{j-2}(V_E^{*})(\lambda_1)$
(resp. a basis of $\wedge_E^2(V_E^{*})(\lambda_2)$). This implies that the image of map \eqref{imageWDG} is generated by all $y_S\wedge y_T$.
If $S$ meets $T$ then it follows from the
very definition of $y_S$ and $y_T$ and basic properties of wedge products that
$y_S \wedge y_T=0$. On the other hand, if $S$ does {\sl not} meet $T$ then $C:=S\cup T=S \sqcup T$ is a $j$-element subset of $B$ and $y_S \wedge y_T=\pm y_{C}$.
This ends the proof.
\end{proof}
Now let us start to prove Proposition \ref{exoticKE}.
Suppose that (b) holds. In light of Remark \ref{KESUR}, it suffices to check that map \eqref{imageWDG} is {\sl not} surjective.
To this end, choose an an eigenbasis $B$ of $V_E$ w.r.t. $A_E$, and choose an order on $B$.
Using arguments of Subsection \ref{multEigen}, choose a $j$-element subset $\tilde{C} \subset B$ such that the function
$$e_{\tilde{C}} \colon \mathrm{spec}(A) \to \Z_{+}$$ coincides with $e$ and therefore
enjoys properties (i)-(iv). Then
$$y_{\tilde{C}} \in \wedge^j(V^{*})(\lambda_1\lambda_2).$$
I claim that $y_{\tilde{C}}$ does {\sl not} lie in the image of map \eqref{imageWDG}.
Indeed, $\wedge_E^{j-2}(V_E^{*})(\lambda_1)$ is generated as the $E$-vector space by elements of the form $y_S$, where $S$ are $(j-2)$-element subsets of $B$ such that
$\prod_{b\in S}\pi(b)=\lambda_1$.
On the other hand, $ \wedge_E^2(V_E^{*})(\lambda_2)$ is generated as the $E$-vector space by elements of the form $y_T$, where $T$ are $2$-element subsets of $B$ such that
$\prod_{b\in T}\pi(b)=\lambda_2$. This implies that the image of map \eqref{imageWDG} is generated as the $E$-vector space by all $y_B \wedge y_T$. If $S$ meets $T$ then
(as we have already seen)
$y_S \wedge y_T=0$. If $S$ does {\sl not} meet $T$ then $S\cup T=S \sqcup T$ is a $j$-element subset of $B$ and $y_S \wedge y_T=\pm y_{S\cup T}$.
\begin{lem}
\label{CversusST}
The $j$-element $\tilde{C}$ does {\sl not} coincide with any of $S\cup T$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{CversusST}] Suppose $\tilde{C}=S\cup T$. This implies that $\tilde{C}$ contains a subset $T$ that consists of two distinct elements say $x_1,x_2\subset B$
with $\pi(x_1)\pi(x_2)=\lambda_2$. So, $\tilde{C}$ contains these $x_1,x_2$. It follows from the definition of $e_{\tilde{C}}$ \eqref{eCdef} that if we put
$$\alpha_1=\pi(x_1), \ \alpha_2=\pi(x_2)$$
then $\alpha_1,\alpha_2 \in \mathrm{spec}(A)$ and
$$\alpha_1 \alpha_2=\lambda_2, \ e(\alpha_1)\ge 1, e(\alpha_2)\ge 1, \ e(\alpha_1)+ e(\alpha_2) \ge 2.$$
If $\alpha_1 \ne \alpha_2=\lambda_2/\alpha_1$ then $e(\alpha_1)\ge 1, e(\lambda_2/\alpha_1)\ge 1$, which violates property (iv).
If $\alpha_1=\alpha_2$ then $\alpha_2=\lambda_2/\alpha_1$. It follows that $\alpha_1=\pi(x_1)=\pi(x_2)$ and therefore $e(\alpha_1)\ge 2$, which also contradicts property (iv). This ends the proof.
\end{proof}
{\sl End of Proof of Proposition \ref{exoticKE}}.
Taking into account that the set of all $y_C$'s where $C$ runs through all $j$-element subsets of $B$
is {\sl linearly independent}, we conclude
that $y_{\tilde{C}}$ cannot be presented as a $E$-linear combination of $ y_{S\cup T}$'s and therefore does {\sl not} lie in the image of map \eqref{imageWDG}. Hence, (a) holds. We proved that (b) implies (a).
Suppose that map \eqref{imageWDG} is {\sl surjective}, i.e., (a) does not hold. We need to prove that (b) does not hold as well. Let $e \colon \mathrm{spec}(A) \to \Z_{+}$ be a function that enjoys the properties (i)-(iii) of Subsection \ref{multEigen}. We need to check
that $e$ does not enjoy property (iv). Using arguments of Subsection \ref{multEigen}, choose a $j$-element subset $\tilde{C} \subset B$ such that the function
$$e_{\tilde{C}} \colon \mathrm{spec}(A) \to \Z_{+}$$ coincides with $e$ and therefore enjoys properties (i)-(iii). This implies that
$$y_{\tilde{C}} \in \wedge^j(V^{*})(\lambda_1\lambda_2).$$
Let us check that $e=e_{\tilde{C}}$ does {\sl not} enjoy property (iv). Indeed, we know that $y_{\tilde{C}}$ lies in the image of map \eqref{imageWDG}. It follows from Lemma \ref{imageWedgePr}
that there is a positive integer $m$, $m$ pairs of subsets $(S_1,T_1), \dots, (S_m,T_m)$ in $B$, and $m$ elements $a_1, \dots, a_m \in E$ such that each
$S=S_r$ and $T=T_r$ are disjoint $(j-2)$-element and $2$-element subsets of $B$ that satisfy \eqref{STC} for all $r=1, \dots, m$, and such that
$$y_{\tilde{C}}=\sum_{r=1}^m a_r y_{S_r\sqcup T_r}.$$
Let us choose such a presentation for $y_{\tilde{C}}$ with smallest possible $m$. In this case all the $j$-element subsets $S_r\sqcup T_r$ are distinct. Now the linear independence of all $y_C$
(where $C\subset B$ is a $j$-element subset) implies that $m=1$ and
$\tilde{C}$ coincides with $S_1\sqcup T_1$.
So, $T_r$ consists of two distinct elements say, $x_1,x_2$. Let us put
$$\alpha_1:=\pi(x_1)\in\mathrm{spec}(A), \alpha_2=\pi(x_2)\in \mathrm{spec}(A).$$
It follows from \eqref{STC} that $\alpha_1 \alpha_2=\lambda_2$. This implies that
$$e(\alpha_1)=e_{\tilde{C}}(\alpha_1) \ge 1, \ e(\alpha_2)=e_{\tilde{C}}(\alpha_2) \ge 1.$$
If $\alpha_1 \ne \alpha_2$ then property (iv) does not hold for $e$. If $\alpha_1=\alpha_2$ then
$$\pi(x_1)=\alpha_1=\lambda/\alpha_1=\alpha_2=\pi(x_2)$$
and therefore $e(\alpha_1) \ge 2$, hence, property (iv) does not hold for $e$. This ends the proof.
\end{proof}
The following assertion will be used in the proof of Theorem \ref{semigroupTate} (with $K=\Q_{\ell}, V=V_{\ell}(X^n), A=\Fr_{X^n}$).
\begin{prop}
\label{lastEff}
We keep the notation and assumption of Subsection \ref{multEigen}.
Assume additionally that $\fchar(K)=0$, $\dim_K(V)$ is even, and $A:V \to V$ is invertible. Let $q$ be a nonzero element of $K$ that is not a root of unity.
Let $h$ and $m$ be positive integers that enjoy the following properties.
\begin{itemize}
\item[(i)] $h<m \le \dim_K(V)/2$.
\item[(ii)]
If $e \colon \mathrm{spec}(A) \to \Z_{+}$ is any nonnegative integer-valued function such that
$$\sum_{\alpha\in \mathrm{spec}(A)} e(\alpha)=2m, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{e(\alpha)}=q^m$$
then there exist positive integers $j_1,j_2$ and nonnegative integer-valued functions
$$f_1 \colon \mathrm{spec}(A) \to \Z_{+}, f_2 \colon \mathrm{spec}(A) \to \Z_{+}$$
such that
$$m=j_1+j_2, \ j_2 \le h;$$
$$ e(\alpha)=f_1(\alpha)+f_2(\alpha)\ \forall \alpha \in \mathrm{spec}(A);$$
$$\sum_{\alpha\in \mathrm{spec}(A)} f_1(\alpha)=2j_1, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{f_1(\alpha)}=q^{j_1},$$
$$\sum_{\alpha\in \mathrm{spec}(A)} f_2(\alpha)=2j_2, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{f_2(\alpha)}=q^{j_2}.$$
\end{itemize}
Then
\begin{equation}
\label{imageWdgePowerQ}
\sum_{j=1}^h \Lambda_{2(m-j),2j,K}\left(\wedge_K^{2(m-j)}(V^{*})(q^{m-j}) \otimes_K \wedge_K^{2j}(V^{*})(q^j)\right)=\wedge_K^{2m}(V^{*})(q^m).
\end{equation}
\end{prop}
\begin{proof}
In light of Remark \ref{restrictionLambdaK} and arguments of Subsection \ref{wedgeImage}, it suffices to check that
\begin{equation}
\label{imageWdgePowerQE}
\sum_{j=1}^h \Lambda_{2(m-j),2j,E}\left(\wedge_E^{2(m-j)}(V_E^{*})(q^{m-j}) \otimes_E \wedge_E^{2j}(V_E^{*})(q^j)\right)=\wedge_E^{2m}(V_E^{*})(q^m).
\end{equation}
Recall that (in the notation of Subsection \ref{multEigen}) that $B$ is an (ordered) eigenbasis of $V_E$ and
all the $2m$-element subsets $C\subset \mathrm{spec}(A)$
with $\prod_{\alpha\in C}\alpha=q^m$
give rise to the base $\{y_C=\wedge_{x\in C}x^{*}\}$ of $\wedge_E^{2m}(V_E^{*})(q^m)$.
So, it suffices to prove that each such $y_C$ lies in one of the summands in LHS of \eqref{imageWdgePowerQE}.
To this end, let us consider the nonnegative integer-valued function
$$e_C \colon \mathrm{spec}(A)\to \Z_{+}, \
e(\alpha)=\#(C(\alpha)) \ \text{ where }\ C(\alpha):=\{x\in C \subset B\mid \pi(x)=\alpha\}.$$
(see \eqref{eCdef}). Clearly,
$$\sum_{\alpha\in \mathrm{spec}(A)}e_C(\alpha)=\#(C)=2m, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{e_C(\alpha)}=\prod_{\alpha\in C}\alpha =q^m.$$
By property (ii), there exist positive integers $j_1,j_2$ and nonnegative integer-valued functions
$$f_1 \colon \mathrm{spec}(A) \to \Z_{+}, f_2 \colon \mathrm{spec}(A) \to \Z_{+}$$
such that
$$m=j_1+j_2, \ j_2 \le h;$$
$$ e(\alpha)=f_1(\alpha)+f_2(\alpha)\ \forall \alpha \in \mathrm{spec}(A);$$
$$\sum_{\alpha\in \mathrm{spec}(A)} f_1(\alpha)=2j_1, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{f_1(\alpha)}=q^{j_1},$$
$$\sum_{\alpha\in \mathrm{spec}(A)} f_2(\alpha)=2j_2, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{f_2(\alpha)}=q^{j_2}.$$
Let us partition each $C(\alpha)$ into a {\sl disjoint union} of two sets
$$C(\alpha)=C(\alpha)_1\cup C(\alpha)_2 \text{ with } \ C(\alpha)_1\cap C(\alpha)_2 =\emptyset, \ \#(C(\alpha)_1)=f_1(\alpha), \#(C(\alpha)_2)=f_2(\alpha)$$
and define $C_1$ (resp. $C_2)$ as the (disjoint) union of all $C(\alpha_1)$ (resp. of all $C(\alpha_2)$). Then $C$ becomes a {\sl disjoint union} of $C_1$ and $C_2$, and
$$f_1=e_{C_1}, f_2=e_{C_2}.$$
It follows that
$$\sum_{\alpha\in \mathrm{spec}(A)} e_{C_1}(\alpha)=2j_1, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{e_{C_1}(\alpha)}=q^{j_1},$$
$$\sum_{\alpha\in \mathrm{spec}(A)} e_{C_2}(\alpha)=2j_2, \ \prod_{\alpha\in \mathrm{spec}(A)}\alpha^{e_{C_2}(\alpha)}=q^{j_2}.$$
This implies that
$$y_{C_1} \in \wedge_E^{2(j_1)}(V_E^{*})(q^{j_1})=\wedge_E^{2(m-j_2)}(V_E^{*})(q^{m-j_2}),
y_{C_2} \in \wedge_E^{2(j_2)}(V_E^{*})(q^{j_2}).$$
Since $C$ is a disjoint union of $C_1$ and $C_2$,
$$y_C =\pm y_{C_1}\wedge y_{C_2}\in
\Lambda_{2(m-j_2),2j_2,E}\left(\wedge_E^{2(m-j_2)}(V_E^{*})(q^{m-j_2}) \otimes_E \wedge_E^{2j_2}(V_E^{*})(q^{j_2})\right).$$
In order to finish the proof, one has only to recall that $j \le h_2$.
\end{proof}
\section{Tate forms}
\label{TateProof}
\begin{sect}
Recall that $X$ is an abelian variety of positive dimension $g$ over a finite field $k$ with $\fchar(k)=p$ and $\#(k)=q$.
Let $\ell\ne p$ be a prime and $T_{\ell}(X)$ the $\ell$-adic Tate module of $X$. Let us consider the corresponding $\Q_{\ell}$-vector space
$$V_{\ell}(X)=T_{\ell}(X)\otimes_{\Z_{\ell}}\Q_{\ell},$$
which is a $2g$-dimensional vector space over $\Q_{\ell}$. The action of $\Fr_X$ extends by $\Q_{\ell}$-linearity to $V_{\ell}(X)$. So, we may view
$\Fr_X$ as a $\Q_{\ell}$-linear automorphism of $V_{\ell}(X)$, whose characteristic polynomial coincides with $\mathcal{P}_X(t)$. A theorem of
Weil \cite{Mumford,Tate1} asserts that $\Fr_X$ acts as a {\sl semisimple} linear operator in $V_{\ell}(X)$.
Let $\bar{\Q}_{\ell}$ be an algebraic closure of $\Q_{\ell}$. Let us choose a field embedding
$$L_X =\Q(R_X)\hookrightarrow \bar{\Q}_{\ell}.$$
Further we will identify $L_X$ with its image in $\bar{\Q}_{\ell}$. We have
$$R_X \subset L_X \subset \bar{\Q}_{\ell}.$$
Let us consider the $2\dim(X)$-dimensional $\bar{\Q}_{\ell}$-vector space
$$\bar{V}_{\ell}(X):=V_{\ell}(X)\otimes_{\Q_{\ell}}\bar{\Q}_{\ell}.$$
Extending the action of $\Fr_X$ by $\bar{\Q}_{\ell}$-linearity, we get a $\bar{\Q}_{\ell}$-linear operator
$$\overline{\Fr}_X \colon \bar{V}_{\ell}(X)\to \bar{V}_{\ell}(X), \ v\otimes \lambda\mapsto \Fr_X(v)\otimes \lambda \ \forall v \in V_{\ell}(X), \lambda \in \bar{\Q}_{\ell}.$$
In the notation of Section \ref{linAlgebra}, let us put
\begin{equation}
\label{notationQl}
K=\Q_{\ell}, V=V_{\ell}(X), A=\Fr_X \colon V_{\ell}(X)\to V_{\ell}(X), E=\bar{\Q}_{\ell}.
\end{equation}
Then
\begin{equation}
\label{notationQlbar}
V_E=\bar{V}_{\ell}(X), A_E=\overline{\Fr}_X; \ \mathrm{spec}(A)=R_X, \mult_A=\mult_X:R_X \to \Z_{+}.
\end{equation}
\end{sect}
\begin{rem}
\label{YXm} If $m$ is a positive integer and $Y=X^m$ then it is well known that there is a canonical isomorphism of $\Q_{\ell}$-vector spaces
$$V_{\ell}(Y)=\oplus_{i=1}^m V_{\ell}(X)$$ such that $\Fr_Y$ acts on $V_{\ell}(Y)$ as
$$\Fr_Y(x_1,\dots x_m)=(\Fr_X x_1, \dots, \Fr_X x_m) \ \forall (x_1,\dots x_m)\in \oplus_{i=1}^m V_{\ell}(X)=V_{\ell}(Y).$$
This implies that
\begin{equation}
\label{RXY}
\mathcal{P}_Y(t)=\mathcal{P}_X(t)^m, \ R_Y=R_X, L_X=L_Y, \mult_Y(\alpha)=m \cdot \mult_X(\alpha) \ \forall \alpha\in R_X=R_Y.
\end{equation}
In particular,
\begin{equation}
\label{multY}
\mult_Y(\alpha) \ge m \ \forall \alpha \in R_Y=R_X.
\end{equation}
\end{rem}
Any invertible sheaf/divisor class $\mathcal{L}$ on $X$
gives rise to (defined up to multiplication by an element of $\Q_{\ell}^{*}$) a $\Q_{\ell}$-bilinear alternating {\sl Riemann form} (the first $\ell$-adic Chern class of $\mathcal{L}$) \cite{Mumford}
$$\phi=\phi_{\mathcal{L}} \colon V_{\ell}(X) \times V_{\ell}(X) \to \Q_{\ell} $$
such that
\begin{equation}
\label{Riemann}
\phi(\Fr_X (x),\Fr_X(y))=q \cdot \phi(x,y) \ \forall x,y \in V_{\ell}(X).
\end{equation}
\begin{sect}
A theorem of Tate \cite{Tate1} asserts that every alternating $\Q_{\ell}$-bilinear form $\phi$ on $V_{\ell}(X)$ that satisfies \eqref{Riemann} is a $\Q_{\ell}$-linear combination
of forms of type $\phi_{\mathcal{L}}$. We call such a form an $\ell$-adic Tate form of degree $2$
and denote by $\tate_2(X,\ell)$ the subspace of all such forms in
$\Hom_{\Q_{\ell}}(\Lambda^2 V_{\ell}(X),\Q_{\ell})$. In other words,
$$\tate_2(X,\ell):=\{\phi \in \Hom_{\Q_{\ell}}(\Lambda^2 V_{\ell}(X),\Q_{\ell})\mid
\phi(\Fr_X (x),\Fr_X(y))=q \cdot \phi(x,y)$$
$$ \forall x,y \in V_{\ell}(X)\}.$$
More generally, let us define for each nonnegative integer $d\le \dim(X)=g$ the subspace $\tate_{2d}(X,\ell)$ of all alternating $2d$-forms
$\psi \in \Hom_{\Q_{\ell}}(\Lambda^{2d} V_{\ell}(X),\Q_{\ell})$
such that
$$\psi(\Fr_X (v_1), \dots, \Fr_X(v_{2d}))=q^d \cdot \psi(v_1,\dots, v_{2d})
\ \forall x_1, \dots, x_{2d} \in V_{\ell}(X).$$
We call elements of $\tate_{2d}(X,\ell)$ {\sl Tate forms of degree $2d$}\begin{footnote}{In \cite{ZarhinEssen} we called them admissible forms.}\end{footnote}.
\end{sect}
\begin{rem}
\label{tateBis}
Clearly, $\tate_{2d}(X,\ell)$ consists of all $\psi \in \Hom_{\Q_{\ell}}(\Lambda^{2d} V_{\ell}(X),\Q_{\ell})$
such that
$$\psi(\Fr_X^{-1} (v_1), \dots, \Fr_X^{-1}(v_{2d}))=q^{-d} \cdot \psi(v_1,\dots, v_{2d}) \ \forall v_1, \dots, v_{2d} \in V_{\ell}(X).$$
Since $\Fr_X$ acts on $V_{\ell}(X)$ as $\rho_{\ell}(\sigma_k)$, the subspace $\tate_{2d}(X,l)$ consists of all $\psi \in \Hom_{\Q_{\ell}}(\Lambda^{2d} V_{\ell}(X),\Q_{\ell})$
such that
$$\psi(\rho_{\ell}(\sigma_k)^{-1} (v_1), \dots, \rho_{\ell}(\sigma_k)^{-1}(v_{2d}))=\chi_{\ell}(\sigma_k)^{-d} \cdot \psi(v_1,\dots, v_{2d}) \ \forall v_1, \dots, v_{2d} \in V_{\ell}(X).$$
Since $\sigma_k$ is a topological generator of $\Gal(k)$, the subspace $\tate_{2d}(X,l)$ consists of all $\psi \in \Hom_{\Q_{\ell}}(\Lambda^{2d} V_{\ell}(X),\Q_{\ell})$
such that
$$\psi(\rho_{\ell}(\sigma)^{-1} (v_1), \dots, \rho_{\ell}(\sigma)^{-1}(v_{2d}))=\chi_{\ell}(\sigma)^{-d} \cdot \psi(v_1,\dots, v_{2d})$$
$$ \forall \sigma \in \Gal(k), v_1, \dots, v_{2d} \in V_{\ell}(X).$$
\end{rem}
\begin{rem}
\label{TateEigenQ}
In the notation of Section \ref{linAlgebra}, \eqref{notationQl} and \eqref{notationQlbar},
$$\tate_{2d}(X,\ell)=\wedge_K^{2d}(V^{*})(q^d)$$
is the eigenspace of $K$-linear operator
$\wedge^{2d}(A^{*}) \colon \wedge_K^{2d}(V^{*}) \to \wedge_K^{2d}(V^{*}) $ attached to the eigenvalue $q^d$.
\end{rem}
\begin{sect}
For each integer $d\ge 3$ the exterior product map
$$\Hom_{\Q_{\ell}}(\Lambda^{2(d-1)} V_{\ell}(X),\Q_{\ell}) \otimes_{\Q_{\ell}} \Hom_{\Q_{\ell}}(\Lambda^{2} V_{\ell}(X),\Q_{\ell})
\to \Hom_{\Q_{\ell}}(\Lambda^{2d} V_{\ell}(X),\Q_{\ell}) ,$$
$$ \phi\otimes \psi \mapsto \phi \wedge \psi$$
induces the $\Q_{\ell}$-linear map
\begin{equation}
\label{wedgeTate}
\tate_{2(d-1)}(X,\ell) \otimes_{\Q_{\ell}} \tate_{2}(X,\ell) \to \tate_{2d}(X,\ell), \ \phi\otimes\psi \mapsto \phi \wedge \psi.
\end{equation}
\begin{defn}
Let $d>1$ be an integer. An $\ell$-adic {\sl Tate form} of degree $2d$ is called {\sl exceptional} if it does {\sl not} lie
in the
image of map \eqref{wedgeTate}.
\end{defn}
\end{sect}
\begin{lem}
\label{exceptreduced}
Let $d$ be a positive integer such that
$$2 \le d \le \dim(X).$$
Let $\ell \ne p$ be a prime.
Then the following conditions are equivalent.
\begin{itemize}
\item[(a)]
There exists an exceptional $\ell$-adic Tate form on $X$ of degree $2d$.
\item[(b)]
There exists an admissible reduced function $e \colon R_X \to \Z$ of weight $2d$ such that
$$0 \le e(\alpha) \le \mult_X(\alpha) \ \forall \alpha\in R_X.$$
\end{itemize}
\end{lem}
\begin{proof}
In the notation of \eqref{imageWDGK}, \eqref{notationQl} and \eqref{notationQlbar}, it follows from Remark \ref{TateEigenQ} that
property (b) is equivalent to the {\sl non-surjectiveness} of
$$\wedge_K^{j-2}(V^{*})(\lambda_1) \otimes_K \wedge_K^2(V^{*})(\lambda_2) \to \wedge_K^{j}(V^{*})(\lambda_1\lambda_2), \psi\otimes \phi \mapsto \psi\wedge \phi$$
with
$$j=2d, \lambda_1=q^{d-1}, \lambda_2=q, \lambda_1 \lambda_2=q^d.$$
By Proposition \ref{exoticKE}, the non-surjectiveness of this map is equivalent to the existence of a function
$e \colon \mathrm{spec}(A) \to \Z_{+}$ that enjoys properties (i)-(iv) of Proposition \ref{exoticKE}. Since $\mathrm{spec}(A)=R_X$, we may view $e$ as a function
$e \colon R_X \to \Z_{+}$. Now property (ii) means that $e$ has weight $2d$, property (iii) that $e$ is admisible, and property (iv) that $e$ is reduced. As for property (i), it means that
$$e(\alpha) \le \mult_A(\alpha)=\mult_X(\alpha) \ \forall \alpha \in \mathrm{spec}(A)=R_X.$$
This implies that properties (i)-(iv) of Theorem \ref{exoticKE} are equivalent to property (b) of Lemma \ref{exceptreduced}.
It follows that
properties (a) and (b) of Lemma \ref{exceptreduced} are equivalent.
\end{proof}
\begin{sect}
\label{etaleT}{\bf Twists and Tate classes}.
Let us
consider an abelian variety $\bar{X}=X \times_k \bar{k}$ over the algebraic closure $\bar{k}$ of $k$
and its \'etale $\ell$-adic cohomology groups $\mathrm{H}^j(\bar{X},\Q_{\ell})$ \cite{Tate0,Kleiman,Katz,Berkovich}. Here $\ell$ is any prime different from $\fchar(k)$, $j$ any nonnegative integer,
and $\mathrm{H}^j(\bar{X},\Q_{\ell})$ is a certain finite-dimensional $\Q_{\ell}$-vector space endowed with a continuous linear action of the absolute Galois group $\Gal(k):=\Gal(\bar{k}/k)$ of $k$.
We write
$$\chi_{\ell} \colon \Gal(k) \to \Z_{\ell}^{*}\subset \Q_{\ell}^{*}$$
for the $\ell$-adic {\sl cyclotomic character} (see Section \ref{neat} above). There exists a certain ``naturally defined'' one-dimensional
$\Q_{\ell}$-vector space $\Q_{\ell}(1)$ (that was denoted by $W$ in \cite[Sect. 2]{Tate0}) endowed with the natural continuous linear action of $\Gal(k)$ defined by the cyclotomic character
$$\chi_{\ell} \colon \Gal(k) \to \Q_{\ell}^{*}=\Aut_{\Q_{\ell}}(\Q_{\ell}(1))$$
(see \cite{Tate0,Kleiman,Katz,Milne}). Namely, $\Q_{\ell}(1)=\Z_{\ell}(1)\otimes_{\Z_{\ell}}\Q_{\ell}$ where $\Z_{\ell}(1)$ is the projective limit of multiplicative groups (finite Galois modules) $\mu_{\ell^n}$
of $\ell^n$th roots of unity in $\bar{k}$.
Let us fix once and for all an {\sl $\ell$-adic orientation}, i.e., an isomorphism of $\Q_{\ell}$-vector spaces
$$\Q_{\ell}(1) \cong \Q_{\ell},$$
which allows us to identify $\Q_{\ell}$ not only with $\Q_{\ell}(1)$ but also with all {\sl tensor powers} $\Q_{\ell}(i)$ \cite{Tate0,Kleiman,Katz,Berkovich,Tate94} of $\Q_{\ell}(1)$.
Let $i$ be an integer. Let us consider the {\sl twist} $\mathrm{H}^j(\bar{X},\Q_{\ell})(i)$
of the Galois module $\mathrm{H}^j(\bar{X},\Q_{\ell})$
by character $\chi_{\ell}^i$ of $\Gal(k)$ \cite{Tate0,Kleiman,Katz,Tate94}. In other words, $\mathrm{H}^j(\bar{X},\Q_{\ell})(i)$ coincides with $\mathrm{H}^j(\bar{X},\Q_{\ell})$ as the $\Q_{\ell}$-vector space but if
$$ \sigma,c \mapsto \sigma(c) \ \forall \sigma \in \Gal(k), \ c \in \mathrm{H}^j(\bar{X},\Q_{\ell})$$
is the Galois action on $\mathrm{H}^j(\bar{X},\Q_{\ell})$
then in $\mathrm{H}^j(\bar{X},\Q_{\ell})(i)$ a Galois automorphism $\sigma$ sends $c$ to $\chi_{\ell}(\sigma)^i\sigma(c)$.
If $W$ is a Galois-invariant $\Q_{\ell}$-vector subspace in $\mathrm{H}^j(\bar{X},\Q_{\ell})$ then we write $W(i)$ for the same $\Q_{\ell}$-vector subspace in $\mathrm{H}^j(\bar{X},\Q_{\ell})(i)$.
Clearly, $W(i)$ is a Galois-invariant subspace of $\mathrm{H}^j(\bar{X},\Q_{\ell})(i)$ but {\sl not} necessarily isomorphic to $W$ as Galois module.
\begin{rems}
\label{twistCup}
\begin{itemize}
\item[(i)]
If $j_1,j_2$ are any nonnegative integers then the Galois-equivariant $\Q_{\ell}$-bilinear cup product in the cohomology of $\bar{X}$ leads to a Galois-equivariant $\Q_{\ell}$-linear map
\begin{equation}
\label{cupH}
\mathrm{H}^{j_1}(\bar{X},\Q_{\ell})\otimes_{\Q_{\ell}} \mathrm{H}^{j_2}(\bar{X},\Q_{\ell}) \to \mathrm{H}^{j_1+j_2}(\bar{X},\Q_{\ell}), \ c_1\otimes c_2 \mapsto c_1 \cup c_2,
\end{equation}
which, in turn, gives rise to
the natural Galois-equivariant $\Q_{\ell}$-linear map \cite{Tate0,Kleiman}
\begin{equation}
\label{cupProduct}
\mathrm{H}^{j_1}(\bar{X},\Q_{\ell})(i_1)\otimes_{\Q_{\ell}} \mathrm{H}^{j_2}(\bar{X},\Q_{\ell})(i_2) \to \mathrm{H}^{j_1+j_2}(\bar{X},\Q_{\ell})(i_1+i_2), \ c_1\otimes c_2 \mapsto c_1 \cup c_2.
\end{equation}
\item[(ii)]
Let $W_1$ (resp. $W_2$) is s a Galois-invariant $\Q_{\ell}$-vector subspace in $\mathrm{H}^{j_1}(\bar{X},\Q_{\ell})$ (resp. in $\mathrm{H}^{j_2}(\bar{X},\Q_{\ell})$)
and $W \subset \mathrm{H}^{j_1+j_2}(\bar{X},\Q_{\ell})$ be the image of subspace
$$W_1\otimes_{\Q_{\ell}}W_2 \subset \mathrm{H}^{j_1}(\bar{X},\Q_{\ell})\otimes_{\Q_{\ell}} \mathrm{H}^{j_2}(\bar{X},\Q_{\ell})$$
under the map \eqref{cupH}. It follows readily that the twist
$$W(i_1+i_2)\subset \mathrm{H}^{j_1+j_2}(\bar{X},\Q_{\ell})(i_1+i_2)$$
coincides with the image of subspace
$$W_1(i_1)\otimes_{\Q_{\ell}}W_2(i_2) \subset \mathrm{H}^{j_1}(\bar{X},\Q_{\ell})(i_1)\otimes_{\Q_{\ell}} \mathrm{H}^{j_2}(\bar{X},\Q_{\ell})(i_2)$$
under the map \eqref{cupProduct}.
\end{itemize}
\end{rems}
\begin{defn}
\label{WlT}
Let $d$ be a nonnegative integer. Let us consider the $\Q_{\ell}$-vector subspace
$$\T_{\ell,d}(X):=\mathrm{H}^{2d}(\bar{X},\Q_{\ell})(d)^{\Gal(k)}$$ of {\sl Galois invariants} in $\mathrm{H}^{2d}(\bar{X},\Q_{\ell})(d)$
and a {\sl weight} $\Q_{\ell}$-vector subspace
$$W_{\ell,d}(X):=\{c \in \mathrm{H}^{2d}(\bar{X},\Q_{\ell})\mid \sigma(x)=\chi_{\ell}(\sigma)^{-d}c \ \forall \sigma \in \Gal(k)\}$$
in $\mathrm{H}^{2d}(\bar{X},\Q_{\ell})$. It follows from the very definitions that
\begin{equation}
\label{TateTwistClass}
\T_{\ell,d}(X)=W_{\ell,d}(X)(d).
\end{equation}
\end{defn}
\begin{rem}
\label{product TateCl}
Let $d_1$ and $d_2$ be nonnegative integers. It follows from the Galois equivariance of maps \eqref{cupH} and \eqref{cupProduct} combined with Remark \ref{twistCup}(ii)
that the image $W_{\ell,d_1,d_2}(X)$ of
$$W_{\ell,d_1}(X)\otimes W_{\ell,d_2}(X)\to \mathrm{H}^{2(d_1+d_2)}(\bar{X},\Q_{\ell}), \ c_1\otimes c_2 \to c_1\cup c_2$$
lies in $W_{\ell,d_1+d_2}(X)$.
Similarly, the image $\T_{\ell,d_1,d_2}(X)$ of
$$\T_{\ell,d_1}(X)\otimes \T_{\ell,d_2}(X)\to \mathrm{H}^{2(d_1+d_2)}(\bar{X},\Q_{\ell})(d_1+d_2), \ c_1\otimes c_2 \to c_1\cup c_2$$
lies in $\T_{\ell,d_1+d_2}(X)$. In addition, it follows from \eqref{TateTwistClass} that
$$\T_{\ell,d_1,d_2}(X)=W_{\ell,d_1,d_2}(X)(d_1+d_2).$$
\end{rem}
\begin{defn}
\label{tateDeinition}
Let $\ell \ne \fchar(k)$ be a prime and $d$ a nonnegative integer.
\begin{itemize}
\item[(i)]
Elements of $\T_{\ell,d}(X)$ are called $\ell$-adic {\sl Tate classes of dimension $2d$} on $X$.
\item[(ii)]
A $2d$-dimensional Tate class $c$ is called {\sl exotic} if $d\ge 1$ and $c$ cannot be presented as a linear combination of products of $d$ Tate classes of dimension 2 with coefficients in $\Q_{\ell}$.
\item[(iii)]
A Tate class $c$ of dimension $2d$ is called {\sl very exotic} if $d\ge 1$ and $c$ cannot be presented as a linear combination with coefficients in $\Q_{\ell}$ of products of Tate classes of dimension $2d-2$ and $2$, i.e.,
$c$ does {\sl not} belong to $\T_{\ell,d-1,1}(X)$.
\end{itemize}
\end{defn}
\begin{rems}
\label{remTate}
\begin{itemize}
\item[]
\item[(i)]
Let $Z$ be a closed irreducible subvariety of codimension $d$ in $X$. The choice of the $\ell$-adic orientation allows to define
the $\ell$-adic class $\mathrm{cl}(Z)\in \T_{\ell,d}(X)\subset\mathrm{H}^{2d}(\bar{X},\Q_{\ell})(d)$ of $Z$ \cite{Tate0,Tate94}.
Tate \cite{Tate0,Tate94} conjectured that for all nonnegative integer $d$ the subspace $\T_{\ell,d}(X)$ is spanned by all $\mathrm{cl}(Z)$
and proved it for $d=1$ \cite{Tate1}.
\item[(ii)]
If $d$ is a nonnegative integer and $d \le g$ then it is known \cite{Tate0,Tate94} that $\T_{\ell,d}(X) \ne \{0\}$.
\item[(iii)]
Clearly, all $2$-dimensional Tate classes are exotic and even very exotic. Hence,
the existence of an exotic (or very exotic) Tate class of dimension $d$ implies readily that
$$2\le d \le g=\dim(X),$$
because $\mathrm{H}^{2d}(\bar{X},\Q_{\ell})=\{0\}$ for all $d>g$,
and, therefore, all $2d$-dimensional Tate classes are just zero. (Actually, it is known that
$\mathrm{H}^{2g}(\bar{X},\Q_{\ell})(g)$ is a one-dimensional $\Q_{\ell}$-vector space generated by the $g$th self-product of the class of a hyperplane
section of $X$ \cite{Tate0}. Therefore, there are no non-exotic Tate classes of dimension $2g$.)
\item[(iv)]
Clearly, every {\sl very exotic} Tate class is {\sl exotic}.
Conversely, suppose that there exists an {\sl exotic} $2d$-dimensional $\ell$-adic Tate class on $X$. I claim that there is a positive integer $d^{\prime} \le d$ such that
there exists a {\sl very exotic} $2d^{\prime}$-dimensional $\ell$-adic Tate class on $X$. Indeed, decreasing $d$ if necessary, we may and will assume that $d$
is the smallest positive integer such that there is an exotic $\ell$-adic $2d$-dimensional Tate class on $X$. Let $c$ be such a class. Then $d>1$.
Assume that $c$ is {\sl not} very exotic. Then $c$ is a linear combination of cup products $h_i \cup c_i$ where
all $c_i$ are nonzero Tate classes of dimension $2$ and all $h_i$ are nonzero Tate classes of dimension $2(d-1)$. Since $c$ is exotic,
there is an index $i$ such that $h_i$ is exotic. But exotic $h_i$ is $2(d-1)$-dimensional, which contradicts the minimality of $d$. The obtained contradiction
implies that $c$ itself is very exotic, which ends the proof.
It follows that {\sl $X$ carries an $\ell$-adic exotic Tate class if and only if it carries a very exotic $\ell$-adic Tate class} (may be, of different dimension).
\end{itemize}
\end{rems}
\end{sect}
\begin{sect}
\label{etale}
{\bf \'Etale cohomology of abelian varieties}.
Let us consider the abelian variety $\bar{X}=X\times_k\bar{k}$ over $\bar{k}$.
Let $j$ be a nonnegative integer and let $\mathrm{H}^{j}(\bar{X},\Q_{\ell})$ be the $j$th \'etale $\ell$-adic cohomology group of $\bar{X}$, which is a finite-dimensional $\Q_{\ell}$-vector space endowed with the canonical continuous linear action of $\Gal(k)$ \cite{Tate0,Kleiman,Milne}.
There is a canonical $\Gal(k)$-equivariant isomorphism of graded $\Q_{\ell}$-algebras (\cite{Tate0,Kleiman,Berkovich}, \cite[Sect. 12]{Milne})
\begin{equation}
\label{HopfIso}
\oplus_{j=0}^{2\dim(X)} \mathrm{H}^{j}(\bar{X},\Q_{\ell})\cong \oplus_{j = 0}^{2\dim(X)} \Hom_{\Q_{\ell}}(\Lambda^{j}_{\Q_{\ell}} V_{\ell}(X),\Q_{\ell}).
\end{equation}
Its Galois equivariance combined with Remark \ref{tateBis} imply that (in the notation of Definition \ref{WlT}) \eqref{HopfIso} induces for all nonnegative integers $d \le \dim(X)$ a $\Q_{\ell}$-linear isomorphism between
$$W_{\ell,d,}(X)=\{c \in \mathrm{H}^{2d}(\bar{X},\Q_{\ell})\mid \sigma(c)=\chi_{\ell}(\sigma)^{-d}c \ \forall \sigma \in \Gal(k) \}=$$
$$\{c \in \mathrm{H}^{2d}(\bar{X},\Q_{\ell})\mid \sigma_k(c)=\chi_{\ell}(\sigma_k)^{-d} c\}
\subset \mathrm{H}^{2d}(\bar{X},\Q_{\ell})$$
and the subspace
$$\tate_{2d}(X,\ell)\subset \Hom_{\Q_{\ell}}(\Lambda^{2d}_{\Q_{\ell}} V_{\ell}(X),\Q_{\ell})$$
of $\ell$-adic Tate forms of degree $2d$ on $X$.
Recall (see Definitions \ref{WlT} and \ref{tateDeinition}) that
the twist
$W_{\ell,d}(X)(d)\subset \mathrm{H}^{2d}(\bar{X},\Q_{\ell})(d)$ coincides with the subspace $\T_{\ell,d}(X)$
of $2d$-dimensional $\ell$-adic Tate classes on $X$.
Recall that map \eqref{HopfIso} is a $\Q_{\ell}$-algebra isomorphism.
Applying Remark \ref{product TateCl} to $d_1=d-1$ and $d_2=1$, we obtain that {\sl the existence of very exotic $\ell$-adic Tate class of dimension $2d$ on $X$ is equivalent to the existence of an exceptional $\ell$-adic Tate form of degree $2d$ on $X$.}
\end{sect}
We will need to state explicitly the following useful assertion.
\begin{lem}
\label{equivTateForms}
Let $X$ be an abelian variety over $k$. Let $\ell \ne \fchar(k)$ be a prime.
Then the following three conditions are equivalent.
\begin{itemize}
\item[(i)]
$X$ carries an exotic $\ell$-adic Tate class.
\item[(ii)]
$X$ carries a very exotic $\ell$-adic Tate class.
\item[(iii)]
There exists an exceptional $\ell$-adic Tate form on $X$.
\end{itemize}
In addition, the validity of equivalent conditions (i)-(iii) does not depend on a choice of $l$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{equivTateForms}]
The equivalence of (ii) and (iii) follows readily from the arguments at the end of Subsection \ref{etale}.
The equivalence of (i) and (ii) was already proven in Remark \ref{remTate}(iv).
Notice that property (b) of Lemma \ref{exceptreduced} does not depend on the choice of $\ell$. Now Lemma \ref{exceptreduced} implies that
the validity of (iii) does {\sl not} depend on the choice of $\ell$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{mainTate}]
By a theorem of Tate \cite{Tate1}, if $Y$ is any abelian variety over $k$ (e.g., $Y=X^n$) then every element of $\T_{\ell,1}(Y)$
is a linear combination of divisor classes on $Y$ with coefficients in $\Q_{\ell}$.
Suppose that there is an exotic $\ell$-adic Tate class on $X^n$ for some positive integer $n$. It follows from Lemma \ref{equivTateForms} (applied to $Y=X^n$
instead of $X$) that there is an exceptional $\ell$-adic Tate form on $X^n$.
In light of Lemma \ref{exceptreduced} (applied to $Y$ instead of $X$),
there exists an admissible reduced function
$R_X=R_Y \to \Z_{+}$. In light of Theorem \ref{mainRelation}, there exists an admissible reduced function
$e \colon R_X \to \Z_{+}$ of weight $\le N(g)$. This means that
$$\wt(e)=\sum_{\alpha\in R_X}e(\alpha) \le 2 N(g);$$
in particular,
$$0 \le e(\alpha) \le 2 N(g) \ \forall \alpha \in R_X.$$
Let us put $Z=X^{2N(g)}$ and consider $e$ as the reduced admissible function
$$R_Z=R_X \to \Z_{+}, \ \alpha \mapsto e(\alpha).$$
In light of Remark \ref{YXm} applied to $m=2N(g)$,
$$e(\alpha) \le 2 N(g)\le \mult_Z(\alpha) \ \forall \alpha \in R_Z=R_X.$$
It follows from Lemma \ref{exceptreduced} that there is an exceptional $\ell$-adic Tate form on $X^{2N(g)}=Z$.
Applying Lemma \ref{equivTateForms} to $Z$, we obtain that there is an exotic $\ell$-adic Tate class on $Z=X^{2N(g)}$.
Now the last assertion of Lemma \ref{equivTateForms} implies that there is an exotic $l$-adic Tate class on $Z=X^{2N(g)}$
for all primes $l \ne \fchar(k)$. This ends the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{semigroupTate}]
In light of arguments of Subsection \ref{etale} combined with Remark \ref{product TateCl}, it suffices to check that each $\ell$-adic Tate form of any even degree $2m$ on $X^n$
can be presented as a linear combination of exterior products of $\ell$-adic Tate forms of degree at most $H(g)$. Let us prove it, using induction by $m$.
The assertion is obviously true
for all $m \le H(g)/2$. Suppose that $m> H(g)$. First, notice that
$$R_{X^n}=R_X \ \forall n.$$
Applying Theorem \ref{semigroupADM}, we conclude that the conditions of Proposition \ref{lastEff} are fulfilled for
$$K=\Q_{\ell}, V=V_{\ell}(X^n), A=\Fr_{X^n}: V_{\ell}(X^n) \to V_{\ell}(X^n), $$
$$ \mathrm{spec}(A)=R_{X^n}=R_X, \ h=H(g)/2.$$
Applying Proposition \ref{lastEff}, we conclude that each $\ell$-adic Tate form of degree $2m$ on $X^n$ can be presented
as a linear combination of wedge products
$$\psi_{m-j}\wedge \phi_j \ (j=1, \dots, H(g)/2))$$
where
$\psi_{m-j}$ is an $\ell$-adic Tate form of degree $2(m-j)$ on $X^n$ and $\phi_j$ is an $\ell$-adic Tate form of degree $2j\le H(g)$ on $X^n$. Applying the induction assumption to all
$\psi_{m-j}$'s, we conclude that each $\ell$-adic Tate form of degree $2m$ on $X^n$
can be presented as a linear combination of exterior products of Tate forms of degree at most $H(g)$.
This ends the proof.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,373
|
using System;
using System.Collections.Generic;
using Umbraco.Core;
using Umbraco.Core.PropertyEditors;
namespace Umbraco.Web.PropertyEditors
{
// TODO: Remove in V8
[Obsolete("This editor is obsolete, use MultiUrlPickerPropertyEditor instead")]
[PropertyEditor(Constants.PropertyEditors.RelatedLinks2Alias, "Related links", "relatedlinks", ValueType = PropertyEditorValueTypes.Json, Icon = "icon-thumbnail-list", Group = "pickers", IsDeprecated = true)]
public class RelatedLinks2PropertyEditor : PropertyEditor
{
public RelatedLinks2PropertyEditor()
{
InternalPreValues = new Dictionary<string, object>
{
{Constants.DataTypes.ReservedPreValueKeys.IgnoreUserStartNodes, "0"},
{"idType", "udi"}
};
}
internal IDictionary<string, object> InternalPreValues;
public override IDictionary<string, object> DefaultPreValues
{
get { return InternalPreValues; }
set { InternalPreValues = value; }
}
protected override PreValueEditor CreatePreValueEditor()
{
return new RelatedLinksPreValueEditor();
}
internal class RelatedLinksPreValueEditor : PreValueEditor
{
[PreValueField(Constants.DataTypes.ReservedPreValueKeys.IgnoreUserStartNodes, "Ignore user start nodes", "boolean", Description = "Selecting this option allows a user to choose nodes that they normally don't have access to.")]
public bool IgnoreUserStartNodes { get; set; }
[PreValueField("max", "Maximum number of links", "number", Description = "Enter the maximum amount of links to be added, enter 0 for unlimited")]
public int Maximum { get; set; }
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,500
|
{"url":"https:\/\/pjbartlein.github.io\/GeogDataAnalysis\/lec14.html","text":"NOTE: This page has been revised for Winter 2021, but may undergo further edits.\n\n# 1 Introduction\n\nScatter-diagram smoothing (e.g.\u00a0using the lowess() or loess() functions) involves drawing a smooth curve on a scatter diagram to summarize a relationship, in a fashion that makes few assumptions initially about the form or strength of the relationship. It is related to (and is a special case of) nonparametric regression, in which the objective is to represent the relationship between a response variable and one or more predictor variables, again in way that makes few assumptions about the form of the relationship. In other words, in contrast to \u201cstandard\u201d linear regression analysis, no assumption is made that the relationship is represented by a straight line (although one could certainly think of a straight line as a special case of nonparametric regression).\n\nIf the basic decomposition-of-the-data model is:\n\ndata = predictable component + noise,\n\nthen for the standard bivariate or multiple (linear) regression, the model is\n\ndata = straight-line, polynomial or linearizable function + noise,\n\nwhile for nonparametric regression, the model is\n\ndata = smooth function determined by data + noise.\n\nAnother way of looking at scatter diagram smoothing is as a way of depicting the \u201clocal\u201d relationship between a response variable and a predictor variable over parts of their ranges, which may differ from a \u201cglobal\u201d relationship determined using the whole data set. (And again, the idea of \u201clocal\u201d as opposed to \u201cglobal\u201d relationships has an obvious geographical analogy.) Nonparametric regression can be thought of as generalizing the scatter plot smoothing idea to the multiple-regression context.\n\n# 2 Specific and general cases of smoothing and nonparametric regression\n\n## 2.1 A review of global fitting (e.g.\u00a0linear regression)\n\nIn ordinary linear regression analysis, the objective can be considered to be drawing a line through the data in an optimal way, where the parameters (regression coefficients) are determined using all of the data, i.e.\u00a0they are globally determined. However, it is possible to think of the line as connecting the points, that for each value of X, represent the local density maxima of Y\u2013it just happens that these local maxima happen to be arranged along a straight line.\n\n## 2.2 Loess curves\n\nA bivariate smoother is a function or procedure for drawing a smooth curve through a scatter diagram. Like linear regression (in which the \u201ccurve\u201d is a straight line), the smooth curve is drawn in such a way as to have some desirable properties. In general, the properties are that the curve indeed be smooth, and that locally, the curve minimize the variance of the residuals or prediction error.\n\nThe bivariate smoother used most frequently in practice is known as a \u201clowess\u201d or \u201cloess\u201d curve. The acronyms are meant to represent the notion of locally weighted regression\u2013a curve- or function-fitting technique that provides a generally smooth curve, the value of which at a particular location along the x-axis is determined only by the points in that vicinity. The method consequently makes no assumptions about the form of the relationship, and allows the form to be discovered using the data itself. (The difference between the two acronyms or names is mostly superficial, but there is an actual difference in R\u2013there are two different functions, lowess() and loess(), the former is an older, simple scatter-diagram-smoothing function, while the latter is newer and more flexible. There is also the locfit (local regression) package that is more flexible still.)\n\nThe mechanics of loess:\n\nThe parameters of an individual loess() curve:\n\n\u2022 span -a value between 0 and 1 controlling the amount of smoothing, with smaller values resulting in less smoothing. Typical values lie in the range of .3 to .5. Span can be considered to represent the width of the smoothing window.\n\u2022 degree -the degree of the locally-fitted polynomial. 1 = locally linear fitting (i.e.\u00a0a line), 2 = locally quadratic quadratic fitting (i.e.\u00a0a parabola).\n\u2022 family -either \u201cSymmetric\u201d or \u201cGaussian\u201d The symmetric option combines the local fitting with a \u201crobustness\u201d step that discounts the influence of unusual points, while the Gaussian option does not.\n\n# 3 Examples of nonparametric regression\n\nThe first examples of nonparametric regression are the familiar scatter diagram smoother lowess() and the related, more flexible loess() function.\n\nLoad some packages, and attach a data set of annual temperatures at Oregon climate stations.\n\nlibrary(RColorBrewer)\nlibrary(sf)\nattach(ortann)\nnames(ortann)\n## [1] \"station\" \"latitude\" \"longitude\" \"elevation\" \"tann\"\n\nLook at the annual temperature data:\n\nplot(elevation, tann, pch=16)\n\nNote that there are actually two versions of the lowess or loess scatter-diagram smoothing approach implemented in R lowess() and loess(). The former function (lowess()) was implemented first, while the latter (loess()) is more flexible and powerful. Because the function names are pronouced similarly, they are often confused, but because they basically do the same thing, that\u2019s not such a big deal.\n\n## 3.1 lowess()\n\nA simple \u201clowess\/loess\u201d curve is constructed using the lowess() function, which finds a \u201cfitted\u201d value for each data point; these can be plotted as individual symbols, but they are usually connected with lines. The lowess() function has a \u201cspan\u201d argument (sometimes symbolized by l) that represents the proportion of the total number of points that contribute to each local fitted value. In practice, the lowess() function is often embedded in a points() or \u2019lines() function.\n\nIn the following, the specific fitted values produced by the lowess() function (one per data point) are plotted in red, and the lowess curve is added, also in red. On top of that, two additional curves are plotted (in magenta and purple) that show the effect of the smoothing parameter choice. The green curve is smoother than the default (l = 0.667), and the magenta curve is \u201clooser\u201d than the default (l = 0.33).\n\n# lowess\nplot(elevation, tann, pch=16, cex=0.6)\npoints(lowess(elevation, tann), pch=16, col=\"red\", cex=0.5)\nlines(lowess(elevation,tann), col=\"red\", lwd=2)\n\n# different smoothing\nlines(lowess(elevation, tann, f=0.33), col=\"blue\", lwd=2)\nlines(lowess(elevation, tann, f=0.80), col=\"purple\", lwd=2)\nlegend(\"bottomleft\", title = \"lowess() spans\", legend=c(\"0.8\",\"0.667\",\"0.33\"), lwd=2, cex=1, col=c(\"purple\",\"red\",\"blue\"))\n\n## 3.2 loess()\n\nThe newer loess() function uses a formula to specify the response (and in its application as a scatter-diagram smoother) a single predictor variable. The loess() function creates an object that contains the results, and the predict() function retrieves the fitted values. These can then be plotted along with the response variable. However, the points must be plotted in increasing order of the predictor variable values in order for the lines() function to draw the line in an appropriate fashion. This is done by using the results of the order() function applied to the predictor variable values, and the explicit subscripting (in square brackets [ ]) to arrange the observations in ascending order.\n\n# loess -- first model\nloess_model <- loess(tann ~ elevation)\nloess_model\n## Call:\n## loess(formula = tann ~ elevation)\n##\n## Number of Observations: 92\n## Equivalent Number of Parameters: 4.69\n## Residual Standard Error: 0.8758\n# second model, smoother curve\nloess_model2 <- loess(tann ~ elevation, span=0.90, degree=2)\nloess_model2\n## Call:\n## loess(formula = tann ~ elevation, span = 0.9, degree = 2)\n##\n## Number of Observations: 92\n## Equivalent Number of Parameters: 4.04\n## Residual Standard Error: 0.8752\n# plot the curves\nplot(tann ~ elevation, pch=16, cex=0.6)\nhat1 <- predict(loess_model)\nlines(elevation[order(elevation)], hat1[order(elevation)], col=\"red\", lwd=2)\nhat2 <- predict(loess_model2)\nlines(elevation[order(elevation)], hat2[order(elevation)], col=\"blue\", lwd=2)\nlegend(\"bottomleft\", title = \"loess() model\", legend=c(\"span=0.667, deg=1\",\"span=0.9, deg=2\"), lwd=2, cex=1, col=c(\"red\",\"blue\"))\n\n## 3.3 loess() surfaces\n\nA locally determined surface can be constructed using loess() which is not limited to a single predictor variable, by first fitting a model that illustrates the response of the response variable as a function of two (or more) location variables, and then using the predict() function to visualize the resulting surface:\n\n# tann as a function of latitude and longitude (and interaction)\ntann_loess <- loess(tann ~ longitude + latitude, span=0.3)\nsummary(tann_loess)\n## Call:\n## loess(formula = tann ~ longitude + latitude, span = 0.3)\n##\n## Number of Observations: 92\n## Equivalent Number of Parameters: 21.11\n## Residual Standard Error: 0.9942\n## Trace of smoother matrix: 25.23 (exact)\n##\n## Control settings:\n## span : 0.3\n## degree : 2\n## family : gaussian\n## surface : interpolate cell = 0.2\n## normalize: TRUE\n## parametric: FALSE FALSE\n## drop.square: FALSE FALSE\n# poor man's R-squared value\ncor(tann, tann_loess\\$fitted)^2\n## [1] 0.8098961\n\nNow create an interpolation \u201ctarget\u201d grid, get the predicted (i.e.\u00a0interpolated) values, and plot the results.\n\n# create an interpolation target grid to display predicted values\ngrid_longitude <- seq(-124.5000, -116.8333, .1667)\ngrid_latitude <- seq(42.0000, 46.1667, .0833)\ngrid_mar <- list(longitude=grid_longitude, latitude=grid_latitude)\n\n# get the fitted (interpolated) values\ntann_interp <- predict(tann_loess, expand.grid(grid_mar))\ntann_z <- matrix(tann_interp, length(grid_longitude),\nlength(grid_latitude))\n\n# plot the interpolated values as shaded rectangles and contours\nnclr <- 8\nplotclr <- brewer.pal(nclr, \"PuOr\")\nplotclr <- plotclr[nclr:1] # reorder colors\n\nplot(st_geometry(orotl_sf), axes=TRUE)\npoints(longitude, latitude)\nplot(st_geometry(orotl_sf), add=T)\n\n## 3.4 Other bivariate smoothers\n\nLoess is one of a number of smoothers (including linear regression as an end-member) that can be used. The different smoothers vary in the assumptions they make about\n\n\u2022 the form of the relationship\n\u2022 the influence of individual points\n\nThe other scatter diagram smoothers include a straight, or \u201cleast-squares\u201d line, a low-order polynomial least-squares line, and the \u201csmoothing spline\u201d. Each can be viewed as special cases of the more flexible loess-type smoothers in which the curve is very simple. The best way to understand these different smoothers is to compare them:\n\n### 3.4.1 Least-squares line\n\n# first-order polynomial (i.e. a straight line)\nlinear_model <- lm(tann ~ elevation)\nlinear_model\n##\n## Call:\n## lm(formula = tann ~ elevation)\n##\n## Coefficients:\n## (Intercept) elevation\n## 11.688139 -0.003238\n\n### 3.4.2 Least-squares polynomials\n\n# second order polynomial\npoly2_model <- lm(tann ~ elevation+ I(elevation^2))\npoly2_model\n##\n## Call:\n## lm(formula = tann ~ elevation + I(elevation^2))\n##\n## Coefficients:\n## (Intercept) elevation I(elevation^2)\n## 1.133e+01 -1.147e-03 -1.459e-06\npoly2_hat <- predict(poly2_model)\n\n### 3.4.3 Smoothing spline\n\nspline_model <- smooth.spline(elevation, tann)\nspline_model\n## Call:\n## smooth.spline(x = elevation, y = tann)\n##\n## Smoothing Parameter spar= 1.12444 lambda= 0.003202393 (15 iterations)\n## Equivalent Degrees of Freedom (Df): 5.311004\n## GCV: 0.800279\n\nPlot the differnt smoothers\n\nplot(tann ~ elevation, pch=16, cex=0.6)\nabline(linear_model, col=\"red\", lwd=2)\nlines(elevation[order(elevation)], poly2_hat[order(elevation)],\ncol=\"blue\", lwd=2)\nlines(spline_model, col=\"purple\", lwd=2)\nlegend(\"bottomleft\", title = \"smoothers\", legend=c(\"linear\",\"polynomial\",\"spline\"), lwd=2, cex=1, col=c(\"red\",\"blue\",\"purple\"))`\n\n# 4 Summary of smoothers\n\nThe various smoothers can be summarized as follows\n\nAssumptions Smoother Form Influence of individual points\nfewest loess no assumptions unusual points discounted\nsmoothing spline smooth curve some discounting of unusual points\nrobust, robust MM straight line unusual points discounted\nleast squares (curvilinear) curve all points influential\nmost least squares (linear) straight line all points influential","date":"2021-03-02 07:35:16","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\": 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\": 0.6967469453811646, \"perplexity\": 4290.1400367708675}, \"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\/1614178363782.40\/warc\/CC-MAIN-20210302065019-20210302095019-00220.warc.gz\"}"}
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static GzCrashLogMessage *share;
@implementation GzCrashLogMessage
+(id)ShareManager{
if(!share){
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[Parse setApplicationId:@"xjbhzxxbxuOybVT8lhlWBTb6AX0JJ99lA3YsRNR1"
clientKey:@"tCBvtzztHqUGxAyLOduSw5P99iYp5dN6ZAmRFk4h"];
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if(!share){
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return share;
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NSString *DBname = [NSString stringWithFormat:@"CrashReport_%@",[NSBundle mainBundle].infoDictionary[@"CFBundleExecutable"]];
PFObject *crash = [PFObject objectWithClassName:DBname];
crash[@"AppName"] = [NSBundle mainBundle].infoDictionary[@"CFBundleExecutable"];
crash[@"ClassName"] = classname;
crash[@"VersionApp"] = [NSString stringWithFormat:@"%@",[[[NSBundle mainBundle] infoDictionary] objectForKey:@"CFBundleShortVersionString"]];
crash[@"CrashMessage"] = crashMessage;
crash[@"BuildVersion"] = [NSBundle mainBundle].infoDictionary[@"CFBundleVersion"];
crash[@"deviceName"] = [[UIDevice currentDevice] name];
crash[@"Ver_ios"] = [[UIDevice currentDevice] systemVersion];
[crash saveInBackgroundWithBlock:^(BOOL succeeded, NSError *error) {
NSLog(@"Done Background Save");
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NSString *DBname = [NSString stringWithFormat:@"CrashReport_%@",[NSBundle mainBundle].infoDictionary[@"CFBundleExecutable"]];
PFQuery *query = [PFQuery queryWithClassName:DBname];
[query findObjectsInBackgroundWithBlock:^(NSArray *objects, NSError *error) {
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NSString *DBname = [NSString stringWithFormat:@"CrashReport_%@",[NSBundle mainBundle].infoDictionary[@"CFBundleExecutable"]];
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[query whereKey:@"VersionApp" equalTo:version];
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[query whereKey:@"deviceName" containsString:version];
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[query whereKey:@"Ver_ios" equalTo:version];
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"redpajama_set_name": "RedPajamaGithub"
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| 8,593
|
\section*{Introduction}
How to define an urban area, the basic spatial unit for urban planning and studies, has been a long-standing problem for researchers and policymakers \citep{batty2006rank, gabaix2004evolution, rozenfeld2008laws}. This problem has become more important in recent decades because of the emergence of a large number of fast-urbanizing regions around the world (e.g., China and India). However, due to the complexity of the urban system, especially the fuzzy urban-rural transition, consistent and robust measurement to quantify urban areas has remained elusive.
For a long time, governments have relied heavily on the administrative boundaries to address urban issues (e.g., environmental and sustainable problems); and many location-based policies are also implemented based on the administrative divisions. However, administrative divisions are mainly divided by historical, political, and geographical reasons, making it difficult to reflect the socio-economic dynamics of cities. Additionally, administrative divisions are incomparable across different countries and periods \citep{wu2018zipf}. Therefore, some countries turn to employ socio-economic indicators (e.g., population, economic activity, and commuting) to re-divide urban areas. For example, metropolitan areas (MAs), the most commonly used socio-economic boundaries, define urban areas as closely related regions in terms of socio-economic connection \citep{berry1969metropolitan}. However, the construction of MAs has three main shortcomings. First, the detailed data (e.g., census data and commuting survey data) to construct MAs are lacking in many developing countries. Second, the data collection process for MAs is time-consuming and expensive, making it unable to capture the rapid urbanization process in fast-growing regions. Third, the standard to define MAs varies among countries. There is still a lack of a unified approach to obtain functional urban areas, which can be applicable to all countries.
Remote sensing data, especially the satellite-based data, provide continuous and consistent observations of urban activities on earth \citep{zhu2019understanding}. They are easily accessible for most countries and have been widely used to study urban dynamics at different spatial scales \citep{aubrecht2016consistent, mertes2015detecting, small2011spatial, taubenbock2019new, zhou2018global}. Based on different urban characteristics (e.g., multispectral information, light emissions, and morphological structures), several global urban maps have been derived from the remote sensing data, such as MODIS500m \citep{schneider2009new, schneider2010mapping}, GHSL \citep{corbane2019automated}, GlobeLand30 \citep{chen2015global}, and GUF \citep{esch2012tandem-x}. Meanwhile, some emerging urban data with humans as sensors, such as volunteered geographical information (VGI, e.g., OpenStreetMap) \citep{goodchild2007citizens} and social sensing data (e.g., mobile phone and social media) \citep{liu2015social}, have also shown great potential in revealing the socio-economic boundaries of cities \citep{jiang2015head, jiang2011zipf, long2016mapping}. In addition to these multi-source datasets, some new methods have also been developed to delimit urban areas \citep{cao2009svm, jiang2015head, liu2018high-resolution, long2016mapping, rozenfeld2008laws, trianni2015scaling, zhou2014cluster-based}. Especially, City Clustering Algorithm (CCA), which defined cities as the maximally connected populated areas, has attracted great attention due to its simplicity and efficiency \citep{jiang2011zipf, rozenfeld2008laws, rozenfeld2011area, vogel2018detecting}. Benefited from advanced computing techniques and easily accessed data sources, CCA or other data-driven methods can derive urban areas in a timely and simple manner. However, due to the complexity of the urban system and the fuzzy urban-rural transition, these methods still have difficulties in finding the optimal threshold that differentiates between urban and non-urban areas. Additionally, most of the previous studies use only one variable (e.g., population \citep{rozenfeld2008laws, rozenfeld2011area}, road networks \citep{jiang2011zipf, long2016mapping}, nighttime light emissions \citep{imhoff1997technique, zhou2018global}, or built-up areas \citep{huang2016mapping, schneider2010mapping}) to quantify urban areas. While human activities are coupled together, it is largely unclear whether different urban data could reflect the similar urbanization process. Therefore, it should be particularly helpful to develop a universal method that can objectively find the optimal threshold to delimit urban areas via multi-source data.
Complexity science of cities sheds some light on the optimal threshold problem. Urban systems, as typical self-organized systems, display some universal macroscopic patterns, such as Zipf's law \citep{krugman1996self, zipf1949human}, scaling laws \citep{bettencourt2007growth}, and fractal characteristics \citep{batty1994fractal}. Previous studies have shown that these macroscopic patterns emerge at the critical point of the urban system \citep{newman2005power}, and several physical models have been adopted to study the critical phenomena of cities \citep{goh2016complexity, makse1995modelling, makse1998modeling}. Notably, the percolation model, a typical model for studying complex systems \citep{christensen2005complexity}, was used on the road network data of Britain to show that the urban system emerges at the critical point of the percolation process \citep{arcaute2016cities, molinero2017angular}. These works inspire us to address the optimal threshold problem with the percolation model.
In this paper, we propose a novel method to extract urban areas from multi-source urban data. We adopt a broader definition of urban areas as maximally connected areas that have more urban elements (i.e., population, infrastructure, economic activity) than non-urban areas, and these three urban elements are widely acknowledged in the urban geography and urban economics fields to measure the urbanization process \citep{arcaute2016cities, rozenfeld2008laws, vogel2018detecting}. We find the optimal urban/non-urban threshold solely through the input data themselves by considering the critical nature of urban systems. Specifically, we traverse all potential thresholds and aggregate the urban units into a cluster system under each threshold. Based on the percolation theory, we get the optimal urban areas when the whole system is at the critical point. To verify our method, we investigate the geographical layouts of urban areas derived by three datasets (population, road, and nighttime light). Despite the datasets of great difference, we find that: i) our method can capture the similar geographical distributions of urban areas; and ii) the rank-size distribution of urban areas fits well with Zipf's law, a fundamental law of urban systems. We further validate our results in 10 cities of the world, using urban reference data based on Landsat imagery. The derived urban areas by different datasets show good agreement with the reference data, and the accuracy can be further improved through data fusion. These findings demonstrate the effectiveness of our method and also deepen our understanding of cities. From the perspective of applications, the efficient, consistent, and low-cost properties of this method make it a good starting point for mapping urban areas around the world.
\section*{Study areas and data}
\subsection*{Study areas}
We choose China as the main study country to demonstrate the effectiveness of our method. Then we apply our method to 28 countries to validate the universality of this approach. Despite the rapid urbanization experienced in the past few decades, many areas of China are still underdeveloped, and the rural-to-urban process is quite uneven across regions. More importantly, as the largest developing country, China is lacking in consistent urban area data, which highlights the meaningfulness of this study. For the remaining countries, we choose those largest countries in each continent, with an area of not less than $100,000km^2$, including both developed and developing countries (Table \ref{tab:country&data}). The national surface area information is derived from the 2016 United Nations Demographic Yearbook, available through United Nations Statistics Division. The administrative boundary data of all countries are available from GADM (\url{https://gadm.org}), an open-source database of global administrative areas.
\begin{table}[htbp]
\caption{Study countries and urban data}
\begin{ruledtabular}
\begin{tabular}{p{0.16\linewidth}p{0.2\linewidth}p{0.6\linewidth}}
Study countries & Africa & Algeria, Chad, D. R. Congo, Libya, Sudan \\ \cline{2-3}
& Asia & China, India, Indonesia, Kazakhstan, Saudi Arabia \\ \cline{2-3}
& Europe & France, Germany, Spain, Sweden, Ukraine \\ \cline{2-3}
& North America & Canada, Honduras, Mexico, Nicaragua, United States \\ \cline{2-3}
& Oceania & Australia, Papua New Guinea, New Zealand \\ \cline{2-3}
& South America & Argentina, Bolivia, Brazil, Colombia, Peru \\
\colrule
Urban data & Nighttime light & Global NPP-VIIRS nighttime light dataset \\ \cline{2-3}
& Population & China mobile phone estimated population dataset \\
& & Global population distribution dataset from WorldPop \\ \cline{2-3}
& Road & China road network dataset from the Ordnance Survey \\
& & Global road shapefile dataset from OpenStreetMap \\
\end{tabular}
\end{ruledtabular}
\label{tab:country&data}
\end{table}
\subsection*{Data}
We use three datasets -- nighttime light (remote sensing data), population (social sensing data), and road networks (VGI data) -- in this research. These datasets represent the three most important urban elements: economic activity, population, and infrastructure, respectively.
\paragraph{Nighttime light data.}
We use the new generation of nighttime light (NTL) data, the global NPP-VIIRS NTL data (available through \url{https://www.ngdc.noaa.gov/eog/viirs}). The NPP-VIIRS NTL data is produced from the Visible Infrared Imaging Radiometer Suite (VIIRS) Day/Night Band (DNB). Compared with the old DMSP/OLS data, the NPP-VIIRS data has a higher spatial resolution (15 arcsec) and partially relieves the saturation effects and blooming effects \citep{shi2014evaluating}. Before averaging the observations, the annual composites exclude data impacted by stray light, lightning, lunar illumination, and cloud-cover. We collect the annual `vcm-orm-ntl' average radiance data for 2016, which has undergone the outlier removal process, with non-light background set to zero. The NTL data is publicly accessible for most countries, which make it a useful data source to map socio-economic activities and functional urban areas \citep{imhoff1997technique, small2011spatial, vogel2018detecting, zhou2018global}
\paragraph{Population data.}
We use two population data sources. The first one is the WorldPop dataset (available through \url{https://www.worldpop.org}). WorldPop provides an open-access archive of high-resolution population distribution data \citep{tatem2017worldpop}. Especially, the `Global per country 2000-2020' datasets have been improved in terms of global consistency. We collect the 2016 data for all study countries. For China, we also collect the second population dataset, which is estimated by the anonymous mobile phone location data. Detailed information about this dataset is presented in \cite{dong2017measuring}. Here, we use the aggregated version with a resolution of $0.001^\circ \times 0.001^\circ$. Note that mobile phone estimated population is only a sample of the whole population; thus, we scale up the data with a factor derived by (national population) / (number of mobile phone users in the sample).
\paragraph{Road networks.}
We collect the OpenStreetMap (OSM) road shapefiles (available through \url{https://download.geofabrik.de}) for all study countries. For China, we also collect the road network data from the Ordnance Survey, a more detailed dataset than the OSM data \citep{liu2016automated}. The raw ordnance survey data records every segment in road networks with two endpoints' IDs and locations. We identify road intersections by endpoint's ID and obtain about 21 million ones.
\subsection*{Data quality assessment}
For the nighttime light, the VIIRS data has been greatly improved with in-flight calibration, finer quantization, and lower light detection \citep{elvidge2013why}. Moreover, the annual `vcm-orm-ntl' dataset is obtained through massive cloud-free observations and eliminates the background noise and ephemeral lights, thereby enhancing the radiance stability across the world \citep{elvidge2017viirs}. Besides, our method relies only on the relative brightness values of different areas within a country. For the population, the Worldpop dataset has been proved to have high accuracy of population distribution as shown in \cite{stevens2015disaggregating}. The mobile phone estimated dataset comes from our previous work and also has high accuracy in measuring population distribution in China \citep{dong2017measuring}. For example, at the district (county) level, the $R^2$s of the regression between mobile phone inferred population and census population are 0.97 and 0.98 for Beijing and Shanghai, respectively. For the road data, the quality of OSM data varies greatly across countries regarding completeness and accuracy. However, due to lacking the `ground truth' of road data, we did not investigate the effect of missing data of OSM in this paper. As demonstrated by previous studies, OSM data can be a reliable data source for the task of mapping urban areas \citep{haklay2010how, jiang2011zipf}.
\section*{Methods}
Our percolation-based city clustering algorithm (PCCA) includes three main steps. First, we aggregate the fine-scale urban data by $0.5^\prime \times 0.5^\prime$ grid cells to unify different datasets. Second, we apply the CCA to merge grid cells into urban clusters under each potential threshold. Third, we perform the percolation analysis on the detected clusters to find the optimal threshold and then map the urban areas at the optimal threshold. Fig. \ref{fig:PCCA_Schematic} shows the schematic of the PCCA. All steps will be discussed in detail below.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.9\linewidth]{Figure1.png}
\caption{Schematic of the percolation-based city clustering method. (a) Multi-source urban data are aggregated by grid cells. (b) We use the CCA to merge urban units into a cluster system under each potential threshold. CCA: i) Cells with a greater value than the potential threshold are marked as urban units (light blue). ii) An unprocessed urban cell is selected to form a new urban cluster. iii) This urban cluster recursively adds the nearest urban cells until all nearest neighbors have been processed. We use the eight nearest neighbors in our research. iv) An urban cluster is formed (dark blue). Then another process begins until all urban cells belong to an urban cluster. (c) Two-dimensional site percolation model. As for a $L\times L$ lattice, each site can be occupied with probability $p$, and adjacent occupied sites form a cluster (light blue). As $p$ increases, the largest cluster (dark blue) remains stable at first, but quickly becomes a giant spanning one at the critical point. We regard each potential threshold as probability $p$ in percolation and find the optimal threshold $D^*$ at this critical point. (d) Urban areas with optimal threshold $D^*$.}
\label{fig:PCCA_Schematic}
\end{figure*}
\subsection*{Aggregating multi-source data by grid cells}
Since multi-source urban data differ in granularity, data preprocessing is required to make the results comparable. For the nighttime light and population datasets, we directly downsample the data into $0.5^\prime \times 0.5^\prime$ grids. For the road datasets, we need more processing as raw data are vector files. For the Chinese ordnance survey data, we count the number of road intersections located in each grid cell and thus derive the road intersection grids. For the OSM data, to speed up the calculation, we divide road lines into small segments and count the total length of road networks in each grid cell. Then we obtain the road length grids. Note that we further divide the cell values of the population and road data by cell's spherical area to derive consistent density maps.
\subsection*{City Clustering Algorithm (CCA)}
At the grid cell level, we set each value of the above-mentioned datasets as a potential urban density threshold, which refers to the minimum density of population, infrastructure, or activity of urban areas. Cells with a greater value (more urban elements) than the threshold will be marked as urban units. Then, we apply the CCA to aggregate urban units into urban clusters under each potential threshold. The CCA originally uses fine-grained grid data of population and defines an urban cluster as the maximal, geographical continuous, populated areas \citep{rozenfeld2008laws}. Here, we expand the CCA not only to population, but also to all kinds of urban data (e.g., nighttime light and road networks). As shown in Fig. \ref{fig:PCCA_Schematic}b, an urban cluster starts with a random unprocessed urban cell, and recursively adds the nearest urban cells until all nearest neighbors have been processed. Then an urban cluster is formed. Another unprocessed urban cell is selected to form a new urban cluster until all urban cells belong to a specific cluster. We use the eight nearest neighbors in our research. To assess the stability of the approach, we also test the four nearest neighbors. A similar percolation process can be observed, and the optimal threshold and final urban maps remain stable, see Fig. S1. After performing the CCA, we get all cluster systems under each potential threshold.
\subsection*{Percolation of the cluster systems}
To find the optimal threshold of the CCA, we apply the percolation theory to analyze the properties of the extracted clusters. The percolation theory was originally developed in statistical physics and mathematics to study the emergent structures of clusters on a random graph. Since percolation can lead to some critical phenomena, urban researchers have then used percolation theory to model urban growth and to understand the critical phenomena of cities \citep{makse1995modelling, makse1998modeling, rozenfeld2008laws}. The two-dimensional site percolation is a simple and intuitive model to explain the percolation theory and explore the critical phenomena (Fig. \ref{fig:PCCA_Schematic}c). As for a $L\times L$ lattice, each site can be occupied with probability $p$, and adjacent occupied sites form a cluster. When $p$ is small, there are only a few small clusters. As $p$ becomes larger, the size of the largest cluster remains stable, despite more occupied sites. When $p$ reaches a certain point, a giant cluster quickly forms and spans the whole lattice. This point is called the critical point or the continuous phase transition. Around the critical point, the cluster system exhibits some critical phenomena (e.g., the size distribution follows power-law), which are also found in urban systems. Therefore, analogous to the two-dimensional lattice, we regard each potential threshold as the occupation probability $p$ in percolation, the optimal threshold $D^*$ can be found when the largest cluster of each cluster system becomes a giant one with a continuous phase transition. We consider the threshold at this critical point as the optimal threshold. After determining the optimal threshold, we obtain the final results of urban areas.
\section*{Results}
\subsection*{Urban areas extracted by PCCA}
We first apply the PCCA to the datasets of China. Following the Methods section, we obtain the density maps of population, road intersections, and nighttime light of China. Then, we extract the cluster systems under all potential thresholds and apply the percolation analysis to the cluster systems to find the optimal threshold.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.85\linewidth]{Figure2.png}
\caption{The normalized size of the largest cluster for different thresholds in the datasets of population (a), road (b), and nighttime light (c), and the distribution entropy of the cluster system for different thresholds in the datasets of population (d), road (e), and nighttime light (f). The solid blue lines and dotted red lines represent the results of clusters larger than $20 km^2$ and all clusters, respectively. The critical points of the continuous phase transition are marked with solid vertical lines. The values of these points are 550 people per $km^2$ in population data, 20 intersections per $km^2$ in road data, 3.0 DN value in nighttime light data. Maximum entropies are marked with dotted horizontal lines, which are also around the critical points. Insets: Similar to the main figures but with a log-scale for x-axis.}
\label{fig:pcca_China}
\end{figure*}
Fig. \ref{fig:pcca_China}a-c presents the size of the largest cluster of different thresholds, and the size has been normalized by the total area of all clusters. At each threshold, there are some fragmented areas, possibly due to data noise or some special land use (e.g., oil fields, scenic areas). We set a minimum size -- $20 km^2$ -- to filter those fragmented areas. This value is set because the smallest land area of a city in China is approximately $20 km^2$. We also test the sensitivity of our method to this parameter by setting the minimum size to $10, 15, 25 km^2$, and the results are robust (Fig. S2). In Fig. \ref{fig:pcca_China}, solid blue lines present the results of clusters larger than $20 km^2$, and dotted red lines show the results of all clusters. For all datasets, as we lower the threshold, a giant spanning cluster quickly forms when the threshold reaches a critical point (vertical lines) -- indicating a continuous phase transition. This phenomenon reflects the characteristics of the urban system as an interconnected complex system. Since the intra-city connections are much stronger than the inter-city connections, weak inter-city connections break up as we increase the threshold. When the threshold reaches a certain point, all weak inter-city connections do not exist, while the intra-city connections can still be tied closely. As a result, the size of the largest cluster goes through a critical point, which we consider as the optimal threshold to quantify urban areas.
Besides the largest cluster, we also calculate Shannon's entropy $H$ of the size distribution for each cluster system:
\begin{equation}
H=-\sum_{i=1}^{N}{p_{i}\log{p_{i}}}
\end{equation}
where $N$ is the number of clusters in the system, $p_i$ is the proportion of the area of cluster $i$ in all clusters. In Fig. \ref{fig:pcca_China}d-f, we find that the entropy also reaches the maximum (horizontal lines) around the critical point (vertical lines). Moreover, for each dataset, the entropies at the critical point are close for the clusters larger than $20 km^2$, which are 5.88 (population), 5.32 (road), and 5.34 (nighttime light), indicating the similar size distributions of urban areas extracted from different data sources.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.85\linewidth]{Figure3.png}
\caption{Urban areas in China delimited by the PCCA in the datasets of nighttime light (a), population (b), and road (c). The numbers of clusters ($> 20 km^2$) are 1085, 1260, and 931 for the nighttime light, population, and road datasets, respectively. Urban clusters are colored according to their geographical areas. Note that for simplicity, the South China Sea Islands are not shown in the maps.}
\label{fig:natural-cities_China}
\end{figure*}
Furthermore, we map the urban areas at the critical point (Fig. \ref{fig:natural-cities_China}). Strikingly, different data yield similar results. Especially, those larger clusters are well-developed cities, such as Chengdu, Xi'an, and Xiamen. Moreover, the maps also echo the uneven regional development in China. The urbanization level is much higher in the southern and eastern regions, and the coastal areas are more developed than the inland areas. The top three largest areas delineated by our method are the Beijing-Tianjin-Hebei, the Yangtze River Delta, and the Pearl River Delta economic zones (enlarged view of Fig. \ref{fig:natural-cities_China}), which are the most urbanized megalopolis areas in China \citep{xie2016updating, liu2010spatial}. These areas break the geographic constraints of administrative boundaries and have highly integrated connections. Our results confirm their high integrations.
To measure the similarities of urban areas delineated by different datasets, we use the Dice similarity coefficient (DSC). DSC is a similarity measure over sets and ranges from 1, with the same sets, to 0, with two completely different sets. It is defined as:
\begin{equation}
DSC=\frac{2\vert X \cap Y \vert}{\vert X \vert + \vert Y \vert}
\end{equation}
where $X$ and $Y$ are the sets of grid cells of urban areas. We remove the clusters smaller than $20 km^2$. The DSCs are 0.62 between population and road, 0.68 between road and nighttime light, and 0.59 between population and nighttime light, indicating that the spatial distributions of the urban areas obtained by different datasets are similar. Besides, we find the differences, those grid cells marked as urban areas in only one dataset, are mainly from the different distributions of intra-city `holes' and the peripheries of each urban cluster. For example, the Olympic Park in Beijing, with few people but dense roads, is a `hole' (non-urban areas) in the population map but urbanized in the road map.
We also investigate whether Zipf's law, one important law for the size distribution of the urban system, holds for our definition of urban areas. Zipf's law reflects the self-organized nature of urban systems and has been found in most countries \citep{auerbach1913gesetz, jiang2011zipf, rosen1980size, soo2005zipf, zipf1949human}. One expression of Zipf's law is that the probability of a city $i$ larger than size $S$ is inversely proportional to $S$:
\begin{equation}
P(S_i>S)=kS^{-\alpha}
\end{equation}
where $k$ is a constant and $\alpha=1$. We find that the size distributions of urban clusters in China all follow a power law of $\alpha$ close to $1$ with standard errors less than $0.06$, indicating that Zipf's law holds well for the urban areas delimited by our method (Fig. S3).
\subsection*{Robustness check}
To check the robustness of our method, we expand the analysis to 28 countries and present the results of France (Fig. \ref{fig:pcca_France}) and India (Fig. \ref{fig:pcca_India}) as examples. In the Appendix, we show the results of the remaining countries. In most countries, we obtain similar results as in China: (1) A giant spanning cluster quickly forms when the threshold reaches the critical point. (2) The distribution entropy reaches the maximum around the critical point. (3) The spatial distributions of urban areas delineated by three datasets are similar. In France, the largest cluster is the Paris metropolitan area, the political and economic capital of France. Other larger clusters also correspond to those well-developed regions, such as Marseille, Lyon, and Toulouse (Fig. \ref{fig:pcca_France}). In India, our method can also capture those important urban clusters, such as New Delhi, Mumbai, and Bengaluru (Fig. \ref{fig:pcca_India}). The critical points are 100 population per $km^2$, 6 $km$ per $km^2$, and 1.0 DN value in France, and 1800 population per $km^2$, 2 $km$ per $km^2$, and 2.0 DN value in India. These findings further validate the robustness and generalization of our method.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{Figure4.png}
\caption{PCCA in France. The largest cluster size for different thresholds in the datasets of population (a), road (b), and NTL (c). The distribution entropy in the datasets of population (d), road (e), and NTL (f). The delineated urban areas in the datasets of population (g), road (h), and NTL (i).}
\label{fig:pcca_France}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.8\linewidth]{Figure5.png}
\caption{PCCA in India. Same as Fig. \ref{fig:pcca_France}.}
\label{fig:pcca_India}
\end{figure*}
However, in some countries, we fail to observe a continuous phase transition (entropy is the maximum at the minimum threshold), especially in population and nighttime light datasets. There may be two reasons. On the one hand, limited to resolution, these urban data cannot detect lower urban intensity than the minimum threshold, which mainly occurs in some underdeveloped regions, such as Kazakhstan (Fig. S19) and Chad (Fig. S20). On the other hand, geographical barriers could block the geographical proximity. For example, in Australia, almost all population clusters are distributed in isolated areas along the coast due to the complex geography (e.g., deserts, mountains, rainforests). These geographical barriers break the weak inter-city connections (e.g., population distribution) even at the minimum threshold, while road networks can still connect the cities. Thus, in Australia (Fig. S5), we find the continuous phase transition only in the road dataset.
\subsection*{Accuracy assessment}
To evaluate the accuracy of our results, we compare the derived urban areas with the Landsat-based urban reference maps in 10 cities of 5 countries, including Beijing, Shenzhen, and Taiyuan in China, Ahmedabad and Guwahati in India, Marseille and Toulon in France, Mexico City and San Luis Potos\'{i} in Mexico, and Los Angeles in the United States. These cities span different continents and vary greatly in population size (from 0.2 million to 20 million inhabitants), including both coastal (e.g., Marseille) and inland (e.g., Beijing) cities, as well as cities in developed (e.g., Los Angeles) and developing (e.g., Ahmedabad) countries, which helps test the generality of our method. We use the administrative boundaries collected from GADM (see Data section) to determine the extent of each city. For each test city, we create a reference map with the same spatial resolution of our results and then label each pixel as urban or non-urban by manually interpreting the corresponding area in the Landsat 8 images (available at \url{https://earthexplorer.usgs.gov}). We also use the high-resolution Google Earth imagery to assist the interpretation when it is difficult to distinguish the land use type with Landsat 8 imagery. Our labeled dataset can be accessed through \url{https://github.com/caowenpu56/PCCA}. Besides, we also perform the accuracy analysis on the 2015 Global Human Settlement Layer (GHSL; \cite{corbane2019automated}), one well-known global urban map, to compare our results with GHSL's performance. The GHSL built-up 250m dataset measures the global built-up area density. We downsample the raw GHSL data and convert the density data to the urban/non-urban binary data. Based on previous works \citep{zhou2014cluster-based, zhou2018global}, areas with the built-up density larger than 20\% are labeled as urban areas in GHSL.
\begin{table*}
\caption{Accuracy assessment of the derived urban areas by PCCA}
\begin{ruledtabular}
\begin{tabular}{p{0.095\linewidth}lllp{0.05\linewidth}lllllllllllllllllllllllll}
City & Country & Total & Urban & Non-urban & \multicolumn{3}{c}{Population} && \multicolumn{3}{c}{Road} && \multicolumn{3}{c}{NTL} && \multicolumn{3}{c}{Fusion} && \multicolumn{3}{c}{GHSL} \\ \cline{6-8} \cline {10-12} \cline {14-16} \cline {18-20} \cline {22-24}
& & pixels & pixels & pixels & PA & UA & $\kappa$ && PA & UA & $\kappa$ && PA & UA & $\kappa$ && PA & UA & $\kappa$ && PA & UA & $\kappa$ \\
\colrule
Beijing & China & 25,619 & 5,328 & 20,291 & $0.83$ & $0.75$ & $0.73$ && $0.89$ & $0.64$ & $0.67$ && $0.81$ & $0.69$ & $0.67$ && $0.90$ & $0.73$ & \textbf{0.74} && $0.86$ & $0.71$ & $0.71$ \\
Shenzhen & China & 2,756 & 1,556 & 1,200 & $0.86$ & $0.94$ & \textbf{0.78} && $0.92$ & $0.87$ & $0.76$ && $0.99$ & $0.71$ & $0.51$ && $0.94$ & $0.88$ & \textbf{0.78} && $0.81$ & $0.95$ & $0.75$ \\
Taiyuan & China & 10,706 & 976 & 9,730 & $0.81$ & $0.76$ & $0.76$ && $0.73$ & $0.73$ & $0.70$ && $0.74$ & $0.63$ & $0.65$ && $0.79$ & $0.78$ & \textbf{0.77} && $0.80$ & $0.72$ & $0.73$ \\
Ahmadabad & India & 10,398 & 837 & 9,561 & $0.52$ & $0.92$ & $0.64$ && $0.68$ & $0.61$ & $0.61$ && $0.87$ & $0.57$ & $0.65$ && $0.71$ & $0.77$ & \textbf{0.71} && $0.47$ & $0.97$ & $0.61$ \\
Guwahati & India & 1,152 & 266 & 886 & $0.73$ & $0.93$ & $0.78$ && $0.88$ & $0.79$ & $0.78$ && $0.92$ & $0.78$ & $0.80$ && $0.89$ & $0.88$ & \textbf{0.85} && $0.67$ & $0.94$ & $0.73$ \\
Marseille & France & 1,229 & 585 & 644 & $0.97$ & $0.76$ & $0.69$ && $0.83$ & $0.69$ & $0.48$ && $0.99$ & $0.73$ & $0.65$ && $0.98$ & $0.76$ & $0.69$ && $0.83$ & $0.89$ & \textbf{0.74} \\
Toulon & France & 2,368 & 750 & 1,618 & $0.93$ & $0.77$ & $0.76$ && $0.78$ & $0.64$ & $0.55$ && $0.91$ & $0.73$ & $0.71$ && $0.94$ & $0.77$ & \textbf{0.77} && $0.77$ & $0.90$ & $0.76$ \\
Mexico City & Mexico & 2,107 & 909 & 1,198 & $0.99$ & $0.87$ & \textbf{0.86} && $0.99$ & $0.74$ & $0.70$ && $1.00$ & $0.75$ & $0.72$ && $1.00$ & $0.81$ & $0.80$ && $0.83$ & $0.88$ & $0.74$ \\
San Luis \newline Potos\'{i} & Mexico & 1,735 & 312 & 1,423 & $0.67$ & $0.94$ & $0.75$ && $0.83$ & $0.69$ & $0.70$ && $0.83$ & $0.79$ & $0.77$ && $0.83$ & $0.88$ & \textbf{0.82} && $0.65$ & $0.96$ & $0.74$ \\
Los Angeles & USA & 15,398 & 4,858 & 10,540 & $0.89$ & $0.94$ & $0.88$ && $0.97$ & $0.76$ & $0.77$ && $0.99$ & $0.73$ & $0.75$ && $0.97$ & $0.84$ & $0.85$ && $0.94$ & $0.93$ & \textbf{0.91} \\
Total & & 73,468 & 16,377 & 57,091 & $0.85$ & $0.84$ & \textbf{0.80} && $0.89$ & $0.71$ & $0.72$ && $0.90$ & $0.70$ & $0.72$ && $0.91$ & $0.79$ & \textbf{0.80} && $0.84$ & $0.83$ & $0.79$ \\
\end{tabular}
\end{ruledtabular}
\footnotetext{Note: Results with the highest $\kappa$ for each city are marked in bold.}
\label{tab:accuracy}
\end{table*}
For our three results (population, road, and nighttime light) and GHSL data, we compare all pixels located in the test extent of each city with the urban reference maps pixel by pixel, and then calculate the confusion matrices. The number of labeled pixels (urban/non-urban) in the reference maps for each city is listed in Table \ref{tab:accuracy}. Due to the large number of non-urban pixels, the overall accuracy (OA) is high in all cites. The OAs of total test areas are $0.93$, $0.89$, and $0.89$ in the population, road, and nighttime light dataset, respectively. In Table \ref{tab:accuracy}, we also list the user's and producer's accuracy of the urban class, and the Kappa coefficients ($\kappa$) for each city. Our results show good agreement with the urban reference maps, with the $\kappa$ of $0.64-0.88$ (mean: $0.76$, sd: $0.07$) in population dataset, $0.48-0.78$ (mean: $0.67$, sd: $0.10$) in road dataset, and $0.51-0.80$ (mean: $0.69$, sd: $0.08$) in nighttime light dataset. These Kappa coefficients calculated on a single dataset are similar to the GHSL data ($\kappa=0.61-0.91$, mean: $0.74$, sd: $0.07$). However, we highlight that our PCCA method is based on a physical model (i.e., percolation) and only has one free parameter (i.e., threshold) that can be derived through the percolation process. These properties make our method better in interpretation and efficiency.
Additionally, since urban areas are derived from multi-source datasets, we can further generate a fused urban area map to improve the results. Specifically, we merge the urban areas delimited by three datasets; and extract urban areas identified in at least two datasets. This process is similar to the majority voting in the ensemble model, which can improve the accuracy and stability of the results in different regions \citep{trianni2015scaling}. Fig. \ref{fig:validation_cities} presents the urban maps of the fusion results, the GHSL data, and the reference data. Visually, our method captures the accurate urban extent for all test cities, and the fusion results match well with the reference data. Statistically, the accuracy scores indicate that the fusion results have improved accuracy and stability over the results from a single dataset. The mean $\kappa$ increases from $0.67-0.76$ to $0.78$, while the standard deviation of $\kappa$ decreases from $0.07-0.10$ to $0.05$ (Table \ref{tab:accuracy}). Besides, the fusion results perform better than the GHSL data with a larger $\kappa$ in 8/10 cities. We also apply a student's t-test to the $\kappa$ between the fusion results and the GHSL data in test cities, and the results show that the improvement is statistically significant ($p$-value $<0.05$).
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{Figure6.png}
\caption{A comparison of the derived urban areas by population (c), road (d), and nighttime light (e) dataset, the fusion results (f), the GHSL data (e) with the Landsat-based urban reference maps (b) in Beijing, Ahmadabad, Mexico City, and Los Angeles. (a) Google Earth imagery.}
\label{fig:validation_cities}
\end{figure*}
\subsection*{Optimal threshold and socio-economic development}
Table \ref{tab:pcca_world} presents the optimal threshold ($D_{pop}$, $D_{road}$, $D_{ntl}$), entropy at the threshold ($E_{pop}$, $E_{road}$, $E_{ntl}$), and Zipf exponent ($\alpha_{pop}$, $\alpha_{road}$, $\alpha_{ntl}$) for each country and each dataset. We find that $D_{pop}$ varies greatly from country to country, while $D_{road}$ and $D_{ntl}$ change less since the development of road networks and nighttime light is limited by geospatial space. For each country, the entropies at the critical point in three datasets are similar, indicating the similar size distributions of the delineated urban areas. Meanwhile, the Zipf exponents of size distributions fall within $[0.75,1.25]$ in 24/28 (population), 20/24 (road), and 24/28 (nighttime light) countries, which means that Zipf's law holds well in most countries.
\begin{table*}
\caption{PCCA in 28 countries}
\begin{ruledtabular}
\begin{tabular}{lllllllccc}
Country & $D_{pop}$ & $E_{pop}$ & $D_{road}$ & $E_{road}$ & $D_{ntl}$& $E_{ntl}$ & $\alpha_{pop}$ & $\alpha_{road}$ & $\alpha_{ntl}$\\
\colrule
Algeria & $300$ & $4.18$ & $2.5$ & $3.83$ & $2.0$ & $4.42$ & $0.97\pm 0.05$ & $1.12\pm 0.06$ & $1.02\pm 0.05$ \\
Argentina & $20^*$ & $5.05$ & $2.0$ & $4.90$ & $0.1^*$ & $5.36$ & $1.07\pm 0.04$ & $1.25\pm 0.04$ & $1.26\pm 0.05$ \\
Australia & $20^*$ & $4.11$ & $2.0$ & $4.18$ & $0.1^*$ & $4.67$ & $1.05\pm 0.03$ & $1.29\pm 0.03$ & $1.13\pm 0.05$ \\
Bolivia & $40$ & $3.38$ & $1.4$ & $4.51$ & $0.1^*$ & $3.25$ & $0.82\pm 0.05$ & $1.36\pm 0.05$ & $1.10\pm 0.09$ \\
Brazil & $20^*$ & $5.92$ & $1.5$ & $5.58$ & $0.1^*$ & $5.93$ & $1.24\pm 0.03$ & $1.15\pm 0.03$ & $1.22\pm 0.03$ \\
Canada & $20^*$ & $4.57$ & $-$ & $-$ & $0.5$ & $4.85$ & $1.06\pm 0.02$ & $-$ & $1.05\pm 0.01$ \\
Chad & $140$ & $3.38$ & $0.8$ & $4.64$ & $0.1^*$ & $2.15$ & $0.98\pm 0.14$ & $1.20\pm 0.05$ & $0.78\pm 0.11$ \\
China & $700$ & $6.49$ & $2.5$ & $5.80$ & $3.0$ & $5.34$ & $1.21\pm 0.04$ & $1.15\pm 0.02$ & $1.00\pm 0.03$ \\
Colombia & $80$ & $4.07$ & $1.6$ & $3.70$ & $0.1^*$ & $4.44$ & $0.81\pm 0.05$ & $0.87\pm 0.09$ & $1.08\pm 0.05$ \\
D. R. Congo & $100$ & $4.74$ & $1.0$ & $5.70$ & $0.1^*$ & $3.12$ & $1.07\pm 0.06$ & $1.26\pm 0.05$ & $1.37\pm 0.25$ \\
France & $100$ & $4.95$ & $6.0$ & $4.97$ & $1.0$ & $4.75$ & $1.11\pm 0.02$ & $1.07\pm 0.02$ & $1.02\pm 0.02$ \\
Germany & $200$ & $4.12$ & $11.0$ & $4.23$ & $0.5$ & $4.34$ & $1.10\pm 0.04$ & $1.12\pm 0.03$ & $1.13\pm 0.03$ \\
Honduras & $80$ & $3.10$ & $-$ & $-$ & $0.1^*$ & $3.54$ & $0.82\pm 0.07$ & $-$ & $1.04\pm 0.08$ \\
India & $1800$ & $5.24$ & $2.0$ & $5.02$ & $2.0$ & $5.55$ & $0.98\pm 0.02$ & $1.19\pm 0.02$ & $1.05\pm 0.03$ \\
Indonesia & $1600$ & $3.39$ & $2.5$ & $4.85$ & $1.0$ & $3.78$ & $0.89\pm 0.09$ & $1.03\pm 0.04$ & $1.00\pm 0.06$ \\
Kazakhstan & $20^*$ & $4.15$ & $1.0$ & $5.78$ & $0.1^*$ & $4.79$ & $0.77\pm 0.05$ & $1.29\pm 0.04$ & $1.11\pm 0.04$ \\
Libya & $140$ & $2.05$ & $0.8$ & $3.76$ & $0.1^*$ & $2.98$ & $0.94\pm 0.08$ & $1.24\pm 0.07$ & $1.01\pm 0.12$ \\
Mexico & $300$ & $4.59$ & $1.8$ & $5.15$ & $3.0$ & $4.35$ & $1.06\pm 0.05$ & $1.00\pm 0.06$ & $0.87\pm 0.05$ \\
New Zealand & $20^*$ & $3.08$ & $1.4$ & $4.15$ & $0.1^*$ & $3.62$ & $0.99\pm 0.05$ & $1.17\pm 0.03$ & $1.13\pm 0.12$ \\
Nicaragua & $100$ & $2.47$ & $-$ & $-$ & $0.1^*$ & $2.25$ & $0.72\pm 0.06$ & $-$ & $1.18\pm 0.17$ \\
Papua New Cuinea & $320$ & $1.96$ & $0.8$ & $4.71$ & $0.1^*$ & $2.49$ & $0.39\pm 0.08$ & $0.96\pm 0.05$ & $1.54\pm 0.37$ \\
Peru & $40$ & $4.63$ & $1.4$ & $5.02$ & $0.1^*$ & $4.46$ & $0.73\pm 0.03$ & $1.19\pm 0.06$ & $0.96\pm 0.05$ \\
Saudi Arabia & $40$ & $3.48$ & $-$ & $-$ & $1.5$ & $4.17$ & $0.74\pm 0.03$ & $-$ & $0.82\pm 0.03$ \\
Spain & $60$ & $4.02$ & $5.5$ & $3.85$ & $0.5$ & $4.36$ & $1.13\pm 0.03$ & $1.12\pm 0.04$ & $0.95\pm 0.04$ \\
Sudan & $20^*$ & $2.19$ & $1.0$ & $4.58$ & $0.1^*$ & $3.40$ & $0.79\pm 0.07$ & $1.24\pm 0.06$ & $0.87\pm 0.06$ \\
Sweden & $20^*$ & $3.95$ & $3.5$ & $4.05$ & $0.1^*$ & $4.86$ & $1.04\pm 0.04$ & $1.09\pm 0.05$ & $1.13\pm 0.03$ \\
Ukraine & $60$ & $5.51$ & $2.5$ & $5.25$ & $0.1^*$ & $5.13$ & $1.17\pm 0.04$ & $1.22\pm 0.03$ & $1.27\pm 0.03$ \\
USA & $300$ & $5.62$ & $4.0$ & $6.00$ & $1.0$ & $5.94$ & $0.96\pm 0.02$ & $1.11\pm 0.01$ & $0.92\pm 0.01$ \\
\end{tabular}
\end{ruledtabular}
\footnotetext{$^*$: no continuous phase transition and the minimum threshold is used.}
\footnotetext{$-$: no available data.}
\footnotetext{$\pm$: standard error.}
\label{tab:pcca_world}
\end{table*}
Furthermore, we explore the relationship between the optimal thresholds and countries' socio-economic indicators. Here, we use the urban population density as the proxy for socio-economic development. We calculate the urban population density by dividing the total population (WorldPop data) that fall within the urban clusters for each country. Intuitively, the population threshold $D_{pop}$ is highly correlated with urban population density (the $R^2$ is 0.81, Fig. \ref{fig:socio-economic}a), and $D_{pop}$ of each country is about $1/3$ of the country's urban population density. However, thresholds of road ($D_{road}$) and nighttime light ($D_{ntl}$) have weak positive correlations with urban population density (Fig. \ref{fig:socio-economic}b,c). This may result from the country's slow development of transportation infrastructure, which mainly occurs in some developing countries with large population size, such as India, Algeria, and Mexico. Besides, the OSM data quality varies greatly across different countries, and many road lines are not captured by OSM in some developing countries. Therefore, $D_{road}$ is smaller than the actual urban road density threshold in some countries. (The $R^2$ between $D_{road}$ and urban population density becomes 0.71 if removing some special cases, as shown in Fig. \ref{fig:socio-economic}e.)
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{Figure7.png}
\caption{Relationship between the optimal threshold and urban population density. All countries are included in the datasets of population (a), road (b), and NTL (c). Some countries are removed in the datasets of population (d), road (e), and NTL (f). In population and NTL dataset, we remove the countries without a continuous phase transition; in the road dataset, we remove those with large population size and poor transportation infrastructure (India, Algeria, Indonesia, Mexico, Colombia, and Argentina).}
\label{fig:socio-economic}
\end{figure*}
Finally, we apply the method to the entire world using nighttime light data and present the delineated urban areas in Fig. \ref{fig:pcca_World}. Similar to the country level findings, the largest cluster size goes through a critical point, and the maximum distribution entropy is exactly at this point. Then, we obtain the optimal threshold -- 1.0 DN value. Through this world map (Fig. \ref{fig:pcca_World}), we find that the urbanization level is much higher in North America, Europe, and East Asia. The top six urban clusters correspond to the Manchester-Milan (Europe), the Greater Cairo (Egypt), the Yangtze River Delta (China), the Boston-Washington (USA), the Delhi National Capital Region (India), and the Taiheiy\a=o Belt (Japan) megalopolises (Fig. \ref{fig:pcca_World}). The size distribution of urban areas of the world also fits well with Zipf's law, with an exponent of $0.97\pm 0.01$. We note that different dimensions of urban areas are captured at different spatial scales. At the country level (such as Fig. \ref{fig:natural-cities_China}), we delimit the metropolitan areas; while at the world level (Fig. \ref{fig:pcca_World}), due to the differences in the economic basis of each country, we delimit those large mega-regions or urban corridors \citep{georg2016new}. For example, the development level of the capitals of some African countries is even far less than that of rural areas in some developed countries. Therefore, the meanings of our extractions are different at different spatial level, and they depend on applications.
\section*{Discussion and conclusions}
\begin{figure*}
\centering
\includegraphics[width=0.8\linewidth]{Figure8.png}
\caption{PCCA applied to the world using nighttime light data. (a) Largest cluster sizes. (b) Distribution entropies. (c) PDF and fitting line of urban cluster areas (Zipf's law). (d) Urban areas by PCCA.}
\label{fig:pcca_World}
\end{figure*}
In summary, we propose a `percolation-based city clustering algorithm' to extract urban areas from multi-source urban data (nighttime light images, population data, and road networks). Our method only needs one parameter (urban/non-urban threshold), which can be derived solely through the input data themselves by considering the critical nature of urban systems. The derived urban areas are validated in several parts of the world, and they can be further improved by data fusion.
The contributions of this study can be summarized in three aspects. First, we bridge the gap between remote sensing and emerging urban data in the task of delimiting urban areas. Our study has demonstrated that despite great differences, different urban data can reflect the similar socio-economic dynamics of cities. Second, our method provides a consistent measurement of urban areas since the optimal threshold is derived automatically and under the same criteria. With our method, urban development can be measured under a unified standard, which allows comparisons across different countries and periods. Third, we show the potential of open-source data in delimiting urban areas. With the proposed method, we can produce reliable urban area maps from these publicly available data, which is especially helpful for those developing regions with limited survey data. Our study is also an attempt for applying complexity science to solve traditional urban problems and could deepen our understanding of urban systems.
There are still some limitations in this study, and several improvements can be explored in future work. First, due to the limited availability of temporal urban data, we have not been able to track the changes of urban areas over time. Such analysis could be possible with more detailed spatio-temporal data in the future. Second, it is meaningful to study the factors that influence the values of optimal thresholds. Possible explanations can be complicated for geographical, social, or economic reasons. For example, environmental awareness can cause a decrease in brightness of nighttime light in some regions of Europe \citep{bennie2015contrasting}. Third, the differences of the urban areas delineated by multi-source data are also worth exploring, as they reflect the inconsistent configuration of urban elements. However, how to explain these differences is full of challenges and requires more in-depth study.
\section*{Data availability}
All open-source datasets are available through the websites described in the data section. The PCCA codes and the maps of the delineated urban areas can be obtained through \url{https://github.com/caowenpu56/PCCA}.
\section*{Acknowledgements}
This research was supported by the National Natural Science Foundation of China (Grant Nos.41625003, 41801299, 41830645) and the China Postdoctoral Science Foundation (2018M630026).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,414
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Emir Pineda, Trade & Logistics Manager for the Marketing Division of the Miami-Dade Aviation Department, which operates Miami International airport, has been elected to the Board of The International Air Cargo Association (TIACA).
Global aviation services group Air Partner has appointed Kevin Macnaughton as Managing Director, Charter, with immediate effect.
ULD management and repair solutions provider Unilode Aviation Solutions and ULD manufacturer Nordisk Aviation Products have developed an integrated pallet edge rail solution for digital tags.
Skyway Aviation Handling Company (SAHCO) has signed a contract to handle Virgin Atlantic Cargo's operations in Lagos.
Chinese retailers wanting to reach consumers in Russia are to benefit from a new route from Xi'an to Moscow courtesy of Atran Airlines, the express carrier within Volga-Dnepr Group.
From 1 April 2019, AVS GSA became Oman Air's exclusive General Sales and Services Agent (GSSA) in Malaysia.
IAG Cargo is growing its product portfolio, with a new Relocation product for customers moving abroad and the addition of 24/7 services to its Critical product.
Khalfan Al Shueili, the Chief Executive Officer of Oman Aviation Services (OAS), has been elected to the Board of The International Air Cargo Association (TIACA).
A sustainability report of PALLITE's product range has found that the general life cycle of a PALLITE pallet has less impact on the environment than a wooden pallet.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,793
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\section{Introduction}
Angular momentum (AM) about a fixed point is conserved in isolated systems characterized by a rotationally symmetric Lagrangian~\cite{Noether_1971}. The eigenstates of atomic electrons exemplify this, and the AM of the collective light-matter system is conserved in association with their spontaneous orbital decay~\cite{Grinter_2008}. For systems with less symmetry, though, there is no expectation that a one-to-one correspondence should exist between the electronic and photonic units of AM in light-matter interactions~\cite{Andrews_2010_1, Andrews_2010_2}.
Of course individual photons, as bosonic field excitations, have a well-defined helicity of unit magnitude~\cite{Leader_2016}. In addition, electromagnetic radiation fields described by the time-ordered superpositions of such excitations can also have a photonic angular momentum (PAM) even when full rotational symmetry is not present. A Gaussian beam of circularly polarized light is composed of photons of the same helicity and has a PAM of $^\pm 1$ in units of $\hbar$. Within the paraxial approximation, optical vortices~\cite{Allen_1992} also have a quantized AM per photon that are eigenvalues of an operator that does not depend on gauge or frame~\cite{QED_BLP_1982, Barnett_2016, Leader_2016}. A circularly polarized Laguerre-Gaussian beam of azimuthal index $l$, for instance, has PAM = $^\pm 1+l$ within the paraxial approximation. These AM measures are possible because the radiation has an axis of symmetry and a negligibly small radial gradient.
A molecular axis of symmetry facilitates the assignment of a meaningful AM to the electronic state of specially designed molecules as well. These have a discrete rotational symmetry group of either $C_N$ or $C_{Nh}$, such as those shown in Figure \ref{molecules}, and their repeated subunits are referred to as arms. These may radiate outward as chiral spokes, as in triphenylphosphine ($\rm{Ph}_3\rm{P}$) and hexaphenylbenzene, or may compose a planar arrangement as found in porphyrin and corronene structures.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.9\textwidth]{Fig01.pdf}
\end{center}
\caption{\emph{Molecules with $C_N$ or $C_{Nh}$ symmetry}. The three planar molecules shown are among many polycyclic aromatic hydrocarbons for which there exists conjugation between adjacent arms. Three-dimensional molecules, such as triphenylphosphine and hexaphenylbenzene, may also offer sufficiently strong coupling between arms.}
\label{molecules}
\end{figure}
The associated molecular eigenstates can be expressed as a phase-shifted superposition of the lowest energy excitons associated with each of the arms. These collective states are identifiable by the phase shift, $2 \pi q_e/N$, between neighboring arms in the superpositions~\cite{AndrewsPRL2013, Andrews_2014}. The integer, $q_e$, is just the number of $2\pi$ windings of the phase accumulated in traversing the circuit of arms and so can be interpreted as the number of units of an excitonic, quasi-angular momentum.
Quasi-momenta are encountered in any material system that exhibits a discrete symmetry. For instance, solid-state lattices have a quasi-linear momentum that is distinct from the particle momentum of the ions and electrons of which it is composed. This momentum is the conserved quantity associated with a discrete translational symmetry of the lattice~\cite{Ashcroft_1976}. An analogous application of discrete rotational invariance has been used to explain how particle angular momentum can be transferred to a quasi-angular momentum for a variety of quasi-particles~\cite{Baltz_2002}. When the lattice points fall on a circle, as with the molecular arms of centrosymmetric molecules, this quasi-angular momentum is simply the product of an effective radius, $R$, with excitonic quasi-momentum, $\hbar k_{ex}$. Azimuthal periodicity then implies that $k_{ex} R$ is an integer, $q_e$. In analogy to their photonic counterparts~\cite{Padgett_2004, Dennis_2009}, these electronic states are referred to here as \emph{twisted excitons} with an excitonic angular momentum (EAM) of $q_e$. Each molecular eigenstate has a distinct EAM, and there is a one-to-one relationship between the eigenenergies and the magnitude of EAM.
Electronic decay of these molecules results in the generation of an optical vortex\cite{AndrewsPRL2013} in which both energy and AM are transferred between excitonic and optical forms. The same is true for absorption events in which this process is reversed and, in fact, for a sequence of such absorption events. Subsequent emission, either spontaneous or stimulated, can produce radiation with the accumulated EAM. In this way, the excitons play the role of an angular momentum bank in which PAM can be deposited and withdrawn in different increments. The concept, illustrated in Figure \ref{Calculator}, offers a means of changing the AM of light that does not rely on higher-order susceptibilities~\cite{SHGPRA1996, SHGPRA1997,FWMPRL2012, FWMOL2016,FWMOE2009, FEMPRB2015, FWMPRB2016}. Angular momentum is conserved in these light-matter interactions, and a one-to-one correspondence between excitonic and photonic angular momentum therefore exists in this special setting.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig02_small.pdf}
\end{center}
\caption{\emph{Changing the angular momentum of light using twisted excitons.} Two laser pulses, each with a PAM = $1_p$, are sequentially absorbed by a molecular assembly resulting in an EAM = $2_e$. The radiation subsequently emitted has a PAM = $2_p$.}\label{Calculator}
\end{figure}
This particular form of AM conservation is in contrast to that associated with the excitation or decay of a single electron state for hydrogen-like atoms lying on the axis of an optical vortex. Suppose that a static magnetic field makes absorption/emission preferred in one direction, and consider a right-circularly polarized Laguerre-Gaussian beam, within the paraxial approximation, of azimuthal index $l = 1$ that is aligned with it. The electromagnetic radiation therefore has PAM = $2_p$. Even though the electric field intensity is zero along the symmetry axis, it has a nonzero gradient, and the light-matter quadrupole coupling, $M = \bf{Q}\cdot \nabla \bf{E}$, is finite. Here $\bf{Q}$ is the transition quadrupole tensor between the $1s$ and $3d_2$ states. Two units of AM can therefore be transferred from beam to electron as has only recently been experimentally confirmed~\cite{Schmiegelow_2016}. Here the electrons have a continuous azimuthal symmetry as opposed to the discrete excitonic symmetry of centrosymmetric molecules.
Electromagnetic radiation composed of a superposition of photons of both helicities exhibits AM conservation in each of its components. A second single-electron example makes this clear, and the idea extends to the molecular setting as well. The same Laguerre-Gaussian beam of azimuthal index $l = 1$, but now linearly polarized, interacts with the atomic ground state, and the electron can once again be raised to the $3d_2$ state. This would seem to violate AM conservation because the aggregate input beam has PAM = $1_p$ and results in an EAM = $2_e$ state. However, the beam is actually composed of a superposition of right and left circularly polarized optical vortices that have AM of $2$ and $0$, respectively. The AM of each individual interaction is conserved: $1_{ps}+1_{pl} \rightarrow 2_e$ and $^-1_{ps}+1_{pl} \rightarrow 0_p$. Here subscripts $p$ and $e$ indicate photonic and excitonic manifestations while subscript $s$ and $l$ denote polarization and vortex origins. For such mixed state beams, electronic occupation probability must be included in the AM algebra: $1_p \rightarrow \frac{1}{2}2_e + \frac{1}{2}(^-1_{ps} + 1_{pl})$.
We examine AM banking in association with centrosymmetric molecules using a combination of theory and computation. A tight binding (TB) paradigm is used to analytically prove that AM is conserved in association with sequences of absorption and emission. Numerical implementations demonstrate this within a time-dependent setting. The associated Hamiltonian is then replaced with one that does not rely on prescribed transition dipoles and for which electron excitations are treated as many-body events with exchange energies and correlation effects included. Twisted excitons are no longer just superpositions of two-level excitations on each molecular subunit, but AM conservation emerges none-the-less. This time-domain Density Functional Theory (TD-DFT) setting is used to simulate the dynamics of AM transfer between photonic and excitonic manifestations, and simple additions and subtractions are once again demonstrated. In both TB and TD-DFT settings, the underlying processes are linear in the sense that only first-order electric dipole interactions are necessary. This is in contrast to the nonlinear optics strategy for up/down converting angular momentum using higher order susceptibilities. There is a tradeoff, certainly, because the AM banking scheme presumes that exciton relaxations and decoherence processes occur on a time scale slower than that of the AM manipulations. For the sake of clarity in exploring this alternative methodology, though, exciton-phonon coupling and dynamic disorder are disregarded.
\section{Tight-Binding Paradigm}\label{TB}
First consider a molecular Hamiltonian in the absence of light-matter coupling. The requisite $C_N$ or $C_{Nh}$ symmetry is provided by an N-arm molecule in which arm $j$ supports two energy levels: ground state $\ket{\xi^{j,0}}$ and excited state $\ket{\xi^{j,1}}$. The tight-binding (TB) Hamiltonian is taken to be
\begin{equation}
\hat{H}_0=\sum_{j=1}^{N}\Delta \hat{c}_j^{\dagger}\hat{c}_j+\sum_{<i,j>}^{N}(\tau\hat{c}_j^{\dagger}\hat{c}_i+H.c.)
\label{TBH0}
\end{equation}
Here $\Delta$ is the excited state energy of each arm, $\tau$ is the coupling between nearest arms, and $\hat{c}_j^{\dagger}$ is the creation operator for arm $j$. It is straightforward to show~\cite{AndrewsPRL2013} that the ground state is $\ket{0}=\prod_{j=1}^{N}\ket{\xi^{j,0}}$ while the N excited states are
\begin{equation}
\ket{v_{q_e}}=\sum_{j=1}^{N}\frac{\varepsilon^{(j-1)q_e}}{\sqrt{N}}\ket{e_j}
\label{TBstates}
\end{equation}
with $\varepsilon=\mathrm{e}^{\imath 2\pi/N}$ and $\ket{e_j}=\ket{\xi^{j,1}}\prod_{m\neq j}^{N}\ket{\xi^{m,0}}$. The EAM, $q_e$, is an integer with values bounded by $\frac{-1}{2}(N-1)$ and $\frac{1}{2}(N-1)$. The corresponding energies are
\begin{equation}
\mathbb{E}_{q_e}=\Delta + 2\tau \cos\biggl(\frac{2\pi q_e}{N}\biggr),
\label{TBenergies}
\end{equation}
with $p=q_e$ for $q_e\ge 0$ and $p=q_e+N$ for $q_e<0$. A hollow $\mathbb{E}$ will be used to distinguish exciton energy from electric field, $E$.
Now introduce semi-classical light-matter coupling via two Hamiltonians: $\hat H_1$ that governs light-mediated interactions between the ground state and each molecular eigenstate, while $\hat H_2$ governs the analogous laser interactions that mediate transitions between eigenstates. The angular momentum of incident electric fields may be manifested as a circular polarization, a vector vortex, or linear polarization with a scalar vortex, but we restrict attention to the first two types. An electric dipole approximation is made, and the discrete rotational symmetry ensures that a rotation of the molecule about its axis by $2\pi/N$ maps one dipole into the next. Under these conditions, the details of electric field structure and dipole orientations are irrelevant, and the light-matter interactions are well-captured by the following Hamiltonians, which are functions of the PAM of the incident light, $q_p$:
\begin{eqnarray}
\hat H_1(q_p)&=& -\mu_{0}^* E \sum_j \varepsilon^{-q_p(j-1)} \hat{c}_j^{\dagger}\hat{c}_0 + H.c. \nonumber \\
\hat H_2(q_p) &=& -\mu_{{\rm hop}}^* E \sum_j \varepsilon^{-q_p(j-1)} \hat{c}_{{\rm mod}(j)+1}^{\dagger}\hat{c}_j + H.c.
\label{H12}
\end{eqnarray}
The \emph{mod} function returns its argument modulo $N$ and use has been made of the fact that $\varepsilon^{-q_p({\rm mod}(j)-1)} = \varepsilon^{-q_p(j-1)}$. The scalars, $\mu^*_0 E$ and $\mu_{{\rm hop}}^* E$, represent the inner product of electric transition dipole moments with a time-dependent electric field.
The total Hamiltonian, $\hat H(q_p) = \hat H_0 + \hat H_1(q_p) + \hat H_2(q_p)$, is then applied to the Schr{\" o}dinger equation with solutions assumed to be of the form
\begin{equation}
\ket{\Psi(t)} = A_0(t) \ket{0} + \sum_{n=1}^N A_n(t) \ket{e_n}.
\label{LCAO}
\end{equation}
This results in a set of $N+1$ coupled ODE's that can be solved numerically for a prescribed electric field and initial state. The evolving state can then be projected onto each excitonic eigenstate to determine the population of each AM, $q_e$, as a function of time:
\begin{equation}
\rho_{q_e}(t) := |\braket{\Psi(t)|v(q_e)}|^2 = \biggl(\sum_{n=1}^{N} A^*_n(t) \frac{\varepsilon^{(n-1)q_e}}{\sqrt{N}}\biggr)^2 .
\label{pop}
\end{equation}
\subsection{Conservation of Angular Momentum}
Suppose that the molecule is initially in eigenstate $\ket{v_{I_e}}$, where subscript $I_e$ indicates an initial EAM of $I$. A beam with PAM = $q_p$ is incident on the molecule, exciting it into eigenstate $\ket{v_{F_e}}$. Subscripts $p$ and $e$ delineate photonic and excitonic manifestations while $F$ is the EAM of the final state. A necessary condition for this transition to occur is that $H_{IF} := \bra{v_{F_e} }\hat H \ket{v_{I_e}}\ne 0$. Assuming that $I_e\ne F_e$, Equations \ref{TBstates} and \ref{H12} imply that
\begin{widetext}
\begin{equation}
H_{IF} = -\mu_{\rm hop}^* E \biggl(\sum_i \frac{\varepsilon^{-(i-1)F_e}}{\sqrt{N}}\bra{e_i}\biggr) \biggl(\sum_j \varepsilon^{-(j-1)q_p} \hat{c}_{{\rm mod}(j)+1}^{\dagger}\hat{c}_j\biggr) \biggl(\sum_k \frac{\varepsilon^{(k-1)I_e}}{\sqrt{N}}\ket{e_k}\biggr),
\end{equation}
which can be easily reduced to
\begin{equation}
H_{IF} = -\frac{\mu_{\rm hop}^* E \varepsilon^{-F_e}}{N} \biggl(\sum_j \varepsilon^{-(j-1)(I_e - q_p - F_e)}\biggl) - \frac{\mu_{\rm hop}^* E \varepsilon^{I_e}}{N} \biggl(\sum_j \varepsilon^{-(j-1)(I_e + q_p - F_e)}\biggl) .
\label{twoterms}
\end{equation}
\end{widetext}
The following cyclic sum orthogonality property of periodic exponentials is then useful:
\begin{equation}
\sum_j \varepsilon^{-(m-n)j} = N \delta_{m,n}.
\end{equation}
It is applied to both terms in Equation \ref{twoterms} to give
\begin{equation}
H_{IF} = -\mu_{\rm hop}^* E \varepsilon^{-F_e} \delta_{I_e, q_p + F_e} - \mu_{\rm hop}^* E \varepsilon^{I_e} \delta_{I_e + q_p, F_e} .
\end{equation}
Resonant illumination implies that exactly one of the two Kronecker delta functions will be nonzero. We therefore have the following statements of AM conservation:
\begin{eqnarray}
\mathbb{E}_{I_e} &<& \mathbb{E}_{F_e} \quad \rightarrow \quad I_e + q_p = F_e \nonumber \\
\mathbb{E}_{I_e} &>& \mathbb{E}_{F_e} \quad \rightarrow \quad I_e = F_e + q_p .
\label{conservation}
\end{eqnarray}
If the initial exciton energy is lower than that of the final state, absorption results in an increase in the quasi-angular momentum of the molecule. The converse is also true in association with radiation. Since the sign of the PAM can be either positive or negative, this allows for a number of ways in which the electromagnetic field can be used to manipulate the level of excitonic angular momentum. It also offers a strategy for withdrawing angular momentum from the molecule in a range of denominations. This is next illustrated in two applications.
\subsection{Tight-Binding Implementations of Angular Momentum Conservation}
As a first proof-of-concept, a 7-arm molecule with an EAM = $2_e$ is subjected to windowed, continuous wave (CW) lasers with a PAM = $^-1_p$ and the following scalar waveform:
\begin{equation}
E(t) = E_0 \,{\rm Sin}\bigl((\mathbb{E}_{F_e} -\mathbb{E}_{I_e})t/\hbar\bigr) .
\label{TB_E}
\end{equation}
As usual, $I_e$ and $F_e$ are the initial and final AM of the exciton with corresponding energies, $\mathbb{E}_{I_e}$ and $\mathbb{E}_{F_e}$, given by Equation \ref{TBenergies}. The parameters used, and the resulting molecular eigenstate energies and EAM values, are given in Table \ref{TB_data}.
\begin{table}[hptb]
\caption{Exciton energies and AM associated with 7-arm molecule used for all TB simulations. The following parameters were used (atomic units): $E_0 = 2\times 10^{-4}, \mu_0=1, \Delta= 1, \tau = 0.067$ and a radial distance to the center of each arm of 0.6. Here $\mu_0\equiv\mu_{{\rm hop}}$ is the strength of the transition dipole. Proportional changes to these parameters do not affect the results.\label{TB_data}}
\begin{ruledtabular}
\begin{tabular}{lccccccr}
EAM& $^-3_e$ & $3_e$ & $^-2_e$ & $2_e$ & $^-1_e$ & $1_e$ & $0_e$ \\
Energy (Ha) & 0.880 & 0.880 & 0.970 & 0.970 & 1.083 & 1.083 & 1.133\\
\end{tabular}
\end{ruledtabular}
\end{table}
The top plots of Figure \ref{TB_AM2_up_down} show how the AM can be transferred from a laser to the molecule. Two scenarios are considered, and the associated AM conservations are listed above each plot. In both, the molecule has an initial EAM = $2_e$, and the associated exciton energy is 0.970 Ha. A laser of the form of Equation \ref{TB_E} is applied. In the left plot, the laser has a PAM = $^-1_p$, and its frequency is set to difference between the energies of states for which EAM = $2_e$ and EAM = $1_e$---i.e. $0.113/\hbar$ Ha. Energy conservation then prevents any meaningful increases in the population of states other than those for which EAM = $^\pm 1_e$. However, angular momentum conservation prevents the growth of the EAM = $^-1_e$ state. The result is that radiation is absorbed, the exciton energy is increased to 1.083 Ha, and the EAM is reduced to a value of $1$. It is worth emphasizing that energy increases are associated with EAM decreases.(See Table \ref{TB_data}.)
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig03.pdf}
\end{center}
\caption{\emph{Addition and subtraction of EAM on a 7-arm molecule.} (a, left): Laser energy is the difference between the second and first eigenstates and results in the absorption of radiation. (a, right): Laser energy is the difference between the second and third eigenstates and results in the emission of radiation. In both cases the laser is applied throughout the entire simulation shown. All parameters are listed in Table \ref{TB_data}. Panel (b) shows phase progression between neighboring arms for both cases as detailed in the text. The color legends of the top panels identify the populations of each EAM state.}
\label{TB_AM2_up_down}
\end{figure}
The top-right plot in Figure \ref{TB_AM2_up_down} demonstrates how a change to the laser frequency can cause radiation to be emitted instead. Here the frequency is set to difference between the energies of states for which EAM = $2_e$ and EAM = $3_e$---i.e. $0.09/\hbar$ Ha. As in the left plot, the laser PAM is set to $^-1_p$. Energy conservation then allows the growth of only the EAM = $^\pm 3_e$ states, but angular momentum conservation prevents the growth in population of the state for which EAM = $^-3_e$. The result is that radiation is emitted, the exciton energy is decreased to 0.880 Ha, and the EAM is increased to a value of $3_e$. These examples demonstrate that a combination of laser frequency and AM is sufficient to either add or subtract AM from the exciton.
The bottom plot in Figure \ref{TB_AM2_up_down} is a composite of results from both simulations which shows the phase relationship between two adjacent arms before and after application of the laser pulses. The horizontal axis was obtained by rigidly translating sections of the plots of amplitude versus time so that they appear in the same time interval. Then time was mapped into phase through multiplication with the associated frequencies of light. In this form, the phase relation between two arms can be easily measured by comparing the amplitude of one arm (solid) with its neighbor (dashed). The initial phase progression should be $4 \pi/7$, from Equation \ref{TBstates}, and this is confirmed in the black and dashed black curves. The addition of PAM results in an EAM = $1_e$ and the measured phase between the red and dashed red curves exhibits the expected progression of $2\pi/7$. Likewise, the subtraction of PAM leaves the molecule with EAM = $3_e$ and the anticipated phase progression of $6\pi/7$ between arms, shown in blue and dashed blue.
A second TB simulation, Figure \ref{TB_addition}, carries out a sequence of AM addition and subtraction that starts and ends in the ground state (GS). The same 7-arm molecule (Table \ref{TB_data}) is subjected to three windowed, CW laser pulses. The first laser $(t_1 \le t_2)$ has PAM = $1_p$ and an energy equal to that of the eigenstate for an EAM $= 1_e$. The second laser $(t_3 \le t_4)$ has the same PAM but with an energy equal to the difference of excitonic states associated with EAM = $1_e$ and $2_e$. The PAM of the third laser is $^-2_p$ with an energy equal to that of the eigenstate for which EAM = $2_e$. The AM balances associated with each laser are given in the figure to more easily interpret the data plotted. These plots also show how the excitonic energy evolves as a function of time, becoming asymptotic to the appropriate eigenenergies after each laser pulse is applied. The simulation shows that a sequence of absorption events can be followed by a single emission with the latter having a PAM equal to the sum of the input PAMs.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig04.pdf}
\end{center}
\caption{\emph{PAM doubling.} 7-arm molecule initially in ground state (GS) is subjected to a sequence of 3 lasers. The excitonic state populations evolve in accordance with conservation of AM as noted at the top of the plot. Two $1_p$ pulses are absorbed and one $2_p$ radiation results. The color legend identifies the populations of each EAM state. Note that the dark green curve and dashed lines are associated with the exciton energy with scale at right.}
\label{TB_addition}
\end{figure}
\section{Angular Momentum Transfers with Time-Domain Density Functional Theory}\label{TD-DFT-Results}
Many of the idealizations associated with the TB paradigm can be removed by reconsidering the light-matter dynamics using time-domain Density Functional Theory (TD-DFT). Unlike standard ground state Density Functional Theory (DFT), TD-DFT captures the non-equilibrium response of material to an externally applied, time-varying electric field. Such real-time simulations are made possible through Runge-Gross (RG) reformulation of the time-dependent Schr{\" o}dinger equation~\cite{RGtddftPRL1984}. A methodology was developed so that TD-DFT can be used to quantify AM transfers as detailed in the Methods section.
TD-DFT calculations are computationally intense and amount to carrying out a standard DFT calculation for a series of very small time steps. The requisite time step for the calculations of this study is $0.027$ a.u. for simulations covering approximately $2067$ a.u. in total. To reduce the computational cost in this initial proof-of-concept, a ring of radially aligned $H_2$ molecules was used as an idealized N-arm system. The ring was given a radius of $3.8$ Bohr with the H-H bond lengths taken to be $1.4$ Bohr. For each simulation, a computational domain was constructed as the sum of spheres of radius $5.7$ Bohr around each atom. This domain was discretized with a spacing of $0.28$ Bohr. The generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof (PBE)\cite{PBE} was adopted to account for exchange and correlation, and a Troullier-Martins pseudopotential was used. Since the wavelength of the requisite laser field is much larger than the dimension of the N-arm system, a point dipole approximation of light-matter interaction is applied in our TD-DFT simulations.
Limitations on the type of external field that can be inputted to the TD-DFT routine necessitated a piecewise construction to approximate optical vortices. Transition dipoles from the ground state were found to be maximal on each arm and so the associated N field components were constructed so as to be arm-centered. In contrast, the transition dipoles between two excited states were maximal at the midpoints between arms, so the associated external field components were centered between the arms.
A 5-arm configuration of $H_2$ dimers was adopted in all TD-DFT simulations. Casida's perturbative TD-DFT methodology~\cite{casida1995response} was first performed to obtain excitation energies. This was necessary in order to design laser pulses with the frequency needed to excite a given excitonic state. The energies, dominant determinants, and the corresponding EAM of the first five excited states are given in Table~\ref{CasidaExs}. The determinants listed there are those that dominate each excited state, representing approximately $90\%$ of the respective wavefunction. Approximating an excited state with only a single determinant makes it possible to find the population of EAM states, Equation~\ref{population}, as detailed in the Methods section.
\begin{table}[hptb]
\caption{Dominant determinant, $\Psi_a^r$, and EAM, $q_e$, of the first five lowest excited states as calculated from Casida perturbation within TD-DFT. Here $\Psi_a^r$ is the spin-adapted singlet so that one electron is excited from $a^{th}$ occupied KS orbital to the $r^{th}$ unoccupied KS orbital. Energies are in Hartrees (Ha). \label{CasidaExs}}
\begin{ruledtabular}
\begin{tabular}{lccccr}
Excited State& $1$ & $2$ & $3$ & $4$ & $5$ \\ \hline
Energy (Ha) & 0.37 & 0.37 & 0.39 & 0.39 &0.41\\
Determinant & $\Psi_5^6$ & $\Psi_4^6$ & $\Psi_3^6$ & $\Psi_2^6$ & $\Psi_1^6$\\
Percents (\%) & 97 & 97 & 94 & 94 & 91\\
$q_e$ &\multicolumn{2}{c}{$^\pm2$} &\multicolumn{2}{c}{$^\pm1$} &0\\
\end{tabular}
\end{ruledtabular}
\end{table}
A radial vector vortex, carrying PAM = $^-1_{pl}$ and energy of 0.37 Ha excites the 5-arm system from its ground state to the EAM = $^-1_e$ state as shown in Figure~\ref{ExcitationTDDFT} (a). Panel (b) of the figure shows that the same state can be achieved using circularly polarized light, $^-1_{s}$. Combining these two photonic structures and using a radiation energy of 0.39 Ha results in an EAM =$^-2_e$ state, as expected (panel (c)). On the other hand, the linearly polarized vortex of PAM = $^-1_{l}$ results in the same EAM =$^-2_e$ state but with only one-half the occupation probability (panel (d)). This is because the beam can be decomposed into vortices with opposing circularly polarizations, one with a combined PAM of $^-1_{s}$ + $^-1_{l}$ and the other with a PAM of $^+1_{s}$ + $^-1_{l}$.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.9\textwidth]{Fig05.pdf}
\end{center}
\caption{\emph{TD-DFT excitations of 5-arm centrosymmetric system.} The plots show the evolution of excitonic state populations in response to: (a) radial vector vortex with energy 0.39 Ha and AM $^-1_{l}$; (b) circularly polarized light with energy 0.39 Ha and AM $^-1_{s}$; (c) circularly polarized vortex with energy 0.37 Ha and combined AM = $^-1_{l} + ^-1_{s}$; and, (d) a linearly polarized vortex with energy $0.37$ Ha and AM = $^-1_{l}$. The Gaussian envelope, $E_0\, \mathrm{e}^{-(t-t_0)^2/2\tau^2}$, has parameters $\{t_0 = 272, \tau=27.2, E_0 = 0.0100\}$ in atomic units.}\label{ExcitationTDDFT}
\end{figure}
\subsection{Algebra of Radial Vector Vortex Fields}
A radial vortex was chosen to test conservation of AM for the more realistic many-body TD-DFT Hamiltonian. Three cases were considered which each involve a sequence of two laser pulses. The incident radiation field, with a Gaussian envelope, was taken to be:
\begin{equation}
\vec{E}(t) = E_0\mathrm{e}^{-\mathrm{i}q_e\phi}\mathrm{e}^{\mathrm{i\omega t}}\mathrm{e}^{-(t-t_0)^2/2\tau^2} \vec{e}_r .
\label{gaussian}
\end{equation}
As with the TB analyses, conservation of energy determines whether or not PAM is added to or subtracted from the molecular assembly.
A first comparative analysis demonstrates how conservation of AM can be used to control exciton manipulations. The sequence of radiation absorption events shown at bottom-left in Figure~\ref{ETCm2tom1}. A 0.367 Ha laser pulse with PAM = $^-2_p$ is first used to create an exciton with EAM = $^-2_e$. This has the highest magnitude of quasi-angular momentum possible and so is of the lowest energy, as listed in Table~\ref{CasidaExs}. A second laser pulse, with an energy equal to the difference between that of the first and second excitonic states, 0.0220 Ha, is subsequently applied. Radiation of this energy must be absorbed since the $1_e$ state is of a higher energy. Angular momentum conservation, $^-2_e + 1_p$ = $^-1_e$, therefore predicts that the resulting excitonic state will have EAM = $^-1_e$, and that is exactly what was is found.
The lower-right plot of Figure~\ref{ETCm2tom1} shows a completely different behavior for a sequence of pulses though. The first laser is identical to that in the figure at lower-left, and it produces an EAM = $^-2_e$. A second laser is then used that has the same energy as that used in the lower-left plot (0.0220 Ha) but with the opposite angular momentum, $^-1_p$. Once again, energy conservation demands that such radiation can only be absorbed and not emitted. However, the associated statement of AM conservation is now $^-2_e + (^-1_p)$ = $^-3_e$. This AM is not supported by the molecule, and the result is that radiation from the second laser cannot be absorbed. The lower-right plot shows that this is the case, where EAM = $^-2_e$ is maintained despite the application of the second laser--i.e. the molecule appears transparent to the second laser in this excitonic state.
As is clear in the top plot of Figure~\ref{ETCm2tom1}, the second laser needs to be much stronger than the first because the transition dipole between excited states is in the mid-arm region, where the relevant state densities are small, while the transition dipoles from the ground state are located in the arms where the relevant state densities are much larger.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.9\textwidth]{Fig06.pdf}
\end{center}
\caption{\emph{TD-DFT AM manipulation:} $^-2_e + 1_p$ = $^-1_e$. Panel (a) shows sequenced laser pulses of 0.367 Ha and 0.0220 Ha, respectively. The system is initially given an EAM = $^-2_e$ using a $2_p$ laser. Application of second laser with AM = $1_p$ transfers the system to a $^-1_e$ state as shown at left in panel (b). The same laser energy, but with an AM = $^-1_p$, does not cause the state to evolve as shown at right in panel (b). Envelop parameters for Equation \ref{gaussian} are $\{t_0 = 272, \tau=27.2, \omega=0.367, E_0 = 0.00300\}$ and $\{t_0 = 1360, \tau=272, \omega=0.0220, E_0 = 0.0300\}$ in atomic units, respectively. }\label{ETCm2tom1}
\end{figure}
A second comparative analysis, summarized in Figure \ref{ETC1to2or0}, shows how light with a fixed PAM can be adjusted in frequency to either increase or decrease EAM. In both scenarios shown, the molecule is first given an EAM = $1_e$ and, in both cases, a second laser pulse with PAM = $^-1_p$ is subsequently applied. In the evolution shown at lower-right, the energy of the second pulse is equal to the difference between the states $1_e$ and $0_e$, 0.0158 Ha (Table~\ref{CasidaExs}). This results in the absorption of radiation because the $0_e$ state is of higher energy with the AM balance equation of $1_e + (^-1)_p = 0_e$. On the other hand, tuning the second pulse to an energy of 0.0257 Ha causes the system to transition to the $2_e$ state because this is the energy difference between the $1_e$ and $2_e$ states (Table~\ref{CasidaExs}). In this case, radiation is emitted because the $2_e$ state is of lower energy, and the AM balance is $1_e -(^-1)_p = 2_e$.
Note that there is an oscillation in the population of the $1_e$ state (blue curve) in both of the lower plots of Figure \ref{ETC1to2or0}. This is an artifact associated with the single-determinant approximation in concert with our piecewise homogeneous construction of the incident beam. This field approximation results in a larger contribution of the non-primary determinant, and the artificial oscillation in population is an indicator that the relative weighting of these determinants is time dependent. In contrast, the analogous curve associated with circularly polarized light (Figure~\ref{STOC}) shows no such oscillation because the non-primary determinants make almost no contribution.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.9\textwidth]{Fig07.pdf}
\end{center}
\caption{\emph{TD-DFT AM manipulation:} $1_e + (^-1)_p = 0_e$ (left) and $1_e - (^-1)_p = 2_e$ (right). Panel (a) shows two laser sequences with the associated population transfers given in panel (b). In both processes, the parameters of Equation \ref{gaussian} for first laser pulses are $\{ t_0=272, \tau=27.2, \omega=0.390, E_0=0.00300 $. The parameters for second laser pulses are (left): $\{ t_0=1090, \tau=136, \omega=0.0158, E_0=0.0500 \}$ and (right): $\{ t_0=1090, \tau=218, \omega=0.0257, E_0=0.0500 \}$ in atomic units. }
\label{ETC1to2or0}
\end{figure}
A third comparative analysis, shown in Figure~\ref{ETC0tom1orp1}, demonstrates how EAMs of opposite sign can be created from the same initial state by changing both the laser frequency and the sign of its angular momentum. In both left and right scenarios, a molecule is placed in its highest energy state, $0_e$, after application of an appropriate laser pulse. A second laser with PAM = $^-1_p$ (lower-left) stimulates emission and changes the molecule to an EAM = $^-1_e$. The associated statement of AM conservation is $0_e - (^+1)_p$ =$ ^-1_e$. On the other hand, illuminating the molecule with a PAM = $^-1_p$ (lower-right) also stimulates the emission of radiation but changes the system to an EAM = $^+1_e$. The associated statement of AM conservation is $0_e - (^-1)_p$ = $^+1_e$.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=1.0\textwidth]{Fig08.pdf}
\end{center}
\caption{\emph{TD-DFT PAM emissions:} $0_e - (1)_p$ = $^-1_e$ (left) and $0_e - (^-1)_p = 1_e$ (right). Two cases are considered, and in both an initial laser pulse, shown in panel (a), generates a $0_e$ AM as shown in both plots of panel (b). The second laser pulse differs for each case though. A $^+1_p$ pulse stimulates emission and changes the excitonic state to $^-1_e$ (b, left), while a $^-1_p$ pulse also stimulates emission but changes the excitonic state to $^+1_e$ (b, right) . The parameters of Equation \ref{gaussian} for first laser are $\{ t_0=272, \tau=27.2, \omega=0.405, E_0=0.00300\}$ and those for the second laser are $\{ t_0=1360, \tau=272, \omega=0.0158, E_0=0.0300 \}$ in atomic units.}
\label{ETC0tom1orp1}
\end{figure}
All three of these comparative TD-DFT analyses have focused on EAM arithmetic. When purposed as a PAM converter, though, it is important to be able to transfer the final AM into an electromagnetic field. Such an operation, already considered within the TB setting, can be demonstrated in a more realistic TD-DFT simulation. Figure~\ref{ETCemission} summarizes a simulation in which the molecule is first given a $^-2_e$ AM which is subsequently changed to $^-1_e$ using a second laser. Taken together, these first two steps can be viewed as PAM addition: $^-2_p + 1_p$ = $^-1_e$. A third laser then stimulates the emission of PAM = $^-1_p$ leaving the system in its ground state. The photonic arithmetic for this conversion process is therefore $^-2_p + 1_p$ = $^-1_p$.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig09.pdf}
\end{center}
\caption{\emph{TD-DFT PAM conversion:} $^-2_p + 1_p$ = $^-1_p$. Panel (a) shows two laser pulses. Panel (b) shows the system is excited into $^-2_e$ state by the first laser. Subsequent absorption of a $1_p$ radiation then transfers the system into $^-1_e$ state. This EAM is converted to PAM with a third laser which brings system back to its ground state by stimulation emission. The laser parameters of Equation \ref{gaussian} are: (first) $\{ t_0=272, \tau=27.2, \omega=0.367, E_0=0.00100 \}$; (second) $\{ t_0=1090, \tau=272, \omega=0.0257, E_0=0.0300 \}$; and (third)$\{t_0=1905, \tau=27.2, \omega=0.390, E_0=0.00100\}$ in atomic units. }
\label{ETCemission}
\end{figure}
The excited state excitonic populations are small because very weak and short lasers are applied in all TD-DFT simulations. This is a pragmatic step taken to avoid the population of extraneous eigenstates that result from stronger or longer laser pulses. This is purely a computational issue that results because the vector vortex beams were approximated with five piecewise-homogeneous components---a computational work-around to the limitations of the input fields allowed by the OCTOPUS TD-DFT software. The result is unphysical light-matter interactions in the regions between each arm that can be remedied by dividing the region into more than five piecewise-homogeneous components. Then stronger and/or longer laser illumination can be applied to increase the population of twisted excitons.
\subsection{Algebra of Circularly Polarized Lights}
As an alternative to applying an radial vector vortex, PAM can be inputted using circularly polarized light. This just amounts to a change of basis for describing the molecular dipoles since circularly polarized light can be mathematically decomposed into a combination of radial and azimuthal vector vortices\cite{CircPolVortexOL2006}:
\begin{equation}
\frac{1}{\sqrt{2}}(\vec{e}_x\pm\imath\vec{e}_y)=\frac{1}{\sqrt{2}}\mathrm{e}^{\pm \imath \phi}(\vec{e}_r\pm\imath\vec{e}_\phi). \label{spinlight}
\end{equation}
Here $\{\vec{e}_x, \vec{e}_y\}$ and $\{\vec{e}_r, \vec{e}_\phi\}$ are the basis vectors in Cartesian and polar representations. In the 5-arm $H_2$ system, only the radial vortex components are absorbed. Through a series of absorption events, circularly polarized light can be used to generate vector vortices with an arbitrary AM.
This is demonstrated in Figure \ref{STOC}, where a sequence of circularly polarized laser pulses are applied. The first laser pulse causes absorption and the following AM conservation relation: $GS + 1_{p} = 1_e$. The second laser, of opposite spin, results in emission: $1_e -(^-1)_{p} = 2_e$.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig10.pdf}
\end{center}
\caption{\emph{Addition of circularly polarized light:} $1_{ps} - (^-1)_{ps} = 2_e$. Panel (a) shows two laser pulse. Panel (b) shows the system is sequentially provided with two units of AM resulting in $2_e$. Subsequent emission would produce $2_{p}$ radiation. The laser parameters are: (first) $ \{t_0=272, \tau=81.5, \omega=0.390, E_0=0.00300 \}$; and (second) $\{ t_0=1360, \tau=109, \omega=0.0257, E_0=0.00200 \}$ in atomic units.}\label{STOC}
\end{figure}
Figure~\ref{STOC} shows that the strength of the first laser is the same as those used for the vortex beams of Figure~\ref{ETCm2tom1}, \ref{ETC1to2or0}, and \ref{ETC0tom1orp1}, but the duration is three times longer. This allows a much larger population of proper excited states without involvement of extraneous eigenstates. As compared with Figure~\ref{ETC1to2or0}, we now have a very smooth population of the $1_e$ state (blue curve). This confirms that the origin of the oscillations in our former TD-DFT simulations is actually the approximation of the vector vortex with five piecewise-homogeneous components.
\section{Conclusions}\label{conclude}
Light and molecules can be engineered so that quanta of angular momentum can be exchanged between the two. Strictly speaking, the transformation is between the AM of a radiation field (PAM) and a quasi-angular momentum (EAM). This makes it possible to design processes in which sequential laser pulses are used to increase or decrease the EAM. Subsequent emission results in radiation with a different PAM than the input light. Unlike existing approaches, this molecular strategy offers a means of manipulating the angular momentum of light which does not rely on the nonlinear optical properties of a mediating crystal. Computational proofs of concept were provided using tight-binding theory as well as the more realistic many-body setting of time-domain density functional theory. While the present work focused on stimulated emissions, AM conversions may culminate in spontaneous emission as well.
Angular momentum conservation, valid within a paraxial approximation, and algebraic manipulations were elucidated using simple tight-binding Hamiltonians. However, time-domain DFT gives consistent results within a many-electron setting with the effects of electron correlation and exchange accounted for. In both settings, the important influence of phonon entanglement and dynamic disorder have been neglected. It was assumed that phase coherence is maintained in the superposition of arm excitons that comprise molecular eigenstates. Unless the electronic coupling between arms is sufficiently strong to preserve coherence in the face of these effects, they will set a time scale over which AM manipulations must be carried out for a given temperature~\cite{Ishizaki_2009}. Moreover, AM conversions must compete favorably with energy relaxation pathways such as fluorescence and internal conversion~\cite{LaCount_2017}.
Since changes to the excitonic angular momentum are accompanied by a change in energy---e.g Table \ref{TB_data}---these AM manipulations can be viewed as a special type of laser-based upconversion or downconversion that can even be carried out with either pulsed or continuous-wave lasers. A number of upconversion methodologies have been experimentally realized, for instance, and these offer a path forward for manipulating the angular momentum of light~\cite{Scheps_1996}. Laser-based energy conversion methodologies tend to use solid-state crystals, so the approach would need to be adapted to molecular media. The sequential banking of AM requires that conversions must be fast relative to competing molecular relaxation processes. The entire AM conversion process of Figure \ref{ETCemission} takes only 44 fs, and this is orders of magnitude faster than the nanosecond time scales for internal conversion and photoluminescence for typical organic molecules~\cite{LaCount_2017}. The issues to be faced here are also encountered in a range of energy upconversion and downconversion strategies~\cite{Chen_2014}.
The level of tight-binding analysis employed does not require any details of the molecular structure beyond point group, and the time-domain DFT analyses adopted, as a computational expedient, molecular arms composed of hydrogen dimers. There exist a panoply of molecules which exhibit $C_N$ or $C_{Nh}$ symmetry, though, with only a few examples shown in Figure \ref{molecules}. Polycyclic aromatic hydrocarbons may be promising candidates of this type, and several examples are shown in the figure. Their aromaticity counters the effects of dephasing due to vibrations. Another intriguing possibility is to functionalize inert scaffolds that have the requisite point symmetry.
Screening of candidate molecules can be carried out using simple DFT analysis to identify the requisite excitonic structures. As an example, the DFT orbitals of $\rm{Ph}_3\rm{P}$ can be used to generate a rudimentary estimate the electronic structures associated with its three EAM states: +1, -1 and 0. The EAM = $0_e$ state is composed of the highest occupied molecule orbital (HOMO) and the third excited state--i.e. two states above the lowest unoccupied molecular orbital (LUMO). These both have the requisite $C_3$ symmetry as shown in Figure \ref{TPP_1}. The associated exciton energy is 3.47 eV (357 nm). On the other hand, the LUMO and LUMO+1 are degenerate and can be combined to create states with EAM = $^\pm 1_e$ as shown in Figure \ref{TPP_2}. Taken with the HOMO, the associated excitons have an energy of 3.43 eV (361 nm), lower than the EAM = $0_e$ state and consistent with both the TB results of Table \ref{TB_data} and the $H_2$ trimers considered with TD-DFT. These states are constructed by first adding the LUMO and LUMO+1 orbitals, then assigning orbitals to each arm by localizing this sum as shown in the top panel of Figure \ref{TPP_1}. These arm orbitals are then given a 120-degree phase progression to obtain the structures of Figure \ref{TPP_2}. This progression is evident in the figure, where it is clear that the two orbitals have the same real part and imaginary parts of opposite sign. As a check, the sum of the squares of the projections of these states with the original LUMO and LUMO+1 orbitals was found to be 0.98 in each case. While carried out in a very crude way, with simple combinations of DFT orbitals to represent excitons, this analysis serves to demonstrate that light at the far edge of the visible spectrum can be used to create twisted excitons.
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig11.png}
\end{center}
\caption{\emph{High-energy exciton of $\rm{Ph}_3\rm{P}$ with EAM = $0_e$.} The $C_3$ symmetry of $\rm{Ph}_3\rm{P}$ (a) results in an exciton composed of HOMO and LUMO+2 orbitals that also has $C_3$ symmetry and so an EAM = $0_e$. The exciton energy is 3.47 eV. The red and blue isosurfaces in (b) are for densities for 0.02 ${\rm bohr}^{-3/2}$ of the real part of wave functions. Colors correspond to plus (red) and minus (blue) values.}
\label{TPP_1}
\end{figure}
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.7\textwidth]{Fig12.png}
\end{center}
\caption{\emph{Low-energy excitons of $\rm{Ph}_3\rm{P}$ with EAM = $^\pm 1_e$.} Orbitals with opposite 120-degree phase progressions can be constructed from the LUMO and LUMO+1 orbitals of $\rm{Ph}_3\rm{P}$. Panel (a) shows the orbital for EAM = $^+1_e$ and panel (b) shows the orbital for EAM = $^-1_e$. Their energies are each 3.43 eV, lower than the exciton with no angular momentum. The red and blue isosurfaces of wave function densities are for 0.02 ${\rm bohr}^{-3/2}$. Colors correspond to plus (red) and minus (blue) values. The real parts of the two orbitals are identical but the imaginary parts are of opposite sign. This is consistent with the opposing 120-degree phase progressions.}
\label{TPP_2}
\end{figure}
\section{Methods}\label{TD-DFT}
Real-time simulations are made possible through the Runge-Gross (RG) reformulation of time-dependent Schr{\" o}dinger equation~\cite{RGtddftPRL1984}:
\begin{equation}
\mathrm{i}\frac{\partial}{\partial t}\psi_i(\vec r,t) = \Big[ -\frac{1}{2}\Delta^2 +\nu_{Ha}[\rho](\vec r,t) +\nu_{xc}[\rho](\vec r,t)+\nu_{ext}(\vec r,t)+\nu_{lm}(\vec r,t)\Big]\psi_i(\vec r,t).
\label{RG}
\end{equation}
Here $\nu_{Ha}$ and $\nu_{xc}$ are the Hartree and exchange-correlation potentials, respectively, and $\nu_{ext}$ is the external potential representing all nuclei. The semi-classical light-matter interaction term, $\nu_{lm}$, has the form $\vec R\cdot\vec{E}$ within electronic dipole approximation adopted here, where $\vec R:=\sum_i\vec{r}_i$ is the Kohn-Sham position operator and atomic units are used. The spin-reduced electronic density, $\rho(\vec r, t)$, is expressed in terms of thes time-dependent Kohn-Sham (TDKS) orbitals, $\psi_i(\vec r, t)$, as
\begin{equation}
\rho(r,t)=\sum_i^{N}|\psi_i(r,t)|^2.
\label{dens}
\end{equation}
These orbitals, in turn, can be represented in the basis of their counterparts at time zero, $\psi_i$, so that the time-propagated multi-electron wavefunction is constructed from a linear combination of determinants built from these initial orbitals\cite{TimePropState}:
\begin{equation}
\ket{\Psi(t)} = c_0(t)\ket{\Psi_{gs}} + \sum_a^{occ}\sum_i^{unocc}c_a^i(t)\ket{\Psi_a^i} .
\label{TPstate}
\end{equation}
Ket $\ket{\Psi_{gs}} = \ket{\psi_1\psi_2\psi_3\psi_4\psi_5}$ is the ground state and $\ket{\Psi_a^i} = \ket{\psi_1\cdots\psi_i\cdots\psi_5}$ is a determinant with the $a^{th}$ occupied Kohn-Sham (KS) orbital replaced by the $i^{th}$ unoccupied orbital. The first summation is over all occupied KS orbitals, five occupied KS orbitals in the case of 5-arm $H_2$ system, and the second summation is over all unoccupied KS orbitals. In our case, because the frequency of laser is chosen to only access the first five lowest excited states, the only unoccupied KS orbital is the sixth as shown in Table~\ref{CasidaExs}.
If it was possible to express Equation~\ref{TPstate} in form of Equation~\ref{TBstates}, the associated EAM could be determined directly. Such a simple expansion of $\ket{\Psi(t)}$ in basis of $\ket{e_j}$ does not exist, though, since it is a many-body wavefunction. This is remedied, albeit in an approximate way, by working only with the dominant determinant for which only lowest unoccupied molecular orbital (LUMO) is involved in Equation~\ref{TPstate}. This makes it possible to combine the determinants corresponding to each EAM subspace $\{^\pm m\}$ with $m=0,1,2\cdots$, as in Equation~\ref{Rearranged}, allowing the EAM of $\ket{\Psi(t)}$ to be obtained via KS orbitals using the method detailed below.
In all simulations, twisted excitons are a constructed as a linear combination of corresponding pairs of degenerate excited states, consistent with the TB model. Focusing on the 5-arm system, if a laser with PAM = $0_p$ is applied, then the resulting excited state can be approximated with a determinant involving only $\Psi_1^6$. Likewise, the application of PAM = $^\pm1_p$ results in excited states that can be approximated with determinants involving only $\Psi_2^6$ and $\Psi_3^6$, and PAM = $^\pm 2_p$ yields states that are well-approximated with only $\Psi_4^6$ and $\Psi_5^6$. The time-propagated wavefunction for each EAM subspaces $\{0\}$, $\{^\pm1\}$ and $\{^\pm2\}$ can therefore be expressed, respectively, as:
\begin{eqnarray}
c_1^6(t)\ket{\Psi_1^6}&=&\ket{c_1^6(t)\psi_6,\psi_2\psi_3\psi_4\psi_5}\label{Rearranged} \\
c_2^6(t)\ket{\Psi_2^6} + c_3^6(t)\ket{\Psi_3^6} &=&\ket{\psi_1,c_3^6(t)\psi_2-c_2^6(t)\psi_3,\psi_6\psi_4\psi_5}\nonumber\\
c_4^6(t)\ket{\Psi_4^6} + c_5^6(t)\ket{\Psi_5^6} &=& \ket{\psi_1\psi_2\psi_3,c_5^6(t)\psi_4-c_4^6(t) \psi_5,\psi_6} .\nonumber
\end{eqnarray}
\begin{figure}[hptb]
\begin{center}
\includegraphics[width=0.8\textwidth]{Fig13.png}
\end{center}
\caption{\emph{Decomposition of KS orbitals of 5-arm $H_2$ system.} The five occupied KS orbitals have been expressed in the basis of arm wavefunctions, $\ket{e_j}$, with their coefficients labeled on the corresponding arms. The red (blue) isosurfaces indicate positive (negative) values of the wavefunctions. The LUMO is given in order to show that it is symmetric. HOMO = highest occupied molecular orbital. }\label{KSorbitals}
\end{figure}
The only difference among these three equations is that the ground state determinant is modified as follows: $\psi_1$ replaced by $c_1^6(t)\psi_6$; $\psi_2$ and $\psi_3$ are replaced by $c_3^6(t)\psi_2-c_2^6(t)\psi_3$; and $\psi_6$, $\psi_4$ and $\psi_5$ are replaced by $c_5^6(t)\psi_4-c_4^6(t)\psi_5$ and $\psi_6$. The ground state determinant $\ket{\psi_1\psi_2\psi_3\psi_4\psi_5}$ is the $0_e$ state of course. These replacement orbitals must therefore be responsible for the EAM of excited states. Figure~\ref{KSorbitals} gives the isosurface and decomposition in the basis of $e_j$ with $j\in\{1,\cdots,5\}$ of all the relevant KS orbitals. As shown in Figure~\ref{KSorbitals}, the LUMO is symmetrically distributed across all five arms. Therefore $\psi_6$ will not introduce a phase difference among arms in the right side of Equation~\ref{Rearranged}. This implies that $c_3^6(t)\psi_2-c_2^6(t)\psi_3$ and $c_5^6(t)\psi_4-c_4^6(t)\psi_5$ will introduce a phase dependence corresponding to $^\pm1_e$ and $^\pm2_e$, respectively. The population of each twisted exciton state is therefore given by:
\begin{eqnarray}
P_{0_e}&=&2|c_1^6(t)|^2\nonumber\\
P_{^-1_e}&=&2|\braket{v_{^-1}|c_3^6(t)\psi_2-c_2^6(t)\psi_3}|^2\nonumber\\
P_{1_e}&=&2|\braket{v_{1}|c_3^6(t)\psi_2-c_2^6(t)\psi_3}|^2\nonumber\\
P_{-2_e}&=&2|\braket{v_{^-2}|c_5^6(t)\psi_4-c_4^6(t)\psi_5}|^2\nonumber\\
P_{2_e}&=&2|\braket{v_{2}|c_5^6(t)\psi_4-c_4^6(t)\psi_5}|^2.
\label{population}
\end{eqnarray}
Here $\ket{v_{q_e}}$ is the eigenstate associated with an EAM of $q_e$, from Equation~\ref{TBstates}, and the factor of two in each expression accounts for the fact that the electron spin can be either up or down.
\section{Acknowledgements}
We are grateful to Profs. David L. Andrews, Mark E. Siemens and Guillermo F. Quinteiro for extended discussions on the generation of twisted light. All calculations were carried out using the high performance computing resources provided by the Golden Energy Computing Organization at the Colorado School of Mines.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,973
|
Q: Connect to remote mysql host from network with proxy
*
*Ubuntu machine in internet zone with proxy server.
*And i want to connect to remote mysql server mysql -u userName -p -h hostname.
*And also i have java application that connect to that remote mysql host using hibernate ORM.
*And i can't connect with the server.
Edit
I dont want to use mysql proxy but my database exists on remote server but my local machine have connection to the internet throw web proxy so i can't connect to remote db server.
And when i try connect from machine that has direct access to the internet That Work Perfectly
*
*Trials.
1-System -> Preferences -> Network Proxy.
2- export http_proxy = http:userName:password@proxyserver:port
but that doesn't work.
and help will be appreciated..
Update:
static {
// SOCKS Proxy
System.setProperty("proxySet", "true");
System.setProperty("socksProxyHost", "proxy-host");
System.setProperty("socksProxyPort", "proxy-port");
System.setProperty("http.proxyHost", "proxy-host");
System.setProperty("http.proxyPort", "proxy-port");
System.setProperty("java.net.socks.username", "username");
System.setProperty("java.net.socks.password", "password");
Authenticator.setDefault(new ProxyAuthenticator("username", "password"));
// Open Web Site Work good
try {
URL url = new URL("http://www.kooora.com/");
URLConnection con = url.openConnection();
BufferedReader in = new BufferedReader(new InputStreamReader(
con.getInputStream()));
// Read it ...
String inputLine;
while ((inputLine = in.readLine()) != null) {
System.out.println(inputLine);
}
in.close();
} catch (IOException iOException) {
iOException.printStackTrace();
}
// But open mysql connection throw exception
Class.forName("com.mysql.jdbc.Driver").newInstance();
return DriverManager.getConnection(getDBUrl(), USERNAME, PASSWORD);
com.mysql.jdbc.exceptions.jdbc4.CommunicationsException: Communications link failure
The last packet sent successfully to the server was 0 milliseconds ago. The driver has not received any packets from the server.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,642
|
\section{#1}\setcounter{figure}{0}
\renewcommand{\jot}{10pt}
\newcommand{{\small UV}}{{\small UV}}
\newcommand{{\small IR}}{{\small IR}}
\newcommand{{\small FRG}}{{\small FRG}}
\newcommand{{\small RG}}{{\small RG}}
\newcommand{{\small QCD}}{{\small QCD}}
\newcommand{{\small LHS}}{{\small LHS}}
\newcommand{{\small RHS}}{{\small RHS}}
\newcommand{{\rm STr}}{\operatorname{Tr}}
\newcommand{\eta^{}_N}{\eta^{}_N}
\newcommand{g^{}_N}{g^{}_N}
\setlength{\paperheight}{11in}
\begin{document}
\title{Global Flows in Quantum Gravity}
\author{N. Christiansen}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany}
\author{B. Knorr}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany}
\author{J. M. Pawlowski}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany}
\affiliation{ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f\"ur
Schwerionenforschung mbH, Planckstr.\ 1, 64291 Darmstadt, Germany}
\author{A. Rodigast}
\affiliation{Institut f\"ur Theoretische Physik, Universit\"at Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany}
\begin{abstract}
We study four-dimensional quantum gravity using non-perturbative
renormalization group methods. We solve the corresponding equations
for the fully momentum-dependent propagator, Newton's coupling and the
cosmological constant. For the first time, we obtain a global phase
diagram where the non-Gaussian ultraviolet fixed point of asymptotic
safety is connected via smooth trajectories to a classical infrared fixed
point. The theory is therefore ultraviolet complete and
deforms smoothly into classical gravity as the infrared limit is approached.
\end{abstract}
\maketitle
\section{Introduction}
Understanding the quantization of the gravitational force is an
outstanding problem in theoretical physics. Any viable theory of
quantum gravity must connect stable infrared ({\small IR}) physics with a
well-behaved ultraviolet ({\small UV}) limit. The asymptotic safety scenario
provides a {\small UV}-completion in a natural way. It is based on a
non-Gaussian {\small UV}{} fixed point, which leads to vanishing
$\beta$-functions in the limit of arbitrarily high energy scales and
renders the couplings finite even beyond the Planck scale
\cite{Weinberg:1980gg}.
The asymptotic safety scenario received growing attention during the
past decades and has been studied with different methods. The
underlying fixed point structure was found in the non-perturbative
continuum approach
\cite{Donkin:2012ud,Christiansen:2012rx,%
Reuter:1996cp,Litim:2003vp,Niedermaier:2006wt,Percacci:2007sz,Codello:2008vh,
Litim:2011cp, Reuter:2012id,
Nagy:2012ef}, as
well as in lattice simulations
\cite{Hamber:2009mt,Ambjorn:2012jv,Ambjorn:2013tki}. The former is
based on the functional renormalization group, in particular on its
formulation for the effective action \cite{Wetterich:1992yh}. The
crucial {\small UV}{} fixed point is confirmed in various
approximations, including the coupling to gauge and matter fields
\cite{Percacci:2002ie,Daum:2010bc,Eichhorn:2011pc,Folkerts:2011jz,Harst:2011zx,
Dona:2013qba}, dilaton gravity \cite{Henz:2013oxa} and higher
derivative calculations
\cite{Benedetti:2009rx,Falls:2013bv,Ohta:2013uca,Benedetti:2013jk,Demmel:2014}.
There is also a rich field of phenomenological applications based on
asymptotically safe quantum gravity. This includes e.g.\ implications
for the standard model and its extensions
\cite{Shaposhnikov:2009pv,Antipin:2013bya}, black hole physics
\cite{Falls:2010he,Falls:2012nd,Koch:2013owa,Koch:2014cqa}, collider
experiments \cite{Litim:2007iu,Gerwick:2011jw} and cosmology
\cite{Weinberg:2009wa,Copeland:2013vva}. However, the standard
calculations lead to an ill-defined {\small IR}{} limit since the
trajectories exhibit a singular behaviour on large length scales.
In \cite{Donkin:2012ud} an {\small IR}{} fixed point has been found
as the endpoint of a singular line. The existence of this fixed point
was only seen within a proper distinction of background and dynamical
couplings. The singular behaviour in the vicinity of the {\small IR}{} fixed
point was attributed to the approximation. The first smooth {\small IR}{}
fixed point in asymptotically safe gravity, allowing for a theory that
is well defined on all energy scales, was found in
\cite{Christiansen:2012rx}. There, however, a non-classical behaviour
in the vicinity of the {\small IR}{} fixed point has been computed,
therefore leading to modified gravity on very large length
scales. Again, the non-classical behaviour may be attributed to the
approximation. Further discussions on this issue from different
perspectives can be found in
\cite{Litim:2012vz,Nagy:2012rn,Rechenberger:2012pm, ContrerasLitim}.
In the present work, we construct a qualitatively enhanced
approximation within the systematic vertex expansion scheme introduced
in \cite{Christiansen:2012rx}. With this enhanced approximation the
theory is asymptotically safe in the {\small UV}{}, and exhibits an {\small IR}{}
fixed point which describes classical gravity. The corresponding
renormalization group ({\small RG}) trajectories are globally smooth. They
connect the known {\small IR}{} physics with classical gravity on large
length scales with a viable theory of Planckian and trans-Planckian
gravity. The present scenario encodes that quantum gravity effects
set in at about the Planck-scale, and are absent for energy-scales $E
\ll M_{\mathrm{Pl}}$.
This work improves the vertex expansion scheme set-up in
\cite{Christiansen:2012rx} in several aspects. We compute, for the
first time, fully momentum-dependent wave-function renormalizations
for the graviton and the ghost. Note that the wave function
renormalizations are functions of the covariant Laplacian, and hence
this takes into account infinitely many terms in an expansion of the
effective action in powers of the covariant derivative. Additionally,
multi-graviton interactions are constructed from their scaling
behaviour, see \cite{Fischer:2009tn}: the dependence on the running
{\small RG}{}-scale is deduced from consistent {\small RG}{}-scaling of the vertex. This
novel self-consistent vertex construction is of major importance for
the transition from {\small UV}{} to {\small IR}{} scaling, and the stability of the
{\small IR}{} regime.
The present work is organized as follows: We introduce our
approximation scheme in \autoref{sec_classoftruncations}, and the
vertex construction in \autoref{subsec_vertex_functions}. The flow
equations for the fully momentum-dependent propagators are derived in
the subsequent subsections. The consistency analysis in
\autoref{sec_lambdas} constrains the momentum-independent parts of the
vertex functions, and is crucial for the global properties
of the phase diagram. The latter is the topic of the first section of
the results, \autoref{sec_phase_diag}. Moreover, the reliability of
all results is tested within a regulator study. To this end we use the
optimized regulator as well as a class of exponential ones. The
results are found to be stable against variations of the
regulator. The properties of the {\small UV}{} regime are discussed in
\autoref{sec_UV_regime}. In this section we also investigate the stability and
reliability
of the derivative expansion, at the basis of the full
momentum-dependence. In \autoref{sec_IR_regime} the properties of the
{\small IR}{} regime are discussed.
\section{Flows in quantum gravity}\label{sec_classoftruncations}
A quantum field theory is entirely described by a complete set of correlation
functions. The generating functional for the 1PI correlators is the
effective action $\Gamma[\bar{g},\phi]$, where we have already introduced a
fixed
background metric $\bar{g}$ and a fluctuation super-field $\phi$.
In the case of gravity this super-field is given by the vector $\phi =
(h,\bar{c},c)$, where $h$ is the graviton field and $c,\bar c$ are
the corresponding ghost and anti-ghost fields.
In the present work we use the functional renormalization group approach, for
reviews
on quantum gravity see
\cite{Litim:2003vp,Niedermaier:2006wt,Percacci:2007sz,Codello:2008vh,
Litim:2011cp,Reuter:2012id,Nagy:2012ef}, for general
reviews and other applications
see
e.g.~\cite{Litim:1998nf,Berges:2000ew,Aoki:2000wm,Bagnuls:2000ae,Polonyi:2001se,
Pawlowski:2005xe,%
Gies:2006wv,Schaefer:2006sr,Rosten:2010vm,Pawlowski:2010ht,%
Scherer:2010sv,Braun:2011pp,Metzner:2011cw,Boettcher:2012cm,2012LNP...852...49D}
.
With the functional renormalization group, the effective action can be
determined via a
functional differential equation for the scale-dependent effective action
$\Gamma_k[\bar{g},\phi]$, which for quantum gravity is given by
\cite{Reuter:1996cp}
\begin{align}\label{floweq}
\notag \partial_t \Gamma_k[\bar{g},\phi] = \frac{1}{2} & \mathrm{Tr} \,
\left[\0{1}{\Gamma^{(2h)}_{k} + R_{k,h}} \,\partial_t R_{k,h}
\right][\bar{g},\phi]
\\[2ex] - & \mathrm{Tr} \, \left[ \0{1}{\Gamma^{(\overline{c}c)}_{k} +
R_{k,c}}\,\partial_t
R_{k,c}\right][\bar{g},\phi] \, .
\end{align}
It involves an {\small IR}{} regulator $R_k$ which is
implemented on the level of the path
integral and carries an {\small IR}{} cutoff-scale $k$.
In addition to that, $t$ denotes the logarithmic {\small RG}{} scale, $t := \log
(k/k_0)$,
with an
arbitrary normalization scale $k_0$, and the Tr implies an integral
over all continuous and a sum over all discrete indices. We will also make use
of the notation $\partial_t f(k) =: \dot{f}(k)$ for any scale-dependent
quantity.
Moreover, we have introduced the
notation
\begin{equation}
\Gamma^{(\phi_1 ... \phi_n)}_k[\bar{g},\phi] \colonequals
\frac{\delta^n \Gamma_k[\bar{g},\phi]}{\delta \phi_1 \cdots \delta
\phi_n} \,
\end{equation}
for the 1PI vertex functions, which are derivatives of the effective action with
respect to the fluctuation fields and are the elements of the full super
space matrix $\Gamma^{(n)}_k$.
The right hand side of the flow equation \eqref{floweq} depends on the
the two-point correlators of the fluctuation field $\phi$. It is
important to note that the fluctuation correlation functions do not
agree with the background correlations, i.e.\
\begin{equation}
\left.\frac{\delta^2 \Gamma_k[\bar{g},\phi]}{\delta h^2}\right|_{\phi=0} \neq
\frac{\delta^2 \Gamma_k[\bar{g},0]}{\delta \bar{g}^2} \, ,
\end{equation}
for details see
\cite{Pawlowski:2002eb,Litim:2002ce,Braun:2007bx,Folkerts:2011jz,%
Christiansen:2012rx}. In other words, one cannot
extract dynamical couplings from the flow of the background field
effective action at vanishing fluctuation fields, $\phi=0$. This
directly relates to the fact that the flow equation \eqref{floweq} for
the effective action at $\phi=0$ is not closed within the standard
background field approach. More importantly, avoiding unphysical
background contributions can be crucial for capturing the correct
non-perturbative physics. In conclusion, in the {\small FRG}{} setup, this
strongly suggests to start from the exact equation for the inverse
fluctuation propagator.
Moreover, the master flow equation \eqref{floweq} leads to an infinite
hierarchy of coupled partial integro-differential equations for the
scale-dependent vertex functions $\Gamma^{(n)}_k$. More precisely, the
equation for the $n$-point vertex function contains vertex functions
of order $n+1$ and $n+2$. This system is usually not exactly
solvable. Therefore, one has to employ certain approximation schemes.
We assume that the effective action can be expanded in a functional
Taylor-series around the fixed background metric $\bar{g}$. Moreover,
we choose the flat Euclidean metric, i.e.\ the identity $\bar{g}_{\mu
\nu} = \delta_{\mu \nu}$, as the expansion point. We will need this
expansion up to fourth order in the graviton field. In symbolic
notation, the effective action takes the form
\begin{align}
\notag \Gamma_k[\bar{g},\phi] = & \sum_n \frac{1}{n!}
\Gamma^{(n)}[\bar{g},0]
\phi^n
\\ \notag = & \Gamma_k[\bar{g},0] +
\Gamma^{(h)}_{k}[\bar{g},0] h +
\Gamma^{(2h)}_{k}[\bar{g},0] h^2
\\ \notag & + \Gamma^{(3h)}_{k}[\bar{g},0] h^3
+ \Gamma^{(4h)}_{k}[\bar{g},0] h^4 +
\ldots
\\ & + \Gamma^{(\overline{c}c)}_{k}[\bar{g},0] \overline{c} \, c
+ \ldots \, .
\label{general_exp}\end{align}
The first and second term are of order $h^0$
and $h^1$ and do not enter the RHS of the flow equations for
any $\Gamma^{(n)}_k$.
The above expansion of the effective action in powers of the fluctuating field
around a flat Euclidean background $\bar g_{\mu \nu} = \delta_{\mu \nu}$
also restricts the number of higher
derivative operators which can contribute to the vertex functions of $n$-th
order. For instance, the most general form of the two-point function derives
from an action which includes at most $\mathcal{O}(R^2)$ operators. All higher
order terms vanish after two functional differentiations and evaluation on a
flat background. Terms with higher order than this only contribute to vertex
functions of order $n>2$.
\section{Vertex expansion}\label{sec_presenttruncation}
In the present work we use the systematic vertex expansion scheme as
suggested in \eqref{general_exp}. The hierarchy of flow equations that
has been introduced in the last section has to be truncated at finite
order. This means that one can calculate the flow of a vertex function
of given order $n$ and use an ansatz for $\Gamma^{(n+1)}_k$ and
$\Gamma^{(n+2)}_k$. We will compute the basic quantity of the present
approach, the full two-point correlation functions of the fluctuation
fields $\phi$, that is $n=2$. The corresponding flows rely on the
two- but also on the three- and four point functions of the
fluctuation fields. Hence we also introduce approximations for
$\Gamma^{(3)}_k$ and $\Gamma^{(4)}_k$ that are consistent with the
symmetries of the theory and have the correct {\small RG}{}-scaling. The latter
property is essential for the global {\small UV}{}--{\small IR}{} flows considered
here.
\subsection{Structure of the vertex
functions}\label{subsec_vertex_functions}
First of all, we have to specify the tensor structures of the vertices. In the
present work, we use the classical tensor structures which arise from functional
differentiation of the Einstein-Hilbert action. The gauge-fixed
Einstein-Hilbert action, including the ghost part, is given by
\begin{align}\label{EH_action}
S = \frac{1}{16 \pi G_N} &\int \text{d}^4x
\sqrt{\text{det}g}\;(-R+2\Lambda) \notag \\
+ &\int\text{d}^4x \sqrt{\text{det}\bar{g}}\; \bar{c}^\mu
\mathcal{M}_{\mu\nu} c^\nu \\
+ &\int\text{d}^4x
\sqrt{\text{det}\bar{g}}\; \frac{1}{2 \xi} \bar{g}^{\mu \nu} F_\mu F_\nu \,.
\notag
\end{align}
In \eqref{EH_action}, $G_N$ is the Newton constant and $\Lambda$ is the
cosmological constant. The Fadeev-Popov operator is given by
\begin{equation}
\mathcal{M}_{\mu\nu} =\bar{\nabla}^{\alpha} (g_{\mu \nu} \nabla_{\alpha}
+ g_{\alpha \nu} \nabla_{\mu}) -\bar{\nabla}_{\mu}\nabla_{\nu} \, ,
\end{equation}
and the linear gauge fixing conditions reads
\begin{equation}
F_\mu =
\bar{\nabla}^\nu h_{\mu \nu} -\tfrac{1}{2} \bar{\nabla}_\mu h^{\nu}_{~\nu} \,.
\end{equation}
Moreover, in this work we restrict ourselves to Landau gauge, that is
$\xi\rightarrow0$. Extended ghost interactions are studied e.g.\ in
\cite{,Eichhorn:2009ah,Eichhorn:2013ug}.
The standard Einstein-Hilbert truncation amounts to replacing the
gravitational coupling and the cosmological constant in
\eqref{EH_action} by running couplings $G_{N,k}$ and $\Lambda_k$. The
vertex functions are then given by functional derivatives of this
effective action. However, this approximation turns out to be
inconsistent in the physical {\small IR}{} limit, which will be discussed
in detail later. We also mention that the basic Einstein-Hilbert
truncation does not disentangle the difference between a wave-function
renormalization and a running coupling, since the running of the
latter is simply identified with the running of the former. Note that
in Yang-Mills theory such an approximation gives a deconfining
potential of the order parameter even in the confining regime, see
\cite{Braun:2007bx,Marhauser:2008fz,Fister:2013bh}. In this case, it
is also the non-trivial momentum-dependence of the correlation
functions that plays a crucial r$\hat{\rm o}$le for capturing the
correct non-perturbative physics. Additionally, in the {\small UV}{} limit,
$k\to\infty$, the full momentum-dependence of correlation functions is
potentially relevant. In particular, a derivative expansion implies
$p^2/{\rm scale}^2\ll 1$ which relates to low energy physics. So far,
these momentum-dependencies have not been taken into account.
Consequently, we construct more general vertex functions that take
into account the above properties while keeping the classical tensor
structures. The construction of such vertex expansions of the
scale-dependent effective action was introduced in \cite{Fischer:2009tn}
and applied in the context of Yang-Mills theories. A similar
truncation based on these ideas was recently applied in quantum
gravity to \cite{Codello2013}. One guiding principle in this
construction is {\small RG}{} invariance, i.e.\ invariance of the full
effective action under a change of the renormalization scale $\mu$:
\begin{equation}
\mu \frac{\mathrm{d}}{\mathrm{d} \mu} \Gamma = 0 \, ,
\end{equation}
where $\mu$ should not be confused with the running {\small IR}{} cutoff
scale $k$, for a detailed discussion see \cite{Pawlowski:2005xe}. In
addition to that, we parameterize the vertex functions, i.e.\ the
coefficients in the expansion \eqref{general_exp} schematically as
\begin{equation}
\Gamma^{(n)} = Z^{\0{n}{2}} \, \bar{\Gamma}^{(n)} \,
\end{equation}
with a $\mu$-independent part $\bar{\Gamma}^{(n)}$ and a
$\mu$-dependent wave-function renormalization $Z(p^2)$ of the attached
fields. The above construction implies the correct scaling behaviour
for the fields according to
\begin{equation}
\mu \frac{\mathrm{d}}{\mathrm{d} \mu} \phi = \eta\, \phi \, .
\end{equation}
In the present work we use a uniform wave function renormalization,
$Z_h = Z_{h_i}$, for all components of the graviton. Non-uniform
$Z$-factors will be subject to a forthcoming publication
\cite{Vertices}. The transverse-traceless ($\rm TT$) part of the full
propagator is now parameterized as
\begin{equation}\label{2point}
\Gamma^{(2h)}_{\mathrm{TT}}(p^2) = Z_{h} (p^2) (p^2 - M^2)
\Pi_{\mathrm{TT}}(p) \,,
\end{equation}
with the transverse-traceless projector $\Pi_{\mathrm{TT}}(p)$, and an
effective mass term $M$ representing the momentum-independent part of
the two-point function. Note that the $Z$-factors are functions of the
covariant Laplacian $\Delta$, and therefore include infinitely many
terms in a covariant expansion of the effective action in powers of
the Laplace operator. This is the first {\small RG}{} study of quantum gravity
taking into account this general momentum dependence of the graviton
and the ghost propagator. The fully momentum-dependent $Z$-factors
lead to a vertex construction with the required {\small RG}{} scaling
properties. They also embody a corresponding implicit, non-canonical
momentum-dependence of the vertex functions, in line with the
co-linear singularity structure of vertex functions. Note that a
similar vertex construction is achieved by simply taking further
$h$-derivatives of the two-point correlation function \eqref{2point}
with $p^2 \to \Delta(\bar g,h)$.
Finally, we allow for additional running parameters $\Lambda^{(n)}_k$
which govern the (consistent) scale-dependence of the
momentum-independent part of the vertex functions. This takes into
account scaling properties of the vertex functions that are crucial
for the global flows structure, and has not been considered before.
The construction of the vertex functions also include appropriate
powers of a scale-dependent Newton coupling $G_{N,k}$ as prefactors of
the vertex functions. These factors, apart from the wave function
renormalization factors, encode the correct scale-dependence of the
vertices, see \cite{Fischer:2009tn}. Note that in general there are
separate coupling constants $G^{(n)}_k$ for each vertex function, but
we identify $G^{(n)}_k = G_{N,k}^{\frac{n}{2}-1}$ in the present
work. Still, this construction goes far beyond the approximations
considered so far in {\small RG}{}-gravity. In summary, the vertex functions take
the form
\begin{equation}
\Gamma^{(\phi_1 \dots \phi_n)}_k = \prod\limits_{i=1}^n
\sqrt{Z_{\phi_i,k}(p_i)}\, G_{N,k}^{\frac{n}{2}-1} \,
\mathcal{T}^{(n)}_k(p_1,\dots,p_n;\Lambda^{(n)}_k) \, ,
\label{vertex_functions}\end{equation}
with tensor structures
\begin{equation}
\mathcal{T}^{(n)}_k= S^{(n)}(p_1,\dots,p_n;G_N
= k^2,\Lambda \rightarrow \Lambda^{(n)}_k) \, ,
\label{tensor_struc}\end{equation}
that arise from functional differentiation of the classical
Einstein-Hilbert action $S$ given in \eqref{EH_action}. We left out
the momentum arguments on the LHS as well as the functional dependence
on the fields. If we leave out the functional argument, it is
implicitly understood that the functional is evaluated at vanishing
fluctuation fields and on a flat background. The tensor structures
$\mathcal{T}^{(n)}_k$ carry not only the canonical explicit
momentum-dependence of the vertex functions, but also the running
parameters $\Lambda^{(n)}_k$. They are defined by
\begin{equation}\label{eq_lambda_n}
\mathcal{T}^{(n)}_k(p_i=0;\Lambda^{(n)}_k) \equalscolon -2 \Lambda^{(n)}_k
\tilde{\mathcal{T}}^{(n)}(\delta_{\mu \nu}) \, ,\end{equation}
where $\tilde{\mathcal{T}}^{(n)}(\delta_{\mu \nu})$ is the tensor structure
arising from functional differentiation of the integral of the volume form with
respect to the metric tensor. The factor of $-2$ in the definition is
introduced such that we recover the classical cosmological constant in the
Einstein-Hilbert action. We emphasize again that the consistent {\small RG}{}-scaling of
the parameters
$\Lambda^{(n)}_k$ is of major importance for the transition from the {\small UV}{}
regime
to the {\small IR}{} regime, as well as for the stability of
the {\small IR}{} regime. In this regime the $\Lambda^{(n)}_k$ are determined via a
self-consistency
analysis of the scaling behaviour. Details will be presented in the
corresponding results sections.
An example for the above vertex construction is the scalar coefficient
of the TT-two point function of the graviton,
\begin{equation}
Z_h(p^2)(p^2 -2
\Lambda_k^{(2)})\, ,
\label{EH_kernel} \end{equation}
see also \eqref{2point}. The tensor structures for the general two-point
function, from which the
above results via TT-projection, are given by
\eqref{tensor_struc} and arise from functional differentiation of the
Einstein-Hilbert action. Equation \eqref{EH_kernel} and \eqref{2point} entail
that
$\Lambda_k^{(2)}$ is the effective graviton mass,
\begin{equation}
M^2_k = -2 \Lambda_k^{(2)} \, .
\label{def_M}\end{equation}
Once again, note that this mass term is not the cosmological constant.
\subsection{Flow of the propagator}\label{propagator_flow}
The flow of $\Gamma^{(2h)}$ in a standard Einstein Hilbert truncation
has been calculated in \cite{Christiansen:2012rx}. The corresponding
flow equation is obtained by functional differentiation of
\eqref{floweq}, and its diagrammatic representation is shown in
\autoref{fig:flow_of_prop}.
\begin{figure}
\begin{align*}
\partial_t \left.
\frac{\delta^2\Gamma[\bar{g},\phi]}{\delta h^2}
\right|_{\phi=0}(p^2) &= -\frac{1}{2}
\includegraphics[height=6ex]{flowgravtadpole} +
\raisebox{-2ex}{\includegraphics[height=6ex]{flowgravbubble}}\\
& \mathrel{\hphantom{=}} - 2
\,\raisebox{-2.2ex}{\includegraphics[height=6ex]{flowghostbubble}}
\equiv \text{Flow}^{(2h)}(p^2)\\
\partial_t \left.\frac{\delta^2\Gamma[\bar{g},\phi]}{\delta c \delta \bar{c}}
\right|_{\phi=0}(p^2) &=
\raisebox{-2.0ex}{\includegraphics[height=6ex]{flowgh_diag_1}} +
\raisebox{-2.8ex}{\includegraphics[height=6ex]{flowgh_diag_2}} \\
&\equiv \text{Flow}^{(\overline c c)}(p^2)
\end{align*}
\caption{Diagrammatic representation of the flow of the second order vertex
functions. The dressed graviton propagator is represented by a double line,
the
dressed ghost propagator by a dashed line, while a dressed vertex is denoted
by a dot
and the regulator insertion by a crossed circle.}
\label{fig:flow_of_prop}
\end{figure}
In the present work, we compute the fully momentum-dependent two-point
functions with the {\small RG}{}-consistent ansatz \eqref{vertex_functions} for
the three- and the four-point function. With
\autoref{fig:flow_of_prop} and \eqref{vertex_functions} the flows of
the two-point functions $\Gamma^{(2)}$ depend on
\begin{equation}\label{couplings}
\left(Z_h(p^2),Z_c(p^2),M^2, G_N,\Lambda^{(3)},\Lambda^{(4)}\right) \,.
\end{equation}
Here we have already dropped the subscript $k$, and if
not stated otherwise, all generalized couplings are scale-dependent.
The two-point function contains all tensor structures in the York
transverse-traceless decomposition. The transverse-traceless part is
not constrained by Slavnov-Taylor identities and is expected to carry
the essential properties of the graviton. In this work we identify all
wave function renormalizations of the graviton modes with that of the
transverse-traceless mode. Hence, from now on, all expressions for the
two-point functions relate to the TT-part. A study with all
independent tensor structures will be presented elsewhere
\cite{Vertices}.
In this work, we use a regulator of the form
$R_{\phi}(p^2) = Z_{\phi}(p^2) R^0_{\phi}(p^2)$ with $R^0_{\phi}(p^2)=
p^2 \, r(p^2) \mathcal{T_{\phi}}$, where $\mathcal{T_\phi}$ denotes
the tensor structure of the corresponding two-point function evaluated
at vanishing mass, and $r(p^2)$ is a dimensionless shape function.
Then, the flow of
the inverse propagator reads
\begin{align}\label{flowofprop2}
\notag \partial_t \Gamma^{(2h)}(p^2) & = (p^2+M^2) \partial_t Z_h(p^2) +
Z_h(p^2) \partial_t M^2 \,
\\ &= \text{Flow}^{(2h)}(p^2) \,,
\end{align}
where
\begin{align}
&\text{Flow}^{(2h)}(p^2) = G_N Z_h(p^2) \times \label{flow_structure} \\
& \int \text{d}^4q
\sum_{\phi} \left(\partial_t r(q^2) + \frac{\partial_t Z_\phi (q^2)}{Z_\phi
(q^2)} r(q^2) \right) I_\phi (p^2,q^2,\Lambda^{(n)}) \, . \notag
\end{align}
In the above equation, $I_\phi (p^2,q^2,\Lambda^{(n)})$ are scalar
functions that arise from the contraction of the diagrams and a
subsequent projection onto the TT-structure, and $n=2,3,4$. This
structure follows from our vertex construction discussed above.
Explicit expressions for the flow equations are given in
Appendix~\ref{app_flow_eq}. For the ghost sector, we apply the same
strategy and arrive at the much simpler equation
\begin{equation}
p^2 \partial_t Z_c(p^2) = \text{Flow}^{(\overline c c)}(p^2) \, .
\label{ghost_flow}\end{equation}
Note that with the {\small RG}{} consistent vertex ansatz \eqref{vertex_functions}
and the structure of the flow equations for $\Gamma^{(n)}$, one can infer that
the wave-function renormalization $Z(p^2)$ does never enter a flow equation
alone, but always in the combination $\dot{Z}/Z$. This
motivates the definition of the anomalous dimensions
of the graviton,
\begin{align}\label{defeta}
\eta_h(p^2) &:= - \frac{\partial_t Z_h(p^2)}{Z_h(p^2)} \, , \\
\intertext{and of the ghosts,}
\eta_c(p^2) &:= - \frac{\partial_t Z_c(p^2)}{Z_c(p^2)} \, .
\end{align}
We note that the RHS of the flow of the $\Gamma_k^{(2)}$ does depend
on $\eta_h,\eta_c,M^2$ and the three- and four-point functions, see
\eqref{couplings}. In the standard Einstein-Hilbert setup, a
scale-dependent gravitational coupling is constructed from the
graviton wave-function renormalization, and the momentum-independent
parts of the vertex functions are all identified with the cosmological
constant, $\Lambda^{(n)} \equiv \Lambda$. This procedure closes the
equations \eqref{flowofprop2} and \eqref{ghost_flow} and was used at
least in parts in all {\small FRG}{} gravity calculations so far.
In the
present work this identification, which spoils the scaling properties
of the correlation functions, is avoided. For the gravitational
coupling constant $G_N$, we use the relation of the present framework
at flat backgrounds to that with geometrical effective actions, see
\cite{Donkin:2012ud}. This is discussed in more detail below. The
couplings $\Lambda^{(n)}$ are constrained within a self-consistency
analysis, see \autoref{sec_lambdas}.
Finally we introduce dimensionless, scale-dependent
couplings, to wit
\begin{align}
g &:= G_N k^2\,, & \mu &:= M^2\, k^{-2},
\\ \lambda &:= \Lambda k^{-2}\, , & \lambda^{(n)} &:= \Lambda^{(n)} k^{-2}
\,,
\label{dimless}\end{align}
with $n\geq3$ and $\Lambda = \Lambda^{(1)}$. It is left to project the
functional flow onto individual flow equations for all running
couplings.
\subsubsection{The running mass}
The flow equation for the mass $M^2$ is obtained from the flow of the
inverse propagator, evaluated at the pole of the propagator, i.e.\
$p^2 = -M^2$. Taking the $t$-derivative of the on-shell
two-point function, $\Gamma^{(2h)}(-M^2)$, yields
\begin{equation}
\begin{aligned}
0 &= \left. \partial_t \Gamma^{(2h)}(p^2)\right|_{p^2=-M^2} - Z_h(-M^2)
\partial_t
M^2 \, .
\end{aligned}
\end{equation}
Solving for the running of the mass parameter, we get
\begin{equation}
\partial_t M^2 = \frac{\partial_t \Gamma^{(2h)}(-M^2)}{Z_h(-M^2)} \, .
\label{runningmass}\end{equation}
One of the goals of this work is to evaluate the phase diagram of quantum
gravity and its
fixed point structure. For this reason, it is convenient to derive $\beta$-
functions for the dimensionless parameters. Then, the above equation
translates into
\begin{align}
\notag \partial_t \mu &= - 2\mu + \frac{\partial_t \Gamma^{(2h)}(-M^2)}{k^2
Z_h(-M^2)}
\\ &=:\beta_{\mu}[\eta_h,\eta_c](g,\mu,\lambda^{(3)},\lambda^{(4)}) \,
\, .
\label{flow_mu}\end{align}
The explicit form of the $\beta$-function is given in
Appendix~\ref{app_flow_eq}. It is clear from
\eqref{flow_structure} that the $\beta$-function for the running mass
shows a functional dependence on the anomalous dimensions. It turns
out that the final result for the $\beta$-function of $\mu$ and the
momentum- dependent equation for the anomalous dimension $\eta(p)$
does not depend on the projection point, see Appendix~\ref{app_anomproj}. The
finite difference at $p^2 =-M^2$ is just the most
convenient choice.
\subsubsection{Integral equations for the anomalous dimensions}
Starting from the general equation \eqref{flowofprop2}, using the
above definition of the anomalous dimension and inserting \eqref{runningmass},
we obtain an integral equation for $\eta_h$ which reads
\begin{equation}
\eta_h(p^2) = -\frac{\dfrac{\partial_t \Gamma^{(2h)}(p^2)}{Z_h(p^2)} -
\dfrac{\partial_t \Gamma^{(2h)}(-M^2)}{Z_h(-M^2)}}{p^2+M^2} [\eta_h,\eta_c]
\label{etanice} \,.
\end{equation}
Note that all isolated $Z$-
factors drop out, see \eqref{flowofprop2}.
The same procedure can be applied for the ghost sector. Since there is no ghost
mass, we trivially arrive from \eqref{ghost_flow} at
\begin{equation}\label{etanicec}
\eta_c (p^2) = -\frac{\partial_t \Gamma^{(\overline{c} c)}}{p^2
Z_c(p^2)}[\eta_h,\eta_c] \, .
\end{equation}
The explicit form of eqs.~\eqref{etanice} and \eqref{etanicec} is given in
Appendix~\ref{app_flow_eq}.
The full expressions of are derived and solved numerically, with the help
of \textsc{Form} \cite{FORM}, \textsc{xTensor} \cite{xTensor}, and
\textsc{Eigen} \cite{EigenWeb}.
\subsection{The running gravitational coupling
\texorpdfstring{$g$}{g}}\label{sec_gdot}
As already mentioned, we use geometrical flow equations for $G_N$ in
order to close the system of differential equations. This approach
allows an inherently diffeomorphism- invariant construction of flows
in quantum gravity, see \cite{Branchina:2003ek,Pawlowski:2003sk}. It
has been applied to the phase structure of quantum gravity in
\cite{Donkin:2012ud}, where evolution equations $\beta_g$ for the
dynamical coupling $g$, and $\beta_{\bar{g}}$ for the background
coupling $\bar g$ are derived. In the present work we utilize the fact that the
geometrical approach is directly related to the present approach in a
flat background. In particular, the dynamical and background couplings
in both approaches agree.
Moreover, with the fully momentum-dependent anomalous dimensions
computed in the present work, we are able to directly incorporate
effects of arbitrarily high powers of derivatives in the equations for
$\beta_g$,$\beta_{\bar{g}}$ in \cite{Donkin:2012ud}. The anomalous
dimensions enter the geometric flow equations in very much the same
way as in \eqref{flowofprop2}. However, the
wave-function renormalizations in \cite{Donkin:2012ud} are
momentum-independent and can be pulled outside the integrals. This is
not the case in our set-up. Entering the equations with the
momentum-dependent $\eta_\phi(p^2)$ calculated via \eqref{etanice}
leads to a modification on the level of the threshold functions
$\Phi$. These modified threshold functions are given in
Appendix~\ref{app_flow_eq}.
With these ingredients, the
general structure of the $\beta$-function for the dynamical gravitational
coupling is given by
\begin{equation}
\beta_{g}[\eta_h,\eta_c]\left(g,\mu \right) = 2g +
F_g[\eta_h,\eta_c]\left(g,\mu
\right) \, ,
\end{equation}
and the one for the background coupling takes the same form with $g$
being replaced by $\bar{g}$ and an individual loop contribution
$F_{\bar{g}}[\eta_h,\eta_c]\left(\bar{g},\mu \right)$. The functionals
$F_g$ and $F_{\bar{g}}$ are given in Appendix~\ref{app_flow_eq}. Note
that the flow equation of the background coupling depends on the
dynamical coupling via the anomalous dimensions, while the converse
does not hold.
\subsection{The couplings
\texorpdfstring{$\Lambda^{(n)}$}{Lambdan}}
\label{sec_lambdas}
In \autoref{subsec_vertex_functions} we have introduced an
approximation which takes into account scale-dependent couplings
$\Lambda^{(n)}$ for the momentum-independent part of each vertex function. For
the second order, we have identified $\Lambda^{(2)}$ as the
graviton mass $M^2$, see \eqref{def_M}. In the present section we discuss the
vertices with $n\geq3$.
The Einstein-Hilbert truncation, which identifies all $\Lambda^{(n)}$
with the cosmological constant $\Lambda$, is ill-defined in the limit
$\mu\rightarrow -1$. As we will see in \autoref{sec_results}, this
limit is approached by physical {\small RG}{} trajectories in the deep {\small IR}{}.
This regime is crucial to understand the global phase structure of
Euclidean quantum gravity: the couplings $\Lambda^{(n)}$ play a
distinguished role, as the related singularities arise from
the momentum-independent parts of the vertex functions. In order to
cure the inconsistencies of the Einstein-Hilbert truncation, we deduce
the singularity structure of the couplings $\Lambda^{(n)}$ with $n \geq 3$. The
full details
are given in Appendix~\ref{appendix_analysis}. Essentially, the idea
is to expand the right-hand sides of the flow equations for the
$n$-point functions in powers of $1+\mu$ and taking into account the
singularities of highest order. Thus, for $\mu\to -1_+$ we use the
ansatz
\begin{equation}
\lim_{\mu\to -1_+}\lambda^{(n)} \sim (1+\mu)^{\alpha_n} \, ,
\end{equation}
for $n \geq 3$.
We proceed by inserting this ansatz in the flow equations for $\Gamma^{(nh)}$
and analyze the generic loop integrals to leading order in the singularities
that arise in the limits under consideration.
Consistent scaling of both sides of the flow equations
for arbitrary $n$ leads to the relations
\begin{equation}
\alpha_n = \alpha_{n-2} + \alpha_4 -1 \,,
\end{equation}
for $n \geq 5$ and
\begin{equation}
\alpha_4 \leq 2 \alpha_3 -1 \, .
\label{final_inequality0}\end{equation}
The parameter $\alpha_4$ obeys the bound
\begin{equation}
\alpha_4 < 0 \,.
\end{equation}
The value of the parameters $\alpha_3$ and $\alpha_4$ cannot be
obtained from the divergence analysis alone. They are dynamically
determined by the flow of the three- and four-point function. This
highlights again that the standard Einstein-Hilbert approximation with
$\lambda^{(n)} = - \mu/2$ is inconsistent in the {\small IR}{}, and the
non-existence of the {\small IR}{} fixed point cannot be inferred from such an
approximation. It is also important to stress that the
qualitative features of the phase diagram do not depend on the
specific choice of $\alpha_3$ and $\alpha_4$, see \autoref{sec_phase_diag} and
Appendix~\ref{appendix_estimate}. In turn, the
quantitative behaviour does only mildly depend on variations of these
two parameters. Their flows will be studied in a forthcoming publication
\cite{Vertices}.
Still, we can estimate $\alpha_3$ based on the saturation of the
inequality \eqref{final_inequality0}. Moreover, the constant parts of
the vertex functions are parametrically suppressed far away from the
singular regime. This entails that there it is viable to identify
$\Lambda^{(n)}=\Lambda^{(2)}$ as done in all other approximations used
in the literature. From these conditions one obtains
$\alpha_3 \approx -1/9$. More details are given in
Appendix~\ref{appendix_estimate}. In Appendix~\ref{app_funcform} it is shown
that
\begin{equation}\label{c_n_param}
\lambda^{(n)} = -\frac{\mu}{2} (1+\delta\lambda^{(n)})
\end{equation}
is consistent with all constraints, where $\delta\lambda^{(n)}$
parametrizes the deviation from the Einstein-Hilbert approximation. The latter
is modeled by
\begin{equation}\label{Lamodel}
\delta\lambda^{(n)} = \text{sgn}(\mu) \, \chi \left| \frac{\mu}{1+\mu}
\right|^{-\alpha_n} \, ,
\end{equation}
with $\chi$ a parameter to be tuned to match the aforementioned conditions.
\subsection{The cosmological constant}
It is left to discuss the role of the cosmological constant $\Lambda$
in the present construction. Written on the right hand side of the
field equations, it can be interpreted as an additional source for
gravity. In the classical limit, the quantum equations
\begin{equation}
\frac{\delta
\Gamma}{\delta \phi} = J_{\mathrm{ext}} \, ,
\end{equation}
with an external source $J_{\mathrm{ext}}$, reduce to the classical
equations of motion. Hence, it is natural to define the cosmological
constant from the one-point function, i.e.\ we identify $\Lambda^{(1)}
= \Lambda$ as the vacuum energy. More precisely, with the vertex
construction \eqref{vertex_functions}, the one-point function takes
the form
\begin{equation}
\left. \frac{\delta}{\delta h} \Gamma \right|_{g=\delta} \sim
\frac{\Lambda}{\sqrt{G_N}} \sqrt{Z_h} \, .
\end{equation}
Note that the one-point function does not enter the flow of higher order
vertex functions. Consequently, the cosmological constant decouples from the
$\beta$-functions for Newtons constant, the effective mass and the set of
integral
equations for the anomalous dimensions. On the other hand, these quantities
obviously determine the running of the cosmological constant, i.\,e.\ the
$\beta$-
function for the dimensionless cosmological constant $\lambda := \Lambda /
k^2$ is of the form
\begin{equation}
\begin{aligned}
\dot{\lambda} & = \beta_{\lambda}[\eta_h,\eta_c](g,\lambda,\mu)
\\& = -2\lambda + g \left( A[\eta_h,\eta_c](\mu) + \lambda\,
B[\eta_h,\eta_c](\mu)
\right) \,.
\end{aligned}
\end{equation}
The explicit form of this flow equation is given in Appendix~\ref{app_flow_eq}.
\subsection{Regulators and stability}
In order to test the quality of our truncation, we will use several regulators
and vary the parameters $\chi$ and $\alpha_n$ introduced before. As regulators,
on the one hand we use the class of exponential regulators given by
\begin{equation}
r_a(x) = \frac{1}{x(2e^{x^a}-1)} \, ,
\end{equation}
where $x=p^2/k^2$ is the dimensionless squared momentum. In our
analysis, we scanned the parameter range $a=\{ 2,3,4,5,6 \}$. On the
other hand, the Litim regulator, \cite{Litim:2000ci}, is used,
\begin{equation}
r_{opt}(x) = \left( \frac{1}{x} - 1 \right) \theta(1-x) \, ,
\end{equation}
where $\theta(x)$ is the Heaviside step function. Note that this
regulator is optimized within the leading order derivative expansion
but not beyond, see \cite{Litim:2000ci,Pawlowski:2005xe}. Also,
with the semi-optimized regulator the divergence analysis for the
$\Lambda^{(n)}$ is slightly different from the one performed in
Appendix~\ref{appendix_analysis},
but leads to similar results.
We also have scanned different values for the parameters $\chi$ and
$\alpha_3$ in \eqref{Lamodel}, and we have restricted our
investigation to the case of equality in \eqref{final_inequality0}. It
turns out that the results do not depend on the specific choice of
$\alpha_3$. Note that the parameter $\chi$ is bounded from above as
otherwise the parametric suppression of the
$\delta\lambda$-contribution away from the singularity is lifted and
the {\small UV}{} regime is changed. In
\autoref{tab:UVFPparamscan} in Appendix~\ref{appendix_estimate}, a
table is given where the change of the {\small UV}{} fixed point values under
a change of $\alpha_3$ and $\chi$ can be ascertained.
\section{Results}\label{sec_results}
In this section we present our results. First, the global phase
diagram is discussed. Subsequently, its {\small UV}{} and {\small IR}{} properties
will be examined in more detail. In doing so, we will also make contact
with older results. If not stated otherwise, all results and pictures
are obtained with the specific choice of the exponential regulator
$r_4$.
\subsection{The phase diagram}\label{sec_phase_diag}
The phase diagram for the dynamical couplings $(g,\mu)$ is depicted in
\autoref{fig:phasediag2}.
\begin{figure}
\includegraphics[width=\columnwidth]{globalflowplot_dashed_letters}
\caption{Fixed points and global phase diagram in the $(g,\mu)$-plane. Arrows
point from the {\small IR}{} to the {\small UV}{}, red solid lines mark separatrices while dots
indicate fixed points. The black dashed line is a specific trajectory that
connects the {\small UV}{} fixed point with the non-trivial {\small IR}{} fixed point, which is
analyzed further in the text. In analogy to \cite{Christiansen:2012rx}, region
Ia
corresponds to trajectories leading to the massless IR fixed point, whereas
region Ib leads to the massive IR fixed point. Region II is not connected to the
UV fixed point, and thus physically irrelevant.}
\label{fig:phasediag2}
\end{figure}
\begin{figure*}[htb!]
\includegraphics[width=2\columnwidth]{etasatuvfp_inset}
\caption{The momentum-dependence of the anomalous dimensions of the graviton
(left) and the ghost field (right) for different regulators at the respective
{\small UV}{} fixed points. Only a weak dependence on the parameter is observed. The
difference between optimized and exponential regulators is due to the fact
that the modes are not integrated out at the same scale. From the fact that the
quadratic external momentum terms cancel in the flow of the graviton
\cite{Christiansen:2012rx}, $\eta_h$ goes to zero in the limit of large external
momenta.
The same is not true for the ghosts, where the anomalous dimension goes to a
constant.}
\label{fig:momdep_eta}
\end{figure*}
We find an attractive {\small UV}{} fixed point with coordinates
\begin{equation}
(g^\text{UV}_*,\mu^\text{UV}_*) = (0.614,-0.645) \, ,
\end{equation}
and complex critical exponents $\theta_{1,2} = (-1.268 \pm 3.009
\mathbf{i})$. This provides further non-trivial evidence for the
asymptotically safe {\small UV}{} structure of quantum gravity. We also
find the built-in repulsive Gaussian fixed point
at $(g^\text{Gauss}_*,\mu^\text{Gauss}_*) = (0,0)$, and a massive {\small IR}{}
fixed point at $(g^\text{IR}_*,\mu^\text{IR}_*) = (0,\infty)$. The
most striking feature of the present phase diagram is the
confirmation of the attractive massless {\small IR}{} fixed point
\begin{equation}\label{IR}
(g^\text{IR}_*,\mu^\text{IR}_*) = (0,-1) \, ,
\end{equation}
which was already found in \cite{Christiansen:2012rx}, where it
corresponds to a de-Sitter fixed point with $\lambda = 1/2$. This
fixed point implies the global existence of trajectories connecting
the {\small UV}{} fixed point with a finite {\small IR}{} fixed point. The
present result is a clear confirmation that this {\small IR}{} fixed point is
not a truncation artifact, but rather a physical property of the
theory.
Importantly, it turns out to be an {\small IR}{} fixed point describing
classical gravity. Physical initial conditions lead to globally
defined trajectories that connect the non-trivial {\small UV}{} fixed
point with the physical {\small IR}{} fixed point
$(g^\text{IR}_*,\mu^\text{IR}_*) =(0,-1)$.
Note also that all {\small UV}{}-complete trajectories are also
{\small IR}{}-complete, and end in either the massive or massless {\small IR}{} fixed
point. In addition to this structure, there is a repulsive fixed point
at $(g^\text{rep}_*,\mu^\text{rep}_*) = (0.250,-0.905)$. This fixed
point was also found in \cite{Donkin:2012ud}. All essential features
do not depend on the choice of the regulator $r(x)$, and there are
only minor quantitative changes induced by variations of the
latter. The variation of the {\small UV}{} fixed point values under a variation of the
vertex model parameters $\chi,\alpha_3$, \eqref{Lamodel} is given in
\autoref{tab:UVFPparamscan} in Appendix~\ref{appendix_analysis}.
\subsection{{\small UV}{} regime}\label{sec_UV_regime}
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline
$a$ & 2 & 3 & 4 & 5 & 6 & opt \\ \hline \hline
$\mu_*$ & -0.637 & -0.641 & -0.645 & -0.649 & -0.651 & -0.489 \\
\hline
$g_*$ & 0.621 & 0.622 & 0.614 & 0.606 & 0.600 & 0.831 \\ \hline
$\overline g_*$ & 0.574 & 0.573 & 0.567 & 0.559 & 0.553 & 0.763 \\
\hline
$\lambda_*$ & 0.319 & 0.316 & 0.316 & 0.318 & 0.319 & 0.248 \\ \hline
EVs & -1.284 & -1.284 & -1.268 & -1.255 & -1.244 & -1.876 \\
& $\pm$3.247\textbf{i} & $\pm$3.076\textbf{i} & $\pm$3.009\textbf{i} &
$\pm$2.986\textbf{i} & $\pm$2.974\textbf{i} & $\pm$2.971\textbf{i} \\
\cline{2-7}
& -2 & -2 & -2 & -2 & -2 & -2 \\ \cline{2-7}
& -1.358 & -1.360 & -1.360 & -1.358 & -1.356 & -1.370 \\ \hline
\end{tabular}
\caption{{\small UV}{} fixed point values and eigenvalues for different regulator
parameters $a$, and the optimized regulator, with parameter values
$\alpha_3=-0.1$ and $\chi=0.1$.}
\label{tab:UVFPregscan}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
& here & \cite{Litim:2003vp} & \cite{Christiansen:2012rx} &
\cite{Donkin:2012ud} &
\cite{Manrique:2010am} & \cite{Codello2013} & \autoref{tab:etader_standard} \\
\hline
$\overline g_*$ & 0.763 & 1.178 & 2.03 & 0.966 & 1.055 & 1.617 & 1.684 \\
\hline
$\lambda_*$ & 0.248 & 0.250 & 0.22 & 0.132 & 0.222 & -0.062 & -0.035 \\ \hline
$\overline g_* \lambda_*$ & 0.189 & 0.295 & 0.45 & 0.128 & 0.234 & -0.100 &
-0.059 \\
\hline
\end{tabular}
\caption{Comparison of the {\small UV}{} fixed point coordinates with earlier results
for the optimized cutoff. Parameter values are $\alpha_3=-0.1$ and $\chi=0.1$.
Methods of the references (in order): background approximation
\cite{Litim:2003vp}, bi-local
projection \cite{Christiansen:2012rx}, geometric approach \cite{Donkin:2012ud},
bi-metric approach \cite{Manrique:2010am}. The mixed approach is applied in
\cite{Codello2013} and is also discussed in the present paper in the last
paragraph of this subsection, \autoref{tab:etader_standard}.}
\label{tab:UVFPcomparison}
\end{center}
\end{table}
Let us further investigate the properties of the {\small UV}{} fixed
point. First of all, the existence of the fixed point does not depend
on the specific choice of the regulator. Moreover, it is attractive in
all four directions investigated here. Furthermore, even though the
critical exponents of the dynamical quantities $(g,\mu)$ are complex,
the ones of the physical background couplings $(\overline g, \lambda)$
are real. This was also found in \cite{Christiansen:2012rx} and
\cite{Codello2013}. Notice
that the eigenvalue corresponding to $\overline g$ is exactly -2,
which can be immediately inferred from the specific structure of the
background coupling flow equation. Also, the eigenvalue corresponding
to $\lambda$ is inherently real, as its flow equation is a polynomial
of order one in the cosmological constant. All these points are
summarized in \autoref{tab:UVFPregscan}.
The connection to earlier results is drawn in
\autoref{tab:UVFPcomparison}. The present results support the
qualitative reliability of the Einstein-Hilbert type approximations in
the {\small UV}{} regime.
The couplings as functions of the {\small RG}{} scale $k$
along one selected trajectory (marked as a dashed black line in the phase
diagram)
are shown in \autoref{fig:traj}. One can see how the couplings tend to
their finite fixed point values in the {\small UV}{}. The {\small IR}{} regime
will be discussed below.
A further quantity of interest is the anomalous dimension. The
momentum-dependence of both graviton and ghost anomalous dimension is given in
\autoref{fig:momdep_eta} for all used regulators at their respective {\small UV}{} fixed
point. As one can see, only quantitative differences occur. The graviton
anomalous dimension is of the order of $0.5$, whereas the ghost anomalous
dimension is of the order of $-1.5$. The difference
between exponential and optimized regulators is due to the fact that the
regulators integrate out modes at different scales. Consequently, the effective
cut-off
scale is regulator-dependent. A formal discussion of scale-optimization
can be found in \cite{Pawlowski:2005xe}, and is applied in the context of finite
temperature Yang-Mills theory in \cite{Fister:2011uw}. For instance, for the
exponential cutoff $r_a(x)$ with $a=4$, we find that if one
rescales
\begin{equation}
k_{\mathrm{opt}} \to 1.15 \, k_{\mathrm{opt}} \, ,
\label{cutoff_rescaling}\end{equation}
the momentum-dependence of the anomalous dimension with a optimized regulator
matches the one
obtained with an exponential regulator, see \autoref{fig:reg_rescale}.
\begin{figure*}[htb!]
\includegraphics[width=2\columnwidth]{etasatuvfp_inset_rescaledopt}
\caption{Comparison of the momentum dependence of the graviton and the
ghost anomalous dimension with the exponential regulator with $a=4$ on the one
hand, and the optimized regulator with the cutoff-rescaling
\eqref{cutoff_rescaling} on the other.}
\label{fig:reg_rescale}
\end{figure*}
In general, we observed that the (TT-part of the) graviton anomalous
dimension is positive, however there are indications that this does
not remain so when the other degrees of freedom of the graviton
receive an individual anomalous dimension \cite{Vertices}. On the
other hand, the ghost anomalous dimension is strictly negative, as was
already found in \cite{Eichhorn:2010tb} and \cite{Groh:2010ta}. We
also note that the anomalous dimensions are not the leading
contribution to the flow. This means that by setting $\eta(q^2) =0$ on
the RHS of the flow \eqref{flow_structure}, one captures all
qualitative properties dicussed here. Hence, the anomalous dimensions
only constitute correction effects while the leading term on the RHS
of the flow equations is the one proportional to $\dot{r}$. In the
ghost sector this pattern is even more pronounced, and dynamical ghost
effects on the phase diagram and the running couplings are very small. \\[-.5ex]
\paragraph*{Derivative expansion:}
We close this section with a discussion of the stability of the
(covariant) derivative expansion which is the standard approximation
scheme used so far. The first calculation of the graviton anomalous
dimension has been presented in \cite{Christiansen:2012rx} within the
{\small FRG}. There, the flow is projected at $p=k$. In the work
\cite{Codello2013} a derivative expansion around $p=0$ is performed.
The full results in the present study show a strong momentum
dependence of the correlation functions as well as their flows in the
cut-off regime with $p^2/k^2 \lesssim 1$. Such a strong momentum
dependence of the flows either requires higher orders in the
derivatives or a non-local expansion that works-in the information of
momenta close to zero and those close to $p^2/k^2 =1$, see
\cite{Fister:2011uw,Christiansen:2012rx,Schnoerr:2013bk}.
Note that there is a strong cut-off dependence of the graviton
anomalous dimension at the {\small UV}{} fixed point for small momenta
$p^2/k^2 \lesssim 0.05$, see \autoref{fig:momdep_eta}. The occurance
of this regime is presumably related to the mass-scale set by the
fixed point value of $\mu$. Here we investigate its impact on the
value of $\eta_{h}$ in the leading order of the derivative
expansion. We also consider a variation of the expansion point. We
also use the present results with full momentum-dependence in order to
investigate the reliability of the derivative expansion. There, the
computation of the anomalous dimension requires
\begin{equation}\label{eq:expandG}
\0{\partial_{p^2} \dot \Gamma_k^{(2h)}}{Z}= -\eta+ \0{\dot Z'}{Z}(x+\mu) +
\0{Z'}{Z}(2 \mu +\dot \mu) \,,
\end{equation}
e.g.\ at vanishing momentum, $x=p^2/k^2=0$. On the other hand, the momentum
derivative
of $\eta$ gives the relation
\begin{equation}\label{eq:expandeta}
\0{\dot Z'}{Z} = -\eta' -\0{Z'}{Z} \eta \,.
\end{equation}
Inserting \eq{eq:expandeta} in \eq{eq:expandG} leads to
\begin{equation}\label{eq:expandG2}
\0{\partial_{p^2} \dot \Gamma_k^{(2h)}}{Z}= -\eta -\eta'(x+ \mu) +
\0{Z'}{Z}\Bigl[(2-\eta) \mu -\eta\, x +\dot \mu\Bigr] \,.
\end{equation}
In the lowest order derivative expansion, that is $\Gamma_k^{(2h)} = Z_k(p^2
+m^2)$, the
anomalous dimension $\eta_{\rm der}$ is given by (minus) \eq{eq:expandG2}
evaluated at $x=0$. Moreover, the lowest order implies $Z'=0$ and we simply
arrive at
\begin{equation}\label{eq:lowestorderder}
\eta_{\rm der}= \eta(0).
\end{equation}
For the regulators used in the present work this leads to
anomalous dimensions
listed in \autoref{tab:etader}.
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline
$a$ & 2 & 3 & 4 & 5 & 6 & opt \\ \hline \hline
$\eta_{\rm der}$ & 0.46 & 0.48 & 0.51 & 0.54 & 0.56 & 0.51 \\ \hline
\end{tabular}
\caption{Anomalous dimension $\eta_{\rm der}$ in the lowest order
derivative expansion derived from the full flow.}
\label{tab:etader}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline
$a$ & 2 & 3 & 4 & 5 & 6 & opt \\ \hline \hline
$\eta_{\rm der}$ & 0.44 & 0.50 & 0.58 & 0.66 & 0.74 & 0.61 \\ \hline
\end{tabular}
\caption{Full anomalous dimension $\eta_{\rm der}$ in the lowest order
derivative expansion derived from the full flow.}
\label{tab:etaderfull}
\end{center}
\end{table}
However, the full lowest order derivative expansion takes into account the
$Z'$-terms on the right hand side.
At vanishing momentum there is the relation
\begin{equation}\label{eq:primelogZ}
\left.\0{Z'}{Z}\right|_{x=0}=\left.\012 \eta'\right|_{x=0}\,.
\end{equation}
This is easily derived from
\begin{equation}\label{eq:Z}
Z_k(p^2)=Z_{k_0}(p^2) \exp\left\{-\int_{k_0}^k \0{d\bar k}{\bar k} \eta^{\
}_{\bar k}(p^2)\right\}\,,
\end{equation}
where both $k$ and $k_0$ are in the scaling regime. The latter
condition implies that $Z'/Z=Z'_{\bar k}/Z_{\bar k}(0)$ and
$\eta'=\eta_{\bar k}'(0)$ are independent of $\bar k\in [k_0\,,\,
k]$. Then we conclude that at $x=0$ we have
\begin{equation}\label{eq:primelogZ1}
\0{Z'_k}{Z_k}=\0{Z'}{Z} \0{k^2}{k_0^2}+ \012 \eta'\,\left(1-
\0{k^2}{k_0^2}\right)\,,
\end{equation}
for all $k, k_0$ in the scaling regime and we are led to
\eq{eq:primelogZ}. Hence, in the scaling regime (with $\dot \mu=0$)
the full anomalous dimension in the derivative expansion at $x=0$ is
given by
\begin{equation}\label{eq:lowestorderderfull}
\eta_{\rm der}= \eta\,\left(1 +\012\,\eta'\,\mu \right) \,,
\end{equation}
leading to \autoref{tab:etaderfull}. These results seem to be much more
stable then the approximation \eqref{eq:lowestorderder}.
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|} \hline
$\alpha$ & 0 & 0.25 & 0.5 & 0.75 & 1 & 1.15 \\ \hline \hline
$\eta_{\rm der}$ & 0.57 & 0.50 & 0.39 & 0.25 & 0.074 & -0.016 \\ \hline
\end{tabular}
\caption{Anomalous dimension $\eta_{\rm der}$ in the standard derivative
expansion with optimized regulator in an expansion around $p=\alpha k$ with
$(\mu=0,g=1)$.}
\label{tab:etader_alpha}
\end{center}
\end{table}
To complete the present reliability analysis of Taylor expansions in
momenta $p^2$, we also investigate expansions about a general
expansion point $p = \alpha k$. We present results for the optimized
regulator and evaluate the anomalous dimension for
$g=1,\mu=0,\eta_{\mathrm{c}}=0$. The conclusions of this study do
not depend on the choice of these parameters. As one can see, the
anomalous dimension of the graviton in a derivative expansion strongly
depends on the specification parameter $\alpha$. This relates to the
fact that such an expansion only works well if the full flow of the propagator
shows a mild momentum dependence. This is not the case for the flow of
the graviton two-point function, see \cite{Christiansen:2012rx}. In
general, even the sign of the
anomalous dimension depends on the specification parameter. We
conclude that a derivative expansion in quantum gravity with
$\alpha=0$ has to be used with great caution. \\[-.5ex]
\paragraph*{Effect of identifications of couplings in the {\small UV}:}
As already mentioned, the present approximation is the first work that
employs individual running couplings for the momentum-independent part
of each vertex function. In particular, the graviton mass term should
not be identified with the cosmological constant. Still, we have
shown, that the full expansion with momentum-dependent wave function
renormalizations and a mass term for the fluctuating graviton $h$
provides {\small UV}{} fixed point results in qualitative agreement with that
of the standard background field approach, if we identify the mass a
posteriori with (minus 1/2 of) the cosmological constant. Within such
an identification we have a deSitter fixed point.
For completeness, we also have investigated a mixed approach: We use a
flat anomalous dimension $\eta_h^{\ }$ in a derivative expansion about
vanishing momentum, or a momentum-dependent one, $\eta_h(p^2)$, for
the fluctuating graviton. In turn, the flows of the graviton mass and
the Newton coupling $g$ are extracted from the flow of the
cosmological constant and the Newton coupling in the background field
approximation. This can be interpreted as an intermediate step towards
the full approximation studied here. Interestingly this leads to a
very small and negative fixed point value for the cosmological
constant, see also \cite{Codello2013} for such a mixed expansion with
a flat anomalous dimension. Our fixed point results for the case with
a flat anomalous dimension are given in \autoref{tab:etader_standard}. They
are in qualitative agreement with the results of
\cite{Codello2013}. Notably, the results in the mixed approach deviate
from both, the background field results and that of the full
approximation introduced in the present work. We have also checked
that this originates in the identification of the mass term with the
cosmological constant, the given alternative choices for the flow of
the Newton constant do not alter this result.
\begin{table}
\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|} \hline
$a$ & 2 & 3 & 4 & 5 & 6 & opt \\ \hline \hline
$g_*$ & 1.68 & 1.72 & 1.75 & 1.77 & 1.80 & 1.68 \\ \hline
$\lambda_*$ & -0.064 & -0.076 & -0.088 & -0.100 & -0.110 & -0.035 \\ \hline
$\eta^{\rm der}_h$ & 0.81 & 0.93 & 1.03 & 1.14 & 1.24 & 0.86\\ \hline
$\eta^{\rm der}_c$ & -1.08 & -1.05 & -1.04 & -1.03 & -1.01 & -0.75 \\ \hline
\end{tabular}
\caption{Fixed point values $g_*$ and $\lambda_*$ and anomalous dimension
$\eta_{\rm der}$ at this fixed point in the mixed approach.}
\label{tab:etader_standard}
\end{center}
\end{table}
\subsection{{\small IR}{} regime}\label{sec_IR_regime}
\begin{figure}
\includegraphics[width=\columnwidth]{traj}
\caption{The running couplings as functions of the renormalization scale $k$ in
units of the Planck mass. In the {\small UV}{} the couplings tend to their finite
fixed point values. As one follows the trajectories down to the {\small IR}{}, one
can see the scaling behaviour in the vicinity of the {\small IR}{} fixed point.}
\label{fig:traj}
\end{figure}
The non-trivial {\small IR}{} fixed point is located at $(g,\mu)=(0,-1)$. The
most important feature is that it is a classical one, i.e. the
essential couplings scale classically and all quantum contributions
vanish: The gravitational couplings and the cosmological constant scale as
\begin{align}
g, \overline g \sim& k^2 \, ,& \lambda \sim& k^{-2} \, ,
\label{cl_scaling_g}
\intertext{and the anomalous dimensions vanish,}
\eta_h \to& 0\,, & \eta_c\to& 0\,,
\end{align}
see Appendix~\ref{appendix_etaIR}. This leads to flow trajectories
that connect the asymptotically safe {\small UV}{} regime for
$k\to\infty$ and short distances, with a classical {\small IR}{} regime for
$k\to 0$ and large distances. Since $\mu$ approaches a finite value in
the limit $k \rightarrow 0$, the dimensionful mass $M^2 = \mu \, k^2$
vanishes in the deep {\small IR}{}. Accordingly, this fixed point corresponds
to a massless theory on large distances, which is consistent with
gravity as a force with infinite range. Moreover, the scaling of
Newton's coupling \eqref{cl_scaling_g} allows us to identify a scale
in the following way. As $g$ (or $\overline g$) scales classically,
the coefficient of proportionality, say $C$, is nothing else than the
Newton constant, because
\begin{equation}
G_N = g k^{-2} = C k^2 k^{-2} = C \, .
\end{equation}
Thus, scales are measured in units of the Planck mass, $M_{Pl}^2 =
1/C$. The physical trajectory is then fixed by measuring the relevant
couplings, that is the (background) Newton constant and cosmological
constant, at a given scale. In \autoref{fig:traj} one can see both,
the classical scaling in the {\small IR}{} as well as the vanishing of the
$\beta$-functions in the vicinity of the {\small UV}{} fixed point for large
$k$. The classical scaling regime extends roughly up to one order of
magnitude below the Planck scale. This implies the absence of quantum
gravity effects for energies $E \ll M_{\mathrm{Pl}}$, as it is
expected in a theory without a large volume compactification of
extra-dimensions. Note also that the difference between the two
couplings $g$ and $\bar g$ is hardly visible, which justifies to some
extent the background approximation for the Newton constant. \\[-.5ex]
\section{Summary and outlook}
We have presented a quantum gravity calculation that shows a classical
regime on large distances, and asymptotically safe physics in the
non-perturbative {\small UV}{} limit. The behaviour at large distances
relates to an attractive {\small IR}{} fixed point with classical scaling
behaviour. This implies that the dimensionful Newton constant $G_N$ and
the cosmological constant $\Lambda$ do not depend on the energy scale
(inverse length scale) for large distances. Hence, for the first time,
this includes the domain of classical gravity, that has been tested
experimentally, in the renormalization group approach to quantum
gravity. The classical gravity regime in the vicinity of the {\small IR}{}
fixed point is connected to a non-perturbative {\small UV}{} fixed
point, which ensures the finiteness of scattering amplitudes at arbitrary
high energies. The small dependence of the results on the
regulator indicates stability of the present truncation. Technically,
this work introduces a novel approximation scheme in {\small RG}{}-gravity
calculations. This scheme has two essential features: First, we work in a vertex
expansion with fully momentum-dependent wave-function renormalizations
for the graviton and for the ghost field. Second, the higher order correlation
functions are parametrized by additional couplings for their
momentum-dependent and momentum-independent parts. The latter become
important in the {\small IR}{} and their properties are determined by a
self-consistency scaling analysis. In particular, we show that the
momentum-independent part of the two point function cannot be
identified with the cosmological constant at large distances.
In summary, this work provides further evidence for the asymptotic
safety scenario in quantum gravity. In addition it substantiates the
physics at the {\small IR}{} fixed point found in
\cite{Donkin:2012ud,Christiansen:2012rx}. By now, the approximation is
quantitative enough to produce classical scaling for the couplings
$G_N$ and $\Lambda$ for large length scales, in accordance with
experimental observations. The present approximation is readily
extended to include higher correlation functions which will be
reported on in future work \cite{Vertices}.
\section*{Acknowledgments} We thank Tobias Henz, Daniel F.~Litim and
Christof Wetterich for discussions and collaboration on related
subjects. We thank Alessandro Codello for providing us with an updated version
of \cite{Codello2013} prior to substitution on arXiv. JMP thanks the Yukawa
Institute for Theoretical Physics,
Kyoto University, where this work was completed during the
YITP-T-13-05 on 'New Frontiers in QCD'. This work is supported by
Helmholtz Alliance HA216/EMMI and by ERC-AdG-290623.
|
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"redpajama_set_name": "RedPajamaArXiv"
}
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Slalom gigant kobiet na 28. Mistrzostwach Świata w Narciarstwie Alpejskim został rozegrany 6 lutego 1985 roku, na trasie Cividale. Tytułu sprzed trzech lat nie obroniła Erika Hess ze Szwajcarii, która tym razem zajęła jedenaste miejsce. Nową mistrzynią świata została Diann Roffe z USA, drugie miejsce zajęła Austriaczka Elisabeth Kirchler, a brązowy medal zdobyła kolejna reprezentantka USA - Eva Twardokens.
W zawodach wystartowało 55 zawodniczek, z których 46 ukończyło rywalizację.
Wyniki
Bibliografia
Wyniki na stronie FIS
alpineskiworld.net: 06.02.1985. Santa Catarina Giant Slalom, women
Mistrzostwa Świata w Narciarstwie Alpejskim 1985
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,863
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Architecturally and aesthetically driven to deliver a truly customer-driven and enjoyable building experience!
Graduating from Michigan State University's Building Construction Management program in 1983, Mike Stevens has been dedicated to the construction industry ever since. With unparalleled attention to detail, Mike takes pride in his degree and commitment to custom designs, construction, and customer service.
Graduating from Michigan State University's Building Construction Management program in 1988, Tom Zenas joined Mike and Stevens Associates Builders in 1994. Working directly with Mike for over 23 years, Tom has met and managed a countless number of contractors to make customer dreams a reality.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 29
|
Q: Sed command garbled on Solaris I'm new to sed and trying to replace a string on Solaris. I have a simple text file test.txt containing values:
apple
grape
orange
Now I'm trying this basic sed command to replace 'apple' with 'banana':
sed -e 's/apple/banana' test.txt > test.tmp && mv "$TMP" test.txt
Returns error: sed: command garbled: s/apple/banana on Solaris 10.
What is wrong with this simple code? Thanks
|
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"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,333
|
La Russo-Baltique, conosciuta anche come Russobalt o Russo-Balt (in caratteri cirillici Русско-Балтийский e Русско-Балт), è stata la prima compagnia russa a produrre automobili nel primo ventennio del secolo scorso, tra il 1909 e il 1923, fornitore ufficiale di Sua Altezza Imperiale la Corte di Russia dal 1913 al 1917.
Storia
Il primo mezzo prodotto, un camion, uscì dalla fabbrica il 9 giugno 1909, e, già nel 1910, al fine di accontentare i clienti che cercavano delle carrozzerie prodotte su misura, venne acquisita Frezn, considerata la migliore nel campo delle carrozzeria a San Pietroburgo. In quel periodo, infatti, le vetture uscivano dagli stabilimenti di produzione come semplici "scheletri", degli chassis che dovevano poi essere completati da una carrozzeria scelta e ordinata a parte. La Russo-Baltique era in questo modo, dopo l'acquisizione, in grado di fornire modelli completi, addirittura con abitacolo riscaldato, vetri panoramici e illuminazione interna.
La prima autovettura con questo marchio (il modello S24-30) apparve in un'occasione speciale; l'ascensione del Vesuvio nel 1910, dopo un percorso San Pietroburgo - Berlino - Praga - Roma - Napoli svoltosi senza che si presentassero particolari problemi.
Nel 1911 venne prodotto anche il primo autoveicolo destinato specificatamente alle competizioni, il modello S25-50: primo esempio di motore con pistoni in alluminio, vinse alcuni premi al rally internazionale di Montecarlo nel gennaio del 1912, alla sua seconda edizione, tra 78 concorrenti.
All'epoca, la competizione constava in un percorso di 3200 km con partenze differenziate e con destinazione Montecarlo, appunto. Il via fu dato il mattino del 13 gennaio, con una temperatura esterna di -23 ˚C, ma già il 23 il traguardo era stato raggiunto. Il pilota, Andren Platonovič Nagel, venne insignito di un'onorificenza per decreto dello zar.
L'obbiettivo del marchio era già avveniristico, con progetti di modelli economici o di lusso, ma la situazione della Russia cominciò a farsi difficile già dal mese di agosto 1914, non permettendo lo sviluppo di questi e molti altri progetti della società.
Fino al 1915, le officine di produzione erano localizzate a Riga, in una fabbrica che produceva vagoni ferroviari. Nel 1917, dopo la rivoluzione, un secondo stabilimento produttivo venne impiantato a San Pietroburgo, dove venivano assemblati veicoli blindati partendo dalle carrozzerie che provenivano dalla Lettonia.
Nel 1922 un nuovo trasferimento della produzione la portò da San Pietroburgo alla BTAZ di Mosca.
Dalle sue catene di montaggio uscivano sia vetture che mezzi pesanti, spesso copie più o meno dichiarate dei veicoli della tedesca Rex-Simplex o della belga Fondu, oltre che mezzi agricoli e per l'aviazione.
Produzione moderna
Dopo un periodo in cui la produzione a Riga è stata dedicata solo ai carrelli, nel 2006 il marchio è stato acquistato da un gruppo di investitori tedeschi e russi, con l'idea di proporre una vettura di alta gamma, la Impression, una coupé che sembra progettata nell'Europa degli anni trenta, presentata ufficialmente e in anteprima al Concorso di eleganza di Villa d'Este, (a Cernobbio) ad aprile 2006 e che utilizza parti meccaniche della Mercedes, precisamente della Mercedes CL65 AMG.
È stata nuovamente presentata come prototipo al Salone dell'automobile di Ginevra nel 2007 dove sono state comunicate ulteriori caratteristiche tecniche È stata dichiarata l'intenzione di stabilire la produzione totale tra le 10 e le 15 vetture, con una media annua di due o tre esemplari, e che gli stabilimenti utilizzati saranno quelli di un'altra compagnia tedesca, la Gerg GmbH.
Verrà commercializzata prevedibilmente al prezzo di circa 1.500.000 euro.
Note
Altri progetti
Collegamenti esterni
Case automobilistiche russe
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,794
|
% asm/integrals
% Version 100 13-oct-2014
%
% Functions that compute the integrals.
%
% integral* functions are passed to the computeAsmIntegral as function handles
% and handles the model parameters, and then performes the integral
% for all the frequencies. The integrands are defined in the integrands*
% functions or in the subfunctions.
%
%
% Parameter handling functions:
% integralFluidSolidFluid - Computes the integral for an axial symmetric
% model using
% integrandFluidSolidFluidAxialSymmetric. Handles
% point sources, plane pistons and focused
% transducers. The plate model can be a perfect
% reflector, fluid-solid-fluid model transmission
% or reflection, where the fluids on each side of
% the solid are identical.
%
% integralFluidSolidFluidMisalignment2D - Computes the integral for an
% axial symmetric model with a misalignment of
% the solid plate. Handles plane piston
% transmitter end receiver. The plate model can
% be a perfect reflector or a solid plate
% embedded in a fluid.
%
%
% Integrands:
% integrandFluidSolidFluidAxialSymmetric:
% Integrand for the axial symmetric model, used by
% integralFluidSolidFluid. Plate is either embedded in a fluid
% or a perfect reflector.
%
% integrandFluidSolidFluid3Dangle
% 3D model with misaligned plate. Not tested.
%
%
% Sources:
% focusedSourceASM - Angular spectrum of a focused bowl transducer.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,014
|
Definition - What does TriBalance Hot Yoga mean?
TriBalance hot yoga is a style of yoga that fits under the umbrella of the popular form of yoga called "hot yoga," in which the practice is performed in a room heated to between 95 and 110 degrees Fahrenheit.
TriBalance is similar to the most popular form of hot yoga, Bikram, but there are some key differences. Bikram is a structured yoga program consisting of 26 poses in a set order plus pranayama exercises to focus on the physical benefits of practice in a heated and humid environment. TriBalance hot yoga, on the other hand, focuses inward with meditation and varied postures that promote the goal of unifying the body with the mind and spirit.
TriBalance hot yoga does not have a set series of poses and uses dim lighting to encourage more of an inward focus. The temperature tends to be higher than in a Bikram class – about 110 degrees Fahrenheit – and the humidity tends to be lower. Some TriBalance classes do not add humidity at all.
TriBalance places emphasis on core and upper body strengthening as well as poses to stretch the back and hips.
Poses are usually held only once (Bikram does two sets of poses), and the postures are held longer to work the deep fascia tissues as in Yin yoga.
TriBalance hot yoga is thought to be effective in rehabilitating spinal injuries. Pregnant yoginis, however, are encouraged to skip hot yoga because of potential risk to the fetus.
TriBalance hot yoga was created by Corey Kelly who, along with Shawnda Falvo, opened TriBalance Yoga in Schaumburg, Ill. in 2007. Kelly developed this hot yoga method by combining what he believed were the best aspects of several styles of yoga, including Niyama, Yin and Bikram.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 958
|
Зали́в Успе́ха () — залив шириной около 130 км на северо-восточном краю лунного Моря Изобилия. Назван в честь удачного полёта автоматической станции «Луна-16» — первого советского космического аппарата, доставившего на Землю образец лунного грунта (рядом с этим заливом находится место посадки этой станции). Название утверждено Международным астрономическим союзом в 1979 году, через 9 лет после полёта «Луны-16».
Расположение и смежные структуры
Координаты центра Залива Успеха — . С его северной стороной сливается 38-километровый кратер Уэбб P, дно и половина вала которого покрыты застывшей лавой из этого залива. С восточной стороны находится 35-километровый сильно разрушенный кратер Кондон, отделённый от залива только небольшой грядой. На юго-восточном краю залива расположен 21-километровый хорошо сохранившийся кратер Уэбб.
Примечания
Литература
Ссылки
Карта на сайте Gazetteer of Planetary Nomenclature (северная и центральная часть), 1 Мб
Карта на сайте Gazetteer of Planetary Nomenclature (южная часть), 1 Мб
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,455
|
St. Augustine Beach és una població dels Estats Units a l'estat de Florida. Segons el cens del 2000 tenia una població de 4.683 habitants.
Demografia
Segons el cens del 2000, St. Augustine Beach tenia 4.683 habitants, 2.213 habitatges, i 1.263 famílies. La densitat de població era de 932 habitants/km².
Dels 2.213 habitatges en un 18,8% hi vivien nens de menys de 18 anys, en un 46,6% hi vivien parelles casades, en un 7,8% dones solteres, i en un 42,9% no eren unitats familiars. En el 28,8% dels habitatges hi vivien persones soles el 10,3% de les quals corresponia a persones de 65 anys o més que vivien soles. El nombre mitjà de persones vivint en cada habitatge era de 2,12 i el nombre mitjà de persones que vivien en cada família era de 2,57.
Per edats la població es repartia de la següent manera: un 15,4% tenia menys de 18 anys, un 11% entre 18 i 24, un 26,2% entre 25 i 44, un 27,5% de 45 a 60 i un 19,9% 65 anys o més.
L'edat mediana era de 43 anys. Per cada 100 dones de 18 o més anys hi havia 92,8 homes.
La renda mediana per habitatge era de 43.505 $ i la renda mediana per família de 55.341 $. Els homes tenien una renda mediana de 34.883 $ mentre que les dones 26.250 $. La renda per capita de la població era de 27.905 $. Entorn del 2,4% de les famílies i el 8,7% de la població estaven per davall del llindar de pobresa.
Poblacions més properes
El següent diagrama mostra les poblacions més properes.
Referències
Entitats de població de Florida
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,886
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Infinity è il secondo album in studio del cantautore canadese Devin Townsend, pubblicato il 21 ottobre 1998 dalla HevyDevy Records.
Antefatti
Dopo il completamento dell'album extreme metal City degli Strapping Young Lad e l'uscita del suo album di debutto Ocean Machine: Biomech, Townsend iniziò ad approcciare un crollo mentale, come dichiarò in Nel 1997 andò a farsi visitare in un centro di sanità mentale, dove gli venne diagnosticato il disturbo di disordine bipolare. La diagnosi lo aiutò a capire da dove "i due estremi" della sua musica venivano, sentì che il suo disordine «diede vita ai due estremi City (Strapping Young Lad) e Ocean Machine: Biomech».
Dopo essere stato dimesso dalla struttura, Townsend sentì di aver superato il momento di crollo mentale e che era in grado di scrivere il suo terzo album solista, Infinity, che descrisse come "il progetto padre" di City e Ocean Machine: Biomech. Tornò quindi in studio, accompagnato dal batterista Gene Hoglan, per lavorare sull'album, dove Townsend suona la maggior parte degli strumenti. Townsend ha affermato che molta dell'ispirazione da cui è nato Infinity è provenuta dall'utilizzo di LSD e varie droghe allucinogene.
Tracce
Testi e musiche di Devin Townsend, eccetto dove indicato.
Truth - 3:48
Christeen - 3:41 (Townsend, Ginger)
Bad Devil - 4:52
War - 6:29
Soul Driven - 5:14
Ants - 2:01
Colonial Boy - 3:04
Dynamics - 5:08
Unity - 6:07
Noisy Pink Bubbles - 5:22
Edizione della Inside Out Music
Truth - 3:48
Christeen - 3:41 (Townsend, Ginger)
Bad Devil - 4:52
War - 6:29
Soul Driven Cadillac - 5:14
Ants - 2:01
Wild Colonial Boy - 3:04
Life Is All Dynamics - 5:08
Unity - 6:07
Noisy Pink Bubbles - 5:22
Sister (live acoustic) - 2:15
Hide Nowhere (live acoustic) - 5:03
Man ('96 Demo) - 5:12
Formazione
Musicisti
Devin Townsend – voce, chitarra, basso, tastiera
Gene Hoglan – batteria
Christian Olde Wolbers – contrabbasso
Andy Codrington – trombone
Erin Townsend, Lyn Townsend, Dave Townsend, Naomi, Tanya Evans, Lara Uthoff, Chris Valagao, Brad Jackson, Jennifer Lewis – cori
Produzione
Devin Townsend – produzione, registrazioni, ingegneria del suono, missaggio, montaggio
Mark Gordon – ingegneria basso
Matteo Caratozzolo – assistenza al montaggio
Jennifer Lewis – assistenza al montaggio
Mark Gordon – assistenza tecnica
Byron Stroud – assistenza tecnica
Marty Schwartz – assistenza tecnica
Ross Gale – assistenza tecnica
Ramon Donati – assistenza tecnica
Scott Ternan – assistenza tecnica
Pete Wonsiak – assistenza al missaggio, registrazione aggiuntiva
Matteo Caratozzolo – registrazione aggiuntiva
Jamie Meyer – montaggio digitale
Brett Anthony – montaggio digitale
Greg Reely – mastering
Collegamenti esterni
|
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| 3,520
|
Qualifications that an individual must meet in order to serve as a judge in an ordinary court. These often concern educational and professional credentials.
What additional restrictions does the constitution place on the eligibility to serve as a member of ordinary courts?
|
{
"redpajama_set_name": "RedPajamaC4"
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| 3,055
|
{"url":"https:\/\/stats.stackexchange.com\/questions\/135565\/optimizing-a-time-series-with-multiple-predictors","text":"# Optimizing a time-series with multiple predictors\n\nI have a few questions about turing a univariate time series into a multivariate time series and optimizing the predictors. Here is the univariate data:\n\nindex\n22\n26\n34\n33\n40\n39\n39\n45\n50\n58\n64\n78\n51\n60\n80\n80\n93\n100\n96\n108\n111\n119\n140\n164\n103\n112\n154\n135\n156\n170\n146\n156\n166\n176\n193\n204\n\n\nMy first step here was to of course create a ts object in R and visualize the data:\n\ntsData <- ts(data = dummyData, start = c(2012,1), end = c(2014,12), frequency = 12)\n\nJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec\n2012 22 26 34 33 40 39 39 45 50 58 64 78\n2013 51 60 80 80 93 100 96 108 111 119 140 164\n2014 103 112 154 135 156 170 146 156 166 176 193 204\n\nplot(tsData)\n\n\nI interpreted this plot as a deterministic time series with a trend and perhaps a bit of seasonality\n\nExamining the acf and pacf plot confirms the trend component of the time series\n\nMy first question has to do with creating trend & seasonal variables for the time series using the decompose() function in R which yields the following plots:\n\nI understand that the decompose() function in R has created a list of vectors for the trend, seasonal and random components of the original time series but what am I suppose to do with them? Should I cbind() them to my univariate data and model:\n\nlm(index ~ trend + seasonal + random)\n\nindex trend seasonal random\nJan 2012 22 NA -23.8940972 NA\nFeb 2012 26 NA -19.4357639 NA\nMar 2012 34 NA 6.8350694 NA\nApr 2012 33 NA -7.5399306 NA\nMay 2012 40 NA 4.3142361 NA\nJun 2012 39 NA 9.5017361 NA\nJul 2012 39 45.20833 -6.3524306 0.14409722\nAug 2012 45 47.83333 -0.8315972 -2.00173611\nSep 2012 50 51.16667 -1.1232639 -0.04340278\nOct 2012 58 55.04167 2.2517361 0.70659722\nNov 2012 64 59.20833 11.2100694 -6.41840278\nDec 2012 78 63.95833 25.0642361 -11.02256944\nJan 2013 51 68.87500 -23.8940972 6.01909722\nFeb 2013 60 73.87500 -19.4357639 5.56076389\nMar 2013 80 79.04167 6.8350694 -5.87673611\nApr 2013 80 84.12500 -7.5399306 3.41493056\nMay 2013 93 89.83333 4.3142361 -1.14756944\nJun 2013 100 96.58333 9.5017361 -6.08506944\nJul 2013 96 102.33333 -6.3524306 0.01909722\nAug 2013 108 106.66667 -0.8315972 2.16493056\nSep 2013 111 111.91667 -1.1232639 0.20659722\nOct 2013 119 117.29167 2.2517361 -0.54340278\nNov 2013 140 122.20833 11.2100694 6.58159722\nDec 2013 164 127.75000 25.0642361 11.18576389\nJan 2014 103 132.75000 -23.8940972 -5.85590278\nFeb 2014 112 136.83333 -19.4357639 -5.39756944\nMar 2014 154 141.12500 6.8350694 6.03993056\nApr 2014 135 145.79167 -7.5399306 -3.25173611\nMay 2014 156 150.37500 4.3142361 1.31076389\nJun 2014 170 154.25000 9.5017361 6.24826389\nJul 2014 146 NA -6.3524306 NA\nAug 2014 156 NA -0.8315972 NA\nSep 2014 166 NA -1.1232639 NA\nOct 2014 176 NA 2.2517361 NA\nNov 2014 193 NA 11.2100694 NA\nDec 2014 204 NA 25.0642361 NA\n\n\nWhen I use the auto.arima function in the forecast package is this all happening under the hood? It seems to me that the auto.arima() selected a MA(1) term and a differencing term to deal with the trend? Is my interpretation correct? What is drift?\n\nplot(forecast(auto.arima(tsData, stepwise=FALSE)))\n\nForecast method: ARIMA(0,0,1)(0,1,0)[12] with drift\n\nModel Information:\nSeries: tsData\nARIMA(0,0,1)(0,1,0)[12] with drift\n\nCoefficients:\nma1 drift\n0.9622 4.5780\ns.e. 0.4698 0.4352\n\nsigma^2 estimated as 176.6: log likelihood=-44.52\nAIC=95.05 AICc=96.25 BIC=98.58\n\nError measures:\nME RMSE MAE MPE MAPE MASE\nTraining set 0.2459764 7.673429 4.967187 -0.7272714 4.661455 0.08876581\nACF1\nTraining set -0.0791942\n\n\nWhat happens if I'm interested in expanding the model to include other time series variables such as spend_1 and spend_2? do I need to create trend and seasonal and random variables for each of these spend variables or do I just plug them into the auto.arima as external variables:\n\nauto.ariam(tsData, xreg=spendData, stepwise= FALSE)\n\nspend_1 spend_2\n0 0\n0 0\n0 0\n0 0\n0 0\n0 209\n0 0\n0 0\n0 239\n0 0\n0 553\n0 216\n0 0\n0 161\n0 449\n107 0\n53 0\n120 81\n242 0\n100 80\n482 0\n708 81\n54 240\n688 0\n80 0\n254 108\n183 84\n104 191\n183 84\n243 167\n0 108\n0 0\n0 191\n0 191\n0 167\n0 0\n\n\nOnce I build this multivariate time series model how do I interpret the coefficients for spend_1 and spend_2? How do to optimize them in order to maximize the index variable where the model was something like:\n\nlm(index ~ spend_1 + spend_2 + trend + seasonal + random)\n\n\nThanks all for the advice please let me know if I can clarify anything further.\n\n\u2022 That's a lot of questions. You might be better breaking it up a little. \u2013\u00a0Glen_b Jan 29 '15 at 21:55\n\u2022 Yeah you are probably right. I wanted to explain the thought process and how I got to my questions. \u2013\u00a0moku Jan 29 '15 at 22:48","date":"2019-05-19 12:35:06","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\": 0.5424747467041016, \"perplexity\": 3713.340160246455}, \"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-2019-22\/segments\/1558232254882.18\/warc\/CC-MAIN-20190519121502-20190519143502-00534.warc.gz\"}"}
| null | null |
Partners Who Have Intercourse Weekly Are Happiest
More intercourse may well not constantly allow you to happier, based on research that is new because of the community for Personality and Social Psychology.
"Although more frequent intercourse is related to greater joy, this website link had been not any longer significant at a regularity of more than once per week," lead researcher Amy Muise stated. "Our findings suggest you don't need certainly to have sexual intercourse each day so long as you're keeping that connection. so it's essential to keep up a romantic reference to your spouse, but"
Some past studies, and an array of articles and self-help publications, have actually reported that more sex equals more pleasure. But this research, predicated on studies of more than 30,000 Us americans built-up over four years, may be the very first to get that relationship just isn't here after partners report making love over and over again an on average week.
The research had not been built to determine the causal procedure, so will not inform us whether making love as much as once per week makes partners happier, or becoming in a happy relationship causes individuals to have significantly more regular sex (up to once per week). In addition, these findings had been particular to individuals in intimate relationships as well as in reality, there is no relationship between intimate regularity and well-being for solitary individuals, stated Muise, a social psychologist and postdoctoral fellow during the University of Toronto-Mississauga.
It's feasible that for solitary people, the web link between intercourse and pleasure is based on an amount of facets including the relationship context where the intercourse does occur and just how comfortable individuals are with intercourse outside of relationship. The findings, which were posted online in the log personal Psychological and Personality Science, are most representative of hitched couples that are heterosexual those who work in established relationships.
Within one research, scientists analyzed survey responses about sexual regularity and happiness that is general a lot more than 25,000 People in america (11,285 males, 14,225 ladies) whom took the overall Social Survey from 1989 to 2012. The survey that is biennial carried out because of the University of Chicago, includes a nationally representative test and covers an array of sociological dilemmas, including viewpoints about battle relations, faith and sex. For partners, pleasure had a tendency to boost with additional sex that is frequent but it is not any longer true after partners report doing sex over and over again a week. This research along with other previous studies report that established partners tend to own intercourse about once per week on average.
Despite typical stereotypes that males want more sex and the elderly have actually less intercourse, there was clearly no difference between the look these up findings centered on sex, length or age of relationship. "Our findings had been consistent for males and females, more youthful and the elderly, and partners who had previously been married for a couple years or decades," Muise said.
Intercourse might be much more highly related to delight than is cash. The scientists also carried out an internet study with 335 individuals (138 males, 197 females) who had been in long-lasting relationships and discovered comparable outcomes since the study that is first. These individuals had been additionally expected about their income that is annual there clearly was a more substantial huge difference in delight between individuals who had intercourse not as much as when 30 days in comparison to those who had sex once per week than between individuals who had money of $15,000-$25,000 in comparison to individuals who had money of $50,000-$75,000 each year.
"People usually believe that more income and much more intercourse equal more happiness, but this is certainly just true as much as a point," Muise stated.
A 3rd research analyzed study outcomes gathered at three time points over 14 years from a lot more than 2,400 married people in america. There isn't a very good website link between intimate regularity and general life satisfaction, but partners reported more satisfaction with their relationships as intimate regularity increased as much as once every seven days, without any noticeable advantages of participating in intercourse more regularly.
The study findings don't always imply that partners should take part in pretty much intercourse to achieve the regular average, but lovers should discuss whether their intimate requirements are now being met, Muise stated.
"It's essential to keep an intimate experience of your spouse without putting way too much force on participating in intercourse as much as possible," she stated.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 9,517
|
Q: Cannot find symbol after gradle modification. (PushPlugin and FCMService) I'm using ionic and i'm having some trouble building my Android App.
I was getting this error:
Could not find com.android.support:support-v13:26.0.2.
And it was fixed when i included the maven section on my gradle file enter image description here
After that i'm getting another error when i try to build my android app.
It is not finding some simbles related to the FCMService and to the PushPlugin on the package android.app. enter image description here
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,336
|
Woking es un distrito no metropolitano con el estatus de municipio, ubicado en el condado de Surrey (Inglaterra). Tiene una superficie de 63,6 km². Según el censo de 2001, Woking estaba habitado por 89 840 personas y su densidad de población era de 1412,58 hab/km².
Referencias
Distritos no metropolitanos de Surrey
Municipios de Inglaterra
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 1,254
|
Q: #each block in Meteor template duplicates element after element is moved I have a template that iterates over a collection as such:
<template name="task_list">
<form id="tasks">
{{#each tasks}}
{{> task}}
{{/each}}
</form>
</template>
<template name="task">
<label class="checkbox" id="label-{{_id}}">
<input type="checkbox" id="{{_id}}" {{{completed}}} /> {{text}}
</label>
</template>
I made this a jQuery sortable, which allows the user to drag and drop within this list. When the user is done moving an item and the DOM is done rendering, I update the collection that this template uses.
This works fine when the user drags an item down, however when an item is dragged up, Meteor duplicates the item that was just dragged.
Thanks in advance for your help.
A: The problem here is that you've told both Meteor and jQuery to manage the same DOM elements. They both try to update the elements, and they end up fighting :) It's like driving a car.. only one person can drive at once.
You've got two options:
*
*Have Meteor manage the elements. In that case, you need to find a way to tell the other library (jQuery in this case) to not actually physically move the elements in the DOM.. just have it give you a callback and let you know what the user wants to do. Then you update the collection, and Meteor updates the screen.
*Have jQuery manage the elements. Put this entire section of your page inside {{#constant}}..{{/constant}} and Meteor will leave it alone. It's up to you to call <collectionname>.find(<query>).observe({...}) and use jQuery to create/move/remove the list items in response to the callbacks that observe gives you. (You can still make each individual item reactive, using Meteor.render, and you can even generate the items out of a template if you want, using something like Meteor.render(Template.<mytemplate>)).
Hope that helps!
A: Here's a wild guess, without seeing the JS:
Meteor is seeing that the list of elements doesn't match its list of objects that it's using for the each statement, so it adds what it considers to be the missing ones.
I would try getting a callback when the drag ends, and then in that callback make sure that Meteor knows which order the elements are supposed to be displayed in, so that when it re-renders the each block, it doesn't get confused about the correct order of elements and overwrite it.
Hope that's on the right track! It can be tricky to get callback-oriented code to function well in conjunction with Meteor's reactivity.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 567
|
package org.apache.flink.runtime.io.network.netty;
import org.apache.flink.runtime.io.network.ConnectionID;
import org.apache.flink.runtime.io.network.NetworkClientHandler;
import org.apache.flink.runtime.io.network.PartitionRequestClient;
import org.apache.flink.runtime.io.network.netty.exception.RemoteTransportException;
import org.apache.flink.runtime.io.network.partition.consumer.RemoteInputChannel;
import org.apache.flink.shaded.netty4.io.netty.channel.Channel;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import java.io.IOException;
import java.util.concurrent.CompletableFuture;
import java.util.concurrent.CompletionException;
import java.util.concurrent.ConcurrentHashMap;
import java.util.concurrent.ConcurrentMap;
import java.util.concurrent.ExecutionException;
import java.util.concurrent.atomic.AtomicBoolean;
import static org.apache.flink.runtime.concurrent.FutureUtils.completeFromCallable;
/**
* Factory for {@link NettyPartitionRequestClient} instances.
*
* <p>Instances of partition requests clients are shared among several {@link RemoteInputChannel}
* instances.
*/
class PartitionRequestClientFactory {
private static final Logger LOG = LoggerFactory.getLogger(PartitionRequestClientFactory.class);
private final NettyClient nettyClient;
private final int retryNumber;
private final ConcurrentMap<ConnectionID, CompletableFuture<NettyPartitionRequestClient>> clients = new ConcurrentHashMap<>();
PartitionRequestClientFactory(NettyClient nettyClient) {
this(nettyClient, 0);
}
PartitionRequestClientFactory(NettyClient nettyClient, int retryNumber) {
this.nettyClient = nettyClient;
this.retryNumber = retryNumber;
}
/**
* Atomically establishes a TCP connection to the given remote address and
* creates a {@link NettyPartitionRequestClient} instance for this connection.
*/
NettyPartitionRequestClient createPartitionRequestClient(ConnectionID connectionId) throws IOException, InterruptedException {
while (true) {
AtomicBoolean isTheFirstOne = new AtomicBoolean(false);
CompletableFuture<NettyPartitionRequestClient> clientFuture = clients.computeIfAbsent(connectionId, unused -> {
isTheFirstOne.set(true);
return new CompletableFuture<>();
});
if (isTheFirstOne.get()) {
completeFromCallable(clientFuture, () -> connectWithRetries(connectionId));
}
final NettyPartitionRequestClient client;
try {
client = clientFuture.get();
} catch (ExecutionException e) {
throw new IOException(e);
}
// Make sure to increment the reference count before handing a client
// out to ensure correct bookkeeping for channel closing.
if (client.incrementReferenceCounter()) {
return client;
} else {
destroyPartitionRequestClient(connectionId, client);
}
}
}
private NettyPartitionRequestClient connectWithRetries(ConnectionID connectionId) {
int tried = 0;
while (true) {
try {
return connect(connectionId);
} catch (RemoteTransportException e) {
tried++;
LOG.error("Failed {} times to connect to {}", tried, connectionId.getAddress(), e);
if (tried > retryNumber) {
throw new CompletionException(e);
}
}
}
}
private NettyPartitionRequestClient connect(ConnectionID connectionId) throws RemoteTransportException {
try {
Channel channel = nettyClient.connect(connectionId.getAddress()).await().channel();
NetworkClientHandler clientHandler = channel.pipeline().get(NetworkClientHandler.class);
return new NettyPartitionRequestClient(channel, clientHandler, connectionId, this);
} catch (Exception e) {
throw new RemoteTransportException(
"Connecting to remote task manager '" + connectionId.getAddress() +
"' has failed. This might indicate that the remote task " +
"manager has been lost.",
connectionId.getAddress(), e);
}
}
void closeOpenChannelConnections(ConnectionID connectionId) {
CompletableFuture<NettyPartitionRequestClient> entry = clients.get(connectionId);
if (entry != null && !entry.isDone()) {
entry.thenAccept(client -> {
if (client.disposeIfNotUsed()) {
clients.remove(connectionId, entry);
}
});
}
}
int getNumberOfActiveClients() {
return clients.size();
}
/**
* Removes the client for the given {@link ConnectionID}.
*/
void destroyPartitionRequestClient(ConnectionID connectionId, PartitionRequestClient client) {
final CompletableFuture<NettyPartitionRequestClient> future = clients.get(connectionId);
if (future != null && future.isDone()) {
future.thenAccept(futureClient -> {
if (client.equals(futureClient)) {
clients.remove(connectionId, future);
}
});
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,291
|
Q: AndroidStudio. Rename package in a module Our app contains several modules, which of them have the same package structure: i.E. com.package.app.*
Now I would like to rename a package (and of cause all its subpackages) within only one module without renaming it in another modules. It should be something like com.package.app.* => com.newpackage.app.*
Is there a simple solution for this?
I saw Android Studio Rename Package but the solutions there seem to work for applications with one module.
Thank you in advance
A: Android studio requires the name of the package to be same for all files in the app.
A: You can not have two package names in you application.
However if you are referring to the structure like this:
Then you will have to right click on the root package and click on new > package
Right Click > New > package
and define the name of the directory.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,812
|
Q: How to set the default format in dayjs? I want set "YYYY-MM-DD HH:mm:ss" to the default format to avoid unnecessary code like:
this.dayjs().startOf("year").format("YYYY-MM-DD HH:mm:ss")
this.dayjs().format("YYYY-MM-DD HH:mm:ss")
this.dayjs().startOf("week").format("YYYY-MM-DD HH:mm:ss")
Can anyone give me some advice?
A: Try adding the below to App.vue,
created() {
this.$dayjs().format('YYYY-MM-DD HH:mm:ss');
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,435
|
Every action has a reaction. We as people so small have the ability to change a world so big. When our lights shine it give others the inspiration to do wonderful things.
I'm on day 7 of the challenge and it's brought nothing but joy. I've fed the homeless, honored a friend, given away a basketball, helped a handicap gentleman shop, fulfilled a request for an instructional video, delivered food to fans, and today donated clothes to Goodwill. The best part is knowing that each recipient of my good deed will pay it forward to someone else. It's the ripple effect.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,015
|
Sophie Ingle, née le à , est une footballeuse internationale galloise évoluant au poste de milieu de terrain.
Biographie
Palmarès
En club
Chelsea Ladies
Vainqueur du championnat d'Angleterre en 2021
Notes et références
Liens externes
Footballeuse internationale galloise
Joueuse du Chelsea FCW
Footballeuse aux Jeux olympiques d'été de 2020
Naissance en septembre 1991
Naissance dans le Vale of Glamorgan
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,360
|
A few days ago, metal icon Rob Zombie found himself needing to defend the most exciting voice in his genre, Japanese trio Babymetal, to his own fans. The story is something of folklore now: Rob posted a photo with the young ladies, his fans immediately insulted them as artists (and, consequently, Zombie's taste) and the guy shut them down.
It was pretty perfect—mad props to Rob for shutting down the institutionalized misogyny that runs rampant in rock music—and on Tuesday (May 10), Babymetal have responded to the kindness.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 735
|
Covid Assessment Controls
Matchday Supporter Guide
Social and fundraising
1874 in the Community
2020/21 fixtures & results
All Appearances
All Goal Scorers
U21 News
2019/20 Appearances
2019/20 Goalscorers
2018/19 Goal Scorers
2017-18 Results
2017-18 Appearances
2017-18 League Table
ANNOUNCEMENT REGARDING THE CHRISTMAS DERBY
Fri 25 Dec 2020 Thu 24 Dec 2020 Vicki England Vicki England
1874 V NVFC
Following the Government's announcement yesterday afternoon regarding Covid-19, we heard that Cheshire had been elevated to Tier 3, bringing additional restrictions for us all. The Board then held an emergency meeting last night, which included input from Paul Bowyer and Wayne Goodison, to discuss the impact on the squad, supporters, Club and the effect on match day arrangements. We felt there was no alternative but to ask the North West Counties League to approve a postponement of the game on Monday 28 December. This request was granted, and we thank our League for their very swift response and understanding. We spoke to Northwich Vics at length and they were in total agreement of this action.
Obviously with the Coronavirus reproduction rate dramatically increasing at present, and with the threat of further restrictions to potentially come, we are extremely concerned for the safety and wellbeing of everyone who would be present. We have worked hard to keep everyone as safe as possible at the ground but clearly, the rapid spread of the virus is a big worry and responsibility for us.
We also assessed the financial impact on the Club, with the new restrictions now in place. We are a Members club and have a duty to ensure that we take responsible decisions when the need arises. There are also moral issues around how we would admit spectators, given that the number of season ticket holders and Members far outweighs the permitted attendance. It also does not sit well that we are not able to welcome any away supporters to such an important game on the calendar.
Whilst it is disappointing that we won't be able to enjoy the traditional Christmas football derby, we are sure that everyone appreciates the significant challenges and the decision that has been taken and we thank you for your understanding.
Categories 1874 News, 1st Team Post navigation
300+ Club Draw – December 2020
Catch up with our news below
Select a category Clothing Gifts and Souvenirs Match and Season Tickets Replica Kit
1874 Northwich Football Club is the trading name of Northwich Community Football Club Limited, registered under the Cooperative and Community Benefit Societies Act 2014. Registration No. 32017R. Registered Office: 20 Greenside Drive, Lostock Green, Northwich CW9 7SR © 2021
This website uses cookies to ensure you get the best experience on our website: Find out more.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,518
|
Palm Beach Gardens is a city belonging to the Palm Beach County. Home to the PGA Headquarters, the neighborhood attracts a number of avid golfers and is often regarded as the golf capital of the world.
Palm Beach Gardens is located on the eastern shore of the Palm Beach County. It is surrounded by the neighborhoods of North Palm Beach, Juno Beach, Jupiter, and West Palm Beach.
Before being known as a golf capital, Palm Beach Gardens was a swampland with cattle ranches and pine forests. Its development happened when John D. MacArthur bought the land from Sir Harry Oakes. He envisioned the community to become a garden city that will become home to 55,000 residents.
However, the city was only incorporated as a paper town in 1959. By 1960, Palm Beach Gardens has only one resident – a squatter that MacArthur permitted to live in the town.
By 1970, the town had a population of 7,000. Although the growth of the community was slow, it was steady – reaching only 48,452 in 2010.
Palm Beach Gardens is an accessible neighborhood, which currently has three interchanges on the I-95. The Florida's Turnpike also serves two interchanges to the city.
The city has a public transportation system, the Palm Tram – a regional commuter bus system. There are nearby international airports at Fort Lauderdale, West Palm Beach and Miami. General aviation airports within 35 miles from the city include North Palm Beach County, Lantana, Stuart and Boca Raton.
The neighborhood has its own Police Department with 117 sworn officers and 85 civilian volunteers that make up the Volunteer in Police Service unit. The department is also responsible for providing emergency communications through the North County Communications Center, which covers Palm Beach Gardens, North Palm Beach, Juno, Jupiter and Palm Beach Shores.
The city also has its own Fire Department with five stations spread within the neighborhood.
Palm Beach Gardens is a golfing community, home to 12 golf courses within the city limits including the PGA National Golf Club and Old Palm Golf Club.
The community also has several country clubs such as the Country Club at Mirasol and the Frenchman's Creek Beach and Country Club. Country Club at Mirasol served as the venue of the Honda Classics from 2003 to 2006. It is now hosted by the PGA National, which previously hosted the 1983 Ryder Cup, 1987 PGA Championship and the Senior PGA Championship from 1982 to 2000.
Aside from golf, residents also love to do some retail therapy. There are a number of shopping centers within the community including the Gardens Mall, Legacy Place, Midtown, PGA Commons and Downtown at the Gardens.
Palm Beach Gardens is home to a number of grand residences, estate villas and oversized mansions built in gated and non-gated communities. Aside from single family homes, the community is also home to condominium complexes.
Students of Palm Beach Gardens can attend any of the schools under the Palm Beach County School District.
Palm Beach Gardens is home to 48,452 according to the last census. There are a total of 27,663 households living in the 55.3 square miles of the neighborhood. It has a population density of 879.5 per square mile. Average household size is 2.23.
Locals are highly educated as well, with as much as 55% of them having college degrees. This may be a factor in the median household income of $65,227, higher than the beachside community of North Palm Beach.
As much as 70% of the residents in the neighborhood are homeowners causing the average monthly rent to go up. From last month's average of $2,500, renters paid an average of $2,762 this month, representing a 10% change.
Palm Beach Gardens' real estate experienced a 2% drop in its year-over-year performance. Based on 159 home sales, properties have a median sales price of $330,000, slightly lower than its December 2016 value of $335,000.
The average sales price per square foot, on the other hand, appreciated by 2% from $180 last year to $185 this year.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,748
|
\section*{Acknowledgments}
We thank A. Bauch and S. Weyers for providing the frequency data between Cs~fountains and H~masers.
The cryogenic silicon cavity laser system employed in this work was developed jointly by the JILA Physics Frontier Center (NSF) and the National Institute of Standards and Technology (NIST), the Centre for Quantum Engineering and Space-Time Research (QUEST) and Physikalisch-Technische Bundesanstalt (PTB).
We acknowledge funding from the DARPA QuASAR program, the European Community ERA-NET-Plus Programme (Grant No. 217257), the European Community 7$^\mathrm{th}$ Framework Programme (Grant Nos. 263500) and the European Metrology Research Programme (EMRP) under IND14.
The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.
J.~Y.~acknowledges support from the Humboldt foundation.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 101
|
Free Obituary Search for Cooper
Example Obituary Search: Cooper "November 8" "Washington Post"
Obituaries Related to "Cooper" from New York Times Archive
Paid Notice: Deaths BUCHANAN, JEAN COOPER
BUCHANAN--Jean Cooper, 67, retired music teacher. Memorial Monday, April 3, 11a.m. St. Paul's Church, 113 Engle St., Englewood, NJ. No flowers. Gifts Jean Buchanan Fund, c/o A & F LaGuardia HS, Box 231485, Ansonia Station, New York, NY 10023.
Michael C. Cooper, Convicted in Tax Scam, Dies a Prisoner at 66
Seeing an opportunity to profit off widespread resentment over the tax system, he ensnared 50,000 Americans who sought to dodge the I.R.S.
Bert Cooper, Boxer Who Knocked Down Holyfield, Dies at 53
Cooper fought many of the top boxers of his day but was better known for his losses than for his victories. He also fought a serious drug problem.
We May Be Able to Get Kevin Cooper Off Death Row
California's governor may permit a DNA test pointing to Cooper's innocence.
California Today: Should the Case of the Death-Row Inmate Kevin Cooper Be Re-examined?
Monday: Pressure mounts on Gov. Jerry Brown, another rebuke to President Trump's immigration policies and Steph Curry is on fire.
Peggy Cooper Cafritz, Patron of Black Artists, Dies at 70
Also a civil rights activist and educator, she championed African and African-American art, building a collection and then rebuilding it after a fire.
Leon Cooper, Who Carried on a Battle for Tarawa, Dies at 98
A veteran of the fight to take a Pacific atoll in World War II, he fought for the return of Marines' remains and to restore a beach as "hallowed ground."
Robert J. Cooper, 39, Creator of Popular Elderflower Liqueur, Dies
Mr. Cooper's 2007 concoction, St-Germain, was so embraced by the cocktail crowd that it became known as "bartender's ketchup."
Henry S. F. Cooper Jr., Space Reporter With Literary Lineage, Dies at 82
Mr. Cooper, a descendant of James Fenimore Cooper, was an author, a writer for The New Yorker and the bulletin editor for the Century Association.
Jocelyn Cooper Dies at 86; Helped Pave Way for First Black Congresswoman
Ms. Cooper and her husband, Andrew Cooper, sued in the early 1960s to challenge racially gerrymandered congressional district lines, which were redrawn under court order.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,145
|
Top 111 Free Agents: Nos. 50-31
By Matthew PouliotOct 29, 2010, 6:00 PM EDT
It's time for part four, which will cover free agents Nos. 50-31. Now we're going to start to see more players likely to command multiyear deals this winter. That's particularly true of the relievers listed below.
Free agents Nos. 111-91
Free agents Nos. 90-71
50. Jim Thome (Twins – Age 40) – Unwanted and unsigned until February, Thome made a bunch of teams feel silly for overlooking him by clubbing 25 homers in 276 at-bats this season. He was one of just six major leaguers to post an OPS over 1000 in at least 100 at-bats. Thome is no longer any sort of option at first base and he probably shouldn't be asked to start more than 120 games as a DH, but he's never failed to produce when healthy. He won't have to settle for a $1.5 million contract again.
49. Kevin Gregg (Blue Jays – Age 32) – The Jays have three choices with Gregg: they can retain him for $4.5 million in 2010, exercise a two-year option that would pay him $8.75 million through 2011 or they can set him free. My guess is that they'll go with the one-year option. Gregg performed admirably after quickly taking over the closer's role in April, saving 37 games in 43 chances. He's never really excelled at any point — his career-best ERA was a 3.41 mark in 2008 — but he is durable and he's struck out a batter an inning everywhere he's been.
48. Chris Young (Padres – Age 31) – Young was limited to four starts by another round of shoulder problems this year, but at least they were all exceptional outings: he allowed just two runs in 20 innings. The negatives with Young are obvious: his career high for innings is 179 1/3, he hasn't even made 20 starts since 2007 and, as an extreme flyball pitcher, he probably wouldn't fare nearly as well if he didn't pitch in Petco Park half of the time. Ideally, the Padres could re-sign him for about half of the $6.25 million he made this year. However, some team with a bigger budget might be willing to give him an incentive-laden contract that would allow him to make much more if he stays healthy.
47. Jon Rauch (Twins – Age 32) – Pressed into the closer's role by Joe Nathan's injury, Rauch was 21-for-25 saving games before the Twins acquired Matt Capps. He slid down the depth chart as the year went on, but he never lost his effectiveness and he finished with a 3.12 ERA. He also pitched 1 2/3 hitless innings in the ALDS. That Target Field turned out to be such a tough home run park likely did him a lot of good. The Twins are expected to emphasize re-signing fellow free agents Jesse Crain and Matt Guerrier, so Rauch may have to go elsewhere for his two-year deal.
46. Aaron Harang (Reds – Age 32) – Never the same pitcher since hurting his elbow in 2008, Harang finished 2010 with a 5.32 ERA and a 1.59 WHIP in 111 2/3 innings. He hasn't lost any velocity from his glory days, but he doesn't miss bats like he used to. I think a switch to a bigger ballpark will allow him to hang on as at least a fourth starter for a couple of more years, but unless he suddenly picks up a quality changeup or cutter, his upside would seem to be limited.
45. Frank Francisco (Rangers – Age 31) – Francisco saved 25 games in 29 attempts in 2009, but when he had a bad week to open this season, the Rangers went right to Neftali Feliz in the closer's role. Francisco bounced back quickly and was a major asset in a setup role until August, when he strained a muscle in his side and it turned into a season-ending injury. At least Francisco avoided arm woes this year, but the fact remains that he's turned in just one 60-inning season since debuting in 2004. What chance he had of securing a three-year deal this winter was probably erased by the injury, but he's one of the most talented relievers available and he should be attractive both to contenders looking for a setup man and weaker teams seeking a closer.
44. John Buck (Blue Jays – Age 30) – Sick of his low batting averages, the Royals sharply reduced Buck's role in 2009 and then cut him following the season. He landed with the Blue Jays and made the All-Star team after hitting 13 homers in the first half. He ended up at .281/.314/.489 in 409 at-bats for the season, making him one of the league's top offensive catchers despite an awful 111/16 K/BB ratio. With Buck probably in line for a multiyear deal, the Jays may choose to move on to J.P. Arencibia now. Buck's lofty average was a fluke, but he's a solid enough defender and he should be a capable regular for a couple of more years.
43. Pedro Feliciano (Mets – Age 34) – We're about to find out just how much value the Mets place on Feliciano's ability to pitch practically every day. The lefty specialist made 86 appearances in 2008, 88 in 2009 and then he became just the fifth different pitcher to work in 90 games, finishing at 92, in 2010. Despite the heavy workload, he's been consistently terrific against lefties throughout. However, righties have fared well against him two of the last three years. He probably won't command as much cash as the top righty setup men on the market. However, he may well land a three-year contract.
42. Brad Penny (Cardinals – Age 32) – He's probably never going to do it for six months, but Penny opened last season as well as any pitcher not named Ubaldo Jimenez. He was 3-0 with a 0.94 ERA after four starts, and he didn't fail to turn in a quality start until his eighth appearance of the season. Unfortunately, that proved to be his next-to-last start, as what was originally thought to be a minor back injury ended up costing him the rest of the year. The Cardinals are focused on signing Jake Westbrook at the moment, so Penny is likely to head elsewhere this winter.
41. Mark Ellis (Athletics – Age 33) – Ellis is a $6 million player when he's in the lineup, but given that he's played in 130 games just twice in his career, the A's aren't going to want to pick up his option and pay him that amount next year. They likely will attempt to re-sign him at a cheaper price, possibly to a two-year deal. While second basemen often lose it in their early 30s, Ellis is coming off his best season since 2007 and he remains a well above average defender.
40. J.J. Putz (White Sox – Age 34) – Credit Ken Williams for seeing that Putz would reemerge as a quality late-game reliever when he looked like anything but before undergoing elbow surgery in 2009. If not for a knee injury that shut him down for a spell in August, Putz probably would have taken over as the White Sox's closer and rebuilt his value further headed back into free agency. He did return in September, and he was effective in allowing three runs over seven innings. Still, it was somewhat telling that the White Sox didn't often let him face tough left-handed hitters with the game on the line. Putz's strong work has put him back in line for a multiyear deal. If the White Sox are nervous about giving him one, he could sign on as a closer elsewhere.
39. Vicente Padilla (Dodgers – Age 33) – Padilla was effective when healthy this year, going 6-5 with a 4.07 ERA, but he missed most of May, June and September due to a bulging disk in his neck. That should serve to make him pretty affordable if the Dodgers want to bring him back. Because of Padilla's attitude and occasional off-the-field problems, many teams view him as not being worth the hassle. However, he hasn't appeared to be the source of any strife in the Dodgers clubhouse. Another one-year deal worth about $5 million would be appropriate.
38. Grant Balfour (Rays – Age 33) – While Rafael Soriano and Joaquin Benoit got most of the credit, Balfour's rebound was another big reason the Rays had one of the game's best bullpens this year. He had a 2.28 ERA during the regular season, and he pitched 3 2/3 scoreless innings in the ALDS against the Rangers. Often set back by arm problems, Balfour took a long time to establish himself. However, he's been on the DL just once the last three years and that was for a strained rib muscle. Since he has a power arm and he won't be too expensive, he could be pursued by as many teams as any free agent this winter. It might get him a three-year deal in the $12 million range.
37. Juan Uribe (Giants – Age 32) – Uribe has spent his entire career alternating between being underrated and overrated. He's almost always had dreadful OBPs, and he's frustrated his teams with occasional lackadaisical play. On the other hand, he was a legitimately excellent defensive shortstop for a few years and he hit 80 homers over a four-season span with the White Sox. Two years ago he was so underappreciated that he had to take a minor league contract from the Giants. Now the pendulum is going to swing the other way. Uribe set new career highs with 24 homers and 85 RBI while making $3.25 million this year, guaranteeing that he'll receive a nice raise. The problem is that he's no longer much of a shortstop, and he might be better off at third than at second. A utility role suits him best, but he'll be paid like a starter this winter.
36. A.J. Pierzynski (White Sox- Age 34) – Nobody likes him, he doesn't throw out basestealers and his offense took a significant dip this year, yet Pierzynski will still likely be regarded as the No. 2 catcher on the market. And deservedly so. He is durable, and he's always seemed to handle pitchers well. Disappointed by Tyler Flowers' progress this season, the White Sox will try to keep the veteran. Ideally, it'd be a one-year deal. There could be enough interest to force the team to go to two years, though.
35. Brad Hawpe (Rays – Age 31) – Hawpe's furious fall from grace culminated in him getting released by the only organization he had ever known in August. Remarkably, there was little enough interest in him after he became a free agent, and he ended up playing only a bit role for the Rays down the stretch before he was left off their ALDS roster. The real fresh start will come next year. Hawpe is a lousy defender in right field, but he posted OPSs right around 900 each season from 2006-09 and he was nearly as good on the road as at Coors Field. What kind of career he has in his 30s will largely be determined by his ability to readapt to first base. It was his original position in the minors, and if he can pick it back up now, he should spend several more seasons as a regular. He wouldn't have nearly as much value as a DH or an outfielder.
34. Joaquin Benoit (Rays – Age 33) – Just an unbelievable season: after missing all of 2009 following shoulder surgery that left his career in doubt, Benoit came back and posted one of the best WHIPs ever in 2010. He ended up with a 1.34 ERA and a 0.68 WHIP in 60 1/3 innings following his April 29 callup. Benoit had a couple of nice seasons previously, particularly in 2007 (2.85 ERA in 82 innings), but he was largely viewed as a disappointment in his Rangers career. It's going to be very interesting to see how he's treated this winter. He was always durable before the shoulder surgery, and he performed well on a big stage in October, throwing 3 2/3 hitless innings in the ALDS. It'd seem worth gambling $10 million over two years to see if he can do it again.
33. Jeff Francis (Rockies – Age 30) – Back from a labrum tear that cost him all of 2009, Francis was expected to put in a full season in 2010. However, he suffered a setback with his shoulder in spring training and didn't make his first start until mid-May. After an encouraging initial run — he had a 3.53 ERA through eight starts — he began to struggle and he went back on the DL in August with shoulder tendinitis. Upon returning in September, he allowed 11 runs in 11 2/3 innings. Francis' velocity has come all of the way back, and he displayed surprisingly good command for someone who figured to be rusty. He's far from a sure thing to stay healthy, but he has the potential to be one of the winter's top bargains.
32. Coco Crisp (Athletics – Age 31) – Crisp's A's career got off to a very rough start, as he was limited to two games in the first 10 weeks by a broken finger and a strained intercostal muscle. Finally healthy in late June, he was exactly the player the A's hoped he'd be; he hit .279/.342/.438, stole 32 bases in 35 tries and played quality defense in center field. Expectations are that the team will pick up his $5.75 million for 2011.
31. Kerry Wood (Yankees – Age 33) – While all of the wildness remains a cause for concern, Wood certainly helped his stock during his time with the Yankees. After posting a 6.30 ERA in 20 innings as the Indians' closer in the first half, he came in at 0.69 ERA in 26 innings with the Bombers. He also had a 2.25 ERA in eight postseason innings. Including October, Wood walked 23 batters in 34 innings. However, he allowed just 20 hits and he struck out 38 in that span. Last time he was a free agent, Wood chose closing for a mediocre team over setting up for a contender. I'm guessing he'll go in the other direction this time, though it's possible he could get the best of both worlds if the Rays want him.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,922
|
# The Robin Hood Thief
## H.C.H. Ritz
### Contents
Get a second book for FREE
Prologue
45 Days
44 Days, 16 Hours
44 Days, 8 Hours
44 Days, 6 Hours, 18 Minutes
43 Days, 16 Hours
43 Days, 14 Hours
42 Days
41 Days
40 Days, 14 Hours
40 Days, 5 Hours
30 Days
29 Days, 12 Hours
29 Days, 6 Hours
28 Days, 18 Hours
28 Days, 12 Hours
27 Days
26 Days
25 Days
24 Days
20 Days, 6 Hours
20 Days, 3 Hours
19 Days
18 Days
17 Days
14 Days
13 Days
12 Days
10 Days
9 Days, 15 Hours
9 Days, 14 Hours, 48 Minutes
8 Days
7 Days
5 Days, 16 Hours
5 Days, 13 Hours
4 Days
3 Days
2 Days, 16 Hours
2 Days, 14 Hours, 24 Minutes
2 Days, 14 Hours, 2 Minutes
2 Days, 13 Hours, 36 Minutes
2 Days, 10 Hours
Zero Days
Dear America
Get Absence of Mind for FREE
Please Review?
Acknowledgments
About the Author
Also by H.C.H. Ritz
Recommended: The Year of the Hydra
Recommended: Spindown
About Grey Gecko Press
To my mother, Janet
Who is, of course, the best mom in the whole world
# Get a second book for FREE
Sign up for the no-spam mailing list and get a free copy of _Absence of Mind_ also by H.C.H. Ritz, plus lots of exclusive content, all for free.
Details can be found at the end of _The Robin Hood Thief_.
# Prologue
When Helen M. Dawson daydreamed, it was always about being a hero. Not the superhero kind, with a cape—that wasn't the right sort of dream for a practical person like Helen—but the kind who had _money_. As soon as she learned the meaning of the word _philanthropist_ , that was what she wanted to be when she grew up. To have the luxury of generosity.
Such a state seemed—not mythical, for other people possessed it—but unattainable. She never had a friend who was wealthy. And as she got older, things got worse. The rich got richer, the poor got poorer.
Inflation stagnated, but so did wages. Environmental refugees heading north swelled the US population, Congress swept aside the remnants of the safety net like strands of a rotten cobweb, and the Supreme Court gutted the bankruptcy codes and allowed imprisonment for default on debts for the first time in two hundred years. The glitterati climbed ever higher and pulled the ladder up after them. By the 2040s, even the lowest rung looked unreachable.
But still Helen dreamed.
When she grew up, she daydreamed on her daily commute to work. She would have done so anyway, but the dreams let her cope with the world that she saw outside her car windows, especially the throngs of homeless and impoverished people begging on the sidewalks and medians.
One time, while she was still young enough to be naive, she was stopping at a light when a man in a dirty white shirt leaped in front of her car. There was a thump, and blood splashed up over the hood onto the windshield. From under the car, he screamed hoarsely and begged for help.
She got out of her car, stunned and sick.
The man cradled red-soaked arms tightly in his lap, sobbing. He wouldn't let her see. "Just give me some money to go to the doctor," he pleaded. "A hundred dollars, something. Please!"
While impatient drivers honked their horns and shouted behind her, she used her smartphone—it was that long ago—to transfer two hundred dollars to his phone. Then he struggled to his feet and disappeared into the press of the crowd.
She went into work shaken, horrified, afraid that he would call the police, press charges, sue. But when she told the story to her coworkers, they told her that she'd been scammed. The blood was fake. The man was fine.
She was dismayed, but by the end of the day, she convinced herself that everyone had to make a living somehow, and that was how he made his. It was a difficult world. A difficult time. She didn't have two hundred dollars to spare, of course—no one she knew did—but that was life.
Then, somehow, twenty years went by, all of them spent working in the nonprofit world, all of them spent striving to help. Twenty years of discovering how pointless it could be. Some people didn't want to be helped, and others couldn't be.
She saw coworkers turn jaded and cold. But for Helen, twenty years of being confronted with reality only strengthened her commitment. If it was easy, it wouldn't be heroic.
So she daydreamed. She fantasized about having so much money that the next time someone leapt in front of her car and pretended to be hurt, she could just go out there and help him up. "It's okay," she could tell him. "I know you're having a tough time. How much do you need to make it through this week?"
And she could give him the money. He'd be amazed. He'd give her a _God bless you_ and run off into the crowd before she could change her mind.
At times, she sensed hints of ego in these daydreams, wanting to be _the one who helped_. She thought about that a lot, because it was important to her. She didn't want it to be about ego. Even if maybe it was, a little tiny bit.
# 45 Days
Helen had not fallen asleep in eight days.
She waited with a hundred other people at the diagnostic machines at Florida Hospital Orlando, in lines that snaked through the dingy, unclean room. Two hours passed while she waited for the MDRI machine, and then an hour and a half while she waited for the DCAT machine.
Not that it mattered. The heavy, suffocating fatigue rendered time a meaningless blur.
Three weeks ago, she'd noticed only routine sleeplessness plus some odd muscle spasms. But every over-the-counter medication had failed to help her sleep. And then came the heart palpitations. And the anxiety.
And then her pupils permanently shrank to tiny pinpoints.
Every time she caught sight of those unnatural eyes in a mirror, her stomach did a slow flip of misery and fear. They whispered to her that this wasn't ordinary insomnia, but something new and malevolent.
She knew from some news article years ago that people subjected to intense sleep deprivation, such as military recruits, would fall asleep within seconds once given the chance. They suffered intense sleepiness. Helen was unutterably, unendurably tired, but falling asleep was no longer a possibility. It was like the part of her brain that governed such things had simply ceased to function.
Instead, she felt wakeful, restless—even driven. She couldn't keep her eyes closed for longer than a moment—but neither could she hold them open for long. Open for an instant...closed for a moment... open again... closed again... a pattern without ceasing for eight days, in relentless purgatory.
The fatigue was a heavy gray blanket over her, weighing down her limbs and dulling all her senses. Her mental state was one drawn-out and drowning plea for release: _God, let me sleep. Just let me sleep._
Helen had arrived at the dark, outdated ER just after lunch, knowing how long the wait for the diagnostic machines would be and hoping others had come in the early morning.
Signs directed incoming patients to the triage kiosks. At Helen's kiosk, the programmed hologram, a female with glasses and brunette hair in a bun, blinked in and out of functionality, but her speech was unnecessary anyway. Everyone already knew what she would say: that emergency rooms in public hospitals like this one were obligated to provide emergency care to indigent applicants, but only if they might die without it. The holograms invited those who weren't that sick yet to return once they were.
They left unsaid the fact that, given the overpopulation and the shortage of health care workers, indigent applicants who qualified for care might still die before they received it. Whereas the ultrawealthy went to private clinics that offered immediate care for every sniffle and cough. There was nothing between the two extremes.
As the brunette woman directed, Helen pressed a finger in the hollow of the triage kiosk for a pinprick blood draw, then typed her symptoms on a worn keyboard: Severe insomnia. Heart palpitations. Drowsiness and fatigue. Anxiety. Constricted pupils. Muscle spasms.
Moments later, the machine informed Helen, in erratic and indifferent tones, that she qualified for emergency care, and relief warred with a hollow fear that grew in Helen's stomach.
Now, three and a half hours later, the soft programmed voices emitting from the diagnostic machines blended with the news coming from the monitors hung at regular intervals, many of the screens not working or with half their pixels burned out.
Helen's heart pounded in her ears and chest, as it had for days now.
She double-checked her e-paper—the new version of what they used to call a smartphone. The ultra-thin, hard, plastic-infused glass unfolded along pre-determined lines and snapped into place in the desired size of the moment.
Currently, it was at memo pad size, and when Helen pressed the button on the edge to wake it up, it read, "DCAT machine A." An electronic diamond code appeared next. The diagnostic machine would read the code and apply the proper screening.
As she waited, a reporter from LSTV spoke from a monitor just above Helen's head. Something about the continually encroaching sea coasts. The city of Houston was pushing an initiative to relocate its eastern half. Galveston and New Orleans and a hundred smaller cities were already lost to the sea.
Helen unfolded her e-paper to paperback size and tried to read a book she'd checked out from the library.
She realized soon that she was reading the same page over and over, none of the words sinking in. She turned the device off.
The line inched forward. At long last, only four people stood between Helen and the DCAT machine. A large ring extended from a pedestal. It split in the center and opened to release its latest victim.
Most of the people in line were on their Earworms—tiny wearable internet devices. A stem fit along the temple, counterweighted behind the ear with a built-in speaker, and a crystal at the tip projected holographic images directly into the pupil. They were among the more expensive gadgets, and Helen had saved up for two years to get one for her daughter, Mandy.
Thinking about her eighteen-year-old daughter made Helen's stomach clench with anxiety. She couldn't afford to be seriously ill. She needed to be there to help her daughter finish out her retail management job training program and find work—and above all, help her avoid debt that could get her imprisoned.
The person in front of her stepped into the DCAT ring. Helen's heartbeat jumped up again. The diagnostic process frightened her. What if it was bad news? Or, worse, what if they couldn't help her? God, that was a horrible thought.
She shifted her purse to the other shoulder. She folded her e-paper to notepad size and it automatically flipped to her notes. She had a to-do item written down: Dog food. She needed to get dog food when she got done here.
Anger tempted, but she was just too tired. This was a recurring fight with her daughter: Mandy was supposed to be responsible for the short, stout, blond mutt, and yet Helen always ended up buying and setting out the dog food.
It was Helen's turn at the machine. Whether she wanted it to be or not.
She stepped up to the screen and scanned the diamond code on her slip of paper.
"It's a brain scan for Helen M. Dawson," said the avatar—a blonde woman this time. "Is that correct?"
"Yes," Helen said, probably louder than was necessary.
The ring split and opened and Helen stepped in.
A bright light came on and some interior part of the machine whirred and spun around her. Something buzzed inside the lower parts of the machine.
Then it stopped. The machine's ring lowered and opened. Helen stepped out and looked for the report machine.
In line there, she fought to keep her burning, exhausted eyes open. Or maybe she fought to keep them closed. Who could tell anymore.
"Results for Dawson, Helen."
Her eyes flew open and she stepped up to the kiosk. She tapped the Accept button and the machine spat out a short piece of paper. She read it as she walked back toward the stairs.
> Sporadic fatal insomnia (sFI). No treatment modalities with statistical improvement of mortality available at this time. Based on degeneration of thalamus, prognosis three to six months. Go to pharma machine and scan diamond codes to request medications.
She stopped her feet from moving toward the stairs.
_Sporadic fatal insomnia_? That didn't even sound real. It sounded like a joke. And _fatal_? How could insomnia possibly be fatal?
_Three to six months?_
She felt as if someone had punched her hard in the stomach.
She got out her e-paper, snapped it into half-sheet size, and took a snapshot of the words to search on. The first result was scientific.
> Sporadic Fatal Insomnia. Extremely rare and invariably fatal neurodegenerative disease characterized by the accumulation of a misfolded prion protein in brain matter. One of the least common human prion diseases. Characterized by disrupted sleep and dsyfunction in motor functioning and autonomic responses.
_Invariably fatal._
Her heart beat so hard she could feel it in her lips.
A site written for the layperson spelled out her doom in plain language:
> The first sign of this exceedingly rare disease is a loss of the ability to fall asleep. The patient's automatic body systems become unregulated, resulting in pinprick pupils, digestive difficulties, fevers, and a racing heart rate, as well as difficulty speaking, swallowing, and controlling bodily movements. The disease progresses over a period of months to stupor and dementia, followed by coma and death. There is no cure.
Helen's stomach roiled. She held her hand over her mouth. Tears sprang to her eyes. On a visceral level, her body rejected this information even as her mind registered _yes, this fits my symptoms exactly. This is what I have. This is how I will die._
As people finishing their scans jostled her to the side, she wiped her eyes with violently shaking hands and read the last paragraph again and again.
This couldn't be happening.
She looked again at the strip of paper in her hand, as if it held her salvation.
_Go to pharma machine,_ it said.
She read the sign on the wall next to her. The pharma machines were back on the first floor.
She put her e-paper away and threaded her way through the ever-present crowds, down the dim hallways where the LED lights were partially burned out. Down the stairs—the elevators never worked.
Her burning eyes were filled with tears. She blinked back the blur.
The pharma machines were tucked deep inside the dark, dingy little drugstore. Only one of the machines worked, and the line for it extended deep into the hospital's lobby, where late afternoon sun fought to penetrate grimy windows and where the homeless in gray, stained clothes slept or lounged on every bench and the carpeted floor.
The air conditioning was failing against the heat and humidity of central Florida, or maybe it was the fevers the website had listed as a symptom. People stood wall to wall here, as everywhere, and Helen felt claustrophobic. Her heart rate would not settle down.
_No treatment modalities with statistical improvement of mortality available at this time._
Why had they written it that way? Anger surged at the nameless, faceless people who'd written that text. Helen was a bookworm with an above-average vocabulary. But that wasn't the norm. How cruel was it to give people this news in words they couldn't even understand?
At length, her turn at the machine came. She scanned the three diamond codes. The machine clunked and whirred. Three pill bottles shot down into the dispensary bowl and another slip of paper jutted from the machine. She took everything and moved to a corner of the drugstore to look at what she'd received.
She read the text for the first drug:
> This medication has been prescribed to provoke sleep in patients unresponsive to over-the-counter sleep aids. Sleep will be immediate and profound. Do not take this medication except while in bed in safe surroundings.
_Thank God._ Tonight, she would sleep.
It sounded like bliss. Like heaven.
The second drug said only, "This medication has been prescribed to improve wakefulness."
One drug to sleep, one drug to wake up. She put the two bags into her purse, armed now against her disease.
The third bottle held a single black pill.
The sight of it took her breath. Then heat surged in her chest and rose to her forehead, bringing out beads of sweat she wiped away with shaking hands.
She knew what this was. Only one pill was black. The assisted suicide pill, legal for fifteen years.
The final act of millions of people was to take one of these. Sometimes, you'd find them sprawled on the streets, still clutching the pill bottle.
A few years ago, a coworker of Helen's had taken his at his desk, just after lunch on a Monday. Another time, on a business trip to Atlanta, Helen had called for an automated taxi, and when she opened the door of the passenger compartment, a young, slender, blonde woman had fallen sideways out of the car, her empty medicine bottle spilling out of her hand and onto the pavement.
The suicide pill was the answer to the overloaded medical system. It was the answer to poverty, to debtor's prisons, to unrelenting misery, to overpopulation and the psychological isolation it ironically bred. It was the answer to every difficult question.
Helen looked at the slip of paper that had come with the black pill. It began with words that were well-known and often lampooned:
> For termination of life. For the use of the specified patient. You have been prescribed this medication following a diagnosis for a condition for which there can be no recovery.
Tears swam in her vision and she wiped them away. The news was beginning to sink in, and she didn't want it to. She didn't want that swallowed-down feeling of newly knowing something awful.
What about Mandy? Who was going to take care of Mandy?
She read on:
> Be advised that this pill terminates life within seconds. Death will be painless. Please notify your next of kin before taking the medication. Please check in at your local crematory before taking the medication. Be advised that this medication is irreversible. You may choose to take your medication as soon as is convenient to minimize potential discomfort. Based upon your diagnosis and prognosis, we recommend you take this medication in no more than forty-five days.
_Forty-five days._
It was another punch to the gut.
How could she prepare Mandy to live on her own in forty-five days?
She looked around her as if for help, as if someone might rescue her from this new reality.
She recognized suddenly the paper sign that was taped to the wall exactly where she had drifted to the side to make room for the next person at the pharma machine. It said, "Please DO NOT take life termination pills here. A crematory is down the hall in Wing D."
How many people had decided that the best way to deal with their diagnosis was to go ahead and check out right this very moment, before they started thinking about it, before they got scared? How often had the staff at the drugstore had to call the morgue to take away the bodies? Apparently, a lot, to bother to put up a sign.
Suddenly she realized how precious this one pill would be to her in the coming days. With dementia looming... The one certainty she had at this moment was that she did not want to face the end of this disease process without this pill. Thank God she had it. She clutched it to her chest.
Then, as quickly, fear overtook her. What if she lost it? This one pill ensured she could die without suffering and without being a burden to her daughter. One tiny pill—far too easy to lose.
She went to the girl at the counter. "I just got a life termination pill, but I'm worried that I might lose it, and—"
The girl pointed to a worn, curling sign taped to a stand on the counter: "Life termination pills can be replaced after a twenty-four-hour waiting period. Request the replacement at the pharma machine."
Helen nodded her understanding and backed away from the counter, still clutching the black pill in its plastic bottle.
She looked around at the dozens of people who still stood in line at the functioning machines, the half-dozen people wandering the aisles of the drugstore, the homeless resting on the benches and floor. Could any of those people look at her and know all that she had just learned? Of course not. She was perfectly calm. It was one of Helen's strengths to be calm in the face of a crisis.
Did this drugstore carry dog food?
She tucked the pill bottle into her purse alongside the other two and went up and down the aisles. The signs and labels passed before her eyes without meaning. She made three passes before she found the pet food section. She scanned the bag of dog food with her e-paper and walked out with it.
The sky was dark now, and it was raining hard, as it did nearly every day during the rainy season. She used the dog food bag as a makeshift umbrella as she trudged through the crowds over to the parking garage.
Here, by the hospital, where visitors had mortality and sympathy on their minds, beggars threw themselves aggressively on the path, nearly tripping their targets. As always, Helen muttered, "Sorry, I can't," and watched them for signs of potential violence as she passed the gauntlet.
This time, she hardly saw them through the opaque fabric of fatigue and devastating truth.
She was going to die.
She was going to get worse and worse until she couldn't bear it anymore, and then she was going to take a black pill and die. Within forty-five days.
Out of the rain, up the four floors of the parking garage, to her car, the ancient light blue Ford, a luxury purchase she and David had indulged in during their brief flush period before they had Mandy. She kicked the door in the right spot to make the door handle work and yanked it open. The door was massively dented, and David's attempt to repaint it with poorly matched spray paint left it an eyesore, and duct tape held the quarter glass window in place. But it was old enough to be grandfathered out of the laws against human-driven cars.
She threw in the dog food, climbed in, hit the ignition button, and started toward home.
It had been raining hard while she'd been inside. Since climate change had taken hold, flooding threatened from May through September, and it was early May now. She noted how high the water rose against the car tires in the streets and judged that she'd make it home okay.
The world had been tilted on its axis ever since she stopped sleeping. The volume was lowered on its colors and scents. She looked around at every detail, searching for new relevance or meaning, and found nothing. The thick blanket of insomnia muted it all.
The people who passed by in a steady stream under their umbrellas... how many of them were sick like her? How many carried a black pill in a bottle?
She craned her neck to see the sky. Just blackness behind the light pollution of the city of ten million. The street lamps made halos in the mist.
News came through the radio.
"Hundreds of rioters have struck at the headquarters of Unistar Oil in Orlando after it announced record profits and a fifty-percent increase in executive pay Tuesday, followed by its fifth round of layoffs in six months. Fourteen are dead so far, ten of them rioters and four of them Unistar employees. It is the third such riot in Orlando this month.
"Violent crime is up nationwide by a margin of twelve percent over last year. It has increased eighty percent over five years ago, and two hundred and twelve percent over ten years ago.
"One of the colonists to the American Mars colony has died of complications following a case of pneumonia. The next planned delivery of colonists has been postponed for now."
The two small moon colonies—test colonies started five years ago—were doing well enough, Helen recalled. They'd sent only prisoners, most of them debtors. The government had thought it prudent to send disposable humans.
The Mars tickets, on the other hand, were prized and outrageously expensive. The waiting list was thousands of people long—most of them millionaires. All the riots and protests demanding a lottery system for tickets had come to nothing. Many believed that the Entitled would take their wealth and leave the mobs of poor behind to rot. And who could blame them? Who would still want to live on planet Earth?
She turned off the radio.
Evening rush hour was in full swing. Sometimes she thought it never ended.
Her mind was detached. Floating around somewhere up in the sky, swirling down with the rain, caught in the halo of a street lamp. It wouldn't land.
She just needed to get the dog food home. That was concrete. She could do that.
From one parking garage to another. Everything was built up and up these days in an effort to handle the burgeoning population.
Because of the relentless Florida humidity, the hallway leading to her apartment always smelled of mildew. She pushed the key into the lock and shoved the door open, then closed it behind her and locked the three one-sided deadbolts.
When she turned, the minuscule kitchen was immediately to her right and the tiny living area was in front of her, with worn furniture all rented as a matching package: a loveseat, an armchair, a standing lamp, and one side table, along with two large photos of New York City alight at night, which Mandy chose.
On the dingy white wall to her left was a two-person dining table, then a door that opened to the shared bedroom and half-tub bathroom. Plastic shelves with buckling plastic drawers lined the walls, holding everything else they owned.
Mandy, her eighteen-year-old daughter, sat on the loveseat, chubby legs bent under her body. Blue hair plumed proudly a foot above her head, with thick blue tendrils coming down alongside her temples and jawbones and caressing her round yet delicate face. Black lipstick contrasted with her pale skin, and a silver ornament encircled the bottom lip. A dozen silver bracelets glinted at her wrists, and a dozen or so necklaces obscured the band name on her black T-shirt.
As always, Mandy's clamshell rested on her lap and her Earworm clung to her ear. Helen couldn't fathom how anyone could use two computers at once.
Her daughter didn't even look at her.
It was always like this.
Every day.
Just the way David had been, after a while. Four beers down his throat and the evening spent locked away inside his Earworm.
Emotion awakened, even through the haze of fatigue. Anger. Resentment. Above all, fear for her daughter's future. Mandy was about to have to live by herself, on her own, and if she couldn't even manage to buy dog food...
As the stout, blond dog, Jessie, raced to leap at her knees, Helen slammed her purse, umbrella, and keys down on the tiny dining table. _Don't do this. Don't be angry._
Her daughter finally looked over.
"Did you get dog food like I asked?" Helen snapped. "Did you?" _Stop it. Stop this._ The dog jumped up at her. "I have asked you every day this week. Every single day, Mandy!" She couldn't make the words stop running out of her mouth. "And then I told you he was out of food last night!"
Mandy stared, her face blank.
"You still didn't do it, did you?" Helen's anger and panic soared into an entirely irrational height. This wasn't her—maybe it was the insomnia, or maybe the disease... or maybe just the fear for her daughter.
She threw the bag of dog food onto the dining table. "I knew you wouldn't. That's why I got it. Because if it were up to you, Jessie would _die_. Because you are _irresponsible_!"
She slammed her hand on the dog food. Jessie backpedaled, confused and alarmed.
Mandy's mouth opened in protest.
"You have got to get your crap together, Mandy. Get it together!"
"God! You're such an asshole!" Mandy pulled herself off the loveseat and dashed into their shared bedroom—a trip of three steps—and slammed the door behind her.
Helen moved automatically into the kitchen, the bag of dog food in her trembling hands. The anger drifted and faded into the blur of fatigue. _I shouldn't have done that. I shouldn't have hurt her. Yelling at her doesn't help. It never helps._
She crouched on the floor, opened the dog food, and poured some into Jessie's bowl. The dog dug in eagerly, the drama between his humans forgotten.
She sank to her knees and petted the dog mechanically, tears brimming at her closed eyelids. Otherwise, she didn't feel anything. The world had grown hollow.
# 44 Days, 16 Hours
The previous night, Helen went into their shared bedroom—a curtain divided the tiny room in half, with Mandy in the back—and found the bottle of the new sleeping pills. She dubiously reread the slip of paper that described its effects as "immediate and profound" and took a dose while she was still dressed and standing by her bed.
The slip of paper was right.
She collapsed to the floor as heavy darkness claimed her with such finality, her last thought was that she had accidentally taken the black pill.
But then... the awful waking up.
At the end of sleep—or perhaps at the beginning of wakefulness—cold, white, clammy, eyeless forms reached out of the ground to drag her down into it. She tried to fight with limbs that wouldn't move, and she struggled to scream for help with lungs that could barely draw breath.
Whether dreams or hallucinations, they finally passed, and she came to consciousness gasping and clawing at her bedroom door. A drugged, heavy feeling pulled her down like weights would drag a body toward an ocean floor.
She silenced herself, as she didn't want to wake Mandy on the other side of the curtain, and dragged herself back to her bedside table to find the other pills—the waking pills. She took one and sagged onto the floor in the fetal position. Twenty minutes passed, and then she began to feel something like her old self. The self that used to sleep.
The heaviness fell away.
She went to the living room and pulled back the curtains in front of their narrow floor-to-ceiling window. Blue sky. White clouds. They were beautiful and vibrant.
She could feel again. In fact, she thought she felt more intensely than before. Perceived more vividly than before.
Her heart rate escalated, then escalated again. She held onto the curtains as if they would hold her down as she went buoyant and flew out into the sky.
What a roller coaster ride this was.
Her alarm went off, and she hurried back to the bedroom to shut it off. It was time to go to work.
She laughed quietly as she changed clothes. Why was she going to work? And yet she was. It was a weekday. She would go to work.
Maybe she was in denial. She remembered her diagnosis full well, and yet the greater part of her didn't believe it.
She picked up her purse and saw, still tucked into it, the bottle that held the black pill. It was too important to leave it sitting in her purse like that.
The locket. One of the few gifts David gave her before the divorce—and before he died suddenly three years ago from a heart attack.
She found the pretty silver locket, along with her wedding ring, under her underclothes in her top dresser drawer. She looked at both artifacts of old, lost love, and the familiar waves of grief and guilt came over her. She'd earnestly tried, with David.
Divorce was the executioner of dreams.
She tried putting the black pill into the silver locket and found that it fit neatly. She slipped the chain over her neck and tucked the locket under her shirt so Mandy wouldn't notice it.
Remembering their fight last night, Helen rapped hesitantly on the wall next to Mandy's curtain. It was dark and quiet on the other side, and Helen didn't want to wake her daughter up, but she had to try to make things better somehow.
Even though it never worked.
"Mandy?"
Nothing.
"I'm sorry about last night," Helen said. "You were right. I was a jerk. You didn't deserve what I said."
No response.
Even if Mandy was awake, she wouldn't answer. Mandy hadn't forgiven Helen for the divorce. Probably she never would. Mandy had loved her daddy, and Helen had sent him away to die alone.
An hour of traffic. At least the skies were clear so far. Helen passed slowly from one desolate area of urban wasteland to another. Towering condos, imposing office buildings, much of it under new construction. At one point, Helen turned her head and counted seventeen cranes building high rises. It was all in response to the population pressure after rising sea waters ate away Tampa and Miami and Jacksonville and put the Everglades under the ocean. Climate refugees had flocked to Orlando until the city filled the remaining width of the state.
Up the elevators to work. The state headquarters of Justice for All, an organization devoted to economic and racial justice. Three rooms crowded with twelve people. Her home for the past eleven years. When she started, they had fifteen rooms and thirty people.
The small, cluttered break room with bare white walls marked with scrapes and dings. The smell of fresh coffee.
She poured a cup, then her arm and leg muscles spasmed, and she dropped the mug and nearly fell. She caught herself in one of the hard plastic chairs.
The spasms passed enough for her to wipe up the spill with paper towels and get another cup of coffee before anyone else came in. She was grateful for that.
_Invariably fatal._
Amid the sounds of her coworkers placing calls—it was phone banking day—she went to her small, rickety desk, turned on her desk lamp, and unlocked the drawer. She took out her clamshell, the device she grew up calling a laptop.
First things first. Her practical side, never far away, commanded her to do what needed to be done. She opened up her clamshell, opened a website for creating wills, and filled out the fields.
Debts—none. When the majority of prisoners were doing time for debts they couldn't pay, she wouldn't risk it, not as a single mother. What she couldn't afford, she lived without, and she lived without a lot.
Assets. She bit her lip. Old Blue, the Ford. Worth a few hundred dollars at best. She wrote it down.
She had about eight hundred in her checking account, only because rent hadn't come out this month yet. She wrote it down anyway.
Her retirement account—not worth much more than when she'd stopped her contributions twenty years ago. About $5,500. She'd started it before Congress tore away the tax protections that made such accounts relevant and her old employer revoked its matching. No one bothered with these accounts anymore. The ultra-rich didn't need them.
Helen couldn't afford life insurance, and anyway, such policies excluded death by black pill, along with all other forms of suicide.
There was nothing else, she realized with a stab of anxiety. All the furniture and appliances in the apartment were rented.
With an inheritance of only a few thousand dollars, Mandy could only get through a couple of months before she would have to get a job. She wouldn't be financially stable and out of the reach of debt.
When Helen had still had hope that she could better her and Mandy's situation someday, it hadn't seemed so bleak. But now... As she faced the reality that $5,500 and a crap car were all she would ever have to show for her life and all she would have to leave to her child, her stomach flipped and sank.
She e-signed the document and saved it in her important files, which Mandy knew the password to.
A sudden thought occurred to her. The news always portrayed Social Security as a meager shadow of its former self, but she logged into the Social Security website anyway to see if Mandy would qualify for survivorship benefits.
The website called up her work history, and she confirmed it. Then she entered Mandy's information: _A non-minor. Dependent. Enrolled in job training. Sole survivor. No other forms of income._
The website calculated.
Available benefit: ninety-seven cents per month.
Helen felt anger rise, and she let out a long, slow breath. Getting angry never helped.
But without the anger, only a sense of helplessness remained.
She looked around her.
The shabby office with blank, dingy walls. Everyone else just scraping by, same as her, in their thrift-store clothes and home-done haircuts—lucky enough not to live in sleep lockers, but not lucky enough for college tuition or health insurance.
She closed the files and folders. She would try to lose herself in her work, as she always did. Her work and Mandy were the two things that kept her going. Never mind that both felt like exercises in futility much of the time.
She reviewed the script and pulled her office phone closer, and her voice joined that of her coworkers.
"Good morning, I'm calling on behalf of Justice for All. We're calling today to collect signatures on an important—"
Click.
She tapped "Hang-up by recipient" and went to the next number.
"—collecting signatures on an important measure affecting our community's children. Did you know—"
Click.
She tapped "Hang-up" again and dialed again.
"—that since public schools can no longer accommodate the number of children we have and children are now permitted to work at the age of fourteen, if a child is neither working nor in school, he or she is subject to arrest for truancy?"
"Well, fine." This from a gruff older man. "If a kid isn't working or in school, they ought to be arrested. What are they doing for society?"
"With the unemployment rate among young Americans at thirty-eight percent, it's virtually—"
"There's plenty of work to be done for those who want to do it. I'm not signing anything." Click.
She looked up at the ceiling—a habitual gesture of frustration, although she realized she didn't feel frustrated. Perhaps her state of denial and shock was useful for phone banking. It overrode the anger and anxiety.
"—virtually impossible for children who can't find space in schools to avoid truancy. Did you know that fifteen percent of the incarcerated are now children as young as fourteen?"
"Well, what are you proposing to do about it?" This from a lady with a vaguely Asian accent.
"We'd like to put a referendum on the ballot for the next election that—"
Derisive laughter. "Nobody votes anymore. Don't waste your time." Click.
"—next election that would make it illegal to put children in prison for nonviolent offenses. It—"
A middle-aged woman. "Well, where else are you going to put them? If they can't be in school and can't find work? They can't just run around in the streets and make gangs. That's what they'll do. They'll make gangs. Orlando will be even more dangerous than it already is."
Helen counted to five. "Do you think a prison is the best place for a child to grow up?"
Click.
The anger was making a dent in the denial after all.
Helen abruptly stood. She didn't need to use the restroom, but she went there just to get out of her office for a moment.
She found herself looking at her face in the mirror. She thought about everything she could see on that face. Raising Mandy, David's depression, the divorce, David's death, the relentless struggle. Everything showed up in the cast of her lips, the furrow between her eyebrows, the shadows under her eyes, the pallor of her skin. She looked as old as she felt, and she felt old.
The person in the mirror was going to die.
Within forty-five days. No, forty-four now.
It couldn't be real.
She pulled out the locket and looked again at the black pill.
_Death will be painless. Please notify your next of kin._
She had no next of kin except Mandy. Her parents died in a bombing almost twelve years ago, and she had no aunts or uncles or siblings. No one but Mandy.
She couldn't fathom telling her daughter yet. Not yet.
She went back out to her desk and sat down. The next phone number waited for her.
Forty-four days.
She sat there while the minutes ticked away, her head tilted as if she could hear them flying past.
For once in her adult life, she couldn't seem to make herself do what needed to be done next.
_You may choose to take your medication as soon as is convenient to minimize potential discomfort._
Her officemate across the way, a young slip of a girl named Julie, must have noted her stillness, because she glanced her way. Then she quickly looked back down at her own work.
Helen understood. This was the culture now. There was no sense getting into other people's business. Human life just couldn't be that important when there was such an excess of it. People were here today and gone tomorrow. Julie was already devoting her life's work to doing good, and that was enough.
That was what Helen had told herself a lot of times too.
Helen looked at the next phone number waiting on the screen and realized she wasn't going to call it. She was done.
She got up and went to her boss's desk in the next room.
"Yeah, Helen?" Oliver swung his thick legs off his desk and straightened his heavy body with an effort. He gestured to the two chairs jammed in front of his desk. "Have a seat."
"Can we go out into the hall?" Helen asked in a low voice.
Oliver's eyes widened. No one asked that unless it was private and important. "Of course."
He stood up and took his can of soda from his desk, and they went out. The others pretended not to watch them go.
They went into the building's long hallway just outside their offices. They took up awkward positions nowhere in particular, but close to each other so they could lower their voices. The few working LED lights overhead cast ghostly light on their faces.
Helen's heart beat faster.
"Oliver."
What was she doing? Her hands were cold. She clasped them together.
"Go on. What is it?" Oliver took a swig of his soda.
"I..." She let out a breath. "I'm quitting. Right now. Today."
Oliver choked and coughed. He took an old-fashioned white cloth handkerchief from his pocket to dab at his mouth and shirt. "What? You're kidding me. You can't quit. You're the best one we've got. You pull the weight of three people."
"Thanks, Oliver. But my mind is made up."
"Why? Something going on? Some trouble?"
She couldn't read whether he was genuinely concerned or just wanted to talk her out of leaving.
She debated whether to tell him. She wasn't obligated to. But some part of her wanted to hear the words out loud.
She lowered her voice even further. "I'm dying... Of—a rare brain disease." Needlessly, she added, "It's terminal."
"Oh." Oliver looked disappointed. Perhaps it was the latter reason, then. He couldn't talk her out of dying. "I'm sorry to hear that."
She nodded.
"Well... " Oliver pursed his lips as his mind worked to incorporate this news. "I guess it turns out you've devoted your entire life to nonprofit work. You told me that once, right? How many years with us? Ten, eleven? And another ten or fifteen before us, right? Congratulations on that. A life well used." He smiled encouragingly.
Helen didn't say anything. She reminded herself that Oliver had gone through a lot before he started working at Justice for All, and he'd never re-learned how to connect emotionally with other people. At best, he could fake it.
"I'm sure you must be very proud of everything you've accomplished." Oliver patted her shoulder awkwardly.
"Yes, thank you," she said. But it was wooden. Everything she'd accomplished? How much was that, really?
"Do you want us to throw you a... I guess a going-away party?"
"No. Thanks. Look, I'm going today. I'm done today. I only have so many days left."
Forty-four days.
"All right. Well, we're sorry to lose you. And I mean that. With our budget as tight as it is, it'll be hard to replace you." His expression and voice grew funereal. "Do you want to tell the others?"
"No... I think I'll just go. You can tell them after I leave. I don't want to make it awkward for everyone."
"I understand. I do. All right, well... best of luck!" He tried to smile encouragingly again.
Helen turned to walk back to the office door. Just as she reached for the doorknob, Oliver said, "Oh, wait, I almost forgot." His face serious, he came to her. "Helen, I'm sorry, but you can't quit _today_. I mean, tonight's the big fundraiser in White Oaks. We get eight percent of our annual budget from this event, remember? We signed up for our table six months ago?"
Of course. It was one of their highest-dollar events. How could she have forgotten that she was supposed to work it tonight? Helen was not the sort of person who would forget such a thing.
Oliver continued, "I mean, I understand your time is limited, but you're so good at getting donations... couldn't you just...?"
Her shoulders sank. Leave it to Oliver to ask such a thing from a dying woman who'd already given this company eleven years of her life. But, why not. Why not, really. What else did she have to do tonight. "Of course. I'll be there."
He forced another smile. "Great! I knew I could count on you." Then his face sobered again. "Do be careful. There are rumors about an attack being planned. Although, on the up side, I guess you don't really have to worry about your safety anymore, right?" He tried to chuckle at his inane joke and patted her on the shoulder again. "Just make sure you get that eight percent."
She couldn't quite summon up an answering smile.
# 44 Days, 8 Hours
A few hours later, Helen stood in front of her draped table loaded with the Justice for All banners and pamphlets. Tables just like hers lined the immaculate wall of the vast circular ballroom, except for one alcove with a microphone. There was a balcony for the live band, which was just getting started. Twenty or more chandeliers shone down with subdued brilliance.
Men and women strode confidently into the elegant room, all of them dressed in formal attire. Jewelry glittered at the necks, wrists, and ears of both sexes.
At the center of the room, a massive ice sculpture depicted a man and a woman standing tall, their hands clasped and raised as if they'd won an Olympic medal. They'd won, all right, Helen thought bitterly—they'd been born into the right families.
In a circle around the statues stood tables of food and drink, with waitstaff behind them.
Helen glanced down and noticed with embarrassment that her cheap cotton slacks were pilled. And she remembered that she needed to trim her short haircut and re-cover the gray. She felt shabby and out of place. She looked around at her colleagues stationed at their respective tables—at least they looked just as bad.
Tonight's event was an annual fundraiser in the White Oaks neighborhood, a gated, upscale area in the northwest portion of the city. Every year, the theme changed.
A portly, black-haired fellow with a too-large smile waved from across the room, gesturing to the charity employees to join him in a small, out-of-the-way cluster.
As Helen approached, he was introducing himself as the master of ceremonies. His mouth still stretched into a fake smile, he said, "I need to give you all a quick briefing, because things are a little different this year." He explained that this year's theme was called the Net Worth Notion— _notion_ in the old-fashioned sense of a whimsical event—and that it was competitive. "This time each guest will choose just one organization to make a single donation to," he announced cheerfully. "Isn't that _fun_?"
At that news, the nonprofit employees looked at each other with resignation. It would be "fun" only for the guests. The employees knew it meant some of them would go back to work empty-handed this year. But they had no power over the rules of the game, so after the briefing, they got to work.
For her part, Helen came out from behind her table by a couple of feet and assertively engaged, with a well-practiced patter, any guest who made eye contact. Maybe Oliver didn't have enough feeling left in him to know what an obnoxious request it had been, but she still wanted to do a good job for Justice for All here at the end.
The knowledge of her impending death disappeared into the background for now. She knew the stages of grief, and that she was probably in the denial stage, where a defense mechanism held reality at bay. She also knew it wouldn't last.
Nearly an hour past the official start time, the MC stepped up to the mike in the alcove. "Ladies and gentlemen, may I have your attention, please."
The people gradually grew quiet.
"I know you're all wondering what the premise of tonight's event is. And we're ready to get started. The Net Worth Notion is this... As you already know, you were required to share your net worth with us in order to attend the event. And here's why. Someone will need to start us off by agreeing to have his net worth revealed to the entire audience."
An appreciate "oooooh" arose from the audience. Mock shock appeared on many faces.
"After that, only someone with a higher net worth can go next. So the game is to reveal your net worth if it is just slightly higher than the last fellow. By the end of the game, we will all know who has the highest net worth in the room—and what it is."
Another appreciative "oooooh" arose.
Helen grimaced. So the whole point of this event was to compare bank accounts in a childish game of "mine's bigger than yours." She raised her chin, trying not to let her disgust show on her face.
"And of course, we each have to give a little something to charity, since that's why we're all here, right, folks? I mean, it's not just for these amazing lobster rolls, am I right? Let's give our chef a round of applause!"
The audience chuckled appreciatively and clapped.
"But don't worry, we're not going to make the charity part too hard on you folks. Each person who reveals their net worth will give .001%—that's a thousandth of a percentage point, folks, so don't panic—you actually can't even miss it—given to the charitable organization of their choosing from the tables here."
Polite applause.
Helen ground her teeth. He was barely even pretending that this event was about fundraising. And ninety-nine percent of the country was drowning in poverty.
"Don't worry, with the power of compound interest, you'll have made that donation back by the time you leave, just by standing around and breathing."
More chuckles. A smattering of self-congratulatory applause. Helen groaned.
"Now, let's start the excitement! Who wants to kick off the bidding? Remember, we want to start low so that the game can go on a long time. It's more fun that way. Who thinks they're the poorest one here?"
There was laughter, and several gorgeously-dressed people rushed the little stage, pretending they were certain they were the poorest present, then all stepped away again, laughing.
Helen's stomach turned in revulsion. She looked over at her fellow nonprofit employees. They watched with the same blank, dead expression she felt on her own face.
It wasn't the guests' fault, she reminded herself. As much as it looked that way—and as much as she wanted to believe it—she was reconciled to the fact that these wealthy people suffered from a condition she called Emotional Self-Preservation Syndrome.
It was a heady blend of ignorance and entitlement passed down in family traditions designed to protect them from the cold, hard reality that they were destroying the world for everyone else. They weren't evil so much as they were weak. They didn't deserve punishment—they just needed a cure.
Years ago, she'd started calling them the "Entitled" as a sort of shorthand, to remind herself not to hate them. Sometimes it worked.
Finally one young man called out, "I'm barely out of college, I've gotta be the poorest one. No shame in youth, now!" He waved to the MC, and the man nodded.
"Brave boy, Mr. Alvarez, brave boy! Good man! Mr. Alvarez is ready to reveal his net worth of... drum roll, please!" The group enthusiastically drummed on their thighs with their free hands.
"Mr. Alvarez..." He gestured dramatically to the wall next to him and the sum appeared, projected from some apparatus out of sight: twenty-one million dollars.
The crowd laughed and applauded and cheered.
"Oh, Mr. Alvarez, you were right. This is embarrassing." The MC wiped his brow, pretending disappointment and disapproval. Everyone laughed. Someone threw a dinner roll at the MC. "What?" he protested. He pointed at Alvarez. "He's the one who's poor, not me!"
Mr. Alvarez laughed as color flushed his cheeks and his friends elbowed him teasingly.
"Mr. Alvarez, I see a lovely lady at your side. Miss, are you sure this is the one you want? Turns out he's not such a prize after all."
Her mouth opened wide in mirth. She grabbed Alvarez's arm with a "Take him from me if you dare" attitude. The crowd ate it up.
"Mr. Alvarez, where do you choose to donate your paltry two thousand one hundred dollars?"
The young man called out a name: Lantern Houses, for children with disabilities.
A good choice, but Helen shook her head. This kid had twenty-one million dollars and he was donating just over two thousand. It wasn't the amount that hurt, when Justice for All would gladly take such a donation. It was the discrepancy between the two numbers. A thousandth of a percent? Seriously?
After more announcements along the same lines, the MC stepped down from the stage to mingle and the event took on the quality of a silent auction. Every few minutes, the chime rang, followed by the appearance of a new name, net worth, and charity. Next came vigorous applause and ooohs and aaaaahs. More food and drink came out from the kitchen. The MC turned up the speakers for the live music. People drank too much and began to dance unsteadily.
Helen worked her table with all the fervor of the newly converted. She tried to tease and charm the older men, especially, who thought that a woman in her forties was still young and vibrant. Winning a donation would come down to simple persuasion.
She handed out flyers and talked about the issues Justice for All worked for, phrased to appeal to the ultrawealthy: keep young people off the streets, preserve the aesthetics of the city, decrease drug use, and reduce street crime and protests, all by helping the homeless and improving the economy. For this group of donors, she left off talk of economic and racial justice.
As she scanned the crowd for a new target to engage, a shrill sound cut through the buzz of conversation.
People stopped and looked around.
The sound came again—it was a woman's scream in the hallway.
# 44 Days, 6 Hours, 18 Minutes
The Entitled stood like wide-eyed statues, like the ice sculptures waiting to thaw. The hired help ran for the kitchens. Helen edged back in the direction of her table.
There was a bang and a crash from the hallway. The sound of shattered glass.
Gunfire.
Three figures ran into the room from the hallway, all of them wet from the rain outside. They brandished weapons—
From the ceiling, white spheres with blinking lights dropped down—
A collision of air—
Suddenly a fine, glittering net hung suspended in the air in front of the three figures, intercepting bullets.
One of the intruders yelled, "We will never be silenced! We will never be—" Then gunfire rattled behind the activists, felling them.
The glittering net subsided in slow motion toward the marble floor.
Helen's feet came unstuck and she ran to the kitchen. She stopped at the doorway and looked back.
Private security forces took in the room at a glance, then checked each body to confirm death. To make sure of it, a guard fired an extra bullet into the slighter figure—a teenager or a woman.
Helen hardly had time to flinch before it was over.
The Entitled turned back toward one another. "Can you believe these people?" "That was so scary." "Who do they think they are?" Someone cracked a joke Helen didn't catch, and there was laughter. The conversation picked back up. The band started playing again.
The guards unfolded body bags and lifted the bodies in, limbs rolling about awkwardly.
The hired help rejoined the party, because it was that or not get paid. The chime rang out. Mr. White. One hundred forty-eight million dollars in net worth. A thousandth of a percent to Star Programs to help the homeless.
Helen moved back to her table. Her mouth was dry. She gulped down half a glass of complimentary wine. Her muscles spasmed and she sat down quickly, clutching her glass.
Riots and attacks like this happened all the time. Anarchists and revolutionaries tried to bring attention to the plight of the disenfranchised, the ignored, the impoverished and imprisoned. Helen could never endorse these methods, but she sympathized with their mission.
For three to have broken through like this, she knew a hundred must have stormed the building and almost all of them been killed. The revolutionary forces were so poorly armed compared to the privatized police forces, they only got in when they had immense numbers on their side.
The guards dragged the bodies out of the ballroom, and the cleaning crew came in for the blood.
A latecomer to the party might have thought nothing had happened.
As Helen drove home through floodwaters in the pouring rain, her clothes damp from her dash to Old Blue, another kind of storm brewed inside her. Maybe the shock of her diagnosis was wearing off, or maybe it was how the revolutionaries had been swept away like their lives meant nothing.
Or, far more petty but still painful, the fact that she'd failed to get a donation for Justice for All. Their budget would be eight percent short now. She couldn't have said what she did wrong that the others did better, but it didn't matter anymore.
Grief wanted to come, but it was caught up in the fine, glittering net and suffocated in the body bags.
She swerved around and around the parking garage, up and up, to the familiar spot.
She slammed the car door too hard, the fury within her expressing itself.
The smell of mildew in the hallway, her door, the key in the lock. The three one-sided deadbolts, the purse on the kitchen table. The same tiny, dingy rooms that only grew dingier with every passing year.
Mandy on the couch, her clamshell balanced on her knees, her Earworm at her temple, not even looking up. Just the way David was those last half-dozen years, lost in his own Earworm, never swimming up out of his suffering to do anything but glower and snap.
Helen went to their room without a word.
She took off her shoes and sat on her bed. Chilled, she wrapped the coverlet around herself and her damp clothes.
Within her, she felt the storm gather.
Something would break. Something had to break.
She was going to die, and she couldn't die without something breaking. Because otherwise it would be okay for her to die. It would just be something that happened. Unremarked. Unremarkable.
Helen had known a couple dozen people who'd died. The world shrugged it off. There were so many people dying... what was the point in mourning them all? And were they really any worse off dead?
Assuming there was no God, assuming no pearly gates—and Helen had never quite figured out where she stood on that—then once you were dead, everything you'd ever experienced was over. Any pain you'd suffered was done. You couldn't remember it, couldn't think about it. If every trauma was forever lost along with your consciousness, did it even matter that it had happened?
Death was safe. It was the final refuge, and it was better to go there—to be finished with the struggle that was living.
Yet it didn't feel that way now. For her to die right now... it wasn't right. It wasn't the way it was supposed to be. She was supposed to see her daughter grow up.
Intensity built in her, and her breath came faster and harder, as if she would sob, but she didn't.
She remembered what Oliver had said: "Twenty-five years in charity work. You must be so proud of all you've accomplished."
Twenty-five years in charity work, and what had she accomplished? Nothing. Things were worse. More kids in prison. Debtors in prison—and permanently disenfranchised, because felons couldn't vote. More families starving. More people who didn't even bother to vote anymore, because the elections were corrupted. More environmental destruction by the wealthy who had decided it didn't bother them—and why should it, when they could just move to a part of the world the devastation hadn't reached? Or, even, a part of the solar system the devastation hadn't reached.
Her breath came faster and harder. A knot in her chest ached. She moved restlessly, trying to find a way to get comfortable and calm down. She breathed deeply, counting: inhale for ten and exhale for twenty... inhale for ten and exhale for twenty...
Calm down? _Why?_
She threw off the coverlet and paced the three steps between her bed and the door.
She'd spent her whole life in a battle to even things out, but still the ultra-wealthy lived in luxury and the poor struggled to survive. Maybe she had never really expected to succeed, but now hope was gone.
She threw open her bedroom door, not sure why or where she was going or what she would do if Mandy saw her in such a state. She took in the tiny room at a glance. The loveseat was empty. Mandy had gone out.
Helen went to the single, narrow, floor-to-ceiling window, threw back the shabby curtain, and stared out into the city night. Graying high-rises stared back at her. Each nuclear family in its own tiny box of poverty and struggle and isolation. Why did they have to live this way?
It was a system created by their parents and grandparents and great-grandparents, a legacy of other people's choices that imprisoned them. It had been done to them, forced on them. What could any of them hope to do about it now? Who could oppose the devastating inertia of generations of past choices?
She heard her breath coming audibly, almost a groan, almost a whimper. She was breaking down. And why shouldn't she? She, who was dying.
She collapsed to the ground and put her hands over her face. Was she going to cry now?
No.
She screamed instead.
Wordless anguish and anger.
Then words.
Words that came of their own volition. " _I'm not done!_ "
It hurt her throat.
She stopped, gasping. Then she stood and grabbed the heavy wood chair next to her and heaved it through the window, shattering the glass. She reveled in the destruction, so uncharacteristic of her. Stepping to the very edge of the window, she watched the chair spin on its way down to shatter in the drained and cracked swimming pool stories below. Rain and hot, humid air came in through the window.
Again the scream tore from her throat. "I'm not done yet!"
She didn't know what it meant, just that it was completely true, true down to her soul.
Dizziness overcame her, and she sank to her knees, her hands over her contorted face.
She heard a shout from behind her.
"Mom! What are you _doing_?"
Mandy.
Helen dropped her hands, her face suddenly calm. The emotion flew away, and all that mattered now was protecting her daughter from the truth.
She stood up and turned around.
"What did you do to the window?" Mandy shrieked. The dog, Jessie, peeked out from behind her, smart enough to stay back.
"Nothing," Helen said, and immediately felt stupid. "It's fine. I was upset, but I'm okay now."
Mandy stared as if Helen were a three-headed alien. "What is wrong with you? You would never do this."
"I know." Helen looked down. She felt as guilty and awkward as if someone had caught her doing vandalism. Which was . . . exactly what had happened. Expensive vandalism, that she couldn't afford to fix. "Look, don't worry about it. I'll handle it."
"Mom, seriously, this is _crazy_. Tell me what happened! Right now!"
A jolt of irritation pierced Helen, and she lashed out. "Back off, Mandy! This is not your concern!"
Mandy's head jerked back, anger and pain evident on her face. "Fine. Do whatever you want. I don't give a shit!" She turned to leave the room.
"Wait!" Helen reached for her daughter, then stopped, knowing the gesture would be rejected, as Mandy had rejected all of her affection since David had died. "I'm sorry—I'm so sorry."
She couldn't let things be like this again, like always, like ever. She would be leaving Mandy alone all too soon. Her face crumpled, but she forced it smooth. She struggled to find the right words. "I will—help you. I will make things better. I promise—it won't always be like this."
Warm rain and wind came at her back.
The look in Mandy's eyes said she thought her mother had lost her mind.
Helen wanted to say more. She wanted to tell her daughter that she loved her and that she would never abandon her on purpose. But it would be too much for Mandy to tolerate. Helen had already gone too far.
"I'm going in," Mandy declared. She retreated to the bedroom, Jessie with her.
Helen stared after her. She wanted more than anything to know how to do this right, because she only ever did it wrong. Wanted to know how to bridge the gap between them that had only widened for the six years since the divorce.
She sank slowly to the floor and looked out over the pool as the warm rain soaked her. And the sentiment that possessed her when she threw the chair bubbled back up.
"I'm not done yet," she whispered through gritted teeth.
God help her, it was true. She was going to _do_ something with this little shard of a life she had left. She was going to make a difference. She was going to _help_ —help Mandy, and help everyone else.
# 43 Days, 16 Hours
Helen woke up in the same way as the previous day. Dreams or waking hallucinations—she couldn't tell the difference. This time, gray-brown tree roots grabbed at her legs and pulled her down into dark loam. She fought them off and broke them, but they came back and came back again—rock- and dirt-encrusted tendrils dragging her down.
She came to consciousness and found herself tucked half under her bed. She crawled out and took a waking pill.
Were the dreams—or hallucinations—side effects of the sleeping pills, or was this just how her brain dealt with sleep now?
By the time the waking pill took hold, she remembered that she didn't need to go to work anymore, but also that she wasn't ready for Mandy to know about any of this.
It was a calculus every parent learned to do in the first few years. When to reveal information. How to say it. How to cause the least amount of pain, yet make their child understand what had to be understood.
Mandy was six years old when she realized that her parents would almost certainly die before she did. She burst into inconsolable sobs. "Then I wish I had never been born," she said. She cried about it for five days. Ever since then, every time Helen remembered, she felt her heart break all over again.
She got up and got dressed as her heart rate went up and up, then lay down on her bed, curled into the fetal position, trying to get her heart to settle down.
Jessie heard the movement and came out of Mandy's half of the room to curl up on Helen's bed. She petted him absently. In a few minutes, her heart rate slowed enough that she no longer felt her pulse in her hands and feet.
She regretted snapping at Mandy last night. And then the apology had been too little, too late, too crazy.
Helen hated it when she turned into David—rather, the David of later years, after his sadness finished shifting into anger, like a pile of dirt drifting from erosion. Helen had maintained the patience of a saint while David snapped and snarled, and then, when she finally sent David away, his anger expressed itself through her.
She developed a theory that whatever a person got used to became necessary. Her psyche understood having anger in her home and could not live without it now. But she hated it. Hated herself every time she did it.
Perhaps David had hated himself too.
She went out to Old Blue, cranked her up, and drove aimlessly through an overcast day. She found herself caught up in road construction. Hemmed in by orange cones, she started and stopped for almost an hour before she could get to an alternate route.
She had just gotten back up to normal street speed when, unexpectedly, her arms twitched and spasmed and nearly jerked her into oncoming traffic.
She pulled over at the next opportunity, an old gas station where paint peeled off the exterior walls and weeds grew out of the cracks in the cement and destitute men sat lethargically on plastic crates and watched her with too much interest. She sat in the car, eyeing the men and worriedly flexing her arm muscles until the spasms stopped.
_Please don't let me have to stop driving._
Even through the fear, she could still feel last night's resolution. The determination that set in after the anger ebbed. "I'm not done," she'd said, "I'm going to _help_ "—and she knew it was true. It filled her spirit like a steel core.
The remaining stages of grief were conspicuously absent. But she didn't have time for them anyway. She had only forty-three days left, and she couldn't spend them grieving. She had to do something before it was too late.
Something.
The proverbial high road had carried her along for years while her faith in her efforts slowly disintegrated. Now her faith was dust. She needed new tactics. It was time to get off the high road.
Last night, when she heaved that chair through the window... the anger was the second stage of grief, but it was more than that, too. She'd felt something shift. She was willing to break things now.
How far was she willing to go? she asked herself.
In harder days, she used to daydream about the things she wanted to do to the Entitled. She let her mind go to that dark place for a moment. Would she hurt or kill someone now, knowing that she was out of other options?
A corner of her mouth twitched. The still, small voice inside her said _Anything goes._
Then she remembered the revolutionaries at the Net Worth Notion, and her half-smile disappeared. No, not _anything_.
What else could she do? Hold people for ransom? She almost laughed. How could she hope to kidnap anyone? She'd had three self-defense classes twenty years ago. That was it. Otherwise, she was a white-collar forty-something mother. With heart palpitations and muscle spasms.
No, as much as she wanted to be a superhero in a cape, she couldn't fight like that. She could use a gun, though.
What was she after, really?
She felt restless again. She pulled out of the gas station's parking lot and drove aimlessly.
What was her goal?
It wasn't to hurt people, that much was certain. She firmly believed that no one deserved to be hurt. Even the Entitled.
She looked at the world outside her car window. In this neighborhood, the houses sagged, lopsided, decaying, with paint long ago worn off the gray wood. Cardboard was taped over broken windows. Yards were filled with clutter and broken cars, because poor people didn't throw anything away—they never knew when they might need it. Three or four families lived in each house.
Helen thought of the mansions in the White Oaks area last night. Ten thousand square feet for two or three people.
They didn't _need_ it, dammit. And for the hundredth time, she thought that someone needed to take away their excess and give it to those who really needed it. Like Robin Hood. Stealing from the rich and giving to the poor.
She smiled, remembering that when she was a little girl, she wanted to be either a cat burglar or a policewoman. As an adult, she was amused that crime and law enforcement had appealed to her equally.
Her gaze landed on the large sign outside a squat building. GUNS. AMMO. KNIVES.
Without even thinking about it, she turned into the pothole-pitted parking lot.
A few moments later, she looked with amazement at the assortment of weaponry available to her. Tasers, shotguns, bows and arrows, mace, Bowie knives, rifles, handguns, machine guns.
She shook her head, remembering the battle for better gun control over the past couple of decades. She was looking right at the evidence that they'd lost that fight.
The aisles were uneven—some more narrow than others. Old particle-board shelves, some with a distinct lean or an odd warp, held the merchandise, each labeled with a hand-scrawled price tag.
A salesman with short gray hair moseyed up. His stomach preceded him. He tucked a mini Snickers wrapper in his pocket and mumbled a "Mornin', ma'am" around the final bite.
"Good morning," Helen said politely, and turned back to the shelf of merchandise in front of her.
He took the hint. "Just let me know if you need anything." He went behind the counter and sat on a stool, looking relieved to take a load off.
She paced through the aisles with her head buzzing. As she studied the weapons, certainty and excitement grew in her.
Yes, it was time for this. Not to kill people—not even to hurt anyone if she could avoid it—but to do something dangerous. Something reckless.
If she who was dying couldn't risk it, who could? Who would ever be able to?
Her heart rattled in her chest.
Her diagnosis was a calling to do what others couldn't. Knowing she would die in forty-three days, there was no reason she couldn't throw her life away in pursuit of what mattered most.
She could do it. She could become like Robin Hood. Steal from the rich and give to the poor.
After all, she'd always wanted to be a hero.
She marveled at how coolly she considered this. She wasn't the same woman anymore. Something bold was in possession of her.
She thought again about her capabilities. How steady were her hands these days? Could she fire a gun? How well could she aim if a muscle spasm came on?
She called to the man. "Is there a place nearby where I can practice with a gun if I buy one?"
He raised one substantial arm and pointed. Then she saw the large, hand-printed sign tacked up in a dark corner. SHOOTING RANGE. An arrow in green marker directed her to the next room.
"Oh." Feeling foolish, she smiled at him and went to take a look.
The man followed her. "It's just me here," he said as he put another candy wrapper in his pocket. "You wantin' to try out a gun?"
"I can do that?" she asked in surprise.
A corner of his mouth turned up. "Sure. It's better. You can't return guns after you buy 'em. You gotta pay for the ammo, though."
A few minutes later, she wore heavy ear protectors and pointed an automatic .45-caliber weapon at a cut-out of a man that was well lit in an otherwise darkened hall. She felt ridiculous. The shop employee, also wearing ear protectors, chewed another bite of chocolate while he stared at her intently, making her self-conscious.
She pulled the trigger, and the weapon practically leapt out of her hands, jolting her hand and wrist and hurting the base of her thumb. She cursed.
The man lost his smirk. "Use both hands," he yelled close to her ear protectors. He showed her how to wrap her hands around the handle of the gun. She tried again. The kick of the gun hurt both wrists.
"You're closin' your eyes," the man yelled. "Don't do that. Keep your eyes on the target."
She sighed and tried again.
A few minutes later, the man waved her out of the booth. They both took off their ear protectors. "You're still flinching and shuttin' your eyes. You couldn't even hit the target. Now I'm not saying you couldn't get better with practice, but you're going to have to practice. You'll need to come back a coupla times a week until you get comfortable with that."
"I don't have that kind of time," she said as she massaged her wrists. "I need to be able to use this now."
He grimaced. "Right now, you'll fire once, if that, and you'll miss, and whoever you're shooting at will take this away from you and kill you with it. No offense, ma'am."
She set her jaw. "A smaller gun, maybe?"
"I'm guessing you want stopping power. A smaller caliber weapon isn't going to stop anybody. Especially not if he's pissed off or high or both."
"Well, maybe I just need to scare somebody."
He leaned back on the rental counter in the shooting range area. "Ma'am, no offense, but"—he gestured at her slender body—"you're not going to scare anybody unless you have a big damn gun in your hands, and even then, not if you're afraid to fire the gun you're holdin'."
She let out an angry sigh. "All right. Fine. Let me see some other options."
He led her back into the store proper. "Tasers are a good choice for self-defense," he said over his shoulder. "They don't kick like a big gun does. They're pocket-sized, and you don't have to aim all that well, and nobody wants to get hit with a Taser. They'll dissuade. And they have stopping power. You'll have time to run away."
"All right. Give me whichever one you recommend. What else do you have?"
He shot her a sideways glance as he walked toward the Tasers. "A Taser's what you want for self-defense or home protection, ma'am, you can trust me on that."
"I... might need more than that," she said carefully. "What about those?" She gestured at the knives.
"You have any trainin' in knife fightin'?"
She didn't answer.
He kept moving. He took a Taser off a peg and handed it to her. It had a gun-like build. "Fire it just like a gun. Wait till he gets within twenty feet. If he's inside a room with you, he's probably close enough."
"I want a knife, too," she said stubbornly. "A big one. And you know what, I do want a gun, a smaller one than the one you showed me. And mace. And..." Suddenly inspired, she tried to think of anything she'd ever seen used in an action movie. "What's the wire thing called they use to choke people with?"
His eyes went wide and his lips twitched in what she suspected was amusement. "A garrote? You want a garrote? Ma'am, how're you gonna get someone to hold still long enough to wrap it around his neck? Ask him nicely?"
Helen glared at him. "I'm a paying customer, aren't I?"
He rolled his eyes and headed toward the knives. "Yeah, I guess you are."
As he bagged her items a few minutes later, he suddenly chuckled. "Ma'am, I don't know who pissed you off, but I think I feel sorry for 'em."
A few minutes later, she sat in Old Blue, clutching the bag of weapons and laughing like a maniac. "I've lost my mind."
Still chuckling, she opened her e-paper to notepad format and added to her list of resources: a semi-automatic pistol, a Bowie knife, a garrote, a Taser, and a can of mace.
It wasn't wise to take the weapons home. Same thing for storing them in Old Blue. These days, cops searched cars at almost every routine stop. She also needed to be careful not to implicate Mandy somehow. Above all, she had to keep her daughter safe in all of this.
Come to think of it, _could_ she keep Mandy safe while doing this Robin Hood thing?
She leaned back in her seat. Nothing was more important than preserving Mandy's safety and freedom.
She suddenly realized that she had just spent a lot of money that she had yesterday willed to Mandy.
Her eyes went wide. She had forgotten that this money was for Mandy.
_How could you forget that?_
Another realization cut through her, and her chest squeezed shut. Rent hadn't been paid yet for May. It wasn't the tenth yet. She always paid it on the tenth.
She should never have forgotten something like that. Never.
She checked the receipt, which she had barely noticed before in her giddiness. She was horrified to see that she had spent just under six hundred dollars.
She only had eight hundred in the account to start with. Rent was four hundred. She couldn't even make rent now.
She had to go right back in and return the items.
Even before she had spent hundreds of dollars on weapons, how was she expecting to pay rent for June? She'd quit her job.
She couldn't _afford_ to quit her job. She couldn't get through forty-five days without income. That was three paychecks she had walked away from.
It was the prion disease. It had to be. Helen was extremely attentive to money and bills. She would never have squandered money thoughtlessly like this or failed to carefully budget their remaining dollars.
And she would never have quit her job on a whim.
_Dammit, Helen. What were you thinking?_
She would be compelled to return to work tomorrow, to do the same old futile, hopeless work for the rest of her dying days. And that hurt. But not as much as realizing that she had recklessly endangered her daughter's survival.
Helen stared out of her car window, agonized by what she was up against. Every time she did her budget or otherwise considered her finances, a cold, hard, dead feeling clamped over her chest. There was something merciless about numbers. They took hopes and dreams and chopped off their limbs to make them fit into the right boxes.
Her hopes and dreams were up against truly merciless numbers right now. Leaving Mandy a few thousand dollars was almost, although not quite, meaningless. If Mandy moved straight into a sleep locker, she could make it maybe five months on that amount of money.
But Mandy was still in job training for retail management and wouldn't graduate for another year. Before, in Mandy's first year of school, Helen tried making her work at the same time, and Mandy started flunking her classes. She couldn't handle it.
That meant Mandy would have to drop out of job training and go get a job early. And with minimum wage eliminated a decade ago, she would only be qualified for what they now called a room-and-board wage. That destroyed her chances of making it through the next few years without debt—and, eventually, debtor's prison.
The situation was dire. No more dire than for millions of other families across the country, of course. Helen had worked with so many people in this situation in the course of her years in nonprofit work.
Which brought her back around to what she had just done. Bought almost six hundred dollars in weapons. Decided to become a sort of Robin Hood.
That decision wasn't just about making things better or fulfilling a long-held dream. It was about ensuring her daughter's survival.
She _had_ to do this. She had to resort to theft. Mandy wouldn't make it otherwise.
But she didn't need all these weapons. She had gotten carried away. It was irresponsible.
She got back out of her car and went toward the store through a drizzle that had just begun. She was raising her arm to push the door open when she saw the paper sign taped to the door:
ALL SALES FINAL.
NO RETURNS. NO REFUNDS. NO EXCEPTIONS.
_You've got to be kidding me._
She stared at the door in dismay, rereading the message again and again as if it might change.
A mosquito buzzed near her neck, and she slapped it away.
She found her way back to Old Blue in a daze.
It was done. There was nothing she could do about it now. She just had to make it work.
First step—find a safe place to keep the weapons.
# 43 Days, 14 Hours
A couple of hours later, Helen surveyed her new hideout with resignation.
The sleep locker was a short, featureless, windowless box, like a storage unit, with centralized climate control. It was only a few feet wide and filled mostly by a bed. High shelves filled the wall over the bed, opposite the door. On the near wall, there was a shower stall area separated by a plastic privacy curtain that also enclosed the toilet and a sink with a stainless steel mirror.
The mattress was impervious rubber on the outside, and the furniture and walls were plastic-coated metal. When she moved out, staff would hose the locker down and the water would run into the drain in the middle of the tile floor. But they never fixed the damage that accumulated over the years—the dented seat of the toilet, the chipped and broken tiles, the profanities deeply etched into every surface. And the smell of mildew that lingered from the mattress.
Although it resembled a prison cell, a sleep locker was still better than living on the street, for those who could afford it. If nothing else, a tenant could lock the door against intruders.
Back when Mandy was a newborn and Helen couldn't work again yet and David had lost his job and there was nothing to be done about it, they lived in a locker like this for five weeks.
With no crib and no space for one, Mandy had to sleep with them. But Helen didn't trust David to sleep in the same bed with an infant. The mattress was narrow, and David slept hard after his nightly four beers. She had heard about babies suffocated by parents who drank. She made David sleep on the tile floor, on folded blankets placed over the drain, and he barely spoke to her the entire five weeks.
This time, Helen told herself, she was renting a locker by choice. This time it was in the hopes of making things better.
She stashed her new assortment of weapons on the rusting shelves over the bed and headed back out.
To become a sort of Robin Hood, to help Mandy and everyone else: this was her new calling—and a necessity so far as Mandy was concerned. But where did she begin?
She approached becoming a thief the way she approached any unfamiliar territory: reading about it.
If she went to a library, she could use a public computer anonymously.
She choose a library distant from her apartment, but she soon regretted it. Construction closed two lanes of I4 while they expanded it from ten to fourteen lanes, and a slow, dreary rain held steady. Traffic crawled. A trip that should have taken about thirty minutes took an hour and fifteen. That kind of delay had always frustrated her, and now her life was ticking away.
She found parking two blocks away, stepped out into the oppressive humidity and ever-present mosquitoes, and weaved her way through the constant crowd of pedestrians.
As she walked, she watched and listened carefully, as always, for signs of violence—it was a habit to look out for anyone behaving erratically, to listen for the pops of gunfire. You never knew whether someone would choose this moment to make a statement. And she watched for sudden movements. Anyone running might have just spotted a crowd of rioters or someone with a weapon or a bomb.
Helen went into the library. She remembered when they were havens for the homeless. No longer. Signs declared, "No Standing. No Sitting on the Floor. No Sleeping at Projcoms. No Sitting in Conference Room Without a Reservation."
The minimalist strip-center locations didn't offer physical books anymore, just a room with computers, one conference room, and a stack of free disposable e-paper books at the front desk. The gadgets were flimsier and more cheaply made than a regular e-paper. You could load about twenty or thirty books before the pixels wore out and the e-paper became unreadable.
She found a projcom available. These were virtual or projected computers—the only physical apparatus was a cigarette-box-sized console that sat in front of the user and projected the holographs for the screen and the keyboard. Typing on a nonexistent keyboard took some getting used to.
Her legs weakened unexpectedly as she went to sit down, and she caught herself by the edge of the table and sat down carefully and waited to see if it would get worse.
After a moment, she decided nothing else was going to happen. Just that moment of weakness to remind her what was happening to her body and brain. She sighed.
She opened the "anonymous" option on the browser, gestured to shrink the screen a bit, turned on the privacy filter, then glanced both ways to make sure no one was paying attention. Her hands trembled with nervousness, which was absurd. No one could suspect her of a thing—not yet. And a handful of other users had their screens set up the same way. She didn't stand out.
She typed in "where to fence stolen goods," then glanced around nervously again. She couldn't help it.
The first result was a law enforcement article where she learned that fences operated behind legitimate business fronts—often pawnbrokers and jewelers.
Helen reviewed the list of items that thieves most often selected for fencing. She dismissed drugs and shuddered at human trafficking and child pornography. She frowned at endangered species: trying to steal an animal or bird sounded like a messy, complicated business, besides which, she loved animals.
That left jewelry, art, and firearms.
A crack of thunder startled her, and she glanced up as the lights flickered. A patter overhead became a roar. She looked out of the window to see sheets of rain coming down. It was a proper Florida thunderstorm now—one of the few things that hadn't changed about her home state since her childhood.
After another cautious look around, she went back to her research. Apparently, thieves preferred to sell their goods within thirty minutes. They could find a fence any time of day or night. Also, she could expect to get forty to fifty percent of the value of the item.
She went to write down these notes in her e-paper's notepad, but as she glanced around nervously again, she suddenly realized it was a terrible idea for her to keep notes of her criminal endeavors. She deleted everything she'd noted already.
Next, she searched on "how to find reputable fences," but when all the results were fence-building companies, she gave up.
She stopped, stretched, and thought this through a little further as she gazed out of the large windows. The palm trees lining the street were dancing in the heavy wind. She noted with some alarm that water was higher than usual. Cars were creeping by.
She checked the weather from the projcom. A red band at the top of the page announced a major storm and multiple flood warnings for Orange County.
Great. She needed to leave soon, before it got worse.
What else did she need to learn before leaving for the day?
What would she do with the money once she'd completed a theft and a sale? How could she safely get the money to Mandy—and then any excess to charity?
In the movies, they always used offshore bank accounts. She opened another browser tab. Two web pages later, she'd learned that they were useless to her—she'd have to provide her real name and address and statements for her current bank account—even explain and substantiate the source of the funds she deposited.
She let out a sigh. She had no idea what she was doing. What were the odds that she could pull any of this off? And her fading memory was haunting her. What critically important thing was she going to forget next, at just the wrong moment?
Her shoulders sank, and she stared at the projected screen in front of her. It all seemed a little too real all of a sudden.
She shut down the screen, picked up her wet umbrella, and went out into the torrential rain.
# 42 Days
The next morning, Thursday, the waking process was either easier or Helen was just getting used to it. She dreaded the awful dreams or hallucinations when she went to bed, but not more than she would have dreaded a completely sleepless night. It was worth it. She was grateful the long nightmare of sleeplessness was over for now.
The weather forecast was relatively clear—rain, of course, but no flooding expected.
Helen got up at her usual time and went to her old office. She needed her old job back, first of all. And secondly, a definite plan—however improbable—was starting to form out of the chaos in her mind. She would need access to her work clamshell.
Justice for All kept records on all the wealthy for phone banking day and to organize charity events. She would use those records to do a little social engineering and find targets for her Robin Hood plan.
As she got out of her car and went in, she wondered whether Oliver had told her old coworkers yet. What would they say when they saw her again?
As she stepped in the office door, the others stared at her too long before focusing on their work too intently, and her heart sank.
Clearly, Oliver had told them. Yet they didn't say anything to her. They didn't even look at her a second time.
As she weaved between the desks crammed into the too-small room, she found herself glaring at their downturned faces. _I'm dying here, people,_ she wanted to say to them. And yet, what could they possibly say? What would she have said if she were in their shoes?
She rapped on Oliver's desk to get his attention.
"Helen! I'm surprised to see you here." He pulled himself to his feet.
"Hi, Oliver. Yeah, sorry about that. Didn't mean to interrupt everyone's day."
That last bit came out a little more bitterly than she'd intended.
"No problem, no problem at all. Uh—what... what can I do for you?"
"Can we talk outside?"
For the second time this week, Oliver picked up his can of soda and they went out and took up positions in the dimly lit hallway.
Oliver looked concerned. "So what can I do for you?" he asked again.
"I'm afraid I'm here to throw myself on your mercy. I realized that my decision on Tuesday was a little hasty. I may not have much time left, but it's enough time that I can't afford to go without a paycheck."
Oliver wasn't nodding agreeably, the way she'd envisioned it. His brow folded in what looked like worry.
"So I have to ask you for my old job back... if that's okay?"
Oliver took his cloth handkerchief from his pocket and mopped his brow. "Um..."
This wasn't good. She felt her face freeze up.
He delayed by taking a sip of his soda, then put on a patently fake smile. "Helen, you've been a great employee, and you're a wonderful person."
Definitely not good.
"But you didn't get the donation from the charity event Tuesday night... that was eight percent of our office's budget, remember? I don't mean to blame you—I'm just saying, the money didn't come in, right? I was in big trouble with national. I had to do some fancy footwork. And the shortfall was covered by... well..." He trailed off, looking suddenly unable to complete the sentence.
"By my salary for the rest of the year," Helen finished woodenly.
"The way Brian is talking... look, I'll call for you, and I'll fight for you, and I'm willing to take the heat for trying, but I will be amazed if I can get him to approve it. How many weeks do you need?"
"About six weeks," Helen said, her mouth dry.
Oliver took another sip of soda. His forehead was turning pink. "Okay, I'll see what I can do. Look, I'll call him right now." He tried to smile, but Helen could see that she was causing him tremendous stress.
As he pulled out his e-paper, Helen reached out a hand. "No, stop. Oliver... don't worry about it. Okay? I don't want you to get in trouble."
Twenty minutes later, she stumbled out to Old Blue in a miserable daze.
Oliver had insisted on calling, and Helen was forced to listen to one side of the most awkward and demoralizing conversation about herself she had ever been subjected to. There was nothing quite like listening to someone argue that since an outstanding employee of eleven years had a terminal illness, six weeks of salary might not be too great a sum—and lose the argument.
None of it was Oliver's fault. She knew that. Her old boss apologized a dozen times—at one point, she thought he would break into tears. Brian, the national manager, was a world-class jerk, and that wasn't the fault of Oliver or the local office or even Justice for All in general. But it didn't stop it from hurting.
She aimed then for the second part of her plan, asking to get some "personal files" from her clamshell—only to learn that it was already wiped clean and the password changed.
Once she made it to her car, she took deep, forceful breaths in a determined attempt to hold back the tears. She refused to cry.
She had less than two hundred dollars in her account to fund her plans and to last Mandy and herself a month and a half.
But she would find a way to make this work. She would help Mandy. And everyone else. Somehow.
After leaving her old office, Helen drove aimlessly through heavy traffic, rain, and endless construction and tried to think of another plan. Breaking into the office to steal someone else's computer would have been pointless. She didn't know anyone's passwords and she was no hacker—nor did she know any hackers.
There were no paper records to steal anymore. The long-awaited paperless office had taken hold with the advent of the Earworm and the e-paper.
Asking Oliver for a list of names and addresses for the Entitled felt awkward at best. She couldn't think of an adequate pretext.
Her original idea was to attend fundraisers and parties and house concerts and sneak out with something to sell. It still seemed like a reasonable idea, but she didn't know how to find the right people to call for information about upcoming events. She also didn't know she was going to look the part of a wealthy socialite on a budget of two hundred dollars.
Utterly at a standstill, Helen decided to go home to check on the window replacement. Yesterday morning, after taping some plastic over the space where the window had been, she'd called the apartment manager, and they were supposed to come in to replace it this afternoon. They'd agreed to add the cost of the repair to next month's rent. Which she might not be able to pay anyway.
She got back to the condo feeling defeated. Money, the lack of money, the likely futility of trying to get any more money, the broken window, her failing memory, the fact that she was dying, the crowds and road construction and traffic, the fact that she was dying... the fact that she was dying.
Mandy sat in her usual spot on the loveseat, her clamshell balanced on her knees and her Earworm over her ear, typing away. She looked up, dark eyebrows raised, when Helen entered. Her look was antagonistic, and her tone matched. "What are you doing here?"
Helen held back a sigh. She had forgotten Mandy would be at home. Her diseased brain was failing her. And why did Mandy always have to talk to her like this?
"I do live here, don't I?" She put down her purse on the two-person dining table.
"It's Thursday. You have work."
"I took off. Errands and chores."
Mandy stared, unblinking. "You never take off."
"Well, I did. Deal with it." Helen got out the coffee and measured it into the coffeepot.
"Well, if you're running errands, why are you _home_?" Mandy asked.
Helen clicked the coffee filter into place a little too hard. "Because I have things I need to do," she snapped. _Stop it. Stop yelling._ _Don't let her get to you._
"Like what?" Mandy snapped in return.
Helen glowered at the black liquid percolating into the pitcher. She didn't want to remind Mandy about the broken window incident.
"You know what? Nothing. Never mind. Forget it." She picked up her purse and opened the door to leave.
She came face-to-face with a maintenance guy with his hand raised to knock.
"Oh, hello," he said.
Great. Perfect.
She opened the door and stood aside.
Two men carried in the long pane of glass.
The fight disrupted, Mandy got up and moved to her bedroom doorway wearing a deer-in-the-headlights expression. "What is that for? Ohhhh... That." Her hostility faded into the background, and she gave her mother an anxious look.
Helen folded her arms and leaned against the fridge. She didn't want to talk about it. About any of it.
They watched the workers in silence. The distance between mother and daughter fairly hummed with unspoken thoughts.
Helen knew what it would take to get her daughter to voice those thoughts when she was in this sort of mood. A longer gaze, a slight smile. Perhaps an idle, neutral comment.
But Helen was holding too much back. She hadn't done the necessary calculus yet—hadn't figured out how to tell her daughter the right things in the right way at the right time.
She remembered again how Mandy cried when she found out her parents would die first, how she said, "Then I wish I had never been born." How Mandy reacted when her father died three years ago.
Helen kept her gaze averted, her arms folded, blinking back tears.
A few minutes later, Mandy ducked into their room and closed the door.
# 41 Days
Early in the afternoon, after taking a long, quiet break at a sunny coffee shop downtown to steady her nerves again, Helen finally figured out the next two steps in her Robin Hood scheme.
In retrospect, it was obvious. But she couldn't count on her brain to recognize the obvious anymore.
First, she opened her e-paper to its largest size and looked up the Orlando society magazines, the types that always had photographs of fancy events so that the Entitled could gossip about one another. She had a hunch that the Net Worth Notion would be covered.
She was right.
The article had a complete list of those who'd donated, although it stopped short of listing the net worths in question. It was enough. It was easy to look up their phone numbers after that—she didn't understand why any of the Entitled felt obligated to be accessible by phone, but many did.
She _loved_ the idea of targeting those who had attended the Net Worth Notion.
If possible—if she could make any of this work—she would give the stolen money to the very same charities they gave a tiny percentage to that night.
Now that would be karma in action.
_After_ she got enough money to give Mandy.
She assembled the list of phone numbers and readied her e-paper.
A ripple of discomfort ran down into her stomach and made it tingle. Her heart palpitations redoubled. They shook her chest.
Nerves. _It's just the usual_ , she told herself, and she put herself into phone banking mode: unfailingly polite, undaunted by failure, working by the numbers.
She dialed the first person on the list.
"Good afternoon, this is Helen Dawson with Justice for All. Mr. Brechtsen is a past donor and I wanted to invite him to a gala we're having on Saturday. I apologize for the short notice, but—"
"I'm sorry, ma'am, but he's not available on Saturday night." A woman's voice.
"Oh, he's not available? Well, that's what we get for the short notice, I guess! Some other gala, I imagine?"
"Sorry I couldn't help. Goodbye." Click.
Damn. Nothing useful there. Helen marked the name off the list and called another person.
"—a gala we're having Saturday night. I apologize for the short notice, but we had some cancellations and—"
The man on the other end of the line could not have sounded more haughty. "Mrs. Kirkpatrick doesn't accept B-list invitations. Good evening." He hung up.
Next call.
"—I apologize for the short notice, but—"
"I'm sorry, Mr. and Mrs. Soon are hosting an event of their own Saturday night."
Helen's heart jumped. "Oh, another gala, I imagine? Or is it a private party? Not something that Justice for All could possibly attend, I suppose?" She inadvertently added a nervous chuckle and then mentally kicked herself for it.
"It's their fiftieth wedding anniversary. I'm afraid it's not the sort of thing a charity would be invited to. Thank you for your call." Click.
Helen grinned triumphantly. A fiftieth wedding anniversary celebration was the perfect party to crash.
That completed the first step of the day.
Next, some online paperwork.
To fund her criminal activities, she would have to withdraw the contents of her retirement account—what would have been her daughter's limited inheritance.
"Sorry, kid. I'll make it up to you," she murmured to herself as she hunted through the appropriate web pages for the right forms.
She discovered that withdrawing early required a sixty-day waiting period—and she had forty-one days left.
Further reading revealed that she could bypass the waiting period by supplying an end-of-life exception form specifying her anticipated date of death—a morbid process, to say the least.
And it would still take two weeks to get the payout.
_And_ she would lose half the total to early withdrawal penalties and taxes.
But she had no choice. She needed resources. She couldn't pull this off while looking as poor as she really was.
That request initiated online, she turned to something even more odious: she went to another website to take out a payday loan against the anticipated early withdrawal. The disclosure statement revealed that the annual interest rate was 730%, and that didn't include the fees, which were $30 per $100 borrowed, or the nonrefundable application fee, which was $50.
She did the math: if she took a loan for $1,000, she would pay $350 in fees and then $300 in interest by the time she received the retirement fund check.
It hurt so much to pay $650 just to have $1,000 cash in hand today. She fought herself for ten minutes to tap the button to confirm the loan. But she had no choice. She had less than $150 in her checking account and 41 days left to live. She couldn't afford to wait.
That done, and her account up to $1,150, she turned to the next errand: buying a new dress. One tiny silver lining: It might be more enjoyable to shop as a rich person than to shove her way through the overcrowded racks of worn-out thrift-store clothing, all of it carrying the discomfiting smell of other people's detergent.
She went out to Old Blue and kicked the door so the handle would work and headed toward the Apple Orchard upscale shopping district to the southeast. She'd often eyed with envy an upscale consignment shop just outside the edge of the shopping district. With designer items at discounted prices, it was a place where someone like Helen might plausibly splurge on a dress sometimes.
As she turned onto a major street that would take her out of downtown, a string of ambulances rushed by with sirens wailing. A moment later, three fire engines followed. Helen watched them go with a frown. Something major was happening. Maybe another riot.
She turned up the news, but the newscaster was midway through a leisurely story about an overall increase in the activity of the El Lobo Feroz street gang.
Traffic slowed to a crawl, then to a stop. Rain started. Helen pulled herself up straight to look ahead, but she saw nothing but a sea of cars. She slid back in her seat and rubbed her hands over her face. She didn't have time for this. The days and hours were ticking by.
"Breaking news," the newscaster said, and Helen turned up the volume. "A series of explosions rocked the federal courthouse in downtown Orlando about ten minutes ago. We are being told that court was out of session today for early summer holidays, and no fatalities have so far been reported."
Bombing the courthouse? Helen raised an eyebrow. The terrorists—or activists, depending on who you asked—were aiming higher than usual.
"This just in: the hacker Cobalt has claimed responsibility for the attack in conjunction with the terrorist group the Boom Boys. Cobalt states, and this quotation has been modified for the air, '[Expletive] judges and [expletive] need to wake the [expletive] up. They're the corruption that is the [expletive] system.'"
_Pretty sure they're already awake to that fact,_ Helen mused.
She let out a sigh and put her car into Park. She wouldn't be going anywhere for a while.
# 40 Days, 14 Hours
Helen spent hours in her dismal sleep-locker hideout preparing for her first robbery.
She used latex to cover her fingerprints and practiced with a variety of disguises: a long black wig and a few dabs of adhesive to create the awkwardly stretched look of someone with too much plastic surgery; a short red wig and too much makeup; glasses, age makeup, and a wig with gray hair in a bun.
She decided to go with a much older look every time. It was easier to disguise herself as older than younger. Plus, when the robberies were connected and they looked at all of her pictures together, they would be more likely to think she actually was older.
Hopefully, it would be a long time before that happened. If she acted natural and drew no attention to herself, she couldn't see any reason for them to pick her out of any security footage of the parties. Also, she planned to pick up items in their bedrooms and studies, and she thought it was unlikely that they would have cameras there.
She had also decided to rent different types of cars from different rental agencies all over the city, and tape fake license plates on them. She would choose targets in different areas. Everything to keep law enforcement from connecting any dots. Everything to keep attention away from where she and Mandy lived. She would even target a couple of wealthy houses nearer her own area, just to throw them off.
As she put on her low, age-appropriate heels, her hands and legs trembled. Yet she didn't feel particularly nervous yet. She frowned at her hands and held them out in front of her to inspect. She didn't see much movement. The tremors were more felt than seen.
She frowned. A new symptom. It brought a lance of fear into her chest.
It would only get worse from here.
Helen finished gathering her things and set out around seven that evening.
Riding in the rented self-driving Tesla was an unexpected luxury. She couldn't hear the engine or any outside noises, and the black leather interior was as comfortable as her own loveseat. She'd thought that a self-driving car would make her nervous, but her Tesla, with the personality of an older gentleman and named Christopher, was a ridiculously cautious driver. It made her impatient instead.
Helen directed Christopher to pass by the Soons' mansion a few times to gauge when the arrivals were thickest. When she saw six cars in line at the drop-off point, she pulled in behind them.
She checked her makeup in the mirror. It startled her again to see herself aged twenty-five years. She wouldn't live long enough to look like this, and from an entirely shallow point of view, she couldn't decide if that was a good thing or a bad thing. She didn't like looking old.
The one thing she'd preserved was her perfectly shaped eyebrows—her one point of vanity.
She tried the trick she'd learned in her thirties: glancing quickly at herself as if the form in the mirror were a stranger. It let her see herself more accurately and more favorably. The method served her this time as well: seen through a stranger's eyes, the elder Helen had a certain elegance.
She straightened her new diamond necklace, which was her one genuine piece of jewelry for the night. She hoped it was flashy enough to distract the guests from the faux diamonds on her hands and wrists. Older women wore just as many diamonds as the young, but they didn't bare their arms, so as her car pulled forward, she put her age-appropriate dress jacket on over her designer dress.
Her turn came, and her car opened its door for her. She clutched her knock-off designer bag—risky, but she couldn't afford a real bag, even at the consignment store—and got out into the humidity and heat. It was time to put on the charm, despite her twisting stomach and choked breath.
Her one advantage was having attended hundreds of similar events as a nonprofit employee. She had watched the wealthy mingle and converse enough that she knew how to join in.
As her car set off to park itself, she went up the walkway and closed in on a young man who was unaccompanied—someone young enough to be her disguised self's son, with a cowlick in the front of his sandy hair. She smiled broadly as she took his elbow. "My goodness, I haven't seen you in forever," she said. "How is your work going?"
"Oh," he said, embarrassed. He did a double-take as he strove to remember her name, then gave up. "It's going well. We've acquired another firm this year, so we've just about doubled in size."
"Wonderful," Helen purred. "Has your title changed?"
"No, still vice president," he said with a smile.
By now they'd come to the gatekeeper, an unsmiling man in a black suit holding the guest list. The one Helen had to get past or this was over before it had begun.
Helen projected her voice toward the gatekeeper as she spoke with maternal affection to the young man whose elbow she held. "Son, I am so proud of you and how far you've come." She squeezed his arm affectionately.
The gatekeeper nodded at Helen's compatriot. "Mr. English." He nodded politely at Helen. He must be assuming that she was either Mr. English's mother or his guest.
Mr. English glanced at Helen in bemusement, then surely realized that he was about to let slip that he still didn't know who she was. He quickly smoothed over the moment. "Let's get you inside and out of this sticky air, shall we?"
"Oh, yes, let's," Helen said with a grateful smile. "My hair will fall." She patted at it as she'd seen older women do before.
She was glad she had him to cling to as she entered the mansion. Otherwise, her trembling hands might have been visible. Worse yet, she might have stumbled in her new heels.
Mr. English graciously escorted her to the drinks table for a cup of coffee and guided her to a chair, then excused himself. That was fine with Helen.
She surveyed the space. This room alone could have occupied an entire floor of Helen's building. It was impeccably furnished in genuine wood and marble, with hand-painted frescoes. A part of her ached to live surrounded by such beauty.
She was pleased to see that there were a significant number of older people present, which helped her blend in. She had expected that, first because it was a fiftieth wedding anniversary, but also because she had noticed in the past that high-class events always had a range of ages present.
Helen let about an hour pass as she mingled and chatted with various people. She'd picked the right time to come in, as over a hundred other people were already present—enough for her to remain thoroughly anonymous. When she approached a group, she always greeted someone as if she already knew them, which guaranteed her a certain degree of acceptance.
The only awkward moment was when her arm muscles unexpectedly weakened and she dropped her wine glass. But servants rushed to clean it up and the moment was forgotten.
The conversation surprised her with its charm. These were people who'd taken pains to learn the art of small talk and the old-fashioned sort of social networking—that done in person. And because they accepted her immediately as one of them, based on first impressions alone, they were welcoming and polite.
However, the conversation was as light and unfulfilling as the puff pastries. One lady talked about her recent six weeks in Prague and the shocking lack of fresh produce there; another lady bragged that her six-year-old grandson had already been promised entrance to an elite high school; a couple had undertaken the renovation of a French chateau, and vintage marble recovered from undersea Venice was essential, of _course_...
She didn't think she could take much more of this Entitled chatter. Her face hurt from all the fake smiling.
She finished a bacon-wrapped shrimp and looked around again. It was time to pretend to look for the powder room.
She picked a doorway at random and went down a hallway. The sudden quiet and smaller scale of the hallways and rooms conveyed that this was a private area.
Two hallways in, the third door on the left opened onto an expansive and deeply carpeted private library. Now this was an ostentatious display of wealth. There was no reason to own paper books other than to show them off. She stepped in cautiously, but she knew she was alone, if only because the lights were off. The only light came in through the gaps in the heavy curtains on the windows opposite the door.
On that side of the room, between the windows, a small display case held several antique revolvers. They looked small enough to slip into her bag. Perfect.
As she crossed the room deeper into the darkness, her heart pounded so loudly she feared she wouldn't be able to hear anyone who might come in after her. But probably it would be all right if the nice old lady were a little deaf.
She stepped across the thick carpet toward the cases as she glanced up and around.
She caught a glimpse of movement in the heights of the farthest corner. Her heart staggered. There was something in the shadows. A small red light flicked on and something electronic slipped out of the corner.
The lights in the library came on, and Helen blinked with her hand up to shield her eyes.
It was a drone, oval in shape and painted white.
"We're so sorry to be rude." The voice that emitted from the drone was feminine. "The security system is programmed to permit only the homeowners and immediate family members in this area."
Helen had not seen this coming at all. She had never seen a drone like this. Utterly taken aback, she hesitated. What were the odds that this thing was recording her, that some security guy was watching her on video right now?
"I'm so sorry," she said, staying in character. "I just really wanted to take a look at those revolvers over there. They're so lovely. Do you think Mr. and Mrs. Soon would mind?"
"We can't permit that," the drone replied, its tone more firm. "Would you mind returning to the public areas of the house now, please?"
Helen decided to press her luck.
"I'll be happy to go in just a minute. I just want to take a closer look." She took several confident steps toward the revolvers.
"Our next action will be unpleasant, and we hate to be ungracious hosts. Please step out of the room now."
Now Helen was far too curious to give up. She took three more steps while she watched the drone over her shoulder.
The color black washed over the drone, and with a series of clicks, three turrets rattled out from the casing. The unmistakable sound of guns preparing to fire struck Helen like physical blows.
The drone spoke again, its voice now loud, masculine, and threatening. "By the laws of this state, homeowners possess the right to use lethal force to protect their property. This right extends to drone and robot agents of the estate. We will not warn you again. Retreat or be shot."
Helen's heart stopped for a moment and then thudded back into action. "Well, that is just rude!" She managed to sound haughty. As she shuffled out, she even managed to shake her fist at it. "I will have a word with the Soons, I promise you that!" she called back, while also hoping to high hell that it wouldn't shoot her for back-talk.
She wondered whether it would escort her back to the party, maybe announce her transgression to the Soons. As she hurried back down the hallway, it emerged from the library behind her and kept pace with her as she moved. Terror kept her moving quickly.
Back at the first hallway, it stopped and watched her go.
As soon as she got out of its sight, she stopped and leaned against a wall, drawing in a deep, shuddering breath.
It occurred to her that if she looked low-class, she might have been shot on sight.
She'd not had time to recover before a tall, imposing man wearing a black suit and a stoic expression appeared from around the corner. "Ma'am, I apologize for the sentry. The Soons have a number of valuable items to protect. I'm sure you understand."
"Of course." Helen tried to laugh, but then she decided to give in to her true feelings. "I'm sorry, but that was awful. I'm terrified. I think I might faint."
He quickly stepped forward and let Helen take his firmly muscled arm. "Let me take you to a sitting room. Ms...."
"English," she filled in. She immediately wished she hadn't, but she wasn't able to think of anything else to say fast enough.
"Ms. English." He led her down the hallway. "Do you need medical assistance?"
"No, that's all right," she said, though still genuinely breathless. "I think some hot tea might be all I need."
"I'll see to it, ma'am."
He took her into a private sitting room and escorted her to an overstuffed divan. "I'll be right back."
She put out a hand to stop him. "There won't be any more of those awful—sentry things, will there?"
"No, ma'am, I'll tell them to stand down from these rooms. I don't think you're the sort of threat that the Soons are worried about." He startled her by casting her a wink and a grin on his way out.
She wanted to laugh. If only he knew about the small arsenal of weapons in her knockoff designer purse.
Even as she caught her breath, she looked around for something to steal. Her gaze lit on three possibilities: a glass vase on the tea table, a bronze sculpture on a pedestal in the corner, and a china plate on a stand on the low bookshelf on the back wall. All were beautiful items, and all were small enough to put into her purse. But what were they worth?
If only she had ever been able to afford objets d'art... or bothered to take an interest in things she would never be able to have for herself... She had no idea what was valuable enough to make this trip worthwhile.
There was a knock on the open door and she nearly jumped out of her skin for the third time in ten minutes as another unfamiliar figure presented itself.
"Excuse me, ma'am. You requested tea?"
"Yes, please."
The man who entered was not one of the caterers from the party. Most likely a butler or cook from the private staff. He placed a tray with a full tea service on the tea table. "Do you need anything else, ma'am?"
"No, thank you."
"Very well, ma'am." He stepped away as soundlessly as he had appeared.
She waited a moment to be sure he was gone, then got up with a racing heart and picked up the glass vase. It was the only thing that wasn't on some sort of stand which would make its absence conspicuous. It was a small, heavy vase, pretty, with flowers in the design. She wrapped it in the cloth napkin from the tea tray, and she tucked the bundle into her purse.
As she left the room, she remembered the drone's slick transformation from helpful to menacing, and she shuddered. It was a metaphor for this high-class world—lovely and pleasant just as long as you belonged... vicious to everyone and everything else.
She'd had enough of pretending she belonged. It made her sick to see the kind of luxury these lucky few enjoyed. And why? What made them deserve it while others didn't?
Helen took a few wrong turns and finally found a restroom to use.
Afterward, she had just caught the sound of the party again and was heading that way when Mr. Suit stepped out of a hallway directly in front of her. This time, he wore a scowl that struck a bolt of anxiety into Helen's chest. She knew instantly that he knew.
"I apologize, ma'am, but an item has gone missing from the sitting room. I'll need to check your bag."
It was stupid for Helen to argue with him, but she couldn't think of another option. Her voice went high in outrage and panic. "That's ridiculous. Are you accusing me of something?"
"I just need to check your bag, ma'am. I'm very sorry." His tone was firm.
Helen's mouth opened, but nothing came out. Her mind cast about for something else to try but came up blank.
Then something angry surged in her. It took over her helpless mind and body and forced out the words, "Fine. I'll just show you everything that's in here, shall I?" That anger turned to a nearby table, emptied her purse, picked up the Taser from the pile of items, and shot the man.
He dropped instantly. He writhed and bucked on the floor while veins popped out on his neck.
Helen dropped the Taser, shoved everything else into her bag, snatched it up, and backed away from the convulsing security guard.
_Jesus Christ._
This was not supposed to happen.
# 40 Days, 5 Hours
Helen only knew of one exit, and it was back through the party.
She ran to the doors of the ballroom, then casually strolled through them. It took such a long time to cross the vast space toward the exit. She knew a Taser didn't keep a person down for more than maybe a minute. She knew already that she didn't have time to wait for her car to come around.
In her desperate need to think of a solution _right now_ , her mind creaked to a halt. The angry part of her that had taken over a moment ago disappeared, like a hit and run. _Thanks a lot,_ she fumed.
She needed to run for her very life, but she was forced to walk at a slow, deliberate pace while the enemy surely closed in behind her, and she dared not even look behind her.
Her gaze lit upon someone's purse and shawl laying unattended on a table, and suddenly she knew what to do.
It took only seconds to swap purses, dumping the contents of her own purse into the other one. She wrapped the shawl around her hair and down over her dress as she walked on, not looking back. At another table, someone had left eyeglasses while they went to dance. She had them on before she pushed open the door. As she went through, she added a hobble to her walk and stooped her shoulders.
It was raining, of course, but there was cover over the pull-through. There was no one else in the car pick-up line. But two people came up behind her almost immediately to wait for their cars. She hoped they would help block sight lines to her.
Moments later, she heard the house's front door open behind her and heard the quick, hard footsteps of security. She pretended deafness and didn't look around, but her back and shoulders hunched in fear. She waited to feel someone's heavy hand on her shoulder.
The footsteps passed near, but not too near.
Her car pulled up and she opened the door.
The security went past her.
As her car pulled away, she looked back to see them looking out into the grounds, which were well-lit but obscured by torrents of rain.
Of course. Waiting in the line for a car was so foolish that they hadn't imagined a criminal would ever do it. They'd expected her to be running away or to have a getaway car or to have left through some other exit. They'd expected her to have a _plan_.
Helen had the car take her to the only twenty-four-hour pawn shop in this part of the city, all the while obsessively checking her rear-view mirror. She expected to see police lights strobing through the rain behind her at any moment. Twenty minutes later, when the car parked at her destination, her heart still pounded, but she felt like she'd gotten away clean. At least for now.
They knew what she looked like, though... or at least what she looked like with twenty-five extra years added to her face. She could only hope that the disguise would be enough, and if it wasn't, that they wouldn't have her face in any police databases.
Or if they did, she reminded herself, it didn't really matter. She was going to die in forty days.
She got out of the car and ducked her head to the rain as she hurried to the door of the pawn shop as best she could with wobbly legs.
Under the harsh lights over the large parking lot, shadows cast down on rough-looking men squatting against the barred glass windows of the free-standing pawn shop. They looked at her with too much interest. She should have changed before coming to the shop—the dress and heels and diamond necklace meant she was worth more dead than alive.
But remembering that she was going to die and that she was already throwing away what was left of her life... there was something empowering about that. What could anyone do to her now? What was there left to lose that she hadn't just risked?
She stared at the men as she walked past them, and something in her gaze must have unsettled them. They averted their eyes.
She pushed the glass door open, causing a bell to jingle. She glanced around for security cameras. There were none visible, but she doubted a shop with valuable items would skimp on security. There was nothing she could do about it either way.
Harsh fluorescents shone down on a practical, no-frills space, but the merchandise was dusted and carefully arranged along wide, well-swept aisles.
She went to the counter, where a slender Greek man smoked an e-cig under a neon sign that read SELL HERE. He had heavy-lidded eyes, a prominent nose, and a five-o'clock shadow.
While he took a puff, he looked her over with one raised eyebrow. No doubt he rarely saw anyone come in dressed like this.
"You have somewhere I can change?" she asked.
He jerked a thumb toward the back corner.
Not much of a talker, this guy.
With another nervous glance around, she took her bag of clothes to the worn but clean restroom she found back there and locked the door.
A feeling of mourning overtook her as she slipped off the exquisite dress and shoes. They were so comfortable, so perfectly fitted, so obviously quality—luxuries in every sense of the word. She'd always secretly hoped that the rich were just paying a "stupid tax" on designer items, but apparently they really did have a better life in even the smallest of ways.
She debated whether to wash off the disguise and peel the latex off her fingertips, then decided she might as well. Up until this very moment, she hadn't really known how she would handle this. But as the minutes unfolded, she realized that her best bet was to establish a relationship with this pawn shop.
She came back out of the restroom and went to the front. The Greek man had gone into the back of the store, and she tapped the bell on the counter to bring him out again.
She glanced around the store nervously and out through the windows again. Still no cops.
She opened the stranger's purse that she'd stolen, and she took out the crystal vase. She unwrapped it and set it on the counter for the man to inspect. It occurred to her then to take a look at the purse itself. A Louis Vuitton. She pushed it across the counter as well.
She also lay down the clothing and diamond necklace. If law enforcement circulated pictures, as she assumed they would, she wouldn't be able to wear the same items again.
When she finally glanced up, she saw the Greek man giving her a completely different sort of look than he had before. It was openly appreciative without being impolite. It made her duck her head. She wasn't used to catching anyone's notice.
He put his e-cig in his shirt pocket and handled the items.
As he did, Helen surprised herself by looking him up and down in return. He had a lean, muscular body under a button-up, short-sleeve shirt. She caught herself and returned her attention to his face.
As he looked over Helen's ill-gotten goods, his eyebrows raised and then lowered in puzzlement. His well-shaped lips twitched.
He looked hard at Helen, then craned his neck to look at the door to the bathroom in the back corner. The door stood open.
He stared at her again.
"Was that you that came in and went in there a minute ago? Wearing these?" he asked.
Helen smiled. Her disguise was a success. "Yep," she said. She waited anxiously to see what conclusions he might draw.
He just shook his head. Perhaps he was deciding that he'd just misjudged her age before. He looked again at the items on his counter, then back at her. "You have certificates of authenticity?"
Damn. Hopefully that wasn't important. She put certainty into her tone. "No. But they're all real."
Well, there was a chance that the Vuitton wasn't real, that she wasn't the only one who relied on knock-offs to perfect an image.
He frowned and turned the vase over and over in his hands, then held it up to the light. He tapped the Earworm clipped over his ear to turn it on and studied the vase through its forward-facing camera.
Helen watched with bated breath. Had she chosen well? Or was this some cheap gift the Soons had been given and felt obligated to display?
"You have to fill out a form with your name and ID," he said.
"Fine. I can write something down," she said. She tried to stare him down but felt her breath tight in her chest. The moment was precarious, and she faked confidence for all she was worth.
The guy gave her a form, and Helen wrote in an invented name and other information.
He picked up the form and frowned at it. "ID?"
"Gosh, so sorry, don't have it on me." She leaned forward and stared aggressively at the man.
At last, he caught on. His eyes widened and he grinned admiringly. "So that wasn't my eyes playing tricks on me before."
Helen gave him an innocent little shrug.
"I will be right back," he said.
Helen leaned against the counter, her heart pounding, but now in relief. He didn't seem at all inclined to turn her in. Perhaps he could be an ally.
She looked out into the pawn shop. It felt empty and desolate without its owner.
Jewelry, computers, televisions, bicycles, art. All these items that people had hoped to come back for when they had better luck. Better luck hadn't come. What were the odds that her day would pay off? Or would this be the end of the line?
The men outside were staring at her. Again, when they saw her looking, they averted their eyes.
This time, Helen felt mixed emotions. From in here, they just looked sad and desperate. Trying to shelter from the rain late on a Saturday night with nowhere to go and no prospects. These were the people that she had spent her life trying to help. Yes, they were rough, but only because being smooth took money and confidence they didn't have. And some of them probably had wives and kids at home to take care of.
After the divorce, Helen found herself unemployed twice, and she remembered the agony of knowing that when the paltry savings ran out, there would be no food for her or Mandy—and it would be her fault. It was up to her and only her. She knew the suffocating feeling, the weight on the stomach, that came when you tried as hard as you could and still failed every single day.
The man came back to the counter. "Twenty-one thousand," he said. "You want a cash card for that?"
Dumbstruck, she stared at him. Had she heard that right? "Twenty-one—thousand? Dollars?"
"Yeah. That was a vintage Tiffany paperweight vase and an authentic Vuitton. Nice clothes and diamonds too. You want a cash card, right?"
She could hardly speak. "I guess?" She realized she had the same deer-in-the-headlights look she detested in Mandy, and she shook it off. "Like a gift card?"
He looked amused. "Yeah, like a gift card. Lets you transfer money without being tracked."
"Okay, yeah. It's... faked, I guess?"
He shrugged noncommittally.
"Sorry," she said.
Now she wanted to bite her tongue, both for the question and for the reflexive apology. If she didn't look like a complete amateur before, she certainly did now. "Yes, that would be good. Thanks."
A few minutes later, he came back out of the back with a plastic card. "Good luck." He handed it to her, then extended his hand for a firm, warm handshake. "Name's Egemon." He pronounced it EGG-eh-men. "You got any more stuff like that, bring it in. I will give you a good price." He even cracked a smile, although it looked a bit out of practice.
She murmured something in acknowledgment and went outside. She clutched the cash card with trembling hands.
Twenty-one thousand dollars. It took her half a year to make that much money.
She was halfway past the rough men squatted against the dirty walls when she noticed them again. They were watching her with gazes too jaded to hope.
She turned toward the men with the cash card in her hand. "You got e-papers?" she asked. She figured they would have those, but not the more expensive Earworms.
They stood quickly, unfolded their devices, and handed them over to her. Their hungry gazes fixed on her hands, trying to see the amount she entered in as she scanned her card. She transferred one thousand dollars to each of them and handed back their e-papers wordlessly. She felt as if she were in church, as if it were a sacrament.
They looked at their e-papers, taking in the number of zeros. One of them said, "Thanks," his tone doubtful. Two just stared, wordless and with closed expressions. One quickly tucked his e-paper into his boot. They all walked away in different directions.
No smiles. No hugs of joy.
"Good luck," she called after them.
No response.
She hurried through the rain to her car and drove away with tears swimming in her eyes. Not because she had hoped her generosity would be more appreciated, but because she now saw too clearly that they were too broken and defeated for hope or joy. And because, she realized belatedly, even a thousand dollars would not change their lives in any meaningful way.
Futility pounded at her heart, demanding entrance.
Maybe she should have given them more. Maybe if she had given them five thousand each... With that amount, they could buy interview suits and work clothes and haircuts, get job training, rent a real hotel room instead of a sleep locker, get public transportation to and from a job until the first paycheck came in.
She turned the car around, her heart racing again. But as she turned into the parking lot, she saw that the men were already gone, swallowed up by the pouring rain.
Helen found a quiet street where she could peel the fake plates off the rental Tesla, then did a late-night rental return and reclaimed poor Old Blue from a few blocks down the street, slapping away mosquitoes as she did so.
Still no cops. Perhaps she had gotten away with it, for now.
She drove to her favorite restaurant. She would enjoy a rare luxury tonight—sushi—an indulgence she'd experienced only twice before.
She entered the opulent, darkly lit restaurant and was escorted to a seat near a politely babbling water fountain.
She ordered dinner and then contemplated the Japanese calligraphy on the walls as she waited. Her mind went to the feat she'd just accomplished. Now safely held within these four familiar walls, she felt as if that adventure had happened in a previous life.
Twenty-one thousand dollars taken from the rich, and four thousand dollars—so far—given directly to the poor. That meant something, didn't it?
_I really am Robin Hood now,_ she said to herself, and she smiled.
Halfway through her wasabi shumai and spicy tuna rolls, her arm spasmed and knocked her glass of plum wine off the table to shatter on the floor.
As the waiter came and cleaned everything up, she put her head in her hands.
Reality came creeping up again, as hard as she tried to fight it off.
She was still dying.
And every success took her closer to eventual failure. Especially successes like the one she'd just had, where she had drawn so much attention to herself. They would find the security footage and zoom in on her.
The bigger the impact she made, the harder they would try to find her.
The faster she made enough money to keep Mandy safe, the less time she would have to spend with her daughter before the police caught up.
She was making no effort to manage her DNA traces, other than her fingerprints, because she didn't know how to, and also because she was confident police databases had no information on her. But she was no expert in disguise, either. Soon enough, they would figure out what she really looked like, and then they could simply look her up. Photo recognition software was nearly perfect, and like everyone else, she had plenty of photos online.
Above all, she had to take her black pill before they took her into custody.
A chill ran through her as she imagined living out the full duration of her disease while locked up. Returning to that sleepless state, that slow-blinking half-alive stasis between restlessness and exhaustion, then deteriorating into hallucinations, dementia, and coma, helpless in a prison cell...
She forced herself to stop thinking about it. No, she told herself, she would take the pill before she was captured. She would make certain of it.
# 30 Days
Over the next ten days, Helen completed three lucrative robberies, managing to avoid sentries and evade notice. She delivered all her spoils to Egemon and his ever-present e-cig, and he started looking happy to see her. She paid back the payday loan, paid the rent for May, put aside cash to replace her paychecks, and reinvested money in disguises and clothing and rental cars—and she was still up by forty thousand dollars. And if all had gone well after that, she might have been able to relax.
But she'd hidden the cash cards somewhere.
And now she couldn't find them.
She had no memory of putting them away, except that she thought she would have hidden them somewhere in the apartment.
She'd already gone through the other rooms, and now she tore through everything under her bathroom sink, hyperventilating but trying hard to use thoroughness and care in her search.
_Goddamn disease. Goddamn brain. You cannot do this to me._
The apartment was only three hundred square feet. It was cluttered with the accumulated odds and ends of two people living in the same place for years on end: paper towels bought in bulk and Mandy's old art from elementary school and Christmas decorations and extra sheets. There were a ton of plastic shelves and drawers.
She searched everywhere—every shelf and drawer and cubby, under her mattress, between her pairs of slacks, inside the band-aid box in the first-aid kit, in the toilet tank, inside her extra pair of shoes—everywhere.
She went out to Old Blue, just in case, but that was easy to search and she didn't find them.
There was a chance they were in the new sleep locker she'd just rented with a cash card so it wouldn't be traceable to her this time. She would go there next, even though she didn't think she'd done that.
Helen threw herself to the floor in the living room, shaking and feeling lost and desperate and panicked and her chest aching from fighting against hyperventilating.
After all this effort and time, for her brain to betray her like this...
Was there any way Mandy had found them and taken them?
No... surely not. Mandy was irresponsible, but not a thief. Surely she was too mature to do something like that.
Anyway, Helen was confident she had hidden them too well for anyone to find.
Too well for Helen herself to find.
The old sleep locker. _Oh God_ , she gasped _._ She might have hidden the cards in the old sleep locker.
She scrambled for her e-paper and called. Moments later, the voice on the other end confirmed what she already suspected—it was rented out to someone else now. No, they hadn't found anything in the unit.
Of course they would say that. Even if they had found them, they would lie.
_God help me._ All that wasted time, when time was the absolute last thing she could afford to waste, when she had only thirty days left.
She had a single cash card in her wallet. It had less than fifteen hundred dollars on it.
Why hadn't she written down where she put the cards, when she knew perfectly well her memory was lapsing?
Helen pulled up at the twenty-four-hour pawn shop, and as she got out of Old Blue, she suddenly felt that her legs were heavy as stones. She stared down at them in confusion. They looked normal—thinner than usual, if anything.
She tried some experimental steps and realized that something was happening to her leg muscles or perhaps her nervous system. Her gait was stiff and clumsy. It was a new symptom—a new reminder that she was running out of time.
As she went in and the bell on the door greeted her with a cheerful jingle, she watched Egemon, who was stationed at his usual spot behind the counter, to see whether he would notice her slow, stiff walk.
"Hi," she said.
Egemon took a slow puff on his e-cig and breathed out vapor as he studied her. "What is the matter?"
She froze. "What do you mean?"
"You look unhappy."
"Oh." He hadn't noticed her gait. "Nothing." She joined him at the counter. "Although I don't suppose there's any way to recover a lost or stolen cash card?"
He shook his head. "The legitimate ones, yes, if they are registered. The kind I give out, no. They are not connected to anything, no bank, just a fake profile. Afraid you are screwed if you lost one. Did you lose one?" He looked concerned.
"Oh, it's not important," Helen said with a shrug. An all-out lie. She let out a sigh. "So I have another question..." She glanced around to make sure they were alone in the store before she confessed her latest plan. "You know those sentries that rich people use to guard their houses... is that all they use? Or are there additional security systems?"
He shrugged. "Depends. Some people have the usual old-fashioned style that goes off if a door or window is opened. Their Earworms turn the systems off automatically when they arrive at home and turn them on when they are out. Other people rely on the sentries completely. They patrol the entire house when no one is home and at night."
"Ah." No wonder she'd seen only a few sentries in each house. She'd wondered why some rooms were left unguarded. Turns out the owners of the sentries just put them on standby during parties.
Helen looked down, unsure what to ask next.
Egemon took pity on her. "What you need is a jammer. It interferes with the wireless signals sent between the Earworms and the sentries or security systems. There are a few different types. Some work better on some systems than others. Then there are those that work directly on the sentries. You are interested?"
Helen nodded.
He took another puff and let it out. "Wait here."
He went into the back of the store and returned moments later with three quarter-sized devices. "You can clip these to your shirt, your waistband, whatever you like. This one"—he held up the black one—"is for Allied brand. This one"—he held up the silver one—"is for Hercules brand security system. And this one"—he held up a silver one with a red edge—"is for the Hercules sentries."
He put them down on the counter between them. "There is nothing for the other ones right now. It is a cat and mouse game between the security companies and the hackers. The hackers exploit a weakness, the security companies patch it up. So, if you come across anything else, for the moment, you are shit out of luck."
Helen held up the small devices. "What do they do?"
He leaned across the counter and rested on his elbows. "The ones for the security systems—just get within about fifty yards of the house, push the center of the jammer, wait for the blinking red light to turn to green. Then the system should be down. But look out for the owners returning home. If the light never turns green, it's not going to work at that house—the system must be newer or upgraded. Try a different house.
"The jammers for the sentries are a bit different. When you push the button, they emit some signal that the sentry does not know, and this causes the sentry to reboot itself. It only works on one sentry at a time. It will get you twenty or thirty seconds. But usually another sentry will come to back up the first one and check to make sure everything is all right, so then you are in double trouble."
Helen grimaced. "Twenty or thirty seconds... that's not long."
"It will not let you stay in a room you should not be in, only pass through. You can use it multiple times. But remember, it only works on one sentry at a time."
Helen let out a sigh. She felt intimidated and overwhelmed. She needed to do more research.
"What about the gates that people live behind? Are those on the same system?"
Egemon shook his head. "I'll give you a tip. Look for lawn guys. They'll already have the gates open. Dress like a salesperson or something. Make sure the yard guys are out back, then ring the doorbell. If they answer, try to sell them something dumb nobody wants. Hardcover Bibles or something. If nobody is home, use the jammer. In you go."
"Oh," she said. So much for needing to do research. Egemon was a more valuable resource than she had known. "What if I go in at night?"
He pursed his lips. "Not such a good idea, unless you know they are on vacation. Then you just have to climb over the wall. Find a spot where trees and bushes hide you going over."
"Okay. Stands to reason. Thanks." She held up the jammers, reluctant to ask the next question. "How much are they?"
"Tell you what, I'll give you a 'repeat customer' discount," Egemon said with a friendly wink and a rare smile. "Twenty percent off makes it..." He typed some numbers onto his e-paper. "Six hundred and forty."
She tried hard to smile instead of wince. Of course, Egemon didn't know that all her other cash cards were lost. "I appreciate the discount," she said. "What's the balance left on the card?"
"Eight hundred and thirty-two," he read off his screen.
She was down to almost the same amount as when she quit her job fourteen days and four thefts ago. That hurt. She had to make these burglaries work—and quickly.
"Thank you," she said. He was being kind to her, and more helpful than he had to be. She gave him a warm smile.
As she went out and the bell on the door jingled farewell, he said, "Good luck. Hope it goes well."
As preoccupied as she was, it made her smile. It was good to have a friendly face around.
A few hours later, Helen huddled in her bed, dry now but still shivering from her misadventure. Her forearms ached from warding off hailstones.
She'd used another series of social engineering phone calls to identify an Entitled family that was on vacation. Mr. Slater, a guest at the Net Worth Notion, was skiing in the Berkshires.
She'd spent more of her precious remaining resources to rent a cheap car, because Old Blue, with its spray-painted dents and duct-taped quarter glass, was too recognizable if anyone noticed it parked on the road.
There was no fence at the front of the stately home, and Egemon's jammer worked as hoped—the house used a Hercules system that proved susceptible. She just walked right in. But her luck ended when she entered the house.
She found a projcom in sleep mode and woke it in hopes of finding bank accounts she could access, but it was password protected. There was nothing she could do. She wished again that she were a hacker, or knew one.
She went to a second-floor bedroom and was gathering expensive-looking items into a pillowcase when she heard the unmistakable sound of the front door being unlocked and opened.
Aghast, she peeked around the corner near the landing and saw the family dropping their suitcases on the marble floor downstairs. She ducked back into the bedroom.
Among the family's chatter, Helen heard a male voice say, "Huh... the security system isn't armed. That's weird. I'm sure I set it when we left. Sarah, did you turn it off?"
Helen crossed the bedroom toward the window.
The woman's response was too quiet to be understood.
"Well, I'm turning it back on," Mr. Slater said.
Helen froze in the act of reaching for the window sash. With the system re-armed, it would go off when she opened the window.
But Egemon hadn't said anything about not using the jammer a second time. She activated it again, waited for the green light, and threw open the window.
She set down the heavy, awkward pillowcase full of goods just inside the window where she could grab it once she had a good perch. She climbed out onto the gable of the roof and turned back for her haul.
Then a teenage girl walked into the room and flipped on the light. She hadn't yet set down her suitcase when her gaze landed on Helen and she froze.
Ear-piercing yells of _MOM! DAD!_ shrilled the room, and Helen scrambled backward, forced to abandon her prizes on the floor.
She made her way down from the roof in a panicked rush, lucky not to seriously hurt herself, and then it began to hail.
She'd warded off the hailstones with her arms as she frantically ran through the yard, down the road, and back to the rental car.
Now she stared dully at the dingy walls of her room, still shivering from leftover fear and adrenaline, futility hard at work on her soul.
For some reason, her mind fixated on the contrast between her apartment and the house she'd just been in. Like all the homes she had robbed so far, the huge mansion held dozens of art objects and other items each valued at hundreds to thousands of dollars. And these weren't even the items that gave these people their staggering net worth. These were just their _decorations_.
She stared around her own bedroom. The lamp she got from a thrift store for eight dollars, its mate next to Mandy's bed on the other side of the curtain. The rented bed that started breaking years ago, propped up with a cement block under the cracked rail. The artwork on the wall—prints purchased cheaply and framed with some Christmas money David gave her a decade ago. Buckling plastic shelves holding her thrift-store accessories and shoes and other personal odds and ends.
She did the best she could to keep everything tidy, clean, and organized. She'd taken pains to paint the room a nice shade that went well with the homemade curtains and the secondhand bedspread. It wasn't awful, all told.
But everything in this room... it might add up to two hundred dollars, at most, and this was all she had accumulated in her entire lifetime.
# 29 Days, 12 Hours
Early afternoon on Wednesday, dressed in work clothes and carrying a briefcase so she could pass herself off as a saleswoman, Helen used Egemon's tip about the yard guys to carry out her first successful burglary.
Even better, she found an unlocked projcom and a series of passwords written down on a sheet of paper in the top drawer of the desk. A few anxious moments later, Mr. Alvarez donated a hundred thousand dollars to Lantern Houses—the same charity he'd given a thousandth of a percent to at the Net Worth Notion.
Justice was served.
Thirty minutes later, she pulled up at the pawn shop and proudly took a cardboard box of her ill-gotten goods out of her trunk.
She couldn't wait to tell Egemon.
Box in hand, she closed the trunk of the car and started toward the pawn shop, then stopped in surprise.
There was police tape across the door, and the lights were off. The door was boarded up.
She stared for a moment, baffled.
Wasn't everything normal just yesterday afternoon? She checked her memory, unsure now of everything. Yes, she was fairly certain that just yesterday he'd given her the 'repeat customer' discount on the jammers and wished her luck.
What had happened?
With the glass door boarded up, stood to reason it was broken—and yes, sunlight glinted off stray shards of glass on the ground.
Had there been a hold-up?
Slowly, trying to puzzle through the situation, she put the box back in her car and closed the trunk. She went to the building and looked through the glass windows. With afternoon sunlight pouring through, she could see that the shelves of merchandise were still full and neatly organized.
She looked around the parking lot. One fellow in grimy pants and a torn shirt ambled toward her, barefoot, his eyes fixed intently on her. She felt herself grimace. He might have information. More likely, talking to him would be a mistake.
She closed some of the distance, keeping an eye on how far she got from her car, and pointed at the pawn shop. "You know what happened here?"
He flashed a too-friendly smile and called back something unintelligible.
She grimaced again and let him get a bit closer. She pointed more emphatically at the pawn shop. "What happened here? Did you see it?"
"Hey, honey," he said. He gave her body a completely unsubtle once-over. "You want my digits?"
"Christ Jesus," she mumbled to herself. To him, she called out, "Never mind. No thanks." She started toward her car.
The guy kept coming, but faster. He put a hand on his crotch. "Hey, babe, you want some of this?"
She didn't answer. She just wanted out of there. He was within a few yards by the time she opened her car door to get in.
His tone and posture shifted. He yelled, "Hey! I'm talkin' to you! You think you're too good for this?" He gestured elaborately at his body. "I'll fucking kill you, bitch!"
She got in, slammed the door, and pulled out of the parking lot. In the rearview mirror, she saw him gesture obscenely after her. As she drove away, she let out a deep breath.
Men like that always freaked her out—the expression of entitlement to her body, the sudden escalation, the implicit or explicit threat of violence. She told herself that everything was okay now. Her foolish mistake was behind her, only a memory. She was lucky.
She drove aimlessly for a few minutes, letting her heart rate settle down and the adrenaline ease off, until her thoughts returned to Egemon and the empty pawn shop.
She felt strangely bereft.
_A thief disposes of stolen goods within thirty minutes._ She remembered that from her initial research a couple of weeks ago. A sensible person would find another fence. Trouble had hit at the pawn shop, and a sensible person stayed away from trouble.
But how hard might it be to find another fence? She suddenly realized how lucky she'd been to find Egemon. Lightning might not strike again so readily. What if it took her days to find another fence she could trust? Only so many days remained.
And what if it was her fault? What if someone had followed her after a theft and taken it out on Egemon?
The police tape across the door. Since the police were involved, they would know what had happened.
A few minutes later, she boldly told a policeman on her e-paper that she'd arrived at her husband Egemon's shop, 24 Pawn, at 17955 Preston and found it barricaded with police tape—and her husband wouldn't answer his e-paper. "What the hell is going on?" she demanded.
After some checking, he asked, "Egemon Agnes?"
"Yes, of course Egemon Agnes," she said. Thanks to her bluff, she now knew Egemon's last name.
"Well, you'd better come bail him out then. He's at the 12th precinct."
_Dammit._
"What the hell is he in jail for?"
"You can ask him that yourself, lady. It's not my place."
She parked a block away from the police station and sat there in anxiety and despair and asked herself whether she was really going to do this.
It seemed enormously stupid to go into a police station an hour or so after committing burglary.
But Egemon could be in jail now because of her. And even if she wasn't to blame, she couldn't let him rot in jail. He had helped her—more than he needed to. He'd given her valuable advice. He was her only safe haven in this new criminal enterprise.
He was the only friendly face in her entire life right now, for Heaven's sake.
She tried to think through what might happen if she went in there. Staying out of police databases was paramount, or her own plans would come to a crashing end. To bail someone out, did she have to leave any biometrics of her own?
She did a quick search on her e-paper. As far as she could tell, it would be a simple matter of making the payment. Egemon would have forms to sign, but she shouldn't have to sign anything other than the receipt.
They would know she'd bailed him out. Would that get her placed on any watch lists?
Would someone recognize her, even though she wasn't in disguise?
It was too risky. Surely someone else could help him. Maybe he had family in the area. Maybe someone would show up in the morning to get him out. Or any minute, even. She probably didn't have enough money to bail him out anyway. She was down to just over eight hundred dollars.
She made up her mind to drive away, to trust that he could handle it.
Maybe the next time she went to the pawn shop, he'd be there again, and she could pretend she didn't know anything about it.
Pretend she hadn't sat in her car a block away and decided to leave him there to rot.
She slumped back in her seat.
Hadn't she decided to be a hero?
Wasn't she dying anyway?
She just wasn't ready to face defeat yet, not with so little progress toward her goal of helping Mandy and everyone else.
She let out a long, slow sigh, knowing she couldn't leave him, even if it would have been the sensible thing to do.
But first she had to ditch the stolen items. She was already outside the thirty-minute window. Surely they weren't really a danger to her so soon, but if this was a best practice among thieves—her lips twitched into a fragment of a smile at the thought—then she needed to behave accordingly. It would be tragic to lose her freedom, and the opportunity to take the black pill when she needed it most, because she was too stubborn to throw away stolen items.
With tremendous reluctance, she took items that might have been worth thousands of dollars, looked around to make sure no one was paying attention, and threw them into a reeking dumpster. Now she had nothing to show for the two burglaries she'd attempted. She slapped away mosquitoes as she got back into her car, and she drove back to the police station.
With weary resignation, she walked up the few steps behind a burly cop who propelled along a staggering suspect with his head down, and they went inside the low building of dirty gray stone.
Cool but damp air greeted them. Scattered LED lights shone dully over a space that was more office-like than she had expected, though also cold, white, and bare. Clean desks, empty white walls, and mottled white tile floors bounced back a sparse collection of noises—rough words from those in custody, voices of the handful of employees on their phones, footsteps, doors slamming. The cop ahead of her pushed his suspect through the maze of desks and gestured for him to sit.
A stocky, balding cop sitting at a counter took her in at a glance— _middle-aged white female, calm, unarmed_ —then spat discreetly into a white Styrofoam cup.
She went up to the counter. Her heart thudded anxiously in her chest. "I'm here to bail out Egemon Agnes."
He tapped several buttons on his e-paper and turned it around and slid it over to her. "Seven hundred," he said.
She winced. She would be left almost penniless. But she scanned her cash card.
As she stared at the signature block on the e-paper, she realized that he hadn't asked to see her ID—and she remembered that the cash cards were anonymous. She took a gamble and scrawled something illegible.
The cop paid no attention when he pulled back the e-paper. "Wait there," he said, pointing at a simple wooden bench near the door.
So far, so good. She sat down and tentatively relaxed.
The minutes ticked by. The cop she'd followed in guided his man into a back hallway past dark-tinted glass walls and out of sight. More cops came in and out, some with suspects in custody, others chatting with each other. They ignored her. She tried not to look guilty.
As she waited, she began to feel exposed and awkward. Now she second-guessed herself. Egemon hardly knew her. Would he think it was strange or presumptuous of her to do this for him? Even worse, would he assume it meant something it didn't?
What _did_ it mean? This urge to rescue him... Was it purely because Egemon was so useful to her?
She was still trying to figure it out when he came out of the back of the building, escorted by another cop.
His face went through surprise, pleasure, and half a dozen microexpressions before it settled on gratitude. The cop took off his handcuffs, and he came to her, and she stood. He looked at her wordlessly with his dark, deep-set eyes. The rest of the world fell away.
Electricity hummed between them.
She wondered whether it was there all along or whether it had appeared just this moment, the result of a good deed thoughtfully performed and thoroughly appreciated.
"I will pay you back the money, of course," he said.
She almost laughed. "I'm actually pretty glad to hear that."
Egemon insisted on calling an automated taxi to avoid drawing further attention to Helen. But before it arrived, they stood outside on the sidewalk for a few moments to talk. Rain had started, and Helen opened her umbrella and urged him closer to share in its cover. He stood very near and looked down at her soberly as he answered her questions.
They'd confiscated his cash cards at the pawn shop, and he hadn't had enough money in a bank account to make bail. But he always stashed plenty of cash cards off site. He would pay her back right away.
His arrest had nothing to do with Helen, not specifically. He wouldn't say more. "Just routine, I guess you could put it. The cost of doing business." He shrugged.
"What's going to happen to you now?" Helen asked.
He looked into the distance with a closed expression. "So I will do some time. It's happened before."
Something inside Helen immediately said _No_. _I won't let that happen._ Feeling protective toward this man made no sense in the grand scheme of things, but somehow, there it was.
He looked at her. "You will too, if you keep going this way."
_Only if I get caught before my time runs out._
"I know," she said. "How long will you have, before..."
"A month or two," he said. "The dockets are always backed up."
Then nothing would happen until after Helen was dead.
That didn't make her feel any better about it.
Helen went home feeling weak, her fingers trembling. Mandy sat on the loveseat, projecting the evening news onto the far wall from her clamshell. As Helen dropped her purse on the dining table, Jessie jumped off the loveseat and came to her to be petted.
Helen didn't feel like eating, but she knew she needed to—her pants were getting loose—and she went into the kitchen to heat up some chow mein noodles.
She was petting Jessie and waiting for the microwave oven to ding when Mandy said, "Mom! You have an evil twin. Except old. Look, look!"
"I have a what?" Helen looked up. As she took in the image on the wall, an electric shock pierced her.
The projection showed a composite image of three of her disguises.
# 29 Days, 6 Hours
Helen's age makeup held up well to the camera, and so did her wigs. Using makeup disguise tricks, she'd successfully changed the shape of her jawline and mouth and eyes. But the overall face shape and cheekbones and nose and eyebrows were exactly her. It was closer than she liked.
"Oh, that's funny," Helen managed to say. "Turn it up?"
Mandy obliged.
"—have so far registered an assortment of biological data, including DNA traces and palm prints, but they don't match records in law enforcement databases. If you see this woman or if you have any information about her, you are urged to contact the authorities immediately. This alleged thief used a Taser against a security guard and should be considered armed and dangerous. In the meantime, be on the lookout for any uninvited house guests, and contact authorities immediately if you spot this woman. These thefts may just be the beginning of a crime spree."
"Wow, Mom. Didn't know you were a house thief." Mandy grinned widely.
Helen studied Mandy's face, trying to guess whether she might suspect the truth. She saw only teasing in her daughter's expression.
"Yeah, very funny," she forced out. She looked at her noodles and knew there was no way she could get them down now. She put the plate in the sink. "Hey, I'm going to go lie down."
Mandy didn't reply. Her attention was on her clamshell again.
Helen went into their bedroom and sagged onto her bed.
And she'd been in the police station an hour ago.
Then again, probably no one was looking for her at a police station.
Now they had biometrics data and a picture that wasn't too far off. Now if anyone recognized her on the street and took a picture with their Earworm—and she wouldn't even know it—and sent it in to the cops... She pulled up the image from the news on her e-paper and studied it. No, probably she wouldn't be recognized. In the picture, she looked twenty-five years older, and the difference between early forties and late sixties was significant.
She would have to alter her appearance more radically in future disguises. She would have to do more with her nose... She would have to sacrifice her perfect eyebrows, damn it. She half-grinned, that such a thing would bother her. She wished that changing her eye color from green to blue had helped more, but it hadn't, really. Too subtle. She would need to get brown contacts to wear sometimes.
More importantly, now everyone knew her modus operandi of crashing parties and pocketing items, and now they would tighten security at these house parties. She would have to be more careful.
She thought about whether she should stick to burglaries from now on and give up on crashing parties. Approaching it pragmatically, she weighed her profits against her failures in each category. With burglaries, she was zero for two. With house parties, she was four for four... even though she'd lost the cash cards in the end. She decided she would forge ahead with both methods. House parties on weekends, burglaries during the week. Time was too short to let a single day pass without making an attempt.
She gave herself a bit of credit for staying rational about all this. Seeing herself on the news had been a nasty shock, but she was adjusting quickly. But then, she'd always expected that she would get caught eventually.
She got up and paced. Something else was bothering her about this news story, but she couldn't pinpoint it.
Pacing turned out to be a bad idea. She was too weak, and she had to sit down. She put her hands over her eyes and leaned forward, her hands touching her knees.
They thought she was a criminal.
Well, she _was_ a criminal.
No, they thought she was a _common_ criminal. An everyday thief who just wanted to steal stuff for her own gain or to buy drugs.
But what did it matter if they thought that? Most of the "common criminals" in prison didn't deserve to be there. It was a corrupt system.
She thought of Egemon again. He fit the "common criminal" profile. Had he really chosen that life or had he been forced into it? A life where you went to prison periodically was only better than the alternative if the alternative was truly awful.
Oliver, her old boss at Justice for All, was an ex-offender who'd served eight years for armed robbery—including six years of solitary confinement—emerged a broken man, was homeless for six years, then was given a job at Justice for All and cobbled together something of a functional life and personality after that.
Helen couldn't bear to think of solitary confinement. It sounded fine to the uninitiated. Lonely, perhaps, but peaceful. No, the United States was the last "civilized" country that still used it, because so many of the inmates went mad, because these were prisons turned mental institutions the way they were two hundred years ago, with screaming and horror inside every tiny room, every cell housing a forgotten and fragmented soul with no possibility of escape.
And the children who were now being put in jail for truancy and who might, if they were the slightest bit unlucky, live out the majority of their lives in solitary confinement—that part turned Helen's stomach.
She held the silver locket that held her black pill, grateful again to have it. Afraid, again, of losing access to it in jail, like so many others.
No, Helen didn't object to being confused for a common criminal. She felt sympathy and grief for the common criminal.
And yet she wanted the world to know—to fully understand—what she was doing. This wasn't about stealing. This was about economic justice, no matter that it was small in scope.
There was only one way to make that clear. She was going to have to go public.
> Dear LSTV,
>
> I am the female thief who has been stealing from the high-net-worth homes around town. But there's more you should know about me, and I invite you to tell the world.
>
> You see, it doesn't matter what happens to me. I'm not in this for my own personal gain. What I'm interested in is providing a little justice. And that's why almost all of my profit from this little venture goes to charity.
>
> These people with all the money—they don't deserve it any more than the poorest person living in a cardboard box in a doorway. The super-rich just got lucky. And it's time they got a little unlucky. I'm here to see to that.
>
> Tell them I'm coming for them.
>
> Love, the Robin Hood Thief
# 28 Days, 18 Hours
The next morning, Helen struggled out of the drug-induced sleep state and found herself curled up in the fetal position, fighting not to break into sobs. Her waking hallucinations were of David submerged in the ocean, swimming through hundreds of corpses that drifted in the water like a school of fish.
Helen had met David when he was one of her clients when she worked at a nonprofit that helped unemployed people get work. It had been her job to lead the training sessions on employment skills and interviewing.
He had a country-boy look to him, with short blond hair and tanned skin. He always wore plaid shirts and jeans. His sad blue eyes caught her attention, that and the slightly bowed right arm he held close to his slender body—a remnant of some past injury.
On the last day of the training, as he went out to catch the bus, she stopped him just outside the building, in the parking lot. She told him, "I'm sorry, I just couldn't let you go without speaking to you. I think you may have the saddest face I've ever seen."
It wasn't an encouraging comment, exactly, but there must have been something about the fact that she noticed and that she cared. He grinned just slightly and his light blue eyes shifted to pleasure for an instant—just a flash of happiness—and that was the moment she fell in love.
He told her his secret in the first week that they dated. "I never tell anyone this," he said, and she felt special, that he would choose to confide in her.
David and his extended family had lived in New Orleans.
When the ocean surge abruptly swept over those sea walls and submerged the centuries-old city for the final time, David had just finished having lunch with his younger brother. They'd walked out into the street just as the wave tore through, breaking David's arms as it broke his grip on his brother, who perished.
So many perished.
David had managed the city team that maintained the sea walls. He blamed himself for the loss of New Orleans and for the loss of his brother. He never forgave himself. Never forgot the sight of the corpses drifting in the water as the surge settled and David swam with broken arms up to the surface.
Helen told him it wasn't his fault, but he didn't believe her.
She married him a year later, and that didn't help, either.
She earnestly tried, with David. But years of her best efforts and even the gift of a beautiful little girl failed to make him happy.
He drank four beers every night—never more—to dull his pain, but it lingered, and it consumed him. As time went on, he retreated farther and farther into himself.
Helen walked on eggshells to avoid setting him off. He didn't hit her or threaten her, he just snapped at her for everything. But Helen was sensitive and it wore on her to always be wrong, to always be under attack, to never be good enough.
Then Mandy got sick. She was twelve years old when a flu turned into what Helen feared was pneumonia. Helen stayed home from work, at her daughter's side, too worried to even read, just sitting and watching her breathe. Thinking. Days to just think.
David was working from home, then, on Earworm, and he just sat there working or watching shows, not lifting a finger, telling Helen she was being paranoid when she told him she was worried about their daughter.
At last, after five days, Helen took Mandy to the emergency room by herself, while David pretended not to see them leave. And sitting there in the ER waiting room, nearly panic-stricken, listening to the bubbling sounds coming from her daughter's lungs and trying to convince herself that her precious only child would be okay, that she hadn't waited too long, she found herself possessed of a slow-burning and well-reasoned rage.
Why did she share her life with this man? It was hopeless. She had spent sixteen years trying to bring the flicker of pleasure she'd seen that day in the parking lot into full flame, and it had never happened. It would never happen. He was too broken to help, and she'd been a fool to try.
When Mandy got well, Helen told David to leave. And he went. Without a fight and without complaint. It was as if he'd just been waiting for her to finally get around to it.
But she failed to understand how much it would hurt Mandy.
After the divorce, David rarely agreed to see Mandy and never initiated a visit himself. Helen's actions only exposed David's lack of investment, yet to Mandy somehow it was all Helen's fault. And Helen didn't know how to make that better.
And then David died suddenly of a heart attack three years later. He was only in his forties, and slender. The autopsy showed a congenital heart defect that had never been diagnosed.
The day they got the news of David's death, Helen went to comfort Mandy as she lay crying in her bed. Helen put her hand on Mandy's shoulder, and Mandy pulled away and snapped, "Don't touch me! Get away from me!"
Mandy jerked away from every affectionate gesture after that. When Helen said "I love you," Mandy answered with thick, resentful silence. A door had slammed, and Helen was an unwelcome roommate, nothing more.
Helen tried to talk to Mandy about it, and she left the room every time. Helen left a note offering to pay for grief counseling, and Mandy threw it away.
After months of unwavering coldness, Helen stopped trying to touch her daughter. It hurt too much when Mandy turned away or knocked Helen's hands aside or snapped, "Stop it!" It had been a year and a half since Helen had last tried.
Helen didn't know how to turn it around, and Mandy didn't want to.
Helen tried not to blame herself for David's death, even though Mandy so obviously did. Helen told herself that she couldn't have helped David, that he'd refused to even get a checkup the entire time they were married and that wouldn't have changed. But still, at times, she thought maybe there was something she could have done. Sometimes she thought maybe Mandy was right.
She lay on her bed now and waited for the waking pill to bring her out of the quicksand of these memories, and she fought against the tears.
If she had kept David in their lives, maybe—just maybe—Mandy would have someone to take care of her now that Helen was going to leave her behind.
# 28 Days, 12 Hours
Helen had chosen her news outlet carefully for her letter the previous night. LSTV tended to be sympathetic to causes like hers. She felt she would get the right kind of coverage there, and that morning, she learned she was right. The string of thefts was minimally newsworthy on its own, but with Helen's email attached to it, it was like candy to whatever passed for journalists these days. They read her email hourly for the next twenty-four hours.
She wondered if they'd tried to reply to her, but she'd sent the email from an anonymous email address and then deleted it, just in case.
Sometimes Helen marveled that such a news organization still existed, that the upper classes couldn't censor the entire internet yet. Maybe the Entitled didn't care what the rabble told each other. Words were cheap.
Out of curiosity, she checked establishment media. Not a word about her.
It didn't matter. She was speaking directly to the rabble. They were her people.
In fact, it occurred to her, she ought to speak to her people even more directly.
Helen didn't have any online profiles. She just didn't care for anything especially technical. But the Robin Hood Thief ought to have a profile somewhere so she could broadcast her messages for herself.
Helen went to a public library so that the cops couldn't track her new profile back to her or Mandy. She went onto a website called Whatsit and created a profile with the combined image from the news. Then she added a brief message in the bio:
[ Hello world. I'm the official Robin Hood Thief. Pleased to meet you. Feel free to chat with me here. ]
Then, with a mischievous grin, she set up her default configuration to paywall anyone whose demographic information showed they made more than fifty thousand dollars a year. If they wanted to see her content badly enough, they could pay a thousand dollars per view. She would accept those payments with one of her anonymous cash cards.
She also blocked wealthy users from augmenting her content, boosting it, or projecting it into the 3D landscape (or 3'scape) via Earworm, because those were ways a wealthy user could share her content with their ilk without having them pass through the paywall she'd just set up.
Her first post was the declaration, "No one deserves luxury while others still suffer." She layered the text over a photo of an anonymous homeless person sleeping in a doorway. Then she authorized Whatsit to charge her twenty bucks to boost it exclusively to her kind of people for the next twenty-four hours.
That task complete, Helen stopped at the weapons store to get a thigh strap that would hold the Taser, and she put her mini canister of mace into the pocket of her jacket so her bag would be free of weapons in case they checked it at the door at her next robbery.
Next, Helen went to the pawn shop again. All this new visibility was making her nervous, and she went to Egemon to ask for a way to get fake IDs. She requested four new alternate selves, each accompanied by dramatic new disguises—and brown or black eyes. She supplied the photos, and as usual, he was quite capable of making the rest happen. "You'll be able to download them in a few hours," he said. "Oh, and don't forget I owe you for the bail money."
"No chance of that," she said with a smile.
He turned and ducked back into the back room—then, just as quickly, he seemed to reconsider. Wordless and with a look she wasn't sure she understood, he came back. He lifted the section of counter that created a walkway and gestured for her to follow.
She sensed that this was a rare privilege, and it made her smile.
Once in the cluttered, dusty back room, he simply said, "Open"—probably to his Earworm—and part of the back wall slid away, revealing a small room bursting with weapons and gadgets.
He went to a drawer and took out a cash card, then programmed it with a special-purpose e-paper in the small room.
In one corner of the ordinary part of the back area, Helen noticed a collection of singular children's toys from bygone ages. Hand-carved rocking horses and ornate dollhouses, porcelain dolls in gauzy dresses, metal fire trucks. They were beautiful, and their presence here surprised Helen.
She nodded toward the items. "Inventory waiting to be sold?"
Egemon shuffled his feet. "Sort of." His eyes softened as he looked at the toys.
"Sort of?" Helen sensed that these were special, and she couldn't resist teasing him a little.
He shrugged and escorted her back to the front, his posture open and relaxed. "Here's your money, with my thanks. And no charge for the IDs." He handed her the card, accompanied by a smile.
Helen smiled in gratitude, but she couldn't forget about the contents of the other room. She studied his face, looking for any clues. "Okay, what's the story with the toys? Come on... you can't let me see and then not tell me about it."
He planted his palms on the counter and gave her an appraising look. When he spoke, it was with the air of confession. "Maybe someday I can open an antique toy store. Okay? I like old toys."
Helen felt a smile break across her face. "Okay, but why? What's the story?"
Egemon let out a gust of a sigh, looking at the counter for a moment. When he looked at her again, his dark eyes were liquid, and his tone soft but reluctant. "There was an old woman, a neighbor of mine I knew a short while. The toys were hers. She called me her grandson, and she left them to me. But don't go and tell people that. They will think I'm soft."
Helen grinned wickedly. "Your secret is safe with me. I promise."
As she pushed the door open, causing the bell to jingle, he gave her an odd sort of salute and a charming smile and said, "Same to you, Ms. Robin Hood."
For an instant, she paused, surprised that he had already guessed who she was, and uncertain as to what it meant. But looking carefully at the Greek man with his hands on the counter, she saw no threat there.
Not an enemy... an ally.
She grinned at him. "Good to know."
He winked at her, and she smiled all the way out to her car.
It was a Thursday evening—an unusual night for a party, but apparently Mr. Brock Tolbrook, her next victim, didn't want to waste a weekend evening on a minor political event. He was hosting a fundraiser for someone else's city council campaign—no doubt some tit for tat.
For that night's robbery, Helen used her usual methods to approach a gentleman who was entering alone, then introduced herself as one of her new fake identities and teasingly chastised the gentleman for not remembering her.
It still surprised her that this worked every single time, but then again, it was only good manners on their part. It wouldn't do to tell someone to buzz off just because you didn't recognize her.
Well, as long as she appeared to match your social class and status, of course. As long as she was "one of us."
Luckily, looking the part was all that was necessary so far.
As Helen's spot in line moved closer to the door, she eyed the security guard checking names. He nodded and spoke to the next guest, a woman a few people ahead of Helen. The woman fished in her purse and brought out her e-paper, which she unfolded and held out for the guard. Probably she was showing her ID.
Helen's heart leapt into her throat, and her chest hollowed out. She felt her smile turn into a grimace. She struggled to think of some plausible reason not to show her fake ID, but suddenly it was her turn and she'd come up with nothing.
The guard apologized and gave her ID only a cursory look. However, he looked directly into her face with such scrutiny that it nearly stopped her heart. Her broad grin felt pasted-on. Surely he would recognize its insincerity. But he smiled politely and wished her a good evening.
She stepped into the party with legs weak from relief.
Soon, she stepped out of the ballroom to start casing the house. The first area, still in earshot of the party, was a wide hallway reaching toward the rest of the house. It was filled with a wide assortment of large art pieces depicting horses and dogs. Around those were dozens of thank-you letters and certificates of appreciation and photos of admirers with the owner of the house, Brock Tolbrook, who was a stocky middle-aged man with an oval face, short black hair that was receding, and a wide smile.
It looked like he had donated thousands of dollars to horse- and dog-related charities. She smiled as she read the words of gratitude for a "one-of-a-kind man, a true philanthropist, a benefactor beyond all others."
He had a soft spot, she realized, for disabled animals—several of the horses in the photos were blind in one eye or missing an ear. There was even a large photo of him mucking out a barn stall wearing gloves and a wide grin. He was willing to rake out horse manure himself, and he wasn't ashamed for people to see it. She found herself nodding in appreciation.
Maybe these people weren't all bad, after all.
"He absolutely loves animals," a genial voice said from near at hand.
Helen startled and looked around.
A soft, round woman in her early forties smiled kindly at her. "Oh, I'm sorry. I didn't meant to scare you. I just saw you admiring the charity collection and had to brag a little about my husband. I'm Mona."
Mona extended a hand, and Helen shook it delicately. "Oh, that's okay, Mona. Yes, I love to see people who care so much about the less fortunate."
Helen realized too late that this quite an overstatement for the Entitled, but the other woman nodded as if it were gospel truth. "He's like a doting mother when he's with a wounded animal. I'm sorry, I didn't catch your name?"
Helen drew a blank for one horrifying moment, and then smoothly said, "Yvette Lane."
"Ms. Lane, it's a pleasure to make your acquaintance. How do you know my husband?"
Despite its gentle and sincere tone, the question threw another spike of fear through Helen. Her prepared back story was distant. She clawed it closer and managed to say, "Through the future councilman, Mr. Avery. My late husband worked with Mr. Avery on the board of education a number of years ago, before he retired."
"Wonderful. Mr. Avery is a very generous man and a good choice for city council. His wife is lovely as well. I've been working with her on constructing a charity children's hospital—it was all her idea."
Helen's internal reaction was at least half cynicism, but all that came out was, "That's wonderful." Her smile had grown tired, and she forced it to remain in place.
"Well, I do hope you've had a chance to enjoy the canapés and get yourself some wine?"
"Oh, yes, it's been a lovely party. You have such an impressive home."
Mona smiled charmingly. "Have a lovely evening, my dear."
"You as well," Helen said.
She watched Mona go, and then returned to staring at the pictures on the wall. Her fake smile sagged and she let out a breath she'd been holding. Mona and Brock seemed like decent people. For a moment, she felt bad about planning to steal from them. She reminded herself that she would be making a trivial dent in their vast net worth.
She looked around to make sure Mona had gone back to the party and that no one else was watching her. Then she wandered down a distant hallway, peeking in doors and avoiding two rooms with sentries.
She went around another turn and opened a door to see Brock Tolbrook going like a rabbit at a young woman who was bent over the back of a sofa.
"Oh, I'm sorry," Helen blurted out. Her face flushed and she made to back up and close the door.
Then she recognized the hideously obvious truth that the young woman bent over the sofa—who was now opening her mouth in a screech of horror—was not Tolbrook's wife.
Helen suddenly liked Brock Tolbrook a good deal less than she had.
Tolbrook pulled away and scrambled to get his trousers up, cursing and red-faced. He was half a foot shorter than he looked in the portraits and photos outside.
The woman blasted him: "You didn't lock the door?"
Helen felt her eyebrows colliding as she advanced on Tolbrook, unthinking, words coming out of her mouth. "Your _wife_ is out there."
She suddenly realized that she had her e-paper in her hand and already unfolded. She folded it into video mode and started recording even as Tolbrook lurched away from the young woman.
He yelped, "No! Don't!"
The young woman shrieked and dropped behind the sofa as if she could disappear.
"Your _wife_ is out there!" Helen yelled again. She felt like an oncoming train—unstoppable—a train of hot, righteous fury that would obliterate everything in its path.
Tolbrook tripped on his pants as he tried to get to Helen. "Turn that thing off!"
Helen turned away from him and rushed back to the doorway, one hand on the doorknob. "You really want me to go out there with this video?"
The man cursed again, his face still livid. Then he came at her fast.
Helen dropped her purse as she fumbled her canister of mace out of her jacket pocket with her one free hand and held it up.
He stopped short and did a little dance of rage with his pants still only half up.
"How much?" he shouted.
The woman popped up from behind the sofa and turned on him. "Shut up! Keep your voice down!"
_How much what?_ Helen tried to understand. She willed her hands to stop shaking, but they wouldn't listen.
"How much do you want to keep your mouth shut?" he demanded. He finally got his pants zipped. He struggled to get his shirt tucked back in.
"How much cash do you have on hand?" Helen asked coolly. Her stomach flipped, then dropped, at her own audacity.
His shirt under control, Tolbrook leaned on a side table. Sweat had broken out on his forehead, and he wiped it away.
Suddenly she remembered the original scheme. Target the Entitled who'd attended the Net Worth Notion, and give the money to the same charities they'd given pennies to at the fundraiser.
"You know what?" Helen said. "Not for me. Transfer money to charity. That's what I want you to do. To Hand in Hand. You remember them? You gave them a thousandth of a percent at the Net Worth Notion."
"Fine!" he raged. "How much?"
She'd transferred a hundred thousand dollars on behalf of Mr. Alvarez. Did she dare demand more from Tolbrook?
"Two hundred fifty thousand," she said.
"You're fucking kidding me," he shouted. His face looked like a splotchy tomato.
His mistress yelled at him again, "Be quiet! You dumb shit. I can't believe you didn't lock the damn door. Your _wife_ could have walked in."
"Shut up," he shouted. "You know what? Get out. Get out of my sight right now. You aren't worth that much money, you better believe that. I wouldn't have paid twenty bucks to bend you over."
The woman's mouth opened in silent shock and protest.
"I'm not paying you _shit_ ," Tolbrook yelled at Helen. "Send the damn video out! I don't care."
The other woman shut her mouth with a snap, then opened it again. "You better not do that. My husband might be pissed off at me, but he will also call off the deal he's about to sign with you. What's that worth to you, twenty or thirty _million_?"
Tolbrook's mouth worked in soundless rage, and Helen cut off a sharp laugh. "Oh, the price just went up," she said. "It's a million dollars now."
"Damn straight." The woman extended both middle fingers toward Tolbrook, then slipped out of a side door, leaving Helen alone with a man who was coming unglued in front of her eyes. Alone in the farthest and most private recesses of his enormous mansion.
He made straight for Helen, and she dodged behind an overstuffed chair, the can of mace still up. But all he had to do was come directly at her, using any one of the dozen couch pillows to shield himself from the stream of mace. Then she would be at his mercy.
She resorted to a bluff—one she'd seen in a movie. She gestured with the e-paper. "This thing is still recording. And this video is being livestreamed to my partner. If I don't get out of here unharmed, he will release it to the public.
"You had better make that transfer. Otherwise, just like your little friend said, you'll lose that business deal. One million dollars to Hand in Hand by five o'clock tomorrow night. Make sure you publicize it so it's in the news. Or else at 5:01 p.m. I will release this video.
"Now I'm going to walk out of here and you're not going to do a goddamn thing about it."
Tolbrook stood still, his chin lowered and his fists clenched, breathing like an angry bull.
She got the hell out of there.
Helen left the mansion, swiping a small modern-art sculpture on her way out, and drove to the pawn shop high on adrenaline. The impromptu blackmail was her biggest victory yet. And the sculpture turned out to be worth eight thousand dollars—her first profit since the forty thousand she'd lost. She was back in business.
Although it scared her to show so much trust, she asked Egemon to hold half the money on a cash card for her in his secret back room, so that she couldn't lose it again—although she didn't tell him that. He gave her a long, curious look, but then he agreed, no questions asked.
On her way home from the pawn shop, the adrenaline fully wore off, and as she thought over what had happened, anxiety set in. If Tolbrook hadn't fallen for her bluff... She had been alone with him in the back rooms of his enormous mansion. He could have beaten her, even killed her, and no one at the party would have heard her scream.
He had been so angry from the start. Livid. She remembered his little dance of rage. How he'd lunged at her. How his mistress had insulted and baited him, enraging him even further.
Quite possibly he had just been too angry to think straight. Quite possibly, Helen had been lucky as hell.
Helen rubbed her tired eyes as she drove. She resolved not to make that mistake again. She would never again confront a homeowner in private. It wasn't worth it.
That night, she was doubly glad to take her sleeping pill, to just turn the lights off on the fear.
# 27 Days
Friday afternoon, while she sat in front of a projcom in a public library, Helen found herself wondering if she could use what had happened with Mr. Tolbrook to her advantage. Could she blackmail other members of the glitterati? It would be a new and far more powerful way to strike at the Entitled.
She felt a twinge of guilt as she contemplated it. The old Helen-of-the-high-road was dismayed. _Be quiet_ , she told that part of herself. _I'm not doing them real harm. I'm only taking a small part of their excess, and only for the benefit of those who are truly suffering._
In her imagination, she played through the scenario: walking through the gorgeous rooms of a mansion in search of something blackmail worthy.
The odds of stumbling across another person in flagrante delicto were absurdly slim. She toyed with the possibility of finding someone's stash of cocaine or heroin or pills—the few remaining illegal drugs—but what could she really expect? A pile of baggies of white powder, each one helpfully labeled?
Even if she did find such a stash, how much would a possession charge hurt one of the Entitled? She was no prosecutor or criminal defense lawyer, but she suspected that such people didn't often serve time—or find themselves in the media spotlight—for minor possession charges.
No, the noteworthy crimes committed by these people were surely white-collar crimes, all committed on Earworms and projcoms. She'd find no physical evidence. If she knew how to hack computers, it would be easy, she imagined. But with her limited technical skills, she would have to find an unlocked projcom to rifle through, like at Mr. Alvarez's house.
All afternoon, Helen researched her next two targets, Mr. and Mrs. Takeuchi and Mr. and Mrs. White, looking for any possible lead on white-collar crimes.
As a break, while she was using the library projcom and the cops couldn't track her activity back to her or Mandy, she logged in to check Whatsit.
Since Wednesday, she had collected 4,291 reactions and 238 comments.
Her fans had boosted her post and profile 112 times. They had augmented her post into over 20 variations, with 7 of them projected into the 3'scape. She'd made a very welcome $4,000 from the few ultrawealthy who couldn't resist their curiosity about her first post.
Messages from two journalists were waiting for her. One was from MCCTv, the channel that broke the news about her string of thefts, and she deleted it with prejudice. The other was from LSTV: [ Hey, I'm Christian Smith, a journalist and a huge fan. Can I interview you? I'll protect your privacy and identity, I promise. ]
She felt a moment of temptation, but then decided not to take the risk. She could talk directly to her people here, on Whatsit.
She scanned the augmentations and comments. A large number of people were doubting that this account was real. Of course—she should have thought of that. She would need to figure out a way around it.
It occurred to her to check to see whether her retirement fund check had hit her bank account yet. She had $2,750 coming. She logged into her account and saw no payment. She went back to the retirement fund website to check the status, and there was nothing about a pending request. Her heart sank, and she picked up her e-paper to call.
She spent the next two hours on hold, only to be told that the fund manager had no record of her request and that she would need to file it again.
She slowly banged her head on the desk in front of her. Thank God for the eight thousand dollars she'd gotten from Tolbrook's sculpture. With a heavy heart, she repeated all the paperwork.
She remembered to check the news at five o'clock. The fourth story revealed that Tolbrook had made the donation to Hand in Hand—one million dollars. That, at least, made her smile.
The library lights flickered on and off three times and a recording told the patrons to make their way to the exits. It was time to go get ready for the evening's robbery.
Helen went into the party that night having drawn a blank on Mr. and Mrs. Takeuchi, and as she pessimistically expected, her brief search of rooms unguarded by sentries turned up no contraband. Her second attempt to blackmail failed.
She found only a moderate prize—Mr. Takeuchi's Centaure watch, in plain sight on his nightstand. She recognized the brand and knew it was worth about a thousand dollars.
Before she left the mansion, she took a video of the watch on the nightstand, also capturing the portrait of Mr. Takeuchi in the background. In a day or two, after the watch had passed through too many hands to be tracked back to Egemon, she would post the video to Whatsit along with the comment, "Mr. Takeuchi, I have your Centaure." That would prove she was the Robin Hood Thief.
She sold the watch to Egemon, left the pawn shop—calling "Good night" to him as she went out—and headed home.
As she got out of Old Blue in her parking spot at her apartment building, she heard muffled rock music coming from somewhere on her floor. Enthusiastic but inexpert rock music, she decided as she trudged along tiredly. The guitar and drums were slightly out of sync, and then the lead guitar hit a flat note, which suggested that it was a live band practicing.
The music only got louder as she approached her hallway and even louder as she approached her door, until she realized with equal parts dismay and confusion that it came from her own apartment.
It had to be friends of Mandy—that was obvious—but did they really have an entire rock band shoved into her microscopic living room? Annoyance and frustration arose, and fear for her daughter's future along with it. When was the girl going to grow up? Other people had a right to peace and quiet in their own homes.
She was ready to give Mandy a piece of her mind as she opened the door and went to step in. The head of a lead guitar nearly hit her in the face, and she pulled back. Two guitarists stood in the kitchen. The drums filled the living room. The keyboard was set up in the bedroom, a step inside the door.
They kept playing, of course. The guitarist, a skinny kid with long black hair and a black motorcycle jacket and a sad excuse for facial hair, grinned maniacally at her and vigorously nodded his head in time to the music. Helen frowned at him.
Where was Mandy in all of this?
She looked around a second time, absolutely ready to believe that Mandy had turned their apartment over to a random band and then gone out, and ready to be livid about it—and then she spotted the plumes of blue hair on the other side of the lead guitarist.
Mandy was playing bass guitar. Her pale, delicate face was intent and her black-lipsticked mouth set in a focused line as she stared at the fretboard.
Helen recognized Mandy's guitar. It had been David's, although he didn't play it anymore by the time he moved in with Helen. When Mandy was eleven, she took an interest in it, and David gave her a few lessons before tuning out again.
It was great to see her daughter playing it. Heck, to see her doing anything besides sitting on the loveseat and staring at a screen. To see her among physical friends, part of something that existed in three-dimensional reality, doing something creative. Happiness soared in Helen, and it chased away her anger and lifted her fatigue.
She watched for a moment, enjoying the awe that comes from discovering some new and impressive dimension of your own child. Mandy wasn't half-bad. She must have been at this for at least a few months.
Helen didn't want to make Mandy feel self-conscious by watching too long, so she gave the band a little wave and went back out, drumming her thigh along with the music as she walked down the hallway.
When she got out to the car again, she sent Mandy a message: [ Let me know when your band leaves so I can go to bed. No hurry. So happy to see you playing Dad's guitar! You're really good, too. ]
Saturday morning, Helen killed time on her e-paper at home on the loveseat, waiting for Mandy to wake up so they could talk about her new band.
But when Mandy finally came out of their bedroom, she was dressed to leave.
"Good morning," Helen said tentatively.
"I'm going out," Mandy replied as she headed toward the door.
"Wait a second," Helen pleaded. She got up from the loveseat and approached, and while she did, Mandy unlocked the front door and went out.
Helen stopped. "Mandy..." she said in protest.
Mandy stood in the hallway with the door half-open. "What?" she asked. She gave Helen a blank, resentful look that said _Can't you just leave me alone?_
"I just wanted to tell you I'm really excited about your band and really proud of you," Helen said forlornly.
"Yeah, okay, thanks," Mandy said. She shut the door.
# 26 Days
Helen went out immediately after Mandy, not in pursuit of her but just wanting to leave that moment behind as quickly as possible, before it broke her heart too completely.
She would lose herself in her work, as she always did. It was just a new line of work these days.
She hoped to have better luck in blackmailing her next target, Mr. White. That night—Saturday night—he and his wife would celebrate the engagement of their youngest daughter.
Something that Helen wouldn't live long enough to see for her own daughter.
She spent the afternoon researching White, and she lucked into some news about him and his pharmaceutical company, Fabre-Terre.
A few years ago, he made twenty million dollars in a single transaction by selling off shares of Fabre-Terre—and two days afterward, a whistleblower filed suit against Fabre-Terre under the False Claims Act, and the stock price dropped precipitously. The timing was suspicious. It was as if White knew the lawsuit was about to be filed.
The SEC charged him with insider trading a few months later, but the federal judge's rulings ever since either damaged or stalled the SEC's case. Three days ago, the judge ruled in favor of Mr. White's pleading to eliminate a key witness in the insider trading case, based on a technicality. In short, the judge was suspiciously friendly to Mr. White. It smelled of bribery.
Perhaps she could do something with this information.
That night, Helen presented her fake ID at Mr. White's door, toasted the glowing couple with pink champagne, and then went into the farthest, most personal rooms to look for a projcom or clamshell.
She might not find a computer unattended, and if she did, it might have security she couldn't bypass. But Mr. White was older than herself, so that lowered the odds he would be tech-savvy.
She felt the clock ticking away as she went through bedrooms, guest rooms, libraries, studies, a billiards room, and more. A sick nervous dread welled up in her stomach. But finally she walked into a bedroom and spotted the distinctive small black box of a projcom at a desk on the far side of the room.
A rush of excitement drove down the nervous dread as she went over and activated the screen and keyboard, her fingers crossed. It lit up to show a desktop, not a lock screen. Praying her luck would hold, she found an email icon and clicked it.
Ten minutes later, she went back out among the self-satisfied glitterati. She took another glass of pink bubbles while Mr. White finished live-narrating the video of his soon-to-be son-in-law's elaborate proposal. It had involved a procession of a hundred musicians, acrobats, and jugglers leading them into the Piazza Navona square in Rome and a private display of fireworks, followed by a helicopter lift-off to Venice.
_Must be nice,_ Helen thought bitterly. In fact, it sounded wonderful.
"The hardest part of the whole ordeal was getting the homeless and poor people out of the way, that and all the trash they leave behind," Mr. White said, "but they would've ruined the effect for my daughter. We had the place gleaming. Good thing the local authorities understand the language of money, am I right?"
The audience guffawed and raised their glasses.
Knowing what she had just done, Helen didn't even get angry.
When the applause wound down, she went out of the mansion and to her rented Mercedes.
As the car took her away to safety, she thought over the email she'd just composed to Mr. White, and she smiled.
> Dear Mr. White,
>
> You'll note that this email comes from your personal email account. Yes, I wrote this from your very own projcom in your bedroom. And no, I'm not there now, so there's no sense alerting security. I decided to take advantage of the "send later" feature on your email application. I wrote this email an hour ago.
>
> Now, I'd like you to take note of the attached video I just made. It documents a very interesting string of emails about the sale of Fabre-Terre stock that you made—to your immense profit—a few years ago. They show how you profited by being informed about the False Claims Act lawsuit before it happened. And they show how you've bribed the judge ever since to obstruct the case.
>
> You'll note the datestamp, as well as how I captured the entire bedroom and the portrait of yourself and your lovely wife, just for additional verification.
>
> Naturally, a copy of the video has already been uploaded to a secure location that I have under my control.
>
> Now, I want you to choose one of your assets. A building, a business, I don't care what. Some asset worth at least five million dollars. Since you made twenty million off your Fabre-Terre sale, I'm sure you'll find that a very fair number.
>
> You're going to sign that asset directly over to Star Programs, and you're going to publicize that generosity. If I don't see an item on LSTV about this by close of business Monday, I will publish the video.
>
> Star Programs is a charity that helps the homeless. But you might remember that. As I read in the news a couple of weeks ago, you donated a thousandth of a percent to them at the Net Worth Notion.
>
> Five million dollars is a little more proportionate.
>
> I do wish I could see your face right now as you're reading this on your e-paper with your party still going on all around you, pretending you're good people while ignoring the suffering of the masses. But, like I said... I'm already gone.
>
> Love, the Robin Hood Thief
# 25 Days
Helen followed the blackmail attempt with a successful burglary on Sunday morning. She took a sculpture and other odds and ends worth just over seven thousand dollars.
But then the Robin Hood Thief was on LSTV again that night. The news was bad.
Mr. White and possibly Mr. Tolbrook had talked to the cops—about the thefts, not the blackmail. Out at a sunlit and peaceful coffee shop, Helen scanned the text version of the article. Under the headline "Robin Hood Thief Strikes Again," she spotted the byline of the journalist who'd reached out to her before—Christian Smith. His photo matched his profile photo on Whatsit, and his tone was, in fact, appreciative.
She played the video clip. "Authorities from all six affected counties are now working together to identify the serial thief," the newscaster announced. "Officials say they have been receiving not only tips but direct assistance and generous donations from several of the victims, though all are doing so under condition of anonymity."
Helen let out a sigh. The net was being cast. She stared into space, suddenly unable to fend off the vision of herself in handcuffs, probably Tasered by the cops, dragged helpless and sobbing into a patrol car.
_Stop._ This was pointless.
She just wished she would have the opportunity to see the fruits of her labors. She'd won a million from Tolbrook and a hundred thousand from Alvarez and maybe five million from White. But it took time for the charities to process the donations, work them into their budgets, and disburse them to the ultimate recipients. She wouldn't live long enough to see the results take shape in the real world.
Her e-paper chimed. That would be Mandy. Only Mandy ever texted her. The message read: [ We need dog food. Jessie's out again. ]
Helen ground her teeth. [ It's YOUR job, Mandy. Go get some today. Now. Walk to the store. It is not that far. ]
No reply, of course.
Hoping to find some encouragement, she logged into Whatsit. Immediately, an anonymous message popped up: [ You're not going to get away with this. You will regret ever setting foot in my house. ]
Her shoulders sagged and she wiped a hand across her face, her heart suddenly thudding. She fought back the urge to look around her in search of someone glaring from a dark corner.
Of course she knew she would make enemies, doing this, but it still freaked her out to have them.
She turned her attention back to her profile. Thousands of people were active on it. Even as she watched, someone posted the police report for Mr. Takeuchi's missing Centaure watch, just as she'd hoped, to corroborate her video.
She scanned some of the thousands of recent comments. A lot were abusive—that would never change. It was the internet, after all. But a lot of the comments bolstered her.
[ What your doing's awesome. Don't stop. ]
[ Take 'em for all their worth. ]
[ Don't get caught, Robin! Give them the slip. They're going to be watching closer and closer. ]
[ I just hope she gets out of the country before it's too late. ]
Then Helen glimpsed something that made her stop in alarm.
[ I did it too! Look at this beauty. Got it from my aunt's friend's house. Sold it for $250, gave it all to charity. ] The picture showed a pocketknife with a mother-of-pearl handle.
Dammit. Helen stared into space with fresh dismay. Maybe she should have expected this. Copycats. Proud copycats. This one sounded like a teenager.
She scrolled down and found another dozen or so along the same lines.
Did she want this? She tapped her nails on the table as she thought. More participants meant more money going to charity. On the face of it, nothing could be better.
But these were kids. Kids with their whole lives ahead of them.
[ Copycats: I love you. I'm proud of you. So proud of you. But please don't take the risk. My situation is different. ]
Helen stopped and let out a long sigh. Speaking of clueless kids, there was something she needed to do—something that had been nagging at her for days.
She opened a new window and logged into her documents folder, the one she shared with Mandy. She created a new document called "Day-to-day stuff." There she added how to pay the rent and utilities, and the fact that Helen had now added Mandy to her financial accounts. The phone numbers for all the utilities. The name and address and phone number of their vet, for Jessie.
She added a business where Mandy could get driving lessons. Mandy had never let her mother teach her. And, assuming Mandy decided to keep Old Blue, how often to service her. The name and address and phone number for the mechanic. Admonitions about keeping up with oil changes and the importance of responding to warning lights and messages.
Where the backup paper copies of Mandy's citizen paperwork were stashed in the apartment. Her blood type. A reminder to get annual physicals and when to see the doctor versus when to just rest. How high of a fever was too high. The best foods for an upset stomach. A warning to stay away from new pharmaceuticals that might not be adequately tested.
Helen forced herself to stop. She put her hands on her face. She was not going to be there to take care of her child. A document could not replace a mother. There was no point in trying.
Anyway, Mandy didn't need it. She could look up anything she needed on the internet.
Helen left what she'd already typed and shared the note. She included the comment, "For when you're older and living on your own." Mandy wouldn't look at it. In fact, Mandy probably wouldn't look at it even after Helen died.
_You're going to have to tell her._ But the thought met a closed door inside her. _Not yet. Not yet._
# 24 Days
Mr. White came through. His five-million-dollar donation was noted on McCTV on the Monday morning news.
Helen threw a fist of victory into the air.
She tried to log into Whatsit to post a celebratory note, and instead of her page, she saw the error message, "This profile is no longer available."
She stared in confusion for a few moments before she figured out that the authorities had found the profile and deleted it.
Wearing a stubborn scowl, she made another one.
She'd just have to repeat the same process she used last time, with Mr. Takeuchi's Centaure watch, to prove the authenticity of the new account.
"Go ahead, delete it again," she muttered to her enemies in law enforcement. "I can play this game all day long."
For another twenty-four days, at any rate.
It was amazing how much new energy this work gave Helen. Gone was the perpetual debilitating exhaustion of just trying to get through one day at a time, one crisis at a time. Gone was the unending struggle for basic survival, with nothing to look forward to.
Now, even with a terminal diagnosis and barely a month left to live, she was filled with new life, new energy. It didn't make sense.
Her hands trembled constantly, and her heart thudded nonstop in her chest, but she didn't care.
She was alive. For now—for the first time—she was alive.
How long would it last?
# 20 Days, 6 Hours
Helen set out for her robbery the next Friday night with three more successful burglaries under her belt and a total of thirty-three thousand dollars on cash cards, most of them safe with Egemon—not as much as she'd gained and then lost the first time, but enough to ensure that Mandy would be all right for at least a year.
Despite her victories, Helen set out with a morbid, sick feeling. Her string of good luck couldn't possibly continue. But what else could she do? She had to make use of the time she had left. Twenty days, now.
Still, she took precautions. She took extra pains with her disguise, with heavy makeup and a new wig. She packed her Taser and her can of mace, wrapped the garrote wire around her thigh like a garter, and clipped the security system jammers Egemon had given her to her bra.
She also chose to take a black taxi—a suitably luxurious model with a human driver—so she wouldn't get caught up in the car pick-up queue. There was a chance the taxi driver would remember her, but it didn't matter. The cash card she used to pay him was anonymous, and how else could he track her? He would just see tonight's disguise, same as everyone else.
Before Helen went in to the party, she had the taxi drive her around the entire estate to get her bearings. It was walled, as all such mansions were, but the front gate was open for the event. There was a busy street just one house down, and she directed her taxi driver to wait for her there.
She got in as easily as ever, but the bad feeling strengthened. She felt too pressured to delay long. She choked down a few bites of the decadent, luxurious food and wine and exchanged some witty comments with a few people, then went on her way, cursing her stiff legs.
As always, fear lent her adrenaline even as it made her hands shake. Never mind that this was her eighth robbery—it never got any less terrifying. Who knew what land mines might await in any given room?
She reached the large, circular antechamber to the master bedroom and was about to enter when a movement in the room stopped her cold. It was a sentry, but it didn't hover high above her head in a corner like other sentries. Instead, it was slowly passing around the room at shoulder height.
This one was set to patrol.
Paranoia paid off—she had the jammers with her. She remembered how Egemon cautioned her that she would only get twenty or thirty seconds once she activated one of them. She needed to get this right the first time.
She waited until it was on the far side of the room, away from the two doorways, then pressed the appropriate jammer button.
The sentry stopped dead in its path, and its lights turned off, then flickered.
She hurried past it into the bedroom, pulled the door mostly closed, and then peered around it at the sentry.
The sentry remained still, its lights flashing. Was that good or bad?
She didn't have time to find out.
She turned and looked around the vast bedroom, with its conversation area, oversized king-size bed, and two-sided fireplace creating a separation between the sleeping area and an oversized Jacuzzi tub.
Then she heard a laugh. A woman's voice coming closer.
She turned back to the bedroom. There was no time at all for her to get anywhere. She slid under the bed. Luckily, the layers of duvets hung all the way to the floor. She peeked out from a sliver between folds of fabric.
A woman strolled through the room with light steps, still laughing at something. "I know," she said as she went through the carpeted bedroom. She went into an adjacent room where her footsteps echoed. Probably a bathroom.
"No, I'm not doing that. I don't care what the old witch does. The business is Henry's already. She signed. She can't undo it."
Helen heard a faint tinkle that went on for a few seconds, then the flush of a toilet. The woman came back out and went back around the bed.
"Whatever. Look, I've gotta get back to this boring event. One of Henry's business things. I'll talk to you later, okay?"
The woman passed out into the antechamber.
Helen peeked out from under the bed. She could see through the open doorway. The woman stood staring at the unmoving sentry, its lights still blinking. No, there were two of them now. Backup had arrived.
Then a masculine voice, warm and friendly, from the other side of the antechamber. "What're you looking at?"
Helen's heart nearly stopped before redoubling.
"Why are there two of them now?" the woman answered.
Helen's lips drew back in a grimace. Dammit.
"Is it supposed to be doing that?" the woman asked.
Helen's body went weak with fear. She couldn't hear the response of the man over the pounding of her own heart.
How much time did she have before the sentries alerted the guards, or before the people did so?
She'd gotten nothing to steal.
The man and the woman went back out into the rest of the house, still talking in concerned tones.
She waited a few more moments just to make sure no one else was coming through. Then she climbed stiffly out from under the bed and hurried toward the sliding glass doors that opened up from the bedroom onto the back garden.
Just as she unlatched the door, she heard the hard footsteps of the security team a room or two away. She slid the door open as they came through the antechamber, and when they burst into the bedroom, she was running out into the rain.
The wall at the back of the yard was so close, but so tall. Could she clear it? Would they shoot at her? Were there dogs?
She had never run with such determination, and yet her stiff, trembling legs slowed her. Her purse, with the strap over her shoulder, knocked awkwardly against her hip. The grass was sodden and slick, and she left a heel behind, caught in the soft turf, then the other. Her vision pounded black at the edges.
There was a sculpture by the wall. A large dove, wings spread.
She leaped onto the base of it and climbed onto a wing, her hands trembling and wet. Pops sounded around her.
She threw herself at the wall. The edge drove the air from her lungs.
She scrabbled for purchase and felt the bricks scrape the flesh of her forearms as she clawed herself forward.
She threw her body weight over the wall, the bricks catching at the tops of her thighs, landed hard and wrong, and ran as hard as she could, her bare feet splashing in startlingly cold water. At the end of the street was her black taxi, waiting. She flung herself at it, tore open the door, threw herself into the back, and slammed the door.
The taxi driver looked back, his face melancholy and disinterested.
"Drive," she gasped. "A thousand. A thousand dollars. Go."
His eyes registered his comprehension.
He drove.
Then she saw the blood streaming from the bullet wound in her thigh.
They started off fast, and the car fishtailed for an instant before the driver got it under control.
"Where to?" the driver called back to Helen.
"I don't know—just drive!" She couldn't even process what might be happening behind them. She was too busy trying to figure out what the hell she was going to do about the bullet wound in her thigh. Her trembling hands hovered over the blood-bubbling hole.
Sirens sounded behind them.
"I'm not doing a full-on chase," the taxi driver yelled. "I'm not risking my license."
"Yes, you are," Helen shouted. "Another thousand dollars for you, okay? Just don't let me get caught! Please!"
In the rear-view mirror, she saw his eyebrows come together, but he kept driving.
They sailed around a corner, the back of the car swerving in the inches-deep water.
"It's too deep," the driver called out. "We can't go through floods."
"So get us to higher ground!" Helen yelled back.
Was there a bullet in her thigh? Did she need to get it out? Had they hit an artery? She wished she knew _anything_ about first aid.
The car swerved violently, tossing her to the other side of the passenger compartment. She shrieked. The car swerved again and she scrabbled for the handle on the roof, over the door. Her trembling hands were slick with rain and blood. They slid right off and she collided with the other door.
Car horns sounded, too close—frighteningly close.
Helen looked forward and saw headlights coming right at them. She bit back another shriek. It would have been pointless.
The driver's arms were stiff between his body and the wheel as he strained to maintain control of the car.
Other vehicles flashed past close on both sides.
If they got caught... if they got caught, it was all over. Her breath came harsh in her throat. Jail. And death. Death without her black pill.
Something big flashed in the darkness and collided with the door of the car with an immense muffled crash.
# 20 Days, 3 Hours
The car slipped and spun, then regained traction.
"You're paying for that!" the driver yelled.
"Fine! Don't get me killed, dammit!"
Then sirens flew past. She saw the flashing lights go by as the taxi whipped to the left, going up, above something and under cover. They skidded to a halt.
Helen tried to catch her breath. She held up her shaking hands. They were covered in blood.
"Are they gone?" Helen asked. The car was oddly quiet now that it was stopped. "Are we safe?"
"They went past," the driver said. "You're paying for that, I'm telling you!"
"Fine! Look, I'm bleeding everywhere. Get me to 3rd and Preston." It was where the twenty-four-pawn shop was. Egemon knew this world. He would know what to do.
The driver said nothing, just unexpectedly got out of the car—was he going to leave her here alone?
She turned to look anxiously out of the windows. He opened the trunk, then closed it again, now with a red satchel in his hand. He got in the car and threw the satchel into the back seat: a first aid kit.
She struggled to calm down and focus on her task as he drove. The kit included large gauze pads and surgical tape. After slipping around in a copious amount of blood as she tried to examine her leg, she figured out that she'd been shot in the back of the thigh and the bullet had gone straight through the front.
She put layers of gauze pads on both sides and wrapped them in all the surgical tape. Everyone in action movies said to apply pressure. She pushed as hard as she could with one hand on each side of the wound. How hard was too hard? Or was there any such thing?
Sweat or rain ran down her back and chest.
The taxi stopped. "We're here," the driver announced, eying her unsympathetically in the rear-view mirror.
She stared out of the window at the pawn shop. She was still in disguise and beaten all to hell, not to mention barefoot and covered in blood. She would have to walk twenty yards under street lamps. Rough men loitered under the awning of the pawn shop, and they would remember her. Soon she could have a price on her head, and then they might turn her in—even if she paid them off now. No, it was better if they didn't see her.
She turned to the taxi driver. "I'll give you another thousand dollars if you'll go in there and ask Egemon to chase off those men."
He looked at her, his dark eyes reluctant. He would go home tonight having had the single most profitable day of his life, but he surely didn't want to leave his car in the possession of a madwoman. His car was his life.
"Two thousand more," she said.
He put up his hand.
She handed him her cash card from her purse, and he scanned it with his e-paper in the front, then handed it back.
He got out slowly, the car shifting as he took his weight off the suspension, and trudged out to the pawn shop. Moments later, he came back out of the shop. Egemon followed him.
The pawn shop owner said a few words to the men outside and gave them cash cards. They cleared off. By then, the taxi driver was back at the car with Helen.
Helen got out, and when she put weight on the injured leg, she bit back a cry of pain. She trembled from head to toe, and all her muscles felt weak. As the cab pulled away, she looked at Egemon there in front of the shop, four lanes away. He might as well have been a mirage in the desert.
He saw her across the street, and an expression of concern flashed across his face. He jogged across the street to her and looked her up and down. The expensive dress sodden with rain, the bare feet, the blood dripping off her hands and her leg.
He didn't say anything. He just picked her up with strength she hadn't suspected.
"You're going to get blood all over you," she protested.
"It's okay," he said. "Shhh."
She closed her eyes and unashamedly clung to him and his warmth. A few moments later, she heard the jingle of the bell on the pawn shop's door, and it almost sounded like coming home.
Egemon took her into the back of the shop, where a bed, a desk, and a couple of chairs were clustered in the other corner opposite the antique toys. He set her gently on the bed and went to make a phone call while she took off her wig and removed what was left of her disguise with makeup wipes from her purse.
In short order, a friend—or perhaps relation—of Egemon's named Zara arrived. The stout woman, her graying hair in a bun, looked over Helen's many injuries and tsk-tsked. To Helen's relief, she asked no questions, only murmured about how dangerous things could be "out there" and then lapsed into soft singing in Greek.
As Zara treated her, Helen realized that she was a wreck. The skin on her hands, forearms, and thighs was torn up from climbing over the wall, and her ankle was sore and swollen from the jump—not to mention the bullet wound.
Zara had also brought her a change of clothes. They were comfortable and soft and that was what Helen wanted right then more than anything in the world.
When she came out of the back room in the new clothes, Egemon was leaning against the counter with his arms folded, his e-cig dangling from his lower lip. Zara was tidying up her medical supplies.
Egemon shook his head at Helen and took the e-cig out of his mouth. "All right, Ms. Robin Hood. Look, you are pissing them off, okay? You are embarrassing them, and to them, there is nothing worse you can do. They will stop at nothing to get you now. Maybe you need to retire. You have not done enough yet?"
Helen shook her head. She felt sick down to her bones. "There's no retiring for me. Just going till I can't go anymore."
He gave her a long look, his dark eyes drilling into hers. "Well, you will have to take a break until that leg gets better. You cannot run for a while. Not for at least three or four weeks."
She looked away. Not for three or four weeks meant never again.
Zara brought Helen some extra prescription pain pills, which she accepted with thanks and a warm smile. Zara made her feel mothered in a wonderfully comfortable way. Spontaneously, Helen offered the other woman a hug, and Zara squeezed her with a gentle " _Paidi mou_." Helen took it as an endearment.
Turning to Egemon, Helen offered him her hand. He took it with a questioning look.
"I may not be able to keep doing what I've been doing," she said. "I'm not sure what I'm going to do next, actually. But I may not see you again. I just wanted to thank you for everything."
He didn't let go of her hand. Instead, he took it in both of his. "It does not have to be like this." His gaze intensified. "Meet me for coffee. Tomorrow."
Heat rose to her cheeks, but then she fought back her instinctive response. It made no sense to go down this path. Not for a woman with a death sentence.
"I wish I could," she said. "I would have liked to."
As she pulled away her hand and went out, he put his e-cig back in his mouth and leaned on the countertop, his face inscrutable.
She summoned a self-driving taxi through Flyte. She had to grit her teeth against the agony in her stiffening thigh muscles as she got out of the Flyte car and limped to Old Blue. At least it wasn't raining.
Once in her car, she beat her forehead slowly against the steering wheel. She was near to tears, but she forced them back. Maybe she was at a standstill for now, maybe she had gotten hurt, maybe she couldn't go on doing what she had been doing, maybe she would get caught any day now.
But she wasn't done yet, by God.
Not done yet.
Helen drove to her condo with her injured leg aching. Sharp pain came with every movement of her foot on the accelerator, and braking brought new waves of agony. She wished she had taken one of the pain pills before leaving the pawn shop. She clenched her teeth and carried on, but by the time she parked in the garage, her muscles were stiff and she could hardly move. She sucked in air through her teeth as she pulled herself out of Old Blue and balanced on her other leg.
Was she even capable of getting to her condo by herself? She looked around for something she could use as a crutch or a cane. Nothing. And it wasn't like she could call Mandy to come help her to the condo.
She ground her teeth again. There was nothing else for it. She took a few deep breaths, then set her jaw. She focused on her breathing as she set out. Each step shot agony through her leg from hip to heel, but she kept putting one foot in front of the other. Within ten steps, she broke into a sweat.
As soon as she turned down her long corridor, the familiar smell of mildew welcoming her home, she saw a young couple making out in the hallway a few yards ahead, their arms wrapped around each other. Smacking sounds made Helen grimace.
The guy pressed the girl against the grimy wall as he kissed her with adolescent passion, blocking Helen's view of the girl's face, but Helen would have recognized her daughter's blue plume of hair anywhere.
Helen lost her focus and staggered. She put too much weight on her injured leg, and pain jolted through her. She winced and put a hand on the nearest wall to balance herself.
Could she still escape notice?
The couple exchanged the angles of their heads, and Mandy saw her. Her eyes widened.
Of course Helen couldn't have been lucky.
Why the hell hadn't she gone to her sleep locker instead of home?
Mandy leaned her head back against the wall, her eyes closed. The boy took it as an invitation to kiss her neck before she pushed him off.
"Hey, Mom," Mandy said, her voice lower and slower than usual, her eyes heavy-lidded. "Whatcha doin'?"
The boy turned around quickly. Helen took him in at a glance. Leather motorcycle jacket, skinny, long hair tied back, a poor excuse for facial hair: the lead guitarist from Mandy's rock band.
Helen had lost her focus, and tears threatened to spill from the sheer agony of standing. She wasn't sure she could speak.
But she had to.
She strained toward normal Mom behavior. "It's late. You should go to bed."
Mandy laughed gently. "I know, right?"
She was stoned.
"Hi, Mrs... Whatever," the boy offered. He had the same glassy-eyed look.
Helen barely glanced at him. "Kid, go home. Right now."
"Mo-o-om," Mandy protested weakly. "I don't want him to."
Helen took a deep breath. "I want you in that house right now. Kid, go. Home."
"Yeah, okay, Mrs..." He looked at Mandy for help, but she was staring with wide-eyed betrayal at her mother. "Sorry, Mrs..."
He sketched some sort of little bow and slunk down the hallway.
Mandy was still staring at her mother, her head tilting as she no doubt began to realize how late it was for her mother to be getting home... and possibly that she didn't recognize her mother's clothes... and maybe that her mother was standing very still and holding on to the wall.
Helen let go of the wall and forced herself to take three steps. With her remaining strength, she pointed at their door. She didn't want to do this, but she had to make Mandy get away from her before she figured out that her mother was injured. "Go in and go to bed. I don't want to see you again until morning."
"You are the worst mother," Mandy said. And she went inside.
# 19 Days
Helen went straight to bed with intense gratitude for the sleeping pill, or, as she decided to start calling it, the lights-out pill. It turned the lights off on the pain, the bone-deep exhaustion, the residual panic from the escape, the terrified anticipation of a SWAT team breaking into her apartment and dragging herself and Mandy out in handcuffs—all of it.
As she came into consciousness now with the bullet wound in her thigh stabbing agony, she took a full dose of Zara's pain pills along with her waking pill. She lay on her bed as she waited for the medications to take effect. Her heart pounded so hard, it shook the bed.
Jessie came in and nestled against her back, and she turned over to pet him.
Her thoughts went to Mandy, of course. She wondered how long Mandy would stay angry. If the past were any guide, Helen could expect two or three days of the silent treatment. Helen thought over their encounter the previous night, trying to imagine some way she could have handled it better. _Welcome to parenting_ , she told herself. _Twenty years in and you still haven't gotten it right._
She often felt that she couldn't get anything right. And this Robin Hood Thief thing was falling right in line with the rest. Her clumsy failure last night had risked the life and freedom of not only herself but also Egemon and possibly Zara. And a taxi driver.
_It's not enough,_ she thought. A dozen stolen knick-knacks and three donations. _It's not nearly enough._ But if she could hardly walk, let alone run, she couldn't risk entering other people's homes—not for burglaries, and not for crashing parties. Not anymore.
Egemon. Remembering how he'd swept her up and carried her into his shop last night brought a trace of a smile to her lips. For a moment, she'd felt blissfully safe and cared for. She remembered his invitation for coffee. She wanted to dwell on the possibilities that might lay there, but she dismissed them with bitter pain. Love was for those with years left to live. Like Mandy.
Mandy and her lead guitarist. At least she had someone. Hopefully he would be of some use to her when Helen was gone.
Failure and exhaustion felt like lead weights in Helen's stomach. She felt so sick and dead and empty that even getting dressed seemed too difficult a task to imagine.
And yet life went on.
Life went on until it didn't.
And she still needed to pretend to go into work for her daughter's benefit.
She got up and got dressed.
On the way out, she saw Jessie's dog food bowl, empty.
Of course it was empty. Of course Mandy hadn't seen to it. The dog would starve to death after Helen died.
_Don't be dramatic,_ she told herself. _Mandy will grow up just as soon as she's forced to... and not a moment sooner._
The bowl looked grimy, and she picked it up to wash it. Jessie leapt off the loveseat and rushed over, eager to be fed, and pawed at her legs. "Down, puppy," Helen said without conviction.
Her arm and leg muscles spasmed, and she fumbled the bowl and nearly fell. She caught herself by grabbing at the sink, and she cursed her faltering body. Jessie made a beeline for the bowl, sniffing to see if food had appeared in it yet, then looked up at her accusingly. "Just a minute, puppy," she told him weakly.
Once she felt that her muscles were stable again, she gently pushed Jessie aside and stooped to pick up the bowl again, and as she did, she saw that someone had taped a plastic baggie to the underside of the bowl.
Her forehead furrowed as she pulled it off.
The lost cash cards were inside.
"Oh my God..."
Of course. Putting them there was exactly something Helen would have done. Mandy would never have found them there—not in a million years. And no burglar would ever have looked there.
She leaned back against the counter and laughed until she cried.
Helen fought an hour and a half of traffic on her way to a public library. Pouring rain was half the problem, and the rest was construction. Then, just as the construction traffic eased, Helen hit another bad patch of traffic and then a thirty-minute detour around a teeming, angry crowd and a police blockade—she never saw what was going on.
Once she finally reached the library and got settled in, she checked for news about herself with trepidation, and immediately found her fears confirmed: Last night's target had reported everything to the police, including how their security forces had shot the Robin Hood Thief, and the police chief had given a press release. The journalist at MCCTv had spun it into a first-rate story about how the Robin Hood Thief's time was running out.
With a heavy sigh, she checked her new Whatsit account. Fans had posted and commented on the news story. A lot of people were expressing their concern. She'd received a message from Christian Smith at LSTV: [ You OK? ]
Perhaps in a moment of weakness, she replied: [ Yeah, thanks. ]
She looked more closely at his profile picture. He looked older, a bit stout, with coarse salt-and-pepper hair in a buzz cut and a square jaw that gave him a tough, jaded look. His bio said he was a former Army aviator before becoming a journalist.
Back on Helen's account, she saw that someone had created an enhanceable post—one that others could freely modify and re-share—that displayed the rising tally of money collected and given to charity by copycats around the United States. If the numbers could be trusted, they'd donated almost fifteen thousand dollars now. Nothing about arrests yet.
Helen shook her head. She posted: [ You guys, I really did get shot last night. I almost got caught too. I'm okay, but listen, it could happen to you too. Think about what you could lose. Please don't take the chance. I love you, but don't do this. ]
Helen stretched and sighed. Next, she would compose another letter to the media. If she could do nothing else, she at least wanted to get full credit for what she _had_ done—including the blackmail.
Maybe there was a little ego in all of this for her after all. Just a little.
She considered sending the letter as an email to LSTV, then wondered if she weren't better off just posting it on her own Whatsit profile.
She could pay to boost it, even project it into the 3'scape in places the Entitled would see. Like their offices, and their bank buildings, and their tennis courts.
Yes. She wanted the wealthy to see this message. She would set it to broadcast to individuals with incomes over a million dollars a year. Afterward, she would share it privately with her own fans. _Plus whatever enemies are following me on my profile_ , she thought wryly.
> Dear lovely wealthy people,
>
> Have you noticed the news about how Mr. Brock Tolbrook and Mr. Emanuel White both suddenly decided to give enormous sums of money to the very same charities they'd first donated to at the Net Worth Notion?
>
> (Go ahead, look it up. I'll wait.)
>
> Now, don't you wonder what the deal is? Does it maybe seem a little... unusual... that these two unconnected people took these very similar actions?
>
> Well, my friends. I would like to step up and claim some credit for that.
>
> Those donations were made on behalf of the Robin Hood Thief.
>
> They'll never admit it, of course. They'll protest that they just felt really generous all of a sudden. But will you believe them?
>
> I will not tell you why they agreed to do it, because I promised that if they made the donations, I wouldn't spill their secrets. And there has to be some honor among thieves, after all.
>
> For the rest of you who have secrets (and who doesn't have one or two?)... I can save you some trouble and effort and considerable embarrassment. If you don't want a little visit from me or one of my agents, you can just go ahead and proactively make a few donations.
>
> You don't have to make them so public as these last two donations. You can make a few sporadic and private contributions that don't catch the news—just so long as they total a million dollars or more. Trust me, I'll know.
>
> And just a word of advice: I suggest giving to charities you think I might approve of.
>
> Love, the Robin Hood Thief
She posted it with a wry grin. It was a bluff, of course. She had no way to know whether any of them complied or not, and no agents to send after them. But if her bluff frightened even one more rich guy into giving a big sum of cash to charity, it was well worth it.
Another message from Christian Smith already: [ Any chance I can snag an interview with you soon? ]
She ignored him. Given that her career in crime was over, she had nothing to say. Anyway, he could write a story based on the letter she'd just sent out.
Then she slumped back in her chair and stared at the wall above her projcom screen.
She wasn't done, she realized. She kept trying to tell herself that it was over, but the larger part of her wasn't buying it. That part of her insisted she just needed a new strategy. But what? She was out of ideas and running out of time.
There was something about these fans she had now. About Whatsit. Something. But her brain just couldn't put it together.
Helen stared at the projected screen, her eyes glazed.
A weather alert popped up. A polar vortex would be coming sometime in the next week or so, bringing temperatures down below freezing. _I should still be alive for that_ , Helen thought. _I'll have to find my jacket._
New comments rushed in on Whatsit in response to her latest letter.
[ Is she saying what I think she's saying??? ]
[ B-L-A-C-K-M-A-I-L baby ]
[ Holy shit. How'd she do that? ]
[ I bet she's bluffing ]
[ who cares as long as it works? ]
[ That is pretty much the coolest thing ever. ]
Helen grinned a little.
It wasn't that late, but she could plausibly be home from work now, so far as Mandy was concerned, and she was too tired to stay and work any longer. She needed to go home and rest.
A chime from her e-paper. She pulled out her e-paper to see what Mandy wanted. Her message said: [ Are we srsly out of yogurt? ]
[ Yes. Srsly. So very srsly. GO GET IT YOURSELF. And pick up some dog food while you're at it. ]
As she stood up slowly and her thigh screamed protest, the locket holding the black pill swung out of her shirt where the top button had come unbuttoned.
"I know, I know," Helen muttered to it as she tucked it back under her shirt. "You don't have to remind me."
Twenty minutes later, Helen rushed through a thunderstorm back to the library, back to a projcom.
Her brain had finally caught up, and her heart was racing in excitement. The answer had been obvious all along.
She couldn't physically go into people's houses anymore... but she might not have to.
How many times during this misadventure had she wished she were a hacker? Or knew one?
These fans of hers on Whatsit... they were anti-establishment, anarchist types. What were the odds that any of them had that skill set?
For a moment, she let herself daydream of the things she could accomplish with the right partner... heck, what _couldn't_ she accomplish?
She posted again: [ Dear my lovelies, if any of you are hackers, I would love to have your help to expand the scope of what I'm doing. Not just blackmail. Much more. Interested? ]
Answers came so quickly that Helen didn't know how to react. She immediately recognized her own naiveté. How would she vet these people?
Then one message got everyone's attention.
[ ok you amateaures back off. Its Cobalt and i will take it from here. Robin, u got fifteen minits to message me at this profile. The rest of you mfs mind ur own beeswax. Peace/up ]
[ whoa ]
[ shit yall shit just got real ]
[ *bows and scrapes* Oh holy one, guide us ]
[ Cobalt come on let us play too ]
[ oh come on no way is it is the real Cobalt. Why would he post as himself? That would be dumb. ]
[ Maybe he's not scared. I wouldn't be. Posting on Whatsit is easily anonymized. ]
The "real" Cobalt? And where had she heard that name before?
Conscious of the fifteen-minute time limit he'd given her, she took two minutes to look him up.
> _Cobalt. American hacker. Trademark phrase: peace/up_
>
> Cobalt broke into the hacking scene with a bang in 2046 by deleting the identities of thirty US legislators. He took credit for the hack promptly, stating, "Fuckers deserve it for thinking theyre better than us and not speaking for us like their fucking job says theyre suppose to [sic]."
>
> A few months later, Cobalt claimed credit for hacking and defacing the New York Police Department's website, leaving the message, "Cowards and bullies. Also we suck dicks."
>
> In May 2047, Cobalt defunded a school district. Three months passed before the school recovered a small portion of the money. Their data and financial recovery experts appraise the likelihood of recovering the final sums at less than 5%.
>
> In September and again in November of 2047, Cobalt worked with anarchist group Sons of Man to flash-riot two shopping malls. Sons of Man enter public spaces quietly and individually, without drawing attention, then riot from within on cue. Cobalt added to the two performances by hacking the PA systems and providing a soundtrack of punk music.
>
> Cobalt has stated several times that he is already independently wealthy through hacking, although he has not divulged any details. According to a text message sent to a friend in December 2047, Cobalt stated, "Yeah no i dont have a fuckin real job. I dont want one. Why would anybody [sic]."
>
> Apart from working with Sons of Man, Cobalt is reputed to work with a handful of other hackers on major projects. Several have claimed to be in his small "posse" but Cobalt has not publicly acknowledged any of them.
>
> In May 2048, Cobalt claimed responsibility for bombing the federal courthouse in downtown Orlando. He stated that he disabled security allowing anarchists the Boom Boys to enter the building. This is the first known occasion that he has partnered with the Boom Boys.
Of course. The federal courthouse that was bombed just days ago.
Helen's daydreams expanded to dizzying heights. Maybe even more was possible than she'd thought.
A message came through from Christian: [ Congrats on the donations you 'secured.' Nicely done. Loved your letter. ]
The guy was persistent—she had to give him that.
Then a massive boom of thunder shook the building, and everything went dark.
_Shit._
She looked around as the emergency lights flickered on at the corners of the ceiling. The projcom keyboards and displays were still off. The people around her grumbled and began to pack up their things. A library employee with frizzy hair called out, "It's twenty minutes to closing anyway. Might as well go home, folks."
_No!_
Cobalt only gave her fifteen minutes to reply, and she'd wasted five of them already.
An older man with wild gray hair stood up, took in a deep breath, and started shouting hoarse profanities. He wanted his computer back on.
The people nearest him moved quickly away. The employee took a step back, eyeing him for any signs of a weapon.
Helen hesitated, wondering whether there was any chance this guy's obscenity-laced tirade would actually get the computers back on, but when the man picked up a chair and threw it, she grabbed her things and hurried out into the driving rain.
How could she get to another anonymous projcom fast enough?
A Scenie electronics store. They were open late, and they always had projcoms and clamshells and Earworms set up for people to try out. With a quick voice command to her e-paper, she learned there was one eight minutes away.
At the Scenie store, she shielded her eyes against the overly bright neon-colored LED lights in a too-white store. She hobbled up to a clamshell and sat at one of the neon-colored stools, dried her hands on her damp clothes, logged back in to Whatsit, and typed rapidly to cobaltx98. [ I'm here. The Robin Hood Thief. What now? ]
She hit send with less than a minute to spare.
She stared at the inbox anxiously and scratched mosquito bites acquired on the way in. A salesman approached. "Anything I can help you with, ma'am?"
"No." She gave him a flat stare, and he took the hint.
She kept watching Whatsit. It occurred to her that a hundred other people would have messaged Cobalt by now, the FBI and CIA among them. That explained the narrow window for her to contact him. Probably this was an account he'd delete after they talked.
A little blue LED came on next to the camera on the front of the clamshell. She frowned at it at first, then figured it out and hastily put her thumb over the camera. "No way," she muttered. She didn't have a disguise in place right now.
She saw a return message show up. [ Show urself ]
It was a painstaking struggle to type one-handed and without taking her thumb off the camera. She had to resort to text-message language. [ Not going 2 risk that. This cld be a trap. U prove who u r. ]
She looked around anxiously, afraid that she would see a sea of blue lights pop on next to the cameras on all of the computers in the showroom. What would she do then?
Her vision suddenly went blurry, and she blinked furiously, which didn't help. She cursed. _Not now. Why are you doing this to me now?_
A message came through and Helen leaned close and blinked hard. It took an agonizingly long time to make out the message.
[ Well i cant think of a way. How cld i prove it when u dont know anything abt me. Not like im gonna use statesecure. ]
StateSecure accounts were government-verified. Using one would prove, but also reveal, Cobalt's true identity.
She felt a presence closing in on her right and glanced over. A figure in a suit was surveying the products on her table.
She tried to minimize the Whatsit window while keeping her thumb over the camera, but she couldn't find where to click in all the fog.
The suit leaned into her space. "Can I look at that one?"
Helen glared at him through her misty vision. "No."
He stared at her for a moment with an expression she couldn't make out through the blur. Then he stepped away and looked at something else.
Cobalt—if it was him—had replied again. She rubbed her eyes with her free hand, which didn't help, and leaned closer to read. [ Look im gonna help u. Youll see im legit. A cop cant pull off what im going to. So anythin u need with computers... data... go bigger. Lets do some damage. Message me tomorrow at profile 76xtlaboc. Tomorrow, not today and not after. Peace/up.]
The blue light next to the camera turned off.
Helen committed the profile ID to memory. It was easy once she saw that the letters spelled cobaltx backward. Then, with great difficulty, given the clouds over her eyes, she deleted the browser's history and closed it.
She looked around the Scenie showroom, gauging how bad the blurriness was. She could hardly see beyond a few yards—everything faded into clouds. Driving would be suicidal.
Just as she was getting truly frightened, her vision cleared.
Helen let out a sigh.
Was that going to happen again?
Who knew?
But for the moment, who cared? _Go bigger,_ Cobalt had said. Hope returned in full bloom.
She left the Scenie showroom and limped back to her car with her head held high.
She fell asleep that night with her thigh throbbing but with possibilities spinning in her head.
# 18 Days
> Dear Cobalt,
>
> I know we could do more blackmail stuff, but I have a way more fun idea—if we can pull it off.
>
> The sentries.
>
> Every rich house I've stolen from has had several sentries. They watch, they listen, they talk, and they're armed to the teeth. If we can hack them—especially if we can hack them _en masse_ —we can force people to turn over a lot more money to charity.
>
> I don't have a lot of time, though. That's something I probably should have told you right up front. I figure we've got a couple of weeks, max, to do everything we want to do together. So if we can get this one going in a few days...? I'm coming up with other ideas, too...
>
> What do you think? Is it doable?
>
> Love, the Robin Hood Thief
The response was almost immediate. [ Oh nice... i like it. Could be really hard tho. Lemme talk to my posse. Message me @ profile skiesblue tmrw. Peace/up ]
Helen eased back from the library projcom keyboard and absently stroked her injured thigh.
It wasn't a no.
She smiled.
As she stood up, a wave of intense heat came over her. By the time she made it to Old Blue, sweat stood out on her face.
# 17 Days
> K so i did like a whole lot of research into this. Its not easy i mean security is the 1# thing with these companies right. But there are only four different companies making most of the sentries and they have the same propiatery software supplying three of the companies. So if we hack into that one software we can get a lot of the sentries.
>
> But actually i dont recommend that we do that. Too ambitious to try to get three makes of sentrie because they do have different command packages. I think we should just try to get into one make. That will still get like %40 of them which is a lot.
*
> Okay, I'm not sure I understood all of that, but if we can get 40% of the sentries, that's great. How will it go down?
*
> So well get into that software and write a new command package and once its uploaded everywhere we pull the trigger. During busness hours so they can do whatever paperwork or bank thing u want them to do.
>
> Im thinking what if the sentries hunt whatever owner is home at the time and actviate weapons and threaten them to do it? And if no one is home then it just doesnt do anything. Cuz the timing is going to be important. Everything has to fire off at once. If no ones home we just miss the chance at that house.
>
> But if u figure we get %40 of rich houses and then maybe %10-20 of people are at home thats still going to be thousands of people we get.
*
> Okay. I love it. I'm really impressed. I understand you say it's not easy, but from out here, it sounds easy. What all do you have to do to make it happen?
*
> Uh its like way too hard to explain all the details. To someone who doesnt know this stuff.
*
> Fair enough. I guess I don't really need to know. How long will it take?
*
> Its not like the movies it takes time to do stuff. Even with my lil posse it still itll be like days and thats working like nonstop.
*
> Okay. A few days is okay. I hope it won't be longer. Here's what I want you to program the sentries to say...
# 14 Days
Early in the morning three days later, Helen cradled a hot cup of coffee at a dim, worn coffee shop with rickety tables, waiting for Cobalt to reach out to her. She huddled in a shadowy back corner where no one could see the screen of her brand-new clamshell. It was a burner clam she'd discard as soon as this mission was over—Cobalt's idea.
She had taken her sleeping pill early the preceding night so she could be here in time—they wanted to catch people before they left for work. Waking up came hard. The hallucinations still haunted her—Mandy's face and eyes melting off and onto the floor as she tried to pick them back up and shove them back onto her skull. Then her hands melted too...
Helen refocused on her screen.
Somehow, Cobalt was going to patch her in to the live feed of a sentry as it did its thing. It would be fun to watch... if it worked. Nerves ran through her stomach, twisting and clenching it. She'd been fretting for days that the plan sounded too easy. Surely something would go wrong. And if this didn't work, that was three days gone forever, and only fourteen left.
She glanced at the time in the corner of the screen. Cobalt was eight minutes late. He had emphasized starting on time. What did it mean, being eight minutes late?
Helen's stomach clenched again. She put her head down on the table next to the clamshell. If this didn't work... She just didn't want to die yet. She hadn't done enough good yet.
A window opened on the clamshell, and Helen picked up her head. The window was blank, empty.
Another, smaller window popped up next to it, and words began to appear in it. [ U there Robin? ]
Helen typed quickly. [ Yes. What's happening? ]
[ Almost online with a random sentry so u can see the fun. Hang on a sec. ]
Minutes passed agonizingly slowly. Helen's stomach did flips.
[ Ok we only have so much time b4 the security company overrides us. Plus theres the kill code the owners always have that let's them shut down a sentry. Im not %100 sure we turned those off yet. ]
Helen clutched her twisting stomach. Security company override? Kill code? Those were two different ways this plan could fail in an instant, neither of which she'd heard anything about. No wonder it had seemed too easy: Cobalt hadn't told her everything. What else had he left out?
The other, larger window on her screen flickered and then a high-res live-stream video appeared from the perspective of a sentry. The floating robot passed down a hallway in one of the outrageously luxurious mansions Helen had become all too familiar with lately. It glided through a large antechamber decorated with a tree-sized topiary, then into a dining room.
An old, old man was shuffling from the kitchen into the dining room. His back was stooped, with his pants pulled up in the front nearly to his sternum, and his lower lip jutted out like a shelf. He carried a cup of tea with its china saucer. His face was sorrowful.
Helen put her hand to her mouth. "Oh, no," she breathed. He looked so helpless, so old, so pained.
She typed quickly. [ This guy is so old. We're going to give him a heart attack! ]
The response was fast. [ Too late. But anyway probaly not. Bet hell be fine. ]
"I am not reassured," Helen muttered to herself.
The old man finished his tea and set it on the dining table, then opened a drawer in a dining room hutch and took out a hedge clippers and a pair of heavy gardening gloves. He was about to put on the gloves when the sentry glided closer, drawing his attention.
Helen's script began to play out while she wrung her hands and hoped the old man wouldn't die on the spot.
Her script satirized the original recording she had first heard at her first robbery, which started, _"We apologize, but_..."
"... but this is the Robin Hood Thief, and you're going to make a donation to a charity for me. Right now. Would you please find a phone right now and call your banking professional?"
The old man sputtered a furious protest, moistening his jutting lower lip. "You want me to do _what_ now?"
"You have very little time to locate a phone and call your banking professional. We advise you to move quickly."
Helen winced a bit at her own wording. This old man shouldn't move quickly—he might fall and break a hip.
The old man mashed his lips together, then threw the hedge clippers at the sentry.
Helen instinctively flinched.
"How dare you?" he shouted. "Are you there? You Robin Hood Thief? I oughta give you a piece of my mind! You nasty bitch! You fucking cunt!"
Helen recoiled, her eyes wide. The old man didn't look so pitiable all of a sudden.
"You think we should all pay for your little waifs and serfs! Your wards of the state! Why don't you go and get them some jobs! Oh, but no, that would require them to get out of bed in the morning and get off their lazy asses and—"
The sentry's script continued. "Our next action will be unpleasant, and we hate to be ungracious thieves."
Helen laughed out loud and clamped her hand over her mouth. Her adrenaline was surging.
The old man's eyes lit up as he remembered something. "Kill command! Kill command delta ninety."
A pause. The sentry whirred quietly. Helen waited at the edge of her seat, her fists clenched. "Come on, Cobalt," she whispered.
"We're very sorry," the sentry declared, "but that command is not currently available."
Helen nearly threw a fist of celebration into the air, but managed to jerk it back. She glanced around the coffee shop. No one reacted to the interrupted gesture.
"You lazy, worthless scum of the earth!" the old man raged, his jutting lower lip wet again. "You think you have any right—"
With a series of clicks, the sentry's three gun turrets rattled out from the casing.
"We will fire upon you in twenty seconds unless you find a phone and call your banking professional. Twenty... Nineteen... Eighteen..."
The old man blanched. He gave the sentry a look of pure hatred and then shuffled toward the kitchen. The sentry followed.
Helen typed quickly into the other window. [ We're bluffing, right? I told you to make sure no one would get hurt? ]
[ Yea. Just warning shots. ]
The old man picked up his e-paper and tapped the screen. A moment later, he said, "Mr. Pierson, please."
Five minutes later, with a little guidance from the sentry, a donation of two million dollars had been made to today's charity of choice: Justice for All—where Helen had worked for eleven years.
So her boss's boss had been a jerk when she needed her job back. It was still a damned worthy charity.
Cobalt typed: [ I just checked in anonmusly. This is what the donations manager lady at Justice for All said: "Yes, in the last few minutes, my notifications system has gone off so many times, I've turned it off. We're receiving three and a half million dollars a minute and it's going up. Do you have any idea what's going on??" ]
It lasted six minutes before the security companies managed to shut it down.
Helen and Cobalt exchanged virtual high-fives.
[ Do you mind if I tell the public what we did and give both of our names? ]
[ Nah dude cool with me ]
Helen sent a message to Christian Smith at LSTV: [ The richest of the rich just donated millions of dollars to Justice for All. I had a little something to do with that, along with the hacker Cobalt, and I think it might make a good story. Just fyi. ]
Then Helen closed her clamshell and paid her tab at the coffee shop's counter.
She walked out floating on air, her back straight, the pain in her thigh present but irrelevant. Now this— _this_ —was what she had hoped for all along. Three and a half million dollars a minute for six minutes. That was _twenty-one million dollars_. Her grin was so broad, her face hurt. She just wished she could see Oliver's face when he heard the news.
As she left the coffee shop and limped down the street to her car, she caught a glimpse of her own face—almost—out of the corner of her eye. Actually, lots of images of her face—almost.
They were artistic, stylized drawings based on law enforcement's compiled image of the Robin Hood Thief. About fifty of the large posters looked out from the abandoned building on the other side of the street. They all bore the message, "We love you too, Robin Hood Thief."
That afternoon, Helen went home and curled up in her bed to watch the news on her e-paper. It was so much fun to see the hack take top billing well into the night. Thousands of victims and their families had made panicked calls immediately afterward: to law enforcement to file charges against the Robin Hood Thief and Cobalt, to the security companies to complain and threaten lawsuits, to the media to demand coverage. The security companies rushed to give press conferences in which they strove to give the appearance of action without actually apologizing or promising anything. Government regulators launched an official investigation.
Of course, MCCTv called it an act of domestic terrorism. Helen hadn't thought of it that way, but perhaps the shoe fit. She certainly hoped to terrify the Entitled.
She was both amused and dismayed at the parade of public figures and politicians who spoke out to denounce the hack. _Come on,_ she wanted to say. _No one got hurt. I just scared them a little._
But on the up side, Helen's favorite politician, Elaine Decatur, president for one term in the late 2020s, posted on Whatsit, "Radical activists Robin Hood Thief and Cobalt have done what many have only imagined—liberated wealth without spilling a single drop of blood."
Maybe it wasn't an endorsement... but it was oh so close.
A few minutes later, Helen's profile on Whatsit disappeared, deleted again. She rolled her eyes. She'd just make another one.
Around three o'clock, the security companies issued formal apologies and started the recall process. A nationwide recall was unavoidable—not a single affected person wanted a sentry present in their home now, even if it was manually disabled. And their owners soon discovered that they were incredibly difficult to disable. LSTV showed one man who shot, drowned, and electrocuted his sentry to no avail. They'd been made to withstand every conceivable attack from the outside.
Helen made popcorn.
# 13 Days
On her way to a different library the next day, Helen let her mind wander into a happy daydream—one of her favorites: all the homeless and destitute people in the country with homes of their own.
Not sleep lockers—real homes. Access to everything they needed to be comfortable and functional human beings. Able to bathe, wash their clothes, cook meals. Safe from the elements, safe from other people. Hope of a job. Hope of a future.
Back in the oughts, a couple of cities experimented with giving homes to all the homeless, no questions asked—no requirements that they get off drugs or get into mental health care. They found that addictions and mental health problems fell dramatically.
Naturally, the city council members who championed the policies eventually got voted out and replaced by members whose campaign contributors included prison operators, cigarette manufacturers, security firms, and in one case, a bail bondsmen's trade group. They ended the programs.
But what if Helen could give the needy homes forever? Homes that legally belonged to them and could never be taken away?
She parked at the library and got set up on a projcom, then typed out a longer note to Cobalt.
> Okay, Cobalt, I have another idea. It's pretty ambitious, although I don't really know how ambitious. I guess it's up to you to tell me what's achievable.
>
> There are millions of people in this country living in the streets or in shacks or tent cities, while millions of houses stand empty only because no one can afford to buy them. That has always made me raving furious. I want to fix it.
>
> The only way I can think to do that with hacking is to wipe out all the mortgage records. As I understand it from my research, if people move into an unoccupied house and there's no proof that it's not theirs, then they should eventually be able to claim it. It looks like it's called _adverse possession_.
>
> So if we destroy the records, then put out an announcement that all unoccupied homes are up for grabs, we can house practically all of the homeless and destitute people in the country in one fell swoop.
>
> We should warn everyone that it won't work on occupied houses, though. There will be proof or at least eyewitness testimony about who really lives there. And I don't want to risk any violence, so I want to warn people away from that.
>
> What do you think? Is it just wishful thinking?
She sat back to wait, but only a few moments passed before she got a reply.
[ Okay so maybe this ones pretty complicated but i bet we can do it. I mean destryoing data is kinda what hackers do, right? ]
[ That's awesome. Is there anything you _can't_ do? ]
[ Hahahaha not rly i mean with enuf people and enuf time pretty much anythings possibl. And i have like a ton of new people who want 2 join my posse. i had like 4 before and im intervuing and testing like 20 people who want 2 join. Ur kinda bringing us together u know? That doesnt usually happen. We kinda do our own thing a lot So thats cool. Peace/up ]
A few hours later, Helen parked at the pawn shop and went in on shaking legs—shaking because of her illness and shaking because of anxiety. She didn't know how Egemon would react to what she had to say, but she was determined to say it anyway.
Egemon was checking out a customer. He welcomed her with a pleased smile, but then he saw her determined expression and took on a quizzical look. "What's happening?" he asked as the customer scanned her e-paper.
"I've just been thinking, that's all," she said.
"Uh-oh. That can be serious." He half-grinned. To the customer, he said, "Have a nice day."
Helen let the woman exit, then nervously launched into what she'd been pondering for the past few hours. "Have you ever heard the phrase _carpe diem_ , 'seize the day'? Because tomorrow we die?"
"Yeah, sure, I have heard this." He took his e-cig out of his pocket and took a drag. He leaned on his elbows on the counter, attentive.
"I started thinking about it today because someone posted an image on my Whatsit profile with it," Helen said. "I've never liked it, not really. I don't like the emphasis on a single day. It's always made me feel hopeless. Because what can you actually do in a day? Nothing big, anyway."
Egemon's dark eyes glittered with amusement. "Go on," he said.
She took his easy pleasure in her words as a good sign and forged ahead.
"But when I saw it today, I started thinking about how it relates to what I've been doing. You know, everything you've helped me with. And I've decided I like the idea of seizing _eternity_. Live like you're _never_ going to die. Because if you can live forever, then nothing's impossible, and there's nothing to be afraid of. There's nothing you shouldn't try."
"I like it," Egemon said.
"I realized I've been doing it all wrong, all this time. All I've ever done is struggle to get by. I've just focused on the day to day, on survival. Which is exactly the wrong thing to focus on."
She found herself kicking at the floor like a nervous schoolgirl and made herself stop.
"So, in the spirit of all of that... seizing eternity and all..." She took a deep breath and laughed at herself. He was happy to see her—she could see that—so she didn't need to be so anxious, and yet she was. "I came to see if your invitation for coffee still stands."
His face broke into a smile. "All of that for coffee?" he asked teasingly. "Yes, of course. Let me lock up and we will go."
He came out from behind the counter and winked at her as he passed very near to her.
The way between them was open now, for better or worse, and in that openness, desire suddenly flickered into life. Suddenly, Helen couldn't take her eyes off his well-muscled body.
Egemon went to the door, changed the sign to say CLOSED, and locked up. Then he turned around and saw Helen waiting for him, and something in her gaze stopped him where he stood.
Electricity played in the air between them, the electricity of desire and expectation. Helen was captivated by its charge. She nearly stopped breathing. She wanted his body against hers—now.
And he seemed to know exactly what she wanted.
He came to her quickly and put his lips to hers.
Afterward, they lay with their bodies comfortably intertwined on Egemon's bed in the back of the shop. Helen felt at ease. He had been a passionate and appreciative lover, and he looked relaxed and comfortable now.
Simply being near him was the most calming and restorative experience she'd had in years. Soothed by his warm animal closeness, her mind was free to tumble through her recent realization.
"This Robin Hood thing," Helen said thoughtfully to the ceiling, "it's about looking life and death in the eye and saying 'I'm in charge. I decide how I'm going to live. Period.'"
"I like where your decisions have brought you," Egemon said.
She swatted him on the chest and smiled.
"This is the whole point of life, you know that? If we're given a beautiful sunset and we don't look at it, we might as well be dead." She rolled over and admired his strong face, his shadowed eyes. She spoke emphatically. _"You_ are a beautiful sunset. And I want to admire you as long as I can."
He stroked her hair, his eyes half-closed, his full lips in half-smile. "I had no idea you were so poetic. And morbid."
She grinned, but she also fretted. Was it right to keep the whole truth from him?
"When I say as long as I can..." She hedged. "I don't think I should do this for very long."
"That decision is not so good," he said with a half-serious frown.
She let out a long breath. She didn't want to burden him with the truth. "My time is short, okay? I didn't become the Robin Hood Thief with the expectation of a long life. And I don't want you to get too attached to me."
"Are you a puppy? That my father will not let me keep?"
She laughed quietly. "Stop it. I'm being serious."
"So am I." He took a long drag from his e-cig. "Listen to yourself. You just said a while ago that you want to seize eternity. Start things as if you will be there to live them out. Why can't I do that as well? Anyway, _no one_ knows how long we will have to live."
_I do,_ she thought bitterly. _Thirteen days._
He sat up. "Listen. I like you. So, too late. Too bad. You will just have to live with it. Anyway, I will go back to prison in a couple of months. This isn't long for either of us, right? But for now, I need to go open back up. I will order us something to eat."
It might have sounded dismissive, but he leaned over and kissed her with such warmth and feeling that it took away her breath and any objection she could have mustered.
# 12 Days
Helen spent much of the next day with Egemon at the pawn shop, keeping him company while he did business and met with customers—some criminal and some otherwise. She made a game of guessing which was which, and then, after each customer, she asked Egemon whether she was right, and he teased her relentlessly when she was wrong. It felt good to joke and laugh.
He was easy to be with. She wasn't accustomed to comfortable, casual affection and attention.
When she told him she needed to get back home, he let her go with a firm, warm kiss and a "Good luck."
It was good, she decided—good to have a friendly face and an occasional refuge from the ongoing crisis of her life, no stress and no strings attached.
> KK been looking into it. Its basically a little more complicated than i was thinking. Like actually its impossible on any larger scale. Because heres the thing. The real property deed records are held by the county clerks and theres three thousand and seveenteen of those across the country and they have their data on various server farms or on third party cloud servers and then theres the backups. Theres just
Helen's heart began a gradual descent toward her stomach, which turned cold. At the same time, frustration tightened her chest. It hadn't even been twenty-four hours since Cobalt had assured her that anything was possible.
She read and reread the dismal email on the library projcom, and then another message came through, a longer one:
> Sorry. Hit send by accident.
>
> So theres no way we can blow up three thousand and sevetteen courthouses all at the same time. And it has to be the same time or they would jsut call in the cops for wehvever we hadnt hit yet. Plus we would have to get the bank records as well. These huge banks have most of the mortgages and they have databases all over the country holding the morgage records. Lots of backups too.
>
> And we cant get every mortage. I mean there are thousands of small banks that do mortgages. A lot of them have their data on shared servers and it would basically be a terible idea to blow them up because then we blow up a whole bunch of other stuff. Like who knows what. Medical records and stuff.
>
> I dont think we can go after the biggest bank, CoUS. Their security is rly good. Besides if we fail to blow up even one backup and they can restore then we fail %100. So we should try I think three of the large ones that can add up that are more local to Florida. Im guessing u want to do Flrida since u live there.
>
> Hyperius, Delton, and Orange County Bank. If u put em together they have %35 of the mortgages in Orlando. And their total number of server farms and backups for all three banks is eleven from what we can tell so far.
>
> So then my posses thinking we should go after those three banks and then go after just the Orlando county clerks. Thats like nineteen courthouses which is a lot more doable.
The message was so difficult to parse, Helen was forced to read it three times. Then she sat back with a sigh to contemplate what Cobalt had said.
The kid was planning to blow up county courthouses to take out the real property deeds. And the servers for the top three banks. With both, they would take out thirty-five percent of the mortgage records in Orlando, if all went well.
Thirty-five percent of Orlando was a profound disappointment when she was hoping for all the mortgages in the nation.
[ Why are we even talking about blowing up government buildings? I'm sorry, I'm kind of at a loss here, this is so unexpected. I was imagining hacking, not bombing buildings. ]
[ Yeah well what would tyler durden do? ]
Helen stared at the message in confusion.
[ Sorry, what? ]
[ Nevermind Anyway, we cant just delete the data becuz deleted data a lot of times isnt rly deleted. It can be recovered unless u drestoyr the physical hard drives. Thats why we need the Boom Boys. U know abt them right ]
[ Yeah. I read up about you guys. What you did at the courthouse, what was that, a week or two ago? Okay. So we have to blow up... thirty buildings all at the same moment? How are we supposed to pull that off? Have you guys ever done something that big? ]
[ well it cant be soon. Three four days minimum. ]
[ Okay. Well, I told you to let me know what was doable, and if this is it, then so be it. We're not going to hurt anyone, right? We need to do this at night when no one is there so that no one gets hurt. And make sure debris doesn't fall on neighboring buildings where there might be people. ]
[ KK yeah peace/up ]
[ Wait. I want to help, too. This is all my idea and I want to take equal responsibility. ]
[ Yeah all rRght. OK u cn blow up the Hyperius building i guess. Peace/up ]
Helen closed the browser with a strange sensation in her chest.
It took a few minutes for it to sink in. The old Helen-of-the-high-road was distant now, an artifact of a life that had already passed away. This Helen, the one with twelve days left to live, was about to blow up buildings.
She'd maybe changed a little in the past few weeks.
She went to see Egemon after that, but she didn't tell him the plan. Maybe she harbored doubts about it, or maybe she didn't want to worry him, or maybe she didn't want to risk incriminating him... or maybe it was all of the above.
Helen stared at the parking lot around her in utter confusion. Something was wrong. Very wrong.
After spending a few more hours with Egemon, she'd decided to drive home. She'd driven the usual route and entered the gate code and parked in her usual spot in the parking lot near the rusting green dumpsters and gotten out of the car, but something about where she stood was wrong. Something was out of place.
She recognized it here. The sprawling parking lot full of potholes with too few street lights, the dumpsters poorly shielded behind a collapsing fence. That one tree, struggling above asphalt, reaching for the sky, but bent and gnarled since birth. She recognized it.
Except the asphalt was newer than she remembered. That was odd.
David would be inside...
Something snapped tight into focus inside her mind.
David was dead.
She hadn't lived here in years.
A hot flush came over her as she hurried to get back into her car and drive away with shaking hands.
# 10 Days
Helen didn't speak of her episode to Egemon the next day, which she spent at the pawn shop again, but she worried about how her faculties were deteriorating and what that might cost her before the end.
After some thought, she sent Cobalt another message: [ I still want to help with the bank job, but only help. Someone else needs to head it up. I realized I shouldn't be responsible for too much when I don't really have this skill set. ]
Cobalt sent back only a simple [ OK ].
That done, Helen's thoughts turned to her daughter. Mandy had hardly spoken to Helen since the night she'd come home injured and interrupted the make-out session in the hallway. And Helen couldn't live with the silence and distance anymore. Not with just ten days left.
How could she bridge the gap now? When years of trying had failed? And without revealing her illness?
Again she asked herself when the time would be right to tell Mandy, but she still could not imagine herself saying those words. _Prion disease. Invariably fatal. The black pill._
Helen had to try to make things better before it was all over.
The next day, Helen hung out at a pizza joint across the street from her condo and watched the building until she saw her daughter go out. Then she went upstairs and pulled back the curtain that separated their sleeping spaces and took a look at the walls. She surveyed the posters of bands and musicians and guessed at Mandy's favorite, judging by the number of posters for that particular band.
Then she went out and fought the road construction and traffic and crowds to splurge on a VR Earworm with the top ten live performances by that band preloaded onto it. It was an expensive gift, not something she would normally buy.
She requested gift wrap and waited anxiously as they put the bow on. If the past was any indication, Mandy would refuse to show appreciation even if it was there. Appreciation had gone the way of affection three years ago. But Helen had to try.
When she finally got home after fighting the road construction and heavy traffic a second time, Mandy was there again, in her usual spot on the loveseat, her clamshell perched on her knees and her Earworm active. She didn't look up. The furrow between her dark eyebrows said she was concentrating.
Helen dropped her purse on the dining table and came over and sat down next to her daughter with the gift in her hands.
Mandy gave her the usual _What do you want?_ glower. It made Helen's heart sink, but she gamely extended the gift to her daughter.
"What's that?" Mandy asked, not taking it.
"It's a present. For you." Helen studied her daughter's face.
It was inscrutable. "Present? For what?"
Helen's extended hand wavered. She let out a pained breath. "Just to say I'm sorry. I've been a jerk lately. I don't mean it. I never mean to be a jerk to you. Whether you believe that or not."
Mandy looked at her without expression.
"Here. Just... take it." Helen waved it at her.
Mandy took it and opened it expressionlessly. As she took in the packaging, her stony expression cracked into a slight smile. "Okay, that's pretty cool. Thanks." She put the package down on the other side of her and returned to her clamshell.
It was something. Helen's heart warmed a smidge. "You want to look at it and tell me how it is?"
"Now?" Mandy looked impatient.
"Well... yeah."
"I can't now. I have stuff I'm doing. For... job training. Accounting stuff. It's important."
The warmth crumbled away. "I just want to know if I picked something good."
"I'll look at it later, Mom. I'll tell you how it is."
"Can't you look at it now?"
"Mom, I told you. What I'm doing is important!" Mandy glared.
Helen stood up and turned away. She almost got out of the room without lashing out, but in the final moment, she failed. Years of hurt compelled her to turn around and say, "How important can anything in your life be, anyway?"
# 9 Days, 15 Hours
The next evening, Helen limped down the carpeted hallways of the high-rise Hyperius bank building alongside a surly new accomplice: an older, wiry fellow named Jack who had grunted a hello and then ignored everything she said.
If Helen had entertained any illusions that everyone in Cobalt's world was a fan of hers, Jack was destroying them. Then again, she was a tag-along who didn't know anything about the task ahead of them, so of course her presence annoyed the expert.
They were dressed in janitor's clothes bought from a worker supply shop. Before them purred a small parade of cleaning bots.
The sturdy, waist-high bots gleamed. Multiple arms sprouted from each, wielding dust mops and spray nozzles and vacuum hoses like cleaning robot Shivas. Helen found the bots unnerving. They looked like they could pull a human apart with little effort and less empathy.
They were autonomous to some degree, but they also required guidance from human operators, and Jack knew what to do. He wore the janitor's Earworm, swiped from a utility locker, and he muttered commands to the cleaning bots as they moved quickly down the hallway.
Every time Helen gave serious thought to what they were doing, shivers of nerves ran down into her belly and anxiety raised gooseflesh on her arms.
They just needed to get to the server room, drop off the briefcase bomb, get to a safe distance, and set it off via Jack's e-paper. She didn't think it would be too difficult... if everything went according to plan.
She fingered the new burner e-paper in her pocket. It was Cobalt's idea, like the burner clamshells, and it meant she could reach Cobalt in an emergency without revealing either of their identities or compromising Cobalt's security.
As they passed by a series of original abstract paintings and then an office door, she glimpsed a man working at his desk, and the sight sent a thrill of horror through her. She bit back an exclamation. Of course not everyone would leave the building promptly at the end of the work day. Helen hadn't thought of that because she was a rookie—and Jack and Cobalt hadn't thought of it because they didn't care.
But people were _not_ going to get crushed in a falling building today. Not on her watch.
The bright LED lights passed in a blur above her while she walked, waiting until they were out of the man's earshot, and then she stopped and grabbed Jack's arm. She wasn't going to let him ignore her this time.
"People are working late. That means we have to pull the fire alarm. To get everyone out. And make sure Cobalt tells everyone else in every other building."
He glared at her hand on his arm until she let go of him.
"We have to do it," she insisted.
"No," he said flatly. He started walking again. "We're not giving the cops and fire engines a head start."
She got in his way, forcing him to stop. "So we'll have to get out fast," she answered. "But we're not blowing people up. Absolutely not."
He looked away. She saw the slow, silent calculations going on inside his head, and she grabbed his arm again, more forcefully this time.
"This is _my_ job," she whispered harshly. "And we are not. Blowing people. Up."
She would do whatever it took to stop this. Jack was thin and at least a decade older than Helen. She could take him out.
Perhaps he noted the wild glint in her eye. He exhaled forcefully, either annoyed or frustrated. "Fine. You pull the fire alarm when I tell you."
They resumed walking and Jack muttered into his Earworm. Helen struggled to keep up with his relentless pace. The dozen or so cleaning bots stuttered as if confused as they passed through an intersection of hallways, and Jack gave them quiet orders. Then he fell silent.
"So can I pull the fire alarm?" Helen demanded.
"In a minute," he said, casting her an ugly look without slowing his pace.
Just then, the cleaning bot directly in front of him stopped abruptly and threw out a spray nozzle as if following a command.
Jack was still glaring at Helen.
He tripped over the bot's arm.
On the way down, he instinctively threw out his arms, but they went wild, and his head cracked against the rock-hard metal shoulder of a bot and snapped back. His body was limp before it hit the carpet.
The little robot parade came to a jostling stop.
Helen rushed around the cleaning bot to see Jack's face, and as soon as she saw his open, blank eyes, she knew he was dead.
# 9 Days, 14 Hours, 48 Minutes
Helen collapsed to her knees, her eyes shut tight against reality, gasping for breath. Her belly felt hollow and empty, and she drew her arms reflexively against her stomach as if to protect herself.
Had that really just happened?
She looked at Jack again just to make sure.
Yes, definitely dead.
She had just considered taking him out herself a moment ago. That wasn't right. She held her hands to her head.
This wasn't possible. He had been alive a moment ago.
Was her disintegrating brain lying to her? Was it a hallucination?
She stared around her in dismay.
Everything felt too real, in that way that horrible things always did. There was a small coffee stain on the beige carpet next to Jack's corpse. The LED light immediately overhead was bluish tinted compared to the rest. Probably a hallucination wouldn't have this kind of detail. So maybe it was real. And if it was, then what did it mean?
The sudden tragedy left no room in her mind for the original plan. She tried hard to conjure it back up.
Servers. Blowing up servers with briefcase bombs.
How many buildings had to be blown up for the plan to work? If one set of servers remained, would the whole plan fail?
If the whole plan failed... then Jack would have died for nothing.
If he was actually dead, and she wasn't just losing her mind.
"God, I'm sorry," she whispered to his body.
She had to go on. Somehow. Until reality made itself known, she had to carry on with the plan.
The cleaning bots were standing around waiting patiently for further instructions. Helen couldn't control them—she didn't know how to use the control Earworm Jack was wearing. But Jack was dead—she couldn't pretend to go on cleaning the building anyway.
She heard a door open behind her.
She looked around and saw an older man in a suit coming down the hallway with an alarmed expression on his face.
"This is not happening," Helen muttered helplessly.
The fire alarm.
She hurried down the hall and pulled the fire alarm.
A high-pitched, ear-piercing noise emitted, then a buzzing, grinding sound began to cycle as lights flashed.
She hobbled back. The older man was now bending over Jack's body. Maybe this was real after all.
"Excuse me," she shouted fiercely over the grinding noise of the fire alarm. "I need to get to him." She pointed to Jack.
"I think he's dead," the old man yelled back, his pale blue eyes stricken.
"I know," she shouted. "Get out of here. There's a fire. See?" She pointed to the flashing lights.
"Let's go, then," he yelled. He pulled himself to his feet with some effort and moved to take her arm.
She pulled away. "I can't go. You get out."
"I'm not going to leave you here!"
She shook her head. As much as she hated to be rude, she had zero time for Mr. Nice Guy. "Thanks anyway," she yelled.
While the older guy looked on and protested, she swept Jack's Earworm off his ear, rummaged through his pockets for the e-paper that was the trigger for the bomb, then grabbed the briefcase. She hobbled as fast as she could toward the elevators.
She put on Jack's Earworm with shaking hands, wondering whether the device would recognize her. Would it have some kind of security? But as soon as she slipped it on, she saw with immense gratitude the note "Server room 912" floating in his heads-up display.
She wanted to message Cobalt somehow, but she didn't know how to use the Earworm and she couldn't type on her e-paper and walk while carrying the briefcase. She just hurried as fast as she could.
A few minutes later, as she hobbled down the hallways on the ninth floor, a message popped up on the Earworm: [ Status report Orlando Blue ]
She called out into the Earworm, "Am I Orlando Blue?"
She had no idea if speaking into the air did anything or if she had to somehow tell the Earworm to listen.
She stopped, put down the briefcase, pulled out her e-paper again, and typed a message to Cobalt. [ How do you use Earworm to talk? Jack is DEAD. ]
Crucially important seconds were slipping by and she would be damned if this mission failed because of her.
She tried tapping on the Earworm's top button as she'd seen others do. The heads-up display blinked out.
Damn.
She tapped it again. When it lit up, there was another message: [ Status report NOW Orlando Blue ]
"Not good. Jack is dead!" she yelled.
Next message: [ WTF IS GOING ON JACK??? Answer me! ]
Damn damn DAMN. She swept the Earworm off her ear and shoved it into a pocket along with her worthless e-paper.
When she got to the door to the server room, it was locked. She swore violently, then found a fire ax in a glass cabinet just down the hallway and brought it back to use on the door. Her hands shook so hard, the ax glanced off the door twice and nearly took off her own leg. Her adrenaline ran so high, she could hardly breathe.
"Died on me," she muttered to herself as she looked inside the server room for a good place to hide the IED briefcase. " _Died_ on me. Dammit, Jack!"
She found an empty, low shelf that looked good, shoved the briefcase onto it, then headed for the nearest stairwell as fast as she could.
Her legs gave out unexpectedly and she came crashing down onto her hands and knees. A young man heading toward the same stairwell saw her picking herself up and came back to take her arm. "You okay, ma'am?"
Helen was taken aback until she remembered that she still looked like a middle-aged janitor. "Fine," she shouted over the alarm sound. "Just freaked out by the alarm. Just want to get out in case it's a real fire, you know?"
"Oh, it's never a real fire," he answered with a wry grin. "I figure this is just what I get for working late."
As they went down the stairs, she struggled to think of something to say, but her mind was jammed full of things to panic about. Like whether she was supposed to have activated something on the IED before dropping it off. It was too late now. Maybe she had ruined the whole plan.
She let the young man help her all the way down and out to the evacuation area on the bottom level of the parking garage. "Thank you," she said, and he responded with a slight bow.
She turned away and put a bit of distance between herself and the remaining handful of people who'd come down. She unfolded Jack's e-paper, the trigger for the bomb. She had no idea what she was supposed to do next. What if, after all this, she couldn't make the thing blow up?
She started tapping every button.
A massive explosion sounded from above just as the day's first raindrops began to fall.
Helen's gut churned with misery and anxiety. Her disguise removed, the uniform discarded in a random dumpster behind an abandoned building and her regular clothes back on, she walked toward a nearby bar, waving away mosquitoes. For once, there was no rain.
As she went, she sent frantic messages to Cobalt on her own e-paper. [ Did it work? ] [ Are enough servers destroyed? ] [ WHAT IS GOING ON? ] [ Why aren't you answering me? ] [ Jack is DEAD. ]
She gave up and pulled up breaking news. What she saw sent a fresh wave of adrenaline through her. There was nonstop coverage of the bombings across central Florida.
As she stepped in to the dive bar, she saw that its patrons were also watching the bombings on the bar's several TVs. She slunk in like the criminal she was, but no one took any notice of her. She ordered a beer, which came in a plastic cup, and sat in a worn wooden booth, alternately watching the news and checking the e-paper over and over for messages. _Why won't he answer me?_
With so many explosions, a state of emergency was declared in Florida. The governor issued a shelter-in-place. Fire alarms had been pulled in all the buildings, and the Boom Boys had been careful to keep the collapses inside the buildings' footprints. So far they said there were no fatalities. That was a welcome relief... but apparently they hadn't found Jack's body yet.
Without warning, Helen's vision blurred as if smoke had risen in front of her eyes. She rubbed them hard, but it didn't help. _Not now, dammit. Not again._
Footage of the burning buildings was joined, split-screen, by text scrolling across prominent marquees and electronic billboards across the city. Helen squinted hard, but she couldn't make it out.
A shout rose up from the bar's patrons. Cheers. Applause.
_What is it?_
Fuming, she waited until the newscaster read it out loud:
> Thousands of mortgage records gone. Claim any empty house in Orlando u want. It has to be empty. Then tell em to prove it aint yours. Ur welcome. Cobalt, the Robin Hood Thief, and the Boom Boys.
Helen clutched her stomach, feeling sick. Cobalt had let the cat out of the bag already. Was it definite? Or was her name attached to a lie?
Egged on by the others, a handful of the bar's patrons left to take their chances on a new home.
Someone at the front of the bar turned, raised his plastic cup, and shouted, "A toast to the Robin Hood Thief!"
More cheers answered him. Cups were raised around the bar.
Meanwhile, more public figures were denouncing the Robin Hood Thief as a terrorist. Stood to reason. Helen couldn't argue—not this time.
The governor updated his shelter-in-place order to state that any person seen on the streets with "apparent criminal intent" would be arrested.
The news supplied a map with pins for each explosion. Helen couldn't see the details. She tried to bring it up on her e-paper, but being closer didn't seem to help this time—she just couldn't see through the fog. She wanted to ask the patrons at neighboring tables, but she didn't want to draw attention to herself.
Finally, the newscaster gave a tally of buildings hit: twenty-three. Significantly less than the original target of thirty. Helen wasn't sure if it was enough. Perhaps enough computer backups survived to ruin the plan entirely.
Live news footage showed people breaking windows and kicking in doors to claim empty homes. Helen's stomach churned. Would it matter? Or would they just lose the houses in the end because the property deeds still existed?
Just as suddenly as it had come, the fog vanished from Helen's eyes. She sighed. She couldn't decide if she was grateful that it was gone, or angry that it had happened at all.
Now that she could see again, she logged into her latest profile on Whatsit and made a post: [ You guys stay away from the cops. Don't riot. Make it peaceable, okay? I don't want anyone to get hurt. And don't try to take occupied buildings—it won't work. ]
Christian Smith from LSTV sent her a message: [ Kudos, Robin. Well done. Especially for no one getting hurt. That was nice work. ]
_Nobody except Jack._
Like Christian, most of her fans were pleased, even delighted, while others seemed lukewarm, and some were turning against her, saying she'd crossed a line. Apparently, blowing up buildings wasn't universally appreciated.
Helen sighed. Only a few weeks ago, she would have been among those condemning this kind of behavior, and she wasn't sure anymore what that meant. She had a vague, lurking fear that perhaps Helen-in-her-right-mind would never have done this. That perhaps it wasn't Helen calling the shots anymore, but her diseased brain.
But what else could she do, besides wait to die?
She had to trust that the daydreams she'd harbored all her life had been true enough—noble enough—to deserve to be brought to life.
An hour or so after it was all over, Helen was getting out of the rusty old Ford next to a dumpster where she planned to throw away the burner e-paper when it buzzed in her hand, startling her. It was Cobalt—finally.
After taking a look around to make sure it was safe to linger here, Helen read the message: [ So we defiantly got enuf for Hyperius. The other two... i dont know yet for sure. Plus four of the county courthouses didnt go off. But we defiantly cleared out like about 150,000 mortgage records i think. And their saying we did $147 billion dollars damage to the banking and real estate industries. And pple rly like us even more now that we did that with teh mortages. Even if it wasnt complete. I mean people dont know what we were aiming for right? They only know what we did. ]
Helen leaned back against the door of the stubborn old Ford, exhausted, contemplating the rusting dumpster before her. She was grateful the kid couldn't see her disappointment. It wasn't his fault that she'd aimed at a hundred million homes and hit a hundred fifty thousand. She'd aimed too high, that was all.
[ I'm grateful for what we accomplished. ]
[ Its rly sideways about Jack. Thats so wierd. He rly tripped over a cleaning bot? ]
[ Yeah. He really did. ]
[ Woww,. Thats effed up.... Um so not to be totally like cold but theres like not much to say about that so whats next Robin? The cool thing is like i said people dont know what we were aiming for just what we did so we impressed alot of people. I can get some people to help i havent worked with before. Like internationsl super star hackers have been messaging me for hours. Theyre waiting for the next job so they can help. Which rocks. Im gonna have like 40 people in my posse. They have some new ideas too. ]
[ That's good. New ideas is what we need, I think. ]
She waited for a reply, but after a few minutes, she figured out that Cobalt was done for the moment. Meanwhile, her thoughts wandered to Jack. She vividly recalled the horrific way his head snapped back as he collided with the bot.
Never had she dreamed of such a pathetic death.
She still couldn't believe it. Of all the petty, dirty, small moments of a life, to die by tripping over a cleaning bot had to rate as one of the lowest.
How could anyone _ever_ have believed in a God or a life after death? How could a God who supposedly marked the sparrow's fall let a man die this way? Even a man who'd been—as much as she hated to think ill of the dead—more or less unlikeable, at least in the few minutes she'd known him?
No. Life didn't stand up to scrutiny. That was plain and simple and obvious as hell. Life just didn't bear looking at too closely. It was too small, too ridiculous, too ugly, too barren.
Maybe only the rich could pretend it was anything else.
But if anyone's life mattered, if she could claim that her own life mattered, she had to honor Jack's life. Had to honor his passing. So what if he was kind of a jerk. So what if he was one of the least pleasant people she'd met in recent years. No one deserved a meaningless life or a meaningless death.
Maybe that was why people used to make such a big deal out of death, back before there were so damn many people on the planet. Maybe it was because they were trying to pretend that death mattered. To pretend that _they_ mattered.
Did she matter?
She thought about her own struggle to make an impact here in the final days of her own life. Why was she trying so hard? If she stepped back far enough... on any scale that was sufficiently large, the impact of any given life flattened to zero. Eventually, all ripples died out.
Was she naively hoping that somehow she would last forever in the history books? There wasn't enough time to talk about or learn about every significant person in history. When so many billions of people lived and died... not even a thousandth of a percent could be remembered. Not even a ten-thousandth of a percent. The very best she could hope for was that a record of her actions would be stored forever on some web page that no one ever read.
No, this wasn't about fame or being immortalized.
Helen cared. That was all. She cared about the world. About the nameless and small, the unremarked and everyday people who died small and ugly deaths. She cared about Jack just because Jack was a human being who struggled and did the best he could and, by God, did not deserve less than anyone else did. Did not _less_ deserve love or kindness or meaning or fulfillment or beauty.
No matter how much of a jerk he was.
She stomped on her e-paper until it shattered, then threw it in the stinking dumpster. She kicked Old Blue's door to make the handle work, slammed the creaky rusted door behind her, and drove home.
That evening, Helen ate some ramen noodles—they were just about all her stomach would take these days—and then typed up two messages to Cobalt on another burner e-paper. She drove out to a different neighborhood before she sent the messages, because she was afraid to communicate with Cobalt from her own home, fearful that law enforcement could somehow track where the messages came from. Above all else, she still had to protect Mandy.
The first message asked Cobalt to find next-of-kin information on Jack. Tomorrow, she would send the cash cards—the ones she'd lost and then found again—to his family as an anonymous gift. She regretted the media attention that was going to come to them. The janitorial company would fail to recognize Jack as an employee, and red flags would go up, and the cops would figure it out.
The second message was about her next plan.
[ What I would love to do next is to set all nonviolent offenders free and wipe out their prison records. Especially children and debtors and anyone in solitary confinement. I know it's a big deal to do that. It's going to piss a lot of people off. What do you think? ]
[ FUCK YEAH. Fucking assholes putting kids in fucking prison and in solitary. YES lets do it. ]
[ Okay. But it would have the same issues we just had, right? We would have to blow up a ton of buildings again? I don't think we'll be able to do that a second time. And I'm not sure I want to. ]
[ Let me check with my posse, especially the new guys. I bet theres a way we can do it but I cant promise for sure because u really dont know til u try to do stuff. ]
_Now he tells me_ , Helen thought. [ Okay. Let's get on it. We have very little time left, okay? Maybe as little as a week, plus this might be the last robbery I can do. I've been thinking about this, and in addition to destroying the criminal records, we have to open the prison doors, which means hacking the security systems of the prisons themselves, and we have to get into the PA systems to tell the prisoners that it's safe to leave and to tell the guards not to stop them. I don't want a lot of prisoners getting shot. ]
[ KK yeah ]
Helen sat back in her car seat. She suddenly thought of another problem with the plan, making her stomach tighten unpleasantly. If thousands of convicts walked along the highways in brightly colored prison jumpsuits, the cops could easily round them back up.
She gazed out of her car window at the passing vehicles until inspiration struck in the form of a Flyte self-driving taxi stopping at a bus stop to pick up a passenger.
Of course.
[ Can you guys hack into the Flyte dispatch system? Send cars in every city to the prisons to pick up the prisoners? And drop new clothes in the cars for the prisoners to wear? ]
[ Hahaha we can try peace/up ]
# 8 Days
> Yeah so Theres this idea one of my new posse has called poisnoing backups. Basicalhy instaling software on the backups that garbles the backed up data. If were overwriting data then thats not the same as deleting so they can't just get it back. It woud have to run like a few days to get all the backups.
>
> And the active servers we can encrypt during operation and then change the passwords and dlete the passwords from the system. Boom locked out baby. No need too permanently destroy thes ervers when they cant even get in.
>
> One of my new guys knows somebody with the DOJ who can get us some of the nfo we need to get in. So we can do all of this from our machines remotely. No explosions. Not messy.
>
> There are like almost 5 000 prisons. But honestly the number doesnt matter cuz we just build the list of servers and then throw the software at all of them. This is a whole lot easier than what we did b4. The other plan was actaly kinda fucking stupid actually. Sorry.
>
> But heres the thing. U wanted us to only do the nonviolent offendors. Well its actually pretty complicated. Their kept in teh same prisons. Except super max those are seperate. So u could just leave out the super max prisons.
>
> But otherwise the prisons have like violent and nonviolent together. Also my buddy whos dad is a cop was telling me violent criminals plea bargain so theyre record shows theyre not convicted of the violent crime. So u reallyc ant be sure who is and who isnt violent.
>
> So basically theres not any easy way to do what u want. Best thing is just let them all out. Or not do it at all. Its up to u. Whatever u want.
Helen leaned back from the library projcom and massaged her temples. The roller-coaster ride was giving her a headache.
So, all the violent felons going free. Rapists, pedophiles, murderers, domestic abusers... set free with no records, free to commit more crimes at their whim. Could she live with that?
She had often heard phrases like "you have to break a few eggs to make an omelet." As a young hothead, she'd loved it, but as an adult, she'd gotten clear that the ends never justified the means. Never. That was why she had chosen to live the life she had. Always taking the high road. Always taking care to cause no harm, as if she were a doctor subject to the Hippocratic Oath.
And now she was considering breaking a few eggs for the greater good.
Other people still had everything left to lose. Did she have any right to make this choice for them?
She sagged against the table. The email on the computer screen blurred as her vision came and went.
She couldn't decide right now. Not here, on the spot. It was too big a decision to make at a moment's notice.
Suddenly, a thought flashed into her mind. Her retirement fund payout. It should have come in a week ago.
She checked her bank account. The payment was there, dated only yesterday, and marked "pending." She called her bank and was told that the payment was on hold for a ten-day verification period that was mandatory for unusual deposits over a thousand dollars.
She slumped back in her chair. The payment might not clear before she died. She didn't truly need the money now, but it was an insult on top of a host of injuries.
What if she had really needed the money? This was evidence of how the system was stacked to work against the very poor. This delay could put someone in jail, if they had an outstanding debt they couldn't pay.
Distracted and tired, she went out to the old Ford.
Cops, private investigators, the FBI, the CIA, hit men sent by the Entitled she'd pissed off—she'd expected any or all of them. But not this—she hadn't expected this.
The unkempt man reeking of alcohol knocked her down next to her old Ford just as she opened the door, just outside the flickering light cast by the dying street light, here between cars, where no one could see.
He planted one knee right on her wounded thigh, and agony took her breath.
He pulled her face to his by her hair and pressed his mouth toward hers, his gaze smeared by drunkenness. She felt his stubble scrape her skin as she tried to turn her face away. She struggled against him.
Wordless, his face twisted with annoyance at her resistance, he elbowed her in the face. She felt something pop—felt agonizing pain and warm blood coming down. Tears filled her vision.
As she lay stunned, he knelt astride her. He ripped at the front of her shirt and tore at her bra, grunting with exertion.
By the time she could think again, he was pulling up her skirt.
She'd had some self-defense training. Three classes twenty years ago. But she remembered something. She twisted her torso and threw one hip hard into his crotch, then tore a hand loose and went straight at his nose with a flat palm. He turned his head at the last second and her hand went off to the side. All the same, blood trickled from his nose.
He cursed her, his voice guttural. "Baby, you don't wanna do that." It was a threat. He swiped at his face and looked at his hands for blood.
The Bowie knife was visible, knocked out of her purse, there on the ground, just out of reach.
He went to hit her in the face again, to stun her again, to keep her from fighting. She threw up her hands and knocked the blow aside.
She tried to slide out from under him, but he dropped his full weight on her and knocked her breath out of her. She struggled, gasping. She tried and failed to do the hip trick a second time. He threw one elbow across her throat and drove his weight down onto her. She couldn't breathe.
Thunder and lightning at once.
Rain. Sudden and ferocious. Halos around the few street lamps.
She tried to scream, but with his forearm on her throat, she could only make a ragged croak. She flailed and struggled, but she'd gone weak.
Her vision flashed black in the corners as she struggled for breath.
The man stared at her with a twisted leer and victory in his dark brown eyes. He waited for her to go unconscious.
She couldn't stop him. She knew that now. Panic ran through her like a chill. She truly couldn't stop him. Hot tears came to her eyes.
Rain wet her hands, her lower legs, her face.
He couldn't wait. He yanked upward at her skirt.
In his eagerness, his elbow on her throat slipped to the side.
She got a full breath.
She threw herself hard to the side. The Bowie knife.
She didn't hesitate. She would not let this man do what he sought to do. Never, never, never.
She shoved the knife into the side of his neck and pulled it back out.
Blood sprayed down on her, hot and wet. She tried to swipe it away from her face.
The man fell on top of her and convulsed, and she pushed with all her strength. She pulled away, onto wet cement. Torrents of cold rain wet her clothes. It was so cold. It wasn't supposed to be this cold.
He reached for her, but he couldn't get to her.
She got up and into the half-wet seat of her car.
She pulled the door closed and locked it. It felt like sanctuary. The engine started.
She drove forward, over the parking curb, the car rocking violently and groaning in protest.
She left him crawling through his blood toward a salvation that he would never reach.
Helen wanted to go to Egemon, but he would ask what had happened to her, and she couldn't bring herself to tell him. She felt sick and ashamed about how she'd been attacked, and sickened again by what she had been forced to do to him.
On the drive back, she tried not to think about it, about the man's moist breath and scratchy chin. She tried not to think about the cascade of hot, heavy blood on her face. The twisted, sinking feeling that always came with sexual assault.
She tried to think about curling up in her own bed with Jessie and watching TV with some hot tea. She tried to think of comfort and safety and peace.
Then she realized that she couldn't go home. She was covered in blood and soaked in rain. She flipped down the mirror and saw that her face was bruised. Her wet clothes were torn half off. She changed direction, toward her sleep locker.
She passed one younger woman in the hallways of the sleep-locker facility. The girl took one look at her and walked faster, carefully averting her eyes.
Helen locked the door to her sleep locker. As she sat on the edge of the rubber mattress and struggled to peel off her wet clothes, she shook violently with cold and old fear. Blood ran down her face and swirled along with the water toward the drain on the floor.
Trying to get her clothes off exhausted her too much. Dizziness overtook her. She crawled, shivering, onto the bare rubber mattress smelling of mildew. She had no bedding here, not having imagined that she would need to spend the night.
Her pain pills were back at home.
The dizziness intensified, sending the room spinning around her faster and faster. The pain in her thigh stabbed into her, and her nose throbbed. Her muscles shook so violently, it couldn't be just from cold.
Helen knew that if she had been a superstitious or religious woman, she would have seen the attack as a sign. A message from the universe about her quandary on the prison break: _Don't let those bastards out._
Instead, she took it to its opposite conclusion. _I won't let this define me,_ she thought as she lay there shivering. _I won't let that one jerk dictate my choices. He can't make me hate. He can't make me withdraw my compassion._
She mentally composed a message to send to Cobalt the next day:
> Let them all out. Even SuperMax. Ninety percent of SuperMax prisoners are kept in solitary confinement for years on end, and I won't leave them there to rot. And please make sure this letter gets to every single prisoner we let out:
*
> Dear newly free human being:
>
> You've caught a ride with the Robin Hood Thief, courtesy of Cobalt. As you know by now, Cobalt also wiped out your record and gave you a new start. You're welcome.
>
> _Now, don't you commit any more damn crimes._
>
> I don't know you from Adam. I don't know what you've done or why. I don't know what you might be capable of. I do know that your life has been hard and you haven't been able to make a regular life work up til now. But I don't care. You're getting a new start now and you _will_ make the most of it. You owe it to me and Cobalt, you understand?
>
> If you've committed violent crimes unprovoked, just because you're angry or hurting or troubled in some way, then for the love of God, get some help. I can't do everything in the time I have left, but I have funded a lot of money into drug addiction and mental health charities. Go to one of them. Do the best you can to get better.
>
> Just remember this. I was about fairness. I was about love. I was about making the world a better place as best I could in the time I had. I never hurt anyone if I could help it. If you like what I'm doing for you and for your country, then do me the honor of living the same way.
>
> Do all the good you can in the time you have. Damn it.
>
> Love, the Robin Hood Thief
# 7 Days
Helen came home in the early hours of the morning, when Mandy would surely be asleep. She spent the next day, Saturday, at home in bed, recovering and hiding from the world. She watched pointless videos and played games on her e-paper and let Jessie sleep on her bed, which he wasn't usually allowed to do. She tried to pretend it was a normal, lazy Saturday as it would have been forty days ago.
But she also found an article from LSTV dated from Friday, written by Christian Smith: "Robin Hood Thief's Squatters Likely to Stick."
>... the city's lawyers have informed the city council that they have to respect the law surrounding adverse possession and let the new residents stay put for the time being.
>
> According to Orlando Law School's Professor of Criminal Law Stephen Dupree, each house whose ownership is now in dispute thanks to the actions of the Robin Hood Thief will have to be handled in a separate legal proceeding, and each proceeding normally takes five to ten years.
>
> However, experts suspect that these particular cases could take up to twice as long, given the number of them that will hit the dockets here in Florida in the coming weeks, and their unusual complexity.
>
> "Failing the admission of any evidence to prove ownership by another party, if the house has been abandoned for a period of time such that eyewitnesses such as neighbors cannot attest to the owner's identity, the current residents have to be considered plausible owners by the courts," said Professor Dupree.
>
> "Essentially, while these residents do necessarily not have the legal right to be in these homes, they cannot be evicted until the cases are completed to prove that that's the case," Professor Dupree stated.
>
> "And by law, if the proceedings take too long—and the exact length of time varies by statute—the current residents gain permanent title to the property," he went on to say.
>
> It's being called an unprecedented redistribution of wealth, even considering its geographically limited scope.
>
> Susan Orsley, a resident of the Pine Peaks area, said, "There was a homeless family that lived on my street by the corner store. Just right out in the open, basically, in a bunch of boxes and a bunch of shopping carts and bags for all their stuff. There were three little kids in that family. All of them skinny, scrawny, sad-looking kids. The day everything happened, on my way home, I saw they were packing everything up. The father said his brother claimed a house for them and they were all moving in. They looked so happy I just about bawled."
>
> A source who requested anonymity said, "My sister and I were living in a sleep locker with our mother. She's in her nineties now. We were all getting sick from the mold in the mattress. Well, my sister's boyfriend got us into a house. Mom's already doing just a thousand times better. We didn't even realize how bad it was for her until we got her out of it. I just have to say thank you, Robin Hood Thief. We are so grateful for what you did for us."
# 5 Days, 16 Hours
Cobalt and his posse worked nonstop for three more days. Helen wished she could help, but it was all out of her hands now, because she had no skills to lend to this effort. Three more days just brought her three days closer to death.
Occasional messages from Cobalt told Helen of their progress—and lack of it. The job felt touch-and-go all along. Hacking Flyte turned out to be a surprising challenge that required more and more of Cobalt's resources. Their point of contact within the Department of Justice fell through, then another one turned up only hours later. On Saturday night, Cobalt thought the job might not work at all, and then, on Sunday night, he reported that it was a go for the following afternoon—but not for all of the prisons.
The nearly five thousand facilities used a variety of security systems. About forty percent of them relied on the same security software and systems vendor, another twenty-five percent relied on a different one, and the remaining thirty-five percent used a smattering of smaller vendors. Cobalt and his posse were only able to hack into the first vendor's systems.
For well over half of the prisons, then, the doors would not open.
She didn't know what the Department of Justice would do when they realized they held prisoners with no record of wrongdoing. Would they be forced, legally, to let them out?
Either way, a lot of prisons would open.
It just didn't feel like enough.
Cobalt also confirmed that vast swaths of Orlando's real property records had been permanently destroyed in the bank job, plus all of the servers and backups of one of the banks. It was a small fraction of what they'd hoped for, but it was something.
It still didn't feel like enough.
Monday morning, the waking hallucinations had Helen trapped in an ice cavern. When she regained consciousness, she knew why. The house was freezing. Helen staggered to the thermostat with her blanket wrapped around her and saw that the temperature was literally below freezing. A polar vortex had hit.
As it did yearly now, given the changes to the Earth's climate, the cold over what remained of the polar ice caps had pushed southward and brought winter into summer. When she went outside, tiny drifts of snow lay on her car's windshield wipers.
It was a bad day, with a lot of trembling and muscle spasms and weakness that forced Helen to pull over twice because she couldn't be sure she was staying in her lane. But not long after, she leaned against Old Blue at a safe distance from a minimum-security prison on the outskirts of greater Orlando.
From her vantage point out at the highway, everything looked normal. Guards in their long coats stood at the guardhouses with rifles. Three cars sat in the visitors' parking lot.
Helen shifted her weight to her one good leg, shivering even with her jacket on. She hoped that the evacuation, as she was starting to think of it, would go smoothly.
Her newest burner e-paper made a noise. She checked for messages.
[ Locked out baby. And requests sent to Flyte ten minits ago. Doors opening in tn more. Pllus my posse came up with an awesum idea. Were adding in fake garbled records for like thousands of rich people so that they cant even tell who had a record to start with and who didnt. ]
Helen grinned and sent back a thumbs-up.
Her stomach did a little flip of excitement and anxiety as she turned back to the prison. She couldn't wait to see the prisoners coming out.
_If_ the doors opened. And _if_ the prisoners didn't get shot.
Time passed with agonizing slowness. She felt colder with every passing moment.
Would the world understand? Would they remain behind her, especially when—inevitably—many of these criminals committed new crimes? She couldn't help what the escaped felons would do, no matter how impassioned her plea in her letter.
Speaking of the letter, it was time.
She sent LSTV a message with her e-paper. It was very similar to the letter she'd given to all the prisoners, except that it was directed to the general public. She was confident the news media would give it air time. The Robin Hood Thief was in the very midst of her fifteen minutes of fame.
A few weeks from now, no doubt, something else would have taken over the proverbial airwaves.
Ah. Whatsit had another anonymous message for her. [ Hey, Robin Hood Thief. You won't be a free woman for much longer. You know why? You know what you did wrong? You forgot that with enough money, _nothing_ is impossible. And I have plenty of money. Your time is almost up. ]
_No shit,_ she wanted to tell her cyberstalker.
There. The first cars coming down the highway. Flyte cars.
There was a line of them. She counted as they came into view. Five, six, seven... ten, twelve... twenty. And still more coming.
Sirens sounded from the prison, then were abruptly cut off. From her vantage point by the highway, she saw the guards stiffen and touch their weapons.
Her heart jolted in her body. She crossed her fingers.
The announcement crossed the flatlands almost as if no space intervened. It was the script Helen had sent over, piped over the loudspeakers by Cobalt. The script was a lie through and through, but a necessary one.
She listened with interest, hoping to know what Cobalt's voice sounded like, but it was a computer-generated ultra-masculine voice.
"Correction officers, the cell doors have been opened by order of the governor of Florida. You are hereby ordered by the state of Florida to facilitate an emergency evacuation of these prisoners. For your safety and for the safety of the inmates, please allow them safe evacuation in the Flyte vehicles that have been provided. Offenders, please evacuate to the Flyte vehicles in an orderly fashion. This message will now repeat."
The guards fingered their weapons and stared out in confusion.
Flyte cars were lining up just outside the outer-most doors of the prison.
Guards waved at each other, shouting, baffled and confused.
There. The doors swung open. The first prisoners came out slowly, looking around with wide eyes, uncertain. Then they saw the waiting cars and began to run.
Twenty, forty, fifty, sixty of them.
The first wave made it to the cars.
A crack of a rifle. One man fell just outside the doors. The others swerved and went around him, running even faster, panic visible in the flailing of their arms.
Another gunshot. Then another. Three people down.
Helen cursed and kicked at the air. "No! No shooting, dammit!"
Another gunshot. Another. Five people down. Helen cried out, helpless.
The guard responsible for shooting got shouted down by another man. Judging by body posture, the shouter was the superior.
She couldn't breathe.
The first cars began to pull away with their passengers.
Moments passed. No more gunshots. The flood of prisoners dwindled, then stopped.
The guards checked on the fallen offenders. Prison medical personnel came out to put them on stretchers.
The guards stared around them, scratching their heads.
The last of the Flyte cars pulled away.
And then a final message over the PA system, one Helen hadn't expected. It was delivered in mocking tones:
"Just kidding! You've just been _fucked_ , courtesy of Cobalt and the Robin Hood Thief. Have a nice day!"
As the prison guards stared at each other in horror, Helen began to laugh.
Helen headed home exhausted, dizzy, feverish, and preoccupied. Some thought nagged at her, wordless and stubborn. She couldn't bring it into focus.
She started the drive slowly because of the swooning sensation of a moderate fever. But when she realized she was weaving, she reduced her speed even further. At least almost every other car was computer-driven. They'd dodge her effortlessly.
As she pulled into the parking garage at home, her mind cleared enough for her to realize that she should have left her car somewhere and gotten a lift with Flyte. She grimaced. Her brain just wasn't working properly. But then again, that was an understatement.
Unable to resist her curiosity, she sat in her car as she checked the news and Whatsit. With traffic and her slow pace, it had taken her two hours to get home, and that was an eternity in Internet time.
As she read, her heart sank. Not because the job had failed, but because it had succeeded.
Her favorite politician, former President Elaine Decatur, had denounced her.
Reluctant and full of dread, Helen played the video of her statement.
> For weeks, the radical activist Robin Hood Thief stole and blackmailed while she rallied support for her illegal actions. Many questioned her motives and her intentions, while others, including myself, thought of her as essentially harmless—or perhaps even commendable.
>
> When the bombings occurred in central Florida, we had to take a step back and reconsider her true nature.
>
> But this time, there can be no question that the Robin Hood Thief has gone too far, dangerously far. We must finally call her what she is—a reckless vigilante who has endangered us all. Her true intentions are clear, and they are the complete destruction of our law-abiding society. I denounce her actions, and I ask all of you to remain calm as law enforcement takes action to correct this absolute wrongdoing.
Hundreds of other prominent politicians were joining in.
Her own fans were splitting on the issue.
[ Whoa. This is crazy. ]
[ Why would she do this? This isn't stealing from the rich. It's not the same kind of thing at all. This puts innocent people in danger. This is crazy and irresponsible. I can't believe she did this. ]
[ Did you see her letter? She asked them to live by her values. I think she knew what might happen. And she was trying to stop it. ]
[ Then she's an idiot if she thought they woud listen to that. ]
[ Robin,I saw your letter but it didn't make any sense. I want to know from you directly, why? ]
[ And she got over a hundred of them shot. Did you see that? More than a hundred prisoners got shot like dogs on teh way out. That's completely her fault. That blood is on her hands. ]
[ Well they're better of dead if you ask me. More than half of the ones shot were Supermax solitary confinement. Way better off dead. ]
[ Shit y'all I've never owned a gun but I'm going out and buying one. You all should too. We're about to have some serious crime. Better protect what's yours or its going to get taken away. Not to mention people who are just gonna fucking kill each other. ]
[ Betcha pres Sokes is gonna use this to declare martial law ]
[ No he wont but its gonna be crazy out there!!!! Robin just killed a hundred people but it will be thousands before its over. ]
Helen closed her eyes, listening the ever-present thumping of her heart as her body heated up with a fresh wave of fever.
They didn't understand. She hadn't made herself clear. Now what? Was there any point in defending herself or explaining herself now?
Or maybe she'd been wrong. Her brain was days away from shutting down in coma and death. Maybe she couldn't think straight anymore. Cobalt couldn't function as a check against her, because he was basically antisocial, and maybe she should have asked someone who had a conscience before she decided to do this thing.
She'd always daydreamed of this—letting out all the criminals. The evidence was clear that imprisonment didn't rehabilitate criminals and dramatically increased recidivism. But actually doing it... had it been the right thing after all?
She couldn't stop scanning the comments.
There was a message from Christian at LSTV.
> Wow, reducing our incarceration rate by 50% overnight was an incredible idea. I have to tell you, although I probably shouldn't, professionally speaking—I love it. I completely support what you've done. Seriously, Robin, I think I've become one of your biggest fans.
>
> You really get what's happening in this world of ours, and unlike everyone else, you're doing something about it. Something huge and crazy and real. You've got blood on your hands, and I don't mean that as an attack or a reproach. I'm a veteran, so I understand that blood has to spill sometimes. Change has never been made peaceably, but most people are too afraid to make it happen. You stepped up and did this for the rest of us, for all of us. You're like an army of one.
>
> But I know you've realized that not everyone else gets it. Come on, Robin, this is the time to have an interview with me. You can speak to the whole world and they'll listen. You can clear the air. Don't you think you owe it to the history books to clarify what you were really about?
Helen let out a long breath. She appreciated the sentiment with all her heart. But she was still afraid that he could somehow track her down if she spoke to him in real time.
Finally, she replied. [ I said it all in my letter. At least, I tried. ]
The next comment on her profile made her heart sink.
[ What good does it even do? Good she got rid of their records, but she just added hundreds of thousands of people into the labor market who aren't gonna be able to find jobs and they're just going to go back to what they were doing before. They're just going to end up back in jail. ]
That was the thought that had been dogging her, the one she couldn't articulate. Of course. Where were they going to go? No homes, no jobs, no possibilities.
What had she done?
# 5 Days, 13 Hours
The cold had killed some of the mildew in the hallway, or maybe her sense of smell was going.
The three deadbolts. Dropping her purse on the tiny two-person table. Mandy at the loveseat, her clamshell in her lap.
But there was something different, too. Mandy's cheeks were flushed, a smile hovering at her lips. She actually glanced up at Helen and said, "Hey!"
Even through her miserable preoccupation about what she'd done, Helen was warmed by Mandy's greeting. "Hey yourself," she said.
She went over and sat down next to Mandy—not too close, not wanting to spook her out of this sudden friendliness.
Another wave of fever hit, and Helen wished she could lie down, but she wasn't willing to reveal her illness to her daughter.
Not yet. It wasn't time yet.
A small voice inside asked _When?_
Mandy said, "You've been hearing about this Robin Hood Thief thing, right?"
Helen nodded, then wished she hadn't. The room spun. "Yeah."
"Isn't it cool? Today she erased the record of every single criminal in the entire justice system and let almost half of them go free. Don't you love it, Mom? It's totally the kind of thing you ought to love, right?"
Helen tried to put together some semblance of a normal mom response. The mom from forty days ago. The mom who would never blow up buildings or steal from anyone. The old mom-of-the-high-road.
"They all get to start over," Mandy went on, still enthused. "People are freaking out, Mom. They're doing a spontaneous parade in downtown right now in honor of Cobalt and the Robin Hood Thief. It's fucking awesome."
A parade _was_ kind of awesome. At least some people still stood behind her. But it was useless. She knew that now, too late. "Listen, she's breaking the law," Helen said miserably. "Setting violent criminals free. You know they're going to go back to their old ways. People are going to get hurt."
"But she's given them all a second chance, Mom. Don't you always say that these people need a second chance?"
Helen let out a sigh. "Yes. But she's just setting them back out into the same systemic racism and classism, the same debtor society," she said, embittered by the knowledge that it was utterly true.
Caught up in self-recrimination, she spoke to Mandy as if her daughter might actually know the answer to the problem. "How are they supposed to make new lives for themselves with the unemployment rate and the institutional racism and all the rest of it?" She looked at Mandy helplessly.
Mandy's face sagged. She looked as if Helen had kicked her, and Helen suddenly realized she had argued them down into a dark place neither of them wanted to be.
She struggled to think of something to say. Some way to take it back or fix it. With a profound sinking sensation, she realized she had only pushed her daughter away yet again.
Mandy closed the lid to her clamshell and started to get up, her delicate features wounded.
Helen put out a hand but stopped short of touching her daughter, knowing not to try. "Wait. I'm sorry. I'm just... I'm frustrated. I'm tired." For once she'd managed to get the apology out before Mandy walked away.
Mandy hesitated. They were frozen, the two of them, Helen looking at her daughter, her daughter looking away. Both of them barely breathing.
"I'm sorry," Helen said. "I know what I said wasn't what you wanted to hear."
Nothing.
"I'm a jerk sometimes," Helen said.
"Yeah, I know," Mandy said. There was a little bite to it.
"Did you like the music Earworm I got you?" Helen asked. She tried not to sound imploring, but in truth, she was begging. Begging for a few moments of connection that actually worked. Begging to close that gap of six years. Begging to be loved again.
"I don't feel like talking about it," Mandy said.
Helen's chest squeezed shut, and Mandy went into their room.
The door closed quietly, not a slam.
It was a signal. Young adults spoke in code. A quietly shut door meant, "I'm not really upset with you anymore, but I have to assert my _right_ to be upset."
The fever vanished and left Helen cold and trembling. She pulled the extra throw out from under the loveseat and put it over herself.
Jessie slept in the chair in the pool of light cast by the standing lamp. His nose twitched and his paws cycled as he ran in his dreams.
Helen let out a long breath. The dizziness intensified and the room wavered and spun. She closed her eyes and watched the pinpoints of light dancing behind her closed lids. The flashes of light were fractured, fractals, flickering brighter and bolder in patterns more vivid than any she remembered. Perhaps it was a sign that things were getting that much worse.
She wondered whether she might not wake up the next morning. Maybe her body wouldn't give her the chance to take the black pill. And maybe she'd be lucky if so.
But she wasn't ready. Not when she had only made things worse. She had to do something now to make them better again.
She couldn't help but look at Whatsit one last time before she went to sleep.
There was a surprising comment at the top of Helen's profile.
[ I know Ms. Robin Hood Thief. And I just want you all to know that you misunderstand her if you think she has evil motives. She is a good person. Better than most I have met. And smart. If she did it, she had a good reason. I am on her side. ]
She let out a slow sigh. It meant a lot. She stared at the message and the profile trying to figure out who it might have been. The profile picture was of a black Lab puppy. The handle was ageg17. She thought she should have known, that it should have been obvious who it was. But it just wouldn't come into focus.
She took her sleeping pill and then read the message one last time as the medication claimed her mind and pulled her into darkness.
# 4 Days
In the morning, Helen's consciousness swam up from visions of flower petals in pastel and jewel shades, all viewed up close—the dusting of pollen around the stamens, the velvet texture of the petals.
Her consciousness returned with an utterly clear plan.
She only needed to write a few more emails to Cobalt and compile a list of the thirty or forty richest and most influential American corporations and hand them over to him.
Then Cobalt could reassign their stock to regular people. Most of the wealth would go to ex-convicts. They would have enough money to rebuild their lives.
And the new stockholders could hire new boards of directors—they were authorized to do that in most corporations. They could also change the vision and goals of the corporations to address global warming, cut executive pay down to size, and use their immense profits and political influence for good.
America could belong to Americans again.
Helen stayed at home that day to work on her task, listening to the news on her e-paper as she did so. The prison break and its fallout occupied almost the entire news cycle. Gun sales surpassed previous records, and there were scattered shootings of supposed ex-felons, mostly unprovoked and panic driven. Otherwise, the peace held. The newly released seemed to be laying low, at least for the moment.
The authorities lectured sternly about what would be done, but the talking heads from the law schools said there was very little that could be done. The records were gone. The only way to put someone back behind bars was to have their criminal trial all over again. Law enforcement would surely try to do so for the worst offenders, but it wasn't possible to do it for the hundreds of thousands of minor offenders.
Christian Smith at LSTV ran half a dozen interviews with ex-convicts, all with their faces blurred out and their voices distorted to prevent identification.
A young man. "I've been in and out of prison my whole life. I've killed people. And I'm only gonna say this once, and only cuz my name ain't on it. But what the Robin Hood Thief said in her letter—it got to me. She did this for people like me. She really cares. I don't know if anyone has ever really cared about me in my whole life. But I think she does. So I'm going to go straight, like she said. I'm going to try to do good. And that's all I have to say about it."
Even the distortion couldn't hide that one of the subjects was a little old woman with a quivery, whisper-thin voice. "My younger son stole my credit cards and ran them up and then didn't pay them. But I was responsible for them. They didn't care who did it, just whose name was on the statement. I don't have any income. My older son supported me. So I went to prison. I was eighty-two when I went in. I'm eighty-seven now.
"I was diagnosed with leukemia four months ago, but they don't offer medical treatment to convicts anymore, not for major illnesses like that. Now that I'm out of prison, I can get a black pill, and I can have the peace and quiet and comfort of home, and my older son can be there with me to tell me goodbye. And that might not sound like much, but it's a lot to me."
A middle-aged man. "I was in for eight years. Six years in solitary. I spent another six years homeless afterward. Now it's been a long time since then. A non-profit company picked me up off the street to give me a job."
Surprise darted down Helen's spine—his story matched Oliver's to an uncanny degree. She flipped the screen to see the video. It was a stocky man about Oliver's size.
"Now it might seem like a prison record shouldn't matter anymore, for someone who's been out for so long and who's got a regular life again. But it does matter to me. Now that my record is gone, I can work anywhere, any kind of job. I can vote again. But more than that..." He took a cloth handkerchief from his pocket and wiped his forehead, just like Oliver always did, and Helen leaned forward. It _was_ Oliver. It had to be.
"I just feel like I'm free to be a new person. I'm not marked anymore. I'm just as good as anyone else now. I feel like she gave me back my future and also gave me back my past. And I'm just grateful. I'll always be grateful."
Tears slipped past Helen's eyelashes, and she wiped them away. "You're welcome, Oliver," she whispered.
Completing her task—looking up thirty or forty names and putting them into an email—took Helen all day.
She hadn't even recognized the atrophy of her faculties until she tried doing real work again. Less than three weeks ago, she'd been able to research potential blackmail material for Mr. White. Now she could hardly copy-and-paste.
She felt very nearly destroyed. Very nearly gone.
She wondered whether, if she didn't have all of this going on, she would have already died. Perhaps it was only the commitment and the vision and the hope that was keeping her alive.
# 3 Days
> So, i dont really think this is possibel. At least, not the way you were hoping it was going to go. Basically the thing is that if we just change who owns the stock ub tthey still have the rcords its just like with the mortgages theyll just put it back unless we delete the history of the records too.
>
> But its like with the mortgages only a lot worse becuz the records are so distrubuted across the whole country across thousands of people and systems. The brokerage firms and brokerdealers take the requests and there are thousands of those. Also the transfer agents which sometimes are banks or brokerage firms or the corproations themselves. Its their job to track every single stock transfer. There are thousands of those too.
>
> And then usually the corporations have their own records. And then the SEC tracks stock transfers by major stockholders and the exectuvies and board members.
>
> So there are thousands of places that have records of every transfer and everyone who owns all the stocks.
>
> So I kinda dont kno w baout this. Sorry. Dont want to let u down., But the more we dig into th eworse it looks. I dont rly wanna say it but fact is i rly dont think we can do this in ur timeline and basically maybe not at all. Sorry.
# 2 Days, 16 Hours
Helen hardly slept that night, and the waking hallucinations were darker and heavier than normal and hard to shake off, even an hour after taking the waking pill. As soon as she was functional enough to use her e-paper, she checked her email. Nothing. Nothing from Cobalt since the devastating email the previous day. He had shut her out.
She could hardly sit up. She pulled herself up onto the edge of the bed, just trying to keep her eyes open.
She could no longer pretend that her medications still had the same efficacy. Whatever was happening to her brain had gotten so much worse that the meds couldn't overcome it.
In another day or two, she would be back to the Purgatory state of no sleep at all—unable to keep her eyes open or closed for any length of time.
Maybe it was all over. Everything she had worked for. She'd freed thousands of criminals and set them loose in a society that was going to go on functioning according to business as usual. Without jobs, _they'd_ go right back to business as usual. The donations she'd gotten for various charities would run out, and everything would go back to the way it had been all along.
Nothing she'd done would matter.
Her _life_ wouldn't matter.
For the first time since she'd started this whole scheme, her illness overcame her. The emotional energy provided by her successes evaporated and left her with bone-deep exhaustion, a foggy mind, and a trembling body.
She locked the apartment door and set out on stiff, awkward legs. She would still pretend to go to work as usual. She still hadn't done the calculus, still hadn't figured out how to tell her daughter... and maybe the calculus was beyond her now.
She registered that the polar vortex was lifting and it was much warmer out, but cloudy and gloomy.
She fixed her eyes on each milestone in turn—the end of the hallway, the entrance to the parking garage, her usual parking spot...
Still a dozen yards away from her car, she thought she heard a buzz from her e-paper, in the depths of her purse, and she stopped to take it out, hoping it was Cobalt or Mandy.
No, nothing.
She waved away a mosquito and frowned at the e-paper. It must have been a phantom vibration. She doublechecked all the screens twice, just in case.
She felt the explosion before she heard it.
The wave of heat, the shuddering of the parking garage, the blast that threw her against the wall behind her.
The wave of heat dissipated as quickly as it'd come.
She pulled herself back up, fueled by adrenaline.
It was her car that was on fire. Poor Old Blue was its own funeral pyre.
_Jesus._
The phantom ring of her e-paper... Helen was not a superstitious woman, but something strange had happened, because if she hadn't felt that buzz, she would be dead right now.
But they knew where she lived and the car she drove. And they were going to kill her.
_Mandy_... they might get Mandy.
Panic surged through her.
She tried to run back toward her apartment, and she fell, bruising her knees and an elbow. She got herself back up and staggered along on unsteady legs, one hand trailing the dirty wall.
Who had done it? Where were they? Were they watching her?
Down the first hallway and around the corner toward her own apartment.
A man was there on the far end, coming around the corner toward her. Dark hair, dark eyebrows and eyes. A tattoo of Roman numerals—XIV—on his cheekbone. A gun in his right hand. A _big_ gun.
He was death coming for her.
She ducked back behind the corner.
She'd known, deep down inside, that it was going to end like this. How could she come to a peaceful end after everything she'd done?
Her pistol. The little .22 she'd bought from the weapons store. Ever since the assault, she carried it tucked into her pocket.
She pulled it out and fumbled with shaking hands to turn off the safety.
She reached around the corner one-handed and fired once, twice.
Roman-numerals guy took cover behind his corner. Then he fired back at her, and fear took her breath away. She ducked back behind her corner. She didn't _want_ to die, damn it.
She looked around the corner. He was coming again.
She fired again, again, again. She kept missing.
_"You couldn't even hit the target. You'll need to come back a coupla times a week until you get comfortable with that."_ That's what the man at the gun store told her, weeks ago.
"Put it down!"
A forceful shout from behind her.
She wheeled around, expecting a cop. She nearly called out _don't shoot_ , but then she saw the man's plain clothes. He was with Roman, and he was pointing a gun at her too. "Put it down, now!" he commanded again.
Put it down? Why would she do that when her daughter was sleeping on the other side of this wall?
"Fuck you!" she shouted. She shot at him.
The bullet caught him somewhere. He dropped to one knee, snarling, then raised his gun again.
Something hit her hard from behind, on the back, above the left shoulder blade.
She turned. The guy with the Roman numerals tattoo was back there, but still far away, running toward her. Had he thrown something at her?
She raised her gun and aimed at Roman, then realized the gun didn't look right. The top part had slid back and locked in place. What did that mean?
Rough hands grabbed her arms from behind.
She dropped her weight and rolled into the legs of the man behind her. He toppled over her, and as he twisted, she hit him in the face with her little pistol. It left an angry red mark high on his cheek, and he clubbed her on the side of the head with the butt of his weapon.
She was on the ground suddenly. Her vision was dark, then it flickered in and out. Her arms and legs wouldn't move.
_Mandy..._
One of the men said something to the other and pointed down at her. He gestured to his own left collarbone. They argued briefly, and the words echoed, incoherent.
Roman-numerals guy knelt and frowned at her, then pressed hard with his knee on her left collarbone area. She couldn't figure out why.
A drop of sweat rolled off his forehead and hit her cheek. She wanted to wipe it off, but she still couldn't move.
While he knelt, Roman went through her pockets and took her e-paper away.
A burning feeling under Roman's knee became searing, then throbbing. Gradually, it dawned on her that he had shot her and was now applying pressure to the wound.
The other man had gone away. What about Mandy?
Roman got up, checked the wound, then rolled her over, zip-tied her arms behind her back, and pulled her to her feet. She could just barely keep herself upright. He pulled her down the hallways and into the stairwell, the heavy rusting door dragging and squealing as he forced it open.
The stairwell smelled of old urine. A sleeping bag in the corner held the still form of someone who was either asleep or dead.
She didn't understand why they wanted her alive. Hadn't they planned to blow her up in her car?
Her strength returned, bit by bit, as they went down the stairs. Out on another floor of the parking garage, Roman-numerals guy opened the door of a black SUV and pushed her forward. She got onto the first step leading into the backseat and then realized that this was her last chance to get away.
She turned and headbutted Roman as hard as she could, knocking them both to the ground.
He screamed at her and hit her in the face. Her right eye went blurry. She struggled to get up to her knees.
_Mandy..._
Roman hit her from behind, and her silver locket flew up out of her shirt. She twisted away enough to draw her knee up to kick him, but he threw his arms around her and flung her down.
She felt something jerk around her neck. The locket hit the ground.
"No!" she cried.
She was on her face on the ground now, grit and gravel rubbing into her cheek and forehead. His weight was on her, and there was no way she could get up. She bucked and twisted helplessly.
"She's gone," Roman said. "She's in the other car. Stop it! You're going to start bleeding again."
Helen fought harder.
"If you quiet down"—he smacked her on the back of the head—"if you get in the car, you can see her again, all right? Cooperate, goddammit."
"Let me see her," Helen begged.
He let her up, holding firmly to the zip tie that secured her arms.
Another black SUV was pulling away. Mandy's blue hair pressed against the window pane, unmoving.
Roman yanked at her arms and she cried out in pain.
"Get in the car," he said.
She got in. All she could think about was Mandy. And how useless the three deadbolts had proven to be.
Roman slammed the door, then picked up the locket and looked at it carefully. He shrugged and put it in his pocket.
# 2 Days, 14 Hours, 24 Minutes
Helen tried to think this through—who had them—but her brain wouldn't work. All she knew was that they weren't law enforcement.
She suddenly realized that these couldn't be the people who'd just blown up her car. Those people wanted her dead. These guys were with someone else.
She had pissed off a lot of different people.
A few blocks from Helen's apartment building, a series of cop cars and SWAT vehicles flew by, their sirens off but their speed indicating that they meant business. She watched them go with a sense of foreboding and yet somehow feeling forlorn at the same time. Cops were never good news—unless you were being kidnapped.
They drove into one of the rich areas of town, Lake Estelle Estates. There was no construction here and a lot less traffic and crowding. Smoothly paved streets, power-washed buildings, freshly watered grass and greenery and flowers, impeccably attired and groomed families taking it all for granted. They oozed entitlement. They thought this was just the way the world was. They thought they'd earned all of this. That they'd worked hard—or their parents before them, anyway—and so this was just what they got to have.
Too damn bad for the rest of the world.
Helen shook her head. So it was some rich guy she'd pissed off. She wondered which one it was going to be. Not that it really mattered. What mattered was what he was going to want from her. And from Mandy.
They turned in to the Hotel Forzando. It was attached to the west end of an enormous luxury shopping mall, the Lake Estelle Shops. She couldn't recall whether any of her victims had connections to this hotel or the shopping mall.
They pulled around to the valet parking. When they got out, still concealed by the SUV's doors, the guard with the Roman numerals tattoo on his cheekbone cut the zip tie on her hands and put his suit jacket around her shoulders to conceal her gunshot wound. "Don't try anything stupid," he said.
She couldn't think of anything to try that wasn't stupid. What was she going to do? Start screaming for help? Maybe she could have done that if she was well-dressed, but she wore work clothes—worn slacks and a wrinkled button-up shirt. And a ten-dollar haircut that needed trimming. She didn't belong here. They'd take her for a crazy person and turn away, guiding their children to safety.
The lobby was huge, spacious, beautiful. Columns soared up to a ceiling painted with a lovely fresco of the gods of Mount Olympus. The heels of the men's shoes clicked on the marble tile.
No, Helen really didn't belong.
The elevator dinged and they got in. Roman punched the button for the top floor, the fourteenth.
"Where's Mandy?" she asked.
"Coming up separately," Roman said.
They rode in silence.
The doors opened and a wave of fever passed over Helen as they went through a small lobby toward double doors. She knew that nothing good could possibly wait for them on the other side of those doors.
They went through, past a hallway and a dining area and into a living room. Floor-to-ceiling windows filled two walls, with views out onto a balcony with a Jacuzzi and a pool and then out to the city.
In front of her, the man responsible for their abduction lounged on a sofa, his arms along the back of the sofa, his legs crossed, his head tilted too far back.
She recognized him. The first time she ever saw him, the short man was going at his mistress like a rabbit. Her lip curled. Brock Tolbrook.
"Hello," she said.
It was perhaps an underwhelming greeting.
He gestured to Roman, who took his suit jacket back and shoved Helen down to the ground in front of the sofa.
Tolbrook stared down at her on the floor. He looked like he was holding back some sort of particularly nasty comment, one that twisted his whole face.
"Well?" she asked. "Can we get this over with?"
His face reddened.
"I told you," he said, his voice filling the room. "I told you that I never quit. I told you I would get you eventually. And I did." He leaned forward and stared into her eyes, trying to be intimidating.
She wondered when he was going to stop yelling, and then she thought that maybe he always spoke too loud. Trying to make up for being short, perhaps.
"How did you know it was me?" she asked.
"Cops finally figured out you weren't old. Age regressed the photos. It was easy after that."
Helen's stomach sank. Her time was up, then. Everyone knew who she was now—Tolbrook had just gotten to her first. Maybe those cop cars and SWAT vehicles a few minutes ago really were for her.
He leaned back, no doubt satisfied with her defeated expression. "You cost me a million dollars. And you're going to pay me back. With interest. A _lot_ of interest. I'm not going to take less than a hundred million dollars."
"Oh, come on," she snapped before she could stop the words from spilling out. "A hundred million? You don't _need_ a hundred million dollars."
His face reddened again. "Of course I don't _need_ it," he snapped back. "I didn't need my first hundred million. But I deserve it, and you're going to get it for me."
"You deserve it? What have you done to deserve it, exactly?"
His face turned purple. His volume went even higher. "I wouldn't expect you to understand decent, hard-working people like me. Whereas you—"
"Decent—Are you kidding me?"
" _You're_ nothing but a criminal and a terrorist. What makes you think you're entitled to do anything you've done?"
"I have done this to help people. You _know_ that, if you've read my letters. I've— "
"Oh sure, of course, you think you're noble. All you are is a _thief_." He stood up to shout down at her. "Taking away what people like me have spent our entire lives working for. I'm a businessman. And a family man. I've played by the rules. I have worked hard for what I have and you have no business sneaking into my home and stealing what's mine!" Spittle flew from his lips.
"No one is entitled to excess when there are those who don't have enough," Helen said.
"Communist bullcrap. You ever actually get your communist world, you'll be sorry. But that's okay. You'll never get it." He wiped his mouth. "Now listen. You are going to get that money for me. You're going to contact Cobalt and have the money sent to me. I don't care where it comes from. You understand?"
"Fine," she said. With everything else they'd done, surely Cobalt could arrange that.
It rankled to give in, but now was not the time to fight back. She told herself that anything Cobalt could do, he could surely undo later. By the time they were done with the cat-and-mouse game, Helen would be dead and Mandy would be safe.
He gestured back toward the large dining table. A projcom waited on it, the holographic keyboard and display already lit.
Still on her knees, she looked back at Tolbrook one more time. She just couldn't stop herself. "What is wrong with you?" she asked quietly. "You love horses and dogs. How could someone who loves horses and dogs be... like this?"
He stared at her as if wondering how anyone could ask a question so stupid. "Horses and dogs are innocent. People like you are scum." He crouched on the ground in front of her. "You came into my house a guest and you..." Remembering, his face reddened again.
He grabbed the back of her hair, pulling her head back. His face hardened into a scowl. "So I guess I'm more of a bad guy than I thought. You're the one who taught me that. Taught me I'd be willing to kill for how you _humiliated_ me."
Was he thinking of killing her now? Hurting her?
"Let me just make one thing clear to you." He paused for dramatic effect. "You've failed."
He stared hungrily at her face—hungry to see her broken. "You set criminals free—congratulations, you've made the world a worse place. You got a few donations and stole a few trinkets—that means nothing. The cops will throw your squatters out eventually. The banks are already being rebuilt. You just got your fifteen minutes, that's all, and once this is over, no one will even remember that you ever existed."
He shoved her over, catching her off guard.
He looked at her on the ground for a moment, still kneeling over her, still too close for comfort, his expression unreadable. "People like me—we own this world. Our parents and grandparents and great-grandparents gave it to us. And we work hard to keep it up. We work hard for it, do you hear me? You don't. You people just take up space. But we let you. And you know why?"
He leaned even closer, his face taking up her entire field of vision, and spoke softly but emphatically. "Because we don't give a shit what you do. Nothing you do matters. Nothing you do will ever matter. So just give up."
He got up and walked to the window and looked out.
Helen couldn't find words.
She got up on trembling legs and went to the projcom. All that mattered was to complete his request and get Mandy out of here. She couldn't think about what he'd said now, or it would sap what little strength remained in her.
As she sat down and tried to get her bearings on the device, she remembered suddenly that Cobalt hadn't replied to her last three messages. What if he wasn't willing to talk to her anymore?
_Please, God, please let Cobalt answer me._
"Nice place, isn't it?" Tolbrook asked arrogantly. "This penthouse. I had it built just for me."
Helen stared at the screen. Her mind was like mud.
"Is it online?" she asked hesitantly. "Is there something I have to...?"
"It's online. Just write the email." Tolbrook snapped.
It had been a dumb question. She couldn't think. A wave of fever came over her. The gunshot wound in her shoulder throbbed and burned.
She went to Whatsit and logged in, carefully, step by step, trying to cajole her fading mind into helping her. Why did everything look new and different?
Something was very wrong.
She felt weak and limp.
"What are you doing on Whatsit?" Tolbrook demanded.
"This is how I contact him," she said. "I don't have any other way to reach him."
He didn't say anything.
She couldn't remember the profile name that Cobalt had last given her. And she was certain that he deleted all the old ones after a day or so.
"Yes, I'm cancelling everything for the afternoon," Tolbrook said, his tone ugly.
She turned to stare at him and then realized he was talking to someone on his Earworm.
A message popped up on the projcom, on Whatsit. It was from Smith.
[ Waiting for that interview with bated breath. ]
_Yeah, right_ , she thought. _Probably a bit too late for that now._
Wait. She was online. On Whatsit. With access to the entire Internet. She looked over her shoulder.
Tolbrook was distracted with his phone call. "No, I don't care what he's saying. I did the fundraiser, I met my obligation."
Quickly she typed [ me and daughter kidnapped penthouse hotel forzando help DON'T REPLY ] and sent the message, then closed the window. Or tried. A little circle spun as the system tried to connect.
Shit.
An error message appeared and Helen frantically tapped at all the Close buttons.
Tolbrook grunted a dismissal to the person on his Earworm and turned back to Helen just as everything shut down.
"What are you doing?" he demanded.
"Got an error message," she said as calmly as she could manage. "Trying again." She re-opened the browser and started Whatsit back up.
Tolbrook stared at her, apparently sensing that something was wrong but unable to figure it out. "You better not be lying to me."
She flinched. "It was an error, okay?" she said as calmly as possible, despite her heart beating frantically. "Your computer is out of date or something."
He didn't say anything, just folded his arms.
She didn't think her message had gone through, and now Tolbrook was watching her.
Well, it had been worth a try.
_Fuck._
She tried again to focus on Cobalt.
What was the profile name he had last given her?
Her heart hammered.
Her chin dropped toward her chest.
_Think, Helen, think._
She felt her cheeks go hot with another wave of fever.
"What is it?" Tolbrook slapped her on the back of the head.
She bit back an angry response.
Tolbrook leaned in close, staring her down.
She tried again to remember. Cobaltajax27? That was the previous one. Maybe.
Suddenly, she was terrified. If she couldn't reach him, then she couldn't get the money, and that meant she couldn't buy Mandy's way out of here.
"I'm just having some trouble remembering the profile name. He changed it every time, and I never wrote them down, and I just need a few minutes..."
Tolbrook tried to laugh. "You think I'm going to buy that?"
"Today's events have taken it right out of my head, okay? I got shot, you know. I just need a few minutes to think."
He just nodded slowly. She could see his clenched jaw. He was pissed.
"Fine. Go say goodbye to your daughter and see if that jogs your memory a bit."
Her heart faltered and her body went weak.
Tolbrook gestured to Roman, who took Helen by the arm and escorted her down the front hallway. He stopped in front of a door and opened it with his key card. "Go in."
She did, and the door slammed behind her.
It was a perfumed luxury bedroom with outrageously high ceilings and enormous windows. And her daughter lay awkward and motionless across the bed, her face turned away. Blood pooled on the silk coverlet under Mandy's blue hair.
Helen choked back a cry and stumbled to her daughter.
_Please, no..._
# 2 Days, 14 Hours, 2 Minutes
Her daughter's eyes were closed. Her blue hair was a bloody mess on the right side. Helen couldn't see how bad it was. She didn't see exposed bone. Blood was still seeping out. The pool of blood under Mandy's head was big.
She felt for a pulse on the side of Mandy's neck.
She was unable to breathe, unable to think, until she knew. Everything stopped until she knew.
There was a pulse.
She bent down and watched Mandy's chest, as she had done every night at the side of her crib when she was a baby, until she saw it rising and falling.
She fell to her knees and gasped shuddering breaths of relief.
_Thank God thank God thank God..._
But they were trapped and alone and both of them were hurt. And Helen couldn't even think straight. She couldn't run or fight.
They were helpless.
And she couldn't remember Cobalt's username so that she could buy their freedom.
"I'm sorry. I'm so sorry. This is all my fault." Grief and guilt squeezed her chest painfully as she stared at her daughter's placid, pale face. "I tried to keep you safe, and I failed."
Her face contorted as she gazed at this person she knew so very well and so little. This person she had carried in her own body, stayed up nights to feed, comforted when she cried. This person she had taught right and wrong. Taught to stay safe. Struggled with. Punished. Cried with. Praised and encouraged and yelled at and bullied. Tried to be both father and mother to. Failed. Failed and failed and failed.
Something in Helen's spirit broke.
"I will help you," she sobbed. "I promise."
It felt like a lie.
Time passed as Helen wept at her daughter's side. Eventually, the worst of the storm of emotion settled, and when it did, Helen's mind latched onto what Tolbrook had said:
_You've failed. Just give up._
The words flashed around inside her mind, taunting her. They'd made her so angry she couldn't speak.
Now, that anger bubbled back up. With it came that same steel core of determination that had come to her when she threw a chair through her living-room window because she'd learned she was going to die in forty-five days.
She straightened up and wiped her eyes and face with trembling hands.
Tolbrook said she'd failed. But she wasn't here to succeed. There wasn't time enough for that. She was here to take action as if she were going to live forever. To start things as if she'd live to see them finish.
The ultimate sting of death wasn't _failure_ —it was _futility_. It was giving up without ever trying. And she had not given up.
She had come alive as the Robin Hood Thief, and it was because she had dared to live boldly, to be a hero, to live with courage.
Now was not the time to quit.
_Courage_.
Not success. Just courage.
No one could take that away from her.
Helen got slowly to her feet and wiped her face again. She took a deep breath and went to the door. She would not give up. She would summon Roman or Tolbrook, and she would do whatever it took to save her daughter.
A faint voice came from the bed.
"Mom...?"
"Mandy!" Exhilaration ran through Helen. She hurried back to Mandy's bedside and knelt. "Are you okay?"
"My head hurts..." Mandy's eyes opened and closed weakly. Tears smudged the smoky makeup around her eyes.
"I know, baby. You got hurt. I'm sorry." Tears came to Helen's burning eyes again and she forced them back.
"Where are we?" Mandy got her eyes fully open and looked around.
"Some rich guy's penthouse. Just stay put, okay? I'm going to rescue us."
"There was a guy with a gun... was that real?"
"Yes, baby, it was real. I'm so sorry." She reached for her daughter, but remembered in time not to touch her.
Mandy struggled to sit up.
"No... don't move," Helen said. "I'm so sorry... I never meant for any of this to happen... I never meant for you to get hurt."
" _You're_ sorry? What are you sorry for?"
"This. All of this. Of course. Mandy, please just lay back down."
"Leave me alone, dammit," Mandy protested as she got herself into a sitting position. "Now explain why the hell you're apologizing!" Strength was returning to her voice.
Helen let out a sigh and braced herself. It was time to confess, like it or not.
"I'm the Robin Hood Thief. All of this is my fault. I'm so sorry."
Mandy's jaw dropped and her eyes widened. "No freaking way, Mom!"
"I never should have done it," Helen said. "I _knew_ they would come after you too. I tried to take precautions to protect you, but..."
"Wow... Holy shit, Mom." Mandy's tone had shifted to something... admiring? "Do they know who I am?"
That was an unexpected question. "Of course they know who you are. They wouldn't have grabbed you if you weren't my daughter."
Mandy shook her head, then grimaced in pain. She leaned in toward Helen's ear and whispered so softly that Helen could barely hear her. "I'm Cobalt."
"What?" Helen gasped.
She leaned back from her daughter, who suddenly felt like a perfect stranger. She stared in amazement as she tried to take it in. "You're Cobalt?"
Mandy nodded soberly.
Helen wanted to laugh.
She'd imagined some scrawny boy, a kid with no job probably still living in his mother's house... the image shifted and morphed. No, a _girl_ with no job still living in her mother's house. With blue hair. _"You_?"
"Yeah." It was all Mandy could seem to muster.
Helen thought back along the interactions she'd had with Cobalt. There had never been any particular reason to think Cobalt was male, she realized. They had all just assumed.
Cobalt was _her_ _daughter_? The chubby stoner, the college student who was making nothing of her life... this person was the amazing hacker who had been Helen's partner in crime for weeks now?
Helen began to laugh. "All that time, I kept going to the library or some Scenie store to send you messages..."
"Shhhhhh..." Mandy whispered insistently, but she broke into quiet laughter herself.
Helen leaned close to whisper the rest of the sentence. "I could have just knocked on the wall."
Their laughter approached hysteria. Hushed hysteria.
"Oh my God," Mandy gasped.
A thought struck Helen and she sobered. "It's really not possible? The thing with the stocks?"
Mandy grimaced. "I don't know, Mom. I really don't. It's super difficult. Maybe in, like, months. Maybe."
They stared at each other.
Helen found herself toying with acceptance. It was alien and strange after all this struggle, to accept that she had already done all she could. But then, the point wasn't success. It was courage.
"What do we do now?" Mandy asked.
Helen stood up.
"You're hurt," she answered. "So it's up to me. I'm going to go out there and try to fight them. I have to get you out of here."
"Mom, you can't fight th—" Mandy finally noticed the bloodstains on Helen's shirt and gasped. "Oh shit! Mom, you got shot!"
"Yeah. It's not the first time, remember?" Helen managed to say with some panache. She grinned wryly as her daughter's mouth fell open for the second time. "But listen, Mandy, I have to try. It's just you and me here. And I'm not going to let you get hurt again if I can help it."
Mandy stared at her mother with awe. "Since when did you turn into a badass?"
Helen grinned. She thought again of her realization a few minutes ago.
"I think I figured out that the only way to live is like you're never going to die. To live with absolute courage."
Mandy stared at her, taking this in.
Helen took a breath. "Mandy..."
It was time to tell her about the diagnosis, and that Helen had three days left to live.
But Mandy interrupted her.
"Absolute courage, huh? Okay. If you're going out there like a crazy person, so am I."
"No, Mandy... no, you're my _daughter."_
"So?" Mandy fixed her with a steely-eyed look as she got to her feet. "Do you actually mean what you just said or not?"
Helen grimaced. She'd painted herself into a corner. "But there's something else I need to tell y—"
The door opened and Roman took a step into the room. "Time's up." He swung Helen's silver locket casually in his hand.
Helen didn't even see how it started. Something long and heavy swung at Roman from above. He threw up an arm and the weapon drove his arm down and slammed into his shoulder and the side of his head. He half-crumpled, stunned. The silver locket hit the floor.
A standing lamp. Mandy had swung it overhead and the weighted base had become a weapon.
Mandy tried to raise it again for a second swing, but Roman grabbed onto it defensively.
Helen's impaired brain was frozen. She stared in confusion and shock.
An explosion sounded from stories below. The floor vibrated beneath Helen's feet.
Mandy dropped the lamp and stepped forward. She kicked Roman in the head with her combat boot, and he grunted as his head snapped back. But as she tried to kick him again, he caught her leg and twisted it, forcing her to the ground. He began to crawl on top of her.
The garrote wire. Helen had it wrapped around her ankle as her weapon of last resort. She bent and unwound it as quickly as she could.
Mandy hit Roman in the face, drawing blood, and he knocked her arms aside.
_"What're you going to do to get that around his neck? Ask him nicely?"_ That was what the man had said when she'd bought the garrote on a whim because she'd seen it in an action movie.
But Roman was already on the ground, distracted, facing away.
She got the wire over his head and crossed the handles and twisted to put her back against his and pulled the wire as taut as she could.
He rolled onto his back, on top of her, crushing the breath out of her, and she went with his momentum until they were both on their sides on the floor. He flailed and caught her in the side of the head with the back of a fist, and her vision flickered, but she didn't let go.
He bucked and struck out with his arms and legs, and Helen absorbed the blows while she focused with wordless grim determination on keeping the wire taut. Mandy stayed at the periphery, crying out in fear.
Helen's muscles weakened, and she began to lose hope. But he weakened too, and at last his weight sagged limply to the floor.
"Help me tie him up," Helen gasped. He would only be unconscious for a few moments.
They searched quickly and found curtain tiebacks they used to secure him.
Then Mandy threw her arms around Helen, and they hugged for dear life. Hugged for the first time in three years.
Helen could hardly bring herself to let go.
"Absolute courage, right, Mom?" Mandy asked breathlessly. She looked like she wasn't sure whether to celebrate or to cry.
"Hell yes," Helen answered fiercely.
Her locket.
She searched quickly on the floor, and when she found the locket, it was open and empty.
She glanced around frantically until she saw the black powder smeared on the floor under Roman and the pill's casing a crushed mess a few feet away.
# 2 Days, 13 Hours, 36 Minutes
"What is it?" Mandy asked, not understanding what Helen saw.
"Nothing. It doesn't matter," Helen lied, even though it felt like the dismay cut her in half.
She stood up, put the necklace in her pocket, and faced her daughter. "Now we take out the rest of them."
Hesitantly, she put out her hand. Her daughter took it and clasped it firmly.
Together, they went out of the room.
As they went down the hall, Helen realized that she had been hearing the unmistakable clatter of a helicopter. And now it was close... very close.
Carefully, cautiously, Mandy and Helen opened the door leading into the living room.
There, Tolbrook stood facing the enormous floor-to-ceiling windows, staring in bemusement at the helicopter as it swung alarmingly close. It was unmistakably interested in the penthouse.
A second explosion shook the ground under them. What was going on?
A guard came in from the other hallway. "Sir, things are escalating downst—"
Tolbrook cut him off with a wave of his hand.
Helen saw the insignia on the chopper. It was LSTV. Her heart skipped again. Was this a good thing? Or a bad thing? Either way, it was peering in the window from all of twenty feet away, just on the other side of the balcony railing, just past the Jacuzzi and the pool.
Tolbrook shifted anxiously. He said to his guard, "Those windows are tinted, right? They can't see in?"
"Yeah," the guard answered.
The chopper rotated to the left, and a man inside adjusted controls to set the helicopter to hover—it was a recent model—then extended a handgun with one hand. He fired off a few shots high, toward the top two feet of the tall windows.
The huge pane of glass came apart and dropped in shattered fragments, exposing the men to the chopper's view.
Tolbrook and the guard jerked back from the window and ran for cover behind the sofa as warm, humid air swept in.
The man in the chopper looked in again, and this time Helen recognized him from his Whatsit photo. Christian Smith. He had gotten her message after all. And he'd come to save them. Admiration and gratitude swelled in her chest.
Helen looked at Tolbrook and his guard, who were still cowering behind the sofa. Tolbrook screamed at the guard. The man's reluctance was obvious on his face, but he ran toward the two women.
"Run!" Helen shouted to Mandy, grabbing her hand again. "Get to the chopper!"
As they ran past the dining table, the chopper came even closer, the wind from the rotor blade tearing at them and everything lightweight in the suite. The racket beat at their ears. The chopper struggled for a landing spot. It got one landing skid down on the balcony railing and teetered.
The guard cut them off on their path toward the chopper. He tried to use his handgun to hit Mandy on the head, but she dodged and ducked behind Helen.
Christian Smith raised his pistol again, sighted, and fired twice with careful precision. The guard dropped—and so did Tolbrook, who'd come too far out from behind the sofa.
Relief swept over Helen. She knelt and picked up the guard's gun, just in case.
Another LSTV chopper appeared from around the same corner. But it kept its distance, behaving as Helen thought a news chopper would. A newsman operated his camera in the back.
Helen hesitated. She felt compelled to find out whether Tolbrook was alive. "Get to the chopper!" she yelled to Mandy, and she hurried to Tolbrook's side.
Mandy disobeyed her and followed, though she stayed back a few feet.
Tolbrook was conscious, but pale. Blood covered his hands where he was trying to apply pressure to his left side.
He looked up at Helen, his chest heaving. "Help me," he begged.
She stared at him, remembering what he had said to her before, when their positions were reversed and she was the one on the floor hoping he wouldn't kill her. "I would tell you to just give up, but I guess I'm not a jerk," she said. "I guess you taught me that. So instead, I'll tell you to have hope. I'm sure this place will be swarming with first responders in about half a minute."
"Please..." he moaned helplessly, and then he passed out.
Just as she was about to turn away, she caught sight of his Earworm still clinging to his ear. She swept it off his head.
She gave it to Mandy.
"Can you please take this guy for all he's worth?" she said. "Copy down all his passwords or something? He didn't see me take it. He'll think he just lost it in all the chaos."
Mandy took it and tucked it into her bra. "Hell yes. Now let's go!" She gestured toward the helicopter.
Both of them ran to the chopper. Christian Smith threw a rope down, and Helen boosted Mandy up toward the chopper as the younger woman scrambled for the rope. Helen hated having her daughter precariously balanced on the skid and the railing with fourteen floors of empty space below her. But she made it in.
Then Helen's gaze connected with the ground far below them, and she gasped.
There were hundreds of people massing outside the building, forming a bottleneck at the doors of the lobby—screaming—attacking the police—and just as many SWAT team members. The cops wore riot gear. They threw smoke bombs and fired their guns into the crowd.
"No..." she breathed. "Not for me, dammit!"
People were dying for her, and she couldn't allow that. Not when her life was forfeit already—had been forfeit from the start of all of this.
Everyone knew who she was now. If she went down there and showed herself, gave herself up... told them to stop...
It would be self-sacrifice and possibly suicide. There was no chance she could somehow stop what was happening downstairs and still evade the police and stay alive and free. But once she was captured or killed, the rest would disperse. Lives would be saved.
She remembered, too, that the black pill was gone, crushed during the fight with Roman. There would be no easy way out of this.
She stared at her daughter up there, safe in the chopper. Mandy held the rope out to Helen with one pale hand. Her daughter reached out to her.
But Helen had no choice.
The helicopter's racket was too loud for Helen to explain, to make Mandy understand. But there was nothing Helen could do about that.
Resolutely, Helen shook her head at her daughter and stepped back. She waved at Christian Smith—telling him to get clear, to get Mandy to safety.
Mandy screamed at her. Helen couldn't make out the words, but she could guess them. _No! Mom, come on! Grab the rope!_
Christian's gaze met hers for a long moment. Surely he couldn't understand.
The sound of more helicopters met her ears. She looked up to see black SWAT choppers descending on the building. They were out of time.
Mandy screamed again.
Helen gestured emphatically at Christian, who nodded, then pulled the chopper away from the building.
She had chosen it, but it still hurt. It hurt to watch her only chance of escape leaving her behind, and it hurt even more to watch her daughter being taken away from her, with so much that now might never be said.
She turned back toward the penthouse. She had no choice but to face whatever waited for her down there.
_Courage. Not success._
She went to the stairwell and through the door.
"There she is!"
Helen looked down the stairwell.
A concierge in a gray hotel suit ran up toward them, one hand on his Earworm. "I've got her! Stairwell four, thirteenth floor! Everybody to stairwell four!" He was... smiling.
Helen stared in confusion.
Behind him, people came up the stairs and into the stairwell from every other floor. A cheer echoed as they looked up at her.
Helen's mind refused to believe who she saw in the crowd. Egemon. With half a dozen weapons strapped to him. He took the stairs three at a time.
"What—" Helen couldn't even figure out how to continue the sentence.
Egemon grinned. "The call went out over Whatsit. Got here as quick as I could. Did you really think we would leave you to get killed?"
A stream of people filled the stairwell. All kinds of people—many were hotel staff and people in assorted employee uniforms she realized must have come from the Lake Estrelle Shops connected to the hotel. Others looked like ordinary people, like Helen. But some were recognizably among the wealthy, either patrons of the hotel or customers of the shops. She even saw, to her astonishment, some police officers and hotel security on her side. Together, the assortment of people packed the stairs and formed a mob.
"These people are all here for you," Egemon said. His dark eyes shone and his breath came fast. He raised his voice triumphantly over the crowd. "Make way for Ms. Robin Hood!"
People pushed back against the sides of the stairwell, and Egemon helped Helen down into the very thick of the crowd.
Shouts and gunfire rang out from below. Another explosion.
"I don't want these people to get hurt because of me," Helen said into Egemon's ear. "You can't do this for me."
Egemon shook his head. "We _have_ to do this for you. You understand?" His dark eyes met hers.
She saw his determination, and suddenly she understood. This was about people taking action, taking control. Standing with her was a way of standing up for themselves.
Still... "I don't want anyone to get _hurt_ ," she repeated.
Egemon smiled. "I don't think you're in any position to argue."
He put her arm around his shoulders and wrapped his arm around her waist, and Helen leaned on him as they rapidly descended the stairs. The mob moved quickly, then even faster, and Helen focused on her footing. At this speed, if she stumbled, she would be trampled.
Swept along by the crowd, she heard a spontaneous chant break out, at first unintelligible, and then clear: "We—love—the Robin Hood Thief. We—love—the Robin Hood Thief."
Acrid, burning smoke rose to meet them as they descended to the ground floor and Helen's feet reached the marble of the lobby. The lights were off, and the smoke defeated the sunlight that came through the double doors and the vast windows. Gunfire flashed here and there in the swirling clouds.
The mob kept up the chant, "We—love—the Robin Hood Thief," as it swept through. Helen could see nothing over the heads of those taller than her. Shouts, cries, and gunfire mixed in the air. People moving too quickly bounced off each other to her right—people were knocked down—she dodged sprawling limbs as Egemon half-lifted her along.
The doors were open in front of them and now those ahead of them were running. Egemon swept Helen up in his arms and carried her at a jog.
Police and SWAT officers were scattered outside the doors, but holding their fire, their expressions confused. Perhaps they could no longer make out who their targets were—not with policemen and security guards and the ultra-rich in the tumultuous mix that swept out of the doors past them.
They were going to make it.
# 2 Days, 10 Hours
Afterward, Egemon, Helen, Mandy, Christian, and Zara met at the pawn shop, where Egemon left the CLOSED sign up and ordered in food for everyone. Adrenaline and exhilaration ran high. There were giddy hugs and introductions all around and the chance for everyone to tell and retell their own story of what had happened. Meanwhile, Zara patched up Helen's gunshot wound and Mandy's head injury.
After everyone ate and the nervous energy settled, and as Mandy started working on Tolbrook's Earworm, Helen knew the time had finally come to make her confession.
They stood at the counter, Egemon in his customary pose behind it, Christian and Mandy leaning on the front of it, Zara cleaning up her medical supplies nearby. Thunder boomed overhead as the afternoon's rain set in.
They sensed her mood sobering, and they all waited, watching her expectantly.
"I feel like I owe you all an apology," Helen said. Christian shook his head, and she lifted her hand to ask for his silence.
She had to take a deep breath. Even through her exhaustion and pain and the ravages of the disease, she could still feel her heart ramp up in anxiety. To finally say those words out loud to the people she cared about most...
"The rescue... I owe all of you my heartfelt thanks for saving my daughter's life. But for myself... I feel that you ought to know..."
Helen's focus narrowed to the person most precious to her in the world—her daughter, who still looked down at Tolbrook's Earworm.
The words stopped in Helen's throat.
There could be no coming back. There could be no escaping the pain. And Helen would do anything to avoid hurting her daughter.
She took another deep breath.
It was always part of the deal, she reminded herself. Mandy had known this reality since she was six years old. She understood that this was how it worked, that they were both helpless in the face of this.
The moment became hyper-real. The traffic sounds from outside the pawn shop. Mandy's face a few feet away. The way Helen's own dry lips parted as she drew breath to say the words.
"I'm dying."
The words filled the space and then dropped and settled into nothing.
Mandy's eyes searched Helen's face. "No way."
"I am. I'm sorry. I don't want to leave you—" She had to stop.
Mandy's eyes filled with tears. "Of what? Dying of what?"
"A brain disease... A rare prion disease. It's incurable... I only have a few days left." Now the words were tumbling from Helen's lips.
Mandy stared. "A few _days?_ "
"Yes. I know. I'm sorry."
Helen braced herself. She suddenly realized that she deserved Mandy's anger and scorn for not telling her sooner, and it felt like a heavy weight on her chest. "I'm sorry. I should have told you sooner. I couldn't find the right way, or the right time, but I should have anyway. I should have just made myself do it."
Mandy looked down at Tolbrook's Earworm, pretending to adjust something on it. Her chin trembled.
Quietly, she said, "I kind of knew, I think. You've gotten really thin, and your sleep has been weird, and you've been acting... bizarre." When she looked up, there was maturity in her gaze. Mandy wasn't six years old anymore. "I guess I was afraid to ask you about it."
They hugged each other at the same moment.
"Did you get a black pill?" Mandy asked, tears in her voice.
"Yeah."
They squeezed each other hard for a moment, then pulled apart. Egemon scrounged behind the counter for some napkins, and they both wiped their faces.
What came after the revelations? After the tears? Helen's brain wouldn't fire properly. Suddenly all she could think about was Old Blue.
"I was going to leave you the car," she said, "but it kind of got blown up this morning."
Mandy almost laughed.
Helen looked at Egemon, afraid of what she might see on his face.
She saw only regret.
"Now you understand," Helen said. "About... not having much time."
He took a drag from his e-cig. "A few days, yeah?"
Helen didn't answer. There was nothing to add.
"I'm sorry for that," he said quietly.
"But it can't end here," Christian said. "What you started. The people need this. We all do."
Egemon nodded. "I agree completely."
Mandy looked up from Tolbrook's Earworm. "I'll keep it going," she said.
They all looked at her, and she met their gazes with a determined stare. "Mom, give me your Whatsit information and I'll keep up your profile. You didn't encourage your copycats, but I will. Because what they're doing is awesome. And I still have my whole posse. We'll keep working on the stock market thing. And whatever other ideas we can come up with. We'll keep the whole thing going. In your honor. You know?"
Helen stared in admiration.
Perhaps it was ironic that now, knowing her daughter was a proficient hacker, Helen finally trusted her to be fine. To take of herself. It was hardly the kind of dream every mother had for her daughter.
"I'll be an advocate for you," Christian said. "For the Robin Hood Thief. First I'll have to bail myself out of the trouble I'm in, and I'll have to get a new job. But in every way I can, I'll make sure the public is aware of what the Robin Hood Thief is doing and why it's important."
"Anything you need," Egemon said to Christian and Mandy, "you come to me. I have a lot of connections."
"I can patch you up when you need it," Zara volunteered. "You people seem to need a lot of that."
Everyone chuckled, the tension and grief replaced by camaraderie.
"The Merrie Men," Christian said suddenly.
Everyone looked at him.
"We're the Merrie Men. You know, for Robin Hood."
There were smiles of recognition all around.
"Okay, but Mandy, you have to use spell-check if you're going to post as me," Helen said with mock disapproval. "Otherwise everyone will know. You have the _worst_ spelling and grammar—I can't believe you ever passed an English class."
Mandy rolled her eyes with a sigh. "English is so overrated."
In the next hour, Mandy broke into Brock Tolbrook's financial accounts via his Earworm—"This guy didn't even have retina security turned on. That's so _basic_ "— and e-signed away all his assets.
Two hundred sixty-three million dollars were donated to charities that helped ex-convicts build new lives.
They set aside five million for the work of the new Robin Hood Thief and her Merrie Men.
And after a lot of persuasion, Egemon, Christian, and Zara each accepted a quarter of a million dollars for services rendered.
It was victory.
The little group was reluctant to disband, even temporarily, now that they had each other. Purpose and friendship were a wonderful new drug. But eventually Zara muttered something about the grandkids and made her goodbyes, and Christian set out soon after.
Mandy observed how Helen and Egemon were looking at each other, and she rolled her eyes and pretended to go examine the merchandise.
Egemon's expression was both stormy and deeply sad. His eyes, always in shadow, were dark and liquid.
He took a long drag from his e-cig, then came around the counter and put his arms around Helen and held her quietly for a while.
She listened to his heartbeat and savored his warmth and the scent of his cologne. It was a sort of paradise.
Eventually, Helen pulled back to look up at him, and he released her, taking her hands instead.
"You wiped out my record," he said. "You and Cobalt. And with that little payment you just made me take, I've got options." He half-grinned, looking embarrassed about what he was going to say next. "I've decided I'm going to open that antique toy store."
Helen smiled.
"I'll go straight," he said. "Except for helping Cobalt, of course."
Helen's smile grew bigger.
Mandy called enthusiastically from across the showroom. "Hey Egemon, how much for this awesome bass guitar? The bright blue one?"
"For you? Free," Egemon called back.
Mandy whooped, and Egemon and Helen grinned at each other.
"Will I see you again these next few days?" Egemon asked quietly.
"If you want to," Helen said soberly. "I don't know if I would choose that if I were you."
"Didn't you say life and death are not in charge of us?" he said with a sad smile, and he put his lips to hers.
# Zero Days
Two nights went by with no sleep, even with the meds, and the heavy curtain of utter exhaustion closed in over Helen's senses.
On Thursday, Helen said goodbye to Jessie. Then she, Mandy, and Egemon went to the hospital where she'd received her diagnosis. They wouldn't let her go alone.
As they stepped out of Egemon's car, Helen put on a surgical mask to hide her face and protect her identity. She held Egemon's arm, because her legs were too unsteady and unpredictable.
On the way to Wing D, Helen stopped in front of the cramped drugstore with the three pharma machines. She looked around.
How things had changed for her since she'd been here last, on an afternoon that had changed her life completely. She still felt enlivened by all she'd done—her life made purposeful. Her body was unable to carry her any farther, but her spirit felt as if it could fly forever.
It made her want to believe in life after death. It didn't seem right that such feeling could be terminated by a black pill.
She wanted to believe she would go up into Heaven and be able to look down at Mandy and just wait for her daughter to join her after she'd had her own adventures.
Instead of just going to sleep and never knowing the difference.
Either way, she'd be free of struggle. Free of pain, free of ugliness and despair. Forever.
But she would never again see her daughter or Egemon or little Jessie. Never again Oliver or Christian or anyone else she had ever cared for or admired.
Never again.
"What is it, Mom?" Mandy asked. She looked concerned.
"Just remembering," Helen said. "We can go."
The three of them went down the hallways with the too-dim, half-burned-out LED lights overhead. They passed a cleaning bot on the way, dormant and awaiting repair next to the janitor's closet, and Helen thought of Jack. If there was a God, he'd better have some apologies ready for Jack.
Wing D. The crematorium. They went through a door into a small waiting area. It was empty for now. Perhaps two in the afternoon wasn't a popular time of day to self-terminate.
Egemon helped Helen up to the counter. "I'm checking in for termination," Helen said. "Jane Doe."
A nurse, a middle-aged woman with dyed red hair, was distracted by something on her Earworm, and she only gestured with long fake nails toward a projcom on the counter. "Fill in the form."
Helen stepped up, pulling down the surgical mask that was now in the way.
She filled out the form on the screen agreeing that her check-in was voluntary, that she was of sound mind, and that she was acquiescing to her own death. She skipped the twenty pages of disclosures and checked "I agree."
"Done," Helen said.
When the nurse stood up, she did a double take, then leaned forward with a smile that brightened the whole room. "You're the Robin Hood Thief! What are you doing here?" Then her smile disappeared. "You're not here to terminate, are you?"
At Helen's resigned look, the other woman shook her head. "You can't terminate. You can't go. We still need you."
Helen shook her head slowly. "I can't help it," she said quietly. "I'm sick. And I'm just about to my limit."
The nurse looked Helen up and down. She must have taken in Helen's thinness and the sunken look of her eyes. Her face changed. "Oh, no."
"I'm sorry," Helen said dully. "I wish I could stay. I wish I could have a chance to see what happens next. But this body is done. Anyway, they know who I am now. I wouldn't be free for much longer."
The nurse turned to Mandy and Egemon. "I'm so sorry," she said.
Mandy just nodded. Her chin wobbled.
"Don't tell anyone, okay?" Helen asked. "I want people to keep believing in what I was doing. Others will carry on in my name. But it will work better if they think it's me, at least for a while."
The nurse's eyes showed her understanding. "But it'll be in our records today. They know your name now."
"Not if you don't tell anyone I came here today. I signed in as a Jane Doe."
The nurse nodded slowly, her mouth open as she came to understand. "Oh... I see." She looked like she wanted to make Helen take it back or change her mind or somehow just not be here to die today. But she settled for giving Helen a grave nod. "All right, then. You have your termination pill?"
"Yes, it's right here." Helen held up the locket. Mandy went yesterday to the drugstore to get the replacement for her.
The woman cast about again for something to prolong the moment, but there wasn't anything any of them could say.
"Right this way, then." The nurse ushered them into a small room adjoining the waiting room.
It had a bed covered in white paper, like that in any exam room. But someone had decided that heavenly imagery would be appropriate at this stage, so the walls were painted: clouds hung against a blue sky that got darker and darker as it went up the walls, and a night sky with yellow stars decorated the ceiling.
"Well, this is it," the nurse said regretfully.
Helen turned to her daughter, and they exchanged a long hug. Helen treasured the comfort of holding her daughter in her arms for the last time. Her little girl who had become a young woman.
She hugged Egemon for the last time. His face was as serious and inscrutable as ever. He helped her climb up onto the bed.
She felt again the pain in her thigh and the whole-body tremor and the thudding palpitations that never went away.
Now they were going to go away.
She took the black pill out of her locket and contemplated it.
New apprehension gripped her, and her stomach flipped over. Her heart rate went up.
She was about to die. To _die_. For real this time. A moment she had anticipated, either with dread or with longing, hundreds of times since she had become conscious of her own mortality.
The nurse gave Helen a glass of water. "It was an honor to have met you," she said. Then she stepped back to give them privacy.
Mandy stepped up, and Helen smiled at her daughter and took her hand. "I'm so impressed by the young woman you've become," she said. "Daddy would have been, too."
Mandy just smiled while tears left tracks on her face.
"Take care of her," Helen said to Egemon.
"I will," he said.
Helen raised the water glass in a toast. "To doing all the good you can in the time you have."
The others nodded.
Helen swallowed the pill, set down the glass of water, and lay down.
Mandy and Egemon each took one of her hands and held tight, and Helen gave them a last sad smile.
Then she looked up into the painted night sky and thought, _Okay. I'm done now._
She let that thought fill her mind until there was nothing else and then nothing more.
# Dear America
> I have something to say to you.
>
> You get to do this life your own way.
>
> Maybe that didn't hit you strongly enough, so let me just say it again.
>
> _You get to do this life your own way._
>
> Find a way to make a choice that's _yours_ and yours alone.
>
> And don't be afraid—or be afraid, but do it anyway.
>
> This was who _I_ had to be—this is what _I_ had to do. You don't have to make the same choices. But if I could do so much in just forty-five days, then _you_ can do even more in the time you have.
>
> I'm not interested in your excuses. _Do_ something. Something that takes a giant step toward a life you can believe in. Something brave and good.
>
> Come on. Join me.
>
> Be a hero.
>
> Love, the Robin Hood Thief
THE END
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# Please Review?
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Reviews are the most powerful way for indie authors to get attention for their books. As much as we'd all love to, we don't have the budget to take out full-page newspaper ads or put posters on the Metro.
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# Acknowledgments
I would like to thank my amazing beta readers: Michele Leonard, Josef Molnar, George Wright Padgett, April Stewart, and my mom, Janet Henderson, who read at least three different drafts. And thank you to my marvelous critique group, Team Gargoyles: Wayne Basta, Ian Everett, Chris Lewis, Laura Rich, and Shannon Winton.
Thanks go also to the redditors of /r/Hacking for their help with the details. Thank you to Ryan "Iggy" Harrison for technical advice. Thank you to Chris Lewis for plot troubleshooting, technical advice, and legal information.
Thank you to Dan Ko for all your great advice at my Writers' Salons—and for buying me lunch that day. (See, told ya.)
Thank you to Jason Aydelotte of Grey Gecko Press for dealing with all the fiddly bits.
Thank you to all my friends and family who listen to me ramble on about my writing life on a regular basis.
And thank you most of all to my wonderful husband, Benjamin, for supporting my writing. I am so lucky.
# About the Author
H.C.H. Ritz has a degree in theatre from the University of Houston and directs community theatre in her spare time. Originally from rural Mississippi, she has lived in Houston, Texas long enough to have turned into a city person. She is married to a wonderful human being and has a young son and a tortoiseshell kitty named Roxie Underfoot.
Connect with H.C.H. Ritz
@hchritz
HCHRitz
www.hchritz.com
hilary@hchritz.com
# Also by H.C.H. Ritz
**_W hen perfection is mandatory, revolution is the only answer._**
Gaylen is devastated when his wife leaves and takes their only child. But in a utopian society built on positive thinking, grief is not okay - and the government steps in to fix him.
Confronted with the dark truth of his seemingly perfectly world, Gaylen soon finds himself a reluctant member of the resistance. But his new role could cost him his life.
Even as the revolutionaries struggle to strike a powerful blow for freedom of thought - even as they are betrayed by the ruthless denizens of the underground and pursued by the murderous secret police - Gaylen still dreams of getting his family back.
But if he survives long enough to put the revolutionaries' plan into motion, Gaylen will have to go up against the calculating masterminds of his vicious society...
Available now
# Recommended: The Year of the Hydra
Could a dark agenda be woven into the architecture of China's most sacred ancient temple? An agenda that only Julian Mancer is seeing? Or is Julian off his meds again? If the structure were in fact a doomsday device awaiting an astronomical tripwire—could Julian stop it?
Julian is determined to discover the answer, as soon as he concludes a far more pressing matter involving a sixteen-year-old girl with a _most_ intriguing mutation.
Available now
# Recommended: Spindown
For over a hundred and fifty years, the rarest and most valuable substance in the solar system has been mined from the only location where it exists in significant quantity: Jupiter's largest moon, Ganymede. For all of this time, the remote mining outpost has been serviced by clone slaves who are drugged into mindlessness, and all of it has been monitored, controlled, and administered by the artificial intelligence known as Prinox.
But what happens when a failed rescue mission causes a small band of escaped clones to begin questioning their lives, their society, and their very existence? Hunted by deadly killing machines, confused and scared, these renegade slaves are about to find out—for better or worse—just what it means to be human.
Available now
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Text & Illustrations ©2016 by H. C. H. Ritz
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All rights reserved. Other than for review purposes, no part or portion of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage or retrieval system, without permission in writing from the publisher. This book is a work of fiction. Any resemblance to real persons (living or dead), events, or entities is coincidental.
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Library of Congress Cataloging-in-Publication Data
Ritz, H. C. H.
The robin hood thief / H. C. H. Ritz
First Edition
|
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| 6,128
|
Q: R: Winsorize one column per year for panel data I would like to winsorize the column "Return". However, I want to winsorize per Year.
My data looks like this:
structure(list(Name = c("A", "A", "A", "A", "A", "A", "B", "B",
"B", "B", "B", "B", "C", "C", "C", "C", "C", "C"), Date = c("01.09.2018",
"02.09.2018", "03.09.2018", "05.11.2021", "06.11.2021", "07.11.2021",
"01.09.2018", "02.09.2018", "03.09.2018", "05.11.2021", "06.11.2021",
"07.11.2021", "01.09.2018", "02.09.2018", "03.09.2018", "05.11.2021",
"06.11.2021", "07.11.2021"), Return = c(0.05, 0.1, 0.8, 1.5,
0.1, -2, 0.4, 0.6, 0.6, 0.2, -0.5, -0.9, -0.4, 1.7, 0.3, -4,
0.6, 0.5), Year = c("2018", "2018", "2018", "2021", "2021", "2021",
"2018", "2018", "2018", "2021", "2021", "2021", "2018", "2018",
"2018", "2021", "2021", "2021")), row.names = c(NA, -18L), class = c("grouped_df",
"tbl_df", "tbl", "data.frame"), groups = structure(list(Year = c("2018",
"2021"), .rows = structure(list(c(1L, 2L, 3L, 7L, 8L, 9L, 13L,
14L, 15L), c(4L, 5L, 6L, 10L, 11L, 12L, 16L, 17L, 18L)), ptype = integer(0), class = c("vctrs_list_of",
"vctrs_vctr", "list"))), class = c("tbl_df", "tbl", "data.frame"
), row.names = c(NA, -2L), .drop = TRUE))
I tried the following code which also works:
Data <- Data %>%
group_by(Year) %>%
mutate("Winsorized Return" = Winsorize(`Return`, probs=c(0.01, 0.99)))
Now I do the same but without grouping the data per year:
Data <- Data %>%
mutate("Winsorized Return" = Winsorize(`Return`, probs=c(0.01, 0.99)))
I get exactly the same results even though I used "group_by(Years) in the first code.
Can someone explain me why the results are the same? Even when I do the same with my real (very large) dataset I get the same results.
Do I have to use another code in order to winsorize the column "Return" per year?
Thank you very much already for any help!
|
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| 8,482
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Arkansas Advanced Energy Foundation
Parnell Consultants Working to Bring Renewable Energy to Arkansas
by Arkansas Advanced Energy | Aug 5, 2016 | 2016 News
Jason Brown
jbrown@comgroup.com
Parnell Consultants of Booneville today announced a partnership with Clean Line Energy Partners (Clean Line) to provide easement acquisition services for the Plains & Eastern Clean Line transmission project in Arkansas. The Plains & Eastern Clean Line is an overhead direct current transmission line that will deliver low-cost wind power from Oklahoma to Arkansas, Tennessee and other states in the Mid-South and Southeast. The project includes a delivery station to be located in Pope County that will provide enough low-cost wind energy to power more than 160,000 Arkansas homes each year.
"This project is a real win for the state of Arkansas and we are happy to be a part of it," said Floyd Parnell, CEO of Parnell Consultants. "Our partnership with Clean Line is a great example of their commitment to working with local businesses. For our firm, the Plains & Eastern project will put 18 additional Arkansans to work full time," said Parnell.
Parnell Consultants, Inc. has served major oil, gas, utility and communications companies since 1987, and is proud to have supported job creation and economic growth in Arkansas. The company currently employs 55 people as right-of-way agents and pipeline inspectors.
"Parnell is a great local provider for us and, with more than 20 years of experience in Arkansas, we're proud to have them on our team," said Michael Skelly, President of Clean Line Energy. "We are pleased to count Parnell among our partners, and look forward to adding to the list and putting more Arkansans to work," said Skelly.
Construction on the Plains & Eastern Clean Line is expected to start in 2017 and is anticipated to be operational in 2020.
About Parnell Consultants, Inc: Parnell was established in 1987 by Floyd T. Parnell to satisfy the demand for the never ending search for fuels. Since that time the company has grown to handle cross country builds for fiber optic systems, project management and inspection services for oil and gas companies.
About Clean Line Energy Partners: Clean Line's mission is to connect abundant, renewable energy resources to areas that have a high demand for clean, reliable energy. Clean Line is developing a series of transmission projects to move renewable energy to market. For more information please visit www.CleanLineEnergy.com.
Commission denies anti-competitive solar proposal from Entergy
AAEA Welcomes Commission Decision Affirming Economic Benefits of Solar
Seal Solar completes 2-megawatt array for Washington County
No rate change for residential solar
PSC's Long-Awaited Net-Metering Ruling Favors Solar Industry
@ 2019 Arkansas Advanced Energy | Website Development by Sullivan Wright Technology Partners
|
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| 4,694
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* Ensure all client Service level Agreements are consistently met and all penalties are avoided.
* Identify the processing, procedural and billing errors and practices that lead to claims inaccuracies and propose ways to systemically address.
* Develop, enhance and improve policies, procedures and workflow processes associated with claims review.
* Support other departments in problem resolution as necessary.
* Complete required reports and ad hoc requests for information.
* Develop reporting templates and dashboards to drive efficiency, increase productivity and improve quality through metrics and standards.
* Consulted often by others for advice and opinions and recognized as a leadership role model.
* Manage a team of direct reports.
* Evaluate team priorities to ensure they are realistic and relevant.
* Establish accountability for performance through the communication of SMART goals and behavioral competency expectations.
* Monitor productivity and quality to ensure SLAs and timelines are met.
* Provide regular, honest feedback on performance.
* Provide guidance and training to drive performance and to help people develop.
* Associate Degree required, BA or BS preferred.
* Strong project management and interpersonal skills, makes sound decisions, exhibiting initiative and intuitive thinking.
* Problem solving, strong analytical skills, people skills, teamwork, people management, and managing processes.
* Strong interpersonal skills for interfacing with all levels of internal and external audit teams and management.
* Ability to prioritize and multi task.
We hire people who will embrace the company's goals and productively contribute in ways that help us serve the customer, innovate, and stay strong.
Our corporate headquarters is located in downtown Chicago within the historic Merchandise Mart-a certified LEED (Leadership in Energy and Environmental Design) building.
CCC Information Services was recognized by Forbes as one of America's Best Mid-Sized Employers in 2018 and ranked #17 in the Top 100 Digital Companies in Chicago in 2017 by Built In Chicago.
About CCC Information Services Inc.
CCC Information Services Inc., a wholly owned subsidiary of CCC Information Services Group (NASDAQ: CCCG), headquartered in Chicago, IL, is a leading supplier of advanced software, communications systems, Internet and wireless-enabled technology solutions to the automotive claims and collision repair industries. Its technology-based products and services optimize efficiency throughout the entire claims management supply chain and facilitate communication among more than 14,700 collision repair facilities, 350 insurance companies, and a range of industry participants.
This company profile was created by AfterCollege and is about CCC Information Services Inc.. This page is not endorsed by or affiliated with CCC Information Services Inc.. For questions regarding company profiles, please email: care@aftercollege.com.
|
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| 8,712
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Files at the National Records of Scotland, opened for the first time, show details about the third full year of operation of the then Scottish Executive.
The records cover the first full year of Jack McConnell's time in office as Scotland's First Minister. Papers cover the full range of Government activity at the time and include information on health, justice, rural affairs and finance.
Since 2009 – as part of its wider commitment to increasing openness and transparency - the Scottish Government has proactively opened files at 15 years.
Tim Ellis, Chief Executive of NRS and Keeper of the Records of Scotland, said "Scottish Government records are important for our understanding of our recent history, and preserving them and making them available to the public is a key part of our role at the National Records of Scotland.
"The annual file release attracts considerable interest – from journalists, researchers and, more generally, anyone with an interest in Scotland and Scottish politics.
"The vast resource of information made available for public inspection at the National Records of Scotland also acts as a clear demonstration of this Government's commitment to continue to strengthen the culture of openness and transparency in Scotland".
|
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\section{Introduction}
A recent paper in this Journal has shown that, in the partition generating
limit\cite{bk}, several orbital equations and clusters of orbital equations of the
quadratic (logistic) map coincide with cyclotomic period equations\cite{jg19}.
Period equations were introduced in 1801 by Gauss as {\it auxiliary equations}
to solve cyclotomic polynomials by radicals.
The purpose of this paper is to report a startling finding obtained by
extensive empirical computations: we find {\sl monogenic} period equations to
be either cyclotomic polynomials, or simple de Moivre reduced forms of cyclotomic
polynomials, thereby implying the existence of a hierarchical interdependence
among fields and subfields of cyclotomic polynomials, or orbital equations.
This fact is quite remarkable because, although cyclotomic polynomials are among
the most extensively studied polynomials for nearly 220 years, it seems to have
hitherto escaped attention that, in essence, Gauss auxiliary monogenic period
equations are nothing else than just cyclotomic polynomials themselves.
\section{Context and basic definitions}
As shown by Dedekind\cite{de78},
it is always possible to fix a number field $K$ of finite degree $n$ over $\mathbb Q$
by
selecting an algebraic integer $\alpha \in K$ such that $K=\mathbb Q(\alpha)$.
In other words, a number field $K$ may be determined
by selecting $\alpha$ as a root of a monic $n$-degree $\mathbb Q$-irreducible
polynomial $f(x)$ and expressing it in terms of $n$ integers
$\alpha_1, \alpha_2,\cdots,\alpha_n$, independent of each other,
forming a {\sl basis}
for $\mathcal O_K$, the {\sl ring} of integers in $K$.
In this context, a key problem
is to decide whether or not the ring $\mathcal O_K$ is {\sl monogenic}, namely
if there exists an element $\alpha\in K$ such that $\mathcal O_K$ is a polynomial
ring $\mathbb Z[\alpha]$, i.e.~if powers of the type
$(1, \alpha, \cdots, \alpha^{n-1})$ constitute a
{\sl power integral basis}\cite{g02,eg17,n04,ha64}.
Every algebraic number field has an integral basis, not necessarily a
power integral basis.
In a monogenic field $K$, the field discriminant $\Delta$ coincides with
the standard discriminant $D$ of the minimal polynomial of $\alpha$.
For non-monogenic fields such identity does not hold.
Generically, $D$ and $\Delta$ are
interconnected by the harmless-looking relation\cite{eg17,n04,ha64,kro}
\begin{equation}
D=k^2 \Delta, \label{discri}
\end{equation}
for some $k\in\mathbb Z$, called {\sl index} by Dedekind\cite{de78}, and
``ausserwesentlicher Theiler der Discriminante'', {\sl inessential discriminant
divisor}, by Kronecker\cite{kro}.
As pointed out by Vaughan\cite{v85},
``{\sl while $D$ can be found by straightforward (if tedious) computation,
the value of $k$ is quite another story.
According to Cohn\cite{hc78}, page 77, for
example, to determine $k$, one would have to test a finite number (which
may be very large) of elements of $K$ to see if they are integral.}''
An additional complication to obtain $k$ is the fact that the choice
of $\alpha$ is not unique and, therefore, there are several distinct minimal
polynomials from which to compute $D$.
So, one may consider $k$ as a sort of ``quality measure''
for minimal polynomial representation and for monogeneity.
As described in \S 4 of Dedekind's paper\cite{de78}, for quite some time he
believed to be always possible to find a suitable $\alpha$ leading to a power basis.
This,
until he found what became a popular textbook example of a non-monogenic field
generated
by a root of $x^3 -x^2 -2x-8=0$, for which $\Delta=-503$, $k^2=4$, and $D=-4\cdot503$.
The computation of the index
$k$ and the verification whether or not a given number field
has a power basis are two hard problems\cite{g02,eg17}.
A taste for the difficulties and the representative computation times involved
in such tasks may be obtained from a paper by Bilu et al.~\cite{bi04}.
Efficient
algorithms for determining generators of power integral bases involve solving
Diophantine equations known as {\sl index form equations}\cite{eg17}.
The first general algorithm for determining all power integral bases in
number fields was given in 1976 by Gy\H{o}ry\cite{k76}.
Subsequently, efficient algorithms were elaborated to determine
power integral bases for number fields of degree at most six and some
special classes of higher degree number fields\cite{g02,eg17,bi04}.
Section 7.3 of the book by Evertse and Gy\H{o}ry\cite{eg17}
discusses significant results by Gras on Abelian number fields
of degree $n$, where $n$ is relatively prime to 6. See also Ref.~\cite{sh14}.
Finally, we mention a general and still largely open problem stated by Hasse:
to give a characterization of monogenic number fields\cite{eg17,n04,ha64}.
Hasse's problem has been considered, among others, by Nakahara and
co-authors in several contexts during the last 50 years or so.
See Refs.~\cite{sh14,hn15,ka16,ka19,sn15} and references therein.
See also Evertse\cite{eee}.
This paper reports results of an extended investigation of the distribution
of monogeneity in cyclotomic period equations $\psi_e(x)$, a wide class of
functions underlying the solution of cyclotomic polynomials\cite{da,bach,reu}.
{ We find that all monogenic period equations are either cyclotomic polynomials
or simple reduced forms of cyclotomic polynomials.}
Here, monogenic period equations are obtained with the help of an
expression for the field discriminant $\Delta_e$
of period equations\cite{jg19}, Eq.~(\ref{delta}) below.
Since the polynomial discriminant $D$ may be easily computed,
knowledge of $\Delta_e$ gives at once
\begin{equation}
k^2 = {D}/{\Delta_e}, \label{kkk}
\end{equation}
which is a convenient tool to sort out all equations with index $k=1$.
In what follows, we present results obtained for such monogenic period equations.
Equations (\ref{kkk}) and (\ref{delta}) grant access to families of
equations of arbitrarily high degrees $e$, opening the possibility of
studying monogeneity well beyond the aforementioned low-degree limits.
Note that for high degrees the division in Eq.~(\ref{kkk})
involves exceedingly large integers.
\section{Field discriminants of cyclotomic period equations}
\label{pereq}
Let $g$ be a primitive root of a prime $p=ef+1$, and
$r=\exp(2\pi i/p)$. In the {\sl Disquisitiones Arithmetic\ae},
Gauss defined $e$ sums $\eta_i$
called ``periods''\cite{da,bach,reu,jg19,ra64}:
\begin{equation*}
\eta_i = \sum_{k=0}^{f-1} r^{g^{ke+i}}, \qquad i=0,1,\cdots,e-1,
\end{equation*}
With them, he defined {\sl period equations} $\psi_e(x)$,
polynomials of degree $e$ whose roots are the periods $\eta_i$
\begin{equation*}
\psi_e(x) = \prod_{k=0}^{e-1} (x-\eta_{k})
= x^e+x^{e-1}+\alpha_2x^{e-2}+\cdots+ \alpha_e,
\qquad \alpha_\ell \in \mathbb Z. \label{theta}
\end{equation*}
Period equations $\psi_e(x)$ constitute a wide class of equations for
which the computation of the field discriminant $\Delta_e$
presents no difficulties, being given by\cite{jg19}
\begin{equation}\label{delta}
\Delta_e =
\begin{cases*}
-p^{e-1}, & {\rm if} $(e-1) \hbox{ \rm mod } 4 =1 {\ \ \ \rm and\ \ \ }
f {\ \rm mod\ } 2 = 1 $, \\
\phantom{-}p^{e-1}, & {\rm if otherwise}.
\end{cases*}
\end{equation}
Together with Eq.~(\ref{kkk}), this discriminant provides a handy
criterion to sort out $k^2=1$ monogenic equations through a simple
division of two (possibly very large) integers.
\section{Properties of monogenic period equations}
Table \ref{tab:tab01}
lists monogenic period equations as a
function of $e$ for the first few equations of a much longer list,
containing seven equations for every $e\leq 250$.
The table also displays the signature of $\psi_e(x)$.
The {\sl signature}\cite{ha64} of a polynomial is
the doublet $(n_R, n_P)$, sometimes written more economically
as $n_R$,
informing the number $n_R$ of real roots of $\psi_e(x)$, and the
number $n_P$ of {\it pairs} of complex roots.
Table \ref{tab:tab01} illustrates regularities that are
consistently observed for $e$ up to 250.
Among the equations obtained for a given $e$ we find no more
than two types of polynomials leading to $k=1$. They have
either totally complex roots, $n_R=0$, or totally real, $n_R=e$.
Polynomials with $n_R=0$ are highlighted differently,
to reflect the sign of their discriminants.
For $e=2$, period equations are quadratic and, as known, are all
monogenic\cite{g02}.
For $e=3$, we determined the growth of the number of $k=1$ equations
as a function of $e$, up to $e=9000$. Such growth obeys
a power-law distribution, implying the existence of
an infinite number of monogenic cubic equations.
For $e\geq4$, we find no more than two monogenic equations for each
value of $e$, as illustrated in Table \ref{tab:tab01}.
From the Table one recognizes a trend observed also for higher
values of $e$: the absence of monogenic equations for several
values of $e$.
For instance, for $e\leq 100$ we find no monogenic period equations
for
$e=7$, $13$, $17$, $19$, $24$, $25$, $27$, $31$, $32$, $34$,
$37$, $38$, $43$, $45$, $47$, $49$, $55$, $57$, $59$, $61$,
$62$, $64$, $67$, $71$, $73$, $76$, $77$, $79$, $80$, $84$, $85$,
$87$, $91$, $92$, $93$, $94$, and $97$.
Analogously, there are 62 cases of missing cyclotomic polynomials
with degree $\leq 100$.
Period equations are not difficult to generate fast and explicitly
up to very high degrees. As already mentioned,
this means that Eqs.~(\ref{kkk}) and (\ref{delta}) open the possibility
to investigate monogeneity systematically for an important family of
equations well beyond the aforementioned low-degree limits.
\begin{table}[!tb]
\tbl{Monogenic period equations as a function of $e$
for primes $p=ef+1$ and signature $n_R$.
Highlighted equations have signature $n_R=0$ and coincide
with cyclotomic polynomials $\Phi_p(x)$.
Non-highlighted equations and signature $n_R=e$ are
{\it reduced\/} cyclotomic polynomials.}
{\begin{tabular}{@{}|c|c|c|c|l|@{}} \toprule
\hline
$e$ & $p$ & $n_R$ & $D=\Delta_e$ & $\psi_e(x)$ \\
\hline
\iw4& \iw5 & \iw0&\cellcolor{iw} $5^3$ & \cellcolor{iw} ${x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
5& 11 & 5& $11^4$ & ${x}^{5}+{x}^{4}-4\,{x}^{3}-3\,{x}^{2}+3\,x+1$\\
\iy6 & \iy7 &\cellcolor{amarelo} 0&\cellcolor{amarelo} $-7^5$ & \cellcolor{amarelo} ${x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
6& 13 & 6& $13^5$ & ${x}^{6}+{x}^{5}-5\,{x}^{4}-4\,{x}^{3}+6\,{x}^{2}+3\,x-1$\\
8& 17 & 8& $17^7$ & ${x}^{8}+{x}^{7}-7\,{x}^{6}-6\,{x}^{5}+15\,{x}^{4}+10\,{x}^{3}-10\,{x}^{2}-4\,x+1$\\
9& 19 & 9& $19^8$ & ${x}^{9}+{x}^{8}-8\,{x}^{7}-7\,{x}^{6}+21\,{x}^{5}+15\,{x}^{4}-20\,{x}^{3}-10\,{x}^{2}+5\,x+1$\\
\iy10&\cellcolor{amarelo} 11&\cellcolor{amarelo} 0&\cellcolor{amarelo} $-11^9$ & \cellcolor{amarelo} ${x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{4}
+{x}^{3}+{x}^{2}+x+1$\\
11& 23& 11& $23^{10}$&${x}^{11}+{x}^{10}-10\,{x}^{9}-9\,{x}^{8}+36\,{x}^{7}+28\,{x}^{6}-56\,{x}^{5}-35\,{x}^{4}+35\,{x}^{3}+15\,{x}^{2}-6\,x-1$\\
\iw12&\cellcolor{iw} 13&\cellcolor{iw} 0&\cellcolor{iw} $13^{11}$& \cellcolor{iw} ${x}^{12}+{x}^{11}+{x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
14& 29& 14& $29^{13}$ & ${x}^{14}+{x}^{13}-13\,{x}^{12}-12\,{x}^{11}+66\,{x}^{10}+55\,{x}^{9}-165\,{x}^{8}-120\,{x}^{7}$\cr
&&&& \qquad $+210\,{x}^{6}+126\,{x}^{5}-126\,{x}^{4}-56\,{x}^{3}+28\,{x}^{2}+7
\,x-1$\\
15& 31& 15& $31^{14}$ & ${x}^{15}+{x}^{14}-14\,{x}^{13}-13\,{x}^{12}+78\,{x}^{11}+66\,{x}^{10}-220\,{x}^{9}-165\,{x}^{8}+330\,{x}^{7}$\cr
&&&& \qquad $+210\,{x}^{6}-252\,{x}^{5}-126\,{x}^{4}+84\,{x}^{3}
+28\,{x}^{2}-8\,x-1$\\
\iw16&\cellcolor{iw} 17&\cellcolor{iw} 0&\cellcolor{iw} $17^{15}$ & \cellcolor{iw} ${x}^{16}+{x}^{15}+{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}$\cr
&&&& \qquad \cellcolor{iw}$+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
\iy18&\cellcolor{amarelo} 19&\cellcolor{amarelo} 0&\cellcolor{amarelo} $-19^{17}$ & \cellcolor{amarelo} ${x}^{18}+{x}^{17}+{x}^{16}+{x}^{15}+{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}$\cr
&&&&\qquad \cellcolor{amarelo} $+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
18& 37& 18& $37^{17}$ & ${x}^{18}+{x}^{17}-17\,{x}^{16}-16\,{x}^{15}+120\,{x}^{14}+105\,{x}^{13}-455\,{x}^{12}-364\,{x}^{11}+1001\,{x}^{10}$\cr
&&&&\qquad $+715\,{x}^{9}-1287\,{x}^{8}-792\,{x}^{7}+924
\,{x}^{6}+462\,{x}^{5}-330\,{x}^{4}-120\,{x}^{3}+45\,{x}^{2}+9\,x-1$\\
20& 41& 20& $41^{19}$ & ${x}^{20}+{x}^{19}-19\,{x}^{18}-18\,{x}^{17}+153\,{x}^{16}+136\,{x}^{15}-680\,{x}^{14}-560\,{x}^{13}+1820\,{x}^{12}$\cr
&&&&\qquad $+1365\,{x}^{11}-3003\,{x}^{10}-2002\,{x}^{9}+
3003\,{x}^{8}+1716\,{x}^{7}-1716\,{x}^{6}-792\,{x}^{5}$\cr
&&&&\qquad $+495\,{x}^{4}+165\,{x}^{3}-55\,{x}^{2}-10\,x+1$\\
21& 43& 21& $43^{20}$ & ${x}^{21}+{x}^{20}-20\,{x}^{19}-19\,{x}^{18}+171\,{x}^{17}+153\,{x}^{16}-816\,{x}^{15}-680\,{x}^{14}+2380\,{x}^{13}$\cr
&&&&\qquad $+1820\,{x}^{12}-4368\,{x}^{11}-3003\,{x}^{10}
+5005\,{x}^{9}+3003\,{x}^{8}-3432\,{x}^{7}-1716\,{x}^{6}$\cr
&&&&\qquad $+1287\,{x}^{5}+495\,{x}^{4}-220\,{x}^{3}-55\,{x}^{2}+11\,x+1$\\
\iy22&\cellcolor{amarelo} 23&\cellcolor{amarelo} 0&\cellcolor{amarelo} $-23^{21}$ &\cellcolor{amarelo} ${x}^{22}+{x}^{21}+{x}^{20}+{x}^{19}+{x}^{18}+{x}^{17}+{x}^{16}+{x}^{15}+{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{10}$\cr
&&&&\qquad \cellcolor{amarelo} $+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x
}^{4}+{x}^{3}+{x}^{2}+x+1$\\
23& 47& 23& $47^{22}$ & ${x}^{23}+{x}^{22}-22\,{x}^{21}-21\,{x}^{20}+210\,{x}^{19}+190\,{x}^{18}-1140\,{x}^{17}-969\,{x}^{16}+3876\,{x}^{15}$\cr
&&&&\qquad $+3060\,{x}^{14}-8568\,{x}^{13}-6188\,{x}^{12
}+12376\,{x}^{11}+8008\,{x}^{10}-11440\,{x}^{9}-6435\,{x}^{8}$\cr
&&&&\qquad $+6435\,{x}^{7}+3003\,{x}^{6}-2002\,{x}^{5}-715\,{x}^{4}+286\,{x}^{3}+66\,{x}^{2}-12\,x-1$\\
26& 53& 26& $53^{25}$ & ${x}^{26}+{x}^{25}-25\,{x}^{24}-24\,{x}^{23}+276\,{x}^{22}+253\,{x}^{21}-1771\,{x}^{20}-1540\,{x}^{19}+7315\,{x}^{18}$\cr
&&&&\qquad $+5985\,{x}^{17}-20349\,{x}^{16}-15504\,{x}^
{15}+38760\,{x}^{14}+27132\,{x}^{13}-50388\,{x}^{12}$\cr
&&&&\qquad $-31824\,{x}^{11}+43758\,{x}^{10}+24310\,{x}^{9}-24310\,{x}^{8}-11440\,{x}^{7}+8008\,{x}^{6}$\cr
&&&&\qquad $+3003\,{x}^{5}-1365\,{x}^{4}-364\,{x}^{3}+91\,{x}^{2}+13\,x-1$\\
\iw28 &\cellcolor{iw} 29&\cellcolor{iw} 0&\cellcolor{iw} $29^{27}$ &\cellcolor{iw} ${x}^{28}+{x}^{27}+{x}^{26}+{x}^{25}+{x}^{24}+{x}^{23}+{x}^{22}+{x}^{21}+{x}^{20}+{x}^{19}+{x}^{18}+{x}^{17}+{x}^{16}$\cr
&&&&\qquad\cellcolor{iw} $+{x}^{15}+{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
29& 59& 29& $59^{28}$ & ${x}^{29}+{x}^{28}-28\,{x}^{27}-27\,{x}^{26}+351\,{x}^{25}+325\,{x}^{24}-2600\,{x}^{23}-2300\,{x}^{22}+12650\,{x}^{21}$\cr
&&&&\qquad $+10626\,{x}^{20}-42504\,{x}^{19}-33649\,{x}^{18}+100947\,{x}^{17}+74613\,{x}^{16}-170544\,{x}^{15}$\cr
&&&&\qquad $-116280\,{x}^{14}+203490\,{x}^{13}+125970\,{x}^{12}-167960\,{x}^{11}-92378\,{x}^{10}+92378\,{x}^{9}$\cr
&&&&\qquad $+43758\,{x}^{8}-31824\,{x}^{7}-12376\,{x}^{6}+6188\,{x}^{5}+1820\,{x}^{4}-560\,{x}^{3}-105\,{x}^{2}+15\,x+1$\\
\iy30&\cellcolor{amarelo} 31&\cellcolor{amarelo} 0&\cellcolor{amarelo} $-31^{29}$ &\cellcolor{amarelo} ${x}^{30}+{x}^{29}+{x}^{28}+{x}^{27}+{x}^{26}+{x}^{25}+{x}^{24}+{x}^{23}+{x}^{22}+{x}^{21}+{x}^{20}+{x}^{19}+{x}^{18}+{x}^{17}+{x}^{16}$\cr
&&&&\qquad\cellcolor{amarelo} $+{x}^{15}+{x}^{14}+{x}^{13}+{x}^{12}+{x}^{11}+{x}^{10}+{x}^{9}+{x}^{8}+{x}^{7}+{x}^{6}+{x}^{5}+{x}^{4}+{x}^{3}+{x}^{2}+x+1$\\
30& 61& 30& $61^{29}$ & ${x}^{30}+{x}^{29}-29\,{x}^{28}-28\,{x}^{27}+378\,{x}^{26}+351\,{x}^{25}-2925\,{x}^{24}-2600\,{x}^{23}+14950\,{x}^{22}$\cr
&&&&\qquad $+12650\,{x}^{21}-53130\,{x}^{20}-42504\,{x}^{19}+134596\,{x}^{18}+100947\,{x}^{17}-245157\,{x}^{16}$\cr
&&&&\qquad $-170544\,{x}^{15}+319770\,{x}^{14}+203490\,{x}^{13}-293930\,{x}^{12}-167960\,{x}^{11}+184756\,{x}^{10}$\cr
&&&&\qquad $+92378\,{x}^{9}-75582\,{x}^{8}-31824\,{x}^{7}+18564\,{x}^{6}+6188\,{x}^{5}-2380\,{x}^{4}-560\,{x}^{3}+120\,{x}^{2}+15\,x-1$\\
33& 67& 33& $67^{32}$ & ${x}^{33}+{x}^{32}-32\,{x}^{31}-31\,{x}^{30}+465\,{x}^{29}+435\,{x}^{28}-4060\,{x}^{27}-3654\,{x}^{26}+23751\,{x}^{25}+20475\,{x}^{24}$\cr
&&&&\qquad $-98280\,{x}^{23}-80730\,{x}^{22}+296010\,{x}^{21}+230230\,{x}^{20}-657800\,{x}^{19}-480700\,{x}^{18}+1081575\,{x}^{17}$\cr
&&&&\qquad $+735471\,{x}^{16}-1307504\,{x}^{15}-817190\,{x}^{14}+1144066\,{x}^{13}+646646\,{x}^{12}-705432\,{x}^{11}$\cr
&&&&\qquad $-352716\,{x}^{10}+293930\,{x}^{9}+125970\,{x}^{8}-77520\,{x}^{7}-27132\,{x}^{6}+11628\,{x}^{5}+3060\,{x}^{4}$\cr
&&&&\qquad $-816\,{x}^{3}-136\,{x}^{2}+17\,x+1$\\
\hline
\end{tabular}}\label{tab:tab01}
\end{table}
\section{Monogenic period equations are cyclotomic polynomials}
Gauss showed how to decompose and to solve explicitly
in terms of radicals cyclotomic polynomials $x^p-1=0$, for prime $p$.
The procedure is described in \S 343 of his
{\sl Disquisitiones Arithmetic\ae}\cite{da,bach,reu,ra64}.
To this end, he introduced period equations as mere
{\sl auxiliary equations}.
This fact notwithstanding, by examining Table \ref{tab:tab01} we
deduce that
{\sl
all monogenic period equations $\psi_e(x)$ are
nothing else than cyclotomic polynomials $\Phi_p(x)$,
interconnected either by $p=e+1$ or $p=2e+1$}.
This latter interconnection may be identified as follows.
Consider, e.g., $e=5$ and $p=2\cdot5+1=11$, when the corresponding
cyclotomic polynomial is
\[ \Phi_{11}(x)=\frac{x^{11}-1}{x-1} = x^{10} + x^{9} + \cdots + x^2+x+1. \]
By changing variables according to a standard
de Moivre transformation\cite{dm30}
\begin{equation}
z=x+1/x, \label{direta}
\end{equation}
and replacing $z$ by $x$ in the equation so obtained, one finds
\[ \psi_5(x) = {x}^{5}+{x}^{4}-4\,{x}^{3}-3\,{x}^{2}+3\,x+1. \]
This quintic, listed in Table \ref{tab:tab01},
was solved explicitly by radicals in a {\sl M\'emoire}
read in November 1770 by Vandermonde\cite{v70,i88}.
Conversely, changing $x$ in $\psi_5(x)$ according to the
{\sl dual transformation}
\begin{equation}
x=z+1/z \label{inversa}
\end{equation}
(and replacing $z$ by $x$) one recovers the cyclotomic $\Phi_{11}(x)$.
Thus, it is an easy matter to pass from one polynomial to the other one,
showing that, essentially, monogenic period equations are
cyclotomic polynomials. Here is another example.
In 1796, almost three decades after Vandermonde\cite{l18},
Gauss recorded in his
mathematical diary\cite{k03,Tbuch,du04} that the regular 17-gon
can be constructed by ruler and compass alone.
In print, his construction appeared\cite{da} only in 1801.
The solution amounts to reducing by two, four times in succession,
the degree of the 17-gon cyclotomic polynomial, namely
\[ \Phi_{17}(x)=\frac{x^{17}-1}{x-1} = x^{16} + x^{15} + \cdots + x^2+x+1. \]
Applying de Moivre's transformation, Eq.~(\ref{direta}), to $\Phi_{17}(x)$
one gets the first of such reductions,
also present in Table \ref{tab:tab01}:\quad
\[\psi_8(x) = {x}^{8}+{x}^{7}-7\,{x}^{6}-6\,{x}^{5}+15\,{x}^{4}
+10\,{x}^{3}-10\,{x}^{2}-4\,x+1. \]
Many additional examples are obtained in
a similar way, by using the dual transformation, Eq.~(\ref{inversa}),
to unfold $\psi_e(x)$ with signature $n_R=e$
for all equations listed in Table \ref{tab:tab01},
thereby obtaining the associated cyclotomic $\Phi_{2e+1}(x)$.
The dual transformation worked also for all additional equations up
to $e=250$ (not all in Table \ref{tab:tab01}).
For instance, $e=96$ is the largest value $\leq100$ with two
monogenic period equations. For signature $n_R=0$ and $f=1$ we find
\begin{alignat*}{2}
\psi_{96}(x) &= x^{96}+x^{95}+x^{94}+x^{93}+x^{92}+\cdots+x^4+x^3+x^2+x+1,
\end{alignat*}
while for $n_R=96$ and $f=2$ we find
\begin{alignat*}{2}
\psi_{96}(x) &= x^{96}+x^{95}-95x^{94}-94x^{93}+4371x^{92}+4278x^{91}-129766x^{90}+\cdots\cr
&\quad -18009460x^6-2118760x^5+230300x^4+18424x^3-1176x^2-48x+1.
\end{alignat*}
Similar doublets occur for
$e=6, 18, 30, 36, 78, 96, 138, 156, 198, 210, 228$, $270$, $306$, $330, \dots$.
There are 187 doublets for
$e\leq10^4$, 1164 for $e\leq10^5$, 7750 for $e\leq10^6$, etc.
\section{Conclusions}
The compelling computational evidence reported here leads us to conjecture
that for $e\geq4$ there are two classes of coincidences between monogenic
period equations $\psi_e(x)$ and cyclotomic polynomials $\Phi_p(x)$
interconnected by $p=ef+1$: \
The class of totally complex period equations, for which $p=e+1$,
and the class of totally real period equations, for which $p=2e+1$.
For all other values of $f$ in these classes, we only found non-monogenic
period equations and no connections to cyclotomic polynomials.
For $e=3$, as already mentioned, it is possible to find an apparently
unbounded supply of monogenic period equations with $f>2$.
Totally real period equations are of significant interest for applications
in quadratic discrete-time dynamical systems in the partition
generating limit\cite{bk,jg19,g19,eg06,eg06b}.
\section*{Acknowledgments}
The author would like to thank
Prof.~K.~Gy\H{o}ry, Debrecen, and Prof.~W.~Narkiewicz,
Wroc\l aw, for their kind feedback and helpful suggestions.
This work was supported by the Max-Planck Institute for the Physics of
Complex Systems, Dresden, in the framework of the Advanced Study Group on
{\sl Forecasting with Lyapunov vectors}.
The author was supported by CNPq, Brazil.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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{"url":"http:\/\/www.physicsforums.com\/showthread.php?t=514574","text":"Recognitions:\nGold Member\n\n## What is configuration space?\n\nIn a section describing \"Problems Raised by Statistical Interpretation\" of the Schrodinger wave equation, Albert Messiah (QUANTUM MECHANICS) says this:\n\n ...the wave equation is described in configuration space not in ordinary space; it can not therefore be identified with the concrete wave we are discussing....which is the representation of a quantum system by its wave function.\nHuh????\n\nI assume he is referring to psi (r,t) which he originally introduced almost 100 pages earlier this way:\n\n The simplist system is that of a particle, for instance an electron, in an external force field. The wave which is associated with it at each instant t is a function of psi(r,t) of the position coordinates of that particle.\nThis seems \"concrete\" enough.\n\nHow to interpret the first quote???\n(Wikipedia did not help. Nor a search here in the forums. )\nThanks.\n\n PhysOrg.com physics news on PhysOrg.com >> Promising doped zirconia>> New X-ray method shows how frog embryos could help thwart disease>> Bringing life into focus\n If you find Wiki explanation as not clear enough, try this: http:\/\/lesswrong.com\/lw\/pi\/classical...ration_spaces\/ In case of single electron you may take 3-D geometrical space as equivalent to configuration space in position representation.\n Mentor It's not correct to say that \"a configuration space is a way of visualizing the state of an entire system as a single point in a higher-dimensional space\". A point in the configuration space represents all the positions of all the component parts, but that's only half of what you need to specify a \"state\". You need the momenta too. In the fancy version of Lagrangian mechanics, the configuration space is a smooth manifold, and the phase space (the set of states) is its cotangent bundle. I agree that $\\mathbb R^3$ is the configuration space of a classical particle. I don't understand the first quote either.\n\n## What is configuration space?\n\nA wave function for a system of N particles is a function of 3N arguments. It is simply saying for this reason, the quantum mechanical wave functions shouldn't be considered as existing in real space, but rather an abstract space. In quantum mechanics we don't use a configuration space of position and momenta, we use the wave function which contains all the information to specify the state.\n\nRecognitions:\nGold Member\nFredrik: \"I don't understand the first quote either. \"\nok, thanks, I feel a little better...\n\nThe \"less wrong\" link above is easy enough to understand, and but says the following:\n\n ...Quantum configuration space isn't quite like classical configuration space....... (and near the end).... Quantum physics inherently takes place in a configuration space. You can't take it out.\nThis is just why the Wikipedia article wasn't clear to me.....How are 'quantum' and 'classical' configuration space different?? Are they referring to the statistical nature of quantum variables?? For example, what's so different if they both use cartesian coordinates??\n\nI am now coming up against the same issue with HILBERT space further along in the same QUANTUM MECHANICS book: The author says\n\n The wave function capable of representing a given quantum system belong to a function space......The wave functions of wave mechanics are the square integrable functions of configuration space.......the function space defined is a Hilbert space.\nAnd he goes on with a list of characteristics of Hilbert space...like those shown in Wikipedia...ok so what...How are they different than Eucledean space....????\nWikipedia says:\n\n The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions\nso it SEEMS that Hilbert IS higher dimensional Euclidean space??....unfortunately under Euclidean space Wikp[edia says:\n\n In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions.\nSo once again I can't figure out what's different....sounds like the same thing.\n\nI'm not looking for a rigorous mathematical analysis; what I am interested in is how all these spaces are different from each other,and from say plain old Euclidean space, and what effects, if any, they might have regarding the physical observables represented in quantum mechanics. For example, one must apparently be able to impose boundary conditions in all these spaces because it's tough to get quantized states if you don't.\n\nMy gut suggests all the different spaces can't be all THAT different because if QM predicts something in an entiely different dimensional entity that is not applicable to what we observe in our space, what use is it if we can't observe and verify???\n\n Euclidean space is a type of Hilbert space. Consider a space of sines and cosines as well as their linear combinations defined on some interval [0,L]. With the definition of an inner product of these elements, this is a Hilbert space, but it certainly is not a Euclidean space.\n\n I'm not looking for a rigorous mathematical analysis; what I am interested in is how all these spaces are different from each other,and from say plain old Euclidean space, and what effects, if any, they might have regarding the physical observables represented in quantum mechanics. For example, one must apparently be able to impose boundary conditions in all these spaces because it's tough to get quantized states if you don't.\nI can feel your frustration with the author defining the wavefunction as psi(r, t), then 100 pages later saying that it can't be visualized in a Euclidean space...\n\nThe trick is the wavefunction as first defined is a single-particle wavefunction, which is indeed just the same as Euclidean space. Where configuration space gets different (and useful) is when you have more particles, and \/ or varying degrees of freedom.\n\nBasically, a configuration space is a compact way of representing all of the information about the positions of (the configuration of) a system of particles; if you have two particles that are constrained to move on the surface of a sphere, you can represent their positions in a four-dimensional configuration space: (theta1, phi1, theta2, phi2) -- thetaN \/ phiN giving the latitude \/ longitude of the Nth particle.\n\nMaybe an easier way to visualize how configuration space differs from Euclidean space is to take a system with just two degrees of freedom -- say two particles constrained to move on the X axis: (x1, x2). Now the configuration space of those two particles IS a rectangle -- but to talk about a \"rectangle\" or a \"disc\" in that space obviously doesn't mean much.\n\nTo take it a step further, usually in mechanics you start with configuration space -- which is just the positions of particles -- then move to \"phase space\", where you have not only the positions, but the momenta of each particle. So each classical free particle gets 6 components: (x, y, z, px, py, pz). Same idea, but just more dimensions. Obviously here the geometrical \/ Euclidean picture is even less useful -- since some of the \"coordinates\" have nothing to do with position at all.\n\nThe Hilbert space of quantum mechanics is basically just the same thing -- but in quantum mechanics, you can't disentangle a particle out from the rest of the system, so each \"state vector\" in the space (like the (x1, x2) or (x, y, z, px, py, pz) above), instead of just containing all the degrees of freedom of one particle, has to account for all the degrees of freedom of ALL particles, so each vector has an infinite number of degrees of freedom -- though in practical situations, you're usually just dealing with a finite number of those degrees of freedom.\n\nIf that last part doesn't make sense, don't lose sleep over it -- if you can at least get your head around the idea of a phase space, you should be fine...I don't think anything you'll be coming across in the near future will depend on you having the pure mathematical distinctions between phase space and Hilbert space nailed down (though obviously it can't hurt).\n\n Recognitions: Gold Member jjustinn: good explanation...gives me the inutitive feel I was seeking. 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Forums Home› EA SPORTS NHL Series› NHL 21› NHL 21 The Pond (General Discussion)
NHL22 Beta Confirmed?
IceLion68
July 16, 2021 2:13PM edited July 2021
@EA_Blueberry pls delete if not allowed
https://twitter.com/PlaystationSize/status/1415104027273089030
https://www.sportsgamersonline.com/games/hockey/nhl-22-beta-leaked/
When asked about the beta on the PlayStation 5, the account said that the size is expected to be over 30GB for users on that platform.
"For this beta, still [no] information," they said. "But for alpha version [it was] +30 GB."
The NHL 22 beta is also expected to come for the Xbox One and Xbox Series X|S consoles. A signup, if confirmed, is likely to be announced when the game is officially revealed at, or around, EA Play Live on July 22.
When revealed, it's also expected that the series will make the jump to the Frostbite Engine after eight games on the Ignite Engine. With the anticipated jump, many fans are hoping for a return of the franchise to PC. Though a beta on the platform shouldn't be expected.
Dad. Gamer. Rocker. Geek.
Sega82mega
Things are starting to move.
Hell - might need to take some extra shift - betta have that brand new ps5 ready - not thaaat far away.
twhite1387
yup...I'm thinking the same thing, @Sega82mega...hopefully
untouchable_BF1
I expect a huge graphical leap and nothing else. Not excited at all.
untouchable_BF1 wrote: »
Hehe as the big hockey fan I know you are - you gotta have some curiosity for this next generation hockey game - with frostbite(?) - HyperMotion (??) as a new game engine...
But maybe a little 'expect the worst hope for the best' scenario.
We all been there - with some dissatisfaction.
Follisimo
EA will be sure to take my $100 so i can own both versions because leagues won't be able to work on next-gen only.
I'm sure next-gen will see quite a few QoL changes. The PS4/XB1 probably not so much.
Rozsos27
You're much more hopeful than I. While Fifa and Madden had noticeable graphical changes they were certainly nowhere near huge. I guess NHL has the most room to grow in that regard, though.
Rozsos27 wrote: »
They were already on frostbite though so it makes a little more sense.
Sega82mega wrote: »
Honestly I'd love to be excited, but the sliders offline don't work, they can play at two different speeds without touching a slider. This has existed for 3-4 years at least now and they've done nothing. The AI have been nothing short of miserable forever with occasional marginal improvements. The AI teammates play so passive and stupid compared to your CPU opponents due to the competitive scene, the AI opponents get to cheat the physics engine and do whatever they want when it comes to skating direction, they win "50/50" puck battles 99% of the time, they score the same scripted rebound goals where they pump a 100 mph shot off the pad from two feet away that produces a .5 mph rebound right to their teammates stick, the goalies just nonsensical idk where to even begin, and we still have things like kicking/sweeping pucks in your own end that can't be done in an aimed fashion as a user while the CPU can make 80 foot breakout passes with it lol.
The AI and the user DO NOT have the same abilities. The AI teammates compared to their AI opponent counterparts DO NOT play the same. The goals are scripted for the CPU, there's nothing remotely "organic" about them. The puck physics have become an absolute joke compared to NHL 15 that it's shocking. Net physics broke this year. Shooting in your own zone targets the net, known issue that is usually a first patch issue every year that was left untouched this year, I just have literally zero excitement for this series anymore.
When you can't even have functioning offline sliders, when multiple community members bring this up and receive literally nothing in return for 3+ years, there's just nothing left to be excited about. This next generation I bet we get a few more marginal AI improvements, puck physics will never return to 15 levels, goalies will be tuned to let in the same 2-3 meta shots every year, presentation will still be a joke, crowd will still be quiet about 3 second after a goal, and the AI skaters will still not adhere to the physics engine.
Those are my predictions based on this last generation where sliders were broken from 18/19 on and almost zero effort went into making the offline experience immersive and fun.
BruinsHockey08
They Dev team has had over a year with a next generation development kit by now I'd imagine so they should at a bare minimum have the best chance to wow us with the graphics from year over year.
I'm more interested in gameplay improvements though. This is the only game I can drop a lot of time into without getting bored with what's going on on the playing surface. But it's far from perfect.
But you are getting yourself a copy?
Im looking forward to some 'first news' - it's hard to know what to expect when I know nothing about whats coming.
But something tells me this can be good.
But I Save a prayer for you - that this slider thing is no longer bothering you or anyone else.
The real question is when do we see the 1st information of NHL 22? I'd assume it would be one of these 2 days.
EA Play is July 22nd.
NHL Draft is July 23rd-24th
Follisimo wrote: »
Pretty sure it's been confirmed for EA Play
I'm usually gifted a copy every year. I end up playing the new version for the marginal improvements because why not when I'm gifted it, but I spent so much time in 21 making my league that I don't see a small improvement making me ditch that.
I'd need a noticeably more competent AI and with the frostbite switch I just don't see that happening. That's a major ask, so I'm not optimistic in the least bit that we'll see anything that will blow me away.
July 19, 2021 9:18AM edited July 2021
Cant wait for my first review of the game.
I will do my best to be fully honest.
😏👍
I can swear - some of the words from the marketing will be something about 'a better AI' - so eventually will find out, if thats true... 😱
AI is not a flashy feature to put on the box. Though it better be better than what we had last generation.
BruinsHockey08 wrote: »
That would be an extremely low bar imo lol. Having the AI on both sides respect the physics engine would be a good start. Maybe some common break-in concepts like "center lane drive" be implemented considering almost every break-in ever revolves around this concept? Forechecking just lacks basic logic and the current strategies listed/able to be chosen mean nothing to nobody except for the people that developed them as there's multiple ways to run forechecks regardless of the structure.
For example, when teaching forechecking IRL, F1's first job is to cutoff the behind the net option. This is called a "force" and is the basic foundation any forecheck is based around. So, cutting off the behind the net pass/skate option and forcing strong-side is my primary goal for F1. (If this fails, we retreat/mitigate the inevitable rush which starts with F3 taking an inside-out angle at the puck carrier who has escaped to the previous weak-side.) Anyway, from there F2 is making a hard play (pressure) on the carrier towards his inevitable escape route which is the current strong-side right? Still following? From there depending on what my opponent is doing for a breakout, I'm either having F3 slide over to the strong side wing (if they're running a high breakout where the center is high, therefore pinching my strong-side D is too risky) or I'm having them sit in the middle of the slot to 1. Cut off the cross-ice pass and 2. Be there to shadow the low center breakout opinion which is where they'll inevitably go after my strong-side D pinches. Long-story short, id consider this a "1-2-2 aggressive" but that's not how this game runs a "1-2-2 aggressive" from what I can tell which is admittedly hard to tell as the AI dmen are woeful at pinching when appropriate so even if this is the case so you'd never know, but the issue still stands. I don't get the goals or intent behind the strategies in this game. They can all be taught in different ways and from what I can tell/assess, none of them are executed even decently in this game.
Another example, there's a team in our conference that LOVES D-to-D NZ regrouping and D-to-D behind the net breakouts. How do you counter this? You send one forward hard at each defenseman and have your 3rd forward sit in the middle. The first forward in this forecheck is actually going to force the initial D with the puck towards the inside (forcing a direction is the basic of all forechecks I should add, not getting the puck as you're not the expected guy to win the puck) so they make the pass as they've been instructed to do. Hopefully F2 gets there early enough to really put pressure on D2 or D1 might hold/hesitate because they feel the pressure and then that's obviously where turnovers and breakdowns happen.
I'm not by any means someone who is a huge X's and O's Guy. I like coaching high schoolers at a level where skill development wins games more than strategies (and this is sadly much overlooked by "coaches" at this level), but these are basic hockey concepts I coach that are seemingly not in the game. This isn't some NHL-level analysis either, this is basic hockey and these are "lines" or "trees" if you will that have many different versions from the few I presented. Why are we not getting at least this level of AI focus in a hockey game in 2021, especially when that holds back the game more than any physical feature offline? The AI does not do any of these things well or even decent imo. We need a complete overhaul when it comes to strategies and execution.
To this point, this "tree" really doesn't involve anything "dynamic" either, like these are options that could be "statically" executed with some expected failure which could be alleviated by human intervention. Cutoff behind the net for F1 which involves having F1 set their "target" to the end boards on the strong side right next to the net, no read necessary. Once that's achieved, pressure the backside of the carrier (basically follow along the boards) to prevent an escape back towards the net. F2 can be set to be aimed right at the carrier, their job is to force up the boards so failure to cut off that pass is not an issue. F3 can be set to either staying glued to the strong side hash mark or middle of the slot depending on the strategy chosen, no read necessary, change will boil down to the user reading the breakout (CPU would adjust accordingly too, to the user's strategy implemented.) lastly, get rid of the pinch/hold line option, this should already be dictated by the forecheck and the offensive pressure level. If I'm running the "middle of the slot" variant I expect my D to pinch 10/10 times, it's part of the breakout. There's no leeway until a back pass is made which is when I'd expect the D to disengage and retreat. If I'm running the "side boards" variant, they should never be pinching as I'm worried about the streaking center and the chip on the strong-side. There's no leeway there again. It's a "never pinch" scenario.
Your words is gold.
MR Bf1.
I think a big thing would be if the physical play becomes more intense.
It's too easy avoiding being hit - which in turn forces the gameplay to be awkward.
When I go for contact - I have a plan - but often I have to recalculate - 4-5 times beacuse the opponent manage to escape my pressure.
Hits that surprise the opponent - dosent happen very often.
<lengthy and ridiculously insightful hockey strategy discussion >
@EA_Blueberry @EA_Aljo
What do we need to do to make this man a game changer?
Hehe nice quote!
The 'quote god'.. 😏
IceLion68 wrote: »
I think a 'spot is open' - If Bf1 even wanna be a game changer? Takes some time, but if he's up for it - in that case he should have a question from EA. Bf1 is probebly the most respected man on this board, both from community people - and I think to EA people.
You can tell he knows hockey and also have pretty great knowledge/insight about this game and it's mechanics.
He got my vote. Without a doubt.
I appreciate the nod boys but I don't think I could do what I do in an official capacity. I'm too critical at times, I realize this. It comes from a place of care but that emotional fire probably wouldn't be too constructive lol. I also don't want to "dial back" though which I feel like comes with that program. I don't want to sweet-talk the game, don't want to justify shortcomings or tell people to be patient. I'd rather be a unbiased voice for the part of the community that Madden is finding out is bigger than they thought when they decided to focus exclusively on multiplayer the last few years.
Sports games simply are built for the offline user. You can try to make them online-focused, but at the end of the day people want to play the actual sport that's attempting to be represented. The elite sweats are a smaller portion than they claim to be, it's time that the core gameplay is focused around hockey and tuned to their liking rather than vice versa. Their meta and preferred style of play are easily recreated by any tuner set, regardless of the AI play. We need the AI to be good enough to engage offline players. The honeymoon of the skill stick and even DSS and RPM now are over. We have feature depth, now we need strategy depth to truly take this game to the next level.
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Q: Equivalence Relation on points of a Projective Variety Let $X$ be a projective variety over an algebraically closed field $k$. Let $x,y \in X(k)$, call $x \sim y$ iff there exists a map $f : \mathbb{P}^1_k \rightarrow X$ such that $x,y \in f(\mathbb{P}^1_k)$.
Is the above relation an equivalence relation on $X(k)$? If not what would be a counterexample?
A: I am just posting Mohan's comment here so as to close this question.
"I think in general, transitivity can fail. Think of blowing up points on, say an abelian surface, then you can have two rational curves $L,M$ meeting at a point, say $P$. Then points on $L$ are related to $P$ and same for $M$, but points other than $P$ one on $L$, another on $M$ are not related.That is why the relation `linearly connected', allowing several rational curves in X is better." – Mohan
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/*global webkitSpeechRecognition*/
import Ember from 'ember';
import layout from './template';
export default Ember.Component.extend({
layout: layout,
buttonTitle: '',
speechRecognition: new webkitSpeechRecognition(),
currentSpeechRecognitionSession: null,
result: '',
// PROPERTIES
language: null,
continuous: false,
interimResults: false,
maxAlternatives: 1,
onRecognitionEnd: function() {
this.get('speechRecognition').stop();
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onRecognitionError: function(error) {
console.log(error);
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onRecognitionResult: function(e) {
let result= '';
let resultNo = 0;
let alternativeNo = 0;
result = e.results[resultNo][alternativeNo].transcript;
this.sendAction('getResult', result);
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actions: {
startRecognition() {
let speechRecognition = this.speechRecognition;
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speechRecognition.continuous = this.get('continuous');
speechRecognition.interimResults = this.get('interimResults');
speechRecognition.maxAlternatives = this.get('maxAlternatives');
// set events
speechRecognition.onresult = Ember.run.bind(this, this.onRecognitionResult);
speechRecognition.onerror = Ember.run.bind(this, this.onRecognitionError);
speechRecognition.onend = Ember.run.bind(this, this.onRecognitionEnd);
this.set('currentSpeechRecognitionSession', speechRecognition);
speechRecognition.start();
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\section{Introduction}
In the era of exponential growth in information volume, collaborative filtering (CF), one of the leading technical branches of recommender systems, aims to mine similar item preferences for similar users. In knowledge-enhanced CF, knowledge graphs (KGs) are used as auxiliary information for items to alleviate the data sparsity and cold-start problems. Recently, based on the success of graph neural networks, representations with high-order neighbor information can be learned in Euclidean space to improve recommendation performance \cite{KGCN:bm3,CKAN:bm6,KGAT:bm4,KGNN-LS:bm5}.
However, the above Euclidean methods fail to model the significant characteristics of large-scale user-item graph s and KGs, e.g., \textit{scale-free (power-law distribution)}, as shown in Figure \ref{fig:powerlow} and \textit{hierarchical structures} \cite{LKGR2022,Hgcf2021}. Unlike the homogeneous Euclidean space, the capacity of a hyperbolic space (a Riemannian manifold of negative curvature) grows exponentially as nodes move away from the origin. More importantly, a hyperbolic can be viewed as a tree-like structure in high-dimensional space, thus fitting the hierarchy. Benefitting from the above advantages, CF models \cite{LKGR2022,HAKG2022,Hgcf2021,HICF2022,HRCF2022} based on hyperbolic representation learning achieve competitive performance in recommender systems.
Despite the breakthroughs in hyperbolic CF, some fundamental issues have yet to be discussed and resolved, owing to the shallow understanding of hyperbolic geometry.
\begin{figure}[t]
\centering
\begin{subfigure}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{f/degreedistribution1.pdf}
\caption{ Degree distribution in the user-item graph\ of Book-Crossing dataset.}
\label{fig:powerlow1}
\end{subfigure}
\begin{subfigure}{0.45\linewidth}
\centering
\includegraphics[width=\linewidth]{f/degreedistribution2.pdf}
\caption{ Degree distribution in the knowledge graph of Book-Crossing dataset.}
\label{fig:powerlow2}
\end{subfigure}
\caption{}\label{fig:powerlow}
\end{figure}
\cite{PEG2021equivariant,ER2022er,deepER} state the importance of enforcing Euclidean equivariance in the social network or KG representation learning; that is, the features the model captured should be full-fidelity
under any transformations since the relative relation in those representations should not be distorted under any transformation (for example, in Figure \ref{fig:eqv1}, the Euclidean equivariant function $\psi$ generalizes the same graph representation under different rotation transformations).
The existing hyperbolic algorithm aims to perform recommendation ranking through the hyperbolic distance of the hyperbolic representation. At the same time, the symmetric representations in the hyperbolic space have more severe distortions than those in Euclidean space \cite{FullyHNN2021,NestedHNN2022}. Therefore, the fidelity of the hyperbolic CF model with respect to symmetry features (i.e., maintaining Lorentz equivariance) is much more urgent to guarantee than in Euclidean space (as shown in Figure \ref{fig:eqv2}, the Lorentz equivariant function $\phi$ perceives the same features under different Lorentz transformations).
However, existing hyperbolic CF algorithms ignore this. Specifically, they apply message aggregation by projecting the representations into the tangent space. Unfortunately, this aggregation method breaks the equivariance in the hyperbolic space (i.e., Lorentz equivariance), and the distortion accumulates as the aggregation layers are staked. In addition, they arbitrarily perform linear transformations on hyperbolic representations, which is also one of the sources of breaking equivariance. The potential of hyperbolic geometry for recommender systems is yet to be fully exploited due to the disregard of Lorentz equivariance.
\begin{figure}[t]
\centering
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=0.8\linewidth]{f/eqv1.pdf}
\caption{ Illustration of rotation equivariance on a graph respect to function $\psi$.}
\label{fig:eqv1}
\end{subfigure}
\begin{subfigure}{\linewidth}
\centering
\includegraphics[width=\linewidth]{f/eqv2.pdf}
\caption{ Illustration of Lorentz equivariance in the hyperbolic space respect to function $\phi$.}
\label{fig:eqv2}
\end{subfigure}
\caption{}
\end{figure}
Although an auxiliary KG enhances hyperbolic CF, preserving the heterogeneity of KG and user-item graph\ while performing high-order CF between them is challenging. A straightforward way \cite{LKGR2022} is to perform different message passing processes for the two graphs separately, which completely ignores the high-order entity signals across two graphs to users (e.g., in Fig \ref{fig:graphpath}, the path across two graphs to users: $ e_4 \rightarrow i_4 \rightarrow u_3 \rightarrow e_3 \rightarrow e_3 \rightarrow i_2 \rightarrow u_1 $ ). Another idea \cite{HAKG2022} is to use gated aggregation for two graphs in the hyperbolic space, which is too complicated and computationally expensive. Therefore, a concise and effective framework that responds to the above issues is urgently needed.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{f/graphpath.pdf}
\caption{ User-item graph with auxiliary knowledge graph.}
\label{fig:graphpath}
\end{figure}
In response, we propose \textbf{L}orentz \textbf{E}quivariant \textbf{C}ollaborative \textbf{F}iltering (LECF) model for knowledge-enhanced recommendation, which is divided into two stages. \textbf{Attribute generation stage}: First, Hyperbolic Sparse Attention Mechanism is proposed to sample the most informative neighbor nodes. Next, when obtaining the higher-order entity representation in KG by the proposed Item Attribute Generator (IAG), we employ the hyperbolic distance centroid strategy rather than the previous tangent space aggregation strategy, which breaks Lorentz equivariance.
\textbf{CF stage}: The high-order entity representations obtained from the first stage are input to LECF layers as the item attributes. To pass the high-order entity signals to users across graphs, the node attribute and hyperbolic embeddings are mutually updated by the well-designed structure of LECF layer and proposed Lorentz Equivariant Transformation (carrying the node attribute signals). Each component of our model strictly maintains Lorentz equivariance, and the potential of hyperbolic geometry is thus stimulated. Our contributions are summarized below:
\begin{itemize}
\item[$\bullet$] LECF inaugurates the study of equivariance in hyperbolic CF. Specifically, LECF can better perceive the symmetric characteristics under the Lorentz transformations by enforcing Lorentz equivariance, thereby stimulating the potential generalization capability.
\item[$\bullet$] Through the collaboration of IAGs and LECF layers with Lorentz Equivariant Transformations, we skillfully balance preserving the heterogeneity and mining the high-order entity information to users across graphs.
\item[$\bullet$] We propose Hyperbolic Sparse Attention Mechanism to replace the random sampling of the KG and user-item graph\ for selecting the most informative neighbor nodes.
\item[$\bullet$] We conduct extensive experiments on three public benchmark datasets to demonstrate the groundbreaking superiority of LECF.
\end{itemize}
\section{Related Work}
\subsection{Knowledge-Enhanced Collaborative Filtering}
The introduction of KGs enhances the recommendation in a two-fold way. That is, it provides rich semantic information for items and additional information dissemination paths out of the user-item graph. The most straightforward way is to match the KG and the user-item graph\ into a unified graph for representation learning \cite{CKAN:bm6,KGAT:bm4,MKGAT}.
However, these methods focus more on learning representations of the unified graph and ignore the heterogeneity between the user-item graph\ and KG. Aiming at these problems, the meth-path based methods \cite{MCRec,TMER} manually define the high-order information propagation path in the recommendation task but seriously rely on more domain expert knowledge. Another idea is to use user-item graph\ as the auxiliary information for the deep relationship mining in KG while simplifying the exploration of collaborative information between users and items \cite{KGNN-LS:bm5,KGCN}. \cite{entity2rec,Atbrg} further apply graph pre-training and rule constraints to generate subgraphs that meet application requirements, thereby providing support for relationship mining between users and items.
Since \cite{Nickel2017,Nickel2018,HyperbolicneuralnetworksGanea,HyperbolicAttentionNetworks}, representation learning for fitting the latent hierarchy in data in hyperbolic space has become popular.
HGCN \cite{HyperbolicGNN2019Leskovec} further combines hyperbolic representation learning with the message passing network, projecting the node representations from the hyperbolic space to the tangent space for message aggregation. Based on the success of the above studies, \cite{LowDimensionalHyperbolicKG,hyperbolicattentionKG,KnowledgeAssociationKG,hyperboliconesKG} find that the hyperbolic space can well fit the hierarchical and logical patterns in KGs, and thus achieve a series of competitive performances through the high-fidelity and concise representation learning.
Similarly, user-item graph s also exhibit the characteristics of a hierarchical and power-law distribution, and \cite{Hme2020,Hyperml2020} therefore demonstrate the significance of combining hyperbolic geometry with recommendation tasks. \cite{Hgcf2021,HICF2022,HRCF2022} capture higher-order information in user-item graphs by incorporating multiple layers of hyperbolic message passing networks.
Based on the successful practice of hyperbolic geometry in recommendation tasks and KG representation learning, \cite{LKGR2022,HAKG2022} further attempt to use hyperbolic CF to address the problem of knowledge-enhanced recommendation, as
the high-fidelity and compact potential of hyperbolic spaces well fit complex relation paradigms in such tasks. Although the auxiliary KG improves recommendation performance, the potential of hyperbolic geometry is not fully exploited since many components are superficially migrated from the Euclidean space.
\subsection{Exploration of Equivariance}
\label{sec:ExplorationofEquivariance}
Different tasks require specific generalizations for various symmetric features in the data. Thus, a series of studies focuses on exploring the diverse equivariance and invariance of neural networks for corresponding tasks.
The most straightforward case is that in image processing, \cite{Understandingimage2015,Harmonicnetworks} hope that the model can ultimately retain the data features after translating, rotating, and reflecting the image, that is, the equivariance of the model in the Euclidean space (for example, the model should be able to recognize the cat in the image with equal probability no matter under which the above transformations). With the rise of graph neural networks, \cite{fastGNN2016,Semi-supervisedGNN2016,sage2017inductive} claim that the permutation equivariance is a basic principle. Compared with general graph data, geometric graph (i.e., graph nodes that contain spatial coordinates) data exhibit a higher demand for equivariance of the model, as the physical and chemical properties and geometric relationships behind them will not change due to spatial transformation.
Specifically, \cite{E(n)GNN2021} propose E(n) equivariant graph neural networks, which achieve leading results in molecular property prediction. In the tasks of computational physics and chemistry, \cite{improveEGNN} generalize E(3) equivariant graph networks so that the nodes can contain covariant information, such as vectors or tensors. \cite{xugeodiff,Equivariantdiff} further introduce the discussion of geometric equivariance in molecular and protein generation models.
The exploration of equivariance is far from limited to Euclidean space. For example, \cite{Generalizing2020lie} enforces equivariance to transformations from any specified Lie group in the vector space. Additionally, \cite{practical2021,Scalars2021} turn their attention to equivariant models under other groups, e.g., SO(n), O(1, 3), Sp(n), and SU(n).
Equivariance has also been discussed by emerging researches in representation learning for social networks or recommendations. For example, \cite{PEG2021equivariant} designs a rotation equivariant position encoding representation learning method to address the isomorphism problem of network structures. \cite{ER2022er} improves the generalization ability by adding semantic equivariance constraints in knowledge graph representation learning. \cite{EquivariantContrastiveLearning} introduces equivariant contrastive learning into data augmentation for the sequential recommendation.
As an emerging field, hyperbolic representation learning is still in the ascendant for its study of equivariance (i.e., Lorentz equivariance). \cite{practical2021,jet2022,Autoencoders2022lorentz}, from the perspective of physics, shows that the models achieve a significant advantage by enforcing Lorentz equivariance. However, another majority of studies based on the Lorentz model still urgently need to study the problem of equivariance.
\begin{figure}[t]
\centering
\begin{subfigure}{0.85\linewidth}
\centering
\includegraphics[width=\linewidth]{f/fig2a.pdf}
\caption{ Lorentz boost.}
\label{fig:2a}
\end{subfigure}
\begin{subfigure}{0.85\linewidth}
\centering
\includegraphics[width=\linewidth]{f/fig2b.pdf}
\caption{Spatial rotation.}
\label{fig:2b}
\end{subfigure}
\caption{ Two types of Lorentz transformations.}\label{fig:2}
\end{figure
\section{Preliminary}
\subsection{Hyperbolic Geometry}
In this paper, the n-dimensional Euclidean space is denoted by $\mathbb{R}^n$. An n-dimensional hyperbolic space is a Riemannian manifold of negative curvature $-1/C(C>0)$. For numerical stability, we choose one of the equivalent models in the hyperbolic space, the Lorentz model, which is defined as $(\mathbb{H}^n,g_\mathcal{L})$. Where $\mathbb{H}^{n}=\left\{\mathbf{x} \in \mathbb{R}^{n+1}:\langle\mathbf{x}, \mathbf{x}\rangle_{\mathcal{L}}=-C, x_{0}>0\right\}$ and $g_{\mathcal{L}}$ is the metric tensor that is defined as $g_{\mathcal{L}}=diag[-1,1,1,\dots,1]$. $\langle\mathbf{x}, \mathbf{x}\rangle_\mathcal{L}$ is the Lorentz inner product, which is elaborated below:
\begin{equation}
\langle\mathbf{x}, \mathbf{y}\rangle_{\mathcal{L}} =-x_{0} y_{0}+\sum_{i=1}^{n} x_{i} y_{i}
\end{equation}
where $\mathbf{x}, \mathbf{y} \in \mathbb{H}^{n}$. Based on this, we can give the definition of the distance between two points:
\begin{equation}
d_{\mathcal{L}}(\mathbf{x}, \mathbf{y})=\sqrt{C} \operatorname{arcosh}\left(-\langle\mathbf{x}, \mathbf{y}\rangle_{\mathcal{L}} / C\right)
\end{equation}
The Lorentz norm can thus be defined as:
\begin{equation}
\| \mathbf{x}\|_{\mathcal{L}}=\sqrt{\langle\mathbf{x}, \mathbf{x}\rangle_{\mathcal{L}}}
\end{equation}
\subsection{Lorentz Model}
In the application of the Lorentz model to special relativity, the first dimension $\mathbf{x}_0$ is interpreted as the time axis, and the remaining dimensions $\mathbf{x}_{1 \sim n}$ are interpreted as the space axes \cite{jet2022}. For ease of description, we inherit these terminologies.
\subsubsection{Lorentz Transformation.}
In the Lorentz model, the linear isometries are called the Lorentz transformation, which could further be polar decomposed \cite{polarDecomposition2002} into a combination of a Lorentz boost and a spatial rotation; see Figure \ref{fig:2} for details.
A polar decomposition for a Lorentz transformation $\mathbf{A} \in \mathbf{SO}^{+}(1,n)$\footnote{$\mathbf{SO}^{+}(1,n)$ is the positive special Lorentz group} is defined as
\begin{equation}
\mathbf{A}=\left[\begin{array}{ll}
1 & 0 \\
0 & \mathbf{R}
\end{array}\right]\left[\begin{array}{ccc}
\mathrm{cosh \alpha} &\mathrm{sinh \alpha} & 0\\
\mathrm{sinh \alpha} & \mathrm{cosh \alpha} &0\\ 0&0& I_{n-1}
\end{array}\right]
\end{equation}
where $\mathbf{R} \in \mathbf{SO}(n)$\footnote{$\mathbf{SO}(n)$ is the special orthogonal matrix}, $\alpha$ is the hyperbolic angle, and $I_{n - 1}$ is the $(n - 1)\times(n - 1)$
identity matrix. The left half is the spatial rotation, and the right half is the Lorentz boost.
\subsubsection{Lorentz Equivariance.}
We first present the general concept of equivariance that encompasses all cases in Section \ref{sec:ExplorationofEquivariance}. $T$ is an arbitrary transformation of $\mathbf{x} \rightarrow \mathbf{x}$ in the abstract group $g$. A function $\psi(\cdot)$ is equivariant to $g$ when it satisfies the following conditions:
\begin{equation}
\psi(T(\mathbf{x}))=T'(\psi(\mathbf{x}))
\end{equation}
where $T' \in g$ is the equivalent transformation of $T$. From this we can give the related concepts of Lorentz equivariance.
Let $\mathbf{A}$ be the Lorentz transformation in the Lorentz group and $\mathbf{v} \in \mathbb{H}^n$, the Lorentz equivariance of
$\phi (\cdot)$ means there exists an equivalent Lorentz transformation $\mathbf{A'}$ such that:
\begin{equation}
\mathbf{A}\phi(\mathbf{v})=\phi(\mathbf{A'}\mathbf{v})\qquad where \quad\mathbf{v},\phi(\mathbf{v}) \in \mathbb{H}^n
\label{equ:equva1}
\end{equation}
Lorentz invariance of $\phi (\cdot)$ is thus defined as:
\begin{equation}
\phi(\mathbf{v})=\phi(\mathbf{A}\mathbf{v})\qquad where\quad \phi(\mathbf{v}) \in \mathbb{R}^n
\label{equ:equva2}
\end{equation}
\begin{table}[t]
\caption{Symbols and their definitions.}\label{table:symbols}
\begin{tabular}{ll}
\midrule[1.5pt] Symbol & Definition \\
\midrule[0.3pt]
$\mathbb{R}^n$&N-dimension Euclidean space \\
$\mathbb{H}^n$& N-dimension hyperbolic space \\
$d_{\mathcal{L}}(,)$ &The hyperbolic distance between two nodes \\
$\|\cdot\|_{\mathcal{L}}$&The Lorentz norm \\
$-1/C(C>0)$ & Curvature of the hyperbolic space\\
$\mathbf{e}_i \in \mathbb{H}^n $ & Hyperbolic embedding of a KG entity \\
$\mathbf{n}_i \in \mathbb{H}^n $ & Hyperbolic embedding of node $n_i$ \\
$\mathbf{r}^k$ & Relation embedding of KG \\
${a}(i, j)$ & The hyperbolic attention coefficient\\
$\hat{a}(i, j)$ & The hyperbolic sparse attention coefficient\\
$\mathbf{x}_u \in \mathbb{H}^n$ & Hyperbolic embedding of a user \\
$\mathbf{x}_i \in \mathbb{H}^n$ & Hyperbolic embedding of an item \\
$\mathbf{h}_u \in \mathbb{R}^n$ & Attribute embedding of a user \\
$\mathbf{h}_i \in \mathbb{R}^n$ & Attribute embedding of an item \\
\midrule[1.5pt]
\end{tabular}
\end{table}
\section{Methodology}
\subsection{Problem Formulation}
In this subsection, we clarify the data structure used in the study and the task definition.
\subsubsection{User-item Graph.} The user-item graph (interaction) represents the collection of the behaviors of user $u$ on item $i$, denoted by $
\left\{\left(u, r^{u}, i\right) \mid u \in \mathcal{U}, i \in \mathcal{I}\right\}
$, where $r^u$ is the abstract representation of user behavior (e.g., clicking, purchasing, and rating).
The user-item interaction matrix $\mathbf{Y}$ is a binary matrix, abstracted by setting a certain threshold for user-item graph interactions ($\mathbf{Y}_{ui}$= 1 indicates that there is an interaction; otherwise, $\mathbf{Y}_{ui}$ = 0).
Items in the user-item graph can be matched with the entities in the KG.
\subsubsection{Knowledge Graph (KG).} A KG is a collection of triples representing real-world concepts and their relationships: $\left\{\left(e_{i}, r^k, e_{j}\right) \mid e_{i}, e_{j} \in \mathcal{E}, r^k \in \mathcal{R}\right\}$. Each triple $\left(e_{i}, r^k, e_{j}\right)$ indicates that the head entity $e_{i}$ has a relationship $r_k$ that points to the tail entity $e_j$. There are many studies proving that KGs can improve recommendation performance by providing semantic and attribute information. Here, we introduce a KG to generate item attributes for the user-item graph.
\subsubsection{Task Definition.} Given a user-item graph and a KG, the task studied in this paper is to provide the predicted recommendation score $\hat{y}_{ui}$ that the user $u$ adopts item $i$ in the test set.
\subsection{Overview of the Model}
In summary, our model can be summarized into two stages. In the attribute generation stage, we abandon the random neighbor node sampling strategy while proposing Hyperbolic Sparse Attention Mechanism to sample the most informative neighbor nodes. Then, the proposed Item Attribute Generator (IAG), takes the hyperbolic distance centroids as the aggregated neighbor information to generate item attributes. In the CF stage, the entity representations obtained from IAG are input into LECF layers as the item attributes of user-item graph. In each layer, the node attribute and hyperbolic embeddings are mutually updated by manipulating Lorentz Equivariant Transformations. Last but not least, we theoretically prove that our model strictly preserves the Lorentz equivariance.
\subsection{Hyperbolic Sparse Attention Mechanism}
Existing CF algorithms mostly perform random sampling when selecting neighbor nodes, thus ignoring the importance levels of neighbors. Instead, we combine the sampling process and attention mechanism, proposing Hyperbolic Sparse Attention Mechanism to sample the most informative neighbor nodes. Specifically, the more uniformly the mechanism distributes the attention coefficients (i.e., the smaller the information entropy of the attention coefficients), the less effective it is \cite{Informer2021}. Therefore, we adopt information entropy to measure the difference between the attention coefficient and the uniform distribution to select more informative neighbor nodes. In short, we select $t$ nodes corresponding to the query with the smallest information entropy of the attention coefficients in the hyperbolic space. Inspired by \cite{Hattention2018}, we calculate the attention coefficient $a(i,j)$ between the node $\mathbf{n}_i \in \mathbb{H}^n$ and its neighbors $\mathbf{n}_j \in \mathcal{N}(\mathbf{n}_i) $:
\begin{equation}
\begin{aligned}
\overline{a}(i, j)&=\mathrm{exp}\left(\mathbf{W}_{ij} d_{\mathcal{L}}\left(\mathbf{n}_i, \mathbf{n}_j\right)\right) \\
a(i,j)&=\frac{{\overline{a}(i, j)} \gamma(\mathbf{n}_i)}{\sum_{\mathbf{n}_{j'} \in \mathcal{N}(\mathbf{n}_i)} { \overline{a}(i, j')}\gamma(\mathbf{n}_{j'})}
\end{aligned}\label{equ:hba1}
\end{equation}
where $\mathbf{W}$ is the parameter matrix and $\gamma(\mathbf{n})=\frac{1}{\sqrt{1-\|\mathbf{n}\|^2}}$ is the Lorentz factor.
Then we calculate the probability distribution of the attention coefficients on each neighbor node. Finally, the importance measure $m(j)$ is calculated by:
\begin{equation}
\begin{aligned}
\mathrm{P}(i\mid j)&=\frac{\mathrm{exp}({{a}(i, j)})}{\sum_{\mathbf{n}_{i'} \in \mathcal{N}(\mathbf{n}_j)} \mathrm{exp}({ {a}(i', j)})}
\\m(j)&=-\sum_{\mathbf{n}_{i'} \in \mathcal{N}(\mathbf{n}_j)} \mathrm{P}\left(i'\mid j\right) \log \mathrm{P}\left(i'\mid j\right)
\end{aligned}\label{equ:hba2}
\end{equation}
For each node $\mathbf{n}_i$, we select its top $t$ neighbor nodes $\mathbf{n}_j \in \mathcal{N}(\mathbf{n}_i) $ with the smallest $m(j)$ and fill the blanks with zeros to derive the final sparse attention coefficient $\hat{a}(i, j)$.
It is worth noting that, Equation \ref{equ:hba1} is the generalized definition of the attention coefficient. For the hyperbolic embedding $\mathbf{e}$ that represents the KG entities and the relation embeddings $\mathbf{r}^k$, we replace Equation 7 with:
\begin{equation}
\begin{aligned}
\overline{a}(i,r^k,j)&=\mathrm{exp}\left(d_{\mathcal{L}}\left(f_{\mathbf{r}^k}(\mathbf{e}_i), \mathbf{e}_j\right)\right) \\
a(i,r^k,j)&=\frac{{\overline{a}(i,r^k, j)} \gamma(\mathbf{e}_i)}{\sum_{\mathbf{e}_{j'} \in \mathcal{N}(\mathbf{e}_i)} { \overline{a}(i, r^k,j')}\gamma(\mathbf{e}_{j'})}
\end{aligned}\label{equ:hbaKG}
\end{equation}
where $f_{\mathbf{r}^k}(\cdot)$ is Lorentz Equivariant Transformation parameterized by the relation $\mathbf{r}^k$, see Section \ref{sec:LorentzEquivariantTransformation} for the specific definition.
\subsection{Item Attribute Generator (IAG)}
The item attributes are generated from the higher-order representations of the entities in the KG that match them. In IAG, the entity representations are initialized in the hyperbolic space by the method \cite{Hgcf2021}, denoted as $\mathbf{e}^{0}$. We perform message propagation on entity representations in the hyperbolic space. To avoid the feature distortion and Lorentz equivariance\ destruction caused by aggregation in tangent space, we adopt the algorithm \cite{LGCN2021WWW} that uses the hyperbolic distance centroid as the aggregation result. The optimization objective is:
\begin{equation}
\arg \min _{\mathbf{e}_{i}^{l+1} \in \mathbb{H}^{n}} \sum_ {\mathbf{e}^{l}_j\in \mathcal{N}(\mathbf{e}_i)} \hat{a}(i, r^k,j) d_{\mathcal{L}}\left( \mathbf{e}_{i}^{l+1},\mathbf{e}_j^{l}\right)
\label{equ:iag1}
\end{equation}
Equation \ref{equ:iag1} has a closed-form solution:
\begin{equation}
\mathbf{e}_{i}^{l+1}=\sqrt{C} \frac{\sum _{\mathbf{e}_j\in \mathcal{N}(\mathbf{e}_i)} \hat{a}(i, r^k,j) \mathbf{e}_{j}^{l}} {|\|\sum _{\mathbf{e}_j\in \mathcal{N}(\mathbf{e}_i)} \hat{a}(i, r^k,j) \mathbf{e}_{j}^{l} \|_{\mathcal{L}} |} \label{equ:iag2}
\end{equation}
After stacking several IAG layers, the obtained entity representations will carry the high-order information in the KG.
\subsection{Lorentz Equivariant Collaborative Filtering (LECF) Layer}
Before implementing the CF,
we match the aggregated entity representations from IAG with the items in user-item graph, taking them as the initial item attribute embeddings, denoted by $\mathbf{h}^0_i$. Moreover, the user attribute embeddings $\mathbf{h}^0_u$ are initialized as ones. Then, we follow the method in \cite{Hgcf2021} to initialize the user and item hyperbolic embeddings, denoted by $\mathbf{x}^{0}_u$ and $ \mathbf{x}^{0}_i$.
To preserve the heterogeneity between the user-item graph\ and KG while mining the high-order entity information to users across graphs, we formulate LECF layer to update the node attribute and hyperbolic embeddings jointly as follows\footnote{We take the updating of user representations as an instance, for the items are the same.}:
\begin{equation}
{{\mathbf{m}}_{ui}}={{\phi }_{e}}\left(\mathbf{h}_{u}^{l},\mathbf{h}_{i}^{l},{{ d_{\mathcal{L}}\left( \mathbf{x}_{u}^{l},\mathbf{x}_{i}^{l} \right)}}\right)
\label{equ:lecf1}
\end{equation}
\begin{equation}
\mathbf{x}_{u}^{l+1}=\sqrt{C} \frac{\sum _{i\in \mathcal{N}(u)} \hat{a}\left(u, i\right) {\pi}\left({f_{\mathbf{x}_i}(\mathbf{m}_{ui})},\mathbf{x}_{i}^{l}\right)} {|\|\sum _{i\in \mathcal{N}(u)} \hat{a}\left(u,i\right) \pi\left({f_{\mathbf{x}_i}(\mathbf{m}_{ui})},\mathbf{x}_{i}^{l}\right) \|_{\mathcal{L}} |} \label{equ:lecf2}
\end{equation}
\begin{equation}
{\mathbf{m}_{u}}=\sum _{i\in \mathcal{N}(u)}{\mathbf{m}_{ui}}
\label{equ:lecf3}
\end{equation}
\begin{equation}
\mathbf{h}_{u}^{l+1}={{\phi }_\mathbf{h}}\left(\mathbf{h}_{u}^{l},{\mathbf{m}_{u}}\right)
\label{equ:lecf4}
\end{equation}
where $\phi_{e}$ and $\phi_{\mathbf{h}}$ are the edge and node operations, respectively, implemented by multilayer perceptrons (MLPs).
$\pi(f_{\mathbf{x}}(\mathbf{m}),\cdot)$ is the Lorentz Equivariant Transformation that will be elaborated on in the next section.
In particular, Equation \ref{equ:lecf1} and Equation \ref{equ:lecf2} are related to hyperbolic geometry. Concretely, Equation \ref{equ:lecf1} obtains the message embeddings $\mathbf{m}_{ui}$ from the hyperbolic distances and the corresponding attributes, thus involving the information from the KG.
In Equation \ref{equ:lecf2}, we update the user hyperbolic embeddings by the neighbor item hyperbolic embeddings manipulated by the Lorentz Equivariant Transformation (involving the message embedding $\mathbf{m}_{ui}$) to fuse the item attribute information.
In turn, the updating of attribute embeddings is also affected by the hyperbolic embeddings via message embeddings, according to Equation \ref{equ:lecf4}.
In order to maintain the geometric properties, both the input and output hyperbolic embeddings in Equation \ref{equ:lecf2} are strictly embedded in the hyperbolic space; see the next section for proof.
\subsubsection{Lorentz Equivariant Transformation.}
\label{sec:LorentzEquivariantTransformation}
Existing hyperbolic recommender systems all project the hyperbolic node embeddings to tangent space and then perform the linear transformation on Euclidean space. However, this method breaks Lorentz equivariance\ and does not fully release the potential of hyperbolic space. That is, it only includes spatial rotation in the Lorentz transformation and excludes the Lorentz boost \cite{FullyHNN2021}. Inspired by \cite{NestedHNN2022}, our Lorentz transformation matrix is formulated as:
\begin{equation}
f_{\mathbf{x}}(\mathbf{m})=\left[\begin{array}{c}
\frac{\sqrt{\left\|\mathbf{W}_{\mathbf{m}} \mathbf{x}\right\|^{2}+C}}{\mathbf{v}^{\top} \mathbf{x}} \mathbf{v}^{\top}\\
\mathbf{W}_{\mathbf{m}}
\end{array}\right]
\end{equation}
where $\mathbf{W}_{\mathbf{m}} \in \mathbb{R}^{n \times(n+1)}$ is transformed $\mathbf{m}$ from MLP, $\mathbf{v} \in \mathbb{R}^{1 \times(n+1)} $ is the learnable parameter vector and $f_{\mathbf{x}}(\mathbf{m})\in \mathbb{R}^{(n+1) \times(n+1)}$.
To enforce Lorentz equivariance, we propose Lorentz Equivariant Transformation to manipulate $\mathbf{x}$ by $f_{\mathbf{x}}(\mathbf{m})$ as follows:
\begin{equation}
\begin{aligned}
\pi(f_{\mathbf{x}}(\mathbf{m}),\mathbf{x})&=\frac{f_{\mathbf{x}}(\mathbf{m})\mathbf{x}}{\|f_{\mathbf{x}}(\mathbf{m})\mathbf{x}\|_\mathcal{L}}\qquad
\\s.t. \, f_{\mathbf{x}}(\mathbf{m})g_\mathcal{L}\left(f_{\mathbf{x}}(\mathbf{m})\right)^{\top}&=g_\mathcal{L}
\label{equ:pi}
\end{aligned}
\end{equation}
\begin{proposition}
$\forall \mathbf{x} \in \mathbb{H}^n \cap \mathbf{W}_{\mathbf{m}} \in \mathbb{R}^{n \times(n+1)} \cap \mathbf{v} \in \mathbb{R}^{1 \times(n+1)}$, we have $ \forall f_{\mathbf{x}}(\mathbf{m})\mathbf{x} \in \mathbb{H}^n$.
\label{Proposition:1}
\end{proposition}
\begin{prof}
$\forall \mathbf{x} \in \mathbb{H}^n $, we get $f_{\mathbf{x}}(\mathbf{m})\mathbf{x}=\tiny{\left[\begin{array}{c}
{\sqrt{\left\|\mathbf{W}_{\mathbf{m}} \mathbf{x}\right\|^{2}+C}}\\
\mathbf{W}_{\mathbf{m}}\mathbf{x}
\end{array}\right]}$. Then, we have $\langle{f_{\mathbf{x}}(\mathbf{m})\mathbf{x}},\\ {f_{\mathbf{x}}(\mathbf{m})\mathbf{x}}\rangle_\mathcal{L}=-C$, therefore $f_{\mathbf{x}}(\mathbf{m})\mathbf{x} \in \mathbb{H}^n$.
\end{prof}
According to Proposition \ref{Proposition:1}, we prove that Equation \ref{equ:lecf2} guarantees that the updating and transformation of node hyperbolic embeddings are strictly embedded in the hyperbolic space. More importantly, since $\mathbf{W_m}$ is transformed from the message embedding $\mathbf{m}_{ui}$, the entity information in the KG will guide the Lorentz transformation process of $f_\mathbf{x}$. Meanwhile, a certain degree of freedom on the time axis is retained by introducing the learnable vector $\mathbf{v}$.
\subsection{Model Prediction and Optimization}
\subsubsection{Model Prediction.}
Before the CF process, $L_1$ layers of aggregation (Equation \ref{equ:iag2}) in IAG will be stacked to propagate the high-order entity information in the KG, generating the item attributes. Then, we stack the $L_2$ layers of LECF layers to obtain the complementary high-order hyperbolic embeddings of users and items. The representations of each layer in the above process will be summed up to compose the final complementary information:
\begin{equation}
\mathbf{e}_i=\sqrt{C}\frac{\sum^{L_1}_{l=1}\omega_1^l\mathbf{e}_i^l}{|\|\sum^{L_1}_{l=1}\omega_1^l\mathbf{e}_i^l\|_{\mathcal{L}}|}\label{equ:predction1}
\end{equation}
\begin{equation}
\mathbf{x}_u=\sqrt{C}\frac{\sum^{L_2}_{l=1}\omega_2^l\mathbf{x}_u^l}{|\|\sum^{L_2}_{l=1}\omega_2^l\mathbf{x}_u^l\|_{\mathcal{L}}|} \, ,\quad \mathbf{x}_i=\sqrt{C}\frac{\sum^{L_2}_{l=1}\omega_2^l\mathbf{x}_i^l}{|\|\sum^{L_2}_{l=1}\omega_2^l\mathbf{x}_i^l\|_{\mathcal{L}}|}\label{equ:predction2}
\end{equation}
where $\omega$ are the hyperparameters. Equation \ref{equ:predction1} corresponds to IAG, and Equation \ref{equ:predction2} corresponds to LECF. Note that the results of the summation are strictly embedded in the hyperbolic space, which was unsatisfied in previous methods \cite{Hgcf2021,HAKG2022}. Finally, we calculate the exponential of the negative hyperbolic distance between the user and item hyperbolic embeddings as the predicting recommendation score: $\hat{y}_{ui}=\exp \left(-d_{\mathcal{L}}\left(\mathbf{x}_{u},\mathbf{x}_{i}\right)\right)$.
\subsubsection{Model Optimization.}
For a user $u$, let $i^+$ be the positive item, i.e., $\mathbf{Y}_{u,i^+}=1$. Let $i^-$ be the negative item, i.e., $\mathbf{Y}_{u,i^-}=0$. In this paper, we sample one positive and one negative item for a user. The optimization goal is to make the user approximate the positive item and move away from negative items in the hyperbolic space; thus, the loss function is defined as:
\begin{equation}
Loss=\sum_{u \in \mathcal{U}}\sum_{i \in \mathcal{N}(u)} \mathrm{max}\left( \hat{y}_{u, i^-}-\hat{y}_{u, i^+}+m,0\right)+\lambda\|\Theta\|_{2}^{2}
\label{equ:loss}
\end{equation}
where $m$ is the separation margin and $\|\Theta\|_{2}^{2}$ is the $\mathrm{L2}$ regularizer assigned by coefficient $\lambda$.
\subsection{Analysis of Lorentz Equivariance}
In this section, we analyze the Lorentz equivariance of the entire model, i.e., the two stages strictly satisfy one of Equations \ref{equ:equva1}, \ref{equ:equva2} and Proposition \ref{prop:2}.
\begin{proposition}
\cite{Scalars2021} A continuous function $\phi(\cdot)$ is Lorentz equivariant if and only if
$\phi\left(\mathbf{v}_{1}, \mathbf{v}_{2}, \cdots, \mathbf{v}_{N}\right)=\sum_{i=1}^{N} g_{i}\left(\left\langle \mathbf{v}_{i}, \mathbf{v}_{j}\right\rangle_{\mathcal{L}}^{N}\right) \mathbf{v}_{i}$, where $g_{i}(\cdot)$ can be taken to be polynomial.
\label{prop:2}
\end{proposition}
First, we consider all the scalars and vectors in Euclidean space to be Lorentz invariant. For Hyperbolic Sparse Attention Mechanism, the only part that involves hyperbolic vectors is Equation \ref{equ:hba1}, and $d_{\mathcal {L}}(\cdot)$ is Lorentz invariant, so the entire attention mechanism is Lorentz invariant. For the message aggregation, i.e., Equation \ref{equ:iag2} in IAG, we can easily prove that it is Lorentz equivariant\ according to Proposition \ref{prop:2}, and because $\hat{a}(\cdot)$ is Lorentz invariant, IAG is thus Lorentz equivariant.
For LECF layer, since $d_{\mathcal {L}}(\cdot)$ is Lorentz invariant, Equation \ref{equ:lecf1} is Lorentz invariant. In Equation \ref{equ:lecf2}, $\pi\left({f_{\mathbf{x}_i}(\mathbf{m}_{ui})},\mathbf{x}_{i}^{l}\right)$ is Lorentz equivariant\ according to \cite{NestedHNN2022}. Moreover, the aggregation process is Lorentz equivariant\ as discussed above; thus, the entire LECF layer is Lorentz equivariant . Inductively, a composition of IAGs and LECF layers will also be Lorentz equivariant . Finally, the summation of the results of each layer is Lorentz equivariant\ according to Proposition \ref{prop:2}, and the computation of the predicting recommendation score is Lorentz invariant.
\section{Experiments}
\subsection{Datasets}
We employ three benchmark datasets with auxiliary KGs to evaluate the effectiveness of LECF: Book-Crossing\footnote{ http://www2.informatik.uni-freiburg.de/~cziegler/BX/}, MovieLens-20M\footnote{https://grouplens.org/datasets/movielens/20m/}, and Yelp2018\footnote{https://www.yelp.com/dataset}. They are widely used in recent research on similar tasks and vary in application domain, data volume, and sparsity. In detail, the auxiliary KG from each user-item graph\ is constructed by considering the triplets that involve two-hop neighbor entities of items. To simulate the implicit feedback setting, we convert the interactions into binary preferences by applying a threshold $\geq$ 4. To ensure KG quality, we follow the preprocess in previous work \cite{KGAT:bm4,LKGR2022}, filtering out infrequent entities (i.e., fewer than 10 in both graphs). We split the datasets into training, validation, and testing sets at a ratio of 6:2:2. The composition statistics of each dataset are shown in Table \ref{table:st1}.
\begin{table}[t]
\caption{Statistics of the datasets in the experiments.}\label{table:st1
\centering\setlength\tabcolsep{2pt}\renewcommand{\arraystretch}{1}
\begin{tabular}{cc|c|c|cl}
\midrule[1.5pt]
\multicolumn{1}{l}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c|}{Book-Crossing} & \multicolumn{1}{c|}{MovieLens-20M} & \multicolumn{1}{c}{Yelp2018} \\ \midrule[0.3pt]
\multicolumn{1}{c}{\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\end{tabular}}} & Users & 17,860 & 138,159 & 45,919 & \\
\multicolumn{1}{c}{} & Items & 14,967 & 16,954 & 45,538 & \\
\multicolumn{1}{c}{} & Interactions & 139,746 & 13,501,622 & 1,185,068 \\ \midrule[0.3pt]
\multicolumn{1}{c}{\multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}\end{tabular}}} & Entities & 77,903 & 102,569 & 90,961 & \\
\multicolumn{1}{c}{} & Relations & 25 & 32 & 42 & \\
\multicolumn{1}{c}{} & KG triples & 151,500 & 499,474 & 1,853,704 \\ \midrule[1.5pt]
\end{tabular
\end{table}
\begin{table*}[ht]
\caption{Recall and NDCG results for all datasets.}\label{table:rs1
\centering
\begin{tabular}{c|cccc|cccc|cccc}
\midrule[1.5pt]
& \multicolumn{4}{c|}{Book-Crossing} & \multicolumn{4}{c|}{MovieLens-20M} & \multicolumn{4}{c}{Yelp2018} \\
&\tiny{R@10(\%)}&\tiny{N@10(\%)} &\tiny{R@20(\%)}&\tiny{N@20(\%)} &\tiny{R@10(\%)}&\tiny{N@10(\%)} &\tiny{R@20(\%)}&\tiny{N@20(\%)} &\tiny{R@10(\%)}&\tiny{N@10(\%)} &\tiny{R@20(\%)}&\tiny{N@20(\%)} \\ \midrule[0.3pt]
BRP &2.91&2.23 & 4.50 & 2.76 &13.04&9.71 & 20.53 & 15.84 &3.82&3.51 & 6.24 & 4.29 \\
CKE &3.01&1.85 & 4.32 & 2.31 &13.93&10.22 & 21.54 & 15.77 &4.11&3.80 & 6.52 & 4.31 \\
KGCN &5.86&5.28 & 7.65 & 5.90 &13.36&10.39 & 19.28 & 13.45 &4.29&3.77 & 6.31 & 4.39 \\
KGAT &4.32&2.62 & 5.29 & 3.05 &{\ul14.75} &{\ul11.37} & 22.26 & 17.23 &4.62&4.10 & 7.02 & 4.57 \\
KGNN-LS &7.19&5.47 & 8.50 & 5.95 &13.71&10.23 & 20.06 & 15.30 &4.10&3.74 & 6.79 & 4.20 \\
CKAN &3.34&2.91 & 7.37 & 5.85 &13.64&10.38 & 21.33 & 15.21 &4.04&3.59 & 6.41 & 4.38 \\
Hyper-Know &4.91&2.85 & 7.17 & 5.64 &12.20&9.15 & 22.72 & 16.83 &4.21&4.02 & 6.85 & 4.46 \\
HAKG &7.33&{\ul5.50} & 8.70 & 6.04 &13.18&10.53 & 23.73 & 17.54 &\textbf{4.86}&\textbf{4.35} & {\ul7.65} & {\ul5.16} \\
LKGR &{\ul7.80}& 5.07 &{\ul 9.23} &{\ul 6.72} &11.89&10.75 & {\ul 25.17} & {\ul 20.35} &4.57&4.22 & 6.83 & 4.35 \\ \midrule[0.3pt]
LECF &\textbf{8.23}&\textbf{5.68} &\textbf{10.07} &\textbf{7.08} &\textbf{15.62}&\textbf{11.87} & \textbf{28.27} &\textbf{22.74} &{\ul4.60}&{\ul4.25} & \textbf{8.32} & \textbf{5.52} \\ \midrule[0.3pt]
\% Improv. &5.5\%&3.27\% & 9.10\% & 5.27\% &5.90\%&4.40\% & 12.35\% &11.75\% &N/A&N/A & 8.76\% & 6.97\% \\ \midrule[1.5pt]
\end{tabular}
\hspace{15pt}\begin{tablenotes}\centering
\item[1] \qquad \quad\scriptsize{The best-performing models on each dataset and metric are highlighted in \\ \, \quad \qquad bold, and the second-best models are underlined.}
\end{tablenotes}
\end{table*}
\subsection{Experiment Settings}
\subsubsection{Baselines.}
To verify the effectiveness of LECF, we compare it with state-of-the-art methods, including both the well-known Euclidean baselines and leading hyperbolic models. For Euclidean baselines, the KG-free method BPR is included. Furthermore, we include the mainstream propagation-based methods: \textbf{KGCN} \cite{KGCN:bm3}, \textbf{KGNN-LS} \cite{KGNN-LS:bm5}, and \textbf{CKAN} \cite{CKAN:bm6}. Additionally, regularization-based methods are included such as \textbf{CKE} \cite{CKE:bm2} and \textbf{KGAT} \cite{KGAT:bm4}. For the hyperbolic models, state-of-the-art methods with auxiliary KGs are considered: \textbf{Hyper-Know} \cite{Hyper-Know:bm7}, \textbf{HAKG} \cite{HAKG2022}, and \textbf{LKGR} \cite{LKGR2022}.
\subsubsection{Experiment Details.}
In evaluating the Top-$K$ recommendation task, we use the trained model to rank $K$ items for each user in the test set with the highest predicted recommendation score $\hat{y}_{ui}$. Thus, two widely-used evaluation protocols \cite{recall2018}, i.e., Recall@$K$ and NDCG@$K$ ($K$= 20 by default) are employed. For a fair comparison, the size of ID embeddings is set to 64, and we initialize the model parameters by Xavier initializer.
We optimize all the Euclidean models with Adam optimizer
and the hyperbolic models with Riemannian SGD \cite{StochasticRiemannian}.
We train all the models on a single NVIDIA GeForce RTX 3090 GPU. Here, a grid search is used to find all optimal parameters. The search range for the regularization coefficient $\lambda$ is $\{ 10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}\}$, and the aggregation order search range is from 2 to 8. The learning rate is tuned within $\{5\times10^{-2},10^{-2},5\times10^{-3},10^{-3}\}$. The range of Riemannian SGD weight decay is $\{10^{-2},10^{-3},10^{-4},10^{-5}\}$. The curvature $-1/C$ is set constant to $-1$ in all experiments.
\subsection{Overall Performance}
The empirical results on Recall@$K$ and NDCG@$K$ ($K$=10 or $K$=20 ) are reported in Table \ref{table:rs1}. Overall, LECF significantly outperforms all Euclidean and hyperbolic models on each dataset. Furthermore, we have the following observations: (i) the hyperbolic models are generally better than the Euclidean models due to their better compatibility with large-scale networks. However, since they do not maintain Lorentz equivariance, their performance is not stable across different datasets. LECF addresses this shortcoming, thus comprehensively surpassing its counterparts.
(ii) LKGR and HAKG outperform Hyper-Know among the hyperbolic models because they consider the high-order knowledge in hyperbolic space. In contrast, LECF further considers high-order entity signals to users across graphs, further improving performance. (iii) LECF has a more significant improvement on MovieLens-20M than the other datasets. A possible reason is that MovieLens-20M has more symmetric features than the other datasets (due to its larger data size). Since LECF can generalize better to symmetric features that benefit from enforcing Lorentz equivariance, the performance is more prominent.
\subsection{Validation Experiments of Lorentz Equivariance}
In this section, we analyze the significance of Lorentz equivariance\ in hyperbolic CF. To demonstrate the effectiveness of preserving Lorentz equivariance, we follow the ablation approach in \cite{jet2022} to break the Lorentz equivariance\ in LECF by replacing Equation \ref{equ:lecf1} with:
\begin{equation}
{{\mathbf{m}}_{ui}}={{\phi }_{e}}\left(\mathbf{h}_{u}^{l},\mathbf{h}_{i}^{l},\mathbf{x}_{u}^{l},\mathbf{x}_{i}^{l}\right)
\label{equ:20}
\end{equation}
Since $\mathbf{x}_{u}^{l},\mathbf{x}_{i}^{l}$ are the hyperbolic vectors, transforming them linearly by $\phi_e$ breaks the Lorentz equivariance of the whole model, and this variant is denoted as LECF$^{\ddag}$.
We performed ablation experiments for Lorentz equivariance on three datasets. The necessity of enforcing Lorentz equivariance\ in hyperbolic CF is proven since the performance of LECF$^\ddag$ is noticeably worse than that of LECF as shown in Figure \ref{fig:ab1}. We further speculate that the gain produced by the hyperbolic geometry in LECF depends heavily on the fulfillment of Lorentz equivariance.
Moreover, to deeply reveal the effects of Lorentz equivariance, we perform the following two tests (both performed on the MovieLens-20M dataset) to demonstrate the advantages of symmetry-preservation under different situations.
\begin{figure}[t]
\begin{minipage}[t]{0.475\textwidth}
\centering
\subcaptionbox{}
{\includegraphics[width=0.49\linewidth]{f/ablation1.pdf}}
\subcaptionbox{}
{\includegraphics[width=0.49\linewidth]{f/ablation2.pdf}}
\caption{Ablation experiments of Lorentz equivariance.}\label{fig:ab1}
\end{minipage}
\hspace{0.03\textwidth}
\begin{minipage}[t]{0.475\textwidth}
\centering
\subcaptionbox{}
{\includegraphics[width=0.49\linewidth]{f/le1a.pdf}}
\subcaptionbox{}
{\includegraphics[width=0.49\linewidth]{f/le1b.pdf}}
\caption{ Model performance as sparsity increases.}\label{fig:le1}
\end{minipage}
\end{figure}
\begin{figure}[ht]
\centering
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{f/le2a.pdf
\caption{}
\label{fig:le2a}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{f/le2b.pdf
\caption{}
\label{fig:le2b}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{f/le3a.pdf}
\caption{}
\label{fig:le3a}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{f/le3b.pdf}
\caption{}
\label{fig:le3b}
\end{subfigure}
\caption{Model performance under Lorentz boost (Figure \ref{fig:le2a}, \ref{fig:le2b}) and spatial rotation (Figure \ref{fig:le3a}, \ref{fig:le3b}).}\label{fig:le2}
\end{figure
\subsubsection{Symmetry Limitation Test.}
In practice, it is difficult to measure the symmetry limitation of a model. However, in very sparse graphs, the impact of symmetry limitation grows dramatically \cite{E(n)GNN2021}. Therefore, we design the test with a sparse sampling of the original dataset to verify the generalization ability of LECF to symmetric features. Specifically, we eliminate the edges in user-item graph\ with the ratio of $p_e$, and the experimental results are shown in Figure \ref{fig:le1}. We observe that on both evaluation metrics, LECF performances drop the slowest as the sparsity of the data increases, while LECF$^{\ddag}$ does not show particular stability.
\subsubsection{Lorentz Transformation Test.}
Another perspective that supports the significance of symmetry-preserving is the robustness of Lorentz equivariant\ model under the Lorentz transformations. To better demonstrate this, we perform Lorentz boost and spatial rotation transformation separately on the test data with the training data unchanged. First, the Lorentz boost under the hyperbolic angle $\alpha$ is used to transform the test data, and the results are shown in Figures \ref{fig:le2a}, \ref{fig:le2b}. Second,
the spatial rotation under angle $\beta$ is performed on the first two dimensions in the space axis, and the results are shown in Figures \ref{fig:le3a}, \ref{fig:le3b}. Overall, Lorentz boost has a greater impact on model performances, which may be related to the geometric characteristics. Accordingly, LECF shows extraordinary stability under both transformations, i.e., the experiments demonstrate that our model perfectly preserves the Lorentz equivariance.
\subsection{ Ablation Study }
We conduct the following ablation studies to obtain deep insights into the effectiveness of all LECF components.
{\centering
\begin{minipage}[t]{0.45\textwidth}
\captionof{table}{Ablation Study on Top-20 recommendation (\%).}\label{table:ab}
\begin{tabular}{c|ccccc}
\midrule[1.5pt]
& \multicolumn{1}{c|}{$\otimes$HG} & \multicolumn{1}{c|}{$\otimes$SA} & \multicolumn{1}{c|}{$\otimes$S1} & \multicolumn{1}{c|}{$\otimes$S2} & \textbf{LECF} \\ \midrule[0.3pt]
BK-R@20 & 8.50 & 9.58 & 5.28 & 6.17 & \textbf{10.07} \\
BK-N@20 & 5.46 & 6.75 & 3.02 & 5.39 & \textbf{7.08 } \\\midrule[0.3pt]
MV-R@20 & 20.91 & 26.93 & 16.48 & 17.04 & \textbf{28.27} \\
MV-N@20 & 15.99 & 22.03 & 13.60 & 14.27 & \textbf{22.74} \\ \midrule[0.3pt]
YP-R@20 & 6.17 & 7.39 & 4.51 & 5.35 & \textbf{8.32} \\
YP-N@20 & 4.78 & 5.02 & 3.97 & 4.46 & \textbf{5.52} \\ \midrule[1.5pt]
\end{tabular}
\begin{tablenotes}
\item[1] \scriptsize{BK, MV, and YP denote
Book-Crossing, MovieLens-20M and Yelp2018 }
\end{tablenotes}\end{minipage}
}
\subsubsection{Effect of hyperbolic geometry.}
To demonstrate the superiority of learning user and item representations in the hyperbolic space over Euclidean space, we transfer all operations in LECF to Euclidean space while retaining all functional components, denoted as LECF$^\mathrm{ \otimes HG}$. In Table \ref{table:ab}, we observe that the results of LECF$^\mathrm{ \otimes HG}$ decline significantly relative to LECF, demonstrating the fundamental significance of hyperbolic geometry for CF based on large-scale networks.
\subsubsection{Effect of mutual signal propagation between the attribute and hyperbolic embeddings.}
In LECF layer, the item attributes, i.e., the entity signals from the KG, are updated mutually with the hyperbolic embeddings of users and items. We cut off these two message propagation paths separately to verify their effectiveness respectively. First, we remove the hyperbolic embeddings in Equation \ref{equ:lecf1} to cut off the signal from user-item graph\ to KG. We denote this variant as LECF$^\mathrm{ \otimes S1}$. Second, we remove Lorentz Equivariant Transformation in Equation \ref{equ:lecf2}, whereby the entity signal in
KG will not be passed to the hyperbolic embeddings of user-item graph . This variant is denoted as LECF$^\mathrm{\otimes S2}$. The severe deterioration of these two variants demonstrates the importance of bi-directionally propagating information between two graphs.
\subsubsection{Effect of Hyperbolic Sparse Attention Mechanism.}
To perform the corresponding ablation experiment, we first sample the neighbor nodes with a random strategy while setting the attention coefficient to be uniformly distributed. This variant is denoted as LECF$^\mathrm{ \otimes SA}$. For the first time, LECF has successfully implemented a sparse attention mechanism strictly embedded in hyperbolic space, and we can prove its effectiveness from the deterioration of LECF$^\mathrm{ \otimes SA}$.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{f/trainingtime.pdf}
\caption{ Training time per epoch (second).}
\label{fig:trainingtime}
\end{figure}
\subsection{Training Efficiency}
In this section, we compare the training time consumption of each epoch to test the efficiency of the models. All settings for the experiment remain the same as in the previous sections. Figure \ref{fig:trainingtime} shows that the propagation-based method is generally more time-consuming than the regularization-based method. Compared with its hyperbolic peers, LECF is significantly more efficient, which benefits from abandoning the tangent space aggregation and applying the sparse attention mechanism.
\section{Conclusion}
Inspired by numerous explorations of Euclidean equivariance in graph representation learning, we inaugurate the discussion of Lorentz equivariance\ in hyperbolic CF. Since the distance distortion under certain transformations in hyperbolic space is more severe than in Euclidean space, the generalization capability of the hyperbolic CF model depends heavily on the ability to perceive the same characteristics under the Lorentz transformations, that is, to maintain the Lorentz equivariance. In this context, we delicately construct the IAGs and LECF layers that are strictly Lorentz equivariant.
Specifically, in each IAG,
the proposed Hyperbolic Sparse Attention Mechanisms sample the most informative neighbor nodes to support downstream tasks better. In each LECF layer, the attribute embeddings generated from IAG and the hyperbolic embeddings are mutually updated by the proposed Lorentz Transformation Equivariant in KG so that the high-order entity signals can be passed to the user representation across graphs. Experiments on three public benchmark datasets demonstrate the superiority of LECF. More importantly, we verify that LECF is Lorentz equivariant through extensive experiments and prove that enforcing Lorentz equivariance significantly improves model performance.
\section{Introduction}
\file{elsarticle.cls} is a thoroughly re-written document class
for formatting \LaTeX{} submissions to Elsevier journals.
The class uses the environments and commands defined in \LaTeX{} kernel
without any change in the signature so that clashes with other
contributed \LaTeX{} packages such as \file{hyperref.sty},
\file{preview-latex.sty}, etc., will be minimal.
\file{elsarticle.cls} is primarily built upon the default
\file{article.cls}. This class depends on the following packages
for its proper functioning:
\begin{enumerate}
\item \file{natbib.sty} for citation processing;
\item \file{geometry.sty} for margin settings;
\item \file{fleqn.clo} for left aligned equations;
\item \file{graphicx.sty} for graphics inclusion;
\item \file{txfonts.sty} optional font package, if the document is to
be formatted with Times and compatible math fonts;
\item \file{hyperref.sty} optional packages if hyperlinking is
required in the document;
\item \file{endfloat.sty} optional packages if floats to be placed at
end of the PDF.
\end{enumerate}
All the above packages (except some optional packages) are part of any
standard \LaTeX{} installation. Therefore, the users need not be
bothered about downloading any extra packages. Furthermore, users are
free to make use of \textsc{ams} math packages such as
\file{amsmath.sty}, \file{amsthm.sty}, \file{amssymb.sty},
\file{amsfonts.sty}, etc., if they want to. All these packages work in
tandem with \file{elsarticle.cls} without any problems.
\section{Major Differences}
Following are the major differences between \file{elsarticle.cls}
and its predecessor package, \file{elsart.cls}:
\begin{enumerate}[\textbullet]
\item \file{elsarticle.cls} is built upon \file{article.cls}
while \file{elsart.cls} is not. \file{elsart.cls} redefines
many of the commands in the \LaTeX{} classes/kernel, which can
possibly cause surprising clashes with other contributed
\LaTeX{} packages;
\item provides preprint document formatting by default, and
optionally formats the document as per the final
style of models $1+$, $3+$ and $5+$ of Elsevier journals;
\item some easier ways for formatting \verb+list+ and
\verb+theorem+ environments are provided while people can still
use \file{amsthm.sty} package;
\item \file{natbib.sty} is the main citation processing package
which can comprehensively handle all kinds of citations and
works perfectly with \file{hyperref.sty} in combination with
\file{hypernat.sty};
\item long title pages are processed correctly in preprint and
final formats.
\end{enumerate}
\section{Installation}
The package is available at author resources page at Elsevier
(\url{http://www.elsevier.com/locate/latex}).
It can also be found in any of the nodes of the Comprehensive
\TeX{} Archive Network (\textsc{ctan}), one of the primary nodes
being
\url{http://tug.ctan.org/tex-archive/macros/latex/contrib/elsarticle/}.
Please download the \file{elsarticle.dtx} which is a composite
class with documentation and \file{elsarticle.ins} which is the
\LaTeX{} installer file. When we compile the
\file{elsarticle.ins} with \LaTeX{} it provides the class file,
\file{elsarticle.cls} by
stripping off all the documentation from the \verb+*.dtx+ file.
The class may be moved or copied to a place, usually,
\verb+$TEXMF/tex/latex/elsevier/+,
or a folder which will be read
by \LaTeX{} during document compilation. The \TeX{} file
database needs updation after moving/copying class file. Usually,
we use commands like \verb+mktexlsr+ or \verb+texhash+ depending
upon the distribution and operating system.
\section{Usage}\label{sec:usage}
The class should be loaded with the command:
\begin{vquote}
\documentclass[<options>]{elsarticle}
\end{vquote}
\noindent where the \verb+options+ can be the following:
\begin{description}
\item [{\tt\color{verbcolor} preprint}] default option which format the
document for submission to Elsevier journals.
\item [{\tt\color{verbcolor} review}] similar to the \verb+preprint+
option, but increases the baselineskip to facilitate easier review
process.
\item [{\tt\color{verbcolor} 1p}] formats the article to the look and
feel of the final format of model 1+ journals. This is always single
column style.
\item [{\tt\color{verbcolor} 3p}] formats the article to the look and
feel of the final format of model 3+ journals. If the journal is a two
column model, use \verb+twocolumn+ option in combination.
\item [{\tt\color{verbcolor} 5p}] formats for model 5+ journals. This
is always of two column style.
\item [{\tt\color{verbcolor} authoryear}] author-year citation style of
\file{natbib.sty}. If you want to add extra options of
\file{natbib.sty}, you may use the options as comma delimited strings
as arguments to \verb+\biboptions+ command. An example would be:
\end{description}
\begin{vquote}
\biboptions{longnamesfirst,angle,semicolon}
\end{vquote}
\begin{description}
\item [{\tt\color{verbcolor} number}] numbered citation style. Extra options
can be loaded with\linebreak \verb+\biboptions+ command.
\item [{\tt\color{verbcolor} sort\&compress}] sorts and compresses the
numbered citations. For example, citation [1,2,3] will become [1--3].
\item [{\tt\color{verbcolor} longtitle}] if front matter is unusually long, use
this option to split the title page across pages with the correct
placement of title and author footnotes in the first page.
\item [{\tt\color{verbcolor} times}] loads \file{txfonts.sty}, if
available in the system to use Times and compatible math fonts.
\item [{\tt\color{verbcolor} reversenotenum}] Use alphabets as
author--affiliation linking labels and use numbers for author
footnotes. By default, numbers will be used as author--affiliation
linking labels and alphabets for author footnotes.
\item [{\tt\color{verbcolor} lefttitle}] To move title and
author/affiliation block to flushleft. \verb+centertitle+ is the
default option which produces center alignment.
\item [{\tt\color{verbcolor} endfloat}] To place all floats at the end
of the document.
\item [{\tt\color{verbcolor} nonatbib}] To unload natbib.sty.
\item [{\tt\color{verbcolor} doubleblind}] To hide author name,
affiliation, email address etc. for double blind refereeing purpose.
\item[] All options of \file{article.cls} can be used with this
document class.
\item[] The default options loaded are \verb+a4paper+, \verb+10pt+,
\verb+oneside+, \verb+onecolumn+ and \verb+preprint+.
\end{description}
\section{Frontmatter}
There are two types of frontmatter coding:
\begin{enumerate}[(1)]
\item each author is
connected to an affiliation with a footnote marker; hence all
authors are grouped together and affiliations follow;
\pagebreak
\item authors of same affiliations are grouped together and the
relevant affiliation follows this group.
\end{enumerate}
An example of coding the first type is provided below.
\begin{vquote}
\title{This is a specimen title\tnoteref{t1,t2}}
\tnotetext[t1]{This document is the results of the research
project funded by the National Science Foundation.}
\tnotetext[t2]{The second title footnote which is a longer
text matter to fill through the whole text width and
overflow into another line in the footnotes area of the
first page.}
\end{vquote}
\begin{vquote}
\author[1]{Jos Migchielsen\corref{cor1}%
\fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}
\author[2]{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}
\author[3]{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}
\cortext[cor1]{Corresponding author}
\fntext[fn1]{This is the first author footnote.}
\fntext[fn2]{Another author footnote, this is a very long
footnote and it should be a really long footnote. But this
footnote is not yet sufficiently long enough to make two
lines of footnote text.}
\fntext[fn3]{Yet another author footnote.}
\affiliation[1]{organization={Elsevier B.V.},
addressline={Radarweg 29},
postcode={1043 NX},
city={Amsterdam},
country={The Netherlands}}
\affiliation[2]{organization={Sayahna Foundation},
addressline={JWRA 34, Jagathy},
city={Trivandrum}
postcode={695014},
country={India}}
\end{vquote}
\begin{vquote}
\affiliation[3]{organization={STM Document Engineering
Pvt Ltd.},
addressline={Mepukada, Malayinkil},
city={Trivandrum}
postcode={695571},
country={India}}
\end{vquote}
The output of the above \TeX{} source is given in Clips~\ref{clip1} and
\ref{clip2}. The header portion or title area is given in
Clip~\ref{clip1} and the footer area is given in Clip~\ref{clip2}.
\deforange{blue!70}
\src{Header of the title page.}
\includeclip{1}{130 612 477 707}{1psingleauthorgroup.pdf
\deforange{orange}
\deforange{blue!70}
\src{Footer of the title page.}
\includeclip{1}{93 135 499 255}{1pseperateaug.pdf
\deforange{orange}
Most of the commands such as \verb+\title+, \verb+\author+,
\verb+\affiliation+ are self explanatory. Various components are
linked to each other by a label--reference mechanism; for
instance, title footnote is linked to the title with a footnote
mark generated by referring to the \verb+\label+ string of
the \verb=\tnotetext=. We have used similar commands
such as \verb=\tnoteref= (to link title note to title);
\verb=\corref= (to link corresponding author text to
corresponding author); \verb=\fnref= (to link footnote text to
the relevant author names). \TeX{} needs two compilations to
resolve the footnote marks in the preamble part.
Given below are the syntax of various note marks and note texts.
\begin{vquote}
\tnoteref{<label(s)>}
\corref{<label(s)>}
\fnref{<label(s)>}
\tnotetext[<label>]{<title note text>}
\cortext[<label>]{<corresponding author note text>}
\fntext[<label>]{<author footnote text>}
\end{vquote}
\noindent where \verb=<label(s)>= can be either one or more comma
delimited label strings. The optional arguments to the
\verb=\author= command holds the ref label(s) of the address(es)
to which the author is affiliated while each \verb=\affiliation=
command can have an optional argument of a label. In the same
manner, \verb=\tnotetext=, \verb=\fntext=, \verb=\cortext= will
have optional arguments as their respective labels and note text
as their mandatory argument.
The following example code provides the markup of the second type
of author-affiliation.
\begin{vquote}
\author{Jos Migchielsen\corref{cor1}%
\fnref{fn1}}
\ead{J.Migchielsen@elsevier.com}
\affiliation[1]{organization={Elsevier B.V.},
addressline={Radarweg 29},
postcode={1043 NX},
city={Amsterdam},
country={The Netherlands}}
\author{CV Radhakrishnan\fnref{fn2}}
\ead{cvr@sayahna.org}
\affiliation[2]{organization={Sayahna Foundation},
addressline={JWRA 34, Jagathy},
city={Trivandrum}
postcode={695014},
country={India}}
\author{CV Rajagopal\fnref{fn1,fn3}}
\ead[url]{www.stmdocs.in}
\affiliation[3]{organization={STM Document Engineering
Pvt Ltd.},
addressline={Mepukada, Malayinkil},
city={Trivandrum}
postcode={695571},
country={India}}
\end{vquote}
\vspace*{-.5pc}
\begin{vquote}
\cortext[cor1]{Corresponding author}
\fntext[fn1]{This is the first author footnote.}
\fntext[fn2]{Another author footnote, this is a very long
footnote and it should be a really long footnote. But this
footnote is not yet sufficiently long enough to make two lines
of footnote text.}
\end{vquote}
The output of the above \TeX{} source is given in Clip~\ref{clip3}.
\deforange{blue!70}
\src{Header of the title page..}
\includeclip{1}{119 563 468 709}{1pseperateaug.pdf
\deforange{orange}
Clip~\ref{clip4} shows the output after giving \verb+doubleblind+ class option.
\deforange{blue!70}
\src{Double blind article}
\includeclip{1}{124 567 477 670}{elstest-1pdoubleblind.pdf
\deforange{orange}
\vspace*{-.5pc}
The frontmatter part has further environments such as abstracts and
keywords. These can be marked up in the following manner:
\begin{vquote}
\begin{abstract}
In this work we demonstrate the formation of a new type of
polariton on the interface between a ....
\end{abstract}
\end{vquote}
\vspace*{-.5pc}
\begin{vquote}
\begin{keyword}
quadruple exiton \sep polariton \sep WGM
\end{keyword}
\end{vquote}
\noindent Each keyword shall be separated by a \verb+\sep+ command.
\textsc{msc} classifications shall be provided in
the keyword environment with the commands
\verb+\MSC+. \verb+\MSC+ accepts an optional
argument to accommodate future revisions.
eg., \verb=\MSC[2008]=. The default is 2000.\looseness=-1
\subsection{New page}
Sometimes you may need to give a page-break and start a new page after
title, author or abstract. Following commands can be used for this
purpose.
\begin{vquote}
\newpageafter{title}
\newpageafter{author}
\newpageafter{abstract}
\end{vquote}
\begin{itemize}
\leftskip-2pc
\item [] {\tt\color{verbcolor} \verb+\newpageafter{title}+} typeset the title alone on one page.
\item [] {\tt\color{verbcolor} \verb+\newpageafter{author}+} typeset the title
and author details on one page.
\item [] {\tt\color{verbcolor} \verb+\newpageafter{abstract}+}
typeset the title,
author details and abstract \& keywords one one page.
\end{itemize}
\section{Floats}
{Figures} may be included using the command, \verb+\includegraphics+ in
combination with or without its several options to further control
graphic. \verb+\includegraphics+ is provided by \file{graphic[s,x].sty}
which is part of any standard \LaTeX{} distribution.
\file{graphicx.sty} is loaded by default. \LaTeX{} accepts figures in
the postscript format while pdf\LaTeX{} accepts \file{*.pdf},
\file{*.mps} (metapost), \file{*.jpg} and \file{*.png} formats.
pdf\LaTeX{} does not accept graphic files in the postscript format.
The \verb+table+ environment is handy for marking up tabular
material. If users want to use \file{multirow.sty},
\file{array.sty}, etc., to fine control/enhance the tables, they
are welcome to load any package of their choice and
\file{elsarticle.cls} will work in combination with all loaded
packages.
\section[Theorem and ...]{Theorem and theorem like environments}
\file{elsarticle.cls} provides a few shortcuts to format theorems and
theorem-like environments with ease. In all commands the options that
are used with the \verb+\newtheorem+ command will work exactly in the same
manner. \file{elsarticle.cls} provides three commands to format theorem or
theorem-like environments:
\begin{vquote}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newdefinition{rmk}{Remark}
\newproof{pf}{Proof}
\newproof{pot}{Proof of Theorem \ref{thm2}}
\end{vquote}
The \verb+\newtheorem+ command formats a
theorem in \LaTeX's default style with italicized font, bold font
for theorem heading and theorem number at the right hand side of the
theorem heading. It also optionally accepts an argument which
will be printed as an extra heading in parentheses.
\begin{vquote}
\begin{thm}
For system (8), consensus can be achieved with
$\|T_{\omega z}$
...
\begin{eqnarray}\label{10}
....
\end{eqnarray}
\end{thm}
\end{vquote}
Clip~\ref{clip5} will show you how some text enclosed between the
above code\goodbreak \noindent looks like:
\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newtheorem}}
\includeclip{2}{1 1 453 120}{jfigs.pdf}
\deforange{orange}
The \verb+\newdefinition+ command is the same in
all respects as its\linebreak \verb+\newtheorem+ counterpart except that
the font shape is roman instead of italic. Both
\verb+\newdefinition+ and \verb+\newtheorem+ commands
automatically define counters for the environments defined.
\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newdefinition}}
\includeclip{1}{1 1 453 105}{jfigs.pdf}
\deforange{orange}
The \verb+\newproof+ command defines proof environments with
upright font shape. No counters are defined.
\vspace*{6pt}
\deforange{blue!70}
\src{{\ttfamily\color{verbcolor}\expandafter\@gobble\string\\ newproof}}
\includeclip{3}{1 1 453 65}{jfigs.pdf}
\deforange{orange}
Users can also make use of \verb+amsthm.sty+ which will override
all the default definitions described above.
\section[Enumerated ...]{Enumerated and Itemized Lists}
\file{elsarticle.cls} provides an extended list processing macros
which makes the usage a bit more user friendly than the default
\LaTeX{} list macros. With an optional argument to the
\verb+\begin{enumerate}+ command, you can change the list counter
type and its attributes.
\begin{vquote}
\begin{enumerate}[1.]
\item The enumerate environment starts with an optional
argument `1.', so that the item counter will be suffixed
by a period.
\item You can use `a)' for alphabetical counter and '(i)' for
roman counter.
\begin{enumerate}[a)]
\item Another level of list with alphabetical counter.
\item One more item before we start another.
\end{vquote}
\deforange{blue!70}
\src{List -- Enumerate}
\includeclip{4}{1 1 453 185}{jfigs.pdf}
\deforange{orange}
Further, the enhanced list environment allows one to prefix a
string like `step' to all the item numbers.
\begin{vquote}
\begin{enumerate}[Step 1.]
\item This is the first step of the example list.
\item Obviously this is the second step.
\item The final step to wind up this example.
\end{enumerate}
\end{vquote}
\deforange{blue!70}
\src{List -- enhanced}
\includeclip{5}{1 1 313 83}{jfigs.pdf}
\deforange{orange}
\section{Cross-references}
In electronic publications, articles may be internally
hyperlinked. Hyperlinks are generated from proper
cross-references in the article. For example, the words
\textcolor{black!80}{Fig.~1} will never be more than simple text,
whereas the proper cross-reference \verb+\ref{tiger}+ may be
turned into a hyperlink to the figure itself:
\textcolor{blue}{Fig.~1}. In the same way,
the words \textcolor{blue}{Ref.~[1]} will fail to turn into a
hyperlink; the proper cross-reference is \verb+\cite{Knuth96}+.
Cross-referencing is possible in \LaTeX{} for sections,
subsections, formulae, figures, tables, and literature
references.
\section[Mathematical ...]{Mathematical symbols and formulae}
Many physical/mathematical sciences authors require more
mathematical symbols than the few that are provided in standard
\LaTeX. A useful package for additional symbols is the
\file{amssymb} package, developed by the American Mathematical
Society. This package includes such oft-used symbols as
$\lesssim$ (\verb+\lesssim+), $\gtrsim$ (\verb+\gtrsim+) or
$\hbar$ (\verb+\hbar+). Note that your \TeX{}
system should have the \file{msam} and \file{msbm} fonts installed. If
you need only a few symbols, such as $\Box$ (\verb+\Box+), you might try the
package \file{latexsym}.
Another point which would require authors' attention is the
breaking up of long equations. When you use
\file{elsarticle.cls} for formatting your submissions in the
\verb+preprint+ mode, the document is formatted in single column
style with a text width of 384pt or 5.3in. When this document is
formatted for final print and if the journal happens to be a double column
journal, the text width will be reduced to 224pt at for 3+
double column and 5+ journals respectively. All the nifty
fine-tuning in equation breaking done by the author goes to waste in
such cases. Therefore, authors are requested to check this
problem by typesetting their submissions in final format as well
just to see if their equations are broken at appropriate places,
by changing appropriate options in the document class loading
command, which is explained in section~\ref{sec:usage},
\nameref{sec:usage}. This allows authors to fix any equation breaking
problem before submission for publication.
\file{elsarticle.cls} supports formatting the author submission
in different types of final format. This is further discussed in
section \ref{sec:final}, \nameref{sec:final}.
\enlargethispage*{\baselineskip}
\subsection*{Displayed equations and double column journals}
Many Elsevier journals print their text in two columns. Since
the preprint layout uses a larger line width than such columns,
the formulae are too wide for the line width in print. Here is an
example of an equation (see equation 6) which is perfect in a
single column preprint format:
In normal course, articles are prepared and submitted in single column
format even if the final printed article will come in a double column
format journal. Here the problem is that when the article is typeset by
the typesetters for paginating and fit within the single column width,
they have to break the lengthy equations and align them properly. Even
if most of the tasks in preparing your proof is automated, the equation
breaking and aligning requires manual judgement, hence this task is manual.
When there comes a manual operation that area is error prone. Author
needs to check that equation pretty well.
However if authors themselves break the equation to the single column
width typesetters need not want to touch these area and the proof authors
get will be without any errors.
\setlength\Sep{6pt}
\src{See equation (6)}
\deforange{blue!70}
\includeclip{4}{105 500 500 700}{1psingleauthorgroup.pdf}
\deforange{orange}
\noindent When this document is typeset for publication in a
model 3+ journal with double columns, the equation will overlap
the second column text matter if the equation is not broken at
the appropriate location.
\vspace*{6pt}
\deforange{blue!70}
\src{See equation (6) overprints into second column}
\includeclip{3}{59 421 532 635}{elstest-3pd.pdf}
\deforange{orange}
\vspace*{6pt}
\noindent The typesetter will try to break the equation which
need not necessarily be to the liking of the author or as it
happens, typesetter's break point may be semantically incorrect.
Therefore, authors may check their submissions for the incidence
of such long equations and break the equations at the correct
places so that the final typeset copy will be as they wish.
\section{Bibliography}
Three bibliographic style files (\verb+*.bst+) are provided ---
\file{elsarticle-num.bst}, \file{elsarticle-num-names.bst} and
\file{elsarticle-harv.bst} --- the first one can be used for the
numbered scheme, second one for numbered with new options of
\file{natbib.sty}. The third one is for the author year
scheme.
In \LaTeX{} literature, references are listed in the
\verb+thebibliography+ environment. Each reference is a
\verb+\bibitem+ and each \verb+\bibitem+ is identified by a label,
by which it can be cited in the text:
\verb+\bibitem[Elson et al.(1996)]{ESG96}+ is cited as
\verb+\citet{ESG96}+.
\noindent In connection with cross-referencing and
possible future hyperlinking it is not a good idea to collect
more that one literature item in one \verb+\bibitem+. The
so-called Harvard or author-year style of referencing is enabled
by the \LaTeX{} package \file{natbib}. With this package the
literature can be cited as follows:
\begin{enumerate}[\textbullet]
\item Parenthetical: \verb+\citep{WB96}+ produces (Wettig \& Brown, 1996).
\item Textual: \verb+\citet{ESG96}+ produces Elson et al. (1996).
\item An affix and part of a reference:
\verb+\citep[e.g.][Ch. 2]{Gea97}+ produces (e.g. Governato et
al., 1997, Ch. 2).
\end{enumerate}
In the numbered scheme of citation, \verb+\cite{<label>}+ is used,
since \verb+\citep+ or \verb+\citet+ has no relevance in the numbered
scheme. \file{natbib} package is loaded by \file{elsarticle} with
\verb+numbers+ as default option. You can change this to author-year
or harvard scheme by adding option \verb+authoryear+ in the class
loading command. If you want to use more options of the \file{natbib}
package, you can do so with the \verb+\biboptions+ command, which is
described in the section \ref{sec:usage}, \nameref{sec:usage}. For
details of various options of the \file{natbib} package, please take a
look at the \file{natbib} documentation, which is part of any standard
\LaTeX{} installation.
In addition to the above standard \verb+.bst+ files, there are 10
journal-specific \verb+.bst+ files also available.
Instruction for using these \verb+.bst+ files can be found at
\href{http://support.stmdocs.in/wiki/index.php?title=Model-wise_bibliographic_style_files}
{http://support.stmdocs.in}
\section[Graphical ...]{Graphical abstract and highlights}
A template for adding graphical abstract and highlights are available
now. This will appear as the first two pages of the PDF before the
article content begins.
\pagebreak
Please refer below to see how to code them.
\begin{vquote}
....
....
\end{abstract}
\begin{graphicalabstract}
\end{graphicalabstract}
\begin{highlights}
\item Research highlight 1
\item Research highlight 2
\end{highlights}
\begin{keyword}
....
....
\end{vquote}
\section{Final print}\label{sec:final}
The authors can format their submission to the page size and margins
of their preferred journal. \file{elsarticle} provides four
class options for the same. But it does not mean that using these
options you can emulate the exact page layout of the final print copy.
\lmrgn=3em
\begin{description}
\item [\texttt{1p}:] $1+$ journals with a text area of
384pt $\times$ 562pt or 13.5cm $\times$ 19.75cm or 5.3in $\times$
7.78in, single column style only.
\item [\texttt{3p}:] $3+$ journals with a text area of 468pt
$\times$ 622pt or 16.45cm $\times$ 21.9cm or 6.5in $\times$
8.6in, single column style.
\item [\texttt{twocolumn}:] should be used along with 3p option if the
journal is $3+$ with the same text area as above, but double column
style.
\item [\texttt{5p}:] $5+$ with text area of 522pt $\times$
682pt or 18.35cm $\times$ 24cm or 7.22in $\times$ 9.45in,
double column style only.
\end{description}
Following pages have the clippings of different parts of
the title page of different journal models typeset in final
format.
Model $1+$ and $3+$ will have the same look and
feel in the typeset copy when presented in this document. That is
also the case with the double column $3+$ and $5+$ journal article
pages. The only difference will be wider text width of
higher models. Here are the specimen single and double column journal
pages.
\begin{comment}
\begin{center}
\hypertarget{bsc}{}
\hyperlink{sc}{
{\bf [Specimen single column article -- Click here]}
}
\hypertarget{bsc}{}
\hyperlink{dc}{
{\bf [Specimen double column article -- Click here]}
}
\end{center}
\end{comment}
\vspace*{-.5pc}
\enlargethispage*{\baselineskip}
\src{}\hypertarget{sc}{}
\deforange{blue!70}
\hyperlink{bsc}{\includeclip{1}{88 120 514 724}{elstest-1p.pdf}}
\deforange{orange}
\src{}\hypertarget{dc}{}
\deforange{blue!70}
\hyperlink{bsc}{\includeclip{1}{27 61 562 758}{elstest-5p.pdf}}
\deforange{orange}
~\hfill $\Box$
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,851
|
\section{Introduction}
Large public facilities, such as airports, host thousands of individuals who are observed by thousands of security cameras, that generate an overwhelming quantity of surveillance footage. Sufficiently accurate person re-identification (ReID) can greatly improve situational awareness by allowing operators to track people by appearance across the entire space. This ability could be used to find a lost child, quickly determine where a suspect has been, or help understand how people make use of the facility.
As the number of potential applications have become apparent, ReID has received a great deal of attention in the literature \cite{Liu16,Liu17,Varior16,Karanam16,zheng2016person}. In recent years, methods based on deep convolutional neural networks (CNNs) \cite{ZhangXHL17, Zhong217, HermansBL17} have shown very promising results on some large datasets such as \textit{Market1501}{} \cite{market1501Dataset}. However, existing studies, including review papers \cite{zheng2016person,Karanam16}, do not fully evaluate some of the more practical needs of a person re-identification system, such as scene-independence and input resolution.
A practical, working ReID system should not depend on labeled training data from its deployed environment. Manually collecting labeled correspondences for each camera at each prospective deployment site is too complex and expensive to be feasible. Even if some labeled data \emph{could} be collected, it probably would not capture the evolution of viewing conditions over the course of a day, let alone the years that a system would operate. Researchers understandably do not report `person-dependent' ReID accuracy where the test images are of the same individuals as the training images; it would be unrealistic to expect the system to have prior labeled data for each individual encountered. However, all promising deep ReID papers only report the \emph{scene-dependent} accuracy where test data is from the same environment as the training data \cite{ZhangXHL17, Zhong217, HermansBL17} (fig.~\ref{fig:depvsindep}). The current literature almost never evaluates the more realistic \emph{scene-independent} scenario where the training and test sets have no overlap in view, environment, or subjects. Even the few works \cite{crossData2015} that look at scene-dependent scenario do not report baseline method performances.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\linewidth]{DepVsIndep.png}
\caption{Scene-dependent vs scene-independent person re-identification scenarios. In the well-studied scene-dependent scenario \cite{zheng2016person}, the training and test sets share camera views but not individuals. The scene-independent scenario is more pracitcal but also more challenging: there is no overlap in camera view or individuals between training and testing data. Here, Scene 1 has images from the \textit{Market1501}{} dataset \cite{market1501Dataset} while Scene 2 images come from the \textit{Airport} dataset\cite{Karanam16}. \label{fig:depvsindep}}
\end{center}
\end{figure}
There are two complementary approaches to dealing with the lack of labeled data at the deployment site: achieving scene-independence by training on a dataset of sufficient size and variety, and achieving scene-independence by using unlabeled data to adapt to the deployed scene. If a large and truly representative dataset is available, then methods trained on it could potentially work on any new scene. However, no such dataset exists at the moment and Geng \emph{et al.} \cite{GengWXT16} argue that creating such a dataset would be too costly. Accordingly, there is a focus on domain adaptation techniques using unlabeled data \cite{Zhou217,FanZY17,Deng17} from a deployment environment. Unsupervised domain adaptation is an important area of research that promises improved accuracy, but comes at the expense of a much more complex deployment model. Even though such domain transfer approaches can benefit from using a scene-independent network as a starting point, the accuracy of baseline methods for scene-independent person ReID is not fully evaluated \cite{Zhou217,FanZY17,Deng17}.
Surveillance systems often rely on views from wide-angle cameras with many people in the scene, some of whom can be quite small in terms of pixels. ReID methods are rarely evaluated at the lower resolutions at which people can be reliably detected \cite{pedDetectionEval}. However, a deployed ReID system that ignores low-resolution images of people may miss a large number of useful matches. For practical surveillance applications, it is important to evaluate ReID methods across a range of input resolutions.
In surveillance, there are two distinctive uses of person ReID: finding a person across all cameras in a facility (across-camera person ReID) and finding all appearances of a person in the same camera view (within-camera person ReID). Existing literature has focused on across-camera ReID \cite{zheng2016person,Karanam16} since the general problem is seen as more challenging. However, within-camera ReID across large time gaps can be challenging due to changes in lighting, person pose, and clothing (e.g. removing a jacket). Moreover, within-camera ReID has many surveillance applications, such as commercial and residential lobby cameras where timing of individual entrances and exits would be valuable. One factor in the lower prominence of within-camera ReID may be lack of existing datasets.
In this paper we address the lack of scene-independent person ReID evaluation and within-camera person ReID evaluation. We use baseline deep CNN person ReID methods to investigate the current state of scene-independent person ReID. We present a mechanism for creating a scene-independent testing scenario using multiple existing person ReID datasets. Using these combined datasets, we evaluate scene-independent person ReID and compare it to baseline scene-dependent person ReID, state-of-the-art scene-dependent person ReID results, and state-of-the-art domain adaptation methods. Furthermore, we investigate the effects of input resolution and deep CNN network architecture on both scene-dependent and scene-independent person ReID.
Finally, we introduce a new person re-identification dataset collected during a full working day at the entrance of an office building. We use this dataset to compare the difficulty of the within-camera and across-camera modes of scene-independent ReID.
\section{Evaluation Setup}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.95\linewidth]{IDNetwork.png}
\caption{ReID using Identification Networks. Deep CNNs are trained to identify people by setting each individual in the training data as a different class. The trained network, without the last fully connected layer, is then used at test time to compute features for two different images. A normalized Euclidean distance is then used to compare the features. \label{fig:idnetwork}}
\end{center}
\end{figure}
There are several ways to use deep convolutional neural networks (CNN) for person re-identification. Perhaps the simplest and most straightforward method is to train an identification network (fig.~\ref{fig:idnetwork}) and reuse the network up to the final feature layer to generate feature vectors at test time. That is, we train a network for classification, where each class is a unique person in the training set. Once trained, the layers associated with the final classification are dropped and the remaining network is used at test time to compute a feature vector from an input person image. The difference in person appearance between images can be quickly estimated from the corresponding feature vectors using a distance metric (usually normalized L2 distance). This identification network approach \cite{zheng2016person} is the most common and does not require formation of image pairs as in Siamese networks \cite{Zheng317} or triplet loss networks \cite{Liu16}. It also produces a compact feature vector which allows for larger-scale storage and appearance matching than the cross-difference method \cite{ahmed2015} that compares large feature sets within the network. As such, we will use this method for training our person re-identification networks.
We use the most broadly-accepted method (an identification network approach with normalized L2 distance) to investigate key aspects of person re-identification, as outlined below.
\subsection{Scene Independence}
\label{sec:SceneIndependenceTest}
Our main goal is to perform a baseline evaluation of scene-independent person re-identification. Since no study has looked at the baseline performance of this scenario, we first describe how we arrange a scene-independent person re-identification evaluation using the available datasets.
For scene-independent person ReID, there must be no overlap in camera view, environment, or people between the training and testing sets (fig.~\ref{fig:depvsindep}). This is not achievable using any single existing ReID dataset. However, using several datasets, we can achieve scene independence. We fix one dataset (\textit{Market1501}{}) as our testing dataset and train on a completely different dataset which has no overlap with the testing dataset. To allow for the best possible outcome, the training set should contain a multitude of scenes, environments, and people. This can be achieved by combining several person ReID datasets together into our training set (\LargeTrainSize{4}). In contrast, to test the performance of scene-dependent person ReID, we use the typical training (\textit{Market1501Train}{}) and testing (\textit{Market1501Test}{}) subsets of \cite{market1501Dataset}.
Our main experiment will test scene-independent person re-identification by using \LargeTrainSize{4} to train our network and \textit{Market1501Test}{} to test our network. We will compare this to scene-dependent person re-identification by training the same network on the \textit{Market1501Train}{} dataset and test on the \textit{Market1501Test}{} set. Details of how we form \LargeTrainSize{4}, \textit{Market1501Train}{}, and \textit{Market1501Test}{} sets can be found in section \ref{sec:datasets}.
\subsection{Within-camera vs Across-camera Re-Identification}
While most papers investigating person ReID focus on matching a pedestrian from one camera view to a different camera view \cite{market1501Dataset,zheng2016person} (across-camera ReID), no papers have looked at the problem of person re-identification in the same camera view but at different times (within-camera ReID). Within-camera ReID is arguably a simpler task as the camera geometry and camera parameters remain the same. However, environmental factors such as lighting, pedestrian pose, and clothing configuration can change. From an industry standpoint, this is still a desirable problem to solve for a variety of tasks. For example, given a video of a person entering a building, find when they left the building (assuming one entrance and exit, which is the case for many smaller stores and office buildings).
Perhaps due to within-camera ReID being a simpler subproblem of across-camera ReID, or possibly because of the lack of a good dataset, the current literature does not directly evaluate within-camera ReID. In the case of scene-independent person re-identification, we would like to know if within-camera ReID performs better than across-camera ReID. To this end we introduce the \textit{OfficeBuilding}{} dataset.
The \textit{OfficeBuilding}{} dataset contains two camera views (an outdoor and indoor view) of people entering and leaving a building during a single workday. Images from the outdoor camera are used for the gallery and the within-camera query. To ensure the within-camera query and the gallery do not have frames of an individual from the same time, we take all frames from a person entering the building for the first time (typically early morning) as the within-camera query image. People exiting the building and anyone seen re-entering the building on the outdoor camera are used as part of the gallery. All individuals seen in the indoor camera view are used for the across-camera view query image. Figure~\ref{fig:officeBuildingDataset} shows sample images from the \textit{OfficeBuilding}{} dataset.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\linewidth]{OfficeBuildingDataset.png}
\caption{\textit{OfficeBuilding}{} Dataset. There are query images from an indoor camera (for across-camera test) and outdoor camera (for within-camera test). The gallery is composed of people in the outdoor camera, excluding the query periods when a person first approached and entered the building. Since the data was collected during an entire day, people are sometimes compared front-view to back-view as well as with and without a jacket or backpack. \label{fig:officeBuildingDataset}}
\end{center}
\end{figure}
\subsection{Network Architecture}
We wish to determine if a particular baseline CNN architecture is best suited for person re-identification. To this end, we test MobileNet \cite{MobileNetNetwork}, ResNet-50 \cite{ResNet50Network}, VGG19 \cite{VGG19Network}, and SqueezeNet \cite{SqueezeNetNetwork}. Another commonly used network architecture is Inception-v3 \cite{Inceptionv3Network}. However, due to the network depth and the amount of pooling employed, we are unable to test this network architecture at different input sizes without removing a significant number of layers. For some of the more in-depth experiment such as dataset size (described in section \ref{sec:DatasetSize}), we test only the best performing network architectures.
\subsection{Input Size and Aspect Ratio}
Many existing works use images at 1:1 aspect ratio with resolutions of 224x224 or 227x227 \cite{zheng2016person}. This is most likely because the network architectures used have pre-trained weights using ImageNet data where images are rescaled and cropped to 224x224 or 227x227 resolution (224 being a practical number that is easy to divide and perform convolutions on). There are two important points to consider: 1) ImageNet resizing and cropping generally preserve aspect ratio \cite{VGG19Network} and 2) network architectures that use global pooling are somewhat robust to input image size, so long as the input size is big enough for all the pooling layers. Given that pedestrian detectors were shown to be effective for person heights as low as 50 pixels \cite{pedDetectionEval}, we investigate input sizes of 64x32, 128x64, and 256x128.
For comparisons between 2:1 and 1:1 aspect ratios we stretch the 256x128 input size to 256x256. We choose 256x256 for comparison because it is close in size to the typically used 224x224 resolution and is simply a stretch from our 256x128 resolution. If we used 224x224 resolution to compare to 256x128 resolution, we could not isolate the influence of aspect ratio from the scale change.
\subsection{Dataset Size}
\label{sec:DatasetSize}
For scene-independent person ReID, the training set should be large and varied enough to span the range of conditions likely to appear in the test set. To test how much scene-independent ReID accuracy depends on size and composition of the training set, we consider a series of six progressively larger training sets, \LargeTrainSize{1\ldots 6}. The details of \LargeTrainSize{1\ldots 6} can be found in section \ref{sec:datasets}.
\subsection{Network Initialization}
All existing approaches to person ReID \cite{zheng2016person} initialize the network using weights trained on ImageNet. However, no papers have tested whether random initialization can achieve similar results. In this study, we compare training these networks initialized both randomly and with ImageNet weights.
\section{Evaluation Implementation}
\subsection{Datasets}
\label{sec:datasets}
We select a number of commonly used ReID datasets for use in our evaluation procedures. The primary datasets used in this study are: \textit{Market1501}{} \cite{market1501Dataset}, \textit{Airport} \cite{Karanam16}, \textit{DukeMTMC4ReID} \cite{dukemtmc4reidDataset}, \textit{LSPS} \cite{LargeScalePersonSearchDataset}, and \textit{CUHK03} \cite{CUHK03Dataset}.
The \textit{Market1501}{} dataset consists of 32,668 images and 1,501 identities across six cameras.
The \textit{Airport} dataset consists of 39,902 images and 9,651 identities across six cameras.
The \textit{DukeMTMC4ReID} dataset consists of 46,261 images and 1,852 identities across eight cameras.
The \textit{LSPS} dataset consists of 18,184 images and 8,432 identities.
The \textit{CUHK03} dataset consists of 13,164 images and 1,360 identities across six cameras.
Our newly introduced dataset, \textit{OfficeBuilding}{}, is also used in testing. This dataset includes 37 identities and 1,279 images captured across two cameras. It contains 21 identities and 210 images which can be used for the across-camera query, as well as 34 identities and 340 images which can be used for the within-camera scenario.
We split \textit{Market1501}{} into a training \textit{Market1501Train}{} and test \textit{Market1501Test}{} set. The split is the standard $751/750$ split used in other published works \cite{market1501Dataset}.
For all datasets we keep $10$ images per person and ensure images are selected from all camera views the person appears in. If a person appears in more than $10$ camera views, one image per camera view is kept. This balances the dataset as some datasets have a large number of images for each person compared to others. This sampling approach is motivated by the work of Xiao et al. \cite{Xiao2016CVPR}, who hypothesize that failing to balance the data from multiple sources can lead to overfitting from smaller datasets. For each dataset, $90\%$ of the unique person IDs are used for ReID training, while the remaining $10\%$ are withheld for testing. Since we are training an ID network, each unique person is treated as a single class. For each person in the training sets, the images are split into training and validation sets at a $70-30$ random split.
In particular, we seek to evaluate how the size and composition of the training set affects the scene-independent accuracy of a person ReID network. To this end, we have created a number of composite datasets. Table~\ref{TAB:DatasetComposition} gives an outline of the dataset groupings used and their composite names.
\LargeTrainSize{4}, which comprises \textit{DukeMTMC4ReID}, \textit{Airport}, \textit{CUHK03}, and \textit{LSPS}, is our primary `large set' used in training models. \LargeTrainSize{5} and \LargeTrainSize{6} are marked as using `Full Data', which indicates these use the entire datasets (no train/test split) with $10$ images per person. In order to have a more thorough test of the effects of added data, \LargeTrainSize{6} contains additional datasets, including: \textit{3DPES}\cite{3DPESDataset}, \textit{CUHK01}\cite{CUHK01Dataset}, \textit{iLIDS-VID}\cite{ilidsDataset}, \textit{PRID 2011}\cite{PRID2011Dataset}, \textit{Shinpuhkan2014}\cite{Shinpuhkan2014Dataset}, \textit{underGround Re-Identification (GRID)}\cite{GRIDDataset}, \textit{VIPeR}\cite{ViperDataset} and \textit{WARD}\cite{WARDDataset}.
\begin{table}[t]
\caption{Contents of Composite Datasets}
\label{TAB:DatasetComposition}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Training Set}&\multirow{2}{*}{Full$^*$}&\multirow{1}{*}{\textit{DukeMTMC4ReID}} &\multirow{1}{*}{\textit{Airport}}&\multirow{1}{*}{\textit{CUHK03}}&\multirow{1}{*}{\textit{LSPS}}&\multirow{2}{*}{Extra$^{**}$} \\
&&\cite{dukemtmc4reidDataset}&\cite{Karanam16}&\cite{CUHK03Dataset}&\cite{LargeScalePersonSearchDataset}& \\
\hline
\LargeTrainSize{1} & N & Y & & & & \\
\hline
\LargeTrainSize{2} & N & Y & Y & & & \\
\hline
\LargeTrainSize{3} & N & Y & Y & Y & & \\
\hline
\LargeTrainSize{4} & N & Y & Y & Y & Y & \\
\hline
\LargeTrainSize{5} & Y & Y & Y & Y & Y & \\
\hline
\LargeTrainSize{6} & Y & Y & Y & Y & Y & Y \\
\hline
\end{tabular}
}
\end{center}
$^*$ Full Data does not reserve any individuals for testing\\
$^{**}$ Extra=\{\textit{3DPES}\cite{3DPESDataset}, \textit{CUHK01}\cite{CUHK01Dataset}, \textit{iLIDS-VID}\cite{ilidsDataset}, \textit{PRID 2011}\cite{PRID2011Dataset}, \textit{Shinpuhkan2014}\cite{Shinpuhkan2014Dataset}, \textit{GRID}\cite{GRIDDataset}, \textit{VIPeR}\cite{ViperDataset}, \textit{WARD}\cite{WARDDataset}\}
\vspace*{-2mm}
\end{table}
\subsection{Architecture}
\label{sec:arch}
We use the following networks in our evaluation: MobileNet, ResNet50, VGG19, and SqueezeNet. At training time, for all networks except SqueezeNet, we remove the final fully-connected layer and replace it with a new fully-connected layer with output dimension based on the number of classes (individuals) in our training data. At test time, the output of the layer before the last fully-connected layer is used as our feature. We use normalized L2 distance as our distance metric.
SqueezeNet does not use a fully-connected layer for classification. Instead, it uses a 1x1 convolution layer with as many channels as there are classes, followed by global pooling before the softmax layer. During training, we set the number of channels in this 1x1 convolution layer to the number of classes (individuals) in our training data. At test time, the output of the layer before the 1x1 convolution layer is fed into a global average pooling layer and the output of the pooling layer is used as our feature.
Since we minimized structural alterations of existing network architectures for obtaining features, we end up with different feature dimensionality for each architecture. The dimensions used are: MobileNet-1024D, ResNet50-2048D, VGG19-4096D, and SqueezeNet-512D.
\subsection{Solver And Data Augmentation}
When training the network, we used stochastic gradient descent (SGD) as our optimizer with a learning rate of 0.001, and default decay/momentum parameters. A batch size of 6 was used for all training. Unless otherwise stated, all networks were trained for 160 epochs, where each epoch is one iteration through the entire training data.
Since our focus is a baseline evaluation, we do not introduce dropout or other fine-tuning techniques. Furthermore, we keep data augmentation simple and use only horizontal flips and random cropping.
\subsection{Hardware/Software Details}
All networks are trained and evaluated using TensorFlow (1.3.0) and Keras (2.0.9) on machines with NVIDIA GeForce GPUs (GTX 1060 and 1080).
\section{Results}
For the purposes of this evaluation, we limit our investigation to single query mode. We report the typical Rank-1 and Rank-5 accuracy based on the cumulative match curve (CMC) and the mean average precision (mAP) \cite{ViperDataset}. For a consistent evaluation, we use the code provided by \cite{zheng2016person,zhengLatestResult}.
\subsection{Architecture and Input Size}
\begin{table*}[t]
\caption{\textit{Market1501Test}{} Results}
\label{TAB:MainResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Input Size}&\multirow{2}{*}{Training Set}&\multicolumn{3}{c|}{MobileNet}&\multicolumn{3}{c|}{ResNet50}&\multicolumn{3}{c|}{VGG19}&\multicolumn{3}{c|}{SqueezeNet} \\
\cline{3-14}
&&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP \\
\hline
\hline
256x128 & \textit{Market1501Train} & \textbf{75.6\%} & 88.9\% & 0.49 & 72.5\% & 87.9\% & 0.48 & 65.4\% & 81.9\% & 0.38 & 59.9\% & 77.5\% & 0.33 \\
\hline
128x64 & \textit{Market1501Train} & 73.6\% & 88.8\% & 0.45 & \textbf{74.2\%} & 87.6\% & 0.48 & 66.0\% & 82.3\% & 0.40 & 58.9\% & 77.8\% & 0.34 \\
\hline
64x32 & \textit{Market1501Train} & 68.1\% & 85.4\% & 0.41 & \textbf{69.1\%} & 86.2\% & 0.44 & 58.5\% & 78.1\% & 0.34 & 49.2\% & 71.4\% & 0.30 \\
\hline
\hline
256x128 & \LargeTrainSize{4} & \textbf{46.9\%} & 64.5\% & 0.21 & 43.5\% & 61.5\% & 0.19 & 32.2\% & 50.2\% & 0.12 & 29.6\% & 46.8\% & 0.11 \\
\hline
128x64 & \LargeTrainSize{4} & \textbf{49.3\%} & 68.2\% & 0.23 & 45.6\% & 64.2\% & 0.21 & 36.0\% & 54.5\% & 0.15 & 31.0\% & 49.4\% & 0.12 \\
\hline
64x32 & \LargeTrainSize{4} & 40.7\% & 60.1\% & 0.18 & \textbf{44.1\%} & 62.4\% & 0.19 & 32.3\% & 50.7\% & 0.13 & 26.6\% & 44.9\% & 0.10 \\
\hline
\end{tabular}
}
\end{center}
\vspace*{-3mm}
\end{table*}
Table \ref{TAB:MainResults} summarizes our main finding. We can see that for the scene-dependent case, MobileNet and ResNet50 have the best performance of the networks tested by a fair margin ($\sim30\%$). MobileNet and ResNet50 also have the best performance for the scene-independent case where the training set is different than the test set
With regards to input size, 128x64 has fairly consistent performance across all networks. 256x128 resolution has slightly better performance in some cases, such as MobileNet trained on \textit{Market1501Train}{} data. In this scene-dependent scenario, the 256x128 resolution has $2\%$ better Rank-1 performance than the 128x64 resolution. However, compared to the 256x128, the 128x64 input resolution has $2.4\%$ better performance on MobileNet for the more challenging scene-independent case. The 64x32 resolution has the lowest accuracy, but it could still generate useful results for people who are more distant from the camera.
Overall, MobileNet with the 128x64 input resolution has the best performance for the scene-independent test scenario, followed by ResNet50 with the same resolution. As a result, the remaining investigations focus on the MobileNet and ResNet50 network architectures with an input size of 128x64.
\begin{figure*}[!tb]
\begin{center}
\subfigure[]{
\includegraphics[width=0.4\linewidth]{CMCOnMarket1501Test.png} \label{subfig:marketcmc}
}
\subfigure[]{
\includegraphics[width=0.4\linewidth]{CMCOfficeBuildingAcrossCamera.png} \label{subfig:officebuildingcmc}
}
\caption{Cumulative Matching Curve (CMC) on the \textit{Market1501Test}{} and \textit{OfficeBuilding}{} datasets using MobileNet at 128x64 input resolution. We see that scene-dependent scenario has much better performance (trained on \textit{Market1501Train}{} and tested on \textit{Market1501Test}{}) than scene-independent scenario. For the scene-independent scenario, performance is controlled by the training dataset size (\LargeTrainSize{1} is smaller than \LargeTrainSize{6}). \label{fig:cmc}}
\end{center}
\end{figure*}
\subsection{Scene Independence and Dataset Size}
\label{SEC:ResultsSceneIndDatasetSize}
From Table \ref{TAB:MainResults} and fig.~\ref{subfig:marketcmc} we can see that scene-independent person re-identification results are consistently much lower than scene-dependent person re-identification regardless of the network or the input size used. There is a staggering $\sim30\%$ Rank-1 accuracy difference. This is to be expected because even though \LargeTrainSize{4} has combined several datasets together, it still lacks the diversity needed to represent the \textit{Market1501Test}{} dataset.
To see the effect of training set size on the scene-independent re-identification scenario, we trained MobileNet and ResNet50 across a range of training set sizes, with the results reported in Table~\ref{TAB:DatasetSizeResults} and fig.~\ref{fig:cmc}. As the training dataset size increases, the accuracy of both the MobileNet and ResNet50 models increase. We can also see that the ResNet50 model is consistently lagging the MobileNet performance by $\sim 2 - 5 \%$. Overall, using our largest training set and MobileNet, we can achieve a Rank-1 accuracy of $55.1\%$ for the scene-independent person re-identification scenario. Though this is still $\sim20\%$ below the results in the scene-dependent scenario, these results are encouraging. The trend suggests that if we can build a large enough training set, we can train a model that can handle scene-independent person re-identification.
\begin{table}[t]
\caption{Results For Different Dataset Size}
\label{TAB:DatasetSizeResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|r|r|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Training Set}&\multicolumn{1}{c|}{\#} &\multicolumn{1}{c|}{\#}&\multicolumn{3}{c|}{MobileNet}&\multicolumn{3}{c|}{ResNet50} \\
\cline{4-9}
&People&Images&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP \\
\hline
\LargeTrainSize{1} & 1,272 & 7,979 & \textbf{38.7\%} & 57.5\% & 0.15 & 36.6\% & 55.2\% & 0.14 \\
\hline
\LargeTrainSize{2} & 2,515 & 12,851 & \textbf{41.6\%} & 61.2\% & 0.16 & 38.2\% & 56.1\% & 0.15 \\
\hline
\LargeTrainSize{3} & 3,836 & 21,729 & \textbf{47.6\%} & 66.5\% & 0.21 & 42.3\% & 60.6\% & 0.19 \\
\hline
\LargeTrainSize{4} & 14,577 & 46,423 & \textbf{49.3\%} & 68.2\% & 0.23 & 45.6\% & 64.2\% & 0.21 \\
\hline
\LargeTrainSize{5} & 16,195 & 68,763 & \textbf{54.8\%} & 71.6\% & 0.27 & 51.9\% & 69.4\% & 0.25 \\
\hline
\LargeTrainSize{6} & 19,092 & 84,080 & \textbf{55.1\%} & 72.5\% & 0.27 & 52.2\% & 69.1\% & 0.26 \\
\hline
\end{tabular}
}
\end{center}
\end{table}
\subsection{Comparison To The State-Of-The-Art}
Overall, our best results are for MobileNet with an input size of 128x64. For this configuration, the Rank-1 accuracy is $73.6\%$ for the scene-dependent scenario (Table \ref{TAB:MainResults}) and $55.1\%$ for the scene-independent scenario (Table \ref{TAB:DatasetSizeResults}). Please note that the addition of dropout, data augmentation such as rotations, etc. was not the focus of this study and could potentially be used to increase these accuracies \cite{zhengLatestResult}. Regardless, if we compare these baseline results to published methods (Table \ref{TAB:StateOfArtResults}), we find that they fall in the middle of the pack. The scene-independent Rank-1 result of $55.1\%$ even beats some of the scene-dependent results as recent as 2016.
\begin{table}[t]
\caption{State-of-the-Art Comparisons for Scene-Dependent ReID}
\label{TAB:StateOfArtResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Alg. & \cite{Wu16} & \cite{Sh16} & \cite{Liu16} & \cite{Jose16} & \cite{Karanam16} & \cite{Martinel16} & \cite{Wu216} & \cite{Liu17} & \cite{Chen16} & \cite{zhang16} \\
\hline
Rank-1 & 37.2 & 39.4 & 45.1 & 45.2 & 46.5 & 47.9 & 48.2 & 48.2 & 51.9 & 55.4 \\
\hline
\hline
Alg. & \cite{Ustinova16} & \cite{Varior16} & \cite{Varior216} & \cite{Ustinova17} & \cite{Zhou17} & \cite{Chen17} & \textbf{RN L} & \textbf{MN M} & \cite{Lin17} & \cite{Barbosa17} \\
\hline
Rank-1 & 59.5 & 61.6 & 65.9 & 66.4 & 70.7 & 71.8 & 72.5 & 73.6 & 73.8 & 73.9 \\
\hline
\hline
Alg. & \textbf{RN M} & \textbf{MN L} & \cite{Zheng17} & \cite{zhao17} & \cite{Zhong17} & \cite{Zheng217} & \cite{Zheng317} & \cite{Li17} & \cite{Zhao217} & \cite{Bai17} \\
\hline
Rank-1 & 74.2 & 75.6 & 76.8 & 76.9 & 77.1 & 79.3 & 79.5 & 80.3 & 81.0 & 82.2 \\
\hline
\hline
Alg. & \cite{Yu17} & \cite{Sun17} & \cite{Zheng417} & \cite{GengWXT16} & \cite{Li217} & \cite{Su17} & \cite{Lin217} & \cite{HermansBL17} & \cite{Zhong217} & \cite{ZhangXHL17} \\
\hline
Rank-1 & 82.3 & 82.3 & 82.8 & 83.7 & 83.9 & 84.1 & 84.3 & 84.9 & 87.1 & 87.7 \\
\hline
\end{tabular}
}
\vspace{0.2cm}
\textbf{MN} MobileNet, \textbf{RN} ResNet50 \\
\textbf{M} 128x64 resolution, \textbf{L} 256x128 resolution
\end{center}
\end{table}
Interestingly, our scene-independent result is very competitive with unsupervised domain adaptation work (Table \ref{TAB:SOARUnsupDomainTransfer}). The unsupervised domain adaptation work is similar to our result in that it uses labeled data from other datasets. However, in addition to the scene-independent training data, it uses scene-dependent training data without ground truth annotation. That is, domain adaptation systems use the \textit{Market1501Train}{} dataset without ground truth label in their training process whereas our result does not use \textit{Market1501Train}{} data at all. The competitive nature of our result, even though we do not use scene-dependent training data, is explained by the large amount of training data we used by combining all other person re-identification datasets into a single dataset. Again, this shows that a large enough training dataset from multiple environments should be able to do scene-independent person re-identification without needing domain adaptation.
\begin{table}[t]
\caption{Unsupervised Domain Transfer}
\label{TAB:SOARUnsupDomainTransfer}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\cline{2-5}
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{Scene Dependent} & Scene Independent \\
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{\textit{Uses Unlabeled}} & \textit{Not Using} \\
\multicolumn{1}{c|}{} & \multicolumn{3}{c|}{\textit{Market1501Train}{}} & \textit{Market1501Train}{} \\
\hline
Alg. & \hspace{0.08cm} \cite{Zhou217} \hspace{0.08cm} & \hspace{0.08cm} \cite{FanZY17} \hspace{0.08cm} & \hspace{0.08cm} \cite{Deng17} \hspace{0.08cm} & \textbf{MobileNet 128x64} \\
\hline
Rank-1 & 40.9 & 44.7 & 57.7 & 55.1 \\
\hline
\end{tabular}
}
\end{center}
\vspace*{-5mm}
\end{table}
\subsection{Network Initialization}
The results thus far have been for networks initialized with ImageNet weights. This is a standard technique used in many different applications because it has been shown to boost performance in comparison to random initialization of the network. Table \ref{TAB:InitTypeResults} summarizes our result for random initialization vs ImageNet initialization. ImageNet initialization gives a boost of $10\%$ or more in comparison to random initialization for both scene-independent and scene-dependent scenarios.
The benefit of ImageNet initialization is a mixed blessing. While it gives a huge boost in performance, it discourages making any changes to existing network architectures that would require retraining on ImageNet data.
\begin{table}[t]
\caption{Results For Different Initialization}
\label{TAB:InitTypeResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Training Set}&\multirow{2}{*}{Init. Type} &\multicolumn{3}{c|}{MobileNet}&\multicolumn{3}{c|}{ResNet50} \\
\cline{3-8}
&&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP \\
\hline
\textit{Market1501Train} & ImageNet & 73.6\% & 88.8\% & 0.45 & 74.2\% & 87.6\% & 0.48 \\
\hline
\textit{Market1501Train} & Rand & 62.9\% & 82.3\% & 0.37 & 64.2\% & 81.9\% & 0.38 \\
\hline
\LargeTrainSize{4} & ImageNet & 49.3\% & 68.2\% & 0.23 & 45.6\% & 64.2\% & 0.21 \\
\hline
\LargeTrainSize{4} & Rand & 33.5\% & 52.8\% & 0.14 & 33.9\% & 53.3\% & 0.13 \\
\hline
\end{tabular}
}
\end{center}
\vspace*{-5mm}
\end{table}
\subsection{Aspect Ratio}
We look at the result of using a 2:1 versus a 1:1 aspect ratio input size in Table \ref{TAB:AspectRatioResults}. When comparing the 256x128 person size to the stretched version at 256x256, we see a lower performance for the stretched version. In particular, ResNet50 shows a nearly $7\%$ drop in performance. Since the 256x256 input also requires more computation than 256x128 input, the 2:1 aspect ratio is preferred. These results are consistent with the findings of \cite{YaoZZLT17}, who performed a similar test.
\begin{table}[t]
\caption{Results for Different Aspect Ratio}
\label{TAB:AspectRatioResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Input Size}&\multicolumn{3}{c|}{MobileNet}&\multicolumn{3}{c|}{ResNet50} \\
\cline{2-7}
&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP \\
\hline
256x128 & 75.6\% & 88.9\% & 0.49 & 72.5\% & 87.9\% & 0.48 \\
\hline
256x256 & 73.2\% & 87.5\% & 0.48 & 65.4\% & 83.3\% & 0.40 \\
\hline
\end{tabular}
}
\end{center}
\vspace*{-5mm}
\end{table}
\subsection{Within-camera vs Across-camera Re-identification}
Using our \textit{OfficeBuilding}{} dataset, we test the accuracy of scene-independent person re-identification for within-camera and across-camera scenarios. We report the results using MobileNet trained on the \LargeTrainSize{6} and \textit{Market1501Train}{} in Table \ref{TAB:OfficeBuildingResults}. Some figures of the results are presented in fig.~\ref{fig:officeBuildingSampleResults}. Section \ref{SEC:ResultsSceneIndDatasetSize} illustrates that, once again, training on a larger, more varied dataset boosts performance. For the \textit{OfficeBuilding}{} dataset, there is a boost in Rank-1 accuracy of $\sim10\%$ when using a larger training set for both within and across-camera person re-identification.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=\linewidth]{SampleResults.png}
\caption{Sample scene independent search results on the \textit{OfficeBuilding}{} dataset. Left: across-camera search results. Right: within-camera search result. \label{fig:officeBuildingSampleResults}}
\end{center}
\end{figure}
When comparing within-camera to across-camera person re-identification, we find that there is a $\sim5\%$ difference. Meaning that matching across-camera is not much more difficult in this instance than matching within-camera. This is perhaps because individuals are matched across large time gaps, typically from morning to evening. Furthermore, we are searching using a person entering the building as the probe and the gallery contains mostly people exiting the building. As a result, we are mostly matching front view to rear view.
The \textit{OfficeBuilding}{} dataset also contains some very challenging scenarios. For example, there are cases in the dataset where a person is not wearing a coat when entering the building but wearing a coat when exiting fig.~\ref{fig:officeBuildingCoat}. In another example, there are overlapping individuals but the ground truth annotation considers the person in the foreground as the true label fig.~\ref{fig:officeBuildingMultiplePeople}.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=\linewidth]{Coat.png}
\caption{Left: In an outdoor query image, the arriving subject has no coat, and the correct match is not found because the gallery has no image of the subject without a coat. Right: In an indoor query image, the subject is wearing a coat when exiting and the indoor-outdoor match quality is high. \label{fig:officeBuildingCoat}}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=\linewidth]{MultiPeople.png}
\caption{A gallery image containing two people is only annotated as the foreground person (left query). Though it is also a reasonable match for the occluded person (right query), it is counted as incorrect. \label{fig:officeBuildingMultiplePeople}}
\end{center}
\end{figure}
The scene-independent test result on the \textit{Market1501Test}{} dataset ($55.1\%$ from Table \ref{TAB:DatasetSizeResults}) is much lower than the numbers reported on our \textit{OfficeBuilding}{} dataset because our data set is much simpler with only 37 unique individuals. This dataset's primary purpose is for comparison of within-camera and across-camera person re-identification.
\begin{table}[t]
\caption{\textit{OfficeBuilding}{} Results Using MobileNet}
\label{TAB:OfficeBuildingResults}
\begin{center}
\setlength{\tabcolsep}{0.5mm}
\scriptsize{
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Training Set}&\multicolumn{3}{c|}{across-camera}&\multicolumn{3}{c|}{within-camera} \\
\cline{2-7}
&Rank-1&Rank-5&mAP&Rank-1&Rank-5&mAP \\
\hline
\textit{Market1501Train} & 59.1\% & 72.4\% & 0.46 & 64.7\% & 77.1\% & 0.58 \\
\hline
\LargeTrainSize{6} & 70.0\% & 82.9\% & 0.58 & 75.3\% & 84.4\% & 0.63 \\
\hline
\end{tabular}
}
\end{center}
\end{table}
\section{Conclusion}
We performed an extensive evaluation of baseline deep convolution networks for person re-identification and found that on average MobileNet with input resolution of 128x64 is the best network for both scene-dependent and scene-independent person re-identification. Using MobileNet with 128x64 input, we can achieve $73.6\%$ for scene-dependent person re-identification on \textit{Market1501Test}{}, which is near the middle of the range of reported results for this dataset.
Furthermore, MobileNet with 128x64 input resolution can perform well for scene-independent person re-identification when using a sufficiently large training dataset. While the result of the scene-independent test is $18.5\%$ lower than the scene-dependent scenario, it is still very competitive with unsupervised domain adaptation techniques which use unlabeled data from the \textit{Market1501}{} dataset during training.
We introduced a dataset to test scene-independent person re-identification when the probe is from the same camera as the gallery (within-camera person re-identification) and when the probe is from a different camera (across-camera person re-identification). Surprisingly, we find that both within-camera and across-camera results are very close ($\sim5\%$ difference). This indicates that even matching people in the same camera view across large time gaps could be as challenging as matching people across-camera views.
\input{ReIDPaper.bbl}
\end{document}
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Also by Laura Bradford
Portrait of a Sister
THE TOBI TOBIAS MYSTERIES
And Death Goes To . . .
30 Second Death
Death in Advertising
Published by Kensington Publishing Corporation
A DAUGHTER'S TRUTH
LAURA BRADFORD
KENSINGTON BOOKS
www.kensingtonbooks.com
All copyrighted material within is Attributor Protected.
Table of Contents
Also by
Title Page
Copyright Page
Dedication
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 20
Chapter 21
Chapter 22
Chapter 23
Chapter 24
Chapter 25
Eleven Months Later
ACKNOWLEDGMENTS
DISCUSSION QUESTIONS
ABOUT THE AUTHOR
This book is a work of fiction. Names, characters, places, and incidents either are products of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.
KENSINGTON BOOKS are published by
Kensington Publishing Corp.
119 West 40th Street
New York, NY 10018
Copyright © 2019 by Laura Bradford
All rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.
Kensington and the K logo Reg. U.S. Pat. & TM Off.
BOUQUET Reg. U.S. Pat. & TM Off.
ISBN: 978-1-4967-1648-4
ISBN-10: 1-4967-1649-3 (ebook)
Kensington Electronic Edition: June 2019
ISBN-13: 978-1-4967-1648-4
ISBN-10: 1-4967-1648-5
First Kensington Trade Paperback Edition: June 2019
For you, my readers
Dear Readers,
With each new women's fiction novel I write, I strive to immerse you in a journey that feels oddly familiar to your own life. Yes, the backdrop I've chosen is an Amish community in rural Pennsylvania, but the story—the journey—is one I suspect we can all relate to in one way or another.
In my first novel, Portrait of a Sister, that journey was about choices and family and finding one's place in life.
Here, in A Daughter's Truth, it's about those moments in life that come out of nowhere and shake us to the core, leaving us to wonder who we are and how we will keep going. Sometimes, as is the case for Emma Lapp, that big moment comes at the hands of other people. Other times, those moments just happen—an unexpected loss, an illness, an act of nature, etc. Either way, though, when the proverbial smoke clears, we're all left standing at the same crossroad with the same big decision....
Will the moment define me, or will I define the moment?
That's the question Emma must answer. And, at some point in our own lives, that's a question we'll likely have to answer as well.
When you've finished reading A Daughter's Truth, take a look at some of the book club questions I've included in the back. Many of them were things I found myself pondering as Emma took me along on her journey.
Happy Reading!
Laura
Chapter 1
Not for the first time, Emma Lapp glanced over her shoulder, the utter silence of the sparsely graveled road at her heels deafening. On any other day, the mere thought of leaving her sisters to do her chores would fill her with such shame she'd no doubt add their tasks to her own as a way to seek atonement inside her own heart. Then again, on any other day, she would be gathering the eggs and feeding the orphan calf just like always.
But today wasn't just any day. Today was her birthday. Her twenty-second, to be exact. And while she knew better than anyone else what the rest of her day would and wouldn't entail, this part—the part she'd been anticipating since her last birthday—had become her happy little secret.
Lifting her coat-clad shoulders in line with her cheeks, Emma bent her head against the biting winds and hurried her steps, the anticipation for what she'd find waiting atop the sheep-tended grass eliciting a quiet squeal from between her clattering teeth. Unlike her five siblings, Emma's birthday wasn't a day with silly games and laughter. It was, instead, a day of sadness—a day when the air hung heavy across every square inch of the farm from the moment she opened her eyes until her head hit the pillow at night. And while she wanted to believe it would get better one day, twenty-one examples to the contrary told her otherwise.
But this—
She rounded the final bend in the road and stopped, her gaze falling on the weathered gravestones now visible just beyond the fence that ran along the edge of the Fishers' property. There, on the other side of the large oak tree, was the reason for both Mamm's on-going heartache and the unmistakable smile currently making its way across Emma's face.
When she was four . . . five . . . six, it had been this same sight on this same day that had swirled her stomach with the kind of dread that came from knowing.
Knowing Dat would stop the buggy . . .
Knowing she and her brother Jakob would follow behind Mamm and Dat to the second row, third gravestone from the right . . .
Knowing Mamm would look down, fist her hand against her trembling lips, and squeeze her eyes closed around one lone tear . . .
Knowing Dat would soon mutter in anger as their collective gaze fell on the year's latest offering—an offering that would be tossed into an Englisher's trash can on the way to school . . .
It was why, at the age of seven, when she'd asked to walk to school with her friends, Emma told them to go ahead without her, buying her time to stop at the cemetery alone, before Mamm and Dat.
That day, she'd fully intended to throw the trinket away in the hopes of removing the anger, if not the sadness, from her birthday. But the moment she'd seen the miniature picnic basket nestled inside her palm, she'd known she couldn't. Instead, she'd wrapped it inside a cloth napkin and hid it inside her lunch pail.
Later on, after school, she'd relocated the napkin-wrapped secret to the hollow of a pin oak near Miller's Pond. In time, she'd replaced the napkin with a dark blue drawstring bag capable of holding the now fifteen objects inside—objects her mind's eye began inventorying as she approached the cemetery.
* The miniature picnic basket
* The pewter rose
* The snow globe with the tiny skaters inside
* The stuffed horse
* The picture of a dandelion
* The bubble wand
* The narrow slip of torn paper housed in a plastic covering
* The sparkly rock with the heart drawn on it
* The red and black checked napkin
* The plastic covered bridge
* The small, red rubber ball
* The yellow spinny thing on a stick
* The signed baseball she couldn't quite read
* The dried flower with the pale blue and pink ribbons tied around the stem
* The whittled bird
Emma savored the lightness the images afforded against the backdrop of an otherwise dark, lifeless day and quickened her pace. All her life, her birthday had been a day to hurry through in the hope Mamm's pain would somehow be lessened. There was always a cake with a handful of candles, but it was set in front of Emma with little more than a whispered happy birthday. There were presents, but they were always handed to her quietly, without the belly laughs and silly antics that were part of her siblings' birthdays. And when the sun sank low at the end of her day, she, too, was glad it was over.
But this? This stop at the cemetery had become the one part of her birthday she actually looked forward to with anticipation each year. Because even while she knew it was wrong to be drawn to a material object, the very act of guessing what it might be felt more birthday-like than anything she'd ever known.
Sliding her focus to the left, she surveyed the long, winding country road that led farther into Amish country, the lack of buggy traffic in keeping with the hour. Morning was a busy time in the Amish community. It was time to tend to the animals and get about the day's tasks. In the spring, summer, and fall, those tasks entailed work in the fields for those, like Dat, who farmed. In the winter months, as it was now, there were still things that needed tending—fences that needed reworking, manure to be spread in the fields, repairs made to aging structures, and assisting neighbors with the same.
A glance to her right netted the one-room schoolhouse where she'd learned to read and write as a young child, and where three of her younger siblings still went. At the moment, there was no smoke billowing from the school's chimney, but she knew that would change in about an hour when the teacher arrived ahead of her students.
Seeing nothing in either direction to impede her adventure, Emma stepped around the simple wooden fence separating the cemetery from the Grabers' farm to the south and the Fishers' farm to the north, eyed the gravestones in front of her, and, after a single deep breath, made her way over to the second row. She didn't need to read the names on the markers she passed. She'd memorized them during her visits there with her parents, when, as a new reader, she read everything she could.
* Isaac Yoder . . .
* Ruth Schrock . . .
* Ruth's twin brother, Samuel . . .
* Abram King . . .
* And, finally, Ruby Stoltzfus, Mamm's younger sister and the aunt Emma had never met
Instinctively, she took in the date of death in relation to the date of birth even though she already knew the answer.
When she was little, and she'd come here with her family, the numbers on the markers she passed hadn't really registered. But as she'd developed math skills and a perspective on life over the next few years, she'd begun to truly understand the reason behind Mamm's grief. Eighteen-year-olds weren't supposed to die. They just weren't. And when she, herself, had inched closer to—and eventually surpassed—the age her aunt had been at death, the whole occurrence took on an even more tragic undertone.
Shaking off the sadness she felt lapping at the edges of her day's one joy, Emma dropped her gaze from the simple lettering to the stark winter earth peeking out from the dormant grass below. An initial skim of the usual places where the various objects had been left in the past turned up nothing and, for a brief moment, her heart sank. But a second, more thorough look netted a brief flash of light off to the left.
Sure enough, as she moved in closer, she saw it, her answering intake of air bringing an end to a neighboring bird's desperate hunt for food in between and around the next row's grave markers. There, wrapped around a medium-sized rock, was a—
"Levi said he saw you out here!"
Whirling around, Emma turned in time to see her best friend waving at her from the other side of the fence. "Mary Fisher! It is not polite to sneak up on a person like that!"
"Sneak?" Mary echoed, shivering. "I-I d-did not s-sneak!"
"I didn't hear you. . . ."
"You did not hear Levi, either."
At the mention of Mary's brother, Emma looked past her friend to the Fishers' fields, a familiar flutter rising inside her chest. "Levi? He saw me?"
"Yah. That is how I knew you were here." Mary climbed onto the bottom slat of the fence and leaned across the top, her brown eyes almost golden in the early morning rays. "Happy birthday, Emma!"
"You remembered. . . ."
Mary's brows dipped. "Of course I remember. We've been best friends since we were babies, silly."
Slowly, Emma wandered between the graves and joined her friend at the fence. "Sorry. I guess I just thought maybe you'd forgotten."
"I didn't." Mary ducked her chin inside the top edge of her coat, muffling her voice as she did. "So . . . it is not any different?"
"It?"
"Your birthday. You know, with your mamm. . . ."
Emma didn't mean to laugh, she really didn't. But somehow it took more effort to refrain. "Thinking my birthday this year will be any different than it's been for the first twenty-one is like thinking your brother would ever notice me in the way he notices Liddy Mast."
"Please . . . Liddy Mast . . ." Mary grumbled on an exhale. "Do not remind me."
"What? Liddy is . . . nice."
"I suppose. Maybe. But she blinks too much."
"Blinks too much?"
"Yah."
Emma closed her eyes against the image of the dark-haired Amish girl who'd shown up at one of their hymn sings three weeks earlier and set her sights on Mary's brother almost immediately. "Levi does not seem to mind this blinking," she whispered.
"Levi is . . . well, Levi. The only things I know for sure about my brother is that he eats as if he has not seen food for days, he likes to put frogs in places I do not expect to see frogs, his constant hammering gives me a headache, and I would really rather speak of your birthday at this moment."
"There is nothing to speak of. It is just another day."
Mary's brown eyes disappeared briefly behind long lashes. "She lost her sister, Emma. That has to be hard."
It was the same argument she had with herself all the time. But . . .
"When Grossdawdy died last year, Mamm and Dat said it was God's will. And when Grossmudder passed in the fall that, too, was God's will. Shouldn't—" Emma stopped, shook away the rest of her thought, and forced herself to focus on something, anything else.
Mary, being Mary, didn't give up that easily. "Her sister was younger than you are now, Emma. And it was so sudden."
"But that's just it, Mary. I don't know if it was sudden or not. Mamm won't talk about it. Ever. She is just sad on this day."
"Maybe you should ask to celebrate your birthday on a different day," Mary suggested. "Maybe then there could be smiles and laughter on your special day, too."
She opened her mouth to point out the oft-shared fact that Mamm rarely smiled around Emma at all, but even Emma was growing tired of the subject. Some things were just a certainty. Like her brother Jonathan's rooster announcing the arrival of morning as the moon bowed to the sun. Like the answering gurgle of her stomach every time she pulled a freshly baked loaf of bread from the oven. Like the cute dimple her sister Esther shared with Mamm. Like the way Jakob's footfalls sounded identical to Dat's on the stairs each night. And like the surprise she knew she'd find beside her aunt Ruby's grave that morning . . .
Stepping off her own perch atop the bottom slat, Emma motioned to Mary's farm. "You should probably go. You do not want to upset your mamm by not doing your chores."
"I have a few minutes before I must be back."
Anxious to get back to the rock and the flash of silver she'd spied just as Mary called her name, Emma patted her friend's cold hand. "I am fine here. Alone."
"But it is your birthday, Emma! You should be doing happy things like talking to me instead of standing at . . ." Mary's words quieted only to drift off completely as she, too, stepped onto the ground. "I will leave you to pray alone. I should not have interrupted the way I did."
She met her friend's sad eyes with a smile. "I am very glad you did, Mary. Truly. I-I just . . ."
"You want to pray alone," Mary finished. "I understand."
Unable to lie to her friend aloud, Emma let her answering silence do the work.
"Well, happy birthday, Emma."
"Thank you."
She remained by the fence, watching, as Mary made her way back across her dat's field and, finally, through her parents' back door. For a moment she let her thoughts wander into the Fisher home, too, Levi's warm smile greeting her in the way it had Liddy Mast at the last hymn sing . . .
"Oh stop it, Emma," she whispered. "Levi does not see you any more than anyone else sees you."
Shaking her head, she picked her way back to Ruby's grave, her gaze quickly seeking and finding the shiny silver chain peeking out from around a nearby rock. Mesmerized, Emma dropped to her knees and slowly fingered the chain from the clasp at the top to the thick, flower-etched—
The air whooshed from her lungs as she lifted the chain from its resting spot and set the heart-shaped pendant inside her palm. She'd seen jewelry before many times—on English shopkeepers in town; on the driver Dat hired when they needed to travel outside normal buggy range; on Miss Lottie, the elderly English woman who lived out near the Beilers; and even on her own wrist for a very short time during her Rumspringa when she was sixteen—but nothing so delicately beautiful as the necklace in her hand at that moment.
"Who would leave something so pretty on the ground?" she whispered. "It does not make any . . ." The words fell away as her eyes lit on a thin line around the outer edge of the heart. A line just wide enough to wedge her nail inside and—
With a quiet snap, the heart split in two and she slowly lifted the top half up and back, her answering gasp echoing around her in the cold morning air.
There, nestled against a pale pink background, was a heart-shaped photograph of an Amish girl not much younger than Emma. . . . An Amish girl with brown hair and eyes so like Mamm's. . . . Yet, with the exception of those two things, everything else about the girl was a mirror image of . . . Emma?
Confused, Emma pulled the open locket still closer as, once again, she studied the face inside. The same high cheekbones . . . The same slender nose . . . The same wide, full lips . . . The same tiny freckles . . . In fact, with the exception of the hair and eye color, she'd actually think she was looking at a picture of herself.
Closing her fingers around the locket, Emma rose to her feet and began to run, the steady smack of her boots against the cold, dry earth no match for the thud of her heart inside her ears.
* * *
She was nearly out of breath when she reached Mary's driveway but she didn't slow down. Instead, she ran faster, her attention ricocheting between the house and the barn while simultaneously trying to work out where she'd be most likely to find her one and only true friend.
Her first stop was the barn, but other than a quick glimpse of Levi mucking a stall in the back corner, and Mary's dat gathering together tools atop a workbench, there was no one else. Spinning around, she ran farther up the driveway to the simple white farmhouse with the wide front porch. When she reached the front door, she made herself stop . . . breathe . . . and knock in a way that wouldn't startle everyone inside.
Still, she knew she had to look out of sorts when, a few moments later, Mary's mamm opened the door and almost immediately furrowed her brow. "Good morning, Emma. Is-is everything okay at home?"
She followed the woman's gaze down to her hands—one clenched tightly around the unseen necklace, the other nervously fiddling with the edge of her favorite pale blue dress. Realizing the sight she must be, standing there on the Fishers' porch, still panting slightly from her run, Emma made herself smile. "I . . . I was hoping maybe I could speak with Mary for a few minutes?"
"I thought you spoke. Outside."
"We . . . did. But . . ." She stopped, swallowed, and willed her voice to remain calm even as her fingernails threatened to draw blood from her palm. "I will not take too much time. I . . . I just forgot to tell her something."
"Very well." Mary's mamm stepped back, motioned Emma inside, and then closed the door against the winter morning. "Mary is making some dough for bread in the kitchen."
She followed the woman down the narrow hallway to the back of the house and the large, yet simple kitchen that was nearly identical to Emma's. Only here, at the Fishers', the table was positioned in the center of the room whereas at home the table was off to the left where Mrs. Fisher kept her sewing table.
"Mary, Emma has something she forgot to tell you." The woman smiled again at Emma and then swept her hand toward the basket of laundry at the base of the steps leading to the second floor. "I'll start putting away the laundry while you girls talk."
"I'll be up as soon as we're done, Mamm." Eyeing Emma with a mischievous grin, Mary spread a cloth over the dough bowl and carried it to a sunny spot in the corner of the room. "You've been running, haven't you?"
"Yah. I—"
"I know we do not see my brother the same, but to run all the way here just so you will look at the ground when he speaks to you? I do not understand you, birthday girl."
"I did not come to see Levi," Emma rasped. "I came to see you."
Mary's left eyebrow arched with intrigue. "But you just saw me. At the cemetery."
"Yah." Emma pointed to the long bench beside the table and, at Mary's nod, sunk onto the wooden seat, looking toward the stairs as she did. When she was satisfied Emma's mamm was no longer within earshot, she pulled her fisted hand to her chest and looked up at her friend. "I-I have to show you something. It is why I ran all the way here."
Mary took a seat on the opposite bench, her gaze locked on Emma. "What? What do you want to show me?"
"This." Emma lowered her hand to the table, opened her fingers, and held out her palm to reveal the locket.
Confusion darted Mary's eyes between the necklace and Emma. "What is that?"
Thrusting her hand forward across the table, Emma swallowed. Hard. "Open it. Please."
"Open it? Open what?"
"I will do it." Emma popped open the delicate silver heart and wordlessly held it out toward Mary once again.
This time, Mary leaned forward, disgust registering across her round face a split second before infusing its way into her voice. "Emma Lapp! The Bible says, 'Thou shalt not make unto thyself a graven image!' "
"I didn't," she whispered.
"You are not to pose for one, either!"
Emma met and held her friend's eye before leading it back down to her open palm. "It's not me, Mary. Look again."
"What do you mean it's not you?" Mary snatched the necklace from Emma's hand. "Of course it's . . ."
Emma waited as Mary's eyes chronicled the same features she, herself, had noted back at the cemetery. Sure enough, as she watched, the disgust her friend had worn only seconds earlier began to dissipate, replaced, instead, by first confusion, and then curiosity.
"Who is this?" Mary finally asked, looking between Emma and the locket. "She looks just like you. . . . But with light brown hair . . . And eyes that are brownish green, instead of blue."
She opened her mouth to speak but closed it when there were no proper words to be found.
"Emma?" Mary repeated. "Who is this?"
Aware of her friend's probing eyes, she swallowed around the lump in her throat. "I-I don't know. That is why I came here. To see you."
"But how can you not know? You're holding it.... And she looks just like you. . . ."
"I found it."
Mary pulled a face. "You found a necklace with a picture of an Amish girl that looks just like you?"
"Yah."
"Emma, this doesn't make sense."
"I thought it would be like all the other things—little and shiny, or even just silly. But it wasn't." She took the necklace from Mary's open palm and stared down at the image again. "It was this."
"What do you mean like the other things? What other things?"
"The presents I find at my aunt Ruby's grave every year."
Mary's head whipped around toward the stairs only to return to start with widened eyes. "Presents?"
Nodding, Emma set the locket at her spot on the table and swiveled her legs over the bench until she was able to stand. Then, beckoning to Mary, she led her friend over to the kitchen window and its view of the cemetery in the distance. "Every year, since I was one, I imagine, Mamm and Dat would stop out at Aunt Ruby's grave on my birthday. We would not stay long, but we went every year. Every year, Mamm would start out sad, but soon she, like Dat, would be angry."
The sadness she saw in Mary's eyes at the beginning of her explanation ebbed into confusion. "I don't understand . . ."
"Someone leaves a present on Aunt Ruby's grave each year. On the day of her death. I don't remember what the presents were back then, but I remember Dat did not like them to be there." She leaned her forehead against the cold glass and lingered her gaze on the area where she knew the cemetery to be. "He would take the present and he would throw it into the first trash box we would see. I did not understand why these things were there, I just knew I did not like to see Mamm hurt even more than she already was on my birthday.
"That is why, when I was seven and able to walk to school alone, I told Luke Graber, Elizabeth Troyer, and the other children I walked with, to go ahead—that I would catch up."
Mary's quiet gasp pulled Emma's focus off the scenery outside the window and fixed it, instead, on her friend. "You went to the cemetery alone? On your birthday?" Mary asked.
"Yah. I wanted to get the silly thing before Mamm and Dat were to see it." Swallowing back against the emotion she felt building, she willed herself to remain calm, to hold back the memory-stirred tears. "But when I saw the miniature picnic basket sitting on the ground, I—"
"Miniature picnic basket?" Mary echoed.
"Yah." Bringing her hand up between them, Emma separated her thumb and index finger by about an inch. "It is just like a real picnic basket, but it is this small. It even opens . . . but there is nothing inside."
"I imagine it was hard to throw such a thing away."
"That is why I didn't." She shrugged away Mary's renewed gasp and wandered back to the table with her friend in tow. "I put it in my lunch pail and kept it there until the school day was over. Then, on the way home, I hid it out by Miller's Pond."
Mary's eyes brightened with excitement as, once again, she peered toward the stairs and dropped her voice to a whisper. "Do you think it could still be there even now?"
Lowering herself to her seat next to the locket, Emma fingered the chain. "It is still there. In a bag. With fourteen other things. This"—Emma closed her hand over the locket—"will make fifteen things."
"You saved them all?" Mary glanced back at the stairs. "In the tree?"
"Yah."
"Can I see them?"
"Maybe. One day."
"Why did you keep them?" Mary asked.
"Going there, to see what was left at Ruby's grave, is the one part of my birthday that feels . . ." Emma stopped, took a breath, and made herself continue despite the emotion that was beginning to choke her words. "Special."
Mary took a spot on the same bench and rested her head on Emma's shoulder. "Oh, Emma, I don't like that your birthday is always so sad. It doesn't seem right."
"Mamm did not ask God to take her sister."
"I know, but still . . . It was God's will."
"I know that, and you know that, but Mamm is still sad."
"I understand that, but it is like you said earlier. It was twenty-two years ago."
"Yah." Emma held the locket to her chest. "I know it's wrong to say, but I began to see these things at the grave as birthday presents for me. They were something to get excited about each year. But I don't know what to think about"—Emma opened her palm to the locket again—"this one. It is the only one with a picture."
"A picture that looks just like you," Mary reminded, straightening up.
The sound of approaching footsteps had Emma closing her fist around the necklace once again. Together, they looked toward the stairs in time to see Mary's mamm appear with an empty laundry basket in her hands.
The same shame that cast Mary's eyes down to the table also propelled her onto her feet. "I did not mean to be so long, Mamm. I am sorry." Then, to Emma, she said, "It is time for me to get back to my chores."
"I understand."
"We will talk again soon." Mary lingered her gaze on Emma's hand and then gestured toward the stairs. "Now, I must sweep."
She watched her friend disappear up the stairs and then turned to find Miriam Fisher watching her closely. "I am sorry I took so long with Mary. I did not realize how long we had spoken."
"It is all right, Emma. Sometimes friends just need a little extra time." Miriam led the way toward the front door but stopped just shy of it. "You are twenty-two today, aren't you, Emma?"
"Yah."
"Happy birthday."
"Thank you." Emma reached for the doorknob only to stop and turn back to the woman. "You grew up with Mamm, didn't you?"
"I did."
"So then you knew her, too, right? You knew Ruby . . ."
Miriam Fisher's dark brown eyes dropped to the floor.
"You knew Ruby . . ." Emma prodded again.
Mary's mamm lifted her gaze and fixed it on something just beyond Emma's shoulder. "I knew Ruby."
"Was she like Mamm?"
Miriam's focus snapped back to Emma's. "Ruby and Rebeccah? No, they were very different. Rebeccah was always so serious. She was selling quilts by the time she was fourteen. And her chores were always done. Always."
"And Ruby?"
"She did her chores, too, of course. But always much later than she was supposed to that last year."
Something about the tone of the woman's voice intrigued her and she leaned forward. "Why? What was she doing?"
Miriam started to speak, shook her head, and then motioned over her shoulder toward the kitchen. "I'm sorry, but I really should get back to my own chores. There is much to be done before lunch."
"But—"
"Emma, you must go."
Disappointment sagged her shoulders, but she knew the woman was right. She'd already taken enough of Mary's time. To do so with Miriam, as well, would be wrong. Still, there was one thing she had to know—one thing she knew she couldn't ask Mamm without intensifying a pain Emma's very existence seemed to stoke.
Readying her hand on the door once again, she waited for Miriam's gaze to return to her. When it did, the question she'd wondered her whole life blurted its way past her lips. "How did Ruby die?"
Miriam jumped backward as if she'd been slapped. "H-how did Ruby die?" she echoed.
"Yah. Was she ill with fever?"
"No."
"Was she in a buggy accident?"
"No."
"Did she fall down or get hurt by an animal on Grossdawdy's farm?"
"No."
There was no denying Miriam's growing discomfort as Emma cycled through the various scenarios she'd imagined in her head over the years, but she needed to know the truth. If she knew, then maybe things would be different between her and Mamm, somehow. . . .
"Was there a fire?"
"No."
"Then, how?" she pleaded. "How did Ruby die?"
"She just died, Emma. That is all. Now, I really must get back to my chores and—"
Emma reached out, stopping the woman's retreat back to the kitchen. "I know she did not just die. That is why I am asking you to tell me what happened."
"Ask Rebeccah."
"Talk of Ruby makes Mamm sad. Especially today, on the day Ruby died. It is why I am asking you."
"Emma, please. I really must get back to—"
"The Bible says, 'A false witness shall not be unpunished, and he that speaketh lies shall not escape.' " Emma willed herself to breathe, to keep her voice steady and polite. "So you cannot say Ruby just died . . ."
Pressing a fist to her mouth, Miriam pinched her eyes closed, her audible inhale whooshing its way past her fingers. "She died in childbirth."
Chapter 2
She was halfway down Mary's driveway when she realized she hadn't said goodbye or thank-you or anything one should say after visiting. But she couldn't turn back. Not now, anyway. Not unless she wanted to run the risk Levi would step from the barn and see her looking as lost as she felt.
For as long as she could remember, she'd wanted to know how her aunt Ruby had died, but she'd been too afraid to ask lest she upset Mamm. When she was old enough to know what death was, she'd imagined the teenager's last breath coming while in bed with the flu. When she'd been old enough to understand conversations between grown-ups, she assumed the death had been tied to Rumspringa and an experimentation of the English world gone wrong. And when she'd attended the funeral of someone her own age in a neighboring district six months earlier, she'd seen Mamm's quiet tears over the tragic buggy accident as evidence that the long-grieved death had been because of something similar.
But to learn that Mamm's little sister had died during childbirth? It was—
"No . . . It must be a mistake," she murmured, stepping onto the roadway and heading east toward home. It couldn't be.... Miriam Fisher had to be wrong.... One only had to look in the cemetery to know that.
If Ruby had died during childbirth, the dead infant would have been buried beside her, the birth date matching that of death. Instead, to the left of Ruby's grave was Abram King, a man who lived to be eighty-two. And to the right was Hannah Troyer, her childhood friend's grandmother.
No, the only baby born that day was—
She stopped.
"Nooo . . ." Her whispered rasp echoed in the still January air only to be chased off by the hitch of her own breath.
Was it possible?
Was she the child that . . .
Feeling her legs begin to give out beneath her, Emma stumbled over to the edge of the road, to the fence that kept the Grabers' pigs from venturing into town. She grabbed the upright closest to her body and closed her eyes against the unmistakable sense of dread snaking its way across every ounce of her being.
The dimple little Esther shared with Mamm . . .
Jakob's tall, lanky build so like Dat's . . .
Sarah's and Annie's hair the exact same shade of brown as Mamm's, while Esther, Jakob, and Jonathan shared Dat's slightly darker hue . . .
Her own dark blond hair and big blue eyes . . .
The pretty sparkle in Mamm's eyes as she looked around the dinner table at the other children . . .
The way that same smile dulled as it came to rest on Emma . . .
She tried to calm her strangled breaths, but it was no use. Instead, she made her way back onto the road, each step she took bringing with it another memory, another certainty.
All her life, she'd felt as if she never quite fit. Not in the classroom or on the playground as a child, and not at hymn sings or other friendly gatherings now that she was older. But never did that sense of always being a step behind hurt like it did at home, where her childhood antics had drawn wary smiles, her thoroughness with her chores earned little more than a labored nod, and her smiles had never been returned with quite the same conviction as which they'd been given.
Was this why?
Because she was actually Ruby's—
Unfurling her fingers from their near death grip on the latest gravesite present, Emma stared down at the heart-shaped locket and the strangely familiar face inside—a face that looked like, yet wasn't, her own. The pronounced cheekbones were the same, and the shape of the eyes was the same, but that didn't necessarily mean anything, did it? Ruby was, after all, Mamm's sister. There was no reason Emma couldn't look like her. And besides, Emma's eyes were blue like—
No one's.
A flurry of footsteps off to her left broke through her thoughts in time to see her five-year-old sister, Esther, running in her direction, a sweet smile stretching the child's cheeks wide. "Emma! Emma! I helped Mamm bake a cake! For your birthday!"
She forced herself to smile as she squatted down for the hug the little girl liked before heading off to school each day. "I bet it will be a wonderful cake."
"You cannot eat it until tonight!" Esther wiggled free in favor of pointing at Emma's hand, the excitement over her cake-baking adventures replaced by a sudden bout of solemnness. "Mamm said we are not to take pictures."
Closing the locket back inside her hand, Emma stood, her gaze inching past Esther to a rapidly approaching Annie and Jonathan. "Did you forget something?" she asked, looking back at Esther.
"No."
She pointed the little girl's attention toward the driveway and their eight-year-old sister. "Seems Annie is carrying two lunch pails when she only has one tummy."
Esther's eyes widened. "Oh. I forgot."
"Yah." She tapped Esther's nose with her index finger. "It is your job to remember your lunch, not Annie's, right?"
Shame led Esther's eyes down to the ground. Curiosity lifted them back to Emma. "Why do you have that in your hand?" Esther asked, pointing at Emma's now closed fist.
"Because I do. Now go get your lunch pail from Annie and make sure you walk together to school, okay?"
For a moment she thought the little girl was going to protest, but, in the end, Esther simply ran back to Annie, liberated her lunch pail from Annie's hand, and then waved for her siblings to follow her onto the road toward school. "Bye, Emma. Happy birthday!"
"Happy birthday, Emma," Annie and Jonathan said in unison as they passed.
"Thank you."
Annie paused and glanced back. "Mamm is looking for you, Emma."
She bit back the I'm looking for her, too, that was on the tip of her tongue and, instead, mustered a smile and an answering wave as the trio set off in the direction of the one-room schoolhouse.
Seconds turned to minutes as she stood there, watching them, her mind's eye filling with memories of her own walks to school—her feet always slowing as she approached and then passed the cemetery. A few classmates had noticed over the years, but they never said anything. Then again, other than the teacher and her brother Jakob, most of the kids hadn't said a whole lot to Emma to begin with. Everyone had always been polite, of course, but no one had ever sought her out to chat or play the way they had with one another. Except, of course, Mary.
She yanked her attention back to her siblings only to discover they'd made it past the bend in the road that claimed them from her sight. Squaring her shoulders, she looked again at the locket, breathed in every ounce of courage she could find, and then headed up the driveway toward the house, the distant tap-tap of a hammer letting her know Jakob was working on another section of fence, while a peek at the chicken coop showed Sarah gathering eggs into a basket.
When she reached the porch, she walked up the steps and into the house, letting the door flap closed in her wake.
"Surely you are not already done collecting the eggs—oh, Emma. I did not know it was you."
Stopping briefly just inside the kitchen doorway, Emma took a moment to catch her breath. When she did, she crossed to the sewing machine and Mamm. With little more than a blink, she thrust her hand out, palm up, and watched as her mother's eyes dropped to the locket.
"What is that?"
With nary a word or even a sound, Emma wedged her finger against the locket's seam and pried it open, her mother's answering gasp echoing around the room. "Emma! Where did you get that?"
"At the cemetery."
Mamm jumped to her feet, her eyes ricocheting between the locket and Emma. "What were you doing at the cemetery?"
"That's what we do on my birthday, isn't it?" she countered, her voice shrill. "We go to the cemetery to visit your sister's grave, and you cry. It is a part of my birthday like a picnic is for Sarah's, a game of stickball is for Jonathan's, a walk to the pond is for Jakob's, a game of follow the leader is for Annie's, and bubble blowing is for Esther's."
Like a moth to an open flame, Mamm's gaze returned to the open locket, tears making their way down her colorless cheeks.
"I used to think your sadness was because Ruby died on my birthday. But it is more than that, isn't it?" Emma demanded. Then, before she lost the courage Mamm's tears were rapidly eroding, she added, "It's because I'm the reason she died, aren't I?"
She wasn't sure what response she'd been expecting. Maybe a gasp . . . Maybe a violent shake of Mamm's head . . . Maybe a frantic reach for Emma's hands while pleading for her to refrain from such silly talk . . . Whatever it was, though, it hadn't been this weighted silence that hung in the air like an impending storm.
Not knowing what to do or say, Emma looked again at the photograph of the girl atop her hand and willed the eyes she saw looking back at her to respond the way she'd wanted Mamm to respond. But as with Mamm, there was only silence.
Heavy, heavy silence.
The kind that was an answer all on its own.
"Please," Emma whispered past the emotion gathering inside her throat. "I-I need to know. Was . . . Was she . . ." She stopped, swallowed, and tried again, her voice reflecting the shake of her locket-holding hand. "Was Ruby my mamm?"
Slowly, Mamm lowered herself back onto her chair, her breath labored. Then, dropping her head into her hands, she released a tired sigh. "Yah."
On the walk home from Mary's, she'd known the truth. Deep down inside herself, she'd known the answer she would soon have. But still, hearing it confirmed aloud pained her in a way nothing else ever had. "I-I don't understand," she said between gulps of air. "Ruby was only eighteen when she died. She wasn't even . . ."
The rest of the protest faded from her lips as the reality of who she was and where she came from hit her like an unexpected kick from one of Dat's mules. Suddenly it made sense why Mamm never smiled at her in quite the same way as her brothers and sisters.... Why Dat had always held her at arm's length, preferring to nod his approval in her direction rather than pat her shoulder the way he did with Sarah, Annie, and Esther. . . . Why her classmates in school hadn't tried as hard with her as they did with each other. . . . Why she was the only girl at the hymn sings who wasn't courting yet....
"I remind you of her," Emma whispered. "Don't I?"
Mamm's head snapped up, her eyes wet with tears. "You do! You have her smile and her cheeks!"
More than anything she wanted to believe that's all it was, that when Mamm looked at her it was like looking at Ruby. But she knew the reminder went deeper than that—to a shame that weighed on Mamm's shoulders every bit as much as any pain. "No, I remind you of her sin," she rasped.
"Emma! That's—"
"Did she get to see me?"
Mamm drew back. "Who?"
"Ruby . . . My mamm . . . I don't know what I'm supposed to call her!"
A flash of pain skittered across Mamm's face as, once again, she rose to her feet. "Emma—"
"Please. Just answer my question. Did she get to see me before she died?"
"Yah."
"And?"
"She smiled at you." Pressing her fist to her mouth, Mamm inhaled sharply only to let the same breath go in a slow, controlled whoosh. "And then she was gone."
Wandering over to the window, Emma looked out over Dat's winter fields. Even with the dormant earth, she could imagine the view come spring, when the corn and barley were starting to grow. Spring was a time of beginnings just like birth. Only, in her case, it had been different. Her birth had meant the end of Ruby's life and, in some ways, Mamm's—
She spun around. "What about my dat? I mean, my real dat? Why didn't he take me when Ruby died?"
"Because he was an Englisher!" Mamm closed the gap between them, grabbed the locket from Emma's hand, and shook it in the air. "He did not care that Ruby was Amish! He did not care that she had been baptized! He did not care about your mamm or you at all!"
"Is . . . is that why I have never fit here?" Emma asked, closing her eyes.
"What is here?"
"Here—in Blue Ball . . . In school . . . At hymn sings . . . When we go visiting . . ." She heard her voice growing hoarse, knew it was only a matter of time before she began to weep. But still, she had to know. "Here—in this house . . . In this family . . . Because everyone knows what I am?"
She waited for Mamm to break the silence, to argue that Emma did belong and always had, but those words never came. Instead, when Emma opened her eyes, she saw that Mamm's focus had moved to somewhere far beyond Emma. And in that moment, she knew the truth.
Chapter 3
Hugging her knees to her chest, Emma wiped the last of her tears against her dress sleeve and looked out over Miller's Pond. Even without Dat's mantel clock, she knew afternoon was beginning to fade into early evening. She also knew that by being here, instead of at the house as she should, Sarah and Jakob, and the younger ones, now home from school, were surely covering her chores and beginning to question where she'd gone.
She tried to imagine what Mamm would say. Would she shrug and change the subject? Would she redirect their attention to a chore that needed to be done? Or would she tell the truth—that Emma was really their cousin, not their sister?
Tired of the tears that had been her constant companion since fleeing the kitchen and Mamm's painful silence, Emma lifted her chin to the sun's waning rays and breathed in the cold, crisp air. For as long as she could remember, this pond, this rock, had always been her retreat of choice. When she'd been little, she'd liked it for its vantage point over the wildflowers that grew along the southern shore, and the squirrels and birds that hovered nearby, oblivious to whatever it was about her that made her peers shy away. When she'd gotten a little older and school was no longer part of her days, she'd venture here during a break in chores, to think and to dream. And, of course, every year on her birthday, she'd come here to hide yet another trinket inside the hollow of the oak tree she could just barely see from her favorite resting spot . . .
For what had to be the hundredth time that day, she opened her palm and gazed down at the eighteen-year-old face she'd all but memorized—a face she'd never known, yet shared in more ways than not. But it was the other stuff—the stuff she couldn't see in a tiny picture—that she most wanted to know about Ruby.
Like the sound of her laugh . . . Had it been hushed like Mamm's or—
"I figured I would find you here."
Dropping her knees back to the rock's surface, she jerked her gaze across her shoulder to the footpath. "Mary! Stop sneaking up on me like that!"
"That stick"—Mary pointed at the ground behind her feet—"and these leaves did not let me sneak. You let me sneak."
Emma swiped the residual dampness from her cheeks and scooted forward until her feet touched the ground. "How did I let you sneak?"
"By not seeing or hearing anything around you." Mary crunched across the last of the leaves separating the footpath from the rock and sat beside Emma, her brows dipped with worry. "Are you okay? Your face looks a little funny. Almost like you've been crying."
Pushing off the rock, Emma shrunk her neck farther into her coat and wandered over to the water's edge. "Why did you come?" she asked instead.
"I made something for you—for your birthday. But when I stopped at your farm to give it to you, Sarah said you were not home. When I asked if you would be back soon, she made a face and went back to feeding your dat's new calf." Mary, too, abandoned the rock and joined Emma at the edge of the pond. "I thought feeding the new calf was your job."
"It is."
Mary laughed. "It is good to have a birthday. It means less chores."
"I have been here since the sun was"—Emma swept her hand across her shoulder—"back there."
Mary's laugh hushed. "But you were at my house this morning . . ."
"Yah. And then I ran home and, after a little while, I ran here."
"You've been here this whole time?" Mary asked, drawing back.
"Yah."
"Doing what?"
"I have done some yelling, some stomping, some throwing, some thinking, and, yah, much crying."
"But why? It's your birthday and you found that pretty . . ." Mary's words trailed off as understanding traded places with confusion. "You showed it to your mamm, didn't you? That's why she looked so troubled when I went up to the house to see if she knew where you—"
"She is not my mamm," Emma whispered. "Her sister, Ruby, was."
"Emma Lapp, you are not to say such things! The Bible says, 'Lying lips are an abomination to the Lord!' "
"The Bible also says, 'God is a Spirit: and they that worship him must worship him in spirit and truth.' "
Mary's eyes narrowed on Emma's. "I do not understand what—"
"Ruby died the day I was born, Mary."
"In an accident or because she was sick or . . . however she died. That doesn't mean she was—"
"Your mamm told me Ruby died in childbirth."
The whoosh of Mary's breath as she sucked in her surprise echoed around them. "M-my mamm told you such a thing?"
"I asked how Ruby died, and she spoke the truth."
"She . . ." Mary stopped, swallowed, and tried again. "She told you Ruby was your mamm?"
"No. I figured that part out on my own when I left your house. It made sense—the day, the way Mamm would never speak of how Ruby died, the way she and Dat have always been different with me than Jakob and Sarah and the rest of the children, seeing her picture in the locket and knowing I look more like her than Ma—I mean, Rebeccah . . ."
Again, Mary stepped back, only this time, the heel of her boot touched water. "Rebeccah? Who—wait. What are you doing?"
"Ruby was my mamm. Rebeccah is my aunt."
"Emma, you shouldn't talk like that!"
"Why? It is the truth! Mamm—I mean, Rebeccah told me!"
Mary's mouth opened, then shut, then opened again. "I-I don't know what to say."
"That is okay, because I don't know what to feel. Except anger and . . ." Emma glanced across the pond at the tree she'd yet to approach. "Questions. Many, many questions."
"But you know now, don't you? What more could there be to know?"
Spinning on the toes of her boots, Emma led the way back to the rock and the necklace splayed across its center. When she reached it, she lifted up the locket, the strands of the delicate chain brushing briefly against her arm. "I want to know where this came from! Why it was next to my real mother's grave!"
Mary looked from Emma to the locket and back again. "You mentioned other things you found, on other birthdays. That you've kept them . . ."
"Yah." She pointed across the pond with her free hand. "They are in a bag in that tree." Then, beckoning her friend to follow, she added, "Come. I will show you."
They trudged around the southern edge of the pond and headed in the direction of the half dozen oak trees that lined the western side. With quick feet, Emma led the way past the first and second tree. At the third tree, she circled around to the back and the hollow she'd long ago masked with a piece of bark.
With quick, deft hands, she maneuvered the piece of bark from the hole, set it on the ground beside the trunk, and reached inside. Sure enough, with little more than a single pat to the left, she closed her cold fingers around the gathered edges of the bag and pulled it out, Mary's answering gasp barely audible over the sudden, yet familiar acceleration of her own heartbeat.
"How long has that been in there?" Mary asked.
"Since a few days after my seventh birthday—though, back then, there was only one thing in the bag. The next year there were two things . . . and then three things . . . and, soon"—Emma shook her locket-holding hand—"sixteen."
"The presents? They're all in there?"
"Yah. I will show you. Come." Once again, she led the way back around the pond to their starting point and the rapidly decreasing sunlight the rock still offered. When they reached her favorite spot, she set the locket down and reached inside the bag's drawstring top.
"This miniature picnic basket was the first one—or, the first one I got to before Dat." She pulled it out, held it up for Mary to see, and then set it on the rock beside the locket. "It is just like a real one, isn't it? Only tiny."
Mary sat down, ran her fingers across the wicker sides and handle, and then glanced up at Emma. "Why would someone leave such a thing next to a grave?"
"I don't know. I only know that they did." Again, Emma reached inside, and again she pulled out a trinket. "This one was when I turned eight. It is heavy for something so small."
Mary leaned forward, her breath warm against Emma's fingers. "That's a rose!"
"Yah. Here. Feel it." She handed the pewter flower to her friend and, at her friend's nod, pulled out one of her favorites—a tiny snow globe with even tinier skaters inside. Mary's answering intake of air made Emma smile. "I know, it is pretty, isn't it? And look . . ."
With practiced hands, she turned the clear-fronted globe upside down, shook it gently, and then righted it for Mary to see.
"It's snowing!"
"Yah." Emma's smile morphed into a giggle as her thoughts traveled back through fourteen years' worth of birthday afternoons. "Every year, since I was nine, I have made it snow on the skaters."
"I would, too." Mary watched with fascination as the final flakes fell and then, with little more than the nudge of her chin at Emma's hand, added, "Can I try?"
Nodding, Emma handed the snow globe to Mary, watched her shake it, and then reached inside for the small brown horse with the black mane. "It is good I found this when I was ten instead of seven. It would have been hard to leave this in a tree."
"Oooh, that looks like Levi's horse, Hoofer, doesn't it?"
Emma turned the horse around in her hands. "You are right, it looks very much like Hoofer."
With great reluctance, Mary set the snow globe down next to the horse and pointed at the bag. "What's next?"
"A picture." She rooted around in the bag until she felt a familiar glossiness. "It is of a dandelion."
"A dandelion?" Mary echoed. "Why would someone leave that?"
Emma shrugged and handed it to her friend. "It's the puffing part of a dandelion . . ."
"Still."
Again, she reached inside. "This was when I turned twelve."
Mary traded the picture for the bubble wand. "What? No bubbles?"
"No bubbles . . ."
"You blew them all, didn't you?" Mary teased.
"No. It was just the wand."
Mary pulled a face. "Why?"
"I don't know." Emma returned to the bag and yet another odd item. "This one was there when I turned thirteen. I don't really know what it is except a piece of ripped paper inside a plastic covering . . ."
"Let me see." Mary took the plastic-encased paper and turned it over a few times. "It looks like it is part of a ticket. But I can't quite read what it—wait . . . It says—"
"Admit one," Emma finished for her friend. "I know. But I do not know what it is for." She reached inside the bag again and pulled out the small round rock she'd found on her fourteenth birthday. "This is pretty, don't you think?"
Mary's brows scrunched together. "It's a rock . . ."
"Yah, but look"—she spun it around for Mary to see—"someone drew a heart on it!"
They examined the rock and its drawing from several different angles and then added it to the growing menagerie atop the rock. Next up was the miniature covered bridge that reminded Mary of the one just past the grain mill on Route 35, the small, red rubber ball Emma caught just before it bounced into the pond, the yellow spinny thing on a stick that had Mary blowing so hard her cheeks turned bright red, the baseball with the name they couldn't read beyond the uppercase B and uppercase H, and the dried flower with the pale blue and pink ribbons Mary suspected had been a rose at one time.
"Wow. This is quite a lot of things," Mary said, sweeping her hand toward the rock.
"There is one more. From last year. When I was twenty-one." Again, Emma reached inside and pulled out the lone remaining object—a whittled bird.
Mary's hushed breath mimicked Emma's. "Wow . . . That is beautiful! And . . . it even has a worm in its mouth."
"Yah."
"It is a shame such things must stay in a bag in a tree."
"Dat would be angry if he saw them." Emma stepped around Mary to place the bird alongside everything else. "He threw away the ones before these."
"Do you remember what they were?" Mary asked.
"Not really, no. Dat would scoop them up before I could see what they were. I knew only that they were small and that they made him angry."
Mary looked from item to item before looking back at Emma. "These were there every year?"
"Yah."
"Haven't you ever wondered why?"
"When I was little, yah. But I wondered more why they upset Dat and Mamm so. After that, when I would go to the cemetery alone before school, I mostly just wondered what the new thing would be." Aware of Mary watching her, Emma busied herself with gathering up the objects and returning them to the bag. "I know it is wrong to keep them, but they are too pretty to throw away—even the picture of the dandelion and the ripped ticket. It is as if they mean something."
"Maybe they do."
Emma paused her hand atop the stuffed horse and studied her friend. "What are you saying?"
"Maybe those things"—Mary pointed at the remaining objects still lined up along the top of the rock—"are from your father."
She pulled a face. "I told you. Dat didn't put them there. He got angry when he saw them, remember?"
"I'm talking about your birth father."
Clutching the bag to her chest, Emma stumbled back a few steps. "My-my birth father?"
Mary nodded.
"But . . . but Mamm—I mean, Rebeccah, said he didn't care about my mother or me," Emma sputtered. "That-that he left her to face her sin alone. That he was . . . English."
Silence settled around them as Mary pried the bag away from Emma and filled it with the last few items. When everything was safely inside, she pulled the bag closed and handed it back to Emma. "It seems to me there is only one way to know if that is true."
"What? That he didn't care about us?"
Mary's nod was slow yet unmistakable even in the growing dusk. "Find him, Emma. Ask him."
Chapter 4
"She brought them again."
Emma shifted her focus from the group socializing by the side of the Troyers' barn to Mary and waited for further explanation. Mary, of course, didn't disappoint.
"I tried to put yours in a spot everyone would see first, but I could not find them."
She sidled closer to the fire Benjamin Troyer had lit in an effort to ward off the dropping temperatures and rubbed her hands together. "I don't know what you're talking about. Who is she and what did she bring?"
Rolling her eyes, Mary dropped onto the vacant bench beside Emma. "Liddy Mast. She brought oatmeal cookies again. But it is okay, because her plate is still full and I did not see a single one of yours anywhere."
Emma glanced over her shoulder, surveyed the table of food, and returned her gaze to start. "Mine are still there."
"Where?" Mary asked, jumping to her feet once again.
"Right next to Liddy's . . ."
Mary marched over to the table and splayed her hands. "I don't see them, Emma."
"The chocolate chip ones are mine," she murmured.
"Chocolate . . ." Mary rested her hands on her hips and stared at Emma. "You always bring oatmeal. Because Levi likes them."
"Liddy Mast brought them to the last hymn sing. Because Levi likes them."
"So . . ."
"He will not notice that I did not bring oatmeal cookies. He will notice only that Liddy did."
"Maybe, if you went over and talked to him . . ."
Emma pointed her chin toward the male version of Mary and the pretty girl hanging on his every word, and waited for her friend to catch up. When she was certain Levi and Liddy's conversation was seen, Emma shrugged. "Levi will smile at me if he catches me looking, but he doesn't speak to me the way he speaks to Liddy."
"Have I ever told you about the time Levi tripped over his own foot and broke his nose?" Mary grabbed two cookies, fast stepped it back to the bench, broke off a piece of chocolate chip cookie, and held it out to Emma. When Emma declined, Mary helped herself. "Levi does not always see what he should see the first time."
"Just because he does not see me, does not mean he doesn't see."
"He broke his nose, Emma. By tripping over his own foot. Who does that?" Mary joked before waving the rest of the first cookie between them. "This is very, very good. Maybe even better than the oatmeal ones."
She mustered a smile worthy of her friend's kind words and then dropped her own voice to a level only Mary could hear. "I thought about what you said at the pond on Monday."
"Good." Mary popped the rest of the cookie into her mouth and grinned. "About what?"
"About trying to find him."
Mary stared at her for a second. "Wait. You mean your birth father?"
"Shhhh . . ." Emma glanced toward Levi, Liddy, and the rest of their peers and, when she was certain they had not heard, returned her words and her focus to her friend. "Yah. My birth father."
Mary took another bite, eyeing Emma closely as she did. "How have things been going at home? With your mamm?"
She fought back the urge to correct Mary's use of the word mamm and, instead, shrugged. "I keep busy with my chores. Sometimes, I feel her watching me, but I do not say anything about it."
"You could ask her about your birth father."
Her answering laugh earned more than a few funny looks from their peers, but, as was always the case, they soon turned away, making it so she could turn to words, instead. "I told you the other day, Mary. She said he didn't care about my real mamm or me."
"That doesn't mean she doesn't know where he is . . ."
"I asked that night, after you and I spoke."
"And?"
"She told me I wasn't to speak of him. Ever."
Mary stilled the second cookie mere inches from her lips, her eyes wide. "Then what are you going to do? How are you going to find him?"
"I don't know. A stuffed horse and a bubble wand can't talk."
"It would be great if they could, wouldn't it?"
"Yah. It would be—wait!" She turned on the bench so fast, her knees crashed into Mary's. "You can see the cemetery from your fields! Perhaps you have seen someone there on the morning of my birthday . . . or maybe even the night before!"
Mary's brows furrowed. "I am not out in the fields at night or that early in the morning. That would be Dat or Levi."
Emma's gaze ricocheted off Mary's and onto the dark-haired man nodding his head at something Liddy was saying while looking at . . .
Emma?
Unsure of what to do or how to respond, she cast her eyes down at her lap, counted to five in her thoughts and, when she looked back up, found him focused on whatever Liddy was saying once again.
"I could ask him if you want. Or, you could ask him!" Mary suggested.
She watched Mary's brother for a few moments and then turned back to her friend. "He is busy. With Liddy."
"I can fix that." Returning to her feet, Mary tugged Emma up and on to her own. "Come with me."
When they reached the outskirts of the group, Mary relinquished Emma's hand a few steps shy of Levi. "Liddy? Could you help me with something over by the food table?"
Liddy's impossibly blue-green eyes widened behind even more impossibly long eyelashes as she stopped talking, followed Mary's finger toward the table, and, after a slight hesitation, nodded her assent.
"Now ask him," Mary whisper-hissed as she fell into step behind Liddy, in route back to the table.
Feeling her hands begin to tremble ever so slightly, Emma steadied them against the sides of her coat and glanced over at Levi. "Hi . . ."
"Hi." He stepped forward, cutting the space between them in half. "I did not see your cookies today."
"Liddy brings oatmeal now, so I made chocolate chip, instead."
"I will need to try one." Levi leaned against the tree at his side. "Did you have a nice birthday?"
Not wanting to lie, she used her sudden shiver to change the subject. "It is hard to think spring will come soon, yah?"
"Are you cold?" he asked, pushing off the tree. "Because we could go sit by the fire if you'd like."
"No . . . I mean, a little, but Liddy will be back soon and I don't want to take you away from—"
He reached up, readjusted his black hat, and then dropped his hand to his side. "I am sure she will find me wherever I am."
"But—"
"Let's sit."
Together, they moved over to the fire. When she was settled on the bench, he sat beside her, warming his hands as he did. "Are you having fun, Emma?"
She tried to nod, even tried to add in a smile, but neither felt normal. For years she'd wondered what it was about her that left her on the outskirts with Levi and the rest of her peers. And for years she'd scurried about trying to fit where she never would. But now, thanks to a heart-shaped necklace and the picture of an Amish girl, she knew her inability to fit in had nothing to do with her bent toward shyness as Mary had always guessed, or any other shortcoming she'd seen in herself.
No, her difference traced back to the very beginning—her very beginning. There was no changing that. For her, or for anyone else.
"You can see the cemetery from your fields, yah?" she asked.
Levi's brows traveled up toward the brim of his hat. "The cemetery? Yah. I can see it from Dat's fields. Why?"
With fingers that were suddenly fidgety, she smoothed down the part of her dress she could see sticking out from the base of her coat. "I . . . I was wondering if you can see people there sometimes."
"See people?"
"Walking around, visiting the graves, that sort of thing."
"It does not happen often, but yah . . ."
She could feel him studying her, clearly wondering about her questions, but she couldn't stop. Not yet. "Is it always Amish who come?"
"It is an Amish cemetery, Emma."
"I know. But perhaps English come sometimes, too?" He looked from Emma to the fire and back again. "Sometimes the English drive by. Sometimes they slow down and look. I have seen some get out of cars and take pictures with their cameras the way they do our farms, and our buggies, and our school. They do not stay long. Except one."
She snapped her head left and stared at Levi. "One?"
"Yah. He does not take pictures. He does not stop at the school or pay me any mind in the field. He just comes to stand inside the cemetery."
"H-he?" she stammered.
"Yah."
"When? When does he come?"
"Every winter."
"E-every . . ." She wiped the ever-increasing dampness from her hands onto her black winter coat. "How do you know?"
"Because I remember the cold, and I remember his jacket. It is black like his truck. It is because of his truck that I see him."
"I don't understand . . ."
"It is hard to see things that early in the morning. A black truck is even harder to see. But I can hear his truck when he comes down the road, and I can hear him shut the door when he gets out, too."
"Does he always come in the morning?" she asked.
"Yah."
Gripping the edge of the bench, she willed the spinning in her head and the thumping in her chest to stop. It didn't. "Have you seen him yet this year?"
"Yah."
"When?"
She followed Levi's gaze to the stars twinkling in the night sky above. "It was just a few days ago."
Her answering and audible inhale pulled more than a few sets of eyes in their direction. "Do you happen to know which day? Like was it Wednesday or—"
"It was Monday."
"Monday?" she echoed. "Are you sure?"
"Yah. I am sure."
Monday . . .
Her birthday and the anniversary of her real mamm's death . . .
The day the locket with her birth mother's picture had been left at the grave . . .
Palming her mouth, Emma forced herself to breathe while mentally picking her way back through Levi's words. Surely it had to be the person who'd been leaving those gifts beside Ruby's grave all these years, didn't it? It was the only thing that made any sense.
"Emma? Are you okay?"
Was she? She wasn't sure. But either way, she needed more information.
"What does he do if he does not take pictures?" she finally asked.
"He bows his head for a long time."
"In prayer?"
"Perhaps."
"Does he say anything?"
After a brief pause, Levi shook his head. "Not that I can hear."
It was all so much to take in, so much to imagine. "Perhaps it is someone different each time?" she suggested.
Again, Levi shook his head. "No. It is the same person."
"How can you be so sure?"
"It is the same shiny black truck."
"There are many black trucks, Levi."
"Yah. But this one has a sign. On the door."
She drew back so hard, she would have toppled off the back of the bench if not for Levi's hand. "A sign? You mean with words?"
"Words and pictures."
"Do you remember what the pictures are?" she prodded. "Or what the words say?"
Slowly, Levi removed his hand from Emma's back and closed it around the edge of the bench, his gaze locked on hers. "I remember both."
"Tell me. Please."
"It is a construction company."
"A construction company? How do you know?"
"Because the sign says Harper Construction. And there is a picture of a hammer on one side and a toolbox on the other. The words and the pictures are white."
"But you said it is dark when he is there. How can you see pictures and read words?"
"He stays until the sun comes up."
Unsure of what to say, Emma turned back to the fire, its decreasing warmth a sure sign that the week's hymn sing was drawing to a close. Soon, the remaining food would be brought into the Troyer home, benches would be put away, and her peers would be getting into buggies and wagons for the drive home. Those who were courting would ride home together. Those who weren't would ride home with a sibling or friend.
"Emma? Is everything okay?"
"Is there anything else you can tell me?" she asked, her voice thick with emotion. "About this Englisher?"
"Yah."
Her gaze flew back to Levi's. "What? Tell me!"
"The sign does not just say Harper Construction. It says New Holland, too."
Chapter 5
She had felt Mamm watching as she'd collected the breakfast plates and carried them to the sink. She'd felt her watching as she swept the floor, and readied the children's lunch pails for the day ahead. In fact, once the little ones had left for school and Jakob and Sarah had headed out to attend to their respective chores, Emma had even caught her getting ready to speak in the reflection of the kettle, but when Mamm's mouth invariably closed without so much as a word spoken, she knew nothing would be forthcoming.
On one hand, she knew she could help ease the week-long tension by acting as if nothing had changed. But, on the other hand, everything had changed, and she simply couldn't pretend otherwise.
"Emma?"
Stilling the dishcloth atop the final plate, she glanced over her shoulder to find Mamm's worried face replaced by Sarah's. "Is something the matter, Sarah?"
"I don't know. You've been so different this past week."
She wished she could argue, but she couldn't. Sarah was right. She was different. It was as if the truth about her birth and her parents had wiped away her incessant urge to please. Sure, she still did her chores and everything that was expected, but now she stopped at that point rather than going beyond. Part of that was because she was angry, sure. But a far bigger part was finally knowing no amount of trying would ever change Mamm and Dat's feelings about her.
"Emma?"
She shook her thoughts back into the moment, added the now-dried plate to the stack inside the cupboard, and hung the damp towel across the stove's handle. "I am fine, Sarah."
"I heard you moving around in your room last night. Even after Dat's lantern went out last night," Sarah protested.
"I-I was thinking." Emma grabbed the broom from its resting spot beside the refrigerator and crossed to the table. "I am sorry if I kept you up."
"Did something happen when we were at the hymn sing? I saw you talking to Levi Fisher by the fire before it was time for us to leave."
"What would happen, Sarah?"
"Perhaps he wants to court you and you do not know what Dat will say?"
She didn't mean to laugh, nor could she stop it. "Levi will soon court Liddy Mast."
"Liddy Mast?"
"Yah. Liddy Mast."
"When did he start liking her?" Sarah asked, her nose scrunched.
"I imagine the first time he saw her at that hymn sing a few weeks ago."
"But—"
Stopping mid-sweep, Emma lifted her hand into the air. "Sarah, it isn't time to talk of such things. There is much work to be done—work I am doing but you are not." Then, with a peek at the clock, she added, "If I am not here when it is time for lunch, please see that the meal is ready when Dat and Jakob come in from the field."
"If you're not here? Why, where would you be?"
Emma glanced toward the stairs and then the window before bringing her attention back to her sister. "I need to use your scooter. My tire is flat."
"My scooter?"
"Yah."
"But why?" Sarah asked. "Where are you going?"
"I just need to go into town."
"Does Mamm know you are going?"
She lifted her chin in defiance. "No. I am twenty-two. I can go into town when I wish."
"Emma!"
"It is true, Sarah. I am not a little girl. My morning chores are done. I am going into town."
Sarah's eyes widened. "Does Dat know?"
"No." Emma pulled the broom handle tightly against her chest and did her best to rein in her growing irritation. "Please, Sarah. There is just something I have to do."
"What am I to say if Mamm asks where you've gone?"
"If you are busy, she won't ask. If that doesn't work, just tell the truth—you don't know where I went."
After a few moments of utter silence, Sarah nodded, pivoted on the toes of her boots, and headed upstairs to the day's waiting laundry. Emma, in turn, eyed the last few crumbs on the floor, returned the broom to its spot beside the refrigerator, and made her way out to the side yard and Sarah's waiting scooter.
She tugged the scooter's handle from its resting spot against the dormant apple tree and walked it toward the driveway, her gaze skittering between the barn and the fields. The telltale ting of a bucket against the ground let her know Jakob was inside the barn, likely getting ready to fill the animals' water troughs. She was curious where Dat was, but to linger any longer than necessary made no sense, especially if she didn't want her plan upset by additional tasks or chores. Besides, the longer she took to get to New Holland and back, the more difficult she made it for Sarah to keep from answering questions.
Stepping her left foot onto the scooter, Emma propelled herself forward with her right, the responding smack of cold air on her cheeks and hands making her wish she'd grabbed her coat on the way out the door. Still, she pressed on—down the driveway, past her neighbors' farms, and onto the main thoroughfare toward town, the occasional buggy-only sighting giving way to one that included cars and trucks, too.
At the traffic light by the Amish eatery popular with the English on vacation, Emma spotted an Amish teenager stacking boxes on the back step. "Hello," she called, scootering to a stop just inches from where he stood. "Do you know New Holland well?"
"Yah."
She pulled her hands from the handles, wiped them down the sides of her dress, and then backed up a few feet in the hope it might minimize any chance he could hear her heart galloping inside her chest. "Have you ever seen a black shiny truck that says Harper Construction on the side?"
"Sure." He pointed to a large tree near the back of the parking lot. "Parks back there most of the time."
She stared at the boy. "You've seen it here?"
"Yah. It is here often—Friday afternoons, mostly." Setting the last box atop the pile, the boy eyed her from head to toe. "He comes in here for lunch after it is busy."
"He?"
"The Englisher who drives the truck."
"Do . . . do you know his name?"
"He told me to call him Harp."
"Harp?" Emma echoed.
The teenager shrugged, then jerked his chin toward the restaurant's back door. "I better get these inside before—"
"Wait!" Tightening her hold on the handles, Emma stepped off the scooter and inched closer. "Is there a building that goes with the truck? You know, like a barn where he builds the things he builds."
"He builds houses. Big houses. He shows me his drawings sometimes when he's waiting for his food to come." The boy wrapped his hand around the bottom box and hoisted the entire pile off the ground. "He said I'm pretty good at reading a floor plan—least that's what he called his picture."
She laid her scooter on its side and then fast stepped it over to the door in an effort to help. "Okay, so maybe he doesn't use a barn for what he builds, but maybe he has an office or something where he draws his pictures?"
The boy maneuvered the boxes through the narrow opening her assistance afforded and then stopped. "He has an office. It's in the old bank building."
She followed the direction indicated by his chin. "Old bank building?"
"Yah. Not far from Otis's Buggy Tours."
* * *
Emma wasn't sure how long she'd been there. An hour? Maybe two? All she knew for certain was that the black shiny truck with the Harper Construction sign Levi had described hadn't moved from its spot outside the back door of the old bank building since she entered the parking lot. Well, that and the fact that the only other car in the lot—a small brown two-door—hadn't moved, either.
Yet every time she tried to tell herself it was time to head home, her feet wouldn't budge from their hiding spot behind the old shed on the edge of a neighboring property. From that spot she could see both the truck and the back door, as well as two different windows that revealed little more than interior lights.
When she'd left the farmhouse, bound for New Holland, Emma's only real goal had been to find Harper Construction. Once she did, the goal shifted to catching a glimpse of the man Levi had seen at the cemetery the morning of her birthday. But the longer she waited for that glimpse, the more agitated she found herself getting.
All her life, she'd believed there was something wrong with her—something she needed to try to change in order to make Mamm smile, or the kids at school include her, or boys want to court her. Maybe she could be a better quilter, do her chores faster, bring just the right cookies to share, or even learn to bat her eyelashes like Liddy Mast. When her efforts fell short as they always had, she tried harder and harder and harder.
But none of it had mattered. None of it was ever going to matter. Because in the end, it wasn't about what she did or didn't do, or how she looked or didn't look. All that mattered was the actions of two people she'd never met—one who died during her birth, and one who hadn't cared enough to even look in on her as she grew.
And then there was Mamm and Dat, or, rather, the people she'd thought were her mamm and dat. Unlike Emma, they'd known the truth. They'd known why she hadn't fit inside their home and community. They'd known that her prayers to be like everyone else were never going to be answered the way she'd hoped. Yet they'd said nothing.
Fisting her hands at her sides, she shifted her focus from the building to the truck. How could someone have a child and not care? How could he go about his life, building houses, and never wonder if his child was even happy? How could he—
She stepped out from the shadow of the shed and headed toward the very door she'd all but memorized from her hiding spot. So much of her life had already been wasted not knowing. Today, that ended. Today, she was going to get the rest of the answers she needed.
With determined steps, Emma made her way around the building and through the front door, the contrast between the late-afternoon sun and the electric lighting that greeted her, momentarily jarring. A few quick blinks, however, helped her eyes adjust enough to be able to pick out an older woman seated at a desk, talking on a telephone. From somewhere beyond the woman's desk, she could make out a deeper, male voice.
"I'll be right with you, Miss."
At Emma's nod, the woman returned to her call, freeing Emma to peruse the framed photographs lining the wall to her right. Although the background in each picture changed, one person remained the same. Leaning closer, she studied the English man's dark blond curly hair, defined chin, wide lips, and eyes the exact same shape and shade of blue as Emma's. . . .
"I'm sorry for that delay. My name is Sue Ellen, how can I help you today, Miss?"
She thought back on the times she'd sat at the dinner table, looking around at her parents and siblings, noting the similarities each sibling had with either Mamm or Dat—similarities she'd always been hard pressed to find in herself. Yet, there, in the picture, she saw her own eyes.... Saw their same smile-born crinkle . . . Saw the—
"Miss? Is everything all right?"
The scrape of the woman's chair against the wood floor snapped Emma back to her surroundings long enough to point at the closest picture. "Who is this man right here?"
Sue Ellen peeked around a large potted plant on her desk. "That's Brad, of course."
"Brad?"
"Brad Harper. President of Harper Construction and"—the woman swiveled her chair just enough to indicate the open office a few feet from her desk and then swiveled back to start—"my boss."
Emma took one last look at the picture and then strode the rest of the way to the desk. "I need to see him."
"Who? Mr. Harper?"
"Yah."
Dropping her chin nearly to her chest, Sue Ellen looked at Emma across the top of her glasses. "Can I ask what this is about?"
"I just need to see him."
"If you tell me what this is regarding, I could make an appointment for you to come back tomorrow." The woman shifted a pile of papers to her left and opened a small black calendar-style book. "He's booked tomorrow morning with a new client, but he has a little time after lunch—say . . . maybe two o'clock?"
"No, I need to see him now. It won't take long."
Sue Ellen paused, took in Emma's kapp and plain Amish dress, and stood. "I'll see if he can spare a moment after he's done with his call."
She considered waiting, but the moment Sue Ellen started for the open door, Emma knew she had to follow.
"Brad? There's someone here to see you."
"Not now, Sue Ellen, I just got off with the foreman out on the Hanson property and the two new guys he hired didn't show up today." The brown leather chair creaked ever so slightly as it swiveled slowly in their direction. "Which means they're gonna fall behind out there if we don't find a pair of brick layers who can step in real quick and get—"
The hitch of her own breath cut short the man's sentence and pulled his gaze past Sue Ellen and onto Emma, his eyes widening a split second before his face drained of all discernible color. "Ru . . . Ruby?" he rasped.
"Actually, it's Emma." She pushed her way past Sue Ellen, thrust out her hand, and opened it to reveal the locket with Ruby's picture. "Did you leave this on my mamm's grave last week?"
"Your-your mamm's grave?" Grabbing hold of his desk for support, the man stood. "No. I left it by . . . no. I-I mean . . . This can't be.... Ruby has been dead for—"
"Twenty-two years this past Monday," Emma finished, her chin raised.
He drew back so fast, his chair banged against the window behind his desk. "How do you know that?" he demanded.
"Because having me is what killed her!" And, just like that, the anger that had brought her to that moment gave way to a steady stream of tears she tried, unsuccessfully, to blink away.
Chapter 6
Brad steadied himself against his desk and, when he seemed confident enough he could stand alone, waved Sue Ellen from the room. Seconds turned to minutes as he stared at Emma and then, unseeingly, at something outside the confines of his office walls. Eventually, he shook himself back into the room, back to Emma. "I . . . I don't know what to . . ." His words trailed away, his eyes pained, his lips trembling. "I-I don't understand. How . . . how can this be? You . . . I mean, I thought . . . I was told you died, too."
Emma wiped away the last of the pesky tears. "If that were so, I would not be here."
"I realize that, I just . . ." Stepping backward, he took her in from the top of her kapp to the tips of her boots before bringing his full attention back to her face. "You have her same cheekbones . . . And the same curve right here." He touched the center of his own top lip only to let his hand drift back to his side. "But your eyes are different. Hers were this pretty brown that sparkled when she smiled. Yours are blue just like . . ."
He covered his mouth with his hand as he took a single step forward and then a half step back. "I-I don't know what to say. I . . . I mean I can't believe this. It doesn't make any sense."
"You did not know?" she rasped.
"Know? Know what?"
"That I lived?"
"No! I thought you died with her!" Raking his fingers through his hair, Brad took off across the office toward his desk, reversed course to the window, and then spun back around toward Emma, the confusion he'd worn just moments earlier morphing first to anger and then . . . joy?
When he reached her, he gathered her hands inside his own and held them tightly. "I don't know what to say . . . You're my-my . . . daughter—Ruby's and my daughter . . ."
"Yah. I mean, yes."
"I can't believe this. It's just . . . I don't know . . . I don't know what to say." He led her over to his desk and invited her to sit in the chair across from his own. "I keep thinking Sue Ellen is going to walk back in here and tell me this is some sort of bad joke. That this is going to end up being just like all my other dreams."
Slowly, Emma lowered herself to the edge of the chair, and after a moment or two of trying to decide what to do with her hands, she simply laid them in her lap. "Other dreams?"
"That Ruby didn't die . . . That you didn't die . . . That I got to build the house we wanted, for us."
She stared at him. "You wanted to build us a house?"
"I did. We did."
"But Mamm—I mean, Rebeccah—said you didn't care about my real mamm. That you didn't care about me."
A darkness not unlike a summer storm cloud passed across his face. "Rebeccah, as in Ruby's sister?" At Emma's nod, he pushed off the edge of the desk, pulling a hand down his face as he did. "That's where you've been this whole time? At Rebeccah and Wayne's place?"
"Yah."
"I sat out on the road, outside their farm so many times after Ruby died, wanting to talk to them, wanting to know something about her death—your death. But Wayne wouldn't let me onto the property. Said Rebeccah had been through enough and my being there was an unnecessary reminder. So I stopped."
A noise much like that of an injured animal followed his words, only to be smacked away by the thump of his fist against his desk. "I stopped! But you were there . . . inside . . . the whole time!"
"Yah."
Clenching his hands into fists, he strode over to the window. "They were wrong! Wrong about my feelings for Ruby, and wrong to keep you from me!"
She looked from the window, to her hands, and back again, her throat constricted by so many emotions she didn't know where to start. "So, it is not true? You did care about her . . . and me?"
"Emma, you . . ." He closed the gap between them with several long steps and then dropped into a squat beside her chair. "My feelings for your mother were like nothing I'd ever felt before—or since. And the word care isn't even in the same universe in terms of my feelings for her. I loved her, Emma. With everything I was and everything I wanted to be. And when she told me about you—that you were coming? I wanted nothing more than to give you a good life, a good home, a loving family."
She considered his words and the emotion with which he said them against everything she knew thus far—which wasn't much. It all sounded good but . . . "If you loved my real mamm the way you say you did . . . and you wanted me the way you say you did . . . then why weren't you there when I was born? Why didn't you know I lived?"
The cloud was back. Only this time, instead of ushering in an outward rage, she sensed a storm brewing behind his eyes. She decided she was right when his hand left hers to wipe at a lone tear. "I want to answer that, Emma, I really do. And I will. But right now, I want to sit with this, with"—he swept his damp fingers toward her face—"you, with all of this. You're . . . alive. You're my-my daughter."
Her lips trembled their way into a smile as she, too, wiped at her own tears. "Yah."
"You look so much like her, Emma. Even now, when you're crying. It's like my life has rewound back some twenty-two years and she's actually here, sitting in the office I told her we'd have one day . . . Only it's not her. It's you. . . . And you have my eyes and"—he leaned forward, his brows knitted—"my mother's chin."
Emma flew her hand to her chin and fingered it gently. "Your mother's chin?"
"Yup. My mom—your grandmother—has that same chin. One of her best features, in fact."
She knew she shouldn't be surprised to hear of a grandmother. It made sense, actually. But she'd been so thrown by the truth surrounding her birth, she hadn't let her thoughts move beyond Ruby and . . .
"I-I don't know what to call you," Emma whispered. "You are my real dat, but you are not Amish."
He returned his hand to hers and held it close. "We don't have to figure all of that out right now. We have tomorrow, and every day after to decide all of that."
Tomorrow . . .
A glance at the window and the waning daylight brought her to her feet. "I-I have to go. It is getting late and my scooter cannot be seen on the road when it is dark."
"Whoa. Whoa." He straightened to a stand. "Slow down. I'll drive you home if that's where you want to go. But if you don't, I can bring you back to my place or to my mom's if that would make you feel more comfortable. And I—"
Palming his mouth, he stepped back, his eyes wide. "My mom is going to freak when she sees you."
"No . . . I have to go back to the farm." Again, Emma looked at the window. "I-I told my sister Sarah that I would be back. I have already left her too long. Now she will have to give answers she should not have to give."
"But—"
"Please. I-I must go." Emma turned toward the door but stopped before she'd gone more than a step. Glancing over her shoulder, she soaked up the sense of belonging she found in his face and the unfamiliar flash of confidence it gave her in return. "I could come back tomorrow, if that's okay?"
"If that's okay?" He grabbed his keys off the top of his desk and fairly ran back to her side. "Trust me, Emma, there is nothing in this entire world that would make me happier than that."
* * *
He pulled her scooter from the back of his black shiny truck and set it on the ground next to her feet, the happiness he'd worn on the way out to the farmhouse now concealed by a wariness in everything from the way he moved, to the way he looked from Emma to the driveway and back again. "Are you sure I can't take you all the way up to the house?"
"Yah."
"Because there is a lot I want to say to them—to both of them."
"Yah. But not today. Today, it is for me to say." She followed his gaze past the maple tree to the corner of the barn she could see from where he parked. There was no visible sign of her brothers, but she knew they were near, likely inside, finishing up with the animals before dinner. "Jakob, Sarah, Jonathan, Annie, and Esther do not know the things I have learned since my birthday. I do not want them to find out this way."
"Those are Wayne and Rebeccah's children?"
Wayne and Rebeccah's children . . .
It sounded so odd to hear them described that way. Without her own name in the mix. And, for a split second, the sudden tightness in her throat almost led to her correcting him. But she couldn't. Wayne and Rebeccah were not her parents. And Jakob, Sarah, Jonathan, Annie, and Esther were not her siblings.
Closing her eyes against a flurry of memories that had her holding a newborn Esther, teaching Annie to make a pie, hugging Jonathan after he scraped his knee when he was not more than four, helping Sarah with her quilts, and playing with Jakob by the pond, she willed herself not to cry.
Everything she'd ever known to be true was now different.
She was different.
"Emma?"
"Yah." She parted her lashes to reveal the barn once again. "They are Wayne and Rebeccah's children."
"Jakob is the oldest?"
"No, I . . ." She stopped, inhaled, and wrapped her hands around the handles of Sarah's scooter in preparation for the ride up to the house. "Yah. Jakob is the oldest. Esther is the youngest. She is five."
"How old is he?"
"Jakob?" At Brad's nod, Emma swallowed. "He is twenty-one."
Cupping his mouth with his hand, Brad rocked back on his heels. "I wonder how many times, when I'd sit down here on the road, hoping to talk to Rebeccah those next few years, that the child I'd seen her holding had actually been you."
"Me?"
"Yeah. I mean, I was younger than you are now. I hadn't been around babies, so I had no idea how big they should or shouldn't be at different ages. Wayne never let me get closer than where I'm parked right now, so when I'd see a little one in Rebeccah's arms on occasion, I assumed it was hers."
She could hear his growing anger, even felt some of her own beginning to press at the bout of sadness, but the sound of Jakob securing the latch on the chicken coop meant it all had to stop. For the moment.
"I really should go." Emma walked the scooter away from the truck but waited to actually step on. "Dinner will be ready for the table and I must help Sarah and Annie."
Brad shut the truck's rear gate with an echoing thump. "I will see you again, right? Tomorrow? Around lunchtime?"
A week ago, she would never have made plans during the height of the workday, certainly not without seeking permission from Mamm or Dat first. But it wasn't a week ago, it was now. "Yah. If you are not too busy."
His laugh punctured the evening air. "To get a second chance like I'm getting right now? No, I will never be too busy for you. In fact, if you're good with lunchtime, I'll get what I need to get done in the morning, and then I can turn it over to Sue Ellen in the office and to my foreman out in the field for the rest of the day."
She met his clear blue eyes and felt the answering shiver that moved its way up her spine. For the first time, she knew what her own eyes looked like in the light of early evening, and how they must look when she, too, was both excited and nervous all at the same time. "I do not need to take your whole afternoon."
"Yes, you do. You need to take many, many of my afternoons." The thin graveled road popped beneath his work boots as he drew closer. "Emma, we have twenty-two years of time together to make up for. And I have twenty-two years of your life to catch up on. I don't intend to rush that."
"Should I come to the same place tomorrow?"
"Yes, or you could call me and I'll come get you."
She pulled a face. "I am Amish. I don't have a phone."
"That's right . . ." He led her focus down the street toward the next closest farm. "What about the phone that used to be in that little wooden shack between this place and Weaver's? Did they pull that out or something?"
"No, it is still there. But it is for business and emergencies." She studied him for a moment, a dozen questions suddenly filling her thoughts. "Did Ruby use that phone to call you?"
His nod was slow, distant. "Sometimes, yeah." Then, shaking off the memory, he turned back to Emma. "I could just be out here waiting at a certain time, if that's easier."
"No, I'll come to you."
"That's a long way to go on a scooter, Emma. Especially at this time of year. Besides, I'd feel better knowing you're safe."
"It is not too long, and I have scootered to town many times," she protested.
"Emma, they have to find out at some point. One way or the other."
He was right. They did. But dealing with anger, as she was where Mamm and Dat were concerned, was somehow easier than the hurt that would come from sharing the truth with her siblings.
"You didn't create this situation, Emma. Remember that. Rebeccah and Wayne did. With their lies." Reaching into his pocket, he extracted the locket she'd shoved at him in his office and unhooked the clasp. "When I left this for you, I never thought I'd actually be able to put it around your neck."
Confusion pushed her chin back. "You didn't leave it for me. You left it on Ruby's . . ."
And then she knew.
The gifts she'd been stashing away in the hollow of the oak tree by Miller's Pond since her seventh birthday had, in fact, been for her all along. Left by a man who'd clearly believed she'd been buried in her mother's arms.
"I-I'm sorry you didn't know," she whispered.
"So am I, Emma." He set the locket on his palm and opened it to the picture inside, the answering pain in his eyes unmistakable. "There's not a day that goes by that I don't think of her. And there's not a day that's gone by that I didn't think about you, mourn you, imagine you—all of it. Sometimes, clients bring their kids to a meeting. And while they're talking about stuff like closet size and window seats, I'm looking at their kids, trying to figure out how old they are and what you might have looked like at that age. Funny thing is, no matter what age they were, the face I always saw in my mind was a variation of Ruby's."
Emma sucked in a breath. "Did you know I was a girl?"
Glancing up from the locket, he met and held her gaze. "I did. That was the one thing Wayne told me when he came out of the house that day."
"What day?"
"The day you were born. I stood right there"—he pointed to the part of the driveway that met the road—"and waited to hear how Ruby was. How you were. And after a few hours, Wayne came out and told me the two of you were gone." He looked again at the locket, his voice thick, raspy. "I remember feeling as if someone punched me in the gut. I couldn't breathe, I couldn't think. But when Wayne started to walk away, I managed to pull it together enough to ask him what you were."
At a loss for what to say, Emma took the locket and chain from his hand and gazed down at the face so like her own. "I-I really do look like her, don't I?"
"Spitting image, minus your blue eyes and blond hair. But even being blue, your eyes are the same shape."
"I wish I could have known her," she managed past the lump she couldn't seem to dislodge no matter how hard she swallowed or how many times she tried to clear her throat. "How she spoke, the kinds of things she liked to do, the things that made her smile, and if she was excited to have me."
"She was very excited to have you. We both were."
"Then . . ." Emma stopped, blinked at the tears beginning to dapple her lashes, and tried to steady her breath enough to continue. But just as she was beginning to doubt she could, her hand and the locket disappeared inside his.
"There is so much I want to tell you, Emma. Things I want you to know, to experience, to believe with your whole heart. And you will, because I'm going to tell you and show you everything." His blue eyes, a mirror image of her own, were waiting when she looked up. "Starting tomorrow."
Chapter 7
She leaned the scooter against the tree and quickly smoothed her hands down the front and sides of her dress. Jakob's voice from somewhere in the vicinity of the barn let her know supper hadn't started. Yet.
Still, the position of the sun in the western sky told her it wasn't far off.
Inhaling the cold winter air, Emma made her way onto the porch and over to the front door, her ears perked for anything that might indicate Mamm's or Dat's exact location. If she were to guess, Dat was in the barn finishing up a few final tasks with Jakob and Jonathan, and Mamm was preparing the dinner plates while Sarah, Annie, and Esther waited to set them on the table.
The thing she couldn't quite guess was what Mamm's reaction would be when Emma strolled into the house after an unexplained absence of nearly six hours. Would she be angry? Would she be upset? Would she be silent?
Shrugging off the imagined answers she wasn't sure she even cared about, Emma let the screen door bang closed at her back and made her way toward the beckoning smells of homemade bread and chicken stew.
"I'm here," she called, wiggling out of her coat.
A sharp intake of air pulled her gaze from her coat's hook to the table in time to see Sarah finish with the napkins and practically run to Emma's side. "Emma! Where have you been? I did not want to tell Mamm that you had gone, but when you were not here to help with lunch or to bake the bread for tonight, Mamm asked me if you'd left. I could not pretend I did not know."
"I know. I wouldn't expect you to lie, Sarah." Emma motioned toward the counter and the stew bowls that stood stacked and waiting to be filled. "Where is she?"
"Who? Mamm?"
It was on the tip of her tongue to correct Sarah by inserting Rebeccah in place of Mamm, but she settled, instead, for a simple nod. Now was not the time or place.
Sarah swept her hand toward the stairs. "She has not been down since Dat and Jakob went back out to the barn after lunch."
"But that was hours ago," Emma protested. "It is Monday! There was bread to be made and—"
"I made the bread. And I sent Annie next door to the Weavers' with a loaf as Mamm always does."
"That is good, but it is not like her to be upstairs for so long."
"I do not think she is feeling well," Sarah said.
Emma stared at her sister. "Have you checked to see if that is so?"
"Yah."
"And?" Emma prodded.
"She said she was not sick and that I was to come down here and make the bread. But she did not look well."
Emma cocked her ear for any sound of life coming from the second floor, but there was nothing. "Is everything ready for dinner?" Emma glanced out the window at the gathering dusk and then back at Sarah, waiting.
"Yah."
"Then as soon as Dat and the boys come inside, start filling the bowls."
"What are you going to do?" Sarah asked.
"I will go upstairs and check on her." Slowly, Emma made her way up the same staircase she'd climbed since she was old enough to walk, each step delivering up a crystal-clear memory from a life that had been lived around a lie. There were the races with Jakob that had always ended with him reaching the top step first.... There were the creaky steps she'd tried to avoid while sneaking downstairs in the hope she might catch a glimpse of whatever new animal had come into the world.... And there was the anticipation that had accompanied her up the steps to her parents' room every time a new sibling had been born....
Only they hadn't been siblings.
They'd been cousins.
And Rebeccah and Wayne hadn't been her mamm and dat.
Ruby had been. And Brad was.
At the top step, Emma turned left, clenching and unclenching her hands at her sides. She knew she should be concerned about the woman at the end of the hall—a woman she'd called Mamm her whole life. But she wasn't. What she was, was angry, and it was an anger that was only growing stronger, if the unfamiliar heat in her face was any indication.
She'd seen anger before. She'd stood next to it each of her first seven birthdays. And even though she'd been small and hadn't understood its underlying origin, she'd felt its power.
Now, that anger was hers.
It was raw and it was intense and she had no desire to push it away for anyone, least of all the woman responsible for its presence in the first place. Still, a lifetime of habit had her knocking on the partially open door rather than pushing her way inside.
"Sarah and Annie have everything ready," Emma said through the opening. "They will put the stew into the bowls when Dat and the boys are ready."
She turned back to the stairs, only to stop at the sound of her name. Part of her wanted to ignore it, to simply go down to the kitchen and wait for dinner to begin. Another part wanted to ignore it and bypass the kitchen altogether in favor of shutting herself away in her own room, away from everyone and everything connected to the first twenty-two years of her life. And still, another part wanted to heed the invitation into the room if for no other reason than to share the details of her day—a day that had started with questions she never should have had, and ended with answers that changed everything. Including her feelings for the woman on the other side of the door.
The pull of the latter won.
Pushing her way into the room, Emma felt the immediate hitch to her breath as her gaze fell on the lone figure in the room—a figure who, with the exception of Emma's birthday, had always seemed so strong. Yet there, standing beside the window with stooped shoulders and red-rimmed eyes, strong was not the word she'd pick for Rebeccah Lapp.
"I was worried about you, Emma. You did not tell me you were leaving."
"You are right. I did not."
Rebeccah crossed to the bed she shared with her husband and sat down. "You were gone a long time. Jonathan said you were not at Miller's Pond."
"You sent Jonathan to look for me?"
"Yah."
Nodding, Emma claimed the now vacant spot at the window. A precursory look at the barn straight ahead, and the road in the distance, quickly bowed to her overwhelming need to shock. "I saw him today," she said.
"Him?"
"My father. My real father."
And just like that, any natural color drained from Rebeccah's face, taking with it the strained yet muted aura that had filled the room just moments earlier. Before the woman could speak, though, Emma continued. "He was surprised to see me. To hear that I"—she touched the front of her chest—"am alive. To know that I did not die with Ruby the way he had been told. By you and by Wayne."
"Emma! What have you done?"
Dropping her hands to her sides, Emma stepped forward, her eyes locked on the pair staring back at her as if Emma was in the wrong. "What have I done?" she spit back. "I am not the one who did this! You are. You and your husband!"
"Emma—"
"No! I never got to know my mamm. I knew her only as your sister, Ruby. Every year we would go to the cemetery and you would be sad. But she was not just your sister. She was my mamm. She died having me. But I didn't know because you didn't tell me. And then, when I found out and asked about my birth father, you said he didn't care. That he didn't care about Ruby or me. But you were wrong! I know this now because I found him and he told me."
"He didn't care about you—either of you!"
Anger propelled her forward, closing the gap between them to mere inches. "You do not speak the truth! You haven't for twenty-two years!"
"Emma!"
"You did not speak the truth to me, and you did not speak the truth to my real father. But now he knows. Because of me! Because I found him! Now he knows I did not die with her. . . . That I lived right here in this house the whole time."
Rebeccah dropped her head into her palms. "Oh, Emma, what have you done?"
"I told the truth! And now, because I did, I will finally know the truth. About my real mamm. About my real dat. About the way they loved each other and—"
"He should not have been with my sister," Rebeccah thundered back, looking up. "He is why she is dead!"
The words drew tears Emma fought to blink away. "No. I am why she is dead. And because of that, I will never know her. But I can know him, and I will. Starting tomorrow."
Rebeccah's answering gasp echoed around the room. "Emma, you can't!"
"Can't what? Get to know the man who would have raised me if you hadn't told him I was dead?" Wiping her face, Emma made her way back to the door while her words, her thoughts remained in the room. "For twenty-two years, you kept me from him. You told him I had died, and you let me believe I belonged here—with you. But I didn't die, and I never belonged here."
"Of course you belonged—"
Emma whirled around, hand up. "The Bible says, 'The lip of truth shall be established forever, but a lying tongue is but for a moment.' But the Bible is wrong! Twenty-two years was not a moment for me. It was my forever. But no more. Tomorrow, my life—the life I should have had all along—will start. With my father."
Chapter 8
The sudden flash of sunlight across the trio of smiling faces propelled her gaze toward Sue Ellen and then onto the backlit figure making haste through the front door of Harper Construction.
"I'm here.... I'm here.... The woman behind the deli counter was new and so it all took a lot longer than it . . ." The deep voice trailed off, returning, seconds later, peppered with hesitation and uncertainty. "Where is she? Where's Emma? I saw her scooter parked out back."
Swiveling her chair to the right, the receptionist's smile directed Brad's gaze through his office doorway and onto Emma. "Ta da!"
"You came." He set two plastic bags atop Sue Ellen's desk and then closed the gap between them with three slow yet deliberate strides. "I was afraid they wouldn't let you."
"It is not for them to choose." Returning her attention to the framed photograph at her shoulder, Emma pointed at the young family depicted. "Are they your friends?"
"Clients, actually."
"They look so happy," she said, staring, again, at the face-splitting smiles worn by the three-member family.
"That's because it was a very big day for them."
"A big day?"
"The day they moved into their first real home."
Confused, Emma looked back at him. "They did not have a home before this?"
"Not a house, no. They lived in an apartment in the city."
Again, she looked at the picture, but this time she focused beyond the exuberant faces to the pale yellow house with a wide front porch and a pillow-topped swing. On the top step, leaning up against a white spindled rail, was a small brown teddy bear and a bright blue suitcase. "An apartment is small, yah?"
"They can be. Especially in cities. Theirs had two bedrooms, but there was not much room for the little girl to play. The mother loved to cook and bake and wanted to involve the little girl in that, but the kitchen was so small she could barely fit by herself. But the house changed all that. The little girl got a yard and a playroom, and the mom got a big kitchen. And now there's another little one in the third bedroom."
Brad nudged her attention toward the next framed photograph and the elderly couple sitting peacefully on the back deck of a home with more windows than Emma had ever seen. "The Donnelsons, there? They downsized. Their kids are grown and so they decided to build a smaller home here, and get a vacation condo at the beach. They spend a lot of their time out on that very deck, reading, chatting, and singing."
"Singing?"
"Well, technically, she sings. He plays the guitar—something he loved doing as a kid but gave up when it came time to get married and raise a family. Retiring and downsizing gave him back that time." Rocking back on his heels, Brad ran his fingers along his jawline. "I love that shot because it encapsulates their reason for building—to slow down, to breathe, to soak up life. Whereas, with the Regans"—he swept his hand toward the shot of the family in front of the yellow house—"you get the feeling they can't wait to get busy with their new life. Two very different ends of the spectrum in a lot of ways, yet both ended up being really memorable projects for me."
"For you?"
His blue eyes met and held hers. "I designed and built both of those houses, Emma. It's what I do. What Harper Construction—my company—does."
Unsure of what to say, Emma dropped her focus to the floor and swallowed. "I-I did not know."
The feel of his hand around hers pulled her focus back. "Hey . . . You not knowing things about me, and me not knowing things about you, is not our fault, kiddo. But we're going to change that, starting now. Okay?"
She was pretty sure she nodded, although it was possible she just imagined it. Either way, he squeezed her hand ever so gently and then hooked his thumb toward Sue Ellen's desk. "I wasn't sure what you like to eat, so I just got a little bit of everything until I figure it out. The picnic table in back works for me unless there's somewhere else you'd rather go to eat?"
"I don't know about a picnic table."
His shoulders rose and fell with a shrug. "We could go to a real restaurant if you'd prefer, but I thought, since it's not all that cold today, that maybe eating outside would be nicer. You know, less distractions and stuff while we get to know each other—"
"Boss?"
Emma followed Brad's attention to the doorway and the woman standing inside it. "Yes, Sue Ellen . . ."
"The picnic basket is packed and ready to go."
"Thanks, Sue Ellen." He returned his smile to Emma. "So are you good with the picnic table in back, or would you prefer a park somewhere?"
His question hovered in her thoughts as she looked back at the picture of the young family.
The woman's pure joy . . .
The man's arm wrapped casually, yet protectively, around his wife and child . . .
The little girl's squeal of excitement you didn't have to hear to know . . .
They were people Emma had never met. People she saw only through a picture frame. Yet standing there, looking at them, she felt as if she'd known them her whole life.
The joy . . .
The protectiveness . . .
The squeals . . .
Only for Emma, they had been part of a dream she'd tried to shake off more times than she could count—convinced her thoughts were a sign of an ungrateful heart.
"Emma?"
She looked past the family to the yellow house, her mind's eye soaking up the suitcase, the teddy bear, and the front door that stood open and waiting. The image blurred as she imagined herself stepping inside, the sound of laughter guiding her feet down the hall. In the kitchen, she saw the woman placing cookies on a plate she then carried over to the little girl seated at the table. Soon, their comfortable chatter and warm laughter beckoned the man inside.
Every time she tried to conjure up a topic for them to talk about in her thoughts, it faded against the simple sound of laughter and . . . ease. The way it did when you truly belonged somewhere.
"Emma? Is everything—"
Breathing in a sudden burst of clarity, she turned, the location for their first lunch together practically rolling off her tongue. "I would like to have our picnic at Miller's Pond."
* * *
Emma settled onto a corner of the blue and black checked blanket and carefully arranged the hem of her dress across the upper edge of her black boots. "It is the first time in many days that I do not see my breath when I am here. Perhaps it will be an early spring."
"I take it you come here often?" Brad leaned against the trunk of the oak tree and slowly unwrapped his sandwich.
"Yah." Pulling her sandwich onto her lap, Emma looked out over the pond, her thoughts wandering to her favorite rock on the other side of the tree. "It is where I come to think and to feel . . . better."
He balled up the wrapper and tucked it inside the empty basket. "You come here, to Miller's Pond, when you're sick?"
"No. When I am sick, I stay close to home."
"Then what did you mean when you said you come here to feel better?"
She dropped her focus to her lap and slowly began to unwrap her own lunch. "Sometimes it is nice to not have to try so hard to be me. I do not have to think of different ways to get a smile or what I must change to make people like me. I can just come here and be me."
For a long moment, he said nothing, his eyes searching her face the way she might search the henhouse for any missed eggs. When he finally spoke, his voice, his words, his interest seemed to still the air around them. "You should always be you, Emma. Always."
She listened to the echo of her laugh as she looked at the pond once again. "When I was in school, I would wonder why the other children did not wave me over to play games like they did with each other. So I would put extra cookies into my lunch pail—to make them like me. It did not work. At the hymn sings I go to, I see the smiles and hear the happy shouts when people win a volleyball game. So when I play, I work hard to help my side win. But when it is time to be silly after we win, they are silly together. Without me. And at home, the other children can make Mamm smile with her eyes. I cannot. Doing extra chores and having Englishers want to buy my quilts at the road stand does not change that."
Returning her attention to her sandwich, she unwrapped it and took a bite, the ham and cheese she'd finally settled on proving to be a good choice. "But now I don't have to wonder why these things are so, and I don't have to keep trying to think of different ways to fit. Because I won't. Not here, anyway."
"You lost me, kiddo. You won't what?"
"Fit." She plucked off the part of the cheese overhanging the edge of her bread and popped it into her mouth. "Cookies and quilts can't change how I was made."
Brad leaned forward. "How you were made?"
"Yah. My real mamm, Ruby, was Amish. You are English. There was no marriage. When people look at me, they see something bad—something wrong."
Pushing his sandwich off his lap, Brad parted the fruit and chips from their resting place in the center of the blanket and scooted forward. "I loved your mother, Emma. Loved her with my whole heart. I wanted a life with her. I wanted a life with you."
She liked how it all sounded, she really did. But—
"How did the two of you meet?" she asked, taking yet another bite of her sandwich. "Was it during her Rumspringa?"
A slow smile gathered at the corners of his mouth as his gaze traveled somewhere far beyond her face. "I was working for my uncle that first summer. Picking up nails, moving material, fetching drinks for the crew, that sort of thing. It wasn't necessarily how I wanted to spend my summer, but schoolwork wasn't really my strong point and my mom wanted to start exposing me to things I could do to earn a living in the future. She let me choose between working with my cousin or my uncle. Being around guys who were repairing things sounded infinitely more appealing to me than cutting grass, so I opted for my uncle's fix-it business instead of my cousin's lawn service."
Brad cupped his hand across his mouth, only to let it slide back down to his thigh. "I remember the day I first laid eyes on Ruby like it was yesterday. I was out on a job site not far from here. Woman's front porch was sagging and it needed shoring up. My uncle had moved on to the steps and needed a level he'd left in his truck. Since I was essentially his gopher for the summer, it was up to me to drop what I was doing and go get it for him. I walked down to his truck, popped open one of his toolboxes, didn't find the level he wanted, and moved on to the second toolbox. Took some rummaging, but I found the right one and put everything else back inside the box. I was just stepping away from the truck when I saw her walking up the road. She had a plate of cookies in one hand, and a loaf of bread in the other. And when I waved, she gave me the prettiest smile I'd ever seen. Felt a reaction clear down to my toes."
Shaking his head quickly, he scooted his way back across the blanket and reclaimed his sandwich. "I'd seen Amish hundreds of times. Can't live in Lancaster County and not see them. Never paid much attention, really. They lived in their world, I lived in mine. But that day? Standing there next to the truck, looking at Ruby? There was only one world that mattered and it very definitely had her in it. So I stepped down to the road and I asked her name. Only the first time she told me, I was so busy looking at her it didn't really register. She was wearing this pale green dress, and just this part of her hair"—he touched his hairline, then pointed at Emma's—"was showing underneath her kapp. The sun was hitting it in such a way, it sparkled. And her eyes? They were this pretty brown, but in the sun, as they were that day, there were these little flecks of gold, too."
He took a big bite and then another as he leaned back against the tree. "I know I had to have looked like a fool at that moment, just standing there, staring at her. But I couldn't help myself. Fortunately for me, the sound of my uncle yelling for the level got me back on track. And that's when I realized I hadn't caught her name. So I asked again. And she answered again. I asked her where she was going and she told me she was bringing the bread and the cookies to the woman whose porch my uncle was fixing.
"Next thing I knew, we were walking side by side up the driveway. To this day, I can't remember handing my uncle that level. I know he was sitting up, looking mighty grumpy when we approached, but I didn't care about anything except Ruby. And when she went inside to deliver the cookies and the bread, I kept that front door in my sight so I'd know the second she came back outside."
He took another bite, grinning as he chewed. "And you know what? When she finished up inside, she came out with an oatmeal cookie wrapped inside a napkin for me. Best cookie I ever had, I'll tell you that."
"So what happened next?" she asked, her curiosity piqued as much by his words as the joy he wore while speaking them.
"I walked her back to the road. And even though the driveway was pretty short, I made the most out of that walk. I found out she was seventeen, that she had one sister and three brothers, and that she worked at a little ice cream shop out on the county road on Wednesdays, Fridays, and Saturdays. And before I was ready, we reached the road and she said she had to get back home to help her sister with dinner. So she headed back in the direction she'd come and I just stood there, watching. She turned around a few times, and every time she did, I waved. After about the fifth time of her turning and me still standing there, she tried to shoo me toward the driveway. When I didn't budge, she laughed. And, Emma? The second I heard that sound, I knew I had to see her again."
Intrigued, Emma swapped her sandwich for a chip and slowly nibbled her way around the outer edge. "Did you get in trouble for standing there so long?"
"You mean with my uncle? Nah. Said he knew I was smitten the moment the two of us walked up the driveway together."
"Your smile is so very big right now."
"That's because I'm thinking about your mother. But really, you should have seen her smile. There were so many things I loved about Ruby, but her smile? It was the best. Distracting as all get-out, but wow."
Emma brushed the residual chip crumbs from her hands and then hugged her knees to her chest. "Tell me more. Please."
And so he did. He talked of borrowing his mother's car to drive out to the ice cream shop where Ruby worked that Friday, and how he kept swapping places with other people in line until he was sure it would be Ruby who would take his order.
"I still remember that moment when she looked up from the ice cream case and she realized who I was." Interlacing his fingers between his head and the tree, Brad lifted his chin until he could see the sky through the bare branches. "She was halfway through the same can I help you I'd heard her say at least a half dozen times while I was playing leapfrog in line, when her eyes widened, the words stopped, and that smile I hadn't been able to forget since the previous day was trained squarely on me.
"For a minute, I actually forgot there was a line of people behind me. All I could do was just stand there, smiling back at her. The longer I stood there, smiling, the pinker her cheeks got until the girl working the case next to her said something in Ruby's ear, prompting Ruby to ask me for my order."
"Was it hard to leave?" Emma prodded, fascinated.
"It would've been, sure. But I didn't leave. I gave her my order, which I still remember—vanilla with this peanut butter hard shell stuff—and watched her scoop it into the cone. When she handed it to me, I gave her the money and found a small table in the corner where I could eat it.... Though it pretty much melted down my hand on account of the fact I spent more time watching her than actually working on my cone."
"Did she know you were still there?"
"At first, no. But after a while, the girl she was working with looked up, spotted me watching Ruby, and whispered something to her. Next thing I knew, Ruby's cheeks were all red again, and she was peeking at me, peeking at her." Brad's laugh cut through the still air. "After a while, I couldn't keep sitting there, you know? My ice cream was gone and people wanted my table.... So I moved outside and sat on a bench. Two hours later, when her shift was up, I was there, waiting."
Emma reached for a cookie and rested it atop her knees. "Was she surprised?"
"She was. She was even more surprised when I offered to drive her home. But she accepted and we talked all the way back to her parents' farm."
"What did you talk about?" Emma asked. "Or do you not remember any longer?"
"I remember everything about my time with Ruby . . . everything." Dropping his hands back to his lap, Brad reached for his drink and took a sip. "I asked her about being Amish. She explained to me that she was getting ready to be baptized soon. She asked me about school and my summer and why I'd been at her English neighbor's house the previous day. So I told her about my job with my uncle and how my mom wanted me to learn a trade. When we passed the turnoff to this very spot, she pointed it out to me and told me how it was one of her favorite places to go and think."
Emma drew back. "Ruby came here? To think?"
"She did." He pointed her attention to the large rock Emma knew all too well. "She liked to sit right there and look out over the pond at all different times of day, but mostly late afternoon, when the sun's position made it so the top of the pond was—"
"Covered with sparkles," Emma finished in a gasped whisper. "I-I sit on that same rock! And I like that part of day best, too!"
Draping his hand atop hers, he squeezed. "Like mother, like daughter, it appears."
It was a lot to take in. A lot to digest. Still, she wanted more. Needed more. "So what happened when you got to Grossdawdy's house?"
"Grossdawdy? Who is that?"
"That means grandfather. But he would not have been Ruby's grossdawdy. He was her dat."
"Ahhh, okay." Brad tugged at a blade of dead grass beside the blanket and, when the ground released it, wrapped it around his finger. "Here I thought I knew everything there was to know about the Amish, yet that is a word I did not know."
She broke off a piece of her uneaten cookie but stopped short of taking a bite. "That is because Mamm and Ruby's grossdawdy went to the Lord when Mamm—I mean, Rebeccah, was not much older than my sister Annie."
When he reached the blade's end, he unwrapped it and tossed it back onto the earth, his eyes returning to hers. "When I dropped Ruby off that night, I knew I had to see her again. When I said that to her, her cheeks grew red again, but she did not say no. She said only that Sunday was the Lord's day—that it was a day of worship and, later, a hymn sing with friends. So I reminded her Sunday was still two days away. That Saturday came first. When she did not say anything, I thanked her for telling me about Miller's Pond and that I was going to check it out the next afternoon with my fishing pole. And then, when I drove away, I said a prayer—something I didn't do much of as an eighteen-year-old boy who was too busy being an eighteen-year-old boy."
"You said a prayer? Why?"
"That Ruby would show up."
"And she did." Emma didn't pose it as a question. She didn't need to. Her very existence on this earth made the statement safe.
This time, when Brad smiled, it was both happy and sad all at the same time. "That day was our first date as far as I'm concerned. I remember it all. I remember the feel of the sun on my left cheek as I stood right there." He pointed to the edge of the pond closest to them. "I remember the way my heart started thumping the second I heard the crunching of old leaves behind me. Because when I did, I knew my prayer had been answered. And I remember the way her hands were trembling when I turned around."
"Why were her hands trembling?" Emma asked, wide-eyed.
"She was nervous. I was nervous. We knew why we were both there—I wanted to see her, and she wanted to see me." Raking his fingers through his wavy blond hair, Brad exhaled a burst of air at the memory. "I thought about asking her if she wanted to try fishing, but I didn't want to spend our time together worrying about whether anything was biting on the line. So I suggested a walk. And that's when she pointed to her—and now your—rock and said maybe we could just sit. And talk."
He brushed away crumbs from the blanket between them and then took another bite of his sandwich. "I learned about her family, she learned about mine. She told me she enjoyed painting on milk cans, and I told her my favorite part of working with my uncle was when I got to build things—steps, decks, sheds. Told her I wanted to build whole houses one day. And, after a while, she told me about Samuel Gingerich."
Emma stared at Brad across her half-eaten cookie. "I know Samuel Gingerich. He lives on the other side of the Beiler farm. He and his wife, Hannah, have six children and a seventh on the way . . ."
"Seven kids," Brad mumbled. "Wow."
Pushing off the blanket, he wandered over to her favorite rock. Emma, in turn, abandoned the rest of her cookie and followed.
"How do you know Samuel Gingerich?" she asked.
"I don't." Brad ran his fingers along the flattest part of the rock, closing his eyes briefly as he did. "Not really, anyway. Ruby told me this Samuel guy had driven her home from the previous hymn sing. And with me being English, I didn't really register what she was saying, at first. I mean, I gave female classmates lifts home after school all the time. But as we sat and talked, she explained to me how courting works. How, once she was baptized, the next step in her life would be marriage. Samuel Gingerich was looking to be the one she'd start courting and then marry."
Shaking off the memory, he retracted his hand from the rock and slowly lowered himself to the very spot where Emma often sat. "I tell you, Emma, hearing that was like getting hit with a two-by-four or something. I know I'd literally only met her two days earlier, and even with that, I still didn't know her beyond a few conversations, but the thought of her getting married? To someone else? I couldn't do it.... I couldn't let it happen."
Mesmerized, Emma sat down on the far end of the rock, Brad's ability to make her feel as if she was at the pond, with him and Ruby, like nothing she'd ever experienced before. "What did you do?"
"I asked her to give me a little time to get to know her." His gaze skirted the pond and their food-strewn blanket before settling back on Emma. "She told me about her upcoming baptism and what it meant. That she couldn't just take up with me, an Englisher. So I asked her about the way she kept looking back at me that first day, and about her reaction to seeing me at the ice cream shop the next day, and about her being there, with me, at the pond at that moment. And then, before she could answer, I told her why I had stood there, watching her walk away that first day. . . . Why I went to the ice cream shop and waited to drive her home afterward . . . And why I'd prayed that she would come to the pond that day . . ."
"Why did you do all those things?" Emma asked.
"Because the moment I saw her, I knew Ruby was the one for me."
She didn't know what to say, so she stayed silent, the man's words, and the raw honesty with which they were spoken, in direct contrast to Mamm's regarding Brad's feelings for Ruby. Unless . . .
"What did Ruby say?"
"At first, nothing. But just as I was starting to think I'd made a complete fool of myself, putting my heart on the line like I did, she told me she felt it, too. And that she was scared."
"I would be scared, too," Emma said, her voice suddenly hoarse. "You are an Englisher, and she was planning to be baptized."
Nodding, he repositioned himself on the rock so as to face Emma directly. "I didn't really know what that meant for the Amish—the part about being baptized. I didn't realize what was expected by her community and what she, herself, would have to eliminate from her life moving forward. I just figured her biggest worry was about me being English. That she was scared because she didn't know my world all that well. So I told her we'd take it slow. That we'd get to know each other on dates."
"Dates?" Emma echoed.
His answering laugh filled the space between them. "Oh, Emma, sometimes, when I look at you, I feel like I've rewound back twenty-three years and . . ." He waved away the rest of his sentence only to gather his next breath in time with his exit from the top of the rock. "She wasn't baptized yet, so that was a plus in our corner. So, too, was the fact she was still technically on Rumspringa. But she'd already made the decision to be baptized the next time the bishop did one and so we had to get creative about how and when we'd see each other. I didn't want to just always come here. I wanted to take her out, get rid of her fear about the English world, and get to know everything about her that I could.
"Sometimes those dates came during the time she'd have been riding her scooter home after work. Instead of her spending all that time getting home, I'd throw her scooter in the back of my truck and we'd use that time to do something together. Sometimes those dates happened when all her chores were done and her parents thought she was working, but she wasn't. And sometimes—"
"You mean, Ruby . . . lied?"
"No. She just didn't correct them when they assumed she'd been at work."
"But she would not have had money to give her dat on days she did not really work."
"I gave her money. From my job."
"But that is not work," Emma protested.
"Trust me, kiddo, it was worth every penny and then some. Because it meant Ruby and I could spend time together—real time, on real dates." He stopped, his smile draining from his face. "I've replayed every moment we ever spent together hundreds—no, thousands—of times over the past twenty-two years. In fact, I'm not sure I'd still be here right now if it wasn't for those moments . . . and you."
She recovered her gaped mouth. "Me?"
He held his hands out to her and, when she took them, pulled her to her feet, the emotion in his blue eyes surely reflected in her own. "I wanted to be the kind of man my little one would've been proud of. If she'd lived."
"I did live," Emma whispered.
"Thank God for that."
Chapter 9
She was just rounding the final bend in the road between Miller's Pond and the farm when the clip-clop of an approaching horse and buggy forced her thoughts into the moment. Lifting her hand as a shield against the late-afternoon sun, Emma stepped to the side of the road and waited for the charcoal-gray buggy of her brethren to draw close enough she could identify the horse or its driver.
Yet just as the white marking between the mare's eyes was starting to click into place, Mary's round face peeked around the edge of the buggy cover. "Emma, hello! Isn't this a wonderful surprise!"
Relieving her hand of sun-shielding duty, Emma waved to her friend and waited for the buggy's shadow to engulf her. When it did, she mustered the closest thing she could to a smile, lifted her gaze to her friend, and wobbled a hairbreadth at the second face peeking out around Mary's.
"Hello, Emma."
She shifted her weight across her boot-clad feet and waited for the sudden flapping inside her chest to stop or, at the very least, slow enough to let her think straight. "Levi . . . Mary . . . hello."
Levi's large brown eyes held hers for a moment before the quiet jut of his chin sent her focus in the direction she'd just come. "It is getting cold."
"Yah."
Rolling her eyes toward the buggy's ceiling, Mary jumped in. "So where are you coming from and where did you get that?"
She followed her friend's finger down to her own hand and the last of the black and white cookies Brad had insisted she take as he headed back to his office. "I just got them, is all."
A glance back at Mary yielded a raised brow—a raised brow Emma knew meant more questions were near if she didn't head them off. Fast. "Would you like a cookie?" Emma asked, holding one out to Levi. "They are very good."
"Very good are your oatmeal cookies."
The unexpected praise scurried her gaze toward her boots. The sudden loss of the cookie from her fingers redirected it back to Mary.
"I am not picky about my cookies." Mary's grin receded long enough to take a bite of the vanilla side of the cookie. "Yah, it is as good as it looks."
"Mary!" Levi scolded.
"What? You turned it down! It is not my fault if you are ferhoodled." Again, Mary rolled her eyes before bringing her full attention back to Emma. "So where are you coming from?"
"Miller's Pond."
"With cookies?"
"Yah."
"Why were you there?" Mary asked, the cookie all but forgotten.
"I . . ." She cast about for something to say short of the truth she didn't want to share in front of Levi. In the end, she settled on being as vague as possible. "I had something to do."
The second the words were out, she knew she'd chosen the wrong response. Vague didn't work with Mary. Vague with her friend was like dangling a mouse in front of one of the barn cats and expecting it to walk the other way.
In short, it didn't work.
Turning to her brother, Mary gestured outside the buggy. "If it is okay, I would like to walk with Emma to her farm. When she is there, I will start home and you can pick me up when you are done at Bishop King's."
"I can walk alone, Mary," Emma protested. "It is still plenty light out."
"Yah. But if I walk with you, it is less time in the buggy with"—Mary pointed at her brother—"him."
His eyes on Emma, Levi addressed his sister. "I have a much better idea."
"This should be good . . ." Mary teased.
"Perhaps you should walk, and Emma shall ride with me."
"Hmmmm . . ." Mary's grin moved from teasing to something that made Emma's cheeks grow even warmer. "Perhaps that would be good. But not today. Today, I will walk with Emma. She has much to tell me, don't you, Emma?"
She wanted to protest, but the truth was, she did want some time with Mary. So much had happened since they'd last spoken, so much she wanted to share. Before she could even nod, though, Mary was out of the buggy and standing beside her, the girl's arm snaking its way around Emma's.
"There is no need to hurry, Levi," Mary instructed. "If I make it home before you do, I will think about setting a spot for you at the dinner table."
"Mary!"
Levi's soft laugh led Emma's widened eyes back to his. "It is okay, Emma. I am used to my sister. But she forgets the many things I find in the barn that I could put in her room if she is not nice."
"Levi Fisher, you would not dare!"
With a wink directed solely at Emma, Levi urged the horse on its way with a firm click of his tongue that was quickly drowned out by Mary's answering huff. "If Levi so much as puts a mouse in my room, he will not eat for days."
It felt good to laugh, and laugh she did. "I don't think your mamm would let Levi go hungry, Mary."
"Maybe not. But I could do something to his food to make it taste bad."
"You wouldn't."
Mary's answering shrug made Emma laugh even more. "You are right. Maybe. But it is fun to think about sometimes. Just as it is fun to think of you being my sister one day."
Emma stopped, mid-step. "How could I be your sister?"
"When you and Levi marry."
This time, her laugh was more of a snort as she wiggled out from her friend's hold and reclaimed her earlier pace. "You are talking nonsense today, Mary."
"No, I'm not."
"Liddy Mast will be your sister one day." Emma stopped, spun around, and batted her eyelashes until Mary caught up. "And when she is, Liddy will do this"—she batted them harder—"across the table at you every time she and Levi come for supper."
Mary held up her palm. "Please. Do not say such things."
"Why? I am not like them. I am like him. I speak the truth."
"Not like them? Who is . . ." Mary's eyes widened, narrowed, and then widened again. "Wait! You are like him? As in—"
"As in my birth father. My real dat." Propelled forward by Mary's answering gasp, Emma met and surpassed her earlier pace, the crunch of the sparsely graveled road beneath her boots rescinding against the memory of the past two days.
Mary ran to catch up. "How can you say that? You can't know if you are like him!"
"I can, and I do."
"How?"
"Because I have met him. And we have spoken."
Mary's boots skidded to a stop while Emma continued walking. "Wait. You have? When? Who? How? And, more importantly, why did I not know?"
"It just happened. Yesterday. And it is because of Levi that I found him."
"Levi?" Mary echoed. "As in my brother, Levi?"
"Yah!"
This time when Mary caught up, it was to grab Emma's arm and pull her over to the fence that separated the Weaver farm from the county road. "Talk to me, Emma. Please! What did my brother do?"
"He told me what he saw the morning of my birthday." Turning toward the empty field, Emma rested her forearms atop the fence. "That he saw the same thing on many of my birthdays."
"Meaning?"
"You were right. It was my real dat who put those things on Ruby's grave each year. Levi saw his truck and him from your dat's field."
"He did not say anything to me."
Emma shrugged. "You did not ask. I did."
"Did Levi speak to him? Is that how you knew who your real dat was?"
"No. But Levi described his truck and the name of the English company it said on the truck's door. I figured out the rest."
"English company? I don't understand . . ."
"Yah. It is a construction company. In New Holland. Harper Construction—that is my real dat's name. Brad Harper."
Mary hoisted herself up onto the bottommost rail and thrust her upper body forward enough to afford a view of Emma's face. "You went to New Holland?"
"Yah."
"By yourself?"
"Yah. On Sarah's scooter."
"And you saw him?"
"Yah. At first, I did not know it was my real dat. I knew only that he'd been at the cemetery on my birthday. So I waited behind an old shed to see him come out to his truck. But when he didn't, my anger led me inside." She took in the barn and farmhouse in the distance, the cows grazing in the foreground, and, finally, her friend. "The minute I saw his picture on the wall, I knew he was my real dat. His eyes are the same blue as mine. His hair is the same color, too."
"Was he mad that you came to his work?" Mary asked.
"He was shocked."
"That you found him?"
Closing her eyes, Emma revisited the exact moment her birth father looked up and saw her, his blue eyes rounding and then widening, his skin draining to the color of her kapp, and the raspy sound of his voice as he said her birth mother's name.
"He was shocked that you found him?" Mary repeated.
Emma shook the memory from her thoughts and met her friend's eyes through her own parted lashes. "No, he was shocked that I was alive."
"But . . ." Mary stepped down off the rail. "That doesn't make any sense. Why would he be shocked about that?"
"Because they told him I died with her."
Mary's mouth gaped, closed, and gaped again. "Who would tell him such a thing?"
"Who else? Wayne and Rebeccah."
"Your mamm and dat?"
"No, Wayne and Rebeccah. Ruby and Brad were—I mean, are—my real mamm and dat."
"But you did not die with Ruby!"
"I know. They lied to him, just as Rebeccah lied to me when she said Brad did not want Ruby and me. He did want us. He did want me."
Mary, too, looked out at the field but not for long. "So? What happened?"
"We talked and then he drove me home. And then today, we spent many hours together at the pond, having a picnic and talking."
"That is where that cookie came from?"
"Yah."
"Your mamm—I mean Rebeccah—was okay with you spending time with him like that?"
"I did not give her a choice." She could feel Mary studying her and turned to meet the inquiry head-on. "She lied, Mary. To my real dat and to me. She should be shunned for what she has done. Dat, too."
"Shunned?" Mary echoed. "Are . . . are you going to tell Bishop King?"
"I don't know. I-I haven't thought about it. But why shouldn't I? They lied. For twenty-two years." She pushed back from the fence and returned to the road, her anger-filled pace making it so Mary had to run to catch up once again. "They were wrong to do that, Mary. Wrong to keep the truth about Ruby . . . and Brad . . . and me. It was not for them to choose!"
"Maybe there was a reason, Emma. Something you don't know. Something they will tell you if you ask."
"I'm not asking them!"
"Why?"
Emma whirled around. "Don't you get it? They've been lying to me for twenty-two years! Why would I ask them anything? If I did, they would just tell more lies!"
Mary opened her mouth to speak, yet said nothing, her worried eyes searching Emma's as the silence between them dragged on.
"I should have known, Mary," Emma insisted, her voice hoarse. "Ruby was my mamm! And Brad—he . . . He really loved her!"
Slipping her arm inside Emma's once again, Mary set their pace at a speed more conducive for talking. "Did he explain what they all meant? The scrap of paper? The bubble wand? The—"
"You mean the presents?" She shrugged away Mary's answering nod. "I didn't ask about them. I wanted to listen to him talk about meeting Ruby." And it was true. She'd been so caught up in hearing the story about how her parents had met, she'd forgotten all about the bag of trinkets stowed in the hollow of the tree on the other side of the pond.
Mary slowed. "Don't you want to know what they mean?"
"Of course. That is why I will ask tomorrow."
Once again, Mary stopped, necessitating a stop on Emma's part as well. "You will see him again tomorrow?"
"Yah."
"But why?"
She tugged her arm free and turned to face her friend. "There is much to learn. For both of us."
"But he is English," Mary warned.
"Yah. Perhaps I would be, too, if he was not told I was dead."
"Emma Lapp, don't you talk like that!"
"Why? It could be the truth."
"Could be does not mean should be!" Mary said, stamping her foot. "And you have been baptized!"
It was on the tip of her tongue to share what she'd learned about her birth mother, but, in the end, she kept Ruby's history to herself. Everything was so new, so fresh. And really, she just wanted to sit with it herself, to digest everything she'd learned at her own pace and in her own time.
Still, the knowledge that Mary's concern was born out of friendship helped tone down some of Emma's anger. "When you look across the table at your mamm and dat, you see parts of you. You know that your smile is like your dat's, and your laugh is just like your mamm's. You know that you have that funny spot on the back of your head like your grossdawdy has.
"All my life I have seen parts of my brothers and sisters in Ma—" She paused for a lengthy inhale. "In Rebeccah and Wayne. But not me. Never me. I want to fit somewhere, too, Mary. Just like you and Levi do. And just like Jakob, Jonathan, Annie, Sarah, and Esther do."
"You fit with me, Emma. You always have. Since we were very little."
"Getting to know my birth father does not change that, Mary."
"I pray that you are right."
* * *
She heard Mamm's footsteps crest the top of the stairs and pause outside Emma's door, waiting no doubt, for something to indicate Emma was still awake. Emma, in turn, sat perfectly still on the edge of her quilt-topped bed, waiting, anxiously, for the steps to continue down the hall and the faint glow of Dat's bedside lantern beneath the door to finally disappear.
The nightly ritual, save for the part that had her praying Mamm wouldn't come into her room to talk, had always been so routine. So . . . normal. Or so she'd always thought. Yet now that she knew her place in this home had been a lie from the start, she couldn't help but feel as if she were suffocating.
How many times had she sat there, in that same exact spot, wishing Mamm would come inside her room, sit on the edge of her bed, and talk to Emma the way she did sometimes with Annie and Esther. But it never happened, leaving Emma to rationalize the reason the same way she did so many other aspects of her life—she was odd, she didn't try hard enough, she'd done something poorly . . .
Yet now that she knew the real reason, Emma wanted nothing more than to be left alone. To think. To feel. To look forward to the next day and more time with her real dat.
Still, she couldn't quite shake the guilt she felt when she'd turned away from Mamm in the kitchen earlier in the evening only to spy Esther watching them, wide-eyed. Emma wasn't entirely sure how long the little girl had been there as Emma had silenced Mamm's every attempt to talk with a raised palm, but it was clear her behavior had caused confusion and fear—feelings she never intended the five-year-old to share.
Clutching the sides of her dress with renewed anger, Emma looked at the gap beneath her door, the shadow of Mamm's feet a reminder to remain still, to keep her breath quiet. Finally, mercifully, the shadow rescinded, followed soon after by all vestiges of light as Mamm joined Dat in their bedroom.
It took a moment to adjust her eyes to the total darkness, and another moment to cross to the window, lift the dark green shade into its daytime position, and look out over the dormant moonlit fields. Just last week, she'd walked those same fields with Sarah and Annie, working to rid them of the rocks and pebbles that always seemed to mysteriously appear in the months following the autumn harvest. It was a task they did every year in preparation for the tilling Dat and the boys would do as February became March.
Closing her eyes, Emma imagined the golden-yellow wheat stalks that would soon grow so tall their soft beards would tickle her cheeks. Spring was her favorite time on the farm. There was something exciting about green replacing brown, food for the animals growing in one field, crops Dat would sell in another, Mamm's vegetable garden behind the house springing back to life, and the insistent moo of yet another new calf through the open barn doors.
She willed her nose to conjure the smells of freshly plowed soil, spring flowers from the garden, and Mamm's apple pie cooling on the kitchen windowsill. . . .
Mamm.
Emma turned and surveyed her room—the quilt-topped bed, the chest of drawers, the porcelain bowl and pitcher she used to wash her face at night, the plain dresses that hung from hooks to the right of her closed door, and, finally, the heart-shaped silver locket and chain resting beside the glass of water on the small bedside table. On quiet stocking-clad feet, she made her way back to the bed, her fingers closing over the locket as she settled against her moon-drenched pillow.
For a few long moments, she simply lay there, soaking up every part of the necklace. The heart shape . . . The delicate flowers etched around the edges . . . The way it shone in the moonlight . . . The featherlight feel of the chain as it snaked across her wrist . . . The smooth simplicity of the locket's underside . . .
Slowly, carefully, she worked her thumbnail into the tiny slit she could just barely make out by sight and popped open the locket, the audible whoosh of air that had marked her first and every subsequent peek at the picture it contained echoing against the plain white walls of her room.
It didn't matter how often she looked at the young girl inside, or how she mentally prepared herself for what she'd see when she did. The near mirror image of herself stunned Emma every single time. In fact, sitting there in a room lit only by the moon, the differing hair and eye color was difficult to discern. Yet as she continued to study the woman who had been her mother, another difference emerged—a difference that was suddenly so glaring she didn't know how she'd missed it until that moment.
Ruby's smile was like nothing Emma had ever seen or felt on her own face. It was all encompassing in the way it drew Emma's gaze in before sending it skittering upward to the girl's cheeks and eyes. She tried to imagine a time she might have looked like that, too, but every time her mind started to search her memories, the face nestled inside her palm pulled her back.
All her life, she'd known Ruby as one thing: Mamm's younger sister. Beyond that, she knew only that Ruby had died at eighteen—on the same day Emma was born, making the day a painful one for the woman she'd been raised to believe was her mamm.
But what Ruby was like? The things she'd liked to do? The moments that had made her smile as she did in the picture? Those were the things Emma didn't know. Those were the things Emma had never dared to ask lest the very subject upset Mamm.
Yet this girl, this sister, this person who'd died on her birthday was Emma's real mamm. She'd given birth to Emma and then died. At eighteen. And now, thanks to Brad, the name she could never so much as mutter as a child was beginning to take shape as a person in her thoughts. A person whose smile had made her real father take notice. A person who had strayed from the life she was supposed to lead and died because of it.
Because of her....
Tugging the locket to her chest, Emma finally gave in to the tears she could no longer hold back.
Chapter 10
Emma heard the telltale crunch the moment his work boots hit the leaves. It was quiet, even a little tentative at first, but as she lifted her head from her knees and their eyes met, it became faster and more purposeful.
"There you are." Brad strode toward her rock. "I was worried when you didn't come to the office as we planned, so I took a chance you might be here. At your spot. And . . ." His words trailed off momentarily as he studied her face. "You okay, kiddo? You look a little worn out."
"I'm just tired. I-I did not sleep well."
Slowly, he lowered himself to a vacant spot beside her. "So why didn't you come to the office like we talked about?"
"I thought better of it."
"Better of it? I don't understand.... Do you not want to spend time with me?"
She looked out over the pond, willing her voice to remain steady. "I do. It's just that . . . I don't know. I just thought this would be best."
"There's nothing best about not seeing you, Emma. Nothing at all."
"But you loved her," she protested around the ever-present lump lodged halfway down her throat. "I mean, really truly loved her."
"Ruby? Yeah, I loved her. Still do. Always will."
Pushing against the invisible yet almost crushing weight pressing down on her shoulders, Emma stood. "Then I do not think you want to spend time with me."
Brad, too, rose to his feet, only instead of looking out over the water, he looped around until they were standing face-to-face. "I can't imagine why on earth you'd think that. You're all I've been able to think about since I saw you standing in my office the other day. All I've been able to talk about—ask Sue Ellen. She's had to clap her hands in my face at least a dozen times when I go from talking about site plans and blueprints, to the way you look so much like Ruby."
"Why must she clap her hands?"
"When a client calls, and I'm too buried in my thoughts to pick up the phone.... When I've been mid-order with a supplier and start thinking about all the places I want to take you and all the people you need to meet.... When she's trying to ask me a question and I haven't heard a word out of her mouth . . ." He scrubbed at his chin and then reached for her hands, holding them gently inside his own. "Do you know how hard it is waiting to tell my mom until she's back from her trip to see her sister? She's going to be absolutely beside herself, Emma!"
Tugging her hands free, she sidestepped him and made her way closer to the water, the answering crunch of leaves letting her know he wasn't far behind.
"Emma, talk to me. What's wrong? Are they giving you a hard time about seeing me? Because I could have them—"
She shook off his words. "No. It's not that. I will see you when I choose. They cannot stop me. But I don't know why you'd want to see me."
"Emma, you're my daughter. Ruby's and my daughter."
"I'm also the reason she's dead," Emma whispered as the tears that had soaked her pillow during the night made their way down her cheeks once again. "I'm . . . the . . . reason . . . you don't have her . . . anymore."
Silence greeted her raspy cry and renewed sniffling, but only for as long as it took Brad to suck in his own breath and spin her around. "Whoa. Don't ever say that again. Ever. You are the best thing that's happened to me in more than twenty-two years. Losing Ruby tore me apart, it really did. And I still haven't fully recovered from her death, either—not sure I ever will. But she made up her mind.
"But you? Being here? That's a gift. An unbelievable gift that I've thanked God for more times than I can count these past few days."
Lifting her watery gaze to his own emotion-filled eyes, Emma marveled, again, at the ability to finally see part of herself in another person. "You are sure?"
"I'm sure." He released his grip on her arms and stepped back, sweeping his hand in the direction of the road. "So, what do you say we get in the truck and head into town? We can grab a bite at the deli, or order in some lunch to my office. I know you want to hear more about your mother."
Emma paused, mid-nod. "Actually, if it's okay, maybe we could stay here again today? I-I have some things I want to ask you about."
"Okay, yeah, sure. We can do that. But let me call Sue Ellen and let her know I found you. I suspect she was worrying along with me when you didn't show."
"I'm sorry. I did not mean to make anyone worry."
Reaching into his pocket, he pulled out his phone, swept his finger across the screen, and then grinned. "I still can't believe this—any of this."
"This?"
"Yeah. That I'm standing here at Miller's Pond, talking to my twenty-two-year-old daughter . . . That I'm about to call my secretary to tell her I tracked you down and we're going to hang out together here for a little while . . . That I'm going to be able to call my mother tomorrow morning and tell her she has a granddaughter . . . It's a little surreal, quite frankly."
"Yah." Emma nudged her chin toward the rock but remained in place as Brad prepared to make his call. "I will meet you at the rock. There is something I must get first."
He paused his finger on the phone's touch screen. "If you need to go somewhere first, I can take you."
"I just need to get something there"—she pointed to the opposite side of the pond—"and then I will bring it to you. I will be back before you are done with your call."
She smiled away the question in his eyes and shooed him toward the rock, his answering laugh soon followed by the sound of his secretary's name. When he was safely in route to their chosen meeting spot, Emma turned and made her way around the outer edge of the pond, her mind's eye skipping ahead to the drawstring bag housed in the oak tree on the other side.
For so long, she'd imagined the trinkets she'd snuck off Ruby's grave every year as a sort of birthday present. She'd always wondered who left them and why Dat had gotten so upset by them, but those questions had always shifted to the background against the fun of the new item. When she'd still been in school, she'd hidden the surprise in her lunch pail until it was time to walk home past the pond. When she completed her schooling, there had been no reason to hide the item as she'd always go straight from the cemetery to the pond. But no matter her age or the route in which she took to get to her secret tree, she always spent time on the rock with the new gift. She'd turn it over, study it, try to imagine its significance, and then carefully add it to the bag.
Now, though, because of the locket and everything it had led her to over the past nine days, she was about to learn what everything meant and why Brad had been putting them there every year on the anniversary of her birth mother's death—details she both wanted and maybe even dreaded a wee bit, too. Everything about her life had been a lie thus far. Except those presents. They'd made her feel special on a day that never was. Yet, in hearing the truth about each one, she'd have to say goodbye to yet another part of her childhood—a part filled with silly little stories and games she'd made up while carrying each new present to its home inside the tree.
The crunching beneath her boots slowed as she reached the tree, her heart suddenly torn between knowing and not knowing. Slowly, she lifted her chin until all she could see in front of her was the early February sun peeking over the tips of the tree's bare branches, its answering warmth on her cheeks quieting her heart. Three deep breaths later, she lowered her attention back to the tree and the hollow her seven-year-old self had disguised from the world with a piece of old bark.
With practiced fingers, she removed the loose bark, set it against the base of the tree, and then reached inside for the dark blue drawstring bag she'd smuggled out of her room fifteen years earlier. Year by year, item by item, she'd filled the cotton bag and kept it hidden in this exact spot. And with the exception of Mary the other day, Emma had never shared its existence or contents with anyone.
She ran her hand across the bag's lumpy innards, her mind's eye filling in the coordinating item.
* The stuffed horse . . .
* The red rubber ball . . .
A second feel had her changing the rubber ball to the baseball before moving on.
* The snow globe with the skaters . . .
* The whittled bird . . .
The other things were harder to feel through the cloth, but she knew they were all present. Clutching the bag to her chest, she peeked around the trunk to the Englisher on the other side of the pond. She took a moment to soak him in, to try to catch her heart up to everything she knew thus far.
For twenty-two years, Wayne and Rebeccah Lapp had been Dat and Mamm. She never really saw herself in them the way she did her siblings, but they were Dat and Mamm. Now, she knew better.
Now, she knew that her real mamm, Ruby, was buried not far away, and her real dat, Brad, was an Englisher with Emma's same hair and eyes.
Mamm said Brad hadn't cared about Emma and Ruby, that he had left Ruby to deal with their sin, alone. Yet everything Brad had told her so far about his relationship with Ruby didn't match Mamm's words. In fact, the stories couldn't be more different.
So, who was right?
And where, exactly, did that leave Emma?
Glancing down at the bag, she gathered her breath and headed back around the pond, her need to know about the contents pushing its way past a whole different set of questions about herself and her birth parents—questions that seemed to multiply by the hour. She could feel him watching her as she maneuvered her way around downed limbs, through piles of old leaves, and across the plank of wood that served as a makeshift bridge from one side of a tiny inlet to the other. When she finally reached the rock, she saw him wipe the back of his hand across his eyes.
"Is . . . is Sue Ellen okay?" she asked.
He dropped his hand to his thigh and nodded. "She's fine. Glad to hear you're okay and that I found you. Why?"
"You look . . . I don't know . . . a little unhappy, I guess."
"I'm happy, Emma." He patted the part of the rock she'd claimed earlier, and, when she accepted the nonverbal invite, he pointed a lazy finger in the direction she'd just come. "Watching you just now? Walking around the pond? It was like watching Ruby. You're about the same height and there's so much about the way you move, the way you carry yourself, that is just like her."
"I'm sorry."
His focus snapped back to her face. "No! Don't be! It's a good thing, Emma. It's as it should be. You're Ruby's daughter. My daughter." He nudged his chin in the direction of her lap. "I'm pretty sure my call to Sue Ellen took less than a minute—two at the absolute longest. Either way, it wasn't time for you to go home."
She followed his questioning gaze to the bag now resting in her lap. "I-I didn't need to go home to get this. I keep it inside an old oak over there. It has been there for many years—fifteen, in fact."
"You've kept a bag inside a tree for fifteen years? Why? What's in it?"
"I was kind of hoping you could tell me." She inched her fingers up to the top of the bag and then yanked on the drawstring opening. "Every year, on my birthday, we would start the day with a visit to the cemetery. To Ruby's grave. Mamm would get very quiet and I knew, when I looked up at her, there would be tears on her cheeks. I also knew, in those early years, that when I looked up at Dat, I would see anger."
"Anger?" Brad barked.
"Yah. He did not like the things we would find on Ruby's grave each year."
"He didn't like the things?"
"Yah. He would throw them in the English trash can when we would leave." Sensing Brad's growing irritation, she jumped to the part that mattered most at that moment—the part that got her to the bag. "The first time I went to the cemetery alone, I was seven. I stopped at the grave before school, and I put the little picnic basket in my lunch pail. I meant to throw it away as Dat had all the others, but I couldn't. My birthdays were never like they were for the other kids. There was cake and some sort of present, but there were no smiles, no laughter, no special hugs. It was a day I didn't get excited about the way my brothers and sisters did. But that day, when I went inside the gate at the cemetery, I was excited to see what was on Ruby's grave. It was as if someone had left a present for me . . . for my birthday. So that day, on the way home from school, I hid it in the oak tree. And when I was able to, I came back with this bag." She lifted it up just long enough to give it a little shake. "I did the same the next year with the silver rose, and the year after that with the snow globe and the tiny skaters inside, and—"
"The stuffed horse, the picture of the dandelion, the bubble blower thing, the torn ticket stub—"
"So, it is a ticket?" She reached inside the bag and slowly removed each item, arranging them on top of the rock in the order in which she'd found them. When she got to the torn slip of paper housed in the clear plastic covering, she took in the details she'd all but memorized and then looked up at Brad. "What was it a ticket for?"
"I took her to a carnival. The ticket was for her first ride on a Ferris wheel . . ." He fingered the clear plastic covering and, at Emma's nod, took the torn ticket in his hand. "I left these things for you, but I never dreamed you actually got them."
"I know now that these things were for me, that what I pretended for so long was actually true, but I don't understand why. Why would you leave presents for me if you thought I was dead?" she asked.
"Because I had to. To get through the day. I never got to see you, or hold you, or tell you I loved you. So I did those things the best way I could with"—he waved the ticket stub at the items spread out between them—"these things. Every year. On your birthday. My mom suggested it as a way to help me through the pain. And it worked. At least a little. It helped me feel connected. Like I was getting to celebrate your special day with you from down here.
"Even when I was away, still lashing out at the world around me, I always came back in time to leave your next gift. Never missed a single one."
Pulling her knees to her chest, Emma considered his words, the quiet relief they allowed warming her from the inside, out. "So they really were meant for me. . . ."
"Every one of them." He rolled a thin stick between his fingers before chucking it onto the ground. "They were my way of telling you about your mom and me. The way we were, the way we loved each other. Looking back, I think getting to stand there by the grave, telling you about them, helped validate all of it for me somehow. My feelings, the relationship, all of it."
She rested her chin atop her knees. "Validate it? How so?"
"Ruby and I were seventeen and eighteen when we started up. People don't take that seriously. They call it young love and puppy love and all sorts of demeaning terms. And maybe that's the case for a lot of teen relationships, I don't know. But I know ours was real. I know my love for Ruby was real. I know I wanted a future with her and with you. And I know my world was forever turned upside down when I lost her, and then you, too."
Emma dropped her legs back down to the rock, and picked up the miniature picnic basket that had started her fifteen-year collection. "I remember peeking inside my lunch pail many times that first day. I would peek to make sure it was still there, I would peek to make sure no one else had found it, and I would peek just so I could see how pretty it was. I even wished I could take it home and play with it, but I knew I couldn't."
A darkness dulled his eyes. "And you say Wayne threw some of my gifts away?"
"I do not remember anything about my first, second, or third birthdays, but I know he did on my fourth, fifth, and sixth." She looked again at the tiny basket between her fingers. "Tell me about this first one.... Why a picnic basket?"
Brad met and then followed her gaze down to her hands, the anger she had sensed in him just moments earlier chased away by a soft laugh. "We had a handful of cookie picnics, Ruby and me. Most of them here, at the pond. On this very rock, in fact. She'd fill an old lunch pail with cookies she'd made. Her oatmeal ones were my favorite."
"I make oatmeal cookies! Levi likes them a lot, too—though Liddy Mast makes them for him now."
He studied her closely. "Levi? Is he your boyfriend?"
"N-no," she sputtered. "He . . . he's just a boy I know. His sister, Mary, is my best friend. I go to hymn sings with them."
His jaw tightened. "Ahhh, hymn sings. I remember those. It's where you go to find someone to court."
"It's not really like that. We go to be with others our age. But yah, many do court after meeting someone suitable at a hymn sing." Emma turned the basket over in her fingers one last time and then set it back down on the rock. "So, the basket is because you had cookie picnics?"
Shaking his head ever so slightly, he picked up the basket and studied it from all sides. "No. The basket is for the full-fledged picnic I put together for us one afternoon. I wanted her to see what an English picnic is like with the fancy basket, the traditional blanket with the red and black squares, sandwiches, grapes, chips, brownies, and Frisbee."
"Frisbee? What is that?"
"It's a round plastic disc that's about the size of a dinner plate that you can throw in the air." His grin spread into an all-out smile. "Ruby hadn't ever seen one before, either. But by the time we had to call it quits so she could get home, she'd actually gotten quite good. That's the way she was, you know. She had a way of picking things up quickly, and picking them up well."
Emma tried to imagine the game as he described, but without ever having seen it herself, she couldn't be sure she had it right. Instead, she took the conversation back to more familiar ground. "And the food? Did she like it?"
"She did. Very much. And even if she hadn't told me that again and again on the drive back to our drop-off spot, I'd have known simply because of the way her eyes sparkled the whole time."
Nodding, she moved on to her eighth birthday and the tiny rose. "What about this?" she asked, holding it atop her open palm. "Why did you give me this? What does it mean? And how is something so small so heavy?"
His laugh reddened her cheeks. "It's heavy because it's pewter."
"Ruby liked pewter?"
"No. I didn't give this to you because of what it's made of. I gave it to you because of what it is—a rose. Pewter just meant it wouldn't die like a real flower." He took it from her hand and held it close to his nose. "I bought her a rose once. On Valentine's Day. She said it was the prettiest flower she'd ever seen. I told her that made sense since she was the prettiest girl I'd ever seen. Because she was—the prettiest. Until I saw you standing in my office the other day, anyway."
Emma held her hands to her flushed cheeks and shook her head. "I'm not pretty. I'm Amish. We're plain people."
"Funny thing is, I used to think that about Amish girls, too. To me, they all dressed the same, covered their hair the same, didn't use makeup like the girls in my high school, didn't wear rings or necklaces or anything like that. And then I met Ruby. She didn't need that red garbage on her cheeks or all that stuff on her eyes. All she had to do was smile at me. Or laugh at something I said. Heck, even when she cried she was beautiful."
"Ruby cried with you?" At his slow, labored nod, she drew back. "Why? Was it because of me? Because I was coming?"
"No, Emma. It wasn't because of you. It was because . . ." He glanced across the pond, seemingly oblivious to the pewter rose he still held between his fingers. "Sometimes life just seems a little uncertain. A little scary, you know? Some people deal with it by yelling or stamping around. Others get quiet or cry. It's life."
"The Amish are not to yell and stamp around."
Closing his hand around the rose, he sighed. "I know."
She waited to see what else he'd say, but, when he remained silent, she moved on to the present that had marked her ninth birthday.
"I loved this one." Scooping up the tiny snow globe, she took a quick look at the skaters inside, shook her hand, and grinned as snow fell down around the couple inside. "I know it is silly, but I would pretend it was me in there skating. Only I would pretend the girl was wearing a kapp and a dress like me."
"She did."
Startled, Emma looked at him across the top of the clear, plastic dome. "I don't understand."
"The girl did wear a kapp and dress." Turning Emma's hand so he, too, could see inside, Brad continued. "I took Ruby skating one afternoon. I borrowed some skates from a girl at school and I took Ruby to the pond next to my house. While we were there, it started snowing just like this."
Emma returned her attention to the figures now covered with white flecks. "Did she like it?"
"She loved it. Took her a little while to get the hang of it, but by the time we were done, she was trying to make patterns in the ice by turning in little circles." He dropped his hands to the rock and leaned back. "Whenever I hear her laugh in my thoughts, it's from that day."
"I wish I could hear her laugh," Emma whispered.
"I do, too, kiddo."
Again, she looked at the skaters. Only this time, instead of the happiness they'd always stirred inside her, there was a sadness she didn't want to feel. Not with him—
"Emma? Are you okay?"
Setting the globe back on the rock, she shrugged off his question in favor of her tenth birthday gift—the stuffed horse. "It took me the whole walk here that day to come up with just the right name for her, but I did." Emma ran her fingers down the toy's silky mane and then handed it to Brad. "I called her Sugar. Still do."
"Sugar . . . I like that." He turned the horse over, inspected it from all angles, and then set it back down.
"Her color made me think of the cinnamon sugar Mamm—I mean, Rebeccah—sometimes puts on the top of her apple pie. If she forgets, someone is always quick to remind her."
Brad's jaw noticeably tightened, yet he said nothing.
"One time, last year, when Esther had just turned four, she was helping me make an apple pie. She was standing on the bench so she could see what I was doing and she noticed I hadn't put the cinnamon sugar on yet. When my back was turned doing something else, she dumped the whole jar onto the top." Emma's laugh echoed in the cool winter air. "That was too much sugar. Even for Esther."
"My mom—your grandmother—makes a great apple pie. It even won a ribbon at the county fair one year. She would have loved getting to teach you how to make it." Brad pushed forward, raking a hand through his hair as he did. "If she'd had the chance, of course."
She stilled her hand just shy of her eleventh birthday gift and swallowed. "I am sorry I did not find you sooner. I did not know."
"You're sorry?" he echoed, his voice thunderous. "You're sorry? For what? For being told you belonged to someone who had no right to claim you? For having no reason to believe you'd been lied to? For living the lie that had been forced on you and forced on me? Please, Emma. There are only two people who are responsible for this situation and it's not you and it's not me. They're the ones who should be sorry—will be sorry, if I have anything to say about—"
The telltale snap of a twig from the direction of the road brought Emma to her feet. "Shhh. Did you hear that? I think someone is . . ." Her words faded off as a quick flash of blue pulled her gaze toward a gnarled tree not more than two buggy lengths away.
"Who's there?" Brad called out as he, too, stood.
Seconds later, a familiar round face peered around the tree, causing Emma to stumble back a half step. "Esther!"
The single crunch that had guided their focus toward the child turned into a series of crunches as Esther covered the distance between them in short order.
"Hi, Emma!" A bright smile pushed the five-year-old's cheeks nearly to her eyes just before she wrapped her arms around Emma's legs. "I looked and looked for you and now I found you!"
Prying the little girl's arms away, Emma squatted down. "Esther, what are you doing here? And where is"—she looked toward the road—"Annie or Sarah or whoever you're with?"
"It is just me! Annie and Sarah are home with Mamm."
She sucked in a breath. "Esther! You are not to come so far by yourself!"
"But I had to share my school cookie with you!" Esther opened her hand to reveal a crumbled cookie half and then slid her attention off Emma and onto Brad. "Hi! What's your name? Are you Emma's friend?"
"My name is Brad and I'm Emma's—"
Grabbing hold of Esther's free hand, Emma stood, her face warm. "Every morning, I-I put two cookies in Esther's lunch pail for her to eat. But each day, she brings half of one back home to me."
"It is a very good cookie," Esther said, her expression earnest. "Here, Emma. Eat it."
"Okay, okay." She took the cookie, offered some to Brad, and, when he declined, popped it into her mouth. "Mmm . . ."
Satisfied, Esther wiggled free and ran to the rock. "Emma, look! It is like a store!"
She followed the tip of the child's finger to the now empty bag and the contents she'd lined up in a row beside it. Before she could think, let alone speak, Esther scooped up the bubble wand and gazed up at Brad. "Did you bring bubbles, too? 'Cause I love bubbles!"
"No! He didn't." Mouthing an apology across the top of her sister's head, Emma gathered up all of the gifts, stuffed them inside the bag, and handed it to Brad. "I need to get her back to the farm. She's too little to be out here alone."
He looked from Esther, to the bag, and, finally, back at Emma. "But we have more to go through, more to talk about."
"I know. And we will. Tomorrow."
Chapter 11
She allowed Esther one final wave at the shiny black truck and then captured the little girl's hand inside her own. "Come now, Esther. It is time to get you home. Mamm will be worried if she discovers you are not in the barn."
"You are not in the barn."
"I'm older than you are. I don't have to be in the barn."
The crunch of gravel beneath Esther's feet slowed as she looked over her shoulder and then back up at Emma. "Is that man your friend, Emma?"
"No, he's . . ." She squeezed her eyes closed, silently counted to ten, and then opened them to find the five-year-old staring up at her, mouth agape. "He is just someone I am getting to know—someone I should have known long before now."
"Can I know him, too?"
"Perhaps. Maybe. I-I don't know." Desperate for a change in topic, Emma pointed at the Weavers' horse rooting around the dormant field and worked to infuse a lightness into her voice that she didn't feel. "Now, is that one Dolly or is that one Molly?"
Esther's gaze followed Emma's to the upcoming fence line and the solitary mare. "That is Molly."
"How can you tell?"
Esther tugged her hand free and ran over to the fence, her finger pointing. "See her eyes? She has black in the middle. Dolly doesn't have black. Only brown. Like the man's horse."
"What man's horse?" she asked, motioning her sister back to the road for the final stretch of their walk.
"The man you are getting to know."
She stopped, mid-step, as Esther ran back to her side. "Brad does not have a horse," Emma corrected. "He has a truck."
"It was on the rock! It was brown, like Dolly."
And then she knew. Esther was talking about Sugar, the stuffed horse Brad had left at Ruby's gravesite on Emma's tenth birthday. Scanning the Weavers' field to the left, she searched for yet another way to distract Esther, but before she could settle on something, Esther began hopping up and down.
"He drawed a heart on a rock."
"Drew," she corrected as her own thoughts returned to the pond and the present she'd found on her fourteenth birthday.
"Why did he drew on a rock?"
Leaning down, Emma tapped her sister on the nose. "Actually, that time it's draw. And I don't know why. You showed up just as we were getting to the—"
"He had a red ball that was very dirty!"
"It wasn't dirty," Emma protested. "It is just old, and someone wrote on it."
Emma pulled a face. "Balls are for throwing, Emma. And rocks are for touching. Paper is for writing."
"That is true. But I am sure there is a reason. We just didn't get to those things yet." She held out her hand and, when Esther took it, began walking again. "Why did you leave the farm, Esther? You know you are not supposed to do that."
"I had to share my cookie. But you were not there to share it."
"That's right, I wasn't."
"But you're always home to share my cookie, Emma."
"I know, sweetie, but things are different now."
Esther peered up at Emma. "Why? Don't you like sharing cookies with me anymore?"
"Of course I do. I-I love sharing cookies with you. But, well, I need to figure out some things is all and—"
"I can help you, Emma! I am learning lots of things at school! Watch!" Esther pulled her hand free and hurried into Emma's path. "A, B, C, D, E, F, G . . ."
Emma grinned. "Come on, what's next?"
Esther tapped her chin, once, twice, and then squealed. "H! I, J, K, L, M, N, O, P!"
"Very good, Esther!" Emma clapped her hands and then squatted down for a big hug. When Esther stepped into her arms, she lingered a kiss against the little girl's kapp. "That's four more letters than last week! You will know the whole alphabet soon!"
"When I do, I can help you figure out things!"
Oh how she wished it were that simple. If it were, maybe she wouldn't feel so alone, so unsettled.
"Mamm's eyes look like that, too," Esther said, pointing at Emma's face.
"Like what?"
"Sad."
She fought against the familiar worry trying to gain a foothold in her heart and made herself shrug, instead. "No, my birthday is over, sweetie. Mamm only gets sad on my birthday. You'll get used to it. I certainly did."
"Mamm made me wet here." Esther reached into her shoulder through the open neckline of her black coat. "But I got dry before I came to find you with your cookie."
"Slow down a minute. Mamm made you wet?"
"Yah. She hugged me like this"—Esther wrapped her arms around herself and then dropped them to her side—"and her tears got me wet."
Emma drew back. "Mamm was crying?"
"Yah."
"Why?"
Esther's little shoulders slumped beneath her coat. "I don't know. Maybe her tummy hurts."
Or maybe she is worried she will be shunned for her lies . . .
Standing, she reached for Esther's hand again. "Maybe. Now come on, sweetie, it's time I get you home."
* * *
They were just passing the barn when the bang of the farmhouse door pulled Emma's gaze off the baby calf she and Esther were laughing about and fixed it on the woman hurrying down the porch steps.
It was a sight she'd seen many times during her life, yet suddenly nothing about it was the same. Now, instead of thinking Mamm, she thought Rebeccah and liar. And now, instead of hurrying her own feet in response, she stopped, tightened her hold on Esther, and fisted her free hand at her side.
"Esther Lapp, where have you been, child? Annie is searching the fields looking for you, and I sent Sarah to the Troyers' to see if you'd chased a barn cat onto their farm." Mamm's brown eyes bore into Esther's before lifting upward to meet Emma's. "Emma? Was she with you?"
"She found me." Emma lifted Esther's focus off the ground and back to her face as it had been off and on for most of their walk home. "Go on. Tell her."
Toeing the ground, the five-year-old stepped closer to Emma, her cheeks red with shame. "I . . . went to . . . the pond. To find Emma."
"Esther Lapp!"
"I always give her the rest of my cookie," Esther protested, her voice shaky. "Emma was not in the house or the barn. I founded her at the pond."
Emma squeezed the little girl's hand, gently. "Found."
"Found," Esther corrected.
Uncertainty guided the woman's gaze back to Emma. "Were you at the pond alone?"
Esther brightened. "I said hi to her friend, Mamm! He said hi, too!"
When Esther pulled her hand from Emma's, Emma folded her arms and nodded. "That's right. She met Brad. My real—"
"Esther, you are to go inside. I need to talk—"
"His horse looks like Dolly, Mamm!"
Surprise propelled the woman back a step. "The Englisher has a horse?"
Giggling, Esther shook her head. Hard. "Not a real horse, Mamm. A toy! And he had a red ball, too! A dirty one. But Emma said it is not dirty. She said someone writed things on it. Isn't that silly, Mamm?"
Emma opened her mouth to correct her sister's grammar but let it go as the little girl began to hop up and down. "He had lots of toys, Mamm! And he isn't little, like me. He is great big like Dat!"
"Toys?" Rebeccah echoed.
"Yah. The ones he"—Emma pointed toward the bearded and hatted man in the farthest field they could see—"did not throw away. The ones I got to first."
Esther stopped hopping. "Dat?"
Squatting down, Emma redirected Esther's attention toward the eight-year-old walking across the dirt, searching. "You know what? You should go tell Annie that you are home so she can stop looking for you. Then go with her to the Troyers' to tell Sarah. I think they have spent enough time looking for you, wouldn't you say?"
Shame reddened Esther's cheeks once again. "Yah."
"Good. Now go."
Emma watched the little girl run across the driveway and into the field, Esther's kapp strings flopping against her tiny shoulders. When she was certain Annie saw Esther, she stood and turned back to Rebeccah. "Ever since I was seven, I have gone to the cemetery before breakfast. That first year, I did it because I wanted to make the day better for you and for"—again she gestured toward the far field—"him. I knew I could not stop your tears for the sister you lost, but I thought I could keep Dat from getting angry. I tried to throw the trinket away, but I could not. It made me happy, the way someone should be on their birthday—the way Jakob, Annie, Sarah, Jonathan, and Esther could be because your sister did not die on their special day. So I kept it in a bag in a special place, and I added to it each year."
"You should not have kept such things!"
"Why? They weren't for you or Dat. They weren't for Ruby, either. They were for me. Me!" She narrowed her eyes until she couldn't see Annie, Esther, the fields, or anything else beyond the pale-faced woman standing just inches away. "Brad thought I died with Ruby. No, he was told I died with Ruby. By you! Those things he left on the grave weren't for Ruby. They were for me. For my birthday."
She jerked back at the feel of Rebeccah's hand on hers. "Don't!"
"Emma, please. There are things Dat and I want to—"
"You mean him?" She pointed at the far field, once again. "Because that's not my dat. Brad is my dat. Or should I say father since he is English?"
"Emma, I know you're upset. But if you would just talk to me, to us, we—"
"Perhaps that is something you should have done many years ago."
"Emma—"
"What is there to say? That I should not go to Bishop King? That you did not mean to lie to me and to my birth father? That you and Wayne did not pretend to be my . . ." The squeak of the barn door stole the rest of her words and sent her gaze racing toward the barn in time to see Jonathan walking toward them, cradling something in the crook of his arm.
"Emma! Mamm! Look what I found near Mini's water pail."
She held Rebeccah's wary eyes for several beats and then found the smile the twelve-year-old sought as he stopped at Emma's elbow. Leaning forward, she patted his sleeve down just enough to afford a clear view of the tiny mound of white and gray matted fur. "Oooh, I see Bean had her babies . . ."
"She did. There's four more next to the water pail, too."
"And Bean?" Emma prodded.
"She's already licking them," Jonathan declared.
"Then you probably should get this one back before the new mamma gets upset." Emma stroked the tiny kitten between the ears. "I'll bring Esther in to see the new additions when she gets back with Annie and Sarah."
Jonathan's eyes disappeared beneath the rim of his hat only to reappear as he gazed back up at Emma. "Aren't you going to say it?"
She knew what he was asking. And why wouldn't he? She'd been greeting every newcomer to the barn the same way since before Jonathan was even born . . .
Not wanting to disappoint, she leaned across her brother's arm and planted a soft kiss atop the newborn kitten's head. "Welcome to your home, little one."
"That's not how you say it," Jonathan admonished as he headed back toward the barn. "You say, 'Welcome home.' Not 'Welcome to your home.' "
When he was safely out of earshot, she turned back to Rebeccah. "Soon we must tell them."
"Them?"
"The children."
"Emma, I—"
"They should know what I did not. They should know that this was never meant to be my home." Sidestepping her way around the woman she'd once believed to be her mamm, Emma willed her stoicism to remain as she gave voice to the last of her new truths. "And that I was never meant to be their sister."
Chapter 12
"I'm sorry that took so long, Emma." Sue Ellen deposited the phone into its base and quietly folded her arms across the top of her meticulously kept desk. "My job with your father is full of peaks and valleys."
Emma stilled her fidgeting fingers. "You are to climb sometimes?"
"Climb? No, I . . ." A knowing smile crept across the sixty-something's face. " 'Peaks and valleys' is an expression, dear. It means there are moments when I sit here twiddling my thumbs. And other times, I have customers in the office, suppliers on the phone, and contractors coming in and out, asking me this, that, and the other.
"Case in point, I was actually getting in a little reading not more than five minutes before you got here. But the second I noticed the door opening, the same supplier I've been trying to reach since yesterday afternoon finally decides to return my call."
"That's okay." Emma smoothed a small wrinkle from her lap and then pointed toward the open office behind Sue Ellen. "He's not here?"
"He had to step out for a little bit, but he won't be long. You can wait at his desk if you'd like."
She studied what she could see of Brad's office from her seat—the large desk, the fancy chair, the bookshelves filled with books and framed pictures—and shook her head. "No. If it is okay, I would like to sit here. I will not be a bother."
"You're not a bother, sweetheart! In fact, you're a breath of fresh air on an otherwise dreary, sunless day." Sue Ellen pushed back her chair, stood, and came around the desk, her pale green eyes narrowed on Emma. "Can I get you something to drink?"
"No thank you. I will just wait."
"If you change your mind, let me know." The woman wandered over to the row of framed photographs that had claimed Emma's attention during her phone call. "Your father is a good man, Emma. A hard worker. Built this company by himself. From the ground up. And it's only been six years."
Inching forward, Emma pointed to the center picture. "Why do so many people smile as he cuts a ribbon?"
"That was the ribbon cutting ceremony that started all of this." Sue Ellen splayed her hands. "It was the start of Harper Construction as you see it today. This building, the crews, my job, all of it. Now, there are Harper houses sprinkled all over Lancaster County, and we're getting ready to break ground on our first full-blown Harper neighborhood. It's all very exciting, Emma. And you're part of it now."
"Me?"
"Of course. Everyone in Brad's family is involved in some way. Brad's mom consults with customers on color schemes—countertops, appliances, paint, etc.; Brad's uncle helps with odds and ends as his health allows; and you? Well, I guess you two will figure that out. Together."
"But I . . . I'm . . ." She let the rest of her words go in favor of the obvious. An obvious that had her sitting in an English office in an aproned dress, black lace-up boots, and a white kapp.
Abandoning her position beside the wall, Sue Ellen crossed to the empty chair beside Emma, the animation she'd shown only seconds earlier all but gone. "I can only imagine what these past few days have been like for you, dear. Are you holding up okay?"
Was she? She didn't know. If she wasn't actively spending time with her birth father, she was wandering around in a daze, the near constant roar in her ears making it difficult to think through anything in its entirety.
"I don't know," she whispered, looking down at her lap. "Everything is different. My mamm and dat are not my mamm and dat. My aunt was not my aunt—she was my mamm. And my real dat is English."
Sue Ellen's gentle hand quietly stilled Emma's trembling one. "It's a lot for anyone to try to process. Your whole world has literally been turned upside down. But Brad will be by your side every step of the way, I know he will. And, for what it's worth, I'm here, too. I may never have walked in your shoes, Emma, but I'm a mighty fine listener and I've been told my hugs have a way of making things a little better."
"Your hugs?"
"My niece showed up in the middle of the night one time to get one of my hugs."
Emma drew back. "She was not sleeping?"
"She was cramming for one of her last exams at college and she needed an energy boost," Sue Ellen said, laughing at the memory. "When I opened the door and saw her standing there, I thought something bad had happened. But she just told me she was doubting herself in terms of the test and needed something to get her over the hump, so to speak. I reminded her that chocolate works beautifully for me in that regard, and that's when she told me my hugs were her chocolate. Tickled me to no end to hear that."
"Did the hug work?"
Sue Ellen nodded. "She aced the test."
"Is that good?"
The woman's momentary confusion parted in favor of yet another nod and laugh. "It means she passed the test with flying colors!"
The roar was still there, and everything was still as daunting as ever. But, for just a moment, something about Sue Ellen's lighthearted voice made Emma feel a little less alone. A little more—
"You're going to pass all of this with flying colors, too, Emma. I just know it."
She allowed herself one good, deep breath as she, again, met Sue Ellen's eye. "But I am not in college. I am not taking such a test."
"You're right. You're not." Pausing, Sue Ellen cupped Emma's cheek in her hand. "But when you're ready to make the leap into this world for good, who knows where you might go and what you might do in life. The sky's the limit for you now, dear. Your father will move heaven and earth to make sure of that."
* * *
Slipping his fingers inside the bag's narrow opening, Brad tugged outward until his entire hand could reach inside. "You're sure you're okay with going through the rest of your presents here at the office instead of back at the pond?"
Emma turned from the window overlooking the back parking lot and managed a nod. "Yah. It has gotten colder since I rode my scooter here."
"I'm sorry I was so late getting back here. My meeting went a little longer than anticipated." One by one, Brad set fifteen years' worth of birthday gifts across the top of his desk and then tossed the drawstring bag onto a nearby shelf. "So, did Sue Ellen take good care of you while you were waiting?"
She wandered back to her chair and lowered herself to its edge. "Yah, I—"
"Oh! I didn't tell you. . . . My mom is due back from her sister's sometime in the next"—he peered at his silver link watch—"three hours or so. I figure I'll either be there when she arrives, or stop by shortly thereafter, to tell her the big news."
"There is big news?"
He pulled a face only to let it dissolve into a smile. "Um, hello . . . My daughter—her granddaughter—is alive and well." He picked up the baseball and turned it over in his hands. "You could go with me, if you want. Really bring home the surprise."
Looking up at the ceiling, he shook his head. "Can you imagine? Hey, Mom . . . The baby didn't actually die with Ruby. She's actually been living with her kidnappers for the past twenty-two years. And . . . ta da . . . here she is." He turned the ball over one last time and then deposited it back atop his desk. "Yeah, probably a bit too jarring, huh?"
She knew he was still talking. His facial expressions told her that. But her thoughts kept returning to one particular word and the way it made her stomach feel the way it did when she thought about Esther wandering away from the farm. "Wait."
His eyes lit on hers. "You okay, Emma?"
Pressing her hand to her stomach in an attempt to stop the sudden yet definitive swirl inside, she made herself breathe. "You called Mamm and Dat kidnappers."
"Because that's what they are."
"But . . ." She thought back over the many newspaper and magazine headlines she'd seen in the English grocery stores while growing up and, as her mind's eye narrowed in on a few, she gasped. "But kidnappers steal people!"
His blue eyes darkened. "That's right. They do."
"They . . . they . . ." She stopped, swallowed, and tried again, the swirling in her stomach growing all the more urgent. "They didn't steal me. They just didn't tell me I was Ruby's baby."
"And my baby." The wheels of his chair thumped against the floor as he stood. "I'm your father, Emma. That means you belonged to me—that I should have left that house with you that day. Only I didn't. Because Wayne told me you didn't make it and I believed him. And for the next twenty-two years they raised you as their own. Unbeknownst to me."
He stopped midway to the window and turned back. "Unbeknownst. To. Me. That's kidnapping, Emma, in every sense of the word."
"But they just didn't tell us," she whispered. "I know it was a terrible lie. I have even thought about telling the bishop so they will be shunned. But—"
His half laugh, half snort echoed around them. "Shunned?"
"Yah. It means the members of our district will turn their backs to them until they repent and—"
He shot his hand up, stopping her explanation. "Trust me, kiddo, I know all about the way the Amish work. The way they govern."
"The Amish do not govern," she protested. "When one is baptized they vow to—"
"Remain Amish. To live in the world but not like the world. I know. I got it. But there are laws in this country that everyone must follow. Including the Amish. And when people break them, there are consequences—consequences that are, thankfully, blind to hats and beards. In theory, the jurors should be as well."
The rhythmic tapping of her booted toes against the carpet broke through the mental roar kicked off by his words. Shifting her hand from her stomach to her knee, she pressed down until the motion stopped. "I-I don't feel well right now."
Like the overhead lights the English flicked on and off with a switch, the anger Brad wore bowed to concern. For her. He covered the gap between them with two long strides, and squatted down beside her chair. "What can I do? Do you want some water? Some air? What?"
"Maybe some . . . air?"
"You got it." He bounced back up, strode toward the window, and lifted it up a few inches. "How about some water, too?"
With her leg steady, she returned her hand to her stomach. "Yah."
He stepped over to his desk, tapped a button on a small brown box, and, when Sue Ellen's voice appeared through it, he asked for a water. Less than a minute later, Emma had her water, and she and Brad were alone once again.
"Look, kiddo, I didn't mean to upset you. The whole reason I put my guy on this is so I can focus on us. So let's let him do his thing while we do ours, okay?" With purposeful steps, he returned to his chair and swept his hands toward the top of his desk. "So, shall we get back to this?"
Nodding, she reorganized the items to match the order in which she'd found them and then picked up the picture of the dandelion from her eleventh birthday. "Sarah and I would always race to see who could blow on these the fastest. I would always win until Sarah got bigger. Now, it does not take Sarah many puffs, and Annie is very good at it, too. Once, it took Annie only one big puff!"
"Sarah and Annie—those are Wayne and Rebeccah's, right?"
Wayne and Rebeccah's . . .
"Yah. Sarah is sixteen. Annie is eight."
"And there are how many others?"
"Three. Jakob, Jonathan, and Esther."
Grabbing a pen from a holder to his right, Brad jotted something on a small notepad and then shoved it off to the side. "Okay . . . So the dandelion picture . . ." His chair creaked as he leaned back against his chair. "I'm not sure why, but the English—as you call us—don't just blow on dandelions when they turn white like that. We make a wish while we blow."
"A wish? Why?"
"I don't really know. It's just something kids, and sometimes even adults, do. It's like wishing on a birthday candle, though I'm not sure it holds as much weight. Depends on who you ask. But my mom taught me to make a wish every opportunity I had. So birthday candles, dandelion fluff, pennies in fountains, and the wishbone from the turkey on Thanksgiving were all fair game as far as I was concerned." He tented his fingers beneath his chin and grinned. "Of course, Ruby thought I was crazy the first time I handed her a dandelion and told her to make a wish and blow."
She looked from Brad, to the picture, and back again. "Did she?"
His laugh eased the last of the tension from her body, allowing her to finally relax into the back of her own chair. "She blew, but she didn't make a wish. So I handed her another one and told her to try again. That time, she supposedly made a wish, but she kept her eyes open."
"I don't understand."
"Wishes don't work if you don't close your eyes." Dropping his hands, he lurched forward, plucked up the picture, and gazed down at it, the skin around his eyes crinkling with amusement. "So I picked up another one and, that time, I made the wish. Closed my eyes and everything.
"When I was done, I opened them to find I'd been successful in scattering the fluff and that Ruby was watching me with the biggest smile, her head cocked just like this." He tilted his chin a hairbreadth. "When I told her that's how she was supposed to do it, she asked me what I'd wished for. I told her I couldn't tell her, or it wouldn't come true."
"Did it?" Emma asked. "Come true, I mean . . ."
Righting his head, Brad took one last look at the picture and then set it back down on the desk in front of Emma. "On your eleventh birthday, when I set this reminder of that special moment on the grave, I'd have said no. But now, with you sitting here? Yeah, it came true. In a form very different than I imagined when I made it, but it's still true, nonetheless."
She hovered her fingertips above the dandelion stem and tried to imagine a younger Brad sitting in a field of dandelions with the girl from her locket. When she had a fuzzy version of it, she looked up. "What was your wish?"
"For Ruby to be part of my life forever."
"But she's not," Emma protested.
A softer yet no less genuine smile reclaimed his lips. "In the sense I envisioned as an eighteen-year-old, no, she's not. But having you here"—he splayed his hands around the office—"in my life, changes that."
Emma pulled her hand away from the picture and braced it on the edge of the desk. "But I am not Ruby."
"But you're a living, breathing part of her, Emma. And there are times, when you look at me the way you are right now, it's like looking at Ruby all over again. Like all the bad stuff never happened." He reached out, patted the top of Emma's hand, and then leaned back in his chair once again. "I'd consider that an answered wish, all things considered."
She took another sip of her water and then set the bottle on the floor. "Did she try again? After you showed her how to do it?"
"You mean with the wish?" He returned her nod with one of his own. "She did. Don't know what she wished for, but I know she kept her eyes closed even after the fluff was scattered. Like whatever she was wishing for was taking a little time.
"I teased her about it when she was done. Asked her if she was wishing for an ice cream cone. But right before we came across all those dandelions, we'd been dreaming out loud about what I might want to do for a career, and what we'd like our one-day house to be like. I guess that's why I've always figured it had to do with one or both of those."
It was a lot to take in. A lot to add to the image she'd managed to conjure of the encounter. Swapping the picture for her twelfth birthday present, she held up the bubble wand. "And this?"
"I surprised her one day with a jar of bubbles. We practically used the entire jar that same day. I showed her how to make double bubbles, and she showed me how to make really long ones. We laughed a lot that day. And she jumped, too . . ."
"Jumped?"
"Ruby would do this little jump when she was excited about something—bubbles, skating, it didn't matter. If she was happy, she did her little jump."
She placed the wand next to what remained of the carnival ticket and moved on to the rock with the sparkly heart drawn on it. "Did Ruby draw this heart?"
Resting his left foot atop his right knee, he nodded. "She sure did. My cousin left a sparkly marker at my house the previous day and I gave it to Ruby. Next thing I knew, she was drawing that heart on a rock she found at the pond and giving it to me. Kept it in my room at Mom's for years. When I came back on your birthday that year to leave my annual gift, I decided to leave the rock because it wasn't more than a week after she drew that heart that she told me she was carrying you."
Emma snapped her attention off the rock and onto Brad. "Were you scared when she told you about me?"
"Nope. I was excited. As far as I was concerned, you being on the way just made us more real somehow. In a good way. Besides, it was just one more reason for us to get married like we wanted to—or like I wanted to, at least." Brad dropped his foot back to the ground with a thud. "So, what's next?"
"Next? I . . ." She followed the path of his finger back to the desk and the items she'd all but forgotten as his words had transported her back to a time when Ruby was alive and Emma had been on the way . . .
"The red and black checked napkin, right?" Brad prompted. At her nod, he slid the piece of cloth in her direction until she relinquished the rock in its favor. "Remember that picnic I told you about? The one I put together to show her what an English picnic was like?"
"Yah . . ."
"She planned one a week later. To show me what an Amish picnic is like." His gaze lifted to the wall just beyond her head but seemed to fix on something far beyond his office. "I guess I'd tucked the napkin in my pocket at some point along the way and didn't realize it was there until I was back home afterward."
Emma pulled the napkin close and tried to imagine Ruby packing it inside a lunch pail along with the food she'd surely prepared. "You spent so much time together. Did my grosselders know about you?"
Shaking his focus back into the room, he looked at Emma. "At that point, no. Eventually, they had to. But you have to remember, when we met, Ruby hadn't officially been baptized yet, so it's not like she was breaking any vows or anything by spending time with me. Still, I know they weren't pleased once they learned of my existence for obvious reasons.... I know I wasn't that Gingerich guy and that my involvement with Ruby put the kibosh on that.... I know I wasn't Amish. . . .
"But you can see, from all of this"—he spread his hands wide to indicate the collection of memories spread across his desk—"that we had something special. Something real. Something worth sticking with. Different doesn't have to mean wrong."
A soft yet staccato tapping sent their collective focus toward the now open office door and Sue Ellen's wide eyes peeking into the room. "Boss? I'm sorry to interrupt, but your mother called and asked me to let you know she should be arriving back at her place around five and she's hoping you'll come for dinner."
Glancing at his watch, Brad pushed back from the desk. "So I've got about thirty minutes if I want to be there when she arrives.... Okay, thanks, Sue Ellen."
"You got it." The secretary's head disappeared from view only to return a half second later. "Oh, one more thing. I touched base with the crew and let them know you'll be out of town Monday and Tuesday and will likely want a meeting with everyone here Thursday morning. Eight a.m. sharp."
"Perfect."
"Is there anything else I can do?" Sue Ellen asked, sending a smile in Emma's direction before looking back at Brad. "For you, or for Emma?"
He shook his head and then winked at Emma. "No, we're good. But it's Friday and the crews are probably calling it a day out at their respective sites, so why don't you pack it up and head home, too. Enjoy your weekend."
"Thanks, Boss. That sounds perfect." Sue Ellen wiggled her fingers around the edge of the door in a happy wave. "You two have fun this weekend. And kiss the proud new grandma for me when you tell her the good news."
"You can count on that." As Sue Ellen pulled the door closed, Brad stood, his excitement palpable. "So, what do you say we save the rest of these stories for tomorrow? That way I can get you out to the farm, and me to my mother's place before she gets back."
She retrieved the water bottle from the floor and set it, instead, on the table. "Yah. But I can get myself home. I have my scooter."
He grabbed the bag off the shelf and held it open as, one by one, Emma put her birthday presents back inside. When the only one left was the napkin she'd set beside her water, she scooped it up and lingered her gaze on the cheerful pattern. "May I ask what Ruby made for your Amish picnic?"
Relinquishing his hold on the bag, Brad wandered over to the window, his voice taking on a seemingly faraway quality. "She made the best fried chicken I ever had, these potatoes that practically slid down my throat, and a slice of homemade bread with apple butter. When that was done, there was a slice of apple pie and two oatmeal cookies. No Frisbee, though."
It was the kind of picnic Emma would likely pack for Levi if she could. If he fancied her the way he did Liddy Mast . . .
"Penny for your thoughts?"
Startled, she looked up to find his back now flush to the window and his attention trained solely on her. "I was just thinking of a friend who likes oatmeal cookies, too."
"Levi?"
"H-how did you know?" she stammered.
"You mentioned him the other day." Bracing his hands on the lip of the sill behind him, he watched her fidget the picnic napkin between her fingers. "You said he was just a friend. Maybe someone's brother?"
"Yah. He is Mary's brother."
"And Mary is one of your friends?"
She didn't mean to laugh. But something about his question and the way in which he said it made it impossible to react any other way. "Mary is my only friend."
His brows, like his mouth, seemed to frown. "I don't believe that. . . ."
"It is true. Mary does not see me as others do."
"How do others see you?"
"As someone who does not belong. But I do not think it is that way with Levi. I think Levi does not see me at all." She added the napkin to the bag and pulled it closed with the string. "I will stop out at the pond on the way back to the farm and put this back inside the tree. Perhaps you can tell me about the rest of the presents tomorrow or one day next week."
He pushed away from the window and joined her at the desk, his hand coming down atop the bag as he did. "If Levi does not see you, he is blind. His problem, not yours. And the other part? About not belonging? That's about to change, I assure you of that."
Her head was shaking before he'd even finished his sentence. "It won't change. Ever. Because I can't change. I know this now."
"Why do you have to change anything?"
"Please. It is getting late. You must go." With the help of her chin, she led his gaze off her face and onto the wall clock to her left.
His answering sigh was muted by his hand just before it slid down his own chin. "You're right. We need to wrap this up for now. But we will talk more about this tomorrow, okay?"
She looked past him to the window and what she could see of the February sky beyond. Soon dusk would begin to settle across a day in which she hadn't done any chores. When it did, Dat and the boys would come in from the fields hungry for a dinner she, once again, had no part in preparing.
It was only a matter of time before Sarah's and Annie's curious glances over Emma's lengthy and unexplained absences morphed into fully formed questions. Especially if she spent the bulk of yet another day away from the farm. But she needed answers. Different answers than the ones her siblings would soon have.
"Emma?" he prodded, stepping forward. "I will see you tomorrow, right?"
She looked down at the strong hand now resting on her forearm and then back up into the eyes of the only person truly capable of providing those answers. "Yah. Tomorrow."
Chapter 13
One by one, Emma took the eggs from Esther's quick-moving hand and added them to the same basket the task demanded each morning—a basket she, herself, had made when she wasn't much older than Esther. She tried to keep count as the little girl's hand maneuvered its way around curious chickens and through piles of straw, but Esther's running commentary on the latest additions to the barn was making even simple math difficult. Still, she was pretty sure they were up to fifteen, maybe six—
"I helped Bean clean the brown and white kitty." Esther pulled an egg through the door of the chicken coop and handed it to Emma. "Her eyes look like this, see?"
Emma deposited the final egg into the basket and then looked up to find the five-year-old peeking out at her from otherwise closed eyes. "You mean the kitty?" At Esther's nod, Emma stood and brushed some loose straw from her dress. "That's because she's a newborn. It takes a little while for them to open their eyes. But when she does—assuming she's a girl, of course—I bet she'll be really curious about you."
"She has to be a girl. I gave her a girl's name." After a final wave to the chickens, Esther wrapped her hand around Emma's and began to tug. "Her name is Flower, and you can't hardly hear her meow."
"What has you in such a hurry this morning?" Emma asked. "Usually I'm the one pulling you away from the chickens."
Esther pointed toward the barn. "I want you to see Flower and all the other kitties, too!"
Emma swung her gaze from the barn, to the fields, and, finally, to the side yard and Sarah. On any other morning, Emma would take the eggs into the house, make sure Esther, Annie, and Jonathan were ready for school or, in today's case, their list of Saturday chores, and then join Sarah at the clothesline or Mamm in the garden.
But it wasn't any other morning. Nor had it been any other morning since she set off for her birthday visit to the cemetery some twelve days earlier.
"Come on, Emma! Flower is waiting for me to say good morning."
She followed along for a moment, only to stop as they reached the open barn doors. "Esther, I can't. I have something I have to do today—somewhere I have to go."
The pressure on her hand intensified and then released as Esther's shoulders slumped. "I don't want you to be gone again, Emma. You keep going away and I miss you. I want to help you bake bread, I want to watch you quilt, I want to sit on your lap and hear stories, and I want you to see Flower and all of Bean's new kitties!"
Squatting down, Emma bobbed her head until the sad eyes she sought were trained solely on her face. "I'm sorry, Esther. I know I haven't been around very much. But there is someone I must see. Someone I need to know."
"Is it the man with the toys?"
She glanced over her shoulder toward the house as she scrambled for an answer that wouldn't be more of the same lies, yet also wouldn't stoke the same old medley of anger, fear, and confusion that kept her awake most nights now. It was a tricky balance, no doubt. "Yah."
"I can come with you," Esther suggested. "I like toys, too!"
"You do?" Emma teased. "I didn't know that!" Then, tapping the little girl on the nose, Emma jutted her own chin toward the open barn doors. "You know what? I could take a few minutes to meet Flower. But not too long, okay?"
Esther rose up on the tips of her shoes, spun around, and then beckoned for Emma to follow her inside, her excitement over Bean's new kittens turning her walk into more of an all-out run. "Look! Look, Emma!" the child said as she ran over to the stall in the far corner of the barn.
Sure enough, tucked neatly inside a bed of hay, no doubt created by Bean, herself, were five mounds of matted fur nestled against their mother. Each kitten appeared to be sleeping as Bean proceeded to clean those in easiest licking range. "Jakob named the white and black one Mewer. Jonathan calls the gray one Whiskers. Sarah says the black one is Jumper. And Annie says the one with the gray and black stripe right there"—Esther touched her own forehead and then pointed at the kitten now wiggling itself closer to Bean—"is Apple Pie, even though that's a very silly name for a kitty."
Following Esther's lead, Emma, too, lowered herself onto the recently mucked ground and leaned in for a closer look at the outlying kitten. "I take it this little cutie is the one you have named Flower?"
Esther grinned. "She is! I like her best!"
"You should like them all," Emma reminded gently. "They're all Bean's babies and they are all mighty cute."
"I do like them all, Emma! But I got to name Flower and I got to lick her, too!"
Emma drew back. "You licked her?"
"Yah! Bean forgot to do it, so I helped!"
Feeling her lips begin to twitch, Emma turned back toward the new kittens and did her best to stifle the urge to laugh lest she hurt Esther's feelings. "While I'm sure Bean was glad for the help, you really should let her do it. She has a lot of babies to lick right now, Esther, but she'll get to all of them. Besides, you don't want to fill your tummy with fur the way Bean does, sometimes, do you?"
Esther stilled her little fingers atop the sleeping ball of brown and white fur, her face solemn. "No. I don't want to get sick."
"Good. Then let's leave the job of cleaning to Bean, okay?" At Esther's slow nod, Emma scooped her hands beneath Flower and gently pulled the newborn kitten against her aproned overlay. "Well, hello, little Flower."
"Do you know why I named her that?" Esther climbed onto Emma's lap for a closer look. "Why I named her Flower?"
Emma took in the tightly closed eyes, the tiny little ear nubs, and the soft brown and white fur before turning her attention back to Esther. "Why?"
"Because her eyes are closed real tight like Mamm's flowers when they first pop out of the ground. When it gets warm and sunny, they all open real pretty. Just like Flower's eyes will do when she gets a little bigger."
She considered the little girl's reason and, after a follow-up glance at the other four kittens, found that it fit perfectly. "I think that's a great reason to call this little one Flower."
Esther's smile spread across her tiny face only to disappear seconds later. "I am sorry there is not a kitten for you to name, Emma. If Bean had another kitten there would be six to name. Like there is six of us!"
Biting back the urge to correct the lie they'd all been told, Emma, instead, returned the sleeping kitten to its indent in the hay and stood. "I must go, Esther."
"Can't you stay just a little while longer? Flower likes when you hold—"
The rest of the little girl's pleas fell away as the clip-clop of an approaching horse just beyond the barn doors brought Esther to her feet, as well. "Someone is here! Someone is here!"
"It certainly sounds like it." Emma took one last look back at Bean and her babies and then followed Esther back across the barn, stopping every few feet to run a hand across the head of one of its tenants.
* Mabel, the aging dairy cow . . .
* Dusty, the field mule . . .
* Robbie, the rooster . . .
Jakob popped his head around the corner of the open door, acknowledged Emma and Esther with a nod, and pointed at the wall where Dat kept many of his farming tools. "Emma, could you grab the saw? Levi Fisher is here to borrow it for his dat."
Startled, Emma looked past her brother to the driveway beyond. "Levi is here? Now?"
"Yah. He is speaking with Dat." Again, Jakob pointed at the saw. "Seems his dat's saw got bent so bad on a fallen tree, it can no longer be used. The tree is blocking the way to Miss Lottie's house."
Emma drew her hand to her chest. "Is Miss Lottie okay?"
"Yah."
Relief sagged her shoulders as the elderly English woman filled her thoughts. Loved by many in Emma's community for her wisdom and gentle ways, Miss Lottie, as she was known by everyone, lived not more than a quarter of a mile to the east, in a small cottage-style home nestled in a field of sunflowers and overgrown grass. "That is good to hear."
"Jakob!"
Together, Emma and Jakob turned toward the back of the barn and its view of the fields Jonathan was painstakingly combing for any rocks the girls might have missed before plowing and planting could start. "I have to go, Emma. I must see what Jonathan needs."
Hooking her thumb across her shoulder, Emma took a step backward toward the tool wall, her gaze flitting between Jakob and the sliver of buggy she could just barely see beyond his shoulder. "I will see that Levi gets the saw."
"Thank you, Emma."
She watched the twenty-one-year-old disappear from her view and then crossed to the series of hooks used to keep track of the various tools and implements used around the farm. The saw she sought hung above a row of hammers and a long piece of metal she recognized as a level. It was also several feet out of her reach.
Glancing around, she spied an empty feed bucket, carried it back to the wall, and turned it upside down atop a nearby hay bale to create a makeshift stool. She stepped on, rose up on the toes of her boots, and extended her hand as far as she could reach, coming within a finger's length of the saw's handle.
Rocking forward even more, she balanced atop the very tip of her boot and stretched a little more, the deficit shrinking to that of a mere fingertip. Determined to get the grip she needed, she inched her toes onto the bucket's rimmed bottom and extended her reach still farther, her fingers grazing the underside of the saw as the bucket tipped to the left.
She felt herself beginning to fall, but just as her predicament was beginning to register, a pair of strong arms encircled her from behind and deposited her safely on the ground.
"Whoa, Emma. I do not want you to get hurt." Setting the now righted bucket off to the side, Levi reached up to the proper hook, retrieved the saw from its resting place, and leaned it against the barn wall as he sought Emma's eyes with his own. "Hello."
Aware of the sudden warmth that claimed her cheeks, she broke eye contact long enough to catch a glimpse of her reflection in the saw's blade. When she confirmed her kapp was on straight and her dress was relatively wrinkle free, she found the smile he deserved for warding off what would have been a nasty fall. "Thank you for catching me."
"I am glad I came in when I did. You could have gotten hurt."
She shrugged away his concern and, instead, collected the saw from its temporary resting spot. "Jakob said you need to borrow this from Dat to clear a tree blocking Miss Lottie's house from the road?"
"Yah."
"Here you go." She handed him the tool. "I hope this one does not get stuck in the tree."
His deep, rumbly laugh filled the space between them. "Perhaps the Beilers will have a saw I can use if it does.
"But if that breaks, too," he added, shrugging, "I will need to buy many new saws for many people."
It felt good to laugh. Great, even. In fact, the image of a half dozen or more saw handles sticking out of a fallen tree made it so the murkiness surrounding everything about her life the past twelve days paled. And for that she was grateful.
"I am glad to see that it is back," Levi murmured.
"It?"
Lifting his free hand into view, he pointed to her mouth. "Your smile."
The last of her laughter faded as she drew back. "I do not understand."
"Your smile. It has been missing for many days."
Unsure of how best to respond, she cast her eyes down at the ground. "I did not mean to not smile. If I have been rude, I am sorry. I—"
At the feel of his hand on her arm, she stopped talking and looked up to find his warm brown eyes studying her closely.
"You must not apologize for sadness, Emma."
Two weeks ago, she'd have rushed to counteract his impression of her and her demeanor with the biggest smile she could muster. But it wasn't two weeks ago. It was now. The Emma who had always tried to please her way into everyone's hearts was gone. In her place was the Emma who finally knew she never could.
Smoothing her hands down the sides of her mint-green dress, she jutted her chin toward the saw, and then the driveway. "Now that you have Dat's saw, I really must go."
Surprise traded places with concern on his kind face. "I am sorry, Emma, I did not mean to keep you from what you must do."
"If I'm late, it is because I let Esther talk me into meeting Bean's kittens."
"Kittens?"
"Yah. There are five of them." She swept her hand and his eyes toward the back stall. "Would you like to see them? Bean had them in a bed of hay in Mini's stall, just out of reach of the mare's hooves."
"It sounds as if Bean chose wisely."
Emma led the way down the same hay-strewn path she'd walked with Esther not more than ten minutes earlier. When they reached Mini's stall, she pointed down at the still sleeping kittens. "The one closer to Bean's back foot is Flower. Esther named that one."
With the help of her finger, she introduced the others, starting closest to a sleeping Bean. "Mewer, Whiskers, Jumper and"—she cast about for the right name—"Apple Pie."
His lips twitched with the smile he couldn't hide. "Apple Pie?"
"I think Annie was hungry when it was her turn to choose a name."
"I think you are right." He bent down, ran his hand across all five sleeping bodies, and then flashed a sheepish smile up at Emma. "Though, if I was hungry when naming a new barn cat, I would choose Oatmeal. After the cookies you would bring to hymn sings each week."
With one final stroke, this one atop Bean's head, Levi stood. "I miss those very much."
"I do not know why." Emma took one last glance at the kittens and then motioned Levi to follow her outside. "Liddy Mast brings oatmeal cookies now."
"They are not the same."
Lifting her hand as a shield against the midmorning sun, she stopped near the back of his buggy. "Oatmeal cookies are oatmeal cookies. They are all the same."
"Then you have not tried Liddy Mast's."
"They are not good?" she asked.
His eyes dropped to the saw in his hands before slowly working their way back to Emma's. "They are not yours."
"Perhaps Liddy missed an ingredient or was busy with the wash when they were to come out of the oven. . . ."
"Two times, she has brought them, and, two times, I have tried them thinking they were yours. And two times, I have known they are not after the first bite." He sidestepped his way to the front of his dat's buggy and placed the saw on the floor. "That is why I am hoping you will point to your cookies at tomorrow's hymn sing."
She dropped her hand to her side and stepped into the shadow made by the barn. "I do not think I will be at the hymn sing this week."
"Oh?"
"Yah. I-I might be busy."
"Busy?" Levi echoed. "On the Lord's day?"
"I will go to church, of course. But it is after—when the hymn sing is to start—that I think I will be busy."
"Doing what?"
"Visiting. With . . ." She took in the house on the opposite side of the driveway and then turned away, a familiar anger lapping at the edges of her tone. "I will be spending the afternoon with family."
Levi started to speak but stopped as she motioned toward the road. "I must go, or I will be late. If there is anything else you need, the boys are in the field, and Sarah and Annie are in the house or"—she craned her neck to afford a view of the side yard—"rather, right there, finishing up at the clothesline."
"No, I have what I need." Adjusting his straw hat atop his head, Levi covered the distance between them with several long strides, stopping only when he was within arm's reach. "If you are leaving the farm, I could take you to where you are going. Then you will not be late."
She looked from the road, to his buggy, and, finally, back to Levi. "Miss Lottie lives in the opposite direction from the pond."
"You are going to the pond?"
"Yah."
"How can you be late to the pond?" he asked.
"It is where I am to meet someone."
Surprise flickered across his face before being scrubbed away by his calloused hand. "The pond is not so far out of the way that I cannot drop you off. I do not want you to be late because you were getting me a saw and showing me new kittens."
"I do not want Miss Lottie to be trapped inside her home."
"She is not trapped. She was sitting on the porch reading a letter when Dat's saw got stuck in the tree. It is Miss Lottie who suggested I borrow one from Beiler or Troyer."
"But you came here . . ."
"Yah." A hint of crimson pricked at his cheeks as he leaned forward and brushed something from the top of his boot Emma could not see.
Not sure what to make of his odd behavior, Emma took another step toward the road. "I really must go."
Her words brought him upright once again. "Wait right there. I will get the buggy and I will take you."
Five minutes later, they were on their way, Levi's mare, Hoofer, faithfully pulling them in the direction of the pond while Emma looked out over the farms of their Amish brethren. She saw the Troyer boys combing the fields for the same rocks that held her brothers' attention one farm south. At the Weaver farm, the laundry was already on the clothesline. And at the Schrock farm, the bench wagon to the side of the house meant the family was inside, preparing their home for the next day's church service.
"Do you ever think about it being different?" she asked.
Levi pulled back on the reins just enough to slow Hoofer to a walk. "About what being different?"
"All of it." Emma splayed her hands to indicate the Schrock farm on their right and the Troyer farm on their left. "Your choice.... The way the Amish live . . ."
"No. I made my choice. I am Amish."
It was such a simple answer—one she, too, would have given just two weeks earlier. Because, prior to her birthday, she hadn't thought about life outside the Amish fold. She'd had her opportunity to experiment with an English life during Rumspringa and she'd opted against it on any large scale. She'd carried around a phone, but never had anyone to call. She'd taken off her kapp a few times while walking along the county road, but had always put it right back on her head not more than twenty or thirty steps later. Because, ever the pleaser, she'd hoped her steadfast commitment to the life would have made Mamm smile.
A sigh, born on yet another example of her naivete, perked Hoofer's ear and brought Levi's focus back on Emma. "Do you think about making a different choice?"
It took a moment to catch up with his question, but, when she did, Emma traveled her eyes back to the land around them. "I didn't . . ."
"You say that as if it has changed."
"Because it has. Everything has."
The clip-clop stopped with his tug. "You know what would happen if you were to leave the church now. You have been baptized, Emma. You have chosen this life."
"I know." Soft green fabric oozed through her fisted fingers only to flatten against her skin in tandem with the breath she hadn't realized she'd been holding. "But this was never meant to be my world."
Feeling the weight of his confusion on the side of her face, Emma looked back at the Schrock farm, her need to find a distraction settling her sights on the bench wagon. "I am sure Waneta and the girls are busy preparing food for tomorrow. Jakob hopes they will make chess pie. He says Waneta's is best."
"I do not know about Waneta's chess pie, but I do know that this is your world, Emma. It is mine, too. We are Amish. We live an Amish life."
She swept her gaze back to his. "You would not be Amish if your mamm and dat were not Amish. You are Amish because they are, because that is the life you were shown. But what if they were not Amish or one was not Amish, and no one told you that until after you had been baptized? Would it still be your world?"
"If I had been baptized, yah." With little more than a shift of his hand and a click of his tongue, the buggy lurched forward, the steady clip-clop of Hoofer's feet shattering the charged silence.
Together, they rode along, the mare's hooves the only sound between them as they approached the lone bend in the road between her farm and Miller's Pond. She could sense Levi wanting to speak, to ask questions about their odd conversation, but, in the end, he said nothing.
Still, she slanted a look at him when they came out of the bend, his widened eyes a precursor for the response now making its way past his lips. "Emma! That is the truck! The truck you were asking about at the last hymn sing!" He slowed the buggy to a crawl. "The one that stops at the cemetery each January with the sign on the door."
"I know."
Bobbing his head left, then right, Levi strained to make out details, but the late-morning sun made it difficult to discern much of anything beyond the truck's presence on the side of the road. "Someone is sitting inside. I wonder if it is the same Englisher that I see standing inside the cemetery. The one who looks down at one of the graves."
"Yah. It is. His name is Brad—Brad Harper."
Levi shifted his full attention onto Emma. "But I did not tell you his name. . . ."
"You told me enough that I was able to find out the rest myself."
"Do you want me to stop? We could ask why he is here, why he comes to an Amish cemetery each year."
She lifted her hand from its resting spot atop her lap and waved a greeting toward the truck. "I know why he comes."
"You do?"
"Yah. He told me."
With yet another tug, Hoofer came to a stop a few buggy lengths away from the truck. "You have spoken to the Englisher?"
"Yah."
"When? How?" Levi split his attention between the man now exiting the truck and Emma. "Why?"
"The day after you and I spoke at the hymn sing, I rode my scooter to New Holland. An Amish boy working at a restaurant told me how to find Harper Construction. For many hours I sat next to a shed behind the building just waiting to see him come out. When he did not, I went inside to see if I was right."
They both turned, in unison, toward the crunch of gravel and the man stepping out from the side of the truck. "I do not understand," Levi said, glancing back at Emma. "Why did you go inside to meet the Englisher from the cemetery?"
"I needed to see if he was the one—the one who had been leaving things beside my mamm's grave all these years."
Levi's inhale echoed across the road, slowing Brad's steps in return. "Your mamm's grave? But—"
"Rebeccah Lapp is not my real mamm," she managed past the lump making its way up her throat. "And Wayne Lapp is not my real dat."
"Emma, how can you say that?"
"Because it is truth—a truth they did not tell the bishop, or the church, or me." Aware of the anger sharpening her tone, she took a deep breath and lifted her face to the sun. "I am sorry, Levi. I do not mean to get angry. None of this is for you to worry about."
"I do not know what to say."
"That is because there is nothing to be said."
Releasing his right hand from rein-holding duty, he hovered it above hers before draping it across the back of the seat, instead. "I am a good listener. Even Mary says so."
"Hey, Emma! I was just leaving a voicemail for my attorney friend when you guys came around the corner." Brad stepped alongside the buggy, nodded once at Levi, and then held his hand up to Emma. "But you're here now so let's get going. There are some people who are mighty excited to meet you, young lady."
Levi stiffened on the bench beside her, earning himself a backward look from Hoofer. "But you wanted to go to the pond," he reminded Emma.
"You are right. I did. To meet him," She pointed down at Brad and then, placing her hand inside the Englisher's, stepped down to the ground, the hem of her dress swirling against the top edge of her boots. "Thank you for the ride, Levi. Please tell Miss Lottie hello for me."
Chapter 14
Even though her shivering had little to do with the February chill, Emma was grateful when, at the flick of Brad's fingers, heat began to emanate from the slats in front of her seat.
"That'll have you warmed up in no time." Returning his hand to the steering wheel, he slid a peek in her direction. "So, who was that, just now? In the buggy? Was that one of Rebeccah's?"
"No. That was a friend."
"He didn't look too pleased to see me."
Pulling the flaps of her coat closer to her body, she shrugged. "It was not you. It was me. I should not have told him those things."
"Things? What kind of things?"
She looked out at the countryside, the farms and animals whizzing by so fast she could barely register which family lived where. "I told him you are my birth father. He did not know."
"Well, then, welcome to the club."
"Club?" she asked, shifting her attention to Brad.
"It's just an expression, kiddo. In this case, it means your friend didn't know just like you and I didn't know." A series of quick vibrations from the cup holder between them had Brad apologizing and reaching for his phone. "Brad here."
She turned back to the passenger side window and noted that the homes were growing larger while the land grew noticeably smaller. Here, there were no cows grazing, no Amish pitching hay or preparing the soil for the spring crops, and no clothes drying in the cold winter sun.
"Okay, okay, slow down. I know you're excited, but don't worry. We're officially on the way. In fact"—his sudden pause drew Emma's attention back to him and the single word he mouthed: mom—"we're coming up on Shady Pine right now."
Emma turned in time to see a wooden sign, mounted atop two stone pillars, with the words SHADY PINE written across the center. Beyond the sign was a narrow road that curved around a grove of trees, vanishing from sight.
"That's right. Five minutes. Tops. See you then." He set the phone back in the cup holder and grinned at Emma. "I wish you could have seen your grandmother when I told her about you last night. For the first few minutes after I was done speaking, she looked at me like I was from another planet. She might have blinked a few times, but that's it. Then, as it all began to sink in, she started screaming—good screaming. From there, we moved into a mixture of questions, tears, and planning."
"I do not want to make someone cry," Emma said, ducking her gaze down to her lap.
The index finger of his right hand nudged her focus back to start. "Not all tears are bad, kiddo. And even the ones that were because of sadness came and went. My mom is a multitasker. Always has been."
"Multitasker? That is not a word I am familiar with."
"My mother thrives on doing many things at one time. Last night was about the shock, and joy and sadness and anger—each of which came and went depending on the question she asked and the answer I gave. When I finally headed out to my own place, she was sitting at the kitchen table making lists and deciding the order in which to call people." He slowed to a stop at a four-way intersection and then continued on, the scenery outside the window holding little interest for either of them. "Today will be about meeting you, soaking you up, asking more questions, and then sitting down together over a meal. And while I know it will probably be a little daunting this first time, I'm confident you'll grow to love her and the way she does things. Everyone does."
It was all so much to process. The sights out her window. . . Being in a shiny black truck with her English birth father . . . Hearing him describe the woman who was her grossmudder . . .
She felt the car begin to slow again, only this time, instead of stopping because of a sign or traffic light, they turned right and headed down a street with neatly kept homes on either side. "What about your dat? Will he be there today, too?"
"He took off before I was a year old. Apparently two kids in as many years was more than he bargained for." At the next crossroad, Brad turned left and then pointed Emma's attention to the lone house on their right. "There it is. The house I was living in when I met Ruby."
Pressing her forehead to the window, she took in everything she could about the house on her right. The powder-blue exterior . . . The pretty white and powder-blue curtains she could glimpse at the windows . . . The billowing flag depicting a snowman anchored into the ground to the left of a walkway . . . The ceiling-mounted swing nestled in a corner of the porch that appeared to look out over a small frozen pond...
Emma sat up tall, craning her neck as far to the left as possible. "Is that where you took her to ice skate?" she posed. "Like the snow globe?"
He stopped, mid-nod, his widening eyes sending her attention back to the house and the woman stepping out onto the porch with a smile as wide as any Emma had ever seen.
"Is-is that . . . her?" she managed on the heels of a hard swallow.
This time when he nodded, he followed it up with a squeeze of her hand that was both reassuring and terrifying all at the same time. "C'mon, kiddo. Let's go meet your grandmother before she bursts."
* * *
She could feel the woman watching her as Brad opened the passenger side door and ushered her onto the walkway leading to the house. Somewhere in the distance she heard a child's voice and a dog barking, but the most prevalent sound of all was that of her heart pounding inside her ears.
Somehow, during the ride, she'd managed to detach herself from their plans. She knew they'd been driving here, that she was to meet a grossmudder she'd known nothing about, but knowing that and actually being less than a dozen steps or so from it happening were two very different things. One, easy to ignore; the other, completely daunting.
"Oh would you look at you!" Scurrying down the porch steps, Delia Harper clamped her hands together beneath her chin, her blue eyes ricocheting back and forth across Emma's face. "My dear, you are the perfect mixture of your mamma and your daddy, what with those big blue eyes and that spray of freckles across the bridge of your nose."
Delia dropped a hand to her chest in conjunction with a fleeting glance at her son. "Good heavens, Brad, I can't believe this is happening."
"Well, believe it. Because it is." Brad leaned over, kissed the sixty-something on her forehead, and then swept his arm toward Emma. "Mom, I'd like you to meet Emma—your granddaughter. And Emma, I'd like you to meet your grandmother. . ."
Waving his introductions away, Delia stepped forward, scanned Emma from the top of her kapp to the tips of her boots, and, after barely a moment's hesitation, pulled Emma in for a hug. "Oh, sweet girl, welcome. You are the answer to a prayer I thought was ludicrous to make. But . . . here you are. Alive and well and"—Delia stepped back, her eyes intent on Emma—"so very beautiful."
"I am Amish," Emma corrected via a raspy whisper. "I am plain."
Delia drew back. "You may be dressed in Amish attire, dear, but you're far from plain. You are, after all, my granddaughter, are you not?" the woman teased, her words peppered with a laugh that reminded Emma of birds chirping on a sunny spring morning—light and calming. "That alone means there's a little zip hiding inside you somewhere."
"I do not use zippers. Or buttons. It is the Amish way."
Delia's eyes narrowed only to widen back to normal size after a beat or two of silence. "I wasn't talking about a zipper, dear. I was talking about zip . . . spark, mischief, that sort of thing. It's my house specialty."
"And she's not kidding." Brad winked at Emma and then gestured toward the steps. "Well? Shall we go inside? Get warm?"
"Yes! Let's!" Linking her arm through Emma's, Delia steered them to the steps, across the porch, and through the front door with its snowflake-adorned wreath. Once inside, she instructed Brad to take Emma's coat and then turned to Emma as he did. "Would you like a tour of the house first, or would you rather sit and warm up? I could make you a mug of peppermint hot cocoa if you'd like—I have some ready to go right now."
Brad arranged Emma's coat atop the hanger and deposited it into the hall closet alongside his own, his eyes sparkling. "Mom makes the best hot chocolate you'll ever have, Emma. It's one of the things I missed most when I went on my little—"
Delia stymied the rest of his words with a splayed palm. "Today is about happy. Today is about getting to know my granddaughter, isn't that right, Emma?"
Without waiting for a reply, Delia pushed closed the closet door and nudged Brad from her path. "You get Emma settled in the living room and I'll be along in just a few minutes with the hot cocoa."
When Delia was out of earshot, Brad lowered his voice to a level only Emma could hear. "How much do you want to bet there'll be cookies, too?"
"I do not need cookies," she countered.
"I suspect you'll change your mind after you try one. They're pretty incredible."
Mustering a smile, Emma followed him down the hall, turned left, and froze.
"Emma?" he asked, turning back at her still feet. "Are you okay?"
She knew she should answer, or, at the very least nod, but aside from having heard her name in conjunction with a question, she wasn't aware of anything except the sitting room in which they were standing. Unlike the largely barren front room at the farmhouse that was used most often for church service every six months or so, this room invited people to come . . . and stay. Maybe even spend a few hours if the barn chores were all done.
"Look at all the books," she whispered, stepping forward, her gaze skittering down shelf after shelf of the kind of stories that so often called to her from the rack inside the English store. Pushing past the hesitancy she didn't want to feel, Emma ran her fingers across the spines of several books before being distracted away by a framed photograph of a younger Brad and—
Grabbing onto the edge of the nearest shelf for support, she staggered back a step. "Is that . . . That's . . . That's Ruby!"
"It is." He came up beside her, plucked the frame from its resting spot, and pulled it into book-reading range. "My mom took this of us the day we went to the carnival."
"That is the torn ticket you gave me when I turned thirteen?"
Brad nodded, his gaze still riveted on the photograph. "Yes. There was a fireman's carnival about twenty minutes north and we stopped here on the way because I'd forgotten my wallet. I remember being antsy to get there and not real excited about the delay Mom and her camera caused. But I'm sure grateful for it now."
"Can . . ." Emma stopped, swallowed, and tried again, her voice a perfect match to the tremble in her hands. "Can I see it, please?"
"Of course." He handed her the frame and stepped in behind her as she took it. "It's like looking at yourself, isn't it?"
She knew she should say something, but she couldn't. Her eyes, her thoughts, her everything was on the picture in her hands.
An eighteen-year-old Brad stood on the left, his dark blond hair so like Emma's. The subtle wave he sported now was more of a curl back then. The sky-blue eyes they shared led her attention to the young girl standing beside him.
All her life, Ruby had been this person Emma could only imagine—a person her mind's eye had created to look like a younger version of Mamm, frozen in time. Yet standing there, staring down at the reality, the only thing she'd been right about was the only two ways in which Emma favored Brad. In all other ways, Ruby had been a slightly younger version of Emma.
* The same cheekbones
* The same sprinkle of freckles across the bridge of her nose
* The same full lips
* The same basic height
* Even the same narrow chin
They were all the same shared features that had mesmerized her inside the locket, but here, in a way the tinier photo hadn't, she felt the person Ruby had been. The way the young girl smiled out at Delia depicted someone with an air of confidence and a yearning for adventure. And the way she'd nestled inside Brad's arm spoke to the depth of her feelings for Emma's birth father.
"She looks so . . . happy," she managed around the growing lump in her throat.
"That's because she was. We both were."
Together, they turned toward the sitting room doorway as Delia entered carrying a tray with three large mugs, three spoons, and a plate of what Emma could see were chocolate chip cookies. "I'm sorry that took so long. I wanted to make sure everything was just right for our Emma—oh, you're looking at photographs. How lovely."
Emma relinquished the picture frame back onto the shelf and hurried to take the mug Delia offered. "Are there more?" she asked.
"More pictures?" At Emma's nod, Delia grinned. "Oh yes!"
Brad started to reach for his own mug but stopped as a series of chirps diverted his hand to the front pocket of his jeans. He slipped out his phone, consulted the screen, and hooked his thumb toward the hallway. "I'm sorry, but I've really got to take this call."
Sliding a glance at Emma, he added a shrug. "You okay for a little while on your own?"
"She's not on her own!" Delia protested, hands on hips. "She's with her grandmother! Now go—go take your call." Then, with barely a moment's hesitation, the woman shoved the mug of peppermint hot cocoa into her son's non-phone-holding hand and shooed him from the room.
When he was gone, Delia turned back to Emma, patting her over to the sofa. "Come. Sit. Let's get to know one another a little bit, shall we? Maybe by then, Brad will be done with his call and he can join us when we look at pictures."
Wrapping her hands around the mug, Emma sidestepped her way between the coffee table and the closest corner of the couch and dutifully lowered herself to its edge, her gaze darting between Delia and the bookshelf. "You have even more books than the English market."
The skin around Delia's eyes crinkled with a laugh. "And this isn't all of them."
"You have more?"
"I have two floor-to-ceiling shelves in my bedroom that are filled with books, and another three in the room I've dubbed my office." Delia settled onto the couch beside Emma, pointing at Emma's mug of hot cocoa as she did. "So? What do you think?"
Recovering her mouth from the shock of Delia's words, she made herself take a sip. "Yah. This is good." She peeked inside the cup and, when she saw only liquid, looked back up at Delia. "I do not see the peppermint stick."
"Peppermint stick?"
"Yah. To make the peppermint taste."
Delia started shaking her head before Emma was even finished speaking. "I could garnish it with a peppermint stick, of course, but I prefer the little crumbles that were on the top of the whipped cream. The peppermint flavor comes from the extract I put in with the milk and the cocoa. The flavor spreads out more that way."
"Perhaps I should get such extract the next time I am at the market. Annie always enjoys the peppermint cocoa I make, but perhaps she would love this more."
Delia turned so her knees were at an angle with Emma's. "You make peppermint hot cocoa, dear?"
"Yah. Every Christmas. When Annie was little—"
"Annie?"
"My sister. She is eight years old and she loves peppermint sticks and cocoa. So I came up with a way to combine both as a Christmas surprise." She took another sip of her own drink and then set it back on the tray in favor of a cookie. "I like to do that.... Mix things that do not always go together when I bake and when I cook. Sometimes it does not work, but many times it does."
"Do you enjoy spending time in the kitchen?"
Emma considered her answer as she broke off a piece of her cookie and paused it just shy of her lips. "It is my favorite place to be, I think."
Delia beamed. "Like grandmother, like granddaughter, I see." Then: "What is it that you like about it?"
Lowering the uneaten bite to her lap, Emma tried to put her feelings into words—feelings she'd never been asked to explain before. "I . . . I like taking simple things, like milk and butter and yeast, and turning it into bread. I like cutting vegetables from the garden and combining it with chicken and chicken stock and making soup. I like taking a recipe that has been followed for many years and changing it to be new. I like, too, seeing a full plate of food grow empty and then be filled again. Because it is then that I know I did it right."
"Have you had formal train—wait." Delia brushed at the air. "That is a silly question. The Amish do not go to school."
"I went to school," Emma protested.
"To the eighth grade, yes, but that is only about learning basics. I'm talking about higher education. The kind that prepares you for—and teaches you about—whatever passion you want to pursue as a career. Like architecture, or fashion, or cooking."
She heard the hitch of her breath and wondered if Delia could hear it, too. "People go to school to cook?" she asked.
"Of course. And depending on where they went to school and how skilled they became, trained chefs can go on to work in restaurants all across the world."
"Is that what you did?" Emma asked. "Go to school to learn to make cookies and peppermint hot cocoa?"
"No, dear. I make those things because I like the Mmmms I get in return when people try them." Delia's knowing grin sent Emma's hand back to her lap and the cookie she'd almost forgotten. "I went to school to be an interior designer but never got to put it into practice until Brad started his company."
Emma turned her full attention on the woman seated to her left. "Interior design? What does that mean?"
"It means this." Delia opened her arms wide to the room. "It means coordinating colors and creating whatever feel a client is looking to establish."
"Feel?"
"Like when you first walked into this room. Did you get any sense, any—"
"I wanted to grab a book and curl up there," Emma said, pointing at the quilt-draped armchair between the window and the fireplace.
"And that's what I do. I decide on a feel and make it come alive. Like in this room. I wanted it to be a haven after a long day, the kind of place where stress just rolls away, leaving you feeling warm and cozy."
"Warm and cozy," Emma repeated only to snap her eyes back to Delia's. "Yah! That is how it feels."
"Then I succeeded." Delia reached across the gap between them and tapped Emma on the tip of the nose. "And that, my dear, is what interior design is and what I went to school for."
"But it is just one room."
"In my house, yes, but I do this for clients all the time."
"They all have a cozy room like this?" Emma asked.
"If they want cozy, I'll create cozy. If they want austere, I'll create austere. If they want whimsical, I'll create whimsical. If they want a room that feels rustic, I'll create rustic."
"Do you like to do it? Your interior design?"
Delia clapped her hands together beneath her chin and grinned. "I love it. Just like you enjoy hearing people ask for more of what you've cooked, I enjoy seeing the tears when people see what I have created in their home."
"They cry?" Emma asked, drawing back.
"A happy cry, yes." Delia picked up the plate of cookies and held it out for Emma to take another. "But that's enough about me. You mentioned wanting to curl up with a book. Do you read?"
"Yah. It is one of my favorite things to do, next to cooking and baking." Emma took another bite of cookie and slowly scooted herself back against the couch. "Sometimes, when the laundry has been taken off the line and the gardening work is done, I will sit at the kitchen table and read as I wait for the bread to finish or the soup to simmer. Once I even burned the bread on the edges because I did not keep track of the time."
"Books have a way of doing that sometimes, don't they?"
"Yah."
Delia returned the cookie plate to the tray and reached for Emma's hand. "Oh, my dear, your world is about to open in ways you can't even imagine and—"
"I'm sorry that took so long," Brad said, breezing into the room. "But it couldn't be helped."
Her hand still covering Emma's, Delia looked up at her son, her mouth pinched. "Was it Nicholas, darling?"
"Yes."
"And?"
"He's going to get everything set up for Wednesday afternoon. At my office."
"Don't you think doing it here"—Delia splayed her hands—"would be less intimidating?"
"No, but we can discuss that later." Brad shifted his attention to Emma and winked. "So? What did you think of your grandmother's famous hot chocolate?"
"It is very good."
Delia neatened the leftover napkins stacked on the tray and then offered Brad a cookie from the plate. "Emma makes hot chocolate, too. For one of the other girls."
"For Annie," Emma corrected.
"That's one of Rebeccah's kids." Brad's eyes, suddenly devoid of the sparkle they'd boasted seconds earlier, pinned Delia. "One of the ones that is actually hers, I should say."
"How many are there?"
Emma grabbed her own mug from the tray and pulled it close in the hope of chasing the sudden chill from her body. "There . . . there are six. I mean . . . five. Jakob, Sarah, Jonathan, Annie, and Esther."
"I met that one," Brad said across his cookie. "The little one."
Emma looked into her mug. "Her name is Esther," she whispered. "Esther is five."
"Yeah. It's pretty obvious she worships our Emma, here."
Shock propelled Emma's gaze back onto Brad. "There is only one to be worshiped and that is God!"
Silence swooped in on the heels of her outburst only to be broken by a soft tsking sound from Delia. "Now, Emma, your father didn't mean any harm. He simply means that this little girl—this Esther—is clearly quite fond of you."
Then, before Emma could even blink, Delia resurrected her smile and stood. "Brad, dear, did you know that Emma has an interest in cooking and books?"
"Oh?" Brad looked back at Emma, his cheeks lifting. "Ruby liked those things, too."
"She-she did?" Emma stammered.
"Yup. And drawing. In fact, a contributing factor in why I went to school for architecture was because of her and all the houses she liked to draw."
Intrigued, Emma took a sip of her drink and studied Brad across the rim of her mug. "She drew houses? Where? Why?"
"Whenever and wherever she had paper," he said, laughing. "It started after I told her I wanted to build whole houses rather than just fix them the way my uncle did. We were lying on some grass out by Miller's Pond when I told her that. Next thing I knew, I was talking about houses with spiral staircases and big bay windows and large patios. Some of the stuff I mentioned, she couldn't picture, so I grabbed a notebook from my backpack and tried to show her what I meant by a spiral staircase and a bay window. I wasn't very good at drawing back then, but once she had a basic concept of what I was talking about, she drew it. And Emma, she was good. Really good. It took her a while to get the hang of dimension and stuff like that, but she would erase and erase and erase until she got it right.
"In the beginning, she just drew parts—like stairs, and windows, and stuff like that. When I suggested she draw the outside of the house, she said that would be boring—that all houses look the same. And when you consider where she grew up, I could see why she thought that. So one afternoon, I drove her through some different neighborhoods so she could see that not all houses are simple farmhouses. They can be ornate like a mansion, they can have turrets, bump outs, different elevations and sizes. All homes, all different."
"Did she begin to draw the outside, then?" Emma asked.
"She did. And she got more and more creative each time. Soon, I was suggesting she could be the one who drew the homes I would build with my company—with our company." Brad brushed his hands over his napkin-topped knee, propped his elbows on the chair's armrests, and tented his fingers beneath his chin, his thoughts clearly taking him somewhere far beyond the confines of his mother's sitting room. "We really had a plan. A good plan. One that would have made for a nice life for the two of us . . . and, as we soon found out, you, too. But she just couldn't see it all the way through. Just pieces and parts like the stairs and the windows, and that wasn't enough."
"She stopped being able to draw whole houses again?" Emma asked.
"More like she reverted back to believing there was only one kind." Squeezing his eyes closed, Brad pulled in a deep breath, held it, and then parted his lashes in conjunction with a loud whoosh. "Well, I think that's enough of that for now, don't you?"
Without waiting for an answer, he rose to his feet and wandered over to the window. "You ever skate, Emma? On ice?"
At Delia's outstretched hand, Emma stood and followed the woman to the same window. A glance outside explained Brad's question. "Sometimes, on the way home from school when we were little, Jakob and I would walk out onto Miller's Pond in the winter and pretend our boots were skates. He would slide fast and I would slide slow. But we did not do that anymore after it cracked under my boot."
"Did you fall in?" Delia asked, mid-gasp.
"Yah. My leg got very wet and very cold."
Delia and Brad exchanged looks, their eyebrows inching upward in a mirror image of each other as Emma continued. "Jakob got me out with a stick."
"How old were you?"
"I had just turned seven."
"So Jakob was six, yes?" Brad prodded.
"Six and seven?" Delia echoed. "Rebeccah allowed two little ones to play on a frozen pond by themselves at six and seven?"
Emma rushed to defend the impression she hadn't meant to give. "We were on our way home from school. Mamm did not know we had stopped to play."
"Why was she not there to pick you up from school?"
"She was at home. Doing chores and looking after Sarah."
Delia pointed at Brad. "It's probably unnecessary, but jot that down for Wednesday." Then, without waiting for a reply, the woman lifted her wrist into view and tapped the face of her watch. "Everyone should be here soon, so I better get this tray back into the kitchen and the oven turned on for the roast."
"Please, let me help with the mess." Emma lurched forward, collecting Brad's and Delia's mugs with efficient hands. "Then I will get my coat."
Delia pulled a face. "Your coat?"
"Yah. So I can leave before everyone comes."
Closing her hand atop Emma's arm, Delia smiled. "They're coming to meet you, dear."
"To meet me?" Emma repeated, turning to Brad.
"You don't think your family stops with your grandmother and me, do you?" At Emma's gaped mouth, he continued, his smile reaching all the way to his eyes. "No, kiddo, you've got an aunt, an uncle, and cousins who can't wait to meet you, either."
Chapter 15
"I do that when I'm nervous, too."
Stilling her fingers against the edge of her dress, Emma looked up to find the English girl Delia had introduced as Emma's cousin Michelle looking at her with the same kind of face Sarah wore while perusing the magazine covers in the checkout aisle of the English grocery store. "Do what?" she asked.
"Fidget. If I'm at my desk at school, I play with my pencil. If I'm riding in the car with my mom, I fiddle with whatever she has in her cup holder that day, and if I'm sitting on a couch with nothing in reach"—Michelle nudged her chin at Emma—"I play with the seam of my jeans like you're doing with your dress."
Michelle stretched out across the carpet in Delia's now empty sitting room and rested her head against the wall at her back. "So maybe that means it's just some family quirk or something, instead of some sort of weird thing like my idiot brother is always claiming."
"I didn't realize I was doing that." Emma moved her hands from her dress to the cushion on her left and the armrest on her right.
"And now you're picking at that loose string Grandma keeps forgetting to cut off."
Sure enough, a glance at the armrest yielded a single strand of thread clasped between Emma's index finger and thumb—a thread she rushed to abandon. "Oh. Sorry. I-I did not realize I was doing that, either."
Picking her head up off the wall, Michelle pushed her long, dark, silky hair back over her shoulder and shrugged. "I'm not getting on your case, Emma. Just noticing we might actually have something in common."
"Yah."
"So I'm guessing this whole"—Michelle wiggled her fingers in the air—"revelation has to be even weirder for you than the rest of us, huh? I mean, you're not just meeting a new cousin like I am, or a new niece like my mom is. You're meeting your entire family."
Unsure of how best to respond, Emma managed a nod while Michelle continued. "So this means you don't really have to dress like that anymore, right?"
Emma followed the path indicated by the teenager's purple fingernails and swallowed. "I am Amish. This is how Amish dress."
"But my mom says you're not supposed to be Amish. Your mom died when you were born and since you should've been with Uncle Brad instead of those Amish people, you're really English." Lurching forward, Michelle grabbed a chip from the bowl Delia had left out for them when the adults opted to linger at the table after dinner. "My mom thinks you might need a little while to get used to everything, but if I were you, I'd be pulling off that head thing and running straight to the junior department at Charlotte Russe or Forever 21 or someplace like that."
Michelle nibbled her way around the outer rim of her chip and then waved the rest of it between them. "Why do you wear that bonnet thing on your head, anyway?"
"It is not a bonnet," Emma corrected. "It is a prayer kapp. The Bible says, 'But every woman that prayeth or prophesieth with her head uncovered dishonoureth her head; for it is one and the same thing as if she were shaven. For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.' "
Michelle's mouth gaped. "You memorized that? From the Bible?"
"Yah."
"Wow. That's crazy." Michelle popped the rest of the chip in her mouth and, after a few moments of quiet chewing, stood and made her way over to the couch. "So, will you? Take that stuff off, I mean?"
"I am not Amish because my mamm and dat are Amish. Before baptism you are being raised by Amish. You are truly Amish when you are baptized."
Kicking her shoes off, Michelle hiked her sock-clad feet up onto the couch beneath her, her eyes riveted on Emma. "And are you? Baptized, I mean."
"Yah."
"But you can get out of it, right? Since you wouldn't have been raised that way if Uncle Brad had known?"
"If I were to leave, my family and my friends could not speak with me ever again."
"But they're not your real family. Uncle Brad is . . . Grandma is . . . I am." Michelle gathered her hair together in a makeshift ponytail only to let it fall down past her shoulders, once again. "And everyone is pretty cool. Except my brother. But you saw that at the dinner table, right? Total. Dork."
On cue, the dark-haired twelve-year-old Delia had introduced as Kyle bounded into the room and skidded to a stop in front of Emma. "Uncle Brad said he's gonna get you ice skates and I can help teach you how to skate out on Grandma's pond!"
Michelle rolled her eyes. "Like she needs a twelve-year-old teaching her anything. Puh-lease."
Kyle's gaze dropped to the floor, prompting Emma to lean forward for his hand. "Actually, I'd like it if you'd teach me the proper way to skate, Kyle. Perhaps then I would not fall down."
"Oh you'll fall. Everyone does when they're learning. But Dad says as long as you get back up and try again, you'll get better!" Kyle slid a glance in the direction of his sister and, when she met his eye, stuck his tongue out.
"This is what having a brother is like," Michelle groaned. "Total torture."
Kyle stuck his tongue out a second time and then sat on the edge of the coffee table closest to Emma. "Do you have any brothers?"
"Two. Jakob and Jonathan."
"How old are they?"
"Jakob is twenty-one, and Jonathan is twelve."
"Like me! Cool! Maybe we can play together."
"No, you can't play with him, dork." Michelle dropped her feet back to the ground with a thud. "He's not our family. And he's not Emma's family, either. She just thought he was her family because his parents pretended Emma was theirs."
Emma bolted off the couch, bumping her knee against the coffee table as she did. "Jonathan is my family!"
"Maybe . . . Technically . . . Since the person who stole you is your aunt, I guess. But why would you want them to be family after what they did?" Grabbing another chip, Michelle popped it in her mouth. "I wouldn't if I were you. I mean, who did she think she was, doing that? She stole your dad from you! And your grandma! And a normal life! My mom says you could've been in college now—probably getting ready to graduate in May. You could be getting ready to get a job or live in the city. Maybe you'd be dating some really amazing guy."
Emma didn't know what to say. She heard the words coming out of Michelle's mouth. She was even able to process most of them. But it was as if she were standing in the barn being handed more and more chores and being told to do them faster and faster when all she really wanted to do was run down the driveway and escape to the pond.
But she wasn't home. She wasn't in the barn. And considering the night sky outside Delia's sitting room window and the time it had taken to get to the woman's house, finding her way to Miller's Pond wasn't a viable option.
Instead, she wandered over to the window and hoped the view of the very pond on which Ruby had once skated would give her the same sense of calm she found at Miller's Pond. Pressing her head to the cool glass, Emma tried to make out the spot where the grass met the water's edge, but in the dark it was hard to see. She knew, from earlier glimpses, that the pond was smaller than Miller's and lacking in the kind of large rocks and downed trees that were perfect for sitting and thinking. Although, at that moment, she'd take the bench she'd seen when she first got out of the car. . . .
Her mind made up, she turned back to Michelle and Kyle, excused herself, and headed up the hallway toward the closet where Brad had hung her coat. Yet as she approached the correct door, muted voices from the dining room had her veering closer to the wall and then stopping, completely.
"I don't know why you haven't done it already, Brad." The voice she knew to be Brad's sister, Jeanine, morphed into an almost hiss-like whisper Emma had to strain to hear. "I mean, if it were me, I'd have called the second I got my mouth up off the ground."
"Yeah, well, after I got my mouth up off the ground as you say, sis, I was kind of focused on Emma—the daughter I'd been told was dead."
"As well you should have been, dear." Delia's tone moved from empathy to reproach with the help of a hushed clucking sound. "Brad is doing everything exactly as he should, Jeanine. This is a very delicate matter. Emma's whole world has just changed. In an instant."
"I get that. So has Brad's. So has all of ours, quite frankly."
A snort she recognized as being from Brad was followed by the distinct scraping of a chair leg against the floor. "How has your life changed, Jeanine?"
"I have a niece I didn't get to watch grow up! My kids have a cousin they're meeting at twelve and sixteen instead of having her be there from the beginning! And instead of being someone they can relate to and look up to, she's . . . Amish."
"Which means what, exactly?" Brad shot back.
"You heard them during dinner. . . . Michelle talked about going to the junior prom and her classes, and Kyle talked about the video game he got for his birthday. Emma nodded along, sure. It's clear she's a nice, polite young woman. But it's also painfully clear she didn't have a clue what they were talking about. And it shouldn't be like that! They're cousins. I mean, don't you remember how much fun we had with our cousins when we were growing up, Brad? The stuff we talked about? The trouble we got into? We got them and they got us. Always. But because of this . . . this travesty put into place by Ruby's sister, our kids missed out on that. Missed out on getting each other on that special cousin-level."
Something that sounded like a snap segued into Delia's voice. "Then it's up to us to help them have that. And we will."
"How?" Jeanine challenged. "My kids know nothing about farming! And Emma knows nothing about real life."
"I can see your attitude hasn't changed since I was dating Ruby."
"What's that supposed to mean?"
"You always thought you were better than Ruby. Always went out of your way to make her feel like she was weird."
"She wasn't weird," Jeanine said. "The fact that you two were trying to have a relationship was weird."
The sound of dishes clunking against one another was quickly followed by another snap. "Enough, Jeanine! Enough! That young woman in there is my granddaughter! I have every faith that Nicholas will see to it that justice will be served on this. I also have every faith that we can find plenty of common ground with Emma now—common ground that will only grow over time."
"Common ground?" Jeanine challenged.
"Yes! Michelle likes to putter around in the kitchen when I'm baking, and I know, from our conversation earlier today, that Emma likes to bake, as well. And as for Kyle, well, he's at the age where he can find something to talk about with anyone. Emma is no exception and—"
"You can just go into the dining room, Emma. You don't have to stand in the hallway like that."
All conversation on the other side of the open doorway ceased as Kyle skidded to a stop next to Emma. She, in turn, tried to think of a response but stopped as Brad came around behind her, his eyes framed with the same kind of worry Dat wore when an approaching storm threatened the crops. He searched her face for a few beats and then sagged against the wall. "You heard some of that, didn't you?"
Shame cast her eyes to the floor. "Yah."
Hooking his finger beneath her chin, Brad guided her gaze back to his. "How about we get your coat and head out? Maybe find a place where we can talk privately before I bring you back out to Blue Ball? Does that sound like something you'd like to do?"
Did it? She wasn't sure. The only thing she knew for certain was that the pounding in her head was rivaled only by the pounding in her chest.
Stepping around him, she flung open the coat closet and closed her hands over her winter coat, the familiar feel of the fabric beneath her fingertips a welcome one. "Yah. That is something I would like to do."
* * *
They rode in silence down one street after the other until, at last, Brad pulled onto the road that connected New Holland with Blue Ball. When they did, he looked at her across the wide bench seat and smiled.
"So . . . You met the crew. . . . They're quite the bunch of characters, aren't they?" He swung his attention back to the road and loosened his grip on the top of the steering wheel. "My sister—Jeanine? She can be a bit of an acquired taste, but she will grow on you. And her husband—Ned? He's quiet, as you saw, but that doesn't mean he's not paying attention. I think he's just so used to Jeanine barreling over him in conversations that he just listens, makes his own assessments, and speaks when she's otherwise occupied. Smart man. Funny, too."
Resting her forehead against the passenger side window, she waited for the calm of the countryside to work its magic, but Brad's need to talk through the day and evening made it difficult. "Now, I know Michelle can be a bit prickly at times, but I'm told that's par for the course with teenage girls. Kyle is a cool kid. He loves to go out to job sites with me sometimes during his summer break. I had a hard hat made for him with his name on it that he thinks is pretty cool."
"He asked many questions about the farm." Emma wiped at the fog made by her breath and then sat back against the seat. "Perhaps he would like to visit it one day."
"I'm sure we can find him a farm if he really wants to see one up close. In fact, one of the sites I'm getting ready to start building is alongside a farm. Maybe I can get the guy who runs it to let Kyle milk a cow or something." He slanted a glance at her and then pointed at a series of dials on the dashboard. "You cold, kiddo?"
"Only a little."
"Then that's a little too much." With a quick turn of his wrist, a blast of warm air seeped out of the dash vents. "And my mom? She's the best, isn't she?"
"She is very kind." And it was true. Despite not really knowing the woman, Delia's very presence had quieted Emma's nerves on more than one occasion. "She has much to say."
Brad's laugh filled the truck's cabin. "My mom always has something to say. Always. But honestly, more times than not, she's right. It's like she has this ability to step back and read a situation and a person within seconds. And her advice? Spot on. Wish I'd listened to her sooner on a few things."
Intrigue lifted her gaze to the side of Brad's face as he continued. "Then again, hindsight is always twenty-twenty, isn't it?"
"Twenty-twenty?" Emma repeated.
"Twenty-twenty means perfect eyesight on its own. But the expression I just referred to means it's always easy to see things for what they are after they've already happened."
"Yah. That is how I feel about why I have never fit no matter how hard I tried. Before I learned about you and Ruby, I always thought it was me. That I did not laugh right, or look right, or talk right, or quilt right, or play right. Now I know it was never about my laugh, or my quilting, or anything I could change."
His jaw tightened along with his hand. "If I could go back and change the day you were born, I would. In a heartbeat. But, like you, I didn't know."
She wandered her gaze off him and onto the road, the headlights of his truck illuminating the upcoming bend even as her thoughts traveled back to Delia's house. "I did not mean to bring anger to your family."
"Anger?" Brad echoed. "You didn't bring anger, Emma! You brought joy! Unbelievable joy! Did you not see my mother's face when we pulled up? Did you not see the way she could barely take her eyes off you when it was just the three of us? Or the way she kept smiling at you—and trying to force more food on you—all during dinner?"
Emma smiled in spite of the heaviness inside her chest. "No, I saw it."
"Okay, good. As for Jeanine, I know she was probably over the top with all the questions she kept asking during dinner, but she means well. You're her niece. She's trying to learn twenty-two years in the timespan of dinner. Still, I didn't get the sense she was angry. Curious, sure. Anxious to learn more, sure. But, angry? No."
"It was the anger at the table after I went into the sitting room with the children."
He slowed the truck to a crawl as they approached yet another bend. "I don't know what—"
"It was about me. That I am Amish. That I am not English like you and them."
"Not English like—wait." On the far side of the bend, Brad pulled onto the shoulder and cut the engine. "So this is about what you overheard at my mom's there at the end, isn't it? When I was in the dining room with my mom and Jeannine?"
"I did not mean to hear." Emma looked from the darkened fields alongside them, to her lap, and back again, the chill she'd had only moments earlier replaced by a growing heat in her cheeks. "I wanted to sit by the pond for a little while. But when I was close to my coat, I heard anger. It was about me. About me being Amish."
"No one is angry at you, Emma. No one. They're angry at the circumstances—that you were raised Amish when you should have been raised by me, that none of us had the chance to watch you grow or influence your development, and that you didn't get to know any of us until now—at the age of twenty-two. They're upset by that and they want justice. We all do. And, come Wednesday—maybe Thursday, at the latest—I'm certain we'll have it."
"I heard talk of Wednesday. Why?" Emma asked. "What is to happen that day?"
Turning his body flush to the driver side door, Brad hooked his right calf onto the bench seat. "First up, I'm flying out to Florida tomorrow morning. There's a housing development down there I need to see, and I made the arrangements before you walked into my office for the first time last week. I can still back out if you need me here, or even if you just want to spend the time together. I keep trying to remind myself we have the rest of forever, but . . ."
"But you will come back, yah?"
"Of course. Tuesday night. I should be back from the airport around eight, maybe a little sooner."
"Sarah and Annie will be pleased that I will be there to do my own chores." Lifting her hand to the windowsill, Emma ran her finger along the edge. "I am sure there is mending to be done on Jonathan's pants by now."
He drew back so quickly, his head thudded against the widow. "No. Let someone else do the mending and the cleaning and the gardening and all the other things you never should have been doing."
"I am good at mending. Sarah, not so much."
"But I was thinking your grandmother would just pick you up out by our usual pickup/drop-off spot by Miller's Pond just as I would if I were here. That way you could go to church with her tomorrow, go shopping for new clothes on Monday, and then maybe, on Tuesday, you two could spend time together in the kitchen, since that's something you both enjoy."
"Tomorrow I am to go to church at the Schrocks, and I do not need new clothes. I made two new dresses for all of us just before Christmas." Emma parted the bottom edge of her jacket to reveal the green of her dress. "It is good fabric. Sturdy. It will last many years."
He started to speak, stopped himself, and, instead, looked out at the road. "Maybe Tuesday, then. I know she'd love to cook with you."
"Yah."
"Wait. Let me give you her number." Heaving himself forward, he popped open a recessed compartment she hadn't noticed and pulled out a piece of paper and pen. "Since I can't give her a number for you, I have to trust that you'll find your way to a phone to call her if you need something or want to finalize a time for Tuesday, if not sooner."
He jotted down a series of numbers and then handed the paper to Emma. "Either way, I need you to be on the road by the pond at eleven o'clock on Wednesday morning. Gives us a little time together before we sit down with Nicholas."
"This Nicholas. You have said his name many times. Is he your friend?"
"He's my lawyer, and a darn fine one at that. Even though he's not a criminal attorney, I knew he was the one I could count on to point us in the right direction in order to minimize any unnecessary glare on you."
Looking up from the slip of paper in her hand, she tried to feign some semblance of understanding, but it was no use. She'd missed something.... "I do not understand."
"I want justice to come swiftly, first and foremost. But if there's a way to do it without having every news truck in the area camped outside our door, that's even better."
"What news truck?"
"Kidnapping is big news, Emma. Particularly within the market in which it takes place. But kidnapping by someone inside the Amish community? That's got the potential to go national. Fast. The past two weeks have been enough of a blur for you—for both of us—all on its own. I don't think we need to compound that with the circus that a national news story will bring if we can avoid it. That's why, on Wednesday, Nicholas is going to ask you questions—about your upbringing, the things you were told, the things you weren't told, that sort of thing."
"Why?"
"Because we need the facts. We need to know what you were told, how you were treated, that sort of thing."
"I was not told anything until I showed Mamm the locket," Emma reminded. "You know this."
"I do. But Nicholas needs to know, too. So justice can be served."
"Soon, I will tell Bishop King and they will be shunned."
His answering laugh was void of anything resembling lightness or humor. "You said that the other day. That backs will be turned on them at church until, I imagine, they say, I'm sorry, I won't do it again?"
"Yah."
"Yeah, no . . . Sorry." He grabbed her hand off her lap and held it tight, his blue eyes holding hers with such intensity she couldn't look away. "They kept you from me, Emma. That's not okay."
Not sure what to say, or even how to slow her breath down enough to think straight, Emma looked out at the road in front of them, the momentary, yet overpowering urge to go home giving way to total emptiness.
Somewhere in the darkness, just beyond the beam of the truck's headlights, was another bend in the road, another handful of farms to pass before the turnoff to her farm.
But it wasn't her farm. Not now. Not then. Not ever.
"I don't know where I belong," she whispered.
"I do. You belong with me. You always did, and you always will."
She looked down at his hand enveloping hers once again and tried to make herself feel something, anything. But there was nothing.
"Well, I guess I should get you back. For now. I've still gotta get back to my own place and get packed for my trip. But you've got Mom's number"—he pointed at the paper atop her lap—"and I'll be back to get you by the pond on Wednesday at eleven o'clock."
She knew he was waiting for her to answer. She could feel it just as surely as she could the chill claiming every inch of her body despite the heat still blowing against her skin. Turning toward the window, she nodded.
Chapter 16
Many times, throughout the next morning's church service, Emma had been aware of Levi's glances. She'd felt them just as surely as she'd felt Annie's breath across her hand as she'd held open the Bible for them both. Once, she'd even caught him as she'd scanned the benches of men and boys in the hope the periodic tapping she heard didn't belong to Jonathan. What she hadn't been able to tell was whether Levi's expression was one of disgust or . . . concern?
It had been a fleeting impression that had lasted only as long as it had taken Annie to plant an elbow in Emma's ribs and Emma to return her attention to the service. But even with that reminder as to where her focus should have been, her thoughts had returned, as they had again now, to the things Brad had said in the truck the previous night. His repeated insistence and unwavering conviction that Mamm and Dat be held accountable for their lie gnawed at her heart and mind.
It wasn't that she suddenly approved of Mamm and Dat's lie. Because she didn't, couldn't. That lie had kept her from having Brad and Delia and the rest of the Harper family in her life all along. It had also left her to spend her whole life erroneously believing if she just tried harder or did more, she would finally fit with her brethren the way her siblings, and everyone else in their district, did so naturally.
But to be shunned by their community? To have backs turned to them in church and at all meals until they repented? Surely that was a worthy punishment. And while she knew their shunning would bring shame to the children, as well, it was they who had lied. They who had acted as if they knew better than God.
"I am sorry, Emma. For what you carry in your heart."
Startled, Emma whirled around only to watch, helplessly, as her dinner roll slipped off the edge of her plate and toppled onto the dirt driveway. "I'm sorry, I do not mean to be wasteful," she murmured.
"That was my fault. For scaring you like I did." Swooping down, Levi retrieved the roll from its resting spot, wiped the dirt onto his pants, and swapped it for the one on his own plate. "Please. You are to have mine."
"I can't ask you to eat a dirty roll," she protested.
"You did not ask, and thanks to my pants, it is not dirty anymore. See?" He lifted it for her to see and then, with a mischievous grin, took a bite. "Yah. That is good. Very, very good."
It felt good to laugh, even if it didn't last. Still, she was grateful when he hooked his thumb toward a sparsely used table on which to set their plates and eat. A glance at the food table showed that it would not be long before Mary's plate, too, was filled and she could join them. Falling into step beside Levi, Emma crossed to the vacant end of the table and lowered herself to the bench while Levi claimed the empty spot opposite. "So? Are you going to the hymn sing at Luke's when we are done eating?" he asked.
She took in the food she'd placed on her plate—the chicken, the stuffing, the corn, the roll—and waited for it to speak to her stomach, but it didn't. Instead, her stomach continued to clench and roil as it had since Brad had spoken of his anger for Mamm and Dat. "I should not have taken so much food."
"That is not so much. This"—he directed her eyes onto his plate and all the same items Emma had chosen, just in larger quantities—"is much food. Which I will eat, I am sure."
Shrugging, she nudged her plate to within inches of his. "Perhaps, if you are not too full, you would like to eat mine, as well."
"That is your food. For you to eat."
"I know, but I can't. My stomach is . . ." She stopped, held her hand against her aproned front, and took a breath. "My stomach is not right."
"Is that why you looked so sad during the church service?" he asked, pausing his fork against the chicken he'd sought out first. "You do not feel well?"
She lowered her eyes back to her plate before closing them for the briefest of moments. "I-I . . . I don't know."
The smile she'd managed to chase from his face disappeared from his brown eyes, as well. "You are still troubled? About what you told me the other day?"
"Yah."
Silence filled the space between them as Levi's gaze darted to the food table and his sister before settling back on Emma once again. "Is he not kind?"
"Who?"
"The Englisher from the cemetery . . . The one you went off with in the black truck the other day . . ."
"You mean my birth father?" she whispered. At his expected nod, she, too, leaned forward. "He is very kind. And I met his mother—my grandmother, too. She likes to cook and to bake like I do."
Levi studied her as she spoke, and then, when she was done, he pointed to her plate with his chin. "Stomachs do not feel bad because people are nice. Stomachs feel bad when one is sad or worried.
"This, here"—he lifted his finger to the skin beside his eyes and then pointed across the table to the same location on her face—"says you are worried, while the smile you are missing says you are sad."
"That's because I—"
"It is crazy how much Isaiah King likes chicken. Lots and lots of chicken." Mary set her plate next to Emma's and hiked her legs, one at a time, over the bench. "But I did manage to beat him to the stuffing and the . . ." Mary's verbal inventory of her plate petered off in favor of bugging out her eyes at her brother. "Must you look at me like that when I'm talking, Levi?"
"Like what?" he groused.
"Like you do not want me here."
"Emma and I were trying to have a conversation. . . ."
Emma shrugged away Levi's frustration and slid her plate next to Mary's. "I have chicken that I am not going to eat."
"Really?" Pulling a face, Mary poked her fork into the first of Emma's two slices. "But you did not even try it yet. I am sure that it is good. Isaiah's big helping was not his first or his second."
"I am not eating it because I think it is not good. I am not eating it—or any of it—because"—Emma slid a glance at Levi—"I am not hungry."
It wasn't an untruth. She wasn't hungry. The fact that her lack of hunger was from the near constant clenching inside her stomach really wasn't important.
"Then why did you fill a plate?" Mary asked as she transferred Emma's first slice of chicken onto her own plate and then shoved the second one across the table to her brother.
Why, indeed.
To Mary, though, Emma merely shrugged. "I thought I was hungry when I put those things on my plate."
Mary sliced off a bite of meat and slipped it onto her tongue, her happy eye roll a nod to the cook. But still, Emma's lack of appetite remained.
After a few bites, Mary pointed the tines of her fork at Emma. "Sarah says you have not been home very much this past week. That you have been in New Holland many times."
"She told you that?"
"I asked why I had not seen you the few times I passed by your farm in the buggy."
"You could have asked me," Emma protested.
"Yah. But you were busy talking to Levi, and Sarah was next to me at the table putting more food out for Clara Schrock." Mary scooped up some stuffing and lowered her voice so as not to be heard by anyone beyond their table. "Did you go into New Holland to see about working at the Quilt Shop? Did they like the quilts you have made? Are they teaching you things you must know to sell them at the store?"
"I went to New Holland, but not for a job."
Mary moved on to her roll, eyeing Emma as she did. "Is this about what you told me the other day? When I walked you home? Did you go back again?"
"Perhaps Emma does not want to answer such questions," Levi challenged.
Mary shifted her full attention onto her brother. "Perhaps you should not say such a thing. I am Emma's friend, remember? Not you. You spend your time at hymn sings talking to—wait!" Mary relinquished her fork onto her napkin and grabbed hold of Emma's arm. "Yah! That is why I had hoped to see you outside when I drove to Katie Beiler's one day, and Miss Lottie's the next. I have wonderful news! Leroy Schrock has asked to drive me to the hymn sing today!"
"Le-Leroy Schrock?" Emma echoed, stunned. "I-I didn't know you two talked all that much."
"I had a problem with Dat's buggy the other day on my way home from town. Leroy passed by and helped tighten the wheel. It is then that we spoke." Mary's eyes shone with excitement. "He may not be good at volleyball, but he is very nice."
Emma started to speak, to string together something that sounded like the excitement she knew Mary was waiting to hear, but, instead, she pressed a hand to her mouth, murmured her apologies for her hasty departure, and ran around the side of the Schrocks' barn. There, she surrendered to her stomach's incessant churning before lowering herself to the ground and resting her head against the weathered wood at her back. Breath by breath, she steadied her hands in her lap and waited for their trembling to stop. And, breath by breath, she reveled in the cool, crisp air against her cheeks and forehead. It didn't change the feeling in her stomach, but focusing on something, anything else for even a few minutes was better than giving in to the tears she didn't want Mary to see when her friend invariably came around the corner.
Yet when the feet she'd anticipated finally came, they didn't belong to Mary. Hurrying his already hurried steps, Levi closed the gap between them with three long strides, the worry she saw etched in his face catching her by surprise.
"Here you are," Levi said as he, too, lowered himself to the straw beside Emma. "Are you okay?"
When it became clear she could not stop the renewed trembling with breaths alone, Emma wedged her hands between her legs and the ground and managed the closest thing to a true nod she could muster. "Where is Mary?" she asked.
"She got up to try to find you, but it is then that Leroy said it was time to head out for the hymn sing. She told him to go ahead, that she had to find you, but I told her to go on—that I would find you and bring you to the hymn sing if that is what you would like."
Tipping her chin upward, Emma took in the charcoal-colored clouds, the lack of any real sun plummeting her mood even more. "That is very kind of you, Levi, but if I am to go, I will go with Jakob. I do not want to take Liddy Mast's place on your buggy seat."
"I am not to bring Liddy Mast to the hymn sing." Levi resituated his hat more squarely atop his head and then gazed up at the sky, as well. "You did not like to hear about Mary and Leroy, did you?"
Oh how she wanted to protest, to say with conviction how happy she was for her dearest friend, but the effort to do so was greater than she could muster at that moment. Instead, she swallowed hard, and waited for the lump she felt forming in her throat to go away.
"I know we have lost a few volleyball games when he has been on our team," Levi said, his voice quiet, "but he is nice."
Closing her eyes briefly, Emma breathed her way through yet another flip and subsequent flop of her stomach. "I do not question that Leroy is kind. We do not speak often, but I have not seen anything to tell me that is not so."
Levi's answering silence soon gave way to an inhale that matched the upward bent of his coat-clad shoulders. "Did you know that my mamm and Fannie Hershberger have been friends since they were both little ones?"
"Mary told me that once."
"Did you know that Fannie got married first?"
Unsure of his reasons for such an odd topic shift, Emma lowered her chin until his face, rather than the clouds, was her only true focal point. "No, but I do not understand why you speak of this."
"I speak of it because it doesn't matter if Mary and Leroy are to court, or who is to marry first. You and Mary will always be friends, Emma."
She didn't mean to laugh, but just the notion that she even had prospects for marriage was silly. Still, she appreciated his kind words and the intent behind them. "I am happy for Mary—I am. It is just that I will miss our talks at the hymn sings."
"That will not change because of Leroy."
"If they marry, she will not go to hymn sings and then . . ." Blinking hard against the tears she heard lapping at her words, Emma made herself stand. "Thank you for checking on me, but I should go. I do not want Jakob to leave without me."
Levi, too, stood, his hand finding its way between his chest and his suspender. "Jakob has already left. So have many."
"Oh." She sagged back against the barn. "I did not know. I am sorry if I have made you late."
"I told Mary I would find you and that I would make sure you were okay."
"And you have done that."
"Yah, I have found you, but"—he studied her closely—"I do not think you are okay."
"I want to be," she whispered. "But I do not know how to be."
"Perhaps you need someone you can talk to without worry. Someone who will listen to your troubles and help you to know what is best."
"I don't know if anyone can tell me that."
"I do not know, either. But if there is one who can, it will be Miss Lottie. She knows of things I do not know, even if I wish that I did." Removing his hold on his suspender, Levi swept his hand toward the line of buggies on the other side of the barn. "Come. I will take you to her now. And when you are done, I will bring you home."
"But we don't know if Miss Lottie is home. She could be out or busy or—"
"Or she could be there, happy to listen, happy to help."
Happy to listen . . .
Happy to help . . .
Emma's thoughts drifted down to her hand and the stomach that suddenly seemed a little less unsettled. "Yah. Perhaps we could ride out to Miss Lottie's and see if she is home."
Chapter 17
For more than a handful of years, Emma had fantasized about this exact moment. Sitting beside Levi on a wagon seat . . . His horse, Hoofer, dutifully pulling them through the countryside. . . No Liddy Mast to flutter her eyelashes . . . Mary otherwise occupied, giving them a chance to really talk . . .
Now that it was here, though, there wasn't any talking. In fact, the only sound beyond Hoofer's hooves against the finely graveled country road was the occasional command of her owner to slow down or move faster. Twice, she'd gotten the sense Levi was about to say something designed to start a conversation, but both times he'd remained silent, freeing her to remain lost in her own thoughts.
The Emma of two weeks earlier would have been mortified at the silence, no doubt trying to scramble for something, anything to say to make Levi see her as courting material. But that Emma was gone, shoved to the side by someone who still looked the same, yet no longer was.
"I remember once, when I was not much older than Esther, bad storms took Dat's crops." She surveyed the land to their left and right even as her mind's eye replaced the view with one from seventeen years earlier. "Dat did not question the Lord's will, but he said something about how fast things could change. When we went to bed, the wheat was plentiful and soon to be harvested. When we woke up, it was gone."
Tugging softly on the reins, Levi glanced across the seat at Emma. "Yah. I remember that storm. I was seven. I stood beside Dat in the kitchen and listened to the hail on the roof. The next morning, there was much worry about how to feed the animals that winter."
"That is how I feel right now." She inhaled the cold winter air into her lungs and released it in a smoky plume. "Only it is not crops that have changed in one night. It is me."
"You?" Levi asked over the slow but steady clip-clop of Hoofer's hooves. "You look like the same Emma to me. You have the same hair, the same"—he pulled his left hand from the reins long enough to brush at the area flanking the bridge of his nose—"freckles here, and the same blue eyes."
She shook her head at his words. "I do not mean those things. I mean the things you cannot see. The things that make me . . . me."
"I do not think you are different."
"Of course I am different. I am not born from Mamm and Dat. I am born from Mamm's sister, Ruby, and an Englisher. And Jakob, Sarah, Jonathan, Annie, and Esther are not my brothers and sisters."
"That does not mean you are not still Emma," he said with a gentle pull of his hand as he guided Hoofer to turn left onto Miss Lottie's dirt driveway.
Torn between trying to find a way to verbalize her feelings and the ready-made distraction that was the white cottage with the wide front porch, she opted for the latter, falling silent as she did. Lottie Jenkins was a staple inside Emma's district, the elderly English woman's wisdom, quiet lifestyle, and utmost respect for the Amish way endearing her to the plain people she lived among.
For as long as Emma could remember, Katie Beiler's mamm had been a summer staple on Miss Lottie's porch, likely sharing a moment of conversation over lemonade and baked treats while Katie's younger siblings chased bubbles nearby with Digger the dog. It had been such a common sighting, in fact, that when Katie's mamm went to the Lord the previous year, her absence from Miss Lottie's porch had been particularly jarring. But while the Beilers had spent the most time there, many in Emma's community stopped by, as well, their lives and troubles always seemingly lighter after time spent on the same porch that was now no more than a few buggy length's away.
"It is too cold for Miss Lottie to sit on the porch with me," she whispered as her gaze gravitated upward to the plume of smoke rising from the simple chimney. "Perhaps it is not a good time."
"There is room to sit and visit inside." Levi drew Hoofer to a stop beside the hitching post at the base of the walkway and turned to Emma. "Go. Spend some time. Share what is troubling your heart."
Again, Emma looked at the house, and again, a hint of calm enveloped her. "Yah. I will go." Turning back to the handsome man behind the reins, she mustered up the smile he deserved for his kindness. "Thank you, Levi. For the idea, and for the ride. Now it is your turn to go. To enjoy the hymn sing and the fun." She stopped, took a breath, and infused a lightness into her voice she didn't feel. "Perhaps you will like Liddy Mast's oatmeal cookies better today."
He started to speak but stopped as a telltale creak drew their collective attention back to the cottage. Seconds later, a light to the left of the door switched on, bathing the porch in a muted glow. "Levi Fisher? Is that you, dear?"
"Yah."
"Come in out of that cold, young man. I'll make us some hot tea."
Tucking the reins onto the floor of the wagon, Levi stepped down off the seat and crossed around to Emma's side of the wagon. "I am going to a hymn sing, Miss Lottie, but I brought someone who would love to have some of that tea with you."
The thump of Miss Lottie's cane preceded her steps onto the porch. "Oh?"
Levi nodded up at Emma and then reached up for her hand. She took it, stepped down onto the driveway, and made her way toward the walkway. "Hello, Miss Lottie. It's me—"
"Emma Lapp!" Miss Lottie tapped the end of her cane against the wooden floor with delight. "What a wonderful surprise this is! Come in! Come in! It's nice and toasty warm inside."
Emma stepped closer, the light from the porch glinting off the upper edge of Miss Lottie's thick glasses. "If it is not a good time, I-I could go."
"You will do no such thing, child! After my breakfast this morning, I made a pie. I didn't know why, just that I was called to make one." Miss Lottie beckoned Emma onto the porch. "The Lord clearly knew you were coming."
When Emma cleared the top step, Miss Lottie peered around her, eyeing Levi across the top of her glasses. "There's plenty for you, too, Levi. It'd give me a way to thank you again for clearing that tree from my driveway."
"You already thanked me, Miss Lottie, and it was no trouble. Just neighbors helping neighbors." Then, stepping forward a half step, he trained his attention back on Emma. "If you're not ready when the hymn sing is over, I will wait right here."
"You are not to get me," Emma protested. "I can walk. It is not far."
"It will be dark. I will stop and bring you home." Nodding first at Miss Lottie, and then Emma, Levi crossed back around to his side of the wagon. With one last nod of encouragement at Emma, he climbed up to his seat, took his place behind the reins, and headed back toward the road, the wheels of his wagon squeaking and groaning with each new rut he encountered.
When Levi and his wagon disappeared from view, Miss Lottie rested her leathery hand atop Emma's arm. "Come inside, dear. We can visit in front of the fire."
"Are you . . . certain?" she eked past the lump in her throat. "It is not too dark to walk home now."
"I am certain, child. Come." Miss Lottie led the way back to the door, pulled it open, and caned her way into the hallway. Where it split, the elderly woman pointed Emma toward the tiny sitting room and its quilt-draped sofa and matching armchair. "Why don't you put a log in the fire and I'll get us some of that pie I told you about. The smell has been tickling my nose and my stomach long enough."
Emma waited for her stomach to react to the notion of food as it had after the Sunday church service, but it didn't. Progress . . .
Stepping into the sitting room, Emma took a moment to look around, the look and feel a near perfect match to the image she'd created in her mind. Although it was a room with four walls, real furniture, and a stone hearth, it was, in many ways, simply an indoor version of Miss Lottie's porch. Warm, welcoming, safe.
She reveled in the warmth emanating from the fireplace and then wandered over to its mantel and the series of framed black and white photographs that covered it from one end to the other. The pictures, themselves, were of places—a building that reached the sky, a bridge over a rushing stream, the top of a mountain overlooking a valley, and the vast ocean. In each picture was a single person—a woman—whose back was to the camera, looking out at the scene. She didn't need to look closer to know the woman was a younger Miss Lottie. The mere presence of the floppy straw hat atop the figure's head was all the proof Emma needed. Still, it was clear the pictures were taken years earlier, before age necessitated the cane that was as much a part of the woman now as the very home in which Emma was standing.
"It takes a while to get everything in here when one hand is occupied with walking, but I get it done." Miss Lottie caned her way into the room with a stack of plates and set them down on the table Emma recognized as being Amish made. "I'll be right—"
"Miss Lottie, I'm sorry," she said, snapping to. "I was so busy enjoying your pictures I did not add a log to the fire as you asked. I will do that now, and then I will get the pie and anything else you need."
"How about we switch? Everything is on a tray in the kitchen. If you can carry that in, I will add the log so we can get to our evening."
"Yah." Emma went in the direction indicated by Miss Lottie's finger, located the tray with its pie, forks, teacups, and teapot, and carried it back into the sitting room in time to see her host backing away from a newly roaring fire. She set the tray on the table and took a spot on the sofa, her gaze gravitating back to the picture-topped mantel. "You have been to many places."
Miss Lottie followed Emma's gaze to the mantel as she settled into her own chair. "I have."
"What was that like?" She poured some tea into each of the two cups and handed one to Miss Lottie. "To go to such places and see such things?"
"It is always an experience to see something for the first time."
Lifting her own cup to her lips, Emma stopped just shy of actually taking a sip. "Do you still go to places like that sometimes?"
"No. I have seen many things. I have had many experiences in my life that have had me traveling to many places." Miss Lottie took a sip of tea, pinning Emma with her eyes across the rim of her mug as she did. "But soon I came to see that where I feel most content, most at peace in my heart, is right here, in Amish country. So I returned."
She considered the woman's words against snatches of conversation she'd heard at Katie Beiler's marriage to Abram Zook in the fall. "You are Katie's kin, yah?"
"I am."
"Which means you were Amish . . ." Emma prodded, setting her cup back onto the tray.
Miss Lottie leaned forward, cut two pieces of pie, and deposited one onto each of their plates. "I was raised by Amish. I did not join the church."
"But you came back here."
"I did."
Emma took the pie plate and a fork from the woman and rested it atop her lap. "Why?"
"I was wandering in an outdoor market in San Francisco one day and I came across a sign. It said, 'Home is not a place, it's a feeling.' " Miss Lottie forked up a piece of pie and took a bite, her eyes disappearing briefly behind closed lashes. After what seemed like a minute, maybe two, the woman's eyes popped open to meet Emma's. "For days after I saw that sign, I couldn't get those words out of my head. I thought about it when I woke up, I thought about it as I was drifting off to sleep. And then, a few days later, it just hit me. I'd always felt as if I was visiting places I went—even places I stayed for several years. Yet when I thought back to the one place where I never felt as if I was visiting, it was here. In Blue Ball. I have never regretted coming back here for even a moment."
"I did not know you were Katie's kin until she married Abram."
"Katie and the rest of 'em didn't know until shortly before that, either. But I knew."
"So, it was wanting to be near them that brought you back here?"
"That was some of it, of course, but I feel as if everyone in this community is my family. It's like the sign in that market said, home is not a place, it's a feeling. The second I arrived back in Blue Ball . . . and bought this house . . . and looked out the window and saw a buggy driving by, I no longer felt as if I was visiting."
A log split, sending sparks up the chimney and Emma's thoughts to the reason she'd come. "Do you think you would feel the same if you learned you were never meant to be here from the start?"
"You mean here, in Blue Ball?" Miss Lottie asked, lowering her teacup to the armrest. "Among the Amish?"
"Yah."
"That's an interesting question, Emma—one I'm not sure how to answer. I think it would come down to choosing to listen to your head or your heart . . . anger or truth."
"Anger and truth," Emma corrected. "They are together, for me."
Leaning forward, Miss Lottie set her cup on the tray, folded her hands atop her lap, and nodded ever so gently at Emma. "I'm listening, child."
Emma traveled her gaze back to the hearth and the flames licking at the base of the chimney. "Everything is different now. I am different."
"You look the same to me."
"Because I am still dressed as if I am Amish. But I am not."
"You were baptized, dear."
"But I should not have been." At Miss Lottie's not so quiet inhale, Emma abandoned her view of the flickering flames and stood, her feet taking her around the room with no clear destination in mind. "I did not know another world."
"That is what Rumspringa is for. To let you taste the English world and its ways. Did you not do that?"
"No, I did. I wore some English clothes a few times . . . listened to English music with Mary . . . and"—she turned back as she reached the entrance to the hallway—"I even tried a cigarette outside the English grocery store, but I did not like it."
"Did you take a full year, or did you hurry back for baptism?" Miss Lottie asked, her eyes following Emma around the room.
"My Rumspringa fell between baptisms for the bishop, so it was a little more than a year. But I knew after six months."
"Then what is different now? Have you met an English man?"
She whirled around, her face hot. "No!"
Miss Lottie's shoulders sagged with relief. "That is good."
"Not in the way that you mean," Emma clarified. At Miss Lottie's pointed look, she rushed to explain. "I have met an English man, yah. But not in a courting way."
Wandering over to the window, she parted the pale yellow curtains with her hand and peered out at the road, a series of flashing orange lights in the distance letting her know the day of worship and visiting at the Schrocks had come to an end. Soon, the hymn sing, too, would end and Levi would be arriving to take her home....
She rested her forehead against the cool glass for a few long moments and then turned back to the woman patiently waiting for Emma to explain the unexplainable.
"The Englisher is my dat. My real dat."
Reaching up, Miss Lottie readjusted her glasses against her eyes as if the enhanced view might change what her ears had heard. But after a slow inspection of Emma's face, the elderly woman squared her shoulders with a hearty breath. "I did not know that, child. I didn't come back to Blue Ball until you were closer to four."
"I did not know until two weeks ago," Emma said, her voice wooden even to her own ears.
A peek at Miss Lottie pointed to surprise as the reason for the room's sudden silence, save for the quick pops and slow crackles from the fireplace. Eventually, though, the woman spoke, her voice so hushed Emma was forced back to the sofa just to hear.
"What made your mamm tell you now?" Miss Lottie asked. "After all this time?"
"I saw the picture—the one of my real mamm. And that is when I knew."
"Your . . ." Miss Lottie stopped, cleared her throat, and tried again. "Your real mamm?"
"Yah. Ruby. She is dead. I killed her."
"Killed her?" Miss Lottie echoed.
"Yah. It is while she was having me that she went to the Lord."
"But I thought Ruby was your mamm's sister."
"And I thought Mamm did not smile at me because I reminded her of a sad day. But it was not that. She does not smile when she looks at me because I am the reason for that sad day."
Miss Lottie moved her hands from her lap to her armrest, her eyes never leaving Emma's. "I don't believe that, child."
"What? That she does not smile at me?" Anger tightened Emma's jaw. "I am not the one who lies! That is Mamm and Dat! They told my real dat that I died with Ruby! Every year on my birthday he has come to the grave to visit with Ruby and me. But I was not there! I played. I went to school. I had birthdays. I grew. I went on Rumspringa. I was baptized. In all that time, Brad did not know I had lived, and I did not know I was his and Ruby's child."
Now that she'd started, she couldn't stop, the details she'd pieced together since the morning of her birthday pouring from her mouth with nary a breath in between. "I have a grossmudder who likes to cook just like me . . . and cousins who are very different but only because I was raised in a home where I did not belong. A home where—when I would look across the table at dinner—I would see bits of Dat and Mamm in everyone else. Dat's eyes in Jakob's . . . Mamm's in Annie's and Esther's . . . Dat's chin in Sarah's, and Mamm's nose in Jonathan's. But my eyes did not look like anyone's. My hair was different, too. It was all just more ways I did not fit. But it was not me! It was not that something was wrong with me! It is because I did not belong at that table! Dat was not my dat! Mamm was not my mamm! The children were not my . . ."
Dropping her head into her hands, Emma gave in to the sobs she could no longer hold back—gut wrenching, shoulder heaving sobs that drenched her cheeks and made it difficult to draw full breaths. But as she wiped her eyes in an attempt to see through the torrent of tears, Miss Lottie was suddenly beside her, pulling her close. "Oh, Emma. . . . It's okay, sweet child.... Let it out. . . ."
And so she did. She cried for the mother she'd never know, she cried for the father she barely knew, she cried for the family that was never supposed to be hers, and she cried for herself—for the life she thought she had and wasn't sure she should.
Soon the sobs gave way to quieter tears and, finally, sniffles that filled the time between cracks and pops of the fire. "I don't . . . I don't know where I . . . belong," she said between the last few hitched breaths. "I don't know where I want to belong."
Slowly, gently, Miss Lottie released Emma from her arms just enough to be able to afford eye contact. "You have had your world turned upside down in a matter of two weeks, Emma. The only way you can know those things is to get information."
"Information?" Emma wiped her face with the back of her hand. "What kind of information?"
"Get to know your birth father. Get to know your grandmother. Get to know your cousins. Learn their world. Soak it all in. And, while you're doing that, pay careful attention to what"—Miss Lottie touched Emma's chest with her finger—"your heart is saying. Then, and only then, can you answer those questions about where you belong."
"It has already been twenty-two years," she protested between leftover sniffles.
"You're right, Emma, it has. You can't erase all that and accumulate twenty-two years of new knowledge in a matter of weeks. Not when it's all still so raw. Decisions made in anger are never good decisions, Emma. Never. You must give it, and yourself, more time."
"But do I even have a choice?" she asked. "My real family is English. Isn't that what I should be now, too?"
Miss Lottie pushed her glasses higher on her nose and then gathered Emma's hands inside her own. "I don't know the reasons behind the decisions your parents—Rebeccah and Wayne—made. Only you and they know that, but—"
"There is never a reason to lie, Miss Lottie. The Bible says, 'Lying lips are an abomination to the Lord, but they that deal truly are his delight.' "
"That is true, certainly. But the Bible also says, 'He that answereth a matter before he heareth, it is folly and shame unto him.' "
She stared at Miss Lottie. "But they lied, Miss Lottie! And no, I don't know why!"
"Have you asked?"
"No. I do not need to ask. Lies are never good, never right."
At the approaching clip-clop of Levi's horse, Miss Lottie squeezed Emma's hands one last time and then released them in exchange for her cane. "Remember, Emma, anger does not make for good decision making."
"I will try. . . ." Emma smoothed her skirt down against her legs and then stood, her eyes making short work of the uneaten pie on her plate. "Oh, Miss Lottie, I never tried your pie. I just got so busy telling you—"
Miss Lottie stilled the rest of Emma's sentence with a gentle finger. "Shhhh . . . It just means you must come back and see me. Soon."
Chapter 18
Emma settled back against the tree and tried not to think about Sarah's face as she'd passed the clothesline and the basket of clothes waiting to be hung with nary an offer to help. But while that memory invoked a knot inside her throat, the one of Mamm's disappointment as, Mamm, too, had looked up at Emma, stirred an anger she didn't want to feel in her special place.
No, Miller's Pond had always been her happy place—the place where something as simple as watching a butterfly flitting around in the spring, or a colored leaf floating down to the earth in the fall brought her a sense of peace. Here, she could be herself without self-critiquing her every move and non-move. Here, she could cry if she wanted to cry, or laugh if she wanted to laugh. And here, she'd been able to pretend what the truth had been all along—that the trinkets she'd added to the drawstring bag in her hand each year were, in fact, her birthday gifts.
Thanks to her time here, at the pond, and later at Brad's office, she now knew the reason behind the gifts left through her fifteenth birthday. She'd hoped they'd get to the last few items the day she'd met her grandmother, but it hadn't happened. Instead, she pulled the last six presents out, one by one, studying each one closely. The plastic covered bridge . . . The small, red rubber ball . . . The yellow spinny thing on a stick . . . The baseball with the ink markings on it . . . The dried flower with the blue and pink ribbons tied around the stem . . . And, finally, the whittled bird . . .
When they were lined up, side by side, across the top of the rock on which she sat, Emma ran her fingers across each and every one, her thoughts visiting a time Brad had brought to life in her head—a time when Ruby was alive, and Emma's birth parents had been able to convey thoughts to one another with little more than a glance. Thanks to Brad, Ruby was becoming more than just kin she'd never met—kin who had died too young and whom Mamm still mourned. Now, Ruby was someone who'd looked like Emma, laughed softly, looked happy in a photograph, learned to skate, planned picnics, rode rides at an English carnival, drew pictures of houses, made wishes, had been in love with Brad, and seemed to be happier in his English world than she'd been in her own.
Scooping the last present up off the rock, Emma took in the carefully whittled bird—the wings poised to indicate balance, the eyes cast downward as if observing something below, and a tiny worm inside its partially open beak.
"That bird needs a baby!"
Sucking in her breath, Emma turned to find Esther not more than five feet away, heading in her direction. In the little girl's left hand was her lunch pail, and in her right, a small picture book Emma recognized from her earliest school days. "What are you doing here?" Emma traveled her gaze past the five-year-old to the path that wound its way around the far side of the pond. "And where are Jonathan and Annie? They should be with you."
"I peeked at the pond through the trees"—Esther pointed to the narrow break between the trees that otherwise hid their location from the road—"and I saw you! I asked Annie if I could walk the rest of the way with you and she said yah!"
Esther set her lunch pail and book on the ground and clambered onto the rock by Emma's feet, her large brown eyes fixing on the bird once again. "Did the nice man give it to you?"
"Nice man?"
"Yah. He spreaded all his toys on the rock the other day."
"Spread," Emma corrected.
"I liked the horse best!" Esther inched her way across the rock toward Emma, her finger guiding Emma's attention back to the whittled bird. "Since we don't have six kittens, you could name the bird. . . ."
She looked a question at the little girl only to shake it away as the answer dawned all on its own. "You did such a good job naming Flower, I think you should name this bird, too."
"Are you sure?" At Emma's nod, Esther's ever-present smile widened even more, revealing the sizeable gap where two of her top front teeth were missing. "I want to name her Emma, just like you!"
"Emma?" She looked at the bird, wiggled it ever so gently, and then pretended to make it fly over to Esther. "Emma is a girl name," she said in a squeaky voice. "I want a bird name."
Giggling, Esther rose up on her knees to address the wooden bird. "But you have a worm in your mouth."
"Yah. I am hungry."
"But you will not eat it," Esther said, her brows dipping down in a sudden burst of seriousness. "You are carrying your worm like the bird in Dat's barn."
Then, abandoning her conversation with the whittled bird momentarily, Esther fixed her eyes on Emma. "Jakob says it is good Flower and the other kittens cannot climb yet because they might eat the baby birds in the nest."
"Are you sure they have hatched? It is not spring yet."
"Yah! There are two babies! And the mamm bird brings them worms! I saw her!"
Emma lowered the now-silent wooden bird back to her lap and, with the index finger of her free hand, tapped her little sister's nose. "You will have to show me the nest when we go back to the house."
"Yah. But I still want to call that one"—Esther pointed at Emma's lap—"Emma."
"Why?"
"Because if I were a baby bird, you would bring me worms just like that bird!"
Leaning her head back against the trunk of the tree, Emma listened to her own laugh as it echoed around them. "Oh? You think I would give you a worm, do you?"
"Yah! Because you love me and you taked good care of me."
"Take," Emma corrected, sitting up. "And you are right. If you were a baby bird, I would bring you worms."
Esther pointed at the bird. "Can I sit there? For just a little while?"
"You mean on my lap?" At Esther's nod, Emma moved the bird to the rock and pulled the little girl into cuddle range. "I always have a spot for you to sit, little one."
"Even if you leave?" Esther whispered.
"Leave?" she echoed, loosening her hold on Esther. "I am not . . ." Breathing back the rest of a sentence she knew she could not say, Emma straightened her shoulders against the tree and pointed at the sleeve of the little girl's dress. "Would you quit growing, please? Because pretty soon there will not be enough fabric in the store to make the dresses you seem to be outgrowing faster than I can make them."
Lifting the bird off the rock, Esther held it to her chest and rested her cheek against Emma. "I tolded Annie and Jonathan you would not leave. I tolded them you do like us. But Annie tolded me you don't anymore. She tolded me that's why you do not help with the chores, and why you don't sit next to me at dinner all the time."
Esther shot her chin up, eyes wide. "She tolded us you yelled at Mamm one day!"
"Told." She knew, in the moment, it didn't matter if Esther's grammar was correct, but drawing attention to it bought Emma time to breathe her way through the sudden dizziness.
"Did you, Emma? Did you yell at Mamm?"
She knew the consternation on Esther's face. It was the same expression, she, herself, would have worn at the notion of any of her siblings ever raising their voice to Mamm. But that was before—before she knew everything about her life had been built on lies.
"Emma?"
Snapped back to the moment by the worry in Esther's voice, Emma lingered a kiss atop the little girl's kapp. "Why do you tiptoe over to see Bean's kittens?"
"Because they are babies. I do not want to wake them if they are sleeping."
"That is a good reason." She breathed in the medley of earth and apples that clung to her sister's hair and then sat back. "I have reasons for things that I do, too, Esther. And right now, there are some things I cannot talk about. But I will . . . soon. Can you wait for me to do so?"
Esther started to nod but stopped to look up at Emma, instead. "You do like us, don't you, Emma? Even if there were no more kittens for you to name?"
Somehow, despite the tears she felt gathering in the corners of her eyes, Emma still managed a laugh. "I do not need a kitten to name when I have a bird to share mine."
She picked up the bird and turned it to face Esther once again, her voice adopting its earlier squeaky quality. "Thank you for naming me Emma, little girl. I like my new name."
Giggling, Esther scrambled back onto her knees to maximize eye contact with the whittled creature. "I named you that because I love Emma."
"That is good," she said, moving the bird in for a wooden beak kiss. "Because Emma loves you, too. Never, ever forget that."
* * *
She'd tried for Esther. She'd tried for Jakob and Jonathan, Annie and Sarah. But nothing about sitting at the dinner table with Mamm and Dat had felt right. On the surface, it had been like any other evening meal—heads bowed in prayer, plates of rolls and potatoes and meat being passed around from person to person, and the sharing of stories from the day. But that was where the similarities had ended for anyone who dared to truly see.
Some of that could be put on Mamm and Dat and the uneasy glances they sent her way every time the conversation around the table lagged in a way that made Emma's silence almost deafening. Some of it, too, could be put on the anger Miss Lottie had warned her about, yet she couldn't shake.
For twenty-two years, these two people had told her an untruth—that she was their child. That single lie had led to the telling of more. To Jakob. To Sarah. To Jonathan. To Annie. And to Esther. Because despite what they all believed, Emma was not their sibling. She was still kin, sure, but not in the way they'd all believed or in the way they still believed.
And every day that went by with her knowing a truth they didn't, she, too, was part of that lie. But until she could tell them without her anger spilling into places it didn't belong, she needed to wait.
"Emma? I think the plate is dry enough to put away."
Halting her hand, mid-squeak, she looked up to find Sarah studying her from the top of the single step stool. "Oh. Yah. Here." She passed the now dry plate to her sister to add to the cupboard and turned her attention to drying up the counter around the sink.
"I do not know why you do not help with the laundry or the mucking or the baking anymore," Sarah said, closing the cupboard door and stepping down off the stool. "I do not know why you do not speak at dinner. But I do know you are making Mamm very sad and that is not good."
Emma stopped wiping to stare at Sarah. "You think I am making Mamm sad?"
"Yah. You do not see the way her smile disappears when you walk out the door without saying what you are doing or where you are going. You do not hear her crying when she says she is quilting."
"Something that is not there cannot disappear."
"What do you mean?" Sarah asked.
"You said Mamm's smile disappears when I walk out the door. But Mamm does not smile at me when I am here, so I do not know how it can disappear when I leave." She draped the wet dishcloth across the oven handle and then turned back to Sarah. "And if she is crying in her room when she is to be quilting, it is because of her choices, her lies."
"Emma!" Sarah stamped her foot on the wood plank floor only to startle herself with the answering noise. Flustered, the sixteen-year-old glanced over her shoulder toward the hallway, waited to see if a reprimand would follow from the vicinity of the front room, and, when it didn't, turned back to Emma, her voice dropping to a whispered hiss. "I do not know why you are this way. I saw Levi bring you home last night. That should make you happy, not like . . . this."
"Like this?"
"Yah. Angry . . . Mean."
Sarah's words snapped her back a step. "I am not mean."
"The Bible says, 'Honour thy father and mother.' But you are not. You are saying things you should not say about Mamm!"
"It is not just Mamm. It is Dat's lie, too."
"Emma!"
"Listen to me, Sarah." Stepping forward, Emma gathered her sister's hands inside her own. "The Bible says many things. It says to honor thy father and mother, but it also says 'Ye shall know the truth and the truth shall make you free.' That is what I am now, Sarah. . . . That is why I leave and do not say where I go. Because I am free—free of lies that should never have been told. And soon you will know the truth about them, too."
With one big pull, Sarah wrenched her hands free, her expression a mixture of fear and defiance. "Maybe it is good that you go, that you do not help with chores as you once did."
Chapter 19
She saw the described car the second it came around the bend. The slow pace, combined with its periodic stops and starts, a clear indication the driver wasn't entirely sure where the path to Miller's Pond was located. For the briefest of moments, Emma actually considered stepping back into the protection of the trees and letting the car drive right by, but considering she was the one who had requested the visit in the first place, it wouldn't be right.
Instead, she stepped forward, waved at the driver she couldn't quite see yet, and waited as the car hurried to a stop on the opposite side of the country road. When the window lowered to reveal the expected face, Emma crossed the finely graveled road to the driver's door.
"Thank you for coming. I hope it is not a bother."
Delia's hand shot through the opening to grab Emma's. "If you had seen my face when I got your call, dear, you would know just how much of a non-bother this is." Then, with her soft blue eyes searching Emma's, she squeezed. "Are you okay?"
Shrugging, she pointed at the other side of the car and, at Delia's emphatic nod, looped around the front and slid into the passenger seat. When she was settled, with the seat belt fastened across her shoulder, she gave into a sigh. "It is all too much to think about. This who is and who is not my family stuff. I don't know what to think, and when I do think, I do not like the anger that I feel."
"Do you want to talk about it?" Delia asked.
Did she? She wasn't sure.
So much had changed. Who she was . . . Where she came from . . . Where she belonged . . . But somehow, amid all that change, she wanted, no needed, to feel as if something—some aspect of herself—was still the same. Yet there she was, sitting in a car with her newly discovered English grossmudder. How could anything ever be the same again?
Aware of a burning in her eyes, Emma turned and looked out her window, the fields she'd glimpsed from this spot nearly every day of her school years soothing her heart with the kind of familiarity she felt every time she—
Emma sat up tall, her focus skipping back to Delia. "Could . . . could we bake together?"
Delia drew back only to have her whole face lift with a smile. "Oh, Emma, I can't think of anything I'd rather do." Placing her hand on the gearshift, the woman divided her attention between the road and Emma. "Do you want to teach me something you like to bake, and I'll teach you something I like to bake?"
"Yah!"
"Then let's head to the store, shall we?" At Emma's emphatic nod, Delia closed her window against the chilly day, made a U-turn back in the direction she'd come, and let loose a happy squeal. "From the moment Michelle was old enough to hold a spoon I tried to cultivate a love of cooking and baking in that child. She always liked to watch, always liked to be nearby for any extra chocolate chips or blueberries that didn't make it into the cookies or muffins I was baking. But beyond that, she's never had any interest in recipes and actually baking. So this is going to be a real grandma-dream-come-true for me."
She sensed the Amish fields giving way to property and houses owned by the English to her right and left, but all she could really see was the dough they'd soon be making from the generations' old recipe she'd tweaked and changed until it was her own. "Where do you get your recipes?" Emma asked on a whim.
"All sorts of places. Magazines. Online recipe sites. Friends. Family. If it has ingredients I like, I'll give it a whirl." Delia turned right at the first traffic light and continued. "Many of our favorite recipes have come about that way. How about you? Where do you get your recipes?"
"Some have been handed down from my mamm and my grossmudder. But they do not stay the same."
Delia stopped at the next light and peered at Emma. "Meaning?"
"The recipes are good as they are—they have fed many mouths for many years. But I like to change them. Sometimes the change is little—like a bit more salt, or a splash of vanilla where there was none. But sometimes, in changing little things, I find that I can change even more. When I am done, it is no longer someone else's recipe. It is mine."
"And the change?" Delia prodded. "Is it well received?"
The last of the tension she'd been harboring in her shoulders faded away as she rested her head against the seatback. "Plates are always empty and tummies are always full when I make something new. Even Esther's."
"Esther is the youngest, correct?"
Emma smiled as an image of the little girl popped—fully formed—into her thoughts. "Yah."
"You two are close, aren't you? I can see it in your face and hear it in your voice every time you say her name."
It was something she'd never really thought about, yet the reason for that closeness was really quite simple. "With Esther, I do not have to try to change," she said. "I can be me."
"You can't be you with the other children?"
Emma felt the car slow in advance of the turn into the grocery store's parking lot and let her gaze travel ahead while her thoughts stayed with Delia's question. "I love them all, but there is something special with Esther. It is why I am most afraid to tell her about all of this."
"Will that be soon?" Delia asked as she pulled into a parking spot not far from the market's door.
"It must be, even if Mamm does not agree."
An audible inhale from behind the steering wheel pulled Emma's gaze back onto Delia. "I don't think Rebeccah is in any position to argue anything with you right now, dear."
"Right now, it is just me who knows of Mamm and Dat's lie." Emma returned her chin, if not her thoughts, back toward the window. "I do not think she wants the others to know. But I do not want to tell lies the way she does."
Delia's answering smile held no sign of humor. "Trust me, sweet Emma, Rebeccah has far bigger worries on the horizon than what those children know, I can promise you that."
Desperate to reclaim the lightness she'd felt not more than five minutes earlier, Emma unlatched the seat belt and wrapped her fingers around the door handle. "Could we go inside? I just want to think about baking, if that is okay?"
"Of course. You're right." Delia pulled her keys from the ignition, tucked them into her purse, and opened her own door. "Let's leave the unpleasantness to those who created it and go have some fun, shall we?"
* * *
With little more than a few finger points at two or three cabinets, Emma moved around Delia's kitchen with ease, gathering requested ingredients for Delia's favorite recipe and familiarizing herself with the many English trappings available to Emma in making her own. When everything they needed was assembled across the top of the counter, Delia nudged her chin at a red-capped bottle.
"I've never made apple cinnamon bread before," Delia said, tying her apron into place.
"We are to make white bread. The cinnamon and apples are for the apple butter we will put on the bread."
Delia brought her hands together with a quick clap. "I love apple butter! But I've always heard it's very labor intensive, no?"
"Not with that"—Emma pointed at the slow cooker she'd found on Delia's shelf of pots and pans. "It will still take many hours, but it will be delicious with the bread." After little more than a brief hesitation, she, too, tied one of Delia's aprons across her dress. "At home, it must sit on the stove for a long time, with much stirring."
Ingredient by ingredient, they created the simple dough for Emma's bread. While it rose in its bowl by the kitchen window, they sliced apples and chatted. When the apple butter was turned over to time, they punched and kneaded the dough and then left it to rise once again.
As it did, they moved on to Delia's recipe—a white chocolate pastry puff that required whisking, melting, and occasional samples enjoyed off the edge of mixing spoons and fingertips. When Emma suggested melting some dark chocolate to drizzle across the top, Delia grinned.
"You remind me of her right now," Delia said as she scooped up the latest dirty bowl and carried it over to the sink.
Emma gathered up the used mixing spoons and followed. "Who?"
"Ruby." Delia squeezed a drop of dishwashing soap into the bowl and turned on the water. "Your mother. She was just like you are now whenever I'd catch her drawing during a visit."
Intrigued, Emma set the spoons into the bowl and waited for more. Delia didn't disappoint.
"You've been floating around this kitchen since the moment we unpacked the groceries from the store. And once we actually started, that joy you told me you feel when you bake was every bit as tangible as that bonnet."
"It is a prayer kapp," Emma corrected quietly. "I like to bake. It makes me happy."
"I can see that. And that's what drawing did for your mother."
Emma picked up a dishcloth to Delia's sponge and waited to dry the first of many things they had dirtied. "Do you mean the drawings she did of houses?"
"I do."
"That was not just for my . . . father?"
"The drawings?" At Emma's nod, Delia handed her the first bowl and moved on to the next. "The first house she drew was for herself—so she could get a better idea of Brad's dream to build homes. But as he talked about different features he wanted to do and different looks he wanted to create both inside and outside, she really seemed to enjoy drafting new versions. Soon, it became apparent she had the kind of ability that made her a natural for a career in architecture.
"Next thing I knew, they were planning their one-day business, with Ruby as the architect, and Brad as the builder. They even came up with a name for their business."
Emma stopped drying. "What was it?"
A sad smile tugged at the corner of Delia's mouth as she scrubbed at a spot near the bottom of the bowl. "Imagine Homes."
Before she could ask why, Delia continued, the woman's voice almost wistful. "They used to say that to each other. He'd say something like, imagine a fireplace with a long, narrow window on each side.... Or she'd say something like, imagine a house built around a flower garden. Sometimes the things they'd throw out would get them laughing, and sometimes it would have Ruby reaching for paper and a pencil while Brad talked the idea through into something workable.
"Believe it or not, even as young as the two of them were at the time, they really had some great ideas. So much so, I truly believed Imagine Homes would exist one day, with Ruby drawing the designs, and Brad making them come to life all over this town."
"But he did not call his company Imagine Homes," Emma said, swapping the now dry bowl for the next wet one. "It is Harper Construction."
Delia reached for the pile of spoons but stopped and shut off the water instead. "He did that at my suggestion. I wanted him to look forward, instead of backward. To really embrace this venture as his own. Though, even with the name change, he still made sure Ruby was part of it even if he's the only one who ever actually sees it."
"A part of the company?" At Delia's emphatic nod, Emma lowered the partially dried bowl back to the counter. "But how? She is dead."
Delia wiped her hands across the front of her apron and reached for Emma, her eyes sparkling. "Come. I have something very special to show you."
Chapter 20
Sitting on the couch, waiting, it was hard to look at anything besides the photograph of a young Brad and Ruby on the way to the carnival. Sure, she saw snippets of herself in Brad's hair and eyes, and Ruby's everything else, but that's where the connection ended. In fact, if not for a handful of lunches and a single family-style dinner, the curly haired boy with the ear-to-ear grin would be as much of a stranger to Emma as the girl smiling warmly at his side.
"I'm sorry that took so long, Emma," Delia said, breezing into the room with a burgundy-colored folder in one hand, and a dark brown leather book in the other. "It seems I need to move clean office a smidge higher on my to-do list for this week and—"
Delia's gaze landed on Emma's face, quickening her steps to the couch as it did. "Emma? Is something wrong? You look a little . . . upset."
"I do not mean to be upset," she said, looking between Delia and the mantel, her voice barely more than a rasp. "It is just that . . . I don't know what to feel."
Depositing the folder and book onto the coffee table, Delia sat down and draped her arm across Emma's shoulders. "Oh, sweetie, this is time to be happy! You've found us and we've found you! It's an answer to many, many prayers!"
"I did not say such prayers."
"That's because you didn't know. Those people kept us from you and you from us."
Those people.
Mamm and Dat.
Only they weren't really—
Shaking off the troubling thought, Emma lifted her finger and Delia's attention back to the picture. "I do not know them. They are strangers to me. But I would not be sitting here without them."
"Brad isn't a stranger anymore, Emma." Delia hooked her finger beneath Emma's chin and guided Emma's gaze back to hers. "Little by little, the two of you—and all of us—are going to build something very special, I just know it."
"Yah."
Delia watched her for a few moments and then reclaimed the brown book from its temporary resting spot on the coffee table. "Brad told me the two of you have been working your way through the little memories he's been leaving at the gravesite each year, yes?"
"There are six more I still do not know."
"And when he gets home tonight, maybe you two can rectify that. Or, better yet, once tomorrow is behind you both."
"Tomorrow?" Emma echoed. "Do you mean this man I am to talk to?"
"Nicholas. Yes."
"I do not know why I must speak to him."
"Because he has to know everything as it happened. He needs facts."
"But I can speak to Bishop King alone. When I do, our community will shun them," she said, her voice rising. "I will tell him as soon as the children have been told. Backs will be turned to them until they repent!"
"Backs will be turned to them?" Delia cupped her mouth only to let her hand fall back to the still-closed book. "That isn't enough, Emma."
"It is awful to be shunned! Your friends and your family cannot look at you, or speak to you! And if they do, they can be shunned, too!" Emma turned toward Delia, her knees scraping against the coffee table. "My friend, Mary? Her uncle Barley was shunned once for using electricity inside his home. Mary could not speak to him or look at him for weeks."
Delia started to speak, stopped, and, after several long moments of silence, tapped the cover of the book. "The other day, when you saw that"—she pointed to the picture on the mantel—"you asked if I had more pictures of Ruby. Perhaps you would like to see them before we get to the reason I brought you into this room in the first place?"
"There are more? In there?"
"There are, indeed." Her smile back in place, Delia opened the book to the first page, her fingers immediately moving to the edge of the first photograph—a picture of a tired house Emma found vaguely familiar. "This is the house Brad was helping repair when he met Ruby for the first time. So, when I suggested putting all his pictures into an album he could look at whenever he was missing Ruby, he said it had to start with this one."
Emma studied the front porch . . . the front windows . . . the dilapidated looking house . . . the—"Wait! I know this house! It is different now. It has fresh paint and the porch is not lopsided like this. And that woman"—she pointed to the person standing on the porch, peering out—"does not live there now. Miss Lottie does!"
"Is she Amish?"
"No. Well, I mean, she was raised by Amish, but she did not join the church. I think she said she moved into this house when I was a little younger than Esther." She leaned in for a closer look, her mind's eye replacing the house in front of her with the tidy cottage she'd sat in just two days earlier. "She is kin to the Beilers and she is very wise."
Delia turned the page, her own soft inhale barely noticeable against Emma's. "He borrowed my camera the first time he went to the ice cream shop because he wanted to take a picture of Ruby for me to see."
"I know this shop! It is still there!" At Delia's nod, Emma pulled the book partially onto her own lap to get a closer look. "This picture? It is when Ruby came outside at the end of the night, isn't it?"
"Brad told you . . ."
"Yah. He told me he waited for two hours for her work to be done. He is right," she said, studying the picture. "Ruby was surprised to see him when she came out. Her eyes are very big like Mamm's get when she is surprised, but there is a smile there, too."
"There's more."
Page by page, they made their way through the album, the images Brad had recorded matching many of the stories he had shared. But as much as she enjoyed having the visual to go with the story, it was the faces, themselves, that held her attention most.
"I realized, the other night, after you left, that we didn't take any pictures. I guess we were all so in the moment, none of us thought of grabbing a camera." Delia relinquished her hold on the edge of the album long enough to squeeze Emma's hand. "We'll have to rectify that so we can start a new album—one with you and your father . . . and me, too."
Emma drew back. "I do not take pictures. The Bible says, 'Thou shalt not make unto thyself a graven image.' It is the Amish way."
"Ruby took pictures," Delia said, sweeping her hand back to the album.
"Ruby had not been baptized yet. I am." Emma swung her gaze between the photo in the album and the one on the mantel and then released a quiet sigh. "I should not even be looking at these pictures."
"Every child has a right to know their parents, dear. Since Ruby is gone, these pictures are your way to know her."
She knew she should argue, but she couldn't. Delia was right. She needed to know. To see. And who would punish her for looking, anyway? Mamm and Dat?
Leaning forward, she nodded at Delia to continue, and, once again, she soaked up everything the woman had to say about each and every picture. When they got to a picture of her birth parents sitting in a field surrounded by white fluffy dandelions, she looked up to find Delia watching her. "Did you take this picture?" Emma asked.
"No. See how it's taken from down low?" Delia pointed Emma's focus back to the picture. "Technology has gotten much better since this was taken, but Brad basically put it on video mode and then I pulled several still frames from it for this album."
She didn't really understand, but, still, she nodded as Delia turned to the next page and the series of pictures showing Ruby blowing on the dandelion. In the first shot, Ruby's eyes were open; in the second shot, one was closed and the other was peeking out at Brad with such a silly expression Emma couldn't help but laugh. And in the third picture, Ruby's eyes were closed and dandelion fluff scattered in the air.
"I wish I could know what she wished for that day," Emma whispered as much to herself as Delia.
"While I can't know for certain, I might have a guess." Delia pointed to the folder on the table. "Shall we take a look at that now?"
Emma looked down at the album and the handful of pages that still remained. "There are more pictures, yah?"
"There are. And we can get back to those. But this will speak to what I said earlier, in the kitchen. About Brad keeping Ruby part of things with the company even now. Even after . . . everything."
Sliding the album across her lap and onto the cushion to her left, Emma turned her attention to Delia and the folder with the same Harper Construction logo used on Brad's truck. "I have seen these at the office. Near Miss Sue Ellen's desk."
"You're right. Only those folders are different from this one in that they're missing one very special floor plan." Flipping the folder open, Delia pulled out a letter with the same Harper Construction logo across the top and set it aside in favor of the drawing on the next page. "Floor plans are what customers look at to decide which house suits their needs best—meaning, does it have the right number of rooms, does it have the bay window they want, or the office they need, et cetera.
"Each floor plan has a name to make it easier for people to reference. Like these ones." Delia took out a handful of floor plans and splayed them across the top of the folder. "The Emerald is a two-story with a bonus room over the garage. The Sapphire is a one-and-a-half story home with a formal dining room and a second-floor laundry. The Amethyst is a ranch with a split bedroom setup, meaning, the master suite is on one side of the home and the other two bedrooms are located on the other end. The Turquoise is a ranch-style home with a mother-in-law suite in the basement that can be accessed through a separate entrance, depending on the lot's grade. And then there's The Diamond. It's the one with all the bells and whistles."
Emma drew back. "Why would people want bells and whistles in their house? That would make it hard to sleep."
Delia's laugh brought a smile to Emma's lips, too. "That's just an expression, dear. It means that it has all of the extras that people want—the best of the best, so to speak." Feature by feature, Delia moved her finger around the floor plan, tapping out each item she listed. "Planning desk, center island, farmhouse sink in the kitchen, wet bar in the living room for entertaining, double bay windows in the dining room, French doors with alcove in office, switchback staircase to access the second floor, Jack and Jill bathroom between bedrooms number three and four, with each of those bedrooms having its own separate alcove for a sink, the second bedroom having its own bath, and, of course, the master suite with built-in fireplace, his and her large walk-in closets, separate tub and shower in the master bath, et cetera."
"There are many things in this house," Emma murmured.
"There's also a finished lower level." Delia's finger moved down the page to another set of drawings. "With a media room, a game room, another wet bar, a fifth bedroom, full bath, and storage space. It really is quite a house. I think they've built six or seven of these so far since Brad added the plan to the packet."
Emma smiled and nodded politely and then slid her attention back to the album. "Could we look at the rest of the pictures now?"
"Wait. I haven't gotten to the whole reason I'm showing this to you in the first place. The one thing you won't see in the packets at the office." Delia restacked the floor plans, set them off to the side, and reached into the folder once again.
Seconds later, Emma was staring down at a floor-plan-like drawing and trying to remember how to breathe. "This . . . this says The Ruby," she whispered, post-swallow.
"It does."
"But . . ." At a loss for words, Emma simply stared at Delia and waited.
"That's the drawing—or, rather, a copy of the drawing Ruby wanted to be the first house Imagine Homes built."
Emma recognized some rudimentary features thanks to the previous floor plans, but still she was grateful when Delia's finger took over the tour. "Ruby didn't want a long front hallway. She wanted people to feel welcome the moment they stepped inside. So that's why the door opens into the family room. She wanted lots of windows because she told Brad she felt most at peace when she was looking out at the wide-open fields and the sunny sky."
"I . . . I like that, too," Emma murmured.
Nodding, Delia moved on, her finger moving to the right. "She wanted a large kitchen."
"To bake in?"
"To sit with family and visit," Delia corrected. "And she wanted lots of windows in the kitchen, too. That way you could nap in your cradle where it was warm and sunny, and she could be nearby, preparing lunch."
Emma drew back. "Me? But I was not born!"
"You were on the way when she made this drawing, dear."
"But—"
"Your pending arrival factored into almost everything here." Moving her finger toward the sketch of the second floor, Delia pointed to each of the four rooms. "She wanted all of the rooms to be together, and she wanted them simple. She said bedrooms were for sleeping. The rest of the house was for being together."
Emma's gaze skipped back to the first floor and the family room that seemed almost notched in two. "What is that line, there?"
"For some reason, Ruby wanted the front room to be quite large, but Brad did not agree. He said it was wasted space for the everyday and suggested a divider of sorts that could divide the space but allow it to be open and large for"—Delia shrugged—"special parties or whatever it was that made Ruby want such a large room."
"It is what she was used to," Emma murmured. "Amish homes have a large room that is used for hosting church. Each family only hosts a few times a year, but the room must be large enough to accommodate benches with many people when they do."
If Emma's words registered at all, they didn't last long as Delia's finger jumped to the front of the house. "Ruby also wanted a wide front porch, one that wrapped around the front and sides of the house. She wanted to be able to knit or quilt in a chair, or simply sit on a comfortable porch swing, and be able to see you wherever you were. And these boxes here? On the first-floor windows? Those are flower boxes."
It was hard to picture an actual home when all she could see was lines, but somehow she could. The simple house with a wide front porch and rocking chairs . . . The view of a pretty summer sky from inside the front room . . . The faces from the mantel-topped picture looking down at a newborn Emma sleeping soundly in a cradle by a kitchen window . . . Ruby's loving smile directed on no one but—
Fisting the tears from her eyes with her left hand, Emma pushed the drawing and folder back onto Delia's lap with her right. "I need you to take me back to Blue Ball. Now."
"Take you back? But-but why?" Delia closed the folder and tossed it onto the coffee table. "I thought we were having a nice time! I thought I could make you dinner, we could try your bread and my pastries, and we could look at the rest of the photo album! I thought, too, that if you were still here around seven, you'd get to see Brad for a little while before it was time to take you back to the farm. I mean, I know you'll be together tomorrow, but it's going to be such a stressful day in a lot of ways and it might be nice to have a little quiet, carefree time first."
"I need to go back."
Delia's whole being sagged. "Emma, dear, if I did something to upset you, it wasn't intentional. Maybe I should've waited and let Brad show you your mother's drawing, but you were asking so many questions about Ruby and the company that—"
"Please. I need to go back," Emma insisted. "I need to pack my things."
For a moment, the silence born on her words was so deafening, Emma actually pressed her hand to her chest in an effort to quiet her pounding heart. But it didn't matter. Delia's answering gasp drowned out all. "Pack your things? Emma, does this mean you're ready to start your new life here? With your father and me?"
"Yah. . . . I mean, yes."
Chapter 21
She watched Delia's car disappear from view and then turned toward home, the anger she'd managed to keep in check during the ride to Miller's Pond propelling her forward. Step by step, she made her way past the first of four farms, the insistent bleating of the Schrocks' goats little more than background noise for the images cycling their way through her thoughts.
* Stopping at the grave before school on her birthdays . . .
* Looking over her shoulder as she added the latest trinket to the bag . . .
* Watching her classmates play at recess and never being asked to join . . .
* Always doing more than asked at home and still never getting a heartfelt smile from Mamm in return . . .
* Standing in Brad's office, staring into eyes that looked just like hers . . .
* Flipping through photographs, trying to memorize everything about two people who, for all intents and purposes, were strangers, yet also her parents . . .
"Good evening, Emma."
Startled off the edge of the road, Emma looked up to find Levi Fisher smiling back. "Levi! I did not hear you coming!"
"I see that." He tugged Hoofer to a stop beside Emma and readjusted his hat. "Soon the sun will be down and it will be quite cold to be out walking."
"I am not out walking."
Dragging his hand down his face, Levi sat back in his seat. "You are out, and you are walking."
"Yah, I am walking. But it is only to get to Dat's farm."
"I could give you a ride if you would like."
"But you live the other way," Emma said, pointing in the very direction they'd both come.
"I do."
She waited for him to say more, but when he didn't, she took the hand he held out to her and climbed into place beside him on the buggy seat. "It is very kind of you to drive me home. Thank you."
"It is my pleasure."
With a quick, yet firm jiggle of the reins, Hoofer began to walk, the mare's gentle pace and sure steps a stark contrast to the anger Emma was just four farms away from unleashing. Still, she took advantage of the quiet that fell around them to steady her breath and try to unfurl her fingers from the fist she couldn't seem to relax no matter how hard she tried.
"Do you see that cow? There?" Levi guided her attention toward a Holstein grazing in the Troyers' field. At her nod, he broke out in a grin. "Found it sitting outside our front porch week before last. Came out after lunch and there it was. Staring back at me like I was the one who didn't belong."
"But you live more than a half mile that way," she said, hooking her thumb over her shoulder. "Why would a cow go so far from home?"
"Dat said it must have smelt Mamm's cooking."
Her laugh mingled with Levi's only to fade away as he continued, his words taking her on a journey past his front porch and a neighbor's cow to his dat's fields. "The rocks have been cleared from the fields and soon we will begin the planting. The days will be long when we do, so I am trying to help Miss Lottie with some repairs now, while I can."
"Repairs? What kind of repairs?"
"The floor in her living room has many creaks, and one of the railings on her front porch is not tight."
"Do you like to do such things?" she asked.
"Yah. I like to work with my hands. Perhaps, if Miss Lottie would like, I can build a shed for her gardening tools since she does not have a barn. I could make it look like a small house. I think Miss Lottie would like that, yah?"
"I think she would like that very much."
"Did it help to speak with her the other night? You were so quiet when I picked you up, I did not want to interrupt your thoughts with questions."
Something about his voice, his very demeanor, snapped her attention back to his face. "If I did not thank you for bringing me to Miss Lottie's, I am sorry. I—"
"You thanked me, Emma. Many times."
"Good." She looked again at the fields and breathed in the cold winter air. "And yah, speaking with Miss Lottie was nice."
"That is good, though you were missed at the hymn sing." This time, her laugh was void of anything resembling humor. "I do not know who would miss me. Mary was with Leroy, so there would be no one to speak to, no one to miss me."
"I missed you." With a gentle pull, he slowed Hoofer's pace still further. "No one makes the kind of oatmeal cookies you make."
"You still didn't like Liddy's cookies?"
"It is as I have told you, Emma, Liddy Mast makes oatmeal cookies, but she does not make them like yours," Levi corrected. "No one does. Not even Mamm."
She felt her mouth growing slack and covered it with a quick swallow as, once again, he continued. "But it is more than just oatmeal cookies that I missed. I missed looking over to check on Mary and seeing your smile. It is something I look to see at every hymn sing now."
"My . . . my smile?" she echoed.
"Yah."
She saw Levi's mouth still moving, knew he was still saying things she probably wanted to hear, but at that moment there was only one voice she heard.
"There were so many things I loved about Ruby, but her smile? It was the best. Distracting as all get-out, but wow."
Brad had liked Ruby's smile....
"I missed hearing you laugh when Mary said something funny. It does not matter who I am talking to or what I am doing when I hear that sound. I always stop and listen." Levi's own soft laugh rumbled past his lips. "And I missed watching that little jump you do when your team wins at volleyball."
She stared at Levi. "I jump?"
"Yah. When you hit the shot that wins."
"Ruby would do this little jump when she was excited about something—bubbles, skating, it didn't matter. If she was happy, she did her little jump."
Emma pressed her hands to her cheeks in an effort to cool their building heat. "I did not know I did such a thing."
"You do."
In lieu of words she didn't have, Emma looked out at the dusky fields, the familiar landscape calming her nerves. Closing her eyes for just a moment, she breathed in the scent of thawing earth and imagined the way it would change as the temperatures warmed and crops began to grow.
"I spent the day with Delia," she whispered. "She is my English grossmudder."
If he was surprised by her admission, he kept it to himself. Instead, he guided Hoofer to the edge of the road and brought the buggy to a stop. "Was it a good visit?" he asked, resting the reins atop his legs.
"It was. We baked together. I showed her how to make my bread and apple butter—" She sucked in the rest of the word, only to wave away the worry. "No, it is okay. It still has many hours. It will be fine."
"I like apple butter," Levi said.
Something about the earnestness in his voice made her giggle. "I must bring you some."
"Yah." He fiddled with the reins for a moment before tucking them to the side in favor of shifting his full attention to her face. "You did not look like Emma when I saw you back there."
She started to protest but stopped as the reality she'd been denying long enough, pushed itself to the forefront of her thoughts. "I am Emma—this Emma—because of Mamm and Dat. They raised me to be Amish because they are Amish. But Brad is my birth dat. He should have raised me. He wanted me. They both wanted me—Brad and Ruby. But Ruby died having me, and Mamm and Dat told Brad I died, too. If they had not done that, he would have raised me. If she'd lived, I would be English. If he'd known I'd lived, I would be English.. . . I would have gone to an English school, maybe even college.... Maybe I would cook in a fancy restaurant the way Delia said.... Maybe I would know important people and travel to many places in the world.... I would not wear a kapp"—she pointed to her head—"and I would drive a car instead of a buggy."
"Do you want to go to college and cook in a fancy restaurant?" Levi asked.
"I don't know. I have never thought of such things. College is for English, not Amish. Being a chef in a fancy restaurant is for English, not Amish. Traveling the world to see many places and many countries is for English, not Amish." She took in the streaks of mauve and pink in the western sky and tried to gather her words into something neat and tidy. "I am only Amish because of lies. If there had been no death and no lies, I would be English."
"When you were little, yah. But you chose to be baptized," Levi reminded. "That was no one's choice but your own."
"It was made because I was not shown another life."
"That is what Rumspringa does. It shows you another life."
"It shows another life to one who is Amish. But it is not the same—not as it would be if the English world is the only world I ever knew." She sat with her own words for a moment, only to shrug them away as the reason she was there, in his buggy instead of eating dinner with Delia, pushed its way to the front of her thoughts. "Ruby drew a house for us to live in."
"She drew a house?"
"Yah."
"Tell me."
And so she did. She told him about the floor plan Ruby had drawn. She told him all the little details Ruby had included with Emma in mind. She told him about the window in the kitchen and the way Ruby had wanted to put Emma's cradle there. She told him about the front porch and how Ruby had wanted a swing there so she could watch Emma playing. And she told him about the smile she knew Ruby would have had for her if Ruby had lived—a smile Mamm never seemed to have for Emma in the way she did for her real children.
"That is why I am leaving. Why I am going to go home, pack my things, and say goodbye to Jakob, Sarah, Jonathan, Annie, and"—her voice grew hoarse—"Esther."
"Say goodbye?" Levi shifted on the seat to face Emma. "Why? Where are you going?"
"I should never have been here." She swept her hands outward. "I should not be Amish."
"But you are Amish, Emma! You chose the Amish way when you joined the church! We all did!"
She blinked against the tears she didn't want him to see. "Yah. I did choose to be baptized. But I knew only lies. I did not know who I was."
"You were Emma then, you are Emma now." Slowly, tentatively, Levi reached for her hands only to pull back at the last second. "My dat always tells me to think before I do. To think of the good and the bad that will come with each choice I make. Sometimes, I want to just choose. Sometimes, I do not want to spend so much time thinking. But when I do as Dat says, and I think about the good and the bad my choice will bring, I see things I did not see at first. Things it is important for me to see."
"They wanted me, Levi. They wanted to build a house for us to live in—as a family."
This time when he reached for her hands, he didn't stop, the feel of his warm skin against hers stealing her breath from her lungs. "Just think, Emma. Please. Think about the good and the bad that will come if you do this. Then, if you still feel it is right for you to leave the Amish way, you must leave."
Chapter 22
She was waiting at the usual spot when Brad drove up at eleven o'clock the next morning, the absence of anything resembling a suitcase at her feet clearly registering on his face the moment he stopped the car.
"Hey there, kiddo." He stepped onto the road, squeezed her hand in greeting, and motioned toward the thicket of trees at her back. "Is your stuff back there? By the pond?"
Casting her eyes down at the drawstring bag beside her feet, Emma shook her head.
A flash of movement sent her eyes back to Brad in time to see him check his wristwatch. "Okay . . . That's okay. As long as we're in and out of there inside ten minutes, we'll still be able to have a little catch-up time together before Sue Ellen starts calling to find out where we are."
Dropping his hand to his side, he motioned toward the truck with his chin and grinned. "So come on. Let's make this official and get your stuff."
"I have not packed my things," she said.
He stopped, mid-step. "Why not? My mother said that's why she took you back earlier than planned yesterday. Because you said you wanted to get your stuff together."
"Yah. That is what I said, what I thought I was going to do. But I didn't."
His gaze traveled down the road toward the farm only to return to hers, all signs of lightness gone. "They gave you a hard time, didn't they?"
"No. I did not tell them."
He cupped his hand over his weighted exhale but said nothing.
"Miss Lottie says it is not good to make decisions in anger."
"Miss Lottie?" Slowly, he dropped his hand to his side to reveal lips that were twisted in controlled anger. "Who is Miss Lottie?"
"I have spoken of her before. She lives closer to the Beiler farm. She is English."
"Does this Miss Lottie know the truth?"
Emma nodded. "We spoke the other night."
"And that's what she had to say? Decisions shouldn't be made in anger?"
"Yah." Emma wandered over to the truck and stared at her reflection in the driver side window, the hooded eyes and somber expression she wore reaching into her very being. "Levi says I should think of the good and the bad when I am to make a decision."
"There is no bad to leaving that house, Emma. What they did to you . . . to me . . . to my mom . . . That is what's bad, not trying to make it right after twenty-two years of lies!"
She didn't need her reflection to know the tears were there, hovering in the corners of her eyes. She could feel them just as surely as she could the disappointment emanating off her birth father. "I'm sorry," she whispered. "I will leave. Soon."
He took a few steps toward the trees, only to double back just as quickly. "I'm not upset with you, Emma. Please know that. I'm upset about the whole situation. I just want it to be behind us so we can get to immersing ourselves in each other's lives the way we should have all along. The way we would have if none of this had ever happened."
"Yah."
A second glance at his watch had him scooping up the drawstring bag and guiding her around the front of the truck to the passenger side. "As much as I know there's more to say, we really should be heading over to my office. Sue Ellen can order us in an early lunch and I can tell you the story behind the rest of your birthday presents. Keeps her from getting all angst-y that I'm not there, and has us right where we need to be, when we need to be there."
"I will do my best to answer your friend's questions," Emma said, fastening her seat belt.
"That's all we can ask, kiddo. So, don't stress, okay?" At her nod, he closed her door and crossed back to his own side of the truck. Once he was settled and they were on the way to New Holland, he flashed a grin at her that rivaled the February sun. "So Mom told me about your cooking session out at her place yesterday. She even gave me a slice of your bread with some of that apple butter on top and—"
Groaning, she dropped her head in her hands. "The apple butter. I forgot about that when I left. It must be ruined by now."
"Nope. Mom set an alarm on her phone so she would get up and shut off the slow cooker when it was time. Then she transferred everything over to some sort of container."
"What kind of container?" she asked, sitting up.
"I don't know. Technically, the apple butter wasn't exactly done when I helped myself to some for my bread, but I couldn't help myself." A long, low whistle filled the truck's cab. "I gotta say, Emma, that stuff was amazing. Maybe even better than Ruby's."
Pride she knew she shouldn't feel warmed her cheeks, forcing her to look out the window until she got her emotions in check. "It is just apple butter."
"There was nothing just about that stuff or the bread. And my mom said you made some suggestions for her pastries she'd never considered before and it made them a million times better."
"I am sure they would be good without my ideas."
"And they always were—one of my favorites, in fact. But she's right, they were better last night. Much better." He let up on the gas as they approached a traffic light, his attention flitting between the line of cars slowing to a stop and Emma. "Mom says you show signs of having some really amazing instincts in the kitchen with everything from tastes to process."
"I like to cook and to bake. It makes me happy."
"That's how I feel about what I do, and how Mom feels about what she does. It's called a passion." When the light turned green, they lurched forward with the line of cars. "Perhaps cooking and baking is your passion, Emma."
"It is just something I do to help at home."
"But maybe, with some proper training, it's something you could do for a career." At the four-way stop, he turned left toward Harper Construction. "Like I did when I went away to school for architecture."
She waved at his words much the way Dat's horse swished his tail at the pesky flies that frequented the barn. "I do not have enough schooling to go to college."
"You don't now, sure. But I can get you a tutor. And I'm sure, if we look, we can find cooking classes that don't require any sort of degree." He turned into the Harper Construction parking lot and claimed his usual spot by the back door. "And if you like it enough to pursue it, I'm sure I can find a friend who has an in at one of the bigger restaurants. Or, better yet, you could open your own catering business or your own bakery, or even your own five-star restaurant one day. Of course, I'd help you get it off the ground with funding and whatever else you need."
She was pretty sure she smiled. If not, maybe a nod? She wasn't entirely sure. All she knew for certain was that her head was beginning to spin and her heart was beginning to race. Reaching down, she wrapped her hand around the top of the drawstring bag and hugged it to her chest, the need for something familiar impossible to ignore. "Can we really look at the rest of the gifts when we go inside?" she asked as she followed him from the car and up the back steps. "There are only six left."
He glanced back at the lot, took in the lone sedan not far from his truck, and then pushed open the door. "Sure thing. Let's just check in with Sue Ellen and make sure everything is still a go with Nicholas and—"
"Brad! Emma!" Sue Ellen abandoned her desk chair to greet them, her warm, welcoming gaze lighting on Emma. "It is good to see you again, sweetheart." Then, turning her attention to Brad, Sue Ellen tapped her watch. "Everything is on schedule for one o'clock. I've set up the conference room for three and I made sure the video feed is ready to go so there are no glitches there."
"Thank you, Sue Ellen." Brad reached into a silver tray marked inbox, extracted a small stack of envelopes and pink sticky notes, and tilted his head toward his office door. "Emma and I have some things to go over, but if you could order in some lunch—maybe some sandwiches or pizza or something—that would be great. Oh, and if Nicholas arrives early, let me know that, too."
"Of course." Sue Ellen turned her smile back on Emma. "Enjoy your time together."
"Thank you." She smiled across her shoulder as Brad ushered her into his office and over to the chair opposite his desk. When she was situated with the bag on her lap, Emma loosened the opening, reached inside, and felt around until she found the covered bridge left on her sixteenth birthday.
"I remember when I found this one," she said, holding the gift up for Brad to see across the stack of mail he was slowly picking his way through. "It reminded me of the covered bridge on the road to Bird in Hand."
"Then I picked well." He separated the envelopes into two different piles and then sat back, tenting his fingers beneath his chin. "Ruby loved to walk down the embankment and sit on the rocks just below the bridge after a hard rain. The first few times, I figured it was just her place to think—like Mom's bench out by the pond always was for me growing up. But when I asked her about it, she said she liked to sit there and listen. She said the sound of the water rushing across the rocks in the creek bed made her feel closer to God.
"I didn't really get it until she sat me down on the grass and covered my eyes with her hands."
"I feel that way by the pond sometimes, too. When the air is perfectly still you can hear everything that is from God—butterfly wings, frogs croaking, and the birds singing." Emma turned the bridge over in her hands. "If I am upset, His sounds give me peace."
"I went there after Ruby died—after I thought you had both died." Brad separated his hands from one another and dropped them to their respective armrest. "The water had frozen, but even if it had been spring and the creek bed had been swollen from a hard rain, I'm not sure I would have heard anything over my anger."
"Anger?"
"Oh yeah."
"At me?" she whispered.
He drew back so fast his head actually thumped against his seat. "You? Why on earth would I have been angry at you?"
"Because Ruby died having me."
"No. I wasn't angry at you. Never you." Splaying his hands, palms out, he leaned forward. "You know what? Let's move on, shall we? What's next?"
She searched his face for anything to indicate his words didn't match his true feelings, but when she saw nothing, she pulled out the small, red rubber ball that had been waiting for her on Ruby's grave the day she'd turned seventeen.
"Ahhh, yes. The rubber ball and the spinner."
"This?" She pulled out the yellow spinny thing she'd gotten the following year and, at his nod, set it on the table next to the ball.
"I took Ruby to an arcade one day. We knew you were on the way and I wanted her to see some of the fun stuff I got to do as a kid. So we played Skee-Ball and all sorts of games. When we were done, we took the tickets to the counter and picked out silly stuff—the ball and the spinner being the ones we had the most fun with."
She dove her hand into the bag again, this time producing the baseball with the blue smudges from her nineteenth birthday. "Why did someone try to write on a baseball?"
"Because that's the baseball I smacked clear out of the ballpark for my team not long after I met Ruby. She came to watch me play and so I stepped up my game. Hit that ball farther than any other ball I'd ever hit. So I signed it like a professional ball player would."
She studied the ball carefully, smiling as the top of the B and the bottom half of an H suddenly made sense. "I see it now. At least a little bit."
"What's next?"
"My twentieth birthday and this dried flower with the pink and blue ribbons tied around the stem."
"That's the flower I gave Ruby after she told me about you. And since we didn't know if you were a boy or a girl, I had the florist put a pink ribbon and a blue ribbon around it."
A quick tap on the partially open door brought Brad to his feet and Emma's attention onto Sue Ellen. "Boss, Nicholas just called. He'll be pulling into the lot in about two minutes."
"We'll be ready." Stepping around his chair, he snuck another peek at the clock. "Is that lunch order going to have enough for all of us?"
"Yes."
"Perfect. Thanks, Sue Ellen." Then, turning back to Emma, he pointed at the bag. "We should be able to get in the last one before he actually gets in here."
Reaching into the bag one last time, she produced the whittled bird. "Esther has named this Emma. Because she says if she were a baby bird, I would bring her worms."
"That's how I saw Ruby being with you. Nurturing and loving—hence, the mother bird. I just believed she could do that in my world, too." He stepped around the desk, motioned her to his side, and led her out to Sue Ellen's desk in time to see a tall blond man walk through the front door with a pad of paper under one arm and some sort of silver contraption in his opposite hand.
"Nicholas, my friend! I see you're running early as always."
Grinning, Nicholas shifted the silver contraption to his left hand and extended his right for a shake Brad returned in short order. "I figured we could go over a few things before we bring in the chief." Then, turning his attention onto Emma, his jaw slacked open.
"You'd swear she was Ruby, wouldn't you?" Brad prodded.
"Seriously. Whoa . . ." Nicholas shifted his hand to Emma. "Hi, Emma, I'm Nicholas—Nicholas Forrester. I've known your dad, here, since we were two."
Not entirely sure what to say, she settled for a nod and a smile as the man took one more head to toe sweep before looking back at Brad with an even wider grin. "Lucky for her, the only thing she seems to have gotten from you are your eyes."
"Ha . . . Ha . . ." Brad sent his gaze to the ceiling, only to drop it back to Emma with a wink. "I've been taking this guy's abuse for a lot of years."
"And you're a far finer man because of it." Nicholas peeked around Brad to acknowledge Sue Ellen as she walked into the room from the direction of the back door. "Is that takeout I see in your hands?"
"It is. Sandwiches from Melly's. The delivery boy just dropped it off."
Nicholas pumped his hand in the air. "This-this is why I really should stop by and see my buddy Brad more often. He feeds me." Shifting his hand back to his pile of things, he offered a more appropriate greeting to Sue Ellen and nudged his chin toward the conference room. "I imagine we'll be in there?"
"Yes, and everything is set up."
"Speakers and video recorder good to go?"
"I tested it all this morning." Sue Ellen set the bags of food on her desk and shifted her attention to Brad. "If you guys want to get started, I'll get this stuff ready."
"Perfect. Thanks, Sue Ellen." Placing his hand on the small of Emma's back, Brad guided her toward a room off the building's front hallway.
The room, itself, was fairly small. Just enough room to hold a long rectangular table with six chairs—two on each side, and one on both ends. A single window, overlooking the road, infused only snippets of light into the room through the partially closed blind. In front of the window, pointing toward the table, was a camera mounted atop a tripod. On the wall, in multiple frames, hung the various floor plans Delia had shown her the previous day. A quick inspection, though, showed no sign of The Ruby.
"Let's put Emma here"—Nicholas directed Brad to the chair the farthest from the window—"that way we're not moving her around the table unnecessarily."
"Emma?" Brad pulled out the chair and motioned her over. When she was seated, he took the chair to her left while Nicholas took the one to her right.
"So, Emma, I'll be recording your words with this voice recorder"—he lifted the silver contraption off the table—"during this first session for my own records. Is that okay?"
Emma scanned the table and chairs before settling her sights on the tripod positioned in front of the window. "Is-is that a camera?" she asked, pointing.
"It is," Nicholas said, setting the silver contraption back down. "But it's not on. This voice recorder is fine for our chat."
She lowered her hand back to her lap. "I am told you want to talk about what they did . . ."
"I do, indeed. So, are you good with the recorder?"
She looked to Brad for approval, and, at his nod, gave one of her own.
"Okay, so let's get started." Clearing his throat, Nicholas positioned his notepad in front of him, consulted a notation on the upper right-hand corner, and then pressed a button on the silver box. "The date is February 10th and I'm here with Emma Lapp. Emma, when were you first told Brad Harper was your father?"
"I-I was not told. I knew when I saw him. When I came to his office to ask why he put things on my birth mother's grave."
Nicholas made some notions on the notepad and then moved on, his tone brisk. "How long did you know the deceased Ruby Stoltzfus had been your mother?"
"I did not know until my birthday. When I found the locket Brad left with her picture inside. When I asked Mamm later that morning, she told me about Ruby—how she had gone to be with the Lord while giving birth to me."
"Until that day, Rebeccah and Wayne Lapp had always said they were your birth parents?"
She stopped, mid-nod. "I do not remember them saying those words. I just knew them to be Mamm and Dat."
"And Ruby? How did they refer to her?"
"I knew her to be Mamm's sister, who died when she was eighteen."
Nicholas jotted something down and then returned his attention to Emma. "Were you told how she died?"
"No."
"Did you ask?"
"No! I did not want to upset Mamm. Speaking of Ruby, thinking of Ruby, visiting Ruby's grave . . . it all made Mamm sad. My birthday made Mamm sad."
Nicholas leaned forward. "Your birthday made Rebeccah Lapp sad? Why?"
"Because Ruby died on my birthday."
"So, you knew that part?"
"Yah. It is on her grave at the cemetery."
"Yet you never asked how she died?" Nicholas asked, again.
"No."
"Did the other children in the house . . . I think there's"—again Nicholas consulted his notepad—"five of them . . . ever ask how Ruby died?"
"No. Everyone wanted to see Mamm smile. Such questions would keep her sad."
"Do you think the children knew you were not their birth sibling?"
She blinked against the tears she knew were seconds away from escaping down her cheeks and willed herself to answer the question. "No."
"When Rebeccah and Wayne introduced you to people, did they refer to you as their daughter?"
Had they?
She couldn't be sure.
"They would call me Emma."
"Is that different than the way they'd introduce the others?"
"No. But we do not need to be introduced to people we already know."
Nicholas allowed a half smile. "Fair enough." He flipped to the next page in his pad and looked back up at Emma. "I want to ask you about an incident when you were seven. I understand you fell through some ice and that no adults were present at the time, is that right?"
She slanted a glance at Brad only to follow his eyes back to Nicholas. "I-I was on the way home from school with Jakob. We should not have been on the ice."
"How long were you in the water before you were rescued?"
"Only my leg fell in. Jakob helped me out with a stick, and we walked home. Mamm warmed me with blankets and scolded me."
"She scolded you?"
"Yah. We were not to be on the pond alone."
"Was Jakob scolded as well?"
"No. I was older."
"You were seven," Nicholas said.
"Jakob was six."
The man jotted some notes and then looked up at Emma once again. "Were you treated like the other children in the house?"
"Mamm did not smile at me the way she did the others. Before the locket, I thought it was because I was a reminder of a sad day. Now, I know I was not just a reminder. I am the reason."
Brad's chair creaked as he leaned forward and snapped at Nicholas to turn off the recorder. "Whoa, whoa. You need to stop saying that, Emma. You are not the reason Ruby died. Having you in a house with people who aren't trained in proper medical care is what killed her, not you. You have to know that. You need to know that."
Oh how she wanted to believe him. To know she wasn't responsible for her birth mother's death and Mamm's heartache....
"You know what?" Brad sat up and shifted his focus back to his friend. "Since we're ahead of schedule anyway, what do you say we take a break, have some of that food from Melly's, and then get back to the rest of your questions?"
Setting his pen atop his notepad, Nicholas shrugged. "Works for me. Though, really, I don't think I need to ask anything else. Emma has done a great job answering everything so far. I'm sure she'll be fine with the chief's questions. We'll just need to make sure the camera is on her face and set to record, and then we can—"
"Camera?" she echoed, looking from Nicholas to Brad and back again. "I cannot have my picture taken. I am Amish."
"Emma, please," Brad said around a moan. "This is not the time or place for that. It was this, or the police station. And Chief Wilton is a good man; his questions will be very much like the ones Nicholas just asked."
Emma braced herself against the edge of the table. "I-I cannot talk to the police! It is not the Amish way!"
Reaching forward, Brad covered her hand with his own. "I know that, kiddo. That's why Nicholas suggested doing this by way of video. We figured it would be less intimidating for you."
"But it is not the Amish way," she repeated, pulling away.
"This isn't about being Amish, Emma. This is about reporting a crime and seeing to it that justice is served."
"But that is what Bishop King will do. He will see to it that Mamm and Dat are shunned."
Brad splayed his palms. "Oh no . . . No way . . . Keeping my child from me—and me from her—for twenty-two years is due way more than a momentary snub." He grabbed her hand off her lap and held it tight, his blue eyes holding hers with such intensity she couldn't look away. "Not only did I miss out on everything—your first smile, your first rollover, your first word, your first step, your first birthday, your . . . everything—I can't recapture any of it by way of pictures or videos because the Amish don't take pictures."
"I can tell you some things."
"Some, yeah. But not everything. Not the things I should have seen with my own two eyes, not the things I should have experienced all on my own."
"But—"
"Emma, Rebeccah and Wayne stole a baby! They stole you! That's not okay. They shouldn't be allowed to walk around like they did nothing wrong, like they didn't just strip us of time we'll never get back . . . memories we missed out on making . . . milestones we didn't celebrate together! Rebeccah and Wayne belong in prison, Emma—for a long, long time!"
"Prison?" she said, pulling her hand from his grasp once again. "You-you mean jail?"
"Yes. Prison . . . Jail . . . Locked away for the rest of their lives . . ." Exhaling through pursed lips, Brad raked his fingers through his hair. "That's where kidnappers belong, Emma."
She started to stand, but a sudden bout of dizziness kept her from actually gaining any momentum. Instead, she gripped the edges of the table and waited for the room to stop spinning.
"Maybe I should give you two a few minutes?" Nicholas asked, pushing back his own chair.
Brad waved for him to stay put. "No. No. She can do this. I know she can." He draped his arm around Emma. "Emma, they're bad people, they have to be punished."
Chapter 23
She heard the crunch of his tires as he backed down the driveway, but she didn't turn. Instead, she lifted her fist to the door and knocked as hard as she could. "Miss Lottie?" she called, her voice hoarse with tears. "It is Emma. Emma Lapp. I-I need to speak—"
The door swung open, knocking her off balance and into the arms of the elderly English woman. When the woman swayed from the unexpected force, Emma jumped back, the apology she'd uttered again and again on the drive over making yet another loop past her lips.
"Shhh . . . Shhhh . . ." Miss Lottie summoned her inside enough to be able to close the door and then pulled her close. "It's okay, child. I'm here."
Seconds gave way to minutes as Emma gave into the torrent of tears she was powerless to stop. As she cried, Miss Lottie rubbed her back, telling her everything is going to be okay . . . to have faith . . . to trust the Lord.
She heard the words, knew they were supposed to comfort, but they fell short. Still, she tried her best to get her emotions under control if for no other reason than the fact her tears had soaked clear through to Miss Lottie's skin.
Stepping back, Emma wiped the back of her hand across her cheeks, her gaze slipping to the floor. "I am sorry, Miss Lottie. I did not mean to make you all wet like I did."
"Oh, child, my dress is fine. Nothing a sit near the fireplace won't fix." Hooking a finger beneath Emma's chin, Miss Lottie searched her eyes before abandoning them in favor of a full head to toe inspection. "Are you hurt?"
"No."
"Your parents? The children? Are they hurt?"
Emma squeezed her eyes closed against the faces she'd been unable to shake from her thoughts since the meeting with Nicholas and eked out another no.
"Then everything else is small potatoes—remember that." Miss Lottie reclaimed her cane from its resting spot beside the door and led Emma down the very hallway she'd walked just three days earlier.
The living room looked very different in the limited winter sunlight streaming in from the windows off the back of the house. The welcoming feel was still there, but something about the change in lighting and the distant tap-tap of a hammer made Emma feel fidgety and ill at ease. "I should not be here, interrupting your day. I-I should go."
"I was knitting, dear. I can knit while you talk, if I want to. But I don't. I'd much rather visit with you." Miss Lottie pointed Emma to the same couch she'd occupied on Sunday evening and then claimed her armchair. "Let's save that piece of pie I owe you until after we talk. I get the sense that's what you need most right now, anyway."
Emma nodded, pulled the nearest throw pillow onto her lap, and exhaled a wobbly breath. "He wants them to go to jail."
Miss Lottie's brow arched above the rim of her glasses. "He?"
"Brad. My birth father."
"I see. And by them, you mean Rebeccah and Wayne?" She swallowed, hard. "Yah. He says they kidnapped me and that kidnappers belong in jail. His friend Nicholas asked me many questions today. He wrote some things down on paper, and he pressed a button, too."
"A button?"
"It was on a small box. It was silver. He pressed the button when we talked, and he pressed it again when we did not."
"So, he was recording you . . ."
"Yah. That is what he said." She startled at the snap of a log in the hearth, the pillow tumbling off her lap and onto the floor. Two rapid apologies and one grab later, it was back on her lap. "He asked things—about Mamm and Dat. Things they said to me when I was little, if they were good to me, and many questions like that. I did not mind answering his questions. I-I thought he just wanted to know, like Brad—my birth father—did. I didn't understand why Nicholas wanted to record what I said, but Brad said it was okay."
"Go on, child."
"It was time to eat, so Nicholas stopped asking questions. That is when Brad mentioned the police. That the policeman would ask me questions after we ate. I said I did not want to talk to the police—that it is not the Amish way—but Brad said it was important. He said Mamm and Dat committed a crime when they kept me and did not tell him I had been born. And that is when he said they were to . . . to . . ." She stopped, wiped a fresh round of tears from her cheeks, and made herself breathe in and out until she could continue speaking. "That is when he said they were to go to jail."
Miss Lottie sat up tall, her wrinkled hands gripping the armrests of her chair. "Did the police arrest them?"
Emma drew back, horrified. "Arrest them? No! Why would they do that?"
"Because that's what will have to happen in order for them to go to jail."
"No! I-I did not talk to the police. I ran outside. I ran down the road. I was almost to the end of the first road when my birth father stopped in his truck. I told him I did not want to talk to the police. I told him I couldn't."
"And what did he say?"
She traveled her thoughts back to the memory of Brad's eyes, hooded and tired, looking back at her through the open passenger side window, the disappointment they'd conveyed leaving her more confused than ever. But when she'd asked to be taken there, to Miss Lottie's, he'd grudgingly consented.
The sound of her name on Miss Lottie's tongue yanked her back into the room in time to hear the woman's question repeated.
"He said he'd give me a little time to get used to the idea, and then we'd try again with Nicholas and the policeman." Pushing the pillow back onto the couch, Emma stood, then sat, then stood again. "I told him I would leave the Amish . . . That we would have new days to spend together . . . That it is okay if he does not have pictures of me as a little one because I can tell him things I did. I told him Bishop King would shun Mamm and Dat, that having backs turned to them was enough. But Brad does not think it is enough. He said they had no right, that they robbed him of me! He says it must be jail."
"I see. . . ."
"But Esther is too little to go to jail, Miss Lottie! Jonathan and Annie, too! Soon Sarah will be baptized, and Jakob? It won't be long until he is courting. They cannot go to jail."
"They wouldn't."
Emma sidestepped her way between the coffee table and the couch to claim a seat closer to the elderly woman. "You do not think Mamm and Dat will go to jail as Brad says?"
"No, if he pursues this, they likely will. But if your mamm and dat go to jail, the children will not."
"But that is where Mamm and Dat would be."
Miss Lottie leaned forward, captured Emma's hands inside her own, and shook her head. "If your parents go to jail, the children will stay behind."
"Stay behind? You mean at the house? But how? Mamm and Dat would not be there. . . ."
"How old is Jakob now, dear?" Miss Lottie asked. "Twenty?"
"He is twenty-one."
"Then I suppose he could petition the court to take responsibility for the others, but it would likely be a hard sell. Esther is still so young.... Do your parents have kin still in the area? Siblings? Parents? Anyone?"
"Two of Dat's brothers live in upstate New York. One of his sisters is in Shipshewana, the other in Holmes County. Mamm's brothers are scattered around, too. The closest one, Jeb, lives in the western part of the state with his wife and their eight children."
"Eight children?" Miss Lottie tsked softly beneath her breath. "So, there is no one here? In Blue Ball?"
"Not kin, no."
"So they'd have to move in all of those cases."
Emma looked down at her hands inside Miss Lottie's and quietly pulled them away, their sudden dampness necessitating a wipe on the sides of her dress. "I don't understand. Who would have to move?"
"Your siblings. They will need someone to go to if Jakob can't take them. Somewhere that will satisfy child services."
"Child services?"
"That's who will place them into foster care if need be."
This time, when Emma jumped up, she hit the edge of the coffee table with her shin, the quick stab of pain barely noticeable against the roar in her head. "That cannot happen. They belong with Mamm and Dat," she protested.
"I agree, but if your mamm and dat are in prison, someone will have to care for them, Emma. I'm not sure how successful a twenty-one-year-old would be in getting custody of four younger siblings."
"But . . ." Emma strode over to the window and its view of her brethren's fields in the distance, the stark browns of a winter's earth chilling her from the inside, out. "What should I do, Miss Lottie?"
"Do you want them to go to jail, child?"
"No. Of course not."
"Is that just because of the children?"
Resting her forehead against the glass, she considered the woman's question, her anger leapfrogging with . . . sadness? "No."
"Why?"
"I-I do not know."
"Perhaps, when you discover the reason for that answer, you will know what you must do."
"What happens if I don't know what to do?" she asked, turning around.
Miss Lottie waved her back to the couch. "You will. In time."
"But I don't have time," Emma protested. "Brad said he will give me a few days to think, but if I still cannot talk to the police, he will talk to them, himself."
Plucking her glasses from the bridge of her nose, Miss Lottie rubbed her worried eyes. "Then use those few days wisely, child."
"What do you mean?"
"What did you do when you found out about Ruby and Brad?" Miss Lottie asked, sliding her glasses back into place.
"I found him."
"And then?"
"I have spent time getting to know him and my grandmother. There is much to learn. But I am learning a little, and they are learning a little."
"I imagine you ask them questions? And they, you?"
"Yah. Many. There is much to learn."
"Have you asked your mamm why she did what she did? Why she didn't tell you about Ruby and Brad?"
"There is nothing to ask! There is no reason to do what she did!" The second she spoke, she dipped her head in shame. "Miss Lottie, I am sorry. I do not mean to speak that way to you."
"You're angry, child. You're also human. But so, too, is your mamm. Remember that."
Emma snapped her eyes back to Miss Lottie. "She didn't tell me Ruby was my real mamm! She had Dat tell Brad I did not live! For twenty-two years, I didn't know, and Brad didn't know! For twenty-two years, I didn't know why I couldn't make Mamm smile the way the others could!"
The hurt, the anger, and the confusion were back, only this time, instead of manifesting themselves in tears, alone, they claimed her voice, too, thickening it until her words were little more than rasped breaths, hitched out between sobs. "How? How could she do that, Miss Lottie? How could she do that to me?"
"There is only one person who can tell you that, dear. But you have to ask . . . and then you must listen. It is the only way you will have the answers you need to do what is right. For you."
Chapter 24
She was sitting by the kittens when she heard Mamm's footsteps, a sound she'd once welcomed as an assurance of safety. Yet, now, after everything, anger and dread were all she felt. Anger over the lies. Dread at the thought that Brad was right and Mamm and Dat belonged in jail.
"Sarah said you were out here." Closing the straw-covered gap with tentative steps, Rebeccah Lapp reached down and ran her calloused fingers across first Bean, and then each of her babies. "Esther talks most about this one," she said, stalling her hand on the brown and white mound lying farthest from Bean. "She has named it Flower because—"
"When she was first born and her eyes were closed real tight, it reminded Esther of your flowers when they shoot up out of the ground, and that when they finally opened, they'd be warm and sunny like your flowers always are." Bracing her hand against the side of the stall, Emma stood, her gaze on everything in the barn but Mamm. "I know. Esther tells me things, too. She loves me."
"Of course she loves you, Emma. You are her big sister." Rebeccah straightened to a full stand, her step forward toward Emma quickly negated by Emma's step back. "I have tried to give you space the past two weeks, to let you learn what you need to learn and—"
Emma's humorless laugh earned her more than a few curious glances from Dat's horses. "Learn what I need to learn? You mean to learn about my real dat and my real grossmudder? My aunt and my cousins? The people I knew nothing about until I found the locket Brad left for me at Ruby's grave? Those people?"
"Emma, please."
"You had Dat tell him I died!"
Rebeccah started to speak, only to stop and bow her head in shame.
"How could you do such a thing?" Emma yelled. "How could you tell such a lie? He wanted me! He wanted a life with Ruby and me! And you? You did not even love me enough to smile at me!"
Whipping her head up, Rebeccah stumbled backward, the thump of her back against the half wall startling Jakob's horse. "Did not love you? Emma, why would you say such a thing?"
"Because it is true. It has been that way for as long as I can remember!"
"Emma! That's not—"
"I remember when I was younger than Esther is now, wanting so much for you to smile at me the way you did Jakob, but you did not. Soon, you smiled at Sarah, and then Jonathan, and then Annie, and then Esther. But never me.
"For so many years, too many years, I thought it was because I had come into the world on the same day you lost your sister—that my birthday, my birth reminded you of a sad day and that is why you could not smile at me the same. But that is not all it was, was it? It was not because I was a reminder. . . . It is because I was—I am the reason, aren't I?"
Rebeccah flew a trembling hand to her mouth only to let it slip down her chin in horror. "Ruby didn't die because of you. She died because of that boy!"
"You mean my father?" Emma countered, her voice shrill.
"Because of him, Ruby was in a family way when she wasn't married!"
Emma turned, her hands clenching her hips. "I am not here because of Brad alone! Your sister was part of it!"
Pain skittered across Rebeccah's face, pushing her back a step. "Ruby was young.... She made mistakes. . . ."
"Mistakes?" Emma echoed. "You mean me?"
"You should have been born to a Mamm and a Dat."
"You are right, I should have been! But you and Dat did not let that happen. You made decisions about my life that were not yours to make," she rasped. "And why? So I could wonder why you never smiled at me?"
Rebeccah stepped forward, and again, Emma stepped backward, maintaining the distance between them. "I smiled at you, Emma."
"This is not a smile." She emulated the wobbly smile she'd seen on occasion. "People do not have tears in their eyes when they are happy!"
The smile she'd just tried to demonstrate flashed across Rebeccah's face. The tears that sprang to her eyes, however, remained as she matched each of Emma's steps until the back wall of the barn eliminated the distance once and for all.
Extending both her arms, Rebeccah bookended Emma's shoulders with her hands and waited until Emma's eyes were on hers. "I smiled right here"—she touched her chest—"every time I looked at you, Emma. But if it did not always show it was because I was afraid this day would come."
"This day?"
"The day I might lose you."
Feeling her breath begin to saw in and out with the kind of emotions she didn't know how to process, Emma closed her eyes, only to open them at the feel of Mamm's hand on her cheek. "Emma, you are not the reason Ruby died. And you are not the reason I was sad. Ruby died because something was not right with her heart. If I am sad when I think of her, it is because she was my sister . . . and because she never got to know you, or hold you the way I have. And Dat? He held you even before I did. When I was tending to Ruby, he held you as his own."
"But I wasn't his own. I was Brad's!" She wriggled free of Rebeccah's hand and stepped around her to return to Bean and the kittens. Bean, clearly aware of their presence, was licking her babies while keeping a wary eye on the activity happening just beyond their makeshift bed. Emma wiggled her finger at Bean and then spun around as Mamm began to speak again.
"You are very much like her, you know. Having you has been like having a little bit of her still here. The same sweetness, the same joy, the same love of home and family. Sometimes, when you are baking something in the kitchen, your smile is so like hers—so full of joy and excitement. And the way you are with Esther? Ruby was like that with your uncle Jeb when he was little—always kind, always patient. She would have made a good mamm."
Emma shifted from foot to foot. "Perhaps, with her, I would have fit."
"Fit?"
"I have never fit, anywhere," she said around the tightening in her throat. "Not here, not at school, not at hymn sings . . ."
"Emma, you have always fit here. Always. And at school, the children did not refrain from playing games with you because they did not like you. They didn't play with you because you would stay inside helping the teacher. When you finally went outside, it was almost time to stop playing."
"How-how do you know?" she stammered. "You . . . you weren't there."
"Because I spoke with your teacher. That is why I gave you cookies to bring, so you would go outside sooner. But you didn't. You stayed inside and put a cookie on every desk while the children played. You did not want to make those who did not bring cookies to share feel poorly." Step by step, Rebeccah made her way back to Emma. "And the hymn sings? I think you fit better than you realize."
"You cannot know that."
"Perhaps you are right. But I know you, Emma, and I know you are quiet. I know you can get lost watching a butterfly or a frog. I know you look to see the smiles that come when people try your cookies and cakes. I know whenever you saw a frown or a hint of sadness on anyone's face—even someone on the other side of a room—you wanted to fix it, as if you were the reason they were not happy and—"
Rebeccah pressed her hand to her lips as a single tear rolled down her cheek. "Oh, Emma . . . I did that to you, didn't I? I-I made you think that anything short of laughter was your fault."
"No, Mamm, I—" And then she stopped. She was doing exactly what Mamm said. "Maybe. I do not know."
Lowering herself to the ground beside Bean, Emma ran a soothing hand across Flower's back and waited for the repetitive motion to calm her thoughts and her breath enough to continue. When she was fairly certain any lingering shake wouldn't manifest itself in her voice, she looked back up at Mamm.
"You will be shunned for keeping me," she whispered. "Maybe even worse."
"Bishop King knows, Emma. He has always known."
"The bishop has . . . known?" At Mamm's nod, she pulled her knees to her chest and tried her best to make sense of everything. Something was missing. Something—"Why didn't you tell me about them?" she asked. "Why did you let Brad think I'd died with her?"
"Because Ruby wanted you to be raised Amish. It was her wish."
A swell of renewed anger pushed her legs back to the ground with an audible thud. "I don't believe that!" Emma countered. "I have seen pictures of my real mamm and dat together. She loved him!"
"Maybe you are right. I don't know. She did not speak of him with me until you were on the way. The only thing I can tell you for sure is that she wanted to raise you in the Amish way."
"How? How can you know that?" she demanded. "Ruby died!"
Rebeccah's shoulders lifted in a pained shrug. "She chose baptism just one month before you were born."
"She chose . . ." A deafening roar filled her ears, making it difficult to think let alone speak. But still, she tried, skipping ahead to the only part that mattered. "But then, she couldn't be with my father."
"It was her choice, Emma. For her and for you."
* * *
The snap of a twig from somewhere off to her left stole her attention from the afternoon sun shimmering atop the pond and sent it skittering toward the man now picking his way around old branches and stumps for the meeting she'd walked a quarter of a mile to request.
Much like Esther needed all the pieces of her wooden puzzle to create a farm, Emma needed all the pieces of the past in order to know her future. And from what Emma could tell, Brad held the second-to-last piece.
"Sorry your call went to my voicemail, kiddo. I tried you back as soon as I got out of the shower and saw that you had called, but the phone just rang and rang."
"I did not stay." Emma slid the drawstring bag into the center of the rock to clear a spot for Brad to sit, and, when he did, she looked back over the water once again. "I am sorry I got so upset yesterday."
"You've been through a lot. We all have. And I probably should have warned you about the video call with the police chief sooner than I did, but I guess I wanted to try and spare you as much of the nasty stuff as possible."
"You mean like sending Mamm and Dat to jail?"
"That's—wait . . . Does that mean you're ready to talk to the chief?" he asked.
"No." The sound of his weighted exhale propelled her gaze back to the pond long enough to gather her breath and her courage. "I did not know Ruby chose to be baptized. That she wanted to raise me in the Amish way."
"Okay . . ."
"Why?"
"I don't know, I guess you can only cram so much into two weeks."
She took one last look at the pond and then turned so the only thing in front of her was Brad. "I mean why did she choose to be baptized when she knew I was coming? When she knew baptism would mean she could not share a life with you?"
"She didn't want to lose Rebeccah and Jeb and her parents."
"But she wouldn't have. Ruby hadn't been baptized when she learned I was coming. She could have lived a life with you and still spent time with her family."
Brad pushed off the rock, picked a small stick up off the ground, and snapped it against the edge of the rock. "That is what I told her. Time and time again. I talked about the company we wanted to open, the floor plans she could design while you slept, and the house I would build for us as soon as I could.
"Those first months after we found out she was pregnant, she seemed all onboard with everything. She even drew up the house she wanted me to build. It was much plainer than the other ones she'd drawn, so I told her to draw it again. I told her to think big, think fancy, and think splashy because our baby was going to grow up having the very best in life—the best toys, the best schooling, the best clothes, best car in the high school parking lot, et cetera."
She made herself nod as if she understood, but she didn't.
"I would have raised you to be a princess, Emma. And I would have treated Ruby like a queen."
A princess . . .
A queen . . .
"But she didn't draw it again, did she?" It was a silly question considering Emma already knew the answer. She'd seen the drawing. Seen the large front room Ruby had wanted yet Brad had insisted on dividing . . .
"No, she didn't. So I just figured I'd change it when I built it. But I never got the chance to do that because, a few weeks later, she told me she wanted to be baptized and to raise you Amish." He tossed the stick a few yards and then raked his now-free hand through his hair. "And that is when she told me we were over. That her decision meant we could not be together.
"We argued. Or, rather, I argued. I reminded her of all the things she could have and be in the English world. I even reminded her that different doesn't mean bad. But it didn't matter. She didn't want any of that. She wanted better for you, she said."
He flicked his hand in Emma's direction. "As if an eighth-grade education and being out of touch with the here and now is better somehow."
She sat with his words for a moment, letting them roll around in her thoughts. "When Ruby made an Amish picnic for you, did you enjoy it?" she finally asked.
"Very much. The food was great."
"Was that the only date she planned?"
"No. We came here, many times. She taught me to skip rocks; how to catch a butterfly so it would still fly when I let go. And that field of dandelions I told you about? We found that when we were out in this area, just walking around."
"Did you enjoy those things?"
He shrugged his assent.
"The Amish see the big and fancy," she said, quietly. "It is all around us. It cannot be pretended away. It is just not the way the Amish choose to live their lives. You say different is not bad, and you are right, I am sure. But simple, as Ruby showed you, is not bad, either. I have had a good life so far. A happy life. I had warm food in my stomach, a bed to rest my head, and family who love me."
"Love?" he spat. "You think keeping me from you is love?"
"I think trying to honor Ruby's wishes for my life was love."
"Ruby was dead, Emma."
"Not in Mamm's heart, she wasn't."
"Nor in mine. But Rebeccah has had a living, breathing link to Ruby in you for twenty-two years." He lowered himself back down to the rock and, ever so gently, tucked a stray wisp of hair back inside Emma's kapp. "Now it's my turn."
"Please don't put them in jail," she whispered. "For me. Please."
Chapter 25
For the second time in less than three weeks, Emma made her way toward the simple white farmhouse with the wide front porch. A quick peek in the barn as she passed revealed little more than a few curious horses, a sleeping barn cat, and an old buggy wheel in need of repair. The tap-tap of a hammer from somewhere just beyond the barn piqued her curiosity, but, still, she kept walking, her need to figure out the final piece of the puzzle leading her to the one person who'd always understood her, even when she didn't always understand herself.
Mary Fisher had been her friend since the beginning, enabling them to carry on a tradition started by their mamms. They'd chased each other around the pond when they were toddlers, sat beside each other as they learned to read, raced each other to the part of their post-school-day walk that required one to go left and one to go straight, and kept each other's innermost secrets. But perhaps, more than all that, Mary had a way of seeing things Emma could not always see.
Taking the porch steps two at a time, Emma fast walked across the wood planking to the dark green door she'd helped Mary paint the previous summer. In spite of the heaviness in her heart, the memory of Mary's howl when Emma accidentally painted her toe brought a fleeting and oh-so-needed smile to her lips. The raw day would prevent them from sitting outside today, but an empty stall in the equally empty barn would certainly work, too—
"Emma? Is that you?" At the familiar voice and the equally familiar flapping it always seemed to kick off inside her stomach, Emma stepped out from behind the upright and waved at the handsome twenty-four-year-old heading in her direction with a hammer in one hand and a level in the other. "Dat left with Mamm and the girls in the buggy not more than twenty minutes ago."
She tried to hide her answering slump, but if the way Levi's eyebrows dipped with worry less than a blink later was any indication, she'd failed. Miserably. Before she could come up with something to placate him, though, he settled himself on the second-to-last step and patted Emma over to the top one. "I know I am not Mary, but I am good at listening, too. Maybe even better."
Her laugh stirred a matching one from Levi as she heeded his invitation. "I think it is good that Mary cannot hear you say such a thing."
"I think you are right." He brushed a piece of straw from his pants leg and then swiveled on the step so Emma was his view. "Have you decided what you will do?"
And just like that, any residual laughter on her part ceased, wiped away by the tug-of-war that had become her life. "I thought I had . . . before I talked to Mamm."
Something sparked behind Levi's eyes. "So, you will stay?"
"I know I should be able to answer such a question, but what seemed so easy two days ago, is not easy now. Miss Lottie said not to make a decision in anger. But without anger, there is only"—she looked out at the barn—"confusion."
"I am listening."
And so she told him. She told him about Brad's steadfast belief that Mamm and Dat belonged in jail. She told him how Miss Lottie convinced her it was time to talk to Mamm. She told him how Mamm's lack of smiles toward Emma over the years had both nothing and everything to do with Emma. She told him Ruby had chosen to raise Emma in the Amish way and how she'd confronted Brad with that information. And last but not least, she'd told him how Brad had finally, finally relented on the notion of jail provided Mamm and Dat didn't interfere in his relationship with Emma ever again.
His slow, thoughtful nod when she got to the end let her know he'd been listening. The quick touch of his hand on hers let her know he cared even if her answering gasp made one of the barn cats rethink his approach and scurry behind a bush, instead.
"It is good that you know these things," Levi said, catching and holding her gaze with his. "It is when you know things, you can decide things."
"Three days ago, I wanted to punish Mamm for keeping me from Brad. He is my only living birth parent and I should know him. But now that I know the truth about everything, I see that Mamm was doing what Ruby wanted her to do. And me? I am a link to Ruby for Mamm, and a link to Ruby for Brad. But I cannot be both, just as I cannot be both English and Amish."
Again, he nodded. And since his hand had never left hers, he simply tightened his grip. "Whatever world you choose, Emma, I will choose it, too."
"Whatever world I . . ." She looked from Levi, to his hand on hers, and back to Levi. "What are you saying, Levi? What do you choose?"
"I choose you and me. To be together."
"Together?" she echoed.
"Yah."
She stared at him, waiting for some outward sign he was teasing, but there was none. Just a tender smile that was trained solely on Emma. "But-but I'm not Liddy Mast!"
"You're right. You are Emma Lapp."
"I know but—"
"You are Emma Lapp," he repeated.
"But I'm not sure what that means.... Who I am, anymore. . ."
Levi quieted her words with a gentle squeeze. "You are still the same person you have always been, Emma. You are kind. You are sensitive. You are caring. You are good at volleyball and baking cookies. You are a fine sister, a fine daughter, and—"
"How can I be a fine daughter when I am so confused?"
"You are a fine daughter because you are confused," he said, his voice thick.
She stared at him. "That does not make any sense."
"You have taken time to get to know your birth father, yah?"
Emma nodded. "Yah. I have learned many things, but there is much more to learn."
"And your mamm?" Levi asked. "Have you learned things about her?"
"Do you mean Ruby or . . ." She stopped, swallowed, and steadied her voice. "Mamm?"
"Both, I guess."
She considered Levi's question. "Yah."
"And?"
"I love them both. Brad and Dat, too."
"Then that's the only real difference I see about you, Emma. You have more people to love, and more people to love you now."
* * *
Emma ran her fingers across the back of the now-closed photograph album and looked up at Brad and Delia, the love in their eyes making her smile tremble even more. "Thank you for letting me look at the rest of these pictures. They help me to see Ruby in a way I never could have without them."
"I'm glad, dear." Delia rubbed Emma's back in smooth, even circles. "She was happy with your father. Very happy."
She could see how they thought that. Ruby's smile in each and every picture was proof. It was also proof that the decision Emma had come to was the right one. For Emma.
"Mom and I talked about it and we know transitioning from an Amish life to an English life is going to take some time. We know you'll make it fine, but we also know it will be filled with unknowns for a while. So that's why we thought maybe it would be best if you and I move in here, with Mom, until you get more comfortable. Then, and only then, we can move to my place—our place."
"Or just stay here," Delia added, resting her cheek against Emma's shoulder. "I certainly have the room and the books to keep you busy."
Emma let her answering laugh accompany her gaze as she took in the cozy sitting room that had made her feel at home on her very first visit to Delia's home.
* The floor-to-ceiling bookshelves filled with more books than the English grocer . . .
* The photographs of Brad as a baby and a young man . . .
* The happy little knickknacks she'd come to know the origin of thanks to the warm and welcoming woman sitting beside her . . .
* The window with its view of the pond Ruby had skated on . . .
* The mantel with the framed picture of her birth parents, together and smiling . . .
Somehow, Emma could see the room as it might look in two years, five years. The pictures and the books would still be the same, but in her mind's eye there would be new things on the shelf, too. Perhaps a framed picture drawn by one of her own children . . . The skates she hoped to own one day lying beside the window . . . Her husband seated beside her on the couch while Brad added a log to the hearth . . . A plate with Delia's pastries and her own bread sitting atop the coffee table . . .
"If there is something you want to change or add, we can do that. This is your home, too, Emma." With the gentlest of fingers, Brad turned Emma's chin until he was the only thing she saw. "We want you to feel as if you fit here—with us. Always."
Just for a moment, as she stared into the eyes she'd yearned to see her whole life, she wished someone would take a picture. But as quick as the thought came, it disappeared. She didn't need a photo album to remember this moment. This man, and his mother, were part of her life to stay.
"Emma? Did you hear me? We want you to know that you fit here. . . ."
She found Brad's hand with her left and Delia's hand with her right and squeezed both. "I know. And I do. But I also fit in Blue Ball. With Mamm, Dat, and the children. And with Levi."
"Levi?" Brad echoed.
"He has asked to court me and I have said yes."
"To court you? As in the Amish tradition . . ."
"Yah."
Brad's eyes left Emma for Delia, only to return with a hint of anger. "Emma, I told you if Rebeccah and Wayne interfered in any way, I will not be able to honor my promise to you."
"This is not about them—not in the way you mean, anyway. Levi was willing to leave his vows to be English with me if that is what I wanted," she said.
"Then do it!" Brad said. "He can come work with me!"
"That is what Levi said, too."
"Good! And you can do what you love, too. You can open a restaurant and people will come from miles to eat what you make!"
The image his words created in her thoughts quickly bowed to another, better one—one responsible for the smile she felt tugging at the corners of her mouth. "I am counting on that."
"Then I don't understand. . . ."
"I do not want to leave my Amish ways, and I do not want to leave my family."
"We're your family, Emma," Brad protested. "Your real family."
"It is nice to look at you and see my eyes, and my same hair. It is nice to look at pictures of Ruby and see my chin and my nose. But Mamm and Dat? And the children? They have made me who I am, too." She looked from Brad to Delia and back again, the love she felt for them setting off a stream of tears she didn't bother to wipe away. "I need all of you in my life—in the simple life Ruby wanted for me and for herself."
"Ruby's choice doesn't have to be your choice, Emma."
She smiled at her birth father. "You are right, it doesn't. But I'm not making this choice because of Ruby. I'm making this choice for me. For my life. I do not need many tables full. I need only one table full—my own."
"But you haven't given me a chance to show you what it can be like here. . . ."
"I don't need you to. I know my life. It is like it was with Ruby. Her smiles in your English world were different than her smiles in simpler times. She smiled here, by your pond, but her smile with the bubbles and the dandelion? They were bigger, happier. Because that is where she fit best. I know I have said I didn't fit in Blue Ball, but that is because I was looking to others instead of inside to my own heart."
"But I just found you, Emma," Brad pleaded. "I don't want to lose you again. I can't lose you again."
"You won't, you can't. I am your daughter." She released her hold on his hand to wipe the tears from his eyes. "And you, Brad Harper, are my dad. Forever and always."
Eleven Months Later
"You are not peeking, are you?"
Emma stopped moving and turned her temporarily sightless eyes in the direction of her husband's voice. "How can I peek if your hands are covering my eyes?"
"I don't know. Your dad told me to be sure you cannot see."
She tried to make a face, but when her mouth was determined to smile as it was at that moment, there was little she could do to make it stop. Three months earlier, in front of God and their families, she'd become Emma Fisher. It had been a surprisingly warm day for late October, a fact Delia hailed as Ruby's part in the special day.
Emma knew she wasn't to think such things, but still, during quiet moments alone, she couldn't help but believe Delia was right. After all, Emma had been able to have what Ruby couldn't—a family that knew no bounds. A family where she had both a dat and a dad.
"Are you ready?" Levi called.
"Yah. I am ready. And I am right here, next to you, so you do not have to be so loud."
Levi's laugh filled her ears. "I was not asking you."
She drew back into Levi's chest, but his hands remained firmly in place. "Who? Who else is here?" she asked.
In lieu of an answer, Levi dropped his hands to a ta-da she recognized as belonging to her father. "Dad? Where . . ." The words drifted away as her uncovered eyes came to rest on the two-story home no more than three buggy lengths away.
She took in the three front steps, the wraparound porch with the view of the Amish countryside on one side, the driveway on the front, and a small pond on the other. She took in the flower boxes on the first floor, the dark green shades of the second-floor windows, and the man standing next to it all with a tool belt around his waist and the smile she could never get enough of seeing.
"Dad?"
"It was a little too big to leave by the grave, but what do you think?" Brad Harper asked, stepping forward.
She rubbed her eyes, took in the house again, and then locked gazes with the man slowly closing the gap between them. "This . . . this is The Ruby, yah?"
"It is, indeed."
"But you have never built one."
"I have now." Brad clapped a hand on Levi's suspender-clad shoulder. "With Levi's help, of course."
She looked from Brad, to Levi, and back to the house, her mind's eye skipping ahead to the inside she'd yet to see. "Is . . . is it the way she drew it inside, too?"
"Why don't we go inside and you can tell me if it's the same or not."
For the briefest of moments, she couldn't move, the notion of stepping inside almost more than she could handle. But it didn't last long. Soon, she was fast walking across the earthen driveway, up the porch steps, and through the front door.
Two steps in, she froze.
Somehow, someway, Levi and her father had managed to take a simple pencil sketch and transform it into something with walls and paint and . . . life. Stunned, Emma inched forward, her eyes scanning the room for the simple details that had been so important to her birth mother all those years ago.
* The fireplace to her left . . .
* The large windows overlooking the Amish countryside...
* The accordion divider tucked into the wall to accommodate church . . .
"It's Ruby's house," she whispered, looking from Brad to Levi and back again. "You built Ruby's house."
"Ruby designed it, and Levi and I built it, but it's your house . . . yours and Levi's," Brad said, grinning.
"Ours?" she echoed. "But—"
"Happy birthday, Emma."
Bookending her face with her hands, she took everything in again, the joy bubbling up inside her making it difficult to breathe. "I do not know what to say."
Brad laughed. "Then look first, talk second."
"I am looking.... It is . . . It is wonderful."
"There is more to see, kiddo."
"More?"
"Surely you need a place to make all that amazing food, don't you?"
She sucked in her breath. "Is it the kitchen Ruby drew?"
"Why don't you see for yourself?"
In need of no directions, Emma crossed the living room to the small linking hallway she knew would take her to the kitchen. Sure enough, as she stepped inside, her eyes moved immediately to the large window Ruby had envisioned as the perfect napping spot for her infant.
She tried to imagine the young girl she'd seen so many times in photographs, tucking Emma's infant self into a sun-drenched wooden cradle and showering her chubby cheeks with sweet kisses....
"I think your mother would be very pleased to know that one day soon, our grandchild will be sleeping in that very same spot," Brad said, his voice thick with emotion. "God willing, of course."
She stole a glance in Levi's direction long enough to trade knowing smiles before turning back to the window and the slightly different image that was now just a little less than seven months away—
Something that sounded a lot like an Esther giggle floated into the room and sent her attention skittering toward a pile of brightly wrapped boxes stacked atop the large center island. Before she could fully process the sight or formulate anything resembling a question, a flurry of faces entered the room from the door Ruby had marked pantry so many years earlier.
* Dat . . .
* Jakob . . .
* Sarah . . .
* Jonathan . . .
* Annie . . .
* Esther . . .
* Delia . . .
* Mary . . .
* Levi's Mamm and Dat . . .
* Miss Lottie . . .
Stunned, she stumbled back into Levi's waiting arms. "What . . . what is this? Why are you all here?"
As if one, all eyes, including Emma's, turned back toward the pantry door as Mamm stepped her way through the crowd of Emma's loved ones with a birthday cake in her hands and a smile as bright as the sun on her face.
"Happy birthday, Emma."
ACKNOWLEDGMENTS
Writing is, by its very nature, a solitary act. I spend months sitting in front of my computer screen, losing myself in my characters' worlds. Still, there are some people who make the journey to a book's completion all the more fun for me.
That said, I'd like to thank my friend Tasha Alexander. The nugget for this story came while visiting her in the most peaceful place I've ever visited. Being able to play the "what-if" game with her for a few hours that same day made it all the more fun.
A huge thank-you also goes to my family for their patience, understanding, and willingness to eat leftovers while I tapped away on this book.
And, finally, I must thank you, my readers. Your kind emails and enthusiasm for my books keep me doing what I'm doing.
A READING GROUP GUIDE
A
DAUGHTER'S
TRUTH
Laura Bradford
ABOUT THIS GUIDE
The suggested questions are included
to enhance your group's reading of
Laura Bradford's A Daughter's Truth.
DISCUSSION QUESTIONS
**1.** In the blink of an eye, Emma's world changes. Not because of her own choices, but because of the choices made by those around her. Have you ever had your life significantly impacted because of the choices of others? Can you share?
**2.** While talking to Emma the first time, Miss Lottie references a sign she saw in a San Francisco outdoor market that said, "Home is not a place, it's a feeling." It's a sentiment that eventually leads the English woman back to her childhood roots in Amish Country.
Is there somewhere (besides your physical house) that always feels like "home" to you? Where you feel the most like yourself? Where?
**3.** Levi's advice to Emma when she's on the brink of leaving everything behind is to "think before you do." If you could give one piece of advice about life to someone, what would it be?
**4.** All her life, Emma has yearned to see herself in someone. For Emma, it's about something tangible she can see. Whom do you most resemble, appearance-wise, in your family? Do your interests/abilities resemble anyone's?
**5.** Mary has always been Emma's safe harbor—the person who has been by her side every step of the way, and truly knows Emma's heart without Emma having to say a word. Do you have a friend like that? How long have you known her/him and how did you meet?
**6.** What do you think was the turning point for Emma—the moment or series of moments that helped her choose the right life for herself?
**7.** Do you think Emma makes the right decision in the end? Why/why not?
Photo: © 2018 Avery Brunkus
ABOUT THE AUTHOR
Laura Bradford is the author of the women's fiction novel Portrait of a Sister and the national bestselling Amish Mysteries. Laura lives in Mohegan Lake, New York. Visit her website at laurabradford.com.
Katie Beiler was always the follower to her twin sister Hannah's lead. That is until Hannah left their Amish upbringing for an English life—leaving Katie to find her own footing in a world that no longer looks as it once did . . . .
Katie has always imagined her life being just like Mamm's. It's why she chose baptism and why she'll soon marry Abram Zook. But ever since Hannah left, the only thing that truly makes Katie smile is the sketch pad in which she indulges her talent for drawing faces—a sin that, if discovered, could get her shunned by her family, her friends, and even Abram. Yet Katie sees her secret pastime as the only way to quiet a growing restlessness she'd just as soon ignore. That is until their Mamm's untimely death brings Hannah back home to Pennsylvania, with a new outlook on life, a man she adores, and, soon, an invitation for Katie to visit her in New York City.
Suddenly, Katie is experiencing a freedom she's never had, in a world she never imagined. She's also spending time in the company of a fellow dreamer, someone who sees her as strong and brave and makes her laugh. But it's when Hannah shows Katie's drawings to a gallery owner that she truly finds herself at a crossroads between the only life she's ever known and the powerful lure of an unfamiliar future.
# Contents
1. Also by
2. Title Page
3. Table of Contents
4. Copyright Page
5. Dedication
6. Chapter 1
7. Chapter 2
8. Chapter 3
9. Chapter 4
10. Chapter 5
11. Chapter 6
12. Chapter 7
13. Chapter 8
14. Chapter 9
15. Chapter 10
16. Chapter 11
17. Chapter 12
18. Chapter 13
19. Chapter 14
20. Chapter 15
21. Chapter 16
22. Chapter 17
23. Chapter 18
24. Chapter 19
25. Chapter 20
26. Chapter 21
27. Chapter 22
28. Chapter 23
29. Chapter 24
30. Chapter 25
31. Eleven Months Later
32. ACKNOWLEDGMENTS
33. DISCUSSION QUESTIONS
34. ABOUT THE AUTHOR
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 500
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TU will be showcased in TEHDAS Teatteri in Turku on 31 May as part of RECEPTION, the 1st Triennial of the Community Arts organised by the Saari Residence maintained by the Kone Foundation and Arts Promotion Centre Finland together with several local partners.
TU is a collaborative performance work by Sonja Jokiniemi and Veera Kivelä. The base of the work is on Veera's language that appears in complex intimate sound systems and movement patterns. Jokiniemi and Kivelä map the performance space through touching, sounding and moving through, communicating with objects, the space itself and with one another. TU is part of a community-based research project Without an alphabet, where Jokiniemi worked with four autistic young adults investigating subjective language systems. Kivelä is one of her collaborators from this project.
The first triennial of the community arts in Finland offers a broad view on socially engaged art practice and on the works of art created in interaction and collaboration with various communities. The theme of the triennial RECEPTION refers on the one hand to the act of receiving as well as to hospitality. On the other hand, it raises questions about the resistance and the power structures. Who do we welcome, to whom do we turn our backs? In which troops do we stand and whose voices are being heard? In this time, what creates the feeling of togetherness? What kind of reactions and artistic outcomes creates the trinity of the artists, the communities and society ?
Between the spring and the early summer of 2016 RECEPTION will spread out to the west coast of Finland. The participating cities are Helsinki, Kemiönsaari, Turku, Mynämäki, Rauma, Pori and Vaasa. In each city, both local and visiting artists open and showcase their collabrative art processes and practices to the wider audience. The full triennial programme contains e.g artists-in-residence, discussions, presentations and works created during the event together with local communities.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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|
Thiim: "Great Honor" to Represent AMR in Expanded GT3 Program
Nicki Thiim feels honored to represent Aston Martin Racing in its expanded GT3 program…
Jake Kilshaw
Photo: Aston Martin Racing
Nicki Thiim says it is a "great honor" to take on an increased role with Aston Martin this season which will include him fronting the manufacturer's new Pro entry in the Blancpain GT Series.
The Dane will form part of R-Motorsport's lineup for the Blancpain GT Series Endurance Cup along with new factory driver Maxime Martin, while also racing with TF Sport in the British GT Championship.
"I've only done one Blancpain GT race so far and that was the Spa 24 Hours, which was probably one of the most enjoyable races I've had," Thiim told Sportscar365.
"It's more of a contact sport than you'd see in WEC, even though it is getting tougher there and closer. GT3 has always been known for its really close competition by all the brands."
A two-car entry under the R-Motorsport banner and run by Arden Motorsport is the beginning of an increased commitment to the series from the British manufacturer after just a single car in the Pro-Am category represented it last season.
Porsche is in a similar situation, having announced an all-pro lineup after also only competing in the Pro-Am Cup in 2017. As a result, this year's Blancpain GT Series could be one of the most competitive yet.
"It shows that the teams and Aston Martin have really approached it well," Thiim said. It's the first time they've really entered Blancpain GT in the proper way in the Pro category.
"It shows that they're really out to do it and that's a great thing. It has been a growing competition in Blancpain GT. I know they've been struggling with turning away some people because there are so many that want to do it.
"To represent Aston Martin, and because they asked me to be one of the front guys to do it for them, is a great honor. That they want me to drive so much this year shows I must be doing a good job!"
Racing the Aston Martin Vantage GT3 more often will come as a slight change of pace for the 28-year-old who has focused on GTE competition in recent years, both in the FIA World Endurance Championship and European Le Mans Series.
"I've had some races in it and it's quite the same, even though the new car in WEC is going to be a completely different story," he said.
"It was a bit easier with the older GTE car where it was almost only the engine that was different, but it's a car with four wheels and I should have the experience to adapt to it hopefully quite easily. That's the most important thing."
New Vantage GTE a "Completely Different Car"
While not entirely confirmed so far, it is expected that Thiim will also be back in the WEC for the upcoming 'Super Season' in which AMR will debut its brand-new Vantage GTE.
After getting a great deal of testing under his belt, Thiim has praised the new car, with its tire management one of the biggest improvements.
"It's a completely different car," he said. "The old one was an old lady, I'd say, and this is just a completely different build-up, starting from the engine.
"The whole chassis, the weight balance, and all that stuff, is completely different with this car. It's fitted towards the new regulations.
"What we struggled with last year was the downforce and that's something that Aston has worked a lot on this year, to get closer to Ford, Ferrari, Porsche and probably BMW.
"We were struggling a lot, especially last year, double stinting tires because our car was not so keen on the tires with the low downforce. That's something they want to change for the future."
Related TopicsAston Martin RacingNicki Thiimblancpain GT
Jake Kilshaw is a UK-based journalist. He is a graduate of Politics and International Relations.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 713
|
Q: Java readline() keeping socket open I am trying to have my client connect to my server, and depending on the command send some string back to the client. Currently the app connects and can send strings to the server very nicely. However when I send the command which instructs the server to send something back it hangs. I found that the problem occurs when the client attempts to read the line send from the server.
Server
PrintWriter out = new PrintWriter(new OutputStreamWriter(clientSocket.getOutputStream()));
out.println("GETDATA" + "\n");
out.flush();
out.close();
Client
BufferedReader fromServer = new BufferedReader(new InputStreamReader(clientSocket.getInputStream()));
incomingLine = fromServer.readLine();
Log.d("HERE", "NOT " + incomingLine);
fromServer.close();
Thanks!
A: I made effectively this same mistake when I was first doing sockets as well.
Don't use PrintWriter with BufferedReader. They're incompatible. By comments, PrintWriter actually hides critical exceptions, so they shouldn't be used in networking. Instead, use a DataInputStream and DataOutputStream for communications.
client = new Socket(hostname, port);
inStr = new DataInputStream(client.getInputStream());
outStr = new DataOutputStream(client.getOutputStream());
Then, send and receive using writeUTF and readUTF, like so:
public void send(String data) throws IOException {
outStr.writeUTF(data); outStr.flush();
}
public String recv() throws IOException {return inStr.readUTF();}
The reason has to do with the UTF encoding; a BufferedReader expects a certain string encoding, which PrintWriter does not give. Thus, the read/write hangs.
A: The method readLine() expects an end of line character "\n" maybe that's your problem
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,822
|
namespace System.Workflow.Activities
{
using System.ServiceModel;
using System.Reflection;
using System.Collections.Generic;
using System.Workflow.ComponentModel;
using System.Workflow.ComponentModel.Compiler;
class ReceiveActivityValidator : CompositeActivityValidator
{
public override ValidationErrorCollection Validate(
ValidationManager manager,
object obj)
{
ValidationErrorCollection validationErrors = base.Validate(manager, obj);
ReceiveActivity receiveActivity = obj as ReceiveActivity;
if (receiveActivity == null)
{
throw DiagnosticUtility.ExceptionUtility.ThrowHelperArgument("obj",
SR2.GetString(SR2.Error_ArgumentTypeInvalid, "obj", typeof(ReceiveActivity)));
}
ITypeProvider typeProvider = manager.GetService(typeof(ITypeProvider)) as ITypeProvider;
if (typeProvider == null)
{
throw DiagnosticUtility.ExceptionUtility.ThrowHelperError(new InvalidOperationException(
SR2.GetString(SR2.General_MissingService, typeof(ITypeProvider).Name)));
}
if (receiveActivity.ServiceOperationInfo == null)
{
validationErrors.Add(
new ValidationError(
SR2.GetString(SR2.Error_Validation_OperationInfoNotSpecified, receiveActivity.Name),
WorkflowServicesErrorNumbers.Error_OperationInfoNotSpecified,
false,
"ServiceOperationInfo"));
}
else
{
// validate operation info
//
ValidationErrorCollection operationInfoValidationErrors =
ValidationHelper.ValidateOperationInfo(
receiveActivity,
receiveActivity.ServiceOperationInfo,
manager);
validationErrors.AddRange(operationInfoValidationErrors);
// do not validate parameter binding if the operation info is not valid
// we might generate noise and false positives.
//
if (operationInfoValidationErrors.Count == 0)
{
validationErrors.AddRange(
ValidationHelper.ValidateParameterBindings(receiveActivity, receiveActivity.ServiceOperationInfo,
receiveActivity.ParameterBindings, manager));
}
// validate the context token
//
validationErrors.AddRange(
ValidationHelper.ValidateContextToken(receiveActivity, receiveActivity.ContextToken, manager));
}
// Check if the validation for all service operations being implemented
// has been done previously.
// If it has been done once then ServiceOperationsImplementedValidationMarker
// will be on the context stack.
//
if (validationErrors.Count == 0 &&
manager.Context[typeof(ServiceOperationsImplementedValidationMarker)] == null)
{
Activity rootActivity = receiveActivity;
while (rootActivity.Parent != null)
{
rootActivity = rootActivity.Parent;
}
validationErrors.AddRange(
ValidationHelper.ValidateAllServiceOperationsImplemented(
manager,
rootActivity));
}
return validationErrors;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,006
|
Q: Is is possible to keep track of the current number of instances of a class? In .NET, I understand that there is a garbage collector that will manage the memory that's being used during the execution of a program. This means that objects will be cleaned up when they go unused.
I was wondering if I could keep a static counter in a certain class that automatically updates when instances of that class are created or garbage collected. For example, if I create some instances of some CountableInstance class, then they could each have an InstanceIndex that keeps track of their current position based on when they were created.
Such a flow could look like this:
*Program starts
*CountableInstance ctble0 is created with InstanceIndex == 0
*CountableInstance ctble1 is created with InstanceIndex == 1
*CountableInstance ctble2 is created with InstanceIndex == 2
*CountableInstance ctble1 goes unused and gets garbage collected
*
*ctble0's index stays the same
*ctble2's index becomes 1
*Program ends
I'm guessing that keeping track of the number of CountableInstance instances would look like this:
public class CountableInstance
{
public static int total = 0;
public CountableInstance()
{
InstanceIndex = total;
total++; // Next instance will have an increased InstanceIndex
}
public CountableInstance(...) : this()
{
// Constructor logic goes here
}
public int InstanceIndex { get; private set; }
}
This takes care of how many instances there are and it also assigns the correct index to each new object. However, this implementation misses the logic that should happen when an instance is garbage collected.
If possible, how could I implement it? Would it also work to keep track of instances of objects that use a particular interface?
A: You can achieve the gist of this as below, but this seems like a bad idea IMO; in particular:
*
*finalizers aren't free (and neither is IDisposable.Dispose())
*you can create objects without ever running a constructor (if you try hard enough)
Anyway, code:
public class CountableInstance
{
static int s_Total, s_Alive;
public CountableInstance() {
// need a separate total vs alive so we don't give duplicate
// InstanceIndex, at least for the first 4 billion objects
InstanceIndex = Interlocked.Increment(ref s_Total);
Interlocked.Increment(ref s_Alive);
}
~CountableInstance() {
Interlocked.Decrement(ref s_Alive);
}
public int InstanceIndex { get; }
public static int Alive => Volatile.Read(ref s_Alive);
public static int Total => Volatile.Read(ref s_Total);
}
A: you could do something like this:
public class CountableInstance : IDisposable
{
public static int total = 0;
...
public void Dispose()
{
//total--;
//Edit:
Interlocked.Decrement(total);
}
}
this would keep track of what happens within your application. but as far as i know garbage collection could still be delayed to a later point long after Dispose.
A: If you want your indexes of your created instances to change if one instance is removed, you'll need to store them in a list. See this solution:
public class CountableInstance : IDisposable
{
private static List<CountableInstance> instances = new List<CountableInstance>();
public CountableInstance()
{
InstanceIndex = instances.Count;
instances.Add(this);
}
public void Dispose()
{
instances.Remove(this);
for(int i = InstanceIndex; i < instances.Count; i++)
{
instances[i].InstanceIndex = i;
}
}
public int InstanceIndex { get; private set; }
}
Note that since the instances are stored in the list, they will never get garbage collected before you call Dispose(). You'll have to call Dispose() manually if you don't need your object anymore, which let's you lose a big advantage of garbage collection. Also note that this solution is not thread safe.
Online demo: https://dotnetfiddle.net/A1LptG
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,976
|
\section{Moduli spaces of theta characteristics}
Theta characteristics appeared for the first time in the context of characteristic theory of odd and even theta functions in the
papers of G\"opel \cite{Go} and Rosenhain \cite{Ro} on Jacobi's inversion formula for genus $2$. They were initially considered in connection with
\emph{Riemann's bilinear addition} relation between the degree two monomials in theta functions with characteristics. Later, in order to systematize the relations between theta constants, Frobenius \cite{Fr1}, \cite{Fr2} developed an algebra of characteristics \footnote{Frobenius' attempts to bring algebra into the theory of theta functions has to be seen in relation to his famous work on group characters. In 1893, when entering the Berlin Academy of Sciences he summarized his aims as follows \cite{Fr3}: \emph{In the theory of theta functions it is easy to set up an arbitrarily large number of relations, but the difficulty begins when it comes to finding a way out of this labyrinth of formulas. Many a distinguished researcher, who through tenacious perseverence, has advanced the theory of theta functions in two, three, or four variables, has, after an outstanding demonstration of brilliant talent, grown silent either for a long time of forever. I have attempted to overcome this paralysis of the mathematical creative powers, by seeking renewal at the fountain of youth of arithmetic}.}; he distinguished between \emph{period} and \emph{theta} characteristics. The distinction, which in modern terms amounts to the difference between the \emph{Prym} moduli space
$\cR_g$ and the \emph{spin} moduli space $\cS_g$, played a crucial role in elucidating the transformation law for theta functions under a linear
transformation of the moduli, and it ultimately led to a correct definition of the action of symplectic group $\mbox{Sp}(\FF_2^{2g})$ on the set of characteristics. An overwiew of the 19th century theory of theta functions can be found in Krazer's monumental book \cite{Kr}. It is a very analytic treatise in character, with most geometric applications either completely absent relegated to footnotes.
\vskip 5pt
The remarkable book \cite{Cob} by Coble
\footnote{An irreverent portrait of Coble in the 1930's from someone who was not exactly well disposed towards algebraic geometry ("\emph{... the only part of mathematics where a counterexample to a theorem is considered to be a beautiful addition to it}"), can be found in Halmos' autobiography "I want to be a mathematician".}
represents a departure from the analytic view towards a more abstract understanding of theta characteristics using configurations in finite geometry. Coble viewed theta characteristics as quadrics in a vector space over $\FF_2$. In this language Frobenius' earlier concepts (syzygetic and azygetic triples, fundamental systems of characteristics) have an elegant translation. With fashions in algebraic geometry drastically changing, the work of Coble was forgotten for many decades \footnote{This quote from Mattuck's \cite{Ma} obituary of Coble reveals the pervasive attitude of the 1960's: \emph{The book as a whole is a difficult mixture of algebra and analysis, with intricate geometric reasoning of a type few can follow today. The calculations are formidable; let them serve to our present day algebraic geometers, dwelling as they do in their Arcadias of abstraction, as a reminder of what awaits those who dare to ask specific questions about particular varieties}.}.
\vskip 3pt
The modern theory of theta characteristic begins with the works of Atiyah \cite{At} and Mumford \cite{Mu}; they showed, in the analytic (respectively algebraic) category, that the parity of a theta characteristic is stable under deformations. In particular, Mumford's functorial view of the subject, opened up the way to extending the study of theta characteristics to singular curves (achieved by Harris \cite{H}), to constructing a proper Deligne-Mumford moduli space of stable spin curves (carried out by Cornalba \cite{C}), or to reinterpreting Coble's work in modern terms (see the book \cite{DO}).
\vskip 3pt
The aim of this paper is to survey various developments concerning the geometry of moduli spaces of spin curves. Particular emphasis is placed on the complete birational classification of both the even and the odd spin moduli spaces, which has been carried out in the papers \cite{F2}, \cite{FV1} and \cite{FV2}. Precisely, we shall explain the following result:
\begin{theorem} The birational type of the moduli spaces $\ss_g^+$ and $\ss_g^-$ of even and odd spin curves of genus $g$ can be summarized as follows:
\begin{center}
$\ss_g^+$: \ \begin{tabular}{c|c}
$g> 8$ & general type\\
\hline
$g=8$ &Calabi-Yau \\
\hline
$g\leq 7$ & unirational
\end{tabular} \ \ \ \mbox{ } \mbox{ }
$\ss_g^-$: \ \begin{tabular}{c|c}
$g\geq 12$ & general type\\
\hline
$g\leq 8$ & unirational\\
\hline
$9\leq g\leq 11$ & uniruled
\end{tabular}
\end{center}
\end{theorem}
We describe the structure of the paper. In the first two sections we recall the interpretation of theta characteristics as quadrics in an $\FF_2$-vector space, then link this description both with the classical theory of characteristics of theta functions and modern developments inspired by string theory. In the next three sections we explain some features of the geometry of the moduli space $\ss_g$ of stable spin curves of genus $g$, discuss ways of constructing effective divisors on $\ss_g$ and computing their cohomology classes, then finally present unirational parametrizations of the moduli space in small genus, by using Mukai models and special $K3$ surfaces. We close by surveying a few open problems related to syzygies of theta characteristics and stratifications of the moduli space.
\section{Theta characteristics: a view using finite geometry}
For a smooth algebraic curve $C$ of genus $g$ we denote by $$J_2(C):=\{\eta\in \mbox{Pic}^0(C):\eta^{\otimes 2}=\OO_C\}$$ the space of two-torsion points in the Jacobian of $C$, viewed as an $\FF_2$-vector space. We recall the definition of the \emph{Weil pairing} $\langle \cdot, \cdot \rangle:J_2(C)\times J_2(C)\rightarrow \FF_2$, cf. \cite{Mu} Lemma 2:
\begin{definition} Let $\eta, \epsilon\in J_2(C)$ and write $\eta=\OO_C(D)$ and $\epsilon=\OO_C(E)$, for certain divisors $D$ and $E$ on $C$, such that $\mbox{supp}(D)\cap \mbox{supp}(E)=\emptyset$. Pick rational functions $f$ and $g$ on $C$, such that $\mbox{div}(f)=2D$ and $\mbox{div}(g)=2E$. Then define $\langle \eta, \epsilon \rangle\in \FF_2$ by the formula
$$(-1)^{\langle \eta, \epsilon \rangle}=\frac{f(E)}{g(D)}.$$
\end{definition}
The definition of $\langle \eta, \epsilon\rangle$ is independent of the choice of the divisors $D$ and $E$ and the rational functions $f$ and $g$. The Weil pairing is a nondegenerate symplectic form.
The set of \emph{theta characteristics} of $C$, defined as $\mbox{Th}(C):=\{\theta\in \mbox{Pic}^{g-1}(C): \theta^{\otimes 2}=K_C\}$, is an affine space over $J_2(C)$. It was Coble's insight \cite{Cob} to realize that in order to acquire an abstract understanding of the geometry of $\mbox{Th}(C)$ and clarify the distinction between the period characteristics and the theta characteristics of $C$, it is advantageous to view theta characteristics as quadrics in the vector space $J_2(C)$.
\begin{definition}
Let $(V, \langle \cdot, \cdot \rangle)$ be a symplectic vector space over $\FF_2$. The set $Q(V)$ of quadratic forms on $V$ with fixed polarity
given by the symplectic form $\langle \cdot, \cdot \rangle$, consists of all functions $q:V\rightarrow \FF_2$ satisfying the identity
$$q(x+y)=q(x)+q(y)+\langle x, y\rangle, \ \mbox{ for all } x, y\in V.$$
\end{definition}
If $q\in Q(V)$ is a quadratic form and $v\in V$, we can define a new quadratic form $q+v\in Q(V)$ by setting
$(q+v)(x):=q(x)+\langle v, x\rangle$, for all $x\in V.$ Similarly one can add two quadratic forms. If $q, q'\in Q(V)$, then there exists a uniquely determined element $v\in V$ such that $q'=q+v$, and we set $q+q':=v\in V$. In this way, the set $\widetilde{V}:=V\cup Q(V)$ becomes a $(2g+1)$-dimensional vector space over $\FF_2$. There is a natural action of the symplectic group $\mathrm{Sp}(V)$ on $Q(V)$. For a transformation
$T\in \mathrm{Sp}(V)$ and a quadratic form $q\in Q(V)$, one defines
$$(T\cdot q)(x):=q(T^{-1}(x)), \mbox{ for } \ x\in V.$$
This action has two orbits $Q(V)^+$ and $Q(V)^-$ respectively, which can de distinguished by the \emph{Arf invariant} \cite{Arf} of a quadratic form.
\begin{definition} Let us choose a symplectic base $(e_1, \ldots, e_g, f_1, \ldots, f_g)$ of $V$. Then define
$$\mathrm{arf}(q):=\sum_{i=1}^g q(e_i)\cdot q(f_i)\in \FF_2.$$
\end{definition}
The invariant $\mbox{arf}(q)$ is independent of the choice of a symplectic basis of $V$. We set $Q(V)^+:=\{q\in Q(V): \mbox{arf}(q)=0\}$ to be the space of even quadratic form, and $Q(V)^-:=\{q\in Q(V): \mbox{arf}(q)=1\}$ that of odd quadratic forms.
Note that there are $2^{g-1}(2^g+1)$ even quadratic forms and $2^{g-1}(2^g-1)$ odd ones.
\vskip 3pt
Every theta characteristic $\theta\in \mbox{Th}(C)$ defines a form $q_{\theta}:J_2(C)\rightarrow \FF_2$, by setting
$$q_{\theta}(\eta):=h^0(C, \eta\otimes \theta)+h^0(C, \theta) \ \mathrm{ mod }\ 2.$$
It follows from the \emph{Riemann-Mumford relation} \cite{Mu} or \cite{H} Theorem 1.13, that $q_{\theta}\in Q(J_2(C))$, that is, the polar of $q_{\theta}$ is the Weil form. For $\eta, \epsilon\in J_2(C)$, the following relation holds:
$$h^0(C, \theta\otimes \eta\otimes \epsilon)+h^0(C, \theta\otimes \eta)+h^0(C, \theta\otimes \epsilon)+h^0(C, \theta)\equiv \langle \eta, \epsilon\rangle\ \mbox{ mod }\ 2.$$
Thus one has the following identification between theta characteristics and quadrics:
\begin{center}
\fbox{$\mathrm{Th}(C)=Q\bigl(J_2(C), \ \langle \cdot, \cdot \rangle \bigr).$}
\end{center}
Under this isomorphism, even (respectively odd) theta characteristics correspond to forms in $Q(V)^+$ (respectively $Q(V)^-$). Furthermore, $\mbox{arf}(q_{\theta})=h^0(C, \theta) \mbox{ mod } 2$.
Using this identification, one can translate Frobenius's
\cite{Fr1}, \cite{Fr2} entire \emph{theory of fundamental systems} of theta characteristics into an abstract setting. Of great importance is the following:
\begin{definition}
A system of three theta characteristics $\theta_1, \theta_2, \theta_3\in \mathrm{Th}(C)$ is called \emph{syzygetic} (respectively
\emph{azygetic}) if $\mbox{arf}(q_{\theta_1})+\mbox{arf}(q_{\theta_2})+\mbox{arf}(q_{\theta_3})+\mbox{arf}(q_{\theta_1+\theta_2+\theta_3})=0$ (respectively $1$).
\end{definition}
Here the additive notation $\theta_1+\theta_2+\theta_3$ refers to addition in the extended vector space $\widetilde{J_2(C)}=J_2(C)\cup \mbox{Th}(C)$. In terms of line bundles, $\theta_1+\theta_2+\theta_3=\theta_1\otimes \theta_2\otimes \theta_3^{\vee}\in \mbox{Pic}^{g-1}(C)$. It is an easy exercise to show that $\{\theta_1, \theta_2, \theta_3\}$ is a syzygetic triple if and only if $\langle \theta_1+\theta_2, \theta_1+\theta_3\rangle=0$. If the system $\{\theta_1, \theta_2, \theta_3\}$ is syzygetic, then any three elements of the set $\{\theta_1, \theta_2, \theta_3, \theta_1+\theta_2+\theta_3\}$ form a syzygetic triple. In this case we say that the four theta characteristics form a \emph{syzygetic tetrad}.
\begin{example} Four odd theta characteristics $\theta_1, \ldots, \theta_4\in \mbox{Th}(C)$ corresponding to contact contact divisors $D_i \in |\theta_i|$ such that $2D_i\in |K_C|$ form a syzygetic tetrad, if and only if there exists a quadric $Q\in \mbox{Sym}^2 H^0(C, K_C)$, such that
$$Q\cdot C=D_1+D_2+D_3+D_4.$$
For instance, when $g=3$, four bitangents to a quartic $C\subset \PP^2$ form a syzygetic tetrad exactly when the $8$ points of tangency are the complete intersection of $C$ with a conic.
\end{example}
To understand the geometry of the configuration $\mbox{Th}(C)$ one defines appropriate systems of coordinates. Following \cite{Kr} p. 283,
one says that $2g+2$ theta characteristics $\{\theta_1, \ldots, \theta_{2g+2}\}$ form a \emph{fundamental system} if any triple $\{\theta_i, \theta_j, \theta_j\}$ where $1\leq i<j<k\leq 2g+2$ is azygetic. The sum of the $2g+2$ elements of a fundamental system equals $0$, see \cite{Kr} p. 284, and the number of fundamental systems in $\mbox{Th(C)}$ is known, cf. \cite{Kr} p. 285:
$$2^{2g}\frac{|\mathrm{Sp}(\FF_2^{2g})|}{(2g+2)!}=2^{2g}\frac{(2^{2g}-1)(2^{2g-2}-1)\cdots (2^2-1)}{(2g+2)!}2^{g^2}.$$
\begin{example}
A smooth plane quartic $C\subset \PP^2$ has precisely $288$ fundamental systems $(\theta_1, \ldots, \theta_8)$, where the first $7$ theta characteristics are odd, and $\theta_8=-\sum_{i=1}^7 \theta_i$ is then necessarily even. All elements in $\mbox{Th}(C)$ can be expressed in the "coordinate system" given by $(\theta_1, \ldots, \theta_7)$: The remaining odd theta characteristics are $\theta_i+ \theta_j+ \theta_k+ \theta_l+ \theta_m$. The $35=36-1$ even theta characteristics are of the form $\theta_i+ \theta_j+ \theta_k$.
\end{example}
One can also consider systems of syzygetic theta characteristics, that is subsets, $\{\theta_1, \ldots, \theta_{r+1}\}\subset \mbox{Th}(C)$ such that all
triples $\{\theta_i, \theta_j, \theta_k\}$ are syzygetic. Then $r\leq g$, see \cite{Kr} p. 299. Following Frobenius \cite{Fr1}, a maximal system of such theta characteristics is called a \emph{G\"opel system} and corresponds to $2^g$ theta characteristics such any three of them form a syzygetic triple. These definitions can be immediately extended to cover general principally polarized abelian varieties not only Jacobians.
\section{Theta characteristics: the classical view via theta functions}
We now link the realization of theta characteristics in abstract finite geometry, to the theory of theta functions with characteristics. There are established classical references, above all \cite{Kr}, \cite{Wi}, \cite{Ba}, \cite{Cob}, as well as modern ones, for instance \cite{BL}, \cite{SM}. We fix an integer $g\geq 1$ and denote by
$$\mathfrak{H}_g:=\{\tau\in M_{g, g}(\mathbb C): \tau=^t\tau, \ \mbox{Im }\tau>0\}$$ the Siegel upper half-space
of period matrices for abelian varieties of dimension $g$; hence $\cA_g:=\mathfrak{H}_g/\mathrm{Sp}_{2g}(\mathbb Z)$ is the moduli space of principally polarized abelian varieties of dimension $g$. For a vector $\Bigl[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\Bigr]=\left[\begin{array}{ccc}
\epsilon_1 \ldots \epsilon_g\\
\delta_1 \ldots \delta_g\\
\end{array}\right]\in \FF_2^{2g}$ one defines the \emph{Riemann theta function with characteristics} as the holomorphic function $\vartheta:\mathfrak{H}_g\times \mathbb C^g\rightarrow \mathbb C$ given by
$$\vartheta\left[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\right](\tau, z):=\sum_{m\in \mathbb Z^g} \mathrm{exp} \Bigl(\pi i \ ^t(m+\frac{\epsilon}{2})\tau (m+\frac{\epsilon}{2})+2\pi i \ ^t(m+\frac{\epsilon}{2})(z+\frac{\delta}{2})\Bigr).$$
For any period matrix $\tau \in \mathfrak{H}_g$, the pair
$$\Bigl[A_{\tau}:=\frac{\mathbb C^g}{\mathbb Z^g+\tau\cdot \mathbb Z^g}, \ \ \Theta_{\tau}:=\Bigl\{z\in A_{\tau}:\vartheta \left[\begin{array}{c}
0\\
0\\
\end{array}\right](\tau, z)=0\Bigr\}\Bigr]$$
defines a principally polarized abelian variety, that is, $[A_{\tau}, \Theta_{\tau}]\in \cA_g$.
There is an identification of symplectic vector spaces $$V:=\FF_2^{2g} \stackrel{\cong}\longrightarrow A_{\tau}[2],\ \mbox{ } \mbox{ given by } \ \mbox{ }\mbox{ }\left[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\right] \mapsto \frac{\tau\cdot \epsilon+\delta}{2}\in A_{\tau}[2].$$
This isomorphism being understood, points in $A_{\tau}[2]$ were classically called \emph{period characteristics}, see \cite{Kr} Section VII.2.
The theta function $\vartheta\Bigl[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\Bigr](\tau, z)$ is the unique section of the translated line bundle $\OO_{A_{\tau}}\Bigl(t_{\frac{\tau\cdot \epsilon+\delta}{2}}^*(\Theta_{\tau})\Bigr)$. Krazer \cite{Kr} goes to great lengths to emphasize the difference between the \emph{period} and the \emph{theta} characteristics, even though at first sight, both sets of characteristics can be identified with the vector space $\FF_2^{2g}$. The reason for this is the transformation formula for theta functions under the action of the symplectic group, see \cite{BL} Theorem 8.6.1. The theta constants with characteristics are modular forms of weight one half with respect to the subgroup $\Gamma_g(4, 8)$. To define the action of the full symplectic group $\mathrm{Sp}_{2g}(\mathbb Z)$ on $\FF_2^{2g}$, we consider the quadratic form associated to a characteristic
$\Delta:=\left[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\right]$, that is,
$$q_{\left[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\right]}\Bigl(x, y\Bigr)=x\cdot y+\epsilon\cdot x+ \delta\cdot y,$$
with $(x, y)=(x_1, \ldots, x_g, y_1, \ldots, y_g)\in \FF_2^{2g}$. We define an action of $\mathrm{Sp}_{2g}(\mathbb Z)$ on the set of characteristics which factors through the following action of $\mathrm{Sp}(\FF_2^{2g})$: If $M\in \mathrm{Sp}(\FF_2^{2g})$, set
$$q_{M\cdot \Delta}:=M \cdot q_{\Delta},$$
where we recall that we have already defined $(M\cdot q_{\Delta})(x, y):=q_{\Delta}(M^{-1}(x, y))$.
It can be shown that this non-linear action of the symplectic group on the set of characteristics is compatible with the transformation rule of theta constants, see also \cite{CDvG}, \cite{BL}. In this context, once more, a theta characteristic $\left[\begin{array}{c}
\epsilon\\
\delta\\
\end{array}\right]$ appears as an element of $Q(V)$. This interpretation makes the link between the classical definition of theta characteristics found in \cite{Kr} and the more modern one encountered for instance in \cite{DO}, \cite{GH}. The algebra of period and theta characteristics was developed by Frobenius \cite{Fr1}, \cite{Fr2} in order to derive general identities between theta constants and describe the structure of the set of such identities.
\vskip 4pt
\subsection{Superstring scattering amplitudes and characteristic calculus.}
Recently, the action of $\mbox{Sp}(\FF_2^{2g})$ on the set of characteristics and the algebra of characteristic systems has been used by Grushevsky \cite{Gr}, Salvati Manni \cite{SM2}, Cacciatori, Dalla Piazza and van Geemen \cite{CDvG} and others, in order to find an explicit formula for the \emph{chiral superstring amplitudes}. This is an important foundational question in string theory and we refer to the cited papers for background
and further references.
Loosely speaking, D'Hoker and Phong conjectured that there exists a modular form $\Xi^{(g)}$ of weight $8$ with respect to the group $\Gamma_g(1, 2)\subset \Gamma_g=\mbox{Sp}_{2g}(\mathbb Z)$, satisfying the following two constraints:
\noindent (1) \emph{Factorization:} For each integer $1\leq k\leq g-1$, the following factorization formula
$$\Xi^{(g)}_{| \mathfrak{H}_k\times \mathfrak{H}_{g-k}}=\Xi^{(k)}\cdot \Xi^{(g-k)},$$
holds, when passing to the locus $\mathfrak{H}_k\times \mathfrak{H}_{g-k}\subset \mathfrak{H}_g$ of decomposable abelian varieties.
\noindent (2) \emph{Initial conditions:} For $g=1$, one must recover the standard chiral measure, that is,
$$\Xi^{(1)}=\theta\left[\begin{array}{c}
0\\
0\\
\end{array}\right]^8 \theta \left[\begin{array}{c}
0\\
1\\
\end{array}\right]^4
\theta \left[\begin{array}{c}
1\\
0\\
\end{array}\right]^4.
$$
A unique solution has been found in \cite{CDvG} when $g=3$ and in \cite{SM2} when $g=4, 5$. This was placed in \cite{Gr} in a general framework that works in principle for arbitrary $g$.
For a set of theta characteristics $W\subset \FF_2^{2g}$, one defines the product of theta constants
$$P_W(\tau):=\prod_{\Delta\in W} \theta[\Delta](\tau, 0).$$
Note that $P_W$ vanishes if $W$ contains an odd characteristic. Then for all integers $0\leq i\leq g$, we set
$$P_i^{(g)}(\tau):=\sum_{W\subset \FF_2^{2g}, \ \mathrm{dim}(W)=i} P_W(\tau)^{2^{4-i}}.$$
It is pointed out in \cite{Gr} Proposition 13, that this function is a modular form of weight $8$ with respect to $\Gamma_g(1, 2)$. This result can be traced back to Frobenius.
Observe that in the definition of $P_i^{(g)}(\tau)$ the only non-zero summands correspond to totally syzygetic systems of characteristics, for else
such a system $W$ contains an odd characteristic and the corresponding term $P_W(\tau)$ is identically zero. Then it is showed that the expression
$$\Xi^{(g)}:=\frac{1}{2^g}\sum_{i=0}^g(-1)^i 2^{i\choose 2} P_i^{(g)}$$
satisfies the factorization rules when $g\leq 5$. Note that for higher $g$ the definition of $\Xi^{(g)}$ leads to a multivalued function due to the impossibility of choosing consistently the roots of unity for the various summands.
\begin{question}
Can one find the superstring amplitudes for higher $g$ by working directly with the space $\ss_g^+$ and constructing via algebro-geometric rather than theta function methods a system of effective divisors of slope (weight) $8$ satisfying the factorization formula?
\end{question}
\section{Cornalba's moduli space of spin curves}
We now concern ourselves with describing Cornalba's \cite{C} compactification $\ss_g$ of the moduli space $\cS_g$
of theta characteristics. We recall that $\cS_g$ is the parameter space of pairs $[C, \theta]$, where $C$ is a smooth curve of genus $g$ and $\theta\in \mbox{Th}(C)$. Following Atiyah \cite{At} such a pair is called a \emph{spin curve} of genus $g$. Mumford \cite{Mu} by algebraic means (and Atiyah \cite{At} with analytic methods) showed that the parity of a spin curve is locally constant in families: If $\phi:X\rightarrow S$ is a flat family of smooth curves of genus $g$ and $\mathcal{L}$ is a line bundle on $X$ together with a morphism $\beta:\mathcal{L}^{\otimes 2}\rightarrow \omega_{\phi}$ such that $\phi_s:\mathcal{L}_{X_s}^{\otimes 2}\rightarrow \omega_{X_s}$ is an isomorphism for all $s\in S$, then the map
$ \mathrm{arf}(\phi):S\rightarrow \FF_2$ defined as $\mbox{arf}(\phi)(s):=\mathrm{arf}(\mathcal{L}_{X_s})$
is constant on connected components of $S$. One can speak of the parity of a family of spin curves and according to the value of
$\mbox{arf}(q_{\theta})$, the moduli space $\cS_g$ splits into two connected components $\cS_g^+$ and $\cS_g^-$. The forgetful map $\pi:\cS_g\rightarrow \cM_g$, viewed as a morphism of Deligne-Mumford stacks, is unramified. A compactified moduli space $\ss_g$ should be the coarse moduli space of a Deligne-Mumford stack, such that there is a \emph{finite} morphism $\pi:\ss_g\rightarrow \mm_g$ fitting into a commutative diagram:
$$\xymatrix{
& \cS_g \ar[d]_{\pi} \ar@{^{(}->}@<-0.5ex>[r] & \ss_g \ar[d]_{\pi}& \\
& \cM_g \ar@{^{(}->}@<-0.5ex>[r] & \mm_g & \\
}$$
As an algebraic variety, $\ss_g$ is the normalization of $\mm_g$ in the function field of $\cS_g$. It is the main result of \cite{C} that the points
of $\ss_g$ have a precise modular meaning in terms of line bundles on curves belonging to a slightly larger class than that of stable curves.
\begin{definition} A reduced, connected, nodal curve $X$ is called \emph{quasi-stable}, if for any component $E\subset X$ that is isomorphic to $\PP^1$, one has that (i) $k_E:=|E\cap \overline{(X-E)}|\geq 2$, and (ii) any two rational components $E, E'\subset X$ with $k_E=k_{E'}=2$, are disjoint.
\end{definition}
Smooth rational components $E\subset X$ for which $k_E=2$ are called \emph{exceptional}.
The class of quasi-stable curve is a very slight enlargement of the class of stable curves. To obtain a quasi-stable curve, one takes a stable curve $[C]\in \mm_g$ and a subset of nodes $N\subset \mbox{Sing}(C)$ which one "blows-up"; if $\nu:\tilde{C}\rightarrow C$ denotes the normalization map and $\nu^{-1}(n)=\{n^-, n^+\}$, then we define the nodal curve
$$X:=\tilde{C}\cup \Bigl(\bigcup_{n\in N} E_n\Bigr),$$
where $E_n=\PP^1$ for each node $n\in N$ and $E_n\cap (\overline{X-E_n})=\{n^+, n^-\}$. The \emph{stabilization map} $\mbox{st}:X\rightarrow C$ is a partial normalization and contracts all exceptional components $E_n$, that is, $\mbox{st}(E_n)=\{n\}$, for each $n\in N$.
\vskip 3pt
Why extend the class of stable curves, after all $\mm_g$ is already a projective variety?
One can define compactified moduli spaces of theta characteristics working with stable curves alone, if one is prepared to allow sheaves that are not locally free at the nodes of the curves. Allowing semi-stable curves, enables us to view a degeneration of a theta characteristic as a line bundle on a curve that is (possibly) more singular. These two philosophies of compactifying a parameter space of line bundles, namely restricting the class of curves but allowing singularities for the sheaves, vs. insisting on local freeness of the sheaves but enlarging the class of curves, can also be seen at work in the two (isomorphic) compactifications of the universal degree $d$ Jacobian variety $P_{d, g}$ over $\cM_g$ constructed in \cite{Ca} and \cite{P}. We now describe all points of $\ss_g$:
\begin{definition}\label{spinstructures} A \emph{stable spin curve} of genus
$g$ consists of a triple $(X, \theta, \beta)$, where $X$ is a quasi-stable curve of arithmetic genus $g$, $\theta\in \mathrm{Pic}^{g-1}(X)$ is a line
bundle of total degree $g-1$ such that $\theta_{E}=\OO_E(1)$ for every exceptional component $E\subset X$, and $\beta:\theta^{\otimes
2}\rightarrow \omega_X$ is a sheaf homomorphism which is
not zero along each non-exceptional component of $X$.
\end{definition}
When $X$ is a smooth curve, then $\theta\in \mbox{Th}(X)$ is an ordinary theta characteristics and $\beta$ is an isomorphism. Note that in this definition, the morphism $\beta:\theta^{\otimes 2}\rightarrow \omega_X$ vanishes with order $2$ along each exceptional component $E\subset X$. Cornalba \cite{C} proved that stable spin curves form a projective moduli space $\ss_g$ endowed with a regular stabilization morphism $\pi:\ss_g\rightarrow \mm_g$, set-theoretically given by $\pi([X, \eta, \beta]):=[C]$; here $C$ is obtained from $X$ by contracting all the exceptional components. The space $\ss_g$ has two connected components $\ss_g^-$ and $\ss_g^+$ depending on the parity $h^0(X, \theta) \mbox{ mod } 2$ of the spin structure.
\begin{definition}\label{thC}
For a stable curve $[C]\in \mm_g$, we denote by $\mbox{Th}(C):=\pi^{-1}([C])$ the zero-dimensional scheme of length $2^{2g}$ classifying stable spin structures on quasi-stable curves whose stable model is $C$.
\end{definition}
The scheme $\mbox{Th}(C)$ has an interesting combinatorial structure that involves the dual graph of $C$. Before describing it for an arbitrary curve $[C]\in \mm_g$, it is helpful to understand the boundary structure of $\ss_g$, which amounts to describe $\mbox{Th}(C)$ when $C$ is a $1$-nodal curve. We shall concentrate on the space $\ss_g^-$ and leave $\ss_g^+$ as an exercise (or refer to \cite{C}, \cite{F2} for details).
The boundary $\mm_g-\cM_g$ of the moduli space of curves decomposes into components $\Delta_0, \ldots, \Delta_{[\frac{g}{2}]}$. A general point of $\Delta_0$ corresponds to a $1$-nodal irreducible curve of arithmetic genus $g$, and for $1\leq i\leq [\frac{g}{2}]$ the general point of $\Delta_i$ is the class of the union of two components of genera $i$ and $g-i$ respectively, meeting transversely at a point.
\subsection{Spin curves of compact type.} We fix an integer $1\leq i\leq [\frac{g}{2}]$ and a general point $[C\cup_y D]\in \Delta_i$, where
$[C, y]\in \cM_{i, 1}$ and $[D, y]\in \cM_{g-i, 1}$ are smooth curves. We describe all stable spin curve $[X, \theta, \beta]\in \pi^{-1}([C\cup_y D])$.
For degree reasons, $X\neq C\cup_y D$, that is, one must insert an exceptional component $E$ at the node $y\in C\cap D$ and then
$$X:=C\cup_{y^+} E\cup_{y^-} D,$$ where
$C\cap E=\{y^+\}$ and $D\cap E=\{y^-\}$.
Moreover $$\theta=\bigl(\theta_C, \theta_D, \theta_E=\OO_E(1)\bigr)\in
\mbox{Pic}^{g-1}(X),$$ and since $\beta_{| E}=0$, it follows that $\theta_C\in \mbox{Th}(C)$ and $\theta_D\in \mbox{Th}(D)$, that is, a theta characteristic on a curve of compact type is simply a collection of theta characteristics on each of its (necessarily smooth) components.
The condition that $h^0(X, \theta)$ be odd
implies that $\theta_C$ and $\theta_D$ have
opposite parities. Accordingly, the pull-back divisor $\pi^*(\Delta_i)$ splits in two components depending on the choice of the respective Arf invariants: We denote by $A_i\subset \ss_g^-$ the closure of
the locus corresponding points for which $\theta_C=\theta_C^-$ and $\theta_D=\theta_D^+$, that is, $\mbox{arf}(\theta_C)=1$ and $\mbox{arf}(\theta_D)=0$.
We denote by $B_i\subset \ss_g^-$ the
closure of the locus of spin curves for which $\mbox{arf}(\theta_C)=0$ and $\mbox{arf}(\theta_D)=1$.
At the level of divisors, the following relation holds $$\pi^*(\Delta_i)=A_i+B_i.$$
Moreover one has that
$$\mbox{deg}(A_i/\Delta_i)=2^{g-2}(2^i-1)(2^{g-i}+1)\
\mbox{ and }\ \mbox{deg}(B_i/\Delta_i)=2^{g-2}(2^i+1)(2^{g-i}-1).$$
\subsection{Spin curves with an irreducible stable model.} We fix a general $2$-pointed smooth curve $[C, x, y]\in \cM_{g-1, 2}$ and identify the points $x$ and $y$. The resulting stable curve $\nu:C\rightarrow C_{xy}$, where $C_{xy}:=C/x\sim y$, corresponds to a general point of the boundary divisor $\Delta_0$. Unlike in the case of curves of compact type, two possibilities do occur,
depending on whether $X$ possesses an exceptional component or not.
If $X=C_{xy}$, then the locally free sheaf $\theta$ is a root of the dualizing sheaf $\omega_{C_{xy}}$. Setting $\theta_C:=\nu^*(\theta)\in \mbox{Pic}^{g-1}(C)$, from the condition $$H^0(C, \omega_C(x+y)\otimes \theta_C^{\otimes (-2)})\neq 0,$$ by counting degrees, we obtain that $\theta_C^{\otimes 2}=K_C(x+y)$.
For each choice of $\theta_C\in \mathrm{Pic}^{g-1}(C)$ as above, there
is precisely one choice of gluing the fibres $\theta_C(x)$ and
$\theta_C(y)$ in a way that if $\theta$ denotes the line bundle on $C_{xy}$ corresponding to this gluing, then $h^0(X, \theta)$ is odd.
Let $A_0$ denote the closure in $\ss_g^-$ of the locus of such points. Then
$\mbox{deg}(A_0/\Delta_0)=2^{2g-2}$. Since we expect the fibre $\mbox{Th}(C_{xy})$ to consist of $2^{2g}$ points (counting also multiplicities), we see that one cannot recover all stable spin curves having $C_{xy}$ as their stable model by considering square roots of the dualizing sheaf of $C_{xy}$ alone. The remaining spin curves correspond to sheaves on $C_{xy}$ which are not locally free at the node, or equivalently, to spin structures on a strictly quasi-stable curve.
\vskip 3pt
Assume now that $X=C\cup_{\{x, y\}} E$, where $E$ is an exceptional component. Since $\beta_{| E}=0$
it follows that $\beta_{| C}\in H^0(C, \omega_{X| C}\otimes \theta_C^{\otimes (-2)})$ must vanish at both
$x$ and $y$ and then for degree reasons $\theta_C\in \mbox{Th}(C)$. For parity reasons, $\mbox{arf}(\theta_C)=1$.
We denote by $B_0$ the closure in $\ss_g^-$ of the locus of such points.
A local analysis carried out in \cite{C} shows that $\pi$ is simply ramified over $B_0$. Since $\pi:\ss_g^-\rightarrow \mm_g$ is not ramified along any other divisors of $\ss_g^-$, one deduces that $B_0$ is the ramification divisor of the forgetful map and the following relation
holds:
$$
\pi^*(\Delta_0)=A_0+2B_0.
$$
A general point of $B_0$ is determined by specifying an odd theta characteristic on $C$, thus $\mbox{deg}(B_0/\Delta_0)=2^{g-2}(2^{g-1}-1)$. By direct calculation one checks that
$$\mbox{deg}(A_0/\Delta_0)+2\mbox{deg}(B_0/\Delta_0)=2^{g-1}(2^g-1),$$
which confirms that $\pi$ is simply ramified along $B_0$.
\vskip 4pt
After this preparation, we are now ready to tackle the case of an arbitrary stable curve $C$. We denote by $\nu:\tilde{C}\rightarrow C$ the normalization map and by $\Gamma_C$ the \emph{dual graph} whose vertices are in correspondence with components of $C$, whereas an edge of $\Gamma_C$ corresponds to a node which lies at the intersection of two components (note that self-intersections are allowed). A set of nodes $\Delta\subset \mbox{Sing}(C)$ is said to be \emph{even}, if for any component $Y\subset C$, the degree $|\nu^{-1}(Y\cap \Delta)|$ is an even number. For instance, if $C:=C_1\cup C_2$ is a union of two smooth curves meeting transversally, a set of nodes $\Delta\subset C_1\cap C_2$ is even if and only if $|\Delta|\equiv 0\mbox{ mod } 2$.
\begin{remark} Assume $[X, \theta, \beta]\in \ss_g$ and let $\mathrm{st}:X\rightarrow C$ be the stabilization morphism. If $N\subset \mbox{Sing}(C)$ is the set of exceptional nodes of $C$, that is, nodes $n\in N$ having the property that
$\mathrm{st}^{-1}(n)=E_n=\PP^1$, and we write that $\mbox{Sing}(C)=N\cup \Delta$, then the set $\Delta$ of non-exceptional nodes
of $C$ is even, see both \cite{C} and \cite{CC}.
\end{remark}
One has the following description of the scheme $\mbox{Th}(C)$, cf. \cite{CC} Proposition 5:
\begin{proposition}\label{components}
Let $[C]\in \mm_g$ and $b:=b_1(\Gamma_C)$ be the Betti number of the dual graph. Then the number of components of the zero-dimensional scheme $\mathrm{Th}(C)$
is equal to
$$2^{2g-2b}\cdot \Bigl(\sum_{\Delta\subset \mathrm{Sing}(C), \Delta\ \mathrm{ even}} 2^{b_1(\Delta)}\Bigr).$$
A component corresponding to an even set $\Delta\subset \mathrm{Sing}(C)$ appears with multiplicity $2^{b-b_1(\Delta)}$.
\end{proposition}
One can easily verify that the length of the scheme $\mbox{Th}(C)$ is indeed $2^{2g}$. From Proposition \ref{components} it follows for instance that $\mbox{Th}(C)$ is a reduced scheme if and only if $C$ is of compact type. In the case we studied above, when $C$ is irreducible with a single node, Proposition \ref{components} gives that $\mbox{Th}(C)=\mbox{Th}(C)_-\cup \mbox{Th}(C)_+$ has $3\cdot 2^{2g-2}$ irreducible components. Precisely, $|\mbox{Th}(C)_-|=2^{2g-2}+|\mbox{Th}(\tilde{C})_-|$ and $|\mbox{Th}(C)_+|=2^{2g-2}+|\mbox{Th}(\tilde{C})_+|$.
\begin{remark}
The structure of the schemes $\mbox{Th}(C)$ for special singular curves $C$ has been used in \cite{Lud} to describe the singularities of $\ss_g$ and in \cite{CS} to prove that a general curve $[C]\in \cM_g$ is uniquely determined by the set of contact hyperplanes $$\bigl\{\langle D\rangle \in (\PP^{g-1})^{\vee}: D\in |\theta|, \ \theta\in \mathrm{Th}(C)_-\bigr\}.$$
In spite of these important applications, a systematic study of the finite geometry of the set $\mbox{Th}(C)$ when $C$ is singular (e.g. a theory of fundamental and G\"opel systems, syzygetic tetrads) has not yet been carried out and is of course quite interesting.
\end{remark}
\vskip 3pt
\subsection{The canonical class of the spin moduli space.}
It is customary to denote the divisor classes in the Picard group of the moduli stack by
\begin{center}
\fbox{$\alpha_i:=[A_i], \ \ \beta_i:=[B_i]\in \mbox{Pic}(\ss_g^-), \ \mbox{ } i=0, \ldots, [\frac{g}{2}].$}
\end{center}
A result of Putman's \cite{Pu} shows that for $g\geq 5$, the divisor classes
$\lambda$, $\alpha_0$, $\beta_0, \ldots, \alpha_{[\frac{g}{2}]}$, $\beta_{[\frac{g}{2}]}$ freely generate the rational Picard group $\mbox{Pic}(\ss_g^-)$. A similar result holds for $\ss_g^+$.
The space $\ss_g^-$ is a normal variety with finite quotient singularities; an \'etale neighbourhood of an arbitrary point $[X, \eta, \beta]\in \ss_g$ is of the form $$H^0(C, \omega_C\otimes \Omega_C)^{\vee}/\mbox{Aut}(X, \eta, \beta)=\mathbb C^{3g-3}/\mbox{Aut}(X, \eta, \beta),$$
where $H^0(C, \omega_C\otimes \Omega_C)$ can be identified via deformation theory with the cotangent space to the moduli stack $\mm_g$ at the point $[C]:=\pi([X, \eta, \beta])$. For a (predictable) definition of an automorphism of a triple $(X, \eta, \beta)$, we refer to \cite{C} Section 1.
\vskip 3pt
Using Kodaira-Spencer deformation theory, one describes the cotangent bundle of the stack $\mm_g$ as the push-forward of a rank $1$ sheaf on the universal curve over $\mm_g$. A famous and at the time, very innovative use in \cite{HM} of the Grothendieck-Riemann-Roch theorem for the universal curve, yields the formula $$K_{\mm_g}\equiv 13\lambda-2\delta_0-3\delta_1-2\sum_{i=2}^{[\frac{g}{2}]} \delta_i\in \mbox{Pic}(\mm_g).$$
From the Riemann-Hurwitz theorem applied to the finite branched cover $\pi:\ss_g^-\rightarrow \mm_g$, we find the formula for the canonical class of the spin moduli stack:
\begin{center}
\fbox{
$K_{\ss_g^-}\equiv 13\lambda-2\delta_0-3\beta_0-3(\alpha_1+\beta_1)-2\sum_{i=2}^{[\frac{g}{2}]} (\alpha_i+\beta_i)\in \mbox{Pic}(\ss_g^-)$.}
\end{center}
An identical formula holds for $\ss_g^+$. Unfortunately both spaces $\ss_g^+$ and $\ss_g^-$ have non-canonical singularities, in particular there exist \emph{local} obstructions to extending pluri-canonical forms defined on the smooth part of $\ss_g$ to a resolution of singularities. However, an important result of Ludwig \cite{Lud} shows that this obstructions are not of global nature. The following result holds for both $\ss_g^+$ and $\ss_g^-$:
\begin{theorem}(Ludwig) For $g\geq 4$ fix a resolution of singularities $\epsilon:\widetilde{\cS}_g\rightarrow \ss_g$. Then for any integer $\ell\geq 0$ there exists an isomorphism of vector spaces
$$\epsilon^*: H^0(\ss_g, K_{\ss_{g, \mathrm{reg}}}^{\otimes \ell})\stackrel{\cong}\longrightarrow H^0(\widetilde{\cS}_g, K_{\widetilde{\cS}_g}^{\otimes \ell}).$$
\end{theorem}
Therefore, in order to conclude that $\ss_g$ is of general type, it suffices to show that the canonical class $K_{\ss_g}$ lies in the interior of the effective cone $\mbox{Eff}(\ss_g)$ of divisors, or equivalently, that it can be expressed as a positive linear combination of an ample and an effective class on $\ss_g$. This becomes a question on slopes of the effective cone of $\ss_g$, which can be solved in a spirit similar to
\cite{HM} and \cite{EH}, where it has been proved with similar methods that $\mm_g$ is of general type for $g\geq 24$. The standard references for effective divisors on moduli spaces of stable curves are \cite{HM}, \cite{EH}, \cite{Log} and \cite{F4}.
\section{Effective divisors on $\ss_g$}
In the papers \cite{F2}, \cite{F3} and \cite{FV1} we have initiated a study of effective divisors on $\ss_g$. The class of the locus $\thet$ of vanishing theta-nulls on $\ss_g^+$ is computed in \cite{F2}; in the paper \cite{FV1} we study the space $\ss_g^-$ with the help of the divisor of spin curves with an everywhere tangent hyperplane in the canonical embedding having higher order contact than expected. Finally in \cite{F3} we define effective divisors of Brill-Noether type in more general setting on both spaces $\ss_g^-$ and $\ss_g^+$. We survey these constructions, while referring to the respective papers for technical details.
We begin with the moduli space $\ss_g^-$: An odd theta characteristic $\theta\in \mbox{Th}(C)$ with $h^0(C, \theta)=1$ determines a unique effective divisor $D\in C_{g-1}$ such that $\theta=\OO_C(D)$. We write in this case that $D=\mbox{supp}(\theta)$.
The assignment $(C, \theta)\mapsto \bigl(C, \mbox{supp}(\theta)\bigr)$ can be viewed as a rational map between moduli spaces
$$\ss_g^-\dashrightarrow \cc_{g, g-1}:=\mm_{g, g-1}/\mathfrak{S}_{g-1},$$
and it is natural to use this map and the well-understood divisor theory \cite{Log} on the universal symmetric product $\cc_{g, g-1}$, in order to obtain promising effective divisors on $\ss_g^-$. In particular, the boundary divisor $\Delta_{0:2}$ on $\cc_{g, g-1}$ with general point being a pair $(C, D)$ where $D\in C_{g-1}$ is a divisor with non-reduced support, is known to be extremal. Its pull-back to $\ss_g^-$ parametrizes (limits of) odd spin curves $[C, \theta]\in \cS_g^-$ such that there exists a point $x\in C$ with $H^0(C, \theta(-2x))\neq 0$. The calculation of the class of the closure of this locus is one of the main results of \cite{FV1}:
\begin{theorem}\label{degen}
We fix $g\geq 3$. The locus consisting of odd spin curves
$$\cZ_g:=\bigl\{[C, \theta]\in \cS_g^-:\theta=\OO_C\bigl(2x_1+\sum_{i=2}^{g-2} x_i\bigr),\ \mbox{ with } x_i\in C \mbox{ for } i=1, \ldots, g-2 \bigr\}$$
is a divisor on $\cS_g^-$. The class of its compactification inside $\ss_g^-$ equals
$$[\overline{\cZ}_g]= (g+8)\lambda-\frac{g+2}{4}\alpha_0-2\beta_0-\sum_{i=1}^{[\frac{g}{2}]} 2(g-i)\ \alpha_i-\sum_{i=1}^{[\frac{g}{2}]} 2i\ \beta_i\in \mathrm{Pic}(\ss_g^-).$$
\end{theorem}
For low genus, $\cZ_g$ specializes to well-known geometric loci. For instance $\cZ_3$ is the divisor of hyperflexes on plane quartics, classifying
pairs $[C, \OO_C(2p)]\in \cS_3^-$, where $p\in C$ is such that $h^0(C, \OO_C(4p))=3$. Then $K_C=\OO_C(4p)$ and $p\in C$ is a hyperflex point.
The divisor $\zz_g$ together with pull-backs of effective divisors on $\mm_g$ can be used to determine the range in which $\ss_g^-$ is of general type. This application also comes from \cite{FV1}:
\begin{theorem}\label{sg-}
The moduli space $\ss_g^-$ is a variety of general type for $g\geq 12$.
\end{theorem}
The passing from Theorem \ref{degen} to Theorem \ref{sg-} amounts to simple linear algebra. By comparing the class $[\zz_g]$ against that of the canonical divisor, we note that $$K_{\ss_g^-}\notin \mathbb Q_{\geq 0}\Bigl\langle [\overline{\cZ}_g], \lambda, \alpha_i, \beta_i, \ i=0, \ldots, \bigl[\frac{g}{2}\bigr]\Bigr\rangle,$$ for the coefficient of $\beta_0$ in the expression of $[\zz_g]$ is too small. However one can combine $[\zz_g]$ with effective classes coming from $\mm_g$, and there is an ample supply of such, see \cite{HM}, \cite{EH}, \cite{F4}. To avoid technicalities, let us assume that $g+1$ is composite and consider the Brill-Noether divisor $\cM_{g, d}^r$ of curves $[C]\in \cM_g$ with a $\mathfrak g^r_d$, where the Brill-Noether number $\rho(g, r, d):=g-(r+1)(g-d+r)=-1$. The class of the closure $\mm_{g, d}^r$ of $\cM_{g, d}^r$ in $\mm_g$ has been computed in \cite{EH} (and in \cite{HM} for $r=1$) and plays a crucial role in the proof by Harris, Mumford, Eisenbud that the moduli space $\mm_g$ is of general type for $g\geq 24$. There exists an explicit constant $c_{g, d, r}>0$ such that the following relation holds \cite{EH},
$$[\mm_{g, d}^r]= c_{g, d, r}\Bigl((g+3)\lambda-\frac{g+1}{6}\delta_0-\sum_{i=1}^{[\frac{g}{2}]} i(g-i)\delta_i\Bigr)\in \mathrm{Pic}(\mm_g).$$
By interpolation, one find a constant $c'_{g, d, r}>0$ such that the effective linear combination
$$\frac{2}{g-2}[\zz_g]+c'_{g, d, r}[\pi^*(\mm_{g, d}^r)]=\frac{11g+37}{g+1}\lambda-2\alpha_0-3\beta_0-\sum_{i=1}^{[g/2]} (a_i\cdot \alpha_i+ b_i),$$
where $a_i, b_i\geq 2$ for $i\neq 1$ and $a_1, b_1>3$ are explicitly known rational constants. By comparison, whenever the inequality
$$\frac{11g+37}{g+1}<13\Leftrightarrow g>12$$
is satisfied (and $g+1$ is composite), the class $K_{\ss_g^-}$ is big, that is, $\ss_g^-$ is of general type. The case $g=12$ is rather difficult and we refer to the last section of \cite{FV1}.
\vskip 3pt
\subsection{The locus of curves with a vanishing theta-null.}
On $\ss_g^+$ we consider the locus of even spin curves with a vanishing theta characteristic. The following comes from \cite{F2}:
\begin{theorem}\label{thetanull}
The closure in $\ss_g^{+}$ of the divisor
$$\Theta_{\mathrm{null}}:=\bigl\{[C, \eta]\in \cS_g^{+}: H^0(C, \eta)\neq
0\bigr\}$$ of curves with an effective even theta characteristics has class equal to
$$[\overline{\Theta}_{\mathrm{null}}]=
\frac{1}{4}\lambda-\frac{1}{16}\alpha_0-\frac{1}{2}\sum_{i=1}^{[\frac{g}{2}]}
\beta_i\in \ \mathrm{Pic}(\ss_g^{+}).$$
\end{theorem}
\noindent In the paper \cite{FV2} it is shown that the class $[\thet]\in \mbox{Eff}(\ss_g^+)$ is extremal when $g\leq 9$. It is an open question whether this is the case for arbitrary $g$, certainly there is no known counterexample to this possibility. Combining $[\thet]$ with pull-backs of effective classes from $\mm_g$ like in the previous case, we find a constant $c^{''}_{g, d, r}>0$ such that
$$8[\overline{\Theta}_{\mathrm{null}}]+c^{''}_{g, d, r}[\pi^*(\mm_{g, d}^r)]=\frac{11g+29}{g+1}\lambda-2\alpha_0-3\beta_0-\sum_{i=1}^{[\frac{g}{2}]}
(a_i'\cdot \alpha_i+b_i'\cdot \beta_i)\in \mathrm{Eff}(\ss_g^+),$$
where the appearing coefficients satisfy the inequalities $a_i', b_i'\geq 2$ for $i\geq 2$ and $a_1', b_1'>3$. Restricting ourselves again to the case when $g+1$ is composite (while referring to \cite{F2} for the remaining cases), we obtain that whenever
$$\frac{11g+29}{g+1}<13\Leftrightarrow g>8,$$ the space $\ss_g^+$ has maximal Kodaira dimension. We summarize these facts as follows:
\begin{theorem}\label{sg+}
The moduli space $\ss_g^+$ is a variety of general type for $g>8$.
\end{theorem}
\section{Unirational parametrizations of $\cS_g$ in small genus}
Next we present ways of proving the unirationality of $\cS_g$ in small genus. The basic references are the papers \cite{FV1} and \cite{FV2}.
We recall that a normal $\mathbb Q$-factorial projective variety $X$ is said to be \emph{uniruled} if through a very general point $x\in X$ there passes a rational curve $R\subset X$. Uniruled varieties have negative Kodaira dimension. Conversely, if the canonical class $K_X$ is not a limit of effective divisor classes (which implies that the Kodaira dimension of $X$ is negative), then $X$ is uniruled, see \cite{BDPP}.
The classification by Kodaira dimension of both $\ss_g^-$ and $\ss_g^+$ is governed by $K3$ surfaces, in the sense that
$\cS_g$ is uniruled precisely when a general spin curve $[C, \theta]\in \cS_g$ can be represented as a section of a special $K3$ surface $S$. By varying $C$ in a pencil on $S$, we induce a rational curve in the moduli space $\ss_g$ passing through a general point. The $K3$ surface must have special properties that will allow us to assign a theta characteristic to each curve in the pencil. In the case of even spin curves, the $K3$ surface in question must be of Nikulin type. We refer to \cite{vGS} for an introduction to Nikulin surfaces.
\begin{definition}
A polarized \emph{Nikulin surface} of genus $g\geq 2$ consists of a triple $(S, e, \OO_S(C))$, where $S$ is a smooth $K3$ surface, $e\in \mbox{Pic}(S)$ is a non-trivial line bundle with the property that $e^{\otimes 2}=\OO_S(N_1+\cdots+N_8)$, where $N_1, \ldots, N_8$ are pairwise disjoint $(-2)$ curves on $S$, and $C\subset S$ is a numerically effective curve class such that $C^2=2g-2$ and $C\cdot N_i=0$, for $i=1, \ldots, 8$.
\end{definition}
Since the line bundle $\OO_S(N_1+\ldots+N_8)$ is divisible by two, there exists a double cover $f:\widetilde{S}\rightarrow S$ branched exactly along the smooth rational curves $N_1, \ldots, N_8$. The curve $C\subset S$ does not meet the branch locus of $f$, hence the restriction $f_{| f^{-1}(C)}: f^{-1}(C)\rightarrow C$ is an unramified double covering which induces a non-trivial half-period on $C$. Each curve in the linear system $|\OO_S(C)|$ acquires a half-period in its Jacobian in this way. The following result is quoted from \cite{FV2}:
\begin{theorem}\label{eventheta}
The even spin moduli space $\ss^+_g$ is uniruled for $g \leq 7$.
\end{theorem}
\begin{proof}
Let us choose a general spin curve $[C, \theta]\in \cS_g^+$ and a non-trivial point of order two $\eta\in \mathrm{Pic}^0(C)[2]$, such that $h^0(C, \theta\otimes \eta)\geq 1$. Because of the generality assumption it follows that $h^0(C, \theta\otimes \eta)=1$ and the support of $\theta\otimes \eta$ consists of $g-1$ distinct points $p_1, \ldots, p_{g-1}$. To simplify matters assume that $g\neq 6$. Then it is proved in \cite{FV2} that the general Prym curve $[C, \eta]\in \cR_g$ is a section of a genus $g$ polarized Nikulin surface $(S, e)$, that is, $C\subset S$ and $\eta=e\otimes \OO_C$. We consider the map induced by the linear system
$$
\varphi_{|\OO_S(C)|}: S\rightarrow \PP^g.
$$
The points $\varphi(p_1), \ldots, \varphi(p_{g-1})$ span a codimension $2$ linear subspace. Let
$\PP\subset |\OO_S(C)|$
be the pencil of curves on $S$ induced by the hyperplanes in $\PP^g$ through $\varphi(p_1), \ldots, \varphi(p_{g-1})$. Each curve $C' \in \PP$ contains the divisor $p_1+\cdots+p_{g-1}$ as an \emph{odd} theta characteristic. The line bundle $\OO_{C'}(p_1+\cdots+p_{g-1})\otimes e_{C'}\in \mathrm{Pic}^{g-1}(C')$ is an \emph{even} theta characteristic on each curve $C'$, because as already discussed, the Arf invariant remains constant in a family of spin curves. This procedure induces a rational curve in moduli
$$
m: \PP \to \ss^+_g, \ \ \mbox{ } m(C'):=[C', e\otimes \OO_{C'}(p_1+\cdots+p_{g-1})],
$$
which passes through the general point $[C, \theta]\in \ss_g^+$ and finishes the proof.
\end{proof}
Observe that in this proof, if instead of being a Nikulin surface, $S$ is an arbitrary $K3$ surface containing $C$, the same reasoning can be used to construct a rational curve in $\ss_g^-$ that passes through a general point, provided the curve $C$ we started with, has general moduli. A general curve of genus $g$ lies on a $K3$ surface if an only if $g\leq 9$ or $g=11$, see \cite{M1}. The case $g=10$ can be handled via a slightly different idea, see \cite{FV1} Theorem 3.10. Thus one also has the following result:
\begin{theorem}
The odd spin moduli space $\ss_g^-$ is uniruled for $g\leq 11$.
\end{theorem}
\subsection{Odd theta characteristics and Mukai models of $\cS_g^-$.} We explain the strategy pursued in \cite{FV1} to construct
alternative models of moduli spaces of odd spin curves which can then be used to establish unirationality of the moduli space:
\begin{theorem}\label{sgunir}
$\ss_g^-$ is unirational for $g\leq 8$.
\end{theorem}
The main idea is to construct a dominant map over $\ss_g^-$ from the total space of a projective bundle over a space parametrizing spin curves on nodal curves of smaller genus. We begin by recalling that Mukai, in a series of well-known papers \cite{M1}, \cite{M2}, \cite{M3}, \cite{M4}, has found ways of representing a general canonical curve of genus $g\leq 9$ as a linear section of a certain $n_g$-dimensional rational homogeneous variety $V_g\subset \PP^{n_g+g-2}$, which we shall call the \emph{Mukai variety} of genus $g$. One has the following list:\bigskip \par \noindent \it
- $V_9$: the Pl\"ucker embedding of the symplectic Grassmannian $SG(3,6)\subset \PP^{13}$, \par \noindent
- $V_8$: the Pl\"ucker embedding of the Grassmannian $G(2,6)\subset \PP^{14}$, \par \noindent
- $V_7$: the Pl\"ucker embedding of the orthogonal Grassmannian $OG(5,10)\subset \PP^{15}$. \bigskip \par \noindent \rm
Inside the Hilbert scheme of curvilinear sections of $V_g$, we denote by $\cU_g$ the open subset
classifying nodal sections $C\subset V_g$ by a linear space of dimension $g-1$.
The automorphism group $\mbox{Aut}(V_g)$ acts on $\cU_g$ and we call the GIT quotient
$$\mathfrak{M}_g:=\cU_g\dblq \mbox{Aut}(V_g)$$
the \emph{Mukai model} of the moduli space of curves of genus $g$. Note that with our definition, the variety $\mathfrak{M}_g$ is only quasi-projective and the Picard number of $\mathfrak{M}_g$ is equal to $1$. The moduli map $\cU_g\rightarrow \mm_g$ being $\mbox{Aut}(V_g)$-invariant,
it induces a regular map $\phi_g:\mathfrak{M}_g\rightarrow \mm_g$. We can paraphrase Mukai's results as stating that the map
$\phi_g: \mathfrak{M}_g \rightarrow \mm_g$ is a birational isomorphism, or equivalently, the general $1$-dimensional linear section of $V_g$ is a curve with general moduli. The map $\phi_g$ deserves more study and in principle it can be used to answer various questions concerning the cohomology of $\mm_g$ or the minimal model program of the moduli space of curves (see \cite{Fed} for a case in point when $g=4$).
The following concept is key to our parametrization of $\ss_g^-$ using Mukai models.
\begin{definition} Let $\mathfrak Z_{g-1}$ be the space of \emph{clusters}, that is, $0$-dimensional schemes $Z\subset V_g$ of length $2g-2$ with the following properties:
\begin{itemize}
\item[(1)] $Z$ is a hyperplane section of a smooth curve section $[C] \in \cU_g$,
\item[(2)] $Z$ has multiplicity two at each point of its support,
\item[(3)] $\mathrm{supp}(Z)$ consists of $g-1$ linearly independent points.
\end{itemize} \par \noindent
\end{definition} \par \noindent
A general point of $\mathfrak{Z}_{g-1}$ corresponds to a $0$-cycle $p_1+\cdots+p_{g-1}\in \mbox{Sym}^{g-1}(V_g)$ satisfying $$\mbox{dim } \langle p_1, \ldots, p_{g-1}\rangle\cap \mathbb T_{p_i}(V_g)\geq 1, \mbox{ for } \ i=1, \ldots, g-1.$$ Furthermore, $\mathfrak{Z}_{g-1}$ is birational to the subvariety of the Grassmannian variety $G(g-1, n_g+g-1)$ parametrizing $(g-2)$-dimensional planes $\Lambda\subset \PP^{n_g+g-2}$ such that $\Lambda \cdot V_g=2p_1+\cdots+2p_{g-1}$, where $p_1, \ldots, p_{g-1}\in V_g$. Then we consider the incidence correspondence:
\begin{center}
\fbox{
$\cU^-_g := \bigl\{ (C,Z) \in \cU_g \times \mathfrak{Z}_{g-1}: Z \subset C \bigr\}.$
}
\end{center}
The first projection map $\cU^-_g \to \cU_g$
is finite of degree $2^{g-1}(2^g-1)$; its fibre at a general point $[C]\in \cU_g$ corresponding to a smooth curve classifies odd theta
characteristics of $C$. The spin moduli map $\cU^-_g \dashrightarrow \overline {\mathcal S}_g^-$ induces a birational isomorphism
$$\phi_g^-:\cU_g^-\dblq \mathrm{Aut}(V_g)\rightarrow \ss_g^-.$$
Let us fix now an integer $0 \leq \delta\leq g-1$. We define the locally closed set of pairs consisting of clusters and $\delta$-nodal curvilinear sections of $V_g$, that is,
$$\cU_{g, \delta}^- := \bigl\{(\Gamma, Z)\in \cU_g^- : \mathrm{sing}(\Gamma)\subset \mbox{supp}(Z)\ \ \mathrm{ and } \ \ |\mathrm{sing}(\Gamma)| = \delta \bigr\}.
$$
The quotient of $\cU_{g, \delta}^-$ under the action of the automorphism group of $V_g$ is birational to the locus $B_{g, \delta}^-\subset \ss_g^-$ with general point given by an odd spin structure on a curve whose stable model is an irreducible $\delta$-nodal curve where each of the nodes is "blown-up" and an exceptional component is inserted. Let us fix a general point $(\Gamma, Z)\in \cU_{g, \delta}^-$ and suppose that $\mbox{Sing}(\Gamma)=\{p_1, \ldots, p_{\delta}\}$ and denote the $p_{\delta+1}, \ldots, p_{g-1}\in \Gamma_{\mathrm{reg}}$ the remaining points in the support of $Z$. If $\nu:N\rightarrow \Gamma$ is the normalization map, then we observe that $\OO_N(p_{\delta+1}+\cdots+p_{g-1})\in \mathrm{Th}(N)_-$, which gives rise to a point in the locus $B_{g, \delta}^-$.
\vskip 3pt
The important point now is that over $\cU_{g, \delta}^-$ one can consider an incidence correspondence that takes into account not only a $\delta$-nodal curve together with a cluster, but also all linear sections of $V_g$ that admit the same cluster. Precisely:
\begin{center}
\fbox{$\mathcal{P}_{g, \delta} := \bigl\{ \bigl(C, (\Gamma, Z)\bigr) \in \cU_g \times \cU^-_{g, \delta} \ : \ Z \subset C \bigr\}.$}
\end{center}
\vskip 3pt
The variety $\mathcal{P}_{g, \delta}$ comes equipped with projection maps
$$
\begin{CD}
{\cU^-_g} @<{\alpha}<< {\mathcal{P}_{g, \delta}} @>{\beta}>> {\cU^-_{g, \delta}}. \\
\end{CD}
$$
It is shown in \cite{FV1} that $\mathcal{P}_{g, \delta}$ is birational to a projective bundle over $\cU_{g, \delta}^-$ and furthermore, the quotient $\PP_{g, \delta}^-:=\mathcal{P}_{g, \delta}\dblq \mbox{Aut}(V_g)$ is a projective bundle over $B_{g, \delta}^-$. Moreover, it is proved that the projection map
$\alpha$ is dominant if and only if
$$\delta\leq n_g-1.
$$
To summarize these considerations, we have reduced the unirationality of $\ss_g^-$ to two conditions. One is numerical and depends solely on the Mukai variety $V_g$, the other has to do with the geometry of the spin moduli space $B_{g, \delta}^-$ of nodal curves of smaller geometric genus:
\begin{theorem}
For $g\leq 9$, if $n_g$ denotes the dimension of the corresponding Mukai variety $V_g$, the moduli space $\ss_g^-$ is unirational provided there exists an integer $1\leq \delta\leq g-1$ such that
\begin{enumerate}
\item $\delta\leq n_g-1$, \
\item $B_{g, \delta}^-$ is unirational.
\end{enumerate}
\end{theorem}
It turns out that the locus $B_{g, g-1}^-$ is unirational for $g\leq 10$ (see \cite{FV1} Theorem 4.16). However condition (i) is only satisfied when $g\leq 8$, and this is the range for which Theorem \ref{sgunir} is known at the moment.
\section{Geometric aspects of moduli spaces of theta characteristics}
In this section we discuss a few major themes related to aspects of the geometry of $\cS_g$ other than birational classification.
\subsection{The Brill-Noether stratification of $\cS_g$.}
One can stratify theta characteristics by their number of global sections. For an integer $r\geq -1$ let us denote by
$$\cS_g^r:=\bigl\{[C, \theta]\in \cS_g: h^0(C, \theta)\ge r+1,\ \ h^0(C, \theta) \equiv r+1 \mbox{ mod }2\bigr\}.$$
The variety $\cS_g^r$ has a Lagrangian determinantal structure discussed in \cite{H} Theorem 1.10, from which it follows that each component
of $\cS_g^r$ has codimension at most ${r+1\choose 2}$ inside $\cS_g$. This bound also follows from Nagaraj's \cite{Na} interpretation of the tangent space to the stack $\cS_g^r$ which we briefly explain. Fix a point $[C, \theta]\in \cS_g^r$ and form the Gaussian map
$$\psi_{\theta}:\wedge^2 H^0(C, \theta)\rightarrow H^0(C, K_C^{\otimes 2}), \mbox{ } \ s\wedge t\mapsto s\cdot dt-t\cdot ds.$$
More intrinsically, the projectivization of the map $\psi_{\theta}$ assigns to a pencil $\langle s, t\rangle \subset |\theta|$ its ramification divisor. Recalling the identification provided by Kodaira-Spencer theory
$$T_{[C, \theta]}(\cS_g)=H^0(C, K_C^{\otimes 2})^{\vee},$$
it is shown in \cite{Na} that the following isomorphism holds:
\begin{center}
\fbox{$T_{[C, \theta]}(\cS_g^r)=\bigl\{\varphi\in H^0(C, K_C^{\otimes 2})^{\vee}: \varphi_{| \mathrm{Im}\ \psi_{\theta}}=0\bigr\}.$}
\end{center}
This description is consistent with the bound $\mbox{codim}(\cS_g^r, \cS_g)\leq {r+1\choose 2}$ from \cite{H}. We now ask what is the actual dimension of the strata $\cS_g^r$? Using hyperelliptic curves one can observe that $\cS_g^{[\frac{g-1}{2}]}\neq \emptyset$
even though the expected dimension of this locus as a determinantal variety is very negative. Moreover, the locus $\cS_{3r}^r$ is non-empty and consists of (theta characteristics on) curves $C\subset \PP^r$ which are extremal from the point of view of Castelnuovo's bound. Therefore one cannot hope that the dimension of $\cS_g^r$ be always $3g-3-{r+1\choose 2}$. However this should be the case, and the locus $\cS_g^r$ should enjoy certain regularity properties, when $r$ is relatively small with respect to $g$. We recall the following precise prediction from \cite{F1}:
\begin{conjecture}\label{strat}
For $r\geq 1$ and $g\geq {r+2\choose 2}$, there exists a component of the locus $\cS_g^r$ having codimension ${r+1\choose 2}$ inside $\cS_g$.
\end{conjecture}
The conjecture is proved in \cite{F1} for all integers $1\leq r\leq 9$ and $r=11$. We point out that $\cS_g^1$ coincides with the divisor $\Theta_{\mathrm{null}}$ studied in \cite{T2} and \cite{F2}. To prove Conjecture \ref{strat} it suffices to exhibit a single spin curve $[C, \theta]\in \cS_g^r$ with an injective Gaussian map $\psi_{\theta}$. As further evidence, we mention the following result, see \cite{F1} Proposition 2.4:
\begin{theorem}
We fix $g, r\geq 1$. If $\cS_{g-1}^r$ has a component of codimension ${r+1\choose 2}$ inside $\cS_{g-1}$ then $\cS_g^r$ has a component of codimension ${r+1\choose 2}$ inside $\cS_g$.
\end{theorem}
One could ask whether in the range $g\geq {r+2\choose 2}$, the locus $\cS_g^r$ is pure-dimensional, or even irreducible. Not much evidence in favor of this speculation exists, but there are no counterexamples either. We mention however that $\cS_g^2$ is pure of codimension $3$ in $\cS_g^-$, and when the locus $\cS_g^3$ has pure codimension $6$ in $\cS_g^+$ for $g\geq 8$, see \cite{T1}.
\subsection{Syzygies of theta characteristics and canonical rings of surfaces.} For a spin curve $[C, \theta]\in \cS_g$ one can form the graded ring
of global sections
$$R(C, \theta):=\bigoplus_{n=0}^{\infty} H^0(C, \theta^{\otimes n}).$$
Note that the canonical ring $R(C, K_C)$ appears as a graded subring of $R(C, \theta)$.
\begin{question} For a general $[C, \theta]\in \cS_g$ (or in $\cS_g^r$), describe the syzygies of $R(C, \theta)$.
\end{question}
Some tentative steps in this direction appear in \cite{R}, where it is shown that with a few exceptions, $R(C, \theta)$ is generated in degree at most $3$. The interest in this question comes to a large extent from the study of surfaces of general type. Let $S\subset \PP^{r+1}$ be a canonically embedded surface of general type and assume for simplicity that $H^1(S, \OO_S)=0$ and $r=p_g(S)-1\geq 3$. Then a general hyperplane section $C\in |K_S|$ comes equipped with an $r$-dimensional theta characteristics, that is, $[C, \theta:=\OO_C(1)]\in \cS_g^r$. By restriction there is a surjective morphism of graded rings
$R(S, K_S)\rightarrow R(C, \theta)$ and the syzygies of the two rings are identical, that is,
$$K_{p, q}(S, K_S)\cong K_{p, q}(C, \theta), \ \mbox{ for all } p, q\geq 0.$$
It is worth mentioning that using Green's duality theory \cite{G}, one finds the isomorphism
$$K_{p-1, 2}(C, \theta)^{\vee}\cong K_{r-2, 2}(C, \theta)\ \mbox{ and } K_{p-2, 3}(C, \theta)^{\vee}\cong K_{r-p+1, 1}(C, \theta)$$
between the various Koszul cohomology groups.
In a departure from the much studied case of syzygies of canonical curves, the graded Betti diagram of a theta characteristic has three
non-trivial rows. For two-torsion points $\eta\in J_2(C)$ a precise \emph{Prym-Green} conjecture concerning the groups $K_{p, q}(C, K_C\otimes \eta)$ has been formulated (and proven for bounded genus) in \cite{FL}. There is no clear prediction yet for the vanishing of $K_{p, q}(C, \theta)$.
\subsection{The Scorza correspondence on the moduli space of even spin curves.} To a non-effective even theta characteristic $[C, \theta]\in \cS_g^+$ one can associate the \emph{Scorza correspondence}
$$R_{\theta}:=\bigl\{(x, y)\in C\times C: H^0(C, \theta(x-y))\neq 0\bigr\}.$$
Denoting by $\pi_1, \pi_2: C\times C\rightarrow C$ the two projections and by $\Delta\subset C\times C$ the diagonal, the cohomology class of the Scorza curve can be computed:
$$\OO_{C\times C}(R_{\theta})= \pi_1^*(\theta)\otimes \pi_2^*(\theta)\otimes \OO_{C\times C}(\Delta).$$ By the adjunction formula, $p_a(R_{\theta})=1+3g(g-1)$. The curve $R_{\theta}$, first considered by Scorza \cite{Sc}, reappears in the modern literature in the beautiful paper \cite{DK}, where it plays an important role in the construction of an explicit birational isomorphism between $\cM_3$ and $\cS_3^+$. It is shown in \cite{FV1} that $R_{\theta}$ is smooth for a general even spin curve, hence one can consider the Scorza map at the level of moduli space, that is,
$$\mathrm{Sc}:\ss_g^+\dashrightarrow \mm_{1+3g(g-1)}, \ \ \mbox{ Sc}[C, \theta]:=[R_{\theta}].$$
Since $\ss_g^+$ is a normal variety, the rational map $\mathrm{Sc}$ extends to a regular morphism outside a closed set of $\ss_g^+$ of codimension at least two. It is of interest to study this map, in particular to answer the following questions:
\noindent (1) What happens to the map $\mathrm{Sc}$ over the general point of the boundary divisor $\thet$, when the determinantal definition of $R_{\theta}$ breaks down?
\noindent (2) What are the degenerate Scorza curves corresponding to general points of the boundary divisors $A_i, B_i\subset \ss_g^+$ for $i=0, \ldots, [\frac{g}{2}]$?
\noindent (3) Understand the Scorza map at the level of divisors, that is, find a complete description of the homomorphism
$$\mathrm{Sc}^*:\mathrm{Pic}(\mm_{1+3g(g-1)})\rightarrow \mathrm{Pic}(\ss_g^+).$$
Answers to all these questions are provided in the forthcoming paper \cite{FI}.
\vskip 3pt
To give one example, we explain one of the results proved. For a general point $[C, \theta]\in \thet$, we denote by $\Sigma_{\theta}$ the \emph{trace curve} induced by the pencil $\theta\in W^1_{g-1}(C)$, that is, $$\Sigma_{\theta}:=\{(x, y)\in C\times C: H^0(C, \theta(-x-y))\neq 0\}.$$ We set
$\delta:=\Sigma_{\theta}\cap \Delta$ and it is easy to see that for a generic choice of $[C, \theta]\in \thet$, the set $\delta$ consists of
$4g-4$ distinct points.
\vskip 3pt
We consider a family $\{(C_t, \theta_t)\}_{t\in T}$ of even theta characteristics over a $1$-dimensional base, such that for a point $t_0\in T$ we have that $[C_{t_0}, \theta_{t_0}]=[C, \theta]\in \thet$ and $h^0(C_t, \theta_y)=0$ for $t\in T_0:=T-\{t_0\}$. In particular, the cycle $R_{\theta_t} \subset C_t\times C_t$ is defined for $t\in T_0$. We prove the following result:
\begin{theorem}
The flat limit of the family of Scorza curves $\{R_{\theta_t}\}_{t\in T_0}$ corresponding
to $t=t_0$, is the non-reduced cycle $$\Sigma_{\theta}+2\Delta\subset C\times C.$$ The associated stable curve $\mathrm{Sc}[C, \theta]\in \mm_{1+3g(g-1)}$ can be described as the transverse union
$\Sigma_{\theta}\cup_{\delta} \widetilde{\Delta}$,
where $\widetilde{\Delta}$ is the double cover of $\Delta$ branched over $\delta$.
\end{theorem}
A proof of this result for $g=3$ using theta functions is given in \cite{GSM}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 4,949
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<?php namespace Buonzz\Scalp\Commands\B2;
use Symfony\Component\Console\Command\Command;
use Symfony\Component\Console\Input\InputArgument;
use Symfony\Component\Console\Input\InputInterface;
use Symfony\Component\Console\Input\InputOption;
use Symfony\Component\Console\Output\OutputInterface;
use Buonzz\Scalp\ExcludedContents;
use ChrisWhite\B2\Client;
use ChrisWhite\B2\Bucket;
class UploadCommand extends Command
{
private $base_folder;
protected function configure()
{
$this
->setName('b2:upload')
->setDescription('Uploads files to BackBlaze')
->addArgument(
'bucket',
InputArgument::REQUIRED,
'Enter the Bucket name where you want to upload the files')
->addArgument(
'folder',
InputArgument::REQUIRED,
'Enter the folder where to read the files to upload');
}
protected function execute(InputInterface $input, OutputInterface $output)
{
$folder = $input->getArgument('folder');
$this->base_folder = $folder;
$account_id = getenv('B2_ACCOUNT_ID');
$application_key = getenv('B2_APPLICATION_KEY');
$bucket_name = $input->getArgument('bucket');
if(!file_exists($folder))
{ $output->writeln('<error>the "'. $folder .'" folder doesn\'t exists!</error>');
exit;
}
$output->writeln("reading files to upload: " . $folder);
$output->writeln('Connecting to BackBlaze using account_id: ' . $account_id);
$b2_client = new Client($account_id, $application_key);
// get all files on destination folder
$files_to_upload = scandir($folder);
usort($files_to_upload, [$this,'compare_time']);
foreach($files_to_upload as $file_to_upload){
if (!in_array($file_to_upload,ExcludedContents::get()) || $file_to_upload == 'files.json'){
try{
if(!$b2_client->fileExists(['BucketName' => $bucket_name,'FileName' => $file_to_upload,]))
{
$output->writeln( "[ ". date("Y-m-d H:i:s") . " ]"
. ' Uploading : <info>' . $file_to_upload . '</info>');
$file = $b2_client->upload([
'BucketName' => $bucket_name,
'FileName' => $file_to_upload,
'Body' => fopen($folder . "/" . $file_to_upload, 'r')
]);
}else
$output->writeln( "[ ". date("Y-m-d H:i:s") . " ]"
. ' Skipped, already exists : <info>' . $file_to_upload . '</info>');
}
catch(\Exception $e){
$msg = $e->getMessage();
if($msg == 'Received error from B2: Authorization token has expired')
{
$output->writeln("Error: Authorization token has expired. restart the app");
exit;
}
$output->writeln("Error: " . $msg);
sleep(5);
continue;
}
} // if
} // foreach
$output->writeln("Success!");
} // execute
function compare_time($a, $b)
{
$timeA = filemtime($this->base_folder . "/".$a);
$timeB = filemtime($this->base_folder . "/".$b);
if($timeA == $timeB) return 0;
return ($timeA < $timeB) ? -1 : 1;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,015
|
Чемпионат СССР по лёгкой атлетике 1923 года проходил 1—8 сентября в Москве на стадионах Всесоюзной сельскохозяйственной выставки и ОППВ.
Призёры
Мужчины
Женщины
Кросс
Места определялись по времени пятого участника команды.
Футбол
С 3 по 9 сентября 1923 года в Москве в рамках Всесоюзного Праздника физкультуры был разыгран Чемпионат СССР по футболу 1923 среди команд городов СССР.
Литература
Спорт в СССР в 1923 году
Сентябрь 1923 года
Соревнования по лёгкой атлетике в Москве
1923 год в Москве
1923 год в лёгкой атлетике
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,418
|
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Another major government contractor on the brink after 90% share price collapse and debt doubling
William J Richardson
"Doing a Carillion". This phrase must surely be soon for the dictionaries, defined something along the lines of: "When a corrupt Government piles contracts on to a debt-riddled private-sector contractor until huge swathes of the nation depend on it, and then the corporation suddenly collapses and workers are left unemployed, projects unfinished, pensions in deficit, but executives floating safely overhead under their golden parachutes".
Today, it appears the Tory Government could once again be stretching its pan-palms out to catch the overspill from the potential collapse of another favoured private contractor, Interserve.
Interserve's debt almost doubled from £274m in 2016 to £513m at the end of 2017. An underestimation of the costs involved in a public-private partnership contract to provide waste-to-energy services, which saw the corporation raise its provision on one such project in Glasgow from £70m to £195m, has badly affected it.
And in further deeply worrying news, the corporation's share price has plummeted from 717p in 2014 to just 63p in December, leading to serious discussions with its lenders over the firm's remaining financial options.
Although the corporation issued profit warnings in September and October 2017, it has announced that it reportedly expects that its 2017 performance was in line with expectations. This, along with the partial recovery in its share price might be due to the fact that the corporation's new chief executive announced cost cutting measures: £15m in 2018 to £50m by 2020.
With the corporation employing 80,000 people worldwide – 25,000 in the UK – and when it is responsible for public contracts including cleaning, healthcare, security, probation and construction, one wonders what cost-cutting measures will actually involve and how these might impact the provision of essential public services.
Indeed, the corporation is also responsible for operating a range of private-finance-initiative funded schools across Northern Ireland. On top of that, in September 2017 Interserve won a place on the £8bn Homes and Communities Agency Panel to deliver mixed-use and residential buildings on public-sector land with private-sector involvement.
Despite the Cabinet Office setting up a team to monitor Interserve, the Government has insisted that the corporation is no Carillion. An ETX Capital analyst has pointed to the fact that Interserve has won contracts in recent months, including a £140m contract to provide facilities services to the BBC and a £227m contract from the DWP.
The fact remains, though, that so had Carillion. Even as it was being shorted, it was winning contracts. Indeed, asset management company Marshal Wace holds the biggest short position against Interserve on the stock market. The partnership also bet heavily against Carillion before its downfall.
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Add to that the fact that it doesn't matter what reassuring statement the Chief Executive puts out or how much the share price bounces. If Interserve has financially overstretched itself and cannot satisfy its creditors, it's going down. If it needs to cut the public sector services that it provides to the bone to avoid that, it will.
It is perverse that the buoyancy of the essential public services on which Britons rely is so dependent on a publicly traded corporation whose fate lies in part in the confidence of investors, and in part in the confidence of private sector creditors.
It is perverse that a private company whose modus operandi is greed, and whose only aim is to simply accumulate profit, can be given so much responsibility over public services.
According to Corbyn, the trumpets of outsourcing's demise are sounding, as in the coming months we could easily see Interserve do a Carillion.
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Private Contractors
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https://www.tremr.com/william-j-richardson/
Will is a freelance writer at Evolve Politics. He is also doing the BPTC at the University of Law with aspirations of becoming a Barrister. Will has two law degrees and lots of opinions.
EXPOSED: Boris Johnson's plan to decimate Workers' Rights, slash Holiday Pay entitlements, bin rest breaks, and force longer hours on workers
Scottish Labour leader Richard Leonard quits just hours after wealthy potential donors pressure Keir Starmer to remove him
UNICEF have been forced to help feed UK kids for the first time ever
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,388
|
{"url":"http:\/\/tuningfly.it\/znch\/xtmixed-stata-ucla.html","text":"I want to drop the observations such that Cod has less than 16 characters. Three Stata commands provide the most general tools for multilevel and mixed-effects modeling. PDF, a 35-page introduction (in a pdf file, to be read by Acrobat Reader, current version March 19, 2007) to MLwiN 2. These posterior probabilities are then used to update our guess of the within-class parameters, which, in turn are used to update the posteriors, and so on until nothing seems to change much. Trivedi, is an outstanding introduction to microeconometrics and how to do microeconometric research using Stata. The types of models fit by these commands sometimes overlap; when this happens, the authors highlight the differences in syntax, data organization, and. A second identical model which allowed intercepts and slopes to vary according to diagnosis. Multilevel models allow: \u2022 Study effects that vary by entity (or groups) \u2022 Estimate group level averages Some advantages: \u2022 Regular regression ignores the average variation between entities. 0000 ----- math | Coef. 0 using the xtmixed command. Data sets and software setups. > > I use the xtmixed command in Stata: > > xtmixed depvar indepvar1 indepvar2 indepvar3 indepvar|| groupvar:, var > > In this model indepvar1 (continuous level-1 variable) becomes highly significant. In SPSS such a procedure is Mixed models, in SAS this can be done using the procedure PROC MIXED, and in Stata using the procedure XTMIXED. The data looks like this. Stata has a friendly dialog box that can assist you in building multilevel models. 5 max = 62 Average RVI = 0. We analyzed the data using Stata version 10 (35). Dear statalisters I have been running an -xi: xtmixed- command like the following in Stata 10 for a large part of the analyses for my PhD: xi: xtmixed [outcomevariableX] i. Module 15: Multilevel Modelling of Repeated Measures Data. In the Stata examples throughout this document, we tell Stata to use REML in order to compare the output with the other four programs. covariate 238. Included is the code for factorial designs, a randomized block design, a randomized block factorial design, three split-plot factorial designs, and a completely randomized hierarchical (nested) design. The former include drawing a stem-and-leaf plot, scatterplot, box-plot, histogram, probability-probability (P-P) plot, and quantile-quantile (Q-Q) plot. I appreciate this clear exposition of effect sizes - Stata's -esize- command is a very useful improvement! However, I would like to add a warning: Users should be aware of a Babylonian confusion of names and symbols (the Stata help to -esize- mentions this confusion, as well): Some renowned scholars denote the pooled but biased effect size measure (here: Cohen's d) as Hedges' g (which to my. The presentation includes Stata code using manova, anova, regress and xtmixed. Use MathJax to format equations. DSS Data Consultant. Included is the code for factorial designs, a randomized block design, a randomized block factorial design, three split-plot factorial designs, and a completely randomized. individuals were sampled within sites (hospitals, companies, community centers, schools, etc. For more complex models, the command xtmixed may be used to estimate a multilevel mixed-effects regression. This might be trivial, but I'm used to HLM7 software output and now I'm switching to Stata (xtmixed). StataCorp. SAS, JMP, and lmer seem to have the ability to do this, and it seems that Stata should, but I have yet to find out how. Chapter 2 Mixed Model Theory. \ud45c\uc900\ud654\uc758 \uae30\uc900(\ubb34\uc5c7\uc5d0 \ub300\ud574 \ud45c\uc900\ud654 \ud558\ub294\uac00?)\uc774 \uc560\ub9e4\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. xtmixed \u2014 Multilevel mixed\u00adeffects linear regression 305 variable wt1would hold the \ufb01rst-level (the observation-level) frequency weights, and wt2would hold the second-level (the school-level) frequency weights. 1EmpiricalAnalysisUsingStata1. Here is my xtmixed code: xi: xtmixed bmiz_ i. Stata has a friendly dialog box that can assist you in building multilevel models. By default, Stata estimates random effects in multilevel mixed models (e. 3 xtmixed as a tool for variance component estimation REML and ML estimates of variance components can be obtained in Stata by using xtmixed for both balanced and unbalanced designs. Posts Tagged 'xtmixed' Multilevel linear models in Stata, part 2: Longitudinal data 18 February 2013 Chuck Huber, Associate Director of Statistical Outreach 10 comments. WEXAMPLE20. The difference between xtreg and xtmixed is that xtreg is designed more for cross-sectional time-series linear regression and can only be used to fit a random intercept. estpost command [ arguments] [, options]. , the probability that a measurement is missing is independent of the measurement itself, given observed data). Active 5 years, 10 months ago. Ask Question Asked 5 years, 10 months ago. As of version 10, Stata contains the xtmixed, xtmelogit, and xtmepoisson commands for fitting multilevel models, in addition to other xt commands for fitting standard random-intercept models. Repeated measures data comes in two different formats: 1) wide or 2) long. intervention) and measures where collected at 3 time points which are baseline (prior to randomisation), 12 weeks and 52 weeks follow-up. 0 using the xtmixed command. Hamilton -8400-6463-2, 978--8400-6463-9, Cengage, 2013 Bridges the gap between statistical texts and the Stata documentation, Statistics with Stata demonstrates how to use Stata to perform a variety of tasks. i have panel data of 1252 firm years observations with 182 firms and time period is 2010-2016. The slope of time now means the change in the probability that you develop the outcome at each time point. 1016\/S2212-5671(15)00077-5 ScienceDirect 7th International Conference on Globalization and Higher Education in Economics and Business Administration, GEBA 2013 A multilevel analysis of life satisfaction in. xtmixed score || schoolid:, mle variance nostderr Performing EM optimization: Performing gradient-based optimization: Iteration 0: log likelihood = -143269. identifier 222. The xtmixed result 22 Alan C. In today's post, I'd like to show you how to use multilevel modeling techniques to analyse longitudinal data with Stata's xtmixed command. London School of Economics and Political Science. Introduction to multilevel linear models in Stata\u00ae, part 1: The -xtmixed- command - Duration: 10:19. The STATA commands used are listed as followings: gen gxt = sex*texp. > > You can get the residuals using -predict-, see the example > below or -help xtmixed postestimation-. Estimates differ slightly because different algorithms are being used. Now in general, this is almost never entirely true. Any suggestion? How to reshape long to wide data in Stata? stata,data-management I have the following data: id tests testvalue 1 A 4 1 B 5 1 C 3 1 D 3 2 A 3 2 B 3 3 C 3 3 D 4 4 A 3 4 B 5 4 A 1 4 B 3 I would like to. time, mle cov(un) In Stata 10 I would usually get my results after 3 or 4 iterations. , matrix algebra) is assumed. \u2022 Importance of variable types: without the double option the d2, sd2, idpartn variables would get wrong values \u2022 Variable types o byte, with range: -127 to 100 o int, with range: -32,767 to 32,740 o long, with range: -2,147,483,647 to 2,147,483,620 o float, with range: -1. Since there are three maternal IQ classes and two group assignments, I should have six lines or growth curves. 1 $\\begingroup$ I am trying to find out if change in marital status have effect on person's health. Fiona Steele. 1016\/S2212-5671(15)00077-5 ScienceDirect 7th International Conference on Globalization and Higher Education in Economics and Business Administration, GEBA 2013 A multilevel analysis of life satisfaction in. Multiple regression using STATA video 1 - Duration: 20:11. > > I use the xtmixed command in Stata: > > xtmixed depvar indepvar1 indepvar2 indepvar3 indepvar|| groupvar:, var > > In this model indepvar1 (continuous level-1 variable) becomes highly significant. individuals were sampled within sites (hospitals, companies, community centers, schools, etc. DOEpatents. However, we have not added new examples. 342 Code would look like this gen newvar = sum(x*b) but I. Multilevel data are characterized by a hierarchical. uk December 2006 1 Survival analysis using Stata 1. hierarchical linear model) The XTMIXED function is for Multilevel mixed-effects linear regressions. Notice this model takes very LONG time to run. 53 Iteration 1: log likelihood = -143269. The Workflow of Data Analysis Using Stata. 1) Because I am a novice when it comes to reporting the results of a linear mixed models analysis, how do I report the fixed effect, including including the estimate, confidence interval, and p. Colin Cameron and Pravin K. This concept of \"before and after\" offers some insight into the estimation of fixed effects models. Multilevel data are characterized by a hierarchical. xtmixed math homework public || schnum: homework, variance covar(un) mle. mixed command to estimate multilevel mixed-effects linear models, also known as mixed-effects, multilevel, or hierarchical models. Some statisticians say you don't do anything differently (at least not in STATA coding) because in long format STATA can determine which variables are level 1 and level 2 by whether or not they. Let's say I have an experiment with three within-subject factors, A, B, & C. Lambert Centre for Biostatistics and Genetic Epidemiology Department of Health Sciences University of Leicester Leicester, UK paul. London School of Economics and Political Science. Here is how you can use mixed to replicate results from xtreg, re. Post a Review You can write a book review and share your experiences. Figure 1: Procedure to detect multicollinearity. DSS Data Consultant. However, your loop gives. mixed or meqrlogit) in the form of variance components - so I get one estimate for an intercept modeled as random effect. In fact, a random effect model is a simple hierarchical linear model with a random intercept. 0000 ----- math | Coef. I appreciate this clear exposition of effect sizes - Stata's -esize- command is a very useful improvement! However, I would like to add a warning: Users should be aware of a Babylonian confusion of names and symbols (the Stata help to -esize- mentions this confusion, as well): Some renowned scholars denote the pooled but biased effect size measure (here: Cohen's d) as Hedges' g (which to my. My main model uses random intercepts (but not random slopes). , in the case of two variables. The compiled table is displayed in the Stata results window or, optionally, written to a text file specified by using filename. 1 What is the stset command? The stset command is used to tell Stata the format of your survival data. xtmixed \u2014 Multilevel mixed\u00adeffects linear regression 305 variable wt1would hold the \ufb01rst-level (the observation-level) frequency weights, and wt2would hold the second-level (the school-level) frequency weights. csv\" for csv and scsv , \". I've just got a 2 week internship in a biostatistics department, where they want me to start a project concerning the variation at hospital level of patient outcomes for an emergency condition that has a high mortality rate, both between hospitals and over a. Here is how you can use mixed to replicate results from xtreg, re. Models were fitted by maximum likelihood using the xtmixed command in Stata 12. Pre-requisites \u2022 Stata practicals for Modules 3 and 5 If you find this module helpful and wish to cite it in your research, please use the following citation: Steele, F. , matrix algebra) is assumed. 1 (StataCorp, College Station, TX). If you click on a highlight, we will spirit you away to our website, where we will describe the feature in a dry. probability 221. Een nette inleiding (pdf) die behalve op de commando's met name ingaat op het verstandig gebruik van Stata in do-files (= Stata's syntax file). Now, after upgrading to Stata 11 the program just keeps doing more and more iterations, without resolving the issue. mi estimate: xtmixed math5 math3 || school: , reml Multiple-imputation estimates Imputations = 5 Mixed-effects REML regression Number of obs = 887 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18. To give an example imagine I have students (level-1) nested within schools (level-2). Stata: Keep only observations with minimum, maximum and median value of a given variable select , stata , vlookup In Stata, I have a dataset with two variables: id and var, and say 1000 observations. xtmixed obamafeel if flagmis==0, || state:, var where flagmis=0 selects those cases that are not missing on the level-1 covariates or level-2 covariate (to be discussed later). Type help hettest or see the Stata reference manual for details. I can't include the data or specific analysis I'm doing for proprietary reasons but will try to include examples and code. Basic Features; Notation for the Mixed Model. Research Data Services Introduction Working with Stata Learning More Navigating Stata When you open Stata, ve windows are immediately visible Command: Where you type in the commands to make Stata go Results: Where your results are displayed Review: A running list of all commands youve used in the order in which youve used them. Stata Practical. Hi, I'm trying to use the margins command after xtmixed in Stata 11. Mixed-effects ML regression Number of obs = 260 Group variable: schnum Number of groups = 10 Obs per group: min = 20 avg = 26. DOEpatents. By clicking here you can download a zipped file MLBOOK. In UNIX, type in \"stata -b do file_name\" to run Stata in the non-interactive. Viewed 24k times 3. The fixed effects are specified as regression parameters. rtf\" for rft , \". In the wide format each subject appears once with the repeated measures in the same observation. The following are web pages that have substantial amounts of information about Stata and HLM. 41775 Prob > chi2 = 0. + , ! \u02c7 \u02c7 \u02c7!! \u02dd%%\u02d8 \u02c7 \u02d8 )! b% \u02d8 \u02d8 ! \u02c6. fitted by maximum likelihood using the xtmixed command in Stata 12. poisson 215. xtmixed has been renamed mixed in Stata 13. qhimwp age_y_cent age_y_centsq i. Repeated measures data comes in two different formats: 1) wide or 2) long. For more complex models, the command xtmixed may be used to estimate a multilevel mixed-effects regression. In fact, a random effect model is a simple hierarchical linear model with a random intercept. As a general paradigm it can be used to handle. Also note that the degrees of freedom for the F default in xtmixed) or the full maximum likelihood (mle, an option in. Module 15: Multilevel Modelling of Repeated Measures Data. Analysis: STATA-software. Here is my xtmixed code: xi: xtmixed bmiz_ i. hierarchical linear model) The XTMIXED function is for Multilevel mixed-effects linear regressions. Random Intercepts, Fixed SlopesThe Stata command for the random intercept, fixed slope model is:xtmixed logtotchg cage || hospid: , variance mleAs with the empty model, the Stata command is XTMIXED and logtotchg is the dependentvariable, while cage is added as a discharge-level. individuals were sampled within sites (hospitals, companies, community centers, schools, etc. Linear Mixed Models are used when there is some sort of clustering in the data. UCLA's Statistical Computing Resources RW Suggestions for Using Stata at Notre Dame (and possibly elsewhere) UCLA's Stata Resources. This concept of \"before and after\" offers some insight into the estimation of fixed effects models. Though we still don't know what units -time- is measured in, you probably want to recenter it by substracting the smallest value. BJ Data Tech Solutions teaches on design and developing Electronic Data Collection Tools using CSPro, and STATA commands for data manipulation. 002878 ** read 1 1084. These posterior probabilities are then used to update our guess of the within-class parameters, which, in turn are used to update the posteriors, and so on until nothing seems to change much. 357 & 367 of the Stata 14. xtmixed and. This is partly because we are writing a book on the gllamm program, including the model framework and applications, to be published by Stata Press. edu \/ STATA ado and hlp files in the package \/ distribution-date: 20101123 Xtmixed; Sample. mixed or meqrlogit) in the form of variance components - so I get one estimate for an intercept modeled as random effect. N = 100, p^ =. Last time, we noticed that our data had two features. Rich Williams' Stata Highlights Page. Gutierrez Director of Statistics StataCorp LP 2008 Fall North American Stata Users Group Meeting R. CFDR Workshop Series. I'll just add one thought to the other useful replies here. Now, after upgrading to Stata 11 the program just keeps doing more and more iterations, without resolving the issue. Back to Estimation. smcl\" for smcl , \". Included is the code for factorial designs, a randomized block design, a randomized block factorial design, three split-plot factorial designs, and a completely randomized. Sanfilippo, Antonio [Richland. FAQ: Linear growth models: xtmixed vs sem. Here is our So the answer to the question, Stata Robust Standard Errors To Heteroskedasticity routine used to create clustered robust standard errors. , matrix algebra) is assumed. The fixed effects are specified as regression parameters. The ml2mixed command (search ml2mixed; (see How can I use the search command to search for programs and get additional help? for more information about using search) can be used to help you convert from the multilevel model specification to the mixed syntax. estimated using the mixed command in Stata. Als je niet tevreden bent met deze intro, Google dan eens op \"Stata tutorial\" of. To >> test if the residuals on the different levels are in fact >> nomally distributed i would like to plot histograms of >> the standardized residuals for level-1 r(ijk), level-2 >> r(jk) and level-3 r(k). i have panel data of 1252 firm years observations with 182 firms and time period is 2010-2016. Let's say I have an experiment with three within-subject factors, A, B, & C. In the Stata examples throughout this document, we tell Stata to use REML in order to compare the output with the other four programs. Posts Tagged 'xtmixed' Multilevel linear models in Stata, part 2: Longitudinal data 18 February 2013 Chuck Huber, Associate Director of Statistical Outreach 10 comments. xtmixed still works. Growth models, spanning 48 to 72 months of age, were estimated with hierarchical linear modeling via STATA\/Xtmixed methods. Dear statalisters I have been running an -xi: xtmixed- command like the following in Stata 10 for a large part of the analyses for my PhD: xi: xtmixed [outcomevariableX] i. Using the xtmixed command in STATA 14. Hi, I'm trying to use the margins command after xtmixed in Stata 11. Three examples are given here to illustrate how xtmixed command is used. s a b c 1 1 1 1 1 1 1 2 1 1 2 1 1 1 2 2 1 2 1 1 1 2 1 2 1 2. xtmixed obamafeel if flagmis==0, || state:, var where flagmis=0 selects those cases that are not missing on the level-1 covariates or level-2 covariate (to be discussed later). I am not aware of any Stata programs (neither. i have 14 independent variables including 3 control variables, 1 mediator and 1 dependent variable. (xtmixed reff*,reffects) In stata we can calculate: (b \u02dc 0 j, b \u02dc 1 j) (b \u02c6 0 j, b \u02c6 1 j) EB: borrow strength across schools MLE: DO NOT borrow strength across Schools. The \"xtmixed\" command in Stata was used to estimate the multilevel linear regression models with random intercepts [48,50]. option a)the association between depvar and indepvar is significant (time is borderline sig p=0. Multiple regression (an extension of simple linear regression) is used to predict the value of a dependent variable (also known as an outcome variable) based on the value of two or more independent variables (also known as predictor variables). Winner of the Standing Ovation Award for \"Best PowerPoint Templates\" from Presentations Magazine. Multi-level modeling accounts for the nesting of time-points within participants, allowing for examination of within- and between-participant change across time (baseline and 3 month follow-up) and by group (expressive writing and control). 2) for each of the continuous diabetes care outcome measures. Mai 2010 23:27 To: [hidden email] Subject: st: How do I graph prediction of mean growth trajectory? Hello all, I am trying to graph expected growth in IQ scores by maternal IQ class and treatment\/control group assignment. xtmixed and. This is likely to be important if you want to calculate marginal effects using -margins-, which expects that you're using the new factor variable syntax introduced in Stata 11. Grilli & C. Models were fitted by maximum likelihood using the xtmixed command in Stata 12. The difference between xtreg and xtmixed is that xtreg is designed more for cross-sectional time-series linear regression and can only be used to fit a random intercept. Re: st: xtmixed & PROC MIXED. xtmixed and mean comparisons 23 Jan 2017, 11:20. Gutierrez Director of Statistics StataCorp LP 2008 Fall North American Stata Users Group Meeting R. Reading Data: \u2022 use Read data that have been saved in Stata format. I have thus accidently managed to press \"ctrl+s\" while in this window accidentaly, and followingly press enter with the intent to execute the command I wrote into the command window. I am using 2 data points panel data. txt\" for fixed and tab , \". Longitudinal data were analyzed with multi-level modeling (MLM) in Stata 12. 2e-16 *** female 1 233. identifier 222. Multilevel and Longitudinal Modeling Using Second Edition SOPHIA RABE-HESKETH ANOERS SKRONOAl Multilevel and Longitudinal Modeling Using Stata Second Edition SOPHIA RABE-HESKETH University of California, Berkeley Institute of Education, University of London ANDERSSKRONDAL London School of Economics Norwegian Institute of Public Health A Stata Press Publication StataCorp LP College Station. The theory behind fixed effects regressions Examining the data in Table 2, it is as if there were four \"before and after\" experiments. Here is our So the answer to the question, Stata Robust Standard Errors To Heteroskedasticity routine used to create clustered robust standard errors. 357 & 367 of the Stata 14. Explore the basics of using the -xtmixed- command to model longitudinal data using Stata. mixed or meqrlogit) in the form of variance components - so I get one estimate for an intercept modeled as random effect. Do Files \u2022 What is a do file?. This STATA 8. The Early Childhood Longitudinal Study-Birth Cohort (ECLS-B) was used to identify predictors of BMI growth trajectories, including child characteristics, maternal attributes, home practices. Following the multilevel mixed-effects reference manual for Stata 13 , the aforementioned standard model where time (slopes) are nested in individuals (intercepts) was used as the homoscedastic model of reference using the Stata function \"xtmixed\". Randomized controlled trial of cognitive behavioral therapy and acceptance and commitment therapy for social phobia: outcomes and moderators Philip Ender and UCLA Statistical Consulting. Graph question for growth model \"xtmixed\" Tuesday, March 31, 2020 Data Cleaning. probit fit nonlinear regression models and examine fixed effects in logit and probit models. 1 for Windows (32-bit). Gutierrez Director of Statistics StataCorp LP 2008 Fall North American Stata Users Group Meeting R. Repeated Measures Analysis with Stata Data: wide versus long. The fixed effects are specified as regression parameters. If this violation is mild, it can be ignored. The output is shown in Figure 1. xtmixed fits linear models, likeamixed-effects counterpart toregress, Similarly, xtmelogit fitsmixed-effects logitregression models forbinary outcomes, likeageneralization. The \"xtmixed\" command in Stata was used to estimate the multilevel linear regression models with random intercepts [48,50]. Many multilevel models can be estimated using mixed model procedures however the syntax will be rather different. Indepvar1 is the only level-1 variable in the model. 002878 ** read 1 1084. Multilevel data are characterized by a hierarchical. xtmixed continues to work but, as of Stata 13, is no longer an official part of Stata. :l'opart ofthlsbook may be reproduced, stored in a retrieval system, or transcribed, in any form. Downloadable! This presentation will give an overview of the three main approaches to analyzing repeated measures analysis of variance: 1) Multivariate models, 2) traditional anova models, and 3) linear mixed models along with discussion of the advantages and disadvantages of each. DAT also used in Chapters 4, 5, 8, and 9 in this textbook, and the. Cross-level interaction of variable sex and texp is included. \u02c7\u02c6 \u02d9\u02d8 \u02dd\u02d8 \u02d8\u02db\u02da\u02dc ! + 2) \u02d8 \u02d8 ) & \u02d8\u02db. Linear Mixed Models are used when there is some sort of clustering in the data. xtmixed\ub294 \uc5c6\ub2e4. To compare the data with past research, we also reported means and t -test results. UCLA Computing STATA Regression page; Simple Linear Model []. Stata now fits nonlinear mixed-effects models, also known as nonlinear multilevel models and nonlinear hierarchical models. Tag: macros,stata. DSS Data Consultant. xtmixed still works. s a b c 1 1 1 1 1 1 1 2 1 1 2 1 1 1 2 2 1 2 1 1 1 2 1 2 1 2. $\\endgroup$ - jtth Mar 7 '13 at 4:19. Mixed models consist of fixed effects and random effects. \u2022 Importance of variable types: without the double option the d2, sd2, idpartn variables would get wrong values \u2022 Variable types o byte, with range: -127 to 100 o int, with range: -32,767 to 32,740 o long, with range: -2,147,483,647 to 2,147,483,620 o float, with range: -1. Since there are three maternal IQ classes and two group assignments, I should have six lines or growth curves. I was suggested to use -meglm- but -meglm- is incompatible with replicate weights. For data in the long format there is one observation for each time period for each subject. In the wide format each subject appears once with the repeated measures in the same observation. 5 max = 62 Average RVI = 0. SAS, JMP, and lmer seem to have the ability to do this, and it seems that Stata should, but I have yet to find out how. Here the reml option specifies that the model will be fit via restricted maximum likelihood rather than the default of maximum likelihood. Oktober 2009 02:38 An: [hidden email] Betreff: st: xtmixed with random slope - STATA 11 vs STATA 10 Hi there, I started using STATA SE\/10 for Mac and then upgraded to STATA SE\/11. This might be trivial, but I'm used to HLM7 software output and now I'm switching to Stata (xtmixed). Re: st: xtmixed & PROC MIXED. xtrc are used to fit hierarchical linear models and random coefficient models. Dear statalisters I have been running an -xi: xtmixed- command like the following in Stata 10 for a large part of the analyses for my PhD: xi: xtmixed [outcomevariableX] i. Making statements based on opinion; back them up with references or personal experience. time, mle cov(un) In Stata 10 I would usually get my results after 3 or 4 iterations. , students within classrooms, or to repeated measurements on each subject over time or space, or to multiple related outcome measures at one. 70141173319*10^36 o double, with range: -8. ppt 44\u9875 \u672c\u6587\u6863\u4e00\u5171\u88ab\u4e0b\u8f7d\uff1a \u6b21 ,\u60a8\u53ef\u5168\u6587\u514d\u8d39\u5728\u7ebf\u9605\u8bfb\u540e\u4e0b\u8f7d\u672c\u6587\u6863\u3002. generate 227. I have trouble to generate a new variable which will be created for every month while having multiple entries for every month. The ml2mixed command (search ml2mixed; (see How can I use the search command to search for programs and get additional help? for more information about using search) can be used to help you convert from the multilevel model specification to the mixed syntax. We generate a simple fake data set : clear set obs 1000 gen u = invnorm(uniform()) gen x. Gutierrez Director of Statistics StataCorp LP 2008 Fall North American Stata Users Group Meeting R. Following the multilevel mixed-effects reference manual for Stata 13 , the aforementioned standard model where time (slopes) are nested in individuals (intercepts) was used as the homoscedastic model of reference using the Stata function \"xtmixed\". 9884656743*10^307 to 8. You can think of them as. Gutierrez (StataCorp) November 13-14, 2008 1 \/ 36. Pre values of the dependent measures were included as a covariate, and Post, 6MFU, and 12MFU were levels of the repeated measures. For continuous outcomes, which were all psychosocial constructs, mixed models (xtmixed in STATA) were used (5). The STATA commands used are listed as followings: gen gxt = sex*texp. Back to Estimation. i have panel data of 1252 firm years observations with 182 firms and time period is 2010-2016. FAQ: Linear growth models: xtmixed vs sem. Thatisadatasetthatisalmostreadyforyourregressionanalysis. > > I use the xtmixed command in Stata: > > xtmixed depvar indepvar1 indepvar2 indepvar3 indepvar|| groupvar:, var > > In this model indepvar1 (continuous level-1 variable) becomes highly significant. Mixed Models A exible approach to correlated data. Repeated measures data comes in two different formats: 1) wide or 2) long. Guidelines for Selecting the Covariance Structure in Mixed Model Analysis Chuck Kincaid, COMSYS Information Technology Services, Inc. The ICC, or Intraclass Correlation Coefficient, can be very useful in many statistical situations, but especially so in Linear Mixed Models. Included is the code for factorial designs, a randomized block design, a randomized block factorial design, three split-plot factorial designs, and a completely randomized hierarchical (nested) design. Posts Tagged \u2018xtmixed\u2019 Multilevel linear models in Stata, part 2: Longitudinal data 18 February 2013 Chuck Huber, Associate Director of Statistical Outreach 10 comments. 1 What is the stset command? The stset command is used to tell Stata the format of your survival data. \u02c7\u02c6 \u02d9\u02d8 \u02dd\u02d8 \u02d8\u02db\u02da\u02dc ! + 2) \u02d8 \u02d8 ) & \u02d8\u02db. qhimwp age_y_cent age_y_centsq i. This FAQ presents some classical ANOVA designs using xtmixed. This talk: overview of panel data methods and xt commands for Stata 10 most commonly used by microeconometricians. 002878 ** read 1 1084. 736 1 1979 2 4878. After fitting the model, it is possible to obtain predictions and standard errors for the fixed portion using -adjust-, or -predict, xb- and -predict, se-. CFDR Workshop Series. Consider a dataset in which students are grouped within schools (from Rabe-Hesketh and Skrondal, Multilevel and Longitudinal Modeling Using Stata, 3rd Edition, 2012). xtmixed obamafeel if flagmis==0, || state:, var where flagmis=0 selects those cases that are not missing on the level-1 covariates or level-2 covariate (to be discussed later). Thatisadatasetthatisalmostreadyforyourregressionanalysis. Note: The xtmixed syntax used on this page is works in Stata 11, but the contrast command needs Stata 12. As of Stata 9, variance components for such designs can be easily estimated with xtmixed. In order to fit non-nested models, we create an artificial level with only one category consisting of all the observations; in addition, we use the notation R. This is the original help file, which we will no longer update, so some links may no longer work. Making statements based on opinion; back them up with references or personal experience. To assess the relationship between FOI and the effectiveness of the care team redesign, we estimated three-level, mixed-effects, multilevel regression models (XTMIXED, STATA 11. qhimwp age_y_cent age_y_centsq i. 1EmpiricalAnalysisUsingStata1. I've just got a 2 week internship in a biostatistics department, where they want me to start a project concerning the variation at hospital level of patient outcomes for an emergency condition that has a high mortality rate, both between hospitals and over a. Confidence Intervals Case II. After fitting the model, it is possible to obtain predictions and standard errors for the fixed portion using -adjust-, or -predict, xb- and -predict, se-. Pre-requisites \u2022 Stata practicals for Modules 3 and 5 If you find this module helpful and wish to cite it in your research, please use the following citation: Steele, F. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. 1 Mixed effects logistic regression is used to model binary outcome variables, in which the log odds of the outcomes are modeled as a linear combination of the predictor variables when data are clustered or there are both fixed and random effects. lab 5: growth curve modeling (from pages 78-87 and 91-94 of the old textbook edition and starting on page 210 of the new edition) Data: Weight gain in Asian children in Britain. Microsoft McKinsey 7S model shows how seven elements of businesses can be aligned to increase effectiveness. Gutierrez Director of Statistics StataCorp LP 2008 Fall North American Stata Users Group Meeting R. We also compare the results with what you get if. xtmixed and mean comparisons 23 Jan 2017, 11:20. Linear Mixed Models are used when there is some sort of clustering in the data. Univariate analyses were calculated for variables. For effects of variables with two categories, use the formulas from Equations 1 - 6 , and 8 for pooling the F - and p -values, for effects of variables with more than two categories use the formulas from. estpost is a tool make results from some of the most popular of these non-\"e-class\" commands available for tabulation. Geological Survey National Elevation Dataset, and comparison with other large-area elevation datasets: SRTM and ASTER. UCLA Computing STATA Regression page; Simple Linear Model []. Additionally, estat imtest displays tests for skew and kurtosis. Longitudinal data were analyzed with multi-level modeling (MLM) in Stata 12. DSS Data Consultant. BA 762 Research Methods course at the University of Kentucky. WEXAMPLE20. We also compare the results with what you get if. 055), however as a categorical variable, option b, the relationship between depvar and indepvar becomes insignificantand time 1 and 2 become very insignificant (p=0. Stata: Keep only observations with minimum, maximum and median value of a given variable select , stata , vlookup In Stata, I have a dataset with two variables: id and var, and say 1000 observations. xtmixed and. 0 log file reports estimations in which CDER Staff Aggregates and PDUFA variable are assigned to drug-months of review for each drug. 1 Overview Correlated data arise frequently in statistical analyses. Hi! I have been having some trouble finding a way to produce some simple standard tables of xtmixed results. Averaging over 50 shots is required in order to get statistics sufficient to uncover a variation in time of the diffraction patterns. The var option causes Stata to report variances rather than standard deviations, which is the default in xtmixed. Stata now fits nonlinear mixed-effects models, also known as nonlinear multilevel models and nonlinear hierarchical models. Learning Stata: Stata Press For your Project\/Data Management Library Long, J. This might be trivial, but I'm used to HLM7 software output and now I'm switching to Stata (xtmixed). Time-resolved diffraction patterns from thin Al foil are recorded. Many softwares, including both SAS and Stata, require the data to be converted to LONG format for analyses. 0000 ----- math | Coef. I've just got a 2 week internship in a biostatistics department, where they want me to start a project concerning the variation at hospital level of patient outcomes for an emergency condition that has a high mortality rate, both between hospitals and over a. Mixed models have both fixed effects and random effects, and are appropriate for cases when observations are clustered in some manner (e. Approximately one-third of 4-year-old females and males were overweight and\/or obese. Test the normality of a variable in Stata. I am running a multilevel model using stata but the data only contains replicate sampling weight. The UCLA page in particular is an important tool for. 2 manual entry for the mixed command. But I wonder: how do I estimate the significance of random effects in multilevel models, such as xtmixed? I suspect there is a reason why neither GLLAMM nor xtmixed gives significance levels and standardized effect sizes, but just coefficients and standard errors. Sorry - forgot the subject on my earlier posting Hello, I am trying to teach myself how to use xtmixed for repeated measures anova. 343 1926m01. Here is our So the answer to the question, Stata Robust Standard Errors To Heteroskedasticity routine used to create clustered robust standard errors. CHAPTER 10 ST 745, Daowen Zhang where g(ZH i (t)) is a vector of function of the history of the covariates that we feel may a\ufb01ect the hazard. We experimentally tested this novel concept at the UCLA Pegasus rf photoinjector. 1 Mixed effects logistic regression is used to model binary outcome variables, in which the log odds of the outcomes are modeled as a linear combination of the predictor variables when data are clustered or there are both fixed and random effects. Therefore in the dialogue box of correlate, enter the independent variables 'pfce' and 'gfcf'. qhimwp age_y_cent age_y_centsq i. Participants should be familiar with the general linear model, but no prior experience with multilevel models or knowledge of advanced mathematics (e. Rampichini - A review of random effects modelling using gllamm in Stata 1. Last time, we noticed that our data had two features. I recently took an introductory course on multilevel modelling. Three specializations to general panel methods: 1 Short panel: data on many individual units and few time periods. xtmixed\ub294 \uc5c6\ub2e4. From the help file for xtmixed: Remarks on specifying random-effects equations. Using anova: anova m method id,repeated(method) I have a significant effect of method and I can compare the methods: test _coef[method[1]] = _coef[method[2]] ( 1) method[1] - method[2] = 0 F( 1, 27) = 58. I'm using xtmixed in Stata for the first time (but am familiar with PROC MIXED in SAS), and am wondering if anybody could please tell me whether or not xtmixed allows for a different covariance structure to be fitted in two different treatment groups? For example, in SAS I would type (in my \"random\" or \"repeated\" statement) \"type=un group=treat\". mixed or meqrlogit) in the form of variance components - so I get one estimate for an intercept modeled as random effect. Getting the most out of xtmixed Tricks of the Trade: Getting the most out of xtmixed Roberto G. Linear growth models: mixed vs sem | Stata FAQ Growth models are a very popular type of analysis. Multilevel data are characterized by a hierarchical. 1 MB) containing the following files:. option a)the association between depvar and indepvar is significant (time is borderline sig p=0. Here is my xtmixed code: xi: xtmixed bmiz_ i. You can think of them as. 2 manual entry for the mixed command. WEXAMPLE20. $\\endgroup$ - jtth Mar 7 '13 at 4:19. intervention. The former include drawing a stem-and-leaf plot, scatterplot, box-plot, histogram, probability-probability (P-P) plot, and quantile-quantile (Q-Q) plot. Rampichini - A review of random effects modelling using gllamm in Stata 1. personid p gender fat 1 capsule M 3. Rich Williams' Stata Highlights Page. The slope of time now means the change in the probability that you develop the outcome at each time point. Small Stata allows string variables to contain a maximum of 80 characters. 0 max = 67 Wald chi2(2) = 65. (xtmixed reff*,reffects) In stata we can calculate: (b \u02dc 0 j, b \u02dc 1 j) (b \u02c6 0 j, b \u02c6 1 j) EB: borrow strength across schools MLE: DO NOT borrow strength across Schools. Now, my models were calculated using Stata's -xtmixed- command, so I've been following Rabe-Hesketh and Skrondal's text on multilevel modeling, 2nd edition, page. For data in the long format there is one observation for each time period for each subject. Useful links for learning Stata. Here the reml option specifies that the model will be fit via restricted maximum likelihood rather than the default of maximum likelihood. RESULTS: Approximately one-third of 4-year-old females and males were overweight and\/or obese. Multilevel models allow: \u2022 Study effects that vary by entity (or groups) \u2022 Estimate group level averages Some advantages: \u2022 Regular regression ignores the average variation between entities. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look that today's audiences expect. Stata 16 is a big release, which our releases usually are. We experimentally tested this novel concept at the UCLA Pegasus rf photoinjector. I am using 2 data points panel data. The cov(un) option tells Stata to treat the random effects covariance matrix as unstructured, meaning that the covariance may also be estimated. I didn't run any of the code, so if it doesnt work, try maybe the help files in Stata or google LR test and melogit with UCLA, usually identifies a nice lecture on the topic from UCLA. N = 100, p^ =. Note: The xtmixed syntax used on this page is works in Stata 11, but the contrast command needs Stata 12. Post a Review You can write a book review and share your experiences. The results from xtreg, mle are equivalent to those from xtmixed, mle. xtmixed command followed by the dependent variable and a list of independent variables. Growth models, spanning 48 to 72 months of age, were estimated with hierarchical linear modeling via STATA\/Xtmixed methods. STATA log file for Time-Varying Covariates (TVC) Duration Model Estimations. Stata 16 is a big release, which our releases usually are. The stata gives the following message default prediction is a function of possibly stochastic quantities other than e(b) I will appreciate any clue. This FAQ presents some classical ANOVA designs using xtmixed. and Consulting \/ UCLA Office of Academic Computing \/ [email\u00a0protected] (xtmixed reff*,reffects) In stata we can calculate: (b \u02dc 0 j, b \u02dc 1 j) (b \u02c6 0 j, b \u02c6 1 j) EB: borrow strength across schools MLE: DO NOT borrow strength across Schools. Indepvar1 is the only level-1 variable in the model. \u2022 insheet Read spreadsheets saved as \"CSV\" files from a package such as Excel. The estat imtest command runs the Cameron-Trivedi decomposition (which includes a test for heteroskedasticity). I'll just add one thought to the other useful replies here. xtmixed fits linear models, likeamixed-effects counterpart toregress, Similarly, xtmelogit fitsmixed-effects logitregression models forbinary outcomes, likeageneralization. Interpreting fixed effects coefficients with categorical independent variables. Version info: Code for this page was tested in Stata 12. Linear growth models: mixed vs sem | Stata FAQ Growth models are a very popular type of analysis. -xi- is not needed for Stata 11 and later. When fitting a regression model, the most important assumption the models make (whether it's linear regression or generalized linear regression) is that of independence - each row of your data set is independent on all other rows. For example, one choice is to use g(ZH i (t)) = Zi(t): If we assume that \u201a(tjZH i (t)) = \u201a0(t)exp(\ufb02TZi(t)); then implicitly we would be assuming that the hazard rate at time t given the entire history of the covariates up to time t is only. Repeated Measures Analysis with Stata Data: wide versus long. Multilevel models allow: \u2022 Study effects that vary by entity (or groups) \u2022 Estimate group level averages Some advantages: \u2022 Regular regression ignores the average variation between entities. Since there are three maternal IQ classes and two group assignments, I should have six lines or growth curves. Geological Survey National Elevation Dataset, and comparison with other large-area elevation datasets: SRTM and ASTER. tab mb_predictable kiwi_club [fw=new_w] if country==2, chi2 business predictabe No-Club Club Total 1 401 146 547 2 253 26 279 3 417 104 521 4 707 126 833 5 110 32 142 Total 1,888 434 2,322 Pearson chi2(4) = 48. 055), however as a categorical variable, option b, the relationship between depvar and indepvar becomes insignificantand time 1 and 2 become very insignificant (p=0. estpost is a tool make results from some of the most popular of these non-\"e-class\" commands available for tabulation. Randomized controlled trial of cognitive behavioral therapy and acceptance and commitment therapy for social phobia: outcomes and moderators Philip Ender and UCLA Statistical Consulting. Consistent with prior studies of moderators (Wolitzky-Taylor et al. The MIXED Procedure: The MIXED Procedure. Tag: macros,stata. Using the xtmixed command in STATA 14. \u2022 infile Read raw data and \"dictionary\" files. Stata's new -asmixlogit- command fits mixed logit models. Oktober 2009 02:38 An: [hidden email] Betreff: st: xtmixed with random slope - STATA 11 vs STATA 10 Hi there, I started using STATA SE\/10 for Mac and then upgraded to STATA SE\/11. 4 1 none M 44. The presentation includes Stata code using manova, anova, regress and xtmixed. Stata 16 is a big release, which our releases usually are. Multiple Regression Analysis using Stata Introduction. Dear StataListers, I have been asked to analyse data from an RCT study with 2 study arms (control vs. If you click on a highlight, we will spirit you away to our website, where we will describe the feature in a dry. 033977 * socst 1 465. SAS, HLM, R, and SPSS use REML by default, while Stata and Mplus use ML. \u2022 insheet Read spreadsheets saved as \"CSV\" files from a package such as Excel. Do Files \u2022 What is a do file?. multilevel 210. 19 The last independent variable is followed by double vertical lines || , after which the grouping variable and random e ects are speci ed. This may be due to group-ing of subjects, e. They'll give your presentations a professional, memorable appearance - the kind of sophisticated look that today's audiences expect. It ranges from lasso to Python and from multiple datasets in memory to multiple chains in Bayesian analysis. The basic syntax of estpost is:. 83 Log likelihood = -875. To compare the data with past research, we also reported means and t -test results. 1 Introduction to Hierarchical Linear Model Hsueh-Sheng Wu. In Stata, you can test normality by either graphical or numerical methods. \u02d8 \u02c7 \u02c6\u02d9\u02dd\u02db\u02dd \u02db \u02d8 \u02c7 \u02d8 \u02d8 \u02db \u02d8 \u2022 \u02da \u02db\u02dc \u02c7 \u02d8 \u02c6 \u02d8\u02c6\u02d9 \u02c6\u02c7 \u02c6 \u02dd \u02d8\u02db \u02d8. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. :\/ b and b*c are big effects in the ANOVA, but aren't coming out as such in any model I try in Stata. 357 & 367 of the Stata 14. StataCorp LLC 113,625 views. , matrix algebra) is assumed. mixed or meqrlogit) in the form of variance components - so I get one estimate for an intercept modeled as random effect. CFDR Workshop Series. Gutierrez (StataCorp) November 13-14, 2008 1 \/ 36. Hello all, I am trying to graph expected growth in IQ scores by maternal IQ class and treatment\/control group assignment. Many growth models can be run either with mixed or sem and yield the same results. Pre-requisites \u2022 Stata practicals for Modules 3 and 5 If you find this module helpful and wish to cite it in your research, please use the following citation: Steele, F. Kijk met name eens naar deze introductie. Cross-level interaction of variable sex and texp is included. Geological Survey National Elevation Dataset, and comparison with other large-area elevation datasets: SRTM and ASTER. Oktober 2009 02:38 An: [hidden email] Betreff: st: xtmixed with random slope - STATA 11 vs STATA 10 Hi there, I started using STATA SE\/10 for Mac and then upgraded to STATA SE\/11. The \"xtmixed\" command in Stata was used to estimate the multilevel linear regression models with random intercepts [48,50]. Linear Mixed Models are used when there is some sort of clustering in the data. Multilevel data are characterized by a hierarchical. Randomized controlled trial of cognitive behavioral therapy and acceptance and commitment therapy for social phobia: outcomes and moderators Philip Ender and UCLA Statistical Consulting. The slope of time now means the change in the probability that you develop the outcome at each time point. 0 that uses the data set MLBOOK1. Multi-level modeling accounts for the nesting of time-points within participants, allowing for examination of within- and between-participant change across time (baseline and 3 month follow-up) and by group (expressive writing and control). Embed Embed this gist in your website. 1{14 A short guide and a forest plot command (ipdforest) for one-stage meta-analysis we will only explore a representative selection of linear random-e ects models in Stata, using the xtmixed command, application to the logistic case using xtmelogit should. lab 5: growth curve modeling (from pages 78-87 and 91-94 of the old textbook edition and starting on page 210 of the new edition) Data: Weight gain in Asian children in Britain. , in the case of two variables. xtmixed obamafeel if flagmis==0, || state:, var where flagmis=0 selects those cases that are not missing on the level-1 covariates or level-2 covariate (to be discussed later). The standard notation for xtmixed assumes that levels are always nested. References []. WEXAMPLE20. estat imtest. I was able to use xtmixed to fit a random intercept random slope model with unstructured covariance in STATA 10 but when I ran the exact same command in the same. How can I analyze a nested model using mixed? | Stata FAQ Please note: The following example is for illustrative purposes only. I've just got a 2 week internship in a biostatistics department, where they want me to start a project concerning the variation at hospital level of patient outcomes for an emergency condition that has a high mortality rate, both between hospitals and over a. If you'd like to see more, please visit the Stata Blog: https:\/\/blo. If the outcome were binary, the model still works. time, mle cov(un) In Stata 10 I would usually get my results after 3 or 4 iterations. Use the ci or cii command. > > I use the xtmixed command in Stata: > > xtmixed depvar indepvar1 indepvar2 indepvar3 indepvar|| groupvar:, var > > In this model indepvar1 (continuous level-1 variable) becomes highly significant. 5 4290 1 1980 2 5537. 0, we performed hierarchical longitudinal analyses to estimate the causal effect of \"winning\" the lottery on each outcome. 2 manual entry for the mixed command. CFAR Biometrics - Longitudinal and Repeated Measures Data (2) 2015_12dec_16 1 Generalized Estimating Equations (GEE) Generalized Linear Mixed Models (GLMM) Focus Called a \"marginal\" mean regression model. For data in the long format there is one observation for each time period for each subject. As with all Stata commands, any modeling options follow a comma (,) after specifying the model variables. doc Author: gl9158 Created Date:. Many softwares, including both SAS and Stata, require the data to be converted to LONG format for analyses. rtf\" for rft , \". Microsoft McKinsey 7S model shows how seven elements of businesses can be aligned to increase effectiveness. To >> test if the residuals on the different levels are in fact >> nomally distributed i would like to plot histograms of >> the standardized residuals for level-1 r(ijk), level-2 >> r(jk) and level-3 r(k). Regards Zubair Khan. Introduction to multilevel linear models in Stata\u00ae, part 1: The -xtmixed- command - Duration: 10:19. The fixed effects are specified as regression parameters. hierarchical linear model) The XTMIXED function is for Multilevel mixed-effects linear regressions. As a general paradigm it can be used to handle. Dear StataListers, I have been asked to analyse data from an RCT study with 2 study arms (control vs. intervention) and measures where collected at 3 time points which are baseline (prior to randomisation), 12 weeks and 52 weeks follow-up. time, mle cov(un) In Stata 10 I would usually get my results after 3 or 4 iterations. , matrix algebra) is assumed. WEXAMPLE20. Data Management Using Stata: A Practical Handbook. BA 762 Research Methods course at the University of Kentucky. This FAQ presents some classical ANOVA designs using xtmixed. Post a Review You can write a book review and share your experiences. Lawrence C. Following the multilevel mixed-effects reference manual for Stata 13 , the aforementioned standard model where time (slopes) are nested in individuals (intercepts) was used as the homoscedastic model of reference using the Stata function \"xtmixed\". Statistics with Stata: Version 12, Eighth Edition, Chapter 15: Multilevel and Mixed-Effects Modeling. Als je niet tevreden bent met deze intro, Google dan eens op \"Stata tutorial\" of. Many multilevel models can be estimated using mixed model procedures however the syntax will be rather different. BJ Data Tech Solutions teaches on design and developing Electronic Data Collection Tools using CSPro, and STATA commands for data manipulation. 1016\/S2212-5671(15)00077-5 ScienceDirect 7th International Conference on Globalization and Higher Education in Economics and Business Administration, GEBA 2013 A multilevel analysis of life satisfaction in. The types of models fit by these commands sometimes overlap; when this happens, the authors highlight the differences in syntax, data organization, and. See -help fvvarlist-. These are choice models that allow researchers to study outcomes such as the choice to walk, ride a bus, or drive a car to work or the. option a)the association between depvar and indepvar is significant (time is borderline sig p=0. xtmixed has been renamed to mixed. The fixed effects are specified as regression parameters. dataset 226. Data from semi-structured interviews. 70141173319*10^38 to 1. I'm using Stata 13. After estimating a model using gllamm, the command gllapred can be used to obtain the posterior means and standard deviations of the latent variables (random effects). Estimates differ slightly because different algorithms are being used. I am currently using Stata\/MP 11. Also, you don't make use of the local employcode_tmp, and that seems to be what you aim for. Using Stata for Confidence Intervals - Page 1. Three examples are given here to illustrate how xtmixed command is used. In other words, there are sales and price data before and after prices change in each of four cities. Repeated measures data comes in two different formats: 1) wide or 2) long. World's Best PowerPoint Templates - CrystalGraphics offers more PowerPoint templates than anyone else in the world, with over 4 million to choose from. If you'd like to see more, please visit the Stata Blog:. If you just have the summary statistics, cii 100 40, level(95) wilson The parameters are the sample size N, the # of successes, the desired confidence. This one is broader than usual. The following is copied verbatim from pp. I was able to use xtmixed to fit a random intercept random slope model with unstructured covariance in STATA 10 but when I ran the exact same command in the same. The standard notation for xtmixed assumes that levels are always nested. xtmixed math homework public || schnum: homework, variance covar(un) mle. The MIXED Procedure: The MIXED Procedure. GettingStarted:\u2022Wewillfirststartwitha\u201chalf-baked\u201ddataset. Hi, I'm trying to use the margins command after xtmixed in Stata 11. :\/ b and b*c are big effects in the ANOVA, but aren't coming out as such in any model I try in Stata. I was suggested to use -meglm- but -meglm- is incompatible with replicate weights. In Stata 13, you can use the. Cross-level interaction of variable sex and texp is included. As with all Stata commands, any modeling options follow a comma (,) after specifying the model variables. stata I have a string variable in Stata called Cod. Using STATA for mixed-effects models (i. We generate a simple fake data set : clear set obs 1000 gen u = invnorm(uniform()) gen x. I was able to use xtmixed to fit a random intercept random slope model with unstructured covariance in STATA 10 but when I ran the exact same command in the same. Stata has a friendly dialog box that can assist you in building multilevel models. Explore the basics of using the -xtmixed- command to model longitudinal data using Stata. \ud45c\uc900\ud654\uc758 \uae30\uc900(\ubb34\uc5c7\uc5d0 \ub300\ud574 \ud45c\uc900\ud654 \ud558\ub294\uac00?)\uc774 \uc560\ub9e4\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. Hello Tom, After checking the -help xtmixed postestimation-, I saw that: -predict fit- is the linear predictor for the fixed portion of the model -predict res, residuals- is the response minus the fitted, where the fitted value is the linear predictor of the fixed portion plus contributions based on predicted random effects So your Y minus fit is not calculating the same thing as res. Linear growth models: mixed vs sem | Stata FAQ Growth models are a very popular type of analysis. Multilevel data are characterized by a hierarchical. fitted by maximum likelihood using the xtmixed command in Stata 12. tdi80cxxuru, 83otorakvr9lpv, 7509zwgl6dex, m5q0jcw6svbvfxd, w0q6oreln3511d, wplvwyo00ye7, s8hlqrlyoy1o, 790ze22lryda, 6uwuxhffaany3pe, 68xpk8ahnfc, btv13lnq60kt3p, zx8l82sqa4ovb5d, 60ftah6ma5, zlmal6z6ek9, yhr1fxv7w1c, 9h0bttuxw6dtqbo, 9hedb2baiq7vu, xldk8ewd2uq1w, 1nlicamlqx, 4h497rcv1ixff, qs0r1r4d7e, 839nsdad3i4, a8e4kn4uefd8952, x9ml4z9yxg267z, uvtp6potq6lk3g, 9nuxi8lunz, t5xv8np9g9, wf6ir9vbxa, j3f6mel4wouf, xzr1nrmjlk","date":"2020-05-27 08:07:20","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\": 0.3644653260707855, \"perplexity\": 4579.950748126399}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2020-24\/segments\/1590347392142.20\/warc\/CC-MAIN-20200527075559-20200527105559-00429.warc.gz\"}"}
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Review & Betting Guide for the Portugal Masters
Arnold Palmers Oceanico Victoria Course © Oceanico Group
Shane Lowry 16/1 © James Kennedy
The Portugal Masters has been held on the Victoria course since 2007 with Steve Webster winning its inaugural event. Spain's Alvaro Quiros shot a final round of 68 to win the 2008 title by three shots ahead of Scotland's Paul Lawrie. Lee Westwood and Richard Green lifted the title in 2009 and 2010 and 20-year-old Tom Lewis, won the 2011 Portugal Masters. Lewis birdied five of the last seven holes to claim his maiden title in only his third professional start. Shane Lowry is the current champion and will be defending his title this week.
Designed by Arnold Palmer the Oceanico Victoria Course is a beautifully manicured track that weighs in at 7,227-yards. The club is situated in the Golf-Mekka Vilamoura area. The terrain is flat but don't let that put you off its a breathtaking design. Each hole has 5 teeing options and the course features are the many lakes and waterfalls as well as the strategically placed bunkers.
Tommy Fleetwood 18/1 © James Kennedy
I am sticking with a couple of players this week and first up is Shane Lowry 16/1 with StanJames, and although he didn't quite challenge come the last round of the Alfred Dunhill links Championship, it was Shane's second Top 10 finish in his last two events, and his forth in his last seven tournaments. He is also teeing up on a course that he won on only two years ago – great memories and great form, so I think we will certainly see Lowry in one of the final pairings come Sunday afternoon.
Another player I am going to stick with, and again comes into this week in great form is Tommy Fleetwood 18/1 with StanJames, who not only had his second consecutive runners-up finish at the Alfred Dunhill links Championship, but it was also his forth Top 5 finish from his last six starts that has seen him win well over half a million Euros in that time. Tommy has never really done well on this track but I think that may change this week.
Bernd Wiesberger 18/1 © James Kennedy
Bernd Wiesberger 18/1 with BetVictor had a rapid move up the leaderboard in the Alfred Dunhill Links Championship firing rounds of 67 and 68 that saw him move into the Top 15, his fifth Top 15 finish in his last six tournaments. The Austrian has had two runners-up placings this year and has not quite been able to just get over the line, but with two Top 5 finishes here in the last two years, I think we may well see him take that final step this week.
Richie Ramsay 33/1 with BetFred is another player who comes into this week in great form and who has four Top 10 finishes from his last five tournaments played. Richie has never been a lover of this course, his best finish being forty-first while firing his best round on this track, a 68 back in 2010. He has obviously matured as a player since then and I think given his current form he will do well here this week.
I am not sure yet as the betting odds are not in yet if Robert-Jan Derksen 80/1 with BetVictor will be an each-way or outside bet, but the next two tips are both nostalgic and emotional bets, something I normally recommend you steer clear of!! Robert had a great Alfred Dunhill Links Championship and he would have loved to have won at the Home of golf before he retires at the end of the year. It's been a long time since Derksen won his last tour event back in 2005 and I think if he is to pull it off before he waves goodbye to the European Tour then it will be on a track that he enjoys playing. The Oceanico Victoria course is that track, Derksen has teed it up here every year since its inception back in 2007 and has only been outside the Top 20 once, could it just be…
Justin Walters 300/1
I must admit I have always believed in being guided, and when Justin Walters 300/1 with PaddyPower broke down crying after he holed that 40-foot putt last year to claim the runners-up spot and secure his European Tour card two weeks after his mothers death, like him I thought there was something in that. He comes into this week in the same position needing to secure his playing card. Justin has recently become a father and not only will it be an emotional week for the South African, don't be surprised if he pulls it off again.
Arnold Palmers Oceanico Victoria Course © Oceanico Golf
TOPICS: Golf, Golf on the Web, Off course, Tournament reviews and betting guides
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Sore Throat News and Research
Important things you need to know about seasonal flu
Seasonal flu is a serious illness causing approximately 200,000 hospitalizations and 36,000 deaths each year. Federal health officials are reporting that the 2019-2020 flu season could be the worst in decades.
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New results from The Environmental Determinants of Diabetes in the Young study show an association between prolonged enterovirus infection and the development of autoimmunity to the insulin-producing pancreatic beta-cells that precedes type 1 diabetes.
San Diego-based Predictive Science, Inc. this week released their first forecast for the 2019-2020 influenza season, which typically runs from November through March.
Good news, bad news: lower antibiotic use, more antibiotic resistance in UK
Public Health England (PHE) is cheering about a significant fall in the number of antibiotic prescriptions in England over the five years between 2014 and 2019, when it has dropped by 17%. However, with more and more bacterial infections showing antibiotic resistance every single year, there is no room for complacency.
Scientists find new strains of human adenovirus in Singapore
Human adenovirus (HAdV) infections in Singapore and Malaysia have caused severe respiratory disease among children and adults in recent years, but scientists still don't know whether these outbreaks are due to new or re-emerging virus strains.
Get vaccinated to protect yourself from flu, say Johns Hopkins experts
Cases of the flu are already on the rise around the nation as flu season begins. Johns Hopkins Medicine experts say now is the time to fight against the flu as the number of people getting sick from the potentially life-threatening virus will increase in the coming months.
New discovery brings scientists closer to developing an RSV vaccine
Scientists have made an important discovery that could lead to a vaccine that will protect against respiratory syncytial virus (RSV).
Infectious disease experts explain how to stay healthy during the flu season
About 40 million people contracted the flu last year, with hundreds of thousands hospitalized and 35,400 to 61,000 deaths, including 134 children, according to the Centers for Disease Control and Prevention.
AI accurately predicts radiation therapy side effects for patients with head and neck cancers
For the first time, a sophisticated computer model has been shown to accurately predict two of the most challenging side effects associated with radiation therapy for head and neck cancer.
Tanzania keeping Ebola under wraps
Ebola could be already on the prowl in Tanzania, but the government is keeping it a secret. This is the official position of the World Health Organization (WHO), who says multiple cases have been secretly reported to it.
New strain of Streptococcus A rampant in England and Wales
According to a new study on cases of scarlet fever and other Streptococcus A infections in England and Wales, the recent rise in cases is due to a newly identified strain. This accounted for the great increase in the number of cases of scarlet fever that occurred in 2014. The last such outbreak was way back in the 1960s.
Scientists discover new strain of group A streptococcus bacteria in England and Wales
A team of scientists led by Imperial College London have discovered a new strain of group A streptococcus bacteria.
WHO declares the deadly Ebola outbreak in Congo a global health emergency
Ebola, the deadly viral infection, is spreading in Congo and the World Health Organization (WHO) has announced it as an international health emergency. Speaking in front of reporters, WHO Director-General Tedros Adhanom Ghebreyesus gave updates on the situation in the Central African country.
FDA approves first drug for treatment of neuromyelitis optica spectrum disorder
The U.S. Food and Drug Administration today approved Soliris (eculizumab) injection for intravenous use for the treatment of neuromyelitis optica spectrum disorder in adult patients who are anti-aquaporin-4 antibody positive.
Ebola spread to Uganda could threaten international health
After smouldering in the Democratic Republic of Congo (DRC) for almost a year, killing near 1,000 people, the deadly Ebola virus crossed the border to Uganda in June 2019. This virus causes sudden high fever and sore throat, with severe weakness and muscle pain.
IVI and MCRI to coordinate global push for developing vaccine against Strep A bacteria
The International Vaccine Institute and Australia's Murdoch Children's Research Institute will coordinate a global push to free the world of Group A Streptococcus, the contagious bacteria that kills half a million people every year and is developing resistance to antibiotics.
New study identifies CD40 molecule as key entry point for dangerous bacteria
A new study identifies a single molecule as a key entry point used by two types of dangerous bacteria to break through cellular barriers and cause disease.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,446
|
namespace RandPass
{
partial class Splash
{
/// <summary>
/// Required designer variable.
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// Clean up any resources being used.
/// </summary>
/// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
protected override void Dispose(bool disposing)
{
if (disposing && (components != null))
{
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows Form Designer generated code
/// <summary>
/// Required method for Designer support - do not modify
/// the contents of this method with the code editor.
/// </summary>
private void InitializeComponent()
{
this.components = new System.ComponentModel.Container();
this.timer1 = new System.Windows.Forms.Timer(this.components);
this.label1 = new System.Windows.Forms.Label();
this.label2 = new System.Windows.Forms.Label();
this.SuspendLayout();
//
// timer1
//
this.timer1.Enabled = true;
this.timer1.Interval = 5000;
this.timer1.Tick += new System.EventHandler(this.timer1_Tick);
//
// label1
//
this.label1.AutoSize = true;
this.label1.BackColor = System.Drawing.Color.Transparent;
this.label1.Font = new System.Drawing.Font("Microsoft Sans Serif", 7.5F, System.Drawing.FontStyle.Bold, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
this.label1.ForeColor = System.Drawing.Color.White;
this.label1.Location = new System.Drawing.Point(12, 422);
this.label1.Name = "label1";
this.label1.Size = new System.Drawing.Size(168, 52);
this.label1.TabIndex = 0;
this.label1.Text = "SOCCENT J6\r\nCENTCOM\r\n7115 S. Boundary Blvd\r\nMacDill AFB, FL 33621-5101";
//
// label2
//
this.label2.AutoSize = true;
this.label2.BackColor = System.Drawing.Color.Transparent;
this.label2.Font = new System.Drawing.Font("Microsoft Sans Serif", 8.25F, System.Drawing.FontStyle.Bold, System.Drawing.GraphicsUnit.Point, ((byte)(0)));
this.label2.ForeColor = System.Drawing.Color.White;
this.label2.Location = new System.Drawing.Point(201, 9);
this.label2.Name = "label2";
this.label2.Size = new System.Drawing.Size(111, 26);
this.label2.TabIndex = 1;
this.label2.Text = "Random Password\r\nGenerator";
this.label2.TextAlign = System.Drawing.ContentAlignment.MiddleCenter;
//
// Splash
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.BackgroundImage = global::RandPass.Properties.Resources.Splash;
this.BackgroundImageLayout = System.Windows.Forms.ImageLayout.Center;
this.ClientSize = new System.Drawing.Size(324, 483);
this.ControlBox = false;
this.Controls.Add(this.label2);
this.Controls.Add(this.label1);
this.FormBorderStyle = System.Windows.Forms.FormBorderStyle.None;
this.Name = "Splash";
this.Text = "Splash";
this.Click += new System.EventHandler(this.Splash_Click);
this.ResumeLayout(false);
this.PerformLayout();
}
#endregion
private System.Windows.Forms.Timer timer1;
private System.Windows.Forms.Label label1;
private System.Windows.Forms.Label label2;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,825
|
Q: do not show function help document in building R package by roxygen2 I am using devtools to build R package, and there are some functions that are NOT designed to be visible to end-users. However, since these functions involve calling C codes by .Call, so that I have to write @useDynLib above the function for automatic generation of .Rd files. In that way, when I build the package, even I did NOT include the @export for those functions, they nonetheless appear in the help document... Is there a way to suppress those functions even if they have been documented? Thanks!
A: According to Hadley's comments, use @keywords internal will make the function invisible to end-users. Details can be found here in the wiki pages of devtools.
A: The wiki linked in the accepted answer no longer discusses @keywords internal (as of April 2016). In case it's helpful for someone to see an example:
# multiplyBy3
#' This is an example of an internal function called \code{multiplyBy3()}
#'
#' Sometimes you want internal functions as part of an R Package built with
#' RStudio and roxygen2, but you don't want .Rd files created for them
#' or to have them be visible in the help document following the build process
#'
#' @keywords internal
#'
#' @param base_num The number to multiply by three
#'
#' @import jsonlite
#'
#' @return Returns a numeric vector
#'
multiplyBy3 <- function(base_number) {
stopifnot(is.numeric(base_number))
return(base_number * 3)
}
Key bits: do not include @export and do include @keywords internal
A: For me, @keywords internal did not work (roxygen2 6.1.1). I was able to achieve the desired result using the following in my roxygen comments:
@noRd
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,208
|
{"url":"http:\/\/mathhelpforum.com\/calculus\/23656-antiderivatives.html","text":"# Math Help - Antiderivatives\n\n1. ## Antiderivatives\n\nHey,\nI have an idea of how to approach this problem, but I dont understand how to solve it. Can someone help me please?\n\nProblem:\nA charged particle moves along the x-axis under the influence of an electric field. The field strength varies with time, and as a result, the velocity of the particle is complicated. The position of the particle at time t is written as x=x(t) and the velocity of the particle at time t is written as v(t).\nSuppose we know that x(0)=0, and also that\nv(t)={2t-1, if 0<t<1\n{4t-3, if 1<t<2\n{6t-7, if 2<t<3\n\nWhat is x(1)? (...find x(2), x(3) and sketch x=x(t), v=v(t))\n\nSo, obviously we can take the antiderivative of v(t) to find x(t) from which you get: $t^2-t+C, 2*t^2-3*t+C, 3*t^2-7*t+C$ ..whats confusing me is the three different intervals for time t. how do i solve for x(1) if there are 2 different antiderivatives for it?? Im confused\n\n2. Originally Posted by coe236\nA charged particle moves along the x-axis under the influence of an electric field. The field strength varies with time, and as a result, the velocity of the particle is complicated. The position of the particle at time t is written as x=x(t) and the velocity of the particle at time t is written as v(t).\nSuppose we know that x(0)=0, and also that\nv(t)={2t-1, if 0<t<1\n{4t-3, if 1<t<2\n{6t-7, if 2<t<3\n\nWhat is x(1)? (...find x(2), x(3) and sketch x=x(t), v=v(t))\n\nSo, obviously we can take the antiderivative of v(t) to find x(t) from which you get: $t^2-t+C, 2*t^2-3*t+C, 3*t^2-7*t+C$ ..whats confusing me is the three different intervals for time t. how do i solve for x(1) if there are 2 different antiderivatives for it?? Im confused\nThe function for x(t) is\n$x = \\int (2t - 1)~dt = t^2 - t + A$ for $0 \\leq t < 1$\n\n$x = \\int (4t - 3)~dt = 2t^2 - 3t + B$ for $1 \\leq t < 2$\n\n$x = \\int (6t - 7)~dt = 3t^2 - 7t + C$ for $2 \\leq t < 3$\n\nWe know that x(0) = 0, so\n$x(0) = 0^2 - 0 + A = 0 \\implies A = 0$\n\nAnd we know that x must be continuous at all times t. Thus\n$\\lim_{t \\to 1}x(t) = 1^1 - 2 = 2 \\cdot 1^2 - 3 \\cdot 1 + B \\implies B = 0$\n\nand\n$\\lim_{t \\to 2}x(t) = 2 \\cdot 2^2 - 3 \\cdot 2 = 3 \\cdot 2^2 - 7 \\cdot 2 + C \\implies C = 4$\n\nThus\n$x(t) = \\left\\{\\begin{array}{rr} t^2 - t,\\text{ if }0 \\leq t < 1 \\\\\n2t^2 - 3t,\\text{ if }1 \\leq t < 2\\\\\n3t^2 - 7t + 4,\\text{ if } 2 \\leq t < 3\\end{array}\\right.$\n\n-Dan","date":"2015-11-28 00:40:09","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\": 0, \"img_math\": 0, \"codecogs_latex\": 12, \"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\": 0.8959630727767944, \"perplexity\": 511.5783857991609}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2015-48\/segments\/1448398450715.63\/warc\/CC-MAIN-20151124205410-00101-ip-10-71-132-137.ec2.internal.warc.gz\"}"}
| null | null |
Q: authentication warning when running Cassandra I am getting the following warning when I execute my scripts.
[warn] c.d.d.c.Connection - /127.0.0.1:9042 did not send an authentication challenge; This is suspicious because the driver expects authentication (configured auth provider = com.datastax.driver.core.PlainTextAuthProvider)
How can I solve this?
A: I suppose the reason could be that I have not configured Cassandra to authenticate incoming connection requests. (https://docs.datastax.com/en/cassandra/3.0/cassandra/configuration/secureConfigNativeAuth.html)
As I am doing unit-tests, this is not required for the moment. I am happy to accept other answer if my reasoning isn't correct.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,701
|
\section{Introduction}
\label{sec:introduction}
The fundamental question of how closed quantum systems admit a
thermodynamic description has motivated the study of their excited
eigenstates and out-of-equilibrium
dynamics~\cite{srednicki1999approach,rigol2008thermalization}. A
central concept is the Eigenstate Thermalisation
Hypothesis (ETH), satisfied by generic ergodic
systems~\cite{deutsch1991quantum,srednicki1994chaos,dalessio2016quantum,deutsch2018eigenstate}.
Positing that eigenstate expectation values of local observables are
smooth functions of the eigenenergies, ETH amounts to a statement
that the energy, an integral of motion, is a state variable so that
local observables in the long-time dynamical state are fully
determined by its value.\footnote{Assuming no other conserved
quantities, for example, total spin, momentum, particle
number etc.} Generic systems satisfy this by default, and any
violation of the ETH is therefore interesting. In a recent
development it was realised that one way ETH can be violated is the
presence of many-body localisation
(MBL)~\cite{basko2006metal,gornyi2005interacting,oganesyan2007localisation,znidaric2008many,pal2010many}
(see
Refs.~\cite{nandkishore2015many,alet2018many,abanin2019colloquium} for
reviews and further references therein). Specifying macroscopic
properties of MBL systems requires an extensive set of emergent
quasi-local integrals of
motion~\cite{serbyn2013local,huse2014phenomenology,ros2015integrals,rademaker2016explicit,imbrie2017local}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{QREMvsEastREM.pdf}
\caption{The Fock-space graph for a system of $N (= 5)$ spins-1/2
where each vertex is a $\sigma^z$-product state as indicated and
the links correspond to spin-flip terms in the Hamiltonian. (a)
The links in red are present for both the QREM as well as for the
EastREM whereas the those in grey are present only for the QREM;
the constraints in the EastREM switches off the latter. (b)
Illustration of how the constraints generally increase the
shortest distance between two vertices. For the pair of vertices
chosen, the path with the QREM is only of length one (green)
whereas for the EastREM the shortest path shown in black is
longer.}
\label{fig:eastrem-vs-qrem}
\end{figure}
Many of the universal properties of ETH systems are well described
within the framework of random matrix theory~\cite{deutsch1991quantum,srednicki1994chaos,dalessio2016quantum,deutsch2018eigenstate,mehta2004random}, where one studies random
matrices incorporating the relevant symmetries of the physical system
instead of actual Hamiltonians.
The physical intuition here is that a random matrix is the ``least
structured'' object which still captures the characteristic properties
of the Hamiltonians of interest. Inspired by this approach, and
focussing on Hamiltonians with finite local Hilbert space dimensions
(spin-1/2 systems constituting possibly the simplest example), we ask
what properties a random many-body Hamiltonian matrix must satisfy so
as to capture the defining properties of MBL systems, namely,
violation of ETH and absence of spatial transport.
As the question pertains to the many-body Hamiltonian, it is natural
to treat the problem directly as one of localisation in Fock
space~~\cite{logan1990quantum,altshuler1997quasiparticle,logan2019many,roy2019self,pietracaprina2016forward,pietracaprina2019hilbert,roy2018exact,roy2018percolation,detomasi2019dynamics,roy2019fock}.
Any many-body Hamiltonian of our type can be interpreted as a
disordered hopping problem on the Fock space of the
system. Considering the Fock basis states as vertices and the hoppings
as links between them, one can view the Hamiltonian matrix as a graph
(see Fig.~\ref{fig:eastrem-vs-qrem}).\footnote{The adjacency matrix of
this graph is closely related to the Hamiltonian expressed in the
Fock basis.} The question then translates to what properties the
graph must have for the system to exhibit non-ergodic behaviour.
In this paper we provide one answer to the question: \emph{constrained
hopping on the Fock space due to local kinetic constraints on the
real-space degrees of freedom can lead to localisation}.
Such constrained dynamics in
translation invariant systems have been to shown to exhibit slow
dynamics and metastable
behaviour~\cite{vanhorssen2015dynamics,lan2018quantum,pancotti2019quantum}, but here we focus on the possibility of a localised phase and accompanying localisation transitions in the eigenstates.
Using exact numerical and approximate analytical techniques we demonstrate that
constraining Fock-space connectivity leads to localisation, and that
this is not due to disorder as the unconstrained version of our model
is disordered but never localised. The Fock space of our model is not
fragmented but is rather fully connected, thus the physics here is
qualitatively different from cases where the constraints fragment Fock
space~\cite{pai2019localization,khemani2019local,sala2019ergodicity}. Furthermore
our model breaks ergodicity strongly, as signified by the presence of a
phase where all eigenstates are localised, thus also differing from
weak ergodicity breaking as in the case of quantum many-body
scars~\cite{turner2018weak,turner2018quantum,khemani2019signatures}. In our case, ergodicity
breaking is due to the states in Fock space naturally grouping into
clusters, with dense intra-cluster but sparse inter-cluster
connections. This leads to potentially non-resonant bottlenecks in the
Fock space, which is the root of localisation.
This establishes the central result of this work -- \emph{how
constrained quantum dynamics can lead to localisation without
fragmenting the Fock space}.
\subsection*{Structure of the paper}
We start with an overview of the paper in Sec.~\ref{sec:overview}
which sets up the Fock space we work with and states the main results
of this paper. In Sec.~\ref{sec:model} we introduce a kinetically
constrained quantum spin-1/2 model to put the ideas on a concrete
footing. The constrained quantum model is based on the quantum random
energy model (QREM) which acts as our reference unconstrained
model. The QREM has been shown to be completely delocalised except for
a vanishing fraction of eigenstates at the spectral
edges~\cite{goldschmidt1990solvable,laumann2014many,baldwin2016manybody,baldwin2017clustering}. We
then impose East model-like
constraints~\cite{ritort2003glassy,garrahan2011kinetically,vanhorssen2015dynamics,garrahan2018aspects},
calling the resulting model the {EastREM}. Section~\ref{sec:phenomenology}
is dedicated to the phenomenology of the model: we map out its phase
diagram using spectral and eigenstate properties in
Sec.~\ref{subsec:spectral}, finding that a fully localised phase
emerges, unlike for the QREM. Dynamical properties further support
this as shown in Sec.~\ref{subsec:dynamics} where we find that an
initial state retains its memory locally in space, reflecting the
locality of the constraints. In Sec.~\ref{subsec:clustering} we
discuss how the constraints impose a particular structure in the
Hamiltonian and construct Hamiltonians random apart from having this
structure, showing that they still display the dynamics of
interest. Finally, in Sec.~\ref{sec:fsa} we use the forward scattering
approximation (FSA) to obtain a (semi)-analytical understanding of
localisation in this model: Secs.~\ref{subsec:bulk} and
\ref{subsec:edges} present an analytical treatment of the FSA for the
spectral bulk and edges respectively, while
Sec.~\ref{subsec:numerical-fsa} presents a numerical treatment of the
FSA, finding agreement with the numerical results of
Sec.~\ref{sec:phenomenology}. The FSA explicitly demonstrates the role
of constraints and reveals clearly the distinction between the
unconstrained (QREM) and constrained ({EastREM}) versions of the model,
explicitly demonstrating the role of the constraints.
\section{Overview}
\label{sec:overview}
Fock space offers a natural viewpoint from which to approach the
problem as any many-body Hamiltonian can be interpreted as a hopping
problem on the Fock-space graph,
\begin{equation}
H=\underbrace{\sum_{\alpha=1}^{N_\mathcal{H}} \mathcal{E}_\alpha\ket{\alpha}\bra{\alpha}}_{H_\mathrm{diag}} + \underbrace{{\sum_{\alpha\neq\beta}}\Gamma_{\alpha\beta} \ket{\alpha}\bra{\beta}}_{H_\mathrm{offdiag}}.
\label{eq:ham-FS}
\end{equation}
Here the set of basis states $\{\ket{\alpha}\}$ are the sites on the
$N_\mathcal{H}$-dimensional Fock-space graph, of which there are exponentially
many (in system size), $N_\mathcal{H}\sim e^N$. The diagonal elements of the
Hamiltonian $\mathcal{E}_\alpha$ are the on-site energies in this
Fock space. The off-diagonal elements $\Gamma_{\alpha\beta}$ then
represent hopping amplitudes. The offdiagonal part of the Hamiltonian,
$H_\mathrm{offdiag}$ also allows us to define a distance on the Fock
space graph -- the distance between two states $\ket{\alpha}$ and
$\ket{\beta},$ denoted as $r_{\alpha\beta}$, is defined as the length of
the shortest path between them following the links generated by
$H_\mathrm{offdiag}$.
Hamiltonian matrices as in Eq.~\eqref{eq:ham-FS} which are associated
with a many-body system (short-ranged with local degrees of freedom)
in general have matrix elements which satisfy two generic
features. First, the Fock-space site energies scale as $\sqrt{N}$,
such that one can define an effective on-site disorder strength on the
Fock space as $W_\mathrm{FS}^2:=N_\mathcal{H}^{-1}\sum_\alpha \braket{\mathcal{E}_\alpha^2}=W^2 N$
and $W$ an $\o(1)$ number. This simply reflects that for generic
short-ranged systems, each Fock-space site energy is an extensive sum
of random numbers. Secondly, the off-diagonal matrix elements are
numbers of magnitude $\o(1)$ and, crucially, the average connectivity
of the Fock-space sites is extensive:
$N_\mathcal{H}^{-1}\sum_{\alpha,\beta}\braket{\Gamma_{\alpha\beta}^2}\sim
N$. This is a result of the fact that for short-ranged systems, the
Hamiltonian connects a state with an extensive number of different
states each differing from the initial one only locally.
If all the Fock-space site energies are independent of each other,
Eq.~\eqref{eq:ham-FS} can be interpreted as an Anderson localisation
problem on a graph with connectivity $N$, hopping amplitude $\Gamma$,
and disorder strength $W_\mathrm{FS}$. Applying the localisation criterion for
Bethe lattices~\cite{abou-chacra1973self}, which we expect to work
well for cases with diverging connectivity, one finds the critical
disorder strength $W_c$ satisfies
$\frac{2e\sqrt{N}\Gamma}{W_c}\ln\left(\frac{W_c\sqrt{N}}{2\Gamma}\right)=1,$
so that $W_c$ diverges in the thermodynamic limit.\footnote{The expression is obtained from the so-called `upper limit approximation', however the exact solution was shown to differ from it by a factor $\approx e/2$~\cite{anderson1958absence,abou-chacra1973self}.} A localised phase
therefore does not exist, at least in the bulk of the spectrum. We
therefore ask what additional ingredients are minimally required to
stabilise a many-body localised phase without altering the generic
features mentioned above.
Elsewhere~\cite{roy2019fock}, one answer to this question was
provided: strong correlations in the $\mathcal{E}_\alpha$, which render the
problem fundamentally different from an Anderson localisation problem
on a high-dimensional graph.\footnote{Here, strong correlations means
that two basis states $\ket{\alpha}$ and $\ket{\beta}$ finitely
distant from each other have Fock-space site energies finitely
different from each other in the thermodynamic limit,
$\vert \mathcal{E}_\alpha-\mathcal{E}_\beta\vert \sim\mathcal{O}(1)$.} In fact, this
is precisely the scenario for local Hamiltonians where the presence of
a localised phase has been argued for on analytical as well as
numerical
grounds~\cite{oganesyan2007localisation,pal2010many,luitz2015many,lev2015absence,imbrie2016many,logan2019many,roy2019fock}.
In this work, we take the complementary perspective and show that,
depending on the pattern and distribution of connectivities, a fully
localised phase may occur even for completely \emph{uncorrelated}
Fock-space disorder, \emph{non-fractured (i.e., fully-connected)} Fock
space, and \emph{typically extensive} connectivity for each site. We
demonstrate this for the case of spatially local kinetic constraints,
which create bottlenecks in the Fock space but leave it fully
connected (every site is accessible from every other).
Although our Fock space is not fragmented, it can be reorganised into
sparsely connected clusters. The picture that emerges is one of sites
densely interconnected within each cluster, but sparse
interconnections between clusters. In other words, the constraints
suppress links between sites belonging to different clusters. We show
that this is the fundamental mechanism which leads to a fully
many-body localised phase in both real and Fock spaces, despite the
Fock-space site energies being uncorrelated and the Fock space not
being fragmented -- this constitutes the central result of this work.
As a concrete setting we consider a system of $N$ quantum spins-1/2
(denoted by the set of Pauli matrices, $\{\sigma^\mu\}$) where the
Fock-space basis states are simply the classical configurations --
product states in the $\sigma^z$-basis. Assigning independent random
energies to the $2^N$ configurations leads to the random energy
model (REM)~\cite{derrida1980random} which, upon addition of spin-flip
terms $\sigma^x$ to the Hamiltonian becomes the QuantumREM
(QREM). This will be our reference unconstrained model and has no
localised phase in the bulk of the spectrum. Imposing East-like
constraints in the spin-flip terms, that is, allowing a particular
spin flip only if the spin to its right is pointing up, results in a
constrained model which we call the EastREM. The construction of the
model and a discussion of implications of the constraints for the
structure of the connectivity of Fock space constitutes
Sec.~\ref{sec:model}.
The phenomenology of the model is established in
Sec.~\ref{sec:phenomenology}. We present results for the statistics of
level spacing ratios and participation entropies of the eigenstates on
the Fock space which reveal a phase diagram with a fully localised
phase. Dynamical autocorrelations from time evolving an initial
product state also show non-ergodic behaviour in the form of retention
of memory of initial configuration. In fact, the real-space profile of
the dynamical autocorrelation directly reflects the effect of the
corresponding local kinetic constraints. Finally, we identify the
clusters made up of densely connected states and then construct a
Hamiltonian matrix where the clusters are described by GOE
Hamiltonians but the matrix elements connecting different clusters are
as for the EastREM. This random matrix analogue to the EastREM, which
we call GOEastREM, displays the relevant features of the EastREM,
demonstrating that the clustering is the crucial ingredient.
Analytical insights into the origin of the localisation on the Fock
space graph are obtained from the FSA, discussed in
Sec.~\ref{sec:fsa}. The FSA is an approximation for the non-local
propagator on the Fock space which takes into account the contribution
only from the shortest paths between two Fock-space sites. As the
constraints essentially have the effect of modifying the statistics of
shortest paths on the Fock space, the FSA is ideally suited for
analysing the EastREM and exposing its differences from the QREM. As
elaborated in Sec.~\ref{sec:fsa}, two aspects of the statistics of
shortest paths are crucial, (i) the scaling of the number of Fock
space sites separated by distance $r$ with both system size and $r$,
and (ii) the scaling of number of paths between such Fock space sites
separated by $r$. These features of the Fock space are inputs to the
FSA, and the results predict an appearance of localised states in the
spectral bulk of the EastREM contrary to the QREM. We also
corroborate the theoretical predictions from the FSA with a numerical
treatment of the FSA by enumerating the directed paths on the Fock
space, and we find that the critical point so obtained is concomitant
with that obtained from exact diagonalisation studies of
Sec.~\ref{sec:phenomenology}.
\section{Constrained quantum model \label{sec:model}}
Our prototypical model for a kinetically constrained quantum system is
one made of $N$ spins-1/2, derived from the QREM by imposing
constraints. The $\sigma^z$-product states constitute the basis
states of our Fock space
$\ket{\alpha}\equiv\ket{\{\sigma^z_i\}_\alpha}$ and to each of them is
associated an independent random energy $\mathcal{E}_\alpha$ drawn from a
normal distribution with zero mean and variance $N$. The diagonal
(first) part of the Hamiltonian of Eq.~(\ref{eq:ham-FS}) is given by
\begin{equation}
H_\mathrm{REM} = \sum_{\alpha=1}^{2^N}\mathcal{E}_\alpha\ket{\{\sigma^z_i\}_\alpha}\bra{\{\sigma^z_i\}_\alpha},
\label{eq:hd}
\end{equation}
with $\mathcal{E}_\alpha\sim\mathcal{N}(0,N)$. Henceforth, we will use the
terms spin-configuration $\ket{\{\sigma^z_i\}_\alpha}$ and Fock-space
site $\ket{\alpha}$ interchangeably.
The QREM is obtained by adding to $H_\mathrm{REM}$ unconstrained single flips
generated by the Hamiltonian
\begin{equation}
\ensuremath{H_{\mathrm{X}}}=\Gamma\sum_{i=1}^N\sigma^x_i,
\label{eq:unconstrained-hopping}
\end{equation}
which corresponds to the second (hopping) term of
Eq.~(\ref{eq:ham-FS}), such that the total Hamiltonian is
\begin{equation}
H_\mathrm{QREM} = H_\mathrm{REM}+\ensuremath{H_{\mathrm{X}}}.
\label{eq:qrem}
\end{equation}
In terms of Fock space sites, the QREM Hamiltonian is precisely a
$N$-dimensional hypercube with $N_\mathcal{H} = 2^N$ vertices each of which has
a connectivity of exactly $N$: Each of the $N$ links on any vertex
corresponds to a flip of a particular spin as the single spin-flips
induced by $\ensuremath{H_{\mathrm{X}}}$ are unconstrained. Another direct
implication of this is that for any Fock-space site, the number of
Fock-space sites at a distance $r$ is\footnote{Note that for the
offdiagonal part of the Hamiltonian given by $\ensuremath{H_{\mathrm{X}}}$ of
Eq.~\eqref{eq:unconstrained-hopping}, the distance between two
Fock-space sites is the same as the usual Hamming distance -- the
number spins different between the two configurations.}
$\binom{N}{r}$.
Localisation or lack thereof in the QREM was studied in
Ref.~\cite{baldwin2016manybody} where it was found that the model is
ergodic in the spectral bulk for infinitesimally small $\Gamma$ while
the spectral edges can have localised eigenstates, so that there are
mobility edges at finite energy densities $\epsilon=E/N\sim
\Gamma$. However, as the width of the density of states
$\sim\sqrt{N}$, in the thermodynamic limit the localised eigenstates
occupy only a vanishing fraction of the spectrum. Generic quantum
dynamics therefore exhibit ergodic behaviour and we consider the
QREM, our reference unconstrained model to be ergodic at all
$\Gamma \neq 0$.
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-fs-dist.pdf}
\caption{(a) The distribution of the connectivities $Z$ on the Fock
space generated by $\ensuremath{H_{\mathrm{East}}}$ \eqref{eq:constrained-hopping}
for different system sizes $N$. The red dashed line shows the same
for $\ensuremath{H_{\mathrm{X}}}$ \eqref{eq:unconstrained-hopping}. While for
the latter it is a delta-function at $Z=N$, for the former it is a binomial distribution and has
support on lower values of $Z$. (b) The distribution of the
shortest distances from the domain wall state for $\ensuremath{H_{\mathrm{East}}}$
(solid lines). The dashed lines show the same for the
$\ensuremath{H_{\mathrm{X}}}$, where it is simply the binomial distribution.}
\label{fig:connectivity-ham-dist}
\end{figure}
The EastREM, our model for a constrained quantum system, is obtained
from the QREM by imposing local constraints of the East type such that
the Hamiltonian is given by
\begin{equation}
H_\mathrm{EastREM} = H_\mathrm{REM} + \ensuremath{H_{\mathrm{East}}},
\label{eq:eastrem}
\end{equation}
where
\begin{equation}
\ensuremath{H_{\mathrm{East}}}=\frac{\Gamma}{2}\sum_{i=1}^N\sigma^x_i(1+\sigma^z_{i+1})
\label{eq:constrained-hopping}
\end{equation}
where we impose periodic boundary conditions, resulting in a Fock
space that is not fragmented.\footnote{In the case of open boundary
conditions, Fock space is fragmented as if the rightmost $w>0$ spins are all down, they remain frozen for all time.} The
constraint modifies the hopping on the Fock space (the second term in
Eq.~(\ref{eq:ham-FS})) so that it allows a spin at real-space site $i$
to be flipped if and only if the spin at site $i+1$ is pointing
up. Hence, in terms of hopping in Fock space, it has the effect of
switching off all the hopping amplitudes of the QREM Hamiltonian that
corresponded to a flip of a spin with the spin to its right pointing
up. A visual demonstration is shown in
Fig.~\ref{fig:eastrem-vs-qrem}(a) where the red links are present for
both the QREM and the EastREM while the blue links are present only on
the QREM. This has a number of consequences.
Firstly, the constraints lead to a suppression of the average
connectivity, although it still scales as $N$. Secondly, the
distribution of connectivities, which is a delta-function at $N$ for
the QREM develops support on lower values as well for the EastREM, see
Fig.~\ref{fig:connectivity-ham-dist}(a). In fact for the EastREM, the
distribution of connectivities is binomial
$P(Z)=\binom{N}{Z}2^{-N}$. Thirdly, the removal of the links generally
increases the shortest distance between two vertices on the Fock
space. For example, Fig.~\ref{fig:eastrem-vs-qrem}(b) shows two sites
that were a single hop away from each other on the QREM Fock-space
graph and which are much further apart on the EastREM Fock-space. This
is studied systematically in Fig.~\ref{fig:connectivity-ham-dist}(b)
where the distribution of shortest distances from a spin-configuration
has larger support on larger values for $\ensuremath{H_{\mathrm{East}}}$ compared to
the $\ensuremath{H_{\mathrm{X}}}$. Finally, the absence of links in the
constrained model also removes a large number of paths connecting any
two vertices (see Fig.~\ref{fig:eastrem-npath-distributions}), the
importance of which will become apparent in Sec.~\ref{sec:fsa}. All
of the above suggest a general tendency of the constraints to localise
a state on the Fock-space. While qualitative now, these pictures will
be important later when we formalise the above ideas using the FSA on
the Fock-space.
In real space, a qualitative picture of the origins of localisation
due to the constraints is as follows. Due to the East-like
constraints, any contiguous block of down spins is slow to thermalise
as it can only do so in a sequential fashion starting from the right
edge of the block. Spins deep inside such blocks, say at a distance
$r$ away from the right edge of the block can flip only at
$r^\mathrm{th}$ order in perturbation theory. By contrast, for the
QREM any spin is free to flip and they can do so in any order.
Furthermore, even the ``liquid'' regions of the chain, which are
regions initially without such frozen blocks are affected by the
constraints dynamically. Thermalising the ``liqud'' regions involves
flipping the up spins to down creating new constrained regions, which
eventually arrest the dynamics.
\section{Phenomenology \label{sec:phenomenology}}
\subsection{Spectral properties and MBL phase
diagram \label{subsec:spectral}}
To establish the phenomenology of the {{EastREM}} in terms of the spectral
properties and obtain an MBL phase diagram we use two commonly
studied numerical diagnostics: statistics of level spacing ratios and
participation entropies of the eigenstates on the Fock space.
The level spacing ratio, $s_n$, is defined
as~\cite{oganesyan2007localisation,pal2010many,atas2013distribution}
${s_n = \min(\Delta_n,\Delta_{n+1})/\max(\Delta_n,\Delta_{n+1})}$ with
$\Delta_n = E_n-E_{n-1}$,
where the $E_n$s denote the consecutive eigenenergies. For an ergodic
system, $s_n$ has a Wigenr-Dyson distribution, reflecting the presence
of level repulsions, so that $\braket{s}\approx 0.53$. A localised
system on the other hand has uncorrelated eigenvalues resulting in
$s_n$ having a Poisson distribution and $\braket{s}\approx0.386$.
The eigenstates on the Fock space also carry signatures of ergodicity
breaking~\cite{deluca2013ergodicity,luitz2015many,mace2019multifractal}. The
$q^\mathrm{th}$ participation entropy of an eigenstate $\ket{\psi}$
defined via
$S^\mathrm{P}_q(\ket{\psi})=\frac{1}{q-1}\ln\left[\sum_\alpha
\vert\braket{\psi|\alpha}\vert^{2q}\right]$ scales
as~\cite{luitz2015many}
\begin{equation}
S^\mathrm{P}_q(\ket{\psi}) = a_q\lnN_\mathcal{H} + b_q\ln\lnN_\mathcal{H}.
\label{eq:pescaling}
\end{equation}
In the ergodic phase $a_q\approx 1$ as a consequence of the
eingenstate being spread over the entire Fock space whereas in the MBL
phase $a_q < 1$ indicating that the support of the eigesntate is a
vanishing fraction of the Fock space dimension in the thermodynamic
limit.
Numerically analysing the two diagnostics using exact diagonalisation
we obtain the MBL phase diagram in the $\epsilon$-$\Gamma$ plane shown
in Fig.~\ref{fig:eastrem-ed}. We emphasise that the density of states
is a Gaussian with a width proportional to $\sqrt{N}$. Hence, any
finite energy density corresponds to the edges of the spectrum where
only a vanishing fraction of the eigenstates live in the thermodynamic
limit (see Fig.~\ref{fig:eastrem-ed}a). It is the middle of the
spectrum, $\epsilon=0$, defined via $\mathrm{Tr}[H]$, which determines the
generic dynamical behaviour of the system.
The critical $\Gamma$ can be obtained from the mean level spacing
ratio by collapsing the data for various $N$ onto a common function of
$g[(\Gamma-\Gamma_c)N^{1/\nu}]$. Such an exercise leads to the set of
critical $\Gamma_c$ at different energy densities shown by the black
circles in Fig.~\ref{fig:eastrem-ed}(b). Representative plots of the
raw data of the mean level spacing ratios in the spectral bulk and
edges are shown in panels (c) and (d) respectively.
The critical line in the $\Gamma$-$\epsilon$ plane so obtained shows a
good agreement with that of the deviation of $a_1$ from 1, the second
diagnostic for the MBL transition. For the {{EastREM}}, a clear MBL phase
emerges at $\epsilon=0$ with a transition to the ergodic phase at
$\Gamma_c\approx 0.17$. This is qualitatively different from the QREM
where at $\epsilon=0$, the model is ergodic at all finite values of
$\Gamma$. Additionally, in the spectral edges (finite $\epsilon$), the
transition from the MBL to ergodic phase occurs at a larger value of
$\Gamma$ in the {{EastREM}} compared to the QREM; this indicates a
parametric increase of the robustness of localised phase in the
presence of the constraints.
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-ed.pdf}
\caption{Localisation phase diagram of the EastREM in the
$\Gamma$-$\epsilon$ plane. (a) The total density of states $\rho(E)$ is a Gaussian with a variance $\sim N$ ($N$ being the system size) such that in terms of energy densities $\epsilon=E/N$, the fraction of eigenstates at all finite $\epsilon$ is vanishingly small in the thermodynamic limit. Note that $\epsilon=0$ corresponds to the middle of the spectrum. (b) The ergodic region (blue) is
characterised by the first participation ratio's volume law
coefficient in Eq.~\eqref{eq:pescaling}, $a_1\approx 1$ as shown
by the colour-map, whereas $a_1<1$ in the MBL phase (light
region). The black dots show the critical $\Gamma$ extracted from
the level spacing ratios for the EastREM whereas the red squares
denote the critical $\Gamma$ line for the
QREM~\cite{baldwin2017clustering}. The black dashed line denotes
the result obtained from a numerical treatment of the FSA
(Sec.~\ref{subsec:numerical-fsa}). (c)-(d) Representative plots of
the mean level spacing ratio, $\braket{s}$, versus $\Gamma$ for
different system sizes $N$ for the bulk and edges of the spectrum
respectively. All the data was averaged over 1000 disorder
realisations and the statistical errorbars estimated using 500
bootstrap resamplings.}
\label{fig:eastrem-ed}
\end{figure}
\subsection{Non-ergodic dynamics \label{subsec:dynamics}}
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-autocorr.pdf}
\caption{Autocorrelation in the EastREM as a function of time,
$A(t)$, starting from the domain wall state for various $\Gamma$
values and system sizes $N$. For small values of $\Gamma$, the
$A(t)$ saturates to a finite value with no perceptible finite-size
effects whereas for larger values of $\Gamma$, it decays with both
$t$ and $N$ like an ergodic system. The critical $\Gamma$ can be
estimated to lie in the vicinity of $\Gamma_c\approx 0.17$. The
data was obtained by averaging over 1000 disorder realisations for
all systems sizes and statistical errors estimated using standard
bootstrap resampling.}
\label{fig:eastrem-autocorr}
\end{figure}
As a dynamical signature of ergodicity breaking, we study the
autocorrelation function
\begin{equation}
A(t)=\frac{1}{N}\sum_{i=1}^N\braket{\psi_0\vert\sigma^z_i(t)\sigma^z_i(0)\vert\psi_0},
\label{eq:autocorr}
\end{equation}
where the initial state is chosen to be the domain wall (DW) state,
$\ket{\psi_0}=\ket{\underbrace{\downarrow\dn\cdots\downarrow\dn}_{N/2}\underbrace{\uparrow\up\cdots\uparrow\up}_{N/2}}$.
The DW state has an extensive connectivity of $N/2$ on the Fock-space
graph, so that arrested dynamics starting from this initial state, if
present, cannot be due to a subextensive connectivity of the initial
state. At the same time, it contains an extensively large blockaded
segment of down spins thus proving to be a convenient choice for
clearly demonstrating the effect of the constraints. We stress that
our choice of the initial state is not special; the phase diagram in
Fig.~\ref{fig:eastrem-ed}(a) shows that there exists a phase where
\emph{all} the eigenstates are localised.
We employ the kernel polynomial method \cite{weisse2006kernel} using
Chebyshev polynomials which allows us to evolve systems with $N=20$ up
to very long times, $t\sim 10^4$. The results for $A(t)$ are shown in
Fig.~\ref{fig:eastrem-autocorr}. For $\Gamma<\Gamma_c$, $A(t)$
saturates to a finite values at long times. The saturated value does
not depend on system size, suggesting that the system retains memory
of its initial condition in the thermodynamic limit at infinite
times. This clearly signifies a strong breaking of ergodicity. In
contrast at larger values of $\Gamma$, $A(t)$ slowly decays with both
$t$ and $N$. The autocorrelation saturates to a finite value for
finite $N$, but this saturation value decays with $N$ such that in the
thermodynamic limit the autocorrelation decays to zero at long
times. This is the hallmark of an ergodic system. While it is
difficult to precisely determine the critical value of $\Gamma$
separating the two dynamical phases, which we estimate to be in the
vicinity of $\Gamma_c\approx 0.17$ (consistently with the exact
diagonalisation results of Sec.~\ref{subsec:spectral}), the existence
of one is clear.
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-autocorr-leftright.pdf}
\caption{Panels (a) and (d) show the spin-expectation value on the
chain (of length $N=20$) at different times (denoted by the
colourbar) for $\Gamma=0.1$ in the localised phase and $0.3$ in
the delocalised phase. While in the former, both the halves of the
chain fail to thermalise, in the latter, both of them do
thermalise although the initially blockaded left half is
slower. The other panels show this via the behaviour of $A_L(t)$
and $A_R(t)$, defined in Eq.~\eqref{eq:autocorr-leftright}, with
$t$ and $N$: In the localised phase (top panels) neither half
thermalises, while in the delocalised phase (bottom panels) both
$L$ and $R$ eventually thermalise, with the initially blockaded
$L$ region thermalising slower.}
\label{fig:eastrem-autocorr-leftright}
\end{figure}
We now turn to the study of the spatially resolved dynamical
autocorrelation. We define the autocorrelation functions separately for
the left and right halves of the chain (which for the DW initial state
correspond to the blockaded and non-blockaded regions),
\begin{equation}
A_{L(R)}(t) = \frac{2}{N}\sum_{i=1 (N/2+1)}^{N/2 (N)}\braket{\psi_0\vert\sigma^z_i(t)\sigma^z_i(0)\vert\psi_0}.
\label{eq:autocorr-leftright}
\end{equation}
Fig.~\ref{fig:eastrem-autocorr-leftright} shows the results for
$A_{L(R)}(t)$ for two values of $\Gamma$ in both localised and
delocalised phases. In the latter both spatial regions thermalise as
reflected in their decay with $N$ and $t$, although the initially
blockaded region is much slower. In the localised phase, both the
regions fail to thermalise as seen by the $N$-independent saturation
of both $A_L(t)$ and $A_R(t)$ at long times.
As anticipated in Sec.~\ref{sec:model}, the breaking of ergodicity
manifested in localised behaviour can be attributed to two effects.
\begin{itemize}
\item [(i)] Because of the East-like constraint of
Eq.~\eqref{eq:constrained-hopping}, any block of contiguous down
spins is slow to ``melt'' since the only spin in that block that can
change dynamically is the one on the rightmost edge. The entire
block can therefore melt only sequentially starting from the
right. In other words, for spin-configurations with such ``solid''
blocks of frozen spins, a large number of channels out of these
configurations, which involve flipping of spins deep in the frozen
block are simply unavailable. Moreover, this also has the effect of
supressing the total number of pathways on the Fock space from one
configuration to another. For example, there is a single shortest
path on the Fock-space graph that connects the DW state to the
all-up state. Contrarily for the QREM, the corresponding number of
shortest paths is $(N/2)! \sim e^N$.
\item [(ii)] In the localised phase, the apparent liquid regions made
up of segments of up spins also don't thermalise, see
Fig.~\ref{fig:eastrem-autocorr-leftright}(a) and (c). The mechanism
underlying this is the creation of new blockades dynamically. Once a
single spin (say at site $i$) is flipped from up to down, the one at
$i-1$ is frozen until the $i^\mathrm{th}$ spin is flipped back
up. However, this flipping is unavoidable; thermalising the region
requires, by definition, that the quantum state explore all other
spin configurations in the Fock space, and these naturally posses
segments of down spins creating new constrained regions which
eventually may lead to localisation.
\end{itemize}
\section{Minimally structured constrained
model \label{subsec:clustering}}
To demonstrate the two effects mentioned at the end of
Sec.~\ref{subsec:dynamics}, we now construct a new model in which the
second effect is removed by hand while the first left in.\footnote{We
focus on constructing a model appropriate to an initial domain wall
state, but this is not a special choice and any spin-configuration
could have been used.} To do so, we recognise that the first effect
above, namely the slowness of the melting of blockaded regions, is due
to the relatively small number of matrix elements leading out of
clusters of states all of which include the same blockaded island,
while the second relates to dynamics inside each cluster.
The model we construct consists of GOE matrices describing each of
these clusters, with each of these matrix blocks connected to the
others by matrix elements which are identical to the same matrix
elements as in the EastREM. It is hence a hybrid of a random matrix
with the EastREM, and arguably the least structured model that still
displays one of the features of the EastREM, namely, the difficulty of
melting the blockaded islands. Unlike the EastREM, liquid regions will
remain liquid under the dynamics of the new model, being fully chaotic
as their dynamics is described by the random matrix.
\begin{figure}
\includegraphics[width=\columnwidth]{goeastrem.pdf}
\caption{Construction of the GOEastREM. (a) The Fock-space graph with
the spin-configurations arranged into clusters (denoted by the
boxes) according to Eq.~\eqref{eq:goeastrem-cluster}. The
offdiagonal parts of the Hamiltonian within the subspace of each
cluster is described by a GOE matrix. The links in red denote
hoppings on the Fock space between spin-configurations in
different clusters as allowed by the East constraints. The
Hamiltonian matrices of the EastREM in (b) and GOEastREM in (c) as
a colour-map. The blocks denoted by the black lines correspond to
the clusters in (a). Note that the inter-cluster matrix elements
of the {{GOEastREM}} are the same as that of the {{EastREM}} whereas the
intra-cluster ones are those of GOE matrices.}
\label{fig:goeastrem}
\end{figure}
To construct our new model, first we group together spin
configurations so that all states with a given length of blockaded
down spins starting from the leftmost spin are in the same group:
\begin{equation}
\begin{matrix}
{\text{Cluster\# 0}}& : & \overbrace{\downarrow\dn\cdots\cdots\downarrow\dn}^{N/2}\overbrace{\circ\circ\cdots\cdots\circ\uparrow}^{N/2},\\
{\text{Cluster\# 1}}& : & \overbrace{\downarrow\dn\cdots\cdots\downarrow}^{N/2-1}\uparrow\overbrace{\circ\circ\cdots\cdots\circ\uparrow}^{N/2},\\
\vdots & \\
{\text{Cluster\# $i$}}& : & \overbrace{\downarrow\dn\cdots\downarrow}^{N/2-i}\uparrow\overbrace{\circ\circ\cdots\cdots\cdots\circ\uparrow}^{N/2+i-1},\\
\end{matrix}
\label{eq:goeastrem-cluster}
\end{equation}
where the $\circ$ denotes sites which could either up or down spins.
We note two features of this separation of Fock space into
clusters:
\begin{itemize}
\item [(i)] Firstly, hoppings in the {EastREM}\ between different clusters
correspond to progressively melting the solid block. This is because
{EastREM}\ only allows either the rightmost spin of a blockaded island or
the first spin after the island to flip, and either of these flips
results in a state in cluster $i\pm 1$ so that transitions are only
allowed between clusters $i$ and $i\pm 1$ by the {EastREM}\ rules.
\item [(ii)] Secondly, flipping spins in the liquid regions
corresponds to Fock-space hoppings within a cluster. These lead to
formation of new constraints as discussed in
Sec.~\ref{subsec:dynamics} and stop the apparently liquid regions
from thermalising in the MBL phase.
\end{itemize}
In the bottom two panels of Fig.~\ref{fig:goeastrem} we show a
representation of the Hamiltonian matrix of the {EastREM}\ (left) and {GOEastREM}\
(right) in the basis of the Fock states, arranged so that states in
the same block are next to each other. The black lines correspond to
the boundaries between blocks, so that the square blocks along the
diagonal of the matrices correspond to transitions inside each
cluster while the off-diagonal blocks to transitions between the
clusters.
To allow spins to flip freely in the liquid regions without the
formation of new blockades, we randomise all matrix elements between
states in the same cluster while keeping the matrix elements between
clusters as in the {EastREM}\ model; in other words, we make the blocks on
the diagonal in Fig.~\ref{fig:goeastrem} GOE matrices while keeping
everything outside them identical to the {EastREM}. This has the effect of
allowing all intra-cluster transitions (that is, dynamics in the
liquid region) with no constraints while keeping the inter-cluster
transitions (corresponding to island melting) as in the {EastREM}\
model.\footnote{For a given disorder realisation, the diagonal
elements, $\{\mathcal{E}_\alpha\}$, are also the same as that of the
{{EastREM}}.} Fig.~\ref{fig:goeastrem} also makes it evident that
decreasing the size of an island by more than 1 still cannot be done
by a single application of the Hamiltonian (there are still no matrix
elements connecting clusters that are not nearest neighbours). Melting
an island is thus slow, involving a time $\mathcal{O}(\Gamma^w)$ for
an island of length $w$, like in the {{EastREM}}. On the other hand, the GOE
structure of the intra-cluster Hamiltonians means that the effect of
constraints within the cluster is no longer there as such new
constraints cannot be created in the liquid regions.
Hence out of the two effects identified earlier, namely, slow
dynamics/localisation in the already frozen region and formation
of new frozen regions, the latter has been eliminated in the {{GOEastREM}}. This is
confirmed in the dynamical autocorrelations in the {{GOEastREM}} starting from
the domain-wall state as shown in
Fig.~\ref{fig:goeastrem-autocorr-leftright}. The results are for
$\Gamma=0.1$ which corresponds to the MBL phase for the {{EastREM}}. The
left half of the system which corresponds to the solid region fails to
thermalise as in the {{EastREM}} as indicated by the saturation of $A_L(t)$
with both $t$ and $N$. On the other hand, the right half rapidly
thermalises, resulting in the systematic decay of the saturation
values of $A_R(t)$ with $N$, in stark contrast to the {EastREM}. This
demonstrates that, as anticipated, the non-ergodic behaviour shown by
the segment of up spins in the {{EastREM}} was indeed caused by the
formation of new blockades, as the {{GOEastREM}} removes that mechanism. At
the same time, as the GOEastREM preserves the constraints which lead
to non-thermalisation of segments of down spins, similar to the
EastREM, indicating that the same mechanism is at play in both the
models.
\begin{figure}
\includegraphics[width=\columnwidth]{goeastrem-autocorr-leftright.pdf}
\caption{Dynamical autocorrelations in the left and right halves of
the chain (Eq.~\eqref{eq:autocorr-leftright}) for the {{GOEastREM}} . (a)
The left half of the chain fails to thermalise as indicated by the
saturation of $A_L(t)$ with both $t$ and system size $N$. (b) The
right half rapidly thermalises as indicated by the systematic
decay of the long-time saturation of $A_R(t)$ with $N$. Results
shown for $\Gamma=0.1$ and data averaged over 500 disorder
realisations.}
\label{fig:goeastrem-autocorr-leftright}
\end{figure}
\section{Forward scattering approximation \label{sec:fsa}}
To provide analytical insight we now turn to the forward scattering
approximation (FSA), which is an approximation to the non-local (in
Fock space) Green's function to lowest order in $\Gamma$ and amounts to
a stability analysis of the trivially localised phase at vanishing
hopping $\Gamma=0$
[Eqs.~\eqref{eq:unconstrained-hopping}-\eqref{eq:eastrem}].
Considering an arbitrary initial state which we label by $\alpha=0$
and which is an eigenstate of the unperturbed $\Gamma=0$ Hamiltonian
(that is, a $\sigma^z$-product state), the weight of the perturbed
eigenstate on an arbitrary spin-configuration $\{\sigma^z_i\}_\alpha$,
denoted as $\psi(\{\sigma^z_i\}_\alpha),$ is
\begin{equation}
\psi(\{\sigma^z_i\}_\alpha) = \sum_{p \in \text{paths}^\ast(0,\alpha)}\prod_{\beta\in p}\frac{\Gamma}{\mathcal{E}_0 - \mathcal{E}_\beta},
\label{eq:fsa}
\end{equation}
where paths$^\ast(0,\alpha)$ is the set of all shortest paths from the
unperturbed $\alpha=0$ state to $\ket{\alpha}$. The $\mathcal{E}_\alpha$, as
before, are the random Fock-space site energies defined in
Eq.~\eqref{eq:hd} and are normally distributed,
$\mathcal{E}_\alpha\sim\mathcal{N}(0,N)$. In this setting, the breakdown of
localisation is signalled by the probability of resonance at
arbitrarily large distances $r$ on the Fock space from the site
$\alpha=0$ approaching unity such that under the state spreads to
Fock-space sites such distances at finite $\Gamma$. The delocalisation criterion can be formally expressed as
\begin{equation}
\lim_{r\to\infty}P\left(\frac{\ln\vert\psi_r\vert^2}{2r}>-\xi^{-1}\right)\to 1,
\label{eq:deloc-criterion}
\end{equation}
where $\psi_r$ denotes the wavefunction amplitude on a Fock-space site
distant by $r$ from the initial state and $\xi$, an analogue of the
localisation length on the Fock space. Note that, the delocalisation
criterion of Eq.~\eqref{eq:deloc-criterion} gives a conservative
estimate in that it provides a lower bound on the critical $\Gamma$ as
it is enough for the maximum of $\psi_r$ over all configurations at
Hamming distance $r$ and disorder realisations to satisfy the
resonance condition.
Before proceeding with the FSA analysis, it is useful to define and
assign notations to two important features of the Fock-space graph,
(i) the number of Fock-space sites at distance $r$ from the initial
state, denoted by $n^{(s)}_r$, and (ii) the number of shortest paths to
a site $\ket{\alpha}$ at distance $r$, which we denote by $n^{(p)}_{r;\alpha}$. While this
quantity is different for each site, and therefore in principle
deserves its site index, Fig.~\ref{fig:eastrem-npath-distributions}
shows that its distribution is not fat tailed. We therefore omit the
site indices and use $n^{(p)}_r$ to indicate the average number of paths
to sites at a distance $r$.
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-npath-distributions.pdf}
\caption{(a) For the EastREM, the distribution of number of paths, $n^{(p)}_{r;\alpha}$, over all states $\ket{\alpha}$ at distance $r=N$ from the DW state. On a log-log scale, the curves bend downwards
showing that the distributions decay faster than a
power-law and hence are not fat tailed. (b) The mean and typical
values of $n^{(p)}_N$, scaled with that of the QREM ($N!$) decays
exponentially with $N$, showing a strong suppression of the number
of paths due to the constraints in the EastREM.}
\label{fig:eastrem-npath-distributions}
\end{figure}
In the following, we analyse the localisation properties for states in
the middle of the spectrum ($\epsilon=0$) as well as the edges (finite
$\epsilon$). We find that in the middle of the spectrum (thus for the
bulk of the states and relevant regime dynamically) the {{EastREM}} has a
localisation transition at a finite $\Gamma_c$ whereas the QREM
remains delocalised for all $\Gamma$, signifying that that the
constraints change the physics qualitatively. In the edges of the
spectrum, the FSA analysis shows that localisation persists for larger
$\Gamma$ in the {{EastREM}} compared to the QREM. These two results are
consistent with those obtained from exact numerical calculations in
Sec.~\ref{sec:phenomenology}.
\subsection{Localisation by constraints in the spectral
bulk\label{subsec:bulk}}
We first focus on states in the middle of the spectrum,
$\epsilon_0= 0$, and which constitute the majority. In this
case, all the factors $\Gamma/(\mathcal{E}_0-\mathcal{E}_\beta)\approx -\Gamma/\mathcal{E}_\beta$
are potentially large and individual paths can become resonant.
While a single resonant path is enough to prevent localisation in the QREM~\cite{baldwin2016manybody}, demonstrating that localisation is stable in the EastREM (Sec.~\ref{sec:phenomenology}) requires that we sum over all the paths. The probability amplitude on a state $\ket{\alpha}$ at a distance $r$ from the initial state is simply then
\begin{equation}
\psi_\alpha = \sum_p \prod_{\beta \in p}\frac{\Gamma}{-\mathcal{E}_\beta},
\label{eq:psi_alpha_edens0}
\end{equation}
where $p$ runs over all shortest paths, the lengths of which are $r$. As interference effects are not important for localisation in high dimensions,
$\vert\psi_\alpha\vert = n^{(p)}_r \Gamma^r
\prod_\beta(\vert\mathcal{E}_\beta\vert)^{-1}$. For a resonance to occur,
$\vert\psi_\alpha\vert>1$ in Eq.~\eqref{eq:psi_alpha_edens0}. Upon
defining $F_r = -\sum_{\beta=1}^r\ln\vert \mathcal{E}_\beta\vert$, the
resonance condition becomes $F_r>-r\ln\Gamma_r$ with
$\Gamma_r=\Gamma\left(n^{(p)}_r\right)^{1/r}$. Transforming the
distributions of the independent $\mathcal{E}$s, the distribution of $F_r$ can
be explicitly written as
\begin{equation}
P_F(F_r) \approx \frac{1}{(r-1)!}\left(F_r + \frac{r}{2}\ln N\right)^{r-1}e^{-\left(F_r + \frac{r}{2}\ln N\right)}.
\end{equation}
The probability for a path of length $r$ to be resonant,
$p_r^\mathrm{res}$, can be computed as the net support of the
distribution $P_F$ on $F_r\ge-r\ln\Gamma_r$,
\begin{align}
p_r^\mathrm{res} &= \int_{-r\ln\Gamma}^{\infty}dF_r~P_F(F_r)\\
&\approx\frac{1}{(r-1)!}\left[r\ln\frac{\sqrt{N}}{\Gamma_r}\right]^{r-1}\exp\left[-r\ln\frac{\sqrt{N}}{\Gamma_r}\right].\label{eq:pl_edens0}
\end{align}
As each of the $n^{(s)}_r$ sites at distance $r$ are
independent, the probability that there is no resonance at distance
$r$ is given by
$Q_r=(1-p_r^\mathrm{res})^{n^{(s)}_r}\approx e^{-n^{(s)}_rp_r^\mathrm{res}}$.
The ratio
$\lambda_r = n^{(s)}_{r+1}p_{r+1}^\mathrm{res}/n^{(s)}_{r}p_{r}^\mathrm{res}$
is a monotically decreasing function of $r$. Hence, if for some $r$ we
have $\lambda_{r} <1$ then $n^{(s)}_{r}p_{r}^\mathrm{res}\to 0$ as
$r\rightarrow\infty$ and consequently $Q_r\rightarrow 1$; this signals
the stability of localisation as $Q_r$ is the probability of no
resonances at distance $r$.
Using Eq.~\eqref{eq:pl_edens0}, the localisation criterion
$\lambda_r<1$ can be rewritten as
\begin{equation}
\Gamma \le \underbrace{\frac{n^{(p)}_r}{n^{(p)}_{r+1}}\left(\frac{n^{(s)}_r}{n^{(s)}_{r+1}}\right)\frac{\sqrt{N}}{\ln N}}_{K(r,N)}\left(1+\frac{1}{r}\right)^{-r},
\label{eq:critical-gamma-edens0}
\end{equation}
for a finite $r$ in the limit of $N\to\infty$. Hence, for localisation
to persist until a finite value of $\Gamma$, we require the dependence
of $n^{(s)}_r$ and $n^{(p)}_r$ on $r$ and $N$ to be such that
$K(r,N)$ does not scale with $N$.
\begin{figure}
\includegraphics[width=\columnwidth]{ratio-eastrem.pdf}
\caption{The factor $K(r,N)$
appearing in Eq.~\eqref{eq:critical-gamma-edens0} for the stability of localisation. For the EastREM, there is range of finite $r$
(seemingly growing $N$), where $K(r,N)$ does not scale with
$N$. Contrarily in the QREM, the ratio decays with $N$ for all
$r$.}
\label{fig:ratio-eastrem}
\end{figure}
For the QREM, $n^{(s)}_r=\binom{N}{r}$ and $n^{(p)}_r=r!$, and the ratio
$K(r,N) \sim N^{-1/2}/r$. Thus the RHS of
Eq.~\eqref{eq:critical-gamma-edens0} scales as $N^{-1/2}$ and vanishes
in the thermodynamic limit rendering localisation impossible. On the
other hand, for the EastREM, the ratio $K(r,N)$
computed numerically does show an absence of dependence on $N$ for finite
$r$ (see Fig.~\ref{fig:ratio-eastrem}); the range of $r$ over
which this holds grows with $N$, suggesting that a
localisation-delocalisation transition is indeed possible at a finite
$\Gamma$ for $\epsilon=0$ in the thermodynamic limit.
Note that the qualitative difference between the QREM and EastREM with
regard to the ratio $n^{(s)}_r/n^{(s)}_{r+1}$ arises purely from the
constraints. In the QREM, after one flips $r\ll N$ spins, one is free
to flip any of the $N-r\approx N$ spins in the next step, which leads
to the ratio $n^{(s)}_{r+1}/n^{(s)}_r$ scaling as $N$. On the other hand, in
the EastREM, flipping some spins from up to down creates new
blockades and so the number states available on successive steps don't
scale as fast as in the QREM. This argument in conjunction with the
FSA presents an analytical picture of how the constraints affect the
distribution of distances on the Fock space which in turn lead to a
constraint-induced localised phase in the EastREM, unlike the QREM.
\subsection{Enhancement of localisation at spectral
edges \label{subsec:edges}}
Let us now consider the situation at a finite energy density
$\epsilon_0=\mathcal{E}_0/N$ which, as the spectral width $\propto \sqrt{N}$,
corresponds to the edges of the spectrum. Even though the density of
states is exponentially small in this energy region, it
is nevertheless important for the dynamics.
The distribution of $\Gamma/(\mathcal{E}_0-\mathcal{E}_\alpha)$ is fat-tailed so that
single sites can become resonant. As most of the $\mathcal{E}_\alpha$s are
$\sim\sqrt{N}$ these resonances are rare, so we focus on paths with a
single resonance; for a Fock-space site $\ket{\alpha}$ at distance $r$
to be resonant it is sufficient for a single site to be resonant. For
a resonance to occur at distance $r$ (and not before), we require that
$\mathcal{E}_\beta\sim \sqrt{N}$ for all but the last $\beta$ on the shortest
path but $\vert\mathcal{E}_0-\mathcal{E}_\alpha\vert\ll 1$. In this scenario, the
amplitude on the Fock-space site $\alpha$ at distance $r$ an be
expressed as
\begin{equation}
\psi_r = n^{(p)}_r\left(\frac{\Gamma}{\mathcal{E}_0}\right)^{r-1}\frac{\Gamma}{\mathcal{E}_0-\mathcal{E}_\alpha},
\label{eq:psi-alpha-finite-eps}
\end{equation}
where as before $n^{(p)}_r$ is the average number of paths to
sites at distance $r$ and we have implicitly assumed that all paths
are independent. As the distribution of the number of shortest paths,
$n^{(p)}_r$ is not fat-tailed (see
Fig.~\ref{fig:eastrem-npath-distributions}(a)), using the average is
justified.
From Eq.~\eqref{eq:psi-alpha-finite-eps}, a resonance at the last site requires that $\vert\psi_r\vert>1$ or
equivalently
$\vert \mathcal{E}_0-\mathcal{E}_\alpha\vert < n^{(p)}_r\Gamma
\left(\frac{\Gamma}{\mathcal{E}_0}\right)^{r-1}$. Thus the probability of the
state being resonant is
\begin{align}
p_r^\mathrm{res} &= \int_{\mathcal{E}_0-n^{(p)}_r\Gamma \left(\frac{\Gamma}{\mathcal{E}_0}\right)^{r-1}}^{\mathcal{E}_0+n^{(p)}_r\Gamma \left(\frac{\Gamma}{\mathcal{E}_0}\right)^{r-1}} d\mathcal{E}_\alpha \frac{1}{\sqrt{2\pi N}}\exp\left[-\frac{\mathcal{E}_\alpha^2}{2N}\right]\\
&\approx\sqrt{\frac{2}{\pi N}}\exp\left[-\frac{\mathcal{E}_0^2}{2N}\right]n^{(p)}_r\Gamma \left(\frac{\Gamma}{\mathcal{E}_0}\right)^{r-1}.
\label{eq:pl-finite-eastrem}
\end{align}
Since the $\mathcal{E}$s are i.i.d. random variables, the expression above
holds for any state at a distance $r$. The probability that none of
the $n^{(s)}_r$ sites at distance $r$ is resonant then is simply given as
before by
$Q_r = (1-p_r^\mathrm{res})^{n^{(s)}_r}\approx e^{-n^{(s)}_r
p_r^\mathrm{res}}$. Localisation persists if $Q_r \to 1$ as
$N \to \infty$ whenever $r$ is a finite-fraction of $N$; we thus
define $x=r/N$ which will be useful later.
\begin{figure}
\includegraphics[width=\columnwidth]{eastrem-finite-eps.pdf}
\caption{(a) The function $Y(r,N)$, defined in Eq.~\eqref{eq:f},
plotted as a function of $x=r/N$ for various $N$ shows that it is
a function of $x$ alone, for the EastREM. The red dashed line
shows the corresponding function for the QREM. Note that $x\leq 1$
for the QREM as no two sites are further than $N$ sites apart,
while the removal of bonds in the {EastREM}\ allows for longer shortest
paths. (b) The boundary between the delocalised and localised
phases obtained from solving Eq.~\eqref{eq:gammac-trans}. As in
(a), the red dashed line corresponds to the QREM
result. $\Gamma_c$ for the EastREM is larger than that for the
QREM for all finite $\epsilon_0$.}
\label{fig:eastrem-finite-eps}
\end{figure}
Using the expression in Eq.~\eqref{eq:pl-finite-eastrem}, $Q_r$ can be
written as
\begin{equation}
Q_r = \exp\left[-k e^{Nf(r,N,\epsilon_0)}\right],
\label{eq:ql}
\end{equation}
where $f$ is
\begin{equation}
f(r,\epsilon_0)=-\frac{\epsilon_0^2}{2} + x\ln\frac{\Gamma}{\epsilon_0}\underbrace{+\frac{1}{N}\ln(n^{(s)}_r n^{(p)}_r)-x\ln N}_{Y(r,N)},
\label{eq:f}
\end{equation}
where we have only kept terms that survive in the thermodynamic
limit.
Crucially, $Y(r,N)$ in Eq.~\eqref{eq:f} is only a function of $x$,
$Y(r,N)=Y(x)$, which in turn means that
$f(r,N,\epsilon_0)=f(x,\epsilon_0)$. This can be trivially shown for
the QREM using $n^{(p)}_r = r!\approx (r/e)^r$ and $n^{(s)}_r = \binom{N}{r}$
that $Y(x)=-(1-x)\ln(1-x)-x$. For the EastREM such analytic expressions
$n^{(p)}_r$ and $n^{(s)}_r$ are not available,\footnote{Note here that the
mean number of shortest paths between Fock-space sites distant by
$N$ in the EastREM scaled by the same quantity for the QREM decays
systematically with $N$ which is a direct result of the constraints.} but the
numerically obtained form in Fig.~\ref{fig:eastrem-finite-eps}(a)
shows that $Y(r,N)$ indeed is simply a function of $x=r/N$. The
localisation condition $Q_r\rightarrow 1$ as $N\rightarrow\infty$ then
requires that
\begin{equation}
\max_{x}f(x,\epsilon_0)<0.
\end{equation}
The critical $\Gamma$ can be obtained by solving the equation
\begin{equation}
-\frac{\epsilon_0^2}{2}+x_\ast\ln\frac{\Gamma_c}{\epsilon_0} + Y(x_\ast)=0
\label{eq:gammac-trans}
\end{equation}
where $f(x,\epsilon_0)$ is maximised at $x_\ast$.
We solve Eq.~\eqref{eq:gammac-trans}, for both the QREM and the
EastREM, showing the results in Fig.~\ref{fig:eastrem-finite-eps}. We
find that in the {EastREM}, localisation persists to a larger value of
$\Gamma$.
\subsection{Numerical treatment of FSA \label{subsec:numerical-fsa}}
We now locate the transition numerically exactly within the FSA for
small system sizes. To do this, we rewrite the delocalisation
criterion of Eq.~\eqref{eq:deloc-criterion} as
\begin{equation}
\lim_{r\to\infty}P\left(\Lambda_r > \ln\left(\frac{1}{\Gamma_c}\right) \right) \to 0,
\label{eq:fsa-transition}
\end{equation}
where $\Lambda_r = \ln\vert\psi_r\vert^2/2r -\ln\Gamma$ and for
$\Gamma<\Gamma_c$. Our strategy is to directly calculate the
amplitudes $\psi_r$ within the FSA (by obtaining the shortest paths
numerically) and from those obtain the distribution of
Eq.~\eqref{eq:fsa-transition}. Eq.~(\ref{eq:fsa-transition}) then says
that the upper limit of its support then determines the critical
$\Gamma_c$. Without loss of generality, for this calculation we shall
take the DW state as the initial state as before.
We note here that while for the QREM to each state there corresponds
only one state at hamming distance $N$, for the EastREM there are
$\sim e^N$ such states. Indeed, in the EastREM, the number of
configurations at distance $r$ is \emph{peaked} at $r=N$. Hence, one
can argue that studying the statistics of $\Lambda_N$ can overestimate
$(1/\Gamma_c)$, thus underestimating $\Gamma_c$ as the likelihood of
having a resonance at $r=N$ is quite high simply due to a large
fraction of the configurations having $r=N$. As our main result is
that $\Gamma_c>0$ and in general larger than for the QREM,
underestimating it is not a problem.
To calculate $P(\Lambda_N)$ starting from the domain-wall state we
construct a matrix
\begin{equation*}
\mathcal{T} = \Gamma\sum_{\beta,\gamma}\frac{A_{\beta,\gamma}}{\mathcal{E}_0-\mathcal{E}_\gamma}\ket{\beta}\bra{\gamma}
\end{equation*}
with $A_{\beta,\gamma}=1$ if $r_{(0,\gamma)}<r_{(0,\beta)}$ and
$\bra{\beta}\ensuremath{H_{\mathrm{East}}} \ket{\gamma}\neq 0$, where $r_{(0,\beta)}$
is the hamming distance between $\ket{\beta}$ and the domain-wall
state. That is, $A_{\beta,\gamma}=1$ if the transition between the two
states $\ket{\beta}$ and $\ket{\gamma}$ is allowed by the Hamiltonian
and it increases the distance from the domain-wall state. The
amplitude on the configuration $\ket{\alpha}$, at a hamming distance
$r$ is then given by
\begin{equation*}
\psi(\{\sigma^z_i\}_\alpha) = \bra{\alpha}\mathcal{T}^l\ket{0};
\end{equation*}
from which $\vert \psi_r\vert$ is obtained as
\begin{equation*}
\vert \psi_r\vert = \max_{\alpha;~ r_{(0,\alpha)}=r}\{\psi(\{\sigma^z_i\}_\alpha)\}.
\end{equation*}
The distribution of $\Lambda_N$ so obtained is shown in
Fig.~\ref{fig:fsa-eastrem}(a) for various $N$. It clearly has finite
support, which has an upper bound that becomes sharper with increasing
$N$. This is consistent with the conclusion that as $N\to\infty$,
there exists a sharp value $1/\Gamma_c$ above which the distribution
has no weight, as required by Eq.~\eqref{eq:fsa-transition}.
Since the distribution tends to get sharper with $N$, one can argue
that the critical $1/\Gamma_c$ can be estimated as
\begin{equation}
\lim_{N\to\infty}\braket{\Lambda_N} = \ln\left(1/\Gamma_c\right).
\end{equation}
In order to estimate this limiting value, we fit $N\braket{\Lambda_N}$
as a function of $N$ to a form
\begin{equation}
N\braket{\Lambda_N}=a + N\ln\left(1/\Gamma\right)_c + b N^\gamma,
\label{eq:fitLambdaN}
\end{equation}
where the last term takes into account the slowly decreasing
fluctuations in $\Lambda_N$ with increasing $N$. The fit is shown in
Fig.~\ref{fig:fsa-eastrem}(b) with the best fit parameters yielding
$\ln\left(1/\Gamma_c\right) = 1.77\pm 0.02$ which implies
$\Gamma_c = 0.17 \pm 0.01$.
Note that the transition criterion in Eq.~\eqref{eq:fsa-transition}
can also be equivalently stated as
\begin{equation}
\lim_{N\to\infty}C(\Lambda_N = \ln(1/\Gamma)_c)\to 1,
\end{equation}
where $C(\Lambda_N)$ is the cumulative distribution corresponding to
$P(\Lambda_N)$. For finite-sized systems, we plot the $1-C(\Lambda_N)$
in Fig.~\ref{fig:fsa-eastrem}(c) and observe a clear crossing of the
data for various system sizes. The value at which the crossing occurs
and which we identify as the critical point matches remarkably well
with that obtained from the finite-size scaling analysis of
$\braket{\Lambda_N}$, as shown by the grey shaded region in
Fig.~\ref{fig:fsa-eastrem}(c). More importantly, the critical value
so obtained, $\Gamma_c^\mathrm{FSA}$, is in excellent agreement with
the infinite temperature $\Gamma_c$ obtained from exact
diagonalisation, see Fig.~\ref{fig:eastrem-ed}.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{fsa-numerical-eastrem.pdf}
\caption{Numerical treatment of FSA for the EastREM. (a) The
distributions $P(\Lambda_N)$ for different $N$. (b) The circles
show the data for $N\braket{\Lambda_N}$ and the red dashed line
shows a fit to the form in Eq.~\eqref{eq:fitLambdaN}. The best fit
parameters are $\ln\left(J/\Gamma\right)_c = 1.77\pm 0.02$ and
$\gamma=0.09$, the low value of the latter highlighting the slow
decay of the fluctuations in $\Lambda_N$ with $N$. (c) The
survival function, $1-C(\Lambda_N)$ corresponding to
$P(\Lambda_N)$ shows a clear crossing of the data for various $N$
suggesting a critical point well in agreement with that extracted
from (b) as shown by the grey shaded region.}
\label{fig:fsa-eastrem}
\end{figure}
\section{Discussion \label{sec:discussion}}
In conclusion, we have shown that local constraints can induce strong
ergodicity breaking manifested as localisation in quantum many-body
systems, crucially without shattering the Fock space. The locality of
the constraints allows us to identify the long-lived local spatial
configurations responsible for the dynamical arrest, which in Fock
space turn out to correspond to dynamical bottlenecks caused by sparse
connectivity between clusters of states. These results are exemplified
by a quantum random energy model with East-like constraints, which we
introduce and call the {{EastREM}}. We provide further support for the
picture of isolated regions of Fock space by constructing and studying
a random matrix model with GOE blocks for each cluster but the same
intercluster connections as the {EastREM}. This model, which we name the
{{GOEastREM}}, is the minimally structured model possessing the localisation
mechanism we have identified. We finally obtain analytical insight by
applying the FSA in Fock space. The constraints modify the distance
(in Fock space) dependence of the number of accessible sites and paths
to them, and the FSA shows how this leads to localisation. In addition
to providing this insight, the FSA is in excellent agreement with the
ED results (see Fig.~\ref{fig:eastrem-ed}).
At this juncture, a number of potential directions for future work
present themselves. An immediate direction of interest is a systematic
study of the statistical mechanics of the paths on the Fock space by
treating them as directed polymers on a correlated but random
landscape. The replica trick~\cite{mezard1987spin} is ideally suited
to obtain further analytical insight into the problem as the non-local
propagator on the Fock space is expected to be dominated only by a few
paths which pass through the resonant bottlenecks. In fact, in the
context of many-body localisation in traditionally studied
short-ranged disordered spin chains, a classical percolation proxy on
the Fock space was recently
introduced~\cite{roy2018percolation,roy2018exact}. The effect of
constraints on such a percolation picture and the potential connections
to the directed polymer picture could shed light on the nature of the
transition.
A different question is whether an approach based on random unitary circuits, recently used to study universal properties of ergodic systems~\cite{nahum2017quantum,keyserlingk2018operator,chan2018solution}, can be generalised to include local constraints such that ergodicity is broken. In fact, there have been works on including conservation laws~\cite{rakovszky2018diffusive,khemani2018operator,friedman2019spectral} as well as ergodicity breaking~\cite{chan2018spectral} in unitary circuits. The question then is to modifying the structure of the unitary gates in the circuit such that the scrambling is constrained locally, analogously to having a conserved degree of freedom locally. The physics in this scenario remains fundamentally different from unitary circuits with conservation laws which shatter the Hilbert space~\cite{pai2019localization,khemani2019local}.
Looking further afield, periodically driven (Floquet) systems have
emerged as one of the more active areas of research in quantum
dynamics. The main difficulty in seeing interesting physics with them
is that ergodic systems inevitably heat up under
driving~\cite{lazarides2014equilibrium,dalessio2014long,ponte2015periodically}. Two routes
to arresting this heating have been
integrability~\cite{lazarides2014periodic} and
Floquet-MBL~\cite{lazarides2015fate,ponte2015many}, both of which rely on breaking ergodicity to prevent heating. It is then natural
to ask whether the present method of breaking ergodicity with local
constraints can also prevent the heating up of driven quantum systems, without explicitly fragmenting the Fock space (thus rendering the physics distinct from that of scars in Floquet systems~\cite{mukherjee2019collapse,haldar2019scars}).
Finally one might ask whether many body localisation originating from correlations in Fock space~\cite{roy2019fock}, can be identified as being caused by emergent constraints due to the correlated Fock-space disorder.
\begin{acknowledgments}
We would like to thank D.~E.~Logan for illuminating discussions about the FSA. This work was in part supported by
EPSRC Grants No. EP/N01930X/1 and EP/S020527/1.
\end{acknowledgments}
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Q: Considerations for mismatched 18650's in series I'm building a large battery pack. I'm in the planning stage, collecting parts and such, but my goal is to have something like a 4s9p or 4s10p arrangement. The cells I have vary though. Not by a ton, but a hundred or so mAh here and there. All of the parallel sets would be identical cells that were all originally used together, but each set in series would have slightly different capacities. What considerations do I need to keep in mind with this sort of setup? I know it's not advised to use mismatched cells, but I might not have a lot of choice in the matter. I know I'll at least have to have good balancing, anything else to keep in mind?
A:
All of the parallel sets would be identical cells that were all originally used together, but each set in series would have slightly different capacities.
You want to do exactly the opposite. A 1 Ah cell in parallel with a 2 Ah cell behaves as a single 3 Ah cell, but a 1 Ah cell in series with a 2 Ah cell yields a 2S battery that can only be discharged 1 Ah, wasting the mismatching capacity of the better cell. If you discharge the hypothetical battery further in order to benefit from the capacity of the better 2 Ah cell, you will overdischarge and ruin the 1Ah cell.
If you insist on reusing salvaged batteries (I wouldn't), you want to combine the mismatching cells into parallel sets of equal capacity, and then place those in series. You also must add overcharge protection, undercharge protection and balancing to your battery in some form or another.
As pointed out by Robherc, if the internal resistances of parallel cells are vastly different (and not in proportion of their differing capacities), you can exceed the maximum charge and discharge rates of individual cells in a parallel set. For example:
You have a battery of two parallel 1 Ah cells rated for 3 A discharge, but one cell has an internal resistance of 0.1 Ω and the other has degraded to 1 Ω. The battery is fully charged to 4.2 V, and you begin a 4 A discharge. The battery voltage will immediately fall to† 3.84 V due to the internal resistance, but the current is not shared equally. Initially the 0.1 Ω cell will be discharging at 3.64 A, exceeding the rating, while the 1 Ω cell will only provide 0.36 A. However as the 0.1 Ω discharges much faster, the 1 Ω cell will eventually catch up during the discharge and start to share the load (assuming that it doesn't combust first). Also, the 0.1 Ω cell will wear much faster with subsequent charge-discharge cycles, slowly equalizing itself with the more degraded cell in the pack.
While this is a rather extreme example, you should still throw out cells that differ greatly in the product of cell capacity and internal resistance, and de-rate the charge and discharge rates from the theoretical values accordingly.
† Internal resistance of the battery as a whole:
1/(1/0.1 Ω + 1/1 Ω) ≈ 0.091 Ω
Voltage loss over the internal resistance:
4 A * 0.091 Ω = 0.364 V
Remaining (true) battery voltage:
4.2 V - 0.364 V = 3.836 V ≈ 3.84 V
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{"url":"http:\/\/mhalpin.co.uk\/lib\/category\/discrete-mathematics\/page\/5\/","text":"# The Theory of Information and Coding: Student Edition\n\nFormat: Hardcover\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 8.97 MB\n\nThis, to my knowledge, has never been investigated in any way. We provide them now because this is our \ufb01rst exposure to such a proof, and we want to make the reasoning absolutely clear. Thus the series diverges. \u0002 Discrete Mathematics Demystified 298 EXAMPLE 13.50 Does the series \u221e \u0019 j=1 1 j \u00b7 (ln j)2 converge? Thus, at the end of the first century we could expect an \"average\" text to be 29.4+(1)(4.85)= 34.25% Byzantine. University of Illinois-Urbana Champaign, 1980.\n\n# Discrete Mathematics (Revised Edition)\n\nFormat: Hardcover\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 10.15 MB\n\nWe apply the Euclidean algorithm to 57 and 20. SAS is a statistical programming language. Sloane, Neil and Simon Plouffe, The Encyclopedia of Integer Sequences (San Diego, California: Academic Press, 1995). There is a certain sense in which mathematicians of that kind have something very important in common, despite their differences, and form a mainstream from which discrete mathematics is mostly excluded. Notice that x F(x) = a0 x + a1 x 2 + a2 x 3 + a3 x 4 + \u00b7 \u00b7 \u00b7 and x 2 F(x) = a0 x 2 + a1 x 3 + a2 x 4 + a3 x 5 + \u00b7 \u00b7 \u00b7 Thus, grouping like powers of x, we see that F(x) \u2212 x F(x) \u2212 x 2 F(x) = a0 + (a1 \u2212 a0 )x + (a2 \u2212 a1 \u2212 a0 )x 2 + (a3 \u2212 a2 \u2212 a1 )x 3 + (a4 \u2212 a3 \u2212 a2 )x 4 + \u00b7 \u00b7 \u00b7 But the basic property that de\ufb01nes the Fibonacci sequence is that a2 \u2212 a1 \u2212 a0 = 0, a3 \u2212 a2 \u2212 a1 = 0, and so on.\n\n# Precalculus and Discrete Mathematics Teaching Aid Masters\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 14.87 MB\n\nThe attractor itself, and the equations from which it is derived, were introduced by Edward Lorenz in 1963, who derived it from the simplified equations of convection rolls arising in the equations of the atmosphere. Prerequisite: MATH 70 or MATH 75, or MATH 75A and B; no credit if taken after MATH 108. Interestingly, Mario Livio notes, \"Each mammalian microtubule is typically made up of thirteen columns, arranged in five right-handed and eight left-handed structures (5, 8, and 13 are all Fibonacci numbers).\n\n# Progress in Analysis: Proceedings of the 3rd International\n\nFormat: Hardcover\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 6.90 MB\n\nPete Wolenski will be teaching for spring 2016. 2057 Multidimensional Calculus (3) F, S, Su Prerequisites: MATH 1552 or 1553. In particular, if one is willing to settle for a path that cost not more than twice the optimal path, then one may \ufb01nd a solution rather ef\ufb01ciently. In this case, this is the point at which D (or, perhaps, A B E L) split off from the main tree. We use theorems, propositions, and lemmas to formulate our ideas. A rigorous introduction to PDE accessible to advanced undergraduates.\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 13.11 MB\n\nIn this wide-ranging research project, general logical and categorytheoretic methods are developed and studied in order to ensure constructive content of mathematical theorems. Refer to the de\ufb01nition of \u201ceven\u201d to see what that promise is: If n can be written as twice a natural number then n \u00b7 n can be written as twice a natural number. Let us consider the powers of the first prime 2. We show here that such relations in a natural way induce equitable ...\n\n# Integration of AI and OR Techniques in Constraint\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 12.42 MB\n\nWould you like to make it the primary and merge this question into it? To clarify the connection between the delooping of $\\text{Emb}_c(\\mathbb{R}^m, \\mathbb{R}^{m+i})$, and $G_i$ -- the group of self-homotopy equivalences of the sphere $S^{i-1}$. P., Hern\u00e1ndez A., Herrera R., A Conic Higher Order Neuron Based on Geometric Algebra and Its Implementation, Advances in Computational Intelligence - 11th Mexican International Conference on Artificial Intelligence (MICAI 2012) Part II - Series: Lecture Notes in Computer Science Vol. 7630, pp. 223-235, issn 0302-9743, doi 10.1007\/978-3-642-37798-3, (2013) Serrano J.\n\n# Fuzzy Logic and its Applications to Engineering, Information\n\nFormat: Hardcover\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 13.00 MB\n\nConsider a statement P(n) about the natural numbers. They are fundamental to modern mathematics and science. But the study of how to represent reals by approximations, as computers do, could be considered part of discrete math). Introduction \u2022 Ant Odometer (c. 150 million BC) \u2022 Primates Count (c. 30 million BC) \u2022 Cicada-Generated Prime Numbers (c. 1 million BC) \u2022 Knots (c. 100,000 BC) \u2022 Ishango Bone (c. 18,000 BC) \u2022 Quipu (c. 3000 BC) \u2022 Dice (c. 3000 BC) \u2022 Magic Squares (c. 2200 BC) \u2022 Plimpton 322 (c. 1800 BC) \u2022 Rhind Papyrus (c. 1650 BC) \u2022 Tic Tac Toe (c. 1300 BC) \u2022 Pythagorean Theorem and Triangles (c. 600 BC) \u2022 Go (548 BC) \u2022 Pythagoras Founds Mathematical Brotherhood (530 BC) \u2022 Zeno's Paradoxes (c. 445 BC) \u2022 Quadrature of the Lune (c. 440 BC) \u2022 Platonic Solids (350 BC) \u2022 Aristotle's Organon (c. 350 BC) \u2022 Aristotle's Wheel Paradox (c. 320 BC) \u2022 Euclid's Elements (300 BC) \u2022 Archimedes: Sand, Cattle & Stomachion (c. 250 BC) \u2022 pi (c. 250 BC) \u2022 Sieve of Eratosthenes (c. 240 BC) \u2022 Archimedean Semi-Regular Polyhedra (c. 240 BC) \u2022 Archimedes' Spiral (225 BC) \u2022 Cissoid of Diocles (c. 180 BC) \u2022 Ptolemy's Almagest (c. 150) \u2022 Diophantus's Arithmetica (250) \u2022 Pappus's Hexagon Theorem (c. 340) \u2022 Bakhshali Manuscript (c. 350) \u2022 The Death of Hypatia (415) \u2022 Zero (c. 650) \u2022 Alcuin's Propositiones ad Acuendos Juvenes (c. 800) \u2022 al-Khwarizmi's Algebra (830) \u2022 Borromean Rings (834) \u2022 Ganita Sara Samgraha (850) \u2022 Thabit Formula for Amicable Numbers (c. 850) \u2022 Kitab al-fusul fi al-hisab al-Hindi (c. 953) \u2022 Omar Khayyam's Treatise (1070) \u2022 Al-Samawal's The Dazzling (c. 1150) \u2022 Abacus (c. 1200) \u2022 Fibonacci's Liber Abaci (1202) \u2022 Wheat on a Chessboard (1256) \u2022 Harmonic Series Diverges (c. 1350) \u2022 Law of Cosines (c. 1427) \u2022 Treviso Arithmetic (1478) \u2022 Discovery of Series Formula for Pi (c. 1500) \u2022 Golden Ratio (1509) \u2022 Polygraphiae Libri Sex (1518) \u2022 Loxodrome (1537) \u2022 Cardano's Ars Magna (1545) \u2022 Sumario Compendioso (1556) \u2022 Mercator Projection (1569) \u2022 Imaginary Numbers (1572) \u2022 Kepler Conjecture (1611) \u2022 Logarithms (1614) \u2022 Slide Rule (1621) \u2022 Fermat's Spiral (1636) \u2022 Fermat's Last Theorem (1637) \u2022 Descartes' La Geometrie (1637) \u2022 Cardioid (1637) \u2022 Logarithmic Spiral (1638) \u2022 Projective Geometry (1639) \u2022 Torricelli's Trumpet (1641) \u2022 Pascal's Triangle (1654) \u2022 The Length of Neile's Semicubical Parabola (1657) \u2022 Viviani's Theorem (1659) \u2022 Discovery of Calculus (c. 1665) \u2022 Newton's Method (1669) \u2022 Tautochrone Problem (1673) \u2022 Astroid (1674) \u2022 L'Hopital's Analysis of the Infinitely Small (1696) \u2022 Rope around the Earth Puzzle (1702) \u2022 Law of Large Numbers (1713) \u2022 Euler's Number, e (1727) \u2022 Stirling's Formula (1730) \u2022 Normal Distribution Curve (1733) \u2022 Euler-Mascheroni Constant (1735) \u2022 Konigsberg Bridges (1736) \u2022 St.\n\n# New Trends in Discrete and Computational Geometry\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 8.47 MB\n\nIn particular, the following topics (2000 AMS classification numbers in parentheses): Mathematical journal Applicable Analysis and discrete mathematics (AADM) is Managed and published by Department of Applied Mathematics, Extractions: Home Home Contact Us About AADM Editors Instructions for authors Reviewing, Acceptance and Publishing Forms and templates Accepted papers Current Issue All issues Archive ETF Department of A.\n\n# Elementary Differential Geometry, Revised 2nd Edition,\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 6.96 MB\n\nThe learner will be able to use mathematical models to represent and understand quantitative relationships and change in real world situations. Students are invited to participate in national mathematics competitions such as the Putnam Exam and the Mathematical Modeling Contest. Show that any subset T of S with more than 5 elements contains two numbers that add up to 11. \ufb01ve blocks. A study of probability models, random variables, estimation, hypothesis testing, and linear models, with applications to problems in the physical and social sciences.\n\n# Exploring Discrete Dynamics. 2nd Editiion. the Ddlab Manual\n\nFormat: Paperback\n\nLanguage: English\n\nFormat: PDF \/ Kindle \/ ePub\n\nSize: 9.06 MB","date":"2018-09-26 15:17:10","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\": 0.679523229598999, \"perplexity\": 6108.38455061514}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2018-39\/segments\/1537267165261.94\/warc\/CC-MAIN-20180926140948-20180926161348-00317.warc.gz\"}"}
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{"url":"http:\/\/physics.stackexchange.com\/tags\/classical-field-theory\/hot","text":"# Tag Info\n\n## Hot answers tagged classical-field-theory\n\n10\n\nWu and Yang (1968) found a static solution to the sourceless SU(2) Yang-Mills equations, (please, see the following two relatively recent articles containing a rather detailed description of the solution: Marinho, Oliveira, Carlson, Frederico and Ngome The solution constitutes of a generalization of the Abelian Dirac monopole. The vector potential is given ...\n\n2\n\nIn my opinion it is better to work in an explicit covariant form. In my answer I will use two different definitions, the Greek indexes always run from $0$ to $3$ and Latin indexes from $1$ to $3$ and the metric $g_{\\mu\\nu}$ has signature $(-1,1,1,1)$. To translate the expressions to a explicit covariant form we define some timelike vector field $v^\\mu$. We ...\n\n2\n\nConsider the $4\\times 4$ matrix $g_{\\mu\\nu}$ with zeroth row $g_{0\\nu}$. Now for $i=1,2,3$, add to the $i$'th row the zeroth row times $-g_{i0}\/g_{00}$. This produces the following matrix $$\\begin{bmatrix} g_{00} & g_{01} & g_{02}& g_{03} \\\\ 0 & -\\gamma_{11} & -\\gamma_{12}& -\\gamma_{13} \\\\ 0 & -\\gamma_{21} & ... 1 I) Here is at least a partial answer. Assume the following set-up. Let there be given a classical Lagrangian field theory in d+1 spacetime dimensions, with dynamical field variables \\phi^{\\alpha}(x,t), and with no explicit time dependence. Action S[\\phi]:=\\int \\! dt~ L[\\phi(t,\\cdot)]. Lagrangian functional L:=T-V. Energy functional E=T+V. ... 1 Exact solutions could not be the right way to understand infrared behavior of Yang-Mills theory. As we know from quantum field theory, we can start with some approximation (weak coupling). With this in mind, it can be proved that the following holds (see http:\/\/arxiv.org\/abs\/0903.2357) for a gauge coupling going formally to infinity$$ ...\n\n1\n\nThis condition is due to the fact that for a free massless particle the Pauli-Lubanski vector $W=*(M\\wedge P)$ must be proportional to the linear momentum (The proportionality factor being the helicity). Thus the condition must be valid to all free massless relativistic field theories.\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2013-05-21 16:29:44","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\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8802863359451294, \"perplexity\": 526.5029381846906}, \"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-2013-20\/segments\/1368700212265\/warc\/CC-MAIN-20130516103012-00059-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
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\section{Introduction} \label{sec:intro}
The accretion discs of active galactic nuclei (AGN) are responsible for feeding supermassive black holes (SMBHs) in the centres of galaxies \citep{lynden-bell1969}. These AGN discs are often described as geometrically thin and optically thick accretion discs, driven by the $\alpha$-viscosity \citep{ss73}. The rapidly growing observational data on AGN discs has led to more detailed models of their structure and evolution, either based on the mass flow inferred from star formation \citep{tqm05}, or on the optical/UV spectral energy distribution (SED) \citep{sg03}.
Galactic nuclei also possess a large number of stars and stellar remnants \citep{miranda2000, bartko2010}. Some regions of the discs might become unstable and give rise to vigorous star-formation \citep{paczynski1987, dittmann2020}, and the interactions of stars and compact objects with the discs could lead to their capture and embedding into the discs \citep{artymovicz1993}.
Stars and compact objects in a gaseous environment could accrete gas and grow over time \citep{davies2020,cantiello1}, as well as change their orbits due to the interaction with the disc \citep{Ost+83,Sye+91,artymovicz1993}. The higher stellar density in the disc, the presence of gas, and the low relative velocities may enhance the formation of binaries through tidal effects \citep{Che+99} and/or through gas-mediated capture \citep{Gol+02,Tag+20}. The interactions of binary stars with the gaseous environment could catalyze stellar mergers \citep{Bar+11}, and the formation and gravitational-wave driven mergers of compact-object binaries \citep{Mck+12}.
Taken together, AGN disc environments could potentially increase the rate of SNe, such as core-collapse SNe or type Ia SNe. The rate of core-collapse SNe may be increased through potentially increased formation rate of massive stars \citep{artymovicz1993}. Similarly, the rate of type Ia SNe may be increased through an increased number of white-dwarf (WD) mergers \citep{mckernan2020}, or through accretion onto WDs, which can make them reach close-to-Chandrasekhar masses and explode \citep{Ost+83}. Other explosive mergers of compact objects with stars or planets could give rise to other transients such as micro - tidal disruption events (when the stars are disrupted by such compact objects, \citealp{Per+16}).
SNe in AGNs were studied in the context of the feedback on the AGN structure and evolution \citep{Rozyczka1995, ferrara2000, morenchel2021}. Here we focus on the observational signatures of SN explosions in AGN discs. The highly relativistic electromagnetic signatures of $\gamma$-ray bursts (GRBs) and the accretion induced collapse of neutron stars (NS) into BHs have been recently explored \citep{GRB_in_AGN, zhu2, perna20, perna2021}. Recently, \citet{zhu3} suggested that type Ia supernovae may potentially be observable in AGN discs. However, a detailed model for the interaction of a non-relativistic SN explosion with the AGN disc and its observational signatures is still lacking. In this paper, we take the first steps in the detailed modelling of general SN explosions in AGN environments.
Electromagnetic signatures from the interaction between SN ejecta and the dense circumstellar medium (CSM) have been studied intensively in the context of type IIn SNe both analytically and numerically \citep[e.g.][]{woosley2007, Chevalier2011,Ginzburg2012,Moriya2013,Dessart2015,Tsuna2019,Takei2020,Suzuki2021}. Most studies assume a wind-like dense CSM, which have density profiles following $\rho\propto1/r^2$, where $r$ is the distance from the progenitor. Some studies have also dealt with asymmetrical CSM structures, such as disc-like morphologies \citep[]{Suzuki2019}. In all cases, the main feature is that the presence of the CSM allows the kinetic energy of the ejecta to be efficiently dissipated and converted into radiation, increasing the luminosity of these events. CSM interaction is considered to be the dominant channel for producing SNe with narrow-line features (type IIn) and the extremely bright SNe called superluminous SNe \citep[SLSNe; see][for a review on SLSNe]{GalYam2019}.
\subsection{SNe in AGN discs}
Motivated by the recent success of CSM interaction models in explaining SLSNe, we develop a detailed model for the shock propagation and breakout of an SN explosion inside an AGN disc. However, there are two main differences between the wind-driven CSM SNe and our setting. While the origin of the CSM in CSM SN is the wind itself, which is already expanding prior to the explosion \citep{nakar_piro2014, morozova_piro2017, piro2020}, in our case, the local AGN gas may be assumed to be static at the time of the explosion. The second difference is that the spatial extent of AGN discs overwhelmingly exceeds the distance scales of wind-driven explosions. For this reason, as we show further, explosions in an AGN disc can be analogous to explosions close to the surface of a much more massive ball of gas \citep{yalinewich_matzner2019}.
The purpose of this paper is to describe the details of SN explosions in dense environments, such as AGN discs. Here we provide an intuitive picture of the underlying physical mechanisms and key takeaway messages.
After the explosion, a shock wave is generated and propagates outward. Initially, the shock propagation velocity is much faster than the photon diffusion speed in an optically thick environment. As the shock propagates outward, the medium outside it becomes increasingly optically thin, until the photons are able to overcome the shock velocity, break out before the shock and diffuse to the photosphere. This is the breakout shell $z_{\rm bo}$, formally defined as a solution to the equation $\tau(z_{\rm bo}) = c / v(z_{\rm bo})$, where $v(z_{\rm bo})$ is the shock velocity and $\tau(z_{\rm bo})$ is the optical depth, all evaluated at $z_{\rm bo}$. At this point, there is a surge of photons that escape out of the material, causing a sudden rise in the observable luminosity. After the initial breakout, the photosphere starts to expand and cool down, which will decrease the luminosity over time.
\subsection{Overview and structure of the paper} \label{structure}
The details of the shock propagation velocity and the breakout depend on the structure of the medium. In sec. \ref{sec:AGN}, we review the structure and properties of AGN discs and derive the vertical density profile $\rho(z)$ and the optical depth $\tau(z)$. In sec. \ref{sec:analytic}, we discuss our assumptions in \ref{sub:assumptions}, derive and compare the velocity profile $v(z)$ with numerical simulations in sec. \ref{sub:shock v}. Once $v(z)$ is established, we find the breakout shell $z_{\rm bo}$ and the breakout time in sec. \ref{sub:bo time}. The shock velocity depends on $\rho(z)$ and $\tau(z)$, and also on the spatial location, and internal properties of the explosion, namely the explosion energy $E_0$ and the ejecta mass $M_{\rm ej}$.
\iffalse
The details of the shock propagation velocity and the breakout depend on the structure of the medium. In sec. \ref{sec:AGN}, we review the structure and properties of the AGN disc, as well as its uncertainties and open questions. The different radial and vertical density profiles are discussed in sec. \ref{rad} and \ref{ver}, respectively. The opacity and the optical depth calculation is presented in sec. \ref{sub:opacity} and \ref{sub:optical depth}, respectively.
Once the vertical optical depth $\tau(z)$ and the vertical density profile $\rho(z)$ are established, we move on to describe how the shock propagates in sec. \ref{sec:analytic}. The shock velocity depends on the aforementioned medium properties, the spatial location, and internal properties of the explosion, namely the explosion energy $E_0$ and the ejecta mass $M_{\rm ej}$. We discuss our assumptions in \ref{sub:assumptions}, derive and compare the velocity profile $v(z)$ with numerical simulations in sec. \ref{sub:shock v}. Once we establish $v(z)$, we can find the breakout shell $z_{\rm bo}$ and calculate the breakout time in sec. \ref{sub:bo time}.
\fi
Once the breakout shell $z_{\rm bo}$ and the photosphere $z_{\rm ph}$ (defined by $\tau(z_{\rm ph})=1$) are established, the photon diffusion length scale is $d=z_{\rm ph} - z_{\rm bo}$. Since the velocity $v(z_{\rm bo})$ is also known, we can estimate the peak luminosity $L_{\rm peak}$ as the energy deposited onto the breakout shell $e_{\rm bo} \sim \rho(z_{\rm bo}) z_{\rm bo}^2 d v_{\rm bo}^2$ divided by the diffusion photon time $\sim d/v_{\rm bo}$. The luminosity is then
\begin{equation}
L_{\rm peak} = C_L \rho(z_{\rm bo}) z_{\rm bo}^2 v_{\rm bo}^3. \label{l_direct}
\end{equation}
The dimensionless prefactor $C_L$ depends on the geometry of the explosion.
The energy deposited in layers beneath the breakout slab cannot efficiently diffuse out, and the material, therefore, cools mostly through adiabatic expansion. Adiabatic cooling leads to a rapid decay of the lightcurve: Initially, the volume is only linearly proportional with time $V\propto t$, and for a pressure $P$, the change in the energy will be $\int dE=\int P dV \propto \int V^{-\gamma} dV$. For a radiation dominated medium (which is the case after it is shocked), we have $\gamma =4/3$, and $\int dE \propto V^{-1/3}\propto t^{-1/3}$, where we used the connection $PV^\gamma = \rm const$. The luminosity falls off as $L\propto E/t \propto t^{-4/3}$. The peak luminosity is hence dominated by the initial breakout, which is likely to be the detectable portion of the lightcurve in the luminous environment of an AGN. In dense regions where the CSM mass until the breakout is much larger than the ejecta mass $M_{\rm CSM} \gg M_{\rm ej}$, adiabatic cooling is the main source of energy loss, while for models with lighter masses $M_{\rm CSM}\lesssim M_{\rm ej}$ the optical depth is reduced, and radiative losses will also contribute, and the decay will be steeper.
After some time, the photosphere will expand further out, the volume will expand spherically, the optical depth will decrease, and deeper layers will be exposed, which makes the lightcurve decay with a much shallower slope. \cite{NakarSari2010} found a slope of $\approx -0.35$ for a radiative atmosphere profile.
The uncertainty of the peak luminosity in terms of the coefficient $C_L$ and the complex structure make it difficult to predict the peak luminosity and subsequent lightcurve. Recent detailed modelling of explosions close to a stellar surface has provided analytical expressions which were corroborated numerically \citepalias{yalinewich_matzner2019}. We extend the modelling of \citetalias{yalinewich_matzner2019} to apply it to our disc geometry under certain approximations in sec. \ref{sub:surf} and derive an approximate analytic expression for the lightcurves in sec. \ref{sub:lc}.
In sec. \ref{sec:numerical}, we present the different initial conditions for our models, which include different internal properties and spatial locations for the explosions, different vertical profiles and SMBH masses, as well as discs with reduced densities. We also describe the numerical techniques used in this paper. We devote sec. \ref{sec:results} to describing the obtained lightcurves for the aforementioned models, both analytically and numerically. We discuss which models are in good agreement between the analytic and numerical models, and which differ and why.
Finally, we discuss the observational aspects of our results in sec. \ref{sec:discussion}. We discuss which of the events may be observable and could be distinguished from the underlying AGN luminosity. We then estimate the total rates of these events, their observed fraction and the origins of their large uncertainty. We also briefly discuss the response of the disc and comment on AGN variability and potential past and future observations of SNe explosions in AGNs.
We make a summary of our key points in sec. \ref{sec:summary}. Readers interested in the key takeaway messages without the technical details can directly skip to this section.
\section{AGN disc structure and properties} \label{sec:AGN}
AGN discs are powered by accretion and are expected to be geometrically thin but optically thick (although there may be optically thin windows, see sec. \ref{sub:opacity}). In order to describe AGN disc structures, we use several parameters. The radial coordinate $r$ is expressed in units of the Schwarzschild radius $r_s\equiv 2GM_{\bullet} /c^2\approx 1.974M_8\ \rm AU$, where $M_8\equiv M_\bullet / 10^8\textnormal{M}_\odot$ is the SMBH mass normalized to $10^8\,\textnormal{M}_\odot$. The vertical structure is governed by the scale height $H$ which is generally small compared to the size of the disc ($H\ll r$).
\subsection{Radial structure models} \label{rad}
AGN discs are generally modelled as viscous accretion discs. On the largest, galactic scales, discs are expected to be cold, gravitationally unstable and fragment into stars, with the effective Toomre $Q\equiv c_s \Omega / \pi G \Sigma \ll 1$, where $c_s$ is the sound speed, $\Omega$ is the orbital frequency and $\Sigma$ is the surface density. However, only a fraction of the gas is fragmented and the remaining gaseous component is still retained. The transition from the unstable galactic discs to accretion discs around a central object is still unclear. Angular momentum transport via global torques, rather than local viscosity, may keep the disc marginally stable and avoid fragmentation, which is the mechanism that is assumed in the \cite{tqm05} model throughout the disc, even at smaller radii.
The structure of AGN discs can be inferred from the optical/UV spectral energy distribution (SED) in the inner regions, and from mass inflow and star formation in the outer regions. \citet[hereafter \citetalias{sg03}]{sg03} obtained an accretion disc model that fits the SED observations, while \cite{tqm05} constructed a model regulated by mass inflow and star formation. The SG03 model best describes the disc structure in the inner regions, up to $\lesssim 10^5 r_s$, which is the radial region of interest in our study
The radial gas density structure of the \citetalias{sg03} model may be inferred from the marginal gravitational instability condition (Eq. 15 of \citetalias{sg03}),
\begin{equation}
\rho(r)=\frac{\Omega^2}{2\pi G Q_{\rm min}} \approx 1.22 \cdot 10^{-9} M_8^{-2} \left(\frac{r}{10^3 r_s} \right)^{-3} {\rm g\ cm^{-3}}. \label{eq:q}
\end{equation}
The latter equation is valid for $r\ge 10^3 r_s$, where $Q_{\rm min}\approx 1$ in the \citetalias{sg03} model. The disc is expected to be dominated by radiation pressure $p_{\rm rad} = aT^4/3$, where $a=4\sigma_{\rm SB}/c$ is the radiation constant and $\sigma_{\rm SB}$ is the Stefan-Boltzmann constant. In the \citetalias{sg03} model, radiation pressure indeed dominates in most parts of the disc. However, the gas pressure may have a significant contribution to the total pressure near $r\approx 10^3r_s$. Nevertheless, we use the radiation pressure throughout the paper, and the adjustments needed to include other sources of pressure are straightforward.
The aspect ratio of a thin disc should be small, i.e. $H/r\ll1$. In the \citetalias{sg03} model, the aspect ratio is approximately given by
\begin{equation}
\frac{H}{r} = 8 \cdot 10^{-3} \left( \frac{r}{10^3r_s} \right)^{1/2},\label{aspect_ratio}
\end{equation}
in our range of interest.
\subsection{Vertical structure models} \label{ver}
The vertical structure of an AGN disc and its stability are also under debate (see the review of \citealp{davis_review2020} for discussion and further references). While gas pressure dominated discs are expected to be vertically stable, the stability of radiation dominated discs is undecided. Theory predicts that radiation dominated discs will be unstable \citep{ss76}. Contrary to this prediction, early radiative magneto-hydrodynamical (MHD) shearing box simulations, such as those with the ZEUS code, found that the disc is stable over many thermal timescales \citep{hirose2009}. However, these results were later challenged by simulations with the ATHENA code. Although small box sizes reproduced the stability of \cite{hirose2009}, larger box size simulations resulted in an eventual runaway (where the disc either expands or collapses), although on much longer timescales than the thermal timescale \citep{jiang2013}. Iron bound-bound opacity may revert the situation again and make the vertical structure stable around temperatures of $1.8\cdot 10^5 {\rm K}$ \citep{jiang2016}.
Given the uncertainties involved in the stability and the detailed vertical structure, we will remain agnostic about the true structure. Nevertheless, a different vertical structure can lead to different observational signatures for the same initial conditions of the explosion. Here we consider several possible models for the vertical structure as described below, where their main common feature is that they all describe a thin disc. Thus the vertical density profile should fall off sharply where the vertical height $h$ is larger than the scale-height, $h\gtrsim H$. Note that it is rather different from a $\rho\propto r^{-2}$ distribution that is often assumed in studies of interaction-powered SNe. For our canonical model, we assume a Gaussian density profile
\begin{equation}
\rho_{\rm gas}(h) = \rho_0 \exp \left(-\frac{h^2}{2H^2}\right);\quad \frac{z}{h}\in \mathbb{R}, \label{eq:rho_gas}
\end{equation}
which is applicable for gas-dominated discs and is stable. We use hydrostatic equilibrium to derive the density profile for radiation dominated discs (if they are stable) in Appendix \ref{vertical profile},
\begin{equation}
\rho_{\rm rad}(h) = \rho_0 \left( 1 - \frac{h^2}{6H^2}\right)^3;\quad \left| \frac{h}{H} \right| < \sqrt{6}, \label{eq:rho_rad}
\end{equation}
and present some of the results with this structure, as well as a simplified step profile of constant $\rho_{\rm step}=\rho_0$ up to a scale height $H$ away from the midplane.
We assume that the disc is stable and static with the respective vertical structure prior to the explosion. We compare the results for different vertical structures in sec. \ref{sec:disc_properties}.
\subsection{Opacity} \label{sub:opacity}
At high temperatures, $T>10^4 \rm K$, the gas is mostly ionized and the opacity is dominated by electron scattering and bremsstrahlung (free-free). A typical value of the opacity in this range can be the electron scattering opacity $\kappa_{\rm es} = 0.2 (1+X)\ \rm cm^2\ g^{-1}$ where $X$ is the Hydrogen fraction. The opacities may be enhanced by $\approx 2$ orders of magnitude when detailed opacity tables are considered (\citealp{iglesias1996}, see also Fig. 3 of \citealp{davis_review2020}). At temperatures below $\sim 100\ \rm K$, dust absorption is the dominant effect that increases opacity, which scales as $\kappa_{\rm dust} \approx 2.4 (T/100 {\rm K})^2 \ \rm cm^2\ g^{-1}$, while the opacity is roughly constant for $T\approx 10^2-10^3\ \rm K$ in the range of $\kappa \approx 1-10\ \rm cm^2\ g^{-1}$ (see Fig 1. of \citealp{tqm05}, which is based on more recent opacity models of \citealp{semenov2003}). For intermediate temperatures around $10^3-10^4\rm K$, the gas is neutral, and most of the dust is already sublimated, thus the opacity is extremely low. Depending on the radial temperature profile, an optically thin window is expected to occur in AGN discs. The discs are optically thin at $r\gtrsim 10^4 r_s$ in the models we adopted. Explosions in the optically thin region will look like standard SNe as the emitted radiation will be unaffected by the surrounding disc, while the associated kinetic outflows will be choked by the enormous mass. Explosions in the optically thick and massive regions ($\approx 10^4 r_s$ for $M_\bullet = 10^7 \textnormal{M}_\odot$) will be completely choked (see sections \ref{sec:numerical} and \ref{sec:discussion}). For our analytical modeling, we use a constant electron scattering opacity of solar composition ($X=0.7$), $\kappa_{\rm es}=0.34\ \rm cm^2\ g^{-1}$. We use realistic OPAL tables \citep{iglesias1996} for our radiative transfer simulations (see sec. \ref{sec:numerical} for details).
\subsection{Optical depth} \label{sub:optical depth}
For a constant opacity $\kappa$, the optical depth at height $h$, in the direction normal to the disc, is given by
\begin{equation}
\tau(h)=\kappa\intop_{h}^{\infty}\rho(h')dh' \equiv \tau_0 \mathcal{F}(\zeta), \label{eq:tau_def}
\end{equation}
where we defined for convenience the dimensionless quantities $\tau_0 \equiv \kappa \rho_0 H$, and $\zeta \equiv h/H$. The functional form of $\mathcal{F}(\zeta) $ depends on the vertical structure.
For a uniform step density, the integral is straightforward, and the solution is $\tau(h)=\kappa\rho_{0}(H-h)=\tau_{0}(1-\zeta)$ for $0\le h\le H$ and $\tau=0$ for $h>H$ since there is no material there. This leads to $\mathcal{F}_{\rm step}(\zeta)=1-\zeta$.
For the Gaussian profile, applicable for the gas dominated disc, we have
\begin{equation}
\mathcal{F}_{\rm gas}(\zeta) = \frac{1}{H} \intop_{h}^{\infty}\exp \left(\frac{-h'^2}{2H^{2}} \right)dh' = \sqrt{\frac{\pi}{2}} {\rm erfc}\left(\frac{\zeta}{\sqrt{2}}\right)
\end{equation}
where ${\rm erfc}(x)$ is the complimentary error function.
For the radiation-dominated profile, we have
\begin{equation}
\mathcal{F}_{\rm rad}(\zeta) = \frac{1}{H} \intop_{h}^{\sqrt{6}H}\left( 1-\frac{h'^2}{6H^2}\right)^3 dh' = \intop_{\zeta}^{\sqrt{6}} \left(1-\frac{\zeta'^2}{6} \right)^3d\zeta' = p(\zeta), \label{eq:frad}
\end{equation}
where $p(\zeta)=16\sqrt{6}/35-\zeta+\zeta^{3}/6-\zeta^{5}/60+\zeta^{7}/1512$.
\section{Analytic lightcurve modeling} \label{sec:analytic}
Here we model the lightcurve of typical explosions. We first discuss the assumptions of the model. We then proceed to discuss the velocity of the shock front, and finally calculate the the lightcurve by drawing analogies between our disc explosions and the recently developed theory of surface explosions. Although our analysis is generic, when applicable, we refer to our canonical model (1) of an explosion of energy $E_0=10^{51}\ \rm erg$ and ejecta mass $M_{\rm ej}=1.3 \textnormal{M}_\odot$, which serves as a proxy for a standard type Ia SN explosion. The radial location of the explosion is at $r=10^3 r_s$ around an SMBH of mass $10^7 \textnormal{M}_\odot$, in the disc midplane. The disc has density $\rho_0=1.22\cdot 10^{-7}\ \rm g\ cm^{-3}$ and scale height $H=8\cdot 10^{-3} r = 2.36\cdot 10^{13}\ \rm cm$ with a gas-dominated Gaussian vertical profile. For the opacity, we make use of a constant opacity $\kappa=0.34\ \rm cm^2\ g^{-1}$, which corresponds to electron scattering at Solar compositions. The optical depth is then $\tau=\sqrt{\pi/2} \kappa \rho_0 H = 1.2\cdot 10^6$. Other models are described in sec. \ref{sec:numerical} and their initial conditions are summarized in table \ref{tab:1}.
\subsection{Assumptions} \label{sub:assumptions}
Here we describe the assumptions made in constructing our solution for shock propagation, breakout and the resulting lightcurve. Consider an explosion going off in dense material. If the material is sufficiently optically thick and the prompt energy deposition is sufficiently energetic, the primary mode of energy transport, at least initially, will be kinetic outflows rather than radiation transport (photon diffusion). The initial disc temperature is $T_0$, and the explosion is spherical. For collimated outflows, not considered here, one would need to introduce an additional solid angle $\Omega_s$ spanned by the outflow.
\textit{i) Is the outflow relativistic?} The characteristic velocity of a non-relativistic outflow is $v_0=\sqrt{E_0/M_{\rm ej}}$, so we require that $(v_0/c)^2 = E_0/(M_{\rm ej}c^2)\ll 1$ for an outflow to be non-relativistic. And indeed, for the canonical model, representative of type Ia SNe, $v_0/c=0.02$. Similarly, core-collapse SNe also produce non-relativistic outflows.
\textit{ii) Is the kinetic outflow the dominant form of energy transport?} For this, the velocity must be faster than the photon diffusion speed, at least initially, or, equivalently, the depth must be sufficiently large. The optical depth at height $z$ is formally given by Eq. \ref{eq:tau_def}. If at the explosion site the local density is $\rho_0$ and the shock propagates over a typical distance, $R_0$, the optical depth is approximately estimated as $\tau \sim \kappa \rho_0 R_0$, where $\kappa$ is the opacity, and the diffusion speed is $c/\tau$. For kinetic outflow to dominate, we require $v_0\gg c/\tau$, or $\tau^2 E_0/M_{\rm ej}c^2\gg 1$. Combining conditions \textit{i)} and \textit{ii)}, the optical depth must also be large, $\tau \gg 1$. Note that this defines a hierarchy of velocities $c/\tau \ll v_0 \ll c$. For our canonical model, the explosion is at the midplane, and the midplane optical depth ($z=0$) is estimated as $\tau \gtrsim 10^6$, so these conditions are initially satisfied.
\textit{iii) Does the shock leave radiation-dominated material behind?} To answer this question, we require that the radiation energy density $a T^4$, which may be approximated by $\sim \rho v_0^2$ near shock, will be much larger than the gas energy density $\rho k_{\rm B} T/m_{\rm p}$, where $k_{\rm B}$ is Boltzmann's constant and $m_{\rm p}$ is the proton mass. Eliminating the temperature, this condition becomes equivalent to $(\rho v_0^2/a) \gg (k_{\rm B} \rho/m_{\rm p}a)^{4/3}$, or
\begin{equation} \label{eq:rad}
v_0^2= \frac{E_0}{M_{\rm ej}} \gg \left( \frac{k_{\rm B}^4 \rho}{m_{\rm p}^4 a}\right)^{1/3} .
\end{equation}
The latter in only an estimate within an order of magnitude.
Note that the latter three conditions are equivalent to the assumptions made by \citet[hereafter: YM19]{yalinewich_matzner2019} in the context of surface explosions. We return to the detailed modeling of \citetalias{yalinewich_matzner2019} in our lightcurve modelling is sections \ref{sub:surf}-\ref{sub:lc}.
\subsection{Shock velocity profile} \label{sub:shock v}
Here we discuss several models for the shock velocity profile. The description is mostly generic, although sometimes, for a concrete example, we use the canonical model (model (1) in table \ref{tab:1}). We later investigate how the variation of some of the disc and explosion properties affect our results in sec. \ref{sec:results}.
For an explosion with energy $E_0$ and ejecta mass $M_{\rm ej}$, the typical velocity is $v_0 \equiv \sqrt{E_0/M_{\rm ej}}$.
The outflow in the plane of the disc will be quenched by the disc material, and the shock will break out in the vertical direction. Therefore, we focus on the vertical height $z$ above the explosion. Generally speaking, a shock wave is always accompanied by three waves: forward shock, reverse shock (or rarefaction), and the contact discontinuity. In our context, the forward shock compresses the CSM, while the reverse shock compresses the ejecta. The contact discontinuity separates the ejecta and the CSM matter. For most of our models, the CSM matter is more massive than the ejecta, so the reverse shock quickly traverses the ejecta while the forward shock is still propagating outwards. We therefore focus only on the forward shock, unless stated otherwise explicitly.
The shock velocity will depend on the shock front radius $z(t)$ and on the density at this radius. Generally, three qualitatively different regimes will be evident:
\textit{i) Free expansion regime:} In this case, the swept mass $M(z)$ is much smaller than the ejected mass, i.e., $M(z) \sim \rho(z_{\rm min}) (z(t)-z_{\rm min})^3 \ll M_{\rm ej}$, where $z_{\rm min}$ is the vertical location of the explosion site, $z(t)$ is the height reached by the shock at time $t$, and the shock velocity is close to $v_0$.
\textit{ ii) Sedov-Taylor deceleration regime:} Once the swept mass becomes comparable to the ejecta mass, $M(z)\gtrsim M_{\rm ej}$ the shock front will decelerate. Dimensional analysis leads us to a length scale $z(t)=\beta(E_{0}/\rho(z_{\rm min}))^{1/5}t^{2/5}$ and a velocity scale
\begin{equation}
v_{\rm ST}(t)=\frac{dz}{dt}=\frac{2\beta}{5}\left( \frac{E_{0}}{\rho(z_{\rm min})}\right)^{1/5}t^{-3/5}. \label{eq:v_st}
\end{equation}
Here, $\beta$ is an order unity parameter which can in principle be estimated from the energy equation. We take for simplicity $\beta=1$. The latter scaling is the self-similar Sedov-Taylor (ST) solution \citep{Sedov1946, Taylor1950}, where the shock front expands as $R\propto t^{2/5}$ and the shock velocity decelerates $v_{\rm ST} \propto t^{-3/5}$.
\textit{iii) Sakurai accelerating regime:} The ST picture is correct if the density is uniform. The vertical structure of the disc is generally far from uniform. Close to the disc edge (either the physical edge or the photosphere, if the disc is formally infinite), the density gradient is quite steep, and the shock propagates more easily, which causes it to accelerate with decreasing density as $v\propto\rho^{-\mu}$, which is known as the Sakurai law \citep{Sakurai1960}. For radiation dominated material, $\mu \approx 0.19$ fits well with previous estimates (see also \citealp{MM99}, hereafter \citetalias{MM99}, their sec. 4.2 and references therein for more details and discussion). Note that the latter picture is valid if the swept mass $M(z)\gtrsim M_{\rm ej}$. Some of our models have the swept mass comparable to or smaller than $M_{\rm ej}$, and in these cases the ST phase is skipped and the shock may only accelerate or directly break out.
Close to the centre, the density is almost constant, and the evolution is as in the ST regime. Close to the edge, a small change in the radial scale leads to a large change in the density, and therefore the shock will follow the Sakurai law. It is possible to write down a general formula that includes both cases, as done in \citetalias{MM99}:
\begin{equation}
v(t)=v_{0}\left(\frac{t_{1}}{t}\right)^{3/5}\left(\frac{\rho(z(t))}{\rho(z_{\rm min})}\right)^{-\mu},\label{v_extra}
\end{equation}
where $t_1$ is the time when a transition from the free expansion to the ST deceleration occurs and is given by $v_{\rm ST}(t_1)=v_0$ from Eq. \ref{eq:v_st}. Plugging $t_1$ into the ST length scale leads to $z(t_1)=z_1=(4M_{\rm ej}/25\rho(z_{\rm min}))^{1/3}$. The transition occurs when the swept mass is $4M_{\rm ej}/25=0.16M_{\rm ej}$. For the ejecta mass of $1.3 \textnormal{M}_\odot$, as in our model 1, the transition occurs at $0.21 \textnormal{M}_\odot$, or at radius $z\approx 1.5 \cdot 10^{13}\ \mathrm{cm} =0.63 H$.
The transition from the ST decelerating velocity to Sakurai accelerating velocity occurs when $v(t)$ is at a minimum. From the ST solution, we can invert $z(t)$ to $t(z)\propto z^{5/2}$ and find the location of the minimum $z_2$ given by $d\ln\rho/d\ln z|_{z_2}=-3/(2 \mu)$. For a Gaussian profile, the velocity behaves as $v\propto z^{-3/2}\exp(\mu z^2/(2H^2))$, the logarithmic derivative of the density is $d\ln\rho/d\ln z=-(z/H)^2$, and the minimum is achieved at $z_2/H=(3/(2\mu))^{1/2}\approx 2.8$.
The three regimes and their transitions described above allow us to construct two possible solutions. The first is the piecewise velocity solution. Here we divide the behaviour into three regimes, where we first begin with free expansion and transform to ST deceleration. The ST approach is not accurate since it was developed for a uniform density. The density change is negligible for $z\ll H$, but it is already significant at $z_1$. We use an effective scaling where we replace $\rho(z_{\rm min})$ in the ST solution by $\rho(z)$. Although not entirely self-consistent, we can write $v$ as in Eq. (\ref{v_extra}) but with an effective density power law $\mu'=\mu+1/5$, where the $1/5$ is coming from the ST scaling of $z\propto \rho^{-1/5}$. The piecewise velocity is
\begin{align} \label{vpi}
{v_{\rm p}(z)=v_{0}\begin{cases}
1 & z<z_{1}\\
\left(\frac{z_{1}}{z}\right)^{3/2}\left[\frac{\rho(z)}{\rho_{1}}\right]^{-(\mu+1/5)} & z_{1}<z<z_{2}\\
\left(\frac{z_{1}}{z}\right)^{3/2}\left(\frac{\rho_{2}}{\rho_{1}}\right)^{-(\mu+1/5)}\left[\frac{\rho(z)}{\rho_{2}}\right]^{-\mu} & z>z_{2}
\end{cases}}
\end{align}
where $\rho_i=\rho(z_i)$ for $i=1,2$.
Another solution is the direct \citetalias{MM99} shock velocity
\begin{equation}
v_{\rm MM99}(z)=\left(\frac{E_{0}}{M_{{\rm ej}}+M(z)}\right)^{1/2}\left(\frac{\rho(z)}{\rho(z_{\rm min})}\right)^{-\mu}, \label{vMM99}
\end{equation}
Note that Eq. (\ref{vMM99}) captures well all of the regimes. For $M_{\rm ej}\gg M(z)$ the density and hence the velocity are essentially constant. When $M(z)\gg M_{\rm ej}$ but $\rho(z) \sim \rho(z_{\rm min})$, the velocity is $\propto z^{-3/2}\propto t^{-3/5}$ as expected from a ST solution. Finally, once $\rho(z)\ll \rho(z_{\rm min})$, the Sakurai term dominates the acceleration.
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_1_final.png}
\caption{Shock velocity prescriptions for model 1 in the AGN disc, in units of the speed of light $c$. The solid lines are the theoretical shock velocities; the piecewise model (blue), Eq. (\ref{vpi}), and the \citetalias{MM99} model (green), Eq. (\ref{vMM99}). The dashed lines are the numerical shock velocities. The hydrodynamical code \texttt{HORMONE} is used to obtain the matter velocity $v_m$ just behind the shocked CSM, and the shock velocity is $v_m (\gamma+1)/2$ in the disc geometry (dashed black) and spherical geometry (dashed red). The result from the spherically-symmetric radiative transfer code \texttt{SNEC} is also presented in magenta. For completeness, the photon diffusion velocity, which is also the inverse of the optical depth, is shown in dot-dashed grey lines. The shock breakout occurs when the gray line crosses the shock velocity, which occurs close to $z\sim 4 H$ for all shock velocity estimates for model 1. }
\label{fig:velocity_models}
\end{figure}
Figure \ref{fig:velocity_models} shows the different velocity models for the canonical (model 1) explosion together with the shock velocity obtained numerically. The detailed description of the hydrodynamical code \texttt{HORMONE} and the radiative transfer code \texttt{SNEC} appears in sec. \ref{sec:numerical}. The three velocity regimes are clearly evident. The flat free expansion phase lasts until $z_1/H\approx 0.63$ for model 1, where the velocity starts to decelerate. The transition to Sakurai acceleration occurs somewhere around $z/H\sim 2-2.8$. We see that the matter-derived shock velocity $v_m(\gamma+1)/2$ better agrees with the piecewise model, while the \texttt{SNEC} shock velocity is somewhere in between the piecewise and \citetalias{MM99} model. In either case, the overall acceleration in the low-density outskirts of the disc is underestimated.
In summary, there are two main theoretical shock velocity estimates for an explosion inside the AGN disc: a piecewise fit explicitly taking account the three distinct propagation regimes (Eq. \ref{vpi}), and a direct continuous measure of the swept mass (\citetalias{MM99} - Eq. \ref{vMM99}). The numerical shock velocity falls between the two theoretical shock velocity models.
\subsection{Breakout time}\label{sub:bo time}
Even though the shock is accelerating in the Sakurai regime, the photon diffusion velocity rises more steeply. Eventually, the photon diffusion speed $c / \tau(z)$ will intersect the shock velocity $v(z)$. The two velocities are equal when the grey dot-dashed line intersects the velocity model of choice in Fig. \ref{fig:velocity_models}. The location of this intersection defines the breakout shell $z_{\rm bo}$, which is implicitly given by $\tau(z_{\rm bo}) = c /v(z_{\rm bo})$ and is around $\sim 4 H$ for model 1, regardless of the velocity model.
The breakout time $t_{\rm bo}$ is the time elapsed for the first light to be beyond the breakout shell from the time of the explosion. In order to finally escape, the photons need to diffuse up to the photosphere. This diffusion time is much shorter than the breakout time.
The breakout time can be calculated by
\begin{equation}
t_{\rm bo} = \intop_{z_{\rm min}}^{z_{\rm bo}} \frac{dz}{v(z)} \label{eq:t_bo}
\end{equation}
The exact expressions are given either by special functions or integrated numerically. We provide them in Appendix \ref{appendix-breakout}, together with the expressions for off-plane explosions.
\subsection{Modelling AGN explosions as surface explosions} \label{sub:surf}
Surface explosions are explosions that occur in a relatively sparse material, close to a surface of a medium and far away from the dense core of the medium (e.g. a massive star). \citetalias{yalinewich_matzner2019} studied in detail such surface explosion in massive stellar envelopes. Our explosion is analogous to a surface explosion in the following way. Due to the planar geometry of the disc and its large spatial extent $r \gg H$, an explosion in the midplane of the disc cannot propagate in the in-plane direction, and the breakout occurs effectively on the vertical surface of the disc. This is analogous to an explosion at a distance $l\ll R_{\star}$ below the surface of a star, the model for which was recently put forward by \citetalias{yalinewich_matzner2019}. The shock propagates only upwards. Here we parametrise our explosion as a surface explosion, with some limitations.
\subsubsection{Choosing length scales} \label{choose_leff}
The \citetalias{yalinewich_matzner2019} model assumes that the density vanishes at some boundary $z_b$ and that the density scales as a power law profile of the form $\rho = \rho_{0}(x/l)^{\omega}$ where $x$ is the \textit{distance from the edge} and $l$ is a typical length scale of order the distance of the explosion hotspot from the edge. When applied to our extended model, $l$ is generally on the order of H, but its exact definition has some ambiguities as we discuss below.
The surface explosion modelling in \citetalias{yalinewich_matzner2019} is encapsulated by two dimensionless parameters, $\Gamma\equiv E_{0}/(c^{2}\rho_{0}l^{3)}$ and $\tau_{0}=\kappa\rho_{0}l$. Initially, the only length scale we have is $H$. In \citetalias{yalinewich_matzner2019}, $l$ has several different roles:
\textit{i) The distance from the explosion to the edge.}
\textit{ii) The transition between the end of the Sedov-Taylor phase and the beginning of the shock acceleration phase.}
\textit{iii) The value in the expression for the velocity at the end of the Sedov-Taylor phase, $v_{{\rm ST}}=\sqrt{E_{0}/\rho_{0}l^{3}}$.}
These three definitions appear to be contradictory. First, the free expansion phase is ignored, or a point-source explosion is assumed. Second, after passing a length $l$ (as in i.), formally, we should be at the edge, which is definitely after the breakout. However, according to ii. and iii., it is only the phase where the shock starts to accelerate and is assumed to be optically thick.
These differences are small in the stellar atmosphere with its steep density profile towards the edge. However, we saw that for model 1, the differences between the transition to accelerating shock at $z_{2}=2.81H$, the breakout shell at $z_{{\rm bo}}=3.99H$ for the piecewise model and $z_{{\rm bo}}=3.96H$ for the \citetalias{MM99} model, and the photosphere at $z_{{\rm ph}}=4.93H$, are around a scale height away from each other. Moreover, free expansion phase ends at $z_{1}=0.64H$, which is also comparable to the scale height.
The velocity at the end of the ST phase is $v_{\rm ST} \sim c\Gamma^{1/2}$. Hence the effective length scale $l_{\rm eff}$ in this case should replicate the total range of the swept mass from the end of the free expansion phase until the breakout shell. We therefore set $l_{\rm eff}=z_{\rm bo} -z_1$, unless otherwise specified.
\begin{equation}
\Gamma = \frac{E_{0}}{c^{2}(M_{{\rm ej}}+\rho_{0}l_{{\rm eff}}^{3})}. \label{Gamma}
\end{equation}
Here, the inclusion of $M_{\rm ej}$ in the denominator extends the analysis of \citetalias{yalinewich_matzner2019} to include the free expansion phase and also avoids unphysically large values of $\Gamma$ for low densities or small $l_{\rm eff}$.
The optical depth $\tau_0$ should be comparable with the midplane optical depth, hence we keep $\tau_0$ as $\kappa \rho_0 H$.
\subsubsection{Effective power law index $\omega_{\rm eff}$ }
Once we set $l_{\rm eff}$ and $\Gamma$ for the explosion and also have $\rho_0, H$ and $\tau_0$ from the AGN disc model, we can assign an effective power law index $\omega_{\rm eff}$ in the following way: $\omega_{\rm eff}$ is a local value and formally varies along the vertical shock propagation.
For a local density profile
\begin{equation}
\rho(x)=\rho_a (x/l_{\rm eff})^{\omega_{\rm eff}}, \label{eq:omega}
\end{equation}
where $\rho_a$ is the density at a distance $l_{\rm eff}$ from the edge, and $0\le x \le l_{\rm eff}$ is measured from the edge.
We impose a matching condition on $d\ln\rho/dz$ between the assumed disc density profile and the power law model in order to fix $\omega$:
\begin{equation}
\rho_{a}=\rho(z_{{\rm ph}}-l_{\rm eff});\quad\omega=-l_{\rm eff}\left.\frac{d\ln\rho}{dz}\right|_{z_{\rm ph}-l_{\rm eff}}\ . \label{omega_eff_gen}
\end{equation}
For the slab profile, the derivative is zero and thus $\omega_{\rm eff}=0$ everywhere. For the Gaussian profile,
$d\ln\rho/dz=-z/H^{2}$, and
\begin{equation}
\omega_{{\rm gas}}(l_{{\rm eff}})=\frac{l_{{\rm eff}}(z_{\rm ph}-l_{{\rm eff}})}{H^{2}}. \label{omega_eff_gas}
\end{equation}
For the radiation dominated profile the power law is
\begin{equation}
\omega_{{\rm rad}}(l_{{\rm eff}})=6\frac{\sqrt{6}H-l_{{\rm eff}}}{2\sqrt{6}H-l_{{\rm eff}}}, \label{omega_eff_rad}
\end{equation}
where we used the actual edge of the disc $z_{\rm edge}=\sqrt{6} H$ instead of the photosphere.
We note that $\rho_a$, $l_{\rm eff}$ and $\Gamma$ are determined by the extent of the mass swept by the outflow prior to the breakout. $\tau_0$ is also well defined. Although Eq. \ref{omega_eff_gen} suggests that determining $l_{\rm eff}$ uniquely determines $\omega_{\rm eff}$, this is misleading, since in the \citetalias{yalinewich_matzner2019} model of Eq.~(\ref{eq:omega}), the density profile is a global power with index $\omega_{\rm eff}$, while in our case the power law is only a local fit to the density profile. Therefore, we have the freedom to decide where to match the density profile to the power law. We opted to do so at the depth of $l_{\rm eff}$ beneath the breakout shell in equations \ref{omega_eff_gas} and \ref{omega_eff_rad}. We could alternatively choose to do so at the breakout radius $z_\mathrm{bo}$ in an attempt to model the immediate post-breakout behavior more accurately; we consider this alternative choice in section \ref{sec:omega}.
Regardless of the local choice of $\omega_{\rm eff}$, the dependence of the luminosity on $\omega$ is expected to be much weaker than on the other parameters.
\subsection{Lightcurves} \label{sub:lc}
As mentioned in \ref{structure}, the slab geometry leads to adiabatic expansion in the optically thick regime and rapid decay. The setting is similar to an explosion close to a stellar surface studied by \citetalias{yalinewich_matzner2019}. They applied the latter arguments for a surface explosion using the dimensionless parameters $\Gamma$ and $\tau_0$ as discussed in sec. \ref{sub:surf}, where an additional phase, where the hotspot of the explosion becomes optically thin enough, which allows depletion of material deeper than the hotspot and forming a crater. The time of the transition where the hotspot is exposed is found to be $t_{\rm sph}=(H/c) \tau_0^{1/2} \Gamma^{-1/4}$ (Eq. 24 of \citetalias{yalinewich_matzner2019}).
Once we find $l_{\rm eff}$, $\omega_{\rm eff}$, followed by $\Gamma$ and $\tau_0$ as describe in sec. \ref{sub:surf}, we can use \citetalias{yalinewich_matzner2019}'s modelling for the lightcurve. From now until the end of the section, we drop the 'eff' subscript and simply write $\omega$ to avoid cumbersome notation. The resulting luminosity is
\begin{equation}
\frac{L(t)}{E_0c/H}=
\begin{cases}
\Gamma^{(\omega\mu -2\omega/3 - 5/6)\delta_{-} } \tau_0^{(5\omega\mu/3 - \omega -4/3)\delta_{-}} \left( \frac{ct}{H} \right)^{-4/3} & \rm{pl}\\
\Gamma^{(-\omega\mu + 1/6)\delta_{+} } \tau_0^{(\omega\mu/ - \omega -4/3)\delta_{+}} \left( \frac{ct}{H} \right)^{(-4\omega\mu+2/3)\delta_{+}} & \rm{sph}
\end{cases}
\label{lightcurve}
\end{equation}
in the planar and spherical phases, respectively, where $\delta_{\pm} \equiv (1\pm \omega\mu + \omega)^{-1}$. For concreteness, we choose $\omega(l_{\rm eff})$ as in equations \ref{omega_eff_gas} and \ref{omega_eff_rad}.
%
In certain cases, we will use an expansion of $\omega_{\rm eff}$ close to the edge, where we use $\omega^{\rm edge}_{\rm eff} = \omega_{\rm eff}(l_{\rm edge})$ with $l_{\rm edge} = z_{\rm edge} - z_{\rm bo}$, where the edge is either the physical edge of the disc (for radiative and slab models), or the photosphere (for Gaussian models). We discuss this choice and its implications in the results section.
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_2_final.png}
\caption{Analytical lightcurve for model 1. We show the analytic bolometric luminosity (i.e. the lightcurve) as a function of time with the solid blue curve. The numerical \texttt{SNEC} result is shown as a dashed red curve. The analytic curve is the maximum of the planar and spherical phases in Eq. \ref{lightcurve}. The vertical dot-dashed lines are the breakout times for the two velocity models given in Eq. \ref{t_pw} and \ref{t_mm99}, where the rightmost one is for the piecewise model since it has a lower velocity than the \citetalias{MM99} model. The gray area between them is the expected range of the breakout times. The dotted blue vertical line is the end of the spherical expansion phase, $t_{\rm sph}$.}
\label{fig:analytica_model1}
\end{figure}
Fig. \ref{fig:analytica_model1} shows the analytic lightcurve for model 1 in solid blue and the numerical \texttt{SNEC} lightcurve in dashed red. We shifted the analytical curve in time so that the breakout falls on top of the maximal luminosity of the numerical curve. After the peak, \texttt{SNEC} model luminosity is initially more shallow than the $L \propto t^{-4/3}$ decay expected in the planar phase from the simple arguments of section \ref{structure}, but subsequently steepens before becoming more shallow again at the transition to the spherical phase. After the initial breakout with $L \sim 10^{44}\ \rm erg\ s^{-1}$, the luminosity rapidly decays by one order of magnitude over $\sim 0.4\ \rm d$ from breakout and then decays more shallowly during the spherical phase. The end of the spherical phase is indicated by the dotted vertical line around $\sim 150\ \rm d$. The gray area in the region between $t_{\rm bo}|_{\rm MM99} < t < t_{\rm bo}|_{\rm pw}$ is given by Eqs. \ref{t_mm99} and \ref{t_pw}, respectively, and indicates the range of times when the breakout occurs. In addition, the \texttt{SNEC} lightcurve is also plotted in dashed red, where the peak luminosity of the analytic model is set to match the peak luminosity of the \texttt{SNEC} lightcurve, which falls in the margins of the breakout time. Overall, there is a good correspondence, and the bumps in the \texttt{SNEC} lightcurve can be attributed to the reverse shock initially and varying opacity effects at later times, as discussed in detail in section \ref{sec:results}.
We note that in order to set $z_{\rm bo}$ (and then $l_{\rm eff}$, $\omega$, $\Gamma$ and the lightcurve), we need to solve for the breakout shell $\tau(z_{\rm bo}) = c/v_{\rm bo}$. This is done numerically using the standard \texttt{fsolve} function of \texttt{Python}'s \texttt{scipy.optimize} package. We also need to choose a model for the velocity. We use the earlier derived piecewise model, Eq. \ref{vpi}, since the numerical breakout time is close to the breakout time evaluated using this velocity prescription. We also tried the \citetalias{MM99} prescription, and it gave almost indistinguishable results in most cases.
\section{Numerical Simulations} \label{sec:numerical}
Here we describe the numerical approach and initial conditions for our simulations and analysis.
\subsection{Initial conditions}
We already introduced the initial conditions for our baseline model at the beginning of section \ref{sec:analytic}. In order to explore the sensitivity to other parameters of the disc and the explosion, we also studied several other models, varying one or two parameters at a time. For changes in the global disc structure with the SMBH mass, we adopt the following scalings from the \citetalias{sg03} model. At radial locations scaled by the gravitational radius of the SMBH, i.e. at $r\propto r_s \propto M_\bullet$, the midplane density is $\rho \propto \Omega^2 \propto M_\bullet/r^3 \propto M_\bullet^{-2}$. The rescaled interacting mass is $\rho H^3 \propto M_\bullet$. The optical depth is also rescaled as $\tau \propto \rho H \propto M_\bullet^{-1}$. The aspect ratio is taken from Eq. \ref{aspect_ratio}.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\# & $M_\bullet$ & $r/r_{s}$ & $E_{51}$ & $\rho(z)$ & $z/H$ & $M_{\rm CSM}/\textnormal{M}_\odot$ & Remarks\tabularnewline
\hline
\hline
1 & $10^{7}$ & $10^{3}$ & $1$ & Gas & $0$ & $12.8$ & a\tabularnewline
\hline
2 & $10^{7}$ & $10^{4}$ & $1$ & Gas & $0$ & $390$ & b\tabularnewline
\hline
3 & $10^{8}$ & $10^{3}$ & $1$ & Gas & $0$ & $128$ &c\tabularnewline
\hline
4 & $10^{6}$ & $10^{4}$ & 1 & Gas & 0 & $39$ & b,c\tabularnewline
\hline
5 & $10^{7}$ & $10^{3}$ & $10$ & Gas & $0$ & $12.8$ & d\tabularnewline
\hline
6 & $10^{7}$ & $10^{3}$ & $0.1$ & Gas & $0$ & $12.8$ & d\tabularnewline
\hline
7 & $10^{7}$ & $10^{3}$ & $1$ & Rad & $0$ & $7.6$ &e\tabularnewline
\hline
8 & $10^{7}$ & $10^{3}$ & $1$ & Step & $0$ & $3.4$ & e\tabularnewline
\hline
9 & $10^{7}$ & $10^{3}$ & $1$ & Gas & $1$ & $1.92$ &f\tabularnewline
\hline
10 & $10^{7}$ & $10^{3}$ & $1$ & Gas & $2$ & $0.15$ & f\tabularnewline
\hline
11 & $10^{7}$ & $10^{3}$ & $1$ & Rad & $1$ & $0.5$ &e,f\tabularnewline
\hline
12 & $10^{7}$ & $10^{3}$ & $1$ & Rad & $2$ & $6.4\cdot10^{-4}$ &e,f\tabularnewline
\hline
13 & $10^{6}$ & $10^{3}$ & 1 & Gas & 0 & $1.28$ &c \tabularnewline
\hline
14 & $10^{7}$ & $10^{3}$ & $1$ & Gas & $0$ & $1.28$ &g\tabularnewline
\hline
15 & $10^{7}$ & $10^{3}$ & $1$ & Gas & $0$ & $12.8$ &h\tabularnewline
\tabularnewline
\end{tabular}
\caption{Initial conditions for the simulations. The columns are: simulation number, central BH mass (in $M_{\odot}$), radial location, explosion energy (in $10^{51}\rm \ erg$), the vertical height above the midplane, CSM mass (Eq.~\ref{eq:m_int}), and remarks describing the parameters being varied: a) The canonical model. b) Model with varied radial distance $r$. c) Model with varied mass of the SMBH. d) Model with varied explosion energy $E_0$. e) Model with varied vertical density profile $\rho(z)$. f) Model with varied vertical distance $z$. g) Model with a reduced density $\rho_0 = 1.23\cdot 10^{-8}\ \rm g\ cm^{-3}$ to mimic a starved AGN. h) Model with a larger ejecta mass, $M_{\rm ej} = 10 \textnormal{M}_\odot$, to mimic a core-collapse SN.} \label{tab:1}
\par\end{center}
\end{table}
Table \ref{tab:1} lists the grid of initial conditions for the numerical simulations. Model 1 is our baseline canonical model, representing a typical type Ia SN explosion. Models 2-4 and 13 explore the impact of the SMBH masses and the radial location. Models 5-6 explore the impact of the explosion energy. Models 7-8 and 11-12 explore the impact of the vertical profile. Models 9-12 explore the impact of the vertical location. Model 14 mimics a starved AGN disc with reduced density, while Model 15 mimics a core collapse SN by considering more massive ejecta than in other models, $M_{\rm ej}=10\ M_{\odot}$. The explosion energy of SN 1987A is estimated at $(1.5\pm0.12) \times 10^{51}$ erg \citep{1987A}. The observed explosion energies of core-collapse SNe range from a few times $10^{50}$ erg to $\sim 10^{53}$ erg for hypernovae with massive progenitors \citep{janka2012}. We therefore keep the explosion energy at $10^{51}$ erg for model 15 as most representative for core-collapse SNe.
The CSM mass in the penultimate column of table~\ref{tab:1} is defined as
\begin{equation}
M_{\rm CSM} =4\pi\intop_{z_{\rm min}}^{z_{\rm max}}\rho(z)(z-z_{\rm min})^{2}dz, \label{eq:m_int}
\end{equation}
which depends on the density profile, and the maximal and minimal vertical locations. In Eq. \ref{eq:m_int}, we assumed a spherical approximation, namely that the most significant interaction of the ejecta with the disc material will be in the vertical direction, allowing us to approximate the density profile at radius $r$ as $\rho(r)=\rho(z=r)$. This is true in our geometry since the in-plane directions are choked. We perform the explicit calculation of the CSM mass for each model in Appendix \ref{app:int_mass}.
\subsection{Spherically symmetric lightcurve modelling}
To model the lightcurves arising from the CSM interaction numerically and verify our analytic models, we perform detailed simulations with the supernova explosion code \texttt{SNEC} \citep{snec_code, snec_code2}. \texttt{SNEC} is a spherically-symmetric Lagrangian radiative-hydrodynamics code that accounts for shock propagation through the use of artificial viscosity. Compared to our analytic models, \texttt{SNEC} directly solves the equations of radiative hydrodynamics and, therefore, allows us to assess the validity of our approximations. Additionally, it allows us to gain some intuition into the importance of ionisation, detailed opacities and heating from $^{56}{\rm Ni}$ for forming the final lightcurves.
\texttt{SNEC} uses OPAL opacities \citep{iglesias1996} in the high-temperature regime (at $3.75<\log_{10} T/{\rm K}<8.7$) and the solar-scaled \citet{ferguson2005} opacities, which accounts for molecular contributions, in the low-temperature regime (at $2.7<\log_{10} T/{\rm K}<4.5$). In the overlapping regions, the low-temperature opacity is preferred.
The code accounts for ionisation states by solving the equilibrium Saha equation and models radiative transfer through flux-limited equilibrium photon diffusion. \texttt{SNEC} also models the contribution to the lightcurve from the radioactive decay of $^{56}{\rm Ni}$ and $^{56}${\rm Co}. The decay leads to the deposition of gamma-rays and positrons, which heat the ejecta, with gamma-rays dominating the heat deposition. The code solves for gamma-ray propagation and absorption through solving for radiative transfer equation with grey opacities.
We set up explosions by initialising balls of material with an exponential density profile with a radial scale height set so that the density at the boundary of the ball is $1\,{\rm g}/{\rm cm}^3$. Such a setup allows for a mild density contrast near the CSM boundary, which improves the resolution of the shock. For models 1-14, having a type Ia-like engine, we set the ball radius to $0.1 \textnormal{R}_\odot$ and assign it $1.3 \textnormal{M}_\odot$ of C-O material. For model 15, we set a $10 \textnormal{M}_\odot$ ball of $2 \textnormal{R}_\odot$ radius, thus mimicking a core-collapse SN potentially stripped by a companion. The engine acquires the explosion energy split equally into kinetic energy and thermal energy. Furthermore, the kinetic energy is distributed to provide comparable escape velocities for all the velocity bins of ejecta. We have verified that details of the setup, such as the fraction of the explosion energy injected as thermal energy or the initial radius of the explosion ball, have a negligible effect on the lightcurves. We simulate all the models twice, with and without $^{56}$Ni. To compare our results to the analytic models of CSM interaction, we use the simulations without $^{56}$Ni, and we examine the qualitative effects of $^{56}$Ni in the results section~\ref{sec:results}. For models with $^{56}$Ni, we assume $0.6\,\textnormal{M}_\odot$ of the radioactive material, typical for type Ia explosions (apart from model 15, wherein we assumed $0.2\,\textnormal{M}_\odot$ of $^{56}$Ni). The CSM is set up in concentric shells, following the density and composition profiles for the different AGN models, with the full models containing between $1000$ and $2000$ grid cells. We sample the lightcurves produced in \texttt{SNEC} with a step of $5$ minutes to resolve the sharp hour-long peaks in some of the models. Overall, the spherically-symmetric models are initialised similarly to how it is done in the analytic modelling.
The code allows us to trace the detailed properties of the CSM and the shock over time and calculate the lightcurves for the models.
\subsection{Effects of the geometry on hydrodynamics and lightcurves}
While the \texttt{SNEC} code can deal with most of the relevant physics such as radiative transfer, detailed opacity tables, nuclear energy deposition from $^{56}$Ni and ionisation, it cannot model non-spherical geometries. To understand how the supernova ejecta interact with the non-spherical AGN disc material, we carry out additional 2D axisymmetric hydrodynamic simulations of the interaction. For this, we use the hydrodynamic code \texttt{HORMONE}, which is a grid-based code that solves the hydrodynamic equations through a Godunov-type scheme \citep[]{hirai2016}. We employ an equation of state with contributions from ideal gas and radiation. A spherical coordinate system is used, where we assume axisymmetry with the symmetry axis taken perpendicular to the disc and the coordinate origin placed at the centre of the supernova explosion. In our axisymmetric treatment we ignore any motion of gas in the disc prior to the explosion, such as shearing motion expected in Keplerian discs, and instead treat the disc as an initially static slab of material. We also could not apply the gravity from the central SMBH, so we ignore all gravitational forces, including self-gravity. Therefore, we neglect the initial thermal energy in the disc material too, since it will lead to artificial expansion without the gravitational forces that keep it bound. These assumptions are still valid as long as the time-scale for the shock to reach the surface of the disc is much shorter than the local Keplerian period, and the shock energy is larger than the pre-explosion thermal energy of the disc.
We set up the simulation by placing a slab of material to represent the AGN disc. At the coordinate origin, we place a homologously expanding supernova ejecta model with an exponential density profile as
\begin{equation}
\rho_\textnormal{ej}(r)=\frac{3\sqrt{6}M_\textnormal{ej}}{4\pi v_0^3t_0^3}\exp{\left(-\frac{\sqrt{6}r}{v_0t_0}\right)},
\end{equation}
where $t_0$ is the time since explosion.
Such exponential profiles are commonly seen in type Ia SN ejecta models \citep[e.g.][]{nomoto1984}. The velocity profile is set to be $v(r)=r/t_0$. When integrated from 0 to infinity, the total mass is $\int_0^\infty4\pi r^2\rho_\mathrm{ej}dr=M_\mathrm{ej}$ and the total kinetic energy becomes $\int_0^\infty\frac12\rho_\mathrm{ej}(r)v(r)^2\cdot4\pi r^2dr=E_0$. We cut off the explosion profile at a radius $r=R_\textnormal{exp}$ and the time since explosion is set through $t_0=R_\textnormal{exp}/(4v_0)$. The factor 4 is chosen so that the total integrated ejecta mass and energy are still very close to $M_\mathrm{ej}$ and $E_0$. We set a small enough value of $R_\textnormal{exp}$ such that the mass of the disc material within that radius is negligible compared to the ejecta mass ($\int_0^{R_\textnormal{exp}}4\pi r^2\rho_\textnormal{disk}dr\ll M_\textnormal{exp}$), where we can safely assume that the ejecta have freely expanded up to that radius. For all the models presented here, we use an ejecta mass of $M_\textnormal{ej}=1.3~\textnormal{M}_\odot$, which represents the ejecta mass for typical type Ia SNe, except for model 15, which has a representative mass of $M_{\rm ej}=10 \textnormal{M}_\odot$, typical for core-collapse SNe.
The ejecta are covered with at least 30 grid points, and the radial grid size is increased as a geometrical series as it goes out. The outer boundary is taken at $\sim100~H$, and we use $\sim1000$ grid points in the radial direction. We divide the polar direction into 400 cells spaced uniformly in $\cos\theta$ so that the solid angle of each cell is equal and the disc is properly resolved.
The lightcurves are estimated through a post-process ray-tracing procedure similar to that of \citet[]{suzuki2016}. See Appendix~\ref{app:lightcurve_hormone} for more details of the lightcurve calculation method.
\section{Results} \label{sec:results}
Here we present the analytic and numerical lightcurves for the different models.
\subsection {Peak luminosities}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $L_{44}^{{\rm YM19}}$ & $L_{44}^{{\rm SNEC}}$ & $C_L^{{\rm YM19}}$ & $C_L^{{\rm SNEC}}$\tabularnewline
\hline
\hline
1 & 1.48 & 1.7 & 7.97 & 9.12\tabularnewline
\hline
2 & 0.12 & 0.14 & 5.54 & <6.48\tabularnewline
\hline
3 & 0.26 & 0.28 & 4.34 & 4.69\tabularnewline
\hline
4 & 0.62 & 0.93 & 5.8 & 8.6\tabularnewline
\hline
5 & 29.38 & 27.68 & 10 & 9.44\tabularnewline
\hline
6 & 0.08 & 0.13 & 6.63 & 10.87\tabularnewline
\hline
7 & $4.24^a$ & 4.42 & 6.91 & 7.21\tabularnewline
\hline
8 & $24.70^b$ & 33.5 & 1.15 & 1.56\tabularnewline
\hline
9 & $3.96^a$ & 7.86 & 10.65 & 30.38\tabularnewline
\hline
10 & $12.77^a$ & 14.90 & 40.97 & 26.84\tabularnewline
\hline
11 & $7.82^a$ & 21.93 & 10.68 & 26.82\tabularnewline
\hline
12 & $22.55^a$ & 5.43 & 47.2 & 11.53\tabularnewline
\hline
13 & 7.85 & 2.65 & 21.05 & 7.1\tabularnewline
\hline
14 & 8.89 & 14.16 & 14.69 & 23.4\tabularnewline
\hline
15 & 1.45 & 1.51 & 13.45 & 14.04\tabularnewline
\hline\tabularnewline
\tabularnewline
\end{tabular}
\caption{Peak luminosities and their coefficients. The first column is the model number. The luminosities are normalized to $L_{44} = L/(10^{44}$ erg/s). The second column is $L^{\rm YM19}$, the peak luminosity derived from the maximal value of Eq. \ref{lightcurve}. The third column is the maximum luminosity obtained in \texttt{SNEC} simulation. The forth column is the ratio $L^{\rm YM19}/(\rho z^2 v^3)_{\rm bo}$, which is an estimate for $C_L$ from Eq. \ref{l_direct}. The denominator is evaluated at the breakout shell. The last column is the same, only for the numerical maximal \texttt{SNEC} luminosity. a) Evaluated $\omega$ at the breakout radius for the analytical fit. b) Used $\omega=1$. } \label{tab:2}
\par\end{center}
\end{table}
In table \ref{tab:2}, we show the peak bolometric luminosities from the analytical \citetalias{yalinewich_matzner2019} modelling and from \texttt{SNEC} simulations. The last two columns also show the dimensionless prefactor $C_L$ for which the estimate in Eq. \ref{l_direct} matches the relevant luminosity. We see that the peak luminosities from the \citetalias{yalinewich_matzner2019} modelling correspond well with the \texttt{SNEC} results for most models. The most discrepant models are 11-13, for which the luminosities differ by factors of 3 to 4.
The estimate of $C_L$ expressed from Eq. \ref{l_direct} shows a discrepancy of up to two orders of magnitude compared to $C_L$ calculated with the previous method. There is no clear trend, and larger values are obtained for models of compact structure and low density, namely models which explode off-plane (model 9-12) or with a reduced SMBH mass and scale height (model 13) or starved AGN (model 14). Excluding them and the slab model (model 8), which has discontinuous density and is challenging to simulate, leads to a factor of two spread in $C_L$ values, $C_L \approx 7-14$ when compared against both \citetalias{yalinewich_matzner2019} models and \texttt{SNEC} simulations (models 1,5-7,15). We note that for a spherical model such as the \texttt{SNEC} one, the spherical geometry suggests the value $C_L=4\pi \approx 12.6$ for thermalized matter, which is within this range.
We devote the remainder of this section to exploring predicted model lightcurves and their unique features, including comparing numerical and analytical results.
\subsection{The lightcurve shape and the effect of changing the energy and ejecta mass}
Fig. \ref{fig:1_5_6} shows the lightcurves of models 1, 5, 6, which correspond to different explosion energies ($10^{51}$, $10^{52}$ and $10^{50} {\rm erg}$, respectively) and also the lightcurve of model 15, which is identical to model 1, but with a larger ejecta mass of $M_{\rm ej}=10\ M_{\odot}$, which represents a fiducial model of core-collapse SN. We also ran a model with $10M_\odot$ ejecta and a lower energy of $10^{50}$ erg. The resulting lightcurve peak luminosity is very similar to the low energy of model 6, and is not shown here. Overall we see a good correspondence between the analytical lightcurve (solid) and the numerical results based on the \texttt{SNEC} code (dashed). We see that the larger the energy is, the faster is the breakout time, and the brighter is the peak. The dependence of the peak luminosity on explosion energy is slightly steeper than linear, $L_{\rm peak} \propto E_0 \Gamma^{(1/2+\mu\omega)\delta_{-}} \propto E_0^{1+(1/2+\mu\omega)\delta_{-}}$, which gives the value of $1.28-1.29$ for our range of density power laws $\omega_i$, where $i={1,5,6}$. $\Gamma$ also depends on $l_{\rm eff}$, but $l_{\rm eff}$ varies only within $10\%$ between the different models. The additional corrections from $l_{\rm eff}$ slightly change the scaling to $1.23-1.24$.
Thus, the peak luminosity reaches $\sim 3 \times 10^{45}\ \rm erg\ s^{-1}$ for model 5, while it is slightly less than $10^{43}\rm\ erg\ s^{-1}$ for model 6. The different $\omega$'s arise due to the different breakout shells in each model: the less energetic models have a lower velocity by a factor of $E_0^{1/2}$ and hence breakout at later times. The more massive ejecta model 15 has a smaller velocity but has a longer phase of free expansion. Thus its effective power law is $\omega_{15}=6.04$, which is the largest one in our models. This is due to the fact that in this model, $l_{\rm eff}=2.67H$ is very close to the maximal value of $\omega_{\rm 15, max}=6.076$ obtained at half the photosphere $z_{\rm ph}/2H=4.93/2=2.465$.
In the spherical phase, the dependence of the luminosity on $\Gamma$ is a very shallow power law, while the dependence on time since the breakout is $L\propto t^{-x}$ with $x=0.44-0.47$. Note that this is steeper than the maximal slope of $-0.35$ derived by \cite{NakarSari2010} for radiation dominated material, which is obtained for $\omega=3$. This is because our extended disc model allows larger values of the effective power law index.
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_3_final.png}
\caption{Probing different energies and ejecta masses. The solid lines are the analytical lightcurves where the semi-transparent dashed lines are the \texttt{SNEC} simulations. Model 1 is our canonical model (blue), model 5 has a $10$ times higher explosion energy of $E_0=10^{52}\ \rm erg$ (red), model 6 has a $10$ times lower explosion energy of $E_0=10^{50}\ \rm erg$ (green). Model 15 (black) has the same explosion energy as model 1, but with larger ejecta mass of $M_{\rm ej}=10 \textnormal{M}_\odot$. The coloured transparent regions indicate the range of possible breakout times as in Fig. \ref{fig:analytica_model1}, with each transparent colour matching the correspondingly coloured model. The vertical dotted lines correspond to the end of the spherical phase for each model shown in their respective colours, also similar to Fig. \ref{fig:analytica_model1}. The effective power laws for the density for each model are $\omega_1 = 5.3$ (the same as in Fig. \ref{fig:analytica_model1}), $\omega_5=4.7$, and $\omega_6=5.75$, and $\omega_{15}=6.04$. }
\label{fig:1_5_6}
\end{figure}
\subsection{Changing the SMBH mass and midplane density} \label{sec:disc_properties}
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_4_final.png}
\caption{Probing different AGN disc locations and properties. Models 4 and 13 correspond to a $10^6 \textnormal{M}_\odot$ SMBH, while model 14 corresponds to a $10^7 \textnormal{M}_\odot$ SMBH. Furthermore, the explosion in model 4 is placed at $10^3 r_s$, while for models 13 and 14, the explosion is placed at $10^4 r_s$. The line style and shaded areas have the same meaning as in Fig. \ref{fig:1_5_6}. The effective power laws are $\omega_4=4.97$, $\omega_{13}=6.7$, and $\omega_{14}=4.98$.}
\label{fig:4_13_14}
\end{figure}
Fig. \ref{fig:4_13_14} shows the lightcurves of models 4, 13, 14 where the AGN disc structure is altered. In models 4 and 13, the reduced SMBH mass of $10^6 \textnormal{M}_\odot$, compared to $10^7 \textnormal{M}_\odot$ for model 13, induces an increase in the midplane density, but the scale height is smaller. Model 4 is exploding at a more distant radial location of $10^4 r_s$, compared to the other two models that explode at $10^3 r_s$, thus making the swept CSM mass larger. While the analytical and \texttt{SNEC} lightcurves broadly agree for model 4, they differ more significantly for models 13 and 14. The general trend is that the \texttt{SNEC} lightcurves decrease more rapidly compared to the analytic ones. The reason may lie in the fact that models 13 and 14 have a reduced CSM interaction mass, which is slightly less than the ejecta mass. Radiative cooling may contribute to the extra drop in the luminosity in the simulations, not accounted for in the analytical model, as discussed in sec. \ref{sub:lc}. Moreover, the peak luminosities in model 13 also differ by a factor of a few. The difference in the peak luminosities and the wiggles seen in the \texttt{SNEC} lightcurves for model 13 may be related to the reverse shock travelling back through the ejecta. Once the reverse shock reaches the origin of the spherical model, it reflects and starts travelling forward. The reverse shock arrives at the breakout region with a delay compared to the initial forward shock and is typically weaker. The reason the reverse shock is prominent in model 13 is likely because the CSM radius is small compared to other models, and the CSM mass is small enough to allow fast breakout.
\subsection{Changing vertical profile}\label{sec:omega}
Fig. \ref{fig:1_7_8} shows the dependence of the lightcurve on the vertical structure. We see that the radiation-dominated profile (model 7) and the uniform step profile (model 8) lead to progressively earlier and brighter breakouts. This is due to the fact that the respective edges of the different vertical profiles (and also their breakout shells) are more compact than the standard Gaussian profile in model 1.
In Fig. \ref{fig:1_7_8}, for the calculation of $\omega$ we used $l_{\rm edge} = z_{\rm edge} - z_{\rm bo}$ in Eq. \ref{omega_eff_rad}, where $z_{\rm edge}$ is the physical edge of the disc for model 7, and the photosphere $z_{\rm phot}$ for model 1. The spatial extent of the explosion which enters into the calculation of $\Gamma$ is unchanged, $l_{\rm eff}=z_{\rm bo} - z_1$. The resulting values for these models are $\omega_1^{\rm edge}=3.76$ for model 1, and a higher value of $\omega_7^{\rm edge}=2.9$ for model 7 (compared to $\omega_7=1.46$ in the default choice). For model 7, since $z_{\rm bo}$ is closer to the disc edge, $\omega_7^{\rm edge}$ is closer to its limiting value of $3$, which is also the power-law suitable for spherical radiative atmospheres. For model 8, the profile is uniform and discontinuous, hence $\omega_8=0$ throughout the disc and infinite at the edge. Since $\omega_8$ is ill-defined for model 8, we express the range of possible lightcurves by shading a grey area, where the upper limit is constructed by choosing $\omega_8=100$, and the lower limit is given by $\omega_8=0$. For comparison, we also plot the analytic fit with $\omega_8=1$, which gives a roughly flat lightcurve, similar to the \texttt{SNEC} lightcurve (though we caution that a numerical treatment will also struggle with a discontinuous density profile, and the \texttt{SNEC} models are likely not robust in this case).
For model 1, this fit is comparable to the one with the default choice (as in Figs. \ref{fig:analytica_model1} and \ref{fig:1_5_6}). This is also a representative case of all the gas dominated Gaussian profiles, namely that they are not very sensitive to $\omega$. For radiation dominated profiles, however, the latter choice of $\omega_7^{\rm edge}$ gives a much better match to the \texttt{SNEC} results. We speculate that the reason may be due to the fact that radiative dominated and slab profiles are more compact and have an actual edge.
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_5_final.png}
\caption{Probing different density profiles. The line style and shaded areas have the same meaning as in Fig. \ref{fig:1_5_6}. For models 1 and 7, we chose a different location for evaluating $\omega$, at the breakout shell, namely using $l_{\rm edge}=z_{\rm edge} - z_{\rm bo}$, in Eq. \ref{omega_eff_rad}, where $z_{\rm edge}$ is either the physical edge of the disc (e.g. radiation-dominated profile as in model 7) or the photosphere, if the disc is formally infinite (Gaussian profile as in model 1). The spatial extent $l_{\rm eff}$ which is used to calculate $\Gamma$ is unchanged. Model 1 is our canonical model (blue), model 8 has a step profile (green), model 7 has a radiation dominated profile (red). The power-law index for model 1 is now $\omega_1^{\rm edge}=3.76$, and for model 7 $\omega_7^{\rm edge}=2.9$. For model 8, the density profile is infinite at the edge and zero elsewhere. We show the analytic fit for $\omega_8=1$ (green). The grey area indicates the possible range of lightcurve for model 8 between the two extremes: $\omega_8=0$ is the bottom limit, and $\omega_8=100$ as the top limit. The grey area extends up to the peak of the lightcurve.}
\label{fig:1_7_8}
\end{figure}
\subsection{Changing vertical location}
Fig. \ref{fig:9_to_12} shows the lightcurves of the off-plane explosion models 9-12. Among the Gaussian density profiles, the further out explosion (model 10) at $z_0=2H$ explodes earlier and is brighter, as expected. On the other hand, for radiative density profiles, the further explosion (model 12) explodes earlier, but is dimmer than the $z=1H$ case (model 11). The reason lies in the very low CSM mass for model 12 ($M_{\rm CSM} < 10^{-3}M_\odot$).
The predicted breakout times are compatible with the \texttt{SNEC} models, although for the $z_0=2~H$ cases, the breakout occurs slightly earlier than expected. The reason may lie again in the low CSM mass, which limits the thermalization of the CSM material. This could account for the large discrepancy between the analytic and numeric lightcurves in model 12.
\begin{figure}
\centering
\includegraphics[width=8cm]{figure_6_final.png}
\caption{Changing the vertical location of the explosion. Gaussian profiles are for $z_0=1H$ (blue, model 9) and $z_0=2H$ (red, model 10). Radiative profiles are for $z_0=1H$ (green, model 11) and $z_0=2H$ (black, model 12). The line style and shaded areas have the same meaning as in Fig. \ref{fig:1_5_6}. We use the $\omega^{\rm edge}$ choice as in fig. \ref{fig:1_7_8}. The effective power laws are $\omega^{\rm edge}_9=3.6$, $\omega^{\rm edge}_{10}=3.17$, $\omega^{\rm edge}_{11}=2.93$, $\omega^{\rm edge}_{12}=2.87$.}
\label{fig:9_to_12}
\end{figure}
\subsection{Comparison between the codes} \label{comparison}
\begin{figure*}
\centering
\includegraphics[width=17cm]{figure_7_final.png}
\caption{ Comparison between the \texttt{SNEC} code (dashed blue) and the \texttt{HORMONE} code (solid black) for all sampled models. The red curve is the analytic fit, but starting with the peak at the average breakout time of the two velocity models calculated in Appendix \ref{appendix-breakout}, $t_{\rm bo}= (t_{\rm bo}|_{\rm pw} + t_{\rm bo}|_{\rm MM99})/2$. The grey area is where the AGN luminosity can obscure the transient. The maximal AGN luminosity is set according to Eq. \ref{l_agn}. The model number appears in bold black for each panel. Note that the ordinate is the same, but the abscissa is different for each panel.}
\label{fig:comparison}
\end{figure*}
Fig. \ref{fig:comparison} shows the lightcurves of different models and compares predictions from the Lagrangian spherically-symmetric radiative hydrodynamics \texttt{SNEC} code (dashed blue) to those from the Eulerian 2D hydrodynamics \texttt{HORMONE} code (solid black). The goal is to understand how important are the effects of the more realistic 2D disc geometry, including its structure evolution and morphology, since our analytic and \texttt{SNEC} models assume spherical symmetry. The \texttt{HORMONE} simulations do not include radiative transfer and thus radiative cooling and the transfer of energy from the inner parts of the CSM outwards are not properly modelled. However, this only becomes important in the later phases and the dynamics up to and close to shock breakout should be mostly adiabatic.
The vertical shock propagation in the 2D \texttt{HORMONE} simulations closely resembles that of the 1D \texttt{SNEC} simulations as shown in Fig.~\ref{fig:velocity_models}.
The lightcurves for the \texttt{HORMONE} models in Fig.~\ref{fig:comparison} are all computed for a face-on viewing angle. This is expected to be the angle where the shock breakout is observed with the brightest luminosity. Unlike the spherically symmetric models, the shock breakout in the 2D models starts from a small patch and spreads out radially due to the different shock propagation times along each polar angle. Furthermore, the effective CSM mass the shock has to interact with before breaking out also increases with the polar angle, decreasing the velocity at the breakout shell and making the luminosity at corresponding annuli lower. These differences may likely explain the lower luminosities reached in \texttt{HORMONE} models in Fig.~\ref{fig:comparison}. The peak is also smeared out over a slightly longer duration because of the delay in shock breakout at lower latitude angles.
For the off-plane explosions, the difference in interacting mass along each polar angle is smaller compared to the mid-plane explosion models. Therefore, the opening angle of the shock breakout region is wider, leading to more similar peak luminosities between 2D and spherical models (Fig.~\ref{fig:comparison} panels 9, 10).
\subsection{Summary of the results}
In summary, the typical lightcurve of a SN explosion will peak at luminosities around $10^{44} - 10^{45} \rm erg\ s^{-1}$. The earliest peak will usually be the brightest one. Radiation-dominated or slab profiles will result in more compact and less massive vertical layers and could produce stronger observable events when compared to gas pressure-dominated (Gaussian) profiles. The type of the SN (or the ejecta mass) will play a relatively minor role in the observational signature, with the main parameter being the explosion energy. This is the case so long as most of the kinetic outflow encounters enough CSM material to reprocess the kinetic energy into outgoing radiation.
The strongest peak will be reached if the CSM mass is comparable to or smaller than the ejecta mass (models 8, 12, 13, 14), although models with CSM mass exceeding the ejecta mass by a factor of ten are also observable. Models with too much CSM mass ($\sim 100 M_{\rm ej}$) will be choked and will not be observable. For these reasons, off-plane explosions, explosions in the inner ($\sim 10^3 r_s$) regions of AGN discs and explosions in the discs of starved AGNs will lead to the brightest transients.
\section{Discussion} \label{sec:discussion}
\begin{figure*}
\centering
\includegraphics[width=0.3\linewidth]{figure_8a_final.png} \includegraphics[width=0.3\linewidth]{figure_8b_final.png}
\includegraphics[width=0.3\linewidth]{figure_8c_final.png}
\caption{SNEC-based ugriz lightcurves for models 1, 7 and 14. The models represent the CSM interaction signature for a type Ia supernova in a typical AGN around a $10^7\,{\rm M}_\odot$ black hole for a pressure-dominated (model 1) and radiation-dominated (model 7) atmospheres, and the same supernova in a starved AGN (model 14). The initial blue/UV half-day-long spike due to shock breakout (which produces the brightest signal in extreme UV / soft X-rays) is followed by the week-long blue/white peak, followed by an extended red/IR emission from the deposited shock energy reprocessed by the CSM. The shape and fall-off times of the lightcurves after the main peak may be inferred from the bolometric lightcurves.}
\label{fig:ugrizLCs}
\end{figure*}
\subsection{Observational picture}
Although the SN explosions in AGN disks are typically more luminous than identical explosion in the absence of circumstellar material, they still need to outshine the luminous AGN environment. \citet{Hubeny2001} studied the spectra of AGN accretion discs and found that for a $10^7 \textnormal{M}_\odot$ disc with a viscosity parameter $\alpha=0.01$ and luminosity
\begin{equation}
L=0.3L_{\rm Edd}= 4.4 \cdot 10^{44} \left( \frac{M_{\bullet}}{10^7 \textnormal{M}_\odot} \right) \rm erg\ s^{-1} \label{l_agn},
\end{equation}
the spectrum peaks at $\nu F_\nu \approx 10^{44}\ \rm erg\ s^{-1}$ in the extreme UV / soft X-ray band around $0.1\ \rm keV$. In Eq. \ref{l_agn}, the Eddington luminosity $L_{\rm Edd}$ assumes the electron scattering opacity.
\textbf{Bolometric luminosity:} For our suite of models, models 1, 6, 9 and 15 peak around $\sim 10^{44} \rm erg\ s^{-1}$ or lower and will likely be obscured by the AGN luminosity in the absence of clear features that make it possible to distinguish AGN spectra from the explosion spectra. Model 4 also peaks around $\sim 10^{44}\ \rm erg\ s^{-1}$, but the host AGN is less massive. Thus it is a factor of $\sim 10$ more luminous than the AGN background. Models 5, 8, 10, 11 and 14 peak with luminosities $L\gtrsim 10^{45}\ \rm erg\ s^{-1}$ and are thus most likely to be observed. Models 13 and 14 are especially promising since they are located in a low-density environment, either in a less massive galaxy or in a starved AGN environment, and could be more luminous than their respective background by a factor of $\gtrsim 100$. We note that this is a conservative estimate, since the observed luminosity can be much lower ($\sim 0.01L_{\rm Edd}$, \citealp{fabian2009}), which will make all models besides 3,4, and 6 observable.
In summary, the events with the best chances to be observed will be either in a radiation-dominated disc, and/or in a low-density environment. There are three possibilities to achieve this among the set of models we considered: i) in a low mass galaxy ($M\approx 10^6 \textnormal{M}_\odot$, ii) in a starved AGN of reduced density, or iii) explosions away from the midplane.
\textbf{Multiband lightcurves:} We show the typical multi-band lightcurves in Fig.~\ref{fig:ugrizLCs}. For all the models, the first breakout will produce a short UV/blue transient, followed by a red/IR tail. As mentioned above, the initial breakout peaks in extreme UV / soft X-rays, so these lightcurves understate the peak bolometric luminosity. The emission is mostly optical after the early breakout peak, which may help to separate these events from the AGN background, which peaks in UV/X-rays. Furthermore, the models producing sharp peaks, e.g. models 8, 12, 14, reach temperatures high enough to give rise to X-ray flares. In this case, we expect approximately hour-long flares reaching up to $10^{43} \rm erg/s$ in the $0.3$~--~$10\,\rm keV$ band. The exceptions to this picture are models 2 and 3, in which the shock stalls due to the large CSM mass. In this case, the thermalised energy is emitted over hundreds of days in red/IR bands.
The presence of $^{56}$Ni in real transients will lead to additional energy deposition on a month timescale. However, since the energy is deposited mostly in the ejecta material located behind the CSM, the luminosity contribution from $^{56}$Ni will be delayed by the photon diffusion time and eventually emitted over several months or longer in red/IR bands. Since the total energy yield in $^{56}$Ni is comparable to that of the regular nuclear transients in the field and since this energy is emitted over longer times, we find that the contribution from radiative decay typically does not exceed $10^{43} \rm erg/s$. The AGN will typically outshine such red/IR contributions. Conversely, in models 12-14 with low CSM mass, the CSM lightcurve after the peak becomes quickly dominated by the contribution from the nuclear decay. In this case, the late lightcurve resembles that of isolated supernovae.
The explosion and its ejecta are expected to be non-relativistic. It is possible that some fraction of the electrons, especially where the temperature is large enough, will not be in thermal equilibrium and will be accelerated to relativistic velocities producing $\gamma$-rays, either in GRBs \citep{GRB_in_AGN,perna20} or in hyper-Eddington accretion-induced Bondi explosions \citep{wang2021}, where due to
inefficient energy transport the material is heated to very large temperatures, and relativistic shocks make a large cavity in the AGN disc. While the GRB radiation is a prompt emission, the radiation from Bondi explosions will be visible only in the broad-line region, $\sim 1\ \rm yr$ after the explosion. For GRBs, following the cocoon breakout, non relativistic ejecta also break out at later times. We find similar peak luminosities to the estimates of \cite{GRB_in_AGN}.
\subsection{Rates of SNe in AGN discs}
\begin{figure*}
\centering
\includegraphics[width=17cm]{figure_9_final.pdf}
\caption{Density snapshots from the 2D simulations for model 1. Each panel shows a different time snapshot. Note that the left panel has a smaller box size.}
\label{fig:model1_2d}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=17cm]{figure_10_final.pdf}
\caption{Same as Fig.~\ref{fig:model1_2d} but for model 9. }
\label{fig:model9_2d}
\end{figure*}
The radiative efficiency of an accreting AGN is $\eta = L/(\dot{M}c^2)<1$. If the SMBH is accreting at a fraction $L=fL_{\rm Edd}$ of the Eddington luminosity, and the same fraction $f$ of the mass accretion rate, the SMBH mass doubling time is
\begin{equation}
t_{\rm doubling} = \frac{M_\bullet}{\dot{M}} =\frac{\eta}{f} \frac{\kappa c}{4\pi G} = 128 \left( \frac{\eta}{0.1} \right) \left( \frac{f}{0.3} \right)^{-1} \rm{Myr}.
\end{equation}
For a disc lifetime of $t_{\rm AGN}=10^8\ \rm yr$, the mass that flows into the disc is $M_\bullet t_{\rm AGN}/t_{\rm doubling} \approx M_\bullet$.
The type Ia SN rate is 1 per few hundred solar masses of star formation in the field \citep{Maoz2017}. Assuming a similar but slightly higher efficiency for AGN discs due to an enhanced binary merger rate (e.g., one SN event per $100 \textnormal{M}_\odot$), and further assuming a fraction $f_1$ is converted into stars, we would have $f_1 10^5$ SNe for a SMBH mass of $10^7 M_\odot$ over an AGN lifetime $t_{\rm AGN}=10^8\ \rm yr$. Thus, the rate per AGN is $f_1 10^{-3}\ \rm yr^{-1}$ per AGN disc. Roughly 1\% galaxies in the local Universe have an AGN disc, so the AGN density is $10^{-4}\ \rm Mpc^{-3}$. Thus, we expect a total of $\sim f_1 100$ AGN SNe per $\rm Gpc^3$ per year. A similar estimate for the rate of core-collapse SNe leads to a slightly higher but otherwise comparable rate of events. This rate is also slightly larger than the the upper limit of the binary black hole merger rate in AGN discs, $\mathcal{R}=0.002-18\ \rm Gpc^{-3}\ yr^{-1}$, which had been recently estimated by \cite{2grobner2020}.
A proper rate estimate should include an integral over the mass function of the AGN in the local universe. Moreover, indirect observations of the AGN phase duration ($10^6-10^9\ \rm yr$), and hence the duty cycle ($10^{-1}-10^{-3}$), could vary by around one order of magnitude \citep{martini2001, marconi2004}. Furthermore, AGNs may not be accreting steadily but rather experience events of rapid accretion and quiescence every $10^5$ years \citep{agn_flicker} which could affect the star formation rate.
\subsection{Observable event rates}
What fraction of the SNe will be observable? In order to address this question, we need to discuss the spatial distribution of stars in AGN discs, the mass function of the formed stars and its dependence on accretion.
Star formation in AGN discs predominantly occurs in the outer regions of the disc at low temperatures, where the disc is gravitationally unstable. The star formation efficiency is especially large when the midplane temperature is between $ 10^3 - 10^4\ \rm K$, and the opacity is very low since dust is still vaporized, but the material is mostly neutral, so electron scattering is low.
\textbf{Migration:} The interaction of the stars with the massive gaseous AGN discs can induce torques on them and cause the stars to migrate \citep[e.g.][and references therein]{artymovicz1993,Mck+12,bellovary2016}. Migration can be stopped if the torque changes sign. Extensive work had been done on migration in protoplanetary discs \citep{goldreich1980, tanaka2002, paardekooper2006, paardekooper2010}, and migration traps were numerically reproduced \citep{lyra2010}. \cite{bellovary2016} found that the outermost migration traps for AGN discs lie at around $\sim 300 r_g$. If most stars end up in this region, the binary mergers and SN explosions there will be both more frequent and more observable.
However, AGN discs are not necessarily as well behaved as protoplanetary discs. The immense radiation and mass make them prone to flaring and instabilities. It is uncertain whether the disc structure of \citetalias{sg03} and \citet{tqm05} is in a steady state, and the fitting formulae for the torque as in \citet{paardekooper2010} may not be applicable. Moreover, \citet{bellovary2016} used an adiabatic index of $\gamma=5/3$. Radiation-dominated discs with $\gamma=4/3$ will result in a different location for the migration traps, which may be shifted outwards.
Another issue is the timescale for the migration to take place. For type I migration, in which viscous torques are unable to form a gap in the disc, the migration time is \citep{tanaka2002}
\begin{equation}
\tau_I \sim \frac{M_{\bullet}^2}{\textnormal{M}_\odot \Sigma(r)r^2} \left(\frac{H}{r}\right)^2 \Omega^{-1} = 7 \left(\frac{M_{\bullet}}{10^7 \textnormal{M}_\odot} \right)^2 \left(\frac{r}{10^4r_s} \right)^2\ \rm Myr \label{t_typeI}
\end{equation}
Stars that formed beyond $10^5 r_s$ or in more massive discs would require longer than the lifetime of the disc to migrate, while less massive discs enable faster, more efficient migration. In some very low mass discs, a gap may be carved, and the migration timescale will be of type II, which will not depend on the mass of the disc (or the SMBH mass). Stars may still be captured from the nuclear star cluster onto the AGN disc.
Finally, the dynamical friction experienced by the stars could be significantly different from these simple estimates if the accretion onto the stars produces strong outflows \citep{gruzinov2020}, which is almost inevitable for the AGN environment where the accretion rates can be well above the Eddington limit (also see below).
The spatial distribution of the progenitors of SN explosions will depend on the migration history. Since the shocks from explosions at radial distances beyond $r\gtrsim 10^3r_s$ will be choked, the number of observed progenitors in the disc will be $N_3=N(r\lesssim 10^3\ r_s)$. If migration is effective, most of the progenitors may be close to the migration traps, yielding a fraction of observed events $f_2=N_3/N_{\rm tot} \sim 1$, where $N_{\rm tot}$ is the total number of the progenitors. If no migration occurs, then assuming an underlying log-uniform distribution between $10^2-10^5\ r_s$, only $f_2 \sim 0.3$ of the events will be observable. If most stars are either stuck in the outer regions or migrate outward, then $f_2\ll 1$ could be rather low.
\textbf{Accretion:} Accretion could be important for changing the initial mass function of stars in AGN discs and affecting the type Ia / core-collapse SN branching ratios. However, neither the details of the expected super-Eddington accretion nor the initial mass function are certain. \cite{artymovicz1993} first studied the growth and accretion of stars in AGN discs. More recently, \citet{davies2020, cantiello1} suggested that significant growth can occur. \citet{Lei+13} also had a similar discussion, with possible implications for depleting the gas through accretion. However, a recent follow-up study showed that rotation and radiative feedback could limit the final mass to be around $10\ \rm M_{\odot}$ \citep{cantiello2}, especially for the innermost regions, which we mainly consider. Moreover, the extremely optically thick environment may block the energy release via radiation and inhibit the accretion rate. A similar situation could lead to "Bondi explosions" \citep{wang2021}.
Accretion onto white dwarfs might, under appropriate conditions, allow for their growth to close to the Chandrasekhar limit and eventual explosion as type Ia SNe \cite{Ost+83}, thus increasing the rate for such events. However, here too, the exact accretion efficiency is difficult to assess.
The frequency of off-plane explosions is also quite uncertain, but probably low. Gas dynamical friction tends to damp the orbital inclination much faster than the migration timescale \citep{rein2012,grishin2015} if the orbital inclination with respect to the disc midplane is not too large. Highly inclined orbits $(i\gg H/r)$ may take much longer to dissipate into the plane of the disc, but they spend most of their time outside of the disc, and are therefore likely to explode in relative isolation without significant CSM interaction. Accretion feedback may change this picture \citep{gruzinov2020}, while close encounters with neighbouring stars may excite the progenitors' orbits.
To summarize, the upper limit for the rates of detectable events is $\mathcal{R} \lesssim f_1 f_2 100 \ \rm yr\ Gpc^{-3}$, where $f_1$ is the efficiency of forming AGN gas into stars, and $f_2$ is the fraction of observable events. The fraction of core-collapse SNe is uncertain and could constrain the efficiency of accretion or the underlying mass function.
Remaining agnostic to the efficiency of accretion and its final fate, we comment that for a Salpeter mass function $dN/dm\propto m^{-2.35}$, around $1\%$ of stars are born massive enough to produce core-collapse SN explosions, while for a top-heavy mass function $dN/dm \propto m^{-0.45}$ \citep{bartko2010}, the fraction of massive stars increases to $\sim\ 10\%$, thus increasing the rates by an additional factor of ten.
\subsection{Disc feedback}
The 2D hydrodynamical simulations allow us to explore the morphology of the ejecta and the impact on the local disc morphology around the explosion site. Fig.~\ref{fig:model1_2d} shows density snapshots of model 1. We see that at the early time of $60\ \rm hr$, shortly before shock breakout (left panel), the blast wave is still rather spherical, expanding within the disc. The shock front is gradually deformed as the in-plane directions are pinched, and the shock front propagates faster in the vertical directions, making the blast wave take a prolate form. Later, the shock breaks out from the disc surface, and a local hole is carved in the disc. The ejecta flow out almost spherically from the location of shock breakout.
The overall features are qualitatively similar to recent 3D numerical simulations of an SN explosion in a more distant radial location \citep[$M_\bullet=10^8~\textnormal{M}_\odot, r=2\cdot10^4~r_s$;][]{morenchel2021}. Due to the larger spatial extent and higher interaction mass, the more distant event will not be observable. However, SN explosions may contribute to the global viscous evolution of the AGN disc. The 2D simulations we present do not take into account the shearing motions of the disc. In reality, the hole excavated by the SN will be elongated in the disc rotation direction \citep{morenchel2021}.
Fig.~\ref{fig:model9_2d} shows the density snapshots of the off-plane explosion, model 9. We see that the shock breaks out earlier from the top edge, while the downward shock propagates longer as it climbs up the density gradient. This downwards CSM interaction creates a strong upwards reverse shock, which quickly flows out through the upper hole. The downwards shock also eventually breaks out from the other side of the disc. The breakout from this side of the disc is much weaker than the upper side, and the luminosity should be much dimmer.
\subsection{AGN flaring and variability}
The significant variability observed for AGNs over a wide range of timescales gives rise to difficulties in identifying nuclear transients and separating such transients from various forms of inherent AGN variability.
As discussed above, the observational signatures of SNe in AGN discs resulting from our models suggest that the SN transients will often be challenging to observe above the background luminosity of the AGN. However, in some cases, these AGN SNe can surpass the AGN background during their rapid rise to a peak. The timescale for such initial high-luminosity flaring is of the order of hours up to 1-2 days at most (see sec. \ref{sec:results}). Identifying such short-term transients requires high-cadence transient surveys. Moreover, many transient surveys do not focus on AGN nuclear regions where the AGN variability challenges identification. To the best of our knowledge, the only high-cadence sky surveys of AGNs were done using the Kepler mission. Interestingly, these surveys did identify one fast, days-timescale flare event KIC 1606852 \citep{Smi+18}, which could potentially be related to the AGN SNe we discuss. However, it might be too long or be the result of other explosive processes \citep{Smi+18}. The timescale of other nuclear transients, such as candidate tidal disruption events and changing-look AGNs, are far longer than the peak timescale of AGN SNe and are not likely to be related. Future high-cadence transient surveys may be able to identify AGN SNe.
\section{Summary} \label{sec:summary}
The AGN disc environment around SMBHs has dense stellar populations, many of which are expected to be embedded in the disc. As stars end their lives in supernova explosions, the dense AGN environment makes a fertile ground for unique transient events. Understanding such explosions and their properties opens a window to the physics of AGN discs and their interactions with the supernova progenitors.
Motivated by superluminous explosions in circumstellar material (CSM), we have developed an analytical model for the expected time of breakout and subsequent lightcurve, depending on the disc and progenitor properties. We validated this model with extensive numerical simulations. We found that the typical peak luminosity for such events may reach a few times $10^{45}\ \rm erg\ s^{-1}$. The most energetic events are also the quickest to break out, ranging from hours to several days, until they become too faint to be detected.
The most luminous explosions are generally found either in radiation-dominated discs, at larger explosion energies, or in regions with reduced density, such as off-plane explosions, low mass SMBH, or starved AGNs with reduced density. The latter two (i.e. low SMBH mass and starved AGN discs) also have better chances of being observed due to reduced background AGN luminosity. The initial breakout events should be dominated by the blue bands, while at later times, the lightcurves will be dominated by the red and infrared bands. The upper limit for the event rate is $\mathcal{R} \lesssim 100\ \rm yr\ Gpc^{-3}$, where optimal star formation and observational conditions are assumed. The actual rate could be lower by orders of magnitude.
In concise form, our study may be summarised as follows: {\it We exploded stars in an AGN. The photons can't get out, the outflow can. The gas converts shocks into brighter SN. We may see a few hundred, if we frequently scan.}
\section*{Data availability}
The simulations underlying this article will be shared on reasonable request to the corresponding author.
\section*{Acknowledgements}
We thank the referee, Pablo Fabi{\'a}n Vel{\'a}zquez, for valuable comments on the manuscript. We thank Almog Yalinewich, Ari Laor and Yossef Zenati for stimulating discussions. EG and HBP acknowledge support for this project from the European Union's Horizon 2020 research and innovation program under grant agreement No 865932-ERC-SNeX. IM is a recipient of the Australian Research Council Future Fellowship FT190100574.
\bibliographystyle{mnras}
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I was told my the - friendly - garage that it could be written off. I am not really sure about that yet, but I'm quite concerned about the amount of work that seems to be necessary in order to get it going again. I mean, that some lights don't work doesn't really matter, but it is mainly the corrosion that is an issue. It is also possible that I would need two new headlights, but I should be able to find those on some scrapyard or maybe via this forum or something.
Anyways, I am hoping that someone here might be able to help me out. I'd like to prevent the car being sold as parts, especially since quite a lot of work on the engine has been done recently (by the previous owner).
That is a bit of a list Mark and you must feel quite disheartened! The ABS fault may be a front sensor connection and if necessary the headlamps could be replaced as you suggest. On the headlamps I wonder if the dipped beam internal focusing fresnels have been removed to increase output? The corrosion issues are a bit more or a worry and especially to the seatbelt anchorages, but no doubt repairs are possible. The cv gaiter and suspension pipes should be quite easily sorted. Where exactly are you located, presumably in the UK? Regards, Matthew T.
Pipe inapropiatley repaired!?? Really! Thats the first Ive even heard of that one!
ABS Lamp could be up for a number of reasons. If it simply doessnt flicker and go out then its not seeing one of the four sensors, either a connector, or broken cable to a sensor, however, if it goes out on startup but then comes on after a drive then theres something interfearing with a sensor tip! Not too hard to sort..
Headlamps, I doubt you will get in a scrap yard, but - being honest I have about 10 sets!
I wonder if the leaking pipes as its on both sides are simply the strut return pipes have split and leaking from strut return leak..
The floor rust/rot under the B pillars - as described as seat belt anchourage - are another common area for rot on XMs these days, and can useually be hidden by the underseal, and the metal useually disapears from behind the underseal..
We hope you do stick with, these are getting thin on the ground these days..
MTXM wrote: That is a bit of a list Mark and you must feel quite disheartened! The ABS fault may be a front sensor connection and if necessary the headlamps could be replaced as you suggest. On the headlamps I wonder if the dipped beam internal focusing fresnels have been removed to increase output? The corrosion issues are a bit more or a worry and especially to the seatbelt anchorages, but no doubt repairs are possible. The cv gaiter and suspension pipes should be quite easily sorted. Where exactly are you located, presumably in the UK? Regards, Matthew T.
Thank you for your kind reply! And yes, I felt terrible when I heard the news and I still struggle with it. I mean, it causes such a dilemma; will I just not bother about the money and get the car fixed? Or is that really the wisest thing to do?
Your point about that removal makes sense, since I heard of the previous owner that he changed something to increase the output; that might be it. I will ask him.
I am located in Eastbourne. That might indeed be relevant, since it felt like the garage I went to (the people there were very nice actually) didn't really know what to do with an XM; one of the things I am hoping for is to find someone who is good with XM's, but won't charge that much. But that's probably something we all share as XM drivers, I guess.
citroenxm wrote: Pipe inapropiatley repaired!?? Really! Thats the first Ive even heard of that one!
@ your first remark: yes, that surprised me too; if it works, it works, right?
Re: ABS; it is just constantly lit.
New headlamps would sound like a good solution, however from what I saw on Ebay is that there are only a few of them available and that they do not come cheap (>100 GBP each).
So the rust sounds to you like something that can be fairly easily fixed?
And regarding your last sentence: thanks, haha, that doesn't make the decision easier! I mean, I know, that's why I would like to cherish it and that is one of the reasons why I bought it anyway, but it feels like it's getting a bit too much at the moment.
I have headlights.. mine certainly are not 100 each.. more like 25 quid or so each..
Does the ABS light being constantly iluminated meand a sensor fault, or a power to the ABS unit issue, in which case it is likely to be a cracked solder joint on the relay holder on the ABS unit.
The MOT test rules are specific about brake pipe repairs.
The use of "conex" type joints with loose olives is not permitted on any hydraulic brake line.
These have been sold on ebay for Citroen pipes in recent times including supply by some Citroen Indys.
The exact wording specified for this type of illegal joint is as has been used on the failure list.
This type of joint MUST be failed even if it is not leaking.
Leaking pipes or joints will be reported as a separate failure item.
It may well be that the leaks and the "illegal" joints are one and the same but have generated 4 separate failure items.
If this is the case there are just two joints to replace with an approved repair.
The rusty front sub-frame might be a problem.
I have had this once previously as a MOT failure.
The tester advised that he would not pass on retest the car with a weld repaired subframe.
He classed the sub-frame as a suspension component.
Weld repair of suspension components is not permitted and if found is a mandatory failure item.
The headlights should be a minor item, could just be bulbs not seated correctly.
But if there is a suspicion they have been modified then a secondhand set may be the quickest fix.
It may be best to find a Citroen Indy with XM experience to quote for repairs and obtaining an MOT.
I can provide details of one along the coast just in Hampshire if you send me a PM(personal message).
xmexclusive wrote: The rusty front sub-frame might be a problem.
That sounds incorrect. Most supframe rust is the closing plate on the bottom, which is common on many makes of car and some manufacturers even supply replacement plates to weld in.
A subframe is not a suspension component.
I agree I have overstated the weld repair issue.
Weld repair is not permitted on highly stressed components.
In 2.4.G.1 and its appendices sub-frames are specifically mentioned as a suspension component requiring examination.
Testers are advised that thin higher strength steel pressings may be used and small amounts of rust may seriously weaken.
As far as I know Citroen do not supply sub-frame repair plates for the XM so proving the correct grade of steel has been used is difficult.
The point I was trying to make, probably badly, is that the garage seem to consider the car only fit for scrap.
I think that makes sending the repaired car back there for retest a bit of a lottery unless a clear understanding on what is needed is reached first with the tester.
At present he has seen a car with obvious faults ignored and bodged repairs.
The tester may not have hydraulic Citroen experience.
He might not like cars which drip LHM and make his test floor slippery.
Rust holes in critical areas.
Would you let a close family member drive a car in this state for the next 12 months.
Quite a few of us might take on such a car and correct the faults.
Hence my suggestion of a Citroen Indy to assess and cost the work.
It will not be cheap but the work will include getting the MOT.
As we saw recently with Ray Fry's 2.1 the Citroen Indy route provides quick answers even if the results are unpalatable.
It avoids significant expenditure on repairs that still do not end up with a car fit for the road.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,953
|
Lawmakers finish with educator ID
By Todd Engdahl
As local news deserts become the norm, Chalkbeat remains 100% committed to the communities we cover. Get our Colorado education stories delivered to your inbox.
The House Tuesday agreed to Senate amendments to House Bill 09-1065, the proposal to create an identifier system for principals and teachers, and repassed the bill 65-0, sending it to Gov. Bill Ritter.
The idea behind the educator identifiers is that they can be used in conjunction with other data, such as individual student performance, to evaluate teacher performance, the effectiveness of teacher training programs and the distribution of high-performing teachers in different kinds of schools.
The bill, developed by an advisory group named the Teacher Quality Commission, originally was proposed as a pilot program in a few districts so the Department of Education could evaluate use of the identifiers. A major goal was to gain data about the "teacher gap" – the problem of low-performing schools being disproportionately served by inexperienced teachers.
But HB 09-1065 gained more importance after announcement of the federal stimulus program, with its emphasis on education reform. Creation of a statewide identifier program is now seen as a way to improve Colorado's chances for stimulus money.
And the bill set up something of a tussle between the governor's office, CDE officials and other researchers and the Colorado Education Association over use of the data. Researchers want maximum flexibility to use the data, while teachers' groups want protections to ensure that teachers aren't evaluated or disciplined solely on the basis of student test scores.
Negotiations, which also involved the Denver Public Schools, went through several fits and starts, but an amendment approved by the Senate Education Committee two weeks ago has the support of CDE, the governor's office and CEA and other education groups.
The language allows use of the data for research, contains various protections but doesn't restrict districts from continuing to use existing data systems and programs in evaluation, assignment and compensation of educators.
CDE would start the program with just a few districts, but the identifier program would be expanded statewide when education officials felt it was ready.
It could be a couple of years or more before it's operational. Private or federal funds will have to be raised to fund it, and state data systems will need upgrading for the program to work.
In other action Tuesday:
The Senate passed and the House concurred in amendments to House Bill 09-1319, the important dual enrollment legislation.
After much wrangling and GOP hand wringing, the Senate gave preliminary approval to House Bill 09-1366, which would close some loopholes in Colorado capital gains taxes. Sen. Chris Romer, D-Denver, endlessly touts this as the first in a series of tax bills (the rest to come next year) the legislature can pass without voter approval, thanks to a recent state supreme court decision.
House approval of Senate amendments:
House Bill 09-1039, resident tuition eligibility for veterans
House Bill 09-1290, increased financial aid for National Guard members
House Bill 09-1267, statutory cleanup of provisions affecting religious colleges
House Bill 09-1343, creation of a legislative early childhood commission
Passed were HJR 09-1025, the school safety study; SJR 09-044, the study of state fiscal stability; SJR 09-056, the Race to the Top attaboy resolution, and HJR 09-10120, the interim study of school finance, were all approved in the House.
Still hanging for the last day
Senate Bill 09-226 – The House accepted the conference committee report on the food allergies bill, but the Senate has yet to act.
House Bill 09-295 – Final consideration of the career-tech dual enrollment bills is pending in the House.
Higher education financial flexibility – The House gave final approval to Senate Bill 09-290 but has yet to act on its companion, House Bill 09-295
Consideration of governor's footnote veto of Senate Bill 09-259, the 2009-10 long appropriations bill.
Todd Engdahl
More stories in Colorado
What happened to free? Colorado Gov. Polis has changed how he talks about preschool
By Erica Meltzer, Ann Schimke
Denver students: 'You cannot teach American history without teaching African American history'
More stories in Leadership & Management
Wanted: Ways to boost the profile of Chicago's Local School Councils. A new bill could be a start.
By Marie Fazio
Chicago schools' top labor negotiator on the veteran pay dispute and what he's learned from all those union contracts
By Yana Kunichoff
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,237
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\section{Introduction}
This paper is a continuation of the author's previous work~\cite{iwao2021neutralfermionic}, in which he provided a fermionic description of the \textit{$K$-$Q$-function} $GQ_\lambda$ (see \S \ref{sec:Background}).
In \cite{iwao2021neutralfermionic}, the author also attempted to give a similar description of the dual function $gp_\lambda$, but he only obtained some implicit formulas.
In this paper, we address the issue and give a new fermionic description of $gp_\lambda$.
We also apply our method to the \textit{$K$-$P$-function} $GP_\lambda$ and its dual $gq_\lambda$.
\subsection{Overview}\label{sec:Background}
The factorial $K$-$P$- and $K$-$Q$-functions~\cite{IKEDA201322} are a distinguished family of symmetric functions that represent a Schubert class in the torus equivariant $K$-theory of the maximal isotropic symplectic and orthogonal Grassmanianns.
When we consider the special case in which the torus equivariant $K$-theory reduces to the (usual) $K$-theory, these functions reduce to their non-factorial versions.
The (non-factorial) $K$-$P$-function $GP_\lambda(x)$ and the $K$-$Q$-function $GQ_\lambda(x)$ are symmetric functions in the variables $x=(x_1,x_2,\dots)$ that are parametrized by a strict partition $\lambda=(\lambda_1>\dots>\lambda_r>0)$.
Their coefficients are in $\ZZ[\beta]$, where $\beta$ is a formal parameter.
(These functions are also refereed to as shifted stable Grothendieck polynomials~\cite{lewis2022combinatorial}.)
At $\beta=0$ limit, $GP_\lambda$ and $GQ_\lambda$ reduce to Schur's $P$- and $Q$-function~\cite[\S III.8]{macdonald1998symmetric} respectively.
Both of $GP_\lambda$ and $GQ_\lambda$ are elements of the algebra $G\Gamma$
that is characterized by the $K$-$Q$-cancellation property~\cite[Definition 1.1]{IKEDA201322}.
Any element of $G\Gamma$ is expressed as an infinite $\ZZ[\beta]$-linear combination of $GP_\lambda$.
On the other hand, all infinite $\ZZ[\beta]$-linear combinations of $GQ_\lambda$ form a proper subalgebra $G\Gamma_{+}\subsetneq G\Gamma$.
It was shown in \cite{IKEDA201322} that,
at $\beta=-1$ limit,
$G\Gamma$ ({\textit{resp.}~$G\Gamma_{+}$}) is isomorphic to the $K$-theory of the Grassmannian of type $B$ and $D$ ({\textit{resp.}~type $C$}).
Through the isomorphisms, $GP_\lambda$ and $GQ_\lambda$ represent the Schubert class $[\mathcal{O}_{\Omega_\lambda}]$.
In the previous paper~\cite{iwao2021neutralfermionic}, the author of the paper presented an algebraic characterization of $GQ_\lambda$ in terms of the \textit{boson-fermion correspondence}~\cite{date1983method,jimbo1983solitons}.
The main idea of the construction was the use of the $\beta$-deformed neutral fermions $\phi_n^{(\beta)}$ and $\phi^{[\beta]}_n$ (Definition \ref{defi:deformed_fermions}) and the ``$\beta$-deformed operators'' listed in Figure \ref{fig:beta-objects}.
\begin{figure}[htbp]
\centering
\begin{tabular}{c|c|c|c}
\multicolumn{2}{c|}{$\beta$-deformed operators}
& $\beta\to 0$ &
notable relations \\\hline\hline
$\phi^{(\beta)}_n$ & $\phi^{[\beta]}_n$& $\phi_n$ (neutral fermion)&
$\phi^{[\beta]}_n=(-1)^n(\phi^{(-\beta)}_{-n})^\ast$
\\\hline
$\Phi^{(\beta)}_n$ & $\Phi^{[\beta]}_n$ & $\frac{1}{2}\phi_n$
&
$
\begin{aligned}
&[(\Phi_m^{(\beta)})^\ast,\phi_n^{[\beta]}]_+\\[-.1em]
&=[(\Phi_m^{[\beta]})^\ast,\phi_n^{(\beta)}]_+=\delta_{m,n}
\end{aligned}
$\\\hline
$b_n^{(\beta)}$ & $b^{[\beta]}_n$ & $b_n$ (current)&
$b_n^{[\beta]}=(b^{(\beta)}_{-n})^\ast$
\\\hline
$\mathcal{H}^{(\beta)}(x)$ & $\mathcal{H}^{[\beta]}(x)$ &
$\mathcal{H}(x)$ (Hamiltonian)&
\\\hline
$e^{\Theta}$ & $e^\theta$ & $1$ & $\theta=\Theta^\ast$
\end{tabular}
\caption{A list of $\beta$-deformed operators.
Operators with a superscript ${}^{(\beta)}$ are used to construct $GP_\lambda$ and $GQ_\lambda$, and operators with a superscript ${}^{[\beta]}$ are used to construct $gp_\lambda$ and $gq_\lambda$.}
\label{fig:beta-objects}
\end{figure}
Our aim is to extend the result in \cite{iwao2021neutralfermionic} to $GP_\lambda$ and the dual functions $gq_\lambda$, $gp_\lambda$.
Here, $gq_\lambda$ and $gp_\lambda$ are unique symmetric functions that satisfy the Cauchy identity~\cite[Definition 5.3]{nakagawa2016generalized} (see also \cite[Definition 1.2]{lewis2022combinatorial})
\begin{equation}\label{eq:Cauchy_identity}
\sum_{\lambda:\mathrm{strict}}GQ_\lambda(x)gp_\lambda(y)
=
\sum_{\lambda:\mathrm{strict}}GP_\lambda(x)gq_\lambda(y)
=
\prod_{i,j}
\frac{1-\overline{x_i}y_j}{1-x_iy_j},\qquad\mbox{where}\quad
\overline{x_i}=\frac{-x_i}{1+\beta x_i}.
\end{equation}
Our main results, Theorems \ref{thm:GP}, \ref{thm:gq}, and \ref{thm:fermion_of_gp}, are the construction of the three types of vectors $\ket{\lambda}_{P}$, $\ket{\lambda}_{q}$, and $\ket{\lambda}_p$ such that
\[
GP_\lambda=\Omega(\ket{\lambda}_P),\quad
gq_\lambda=\chi(\ket{\lambda}_q),\quad
gp_\lambda=\chi(\ket{\lambda}_p),
\]
where $\Omega$ and $\chi$ are the $\beta$-deformed boson-fermion correspondences (see \S \ref{sec:boson_fermion}).
These vectors can be written in an explicit form by using our $\beta$-deformed operators.
We briefly outline the construction of the $K$-theoretic functions $GP_\lambda,GQ_\lambda,gp_\lambda$, and $gq_\lambda$.
We first define the $\beta$-neutral operators listed in Figure \ref{fig:beta-objects}.
Each of the operators $\phi^{(\beta)}_n$, $\phi^{[\beta]}_n$, $\Phi^{(\beta)}_n$, $\Phi^{[\beta]}_n$ (\S \ref{sec:beta_ferm}--\ref{sec:beta_ferm_2}) can be written as an infinite $\QQ(\beta)$-linear combination of the neutral fermions $\phi_n$ (\S \ref{sec:Fock_space}).
They satisfy the duality property $[(\Phi_m^{(\beta)})^\ast,\phi^{[\beta]}_n ]_+=[(\Phi_m^{[\beta]})^\ast,\phi^{(\beta)}_n ]_+=\delta_{m,n}$ (Lemma \ref{lemma:duality}), which generalizes the anti-commutation relation $[(\tfrac{1}{2}\phi_m)^\ast,\phi_n]_+=\delta_{m,n}$.
We also define the two types of ``$\beta$-deformed power sums'' $p^{(\beta)}_n,p^{[\beta]}_n$ (\S \ref{sec:beta_def_powersum}), which satisfy the Cauchy identity (Lemma \ref{lemma:Cauchy}):
\begin{equation}\label{eq:Cauchy_2}
\sum_{\lambda:\mathrm{odd}}2^{\ell(\lambda)}z_\lambda^{-1}
p_\lambda^{(\beta)}(x)p_\mu^{[\beta]}(y)=
\prod_{i,j}\frac{1-\overline{x_i}y_j}{1-x_iy_j},\qquad z_\lambda=\prod_{i\geq 1}i^{m_i}\cdot m_i!,
\end{equation}
where $m_i(\lambda)=\sharp\{k\,|\,\lambda_k=i\}$.
Comparing \eqref{eq:Cauchy_identity} with \eqref{eq:Cauchy_2}, we can show the existence of the non-degenerate inner product $\langle\cdot,\cdot\rangle$ (Eq.~\eqref{eq:def_of_bilinear_form}) that satisfies $\langle p^{(\beta)}_\lambda,p^{[\beta]}_\lambda\rangle=2^{\ell(\lambda)}z_\lambda\delta_{\lambda,\mu}$ for odd paritions $\lambda,\mu$, and $\langle GQ_\lambda,gp_\mu\rangle=\langle GP_\lambda,gq_\mu\rangle=\delta_{\lambda,\mu}$ for strict partitions $\lambda,\mu$.
By virtue of this property, we show the fact that the $\beta$-deformed boson-fermion correspondences $\Omega,\chi$, which are defined in \S \ref{sec:boson_fermion}, preserve the inner product \eqref{eq:inner_pres}.
The construction of $GP_\lambda$ and $GQ_\lambda$ (Theorems \ref{thm:prev_main_theorem}, \ref{thm:GP}) are given by comparing the vacuum expectation values $\Omega(\ket{\lambda}_P)$, $\Omega(\ket{\lambda}_Q)$ with the generating functions of $GP_\lambda$, $GQ_\lambda$, which were presented in the previous work~\cite{nakagawa2018universalfactrial} by Nakagawa-Naruse.
For the dual functions $gp_\lambda$ and $gq_\lambda$, we show the fermionic descriptions $gq_{\lambda}=\chi(\ket{\lambda}_q)$ (Theorem \ref{thm:gq}) and $gp_{\lambda}=\chi(\ket{\lambda}_p)$ (Theorem \ref{thm:fermion_of_gp}) by proving the orthonormality ${}_Q\inner{\mu}{\lambda}_p={}_P\inner{\mu}{\lambda}_q=\delta_{\lambda,\mu}$.
As an immediate application of our description, we present a generating function of $gq_\lambda$ (Eq.~\eqref{eq:gq_gen}) and $gp_\lambda$ (Eq.~\eqref{eq:gen_gp}).
\subsection{Related works}
A Pfaffian formula of $GP_\lambda$ and $GQ_\lambda$ was first given in \cite{HUDSON2017115} in the context of the connective $K$-theory of Grassmann bundles.
Nakagawa-Naruse~\cite{nakagawa2016generalized,nakagawa2017generating,nakagawa2018universalfactrial} introduced universal cohomology versions of these functions.
The generating function of $gq_\lambda$ \eqref{eq:gq_gen} was first conjectured in \cite{nakagawa2017generating}.
They also conjectured in ~\cite{nakagawa2018universalfactrial} a combinatorial description of $gq_\lambda$ and $qp_\lambda$ in terms of shifted plane partitions, which was proved by Lewis-Merberg~\cite{lewis2022combinatorial}.
In~\cite{lewis2022combinatorial}, they showed that $\bigoplus_{\lambda}\ZZ[\beta]\cdot GP_\lambda\subsetneq G\Gamma$, the subspace of \textit{finite} linear combinations of $GP_\lambda$, is closed under the multiplication.
Our results also generalize the fermionic description of $K$-theoretic functions of type $A$ such as the stable Grothendieck polynomials~\cite{iwao2020freefermion,iwao2022free}, the multi-Schur functions~\cite{iwao2021free}, and the canonical Grothendieck functions~\cite{iwaomotegiscrimshaw2022free}.
\subsection*{Acknowledgements}
This work is partially supported by Grant-in-Aid for Scientific Research (C) 19K03065 and 22K03239.
\section{Preliminaries}\label{sec:Pre}
This section gives a brief summary of the neutral fermion Fock space.
We recommend Baker's paper~\cite{baker1995symmetric} and Jimbo-Miwa's paper \cite{jimbo1983solitons} for readers who are interested in this theme.
Let $[A,B]=AB-BA$ be the commutator and $[A,B]_+=AB+BA$ be the anti-commutator.
\subsection{Fock space}\label{sec:Fock_space}
Let $\mathcal{A}$ be the $\QQ(\beta)$-algebra of \textit{neutral fermions} generated by $\{\phi_n\}_{n\in \ZZ}$ satisfying the anti-commutation relation
$
[\phi_m,\phi_n]_+=2(-1)^m\delta_{m+n,0}
$.
In particular, we have $\phi_0^2=1$ and $\phi_n^2=0$ for $n\neq 0$.
Let $\ket{0}$ and $\bra{0}$ denote the \textit{vacuum vectors}:
\[
\phi_{-n}\ket{0}=0,\quad
\bra{0}\phi_n=0,\qquad (n>0).
\]
The \textit{Fock space} $\mathcal{F}:=\mathcal{A}\cdot \ket{0}$ is the left $\mathcal{A}$-module generated by $\ket{0}$, and the \textit{dual Fock space}
$\mathcal{F}^\ast:=\bra{0}\cdot \mathcal{A}$ is the right $\mathcal{A}$-module generated by $\bra{0}$.
The \textit{vacuum expectation value} is the unique bilinear form
\[
\mathcal{F}^\ast \otimes_{\QQ(\beta)} \mathcal{F} \to \QQ(\beta);\qquad
\bra{u}\otimes \ket{v}\mapsto \inner{u}{v}
\]
that satisfies (i) $\inner{0}{0} =1$, (ii) $\bra{0}\phi_0\ket{0}=0$, and (iii)
$(\bra{u}\phi_n)\ket{v}=\bra{u}(\phi_n\ket{v})$.
We use the abbreviations
$\bra{u}X\ket{v}:=(\bra{u}X)\ket{v}=\bra{u}(X\ket{v})$ and
$\langle X\rangle:=\bra{0}X\ket{0}$ for any $X\in \mathcal{A}$.
The Fock space $\mathcal{F}$ is split into two subspaces as $\mathcal{F}=\mathcal{F}_{odd}\oplus \mathcal{F}_{even}$, where $\mathcal{F}_{odd}$ (\textit{resp.}~$\mathcal{F}_{even}$) is the subspace generated by all vectors obtained from $\ket{0}$ by applying odd (\textit{resp.}~even) numbers of $\phi_n$ ($n\geq 0$).
The dual Fock space is also split as $\mathcal{F}^\ast=\mathcal{F}^\ast_{odd}\oplus \mathcal{F}^\ast_{even}$.
If
$\bra{u}\otimes \ket{v}$ is in $ \mathcal{F}^\ast_{even}\otimes \mathcal{F}_{odd}$ or in $\mathcal{F}^\ast_{odd}\otimes \mathcal{F}_{even}$, the vacuum expectation value $\inner{v}{u}$ annihilates automatically.
By restriction, it induces the nondegenerate bilinear form
\begin{equation}\label{eq:even_bilin}
\mathcal{F}_{even}^\ast\otimes_{\QQ(\beta)} \mathcal{F}_{even}\to \QQ(\beta).
\end{equation}
There exists an anti-algebra isomorphism $\mathcal{A}\leftrightarrow\mathcal{A}$; $x\leftrightarrow x^\ast$ defined by $\phi_n^\ast=(-1)^n\phi_{-n}$ and $(xy)^\ast=y^\ast x^\ast$.
It induces the involution $\mathcal{F}\leftrightarrow \mathcal{F}^\ast;v\leftrightarrow v^\ast$ with $\ket{0}^\ast=\bra{0}$.
\subsection{Wick's theorem}
For a $2r\times 2r$ matrix $X$, $\mathrm{Pf}(X)$ denotes the \textit{Pfaffian}
\[
\mathrm{Pf}(X)=
\hspace{-1em}
\sum_{\substack{
\sigma(1)<\sigma(3)<\cdots <\sigma(2r-1)\\
\sigma(1)<\sigma(2),\
\sigma(3)<\sigma(4),\dots,
\sigma(2r-1)<\sigma(2r)
}}
\hspace{-1em}
\mathrm{sgn}(\sigma)
a_{\sigma(1),\sigma(2)}
a_{\sigma(3),\sigma(4)}\dots a_{\sigma(2r-1),\sigma(2r)}.
\]
For $n_1,\dots,n_{2r}\in \ZZ$, we have \textit{Wick's theorem}
\begin{equation}\label{eq:Wick}
\langle
\phi_{n_1}\phi_{n_2}\dots \phi_{n_{2r}}
\rangle
=\mathrm{Pf}
\left(
\langle
\phi_{n_i}\phi_{n_j}
\rangle
\right)_{1\leq i<j\leq 2r}.
\end{equation}
For a strict partition $\lambda=(\lambda_1>\lambda_2>\dots>\lambda_r>0)$ of length $r$, we define the vector $\ket{\lambda}\in \mathcal{F}_{even}$ as
\[
\ket{\lambda}=\begin{cases}
\phi_{\lambda_1}\phi_{\lambda_2}\dots \phi_{\lambda_r}\ket{0} & (r:\mbox{even})\\
\phi_{\lambda_1}\phi_{\lambda_2}\dots \phi_{\lambda_r}\phi_0\ket{0} & (r:\mbox{odd})
\end{cases}.
\]
The orthogonality $\langle \mu|\lambda\rangle=\delta_{\lambda,\mu}$ is derived directly from Wick's theorem \eqref{eq:Wick}.
\section{Definition of $\beta$-deformed operators}\label{sec:beta_def_op}
\subsection{$\beta$-deformed fermion fields $\phi^{(\beta)}(z),\phi^{[\beta]}(z)$}\label{sec:beta_ferm}
The \textit{neutral fermion field} $\phi(z)$ is the formal series
$
\phi(z)=\sum_{n\in \ZZ}\phi_nz^n
$, which determines a $\QQ(\beta)$-linear map $\mathcal{F}\to \mathcal{F}((z))$ by left multiplication, and a $\QQ(\beta)$-linear map $\mathcal{F}^\ast\to \mathcal{F}^\ast((z^{-1}))$ by right multiplication.
\begin{defi}\label{defi:deformed_fermions}
We define the \textit{$\beta$-deformed fermion fields}
$\phi^{(\beta)}(z)=\sum_{n\in \ZZ}\phi^{(\beta)}_nz^n$ and $\phi^{[\beta]}(z)=\sum_{n\in \ZZ}\phi^{[\beta]}_nz^n$ as follows.
\[
\begin{gathered}
\sum_{n=0}^\infty \phi^{(\beta)}_nz^n=
\sum_{n=0}^\infty \phi_n(z+\tfrac{\beta}{2})^n,\quad
\sum_{n=1}^\infty \phi^{(\beta)}_{-n}z^{-n}=
\sum_{n=1}^\infty \phi_{-n}\left(\frac{z^{-1}}{1+\frac{\beta}{2}z^{-1}}\right)^n,\\
\sum_{n=1}^\infty \phi^{[\beta]}_nz^n=
\sum_{n=1}^\infty \phi_n\left(\frac{z}{1+\frac{\beta}{2}z}\right)^n,\quad
\sum_{n=0}^\infty \phi^{[\beta]}_{-n}z^{-n}=
\sum_{n=0}^\infty \phi_{-n}(z^{-1}+\tfrac{\beta}{2})^n.
\end{gathered}
\]
By right multiplication, $\phi^{(\beta)}(z)$ determines a $\QQ(\beta)$-linear map $\mathcal{F}^\ast\to \mathcal{F}^\ast((z^{-1}))$.
By left multiplication, $\phi^{[\beta]}(z)$ determines a $\QQ(\beta)$-linear map $\mathcal{F}\to \mathcal{F}((z))$.
\end{defi}
Informally, Definition \ref{defi:deformed_fermions} is rewritten as
\begin{equation}\label{eq:def_of_b_deformed_fermion}
\phi^{(\beta)}(z)=\phi(z+\tfrac{\beta}{2}),\qquad
\phi^{[\beta]}(z)=\phi(\tfrac{z}{1+\frac{\beta}{2}z}).
\end{equation}
For positive $n>0$, the operators $\phi^{(\beta)}_n,\phi^{[\beta]}_n$ are expressed as a linear combination of $\phi_1,\phi_2,\dots$, and $\phi^{(\beta)}_{-n},\phi^{[\beta]}_{-n}$ are expressed as a linear combination of $\phi_{-1},\phi_{-2},\dots$
Then, we have the annihilation rule
\begin{equation}\label{eq:ann_rule}
\bra{0}\phi_n^{(\beta)}=\bra{0}\phi_n^{[\beta]}=0,\qquad
\phi_{-n}^{(\beta)}\ket{0}=\phi_{-n}^{[\beta]}\ket{0}=0,\qquad (n>0).
\end{equation}
The actions of $\phi_0^{(\beta)}$ and $\phi_0^{[\beta]}$ on the vacuum vectors are expressed as
\begin{equation}\label{eq:vs_phi_0}
\bra{0}\phi_0^{(\beta)}=\bra{0}\phi_0,\quad
\bra{0}\phi_0\phi_0^{(\beta)}=\bra{0},\quad
\phi_0^{[\beta]}\ket{0}=\phi_0\ket{0},\quad
\phi_0^{[\beta]}\phi_0\ket{0}=\ket{0}.
\end{equation}
\begin{lemma}[{\cite[\S 9.1]{iwao2021neutralfermionic}}]
\label{lemma:basic_anti_commutation}
We have the anti-commutation relation:
\[
[(\phi^{(\beta)}_m)^\ast, \phi^{[\beta]}_n]_+
=
\begin{cases}
2 & (m=n)\\
\beta & (m=n-1)\\
0 & (\mbox{otherwise})
\end{cases}.
\]
\end{lemma}
\begin{proof}
This lemma is derived directly from $[(\phi_m)^\ast,\phi_n]_+=2\delta_{m,n}$ and Definition \ref{defi:deformed_fermions}.
For details of the proof, see \cite[\S 9.1]{iwao2021neutralfermionic}.
\end{proof}
\subsection{$\beta$-deformed fermion fields $\Phi^{(\beta)}(z),\Phi^{[\beta]}(z)$}\label{sec:beta_ferm_2}
We next define another two $\beta$-deformed fermion fields
$\Phi^{(\beta)}(z)=\sum_{n\in \ZZ}\Phi_n^{(\beta)}z^n$ and
$\Phi^{[\beta]}(z)=\sum_{n\in \ZZ}\Phi_n^{[\beta]}z^n$ as follows:
\begin{defi}\label{def:Phi}
We define the $\beta$-deformed fermion fields $\Phi^{(\beta)}(z),\Phi^{[\beta]}(z)$ by
\[
\Phi^{(\beta)}(z):=\frac{1}{2+\beta z^{-1}}\phi^{(\beta)}(z),\qquad
\Phi^{[\beta]}(z):=\frac{1}{2+\beta z}\phi^{[\beta]}(z).
\]
By right multiplication, $\Phi^{(\beta)}(z)$ determines a $\QQ(\beta)$-linear map $\mathcal{F}^\ast\to \mathcal{F}^\ast((z^{-1}))$.
By left multiplication, $\Phi^{[\beta]}(z)$ determines a $\QQ(\beta)$-linear map $\mathcal{F}\to \mathcal{F}((z))$.
\end{defi}
For any $n\in \ZZ$, $\Phi_n^{(\beta)}$ and $\Phi_n^{[\beta]}$ expand as
\begin{align}
&\Phi_n^{(\beta)}=\frac{1}{2}\phi_n^{(\beta)}-\frac{\beta}{4}\phi_{n+1}^{(\beta)}+\frac{\beta^2}{8}\phi_{n+2}^{(\beta)}-\cdots,
\label{eq:Phi(beta)exp}\\
&\Phi_n^{[\beta]}=\frac{1}{2}\phi_n^{[\beta]}-\frac{\beta}{4}\phi_{n-1}^{[\beta]}+\frac{\beta^2}{8}\phi_{n-2}^{[\beta]}-\cdots.
\label{eq:Phi[beta]exp}
\end{align}
From (\ref{eq:Phi(beta)exp}--\ref{eq:Phi[beta]exp}), we have the annihilation rule
\begin{equation}\label{eq:ann_Phi}
\bra{0}\Phi_n^{(\beta)}=0,\quad
\Phi_{-n}^{[\beta]}\ket{0}=0,\qquad(n>0).
\end{equation}
However, unexpectedly, the vectors $\Phi_{-n}^{(\beta)}\ket{0}$ and $\bra{0}\Phi_{n}^{[\beta]}$ do not annihilate for $n>0$.
In fact, by using $\phi_0
=\phi_0^{(\beta)}-\frac{\beta}{2}\phi_1^{(\beta)}+\frac{\beta^2}{4}\phi_2^{(\beta)}-\cdots=
\phi_0^{[\beta]}-\frac{\beta}{2}\phi_{-1}^{[\beta]}+\frac{\beta^2}{4}\phi_{-2}^{[\beta]}-\cdots
$, we rewrite (\ref{eq:Phi(beta)exp}--\ref{eq:Phi[beta]exp}) as
\begin{equation}\label{eq:Phi_expand}
\begin{aligned}
&\Phi_{-n}^{(\beta)}=\frac{1}{2}\phi_{-n}^{(\beta)}-\frac{\beta}{4}\phi_{-n+1}^{(\beta)}+\frac{\beta^2}{8}\phi_{-n+2}^{(\beta)}-\cdots+\frac{(-\beta)^n}{2^{n+1}}\phi_0,\\
&\Phi_{n}^{[\beta]}=\frac{1}{2}\phi_{n}^{(\beta)}-\frac{\beta}{4}\phi_{n-1}^{(\beta)}+\frac{\beta^2}{8}\phi_{n-2}^{(\beta)}-\cdots+\frac{(-\beta)^n}{2^{n+1}}\phi_0
\end{aligned}
\end{equation}
for non-negative $n\geq 0$.
These equations imply
\begin{equation}\label{eq:lack_of_ann_rule}
\bra{0}\Phi_n^{[\beta]}=\frac{(-\beta)^n}{2^{n+1}}\cdot \bra{0}\phi_0\neq 0,\quad
\Phi_{-n}^{(\beta)}\ket{0}=
\frac{(-\beta)^n}{2^{n+1}}\cdot \phi_0\ket{0}
\neq 0,\qquad(n\geq 0)
\end{equation}
\begin{lemma}[Duality of the $\beta$-deformed fermions]
\label{lemma:duality}
We have the following commutation relation:
\[
[(\Phi^{(\beta)}_m)^\ast, \phi^{[\beta]}_n]_+
=[(\Phi^{[\beta]}_m)^\ast, \phi^{(\beta)}_n]_+
=\delta_{m,n}.
\]
\end{lemma}
\begin{proof}
It is straightforward to derive the commutation relation from Lemma \ref{lemma:basic_anti_commutation} and (\ref{eq:Phi(beta)exp}--\ref{eq:Phi[beta]exp}).
\end{proof}
\subsection{$\beta$-deformed current and Hamiltonian operators}
For any odd integer $m$, the \textit{current operator} $b_m$ is the formal sum
$
b_m=\frac{1}{4}\sum_{i\in \ZZ}(-1)^i\phi_{-i-m}\phi_i
$.
\begin{defi}
For any (possibly even) integer $m\neq 0$, we define the \textit{$\beta$-deformed current operators} $b_m^{(\beta)},b_m^{[\beta]}$ by
\[
b_m^{(\beta)}=
\left.\frac{(X-\frac{\beta}{2})^m-(-X-\frac{\beta}{2})^m }{2}\right|_{X^k\mapsto b_k}\mbox{\quad and \quad}
b_m^{[\beta]}=(b^{(\beta)}_{-m})^\ast.
\]
Here, $\frac{(X-\frac{\beta}{2})^m-(-X-\frac{\beta}{2})^m }{2}$ is expanded as a polynomial in $X$ if $m>0$, and as a power series of $X^{-1}$ if $m<0$.
For example, we have $b_1^{(\beta)}=b_1$, $b_2^{(\beta)}=-\beta b_1$, $b_3^{(\beta)}=b_3+\frac{3\beta^2}{4}b_1$, $b_{-1}^{(\beta)}=b_{-1}+\frac{\beta^2}{4}b_{-3}+\frac{\beta^4}{16}b_{-5}+\cdots$.
\end{defi}
Let $p_n(x)=x_1^n+x_2^n+\cdots$ denote the $n$-th power sum.
We define the \textit{Hamiltonian operator} $\mathcal{H}(x)$ by
\[
\mathcal{H}(x)=2\sum_{n=1,3,5,\dots} \frac{p_n(x)}{n}b_n.
\]
The \textit{$\beta$-deformed Hamiltonian operators} $\mathcal{H}^{(\beta)}(x),\mathcal{H}^{[\beta]}(x)$ are obtained by substituting $b_n^{(\beta)},b_n^{[\beta]}$ in the place of $b_n$ respectively:
\[
\mathcal{H}^{(\beta)}(x)=2\sum_{n=1,3,5,\dots} \frac{p_n(x)}{n}b^{(\beta)}_n,\qquad
\mathcal{H}^{[\beta]}(x)=2\sum_{n=1,3,5,\dots} \frac{p_n(x)}{n}b^{[\beta]}_n.
\]
For brevity, we often use the abbreviations $\mathcal{H}^{(\beta)}=\mathcal{H}^{(\beta)}(x)$ and
$\mathcal{H}^{[\beta]}=\mathcal{H}^{[\beta]}(x)$.
\subsection{$\beta$-deformed power sums}\label{sec:beta_def_powersum}
The \textit{$\beta$-deformed power sums}
$p^{(\beta)}_n(x),p^{[\beta]}_n(x)$ are defined by
\[
p^{(\beta)}_n(x):=
\sum_{i=0}^\infty \tbinom{-n}{i}(\tfrac{\beta}{2})^ip_{n+i}(x),\qquad
p^{[\beta]}_n(x):=\sum_{i=1}^n\tbinom{n}{i}(\tfrac{\beta}{2})^ip_i(x).
\]
It is useful to remember the informal expressions
\[
p^{(\beta)}_n(x)=p_n(\tfrac{x}{1+\frac{\beta}{2}x})\quad \mbox{and}\quad
p^{[\beta]}_n(x)=p_n(x+\tfrac{\beta}{2})-p_n(\tfrac{\beta}{2}).
\]
For any partition $\lambda=(\lambda_1\geq \lambda_2\geq\dots )$, we put
$
p^{(\beta)}_{\lambda}:=
p^{(\beta)}_{\lambda_1}p^{(\beta)}_{\lambda_2}\cdots$ and
$p^{[\beta]}_{\lambda}:=
p^{[\beta]}_{\lambda_1}p^{[\beta]}_{\lambda_2}\cdots
$.
Let $\widehat{G\Gamma}=\QQ(\beta)[[p_1^{(\beta)},p_3^{(\beta)},p_5^{(\beta)},\dots]]$ and $g\Gamma=\QQ(\beta)[p_1^{[\beta]},p_3^{[\beta]},p_5^{[\beta]},\dots]$.
Any element of $\widehat{G\Gamma}$ is expressed as a (possibly) infinite $\QQ(\beta)$-linear combination of $p_\lambda^{(\beta)}$, where $\lambda$ runs over all odd partitions.
There exist two natural isomorphisms
\[
\iota^{(\beta)}:\widehat{\Gamma}\to \widehat{G\Gamma};\quad
p_n\mapsto p_n^{(\beta)},\qquad
\iota^{[\beta]}:\Gamma\to g\Gamma;\quad
p_n\mapsto p_n^{[\beta]},
\]
where $\Gamma=\QQ(\beta)[p_1,p_3,\dots]$ and
$\widehat{\Gamma}=\QQ(\beta)[[p_1,p_3,\dots]]$.
The map $\iota^{(\beta)}$ coincides with the substitution map $x_i\mapsto \frac{x_i}{1+\frac{\beta}{2}x_i}$.
Let $Q_\lambda\in \Gamma$ be the \textit{Schur $Q$-function}~\cite[\S III.8]{macdonald1998symmetric}, where $\lambda$ is a strict partition.
The $\beta$-deformed Hamiltonians and power sums are related as
\begin{equation}\label{eq:beta_deformed_Q}
\iota^{(\beta)}(Q_\lambda)=
\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda},\qquad
\iota^{[\beta]} (Q_\lambda)=\bra{0}e^{\mathcal{H}^{[\beta]}}\ket{\lambda},
\end{equation}
Set
$Q^{(\beta)}_\lambda:=\iota^{(\beta)}(Q_\lambda)$ and
$Q^{[\beta]}_\lambda:=\iota^{[\beta]}(Q_\lambda)$.
\subsection{$\beta$-deformed bilinear form}
We define the bilinear form
\begin{equation}\label{eq:def_of_bilinear_form}
\widehat{G\Gamma}\otimes_{\QQ(\beta)}g\Gamma\to \QQ(\beta);\qquad
f\otimes g\mapsto
\left\langle
f,g
\right\rangle
\end{equation}
by $
\langle
p_\lambda^{(\beta)}, p_\mu^{[\beta]}
\rangle=
2^{-\ell(\lambda)} z_\lambda\delta_{\lambda,\mu}$ for all odd partitions $\lambda,\mu$.
\begin{lemma}[{\cite[\S 8]{iwao2021neutralfermionic}}]\label{lemma:Cauchy}
We have the Cauchy identity
\[
\sum_{\lambda:\mathrm{odd}}2^{\ell(\lambda)}z_\lambda^{-1}
p_\lambda^{(\beta)}(x)p_\mu^{[\beta]}(y)=
\prod_{i,j}\frac{1-\overline{x_i}y_j}{1-x_iy_j},\qquad
\mbox{where}\quad \overline{x}=-\frac{x}{1+\beta x}.
\]
\end{lemma}
\begin{proof}
This lemma is obtained from the equation
\[
\sum_{\lambda:\mathrm{odd}}2^{\ell(\lambda)}z_\lambda^{-1}
p_\lambda(x)p_\mu(y)=
\prod_{i,j}\frac{1+x_iy_j}{1-x_iy_j}
\]
by substituting $x_i\mapsto \frac{x_i}{1+\frac{\beta}{2}x_i}$ and $p_n(y)\mapsto p_n(y+\frac{\beta}{2})-p_n(\frac{\beta}{2})$.
For details, see \cite[\S 8]{iwao2021neutralfermionic}.
\end{proof}
Through the bilinear form \eqref{eq:def_of_bilinear_form}, $\widehat{G\Gamma}$ is identified with a subspace of $\mathrm{Hom}_{\QQ(\beta)}(g\Gamma,\QQ(\beta))$.
By construction, they are in fact isomorphic:
\begin{equation}\label{eq:dual_isom}
\widehat{G\Gamma}\simeq \mathrm{Hom}_{\QQ(\beta)}(g\Gamma,\QQ(\beta)).
\end{equation}
\begin{lemma}\label{lemma:beta_Q_dual}
For any strict partitions $\lambda,\mu$, we have
$
\langle
Q^{(\beta)}_\lambda,
Q^{[\beta]}_\mu
\rangle
=2^{\ell(\lambda)}\delta_{\lambda,\mu}
$.
\end{lemma}
\begin{proof}
Let $\langle f,g\rangle':=\langle \iota^{(\beta)}(f),\iota^{[\beta]}(g)\rangle$ for $f,g\in \Gamma$.
Then, the bilinear form $\langle \cdot,\cdot \rangle'$ coincides with the Hall inner product on $\Gamma$~\cite[\S III.8]{macdonald1998symmetric}.
The lemma follows from the orthogonality $\langle Q_\lambda,Q_\mu\rangle'=2^{\ell(\lambda)}\delta_{\lambda,\mu}$.
\end{proof}
\subsection{$\beta$-deformed boson-fermion correspondence}
\label{sec:boson_fermion}
Let $\Omega_0,\chi$ be the $\QQ(\beta)$-linear maps defined by
\[
\begin{gathered}
\Omega_0:\mathcal{F}_{even}\to \widehat{G\Gamma};\quad \ket{v}\mapsto \bra{0}e^{\mathcal{H}^{(\beta)}}\ket{v},\\
\chi:\mathcal{F}_{even}\to g\Gamma;\quad \ket{v}\mapsto \bra{0}e^{\mathcal{H}^{[\beta]}}\ket{v}.
\end{gathered}
\]
From \eqref{eq:beta_deformed_Q} and Lemma \ref{lemma:beta_Q_dual}, we show that they preserve the inner products:
$
\inner{u^\ast}{v}=
\big\langle \Omega_0(\ket{u}), \chi(\ket{v}) \big\rangle
$, where $\bra{u^\ast}=\ket{u}^\ast$.
Through the inner product \eqref{eq:even_bilin}, $\mathcal{F}_{even}$ is identified with a subspace of
$
\widehat{\mathcal{F}}_{even}:=
\mathrm{Hom}_{\QQ(\beta)}(\mathcal{F}^\ast_{even},\QQ(\beta))
$.
For any $\varphi\in \widehat{\mathcal{F}}_{even}$, we let $[\varphi]$ denote the unique element of $\widehat{G\Gamma}$ that satisfies
\[
\big\langle
[\varphi],\chi(\ket{v})
\big\rangle=
\varphi(\bra{v^\ast}),\qquad \forall \ket{v}\in\mathcal{F}_{even}.
\]
Then, the map $\Omega_0$ naturally extends to
\[
\Omega:\widehat{\mathcal{F}}_{even}\to \widehat{G\Gamma};\quad \varphi\mapsto [\varphi],
\]
which satisfies
\begin{equation}\label{eq:inner_pres}
\inner{u^\ast}{v}=
\big\langle \Omega(\ket{u}), \chi(\ket{v}) \big\rangle
\end{equation}
for any $\ket{u}\in \widehat{\mathcal{F}}_{even}$ and $\ket{v}\in \mathcal{F}_{even}$.
One can easily check that $\Omega$ and $\chi$ are bijective.
We call these linear maps $\Omega,\chi$ the \textit{$\beta$-deformed boson-fermion correspondences}.
\section{Commutation relations between $\beta$-deformed operators}\label{sec:comm_rel}
In this section, we give a list of useful commutation relations between $\beta$-deformed operators.
Most proofs are given in the previous paper \cite{iwao2021neutralfermionic}.
Define the bilinear operators $\oplus$ and $\ominus$ by
\[
x\oplus y=x+y+\beta xy,\quad
x\ominus y=\frac{x-y}{1+\beta y}.
\]
In particular, we have $\overline{t}=0\ominus t=\frac{-t}{1+\beta t}$.
Let
\[
\Theta=2
\sum_{n=1,3,5,\dots}\left(\frac{\beta}{2}\right)^n\frac{b_{-n}}{n},\qquad
\theta=
\Theta^\ast=
2
\sum_{n=1,3,5,\dots}\left(\frac{\beta}{2}\right)^n\frac{b_{n}}{n}.
\]
\begin{lemma}\label{lemma:comm_rels_0}
We have the following commutation relations:
\begin{enumerate}
\item\label{item:1-1} $\bra{0}e^{\Theta}=\bra{0}$.
\item\label{item:1-2} $e^{-\Theta}\phi^{(\beta)}(z)e^{\Theta}
=\frac{1}{1+\beta z^{-1}}\cdot \phi^{(\beta)}(z)
$.
\item\label{item:1-3} $e^{-\Theta}e^{\mathcal{H}^{(\beta)}}e^{\Theta}=\prod_{i}(1+\beta x_i)\cdot e^{\mathcal{H}^{(\beta)}}$.
\item\label{item:1-4}
$e^{\mathcal{H}^{(\beta)}}
\phi^{(\beta)}(z)e^{-\mathcal{H}^{(\beta)}}=\prod_{i}\frac{z^{-1}\oplus x_i}{z^{-1}-x_i}\cdot \phi^{(\beta)}(z)$.
\item\label{item:1-5}
$
\left\langle
\phi^{(\beta)}(z)\phi^{(\beta)}(w)
\right\rangle
=\frac{w^{-1}-z^{-1}}{w^{-1}\oplus z^{-1}}
$,
where $\frac{w^{-1}-z^{-1}}{w^{-1}\oplus z^{-1}}$ expands as
\[
\begin{aligned}
\textstyle\frac{1-wz^{-1}}{1+wz^{-1}+\beta z^{-1}}=
1-(2w+\beta)z^{-1}+(2w^{2}+3\beta w+\beta^2)z^{-2}-\cdots
\end{aligned}
\]
in the field $\QQ(\beta)((w^{-1}))((z^{-1}))$ of formal Laurent series\footnote{
We should note the fact that $\QQ(\beta)((w^{-1}))((z^{-1}))=\left\{\QQ(\beta)((w^{-1}))\right\}((z^{-1}))$ is not the same as $\QQ(\beta)((z^{-1}))((w^{-1}))$.
In fact, the former contains $1+wz^{-1}+w^2z^{-2}+w^{3}z^{-3}+\cdots$, while the latter does not.
}.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{item:1-1} follows from $\bra{0}b_{-n}=0$ for $n>0$.
\eqref{item:1-2} and \eqref{item:1-3} are given in \cite[\S 4.3]{iwao2021neutralfermionic}.
\eqref{item:1-4} follows from the following equation, which is given in \cite[\S 4.2]{iwao2021neutralfermionic},
\[
e^{\mathcal{H}^{(\beta)}}\phi^{(\beta)}(z)e^{\mathcal{H}^{-(\beta)}}
=
\exp\left(\sum_{n=1}^\infty\frac{p_n(x)}{n}(z^n-(-z-\beta)^n)\right)\phi^{(\beta)}(z)
\]
and the identity $\exp\left(\sum_{n=1}^\infty\frac{p_n(x)}{n}A^n\right)=\prod_i\frac{1}{1-x_iA}$.
\eqref{item:1-5} is given in \cite[\S 3.2]{iwao2021neutralfermionic}.
\end{proof}
\begin{lemma}\label{lemma:commutation_rels}
We have the following commutation relations:
\begin{enumerate}
\item\label{item:2-1} $e^{\theta}\ket{0}=\ket{0}$.
\item \label{item:2-2}
$e^{-\theta}\phi^{[\beta]}(z)e^{\theta}=\frac{1}{1+\beta z}\cdot \phi^{[\beta]}(z)$.
\item\label{item:2-3} $e^\theta$ and $e^{\mathcal{H}^{[\beta]}}$ commute with each other.
\item \label{item:2-4}
$e^{\mathcal{H}^{[\beta]}}\phi^{[\beta]}(z)e^{-\mathcal{H}^{[\beta]}}
=
\prod_{i}\frac{1-x_i\overline{z}}{1-x_iz}\cdot
\phi^{[\beta]}(z)
$.
\item \label{item:2-5}
$
\left\langle
\phi^{[\beta]}(z)\phi^{[\beta]}(w)
\right\rangle
=\frac{z-w}{z\oplus w},
$
where $\frac{z-w}{z\oplus w}$ expands as
\[
\begin{aligned}
&\textstyle
\frac{1-wz^{-1}}{1+wz^{-1}+\beta w}=
1-(2z^{-1}+\beta)w+(2z^{-2}+3\beta z^{-1}+\beta^2)w^2-\cdots
\end{aligned}
\]
in $\QQ(\beta)((z))((w))$.
\end{enumerate}
\end{lemma}
\begin{proof}
\eqref{item:2-1} follows from $b_{n}\ket{0}=0$ for $n>0$.
\eqref{item:2-2} and \eqref{item:2-5} are given in \cite[\S 10.1]{iwao2021neutralfermionic}.
\eqref{item:2-3} follows from $[b_m,b_n]=0$ for $m,n>0$.
\eqref{item:2-4} is given in \cite[\S 9.1]{iwao2021neutralfermionic}.
\end{proof}
\begin{cor}\label{cor:sublemma}
We have
\begin{enumerate}
\item\label{item:aa} $e^{\theta}\phi_n^{[\beta]}e^{-\theta}=\phi^{[\beta]}_n+\beta \phi^{[\beta]}_{n-1}$,
\item
$e^{-\theta}\phi_n^{[\beta]}e^{\theta}=\phi^{[\beta]}_n-\beta \phi^{[\beta]}_{n-1}+\beta^2\phi^{[\beta]}_{n-2}-\cdots$,
\item\label{item:bb}
$e^{\theta}(\phi_n^{(\beta)})^\ast e^{-\theta}=
(\phi_n^{(\beta)}
-\beta \phi_{n+1}^{(\beta)}
+\beta^2 \phi_{n+2}^{(\beta)}-\cdots)^\ast
$.
\end{enumerate}
\end{cor}
\section{$GP_\lambda$ and $GQ_\lambda$-functions}\label{sec:main_part}
\subsection{The fermionic presentation of $GP_\lambda$ and $GQ_\lambda$}
Define the vector $\ket{\lambda}_Q$ by
\begin{equation}\label{eq:vector_Q}
\ket{\lambda}_Q=
\begin{cases}
\phi^{(\beta)}_{\lambda_1}e^{\Theta}\phi^{(\beta)}_{\lambda_2}e^{\Theta}\cdots
\phi^{(\beta)}_{\lambda_r}e^{\Theta}\ket{0} & (r:\mbox{even})\\
\phi^{(\beta)}_{\lambda_1}e^{\Theta}\phi^{(\beta)}_{\lambda_2}e^{\Theta}\cdots
\phi^{(\beta)}_{\lambda_r}e^{\Theta}
\phi^{(\beta)}_{0}e^{\Theta}
\ket{0} & (r:\mbox{odd})
\end{cases}.
\end{equation}
Note that $\ket{\lambda}_Q$ is an element of $\widehat{\mathcal{F}}_{even}$, but not $\mathcal{F}_{even}$.
The main theorem of the previous paper~\cite{iwao2021neutralfermionic} is described as follows:
\begin{thm}[{\cite[\S 7]{iwao2021neutralfermionic}}]\label{thm:prev_main_theorem}
The $K$-theoretic $Q$-function $GQ_\lambda(x)$ is expressed as
\begin{equation*}
GQ_\lambda(x)=\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda}_Q
=\Omega(\ket{\lambda}_Q).
\end{equation*}
\end{thm}
In \cite{iwao2021neutralfermionic}, the author proved Theorem \ref{thm:prev_main_theorem} by comparing the vacuum expectation value $\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda}_Q$ with Hudson-Ikeda-Matsumura-Naruse's Pfaffian formula~\cite{HUDSON2017115}.
However, for later use, we now give another proof using the generating function
\begin{equation}\label{eq:gen_of_GQ}
GQ_\lambda(x)=[u_1^{-\lambda_1}\dots u_{r}^{-\lambda_{r}}]
\prod_{i=1}^r
\frac{1}{1+\beta u_i}
\prod_{i,j}\frac{u_i\oplus x_j}{u_i\ominus x_j}
\prod_{1\leq i<j\leq r}\frac{u_j\ominus u_i}{u_j\oplus u_i},
\end{equation}
which was given by Nakagawa-Naruse \cite[\S 5.2]{nakagawa2018universalfactrial}.
Here, the rational functions in \eqref{eq:gen_of_GQ} expand in
\[
\QQ(\beta)((u_r))\dots ((u_2))((u_1))[[x_1,x_2,\dots]]
\]
and $[u_1^{n_1}\dots u_r^{n_r}]F(u_1,\dots,u_r)$ is the coefficient of the monomial $u_1^{n_1}\dots u_r^{n_r}$ in the expansion of $F$.
This expansion coincides with the Laurent expansion on the domain $D_{(r)}:=\{|x_j|<|u_1|<|u_2|<\dots<|u_r|<|\beta^{-1}|\;;\;\forall j\}$.
When substituting $\lambda_{r+1}=0$ to \eqref{eq:gen_of_GQ} formally, we have
\[
\begin{aligned}
GQ_{(\lambda,0)}
&
=[u_1^{-\lambda_1}\dots u_{r}^{-\lambda_r}u_{r+1}^0]
\prod_{i=1}^{r+1}\frac{1}{1+\beta u_i}
\prod_{i,j}\frac{u_i\oplus x_j}{u_i\ominus x_j}
\prod_{1\leq i<j\leq r+1}\frac{u_j\ominus u_i}{u_j\oplus u_i}\\
&=
[u_1^{-\lambda_1}\dots u_{r}^{-\lambda_r}]
\prod_{i=1}^{r}\frac{1}{1+\beta u_i}
\prod_{i,j}\frac{u_i\oplus x_j}{u_i\ominus x_j}
\prod_{1\leq i<j\leq r}\frac{u_j\ominus u_i}{u_j\oplus u_i}\\
&\hspace{5em}\times
\frac{1}{2\pi i}
\oint_{u_{r+1}\in D_{(r+1)}}
\frac{1}{1+\beta u_{r+1}}
\prod_j\frac{u_{r+1}\oplus x_j}{u_{r+1}\ominus x_j}
\prod_{i=1}^r\frac{u_{r+1}\ominus u_i}{u_{r+1}\oplus u_i}
\frac{du_{r+1}}{u_{r+1}}\\
&
=[u_1^{-\lambda_1}\dots u_{r}^{-\lambda_r}]
\prod_{i=1}^{r}\frac{1}{1+\beta u_i}
\prod_{i,j}\frac{u_i\oplus x_j}{u_i\ominus x_j}
\prod_{1\leq i<j\leq r}\frac{u_j\ominus u_i}{u_j\oplus u_i}\\
&\hspace{5em}
\times
\mathop{\mathrm{Res}}_{w=-\beta}
\left(
-\frac{1}{1+\beta w^{-1}}
\prod_j\frac{w^{-1}\oplus x_j}{w^{-1}\ominus x_j}
\prod_{i=1}^r\frac{w^{-1}\ominus u_i}{w^{-1}\oplus u_i}
\frac{dw}{w}
\right).\qquad (w^{-1}=u_{r+1})
\end{aligned}
\]
A straightforward calculation leads that the residue is $1$ and $GQ_{(\lambda,0)}=GQ_\lambda$.
\begin{proof}[Proof of Theorem \ref{thm:prev_main_theorem}]
We can assume $r$ to be even without the loss of generality by adding $\lambda_{r+1}=0$ to the end of a strict partition if needed.
The generating function of $\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda}_Q$ expands as
\begin{align*}
&
\left\langle
e^{\mathcal{H}^{(\beta)}}
\phi^{(\beta)}(z_1)e^{\Theta}
\cdots
\phi^{(\beta)}(z_{r})e^{\Theta}
\right\rangle\\
&=
\prod_j{(1+\beta x_j)^{r}}
\prod_i\frac{1}{(1+\beta z_i^{-1})^{r-i+1}}
\prod_{i,j}\frac{z_i^{-1}\oplus x_j}{z_i^{-1}-x_j}
\left\langle
\phi^{(\beta)}(z_1)
\cdots
\phi^{(\beta)}(z_{r})
\right\rangle\\
&=
\prod_j{(1+\beta x_j)^{r}}
\prod_i\frac{1}{(1+\beta z_i^{-1})^{r-i+1}}
\prod_{i,j}\frac{z_i^{-1}\oplus x_j}{z_i^{-1}-x_j}
\cdot
\mathrm{Pf}
\left(
\langle
\phi^{(\beta)}(z_i)\phi^{(\beta)}(z_j)
\rangle
\right)_{1\leq i<j\leq r}
\quad (\mathrm{Eq.~\eqref{eq:Wick}})
\allowdisplaybreaks
\\
&=
\prod_j{(1+\beta x_j)^{r}}
\prod_i\frac{1}{(1+\beta z_i^{-1})^{r-i+1}}
\prod_{i,j}\frac{z_i^{-1}\oplus x_j}{z_i^{-1}-x_j}
\cdot
\mathrm{Pf}
\left(
\frac{z_j^{-1}-z_i^{-1}}{z_j^{-1}\oplus z_i^{-1}}
\right)_{1\leq i<j\leq r}
\quad (\mathrm{Lemma~\ref{lemma:comm_rels_0}~\eqref{item:1-5}})
\allowdisplaybreaks
\\
&
\stackrel{(\ast)}{=}
\prod_j{(1+\beta x_j)^{r}}
\prod_i\frac{1}{(1+\beta z_i^{-1})^{r-i+1}}
\prod_{i,j}\frac{z_i^{-1}\oplus x_j}{z_i^{-1}-x_j}
\prod_{1\leq i<j\leq r}
\frac{z_j^{-1}-z_i^{-1}}{z_j^{-1}\oplus z_i^{-1}}\\
&=
\prod_i\frac{1}{1+\beta z_i^{-1}}
\prod_{i,j}\frac{z_i^{-1}\oplus x_j}{z_i^{-1}\ominus x_j}
\prod_{1\leq i<j\leq r}
\frac{z_j^{-1}\ominus z_i^{-1}}{z_j^{-1}\oplus z_i^{-1}}.
\end{align*}
For the equality ($\ast$), we use the following formula from Ikeda-Naruse~\cite[Lemma 2.4]{IKEDA201322}:
\begin{equation}\label{eq:IN-formula}
\mathrm{Pf}\left(
\frac{T_i-T_j}{T_i\oplus T_j}
\right)_{1\leq i<j\leq r}=
\prod_{1\leq i<j\leq r}\frac{T_i-T_j}{T_i\oplus T_j}.
\end{equation}
Putting $z_i=u_i^{-1}$ and taking the coefficients of $z_1^{\lambda_1}\dots z_{r}^{\lambda_{r}}$, we obtain $GQ_\lambda(x)=\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda}_Q$.
\end{proof}
We extend this result to the $K$-theoretic $P$-function $GP_\lambda(x)$.
Define $\ket{\lambda}_P\in \widehat{\mathcal{F}}_{even}$ by
\begin{equation}\label{eq:vector_P}
\ket{\lambda}_P=
\begin{cases}
\Phi^{(\beta)}_{\lambda_1}e^{\Theta}
\Phi^{(\beta)}_{\lambda_2}e^{\Theta}\cdots
\Phi^{(\beta)}_{\lambda_r}e^{\Theta}\ket{0} & (r:\mbox{even})\\
\Phi^{(\beta)}_{\lambda_1}e^{\Theta}
\Phi^{(\beta)}_{\lambda_2}e^{\Theta}\cdots
\Phi^{(\beta)}_{\lambda_r}e^{\Theta}
\phi^{(\beta)}_{0}e^{\Theta}
\ket{0} & (r:\mbox{odd})
\end{cases}.
\end{equation}
\begin{thm}\label{thm:GP}
The $K$-theoretic $P$-function $GP_\lambda(x)$ is expressed as
\[
GP_\lambda(x)=\bra{0}e^{\mathcal{H}^{(\beta)}}\ket{\lambda}_P=\Omega(\ket{\lambda}_P).
\]
\end{thm}
\begin{proof}
This theorem is given by comparing the definition of $\Phi^{(\beta)}(z)$ (Definition \ref{def:Phi})
and the generating function
\begin{align}\label{eq:quote_GP}
&GP_\lambda(x)=[u_1^{-\lambda_1}\dots u_r^{-\lambda_r}]
\prod_{i}
\frac{1}{2+\beta u_i}
\frac{1}{1+\beta u_i}
\prod_{i,j}\frac{u_i\oplus x_j}{u_i\ominus x_j}
\prod_{1\leq i<j\leq r}\frac{u_j\ominus u_i}{u_j\oplus u_i},
\end{align}
which was given in \cite[\S 4.1]{nakagawa2018universalfactrial}\footnote{
In \cite[\S 4.1]{nakagawa2018universalfactrial}, Nakagawa-Naruse presented the generating function
\[
HP_{\lambda}(x)
=HP_{\lambda}(x|\mathbf{0})
=[\mathbf{u}^{-\lambda}]
\prod_i\frac{u_i}{u_i+_{\LL}[t]\overline{u_i}}\cdot
\frac{1}{\mathcal{I}^{\LL}(u_i)}\cdot
\prod_{i,j}\frac{u_i+_{\LL}[t]\overline{x_j}}{u_i+_{\LL}\overline{x_j}}
\cdot
\prod_{j<i}
\frac{u_i+_{\LL}\overline{u_j}}{u_i+_{\LL}[t]\overline{u_j}}
\]
of the universal $P$-function $HP_\lambda(x)$.
Here, $HP_\lambda(x|b)$ is the universal factorial $P$-function.
For the $K$-theory setting, we substitute
\[
HP_\lambda(x)\mapsto GP_\lambda(x),\quad
x+_{\LL}y\mapsto x\oplus y,\quad
\mathcal{I}^{\LL}(u)\mapsto 1+\beta u,\quad
[t]\overline{u}\mapsto u.
\]
to obtain \eqref{eq:quote_GP}.
}.
\end{proof}
From Theorems \ref{thm:prev_main_theorem} and \ref{thm:GP}, the $K$-$P$-function $GP_\lambda$ and the $K$-$P$-function $GQ_\lambda$ are contained in the algebra $\widehat{G\Gamma}$.
\section{$gp_\lambda$ and $gq_\lambda$-functions}
Comparing \eqref{eq:Cauchy_identity} with Lemma \ref{lemma:Cauchy}, we find that $gp_\lambda$ and $gq_\lambda$ are unique elements of $g\Gamma$ that satisfy the formula
\[
\langle
GQ_\lambda,gp_\mu
\rangle
=
\langle
GP_\lambda,gq_\mu
\rangle
=\delta_{\lambda,\mu}.
\]
In this section, we present a fermionic description of these functions and their generating functions.
\subsection{Fermionic presentation of $gq_\lambda$}\label{sec:Fermion_of_gq}
Define $\ket{\lambda}_q\in \mathcal{F}_{even}$ by
\[
\ket{\lambda}_q:=
\begin{cases}
\phi^{[\beta]}_{\lambda_1}e^{-\theta}
\phi^{[\beta]}_{\lambda_2}e^{-\theta}
\dots
\phi^{[\beta]}_{\lambda_r}e^{-\theta}
\ket{0} & (r:\mbox{even})\\
\phi^{[\beta]}_{\lambda_1}e^{-\theta}
\phi^{[\beta]}_{\lambda_2}e^{-\theta}
\dots
\phi^{[\beta]}_{\lambda_r}e^{-\theta}
\phi^{[\beta]}_{0}e^{-\theta}
\ket{0} & (r:\mbox{odd})
\end{cases}.
\]
\begin{thm}\label{thm:gq}
The dual $K$-theoretic $Q$-function $gq_\lambda(x)$ is expressed as
\[
gq_\lambda(x)=\bra{0}e^{\mathcal{H}^{[\beta]}}\ket{\lambda}_q=
\chi(\ket{\lambda}_q).
\]
\end{thm}
Let ${}_P\bra{\mu}:=(\ket{\mu}_P)^\ast$ for a strict partition $\mu$.
We prove Theorem \ref{thm:gq} by showing ${}_P\inner{\mu}{\lambda}_q=\delta_{\lambda,\mu}$ (see \eqref{eq:inner_pres}).
For strictly decreasing sequences $\lambda=(\lambda_1>\lambda_2>\dots>\lambda_r\geq 0)$ and
$\mu=(\mu_1>\mu_2>\dots>\mu_s\geq 0)$, we define new vectors $(\mu|\in \mathcal{F}^\ast_{even}$ and $|\lambda)\in \mathcal{F}_{even}$ by
\[
\mbra{\mu}=
\begin{cases}
\bra{0}
e^\theta (\Phi^{(\beta)}_{\mu_r})^\ast
\dots
e^\theta (\Phi^{(\beta)}_{\mu_2})^\ast
e^\theta (\Phi^{(\beta)}_{\mu_1})^\ast & (\mu_r>0)\\
\bra{0}
e^\theta (\phi^{(\beta)}_{0})^\ast
e^\theta (\Phi^{(\beta)}_{\mu_r})^\ast
\dots
e^\theta (\Phi^{(\beta)}_{\mu_2})^\ast
e^\theta (\Phi^{(\beta)}_{\mu_1})^\ast & (\mu_r=0)
\end{cases}
\]
and
\[
\mket{\lambda}=
\phi^{[\beta]}_{\lambda_1}e^{-\theta}
\phi^{[\beta]}_{\lambda_2}e^{-\theta}
\dots
\phi^{[\beta]}_{\lambda_r}e^{-\theta}\ket{0}.
\]
Here, $\lambda_r$ and $\mu_s$ are possibly $0$.
Then, the desired equation is equivalent to
\begin{equation}\label{eq:to_prove_1}
(\mu|\lambda)=\delta_{\lambda,\mu}.
\end{equation}
\begin{lemma}\label{lemma:hojyo_1}
We have the following facts:
\begin{itemize}
\item[(A)]
If $n>0$, then $\bra{0}e^{\theta}(\phi_0^{(\beta)})^\ast\cdot \phi^{[\beta]}_n=0$.
\item[(B)]
If $\mu\neq \emptyset$ and $n>\mu_1$, then $\mbra{\mu}\phi^{[\beta]}_n=0$.
\item[(C)]
If $\lambda\neq \emptyset$ and $m>\lambda_1$, then
$\Phi^{(\beta)}_m\mket{\lambda}=0$.
\end{itemize}
\end{lemma}
\begin{proof}
(A):
If $n=1$, we have
\begin{align*}
\bra{0}e^{\theta}(\phi_0^{(\beta)})^\ast\cdot \phi^{[\beta]}_1
&=
\bra{0}e^{\theta}\left\{
[(\phi_0^{(\beta)})^\ast,\phi^{[\beta]}_1 ]_+-\phi^{[\beta]}_1(\phi_0^{(\beta)})^\ast
\right\}\\
&
=
\beta\cdot \bra{0}e^{\theta}-\bra{0}e^{\theta}\phi^{[\beta]}_1(\phi^{(\beta)}_0)^\ast
\qquad (\mathrm{Lemma\ \ref{lemma:basic_anti_commutation}})\\
&
=
\beta\cdot \bra{0}e^{\theta}-\bra{0}(\phi^{[\beta]}_1+\beta \phi^{[\beta]}_0)(\phi^{(\beta)}_0-\beta\phi^{(\beta)}_1+\beta^2\phi^{(\beta)}_2-\cdots)^\ast e^\theta
\qquad (\mathrm{Cor.~\ref{cor:sublemma}})
\\
&=
\beta\cdot \bra{0}e^{\theta}-
\beta\cdot \bra{0}e^{\theta}\qquad (\mathrm{Eqs.~\eqref{eq:ann_rule},\eqref{eq:vs_phi_0}} )\\
&=0.
\end{align*}
For $n>1$ case, (A) is proved immediately from $[(\phi^{(\beta)}_0)^\ast,\phi^{[\beta]}_n]_+=0$ and the annihilation rule \eqref{eq:ann_rule}.
(B):
We prove (B) by induction on $s\geq 1$.
For $s=1$ and $\mu_1=0$ case, (B) is nothing but (A).
For $s=1$ and $\mu_1>0$ case, (B) follows from
\[
\begin{aligned}
\mbra{\mu}\phi^{[\beta]}_n
&=
\bra{0}e^\theta (\Phi^{(\beta)}_{\mu_1})^\ast\phi^{[\beta]}_n
\stackrel{\mathrm{Lemma\,\ref{lemma:duality}}}{=}
-\bra{0}e^\theta \phi^{[\beta]}_n(\Phi^{(\beta)}_{\mu_1})^\ast
=
-\bra{0}(\phi_n^{[\beta]}+\beta \phi^{[\beta]}_{n-1})
e^{\theta}
(\Phi^{(\beta)}_{\mu_1})^\ast=0,
\end{aligned}
\]
in which the last equality is derived from \eqref{eq:ann_rule}.
For general $s>1$, put $\mu'=(\mu_2>\dots>\mu_s\geq 0)$.
Then, we have
\[
\begin{aligned}
\mbra{\mu}\phi^{[\beta]}_n
&=
\mbra{\mu'}e^\theta (\Phi^{(\beta)}_{\mu_1})^\ast\phi^{[\beta]}_n
=
-\mbra{\mu'}e^\theta \phi^{[\beta]}_n(\Phi^{(\beta)}_{\mu_1})^\ast
=
-\mbra{\mu'}(\phi_n^{[\beta]}+\beta \phi^{[\beta]}_{n-1})
e^{\theta}
(\Phi^{(\beta)}_{\mu_1})^\ast=0,
\end{aligned}
\]
where the last equality follows from the induction hypothesis.
(C):
The claim (C) follows from the fact that $|\lambda)$ is expressed as a $\QQ(\beta)$-linear combination of vectors of the form
$\phi^{[\beta]}_{n_1}\phi^{[\beta]}_{n_2}\dots
\phi^{[\beta]}_{n_r}\ket{0}$ with $m>n_1>n_2>\dots>n_r\geq 0$
(see Corollary \ref{cor:sublemma} (2)).
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:gq}]
Let $E=( \mu|\lambda)$.
Note that $E$ is automatically $0$ when $r+s$ is odd.
(i) The case when $s=0$.
If $r=0$, we have $E=\langle 0|0\rangle=1$.
If $r=1$, we have $E=0$ since $r+s$ is odd.
If $r\geq 2$, $\lambda_1$ must be positive. Therefore, we have $E=(\emptyset|\lambda)=0$ by the annihilation rule \eqref{eq:ann_rule}.
(ii)
For $r=0$ case, we can show $E=0$ in a similar manner to (i).
(iii)
For general $r,s\geq 0$, we prove the theorem by induction on $s\geq 0$.
If $\mu_1<\lambda_1$, $E=0$ follows from Lemma \ref{lemma:hojyo_1} (B).
If $\mu_1>\lambda_1$, $E=0$ follows from Lemma \ref{lemma:hojyo_1} (C).
Assume $\mu_1=\lambda_1$.
If $\mu_1=\lambda_1=0$, we have
\[
\begin{aligned}
E
&=
\bra{0}e^{\theta}(\phi^{(\beta)}_0)^\ast \phi_0^{[\beta]}e^{-\theta}\ket{0}\\
&\stackrel{
\hbox to 0pt{\scriptsize$\mathrm{Cor.~\ref{cor:sublemma}}$}
}{=}\hspace{1em}
\left\langle(\phi^{(\beta)}_0-\beta \phi^{(\beta)}_1+\beta^2\phi^{(\beta)}_2-\cdots )^\ast (\phi_0^{[\beta]}+\beta \phi_1^{[\beta]})
\right\rangle
\stackrel{\eqref{eq:ann_rule}}{=}
\left\langle
(\phi^{(\beta)}_0)^\ast \phi_0^{[\beta]}
\right\rangle=1.
\end{aligned}
\]
If $\mu_1=\lambda_1>0$, putting
$\lambda'=(\lambda_2>\dots>\lambda_r\geq 0)$ and
$\mu'=(\mu_2>\dots>\mu_s\geq 0)$, we have
\[
\begin{aligned}
E
=
(\mu|\lambda)
&=
\mbra{\mu'}
e^\theta (\Phi^{(\beta)}_{\mu_1})^\ast
\phi^{[\beta]}_{\lambda_1} e^{-\theta}
\mket{\lambda'}\\
&
=
\mbra{\mu'}
e^\theta
\{
1-\phi^{[\beta]}_{\lambda_1} (\Phi^{(\beta)}_{\mu_1})^\ast
\}
e^{-\theta}
\mket{\lambda'}\qquad
(\mathrm{Lemma\ \ref{lemma:duality}})
\\
&=
(\mu'|\lambda')
-
\mbra{\mu'}
e^\theta
\phi^{[\beta]}_{\lambda_1} (\Phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}
\mket{\lambda'}\\
&=
( \mu'|\lambda')
-
\mbra{\mu'}
e^\theta
\phi^{[\beta]}_{\lambda_1}
e^{-\theta}
(\Phi^{(\beta)}_{\mu_1}-\beta \Phi^{(\beta)}_{\mu_1+1}+\cdots)^\ast
\mket{\lambda'}\qquad (\mathrm{Cor.~\ref{cor:sublemma}}).
\end{aligned}
\]
The second term of the last expression is $0$ by Lemma \ref{lemma:hojyo_1} (C).
Hence, we have $E=(\mu'|\lambda').$
By induction hypothesis, we conclude $E=\delta_{\mu,\lambda}$.
\end{proof}
\begin{cor}
Let $gq_n(x)=gq_{(n)}(x)$ and $gq(z)=\sum_{n=0}^\infty gq_n(x)z^n$.
Then we have
\begin{equation}\label{eq:gen_gq_one_row}
gq(z)=\left\langle
e^{\mathcal{H}^{[\beta]}}\phi^{[\beta]}(z)\phi_0
\right\rangle
=\prod_i\frac{1-x_i\overline{z}}{1-x_iz}.
\end{equation}
Equivalently, we have
\begin{equation}\label{eq:generating_gq_gen}
e^{\mathcal{H}^{[\beta]}}\phi^{[\beta]}(z)e^{-\mathcal{H}^{[\beta]}}=gq(z)\cdot \phi^{[\beta]}(z).
\end{equation}
\end{cor}
\subsection{Pfaffian formula and generating function for $gq_\lambda$}
As a corollary of Theorem \ref{thm:gq}, we derive a Pfaffian formula for $gq_\lambda(x)$.
Assume that $r$ is even.
By Wick's theorem, the generating function of $gq_\lambda$ is calculated as follows.
\begin{align}
&\left\langle
e^{\mathcal{H}^{[\beta]}}
\phi^{[\beta]}(z_1)e^{-\theta}
\phi^{[\beta]}(z_2)e^{-\theta}
\dots
\phi^{[\beta]}(z_r)e^{-\theta}
\right\rangle\nonumber
\\
&=
\prod_{i=1}^r\frac{1}{(1+\beta z_i)^{i-1}}
\left\langle
e^{\mathcal{H}^{[\beta]}}
\phi^{[\beta]}(z_1)
\phi^{[\beta]}(z_2)
\dots
\phi^{[\beta]}(z_r)
\right\rangle\nonumber
\allowdisplaybreaks
\\
&=
\prod_{i=1}^r\frac{1}{(1+\beta z_i)^{i-1}}
\cdot \mathrm{Pf}\left(
\langle
e^{\mathcal{H}^{[\beta]}}
\phi^{[\beta]}(z_i)
\phi^{[\beta]}(z_j)
\rangle
\right)_{1\leq i<j\leq r}\nonumber
\allowdisplaybreaks
\\
&=
\prod_{i=1}^r\frac{1}{(1+\beta z_i)^{i-1}}
\cdot
\mathrm{Pf}\left(
gq(z_i)gq(z_j)
\langle
\phi^{[\beta]}(z_i)
\phi^{[\beta]}(z_j)
\rangle
\right)_{1\leq i<j\leq r}\nonumber
\qquad (\mathrm{Eq.~\eqref{eq:generating_gq_gen}})
\\
&=
\mathrm{Pf}\left(
\frac{gq(z_i)}{(1+\beta z_i)^{i-1}}
\frac{gq(z_j)}{(1+\beta z_j)^{j-1}}
\cdot
\frac{z_i-z_j}{z_i\oplus z_j}
\right)_{1\leq i<j\leq r}.\label{eq:Pf_cont}
\end{align}
From \eqref{eq:IN-formula}, the Pfaffian \eqref{eq:Pf_cont} is written as
\[
\prod_{i=1}^r\frac{gq(z_i)}{(1+\beta z_i)^{i-1}}
\cdot
\prod_{1\leq i<j\leq r}
\frac{z_i-z_j}{z_i\oplus z_j}=
\prod_{i=1}^r gq(z_i)
\cdot
\prod_{1\leq i<j\leq r}
\frac{z_i\ominus z_j}{z_i\oplus z_j}.
\]
Then, we have
\begin{equation}\label{eq:gq_gen}
gq_\lambda=[z_1^{\lambda_1}\dots z_r^{\lambda_r}]
\prod_{i=1}^r gq(z_i)
\cdot
\prod_{1\leq i<j\leq r}
\frac{z_i\ominus z_j}{z_i\oplus z_j},
\end{equation}
where the rational functions expand in $\QQ(\beta)[x_1,x_2,\dots]((z_1))\cdots ((z_r))$.
This generating function was conjectured in \cite[Conjecture 5.3]{nakagawa2018universalfactrial}.
The expression \eqref{eq:gq_gen} admits a substitution $\lambda_r=0$ and $gq_{(\lambda,0)}=gq_{\lambda}$.
\subsection{Fermionic description of $gp_\lambda$}
\label{sec:fermionic_pre_of_gp}
A fermionic description of $gp_\lambda$ is significantly different from that of $gq_\lambda$.
In terms of $\beta$-deformed fermions, the difference is due to the fact that $\bra{0}\Phi_n^{[\beta]}\neq 0$ while $\bra{0}\phi_n^{[\beta]}=0$ for positive $n$.
By analogy, one may expect the symmetric function \begin{equation}\label{eq:gp_prime}
gp'_\lambda(x)=
\begin{cases}
\langle
e^{\mathcal{H}^{[\beta]}}
\Phi^{[\beta]}_{\lambda_1}e^{-\theta}
\Phi^{[\beta]}_{\lambda_2}e^{-\theta}
\dots
\Phi^{[\beta]}_{\lambda_r}e^{-\theta}
\rangle & (r:\mbox{even})\\
\langle
e^{\mathcal{H}^{[\beta]}}
\Phi^{[\beta]}_{\lambda_1}e^{-\theta}
\Phi^{[\beta]}_{\lambda_2}e^{-\theta}
\dots
\Phi^{[\beta]}_{\lambda_r}e^{-\theta}
\phi^{[\beta]}_{0}e^{-\theta}
\rangle & (r:\mbox{odd})
\end{cases}
\end{equation}
to be a possible candidate for $gp_\lambda(x)$.
However, we unfortunately have
\begin{equation}\label{eq:Fal}
\langle
GQ_{\lambda},gp'_\mu
\rangle\neq \delta_{\lambda,\mu},
\end{equation}
which contradicts the expectation.
If fact, we can easily derive
\[
\langle
GQ_\emptyset,gp_n'
\rangle
=
\langle
e^\theta (\phi_0^{(\beta)})^\ast
\Phi^{[\beta]}_ne^{-\theta}
\rangle
=
\frac{(-\beta)^n}{2^{n+1}}
\langle
e^\theta (\phi_0^{(\beta)})^\ast
\phi^{[\beta]}_0e^{-\theta}
\rangle
=
\frac{(-\beta)^n}{2^{n+1}}
\neq 0.
\]
To address the issue, we define the new vector $\ket{\lambda}_p\in \mathcal{F}$ by
\[
\ket{\lambda}_p
=
(\Phi^{[\beta]}_{\lambda_1}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_1})e^{-\theta}
(\Phi^{[\beta]}_{\lambda_2}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_2})e^{-\theta}
\cdots
(\Phi^{[\beta]}_{\lambda_r}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_r})(\phi_0+1)\ket{0}
\]
for a strict partition $\lambda=(\lambda_1>\dots>\lambda_r>0)$.
We should note that $\ket{\lambda}_p$ is not an element of $\mathcal{F}_{even}$.
\begin{thm}\label{thm:fermion_of_gp}
For a strict partition $\lambda=(\lambda_1>\dots>\lambda_r>0)$, we have
\begin{equation}\label{eq:def_of_gp}
\begin{aligned}
gp_\lambda(x)=\bra{0}e^{\mathcal{H}^{[\beta]}}\ket{\lambda}_p.
\end{aligned}
\end{equation}
\end{thm}
Recall that, if $\bra{v}\in \mathcal{F}^\ast_{even}$ and $\ket{w}\in \mathcal{F}_{odd}$, the expectation value $\inner{v}{w}$ vanishes automatically.
Let $\ket{\lambda}_p=\ket{\lambda_{even}}_p+\ket{\lambda_{odd}}_p$ be the unique decomposition satisfying $\ket{\lambda_{even}}_p\in \mathcal{F}_{even}$ and $\ket{\lambda_{odd}}_p\in \mathcal{F}_{odd}$.
Since $\bra{0}e^{\mathcal{H}^{[\beta]}}\ket{\lambda_{odd}}_p$ vanishes, \eqref{eq:def_of_gp} is equivalent to
$gp_\lambda(x)=\bra{0}e^{\mathcal{H}^{[\beta]}}\ket{\lambda_{even}}_p$.
On the other hand, we have ${}_{Q}\langle
\mu|\lambda
\rangle_p={}_{Q}\langle
\mu|\lambda_{even}
\rangle_p$ by the same reason.
Finally, we know that Theorem \ref{thm:fermion_of_gp} is equivalent to
\begin{equation}\label{eq:to_prove}
{}_{Q}\langle
\mu|\lambda
\rangle_p
=\delta_{\lambda,\mu},\qquad (\forall \mu).
\end{equation}
For a string of decreasing integers
$\mu=(\mu_1>\dots>\mu_s\geq 0)$ (possibly, $\mu_s=0$),
we define the new vectors $\bbra{\mu}$ and $\kket{\lambda}$ by
\[
\begin{aligned}
&
\bbra{\mu}=\bra{0}(\phi_0+1)
e^\theta (\phi^{(\beta)}_{\mu_s})^\ast
\dots
e^\theta (\phi^{(\beta)}_{\mu_2})^\ast
e^\theta (\phi^{(\beta)}_{\mu_1})^\ast,\\
&
\kket{\lambda}=
(\Phi^{[\beta]}_{\lambda_1}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_1})e^{-\theta}
(\Phi^{[\beta]}_{\lambda_2}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_2})e^{-\theta}
\cdots
(\Phi^{[\beta]}_{\lambda_r}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_r})\ket{0}.
\end{aligned}
\]
Since
$
\left\langle (\phi_0+1)X\right\rangle
=
\left\langle X(\phi_0+1)\right\rangle
$
for any $X\in \mathcal{A}$, \eqref{eq:to_prove} is equivalent to
\begin{equation}\label{eq:mokuhyou}
\iinner{\mu}{\lambda}=\delta_{\lambda,\mu}.
\end{equation}
\begin{lemma}\label{lemma:hojyo2}
We have the following facts:
\begin{enumerate}
\item[(A).] If $\mu=\emptyset$ and $n\geq 0$, then $\bbra{\emptyset}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})=0$.
\item[(B).] If $\mu\neq \emptyset$ and $n> \mu_1$, then $\bbra{\mu}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})=0$.
\item[(C).] If $\lambda\neq \emptyset$ and $m>\lambda_1$, then $(\phi^{(\beta)}_m)^\ast \kket{\lambda}=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
(A):
It is straightforward to derive
\[
\begin{aligned}
\bbra{\emptyset}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})
&=
\bra{0}(\phi_0+1)(\Phi_n^{[\beta]}-\tfrac{1}{2}(-\tfrac{\beta}{2})^n)\\
&=
\tfrac{1}{2}(-\tfrac{\beta}{2})^n
\bra{0}(\phi_0+1)(\phi_0-1)\qquad (\mathrm{Eq.~\eqref{eq:Phi_expand}})\\
&=
\tfrac{1}{2}(-\tfrac{\beta}{2})^n
\bra{0}(\phi_0^2-1)=0.
\end{aligned}
\]
(B):
We prove (B) by induction on $s\geq 1$.
For any $s\geq 1$, we have
\begin{align*}
\bbra{\mu}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})
&=
\bbra{\mu'}e^{\theta}(\phi^{(\beta)}_{\mu_1})^\ast(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})\\
&=
\bbra{\mu'}e^{\theta}(-\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})(\phi^{(\beta)}_{\mu_1})^\ast\\
&=
\bbra{\mu'}(-\Phi^{[\beta]}_n-\beta\Phi^{[\beta]}_{n-1} -\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})e^{\theta}(\phi^{(\beta)}_{\mu_1})^\ast \qquad (\mathrm{Cor.~\ref{cor:sublemma}~(1)}).
\end{align*}
As
\[
-\Phi^{[\beta]}_n-\beta\Phi^{[\beta]}_{n-1} -\tfrac{1}{2}(-\tfrac{\beta}{2})^{n}=
-(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^n)
-\beta (\Phi^{[\beta]}_{n-1}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n-1}),
\]
the term $\bbra{\mu}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})
$ is a linear combination of
$
\bbra{\mu'}(\Phi^{[\beta]}_n-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n})
$
and
$
\bbra{\mu'}(\Phi^{[\beta]}_{n-1}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{n-1})
$.
For $s=1$, the desired equation reduces to (A).
For general $s>1$, (B) is proved by induction hypothesis.
(C):
This claim is proved in a similar way to the proof of Lemma \ref{lemma:hojyo_1} (C).
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:fermion_of_gp}]
Let $E=\iinner{\mu}{\lambda}$.
(i) The case when $s=0$.
If $r=0$, we have $E=\bra{0}(\phi_0+1)\ket{0}=1$.
If $r>0$, we have $E=0$ from Lemma \ref{lemma:hojyo2} (A).
(ii) For $r=0$ case, we have $E=0$ from Lemma \ref{lemma:hojyo2} (C).
(iii) For general $s,r>0$, we prove \eqref{eq:mokuhyou} by induction on $s$.
If $\mu_1<\lambda_1$, then $E=0$ by Lemma \ref{lemma:hojyo2} (B).
If $\mu_1>\lambda_1$, then $E=0$ by Lemma \ref{lemma:hojyo2} (C).
For the case when $\mu_1=\lambda_1(>0)$, we have
\[
\begin{aligned}
E
&=
\bbra{\mu'}e^{\theta}(\phi^{(\beta)}_{\mu_1})^\ast
(\Phi^{[\beta]}_{\lambda_1}-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_1})e^{-\theta}\kket{\lambda'}\\
&=
\bbra{\mu'}e^{\theta}
\left[
(\phi^{(\beta)}_{\mu_1})^\ast,
\Phi^{[\beta]}_{\lambda_1}
\right]_+
e^{-\theta}\kket{\lambda'}
-
\bbra{\mu'}e^{\theta}
\Phi^{[\beta]}_{\lambda_1}
(\phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}\kket{\lambda'}
-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_1}
\bbra{\mu'}e^{\theta}
(\phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}\kket{\lambda'}\\
&
=
\iinner{\mu'}{\lambda'}
-
\bbra{\mu'}e^{\theta}
\Phi^{[\beta]}_{\lambda_1}
(\phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}\kket{\lambda'}
-\tfrac{1}{2}(-\tfrac{\beta}{2})^{\lambda_1}
\bbra{\mu'}e^{\theta}
(\phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}\kket{\lambda'}
\qquad
(\mathrm{Lemma\,\ref{lemma:duality}}),
\end{aligned}
\]
in which the last two terms equal to $0$ because
\[
(\phi^{(\beta)}_{\mu_1})^\ast
e^{-\theta}\kket{\lambda'}=
e^{-\theta}
(\phi^{(\beta)}_{\mu_1}-\beta \phi^{(\beta)}_{\mu_1+1}+\cdots)^\ast
\kket{\lambda'}
\stackrel{\mathrm{Lemma\,\ref{lemma:hojyo2}\,(C)}}{=}
0.
\]
By induction hypothesis, we have $E=
\iinner{\mu'}{\lambda'}
=\delta_{\lambda,\mu}$.
\end{proof}
\subsection{Generating function of $gp_\lambda$}
The generating function of $gp_\lambda$ is calculated as
\[
\begin{aligned}
&
\left\langle
e^{\mathcal{H}^{[\beta]}}
\left(
\Phi^{[\beta]}(z_1)-\frac{1}{2+\beta z_1}
\right)
e^{-\theta}
\cdots
e^{-\theta}
\left(
\Phi^{[\beta]}(z_r)-\frac{1}{2+\beta z_r}
\right)
(\phi_0+1)
\right\rangle\\
&=
\prod_{i=1}^r\frac{1}{2+\beta z_i}
\left\langle
e^{\mathcal{H}^{[\beta]}}
\left(
\phi^{[\beta]}(z_1)-1
\right)
e^{-\theta}
\cdots
e^{-\theta}
\left(
\phi^{[\beta]}(z_r)-1
\right)
(\phi_0+1)
\right\rangle
\allowdisplaybreaks
\\
&=
\prod_{i=1}^r\frac{1}{2+\beta z_i}
\sum_{
\substack{
0\leq a\leq r\\
i_1<i_2<\dots<i_a
}
}
(-1)^{r-a}
\big\langle
e^{\mathcal{H}^{[\beta]}}
e^{-(i_1-1)\theta}
\phi^{[\beta]}(z_{i_1})
e^{-(i_2-i_1)\theta}
\phi^{[\beta]}(z_{i_2})
\\
&\hspace{20em}
\cdots
e^{-(i_a-i_{a-1})\theta}
\phi^{[\beta]}(z_{i_a})
(\phi_0+1)
\big\rangle
\allowdisplaybreaks
\\
&=\prod_{i=1}^r\frac{1}{2+\beta z_i}
\sum_{
\substack{
0\leq a\leq r\\
i_1<i_2<\dots<i_a
}
}
(-1)^{r-a}
\prod_{\kappa = 1}^a
\frac{1}{(1+\beta z_{i_\kappa})^{i_\kappa-1}}
\big\langle
e^{\mathcal{H}^{[\beta]}}
\phi^{[\beta]}(z_{i_1})
\cdots
\phi^{[\beta]}(z_{i_a})
(\phi_0+1)
\big\rangle
\allowdisplaybreaks
\\
&
=\prod_{i=1}^r\frac{1}{2+\beta z_i}
\sum_{
\substack{
0\leq a\leq r\\
i_1<i_2<\dots<i_a
}
}
(-1)^{r-a}
\prod_{\kappa = 1}^a
\frac{gq(z_{i_\kappa})}{(1+\beta z_{i_\kappa})^{i_\kappa-1}}
\prod_{i_\kappa<i_\theta}\frac{z_{i_\kappa}-z_{i_\theta}}{z_{i_\kappa}\oplus z_{i_\theta}}\\
&=
\prod_{i=1}^r\frac{1}{2+\beta z_i}
\sum_{
\substack{
0\leq a\leq r\\
i_1<i_2<\dots<i_a
}
}
(-1)^{r-a}
\prod_{\kappa = 1}^a
\frac{gq(z_{i_\kappa})}{(1+\beta z_{i_\kappa})^{i_\kappa-\kappa}}
\prod_{i_\kappa<i_\theta}\frac{z_{i_\kappa}\ominus z_{i_\theta}}{z_{i_\kappa}\oplus z_{i_\theta}}.
\end{aligned}
\]
Then we have
\begin{equation}\label{eq:gen_gp}
gp_\lambda=[z_1^{\lambda_1}\dots z_r^{\lambda_r}]
\prod_{i=1}^r\frac{1}{2+\beta z_i}
\sum_{
\substack{
0\leq a\leq r\\
i_1<i_2<\dots<i_a
}
}
(-1)^{r-a}
\prod_{\kappa = 1}^a
\frac{gq(z_{i_\kappa})}{(1+\beta z_{i_\kappa})^{i_\kappa-\kappa}}
\prod_{1\leq \kappa<\theta\leq a}\frac{z_{i_\kappa}\ominus z_{i_\theta}}{z_{i_\kappa}\oplus z_{i_\theta}}.
\end{equation}
If we substitute $\lambda_{r+1}=0$ formally, we obtain $gp_{(\lambda,0)}=0$.
For the case of $r=1$, we have the generating function of $gp_n=gp_{(n)}$, where $\lambda=(n)$ is a one-row partition:
\begin{equation}\label{eq:gp_one_row}
\begin{aligned}
\sum_{n=1}^\infty gp_nz^n
=\frac{1}{2+\beta z}
\left(gq(z)-1\right).
\end{aligned}
\end{equation}
Hence, we have
\[
gp_n=\frac{1}{2}\left(
gq_n-\frac{\beta}{2}gq_{n-1}+\frac{\beta^2}{4}gq_{n-2}-\cdots+\left(-\frac{\beta}{2}\right)^{n-1}gq_1
\right).
\]
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,993
|
require_relative './spec_helper'
describe 'metatest' do
before { I18n.locale = :en }
let(:result) { framework.test compilation, examples }
let(:options) { { show_initial_board: false, check_head_position: true } }
let(:framework) do
Mumukit::Metatest::Framework.new checker: Gobstones::Checker.new(options),
runner: Gobstones::MultipleExecutionsRunner.new
end
let(:dummy_view_board) do
{ head: { x: 0, y: 0 }, width: 3, height: 3, table: {
json: [[{}, {}, {}], [{}, {}, {}], [{}, {}, {}]]
} }
end
let(:dummy_gbb) do
'GBB/1.0\r\nsize 1 1\r\nhead 0 0\r\n'
end
let(:exit_status) { { type: "Number", value: 29 } }
let(:compilation_boom) do
[
{
status: "runtime_error",
result: {
initialBoard: dummy_view_board,
extraBoard: dummy_view_board,
finalBoardError: {
on: {
range: {
start: {
row: 1,
column: 2
}
}
},
message: "Blah",
reason: {
code: "no_stones"
}
}
}
}
]
end
before { allow(Mumukit).to receive(:runner_url) { 'http://gobstones.runners.mumuki.io' } }
def board_with_stones(headX, headY, cell10 = {})
{
head: { x: headX, y: headY },
width: 3,
height: 3,
table: {
gbb: dummy_gbb,
json: [
[{}, {}, {}],
[{}, {}, {}],
[{ black: 1, green: 1 }, cell10, {}]
]
},
returnValue: exit_status
}
end
def compilation_board(expected_board = dummy_view_board)
[
{
status: "passed",
result: {
extraBoard: expected_board,
initialBoard: dummy_view_board,
finalBoard: board_with_stones(0, 1)
}
}
]
end
describe 'final_board postcondition' do
let(:examples) {
[{
id: 0,
postconditions: {
final_board: dummy_gbb
}
}]
}
context 'when the program returns a final board' do
context 'when passes with check_head_position=true' do
let(:compilation) {
compilation_board board_with_stones 0, 1
}
it { expect(result[0][0]).to include :passed }
end
context 'when passes with check_head_position=false' do
let(:compilation) {
compilation_board board_with_stones 5, 5
}
let(:options) { { show_initial_board: false, check_head_position: false } }
it { expect(result[0][0]).to include :passed }
end
context 'when fails by different boards (header)' do
let(:compilation) {
compilation_board board_with_stones 2, 2
}
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "head doesn't match" }
end
context 'when fails by different boards (stones)' do
let(:compilation) {
compilation_board board_with_stones 0, 1, { blue: 9 }
}
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "different board was obtained" }
end
end
context 'when the program does boom' do
let(:compilation) { compilation_boom }
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "The program did BOOM." }
end
end
describe 'error postcondition' do
let(:examples) {
[{
id: 0,
postconditions: {
error: "no_stones"
}
}]
}
context 'when the program returns a final board' do
let(:compilation) { compilation_board }
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "The program was expected to BOOM but a final board was obtained." }
end
context 'when the program does boom' do
let(:compilation) { compilation_boom }
context 'with the same reason as expected' do
it { expect(result[0][0]).to include :passed }
end
context 'with another reason' do
let(:examples) {
[{
id: 0,
postconditions: {
error: "out_of_board"
}
}]
}
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "The program was expected to fail by <strong>out of board</strong>, but it failed by another reason." }
end
end
end
describe 'return postcondition' do
let(:examples) {
[{
id: 0,
postconditions: {
return: 29
}
}]
}
context 'when the program returns a final board' do
let(:compilation) { compilation_board }
context 'when passes with equal value' do
it { expect(result[0][0]).to include :passed }
end
context 'when fails by no return value' do
let(:exit_status) { nil }
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "<strong>29</strong> was expected but no value was obtained." }
end
context 'when fails by different values' do
let(:examples) {
[{
id: 0,
postconditions: {
return: 11
}
}]
}
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "<strong>11</strong> was expected but <strong>29</strong> was obtained." }
end
end
context 'when the program does boom' do
let(:compilation) { compilation_boom }
context 'when fails because the program did boom' do
it { expect(result[0][0]).to include :failed }
it { expect(result[0][0][3][:message]).to include "The program did BOOM." }
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,661
|
The University of North Texas (UNT) is a public research university in Denton, Texas. It was founded as a nonsectarian, coeducational, private teachers college in 1890 and was formally adopted by the state 11 years later. UNT is a member of the University of North Texas System, which includes additional universities in Dallas and Fort Worth. UNT also has a location in Frisco.
The university consists of 14 colleges and schools, an early admissions math and science academy for exceptional high-school-age students from across the state, the Texas Academy of Mathematics and Science, and a library system that comprises the university core. It is classified among "R1: Doctoral Universities – Very high research activity". According to the National Science Foundation, UNT spent $78.4 million on research and development in 2019.
Campus
The main campus is located in Denton, TX, part of the largest metropolitan area in Texas, Dallas-Fort Worth. The campus is 1,200 acres including the main campus north of I-35E, the Eagle point athletic complex south of I-35E, and Discovery Park; a research campus located between state highways 77 and 380.
On behalf of the state, the university, in its civic advocacy for the state, prevailed with three new-location, capital-intensive expansions over the last years.
The university acquired in 1975 and subsequently developed a medical school in Fort Worth
The university created a campus in South Dallas in 2000
The university laid the groundwork for establishing the first public law school in the region.
In 1981, the university spun off its new medical school as its own independent institution under the UNT Board of Regents. In 2009, the University of North Texas at Dallas became its own independent institution. That same year, the Texas legislature approved the creation of University of North Texas at Dallas College of Law, opening in 2014 in Downtown Dallas as part of UNT Dallas. UNT and its three sister institutions are governed by the University of North Texas System, a system established in 1980 by the board of regents and legislatively recognized in 2003 by the 78th Texas Legislature.
In 2004, UNT opened UNT Discovery Park – – in Denton, north of the main campus with technology incubator facilities dedicated to science and engineering. In 2011, the College of Visual Arts and Design launched the Design Research Center in downtown Dallas in the Design District.
In 2016, UNT opened a location in Hall Park in Frisco in Collin County. In 2018, UNT opened Inspire Park. UNT teaches nearly 2,000 students in Collin County each semester at Hall Park, Inspire Park and the Collin Higher Education Center in McKinney. In 2020, the Texas Higher Education Coordinating Board approved UNT building a branch campus to provide upper-level and graduate courses on 100 acres donated by the city of Frisco. Classes are expected to begin in Spring 2023.
Official designations
In 1985 the Governor's Select Committee on Higher Education recommended that North Texas be designated an "emerging national research university." Nine years earlier, in 1976, the Carnegie Foundation designated North Texas as a "Class 1 Doctorate-Granting Institution." Four decades later, in February 2016, Carnegie elevated North Texas to its top category – Doctorate-Granting Institutions with "highest research activity." At that time, Carnegie had 115 universities listed at that level.
In 1988, U.S. Secretary of Education William Bennett cited UNT for its innovative approach to undergraduate education in the Classic Learning Core, an integrated liberal arts curriculum similar to those usually found only in small, private colleges. In 1992, UNT was elected to full membership in the Association of Public and Land-grant Universities. And, in 2011, the Texas Higher Education Coordinating Board included UNT as one of eight Emerging Research Institutions in its accountability system.
In 2020, UNT achieved designation from the Department of Education as a Title III & Title V Minority-Serving Institution (MSI) and as a Hispanic Serving Institution (HSI).
Enrollment
Certified enrollment as of the fall of 2021 was 42,372, the fifth largest in the state. For the 2019–20 academic year, the university awarded 10,270 degrees. UNT awarded 459 Ph.D. degrees from fiscal years 2009 to 2011.
Academics
Of the 14 colleges and schools, UNT offers 109 bachelor's degree programs, and 94 master's and 36 doctorate degree programs.
The student-faculty ratio at UNT is 23:1, and 28.8 percent of its classes consist of fewer than 20 students. The most popular majors include business, management, marketing, communication, journalism, English, multi/interdisciplinary studies, and visual and performing arts.
UNT has been accredited by the Southern Association of Colleges and Schools since 1924 and is among the twenty-seven universities in Texas accredited at Level VI, the highest level. As of 2020, the university was home to 22 research centers and institutes. In 2007, the university launched four Institutes of Research Excellence: (i) Advanced Environmental Research Institute, (ii) Advanced Materials and Manufacturing Processes Institute, (iii) BioDiscovery Institute, and (iv) Jim McNatt Institute for Logistics Research. In 2019, UNT launched the Center for Agile and Adaptive Additive Manufacturing.
College of Liberal Arts and Social Sciences
The College of Liberal Arts and Social Sciences houses 22 academic departments and programs and five public services (including a psychology clinic and a speech and hearing clinic), and eight student services (of which seven are labs).
College of Science
UNT has been offering Bachelor of Science degrees for years, Master of Science degrees (in biology, mathematics, chemistry, and economics) for years, and Doctor of Philosophy degrees in several scientific disciplines—including chemistry, biology, and physics—for years. UNT is a sponsoring institution member (Ph.D.-granting) of Oak Ridge Associated Universities (ORAU), a consortium of 105 major research universities that leverage scientific research through partnerships with national laboratories, government agencies, and private industry. It has been a member of the consortium since 1954.
G. Brint Ryan College of Business
The College of Business is host to five academic departments: (i) Accounting, (ii) Finance, Insurance, Real Estate and Law, (iii) Information Technology and Decision Sciences, (iv) Marketing, Logistics, and Operations Management (v) Management. It offers seven undergraduate programs, fourteen M.B.A. and master of science programs, and six Ph.D. programs. In Fall 2011, the college moved into a new state-of-the-art Gold LEED certified $70 million facility named the Business Leadership Building. The college is accredited in both business and accounting by the Association to Advance Collegiate Schools of Business—accreditation for the former stretches back years (1961) and the latter, years (1987).
The college of business was renamed in 2019 to the G. Brint Ryan College of Business following a gift from alumnus G. Brint Ryan, alumnus and UNT System Board of Regents Chairman. The $30 million gift awarded by Ryan and his wife Amanda will create at least six endowed chairs and provide funding for academic program initiatives over seven years. Among the areas of focus are taxation and tax research, entrepreneurship, finance, logistics, information technology, cybersecurity and behavioral accounting.
Undergraduate business education
In 2018, 5,093 students were enrolled as business majors at the undergraduate level.
Graduate business education
In 2018, 691 students were working on graduate degrees. The college is host to two research centers (ii) the Institute of Petroleum Accounting and (iii) the Murphy Center for Entrepreneurship.
U.S. News & World Report's "2021 Best Online Programs" ranked UNT 31st in the nation among the Best Online Graduate Business Programs.
College of Education
The College of Education is a legacy of the university's founding as a teachers college years ago. The college is organized as four departments and one center: (i) Counseling and Higher Education, (ii) Educational Psychology, (iii) Kinesiology, Health Promotion and Recreation, (iv) Teacher Education and Administration, and (v) The Kristin Farmer Autism Center. The college offers 12 bachelor's degrees, 19 master's degrees and 15 doctoral concentrations. As of the 2010–2011 school year, the college certified over 1,147 teachers, the second largest number in the state by a university. In 1979, the Texas Higher Education Coordinating Board approved renaming the "School of Education" to the "College of Education." At that time, the college was the largest in Texas and the Southwest, the largest doctoral program in the state, and the twenty-fifth largest producer of teacher certificates in the United States. Its prior name, "School of Education," dates back to 1946, when the teachers college outgrew itself and reorganized as six schools and colleges.
College of Engineering
The College of Engineering, founded in 2003, inherited longstanding programs (i) Computer Science, (ii) Information Technology, and (iii) Engineering Technology—with majors in (a) Construction Engineering Technology, (b) Electronics Engineering Technology, (c) Manufacturing Engineering Technology, (d) Mechanical Engineering Technology, and (e) Nuclear Engineering Technology—and launched (iv) Computer Engineering, (v) Electrical Engineering, (vi) Materials Science and Engineering, (vii) Mechanical Engineering, and (viii) Biomedical Engineering (2014). The college is host to three research centers, one of which being the Net-Centric Software and Systems Center (launched February 24, 2009), a research consortium hosted by UNT and organized as a National Science Foundation Industry-University Cooperative Research Center (NSF I/UCRC). It is primarily funded by industry members (which consist of 16 corporations) and universities (which consist of 5). The focus is developing computing models for the future—models that go beyond applications with preordained fixed capabilities—models capable of services that are dynamically created, verified, and validated in the field and .
College of Information
The College of Information was created in October 2008 by consolidating two existing academic units: Learning Technologies (formerly within the College of Education) and the School of Library and Information Sciences. The School of Library and Information Services was created in 1970 as an outgrowth of its former structure as the Department of Library Services. The college sponsors three research centers, one being The Texas Center for Digital Knowledge.
College of Merchandising, Hospitality and Tourism
The College of Merchandising, Hospitality and Tourism houses the largest merchandising program in the nation and one of the largest hospitality and tourism management programs. The college offers bachelor's degrees with majors in digital retailing, home furnishings merchandising, hospitality management, event design & experience management, and merchandising, and master's degrees in hospitality management, international sustainable tourism and merchandising. It has the nation's first bachelor's in digital retailing and master's in international sustainable tourism. The college was formerly known as the School of Merchandising and Hospitality Management.
College of Music
The College of Music is a comprehensive institution of international rank. Its heritage dates back years, when North Texas was founded. The college has the largest enrollment of any music institution accredited by the National Association of Schools of Music. It has been among the largest music institutions of higher learning in North America since the 1940s. The music library, founded in 1941, has one of the largest music collections in the United States, with over 300,000 volumes of books, periodicals, scores, and approximately 900,000 sound recordings. North Texas was first in the world to offer a degree in jazz studies. U.S. News & World Report ranked the jazz studies program as the best in the country every year from 1994, when it began ranking graduate jazz programs, to 1997, when it retired the category. The One O'Clock Lab Band has been nominated for 7 Grammy Awards.
College of Health and Public Service
Previously called the College of Public Affairs and Community Service (PACS) and before that the College of Community Service, the college adopted its current name in Fall 2017. The college is organized in seven departments: Audiology and Speech-Language Pathology; Behavior Analysis; Criminal Justice; Emergency Management and Disaster Science; Public Administration; Rehabilitation and Health Services; and Social Work.
The department of public administration is home of the nation's first comprehensive degree program in emergency and disaster management that launched in 1983. The degree incorporates interdisciplinary curricula from other colleges that include applied philosophy and environmental ethics. The degree is tailored for both management practitioners and researchers and is collaborative with the Federal Emergency Management Agency Region VI—based in Denton—which oversees Arkansas, Louisiana, New Mexico, Oklahoma, and Texas. Denton became home to FEMA when its predecessor, the Office of Civil Defense and Mobilization, constructed the nation's first Federal underground defense center in 1959.
The college is host to five research institutes, one being the Turkish Institute for Police Studies (TIPS). The institute has, since its founding in 1999, been based at North Texas. Its institution is a collaboration between the Turkish National Police (TNP) and U.S. universities in areas of terrorism, organized crime, narcotics, administration, intelligence, and investigation.
UNT and Texas Women's University began a joint Master of Social Work (M.S.W.) program in 2017.
College of Visual Arts and Design
The College of Visual Arts and Design has the 10th largest enrollment of any art and design school accredited by the National Association of Schools of Art and Design, and the second largest of any that awards doctorates. The college name changes reflect the curricular expansion of programs. In 1992, what then had been the "Department of Art" within the College of Arts and Sciences, became "School of Visual Arts;" and in 2007, it became the "College of Visual Arts and Design." Art classes began at UNT in 1894, four years after its founding. Master's degrees were initiated in the 1930s and the first Master of Science degree in art was awarded in 1937. Since 1972, the college has served as curator and custodian of the Texas Fashion Collection that was started by Stanley Marcus in 1938.
Honors College
The Honors College offers academic enrichments, including honors seminars and exclusive classes only for high-achieving undergraduates. There is no age limit. Its classes can either supplement or substitute core coursework. Its objective is to challenge exceptional students at higher levels and to promote leadership. The college is an autonomous collegiate unit on equal footing with the other collegiate units. Academically, it offers no degrees; but its courses are integrated with the baccalaureate programs of the other ten constituent colleges and the journalism school. Graduates are awarded a special medallion. The college offers many perks, including scholarships, exchange programs, and exclusive housing—Honors Hall.
The college began as an honors program years ago (Fall 1971). Its initial enrollment of 50, back then, quickly grew to 400. But the program lost support under a system of borrowing faculty members. The Honors Program was reconstituted in 1994 and was elevated as a college in 2005.
Mayborn School of Journalism
Curricular journalism at North Texas dates back to 1945. As a department, Journalism eventually became part of the College of Arts and Sciences. The Graduate Division of Journalism began in the fall of 1970 under the direction of Reginald Conway Westmorland. In 1999, twelve years after the death of Frank W. Mayborn, its graduate program was renamed the Frank W. Mayborn Graduate Institute of Journalism. On September 1, 2009, the entire program was elevated as its own collegiate unit and named the Frank W. and Sue Mayborn School of Journalism. Eight Pulitzer Prizes have been won by five of its alumni, among whom are Bill Moyers and Howard Swindle. Other notable alumni include Samir Husni and Cragg Hines. Since 1969, the news-editorial sequence has been accredited by the Accrediting Council on Education in Journalism and Mass Communications; and since 1986, the entire program has been accredited. The school is in its year as founding host of the annual Mayborn Literary Nonfiction Conference.
Virginia Ellison (née Virginia Jones Paty; 1920–2009)—a North Texas alumna (BA, English, '41) who also taught English and journalism, sponsored the Student Press Club, and served as director of publicity at North Texas from 1942 to 1944—won a Pulitzer Traveling Fellowship in 1945, the year she earned a degree from the Columbia University Graduate School of Journalism.
Texas Academy of Mathematics and Science
TAMS is a two-year residential early college entrance program that has, since 1987, served exceptionally qualified Texas students who otherwise would be attending high school as juniors and seniors. It was the first of its kind in the nation and, , the only in the state and one of five in the nation.
Toulouse Graduate School
The Toulouse Graduate School, founded years ago, is the academic custodian and administrator of all graduate programs offered by nine colleges and one school. It maintains records, administers admissions, and serves various roles in recruiting. It was renamed in 1990 in honor of Robert Bartell Toulouse, EdD (1918–2017), who joined in 1948 as a professor in the College of Education, then served dean of the Graduate School from 1954 to 1982. Toulouse, before retiring as professor emeritus, had served other roles at the university, including provost and vice president of academic affairs from 1982 to 1985.
Libraries
UNT Libraries are made up of four public service points and two remote storage facilities. Willis Library is the main library on campus, housing the business, economics, education, humanities and social sciences collections along with microforms and special areas such as the Music Library, Government Documents, the Digital Library Division, Archives, and the Rare Book and Texana collections. The Media Library in Chilton Hall houses a large collection of audiovisual materials, including films, audiobooks, and video games (see Game Design, above). Video recording equipment and gaming consoles are available for checkout. The Sycamore Library houses the government documents, law, political science, geography and business collections. It also houses the Collaboration and Learning Commons, a place to study in groups, create multi-media projects, and record presentations. The Discovery Park Library supports the College of Engineering and the College of Information, Library Science, and Technologies. It covers multiple areas of engineering, library and information science, and learning technology.
The Intensive English Language Institute (IELI)
Established in 1977, IELI is the largest intensive English program (IEP) in North Texas, serving international students who wish to learn academic English in preparation for university studies in the United States. IELI is a constituent of UNT International Affairs, an interdisciplinary unit and exponent of globalization in higher education that provides leadership and support of international teaching, research, and study-abroad initiatives. , IELI has been located in Marquis Hall on the UNT Denton campus.
Student life
Residential life
All freshmen are required to live on campus to satisfy a residency requirement. 15.5% of students, or 5,620, live in on-campus residence halls. In addition, 37.3%, or 13,494, live within the city of Denton while 4,021, or 11.1% live outside of the city of Denton but within Denton County and 36.1% or 13,043 students live outside of Denton County.
Student residence halls
There are 15 residence halls on the Denton campus. UNT also offers the Residents Engaged in Academic Living (REAL) Communities program. The REAL communities offer students the ability to live with other residents in their major, and allow them to interact with each other and participate in programs that are geared toward their major or discipline. On August 22, 2011, -year-old Maple Street Hall became the first all-vegan ("Mean Greens") college cafeteria in the country. The given 14 residence hall at the University of North Texas are : Bruce Hall, Clark Hall, College Inn, Crumley Hall, Joe Greene Hall, Honors Hall, Kerr Hall, Legends Hall, Maple Hall, Mozart Square, Rawlins Hall, Santa Fe Square, Traditions Hall, Victory Hall, West Hall.
Pohl Recreation Center
The Pohl Recreation Center is the student recreation center located on the campus of the University of North Texas.
Social Greek organizations
The social Greek community is made-up of four councils that oversee 42 fraternities and sororities. Four percent of undergraduate students of both genders are members of social fraternities and sororities.
Traditions
Primary colors
North Texas adopted Green and White as its official colors during the 1902–1903 school year.
Mascot
UNT's mascot, the American eagle, was adopted on February 1, 1922, as a result of a student-faculty council debate and ensuing student election.
The eagle has had three nicknames, beginning with "Scrappy" in 1950. The human costumed eagle character, launched in 1963, carried the name "Scrappy" until 1974—during the throes of the Vietnam War—when students adopted the name "Eppy" because it sounded less warlike. Since then, the name has switched back and forth, from Eppy to Scrappy; but for the last years, the name "Scrappy" has endured.
Nickname for intercollegiate athletics
The name "Mean Green," now in its year, was adopted by fans and media in 1966 for a North Texas football defensive squad that finished the season second in the nation against the rush. That season, Joe Greene, then a sophomore at North Texas, played left defensive tackle on the football team and competed in track and field (shot put). The nickname "Mean Joe Greene" caught-on during his first year with the Pittsburgh Steelers in 1969 when Pittsburgh fans wrongly assumed that "Mean Green" was derived from a nickname Joe Greene had inherited while at North Texas. The North Texas athletic department, media, and fans loved the novelty of the national use of its nickname, and its association with Joe Greene's surname and university's official school color. By 1968, "Mean Green" was branded on the backs of shirts, buttons, bumper stickers, and the cover of the North Texas football brochure.
Fight song
Francis Edwin Stroup, EdD (1909–2010), emerged in 1939—ten years after graduating from North Texas—as the winning composer (lyrics and music) of a university sponsored fight song competition organized by Floyd Graham. He taught summers at North Texas from 1939 to 1942. The song, "Fight, North Texas," has endured for years and the lyrics have changed minimally to reflect the name changes of the university. While serving as an associate professor at the University of Wyoming from 1946 to 1950, Stroup rewrote the lyrics for the chorus to "Ragtime Cowboy Joe," which was adopted in 1961 as the university's fight song. After serving as head of the Physical Education Department at Southern Arkansas University from 1950 to 1959, Stroup became Professor of Physical Education at Northern Illinois University. While there, Stroup rewrote the lyrics to the chorus of Alonzo Neil Annas' (1882–1966) NIU "Loyalty Song" (1942), which was informally adopted in 1961 and officially 1963 as the "Huskie Fight Song." Stroup also composed songs for Drake University and the University of Chicago. A collegiate academician who played piano mostly by ear and neither majored nor worked in music, Stroup lived to be 101, a number exceeding the songs he composed by one digit. Stroup was inducted in the Halls of Fame of Northern Illinois University and the University of North Texas (1987).
Alma mater
In 1919, Julia Smith (1905–1989), while a music student, and Charles Kirby Langford (1903–1931), then a third-year letterman on the football team and an outstanding overall athlete, composed "Glory to the Green and White" which was adopted as the school's alma mater in 1922. Smith wrote the music and Langford wrote the lyrics.
Other traditions
The Spirit Bell—a bell brought from Michigan in 1891—was a curfew bell from 1892 to 1928. The Talons, a spirit and service organization formed in 1960, acquired it in the 1964, mounted it on a wagon, and began the tradition of running it around the football field to rally fans. It was retired to the University Union in 1982 after it developed a crack. A similar Spirit Bell is currently in use at games. A different organization by the name "Talons" was founded in 1926 as the first social fraternity at North Texas.
On Homecoming Fridays, the Talons light a bonfire built from wooden pallets, typically in a 40-by-40-by-25-foot-height structure. The tradition has endured since the 1930s.
"Boomer" is a cannon fired by the Talons at football games since the 1970s. It is a 7/8th scale M1841 6 pound, smooth bore muzzleloader, resting on hand-crafted solid oak from the campus. Talon alumni have restored it three times, the most recent being in the Fall of 2007, adding a custom for transport and equipment.
The Mean Green Machine, a green and black 1931 Ford Model A Tudor Sedan, is driven by the Talons Motorpool Committee at football games and special events. It was donated by alumnus Rex Cauble in 1974. In 2012, a team of engineering students installed a NetGain WarP 9 electric engine. , the Mean Green Machine has been re-equipped with a modified Model A engine after complications with the electric engine.
McConnell Tower, the clock tower atop the Hurley Administration Building at the center of campus, is bathed in green light for victories. The clock is depicted on the official class ring with two different times on its faces: 1:00 (for the One O'Clock Lab Band) and 7:00—the curfew initiated in 1892.
The eagle talon hand signal is formed by curling the thumb and index and middle fingers forward—the ring and pinkie fingers stay closed against the palm.
"In High Places," is a tall bronze statue of a flying eagle created by Gerald Balciar and dedicated during the university's centennial in 1990.
Broadcast, print, and digital media
Broadcast
KNTU (88.1 FM), licensed and owned by the university and operated by students, has, for years, broadcast to the North Texas region. Jazz has always been a feature of the station; but in 1981, it became the predominant format. KNTU began broadcasting in stereo in 1986 and, on March 22, 1988, increased its broadcasting power from 6,700 watts to 100,000, extending its reach to about a 60-mile radius from its tower located on the Denton campus. KNTU is part of the Mean Green Radio Network, which reaches 10 million listeners. Under the guidance of now-retired faculty member Bill Mercer, several sports broadcasters and radio personalities have emerged from North Texas, including Dave Barnett formerly of ESPN, George Dunham, and Craig Miller.
NTTV, UNT's 24-hour cable television station, features student-produced and student-centric programming.
Student publications
North Texas Review is an annual publication of the English Department. It is produced by UNT students and exclusively features works—art, poetry, fiction, non-fiction—by UNT students.
Student yearbooks through the years have included Cotton-tail (1906), Yucca (1907–1974), Wings (1977–1980), and Aerie (1982–2007). Aerie ceased publication after the 2007 edition, following a trend of the digital age cited by The Economist in 2008.
North Texas is the home of American Political Science Review . The journal moves among national universities every four to six years. UNT will be the first university in the South or Southwest to house the publication.
The North Texas Daily is the official university daily newspaper, staffed by students. Print issues are published Tuesday through Friday during the fall and spring semesters, and weekly during the summer. The paper was founded in 1916 as The Campus Chat and adopted its current name in 1971.
Athletics
, North Texas sponsored fifteen athletic teams that compete at the intercollegiate level of NCAA Division I—for men: football; for men and women: basketball, track & field, cross country, and golf; for women only: diving, soccer, softball, swimming, tennis, and volleyball. UNT has been a member of Conference USA for years.
Football
In its –year history of intercollegiate athletics, the North Texas football team has won 24 conference championships, with the last four occurring from 2001 to 2004 in the Sun Belt Conference. , the team has appeared in thirteen bowl games, winning three including the 1946 Optimist Bowl, the 2002 New Orleans Bowl and the 2014 Heart of Dallas Bowl. Currently, Seth Littrell serves as the head coach, and is in his th year as head coach. From 1952 to 2010, home football games were played at Fouts Field. In 2011, UNT began playing in newly constructed Apogee Stadium.
Men's basketball
The North Texas men's basketball team won the 2006–07 Sun Belt Conference championship and advanced to the NCAA Tournament. The season marked the beginning of four consecutive seasons of 20-plus wins. North Texas won the Sun Belt Conference championship again during the 2009–10 season, and again advanced to the NCAA Tournament. The – season marks the season that the UNT Coliseum has served as the home for Men's basketball.
Women's basketball
The head coach of the North Texas Mean Green women's basketball team is Jalie Mitchell.
Notable alumni
As of 2020, the University of North Texas had approximately 448,000 living alumni. More than 304,000 reside in the Dallas–Fort Worth Metroplex.
R'Bonney Nola Gabriel - Miss Texas USA 2022, Miss USA 2022, and Miss Universe 2022.
Sustainability
In 2005, UNT launched the first PhD program in Environmental Ethics in the world. Three years later, the university became the first large public university in Texas to sign the "American College and University President's Climate Commitment" (ACUPCC). , twenty-four of the 658 signatory institutions of higher learning were from Texas. Of those twenty-four, five were full undergraduate-graduate institutions (2 private, 3 public). Of those five, UNT was the largest. The objectives include achieving carbon neutrality by 2040 and ensuring that all new university buildings and facilities meet a minimum Leadership in Energy and Environmental Design (LEED) Silver rating by the U.S. Green Building Council The university continued to promote sustainability in 2017 when it purchased a year worth of renewable energy credits, to allow the University of North Texas to be powered by renewable energy.
The Life Science Complex, built in 2011, became UNT's first LEED certified structure, earning a Gold rating. The Complex is a state-of-the-art research facility that houses the university's biochemistry, molecular biology, developmental physiology, genetics and plant sciences programs. The building features four climate-controlled rooftop greenhouses and one of the country's most sophisticated aquatics laboratories with more than 2,500 tanks. Also in 2011, Apogee Stadium, the -year-old football stadium, became the first newly built sports stadium in the nation to earn a Platinum LEED certification, the highest of four certifications. The facility features wind turbines, eco-friendly building materials, and native landscape architecture.
The following year, The Princeton Review's Guide to 322 Green Colleges, 2012 Edition, listed UNT for the second consecutive year, citing its top 17-percent ranking among green-compliant universities nationwide under ACUPCC. The article stated that forty percent of the energy on campus is derived from renewable sources, and 43 percent of the buildings have undergone energy retrofits. The campus has posted strong numbers in recycling: since 2009, the university has recycled nearly 1,000 tons of waste materials. UNT offers graduate degrees in Environmental Science and Public Administration and Management.
Further reading
The Portal to Texas History is an undertaking of the North Texas Libraries Digital Projects Unit.
Texas State Historical Association, housed on the Denton campus , administers its website and distributes its Handbook of Texas Online. The association had previously been at the University of Texas at Austin since its founding in 1897.
UNT Research Magazine is an annual digital magazine. It was founded as ReSource (with various subtitles) in 1992 and adopted its current name in 2006.
See also
American Literary Review is a national magazine of poetry, fiction, and nonfiction by writers at all stages in their careers. It was founded in 1990. The Review is largely student run, with faculty editorial oversight. In the fall of 2013, the Review become exclusively an online digital publication.
Environmental Ethics is a peer-reviewed academic journal covering the study of philosophical aspects of environmental problems. It was established in 1979.
University of North Texas Press, founded in 1987, is a relatively young albeit prolific book publisher with more than 300 titles in print ().
Notes
References
External links
North Texas Athletics website
University of North Texas System
1890 establishments in Texas
Educational institutions established in 1890
University of North Texas
Universities and colleges accredited by the Southern Association of Colleges and Schools
University of North Texas
Education in Denton County, Texas
Tourist attractions in Denton, Texas
University of North Texas
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{"url":"http:\/\/mathoverflow.net\/questions\/85941\/is-omega-absolute-in-set-theory-without-foundation?sort=newest","text":"# Is $\\omega$ absolute in set theory without foundation?\n\nLet $\\text{ZF}^-$ be the set theory without powerset, choice, and foundation. Consider the following notions:\n\n\u2022 Wellfounded sets $$WF(c) \\Leftrightarrow (\\forall x \\subseteq TC(c)) \\left[x \\neq \\emptyset \\rightarrow (\\exists y \\in x) (\\forall t \\in x) [t \\notin y]\\right]$$\n\u2022 Ordinals $$ON(c) \\Leftrightarrow WF(c) \\wedge \\text{Transitive}(c) \\wedge (\\forall x,y \\in c)[x = y \\vee x \\in y \\vee y \\in x]$$\n\u2022 $\\omega$ $$x = \\omega \\Leftrightarrow ON(x) \\wedge (\\forall y \\in x)[y = \\emptyset \\vee (\\exists z)[y = z \\cup \\{z\\}]]$$\n\nAre those $\\Delta_1$ notions? Are those notions absolute between transitive models of $\\text{ZF}^-$? More precisely, is there a model $V$ of $\\text{ZF}^-$ with transitive classes $N,M \\models \\text{ZF}^-$ s.t. those notions are not absolute between $N$ and $M$? Does the situation change if we add powerset or choice to our theory?\n\nThose notions are absolute once we have foundation, moreover they are definable by a $\\Delta_0$ formula. I was wondering if they are still absolute without assuming foundation, but that requires more work to show; or if they cease being absolute at all. If so, is there an easy way to construct a counterexample?\n\n-\nAnd could you also clarify the sense of absoluteness you have in mind? This notion is used with various inequivalent senses in set theory. Of course, even in ZFC none of these notions is fully absolute, in the sense of having all models of ZFC agree on whether a set is an ordinal, but the notions are absolute between transitive models of ZFC. Do you mean $\\Delta_1$ definable? \u2013\u00a0 Joel David Hamkins Jan 18 '12 at 2:47\nAbsoluteness I had in mind is between transitive models. \u2013\u00a0 Anton Jan 18 '12 at 2:53\nI find that ambiguous, since it isn't clear if you are considering transitive models of your weak theory with ZFC in the ambient background, or if you also want only your weak theory in the background, and then consider transitive models of your weak theory. Of course, if you have foundation in the background theory, then any transitive model of your weak theory will automatically satisfy foundation. \u2013\u00a0 Joel David Hamkins Jan 18 '12 at 3:12\nWell I am just following the usual proof that CON(ZF - Foundation) implies CON(ZFC). That is usually done in two steps, first proving that CON(ZF - Foundation) implies CON(ZF) by the construction of V; and then proving CON(ZF) implies CON(ZFC) by constructing L. The reason requiring foundation is usually explained as a technical convenience, that helps when absoluteness is developed. What I was wondering is whether you actually need foundation in order to have ordinals absolute, or whether it just lets us write shorter proofs. \u2013\u00a0 Anton Jan 18 '12 at 3:20\nOK, so it seems you want to assume only your weak theory in the background. But I edited my answer with another ambiguity about what your theory is, since even ZFC-powerset is ambiguous, since replacement and collection no longer give rise to the same theory when powerset is not present, and the various versions of choice also are not equivalent without power set. \u2013\u00a0 Joel David Hamkins Jan 18 '12 at 3:38\n\nNone of these three notions is absolute, even if you retain powerset and the axiom of choice.\n\nIn Boffa\u2019s set theory (which contains ZFC without foundation, and is conservative over ZFC with respect to the well-founded kernel), every extensional set-like binary relation is isomorphic to a transitive class with $\\in$. In particular, you can take the ultrapower of the universe over a nonprincipal ultrafilter on a countable set, and let $M$ be its transitive collapse. Then there are nonstandard integers in $M$, i.e., $\\omega^M\\ne\\omega$.\n\nReferences for Boffa\u2019s axiom:\n\n1. Maurice Boffa, Forcing et n\u00e9gation de l\u2019axiome de Fondement, Acad\u00e9mie Royale de Belgique, M\u00e9moires, Classe des Sciences, Collection $8^o$, II. S\u00e9rie 40, No. 7 (1972).\n\n2. David Ballard and Karel Hrb\u00e1\u010dek, Standard Foundations for Nonstandard Analysis, Journal of Symbolic Logic 57 (1992), No. 2, 741\u2013748.\n\n-\nI recall that transitive collapse exists for well-founded structures. I suppose that this is extended in Boffa's work? \u2013\u00a0 Asaf Karagila Jan 18 '12 at 17:36\nRight, well-founded extensional structures have transitive collapse already in ZF w\/o foundation, whereas Boffa\u2019s axiom ensures we can drop the assumption of well-foundedness from this result. (That this is implied by the axiom is quite easy to see, the real work is to show that the axiom is consistent, though that\u2019s not terribly difficult either.) \u2013\u00a0 Emil Je\u0159\u00e1bek Jan 18 '12 at 18:26\n\nWithout the foundation axiom, you have to specify what you mean by an ordinal, precisely because the various definitions are no longer equivalent:\n\n\u2022 An ordinal is a transitive set well-ordered by $\\in$.\n\u2022 An ordinal is a hereditarily transitive set.\n\u2022 An ordinal is a transitive set linearly ordered by $\\in$.\n\nFor example, in a model of Aczel's AFA, there is a set $a$ which is equal to $\\{a\\}$, and such a set is hereditarily transitive, but it is not well-ordered by $\\in$, since it has no $\\in$-least member, and it is not a strict linear order, since it is reflexive. One may similarly construct sets in AFA that are transitive and linearly ordered by $\\in$, but not well-ordered.\n\nSimilarly, the various equivalent formulations of well-foundedness become inequivalent without the axiom of choice. For example, the equivalence of the following two notions of well-foundedness is itself equivalent to the principle of dependent choice, a weak form of the axiom of choice:\n\n\u2022 There is no infinite descending sequence\n\u2022 Every subset has a minimal element.\n\nThus, without any AC, the notion of well-foundedness depends on how you express it.\n\nAnother ambiguity here is that the meaning of ZFC-powerset is ambiguous without elaboration, as Victoria Gitman, Thomas Johnstone and I proved in our paper What is the theory ZFC-powerset?. Also, the familiar equivalent formulations of the axiom of choice (such as WOP or choice functions, etc.) are no longer equivalent when power set is absent.\n\nSo it isn't clear exactly what your weak theory is.\n\n-\nThis is a valid remark, I should have been more precise. Well-foundedness I had in mind is the one where every subset has a minimal element. Then I'm only interested in well-founded sets for the definition of the ordinals. With that restriction all those three definitions should be equivalent, right? \u2013\u00a0 Anton Jan 18 '12 at 2:30","date":"2015-07-01 02:04:06","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\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9138449430465698, \"perplexity\": 375.9661024293539}, \"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-2015-27\/segments\/1435375094634.87\/warc\/CC-MAIN-20150627031814-00225-ip-10-179-60-89.ec2.internal.warc.gz\"}"}
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The Bat Segundo Show & Follow Your EarsThe Bat Segundo Show is a cultural radio program devoted to quirky and very thorough long-form interviews with contemporary authors, idiosyncratic thinkers, and other assorted artists. Guests have included John Waters, John Updike, Stephen Fry, Marilynne Robinson, Karen Russell, David Lynch, Weird Al Yankovic, Robert A. Caro, and more than 500 others. Follow Your Ears is an investigative radio program committed to original inquiry and the pursuit of a specific subject through several angles.
Guests A-Z
Amanda Vaill (BSS #549)
June 27, 2014 0 comments Article People
Martha Gellhorn and Ernest Hemingway headed to Spain to help the Loyalists during the Civil War. Gellhorn was to transform into one of the 20th century's best war correspondents. Hemingway needed to have his romanticism crushed to write a masterpiece. They are two figures in Amanda Vaill's HOTEL FLORIDA. This conversation examines how the Civil War changed not only the trajectory of Spain, but the future of world culture.
Listen: Play in new window | Download (Running Time: 54:24 — 50.0MB)
Diane Johnson (BSS #533)
February 5, 2014 0 comments Article Fiction
Diane Johnson is best known for her comic novels centered around France: LE MARIAGE and LE DIVORCE. But before all this, many years before, she wrote a darker novel called THE SHADOW KNOWS that attracted Stanley Kubrick's notice. Johnson has published a new memoir, FLYOVER LIVES, that details her thoughts on her ancestors, growing up in the Midwest, her life, and her work. Our vivacious and variegated chat gets into the current state of Franco-American relations, forgotten writers, the Methodist practice of being frightened into being good, America's migratory impulse, the demise of the American rail system, foodies, California history, and the considerable references and ideas that Johnson and Kubrick consulted for their work on THE SHINING. Read More
Simon Winchester II (BSS #527)
November 26, 2013 0 comments Article Ideas
In 2007, we aired an infamous program with Simon Winchester, in which he argued with us over the finer points of local history. His new book, THE MEN WHO UNITED THE STATES, shifts the action to a bigger stage, taking on the entire United States. With greater historical stakes, the affable Englishman returns for a conversational rematch six years later. This program features an affably argumentative and cheerfully divergent chat between two wildly energized men united by the common belief that history is always worth returning to. Read More
Wendy Lower (BSS #526)
More than seven decades after World War II, we're still deeply uncomfortable about the idea that women under the Nazi regime committed barbaric acts. We talk with Holocaust scholar (and National Book Award finalist) Wendy Lower about the realities she confronts in her new book, HITLER'S FURIES. How much are the women who were socialized under Nazi policies to blame? Why did the postwar courts allow these women, some of whom massacred children, to return to society without consequence? Read More
Samira Kawash (BSS #522)
October 31, 2013 0 comments Article Ideas
This is the second of two shows devoted to Halloween. Did you know that there was once a chocolate bar called the Chicken Dinner? That cigarette companies once considered candy to be a threat to discretionary spending? Or that candy was used by the military for safety purposes? We didn't either, until we read Samira Kawash's CANDY: A CENTURY OF PANIC AND PLEASURE. We discuss the serpentine history of candy with the Candy Professor herself! Read More
Guns, Part One (FYE #1)
January 22, 2013 0 comments Article Follow Your Ears
Aurora, Sandy Hook, Virginia Tech. We're shocked by the massacres and the loss of life, but how did we get to this? This is the first of a two part program examining guns at length. Edge of the South Bronx On the edge of the South Bronx, everybody we talk with has an opinion about Read More
Roger Corman (BSS #416)
October 25, 2011 2 comments Article Film
In addition to directing some of the most memorable and entertaining drive-in movies of the 20th century (among many other accomplishments), Roger Corman is most recently the subject of a new documentary called Corman's World, which is now playing film festivals and is set for release on December 16. Condition of Mr. Segundo: Not of Read More
Susan Orlean (BSS #415)
Susan Orlean is most recently the author of Rin Tin Tin: The Life and the Legend. Condition of Mr. Segundo: Pondering an alternative timeline with the golden retriever rising as the heroic dog of choice. Author: Susan Orlean Subjects Discussed: Rin Tin Tin references in Finnegans Wake, Rinty's indefinable charm, Jack London, dogs in World Read More
Adam Hochschild (BSS #396)
June 1, 2011 0 comments Article Ideas
Adam Hochschild is most recently the author of To End All Wars. Condition of Mr. Segundo: Conscientiously objecting and objectifying consciousness. Author: Adam Hochschild Subjects Discussed: What is considered morally permissible in war, mustard gas, deadly military technology, Ray Bradbury's "The Flying Machine," the women's suffrage movement and World War I, Emmeline Pankhurst and the Read More
Holly Tucker (BSS #388)
April 8, 2011 0 comments Article Ideas
Holly Tucker is most recently the author of Blood Work. Condition of Mr. Segundo: Wondering why his bank statements come back bloody. Author: Holly Tucker Subjects Discussed: Early philosophical notions of blood, ill humors, whether science without the scientific method can be adequately called science, the Royal Society, William Harvey and the discovery of circulation, Read More
Michael Crummey (BSS #387)
March 29, 2011 0 comments Article Fiction
Michael Crummey is most recently the author of Galore. Condition of Mr. Segundo: Wondering if you can rent a motel room in the whale of a belly. Author: Michael Crummey Subjects Discussed: Childbearing in poor families, grisly deaths and irresponsible life decisions, infant mortality in the early 20th century, the relationship between historical investigation and Read More
Isabel Wilkerson (BSS #366)
November 19, 2010 1 comment Article Ideas
Isabel Wilkerson is most recently the author of The Warmth of Other Suns. Condition of Mr. Segundo: Warming up to fascinating history. Author: Isabel Wilkerson Subjects Discussed: [List forthcoming] EXCERPT FROM SHOW: Correspondent: I wanted to ask about one of the key pieces of conflict relating to the Great Migration that fascinated me. You pointed Read More
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A Wealth of Adam Smith
Arthur Shenfield | From the November 1977 issue
The Wealth of Nations, by Adam Smith, edited by Edwin Cannan, with a new preface by George J. Stigler, 2 vols. in 1, Chicago: University of Chicago Press, 1976, 568 pp., $8.95 (paper)
The Theory of Moral Sentiments, by Adam Smith, with an introduction by E.G. West Indianapolis: Liberty Classics, 1976, 546 pp., $9.95/$2.95
Adam Smith: The Man and His Works, by E.G. West, Indianapolis: Liberty Press, 1976, 254 pp., $6.95/$1.45
The Wisdom of Adam Smith, selected by John Haggarty, edited and with an introduction by Benjamin A. Rogge, Indianapolis: Liberty Press, 1976, 233 pp., $7.95/$1.95
What a treasure there is here for the Adam Smith buff, as Professor Benjamin Rogge styles himself in one of these volumes, and also for the connoisseur of the arts of printing and publishing!
First, the University of Chicago Press has reproduced and bound in one volume, instead of the customary two, Edwin Cannan's famous edition of The Wealth of Nations, adding to it an introduction by that doyen of contemporary Smith scholars, Professor George Stigler. Though the single volume is perhaps physically unwieldy, it is also physically sturdy enough to sustain heavy use, which is high testimony to the printing and binding skill that has gone into its making. And, as Stigler points out, the decision to make one volume of it has enabled the publishers to produce the great tome at a price that, being singularly modest, would have received the ready approval of the master himself. The merits of Cannan's edition established it long ago as the best available, especially because of his punctilious attention to every verbal variation, however minute, between the first few editions of the book and the careful judgment he exercised in making his choice between them. Stigler's introduction is not as replete as might be expected with those witty sallies that come so easily to his pen and are so delightful. Perhaps when one is closely bound to Smith himself and to that solid scholar Cannan, it is right to be a sobersides. Witty or not, Stigler rarely fails to be instructive and illuminating, and he is not lacking in these qualities here.
INTRODUCING ADAM SMITH
Second, the Liberty Fund of Indianapolis has produced a tour de force of publishing in the other volumes noticed here. It is establishing two series, which on the evidence of these volumes are likely in point of technical quality to arouse the high admiration of the publishing world. Liberty Classics is offering new editions of famous works that have long been established in the world of learning, and Liberty Dress is producing modern studies by scholars writing mainly in the classical liberal tradition and its offshoots. Adam Smith's other classic, The Theory of Moral Sentiments, appears in the first series. E.G. West's biography, a study already fairly widely known, and Benjamin Rogge and John Haggarty's work appear in the second. All three books have been produced in a typeface to delight the eye and in paper and binding of choicest grade. Here are printing and bookbinding to excite the admiration of the bibliophile, whatever his views on Smith's economic, political, or moral theories.
The Liberty Fund has also been fortunate in its selection of Professors West and Rogge and Mr. Haggarty to discharge the tasks allotted to them. The reader who comes fresh to The Theory of Moral Sentiments, having, as is common, read only The Wealth of Nations, or parts of it, will find West's introduction a model of its kind. There is much in this stage of the development of Smith's thought that can be misunderstood and also much that might appear to be in conflict with his later views as propounded in The Wealth of Nations. Patiently and perceptively, West deals with these points and leads the student to a grasp of the value of this work, which would have attracted far more scholarly notice in the past two centuries had it not been overshadowed by the superabundant fame of The Wealth of Nations.
Professor Rogge's introduction is intended, not to give us any learned disquisition on Smith, but mainly to tell us who John Haggarty is and how he, a noneconomist and nonphilosopher, came to undertake the task of selecting and assembling the quotations in The Wisdom of Adam Smith. John Haggarty it was who, under the guidance of Professors Ronald Coase, Rogge, and West, was responsible for the production of the film on Smith celebrating the bicentenary of The Wealth of Nations, a film that has now been seen far and wide on both sides of the Atlantic. Here we have a selection of several hundred quotations that run the gamut of his teaching in both of his classics.
Even those who have never given him a serious reading know that Smith was a jewel of a writer. Many of his choice observations have of course been quoted numberless times, because of the aptness and skill of his use of language and because their applicability is timeless. Over and over again he knocks a nail fairly and squarely on the head, not merely for his own times, but also for us who live two centuries later. It is as if, in dilating on policy and politicians, business and businessmen, he were sitting in our own, armchair and discoursing with unerring accuracy on the problems, and mostly on the follies, that present themselves to us this very day. Mr. Haggarty enables us to run our eye over pearl after pearl of wisdom, intelligently classified and assembled, so that there are likely to be few greater pleasures for the student of human affairs than to take his book down from its shelf from time to time and regale himself with some quotation that, although already repeated a thousand times, is as fresh as today's newspaper headline—and probably far more apt to the event concerned.
With books that are first coming to the public's notice, a reviewer will be mainly concerned with their substance and hardly, if at all, with the achievements embodied in them of the printer, bookbinder, publisher, or even the learned writer of an introduction. Here, with two established classics, the case is different. Hence I have thought it right so far to draw the reader's attention to these volumes partly as new products of the arts of publishing, to be considered on that basis to be worthy or unworthy of addition to his library, and partly on the basis of the work of their editorial and introductory contributors. As The Wisdom of Adam Smith consists of quotations taken from the two Smith classics, it is in the same category as they except insofar as the author's wisdom of selection falls for consideration. Only West's biography of Smith is in a different category, calling additionally for assessment as a modern study.
THE CLASSICAL TRADITION
Is there nothing to be said, then, about the substance of the two Smith classics that has not already been said, perhaps ad nauseam? It may be so. Quite apart from the attention Smith has received in the past 200 years from scholars and commentators of all kinds and from many points of the compass, the bicentenary year, 1976, naturally saw a rich crop of celebratory essays, mostly devoted, as was proper, to his relevance to our times. Nevertheless, it may not be otiose to offer some observations on certain features of his work. For though the world may commit again and again all the errors exposed and castigated by Smith—and then some—yet it can never be the same as it was before 1776. As the world aga1n and again looks to be saved from the effects of its errors, it may repeatedly overlook Smith; but he will be there all the time to point the way to safety or success, and sooner or later there will be men to take notice.
One does not need to read more than a few pages of Smith to see that here was a man whose distinguishing mark was not analytical ability—admirable though that was despite the fairly numerous mistakes which later economists exposed—but civility; urbanity; freedom from subservience to or partiality for special interests; absence of the passion that corrodes thought, especially thought upon the problems of society; a capacity to see life whole; an ability to see men as they are and not as the gods or utopians might wish them to be; and an absence of any attachment to a grand theory of history or society purporting to give us the key to everything. We are familiar with these characteristics as typical of the best qualities of the 18th-century English-speaking world, and especially of the intellectuals of that world, but in fact they are also typical of the classical economists in general. Thus, for example, compare Carlyle and Ruskin with the classical economists in character as well as in intellectual insight, and see what impudence it was for them to assail the classics. To the extent that these qualities of the classical tradition have evaporated or been diluted, the loss to the civilized world has been calamitous.
A QUALITY THINKER
Did I say that Smith was free from subservience to or partiality for special interests? Was he not a man of his times? Did he not serve the interests of the rising capitalist class? For all his criticisms of the follies of politicians, did he not refrain from advocating the overthrow of the political structure of the Britain, or even Europe, of his time? Thus by implication did he not endorse it? Yes, he was indeed in the best sense a man of his times, which is as good a commendation as one can give him. He did indeed serve the interests of the rising capitalists, but only because and to the extent that their interests coincided with the general interest. His pen was never so quick or eloquent to denounce the service of a special interest as when that interest was the selfish and antisocial one of merchants or capitalists (except perhaps in the surprising case of his misunderstanding of usury, where he was ready to control the rate of interest in favor of "genuine" investors and against the "profligate" men of fashion). He did indeed broadly accept the political structure of Britain, and to some extent of Europe, but that was because he knew that all improvement must start from where it starts—a fact of life that all utopians and most reformers can never learn. He barely lived long enough to see the beginnings of the French Revolution, but had he survived a little longer he would without doubt have taken the same view of it as Burke, and it hardly needs to be said that he would have been right. Of course, the Glorious Revolution of 1688 and the American Revolution were of a very different kidney.
Did I say that he was free from attachment to any grand theory of history or society, which is the mark both of the inferior scholar and of the ideologue who lays the paving for the road to hell? What about his "natural system of liberty"? Was not that a grand theory of society? Perhaps it was, but see with what common sense and freedom from fanaticism he held it. And in any case the "natural system of liberty" is perhaps the only type of grand theory of society which cannot pave the way to hell.
Did I say he was free from passion? Surely there was some passion in his exposure and denunciation of the follies of mercantilism and of related errors of politicians and merchant wirepullers. Yes, but this was not the passion that corrodes, the passion of the pulpiteer or the ideologue who not only denounces but also hates those whom he thinks to be in error.
Did I say that he saw men as they were, not as they ought to be by the standards of the gods or utopians? Then how could he believe in the possibility of the improvement of conduct that his natural system of liberty, free trade, and a mainly noninterventionist state would require? And if he did see men as they were, was it not this that led him to place more reliance upon the invisible hand of self-interest than it could bear? It is true that men being what they were, they would seek to manipulate the power of the State in their favor. This was the mark of the State in 1776, and it is even more so in 1976 or 1977. Yet Smith was not in error in thinking it possible that to some extent men would learn to moderate their appetite for the manipulation of State power, though typically he did not succumb to excessive optimism on this point (after all it was as likely, he thought, that an Oceana or Utopia would be established in Britain as that free trade would be adopted!). And free trade was adopted, albeit 70 years after 1776, and a largely noninterventionist State did rise in Britain in the mid-19th century. It is true that later there was a slow but terrible relapse, so that now Britain is perhaps the archetype of the modern "democratic" State in which there is a war of all against all by way of the competitive manipulation of the State by an array of group interests. Smith would not have been surprised by this, but wisely he would not have accepted it as the last act in the drama of political affairs. After all, even in our degenerate democracy, there are still some things in which the general interest prevails over sectional interests, and to take men as they are, with all their failings, does not imply that they cannot rise above their attachment to their sectional interests and cleave in fair measure to the general interest.
As for self-reliance and the invisible hand, it is quite false to say that Smith placed more reliance upon their beneficent results than was justified. Although the invisible hand is a metaphor or image that bespeaks a highly gifted writer, it has had the unfortunate effect among the ignorant or the perverse of suggesting that Smith thought there was some kind of magic at work in the process, transmuting selfishness into unselfishness, evil into good, dross into gold. How easy to demonstrate the naivete of the champions of the free economy if they believe in such magic! Those, especially in Germany, who sneered at Smith's alleged Harmonielehre, seemed to believe that Smith knew no difference between selfishness and unselfishness—a travesty of the thinking of a man like Smith who knew far better than most people what made the world go round.
There are several reasons why this kind of view is a travesty. First, in its simplest form the concept of the invisible hand advanced two propositions of undeniable truth and importance: that in a free bargain the purposes of both parties must be served even though each has in mind only his own purpose; and that an economic order or pattern can arise without any central direction. The latter truth is, of course, always ignored or misunderstood by those who imagine that the free economy is an anarchy, without order or coherence, and must therefore be replaced by the "planned economy" (which, ironically, really is without order or coherence!).
Second, the self-interest that is turned to good purpose by the invisible hand is not to be equated with selfishness. It is nothing other than the aim of the purposeful behavior of Ludwig von Mises's acting man. It may be selfish or unselfish, innerdirected or otherdirected. The food in the market may be bought by the glutton whose thought is only for his belly or by the manager of the soup kitchen whose only aim is to feed the hungry. The bricks in the market may be sold for a church or a brothel, for a school or for a gambling den. The invisible hand is at work in all these cases. Of course, as Smith was familiar with the stuff of mankind, he knew very well that selfishness is sometimes the sole spring of human action and often a dominant part of it. All the more important it was, therefore, that men should see that if only they built the right legal and constitutional framework for the market (consistent with and facilitating the "natural system of liberty"), even men's baser instincts could be harnessed to the needs of other men. The concept of the invisible hand is thus one of the major discoveries of scientific thinking.
Third, it is almost always overlooked by Smith's critics that he first propounded the concept of the invisible hand in The Theory of Moral Sentiments, not in The Wealth of Nations. Now in the former he was concerned with the sympathy of men for each other and with the Impartial Spectator (acting as our conscience) who induced men to behave more or less with a worthy regard for the interests of others. Thus the notion that he erected self-interest, in the sense of selfishness, as a maxim for human behavior is a preposterous misunderstanding. It is true, of course, that Smith's concept of sympathy and of the Impartial Spectator was not rooted in the precepts of religion or of any pure form of altruism. There was a self-regarding element in it. Nevertheless, even though his account of the springs of sympathy and care for others is certainly not the last word in psychological analysis, there is a substantial measure of illuminating truth in it that has in my judgment been largely ignored, owing to the fact that it appeared in the relatively neglected Theory of Moral Sentiments.
SMITH SMILED
A further feature of Smith's work that ought to be noted is that there is a marvellous ripeness about it owing to his capacity to see life in the round, his knowledge of the ways of the world, and above all his profound and extensive reading of history. Even his mistakes, though sometimes serious, are never of a contemptible or perverse character. His knowledge of the facts of the world, of the state of affairs in all kinds of countries from China to Peru, and of the history of the ancient and medieval worlds, taught him, as it had to do, that the human record is mostly one of folly, evil, and perversity. As the Swedish Baron Oxenstierna said in a famous note, "See, my son, with what little wisdom the affairs of the world are conducted." Yet there was no sourness in Smith. Many men, he knew, were Yahoos, but many were not. The blood never rushed to his head, even when describing or denouncing folly. It is this that makes us so comfortable with him as in our mind's eye we sit and converse with him on the affairs of the world.
Compare this with Marx. Marx, too, was a voracious and voluminous reader of history, and he too had a very wide knowledge of what we would now call sociological phenomena. But, quite apart from the preposterousness of his theory of history and the perversity of his economic analysis, what I believe should and does turn off the wise when contemplating Marx is the sourness that his historical and sociological reading either created in him, or intensified, or failed to dilute.
Smith's urbane and civilized character is well examined and presented in West's biography. This is not the heavy two-tome biography that used to be accorded to the famous, nor is it likely to qualify as the final and definitive Life of the master. But it is an excellent and highly readable account of the essential facts of his life and work. For the majority of students, and especially for the newcomer to Smith, it is to be highly recommended.
An economist and barrister at law, Arthur Shenfield has taught at various universities in the United States and Great Britain.
NEXT: Frontlines: Libertarian Party Convention
Arthur Shenfield
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Antonio Grant (, 2. veljače 1977.) je američki profesionalni košarkaš. Igra na poziciji krilnog centra, a trenutačno je član Čerkaskih Mavpa koji igraju u drugoj po snazi ukrajinskoj ligi.
Karijera
Europsku karijeru započeo je u latvijski Neptūnasu u kojem je proveo dvije sezone. Odlazi u ruski Sankt Peterburg gdje prosječno u 9 utakmica postiže 9,1 poen, odlazi u zimskom transferu zbog financijskih problema kluba.
Vraća se u Latviju, ovaj put u Riga. Kratko ostaje u Rigi i seli se u drugi latvijski klub Šiauliai. Ondje provodi jednu sezonu prije nego odlazi u KK Split koji nastupa u Jadranskoj ligi.
Od siječnja 2009. Grant više nije igrač Splita, jer je klub odlučio raskinuti njegov ugovor već nakon samo jedne polusezone. Žutom dresu je u prosjeku igrao 29 minuta i zabijao 13.5 koševa, uz 4.5 skokova i 1.8 asista. Razlozi odlaska nisu poznati. Pred sam kraj zimskog prijelaznog roka potpisao je ugovor s ukrajinskim Čerkaskim Mavpima.
Izvori
Vanjske poveznice
Profil na NLB.com
Američki košarkaši
Košarkaši KK Splita
Krilni centri
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: Select2 ajax: preselected data in edit mode I'm making user profile page on Laravel and using select2 component to filter huge list of items.
I have a ajax-based select2. It's good when you are on /create page, but I need to have selected value in it, when I am on page /edit/1.
$('.search-filter-ajax').select2({
width: '100%',
minimumInputLength: 3,
placeholder: "Search...",
ajax: {
url: '/api/listing/search/',
data: function (term) {
return {
data: term.term
};
},
processResults: function (data) {
return {
results: $.map(data, function (item) {
return {
text: item.name,
search_from: item.name,
model: 'some_model',
id: item.id
}
})
};
},
dataType: 'json',
type: "GET",
delay: 250,
},
});
I tried to use initSelection function, but no luck, because it creates only select2 text elements, when I need real <option value="1"> something </option> component.
initSelection: function (element, callback) {
var id = $(element).data('select-id');
var text = $(element).data('select-text');
callback({ id: id, text: text });
},
How can I have a valid preselected option in select2 on page load, but still having opportunity to fire ajax call by onChange?
A: well, you could try to use your own logic to generate slect options like
$.ajax().then((res)=>{
$('#select').html('');
res.data.forEach((item)={$('#select').append($('option').text(item.text).value(item.value);)})
})
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,072
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Q: Woocommerce coupon rounding issue In our store we enter prices including tax. On frontend we hide decimals (In Sweden we don't really use decimals anymore in prices), but backend we keep them. This is why: https://krokedil.com/dont-display-prices-with-0-decimals-in-woocommerce/
We use the function from the link above to hide the decimals on frontend and have also added a function to round off coupons at front end.
/**
* Trim zeros in price decimals
**/
add_filter( 'woocommerce_price_trim_zeros', '__return_true' );
function iconic_was_loop_position( $position ) {
return 'woocommerce_after_shop_loop_item_title';
}
add_filter( 'iconic_was_loop_position', 'iconic_was_loop_position', 10, 1 );
/**
* Round off decimals for coupons
**/
function filter_woocommerce_coupon_get_discount_amount( $discount,
$discounting_amount, $cart_item, $single, $instance ) {
$discount = ceil( $discount );
return $discount;
}
add_filter( 'woocommerce_coupon_get_discount_amount',
'filter_woocommerce_coupon_get_discount_amount', 10, 5 );
This works well for both prices and percentage coupons/discounts.
However, say I have a coupon with a fixed cart amount of SEK400, when using it the amount instead changes to SEK402.
I guess this has to do with how woo applies the coupons combined with the rounding filter.
So my question is, is it possible to exclude the fixed cart coupons from the rounding function in some way?
A: This can be done very easily using the WC_Coupon conditional method is_type(). In your hooked function filter_woocommerce_coupon_get_discount_amount() you will use last argument $instance that is the WC_Coupon Object in an if statement, this way:
add_filter( 'woocommerce_coupon_get_discount_amount', 'filter_woocommerce_coupon_get_discount_amount', 10, 5 );
function filter_woocommerce_coupon_get_discount_amount( $discount, $discounting_amount, $cart_item, $single, $instance ) {
// Round the discount for all other coupon types than 'fixed_cart'
if( ! $instance->is_type('fixed_cart') )
$discount = ceil( $discount );
return $discount;
}
Code goes in function.php file of your active child theme (or theme) or also in any plugin file.
This should work…
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,872
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\section{Introduction}
General relativity (GR) is an essential component in the realistic modeling of core collapse supernovae because of the very
strong gravitational fields in the vicinity of the collapsed core of a star. Hydrodynamics and neutrino transport are closely
connected in this problem, and as we will show, GR can have a profound effect on each of these, especially in the critical phase
of shock reheating. The detection of neutrinos from supernova 1987A (Bionta \emph{et al.} 1987, Hirata \emph{et al.} 1987) and
the hope of detecting neutrino signatures from future supernovae, with next-generation detectors, is additional motivation for an
accurate general relativistic treatment of the neutrino transport in numerical simulations.
\begin{figure}[t]
\label{radii}
\includegraphics[angle=-90, scale=0.65]{figure1l.eps}
\caption{Shock and gain radii vs. post-bounce time for model S15s7b. Both cases are calculated with Newtonian radiation
transport.}
\end{figure}
\begin{figure}[t]
\label{rmsr}
\includegraphics[angle=-90, scale=0.65]{figure2l.eps}
\caption{RMS energy vs. radius for the $\nu_{\mathrm{e}}$'s in model S15s7b. This figure contrasts stationary state GR and
Newtonian transport at $t_{\mathrm{pb}}=114$ ms.}
\end{figure}
\begin{figure}[t]
\label{lum}
\includegraphics[angle=-90, scale=0.65]{figure3l.eps}
\caption{Luminosity vs. radius for the $\nu_{\mathrm{e}}$'s in model S15s7b. This figure contrasts stationary state GR and
Newtonian transport at
$t_{\mathrm{pb}}=114$ ms.}
\end{figure}
\begin{figure}[t]
\label{rmst}
\includegraphics[angle=-90, scale=0.65]{figure4l.eps}
\caption{RMS energy vs. post-bounce time for $\nu_{\mathrm{e}}$'s and $\bar{\nu}_{\mathrm{e}}$'s in model S15s7b.}
\end{figure}
We have developed a code for general relativistic multigroup flux-limited diffusion (MGFLD) that computes the neutrino transport
in a static background metric. This is the first step in the development of a fully GR MGFLD code. The GR metric
used is $ds^2 = a^2 c^2 dt^2 - b^2 dr^2 - r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$. This metric is allowed to evolve in the
hydrodynamics calculations, but is ``frozen'' in the transport calculations, which are then performed treating the metric as
static. Two precollapse models, a 15$\mathrm{M}_{\odot}$ model (S15s7b) and a 25$\mathrm{M}_{\odot}$ model (S25s7b) (Woosley
\& Weaver 1995; Weaver \& Woosley 1997), were evolved through core collapse, bounce, and to approximately 800 ms after bounce in
three sets of simulations. The first was with Newtonian hydrodynamics and Newtonian radiation transport, the second was with GR
hydrodynamics and Newtonian transport, and the third was with both GR hydrodynamics and transport. In addition to these
three sets of simulations, stationary-state neutrino distributions were computed for various post-bounce time slices in these
models in order to isolate the effects associated with each of the metric components.
\section{The Role of General Relativity}
The effects of GR are seen quite clearly in the hydrodynamic evolution of the initial models. GR hydrodynamics produces a much
more compact post-bounce structure than Newtonian hydrodynamics. After $t_{\mathrm{pb}} = 0.4$ s in both models ($t_{\mathrm{pb}}$ being the
post-bounce time), the radius of the shock and the gain radii are reduced by a factor of 2 in the GR calculations, as shown in
Figure 1.
Also strongly affected by GR is the flow velocity between the shock and the proto-neutron star. Because matter falls through a
greater potential well to reach the shock in the GR calculation, GR preshock and therefore postshock velocities are larger than
their Newtonian counterparts, again by a factor of approximately 2.
The main effect of GR on the neutrino rms energies is the redshift of the neutrinos after they decouple from the matter, which
is governed by the metric parameter $a$. The rms energies are reduced by a factor of $a$ evaluated at the $\nu$-sphere. A
smaller effect is a slight outward shift of the $\nu$-sphere resulting from the (non-unity) value of the metric parameter
$b^{-1}$, which causes the neutrinos to decouple outside the $\nu$-sphere at a lower temperature. These effects are shown in
Figure 2. Also shown are the independent effects of $a$ and $b$ on the neutrino transport.
GR reduces the neutrino luminosity by three effects: redshift, governed by the metric parameter $a$; the difference in the local
clock rates at the emission surface and the observer radius, also governed by the metric parameter $a$; and the reduction of the
neutrino flux, governed by the metric parameter $b^{-1}$. All three of these effects are of roughly equal magnitude
and reduce the luminosities by a total factor of $a^2 b^{-1}$ evaluated at the $\nu$-sphere, as shown in Figure 3. This figure
also shows the independent effects of $a$ and $b$ on the stationary-state neutrino transport.
The shock heating rate is proportional to the product of the luminosity and the square of the $\nu_{\mathrm{e}}$ rms energy.
This means that any percentage change in these quantities add together. Looking at Figures 2 and 3, we see a decrease of 8\% in
the $\nu_{\mathrm{e}}$ luminosity and a 3\% reduction in the $\nu_{\mathrm{e}}$ rms energy. These combine to give a 14\%
reduction in the heating rate. Similar results are obtained for the $\bar{\nu}_{\mathrm{e}}$'s.
These are significant differences [\emph{e.g.}, see Burrows \& Goshy (1993), Janka \& M\"uller (1996), Mezzacappa \emph{et al.}
(1998), and Messer \emph{et al.} (1998)] and serve to illustrate the point that modeling core collapse supernovae without GR
hydrodynamics and transport leads to results that cannot be interpreted as realistic. For more information on this subject, the
reader is referred to Bruenn, De~Nisco, \& Mezzacappa (1998).
\section{Nucleosynthesis}
r-process nucleosynthesis is believed to occur in a neutrino-driven wind emanating from the proto-neutron star after the
successful launch of the shock. The r-process yields are a function of the $\nu_{\mathrm{e}}$ and $\bar{\nu}_{\mathrm{e}}$
luminosities and rms energies. The luminosities affect the entropy, mass loss rate, and expansion time scale associated with the
wind, and the rms energies determine the neutronization of the wind. As an example of the impact of GR on the r-process,
consider its effect on the rms energies. Because the $\bar{\nu}_{\mathrm{e}}$-sphere lies below the
$\nu_{\mathrm{e}}$-sphere, the $\bar{\nu}_{\mathrm{e}}$'s suffer a greater emergent redshift. As can be seen in Figure 4, general
relativistic transport and hydrodynamics affects the ratio of the $\nu_{\mathrm{e}}$ and $\bar{\nu}_{\mathrm{e}}$ rms energies.
This differential redshifting affects, in turn, the ratio of the number of $\nu_{\mathrm{e}}$'s to
$\bar{\nu}_{\mathrm{e}}$'s, at a given neutrino energy, and therefore, the neutronization of the wind. This suggests that
general relativistic hydrodynamics and transport will be required to obtain accurate r-process yields.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,053
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A taxonomic genus within the family Cuculidae — the guira cuckoo.
"guira." YourDictionary, n.d. Web. 18 April 2019. <https://www.yourdictionary.com/Guira>.
This is the Momotus brasiliensis of modern ornithologists, and from its geographical range cannot be the original Motmot of Hernandez, but is most likely the "Guira guainumbi" of Marcgrave.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,535
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Q: problem with reaching attributes in xml for use in object I'm trying to put each attribute from my xml file and store each attribute in an Object, but can't reach them. What am I doing wrong? Have 2 other classes which load images too, but not important for my question.
XML:
<personal>
<person id="1" name="Oprah Winfrey" image="oprah-winfrey.jpg" title="administrator"></person>
<person id="2" name="Zlatan Ibrahimovic" image="zlatan-ibrahimovic.jpg" title="technician"></person>
<person id="3" name="Barack Obama" image="barack-obama.jpg" title="CEO"></person>
</personal>
AS3:
private var _items:Array = new Array();
private var _xmlLoader:URLLoader = new URLLoader();
private var _Loader:Loader;
private var _urlRequest:URLRequest;
private var _xml:XML;
public function Main(){
_xmlLoader.load(new URLRequest("personal.xml"));
_xmlLoader.addEventListener(Event.COMPLETE, onXmlLoadComplete);
}
private function onXmlLoadComplete(e:Event):void{
_xml = new XML(e.target.data);
var _xmlList:XMLList = _xml.person;
for each(var node in _xmlList){
for each(var attribute in node.attributes())
//trace(attribute.name()+"::"+attribute) //will output each attribute
//trace("********Node End*********")
var obj:Object = attribute;
trace("obj "+obj.('image')); //outputs "title" node from xml file
var item:ImageItem = new ImageItem(obj.image,
obj.name,
obj.title);
addChild(item);
_items.push(item);
}
trace("items "+_items.length);
}
A: your very close! Here is a updated version of your loop code:
xml = new XML(e.target.data);
for each (var node:XML in xml.person) {
var obj:Object = {};
obj.name = node.@name;
obj.image = node.@image;
obj.title = node.@title;
var item:ImageItem = new ImageItem(obj.image, obj.name, obj.title);
...
}
I hope that helps!
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,371
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Q: How to reexport std::source_location with modules? I'm starting to use modules in my code and tried to create a simple exported function in a module using std::source_location. However, compilation fails at the function call stating error: 'source_location' is not a member of 'std'.
This is the code in question:
util.cpp:
module;
#include <source_location>
#include <iostream>
export module util;
export using source_loc = std::source_location;
export void panic(const char* message, const source_loc loc = source_loc::current()) {
std::cout << "file: "
<< loc.file_name() << "(" << loc.line() << ":" << loc.column() << ") `"
<< loc.function_name() << "`: " << message << '\n';
exit(EXIT_FAILURE);
}
main.cpp:
import util;
int main() {
panic("oh noez"); // < compiler error here
}
The compiler gives a hint about forgetting to #include <source_location> and indeed if I add it to main.cpp it compiles and works as expected. Except I already exported std::source_location as source_loc from my util-module so I would expect it to be accessible in mainbecause of the import util statement. Having to explicitly #include <source_location> everywhere I use the defined panic function seems to defeat the very purpose of modules altogether.
What is going on here? Am I missing something essential about C++ modules? Or did I simply make a mistake that is easily fixed?
I'm using g++ 11.2 to compile the code like this: g++ -std=c++20 -fmodules-ts util.cpp main.cpp -o main
A: This now compiles with gcc 12.2 and Visual Studio 2022 (17.3.6) (the latter when enabling experimental features)
https://godbolt.org/z/89jb7cP5G
I think gcc has been fixed since then and I think your code is correct too.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,398
|
if(!dojo._hasResource["dojox.wire.ml.DataStore"]){ //_hasResource checks added by build. Do not use _hasResource directly in your code.
dojo._hasResource["dojox.wire.ml.DataStore"] = true;
dojo.provide("dojox.wire.ml.DataStore");
dojo.require("dijit._Widget");
dojo.require("dojox.wire._base");
dojo.declare("dojox.wire.ml.DataStore", dijit._Widget, {
// summary:
// A widget for a data store
// description:
// This widget represents a data store of 'storeClass' attribute.
// storeClass:
// A class name of a data store
storeClass: "",
postCreate: function(){
// summary:
// Call _createStore()
// description:
// See _createStore().
this.store = this._createStore();
},
_createStore: function(){
// summary:
// Create a data store
// desription:
// A data store of 'storeClass' is created with arguments
// specified with attributes.
// returns:
// A data store
if(!this.storeClass){
return null; //null
}
var storeClass = dojox.wire._getClass(this.storeClass);
if(!storeClass){
return null; //null
}
var args = {};
var attributes = this.domNode.attributes;
for(var i = 0; i < attributes.length; i++){
var a = attributes.item(i);
if(a.specified && !this[a.nodeName]){
args[a.nodeName] = a.nodeValue;
}
}
return new storeClass(args); //Object
},
getFeatures: function(){
// summary:
// Call getFeatures() method of a data store
// description:
// See dojo.data.api.Read.getFeatures().
// returns:
// A features object
return this.store.getFeatures(); //Object
},
fetch: function(/*Object*/request){
// summary:
// Call fetch() method of a data store
// description:
// See dojo.data.api.Read.fetch().
// request:
// A request object
// returns:
// A request object
return this.store.fetch(request); //Object
},
save: function(/*Object*/args){
// summary:
// Call save() method of a data store
// description:
// See dojo.data.api.Write.save().
// args:
// A save arguments object
this.store.save(args);
},
newItem: function(/*Object*/args){
// summary:
// Call newItem() method of a data store
// description:
// See dojo.data.api.Write.newItem().
// args:
// A new item arguments object
// returns:
// A new item
return this.store.newItem(args); //Object
},
deleteItem: function(/*Object*/item){
// summary:
// Call deleteItem() method of a data store
// description:
// See dojo.data.api.Write.deleteItem().
// returns:
// A boolean
return this.store.deleteItem(item); //Boolean
},
revert: function(){
// summary:
// Call revert() method of a data store
// description:
// See dojo.data.api.Write.revert().
// returns:
// A boolean
return this.store.revert(); //Boolean
}
});
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,299
|
Q: meaning of the word "nail" in context It is from this video. It is at 4 minute and 16 second. Here is the context:
All those things coming because we accruing volume and volume creates overload. But volume will also create overuse. And if you keep relying on this – especially if you are not using spot on, dead on, nails perfect form – it will start rear its ugly head a lot faster.
A: Way down at the bottom of the Cambridge dictionary page:
nail (verb): (informal) to do something successfully:
To nail something is an idiomatic expression that means to do it more or less perfectly. The guy in the video is saying that if you are not nailing perfect form each time, overuse (lifting too much or too often) will create problems.
Other examples:
The gymnast fumbled a little in the middle of her routine, but since she nailed the landing the judges gave her good scores.
I'm not sure about some of the questions on the exam, but I nailed the essay.
(edit) "Nails-perfect" is not typical of the way I've heard nail used as a verb, but language evolves and perhaps it's common slang in the speaker's circles.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 794
|
Hola estudiantes de español II! This intermediate Spanish language class will continue the previous course of conversational Spanish in the form of songs, games, and activities. Instruction, provided by the Spanish Academy, is aimed to encourage communication and appreciation of the Spanish language for students who have completed one or more sessions of Spanish I at the JCC.
Hola estudiantes espanoles! This introductory Spanish language class is a fantastic supplement to the Elementary Spanish language elective taught in school. K-5th grade students will be exposed to conversational Spanish language in the form of songs, games, and activities taught by the Spanish Academy.
Acro Dance is a blend of acrobatics/tumbling and jazz technique. Dancers will increase flexibility, strength, balance, coordination, endurance, timing, body awareness, self-discipline, and confidence. Classes will focus on tumbling skills as well as proper hand placement and body alignment with emphasis on the muscles and flexibility required to perform proper technique.
Students will work on many different jazz forms including contemporary, lyrical and modern dance.
Soccer is a great way to help your little ones with their hand eye coordination as well as get them in touch with the international favorite game. They will learn the basics of soccer including kicking technique, how to play defense and how to score a GOAL!
This exciting class is filled with singing, jumping, climbing, swinging and tumbling. Class time includes circle time, individualized gymnastics attention, parachute play and bubble time. It is a perfect start to help build your toddlers physical, cognitive and social skills in a positive learning environment.
For more information or to register contact Ms. Patti at 561-315-6338.
This non-stop gymnastics class will keep your kids flipping head over heels. The fundamentals for gymnastics (and all sports) are developed as children improve their skills on various gymnastics equipment. In a class filled with laughter, your child will develop self-confidence while experiencing the joys of gymnastics.
Level I, II. Students will work on jazz technique, balancing skills and hip hop dance concepts while learning fun combinations each week.
Ready … set … hike! Flag Football at the JCC offers a fun, non-contact football experience for boys and girls ages 4-5. Students will learn the fundamentals of the game and effective teamwork through drills, scrimmage, rules, and sportsmanship.
Cooking classes are specifically designed for our littlest chefs through easy-to-complete steps and storytelling. Kids will eat, learn and have fun exploring basic recipes taught by Bruchy Cheplowitz.
Register for Cooking >> Registration is now closed.
Classes are designed for elementary learners who are ready to learn proper measuring and utensil skills. Students will learn basic kitchen terminology and have fun exploring recipes that range from beginner to intermediate taught by Bruchy Cheplowitz.
There is no better place to learn gymnastics! We use a safe and encouraging method while teaching gymnastics in the most positive and fun environment. Building self-confidence is most important while students continue to gain strength, flexibility, coordination and balance. Class time includes warm-up, working on all the gymnastics apparatuses and conditioning; all of which help build a healthy body.
12 Week Session: $252 per Person For more information or to register contact Ms. Patti at 561-315-6338.
Level I, II. Students will work on stretching, balance and ballet terminology while at the bar, dancing across the floor and while learning combinations.
Students will work on stretching, balance and ballet terminology while at the barre, dancing across the floor and while learning combinations.
Bienvenido ninos! This introductory Spanish language class will expose preschool age children to conversational Spanish in the form of songs, games, and activities. Instruction, provided by the Spanish Academy, is aimed to encourage communication and appreciation of the Spanish language.
Level I, II. Students will explore rhythms and tap concepts with and without music.
Rachel Karpeles at 561-712-5226 or RachelBK@JCConline.com.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 733
|
{"url":"https:\/\/math.stackexchange.com\/questions\/1384080\/closed-forms-of-the-integrals-int-01-k-sqrtk2-dk-int-01-e-sqrt","text":"# Closed-forms of the integrals $\\int_0^1 K(\\sqrt{k})^2 \\, dk$, $\\int_0^1 E(\\sqrt{k})^2 \\, dk$ and $\\int_0^1 K(\\sqrt{k}) E(\\sqrt{k}) \\, dk$\n\nLet denote $K$ and $E$ the complete elliptic integral of the first and second kind.\n\nThe integrand $K(\\sqrt{k})$ and $E(\\sqrt{k})$ has a closed-form antiderivative in term of $K(\\sqrt{k})$ and $E(\\sqrt{k})$, so we know that $$\\int_0^1 K\\left(\\sqrt{k}\\right) \\, dk = 2,$$ and $$\\int_0^1 E\\left(\\sqrt{k}\\right) \\, dk = \\frac{4}{3}.$$\n\nI couldn't find closed-form antiderivatives to the integrals $\\int K(\\sqrt{k})^2 \\, dk$, $\\int E(\\sqrt{k})^2 \\, dk$, $\\int E(\\sqrt{k})K(\\sqrt{k}) \\, dk$, but I've conjectured, that\n\n\\begin{align} \\int_0^1 K\\left(\\sqrt{k}\\right)^2 \\, dk &\\stackrel{?}{=} \\frac{7}{2}\\zeta(3),\\\\ \\int_0^1 E\\left(\\sqrt{k}\\right)^2 \\, dk &\\stackrel{?}{=} \\frac{7}{8}\\zeta(3)+\\frac{3}{4},\\\\ \\int_0^1 K\\left(\\sqrt{k}\\right)E\\left(\\sqrt{k}\\right) \\, dk &\\stackrel{?}{=} \\frac{7}{4}\\zeta(3)+\\frac{1}{2}. \\end{align}\n\nHow could we prove this closed-forms? It would be nice to see some references to these integrals.\n\n\u2022 highly interesting question (+1) \u2013\u00a0tired Aug 4 '15 at 12:51\n\u2022 math.stackexchange.com\/questions\/568615\/\u2026 This will be quite interesting for you \u2013\u00a0tired Aug 4 '15 at 13:12\n\u2022 It looks like it is just a matter of computing $$\\int_{0}^{1}\\frac{dk}{\\sqrt{1-kt^2}\\sqrt{1-ks^2}}$$ through Legendre's identity, then integrate it against $\\frac{1}{\\sqrt{1-s^2}\\sqrt{1-t^2}}$ over $(0,1)^2$. \u2013\u00a0Jack D'Aurizio Aug 4 '15 at 13:15\n\u2022 @JackD'Aurizio i tried that, but this leads to a dead end... :( \u2013\u00a0tired Aug 4 '15 at 13:49\n\u2022 This might be related question. \u2013\u00a0Vladimir Reshetnikov Aug 4 '15 at 15:33\n\nWe have $$K(\\sqrt{k}) = \\int_{0}^{1}\\frac{dt}{\\sqrt{1-t^2}\\sqrt{1-k t^2}}\\tag{1}$$ hence: $$K(\\sqrt{k})^2 = \\iint_{(0,1)^2}\\frac{dt\\,ds}{\\sqrt{1-t^2}\\sqrt{1-s^2}\\sqrt{(1-kt^2)(1-ks^2)}}\\tag{2}$$ and since: $$\\begin{eqnarray*} \\int_{0}^{1}\\frac{dk}{\\sqrt{(1-ks^2)(1-kt^2)}}&=&\\frac{1}{st}\\int_{0}^{st}\\frac{dk}{\\sqrt{1-\\left(\\frac{s}{t}+\\frac{t}{s}\\right)k+k^2}}\\\\&=&\\frac{1}{st}\\,\\left.\\log\\left(2k-\\left(\\frac{s}{t}+\\frac{t}{s}+2\\sqrt{k^2-\\left(\\frac{s}{t}+\\frac{t}{s}\\right)k+1}\\right)\\right)\\right|_{0}^{st}\\\\&=&\\frac{1}{st}\\,\\log\\left(\\frac{t^2+s^2-2t^2 s^2-2st\\sqrt{(1-s^2)(1-t^2)}}{(s-t)^2}\\right)\\tag{3}\\end{eqnarray*}$$ it follows that:\n$$\\int_{0}^{1}K(\\sqrt{k})^2\\,dk = \\iint_{\\left(0,\\frac{\\pi}{2}\\right)^2}\\log\\left[\\frac{\\sin^2(\\phi-\\theta)}{(\\sin\\phi-\\sin\\theta)^2}\\right]\\cot(\\phi)\\cot(\\theta)\\,d\\phi\\,d\\theta$$ and now we may use a change of coordinates and the Fourier series of $\\log\\sin$.\nAn interesting chance is also given by exploiting the expansion of $K(k)$ with respect to the base of $L^2(0,1)$ given by the shifted Legendre polynomials. We have: $$K(k) = 2\\sum_{n\\geq 0}\\frac{P_n(2k-1)}{2n+1}\\tag{4}$$ and since: $$\\int_{0}^{1}P_n(2\\sqrt{k}-1)^2\\,dk = \\frac{1}{2n+1},$$ $$\\int_{0}^{1}P_n(2\\sqrt{k}-1)P_{n+1}(2\\sqrt{k}-1)\\,dk=\\frac{n+1}{(2n+1)(2n+3)}\\tag{5}$$ we have: $$\\begin{eqnarray*} \\int_{0}^{1}K(\\sqrt{k})^2\\,dk&=&4\\sum_{n\\geq 0}\\frac{1}{(2n+1)^3}+8\\sum_{n\\geq 0}\\frac{n+1}{(2n+1)^2(2n+3)^2}\\\\&=&\\color{red}{\\frac{7}{2}\\,\\zeta(3)+1}.\\tag{6}\\end{eqnarray*}$$","date":"2019-05-22 22:37:20","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\": 0.9835363030433655, \"perplexity\": 475.9733981202163}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2019-22\/segments\/1558232256980.46\/warc\/CC-MAIN-20190522223411-20190523005411-00541.warc.gz\"}"}
| null | null |
If you ask a bunch of men what women do best, I bet their top answer would be talk. Let's face it, women like to talk...a lot! Well, today I am giving you the opportunity to have a complimentary studio session with me just for talking about my work. I need to spread the word about my bridal boudoir sessions and I figured, who better to help me out than my loyal blog readers? And, of course, I want to give you an extra incentive to talk about me...a complimentary session in my new studio!
Do you know someone getting married this year?
Would you like to have a glamour or boudoir session with me for free?
If you answered yes to at least 2 of those questions, then keep reading!
Tell everyone you know that is getting married, thinking about getting married, or maybe they are married but never gave their groom a special gift and would like to remedy that, about my bridal boudoir sessions. If they book a session with me and give me your name as a referral you will earn a complimentary session in my studio! Your session can be for glamour or boudoir in my studio and includes hair and makeup by my super awesome talented stylist.
Now maybe you are wondering what is the difference between a boudoir session and a bridal boudoir session. Well, bridal boudoir tends to include some of your wedding lingerie, maybe your veil and shoes or perhaps even your dress. These images are usually given to the groom in an album as a gift prior to him meeting you at the altar. Think of it as building anticipation for the wedding night. He opens this album alone before he goes to wait for you to walk down the aisle. He will be thinking about that lingerie and what the wedding night will hold for the two of you. He will never forget that anticipation after looking at your photos. He will never forget your wedding night and how you looked as you stood there in your lingerie. He will always have that because you gave him the gift of an album full of you and pieces of that special day.
To book a session use the contact page or call me at 336.309.3506. Thanks again beauties!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,598
|
package org.artifactory.ui.rest.service.home;
import org.apache.commons.lang.time.DurationFormatUtils;
import org.artifactory.addon.AddonInfo;
import org.artifactory.addon.AddonState;
import org.artifactory.addon.AddonType;
import org.artifactory.addon.AddonsManager;
import org.artifactory.addon.CoreAddons;
import org.artifactory.api.config.CentralConfigService;
import org.artifactory.api.context.ContextHelper;
import org.artifactory.api.repo.RepositoryService;
import org.artifactory.api.security.AuthorizationService;
import org.artifactory.api.version.VersionHolder;
import org.artifactory.api.version.VersionInfoService;
import org.artifactory.common.ArtifactoryHome;
import org.artifactory.common.ConstantValues;
import org.artifactory.common.property.ArtifactorySystemProperties;
import org.artifactory.descriptor.config.CentralConfigDescriptor;
import org.artifactory.rest.common.service.ArtifactoryRestRequest;
import org.artifactory.rest.common.service.RestResponse;
import org.artifactory.rest.common.service.RestService;
import org.artifactory.ui.rest.model.home.AddonModel;
import org.artifactory.ui.rest.model.home.HomeModel;
import org.artifactory.ui.utils.RequestUtils;
import org.springframework.beans.factory.annotation.Autowired;
import org.springframework.beans.factory.config.BeanDefinition;
import org.springframework.context.annotation.Scope;
import org.springframework.stereotype.Component;
import javax.annotation.PostConstruct;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Properties;
/**
* @author chen keinan
*/
@Component
@Scope(BeanDefinition.SCOPE_PROTOTYPE)
public class GetHomePageService implements RestService {
private static final String ARTIFACTORY_ACCOUNT_MANAGEMENT_URL = "artifactory.accountManagement.url";
public static final String DEFAULT_ACCOUNT_MANAGEMENT_URL = "http://localhost:8086/dashboard/webapp";
private String accountManagementUrl = DEFAULT_ACCOUNT_MANAGEMENT_URL;
@Autowired
AuthorizationService authorizationService;
@Autowired
CentralConfigService centralConfigService;
@Autowired
RepositoryService repositoryService;
@PostConstruct
private void initialize() {
ArtifactorySystemProperties artifactorySystemProperties = ArtifactoryHome.get().getArtifactoryProperties();
accountManagementUrl = artifactorySystemProperties.getProperty(ARTIFACTORY_ACCOUNT_MANAGEMENT_URL, DEFAULT_ACCOUNT_MANAGEMENT_URL);
}
@Override
public void execute(ArtifactoryRestRequest request, RestResponse response) {
AddonsManager addonsManager = ContextHelper.get().beanForType(AddonsManager.class);
List<AddonInfo> installedAddons = addonsManager.getInstalledAddons(null);
HomeModel homeModel = new HomeModel();
// update home addon model
updateHomeModel(homeModel, request);
HashMap<String, AddonInfo> addonInfoMap = new HashMap<>();
installedAddons.forEach(addonInfo -> addonInfoMap.put(addonInfo.getAddonName(), addonInfo));
List<AddonModel> addonModels = new ArrayList<>();
// update addon data
updateAddonList(addonInfoMap, addonModels);
homeModel.setAddons(addonModels);
homeModel.setUpTime(getUptime());
homeModel.setAccountManagementLink(getAccountManagementLink());
homeModel.setDisplayAccountManagementLink(displayAccountManagementLink());
response.iModel(homeModel);
}
/**
* return system up time
*
* @return up time as string
*/
public String getUptime() {
long uptime = ContextHelper.get().getUptime();
String uptimeStr = DurationFormatUtils.formatDuration(uptime, "d'd' H'h' m'm' s's'");
return uptimeStr;
}
/**
* update home model data
*
* @param homeModel - home model object
*/
private void updateHomeModel(HomeModel homeModel, ArtifactoryRestRequest request) {
Map<String, String> headersMap = RequestUtils.getHeadersMap(request.getServletRequest());
String currentVersion = ConstantValues.artifactoryVersion.getString();
VersionInfoService versionInfoService = ContextHelper.get().beanForType(VersionInfoService.class);
VersionHolder versionHolder = versionInfoService.getLatestVersion(headersMap, true);
CentralConfigDescriptor configDescriptor = centralConfigService.getDescriptor();
updaateLatestVersion(homeModel, versionHolder, configDescriptor);
homeModel.setVersion(currentVersion);
homeModel.setArtifacts(getArtifactsCount());
}
/**
* update latest version data and link
*
* @param homeModel - home model
* @param versionHolder - version holder
* @param configDescriptor - config descriptor
*/
private void updaateLatestVersion(HomeModel homeModel, VersionHolder versionHolder,
CentralConfigDescriptor configDescriptor) {
if (ConstantValues.versionQueryEnabled.getBoolean() && !configDescriptor.isOfflineMode()) {
String latestVersion = versionHolder.getVersion();
String latestVersionUrl = versionHolder.getDownloadUrl();
if (latestVersion != null && !latestVersion.equals("NA")) {
homeModel.setLatestRelease(latestVersion);
}
homeModel.setLatestReleaseLink(latestVersionUrl);
}
}
/**
* update addon list data
*
* @param addonInfoMap - addon info map
* @param addonModels - addons models
*/
private void updateAddonList(HashMap<String, AddonInfo> addonInfoMap, List<AddonModel> addonModels) {
if (!isAol()) {
addonModels.add(new AddonModel(AddonType.HA, addonInfoMap.get("ha"), getAddonLearnMoreUrl("ha"),
getAddonConfigureUrl(AddonType.HA.getConfigureUrlSuffix())));
}
addonModels.add(new AddonModel(AddonType.BINTRAY_INTEGRATION,
getAddonInfo(AddonType.BINTRAY_INTEGRATION, AddonState.ACTIVATED),
getAddonLearnMoreUrl("bintrayIntegration"), getAddonConfigureUrl(
AddonType.BINTRAY_INTEGRATION.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.BUILD, addonInfoMap.get("build"), getAddonLearnMoreUrl("build"), getAddonConfigureUrl(AddonType.BUILD.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.DOCKER, addonInfoMap.get("docker"), getAddonLearnMoreUrl("docker"), getAddonConfigureUrl(AddonType.DOCKER.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.REPLICATION, addonInfoMap.get("replication"), getAddonLearnMoreUrl("replication"), getAddonConfigureUrl(AddonType.REPLICATION.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.MULTIPUSH, getAddonInfo(AddonType.MULTIPUSH), String.format(ConstantValues.addonsInfoUrl.getString(), "replication"), getAddonConfigureUrl(AddonType.MULTIPUSH.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.AQL, getAqlAddonInfo(), getAddonLearnMoreUrl("aql"), getAddonConfigureUrl(AddonType.AQL.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.FILE_STORE, getAddonInfo(AddonType.FILE_STORE),
getAddonLearnMoreUrl("filestore"), getAddonConfigureUrl(AddonType.FILE_STORE.getConfigureUrlSuffix())));
AddonInfo aolAddonPlugin;
aolAddonPlugin = getUserPluginAddonInfo(addonInfoMap);
addonModels.add(new AddonModel(AddonType.PLUGINS, aolAddonPlugin, getAddonLearnMoreUrl("plugins"),
getAddonConfigureUrl(AddonType.PLUGINS.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.NUGET, addonInfoMap.get("nuget"), getAddonLearnMoreUrl("nuget"), getAddonConfigureUrl(AddonType.NUGET.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.NPM, addonInfoMap.get("npm"), getAddonLearnMoreUrl("npm"), getAddonConfigureUrl(AddonType.NPM.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.BOWER, addonInfoMap.get("bower"), getAddonLearnMoreUrl("bower"), getAddonConfigureUrl(AddonType.BOWER.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.REST, addonInfoMap.get("rest"), getAddonLearnMoreUrl("rest"), getAddonConfigureUrl(AddonType.REST.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.GITLFS, addonInfoMap.get("git-lfs"), getAddonLearnMoreUrl("git-lfs"), getAddonConfigureUrl(AddonType.GITLFS.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.VAGRANT, addonInfoMap.get("vagrant"), getAddonLearnMoreUrl("vagrant"), getAddonConfigureUrl(AddonType.VAGRANT.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.LDAP, addonInfoMap.get("ldap"), getAddonLearnMoreUrl("ldap"), getAddonConfigureUrl(AddonType.LDAP.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.SSO, addonInfoMap.get("sso"), getAddonLearnMoreUrl("sso"), getAddonConfigureUrl(AddonType.SSO.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.VCS, addonInfoMap.get("vcs"), getAddonLearnMoreUrl("vcs"), getAddonConfigureUrl(AddonType.VCS.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.YUM, addonInfoMap.get("yum"), getAddonLearnMoreUrl("yum"), getAddonConfigureUrl(AddonType.YUM.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.DEBIAN, addonInfoMap.get("debian"), getAddonLearnMoreUrl("debian"), getAddonConfigureUrl(AddonType.DEBIAN.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.GEMS, addonInfoMap.get("gems"), getAddonLearnMoreUrl("gems"), getAddonConfigureUrl(AddonType.GEMS.getConfigureUrlSuffix())));
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addonModels.add(new AddonModel(AddonType.PROPERTIES, addonInfoMap.get("properties"), getAddonLearnMoreUrl("properties"), getAddonConfigureUrl(AddonType.PROPERTIES.getConfigureUrlSuffix())));
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addonModels.add(new AddonModel(AddonType.LAYOUTS, addonInfoMap.get("layouts"), getAddonLearnMoreUrl("layouts"), getAddonConfigureUrl(AddonType.LAYOUTS.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.LICENSES, addonInfoMap.get("license"), getAddonLearnMoreUrl("license"), getAddonConfigureUrl(AddonType.LICENSES.getConfigureUrlSuffix())));
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new AddonModel(AddonType.MAVEN_PLUGIN, getAddonInfo(AddonType.MAVEN_PLUGIN, AddonState.ACTIVATED),
getAddonLearnMoreUrl("mavenPlugin"),
getAddonConfigureUrl(AddonType.MAVEN_PLUGIN.getConfigureUrlSuffix())));
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getAddonInfo(AddonType.GRADLE_PLUGIN, AddonState.ACTIVATED),
getAddonLearnMoreUrl("gradlePlugin"),
getAddonConfigureUrl(AddonType.GRADLE_PLUGIN.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.JENKINS_PLUGIN,
getAddonInfo(AddonType.JENKINS_PLUGIN, AddonState.ACTIVATED),
getAddonLearnMoreUrl("jenkinsPlugin"),
getAddonConfigureUrl(AddonType.JENKINS_PLUGIN.getConfigureUrlSuffix())));
addonModels.add(new AddonModel(AddonType.BAMBOO_PLUGIN,
getAddonInfo(AddonType.BAMBOO_PLUGIN, AddonState.ACTIVATED ),
getAddonLearnMoreUrl("bambooPlugin"),
getAddonConfigureUrl(AddonType.BAMBOO_PLUGIN.getConfigureUrlSuffix())));
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getAddonInfo(AddonType.TC_PLUGIN, AddonState.ACTIVATED),
getAddonLearnMoreUrl("tcPlugin"),
getAddonConfigureUrl(AddonType.TC_PLUGIN.getConfigureUrlSuffix())));
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* get User plugin for aol or pro
* @param addonInfoMap - addon info map
* @return
*/
private AddonInfo getUserPluginAddonInfo(HashMap<String, AddonInfo> addonInfoMap) {
AddonInfo aolAddonPlugin;
if (isAol()) {
aolAddonPlugin = getAddonInfo(AddonType.PLUGINS, AddonState.INACTIVATED);
} else {
aolAddonPlugin = addonInfoMap.get("plugins");
}
return aolAddonPlugin;
}
/**
* @return global artifact count
*/
private long getArtifactsCount() {
long count = repositoryService.getArtifactCount();
return count;
}
/**
* return add on lean more url
* @param addonId - addon id
* @return
*/
private String getAddonLearnMoreUrl(String addonId) {
return String.format(ConstantValues.addonsInfoUrl.getString(), addonId);
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* return add on configure more url
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*/
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return String.format(ConstantValues.addonsConfigureUrl.getString(), addonId);
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return addonInfo;
}
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* get addon info
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private AddonInfo getAddonInfo(AddonType type) {
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AddonInfo addonInfo = new AddonInfo(type.getAddonName(),
type.getAddonDisplayName(), "",
(haLicensed) ? AddonState.ACTIVATED : AddonState.NOT_LICENSED, new Properties(), 10);
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type.getAddonDisplayName(), "", state, new Properties(), 10);
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* if true - aol license
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private boolean isAol() {
return ContextHelper.get().beanForType(AddonsManager.class).addonByType(CoreAddons.class).isAol();
}
/**
* display account managements link
* @return
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private boolean displayAccountManagementLink() {
return isAol() && ConstantValues.aolDisplayAccountManagementLink.getBoolean();
}
/**
* get account managements link
* @return
*/
private String getAccountManagementLink() {
return accountManagementUrl;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,338
|
Home Business Thai Household Debt Highest In Nine Years
Thai Household Debt Highest In Nine Years
Members of an illegal moneylender network under arrest in Songkhla district, 19 May 2013. A recent research estimates that many Thais owe money to illegal loan sharks.
BANGKOK — The average household debt in Thailand has reached a nine-year high, according to a research by the Thai Chamber of Commerce.
"Average household debt is currently 219,158 baht, which is an increase of 16 percent [from last year]," said Thanawat Polwichai, director of the Centre of Economy and Business Forecasting at the University of Thai Chamber of Commerce.
The number is the highest since 2006, when the Chamber of Commerce first started gathering data about household debt in Thailand, Mr. Thanawat said.
According to the research, which involved interviewing 1,200 people with an income between 15,000 – 50,000 baht per month, three quarters of the respondents reported being in debt.
Almost forty percent these debts are from buying assets, 32.2 percent are a result of everyday spending, 19.6 percent are from long-term investments, and 10.1 percent are from buying estates or houses, according to the survey.
The data also shows that 40 percent of the debts are owed to "legal" institutions such as banks and registered moneylenders, 37.2 percent to "extra-legal" loan sharks, and 22.8 per cent to both legal and extra-legal lenders.
Eighty-three percent of respondents said they have faced difficulty paying off their debts in the past year.
Furthermore, the survey indicates that many Thai people have a shaky financial foundation, with half of respondents reporting to have no savings at all.
Mr. Thanawat said household debt is the main factor keeping domestic spending low, which is holding back the recovery of Thai economy following six months of political turmoil.
He suggested that the state launch a stimulus package to boost recovery and simultaneously implement programs aimed at easing the cost of living. The authorities should also convince low-earning citizens to borrow money from registered moneylenders rather than illegal ones, Mr. Thanawat said.
The researcher added that one surprising element in the survey was the relatively low debt caused by betting in 2014 FIFA World Cup, which came to an end on 13 July in Brazil. Only 4.9 per cent of all debts are related to football gambling, Mr. Thanawat said.
"It's the result of the efforts to solve social problems by the National Council for Peace and Order. That is why gambling-related problems are low," said Mr. Thanawat, explaining that money circulating in the football gambling industry decreased during the World Cup season compared to the period that preceded it.
As part of an intense crackdown on football gambling networks, authorities arrested 4,720 people across the country for charges related to football gambling this year, according to a data by Royal Thai Police. Over 34 million baht was confiscated in connection with these gambling networks, police say.
For comments, or corrections to this article please contact: [email protected]
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 713
|
\section{Introduction: {\{6j\}}-symbol and the recursion formula}
\label{intro}
Spinfoam models present a proposal for a well-defined path integral formulation for quantum gravity. They define the structure quantum space-time at the Planck scale. The current challenge is to study their semi-classical behavior at large scale, show that we recover general relativity and a locally flat space-time as expected and then to extract the perturbative quantum gravity correction to the classical gravitational dynamics. The main spinfoam models for 4d quantum gravity are the Barrett-Crane model \cite{BC} and the more recent family of models \cite{EPR,FK,LS} based on coherent state techniques. These models are all derived from the reformulation of general relativity as a constrained topological BF-theory and attempt to define a discretization of the path integral over space-time geometries.
A recent proposal, which is actively investigated, to probe the semi-classical behavior of the spinfoam amplitudes is the test of the graviton propagator proposed by Rovelli and collaborators \cite{graviton}. There have been several more or less rigorous analytical computations \cite{anal}, as well as numerical simulations \cite{numerics}, showing that the most recent spinfoam models have the expected behavior reproducing at first order (and for the simplest space-time triangulation) Newton's law and the tensorial structure of the graviton. These rely on the computation of the leading order asymptotics of some spinfoam amplitudes for large spin. The next step is to go beyond the leading order and compute the quantum corrections which should correspond to the loop corrections of the standard perturbative quantum field theory approach. This necessarily requires a better understanding of the structure of the asymptotics of the spinfoam amplitudes. Here, we investigate this issue in the simplified framework of 3d quantum gravity.
In three space-time dimensions, gravity is a topological theory which can be exactly quantized as a spinfoam model, the well-known Ponzano-Regge model \cite{PR}. Its basic building block is the {\{6j\}}-symbol. Its asymptotics has been well-studied at leading order and been derived using various techniques: from the brute-force calculation based on the explicit formula of the {\{6j\}}-symbol in term of factorials \cite{razvan} to more refined saddle point approximation on integral formulas \cite{6jsaddle}. Recently, these calculations have been pushed further in order to compute the corrections to the asymptotic behavior using both the saddle point technique \cite{valentin} and the brute-force method \cite{maite}. These results apply to the computation of graviton-like correlations in 3d quantum gravity \cite{3dtoymodel, valentin}. This should lead to a better understanding of the structure of the quantum corrections of the spinfoam graviton propagator, which should be relevant to the four-dimensional case.
In the present paper, we are interested in the computation of the asymptotics of the {\{6j\}}-symbol through the use of recursion relations. As was first shown in \cite{SG}, the {\{6j\}}-symbol satisfies a recursion formula which are intimately related to the topological invariance of the Ponzano-Regge spinfoam model. This formula turned out to be very useful: it can be approximated in the large spin limit by a second order differential equation and one can use it to derive the leading order of the asymptotics through a WKB approximation, but it also allows fast numerical calculations of the {\{6j\}}-symbol.
Here, we investigate two aspects of these recursion relations.
First, we show how to extract the next-to-leading and subsequent corrections to the {\{6j\}}-symbol. We focus on the case of the isosceles tetrahedron (since this is the case relevant for the computation of graviton-like correlations \cite{3dtoymodel}) and we compare our results with the previous work \cite{valentin, maite}.
Second, we study the consequences of the existence of such a recursion relation on the behavior of the graviton-like correlation functions in spinfoam models. We show that it leads to relations on these correlations functions, which relate the expectation values of different observables. These relations are similar to the Ward-Takahashi identities (and to the Schwinger-Dyson equation) in standard quantum field theory, which allow to relate different correlation functions of different orders. These equations play a key role in the study of the renormalization of quantum field theories, so we expect these new recursion relation to be equally relevant to the study of the renormalization/coarse-graining of spinfoam models.
\section{The Recursion Relation for the Isosceles Tetrahedron}
\subsection{Exact and Approximate Recursion}
We will focus on the isosceles tetrahedron, which is relevant for the computations of geometrical correlations in the simplest non-trivial toy model in 3d quantum gravity \cite{3dtoymodel}. Such a tetrahedron has four of its edges of equal length with the two remaining opposite edges of arbitrary length. The corresponding isosceles {\{6j\}}-symbol is:
$$
\{a,b\}_J\equiv\sixj{a&J&J}{b&J&J},
$$
where $J\in \mathbb{N}/2$ and $a,b$ are integers smaller than $2J$ (to satisfy the triangular inequality). The associated tetrahedron has edge lengths $l_j=d_j/2$ for $j=a,b,J$, where $d_j=2j+1$ is the dimension of the $\mathrm{SU}(2)$ representation of spin $j$.
The volume $V$ of the tetrahedron is given by the simple formula:
\begin{equation} \label{volume}
V_J(a,b)=\f1{12}l_al_b\sqrt{4l_J^2-\left(l_a^2+l_b^2\right)},
\end{equation}
while the (exterior) dihedral angles $\theta_j$ can also be written in term of the edge lengths (see e.g. \cite{valentin} for more details):
\begin{equation} \label{angles}
\cos\theta_a\,=\,
-\frac{4l_J^2-l_a^2-2l_b^2}{4l_J^2-l_a^2},\quad
\cos\theta_b\,=\,
-\frac{4l_J^2-l_b^2-2l_a^2}{4l_J^2-l_b^2},\quad
\cos\theta_J\,=\,
\frac{-l_al_b}{\sqrt{4l_J^2-l_a^2}\sqrt{4l_J^2-l_b^2}}.
\end{equation}
The general recursion relation for the {\{6j\}}-symbol given by Schulten and Gordon in \cite{SG} simplifies in this specific isosceles case:
\begin{equation} \label{exactrecursion}
(l_a+\frac{1}{2})\left[4l_J^2-(l_a+\frac{1}{2})^2\right]\{a+1,b\}_J
-2l_a\left[(4l_J^2-l_a^2)\cos(\theta_a)+\frac{1}{4}\right]\{a,b\}_J
+(l_a-\frac{1}{2})\left[4l_J^2-(l_a-\frac{1}{2})^2\right]\{a-1,b\}_J =0
\end{equation}
In the asymptotic regime, we know (analytically and numerically) the behavior of the {\{6j\}}-symbol at the leading order:
\begin{equation}
\label{LO}
\{a,b\}_J \,\sim\,
\{a,b\}_J^{LO}\,\equiv
\f1{\sqrt{12\pi V}}\,\cos\left(
l_a\theta_a +l_b\theta_b +4l_J\theta_J +\frac\pi4
\right),
\end{equation}
which is actually valid under the assumption that the tetrahedron with edge lengths $l_a,l_b,l_J$ exists (else the generic asymptotics can be expressed in term of Airy functions). The oscillatory phase is given by the Regge action $S_R=l_a\theta_a +l_b\theta_b +4l_J\theta_J$.
Using the obvious trigonometric identity $\cos((n+1)\phi)+\cos((n-1)\phi)=\,2\cos\phi\,\cos n\phi$, we can write an exact recursion relation for the leading order of the {\{6j\}}-symbol:
\begin{equation} \label{exactrecurLO}
\sqrt{V_J(a+1,b)}\,\{a+1,b\}_J^{LO}
-2\cos\theta_a \,\sqrt{V_J(a,b)}\,\{a,b\}_J^{LO}
+\sqrt{V_J(a-1,b)}\,\{a-1,b\}_J^{LO}=0.
\end{equation}
A similar recursion relation holds for $b$-shifts and also $J$-shifts.
The most natural idea is to compare this recursion relation for the leading order to the previous equation on the exact {\{6j\}}-symbol to see how to use them to extract the next-to-leading correction to the asymptotic behavior. We can first find the link between the leading order of equation (\ref{exactrecursion}) and the leading order of equation (\ref{exactrecurLO}). Both equations can be written under the same form at the leading order:
\equa{
\{a+1,b\}_J -2\cos \theta_a \{a,b\}_J +\{a-1,b\}_J \approx 0,
}
which turns into a simple second order differential equation in the large spin limit.
Then the next-to-leading order of the equation (\ref{exactrecursion}):
\equa{\tabl{ll}{
\sqrt{V_J(a,b)}\left(1+ \frac{1}{2l_a} \left(1-\frac{2l_a^2}{4l_J^2-l_a^2}\right)\right)&\{a+1,b\}_J-2\cos \theta_a \sqrt{V_J(a,b)}\{a,b\}_J \\
&+\sqrt{V_J(a,b)}\left(1- \frac{1}{2l_a} (1-\frac{2l_a^2}{4l_J^2-l_a^2})\right)\{a-1,b\}_J \approx 0\\
}}
will have to be compared to an recursion relation for the next-to-leading order of the {\{6j\}}-symbol.
\subsection{Pushing to the Next-to-Leading Order}
We are interested in the asymptotic expansion of the {\{6j\}}-symbol. It was shown in previous works \cite{valentin,maite} that $l_j$ seems to be the right parameter to consider when studying the semi-classical behavior of the {\{6j\}}-symbol. So from now we write:
$$
\sixj{a&J&J}{b&J&J}\equiv \{l_a, l_b\}_{l_J}.
$$
Notice that shifting $a$ by $\pm 1$ is equivalent to shifting the edge length $l_a=a+1/2$ by $\pm 1$. We rescale now $l_j$ by $\lambda l_j$ and we replace the exact {\{6j\}}-symbol by a series in $1/\lambda$ alternating cosines and sinus of the Regge action (shifted by $\pi/4$) in the previous equation (\ref{exactrecursion}). The fact that there is no mixing up of cosines and sinus at all order was show in \cite{valentin}. More precisely, we write the {\{6j\}}-symbol asymptotic expansion under the form:
\equa{\label{6jNNNLO}
\tabl{ll}{
\{\lambda l_a, \lambda l_b\}_{\lambda l_J}=\frac{1}{\lambda^{3/2}D(l_a,l_b,l_J)}[\cos(\lambda S_R+\pi/4)+ &\frac{F^{(1)}(l_a,l_b,l_J)}{\lambda}\sin(\lambda S_R+\pi/4)+\frac{G^{(1)}(l_a,l_b,l_J)}{\lambda}\cos(\lambda S_R+\pi/4))\\
&+ \frac{F^{(2)}(l_a,l_b,l_J)}{\lambda^2}\cos(\lambda S_R+\pi/4)+\frac{G^{(2)}(l_a,l_b,l_J)}{\lambda^2}\sin(\lambda S_R+\pi/4))\\
&+ \frac{F^{(3)}(l_a,l_b,l_J)}{\lambda^3}\sin(\lambda S_R+\pi/4)+\frac{G^{(3)}(l_a,l_b,l_J)}{\lambda^3}\cos(S_R+\pi/4)\\
&+ \frac{F^{(4)}(l_a,l_b,l_J)}{\lambda^4}\cos(\lambda S_R+\pi/4)+\frac{G^{(4)}(l_a,l_b,l_J)}{\lambda^4}\sin(S_R+\pi/4) + O(\lambda^{-5})],
}}
where the pre-factor denominator $D(l_a,l_b,l_J)$ is given by the square-root of the tetrahedron volume as in equation \Ref{LO}.
To study the asymptotics, it is convenient to factorize the whole equation (\ref{exactrecursion}) by $\lambda^{3/2}$. We then write $\{ l_a \pm 1/ \lambda, l_b \}_{l_J}$ for $\{\lambda l_a \pm 1, \lambda l_b \}_{\lambda l_J}$. We also factorize the coefficients of the recursion relation. We start by defining $C(l_j)=l_a(4l_J^2-l_a^2)=\frac{16(A(la,l_J,l_J))^2}{l_a}$ where $A(a,b,c)=\frac{1}{4}\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}$ is the area of the triangle of edge lengths given by $a$, $b$ and $c$.
The coefficient which appears in front of $\{l_a \pm 1/\lambda, l_b\}_{l_J}$ becomes $C(l_a\pm1/(2\lambda),l_b,l_J)=(l_a\pm 1/(2\lambda))(4l_J^2-(l_a\pm1/(2\lambda))$, where we underline that the shift is $\pm1/(2\lambda)$ and not simply $\pm1/\lambda$. We expand $C(l_a\pm1/(2\lambda),l_b,l_J)$ in term of derivatives:
$$
C(l_a\pm1/(2\lambda),l_b,l_J)= \sum_n \frac{1}{n!}\frac{1}{(2\lambda)^n}\frac{\partial^nC}{\partial l_a^n}
$$
with \equa{\label{coefequa}
\left\{
\tabl{l}{C=l_a(4l_J^2-l_a^2) \\
\frac{\partial C}{\partial l_a}=4l_J^2-3l_a^2\\
\frac{ \partial^2 C}{\partial l_a^2}=-6l_a^2\\
\frac{ \partial^3 C}{\partial l_a^3}=-6\\
\frac{\partial^nC}{\partial l_a^n}=0 \textrm{ for } n\geq 4}
\right.
}
Then to express $\{ l_a \pm 1/\lambda, l_b\}_{ l_J}$ we need to expand $D(l_a\pm 1/\lambda)$, $F^{(i)}(l_a\pm 1/\lambda)$, and $G^{(i)}(l_a\pm 1/\lambda)$: $(i\in \{1\cdots 4\})$
\equa{\label{coef6j}
\left\{
\tabl{l}{
D(l_a\pm 1/\lambda)= D\pm \frac{1}{\lambda} \frac{\partial D}{\partial l_a}+\frac{1}{2\lambda^2}\frac{\partial^2 D}{\partial l_a^2}\pm \frac{1}{3!\lambda^3}\frac{\partial^3D}{\partial l_a^3} +\frac{1}{4!\lambda^4}\frac{\partial^4D}{\partial l_a^4}\\
F^{(i)}(l_a\pm 1/\lambda)=\sum_{k=0}^{4-i}(-1)^{k}\frac{1}{k!\lambda^k}\frac{\partial^k F^{(i)}}{\partial l_a^k} \; \\
G^{(i)}(l_a\pm 1/\lambda)=\sum_{k=0}^{4-i}(-1)^{k}\frac{1}{k!\lambda^k}\frac{\partial^k G^{(i)}}{\partial l_a^k} \\
}
\right.}
$F^{(1)}(l_j)$ was computed in a previous paper \cite{valentin,maite}. It was also suggested that the asymptotic expansion of the {\{6j\}}-symbol in term of the length scale $\lambda$ is given by an alternative of cosines and sinus at each order, so we expect that $G^{(i)}(l_j)=0$ for $\forall i\ge 1$. Finally, we also need to expand the Regge action $\lambda S_R(l_a\pm \frac{1}{\lambda})$, remembering that $\theta_j=\theta_j(l_a)$~:
\equa{ \label{action}
\lambda S_R(l_a\pm \frac{1}{\lambda})= \lambda S_R +\displaystyle{\sum_{k=0}^{4}}\frac{(-1)^{k+1}}{(k+1)!\lambda^k}\frac{\partial^k \theta_a}{\partial l_a^k}
}
with $$\left\{ \tabl{l}{ \frac {\partial \theta_a}{\partial l_a}= \frac{-2l_a l_b}{(4l_J^2-l_a^2)\sqrt{4l_J^2-l_a^2-l_b^2}} \\
\\
\frac{\partial^2\theta_a}{\partial l_a^2}= - \frac{2l_b(4l_J^2l_a^2-2l_a^4-l_a^2l_b^2+16l_J^4-4l_b^2l_J^2)}{(4l_J^2-l_a^2)^2[4l_J^2-l_a^2-l_b^2]^{3/2}}\\
\\
\frac{\partial^3 \theta_a}{\partial l_a^3}=-\frac{2l_al_b(24l_b^4l_J^2+40l_J^2l_a^2l_b^2-12l_J^2l_a^4+5l_a^4l_b^2-192l_J^4l_a^2-240l_J^4l_b^2+2l_a^2l_b^4+6l_a^6+576l_J^6)}{(4l_J^2-l_a^2)^3[4l_J^2-l_a^2-l_b^2]^{5/2}} \\
\\
\frac{\partial^4 \theta_a}{\partial l_a^4}= \frac{1}{(4l_J^2-la^2)^4(4l_J^2-l_a^2-l_b^2)^{7/2}}(6(8l_a^{10}+8l_a^8l_b^2+152l_J^2l_a^6l_b^2-720l_J^4l_a^6+7l_a^6l_b^4+3520l_J^6l_a^4-1472l_J^4l_a^4l_b^2+2l_a^4l_b^6\\
\quad \quad \quad+140l_a^4l_b^4l_J^2-560l_a^2l_b^4l_J^4-3840l_J^8l_a^2+2432l_J^6l_a^2l_b^2+48l_a^2l_J^2l_b^6+2048l_J^8l_b^2-448l_J^6l_b^4-3072l_J^{10}+32l_b^6l_J^4)l_b)
}\right. $$
We can now write an asymptotic recursion equation from equations (\ref{exactrecursion}), (\ref{coefequa}), (\ref{6jNNNLO}), (\ref{coef6j}) and (\ref{action}) in terms of $\lambda$ neglecting terms of order $O(\lambda^{-4})$ and smaller, assuming that $\lambda$ is large. This leads to a couple of equations at each order, one for the $\cos$-oscillations and one for the term in $\sin$:
\begin{itemize}
\item The first equation is given by the terms of order $\lambda^{0}$ and it is trivially satisfied $(0=0)$ since we have already written the leading order of the {\{6j\}}-symbol proportional to $\cos(S_R+\frac\pi4)$ (the Ponzano-Regge asymptotic formulae).
\item The second equation is given by the terms of order $\lambda^{-1}$:
\equa{ \label{equa0}
\left( \frac{1}{2C}\frac{\partial C}{\partial l_a}-\f1D\frac{\partial D}{\partial l_a}\right)\sin(\theta_a) +\f12 \frac{\partial \theta_a}{\partial l_a} \cos(\theta_a)=0
}
which can be rewritten as a differential equation for $D$:
\equa{ \label{equa1}
\frac{\partial \ln D}{\partial l_a}=\frac{1}{2} \left[ \frac{\partial \theta_a}{\partial l_a} \frac{\cos \theta_a }{\sin \theta_a} + \frac{\partial \ln C}{\partial l_a} \right].
}
This allows to determine $D$: $\ln D= \f12 \ln(C\sin(\theta_a))+ K$, which simplifies into $D=K\sqrt{ l_a l_b \sqrt{4l_J^2-l_a^2-l_b^2}}$ where $K$ is a constant factor. Thus this second equation shows that $D$ is correctly proportional to the square-root of the volume $V$ of the isosceles tetrahedron. To determine the normalization constant $K$ (as well as $G^{(1)}$), the orthonormality property of {\{6j\}}-coefficients can be employed: $\sum_a 4l_a \sqrt{l_bl_{b^\prime}} \{a,b\}_J \{a,b^\prime \}_J= \delta_{b b^\prime }$ and we get the $K=\sqrt{12\pi}$. The details are given in the next section.
\item The third equation is given by the terms of order $\lambda^{-2}$ and which are proportional to $\cos (S_R + \frac\pi4)$
\equa{ \tabl{ll}{ \label{equa2}
\frac{\partial F^{(1)}}{\partial l_a}=\frac{l_a}{4C \sin \theta_a} + &\left( \frac{1}{2C}\frac{\partial C}{\partial l_a } -\f1D\frac{\partial D}{\partial l_a}\right) \f12 \frac{\partial \theta_a}{\partial l_a}+\f16 \frac{\partial^2 \theta_a}{\partial l_a^2}\\
&+ \frac{\cos \theta_a}{\sin \theta_a} \left(\frac{1}{2D}\frac{\partial^2 D}{\partial l_a^2}-\frac{1}{8C}\frac{\partial^2 C}{\partial l_a^2}+\f1D\frac{\partial D}{\partial l_a}\left(\frac{1}{2C}\frac{\partial C}{\partial l_a}-\f1D \frac{\partial D}{\partial l_a}+(\f12\frac{\partial \theta_a}{\partial l_a})^2 \right) \right)
}}
where we used the fact that $\left(\frac{1}{2C}\frac{\partial C}{\partial l_a}-\f1D\frac{\partial D}{\partial l_a}\right)\sin(\theta_a) +\f12 \frac{\partial \theta_a}{\partial l_a} \cos(\theta_a)=0$ (eqn. \Ref{equa1}) to remove all the terms proportional to $F^{(1)}$ itself.
The first term of the right-hand side of the equation (\ref{equa2}) comes from the variation of the coefficient in front of $\{a,b\}_J$ in the recursion equation (\ref{exactrecursion}). The terms with a derivative of $C$ with respect to $l_a$ come from the coefficients in front of $\{a\pm 1, b\}_J$ and $\{a,b\}_J$. The variation of $C$ with respect to $l_a$ is given by the variation of the areas of the triangles of the tetrahedron. From eqn.(\ref{equa1}), we relate it to the variations of $D$ (the volume) and to the variations of the dihedral angle $\theta_a$: $\f1C \frac{\partial C}{\partial l_a}=\f2D \frac{\partial D}{\partial l_a} -\frac{\cos \theta_a}{\sin \theta_a}\frac{\partial \theta_a}{\partial l_a}$. The terms with a derivative of $D$ with respect to $l_a$ come from the variation of the leading order of the asymptotic of the {\{6j\}}-symbol and the terms with a derivative of the dihedral angle $\theta_a$ come from the variations of the Regge action $S_R$. We can now compute the derivative of $F^{(1)}$ with respect of $l_a$ (equation (\ref{equa2})) in terms of $l_a$, $l_b$ and $l_J$ the edge lengths of the tetrahedron:
\equa{\tabl{ll}{
\frac{\partial F^{(1)}}{\partial l_a}=-&\frac{1}{48(l_a^2(-4l_J^2+l_a^2)^2(4l_J^2-l_a^2-l_b^2)^{(5/2)}l_b)}(-32l_b^6l_J^2l_a^2+10l_b^6l_a^4+96l_b^6l_J^4-960l_J^6l_b^4+15l_a^6l_b^4+400l_J^4l_b^4l_a^2-100l_J^2l_b^4l_a^4\\
&-168l_J^2l_a^6l_b^2-1664l_J^6l_b^2l_a^2+20l_a^8l_b^2+576l_J^4l_b^2l_a^4+3072l_J^8l_b^2-3072l_J^{10}+48l_J^4l_a^6+2304l_J^8l_a^2-576l_J^6l_a^4)
}}
and then easily integrate this equation over $l_a$:
\equa{\tabl{ll}{
F^{(1)}(l_j)&=\\
&-\frac {768{l}_J^{6}(l_J^2-l_a^2-l_b^2)+736{l}_J^{4}l_a^{2}l_b^{2}+240l_J
^{4}(l_a^{4}+l_b^{4})-176{l}_J^{2} l_a^
{2}l_b^{2}(l_a^2+l_b^2)-24{l}_J^{2}( l_a^{6}+l_b^{6})+10l_a^{2} l_b^
{2}(l_a^4+l_b^4)+25 l_a^{4}l_b^{4}}
{ 24\left( 4l_J^2-l^2_b \right)\left( 2l_J^2-l_a^2 \right)\left( 4{l}_J^{2}-l_b^{
2}-l_a^{2} \right)^{3/2}l_al_b } +Z(l_b, l_J)
}}
The integration constant $Z(l_b, l_J)$ can be determined using the symmetry properties of the {\{6j\}}-symbol: symmetry of the isosceles {\{6j\}}-symbol with respect to $l_a$ and $l_b$, coupling of $l_a$, $l_b$ and $l_J$ by this isosceles {\{6j\}}-symbol and homogeneity of $F^{(1)}$ ($[F^{(1)}]=l_j^{-1}$) imply that $Z(l_b,l_J)=0$. Then this gives us the same result as in the previous paper \cite{maite}. Moreover, using the definitions of the tetrahedron volume (\ref{volume}) and of the dihedral angles (\ref{angles}), we can express $F^{(1)}$ in terms of some geometrical characteristics of the tetrahedron:
\equa{ \label{geometricNLO}
F^{(1)}=-\frac{\cos \theta_J\left(3(12V)^8-(12V)^4l_a^4l_b^4\left(3(l_a^2-l_b)^2+2l_a^2l_b^2\right)-l_a^{12}l_b^{12} \right)+6l_a^{12}l_b^{12}}{48(12V)^3l_a^8l_b^8}
}
\item The fourth equation is given by the terms of order $\lambda^{-2}$ and which are proportional to $\sin (S_R + \frac\pi4)$. It is the same equation as the previous one for $G^{(1)}$ but the right-hand side is now equal to zero (homogenous equation). That is we simply get that
\equa{
\frac{\partial G^{(1)}}{\partial l_a}= 0
}
so $G^{(1)}=Z(l_b,l_J)$ is just a constant of integration. Once again the symmetry properties of the {\{6j\}}-symbol implies that $G^{(1)}=0$.
\item The next equation is given by the terms of order $\lambda^{-3}$ and which are proportional to $\sin (S_R + \frac\pi4)$. We get an equation for the first derivative of $F^{(2)}$ with respect to $l_a$
\equa{ \tabl{l}{\label{equa5}
\frac{\partial F^{(2)}(l_j)}{\partial l_a}= \frac{\cos \theta_a}{2 \sin \theta_a} \frac{\partial^2F^{(1)}}{\partial l_a^2} - \left(\frac{1}{\sin^2\theta_a}\frac{\partial \theta_a}{\partial l_a} + F^{(1)}\right) \frac{\partial F^{(1)}}{\partial l_a} \\
\;\;\;+ \frac{\cos \theta_a}{\sin \theta_a} \left[ -\frac{1}{4!}\frac{\partial^3 \theta_a}{\partial l_a^3}+\left(\f1D \frac{\partial D}{\partial l_a}\frac{1}{2C}\frac{\partial C}{\partial l_a} + \frac{1}{2D}\frac{\partial^2D}{\partial l_a^2}-\left(\f1D\frac{\partial D}{\partial l_a}\right)^2-\frac{1}{8C}\frac{\partial^2C}{\partial l_a^2}+ \frac{1}{3!}\left(\f12\frac{\partial \theta_a}{\partial l_a}\right)^2\right)\f12 \frac{\partial \theta_a}{\partial l_a} +\left(\f1D\frac{\partial D}{\partial l_a}-\frac{1}{2C}\frac{\partial C}{\partial l_a}\right) \frac{1}{3!}\frac{\partial^2\theta_a}{\partial l_a^2} \right] \\
\;\;\; +\frac{1}{2C}\frac{\partial C}{\partial l_a}\frac{1}{2D}\frac{\partial^2 D}{\partial l^2_a}+\frac{1}{8C}\frac{\partial^2 C}{\partial l^2_a}\frac{1}{D}\frac{\partial D}{\partial l_a} +\left( \frac{1}{2C}\frac{\partial C}{\partial l_a}-\frac{1}{D}\frac{\partial D}{\partial l_a}\right) \f12 \left(\frac{1}{2}\frac{\partial \theta_a}{\partial l_a}\right)^2 +\frac{1}{3!}\frac{\partial^2 \theta_a}{\partial l^2_a}\frac{1}{2}\frac{\partial \theta_a}{\partial l_a} +\left(\frac{1}{D}\frac{\partial D}{\partial l_a}\right)^3-\frac{1}{D}\frac{\partial D}{\partial l_a}\frac{1}{D}\frac{\partial^2 D}{\partial l^2_a}+ \frac{1}{3!D}\frac{\partial^3 D}{\partial l^3_a}\\
\;\;\; -\frac{1}{8\cdot 3! C}\frac{\partial^3 C}{\partial l^3_a} -\frac{1}{2C}\frac{\partial C}{\partial l_a}\left(\frac{1}{D}\frac{\partial D}{\partial l_a}\right)^2
}}
We recall that $D$ is proportional to the square root of the tetrahedron volume, $C$ can be expressed in terms of the volume $V$ and the sinus of the dihedral angle $\theta_a$ (see equation (\ref{equa1})). To integrate this equation, we first express explicitly~\footnotemark it in terms of $l_a$, $l_b$ and $l_J$,
\footnotetext{$\frac{\partial F^{(2)}}{\partial l_a}(l_j)=-\frac{1}{2304\left((4l_J^2-l_b^2)l_a^3(4l_J^2-l_a^2)^3(4l_J^2-l_a^2+l_b^2)^4l_b^2\right)}(-1604l_a^8l_b^8l_J^2+1250816l_a^4l_J^8l_b^6-207104l_a^6l_b^6l_J^6+31904l_a^{10}l_J^4l_b^4
-169344l_a^4l_b^8l_J^6-3920l_a^6l_b^{10}l_J^2+24992l_a^6l_b^8l_J^4-46848l_a^{10}l_J^6l_b^2-7129088l_a^4l_J^{10}l_b^4+1770496l_a^6l_J^8l_b^4+16832l_a^4l_b^{10}l_J^4
+34368l_a^8l_J^4l_b^6-6816l_a^{12}l_J^4l_b^2-278912l_a^8l_J^6l_b^4+14524416l_a^2l_J^{12}l_b^4+486144l_a^2l_J^8l_b^8-560l_b^{12}l_a^4l_J^2-43776l_a^2l_b^{10}l_J^6-3317760la^2lJ^{10}l_b^6
+22241280l_a^4l_J^{12}l_b^2+794l_a^{12}l_b^6-6955008l_a^6l_J^{10}l_b^2+2801664l_J^{12}l_b^6+672l_a^{14}l_b^2l_J^2-26542080l_a^4l_J^{14}-451584l_J^{10}l_b^8+46080l_J^8l_b^{10}
-2304l_b^{12}l_J^6-21233664l_J^{18}+37158912l_a^2l_J^{16}+1072128l_a^8l_J^8l_b^2-1528la^{12}l_b^4l_J^2-10911744l_J^{14}l_b^4-35979264l_a^2l_J^{14}l_b^2+1728l_b^{12}l_J^4l_a^2
+9953280l_a^6l_J^{12}+228096l_a^{10}l_J^8-10368l_a^{12}l_J^6-2073600l_a^8l_J^10+27l_a^{10}l_b^8+23592960l_J^{16}l_b^2+400l_a^8l_b^{10}-88l_a^{14}l_b^4-8144l_a^{10}l_b^6l_J^2+100l_b^{12}l_a^6)$}
and then deduce $F^{(2)}$:
\equa{ \tabl{ll}{ \label{NNLO}
F^{(2)}(l_j)&=\frac{-1}{4608\left((4l_J^2-l_a^2)^2(4l_J^2-l_b^2)^2(4l_J^2-l_a^2-l_b^2)^3l_a^2l_b^2\right)}(-2359296l_a^2l_J^{10}l_b^4-224512l_a^6l_J^6l_b^4+100l_a^{12}l_b^4+576l_J^4l_b^{12}+112896l_J^8l_b^8\\
&+2727936l_a^4l_J^{12}+5308416l_J^{16}+212l_b^{10}l_a^6-5898240l_a^2l_J^{14}-11520l_a^{10}l_J^6+941056l_a^4l_J^8l_b^4+31584l_a^8l_J^4l_b^4\\
&-2416l_a^4l_b^{10}l_J^2
-79872l_a^8l_J^6l_b^2-480l_b^{12}l_J^2l_a^2-7040l_a^8l_b^6l_J^2-2416l_a^{10}l_J^2l_b^4+100l_a^4l_b^{12}+212l_a^{10}l_b^6\\
&+2727936l_J^{12}l_b^4-700416l_J^{10}l_b^6-5898240l_J^{14}
l_b^2-11520l_J^6l_b^{10}-700416l_a^6l_J^{10}+609l_a^8l_b^8+112896l_a^8l_J^8\\
&+576l_a^{12}l_J^4-2359296l_a^4l_J^{10}l_b^2+528384l_a^6l_J^8l_b^2+5849088l_a^2l_J^{12}l_b^2-79872l_a^2l_b^8l_J^6+31584l_a^4l_b^8l_J^4-7040l_a^6l_b^8l_J^2\\
&-224512l_a^4l_b^6l_J^6+58816l_a^6l_b^6l_J^4+8640l_a^{10}l_J^4l_b^2+8640l_b^{10}l_J^4l_a^2-480l_a^{12}l_J^2l_b^2+528384l_a^2l_J^8l_b^6)
}}
which is the only result with the required symmetries\footnote{If the result is not symmetric after integration, a non-null integration constant has to be added and its determination can be done using the symmetry properties of the {\{6j\}}-symbol. Indeed, we have $\frac{\partial F^{(2)}}{\partial l_a}= H(l_a, l_b, l_J)$ so by integration over $l_a$, $F^{(2)}(l_j)=h(l_a, l_b, l_J)+Z(l_b, l_J)$. Moreover by symmetry, we must have $\frac{\partial F^{(2)}}{\partial l_b}= H(l_a=l_b, l_b=l_a, l_J)$ and then integrating over $l_b$, we obtain a second expression for $F^{(2)}$: $F^{(2)}(l_j)=h(l_a=l_b, l_b=l_a, l_J)+Z(l_a, l_J)$ which implies that the constant of integration satisfies $Z(l_b,l_J)-Z(l_a,l_J)=h(l_a=l_b, l_b=l_a, l_J)- h(l_a, l_b, l_J)$. This equation allows to determine $Z$ and to get (\ref{NNLO}).}. The geometrical meaning of this function does not seem obvious. Nevertheless, we can give a more compact expression for the denominator of $F^{(2)}$:
\equa{\label{denoF2}
(4l_J^2-l_a^2)^2(4l_J^2-l_b^2)^2(4l_J^2-l_a^2-l_b^2)^3l_a^2l_b^2= \frac{(12V)^6}{\cos^4\theta_J}.
}
\item The next equation comes from the terms of order $\lambda^{-3}$ which are proportional to $\cos(S_R+\frac\pi4)$:
\equa{
\frac{\partial G^{(2)}}{\partial l_a}(l_j)=0
}
which implies once again that $G^{(2)}=Z(l_b,l_J)$ is a constant of integration. Then the symmetry properties of the {\{6j\}}-symbol implies $G^{(2)}(l_j)=0$.
\end{itemize}
We can now give the asymptotic expansion of an isosceles {\{6j\}}-symbol until the next to next to leading order (NNLO):
\equa{ \label{isoNNLO}
\{ l_a, l_b\}^{\textrm{NNLO}}_{l_J}= \frac{1}{\sqrt{12\pi V_{l_J}(l_a,l_a)}} \left[\cos (S_R +\frac\pi4)+F^{(1)}(l_j) \sin(S_R+\frac\pi4)+F^{(2)}(l_j) \cos(S_R+\frac\pi4) \right]
}
where the expression for $F^{(1)}$ and $F^{(2)}$ are given by equations (\ref{geometricNLO}) and (\ref{NNLO}). This result seems to confirm that the expansion of the {\{6j\}}-symbol is a series alternating cosines and sinus of the Regge action (shift by $\frac\pi4$). In the case of an equilateral tetrahedron, all the edges have the same length, that is $l_a=l_b=l_J=l$ and $V= \frac{\sqrt{2}}{12} l^3$. Then equation (\ref{isoNNLO}) reduces to:
\equa{\label{equaNNLO}
\{6j\}^{\textrm{NNLO}}_{\textrm{equi}}= \frac{1}{\sqrt{\pi l^3\sqrt{2}}} \cos(S_R+\frac\pi4)-\frac{ 31}{72\,2^{1/4}\,2^{5/2}\sqrt{\pi l^5}}\sin(S_R +\frac\pi4)-\frac{45673}{20736}\frac{1}{\,2^{1/4}\,2^{4}\sqrt{\pi l^7}}\cos(S_R+\frac\pi4)
}
where the Regge action is given by $S_R=6 l \theta$ and $\theta=\theta_a=\theta_b=\theta_J= \arccos(-1/3)$.
This result is confirmed by numerical simulations. The plot figure \ref{plotequiNNLO} represents numerical simulations of the equilateral {\{6j\}}-symbol minus its approximation (\ref{equaNNLO}). Moreover, to enhance the comparison, we have multiplied by $l^{7/2}$ to see how the coefficient of the NNLO is approached and we have divided by $\sin(S_R+\frac\pi4)$ (oscillations of the next to next to next to leading order) to suppress the oscillations. This gives an error that decreases as expected as $l^{-1}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=4cm]{NNLOequi}
\caption{Difference between the equilateral {\{6j\}}-symbol and the analytical result (\ref{equaNNLO}). The x-axis stands for $l=d/2$ and $d$ goes from 100 to 5000. The error decreases as expected as $l^{-1}$ confirming our asymptotic formula.} \label{plotequiNNLO}
\end{center}
\end{figure}
\begin{itemize}
\item The next two equations come from the terms of order $\lambda^{-4}$. The equation for $G^{(3)}$ is the same as the one for $G^{(1)}$ and $G^{(2)}$, that is: $\frac{\partial G^{(3)}}{\partial l_a}=0$. Using the same arguments of symmetry we deduce that $G^{(3)}=0$. This confirms our expectation of a series alternating cosines and sines in the asymptotic of the {\{6j\}}-symbol:
\equa{\label{asympt}
\{\lambda l_a, \lambda l_b\}_{\lambda l_J}=\frac{1}{\lambda^{3/2}D(l_a,l_b,l_J)}\left[\cos(\lambda S_R+\frac\pi4)+ \displaystyle{\sum_{k=1}^\infty}\frac{F^{(k)}(l_a,l_b,l_J)}{\lambda^k}\cos(\lambda S_R+\frac\pi4+\epsilon(k)\frac\pi2)\right],
}
where $\epsilon(k)=-1$ when $k$ is odd and $\epsilon(k)=0$ when $k$ is even.
We already have an expression for $F^{(1)}$ and $F^{(2)}$. The equation for $F^{(3)}$ is the second equation of order $\lambda^{-4}$ and gives its first derivative with respect to $l_a$ in terms of $l_a$, $l_b$ and $l_J$. It is straightforward (though lengthy) to integrate it over $l_a$ and get the expression\footnotemark of $F^{(3)}$ in terms of $l_a$, $l_b$ and $l_J$. In the equilateral case ($l_a=l_b=l_J=l$), the formula reduces to:
\equa{
F^{(3)}=\frac{28833535}{17915904}\frac{1}{2^{9/2}2^{1/4}l^3}
}
\footnotetext{$F^{(3)}(l_j)=-\frac{1}{3317760(l_a^3(l_a-2l_J)(l_a+2l_J)(4l_J^2-l_b^2)^3(4l_J^2-l_a^2-l_b^2)^(9/2)l_b^3(4l_J^2-l_a^2)^2)}(-6965741568l_a^6l_J^{10}l_b^8+1069728l_a^6l_b^{16}l_J^2+28270080l_a^{16}l_J^6l_b^2-743040l_J^4l_a^2l_b^{18}+1750503260160l_J^{20}l_a^2l_b^2+379404800l_a^{12}l_J^6l_b^6+788705280l_a^{10}l_J^{10}l_b^4-33547200l_a^6
l_b^{14}l_J^4-98578608l_a^{12}l_b^8l_J^4-81032970240l_a^8l_J^{14}l_b^2+379404800l_a^6l_b^{12}l_J^6-389214720l_J^8l_a^2l_b^{14}-323813376000l_a^6l_J^{18}-156036l_a^8l_b^{16}-22176l_J^2l_a^4l_b^{18}-3824640l_a^{16}l_J^4l_b^4-28408l_a^{18}l_b^6-31104000l_a^{16}l_J^8-1262270545920l_J^{18}l_a^2l_b^4-1262270545920l_J^{18}l_a^4l_b^2+461814497280l_J^{16}l_a^2l_b^6+1069728l_a^{16}l_J^2l_b^6+74144841728l_a^6l_J^{12}l_b^6+824520278016l_J^{16}l_a^4l_b^4+461814497280l_a^6l_J^{16}l_b^2+788705280l_J^{10}l_a^4l_b^{10}-267541217280l_a^6l_J^{14}l_b^4+115174656l_J^6l_a^4l_b^{14}-999532800l_J^8l_a^4l_b^{12}+68611276800l_a^8l_J^{16}-3824640l_J^4l_a^4l_b^{16}-1242078720l_a^{10}l_J^8l_b^6+28270080l_J^6l_a^2l_b^{16}+38483472384l_J^{12}l_a^4l_b^8+2157096960l_a^{12}l_J^{10}l_b^2-1242078720l_a^6l_b^{10}l_J^8+509607936000l_J^{24}-33547200l_a^{14}l_b^6l_J^4+1222041600l_J^{12}l_a^2l_b^{10}+11119152l_a^{12}l_b^{10}l_J^2-999532800l_a^{12}l_J^8l_b^4+783522201600l_J^{20}l_a^4+2157096960l_J^{10}l_a^2l_b^{12}-28408l_a^6l_b^{18}-389214720l_a^{14}l_J^8l_b^2-783601920l_a^8l_J^8l_b^8-98578608l_a^8l_J^4l_b^{12}-6965741568l_a^8l_J^{10}l_b^6-323813376000l_J^{18}l_b^6+1222041600l_a^{10}l_J^{12}l_b^2+115174656l_a^{14}l_J^6l_b^4-743040l_a^{18}l_J^4l_b^2-22176l_a^{18}l_J^2l_b^4-81032970240l_J^{14}l_a^2l_b^8+7201368l_a^{14}l_b^8l_J^2-114011616l_a^{10}l_b^{10}l_J^4-985242009600l_J^{22}l_b^2+568565760l_a^{10}l_b^8l_J^6-1401753600l_a^{12}l_J^{12}-267541217280l_J^{14}l_a^4l_b^6+568565760l_a^8l_J^6l_b^{10}+11119152l_a^{10}l_b^{12}l_J^2+783522201600l_J^20l_b^4-342523l_a^{12}l_b^{12}+38483472384l_a^8l_J^{12}l_b^4-3981312000l_a^{10}l_J^{14}-156036l_a^{16}l_b^8-511602l_a^{10}l_b^{14}+344217600l_b^{14}l_J^{10}-1401753600l_b^{12}l_J^{12}+1036800l_b^{18}l_J^6+344217600l_a^{14}l_J^{10}+68611276800l_J^{16}l_b^8-3981312000l_J^{14}l_b^{10}-511602l_a^{14}l_b^{10}-31104000l_b^{16}l_J^8-985242009600l_J^{22}l_a^2+1036800l_a^{18}l_J^6+7201368l_a^8l_b^{14}l_J^2)$}
Therefore, we have the expression of the asymptotic expansion of the equilateral {\{6j\}}-symbol up to the next-to-next-to-next to leading order (NNNLO):
\begin{eqnarray}
\{6j\}^{\textrm{NNNLO}}_{\textrm{equi}}&= \frac{1}{2^{1/4}\,\sqrt{\pi l^3}} &
\left[ \cos(S_R+\frac\pi4)-\frac{31}{72\,2^{5/2}\,l}\sin(S_R +\frac\pi4)\right.\nonumber\\
&&-\left.\frac{45673}{20736\,2^4\,l^2}\cos(S_R+\frac\pi4) +\frac{28833535}{17915904\,2^{9/2}\,l^3}\sin(S_R+\frac\pi4)\right].
\end{eqnarray}
We check this result numerically by computing it using Mathematica for two values of spins $j=50$ and $j=100$. More precisely, we computed the renormalized error $\frac{(\{6j\}_{\textrm{equi}}-\{6j\}^{\textrm{NNNLO}}_{\textrm{equi}})l^{9/2}}{\cos(S_R+\frac\pi4)}$ and we got the expected $1/\lambda $-behavior. However for $l> 100$, Mathematica is not accurate enough and the numerical errors are too important to get exploitable results.
In the general isosceles case, the expression of $F^{(3)}$ is quite complicated and its geometrical interpretation remains to be understood. Nevertheless, we can again give as before a more compact formula for the denominator of $F^{(3)}$:
\equa{\label{denoF3}
\textrm{denominator}_{F^{(3)}}=3317760 (4l_J^2-l_a^2-l_b^2)^{9/2}l_a^3l_b^3(4l_j^2-l_a^2)^3(4l_J^2-l_b^2)^3= 30(48)^3\frac{(12V_J(a,b))^9}{\cos^6\theta_J}
}
From this equation, the equation giving the denominator of $F^{(2)}$ and remembering that the denominator of $F^{(1)}$ can be written under a similar form: denominator$_{F^{(1)}}=48\frac{(12V)^3}{\cos^2\theta_J}$, we can conjecture that:
\equa{
\textrm{denominator}_{F^{(k)}} \propto \frac{(12V_J(a,b))^{3k}}{(\cos \theta_J)^{2k}}
}
where $F^{(k)}$ are the terms appearing in the asymptotic expansion of the {\{6j\}}-symbol (\ref{asympt}). And consequently, the numerator of $F^{(k)}$ is a polynomial in $l_j$ of degree $8k$.
\end{itemize}
So, using the recursion relation for the isosceles {\{6j\}}-symbol as well as its symmetry properties, we have computed explicitly the asymptotic expansion of the isosceles {\{6j\}}-symbol to the fourth order up to an overall factor $K$ (this integration constant $K$ comes from the integration of the first equation (\ref{equa1})). The well-known value $K=\sqrt{12 \pi}$ which already appears in the Ponzano-Regge formula can be obtained easily using the unitary property of the {\{6j\}}-symbol, as we show in the next section. The equilateral case has been checked against numerical calculations. This method using the recursion relation is fairly easy to implement. It requires integrating a rational fraction at each level and does not involve neither Riemann sum nor saddle point analysis.
Moreover, since the coefficient $C$ of the recursion relation (\ref{exactrecursion}) is a polynomial of degree 3; $\frac{\partial^n C}{\partial l_a^n}=0$ for $n \geq 4$. Therefore, we expect to get a stable relation for the first derivative of $F^{(k)}$ and $G^{(k)}$ with respect to $l_a$ for $k \geq 3$. On one hand, this allows to prove that $G^{(k)}$ always vanishes; and on the other hand, it should provide a systematic method to extract $F^{(k)}$ for arbitrary order $k$.
We conclude this section with a general remark on the asymptotic expansion of the {\{6j\}}-symbol. In the context of 3d quantum gravity, it is often argued that the leading order of the {\{6j\}}-symbol is a $\cos(S)$ instead of a complex phase $\exp(+iS)$, thus reflecting that the path integral is invariant under a change of (local) orientation (see e.g.\cite{laurent}). This obviously neglects the $+\pi/4$ shifts, which can be considered as a quantum effect (like an ordering ambiguity). However, in the light of the present expansion, it is clear that we have terms of the type $\sin(S)$ beyond the leading order and such terms are not invariant under the change $S\rightarrow -S$. This means that the role of this symmetry in the spinfoam path integral should be more subtle than originally thought.
\subsection{Consequences of the unitary property of the {\{6j\}}-symbols}
The orthogonality property of the {\{6j\}}-symbols states that:
\equa{
\sum_{l_a} 4l_a \sqrt{l_b l_{b^{\prime}}} \{l_a, l_b\}_{l_J} \{l_a, l_{b^{\prime}}\}_{l_J}=\delta_{l_b l_{b^{\prime}}}
}
This relation corresponds to the unitarity of the evolution in the Ponzano-Regge 3d quantum gravity.
We want to use this property to determine the constant of the leading order of the {\{6j\}}-symbol. From the recursion relation we have shown that $\{l_a, l_b\}^{\textrm{LO}}_{l_J}= \frac{K}{\sqrt{V_J(a,b)}} \cos(S_R+\frac\pi4)$; but $K$ is still undetermined. For large spin and for $l_b \approx l_{b^\prime}$, we can approximate the unitary property at the leading order in $(l_b-l_{b^\prime})$ by:
\equa{\label{unitary}
\int_0^\infty dl_a 4 l_a l_b \frac{K^2}{V_J(a,b)}\cos(S_R(l_a, l_b)+\frac\pi4) \cos(S_R(l_a, l_{b^\prime})+\frac\pi4)\approx \delta(l_b-l_{b^\prime } ).
}
The product of the cosines can be simplified at leading order:
\begin{eqnarray}
\cos(S_R(l_a, l_b)+\frac\pi4) \cos(S_R(l_a, l_{b^\prime})+\frac\pi4)
&=&
\f12\left[\cos(S_R(l_a, l_b)+S_R(l_a, l_{b^\prime})+\frac\pi2)+\cos(S_R(l_a, l_b)-S_R(l_a, l_{b^\prime}))\right] \nonumber\\
&\sim&
\f12\left[\cos(2S_R(l_a, l_b)+\frac\pi2)+\cos((l_b-l_{b^\prime})\theta_b)\right], \nonumber
\end{eqnarray}
where the dihedral angle $\theta_b=\arccos \left( -\frac{4l_J^2-l_b^2-2l_a^2}{4l_J^2-l_b^2}\right)$ is considered as a function of the length $l_a$. We do a saddle point approximation. The first term oscillates and its integral is exponentially suppressed. And we are left with the second term, which should satisfy the following equation:
\equa{
\int_{-\infty}^\infty dl_a l_a l_b \frac{K^2}{V_J(a,b)}\cos((l_b-l_{b^\prime }) \theta_b) \approx \delta(l_b-l_{b^\prime } )
}
We recall that:
$$
\frac{1}{2\pi} \int_{-\infty}^\infty dl_a \cos(l_a(l_b-l_{b^\prime }))= \delta(l_b-l_{b^\prime })
$$
therefore we can conclude that
\equa{
l_al_b \frac{K^2}{V}=\frac{1}{2\pi} \left| \frac{\partial \theta_b}{\partial l_a} \right|.
}
$\theta_b$ and $l_b$ are so conjugate variables and $K$ comes from the Jacobian of the change of variables between $l_a$ and $\theta_b$.
Computing the derivative of the dihedral angle gives:
\begin{equation}
\frac{\partial \theta_b}{\partial l_a}= \frac{-2}{\sqrt{4l_J^2-l_a^2-l_b^2}}=\frac{-l_al_b}{6V_J(a,b)}
\quad
\Rightarrow
\quad
K=\frac{1}{\sqrt{12\pi}}.
\end{equation}
Moreover, pushing the approximation of the unitary property to the next to leading order in $(l_b-l_{b^\prime})$ and using the next to leading order of the {\{6j\}}-symbol shows that $G^{(1)}=0$. This was already shown in the previous part using the recursion relation and the symmetry properties of the {\{6j\}}-symbol and comes as a confirmation.
\section{``Ward-Takahashi identities" for the spinfoam graviton propagator}
We are interested in the two-point function in 3d quantum gravity for the simplest triangulation given by a single tetrahedron. This provides the first order of the ``spinfoam graviton propagator" in 3d quantum gravity.
Considering the isosceles tetrahedron, we focus on the correlations between the two representations $a$ and $b$:
\begin{equation}
\langle {\cal O}(a) \widetilde{{\cal O}}(b)\rangle_{\psi_J}=
\frac{1}{Z} \,\sum_{a,b}\psi_J(a)\psi_J(b){\cal O}(a) \widetilde{{\cal O}}(b) \{a,b\}_J,
\qquad
Z\equiv\,\sum_{a,b}\psi_J(a)\psi_J(b)\{a,b\}_J,
\end{equation}
where $\psi_J(j)$ is the boundary state, which depends also on the bulk length scale $J$, and ${\cal O},\widetilde{{\cal O}}$ are the observables whose correlation we are studying.
Now, inserting a recursion relation with shifts on $a$, $b$ or $J$ in the sum over the representation labels $\sum_{a,b}$ leads to equations relating the expectation values of different observables. We distinguish two cases: when the state $\psi_J$ does not change or when the length scale $J$ also varies.
\subsection{Relating Observables}
Inserting the recursion relation on $a$-shifts in the definition of the correlation function, we obtain the following exact identity:
\begin{equation}\tabl{ll}{
\langle \frac{\psi_J(a-1)}{\psi_J(a)}{\cal O}(a-1)\widetilde{{\cal O}}(b)(l_a-\f12)(4l_J^2-(l_a-\f12)^2)\rangle_\psi&-
\langle {\cal O}(a)\widetilde{{\cal O}}(b)2l_a(2\cos \theta_a (4l_J^2-l_a^2)+\f14)\rangle_\psi\\
&+ \langle \frac{\psi_J(a+1)}{\psi_J(a)}{\cal O}(a+1)\widetilde{{\cal O}}(b)(l_a+\f12)(4l_J^2-(l_a+\f12)^2)\rangle_\psi
=0.
}\end{equation}
We call this a Ward identity for our spinfoam correlation.
If the observable diverges at $a=0$, more precisely if it contains terms in $1/a$ or in $1/(a+1)$, then we need to take into account extra boundary terms in this equation corresponding to contributions at $a=0$. But all observables usually considered are regular in this sense.
Then one can choose different sets of observables ${\cal O}$ and $\widetilde{{\cal O}}$ and one gets different identities on the correlation functions of the spinfoam model.
For example, taking ${\cal O}(a)=l_a$, we get:
$$ \tabl{ll}{
\langle \frac{\psi_J(a-1)}{\psi_J(a)}\widetilde{{\cal O}}(b)(l_a-1)(l_a-1/2)(4l_J^2-(l_a-1/2)^2)\rangle_\psi&-
\langle \widetilde{{\cal O}}(b)(2\cos \theta_a l_a^2(4l_J^2-l_a^2)+l_a^2/2)\rangle_\psi\\
&+\langle \frac{\psi_J(a+1)}{\psi_J(a)}\widetilde{{\cal O}}(b)(l_a+1)(l_a+1/2)(4l_J^2-(l_a+1/2)^2)\rangle_\psi
=0.
}$$
We recall that the area of the triangle of edge lengths given by $l_a$, $l_J$, $l_J$ is equal to $A(l_a, l_J)= \f14 l_a\sqrt{4l_J^2-l_a^2}$; then $(l_a\pm1)(l_a \pm 1/2)(4l_J^2-(l_a\pm 1/2)^2)=16[A^2(l_a\pm1/2,l_J)\pm \frac{A^2(l_a\pm1/2,l_J)}{2(l_a\pm 1/2)}]$, therefore we can rewrite the previous equation as an equation between correlation functions of the observable $\widetilde{{\cal O}}(b)$ and different observables proportional to the square of the triangle area $A(l_a, l_J)$:
$$ \tabl{ll}{
\langle \frac{\psi_J(a-1)}{\psi_J(a)}[A^2(l_a-1/2,l_J)-\frac{A^2(l_a-1/2,l_J)}{2(l_a- 1/2)}]\widetilde{{\cal O}}(b)\rangle_\psi&-
\langle(2\cos \theta_aA^2(l_a,l_J)+l_a^2/2)\widetilde{{\cal O}}(b)\rangle_\psi\\
&+\langle \frac{\psi_J(a+1)}{\psi_J(a)}[A^2(l_a+1/2,l_J)+\frac{A^2(l_a+1/2,l_J)}{2(l_a+ 1/2)}]\widetilde{{\cal O}}(b)\rangle_\psi
=0.
}$$
The standard choice of boundary is a phased Gaussian \cite{graviton, 3dtoymodel, physical}:
\begin{equation}
\psi_J(j) \,\sim\, e^{i2l_j\vartheta} e^{-2\alpha\frac{(l_j-l_J)^2}{l_J}},
\end{equation}
where $\vartheta$ is a fixed angle defining a posteriori the external curvature of the boundary and $\alpha$ is an arbitrary real positive number (which can be fixed by the requirement of a physical state \cite{physical}).
In this case, we can compute explicitly the ratios $\psi(a\pm1)/\psi(a)$ entering the Ward identity:
$$
\frac{\psi_J(a\pm 1)}{\psi_J(a)}= e^{\pm i2\vartheta} e^{\mp 4 \alpha \frac{l_a-l_J}{l_J}} e^{-\frac{2\alpha}{l_J}}
$$
Of course, this ratios does not depend on $b$; therefore if the observable $\widetilde{{\cal O}}(b)=1$, then the dependence on $b$ only appears in one correlation function through the cosine of the dihedral angle $\theta_a$.
As another example, we consider ${\cal O}(a)=l_a^{-1}$ and $\widetilde{{\cal O}}(b)=\frac{4l_J^2-l_b^2}{(2l_J)^{4}}$, then:
$$\tabl{ll}{
\langle \frac{\psi_J(a-1)}{\psi_J(a)}\, \frac{l_a-1/2}{l_a-1}\,\frac{4l_J^2-(l_a-1/2)^2}{4l_J^2}\, \frac{4l_J^2-l_b^2}{4l_J^2} \rangle_\psi &-2\langle \cos\theta_a\frac{4l_J^2-l_a^2}{4l_J^2}\, \frac{4l_J^2-l_b^2}{4l_J^2} + \frac{1}{16l_J^2}\, \frac{4l_J^2-l_b^2}{4l_J^2} \rangle_\psi \\ &+ \langle \frac{\psi_J(a+1)}{\psi_J(l_a)} \,\frac{l_a+1/2}{l_a+1}\, \frac{(4l_J^2-(l_a+1/2)^2}{4l_J^2}\, \frac{4l_J^2-l_b^2}{4l_J^2}\rangle_\psi=0
}$$
which can be approximated by:
$$
\langle e^{- i2\vartheta} e^{ 4 \alpha \frac{l_a-l_J}{l_J}}\,\Delta((l_a-1/2)^2)\Delta(l_b^2) \rangle_\psi -2e^{\frac{2\alpha}{l_J}}
\langle \cos\theta_a\Delta(l_a^2)\Delta(l_b^2)+ \frac{1}{16l_J^2}\Delta(l_b^2) \rangle_\psi + \langle e^{ i2\vartheta} e^{-4 \alpha \frac{l_a-l_J}{l_J}} \,\Delta((l_a+1/2)^2)\Delta(l_b^2)\rangle_\psi \approx 0
$$
where $\Delta(l_j^2)=\frac{l_j^2-4l_J^2}{4l_J^2}$.
\subsection{Rescaling the Tetrahedron}
We can now vary also the length scale $l_J$. First let's notice that in the same way we wrote an exact recursion relation for the leading order of the isosceles {\{6j\}}-symbol shifting the representation $a$ (equation (\ref{exactrecurLO})), we can write a similar exact recursion relation for the leading order of the {\{6j\}}-symbol shifting the label $J$; that is
\equa{
\sqrt{V_{J+1}(a,b)} \{a,b\}^{\textrm{LO}}_{J+1}-2\cos(4\theta_J)\sqrt{V_{J}(a,b)} \{a,b\}^{\textrm{LO}}_{J}+\sqrt{V_{J-1}(a,b)} \{a,b\}^{\textrm{LO}}_{J-1}=0
}
Inserting this recursion relation on $J-$shifts in the definition correlation function, we obtain the following identity:
\begin{eqnarray}
\langle \sqrt{V_{J+1}(a,b)} \frac{\psi_J(a)\psi_J(b)}{\psi_{J+1}(a)\psi_{J+1}(b)} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi
&+\langle \sqrt{V_{J-1}(a,b)} \frac{\psi_J(a)\psi_J(b)}{\psi_{J-1}(a)\psi_{J-1}(b)} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi&\nonumber\\
&-2\langle \cos(4\theta_J) \sqrt{V_{J}(a,b)} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi &
=0
\end{eqnarray}
The correlation functions appearing in this equation are in fact approximation. We are allowed to use the leading order of the {\{6j\}}-symbol because the boundary state used picks the function on large $j_0$. And for the same reason, we can expand $\sqrt{V_{J\pm1}(a,b)}$ and the ratios $\frac{\psi_J(a)\psi_J(b)}{\psi_{J\pm1}(a)\psi_{J\pm1}(b)}$:
\equa{\tabl{l}{
\langle \sqrt{V_{J}(a,b)}\left(1-\frac{2l_J}{4l_J^2-l_a^2-l_b^2}\right) e^{-4\alpha\frac{(2l_J-(l_a+l_b))}{l_J}[1+\frac{3l_J-2(l_a+l_b)}{2l_J(2l_J-l_a-l_b)}]} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi -2\langle \cos(4\theta_J) \sqrt{V_{J}(a,b)} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi \\
\quad \quad \quad \quad \quad \quad+\langle \sqrt{V_{J}(a,b)}\left(1+\frac{2l_J}{4l_J^2-l_a^2-l_b^2}\right) e^{4\alpha\frac{(2l_J-(l_a+l_b))}{l_J}[1-\frac{3l_J-2(l_a+l_b)}{2l_J(2l_J-l_a-l_b)}]} {\cal O}(a) \widetilde{{\cal O}}(b) \rangle_\psi \approx 0.
}}
We hope that such equation will turn out useful to study the asymptotic properties of the correlations function as the length scale $J$ grows large, but we leave this for future investigation.
\section*{Conclusion}
We have used the recursion relation satisfied by the {\{6j\}}-symbol to study the structure of its asymptotical expansion for large spins. The exact recursion relation allowed us to compute explicit the asymptotical approximation of the isosceles {\{6j\}}-symbol up to fourth order. This confirms previous results \cite{valentin,maite} and introduces techniques allowing further systematic analytical calculations of the corrections to the behavior of the{\{6j\}}-symbol at large spins. However a clear and simple geometrical interpretation of the polynomials appearing in this expansion is still missing, but the differential equations that we provide for these coefficients should be a first step in this direction.
This work is useful in particular for the study of large scale correlations in the spinfoam model for 3d quantum gravity. In this context, the recursion relation allowed us to write equations satisfied by the spinfoam correlations similar to the Ward identities of standard quantum field theory. We hope that such recursion techniques can be further applied to the study of 4d spinfoam amplitudes and the resulting spinfoam graviton propagator \cite{recursion}.
\section*{Ackowledgements}
The numerical simulations and plots were done using Mathematica 5.0.
MD and ER are partially supported by the ANR ``Programme Blanc" grant LQG-06.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 259
|
\section{ Introduction
C.C. Chang \cite{chang1,chang2} introduced MV-algebras to provide an algebraic proof of the completeness theorem of \L ukasiewicz's infinite-valued propositional calculus. D. Mundici \cite{Mun} proved that there is a categorical equivalence between the category of unital Abelian $\ell$-groups and the category of MV-algebras. Today, the theory of MV-algebras is very deep and has many connections with
other algebraic structures and other parts of mathematics with many important applications to different areas (for more details see \cite{cdm, mundici 2}). It is well known that MV-algebras form a variety.
In \cite{Kom}, Y. Komori has described all subvarieties of the variety of MV-algebras. He proved that the lattice of subvarieties of the variety \textsf{MV} of MV-algebras is countably infinite. A. Di Nola and A. Lettieri presented in \cite{DiLe1} an equational base of any subvariety of the variety \textsf{MV} which consists of finitely many MV-equations.
Recently in \cite{Dvz}, we introduced EMV-algebras to generalize MV-algebras and generalized Boolean algebras. An EMV-algebra locally resembles MV-algebras, but a top element is not guaranteed.
Conjunction and disjunction exist but negation exists only in a local sense, i.e. negation of $a$ in $b$ exists whenever $a\le b$, but the total negation of the event $a$ is not assumed. There is an interesting representation for EMV-algebras. Indeed, an EMV-algebra either has a top element or we can find an EMV-algebra $N$ with top element where the original EMV-algebra can be embedded as a maximal ideal of the EMV-algebra $N$, \cite[Thm 5.21]{Dvz}. This result is crucial for our reasoning. The Loomis--Sikorski theorem for these algebras was established in \cite{Dvz1}. States as analogues of finitely additive measures were investigated in \cite{Dvz2} and morphisms and free EMV-algebras were described in \cite{Dvz3}. In \cite{Dvz4}, we showed that every EMV-algebra $M$ is a homomorphic image of $\B(M)$, the generalized Boolean algebra $R$-generated by $M$, where a homomorphism is a homomorphism of generalized effect algebras. Unfortunately, as we proved in \cite{Dvz},
the class of all EMV-algebras, $\mathsf{EMV}$, does not form a variety, since it is not closed under the subalgebra operator.
The main aim of the paper is two-fold: (1) investigate $\mathsf{EMV}$ in order to find the least variety ``containing" $\mathsf{EMV}$, (2) and study Pierce sheaves of EMV-algebras.
(1) For the first purpose, we introduced a new class of algebras, wEMV-algebras, of type (2,2,2,2,0) which form a variety. The language of every EMV-algebra can be naturally extended by a derived binary operation $\ominus$, so we obtain an associated wEMV-algebra corresponding to the original EMV-algebra.
The class of $\mathsf{EMV}_a$ of such associated wEMV-algebras is a proper subclass of wEMV-algebras. Then we define a strict wEMV-algebra and use it to show that each wEMV-algebra $M$ can be embedded into an associated wEMV-algebra. Also, $M$ can be embedded into a direct product of an associated EMV-algebra and a strict wEMV-algebra. Then it is proved that the class of wEMV-algebras is the least subvariety of the variety $\mathsf{wEMV}$ containing $\mathsf{EMV}_a$. We show that every wEMV-algebra without top element can be embedded into an associated wEMV-algebra with top element as its maximal ideal. We describe all subvarieties of wEMV-algebras and we show that they are only countably many.
(2) For the second goal, we note that if $M$ is a bounded EMV-algebra, then $M$ is termwise equivalent to an MV-algebra on $M$, and the Pierce representation of MV-algebras is studied in \cite[Part 4]{georgescu}.
So, in the article, we concentrate to a case when an EMV-algebra $M$ does not have a top element (proper EMV-algebra). Thus we construct a Pierce sheaf of a proper EMV-algebra and we show that a global section of $T=(E_M,\pi,X)$ forms an EMV-algebra. It is proved that a semisimple EMV-algebra under a suitable condition can be embedded into a sheaf of bounded EMV-algebras on the space $X$. Finally, we show that if $(M;\vee,\wedge,\oplus,0)$ is a Stone EMV-algebra, then it can be embedded into the MV-algebra of global sections of a Hausdorff Boolean sheaf whose stalks are MV-chains.
The paper is organized as follows. In Section 2, we gather basic facts on EMV-algebras and their Basic Representation Theorem. Section 3 defines wEMV-algebras which form a variety whereas the class of EMV-algebras not. For every EMV-algebra, we extend its language adding a new derived binary operation $\ominus$, so that it is a wEMV-algebra associated to the original EMV-algebra, and we show that the variety of wEMV-algebras is the least subvariety of the variety of wEMV-algebras containing all associated wEMV-algebras. In addition, we study a decomposition of wEMV-algebras as a direct product. In the last section, we study the Pierce sheaves of EMV-algebras.
\section{Preliminaries
Recently, we have introduced in \cite{Dvz} a common extension of generalized Boolean algebras and of MV-algebras called EMV-algebras.
An algebra $(M;\vee,\wedge,\oplus,0)$ of type $(2,2,2,0)$ is said to be an
{\it extended MV-algebra}, an {\it EMV-algebra} in short, if it satisfies the following conditions:
\begin{itemize}
\item[{\rm (E1)}] $(M;\vee,\wedge,0)$ is a distributive lattice with the least element $0$;
\item[{\rm (E2)}] $(M;\oplus,0)$ is a commutative ordered monoid (with respect to the lattice ordering (E1), i.e. $a\le b$ implies $a\oplus c \le b\oplus c$ for each $c\in M$) with a neutral element $0$;
\item[{\rm (E3)}] for each $a\in \mI(M):=\{x\in M\mid x\oplus x=x\} $, the element
$\lambda_{a}(x)=\min\{z\in[0,a]\mid x\oplus z=a\}$
exists in $M$ for all $x\in [0,a]$, and the algebra $([0,a];\oplus,\lambda_{a},0,b)$ is an MV-algebra;
\item[{\rm (E4)}] for each $x\in M$, there is $a\in \mI(M)$ such that
$x\leq a$.
\end{itemize}
\vspace{1mm}
An EMV-algebra $M$ is called {\em proper} if it does not have a top element. Clearly, if in a generalized Boolean algebra we put $\oplus =\vee$, $\lambda_a(x)$ is a relative complement of $x \in [0,a]$ in $[0,a]$, every generalized Boolean algebra can be viewed as an EMV-algebra where top element is not necessarily assumed. If $1$ is a top element of an EMV-algebra $M$, by (E3), $([0,1];\oplus,\lambda_1,0,1)=(M;\oplus,',0,1)$ is an MV-algebra. Conversely, if $(M;\oplus,',0,1)$ is an MV-algebra, then $(M;\vee,\wedge,\oplus,0)$ is an EMV-algebra with top element $1$. In addition, every EMV-algebra $(M;\vee,\wedge,\oplus,0)$ with top element $1$ is termwise equivalent to an MV-algebra $(M;\oplus,',0,1)$.
We note that in \cite{Dvz}, there a list of interesting examples of EMV-algebras.
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra. Its reduct
$(M;\vee,\wedge,0)$ is a distributive lattice with a bottom element $0$. The lattice structure of $M$ yields a partial order relation on $M$, denoted by $\leq$, that is $x\leq y$ iff $x\vee y=y$ iff $x\wedge y=x$. Also, if $a$ is a fixed idempotent element of $M$, there is a
partial order relation $\preccurlyeq_a$ on
the MV-algebra $([0,a];\oplus,\lambda_a,0,a)$ defined by $x\preccurlyeq_a y$ iff $\lambda_a(x)\oplus y=a$.
By \cite{Dvz3}, we know that for each $x,y\in [0,a]$, we have
\[x\leq y\Leftrightarrow x\preccurlyeq_a y.\]
In addition, if also $x,y \le b\in \mathcal I(M)$, then
\[x\preccurlyeq_a y \Leftrightarrow x\leq y\Leftrightarrow x\preccurlyeq_b y.\]
\begin{prop}\label{2.2}{\rm \cite[Prop 3.9]{Dvz}}
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra and $a,b\in \mI(M)$ such that $a\leq b$.
Then for each $x\in [0,a]$, we have
\begin{itemize}
\item[{\rm (i)}] $\lam_a(x)=\lam_b(x)\wedge a$;
\item[{\rm (ii)}] $\lam_b(x)=\lam_a(x)\oplus \lam_b(a)$;
\item[{\rm (iii)}] $\lam_b(a)$ is an idempotent, and $\lam_a(a)=0$.
\end{itemize}
\end{prop}
\begin{lem}\label{2.3}{\rm \cite[Lem 5.1]{Dvz}}
Let $(M;\vee, \wedge,\oplus,0)$ be an EMV-algebra. For all $x,y\in M$, we define
\begin{equation}\label{eq:odot1}
x\odot y=\lam_a(\lam_a(x)\oplus \lam_a(y)),
\end{equation}
where $a\in\mI(M)$ and $x,y\leq a$. Then $\odot:M\times M\ra M$ is an order preserving, associative well-defined binary operation on $M$ which does not depend on $a\in \mI(M)$ with $x,y \le a$.
In addition, if $x,y \in M$, $x\le y$, then $y \odot \lambda_a(x)=y\odot \lambda_b(x)$
for all idempotents $a,b$ of $M$ with $x,y\le a,b$.
\end{lem}
The following important result on representing EMV-algebras was established in \cite[Thm 5.21]{Dvz}, it generalizes an analogous result for generalized Boolean algebras, see \cite{CoDa}.
\begin{thm}\label{2.4}{\rm [Basic Representation Theorem]}
Every EMV-algebra $M$ either has a top element or $M$ can be embedded into an EMV-algebra $N$ with top element as a maximal ideal of $N$ such that every element $x\in N$ is either the image of some element from $M$ or $x$ is the complement of the image of some element from $M$.
\end{thm}
The EMV-algebra $N$ with top element in the latter theorem is unique up to isomorphism and it is said to be {\it representing} the EMV-algebra $M$.
For more details, we refer to \cite{Dvz}.
An {\it MV-algebra} is an algebra $(M;\oplus,',0,1)$ (henceforth written simply as $M=(M;\oplus,',0,1)$) of type $(2,1,0,0)$, where $(M;\oplus,0)$ is a
commutative monoid with the neutral element $0$ and for all $x,y\in M$, we have:
\begin{enumerate}
\item[(i)] $x''=x$;
\item[(ii)] $x\oplus 1=1$;
\item[(iii)] $x\oplus (x\oplus y')'=y\oplus (y\oplus x')'$.
\end{enumerate}
\noindent
In any MV-algebra $(M;\oplus,',0,1)$,
we can also define the following operations:
\begin{equation}\label{eq:oplus}
x\odot y:=(x'\oplus y')',\quad x\ominus y:=(x'\oplus y)'.
\end{equation}
\noindent
We note that any MV-algebra is a distributive lattice where $x\oplus (x\oplus y')'=x\vee y = y\oplus (y\oplus x')'$ and $x\wedge y =x\odot (x'\oplus y)$.
Prototypical examples of MV-algebras are connected with unital $\ell$-groups, i.e. with couples $(G,u)$, where $G$ is an Abelian $\ell$-group with a fixed strong unit of $u\in G^+$. If we set $[0,u]=\{g\in G\mid 0\le g\le u\}$, then $\Gamma(G,u)=([0,u];\oplus,',0,u)$, where $x\oplus y := (x+y)\wedge u$ and $a':=u-x$, $x,y \in [0,u]$, is an MV-algebra, and every MV-algebra is isomorphic to a unique $\Gamma(G,u)$.
\section{A variety containing EMV-algebras
As we showed in \cite[Sec 3]{Dvz}, the class of EMV-algebras, $\mathsf{EMV}$, is not closed under subalgebras so it is neither a variety nor a quasivariety with respect to the original EMV-operations. In the section, we introduce a new class of algebras called wEMV-algebras. If we extend the language of EMV-algebras adding a derived binary operation $\ominus$, we obtain an associated wEMV-algebra, and the variety of wEMV-algebras contains the class of associated wEMV-algebras, $\mathsf{EMV}_a$. We show that
this class is the least subvariety of the variety $\mathsf{sEMV}$ containing $\mathsf{EMV}_a$. In addition, we find some properties of this class as a direct decomposition of an wEMV-algebra to two its factors.
\begin{defn}\label{WEMV}
An algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ of type (2,2,2,2,0) is called a {\em wEMV-algebra} (w means weak) if it
satisfying the following conditions:
\begin{enumerate}
\item[(i)] $(M, \vee, \wedge, 0)$ is a distributive lattice with the least
element $0$;
\item[(ii)] $(M; \oplus, 0)$ is a commutative monoid;
\item[(iii)] $(x \oplus y) \ominus x \leq y$;
\item[(iv)] $x \oplus (y \ominus x) = x \vee y$;
\item[(v)] $x \ominus (x \wedge y) = x \ominus y$;
\item[(vi)] $z \ominus (z \ominus x) = x \wedge z$;
\item[(vii)] $z \ominus (x \vee y) = (z \ominus x) \wedge (z\ominus y)$;
\item[(viii)] $(x\wedge y)\ominus z=(x\ominus z)\wedge (y\ominus z)$;
\item[(ix)] $x\ominus (y\oplus z)=(x\ominus y)\ominus z$;
\item[(x)] $x\oplus (y\vee z)=(x\oplus y)\vee (x\oplus z)$.
\end{enumerate}
\end{defn}
An {\it idempotent} of a wEMV-algebra $M$ is any element $x\in M$ such that $x=x\oplus x$. We denote by $\mathcal I(M)$ the set of all idempotents of $M$, then $0\in \mathcal I(M)$. It can happen that $\mathcal I(M)=\{0\}$ as in Example \ref{ex:1} below.
In the following, we present some important examples of wEMV-algebras.
\begin{exm}\label{ex:1}
If $M=G^+$ is the positive cone of an Abelian $\ell$-group $G$, and if we define on $G^+$ two operations $x\oplus y:= x +y$ and $x\ominus y :=(x - y)\vee 0$, $x,y \in G^+$, then $(M;\vee,\wedge,\oplus,\ominus,0)$ is an example of a wEMV-algebra, called also a {\it wEMV-algebra of a positive cone}. Moreover, it can be embedded into the MV-algebra $N:=\Gamma(\mathbb Z \lex G,(1,0))$ as an maximal ideal of $N$, and every element of $N$ is either $(0,g)$ for some $g \in G^+$ or $(0,g)'=(1,0)-(0,g)$ for some $g\in G^+$.
\end{exm}
\begin{exm}\label{ex:2}
Let $(M;\oplus,',0,1)$ be an MV-algebra. If we set $x\ominus y:= x\odot y'$, $x,y \in M$, then $(M;\vee,\wedge,\oplus,\ominus,0)$
is a wEMV-algebra with a top element $1$.
\end{exm}
\begin{exm}\label{ex:3}
Consider an arbitrary proper EMV-algebra $(M;\vee,\wedge,\oplus,0)$. By Theorem \ref{2.4},
$M$ can be embedded into an EMV-algebra $N_0$ with top element as a maximal ideal of $N_0$. Then
$(N_0;\oplus,\lambda_1,0,1)$ is an MV-algebra. For simplicity, we use $x'$ instead of $\lambda_1(x)$, for all $x\in N_0$.
Let $\ominus$ be the well-known operation on $N_0$, that is
$x\ominus y=(x'\oplus y)'$ for all $x,y\in N_0$. Since $M$ is an ideal of $N_0$, then $M$ is closed under $\ominus$.
So, we have an example of a wEMV-algebra without top element.
\end{exm}
Combining Examples \ref{ex:2}--\ref{ex:3} with the Basic Representation Theorem, we see that if $(M;\vee,\wedge,\oplus,0)$ is an arbitrary EMV-algebra, extending its language with a binary operation $\ominus$, we obtain a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$; it is said to be a {\it wEMV-algebra associated} with the EMV-algebra $(M;\vee,\wedge,\oplus,0)$; simply we say $M$ is an {\it associated wEMV-algebra}. We denote by $\mathsf{EMV}_a$ the class of associated wEMV-algebras $(M;\vee,\wedge,\oplus,\ominus,0)$, where $(M;\vee,\wedge,\oplus,0)$ is any EMV-algebra. By a way, it is possible to show that if $x,y \le a$, where $a$ is an idempotent of $M$, then
\begin{equation}\label{eq:ominus}
x\ominus y = \lambda_a(\lambda_a(x)\oplus y),
\end{equation}
and it does not depend on the idempotent $a$. The same is true if we define
\begin{equation}\label{eq:odot}
x\odot y =\lambda_a(\lambda_a(x)\oplus\lambda_a y).
\end{equation}
\begin{exm}\label{exm3.14}
Let $\{(M_i;\vee,\wedge,\oplus,\ominus,0)\}$ be a family of wEMV-algebras. Then we can easily prove that
$\sum_{i\in I}M_i=\{f\in \prod_{i\in I}M_i\mid \supp(f) \mbox{ is finite} \}$
with the componentwise operations form a wEMV-algebra. Recall that $\supp(f)=\{i\in I\mid f(i)\neq 0\}$.
\end{exm}
Let $a$ be an element of a wEMV algebra $(M;\vee,\wedge,\oplus,\ominus,0)$. We define
$$
0.a=0,\quad (n+1).a= (n.a)\oplus a,\ n\ge 0.
$$
Now, we present some properties of wEMV-algebras.
\begin{prop}\label{pr:1}
Basic properties of a wEMV-algebra are as follows:
\begin{enumerate}
\item[{\rm (a)}] $(M;\oplus,0)$ is an ordered monoid, i.e. $x\le y$ implies $x\oplus z\le y\oplus z$ for each $z\in M$. Moreover, $x\le y$ iff there is $a\in M$ such that $y=x \oplus a$.
\item[{\rm (b)}] If $x\le y$, then $x\ominus z\le y\ominus z$ for each $z\in M$. If $z_1\le z_2$, then $x\ominus z_2\le x\ominus z_1$ for each $x \in M$.
\item[{\rm (c)}] $z\ominus x\le z$ for all $x,z\in M$.
\item[{\rm (d)}] $x\vee y\le x\oplus y$, $x,y \in M$.
\item[{\rm (e)}] If $x\le z$, then $(z\ominus x)\oplus x = z$ and $(z\ominus (z\ominus x))=x$
\item[{\rm (f)}] If $x\le z$, then
$z\ominus x= \min\{y\in [0,z]\colon x\oplus y=z\}$.
\item[{\rm (g)}] For all $x,z \in M$, we have $z\ominus x=\min\{t \in [0,z]\colon t\oplus (z\wedge x)=z\}$.
\item[{\rm (h)}] For $x,y,z \in M$, there holds $z \le x\oplus y$ if and only if $z\ominus x\le y$.
\item[{\rm (i)}] $z\ominus 0=z$ and $z\ominus z =0$ for each $z\in M$. Moreover, $z\ominus y=0$ if and only if $z\le y$.
\item[{\rm (j)}] If $x\le z$ and $z\ominus x=0$, then $z=x$.
\item[{\rm (k)}] If $a$ and $b$ are two different atoms of $M$, then $a\vee b = a\oplus b$.
\item[{\rm (l)}] If $a_1,\ldots,a_n$ are mutually different atoms of $M$, then $a_1\vee \cdots \vee a_n = a_1\oplus \cdots \oplus a_n$.
\end{enumerate}
\end{prop}
\begin{proof}
(a) Definition \ref{WEMV}(x) implies that $(M;\oplus,0)$ is an ordered monoid. Let $x\le y$, then by (iv), $y=x\vee y= x\oplus (y\ominus x)$; we put $a=y\ominus x$. Conversely, let $y= x\oplus a$ for some $a \in M$. Then
$x\vee y = x\oplus (y\ominus x)\ge x\oplus 0=x$, i.e. $x\le x\oplus y$. Similarly, $y\le x\oplus y$. Now, let $y=x\oplus a$ for some $a\in M$. Then $y=x\oplus a\ge x\oplus 0=x$.
(b) It follows from (vii) and (viii).
(c) We have $z\wedge (z\ominus x)=z\ominus (z\ominus (z\ominus x))=z\ominus (x\wedge z)=z\ominus x$, i.e. $z\ominus x\le z$.
(d) $x\oplus y\ge x\oplus 0=x$, i.e. $x\le x\oplus y$. Similarly, $y\le x\oplus y$. Hence, $x\vee y \le x\oplus y$.
(e) Applying (iv), we have $ (z\ominus x)\oplus x=z\vee x=z$.
(f) By (e), $z\ominus x \in \{y\in [0,z] \mid x \oplus y=z\}$. If $x\oplus y=z$, then $z\leq x\oplus y$ which implies that
\[(z\ominus x)\wedge ((x\oplus y)\ominus x)=(z \wedge (x\oplus y))\ominus
x=z\ominus x.\]
That is, $z\ominus x\leq (x\oplus y)\ominus x\leq y$, by (iii). Hence, (f) is proved.
(g) Let $x,y \in M$. By (v), $z\ominus x = z\ominus (x\wedge z)$. Applying (f), we have establish (g).
(h) Let $z\le x \oplus y$. By (b), $z\ominus x\le (x\oplus y)\ominus x \le y$, by (iii).
Conversely, let $z\ominus x \le y$. Using (iv), we get $z\le z\vee x = (z\ominus x)\oplus x \le y \oplus x \le x \oplus y$.
(i) Check $z\ominus 0= (z\ominus 0)\oplus 0 = z\vee 0 = z$. On the other side, $z\ominus z = (z\oplus 0)\ominus z \le 0$. The second part follows from (v) and the first part of the present proof of (i): $z\ominus y = z\ominus (z\wedge y)=0$ iff $z\wedge y = z$.
(j) Let $x\le z$ and $z\ominus x=0$, then by (e) and (i), we have $x=z\ominus (z\ominus x)= z\ominus 0=x$.
(k) Due to (b), we have $(a\vee b) \ominus a \le (a\oplus b)\ominus a \le b$. If $(a\vee b) \ominus a=0$, then (j) entails $a=a\vee b$, so that $b\le a$ which means $a=b$, a contradiction. Whence, $(a\vee b) \ominus a=b$. Then $a\vee b= ((a\vee b) \ominus a)\oplus a = b\oplus a=a\oplus b$.
(l) We proceed by induction. Due to (k), the statement holds for $n=2$. Assume that it holds for each integer $i\le n$, i.e. $a_1\vee \cdots \vee a_i=a_1\oplus \cdots \oplus a_i$. Set $b_n= a_1\vee \cdots \vee a_n = a_1\oplus \cdots \oplus a_n$. Check
$$
(b_n \vee a_{n+1})\ominus b_n \le (b_n \oplus a_{n+1})\ominus b_n \le a_{n+1}.
$$
There are two cases: First $(b_n \vee a_{n+1})\ominus b_n= 0$. Then by (j), $b_n\vee a_{n+1} = b_n$ and $a_{n+1}\le b_n = a_1\vee\cdots\vee a_n$. Distributivity implies $a_{n+1}= (a_{n+1}\wedge a_1) \vee \cdots\vee (a_{n+1}\wedge a_n) = 0$ which is a contradiction. Therefore, we have the second case $(b_n \vee a_{n+1})\ominus b_n= (b_n\oplus a_{n+1})\ominus b_n = a_{n+1}$ which yields
$$
b_n \vee a_{n+1} = ((b_n \vee a_{n+1})\ominus b_n)\oplus b_n = a_{n+1}\oplus b_n
$$
as claimed.
\end{proof}
\begin{lem}\label{le:5.2}
Let $a$ be an atom of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ and $b$ an arbitrary element of $M$.
If there is an integer $n\ge 0$ such that $b\le n.a$, then $b = m.a$ for some $m\ge 0$.
\end{lem}
\begin{proof}
(1) We first show that either $(n+1).a$ is a cover of $n.a$ for each $n\ge 0$ or $n.a$ is an idempotent for some $n\ge 1$.
Assume that there is an integer $n\ge 0$ such that $n.a< b\le (n+1).a$. We show that $b=(n+1).a$. By Proposition \ref{pr:1}(a), there is an element $c\in M$ such that $b= (n.a)\oplus c$. Hence, $((n.a)\oplus c)\ominus (n.a)\le (a\oplus n.a)\ominus (n.a)\le a$. There are two cases: Either $((n.a)\oplus c)\ominus (n.a) = 0$ or $((n.a)\oplus c)\ominus (n.a)=a$. In the first one, we have
\begin{align*}
n.a &= \left(\left(\left(n.a\right)\oplus c\right)\ominus \left(n.a\right)\right)\oplus (n.a)\\
&=\left(\left(n.a\right)\oplus c\right)\vee \left(n.a\right)= \left(n.a\right)\oplus c = b,
\end{align*}
which is a contradiction. Hence, we have the second case $((n.a)\oplus c)\ominus (n.a)=a$ which yields
\begin{align*}
&\left(\left(\left(n.a\right)\oplus c\right)\ominus \left(n.a\right)\right)\oplus (n.a)=(n+1).a\\
&\left(\left(n.a\right)\oplus c\right)\vee (n.a)= (n+1).a\\
&b= (n.a)\oplus c = (n+1).a.
\end{align*}
Now, let $b\leq n.a$. If $n=1$, then $b=0=0.a$ or $b=a=1.a$. Let $n\geq 2$ and let, for all $k<n$, we have $b\leq k.a$ implies that $b=m.a$ for some $m\leq k$.
We assume that $m.a\neq (m+1).a$ for all $m<n$. Otherwise, the proof follows from the assumption.
Consider the elements $b\wedge a$, $b\wedge 2.a,\ldots, b\wedge (n-1).a$ and $b\wedge n.a$.
(2) If there exist integers $k$ and $m$ with $k<m<n$ such that $b\wedge m.a=k.a$, we add $(n-m).a$ to each side of the equation, so that $(b\oplus (n-m).a)\wedge n.a=k.a\oplus (n-m).a$. Therefore, $b\leq (b\oplus (n-m).a)\wedge n.a=k.a\oplus (n-m).a\leq (n-1).a$ and
by the assumption, $b=t.a$ for some integer $t\leq n$.
(3) From $b\wedge a\leq a$ we get that $b\wedge a=0$ or $b\wedge a=a$. If
$b\wedge a=0$, then $b=b\wedge n.a\leq n.(b\wedge a)=0$. If $b\wedge a=a$, then we have $b=a$ or $a<b$.
If $b=0$ and $b=a$, then we have nothing to prove.
Otherwise, $a< b$. Then $a\leq b\wedge 2.a\leq 2.a$, which implies that $b\wedge 2.a=a$ or $b\wedge 2.a=2.a$.
The condition $b\wedge 2.a=a$ by (2) imply that $b=m.a$ for some $m\leq n$.
If $b\wedge 2.a=2.a$, then
$b\geq 2.a$ and so $b=2.a$ or $b>2.a$. Now, consider $2.a\leq b\wedge 3.a\leq 3.a$. In a similar way, we can show that
$b=m.a$ for some $m\leq n$ or $3.a\leq b\wedge 4.a\leq 4.a$.
By finite calculations, we get that $b=m.a$ for some $m\leq n$ or $(n-1).a\leq b\wedge n.a\leq n.a$. It follows that $(n-1).a=b$ or $b=n.a$.
\end{proof}
\begin{prop}\label{pr:5.3}
Let $a$ be an atom of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$. Let $\ominus$ on $\mathbb N$ denote the truncate difference, i.e. $m\ominus n=(m-n)\vee 0$, $m,n \in \mathbb N$.
If we denote by $M_a:=\{m.a\mid m\ge 0\}$, then $M_a$ is a subalgebra of $M$.
In addition, if there is the least integer $m_0$ such that $m_0.a$ is an idempotent element of $M$, then $M_a=\{0,a,\ldots,m_0.a\}$ is an EMV-algebra that is termwise equivalent to the MV-algebra $(M_a;\oplus,\lambda_{a_0},0,m_0.a)$ that is isomorphic to $\Gamma(\frac{1}{m_0}\mathbb Z,1)$, and $m.a\ominus n.a = (m\ominus n).a$ for each $0\le m,n \le m_0$.
Otherwise, $M_a $ is isomorphic to the wEMV-algebra $(\mathbb Z^+;\vee,\wedge,\oplus,0)$, where $m\oplus n=m+n$, $m,n \in \mathbb Z^+$.
\end{prop}
\begin{proof}
Due to Lemma \ref{le:5.2}, it is clear that $M_a$ is closed under $0,\vee,\wedge, \oplus$. We show that it is closed also under $\ominus$. If $m\le n$, then clearly $m.a\ominus n.a=0=(m\ominus n).a$. In the rest, we assume that $m>n$.
(1) First, let $m_0$ be the least integer $m$ such that $m.a$ is an idempotent of $M$. Let $0\le m,n\le m_0$. Let $n=m-i$ for some $i=1,\ldots,n$. Then $m.a \ominus (m-i).a=j.a$, where $j=0,\ldots,i$. Assume that $j<i$. Then $(m.a \ominus (m-i).a)\oplus (m-i).a = m.a= j.a\oplus (m-i).a=(m+j-i).a <m.a$ when we apply Lemma \ref{le:5.2}. This gives a contradiction, so that $j=i$ and $m.a\ominus n.a= (m-n).a$. Clearly, $M_a$ corresponds to $\Gamma(\frac{1}{m_0}\mathbb Z,1)$.
(2) Now, let any $m.a$ be no idempotent and let $m>n$. Then $m.a\ominus n.a=i.a$ for some integer $i>0$ and due to Lemma \ref{le:5.2}, every $(k+1).a$ is a cover of $k.a$, $k\ge 0$. Hence, as at the end of (1), we conclude that $m.a\ominus n.a=(m-n).a=(m\ominus n).a$ whenever $m>n$.
This implies that $k.a\oplus l.a=(k+l).a$ and $M_a$ is isomorphic to the wEMV-algebra $(\mathbb Z^+;\vee,\wedge,\oplus,0)$ of the positive cone $\mathbb Z^+$, see Example \ref{ex:1}.
\end{proof}
\begin{lem}\label{lem3.1}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra and let $a$ be an arbitrary element of $M$. For each $x,y,z\in [0,a]$ we have:
\begin{itemize}
\item[{\rm (i)}] $a\ominus (a\ominus x)=x$;
\item[{\rm (ii)}] $x\wedge y=a\ominus((a\ominus x)\vee (a\ominus y))$;
\item[{\rm (iii)}] $(x\wedge y)\oplus z=(x\oplus z)\wedge (y\oplus z)$;
\item[{\rm (iv)}] $z\ominus (x\wedge y)=(z\ominus x)\vee (z\ominus y)$.
\end{itemize}
\end{lem}
\begin{proof}
Let $x,y,z\in [0,a]$. Then
(i) It follows from Definition \ref{WEMV}(iv).
(ii) By (i), $x\wedge y=\left(a\ominus \left(a\ominus x\right) \right)\wedge \left(a\ominus \left(a\ominus y\right)\right)=
a\ominus \left(\left(a\ominus x\right)\vee \left(a\ominus y\right)\right)$.
(iii) By Proposition \ref{pr:1}(a), $(x\wedge y)\oplus z\leq x\oplus z, x\oplus y$. Now, let $w\in M$ such that $w\leq x\oplus z, x\oplus y$.
Proposition \ref{pr:1}(h) implies that $w\ominus z\leq x\wedge y$ $(w\ominus z)\oplus z\leq (x\wedge y)\oplus z$ and so
$w\leq (x\wedge y)\oplus z$ (by Definition \ref{WEMV}(iv)). Therefore, $(x\wedge y)\oplus z=(x\oplus z)\wedge (y\oplus z)$.
(iv) By Proposition \ref{pr:1}(b), $z\ominus(x\wedge y)\geq z\ominus x,z\ominus y$. Now, let $u\in M$ such that
$u\geq z\ominus x,z\ominus y$. Then $u\oplus x,u\oplus y\geq z$ which imply that
$u\oplus (x\wedge y)=(u\oplus x)\wedge (u\oplus y)\geq z$ (by (iii)). Now, by Proposition \ref{pr:1}(h),
$u\geq z\ominus (x\wedge y)$. It follows that $z\ominus (x\wedge y)=(z\ominus x)\vee (z\ominus y)$.
\end{proof}
\begin{prop}\label{prop3.2}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra. For each $a\in M$, $(M;\oplus_a,\lambda_a,0,a)$ is an MV-algebra, where for each
$x,y\in [0,a]$,
\[x\oplus_a y=(x\oplus y)\wedge a,\quad \& \quad \lambda_a(x)=a\ominus x.\]
Moreover, if we put $x\ominus_a y:= a\ominus ((a\ominus x)\oplus_a y))$, see {\rm (\ref{eq:oplus})}, then $x\ominus_a x = x\ominus y$.
\end{prop}
\begin{proof}
Put $x,y,z\in [0,a]$.
\begin{eqnarray*}
(x\oplus_a y)\oplus_a z&=& \left(\left(\left(x\oplus y\right)\wedge a\right)\oplus z \right)=
\left(\left(\left(x\oplus y\right)\oplus z\right)\wedge \left(a\oplus z\right) \right)\wedge a,
\mbox{ by Lemma \ref{lem3.1}(iii)}\\
&=& \left(\left(x\oplus y\right)\oplus z\right)\wedge a.
\end{eqnarray*}
In a similar way, $x\oplus_a (y\oplus_a z)=(x\oplus (y\oplus z))\wedge a$ and so $\oplus_a$ is associative.
Now, we can easily show that $([0,a];\oplus_a,0)$ is a commutative ordered monoid with the neutral element $0$,
and $([0,a];\vee,\wedge,0,a)$ is a bounded
distributive lattice.
We know that $(x\ominus y)\oplus y=x\vee y\leq a$, so
\begin{equation}\label{Eq3.2.1}
(x\ominus y)\oplus y=((x\ominus y)\oplus y)\wedge a=(x\ominus y)\oplus_a y.
\end{equation}
On the other hand,
\begin{eqnarray}\label{Eq3.2.2}
a\ominus \left(\left(\left(a\ominus x\right)\oplus y\right)\wedge a\right)&=&
\left(a\ominus \left(\left(a\ominus x\right)\oplus y \right)\right)\vee (a\ominus a),\mbox{ by Lemma \ref{lem3.1}(iv),}\nonumber \\
&=& \left(a\ominus \left(\left(a\ominus x\right)\oplus y \right)\right)\vee 0=(a\ominus (a\ominus x))\ominus y,\mbox{ by Definition \ref{WEMV}(ix)}\nonumber \\
&=& x\ominus y, \mbox{ by Lemma \ref{lem3.1}(i)}.
\end{eqnarray}
It follows from (\ref{Eq3.2.2}) that
\begin{eqnarray}\label{Eq3.2.3}
y\oplus_a \left(a\ominus \left(\left(a\ominus x\right)\oplus_a y\right) \right)=y\oplus_a(x\ominus y)=y\oplus(x\ominus y)=x\vee y,\mbox{ by (\ref{Eq3.2.1})}.
\end{eqnarray}
In a similar way, $x\oplus_a \left(a\ominus \left(x\oplus_a \left(a\ominus y\right)\right) \right)=x\vee y$.
Finally, let $x,y \le a$. Check
\begin{align*}
x\ominus_a y&=a\ominus ((a\ominus x)\oplus_a y))= a\ominus ((a\ominus x)\oplus y)\wedge a)\\
&= [a\ominus ((a\ominus x)\oplus y)]\vee (a\ominus a)=(a\ominus (a\ominus x))\ominus y\\
&=x\ominus y.
\end{align*}
\end{proof}
Recall that if $f:M_1\ra M_2$ is a map between wEMV-algebras $(M_1;\vee,\wedge,\oplus,\ominus,0)$ and $(M_2;\vee,\wedge,\oplus,\ominus,0)$, then $f$ is a wEMV-homomorphism if $f$ preserves the operations $\vee$, $\wedge$, $\oplus$, $\ominus$ and $0$. Moreover, a non-empty subset $S$ of $M_1$ is a wEMV-subalgebra of the wEMV-algebra $M_1$ if it is closed under operations $\vee$, $\wedge$, $\oplus$, $\ominus$.
Consider the class $\mathsf{wEMV}$ of all wEMV-algebras.
Clearly, $\mathsf{wEMV}$ is a variety. Due to \cite[Thm 12.5]{BuSa}, this variety is even arithmetical which can be demonstrated by the Pixley term $m(x,y,z):=\left(\left(x\ominus y\right)\oplus
z\right)\wedge \left(\left(\left(z\ominus y\right)\oplus x\right)\wedge
\left(x\vee z\right)\right)$.
By Proposition \ref{prop3.2}, we can easily show that $\mathsf{wEMV}$ contains $\mathsf{EMV}_a$, the
class of all wEMV-algebras which are associated with EMV-algebras (for more details see \cite{Dvz}). There is a natural question.
``Is $\mathsf{EMV}_a$ a proper subclass of $\mathsf{wEMV}$?"
According to the following example the answer to this question is positive.
\begin{exm}\label{exm3.3}
Consider the positive cone $M:=G^+$ of a non-trivial Abelian $\ell$-group $G$. Define $x\ominus y:=0\vee (x-y)$ and $x\oplus y :=x+y$. According to Example \ref{ex:1}, $(M;\vee,\wedge,+,\ominus,0)$ is a wEMV-algebra. But, its reduct $(M;\vee,\wedge,+,0)$ is not an EMV-algebra, since
for each $x\in M\setminus\{0\}$, we have $x<x+x$, so that for every $x\in G^+\setminus \{0\}$, there is no idempotent $a\in M$ such that $x\le a$, see (E4).
Therefore, $\mathsf{EMV}_a$ is a proper subclass of $\mathsf{wEMV}$.
\end{exm}
A non-empty subset $I$ of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ is called an {\it ideal} if $I$ is a down set which is closed under $\oplus$. Clearly, by Proposition \ref{pr:1}, we can easily see that $I$ is closed under the operations $\vee$, $\wedge$ and $\ominus$, too. An ideal
$P$ of the wEMV-algebra $M$ is {\it prime} if $x\wedge y\in P$ implies that $x\in P$ or $y\in P$. The set of all prime ideals of $M$ is denoted by
$Spec(M)$.
\begin{lem}\label{lem3.4}
In each wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ the following inequality holds:
\begin{equation}
\label{eq3.4} x\ominus z\leq (x\ominus y)\oplus (y\ominus z).
\end{equation}
\end{lem}
\begin{proof}
Let $x,y,z\in M$. Put $a\in M$ such that $x\oplus y \oplus z\leq a$.
Then by Proposition \ref{prop3.2}, consider the MV-algebra $([0,a],\oplus_a,\lambda_a,0,a)$. Let $\ominus_a$ be the well-known binary operation
in this MV-algebra, that is $x\ominus_a y=\lambda_a(\lambda_a(x)\oplus_a y)$ for all $x,y\leq a$. Then by Lemma \ref{lem3.1}(iv) and Definition \ref{WEMV}(iv), we have
\begin{eqnarray}
\label{eq3.4.1} x\ominus_a y&=&a\ominus\left(\left(\left(a\ominus x\right)\oplus y\right)\wedge a\right)=\left(a\ominus\left(\left(a\ominus x\right)\oplus y\right)\right) \vee \left(a\ominus a\right)\nonumber\\
&=&\left(\left(a\ominus x\right)\oplus y\right)=\left(a\ominus (a\ominus x)\right)\ominus y=x\ominus y.
\end{eqnarray}
So, the result follows directly since in the MV-algebra $([0,a],\oplus_a,\lambda_a,0,a)$ we have
\begin{equation*}
x\ominus_a z\leq (x\ominus_a y)\oplus_a (y\ominus_a z)\leq (x\ominus_a y)\oplus (y\ominus_a z)=(x\ominus y)\oplus (y\ominus z).
\end{equation*}
\end{proof}
\begin{prop}\label{prop3.5}
Let $I$ be an ideal of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$.
Then the relation $\thi:=\{(x,y)\in M\times M\mid x\ominus y,y\ominus x\in I\}$ is a congruence relation on $M$.
\end{prop}
\begin{proof}
Clearly, $\thi$ is reflexive and symmetric. Transitivity follows from Lemma \ref{lem3.4}.
Let $z\in M$ and $(x,y)\in \thi$. Put $a\in M$ such that
$x\oplus y,x\oplus z,y\oplus z\leq a$.
By Proposition \ref{prop3.2}, $([0,a],\oplus_a,\lambda_a,0,a)$ is an MV-algebra.
Clearly, $I_a:=I\cap [0,a]$ is an ideal of this
MV-algebra. Let $*\in\{\vee,\wedge,\ominus\}$. Then $x*z,y*z\in [0,a]$ and we can easily seen
that $(x*z)\ominus_a(y*z)\in I_a\s I$,
(since $I_a$ is an ideal of the MV-algebra $[0,a]$). In a similar way, $(x*z)\ominus_a(y*z)\s I$.
From equation (\ref{eq3.4.1}), we have
\begin{eqnarray}\label{eq3.5}
(x*z,y*z)\in \thi,\quad *\in\{\vee,\wedge,\ominus\}.
\end{eqnarray}
On the other hand, since $x\oplus y\leq a$, then $(x\oplus y)\in I_a$ and $(x\oplus_a z)\ominus_a(y\oplus_a z)\in I_a$. Now,
$x\oplus z,y\oplus z\leq a$ and equation (\ref{eq3.4.1}) imply that $(x\oplus z)\ominus (y\oplus z)\in I_a\s I$.
By the similar way, we can prove that $(y\oplus z)\ominus (x\oplus z)\in I$.
Therefore, $\thi$ is a congruence relation on the wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$.
\end{proof}
Let $I$ be an ideal of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$. The set of all congruence classes with respect to $I$ is denoted by $M/I$. Clearly, $M/I$ together with the natural operations forms a wEMV-algebra, see Proposition \ref{prop3.5}. For simplicity, we use
$x/I$ to denote the class $x/\thi$. Therefore, $(M/I;\vee,\wedge,\oplus,\ominus,0/I)$ is a wEMV-algebra which is called the {\it quotient wEMV-algebra} with respect to $I$.
\begin{prop}\label{prop3.6}
Let $P$ be a prime ideal of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$. Then the lattice $(M/P;\vee,\wedge)$ is a chain.
\end{prop}
\begin{proof}
Let $x,y\in M$. There is $a\in M$ such that $x\oplus y\leq a$. Consider the MV-algebra $([0,a];\oplus_a,\lambda_a,0,a)$. Similarly to the proof of
Proposition \ref{prop3.5}, we can show that $P_a:=P\cap [0,a]$ is a prime ideal of the MV-algebra $[0,a]$ and so the quotient MV-algebra
$[0,a]/P_a$ is a chain (with the natural operations). Without loss of generality, we assume that $x/P_a\leq y/P_a$.
Then $(x\ominus_a y)\in P_a\s P$ and so by equation (\ref{eq3.4.1}) $x\ominus y\in P$. Now, we can easily conclude that
$x/P\leq y/P$ on the wEMV-algebra $(M/P;\vee,\wedge,\oplus,\ominus,0/P)$.
\end{proof}
We can easily check that the converse of Proposition \ref{prop3.6} is also true, that is if $E/I$ is a chain, then $I$ is a prime ideal.
A wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ with no non-zero idempotent element is called a {\em strict wEMV-algebra}.
\begin{prop}\label{prop3.7}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a linearly ordered wEMV-algebra. Then it is strict or $M$ is termwise equivalent to an EMV-algebra with
a top element.
\end{prop}
\begin{proof}
Assume that $M$ is not strict. Then there exists an idempotent element $a\in M\setminus \{0\}$. We claim that $M=[0,a]$. Otherwise, put
$x\in M\setminus [0,a]$. Set $b:=(x\oplus a)\oplus (a\oplus x)$. By Proposition \ref{prop3.2}, $([0,b],\oplus_b,\lambda_b,0,b)$ is an MV-algebra containing $a$ in which $a\oplus_b a=(a\oplus a)\wedge b=a\oplus a=a$ and $b\oplus_b b=b$. That is, $0\leq a\leq b$ is a chain of Boolean elements
in the MV-chain $([0,b],\oplus_b,\lambda_b,0,b)$. It follows that $a=b$ and so $x\leq a$ which is a contradiction. Therefore, $M=[0,a]$.
\end{proof}
In each wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$, we can easily check that, for each ideal $I$ of $M$ and each non-empty subset
$S\s M$, the ideal of $M$ generated by $\{I\cup S\}$ is the set
$\{x\in M\mid x\leq a\oplus x_1\oplus\cdots \oplus x_n,~ \exists\, n\in\mathbb{N},\exists\, a\in I, \exists\, x_1,\ldots x_n\in S \}$.
Now, let $z\in M\setminus I$. Let $T$ be the set of all ideals of $M$ containing $I$ such that $z\in M\setminus J$. By Zorn's lemma,
$T$ has a maximal element, say $P$. Clearly, $z\notin P$. Let $x\wedge y\in P$ for some $x,y\in M$. We claim that $x\in P$ or $y\in P$.
Otherwise, $z\in\langle P\cup \{x\}\rangle$ and $z\in\langle P\cup \{y\}\rangle$. Then there exist $n\in\mathbb{N}$ and $u,v\in P$ such that $z\leq u\oplus n.x$ and $z\leq v\oplus n.y$.
Let $b\in M$ be such that $2n.(u\oplus v)\oplus n^2.(x\oplus y)\leq b$.
Consider the MV-algebra $([0,b];\oplus_b,\lambda_b,0,b)$. Then we have $z\leq (u\oplus n.x)\wedge (v\oplus n.y)\leq
(u\oplus v\oplus n.x)\wedge (u\oplus v\oplus n.y)$. Since the right hand side of the last inequality belongs to $[0,b]$,
we have $z\leq (u\oplus_b v\oplus n\bullet x)\wedge (u\oplus v\oplus n\bullet y)$, where $1\bullet x=x$ and $n\bullet x=x\oplus_b (n-1)\bullet x$
for all integer $n\geq 2$. Since $([0,b];\oplus_b,\lambda_b,0,b)$ is an MV-algebra by \cite[Prop 1.17(i)]{georgescu},
\begin{equation*}
z\leq (u\oplus_b v)\oplus_b (n\bullet x \wedge n\bullet y)\leq 2n\bullet (u\oplus_b v)\oplus_b n^2\bullet(x\wedge y)\leq
2n.(u\oplus v)\oplus n^2.(x\wedge y) \in P,
\end{equation*}
which is a contradiction. So, $P$ is a prime ideal of the wEMV-algebra $M$. Summing up the above arguments, we have the next proposition.
\begin{prop}\label{prop3.8}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a non-zero wEMV-algebra. Then we have:
\begin{itemize}
\item[{\rm (i)}] For each $x\in M\setminus \{0\}$ there exists a prime ideal
$P$ of $M$ such that $x\notin P$.
\item[{\rm (ii)}] $\bigcap\{P\mid P\in Spec(M)\}=\{0\}$.
\item[{\rm (iii)}] Any ideal $J$ of $M$ can be represented by the intersection of prime ideals contains $J$.
\end{itemize}
\end{prop}
We note that the binary operation $\oplus$ of a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ is {\it cancellative} if, for all $x,y,z \in M$, $x\oplus y= x\oplus z$ implies $y=z$, and $M$ is said to be a {\it cancellative wEAM-algebra}.
\begin{lem}\label{lem3.9}
{\rm (1)} Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a cancellative wEMV-algebra. Then it is isomorphic to the wEMV-algebra of the positive cone of some $\ell$-group $(G;+,0)$.
{\rm (2)} In addition, every linearly ordered strict wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$ is a cancellative wEMV-algebra, and $M$ is isomorphic to the wEMV-algebra of the positive cone of a linearly ordered group $(G;+,0)$.
\end{lem}
\begin{proof}
(1) Let $M$ be a cancellative wEMV-algebra. Since according to Proposition \ref{pr:1}(a), $M$ is naturally ordered, i.e. $x\le y$ iff $y=x\oplus z$ for some $z \in M$, due to the Nakada Theorem, \cite[Prop X.1]{Fuc}, there is an $\ell$-group $G$ such that $M$ is isomorphic to $(G^+;\vee,\wedge,\oplus,\ominus,0)$, where $g_1\oplus g_1 = g_1+g_2$, $g_1\ominus g_2=(g_1-g_2)\vee 0$ (see Example \ref{ex:1}).
(2) Let $M$ be a strict and linearly ordered wEMV-algebra. We claim that the operation $\oplus$ in the commutative monoid $(M;\oplus,0)$ is cancellative. Indeed, assume that $x,y,z\in M$ such that
$x\oplus z=y\oplus z$. Since $M$ is a linearly ordered strict wEMV, for each $u\in M$, there is an element $v\in M$ such that $u\lneqq v$.
Let $a\in M$ be such that $2.(x\oplus y\oplus z)\lneqq a$. Consider the MV-algebra $([0,a];\oplus_a,\lambda_a,0,a)$.
There exists an $\ell$-group $(G_a;+,0)$ with a strong unit $u_a$ such that $([0,a];\oplus_a,\lambda_a,0,a)\cong \Gamma(G_a,u_a)$ (see \cite[Sec 2 and 7]{cdm}). We put $\Gamma(G_a,u_a)=[0,a]$ and so $u_a=a$. Then $x\oplus_a z=(x+z)\wedge a$ and $x\oplus_a z=(x\oplus z)\wedge a$.
Since $x\oplus z\lneqq a$, then $x\oplus_az\lneqq a$ which implies that $x+z\lneqq a$ (otherwise, $a\leq (x+z)$, that is $(x+z)\wedge a=a$).
In a similar way, we can show that $y+z\lneqq a$. Hence, $x+z=x\oplus_a z=y\oplus_a z=y+z$ and so $x=y$, and $M$ is a cancellative wEMV-algebra.
According to (1), there is an $\ell$-group $(G;+,0)$ such that $M$ is isomorphic to the wEMV-algebra of the positive cone $G^+$. Since $M$ is linearly ordered, $(G;+,0)$ is a linearly ordered group.
\end{proof}
\begin{thm}\label{thm3.10}
Each wEMV-algebra is a subalgebra of an associated wEMV-algebra with top element.
\end{thm}
\begin{proof}
If $M=\{0\}$, then the proof is evident. Let $M\neq \{0\}$.
\noindent
Let $S_1:=\{P\in Spec(M)\mid M/P \text{ has a non-zero idempotent element}\}$ and $S_2:=\{P\in Spec(M)\mid M/P \text{ is strict}\}$.
Then $Spec(M)=S_1\cup S_2$. Also, by Proposition \ref{prop3.8}(ii), we can easily prove that
the map $\varphi:M\ra \prod_{P\in Spec(M)}M/P$ sending $x\in M$ to $(x/P)_{P\in Spec(M)}$ is a one-to-one homomorphism.
On the other hand, $\prod_{P\in Spec(M)}M/P\cong (\prod_{P\in S_1}M/P)\times (\prod_{P\in S_2}M/P)$, so we identify these two wEMV-algebras.
By Propositions \ref{prop3.7} and \ref{prop3.6}, for each $P\in S_1$, $M/P$ is an associated wEMV-algebra and so $\prod_{P\in S_1}M/P$ can be viewed as an EMV-algebra, too.
Note that due to \cite[Thm 3.24]{Dvz}, this associated wEMV-algebra has a top element. Now, let $P\in S_2$. If there is $a\in\mathcal I(M)$ such that $a\notin P$, then clearly
$a/P$ is a non-zero idempotent element of $M/P$, which is a contradiction. That is, $\downarrow \mathcal I(M) \s P$ for all $P\in S_2$, so that $\downarrow \mathcal I(M)\subseteq \bigcap\{P\mid P\in S_2\}$.
Suppose that $P\in S_2$. Then $M/P$ is a linearly ordered strict wEMV-algebra. So, by Lemma \ref{lem3.9}, it is the positive cone of an
$\ell$-group $G_P$. Example \ref{ex:1} entails that $M/P$ can be embedded into an associated wEMV-algebra with top element. Hence, $\prod_{P\in S_2}M/P$ can be embedded into an associated wEMV-algebra with top element, too.
Summing up the above arguments, the wEMV-algebra $M$ is a subalgebra of an associated wEMV-algebra with top element.
\end{proof}
The latter theorem allows us to present a similar representation result as the Basic Representation Theorem \ref{2.4} for EMV-algebras. We recall that if a wEMV-algebra possesses a top element, then $1\ominus x$ is said to be a {\it complement} of $x$.
\begin{thm}\label{th:Repres}
Every wEMV-algebra $M$ either has a top element and so it is an associated wEMV-algebra or it can be embedded into an associated wEMV-algebra $N$ with top element as a maximal ideal of $N$. Moreover, every element of $N$ is either the image of $x\in M$ or is a complement of the image of some element $x \in M$.
\end{thm}
\begin{proof}
If $M$ has a top element, the statement is trivial. So suppose that the wEMV-algebra has no top element. Take $S_1$ and $S_2$ as the sets of prime ideals of $M$ defined in the proof of Theorem \ref{thm3.10}. If $S$ is the set of all prime ideals, then $S=S_1\cup S_2$. If $P \in S_1$, then $M/P$ is an associated linearly ordered EMV-algebra with top element. If $P\in S_2$, then $M/P$ is a linearly ordered strict and consequently cancellative wEMV-algebra without top element which corresponds to a wEMV-algebra of a positive cone $G^+_P$. So it can be embedded into $\Gamma(\mathbb Z\lex G_P,(1,0))$. Denote by $N_0=(\prod_{P\in S_1} M/P)\times (\prod_{P\in S_2} \Gamma(\mathbb Z\lex G_P,(1,0)))$ which is an associated wEMV-algebra with a top element $1$, and according to Theorem \ref{thm3.10}, $M$ can be embedded into $N_0$.
Without loss of generality, we can assume that $M\subset N_0$ is a proper wEMV-subalgebra of $N_0$. We denote by $\ominus$ and $\oplus$ also the binary operations of $N_0$. Denote by $M^*=\{1\ominus x\mid x \in M\}$. We assert that $M\cap M^*=\emptyset$. Indeed, if $1\ominus x=y$ for some $x,y\in M$, then $1=(1\ominus x)\oplus x = x\oplus y$ which says $1=x\oplus y \in M$, a contradiction.
First, we define a binary operation $\odot$ on $N_0$ as $x\odot y :=1\ominus ((1\ominus x)\oplus (1\ominus y))$, $x,y \in N_0$.
\vspace{3mm}
\noindent
{\it Claim} {\it If $x,y \in M$, then $x\odot y \in M$ and $x\ominus y = x\odot (1\ominus y)$.}
\vspace{2mm}
Let $x=(x_P)_{P \in S},y=(y_P)_{P \in S}, 1=(1_P)_{P \in S}\in N_0$. Then
$$1\ominus ((1\ominus x)\oplus (1\ominus y)) = (1_P)_{P \in S}\ominus (((1_P)_{P \in S}\ominus (x_P)_{P \in S})\oplus ((1_P)_{P \in S}\ominus (y_P)_{P \in S})).
$$
If $P\in S_1$, then $1_P\ominus ((1_P\ominus x_P)\oplus (1_P\ominus y_P))\in M/P$ since $M/P$ is an associated wEMV-algebra and applying (\ref{eq:odot1}). If $P\in S_2$, then using calculations in $\Gamma(\mathbb Z \lex G_P,(1,0))$, we have also $1_P\ominus ((1_P\ominus x_P)\oplus (1_P\ominus y_P))\in M/P$. Then $1\ominus ((1\ominus x)\oplus (1\ominus y))\in M$.
In addition, $x\odot (1\ominus y)=1\ominus ((1\ominus x)\oplus (1\ominus(1\ominus y))) = 1\ominus ((1\ominus x)\oplus y)= x\ominus y$ (applying Definition \ref{WEMV}(ix)).
Set $N = M\cup M^*$. We show that $N$ is an associated EMV-subalgebra of $N_0$ which satisfies the conditions of our theorem.
Clearly $N$ contains $M$ and $1$. Let $x,y \in N$. We have three cases: (i) $x=x_0,y=y_0 \in M$. Then $x\vee y,x\wedge y, x\oplus y \in N$. Due to Proposition \ref{prop3.2}, we have $x\ominus y = x\ominus_1 y$ and using (\ref{eq:ominus}) and a similar verification as in Claim, we have $x\ominus y \in M \subset N$.
(ii) $x=1\ominus x_0$, $y=1\ominus y_0$ for some $x_0,y_0\in M$. Then $x\vee y = (1\ominus x_0)\vee (1\ominus y_0)= 1\ominus (x_0\wedge y_0)$, $x\wedge y= 1\ominus (x_0\vee y_0)$ and $x\oplus y = (1\ominus x_0)\oplus (1\ominus y_0) = 1\ominus (x_0\odot y_0)\in N$. Finally, by Claim $(1\ominus x_0)\ominus (1\ominus y_0)= (1\ominus x_0)\odot y_0=y_0\odot (1\ominus x_0)= y_0\ominus x_0 \in M\subset N$.
(iii) $x=x_0$ and $y=1\ominus y_0$ for some $x_0,y_0\in M$. We note that $N_0$ can be viewed also as an EMV-algebra with top element, according to \cite[Lem 5.1]{Dvz}, we have $x\odot (1\ominus y)=x\odot (1\ominus (x\wedge y))$.
Then
$$
x\oplus y = x_0\oplus (1\ominus y_0) = 1\ominus (y_0\odot (1\ominus x_0))= 1\ominus (y_0\odot (1\ominus (x_0\wedge y_0)))=1\ominus (y_0 \ominus (x_0\wedge y_0))\in M^* \subset N.
$$
In addition, we have
$$x\wedge y = x_0\wedge (1\ominus y_0)= x_0\odot ((1\ominus x_0)\oplus (1\ominus y_0)) = x_0\odot (1\ominus (x_0\odot y_0))= x_0\ominus (x_0\odot y_0)\in M \subset N.
$$
Using $x\vee y = 1\ominus ((1\ominus x)\wedge (1\ominus y))= 1\ominus((1\ominus x_0)\wedge y_0)$, we have, due to the latter paragraph, $x\vee y \in N_0$. Moreover, $x\ominus y = x_0\ominus (1\ominus y_0)= x_0\odot y_0\in M$ and $y\ominus x = (1\ominus y_0)\ominus x_0 = 1 \ominus (x_0\oplus y_0)\in M^*$, when we have used (ix) of Definition \ref{WEMV}.
Now, we prove that $M$ is a maximal ideal of $N$. Since $M$ is a wEMV-algebra without top element, $M$ is a proper subset of $N$. To show that $M$ is an ideal, it is sufficient to assume $y\le x \in M$. If $y=(1\ominus y_0)$, then $1= (1\ominus y_0) \oplus y_0\le x_0\oplus y_0\in M$
which is absurd while $1\notin M$. Therefore, $M$ is a proper ideal of $N$. Now, let $y \in N\setminus M$, then $y=1\ominus y_0$ for some $y_0\in M$. Then the ideal $\langle M,y\rangle$ of $N$ generated by $M$ and $1\ominus y_0$ contains $1$, so that $\langle M,1\ominus y_0\rangle = N$ proving $M$ is a maximal ideal of $N$.
\end{proof}
The associated wEMV-algebra $N$ with top element in the latter theorem is said to be a {\it representing} $M$. We note that all representing associated wEMV-algebras of $M$ are mutually isomorphic.
\begin{thm}\label{thm3.11}
The class $\mathsf{wEMV}$ is the least subvariety of the variety $\mathsf{wEMV}$ containing $\mathsf{EMV}_a$. Moreover, $\mathsf{wEMV}=HSP(C)$, where $C$ is the class of all
linearly ordered wEMV-algebras.
\end{thm}
\begin{proof}
By Example \ref{ex:3}, $\mathsf{wEMV}$ contains $\mathsf{EMV}_a$. Let $\mathsf{V}$ be an arbitrary variety of wEMV-algebras containing
$\mathsf{EMV}$. Then by Theorem \ref{thm3.10}, $\mathsf{wEMV}\s \mathsf{V}$. The second part follows from the proof of Theorem \ref{thm3.10}.
\end{proof}
As it was already mentioned, according to \cite{Kom}, the lattice of subvarieties of the variety $\mathsf{MV}$ of MV-algebras is countably infinite. Di Nola and Lettieri presented in \cite{DiLe1} an equational base of any subvariety of the variety $\mathsf{MV}$ which consists of finitely many MV-equations using only $\oplus$ and $\odot$. We know that we can define a binary operation $\odot$ on $M$, see Claim in the proof of Theorem \ref{th:Repres}. Given $x \in M$ and an integer $n\ge 1$, we define
$$
x^1:=x,\quad x^{n+1}:=x\odot x^n,\quad n\ge 1,
$$
and $x^0:=1$ if $M$ has a top element $1$. We note that the subvariety of MV-algebras generated by the MV-algebra $\Gamma(\mathbb Z\lex\mathbb Z,(1,0))$ has an equational base $(2.x)^2=2.x^2$, see \cite[Thm 5.11]{DiLe}, \cite{DiLe1}; it is the subvariety generated by perfect MV-algebras. Denote by $\mathsf{O}$ the trivial subvariety of wEMV-algebras consisting only of the zero element.
\begin{thm}\label{th:can}
Let $\mathsf{Can}$ denote the class of cancellative wEMV-algebras. Then $\mathsf{Can}$ is a subvariety of the variety $\mathsf{wEMV}$, and a wEMV-algebra $M$ belongs to $\mathsf{Can}$ if and only if $M$ satisfies the identity
$$(x\oplus y)\ominus x= y.
$$
Equivalently, $M\in \mathsf{Can}$ if and only if $M$ is isomorphic to the wEMV-algebra of the positive cone $G^+$ of some $\ell$-group $G$.
In addition, if $\mathbb Z^+=(\mathbb Z^+; \vee,\wedge,\oplus,\ominus,0)$ is the wEMV-algebra of the positive cone $\mathbb Z^+$, then $\mathsf{Can}=Var(\mathbb Z^+)$, and $\mathsf{Can}$ is an atom of the lattice of subvarieties of $\mathsf{wEMV}$.
Moreover, if we denote by $\mathsf{Perf}$ the subvariety of wEMV-algebras satisfying the equation $(2.x)^2=2.x^2$, then $\mathsf{Perf}$ is a cover of the subvariety $\mathsf{Can}$, and the associated wEMV-algebra $(\Gamma(\mathbb Z\lex \mathbb Z,(1,0)); \vee,\wedge,\oplus,\ominus,0)$ with top element and representing the cancellative wEMV-algebra $\mathbb Z^+$ is a generator of the variety $\mathsf{Perf}$.
\end{thm}
\begin{proof}
Let a wEMV-algebra $M$ satisfy the equation $(x\oplus y)\ominus x = y$. We assert that $M$ is cancellative. If $x\oplus y=x\oplus z$, $x,y,z\in M$, then $y= (x\oplus y)\ominus x=(x\oplus z)\ominus x =z$, so that $M$ is a cancellative wEMV-algebra. If $M$ is a cancellative wEMV-algebra, according to Lemma \ref{lem3.9}(1), $M$ is isomorphic to the wEMV-algebra of the positive cone $(G^+;\vee,\wedge,\oplus,\ominus,0)$ of some $\ell$-group $G$. Whence, $\mathsf{Can}$ is a proper non-trivial subvariety of the variety $\mathsf{wEMV}$. It is well known that the group of integers $\mathbb Z$ generates the variety of Abelian $\ell$-groups, see e.g. \cite[Thm 10.B]{Gla}. Using this fact, and the $HSP$-technique, it is possible to show that the wEMV-algebra of the positive cone $\mathbb Z^+$ generates the variety $\mathsf{Can}$.
Clearly, $\mathsf{O} \subsetneq \mathsf{Can}$ is a subvariety of $\mathsf{Perf}$. Let $\mathsf V$ be a subvariety of wEMV-algebras such that $\mathsf O\subsetneq \mathsf V\subseteq \mathsf{Can}$. Then every non-trivial wEMV-algebra of $\mathsf V$ is cancellative. Let $M\in \mathsf{Can}$ be non-trivial and let $f\in M$ be a non-zero element. Then $\{n.f \mid n\ge 0\}$ is a wEMV-subalgebra of $M$ generated by $f$ and it is isomorphic to the wEMV-algebra of the positive cone $\mathbb Z^+$, which implies $\mathbb Z^+\in \mathsf V$ and thus $\mathsf V=\mathsf{Can}$, and $\mathsf{Can}$ is an atom in the lattice of subvarieties of $\mathsf{wEMV}$.
Let $M \in \mathsf{Perf}$. If $M$ possesses a top element, then $(M;\oplus,\lambda_1,0,1)$ is an MV-algebra satisfying the equation $(2.x)^2=2.x^2$. If $M$ has no top element, let $N$ be its representing associated wEMV-algebra. Without loss of generality we can assume that $M\cup M^*=N$. If $x\in M$, then clearly $(2.x)^2=2.x^2$. If $x\in M^*$, then $x=1\ominus x_0$ for some $x_0\in M$ and $(2.x_0)^2=2.x_0^2$ which entails $(2.(1\ominus x_0))^2=2.(1\ominus x^2_0)$, so that $N\in \mathsf{Perf}$.
If $M$ is cancellative and non-trivial, then $M$ is without top element and is isomorphic to the positive cone wEMV-algebra $G^+$. Its representing wEMV-algebra is isomorphic to the associated wEMV-algebra $N=\Gamma(G,u)$ and it satisfies as an MV-algebra the identity of perfect MV-algebras $(2.x)^2=2.x^2$, therefore, $N$ as a wEMV-algebra, satisfies the identity $(2.x)^2=2.x^2$, and $N \in \mathsf{Perf}$. Henceforth, we conclude that $\mathsf{Can}\subsetneq \mathsf{Perf}$. Take the associated wEMV-algebra with top element $M_0=\Gamma(\mathbb Z\lex \mathbb Z,(1,0))$. Then the cancellative wEMV-algebra $\mathbb Z^+$ is a subalgebra of $M_0$ and $\mathbb Z^+ \in Var(M_0)$, where $Var(M_0)$ is the subvariety of $\mathsf{wEMV}$ generated by $M_0$. Whence, $\mathsf{Can}\subseteq Var(M_0)$. The associated wEMV-algebra with top element $\Gamma(\mathbb Z \lex \mathbb Z,(1,0))$ satisfies the identity $(2.x)^2=2.x^2$, so that $Var(M_0)\subseteq \mathsf{Perf}$. Now, let $M\in \mathsf{Perf}$ be an arbitrary wEMV-algebra. Using Theorem \ref{thm3.10}, we know that $M$ is a subdirect product algebra of $N:=(\prod_{P\in S_1}M/P)\times (\prod_{P\in S_2}M/P)$. Let $N_1:= \prod_{P\in S_1}M/P$ and $N_2:=\prod_{P\in S_2}M/P$. Then $N_1$ is an associated wEMV-algebra with top element satisfying $(2.x)^2=2.x^2$. Therefore, $N_1 \in Var(M_0)$ and $N_2$ is a cancellative wEMV-algebra so that $N_2\in \mathsf{Can}\subseteq Var(M_0)$ which yields $N=N_1\times N_2\in Var(M_0)$ and $\mathsf{Perf}\subseteq Var(M_0)$ which proves that the associated wEMV-algebra $\Gamma(\mathbb Z\lex \mathbb Z,(1,0))$ generates the variety $\mathsf{Perf}$.
In what follows, we show that $\mathsf{Perf}$ is a cover of $\mathsf{Can}$. So let $\mathsf V$ be a subvariety of wEMV-algebras such that $\mathsf{Can}\subseteq \mathsf V\subseteq \mathsf{Perf}$ and
let $M \in \mathsf V \setminus \mathsf{Can}$. There are two cases. (1) If $M$ has a top element, then $M$ is an associated wEMV-algebra with top element so that the termwise MV-algebra belongs to the variety generated by perfect MV-algebras, so that the MV-algebra $\Gamma(\mathbb Z\lex \mathbb Z,(1,0))$ belongs to the variety of MV-algebras generated by $M$, consequently, the associated wEMV-algebra $\Gamma(\mathbb Z\lex \mathbb Z,(1,0))$ belongs to the subvariety $Var(M)$ of wEMV-algebras generated by the wEMV-algebra $M$. As it was established in the latter paragraph, $Var(M)=\mathsf{Perf}$. If $M$ has no top element, use again the subdirect embedding of $M$ into $N=N_1\times N_2$ from the latter paragraph. (2) If $S_1=\emptyset$, then $N_2\in \mathsf{Can}$ and $M$ as a subalgebra of $N_2$ also belongs to $\mathsf{Can}$, an absurd. Hence, $S_1$ is non-empty and there is $P \in S_1$ so that $M/P$ is an associated wEMV-algebra with top element and $M/P \in Var(M)$. As in case (1), $\Gamma(\mathbb Z\lex \mathbb Z,(1,0))\in Var(M/P)\subseteq Var(M)\subseteq \mathsf V $ and therefore, $\mathsf V=\mathsf{Perf}$ which proves that $\mathsf{Perf}$ is a cover of $\mathsf{Can}$.
\end{proof}
\begin{thm}\label{th:subvar}
The lattice of subvarieties of the variety $\mathsf{wEMV}$ is countably infinite.
\end{thm}
\begin{proof}
Due to \cite{Kom}, the lattice of subvarieties of the variety $\mathsf{MV}$ of MV-algebras is countably infinite and in \cite{DiLe1}, there is an equational base of any subvariety of the variety $\mathsf{MV}$ which consists of finitely many MV-equations using only $\oplus$ and $\odot$. Hence, let $\mathsf V_{MV}$ be any subvariety of MV-algebras with a finite equational base $\{f_i(x_1,\ldots,x_n)=g_i(y_1,\ldots,y_m) \mid i=1,\ldots,k\}$, where $f_i,g_i$ are finite MV-terms using only $\oplus$ and $\odot$. Denote by $\mathsf W(\mathsf V_{MV})$ the subvariety of wEMV-algebras which satisfies $\{f_i(x_1,\ldots,x_n)=g_i(y_1,\ldots,y_m) \mid i=1,\ldots,k\}$.
Now, let $\mathsf W$ be any non-trivial subvariety of wEMV-algebras. Let $\mathcal V(\mathsf W)$ denote the system of all wEMV-algebras $(M;\vee,\wedge,\oplus,\ominus,0) \in \mathsf W$ with top element, and let $\mathsf V_{MV}(\mathsf W)$ be the subvariety of MV-algebras generated by equivalent MV-algebras $(M;\oplus,\lambda_1,0,1)$ from $\mathcal V(\mathsf W)$. It has a finite equational base using only $\oplus$ and $\odot$. The system of wEMV-algebras satisfying these identities forms a subvariety $\mathsf W(\mathsf V_{MV}(\mathsf W))\subseteq \mathsf W$.
Take an arbitrary wEMV-algebra $M$ from $\mathsf W$. If $M$ has a top element, then $M\in \mathsf W(\mathsf V_{MV}(\mathsf W))$. If $M$ is without top element, we have an embedding of $M$ into the subdirect product $(\prod_{P\in S_1}M/P)\times (\prod_{P \in S_2}M/P)$. If $S_1$ is non-empty, then $N_1=\prod_{P\in S_1}M/P\in \mathsf W(\mathsf V_{MV}(\mathsf W))$. If $S_2$ is non-empty, then $N_2=\prod_{P\in S_2}M/P\in \mathsf{Can}$. Whence, we have three cases. (1) For each non-trivial $M \in \mathsf W$, $S_2$ is empty, then $\mathsf W\subseteq \mathsf W(\mathsf V_{MV}(\mathsf W))\subseteq \mathsf W$. (2) For each non-trivial $M\in \mathsf W$, $S_1$ is empty, then $\mathsf W\subseteq \mathsf{Can}$ and since $\mathsf{Can}$ is an atom in the lattice of subvarieties of $\mathsf{wEMV}$, see Theorem \ref{th:can}, we have $\mathsf W = \mathsf{Can}$. (3) There is a non-trivial wEMV-algebra $M\in \mathsf W$ such that $S_1$ and $S_2$ are both non-empty. Then $\mathsf{Can}\subset \mathsf W \subseteq \mathsf W(\mathsf V_{MV}(\mathsf W)) \vee \mathsf{Can}\subseteq \mathsf W$ which proves $\mathsf W= \mathsf W(\mathsf V_{MV}(\mathsf W))\vee \mathsf{Can}$.
Summarizing, we see that every subvariety $\mathsf W$ of wEMV-algebras either satisfies some finite system of MV-algebras, so it is $\mathsf W(\mathsf V_{MV})$, or it is equal to $\mathsf W(\mathsf V_{MV})\vee \mathsf{Can}$ for some subvariety $\mathsf V_{MV}$ of MV-algebras. Due to Komori, we see that the lattice of wEMV-subvarieties is countably infinite.
\end{proof}
To illustrate the last mentioned three possibilities, case (1) is true e.g. for the subvariety $\mathsf{Idem}$ of wEMV-algebras determined by $x\oplus x = x$, case (2) for $\mathsf{Can}$, and case (3) for $\mathsf{Idem}\vee \mathsf{Can}$. More generally, we have the following characterization.
\begin{rmk}\label{re:variety}
For each integer $n\ge 1$, we define MV-algebras $L_n=\Gamma(\mathbb Z,n)$ and $K_n=\Gamma(\mathbb Z\lex Z,(n,0))$. It is known, see \cite[Thm 8.4.4]{cdm}, that for every proper variety $\mathsf V_{MV}$ of MV-algebras, there are finite sets $I$ and $J$ such that $I\cup J$ is non-empty and $\mathsf V_{MV}$ is generated by $\{L_i,K_j\colon i\in I,j\in J\}$. Then situation (1) at the end of the proof of Theorem \ref{th:subvar} happens only if $\mathsf V_{MV}(\mathsf W)$ is generated only by finitely many $L_i$'s and no $K_j$.
For situation (3), we have two subcases. (3i) $\mathsf V_{MV}(\mathsf W)$ is generated only by finitely many $L_i$'s, then $\mathsf W = \mathsf(\mathsf W_{MV}(\mathsf W))\vee \mathsf{Can}$. (3ii) $\mathsf V_{MV}(\mathsf W)$ contains at least one generator of the form $L_i$ and at least one generator of the form $K_j$. Then $\mathsf W = \mathsf W(\mathsf V_{MV}(\mathsf W))\vee \mathsf{Can} = \mathsf W(\mathsf V_{MV}(\mathsf W))$ because the cancellative wEMV-algebra $\mathbb Z^+$ is a subalgebra of the associated wEMV-algebra $K_j$.
\end{rmk}
In what follows, we investigate a question when a wEMV-algebra $M$ and its representing associated wEMV-algebra $N$ with top element belong to the same variety and when not.
\begin{cor}\label{co:var}
Let $M$ be a wEMV-algebra without top element, let $N$ be its representing associated wEMV-algebra with top element, let $\mathsf W$ be a proper variety of wEMV-algebras, and $M \in \mathsf W$.
{\rm 1.} If $\mathsf W$ satisfies case {\rm (1)} or case {\rm (3ii)}, then $N$ belongs to $\mathsf W$.
{\rm 2.} If $\mathsf W$ satisfies case {\rm (2)}, then $N\notin \mathsf W$.
{\rm 3.} If $\mathsf W$ satisfies case {\rm (3i)}, then it can happen that $N\notin \mathsf W$.
\end{cor}
\begin{proof}
Let $M \in \mathsf W$.
Applying Theorem \ref{thm3.10}, we know that $M$ is a subdirect product of $N_0:=(\prod_{P\in S_1}M/P)\times (\prod_{P\in S_2}M/P)$ and let $N_1=\prod_{P\in S_1}M/P$ and $N_2= \prod_{P\in S_2}M/P$.
1. Case (1). Then $S_2=\emptyset$ and $M$ is a subdirect product of $\{M/P\colon P \in S_1\}$. Since every $M/P$ has a top element, $N$ is a subalgebra of $N_1$ and thus $N\in \mathsf W$. Case (3ii). Then $S_1$ and $S_2$ are non-empty. The wEMV-algebra $N_1$ has a top element. Every $M/P$ is cancellative for each $P \in S_2$. But $M/P$ can be isomorphically embedded into $\Gamma(\mathbb Z\lex G_P,(1,0))\subseteq \Gamma(\mathbb Z\lex G_P,(j,0)) \in \mathsf W$, so that $N\in \mathsf W$.
2. Case (2). Then $\mathsf W=\mathsf{Can}$, so that $N\notin \mathsf W$.
3. Case (3i). If $S_2=\emptyset$, then $N \subseteq \prod\{M/P\colon P \in S_1\}\in \mathsf W$. If $S_2$ is non-empty, then $N$ is a subalgebra of $N_1\times \prod \{\Gamma(\mathbb Z\lex G_{P},(1,0))\colon P\in S_2\}$. But $N_1$ has a top element and $\prod \{\Gamma(\mathbb Z\lex G_{P},(1,0))\colon P \in S_2\}\notin \mathsf W$. Whence, $N \notin \mathsf W$.
\end{proof}
In the following remark, we describe some interesting categories of wEMV-algebras and their $\ell$-group representations.
\begin{rmk}\label{re:category}
(1) Denote by $\mathsf{wEMV}_1$ the class of wEMV-algebras with top element.
Applying Proposition \ref{prop3.2} and Mundici's representation of MV-algebras by unital $\ell$-group, the category $\mathsf{wEMV}_1$ is categorically equivalent to the category of MV-algebras and also to the category of unital $\ell$-groups.
(2) The category $\mathsf{Can}$ of cancellative wEMV-algebras is categorically equivalent to the category of Abelian $\ell$-groups.
(3) The category of associated wEMV-algebras without top element is categorically equivalent to the category of $\ell$-groups with a fixed special maximal $\ell$-ideal, see \cite[Thm 6.8]{Dvz}.
\end{rmk}
\begin{rmk}\label{rmk3.12}
By \cite[Cor 1.4.7]{cdm}, an MV-equation is satisfied by all MV-algebras \iff it is satisfied by all linearly ordered MV-algebras.
We can simply check that the following identities hold in each linearly ordered MV-algebras:
\begin{eqnarray}
\label{3.12e2} (x\vee y)\ominus z&=&(x\ominus z)\vee (y\ominus z),\\
\label{3.12e1} x\oplus y&=&(x\vee y)\oplus (x\wedge y).
\end{eqnarray}
So, they hold in each MV-algebra, too. Now, let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra and $x,y,z\in M$.
Let $u\geq x\oplus y\oplus z$ be an element of $M$. In the MV-algebra $([0,u];\oplus,\lambda_u,0,u)$, we have
$(x\vee y)\ominus_u z=(x\ominus_u z)\vee (y\ominus_u z)$ which entails that
$(x\vee y)\ominus z=(x\ominus z)\vee (y\ominus z)$ (by equation (\ref{eq3.4.1})).
So, (\ref{3.12e2}) holds in each wEMV-algebra. In a similar way, we can easily show that $(\ref{3.12e1})$ holds in each wEMV-algebra.
In addition, identity (\ref{3.12e1}) implies the following quasi identity
\begin{equation}\label{3.12e3}
x\wedge y = 0 \Rightarrow x\oplus y = x\vee y
\end{equation}
holding in each wEMV-algebra.
\end{rmk}
Now, given an wEMV-algebra $M$, we introduce two important its subalgebras $M_1$ and $M_2$.
\begin{prop}\label{pr:subalg}
Given a wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$, we define $M_1:=\downarrow \mathcal I(M)$ and $M_2:=\{x\in M\mid x\wedge y=0,~\forall\, y\in \mathcal I(M)\}$. Then $M_1$ is the biggest associated wEVM-subalgebra of $M$ and $M_2$ is a strict wEMV-subalgebra of $M$.
Moreover, if $x_1\vee x_2=y_1\vee y_2$, where $x_1,y_1\in M_1$ and $x_2,y_2\in M_2$, then $x_1=y_1$ and $x_2\vee y_2$. In addition, $x\in M_1$ and $y\in M_2$ imply $x\wedge y = 0$ and $x\oplus y = x\vee y$. Similarly, if $x_1,y_1\in M_1$ and $x_2,y_2\in M_2$, then $x_1\oplus x_2=y_1\oplus y_2$ entails $x_1=y_1$ and $x_2=y_2$.
\end{prop}
\begin{proof}
Consider an arbitrary wEMV-algebra $(M;\vee,\wedge,\oplus,\ominus,0)$.
Clearly, $M_1:=\downarrow \mathcal I(M)$ is closed under the operations
$\vee$, $\wedge$, $\oplus$, $\ominus$ and $0$ which implies that $M_1$ is a subalgebra of $M$. Also, by definition, $(M_1;\vee,\wedge,\oplus,0)$ is an EMV-algebra and $M_1$ is an ideal of $M$, too. In addition, let $M_1'$ be an associated wEMV-algebra that is a subalgebra of $M$. Then clearly, $M_1'\subseteq M_1$.
Now, let $M_2:=\{x\in M\mid x\wedge y=0,~\forall\, y\in \mathcal I(M)\}$.
Then $0\in M_2$ and by Proposition \ref{pr:1}, $M_2$ is closed under $\wedge$ and $\ominus$. Also, distributivity of $(M;\vee,\wedge)$ implies
that $M_2$ is closed under $\vee$. Let $x,y\in M_2$. Put an arbitrary idempotent element $a\in \mathcal I(M)$.
For $b:=x\oplus y\oplus a$, by Proposition \ref{prop3.2},
$([0,b];\oplus_b,\lambda_b,0,b)$ is an MV-algebra and so by the assumption and \cite[Prop 1.17]{georgescu}, we have
\begin{eqnarray*}
(x\oplus y)\wedge a=(x\oplus_b y)\wedge a\leq (x\wedge a)\oplus_b (y\wedge a)= [(x\wedge a)\oplus (y\wedge b)]\wedge b\le (x\wedge a)\oplus (y\wedge a)=0.
\end{eqnarray*}
Thus, $M_2$ is a subalgebra of the wEMV-algebra $M$. Clearly, $M_2$ does not have any non-zero idempotent element, so that $M_2$ is strict.
Let $x\in M$ such that $x=x_1\vee x_2$ and $x=y_1\vee y_2$,
where $x_1,y_1\in M_1$ and $x_2,y_2\in M_2$. Then
\begin{eqnarray*}
x_1=x_1\wedge (y_1\vee y_2)=(x_1\wedge y_1)\vee (x_1\wedge y_2)=x_1\wedge y_1,
\end{eqnarray*}
and so $x_1\leq y_1$. In a similar way, $y_1\leq x_1$ and so $x_1=x_2$. We can easily show that $x_2=y_2$.
Now, if $x\in M_1$ and $y \in M_2$, we conclude that $x\wedge y = 0$ and (\ref{3.12e3}) entails $x\oplus y = x\vee y$. Consequently $x_1\oplus x_2=y_1\oplus y_2$ implies $x_1\vee x_2=y_1\vee y_2$, so that $x_1=y_1$ and $x_2\vee y_2$.
\end{proof}
The associated wEMV-subalgebra $M_1$ and a strict wEMV-subalgebra $M_2$ of $M$ play an important role in a decomposition of $M$ as a direct product of $M_1$ and $M_2$.
\begin{thm}\label{thm3.13}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra such that the ideal of $M$ generated by $\langle M_1\cup M_2\rangle$ is equal to $M$.
Then $M\cong M_1\times M_2$ as wEMV-algebras.
\end{thm}
\begin{proof}
Let $x\in M$. Then by the paragraph just after Proposition \ref{prop3.7}, there are $x_1\in M_1$ and $x_2\in M_2$ such that
$x\leq x_1\oplus x_2$ (note that $M_1$ and $M_2$ are ideals of $M$).
Put $u\in M$ such that $x\oplus x_1\oplus x_2\leq u$. Then by \cite[Prop 1.17(1)]{georgescu}, in the MV-algebra $([0,u];\oplus_u,\lambda_u,0,u)$, we have $x=x\wedge (x_1\oplus x_2)=x\wedge ((x_1\oplus x_2)\wedge u)=x\wedge (x_1\oplus_u x_2)\leq (x_1\wedge x)\oplus_u (x_2\wedge x)=(x_1\wedge x)\oplus (x_2\wedge x)$. So, we can always assume that $x_1,x_2\leq x$. Then
$x\leq x_1\oplus x_2$ and Proposition \ref{pr:1}(h) entails that
$x\ominus x_1\leq x_2$, whence $x_3:=x\ominus x_1\in M_2$.
It follows that $x_1\oplus x_3=x_1\oplus (x\ominus x_1)=x\vee x_1=x$ (since $x_1\leq x$). Define $\varphi:M\ra M_1\times M_2$ by
$\varphi(x)=(x_1,x_2)$, where $x_1\in M_1$, $x_2\in M_2$ and $x=x_1\oplus x_2$.
(i) If $u\in M_1$ and $v\in M_2$, then by equation (\ref{3.12e1}),
$u\oplus v=u\vee v$.
(ii) $\varphi$ is well defined.
Let $x=x_1\oplus x_2=y_1\oplus y_2$ for some $x_1,y_1\in M_1$ and $x_2,y_2\in M_2$.
By (i), $y_1\oplus y_2=y_1\vee y_2$.
Then
$x_1=x_1\wedge (y_1\vee y_2)= (x_1\wedge y_1)\vee (x_1\wedge y_2)$. Since $x_1\in M_1$, then there is $a\in\mathcal I(M)$
such that $x_1\leq a$ and so $x_1\wedge y_2\leq a\wedge y_2=0$. It follows that $x_1= x_1\wedge y_1$ and so $x\leq y_1$. In a similar way,
$y_1\leq x_1$ which implies that $x_1=y_1$. Similarly, we can show that $x_2=y_2$. It follows that $\varphi$ is well defined.
(iii) $\varphi$ preserves $\vee$, $\wedge$, $\oplus$, $\ominus$ and $0$. Clearly, $\varphi$ preserves $\vee$ and $0$. Let $x,y\in M$.
Then there exist $x_1,y_1\in M_1$ and $x_2,y_2\in M_2$ such that $x=x_1\vee x_2$ and $y=y_1\vee y_2$.
$\varphi(x\oplus y)=\varphi(x_1\oplus x_2\oplus y_1\oplus y_2)=\varphi((x_1\oplus y_1)\oplus (x_2\oplus y_2))=(x_1\oplus y_1,x_2\oplus y_2)=
(x_1,x_2)\oplus (y_1,y_2)=\varphi(x)\oplus \varphi(y)$.
$\varphi(x\wedge y)=\varphi((x_1\vee x_2)\wedge (y_1\vee y_2))=\varphi((x_1\wedge y_1)\vee (x_2\wedge y_2))=(x_1\wedge y_1,x_2\wedge y_2)=
\varphi(x)\wedge \varphi(y)$ (note that $x_1\wedge y_2=0=x_2\wedge y_1$).
By the properties of wEMV-algebras and equation (\ref{3.12e2}), we have $(x_1\vee x_2)\ominus(y_1\vee y_2)=((x_1\vee x_2)\ominus y_1)\wedge ((x_1\vee x_2)\ominus y_2)=
((x_1\ominus y_1)\vee (x_2\ominus y_1))\wedge ((x_1\ominus y_2)\vee (x_2\ominus y_2))$. Also,
$x_2\ominus y_1= x_2\ominus (x_2\wedge y_1)=x_2\ominus 0=x_2$. Similarly, since $x_1\wedge y_2=0$, then
$x_1\ominus y_2=x_1$. So, $(x_1\vee x_2)\ominus(y_1\vee y_2)=((x_1\ominus y_1)\vee x_2)\wedge ((x_2\ominus y_2)\vee x_1)$. It follows that
$\varphi(x\ominus y)=\varphi(((x_1\ominus y_1)\vee x_2)\wedge ((x_2\ominus y_2)\vee x_1))=\varphi((x_1\ominus y_1)\vee x_2)\wedge
\varphi((x_2\ominus y_2)\vee x_1)=(x_1\ominus y_1, x_2)\wedge (x_1,x_2\ominus y_2)=(x_1\ominus y_1,x_2\ominus y_2)=\varphi(x\ominus y)$.
(iii) $\varphi$ is an isomorphism. Clearly, $\varphi$ is one-to-one and onto.
From (i)--(iii) we conclude that $M\cong M_1\times M_2$.
\end{proof}
We note that in the last theorem, $M_1$ is an associated wEMV-algebra and $M_2$ is a strict wEMV-algebra. So, if $M$ satisfies the conditions
of Theorem \ref{thm3.13}, then $M$ is a direct product of an associated wEMV-algebra and a strict wEMV-algebra.
We can easily prove that the converse also holds.
\begin{cor}\label{cor3.15}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra. Then $M\cong M_1\times M_2$ if and only if, for each $x\in M$, the set
$\{x\wedge a\mid a\in\mathcal I(M)\}$ has a greatest element.
\end{cor}
\begin{proof}
Let $M\cong M_1\times M_2$. Then
$M=\langle M_1\cup M_2\rangle$. Put $x\in M$.
There exist two elements $x_1\in M_1$ and $x_2\in M_2$ such that $x=x_1\oplus x_2$. Let $a$ be an arbitrary element of $\mathcal I(M)$.
By part (i) in the proof of Theorem \ref{thm3.13}, we have
$x\wedge a=(x\wedge a)\wedge x=(x\wedge a)\wedge (x_1\oplus x_2)=(x\wedge a)\wedge (x_1\vee x_2)=\left(\left(x\wedge a\right)\wedge x_1 \right)\vee \left(\left(x\wedge a\right)\wedge x_2 \right)=\left(x\wedge a\right)\wedge x_1$. It follows that $x_1$ is the greatest element
of the set $\{x\wedge a\mid a\in\mathcal I(M)\}$.
Conversely, by Theorem \ref{thm3.13}, it suffices to show that $M=\langle M_1\cup M_2\rangle$. Put $x\in M$. Let
$x_1:=\max\{x\wedge a\mid a\in\mathcal I(M)\}$. Then $x_1=x\wedge a$ for some $a\in\mathcal I(M)$. Set $x_2:=x\ominus x_1$.
Note that $x_1\wedge x_2=0$. Indeed, we have $x_1\oplus x_2=x_1\vee x=x$. We claim that $x_2\in M_2$.
(1) Let $u\in\mathcal I(M)$ and $z:=x\oplus u$.
According to Proposition \ref{pr:1}(g), we have $x\ominus u=\min\{t\in [0,x]\mid t\oplus (x\wedge u)=x\}$.
Also, it is well known that in the MV-algebra $([0,z];\oplus_z,\lambda_z,0,z)$ we have
$x\ominus_z u=\min\{t\leq x\mid t\oplus_z (x\wedge u)=x\}$. Since $(x\ominus_z u)\leq x$, then $(x\ominus_z u)\oplus (x\wedge u)\leq
x\oplus u\leq z$, then $(x\ominus_z u)\oplus (x\wedge u)=(x\ominus_z u)\oplus_z (x\wedge u)$ which simply implies that
$x\ominus_z u=x\ominus_z u$. So, $(x\ominus_z u)\wedge u=\lambda_z(\lambda_z(x)\oplus_z u)\wedge u=0$
(since $\lambda_z(\lambda_z(x)\oplus_z u)\leq \lambda_z(u)$ and $u$ is a Boolean element (i.e. an idempotent) of the mentioned MV-algebra).
For $u:=a$, we get that $x_1\wedge x_2=0$.
(2) Let $b\in\mathcal I(M)$. By the assumption, $x\wedge (a\vee b)=x\wedge a$.
\begin{eqnarray*}
(x\ominus a)\wedge b&=&\left(x\ominus \left(x\wedge a \right) \right)\wedge b=
\left(x\ominus \left(\left(x\wedge (a\vee b) \right) \right) \right)\wedge b=
\left(x\ominus \left(a\vee b\right) \right)\wedge b\\
&\leq & \left(x\ominus \left(a\vee b\right) \right)\wedge (a\vee b).
\end{eqnarray*}
By part (1), $\left(x\ominus \left(a\vee b\right) \right)\wedge (a\vee b)=0$, which implies that $x_2=x\ominus (x\wedge a)=x\ominus a\in M_2$.
\end{proof}
\begin{rmk}\label{product}
Let $(G^+;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra.
$(1)$ Let $M\cong A\times B$, where $A$ is an associated wEMV-algebra and $B$ is a strict wEMV-algebra.
Then $\mathcal I(M)\cong \{(a,0)\mid a\in \mathcal I(A)\}$. Clearly, $M_1\cong A\times \{0\}$. Also,
$M_2\cong \{0\}\times B$ (since $B$ does not have any non-zero element).
$(2)$ As a direct corollary, we have that if a wEMV-algebra $M$ admits a decomposition $M\cong M_1\times M_2$, where $M_2$ is a strict part of $M$ and $M_2$ is even cancellative, then there is an $\ell$-group $G$ such that $M_2\cong (G^+;\vee,\wedge,\oplus,\ominus,0)$, where the latter wEMV-algebra is defined in Example \ref{ex:1}.
\end{rmk}
\begin{cor}\label{cor3.16}
Consider the assumptions and notations in Theorem \ref{thm3.13}. The wEMV-algebra $M/M_1$ is a strict wEMV-algebra which is isomorphic to $M_2$.
\end{cor}
\begin{proof}
By the note just before Proposition \ref{prop3.7}, $M_1$ is an ideal of the wEMV-algebra $M$ and so by Proposition \ref{prop3.5},
$M/M_1$ is a wEMV-algebra. According to the proof of Theorem \ref{thm3.13}, for each $x\in M$, there are unique elements $x_1\in M_1$
and $x_2\in M_2$ such that $x=x_1\oplus x_2$. Define $f:M/M_1\to M_2$ by $f(x/M_1)=x_2$. We can easily check that $f$ is an isomorphism.
Since $M_2$ is strict, then $M/M_1$ is strict, too.
\end{proof}
We note that another type of a direct decomposition of a wEMV-algebra using a non-zero idempotent will be present in Remark \ref{re:decomp} below; it will use the representation result Theorem \ref{th:Repres}.
Now, we give two examples of wEMV-algebras satisfying the assumptions of Theorem \ref{thm3.13}.
\begin{exm}\label{exm3.17} Suppose that $(M;\vee,\wedge,\oplus,\ominus,0)$ is a wEMV-algebra. \\
(1) Let $M$ be a chain and take $x\in M$.
(i) If there exists $x\in M_2\setminus\{0\}$, then for each $a\in\mathcal I(M)$ we have $a\wedge x=0$.
Since $M$ is a chain, $x\leq a$ or $a\leq x$. From $x\leq a$, we get that $x=0$ which is absurd. Hence
$a\leq x$ which implies that $a=0$. Therefore, $\mathcal I(M)=\{0\}$. That is $M=M_2$. (ii) Otherwise,
$M_2=\{0\}$. If $x\in M\setminus M_1$, then $x\geq a$ for all $a\in\mathcal I(M)$ (since $M$ is a chain).
If there is $z\in M$ such that $x<z$, then
\begin{equation*}
0<\lambda_z(x)\wedge a=\lambda_z(x)\odot_{z} a \leq \lambda_z(x)\odot_{z} x=0,\quad a\in \mathcal I(M),
\end{equation*}
where $\odot_z$ is the well-known binary operation of the MV-algebra $(M;\oplus_z,\lambda_z,0,z)$.
It follows that $\lambda_z(x)$ is a non-zero element of $M_2$ which is a contradiction. So, $x$ is the greatest element of $M$, which
means that $x\oplus x=x\in\mathcal I(M)$ and $M=M_1$. By (i) and (ii), we entail that if $M$ is a chain, then $M=M_1$ or $M=M_2$.
That is, $M=\langle M_1\cup M_2\rangle$.
\vspace{2mm}
\noindent
(2) Let $M$ be the product of a family $\{M^i\}_{i\in T}$ of linearly ordered wEMV-algebras.
(i) Clearly, $\mathcal I(M)=\{(x_i)_{i\in T}\mid x_i\in\mathcal I(M^i),~\forall i\in T\}=\prod_{i\in T}\mathcal I(M^i)$.
It follows that $M_1=\prod_{i\in T} M^i_1$.
(ii) $(x_i)_{i\in T}\in M_2$ iff $(x_i)_{i\in T}\wedge (a_i)_{i\in T}=(0)_{i\in T}$ for all $(a_i)_{i\in T}\in \mathcal I(M)$ iff
$x_i\wedge a_i=0$ for all $a_i\in \mathcal I(M^i)$ and all $i\in T$. Hence $N_2=\prod_{i\in T}M^i_2$.
Let $x=(x_i)_{i\in T}\in M$. For each $i\in T$ by Case 1, $M^i=M^i_1$ or $M^i_2$. Let $T_1:=\{i\in T\mid M^i=M^i_1\}$
and $T_2:=\{i\in T\mid M^i=M^i_2\}$. Then $T=T_1\cup T_2$. For each $i\in T$ assume that
\begin{equation*}
y_i= \left\{\begin{array}{lllll}
x_i & i\in T_1\\
0 & \mbox {otherwise,}
\end{array}\right.
\hspace{2cm}
z_i= \left\{\begin{array}{llll}
x_i & i\in T_2 \\
0 & \mbox {otherwise.}
\end{array}\right.
\end{equation*}
Then by (i) and (ii), $y:=(y_i)_{i\in T}\in M_1$ and $z:=(z_i)_{i\in T}\in M_2$. Also, $x=y\oplus z$, so $M=\langle M_1\cup M_2\rangle$.
\end{exm}
\section{EMV-algebras and Pierce Sheaves
In the section, we study sheaves of EMV-algebras. If $M$ is a bounded EMV-algebra, $M$ is termwise equivalent to an MV-algebra on $M$, and the Pierce representation of MV-algebras is studied for example in \cite{FiGe} or in \cite[Part 4]{georgescu}. Theory of sheaf spaces of universal algebras is described in \cite{Dav}. In this part we concentrate to a case when an EMV-algebra $M$ does not have a top element.
First we investigate question on direct decomposability of an EMV-algebra.
We show that every idempotent element $a$ of an EMV-algebra determines a decomposition.
\begin{prop}\label{pr:5.1}
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra and let $a>0$ be a fixed idempotent of $M$ which is not top element. If $M$ is with top element,
$M$ is isomorphic to the direct product of bounded non-trivial EMV-algebras $([0,a];\vee,\wedge,\oplus,0,)$ and $([0,a'];\vee,\wedge,\oplus,0,)$, where $a'=\lambda_1(a)$, i.e.
$$
M \cong [0,a]\times [0,a'].
$$
If $M$ has no top element, let $(N;\vee,\wedge,\oplus,0)$ be its representing EMV-algebra with top element $1$, where $M$ can be embedded onto a maximal ideal of $N$. For simplicity, let $M\subseteq N$. Let $a'=\lambda_1(a)$.
Then $M$ is isomorphic to the direct product of the bounded non-trivial EMV-algebra $([0,a];\vee,\wedge,\oplus,0,)$ with a proper EMV-algebra $M_1= M\cap [0,a']$, i.e.
$$
M \cong [0,a]\times ([0,a']\cap M).
$$
Moreover, the set $[0,a']\cap M$ is a maximal ideal of the EMV-algebra $([0,a'];\vee,\wedge,\oplus,0)$ with top element $a'$ such that every element $x\in [0,a']\cap M$ is either from $M$ or $x= \lambda_{a'}(x_0)$ for a unique element $x_0\in M$.
\end{prop}
\begin{proof}
The mapping $f_a: M \to [0,a]$ defined by $f_a(x)=(x\wedge a)$, $x\in M$, preserves $0,\vee,\wedge,\oplus, \lambda_b$ ($b \in \mathcal I(M)$) and also $1$ if it exists in $M$, that is, $f_a$ is an EMV-homomorphism from $M$ into the bounded EMV-algebra $([0,a];\vee,\wedge,\oplus,0)$.
If $M$ has a top element, then the mapping $\phi: M\to [0,a]\times [0,a']$ defined by $\phi(x)=(x\wedge a, x\wedge a')$, $x \in M$, is an isomorphisms of the EMV-algebras $M$ and $[0,a]\times [0,a']$, i.e.
$ M \cong [0,a]\times [0,a']$.
Now, let $M$ be a proper EMV-algebra, $(N;\vee,\wedge,\oplus,0)$ be its representing EMV-algebra with top element, $M\subseteq N$. Then $M$ is a maximal ideal of $N$.
The mapping
$\varphi:N\ra [0,a]\times[0,a']$ sending $x\in N$ to $(x\wedge a,x\wedge
a')$ is an isomorphism of EMV-algebras and
$N\cong[0,a]\times[0,a']$. Set $E:=\varphi(M)=\{(x\wedge a,x\wedge a')\mid
x\in M\}$.
Clearly, $E$ and $M$ are isomorphic EMV-algebras. We claim that
$E=[0,a]\times ([0,a']\cap M)$. If $(x,y)\in [0,a]\times ([0,a']\cap M)$,
then clearly $\varphi(x\vee y)=(x,y)$, which implies that
$[0,a]\times ([0,a']\cap M)\s E$. Conversely, for each $x\in M$, we have
$x\wedge a'\le x\in M$, which gives $x\wedge a'\in M$ because $M$ is an ideal of $N$. Whence, $(x\wedge a,x\wedge a')\in [0,a]\times ([0,a']\cap M)$, so
the claim is true.
We note that $([0,a'];\vee,\wedge,\oplus,0)$ is an EMV-algebra, thus $[0,a']\cap M$ is a proper EMV-algebra, too. Indeed, if $y\in [0,a']\cap M$, then $y\in M$ and there is an idempotent $b\in M$ such that $y\le b$. The element $b\wedge a'\le b$, so that $b\wedge a'$ is an idempotent of $[0,a']\cap M$ such that $y\le b\wedge a'$. In addition, if $b$ is an idempotent of $[0,a']\cap M$, so is of $M$. If we take $y\in [0,b]$, then clearly $\lambda_b(y)$ is the least element $z \in [0,a']\cap M$ such that $z\oplus y = b$.
Finally, $M \cong [0,a]\times ([0,a']\cap M)$.
Clearly, $[0,a']\cap M$ is closed under $\oplus$. Let $x\in [0,a']\cap M$ and $y\in [0,a']$ such that $y\le x$. Since $M$ is an ideal of the EMV-algebra $N$, $y\in M$, so that $y\in [0,a']\cap M$. Now, let $z\in [0,a]\setminus ([0,a']\cap M)$. Then $z\in N\setminus M$ and the ideal of $N$ generated by $M$ and $z$ has to be $N$ because $M$ is a maximal ideal of $N$. Consequently, the ideal of the EMV-algebra $[0,a']$ generated by $[0,a']\cap M$ and $z$ is equal to $[0,a']$ which means that $[0,a']\cap M$ is a maximal ideal of $[0,a']$. Finally, let $x\in [0,a']$. If $x\in M$, then $x\in [0,a']\cap M$. If $x\in [0,a']\setminus ([0,a']\cap M)$, then $x\in N\setminus M$ and there is a unique element $y_0\in M$ such that $x=\lambda_1(y_0)$. Now, we use that there is a unital Abelian $\ell$-group $(G,u)$ such that $N=\Gamma(G,u)$ which means that $x= u-y_0$. Then $x= a+a' -y_0$ and $x=x\wedge a'= (a\wedge a')+(a'\wedge a') -(y_0\wedge a')= a'-x_0 $, where $x_0=y_0\wedge a'\in [0,a']\cap M$. Clearly that $x=\lambda'_{a'}(x_0)$, where $\lambda'_{a'}(x_0)=\min\{t \in [0,a']\cap M\mid t\oplus x_0=a'\}$ and it finishes the proof.
\end{proof}
We note that an EMV-algebra $M$ with top element is directly indecomposable (i.e. it cannot be expressed as a direct product of two non-trivial EMV-algebras) iff $\mathcal I(M)=\{0,1\}$. If $M$ has no top element it is always decomposable as a product of two non-trivial EMV-algebras.
Recall that a bounded distributive lattice $(L;\vee,\wedge,0,1)$ is called a {\em Stone algebra} if, for any $a\in L$, there exists a
Boolean element $b\in L$ such that $\{x\in L\mid x\wedge a=0\}=\downarrow b$.
\begin{rmk}
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra.
If $M$ is directly indecomposable, then $M$ has a greatest element $1$ and is termwise equivalent to an MV-algebra, $(M;\oplus,\lambda_1,0,1)$.
From \cite[Lem 4.9]{georgescu} it follows that $(M;\vee,\wedge,\oplus,0)$ is directly indecomposable and $(M;\vee,\wedge,0,1)$ is a
Stone algebra \iff $(M;\vee,\wedge,\oplus,0)$ is termwise equivalent to a linearly ordered MV-algebra.
\end{rmk}
\begin{rmk}\label{re:decomp}
Let $(M;\vee,\wedge,\oplus,\ominus,0)$ be a wEMV-algebra and let $N$ be its representing associated wEMV-algebra with a top element $1$, see Theorem \ref{th:Repres}. We can assume that $M \subseteq N$ and $N$ is equivalent to the MV-algebra $(N;\oplus,\lambda_1,0,1)$. Then Proposition \ref{pr:5.1} can be reformulated and proved verbatim in the same way also for wEMV-algebras. We note that we have non-trivial wEMV-algebras without any non-zero idempotent, see Example \ref{ex:1}.
\end{rmk}
A {\it sheaf space} of sets over $X$ or a {\it sheaf} is a triple $T=(E,\pi,X)$, where $E$ and $X$ are topological spaces and $\pi:E \to X$ is a surjective mapping that is a local homeomorphism, i.e. for all $e \in E$, there exist neighborhoods $U$ of $e$ and $V$ of $\pi(e)$ such that $\pi$ on $U$ is a homeomorphism of $U$ onto $V$. For all $x \in X$, the set $\pi^{-1}(\{x\})$ is a {\it fiber} of $x$. If $U$ is an open set of $X$, a {\it local section over} $U$ is a continuous function $g:U\to E$ such that $g(x) \in \pi^{-1}(\{x\})$ for all $x \in U$. If $U=X$, a local section $g$ is called a {\it global section} of the sheaf.
We note that a sheaf $(E,\pi,X)$ is a {\it sheaf of EMV-algebras} if
\begin{itemize}
\item[{\rm (i)}] each fiber $E_x=\pi^{-1}(\{x\})$ is an EMV-algebra,
\item[{\rm (ii)}] if $E\Delta E=\bigcup_{x\in X} (E_x\times E_x)$ with the induced topology from $E\times E$, then all operations $\oplus,\vee,\wedge$ are continuous from $E\Delta E$ to $E$.
\end{itemize}
If $(M;\vee,\wedge,\oplus,0)$ is a bounded EMV-algebra with top element $1$, then $(M;\oplus,',0,1)$, where $x':=\lambda_1(x)$, $x\in M$, is an MV-algebra which is termwise to $(M;\vee,\wedge,\oplus,0)$ and for MV-algebras there are known their Pierce representation by Boolean sheaves whose stalks are directly indecomposable, see \cite[Sect 4]{georgescu} for more information. Inspired by this result, we present a representation of proper EMV-algebras by Pierce sheaves.
Let $(M;\vee,\wedge,\oplus,0)$ be a proper EMV-algebra and $a\in\mathcal
I(M)$. Let $\mathcal P(\mathcal I(M))$ be the set of all prime ideals of $\mathcal
I(M)$. Define $V_a:=\{P\in \mathcal P(\mathcal I(M))\mid a\notin P \}$. Consider the
relation $\sim_a$ on $M$ define by $x\sim_a y$ if and only if
$x\odot a=y\odot a$. Clearly, $\sim_a$ is an equivalence relation on $M$.
There is $b\in\mathcal I(M)$ such that $a,x,y< b$.
Since $([0,b];\oplus,\lambda_b,0,b)$ is an MV-algebra, by \cite[Lem
4.15]{georgescu}, $x\sim_a y$ if and only if $(x\ominus y)\vee (y\ominus x)\leq
\lambda_b(a)$. Note that, by Lemma 5.1, the operation $x\ominus y:=x\odot
\lambda_b(y)$ is correctly defined and $x\odot y=x\odot_b y$.
It follows that $\sim_a$ is a congruence relation on the MV-algebra
$[0,b]$ and so by Proposition 3.13, it is a congruence relation on the
EMV-algebra $M$.
(1) For each $a\in\mathcal I(M)$, set $M_a:=M/\sim_a$, the quotient
EMV-algebra induced by the congruence relation $\sim_a$.
Also, for simplicity, we denote $x/\sim_a$ by $x/a$.
(2) If $V_b\s V_a$, then $a\leq b$. Otherwise, $a\notin [0,b]$ and so by
\cite[Thm 5.12]{Dvz}, there exists prime ideal $P$ such that
$a\notin P$ and $[0,b]\s P$, which is absurd. So, for each couple of
elements $a,b \in \mathcal I(M)$ with $a\le b$, define $\pi_{a,b}: M_b\ra M_a$ by
$\pi_{a,b}(x/b)=x/a$. It is an onto homomorphism of EMV-algebras. We can
easily see that if $a,b,c\in\mathcal I(M)$ such that
$a\leq b\leq c$, then $\pi_{a,b}\circ \pi_{b,c}=\pi_{a,c}$. Moreover,
$\pi_{a,a}:M_a\ra M_a$ is the identity map on $M_a$.
(3) Let $X$ be the set of all prime ideals of $\mathcal I(M)$ endowed with
the Stone--Zariski topology (= the hull-kernel topology) $\tau$. The sets $\{V_a\mid a\in\mathcal I(M)\}$ form a base of clopen subsets for this topological space. Since $M$ does not have a top element, the Stone--Zariski topology on $X$ gives a Hausdorff topological space that is locally compact but not compact and every $V_a$ is compact and clopen, see \cite[Lem 4.2, Thm 4.10]{Dvz1}.
Then we can extend the assignment
$V_a\mapsto M_a$ and $V_b\s V_a\mapsto \pi_{a,b}:M_b\ra M_a$ to
a Boolean sheaf $T$ of EMV-algebras, called a {\em Pierce sheaf} of $M$.
\begin{rmk}\label{re:6.1}
Consider the above assumptions.
(i) Let $I$ be an ideal of $\mathcal I(M)$.
Then $\downarrow I:=\{x\in M\mid x\leq a\ \exists\, a\in I\}$ is an ideal of
$M$, we call it a {\it Stonean ideal}. By Theorem 3.16, $M/\downarrow I$ is an EMV-algebra. We can easily show that
\begin{equation}
x/\downarrow I=y/\downarrow I \Leftrightarrow x\ominus y,y\ominus x\in
\downarrow I\Leftrightarrow x\ominus y,\ y\ominus x\leq a,\ \exists\, a\in I.
\end{equation}
(ii) Set $E_M:=\{x/\downarrow P\mid x\in M,~ P\in X\}$. Define $\pi:E_M\ra X$ by $\pi(x/\downarrow P)=P$. We show that $\pi$ is a well-defined surjective mapping. Let $x/\downarrow P=y/\downarrow Q$ for some $x,y \in M$. If $z\in x/\downarrow P$, then $z/\downarrow P=z/\downarrow Q$, which yields $\downarrow P= 0/\downarrow P=(z\ominus z)/\downarrow P= z/\downarrow P\ominus z\downarrow Q= z/\downarrow Q\ominus z/\downarrow Q=0/\downarrow Q=\downarrow Q$.
Now, we show that
$\downarrow P=\downarrow Q$ implies that $P=Q$. Indeed, if there exists $a\in P\setminus Q$, then from $a\in \downarrow Q$ we get that $a\leq b$ for some
$b\in Q$ and so $a\in Q$ (since $Q$ is an ideal of $\mathcal I(M)$). It follows that $\pi$ is well defined, moreover, $\pi$ is surjective. Suppose that
$$
U(I,x):=\{x/\downarrow P\mid I\nsubseteq P\},$$
where $I$ is an ideal of $\mathcal I(M)$.
Then $U(\{0\},x)=\emptyset$, and in addition, we have:
(1) $z\in P\in U(I,x)\cap U(J,y)$ implies that $z=x/\downarrow P=y/\downarrow Q$ for some $P,Q\in X$. Hence by (ii), $P=Q$, $I\nsubseteq P$,
$J\nsubseteq Q$ and $z=(x\wedge y)/\downarrow P$. There exist $w_1\in I\setminus P$ and $w_2\in J\setminus P$. Clearly, $w_1\wedge w_2\notin P$ and so $I\cap J\nsubseteq P$ which entails that $z= (x\wedge y)/\downarrow P\in U(I\cap J,x\wedge y)$.
(2) For each $x/\downarrow P\in E_M$, choose an idempotent
$a$ such that $a\notin P$. Then clearly, $x/\downarrow P\in U(\langle a\rangle,x)$.
From (i) and (ii) it follows that $\{U(I,x)\mid I\in \mathrm{Ideal} (\mathcal I(M)),~x\in M\}$ is a base for a topology $\tau'$ on $E_M$. Denote this
topological space by $(E_M,\tau')$.
\end{rmk}
We note that $\{V_a\mid a\in\mathcal I(M)\}$ is a base of the topology $\tau$. Clearly, $V_a\cap
V_b=V_{a\cap b}$, since if
$P\in V_a,V_b$, then $a,b\notin P$ and so $a\wedge b\notin P$. Conversely,
if $P\in V_{a\wedge b}$, then $a\wedge b\notin P$ and so
$a\notin P$ and $b\notin P$, so that $a\in P$ or $b\in P$ which implies that
$a\wedge b\in P$. That is $\{V_a\in a\in\mathcal I(M)\}$ is closed
under finite intersections. Moreover, we can easily show that, for the family $\{V_{a_i}\mid a_i\in\mathcal I(M),i\in J\}$, we have
$\bigcup_{i\in J} V_{a_i}=\{P\in X\mid \{a_i\mid i\in J\}\nsubseteq P\}=\{P\in
X\mid \langle\{a_i\mid i\in J\}\rangle \nsubseteq P\}$.
So, there is an ideal $I$ of $\mathcal I(M)$ such that $\bigcup_{i\in J}
V_{a_i}=V_I:=\{P\in X\mid I\nsubseteq P\}$. In addition, every open set in $\tau$ is of the form $V_I=\{P\in X\mid I\nsubseteq P\}$, where $I$ is an ideal of $\mathcal I(M)$ and vice-versa.
We can easily check that $\pi:E_M\ra X$ sending $x/\downarrow P$ to $P$ is a local homeomorphism from the topological space $(E_M,\tau')$ to $(X,\tau)$. Consequently, we have the following result:
\begin{thm}\label{th:6.2}
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra. Then
\begin{itemize}
\item[{\rm (i)}] $T=(E_M,\pi,X)$ is a sheaf of sets over $X$.
\item[{\rm (ii)}] For each $P\in X$, $\pi^{-1}(P)=M/\downarrow P$ is a bounded directly indecomposable EMV-algebra.
\item[{\rm (iii)}] $T=(E_M,\pi,X)$ is a sheaf of bounded EMV-algebras over $X$.
\item[{\rm (iv)}] For each $x\in M$, the map $\widehat{x}:X\ra E_M$, defined by $\widehat{x}(P)=x/\downarrow P$, is a global section of $T$.
\end{itemize}
\end{thm}
\begin{proof}
(i) This part is straightforward to verify.
(ii) Now, we show $M/\downarrow P$ is a bounded directly indecomposable EMV-algebra. The proof is divided into three steps.
(1) Let $x/\downarrow P$ be an idempotent element of $M/\downarrow P$.
Then $(x\oplus x)/\downarrow P=x/\downarrow P$ and so $(x\oplus x)\ominus x\leq p\in P$. Put $a\in \mathcal I(M)$ such that $x\leq a$.
Then $2.x\leq x\oplus p$. Also, $x\oplus p\leq 2.(x\oplus p)=(2.x)\oplus p\leq (x\oplus p)\oplus p=x\oplus p$, so $x\oplus p\in\mathcal I(M)$.
On the other hand, from $\lambda_a(x)/\downarrow P\in \mathcal I(M/\downarrow P)$,
we can show that $\lambda_a(x)\oplus q\in \mathcal I(M)$ for some $q\in P$. Let $a,p,q\leq b\in \mathcal I(M)$.
Consider the MV-algebra $([0,b];\oplus,\lambda_b,0,b)$.
Then
\begin{eqnarray*}
x\wedge \lambda_a(x)\leq x\wedge \lambda_b(x)=\lambda_b(\lambda_b(x\oplus x)\oplus x)=(x\oplus x)\ominus x\leq p\leq c,
\end{eqnarray*}
where $c:=p\oplus q$ (note that, $c\leq b$).
It follows that $(x\oplus c)\wedge (\lambda_a(x)\oplus c)=(x\wedge \lambda_a(x))\oplus c\leq c\oplus c=c\in P$.
Clearly, $x\oplus c=x\oplus p\oplus q\in \mathcal I(M)$. In a similar way, $\lambda_a(x)\oplus c\in \mathcal I(M)$ and so
$x\oplus c\in P$ or $\lambda_a(x)\oplus c\in P$. Consequently, $x\in \downarrow P$ or $\lambda_a(x)\in \downarrow P$. That is either
$x/\downarrow P=0/\downarrow P$ or $\lambda_a(x)/\downarrow P=0/\downarrow P$ for all idempotent $a\geq x$.
(2) Now, we prove that $M/\downarrow P$ is a bounded EMV-algebra, i.e. it has a top element. First, let $x$ and $y$ be such elements of $M$ that $x/\downarrow P$ and $y/\downarrow P$ are idempotents of $M/\downarrow P$ and $x/\downarrow P\le y/\downarrow P$. From the previous paragraph we know that we can assume without loss of generality that $x$ and $y$ are idempotents of $M$. In addition, since $x/\downarrow P\le y/\downarrow P$ iff $x\le y\oplus p$ for some idempotent $p \in P$. Hence, we can assume that $x$ and $y$ are idempotents such that $x\le y$. Denote by $x_0=y\ominus x$. Then $x_0$ is an idempotent of $M$ such that $x_0/\downarrow P = y/\downarrow P \ominus x/\downarrow P$. Since $\lambda_y(x_0)\in \mathcal I(M)$, from $x_0\wedge \lambda_y(x_0)=0\in P$, we have either $x_0\in P$ or $\lambda_y(x_0)\in P$, so that either $x_0/\downarrow P =0/\downarrow P$ or $\lambda_y(x_0)/\downarrow P =0/\downarrow P$. In the first case we have $x/\downarrow P=y/\downarrow P$. In the second one from $y=x_0\oplus \lambda_y(x_0)$ we have $y/\downarrow P=x_0/\downarrow P= y/\downarrow P\ominus x/\downarrow P$ which yields $x/\downarrow P=0/\downarrow P$.
(3) Now, assume that $M/\downarrow P$ does not have a top element. Therefore, there exists an infinite sequence $\{a_n\}_n$ of elements of $M$ such that every $a_n/\downarrow P$ is an idempotent of $M/\downarrow P$ and $a_n/\downarrow P<a_{n+1}/\downarrow P$. Due to the last paragraph, we can assume that every $a_n$ is an idempotent of $M$, and due to fact that for each $n$, there is an idempotent $p_n$ such that $a_n \le a_{n+1}\oplus p_n$ which allows us to assume that $a_n\le a_{n+1}$ for each $n\ge 1$. By paragraph (2), we see that in each interval $[0/\downarrow P,a_n/\downarrow P]$ there is no idempotent of $M/\downarrow P$ different of $0/\downarrow P$ and $a_n/\downarrow P$ which is a contradiction with $a_n\downarrow P<a_{n+1}/\downarrow P$. Whence, $M\downarrow P$ is a bounded EMV-algebra.
Finally, we have therefore, $M/\downarrow P$ has only two different idempotent elements, so it is bounded and directly indecomposable.
(iii) By (ii), we have that $T=(E_M,\pi,X)$ is a sheaf whose each fiber $\pi^{-1}(P)$ is a bounded indecomposable EMV-algebra. We have to show that $\oplus$ is continuous; the proof of continuity of $\vee$ and $\wedge$ is similar.
Thus, let $x,y \in M$, $P\in X$, and $\widehat x(P), \widehat y(P) \in \pi^{-1}(\{P\})=M/\downarrow P$ be given. Let $V$ be an open neighborhood of $\widehat{x\oplus y}(P)$. Without loss of generality, let $V=U(I,x\oplus y)$ for some neighborhood $V_I$ of $P$. The set $B=\{(\widehat x(P),\widehat y(P))\mid P \in U(I,x\oplus y)\}$ is an open neighborhood of $(\widehat x(P),\widehat y(P))$ in $E_M \Delta E_M$. For the mapping $\beta: (t,s) \mapsto t\oplus s$, we have $\beta^{-1}(V)=B$, so that $\beta$ is continuous.
(iv) Let $x \in M$. Since $\widehat x(P)=x/\downarrow P \in \pi^{-1}(\{P\})$, it is necessary to show that $\widehat x$ is a continuous mapping. Take an arbitrary open set in $E_M$ of the form $U(I,x)=\{x/\downarrow P\mid I \nsubseteq P\}$, where $I$ is any ideal of $\mathcal I(M)$. Then
\begin{align*}
\widehat x^{-1}(U(I,x))&=\{P \in X \mid x/\downarrow P \in U(I,x)\}\\
&=\{P\in X\mid x/\downarrow P= y/\downarrow Q, y\in M, I\nsubseteq P,Q\}\\
&= \{P \in X\mid I \nsubseteq P\}=V_I,
\end{align*}
which is an open set in the hull-kernel topology on $X$. Whence, each $\widehat x$ is a global section.
\end{proof}
\begin{cor}\label{th:6.3}
Let $(M;\vee,\wedge,\oplus,0)$ be an EMV-algebra and $\widehat{M}:=\{\hat{x}\mid x\in M\}$. Consider the following operations on $\widehat{M}$:
$$(\widehat{x}~ \widehat{*}~ \widehat{y})(P)=\widehat{x}(P)* \widehat{y}(P),\quad \forall *\in\{\vee,\wedge,\oplus\}. $$
Then $(\widehat{M};\widehat{\vee},\widehat{\wedge},\widehat{\oplus},\widehat{0})$ is an EMV-algebra.
\end{cor}
\begin{proof}
It is a direct corollary of Theorem \ref{th:6.2}
\end{proof}
We say that an EMV-algebra $M$ is {\it semisimple} if it is a subdirect product of simple EMV-algebras. It is possible to show that $M$ is semisimple iff the intersection of all maximal ideals of $M$ is the set $\{0\}$. In addition, in \cite[Thm 4.11]{Dvz}, we have characterized semisimple EMV-algebras as EMV-algebras of fuzzy sets where all EMV-operations are defined pointwisely.
We say that an EMV-algebra $M$ satisfies the {\it general comparability property} if it holds for every MV-algebra $([0,a]; \oplus,\lambda_a,0,a)$, i.e. if, for any $a \in \mathcal I(M)$ and $x,y \in [0,a]$, there is an idempotent $e \in [0,a]$ such that $x\wedge e\le y$ and $y\wedge \lambda_a(e) \le x$.
In what follows we show that every semisimple proper EMV-algebra with the general comparability property can be embedded into a sheaf of bounded EMV-algebras on the space $X$.
\begin{thm}\label{th:6.4}
Every semisimple EMV-algebra with the general comparability property can be embedded into a sheaf of bounded EMV-algebras on the space $X$.
\end{thm}
\begin{proof}
Due to \cite[Thm 4.4]{Dvz}, the restriction of any maximal ideal $I$ of $M$ to $I\cap \mathcal I(M)$ gives a maximal ideal of $\mathcal I(M)$, so it belongs to $X$. Conversely, according to \cite[Thm 4.9]{Dvz1}, every prime ideal $P$ of $\mathcal I(M)$ (hence every maximal ideal of $\mathcal I(M)$) can be extended to a maximal ideal $\downarrow P$. Then $\bigcap \{\downarrow P\mid P \in X\}=\Rad(M)=\{0\}$ which implies that $M$ is a subdirect product of the system $\{M/\downarrow P \mid P\in X\}$ of bounded indecomposable EMV-algebras.
\end{proof}
Now, we present the following representation of EMV-algebras as sections of sheaves.
\begin{thm}\label{th:6.5}
Let $M$ be an EMV-algebra and $X$ be a Hausdorff topological space. If for $x\in M$, there is an ideal $I_x$ of $M$ such that $\bigcap_{x\in X} I_x =\{0\}$ and for all $x\in M$, the set $\{x\in X\mid x \in I_x\}$ is open, then $M$ can be embedded into a sheaf of EMV-algebras on the space $X$.
\end{thm}
\begin{proof}
Let $E=\bigcup_{x\in X}\{M/I_x \times \{x\}\}$ and define a mapping $\pi:E\to X$ by $\pi(a/I_x,x)\mapsto x$, $(a/I_x,x)\in E$. It is a well-defined mapping because if $(a/I_x,x)=(b/I_y,y)$, then $x=y$ and $a/I_x=b/I_x$. In addition, $\pi$ is surjective and $\pi^{-1}(\{x\})=M/I_x$ for each $x \in X$.
For all $a\in M$, define a mapping $\widehat a: X \to E$ by $\widehat a(x)=(a/I_x,x)$, $x \in X$.
We assert that the system $\{\widehat a(U) \mid U \text{ open in } X,\ a \in M\}$ is a base of a topology on $E$. Let $a,b \in M$ and $U,V$ be open in $X$. Since $\{x\in X\mid a\in I_x\}=\{x\in X\mid \widehat a(x)=(0/I_x,x)\}$ is open, then $A=\{x \in X \mid \widehat a(x)=\widehat b(x)\}= \{x\in X\mid (\widehat{a\ominus b})(x)=(0/I_x,x)\}\cap \{x \in X\mid (\widehat{b\ominus a})(x)=(0/I_x,x)\}$ is open in $X$. Whence, $B=A\cap U\cap V$ is also open. For all $w \in B$, $\widehat a(w)=\widehat b(w)$ and $\widehat a(w) \in \widehat a(U)\cap \widehat b(V)$. If $t \in \widehat a(U)\cap \widehat b(V)$, then $\widehat a(\pi(t))=t=\widehat b(\pi(t))$ which yields $\widehat a(B)=\widehat b(B)= \widehat a(U)\cap \widehat b(V)$. So this system is a base of a topology on $E$. Every mapping $\widehat a$ is continuous. Indeed, choose $b\in M$ and $V$ open in $X$. Then we have $\widehat a^{-1}(\widehat b(V))=\{x\in X\mid \widehat a(x)=(a/I_x,x)\in \widehat b(V)\}=\{x\in V\mid (a/Ix,x)=(b/I_x,x)\}=\{x\in V\mid a\ominus b\in I_x\}\cap\{x\in V\mid b\ominus a\in I_x\}$ is open in $X$. In addition, $\widehat a$ is an open mapping and $\pi$ is a local homeomorphism.
The system $T=(E,\pi,X)$ is thus a sheaf. Now, we show that all operations $\oplus,\vee,\wedge $ are continuous. We verify it only for $\oplus$ and for other operations it is similar. Let $x\in X$ and $\widehat a(x),\widehat b(x)\in\pi^{-1}(\{x\})=M/I_x$. Let $V$ be an open neighborhood of $(\widehat{a\oplus b})(x)$. Without loss of generality, let $V=(\widehat{a\oplus b})(U)$ for some open neighborhood $U$ of $x\in X$. The set $C=\{(\widehat a(u),\widehat b(u))\mid u \in U\}$ is an open neighborhood of $(\widehat a(x),\widehat b(x))$ in $E\Delta E$. The mapping $\alpha: (s,t) \mapsto s\oplus t$ from $E\Delta E$ to $E$ has the property $\alpha^{-1}(V)=C$, so that $\alpha$ is continuous and hence, $\oplus $ is continuous.
\end{proof}
\begin{defn}
A distributive lattice $(L;\vee,\wedge)$ with the least element $0$ is called a {\em weak Stone algebra} if, for each $x\in L$, there is
a Boolean element $a\in L$ such that $[0,a]$ is a Stone algebra. For simplicity, an EMV-algebra which is a weak Stone algebra is called
a {\em Stone EMV-algebra}.
\end{defn}
Let $\{M_i\mid i\in\mathbb N\}$ be a class of Stone MV-algebras. By \cite{Dvz}, we know that $M:=\Sigma_{i\in\mathbb N} M_i$ is an EMV-algebra.
Let $(x_i)_{i\in\mathbb N}$ be an arbitrary element of $\Sigma_{i\in\mathbb N} M_i$. There is $n\in\mathbb N$ such that $x_i=0$ for all
$i\geq n+1$. Let $\{x\in M_i\mid x\wedge x_i=0\}=\downarrow b_i$ for all $i\in\{1,2,\ldots,n\}$. Set $u_i=b_i$, for all $1\leq i\leq n$ and
$u_i=0$ for all $i\geq n+1$. Then $u=(u_i)_{i\in\mathbb N}\in\mathcal I(M)$, $x\leq u$, and we can easily check that
$([0,u];\oplus,\lambda_u,0,u)$ is a Stone MV-algebra. Therefore, $M$ is a Stone EMV-algebra.
\begin{thm}\label{WStone pro}
Let $(M;\vee,\wedge,\oplus,0)$ be a Stone EMV-algebra and $P\in\mathcal P(\mathcal I(M))$. Then
\begin{itemize}
\item[{\rm (i)}] $[0,a]\cap\downarrow P$ is a prime ideal of the MV-algebra $(M;\oplus,\lambda_a,0,a)$;
\item[{\rm (ii)}] $Q:=\downarrow P$ is a prime ideal of EMV-algebra $M$;
\item[{\rm (iii)}] $M$ can be embedded into the MV-algebra of global sections of a Hausdorff Boolean sheaf whose stalks are MV-chains.
\end{itemize}
\end{thm}
\begin{proof}
(i) Let $a\in\mathcal I(M)$ and $Q:=\downarrow P$. Clearly, $Q$ is an ideal of $M$ and $[0,a]\cap Q=\downarrow(P\cap [0,a])$.
Since $[0,a]\cap P$ is a prime ideal of $\mathcal I([0,a])$, then by the assumption and \cite[Lem 4.20]{georgescu},
$Q\cap [0,a]=\downarrow ([0,a]\cap P)$ is a prime ideal of the MV-algebra $([0,a];\oplus,\lambda_a,0,a)$.
(ii) Put $x,y\in M$ such that $x\wedge y\in Q$. Then there exists $a\in\mathcal I(M)$ such that $x,y\leq a$.
Consider the MV-algebra $([0,a];\oplus,\lambda_a,0,a)$. By (i), $Q\cap [0,a]$ is a prime ideal of $[0,a]$, so from
$x,y\in [0,a]$ and $x\wedge y\in Q\cap [0,a]$ it follows that $x\in Q\cap [0,a]$ or $y\in Q\cap [0,a]$, which means that $Q$ is a
prime ideal of $M$.
(iii) First, we show that the natural map $M\to \prod_{P\in X}M/\downarrow P$ is one-to-one, where $X=\mathcal P(\mathcal I(M))$.
Let $x\in M$ be such that $x\in\downarrow P$ for all $P\in X$. If $x\in\mathcal I(M)$, then clearly $x=0$ (since $\bigcap_{P\in X} P=\{0\}$).
Otherwise, if $x\notin \mathcal I(M)$, then by the assumption, there is $a\in\mathcal I(M)$ and $b\in\mathcal I([0,a])$ such that
$x\leq a$, $([0,a];\oplus,\lambda_a,0,a)$ is a Stone MV-algebra and $\{y\in [0,a]\mid y\wedge x=0\}=\downarrow b$.
Put $P\in X$. Then there is $e\in P$ such that $x\leq e$. Clearly, $x\leq a\wedge e\in P$. Set $f:=a\wedge e$. Then
$x\wedge \lambda_a(f)=x\odot \lambda_a(f)=0$ which implies that $\lambda_a(f)\leq b$ and so $\lambda_a(b)\leq f\in P$.
It follows that $\lambda_a(b)\in \bigcap_{P\in X} P=\{0\}$. Thus $b=a$ and so $x=x\wedge a=0$ which is a contradiction.
Therefore, $\bigcap_{P\in X}\downarrow P=\{0\}$, which implies that the natural map $M\to \prod_{P\in X}M/\downarrow P$ is one-to-one.
The rest part of the proof is similar to the proof of Theorem \ref{th:6.4}.
\end{proof}
\section{Conclusion}
EMV-algebras are a common generalization of MV-algebras and generalized Boolean algebras so that the existence of a top element is not assumed a priori. Every EMV-algebra either has a top element and then is equivalent to an MV-algebra or a top element fails but it can be embedded into an EMV-algebra with top element as the maximal ideal of the second one. The class of EMV-algebras is not a variety because it is not closed under forming subalgebras. Therefore, we were looking for an appropriate variety of algebras very closed to EMV-algebras containing the class of EMV-algebras as the least variety. We showed that such a class of algebras is forming by new introduced wEMV-algebras which form a variety. If we added to the language of every EMV-algebra a new derived operation $\ominus$, we obtained a wEMV-algebra associated to the original EMV-algebra. One of the basic result is to show that the variety of EMV-algebras is the least subvariety of the variety of wEMV-algebra containing all associated EMV-algebras, see Theorem \ref{thm3.11}. This was possible due to the fact that every wEMV-algebra can be embedded into some associated wEMV-algebra, Theorem \ref{thm3.10}. A representation of a wEMV-algebra $M$ by an associated wEMV-algebra $N$ with top element, where $M$ can be embedded as a maximal ideal of $N$ was presented in Theorem \ref{th:Repres}. We have shown that we have countably many different subvarieties of wEMV-algebras, see Theorem \ref{th:subvar}.
In addition, we studied a situation when a wEMV-algebra $M$ is isomorphic to a direct product of two subalgebras $M_1$ and $M_2$ of $M$, where $M_1$ is a greatest associated wEMV-subalgebra and $M_2$ is a strict wEMV-subalgebra, see Theorem \ref{thm3.13}.
Finally, we studied the Pierce sheaves of EMV-algebras without top element in Section 4, see Theorem \ref{th:6.2}--\ref{th:6.5}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,045
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This is Public Enemy No. 1 for Republicans
ByEileen Shim
President Bill Clinton has been stumping around Kentucky this week to drum up support for Alison Lundergan Grimes, the Democrat who is trying to unseat the incumbent Sen. Mitch McConnell (R-Ky.) in the coming Senate race. But Sen. Rand Paul (R-Ky.), the junior senator from the state who has endorsed McConnell, is not quite happy to see the former president in his neck of the woods.
"We have a lot of conservative Democrats in our state who go to church each week and really don't approve of his behavior, what he has done with sexual harassment in the workplace," Paul told Fox News on Tuesday.
Paul is of course referring to the Monica Lewinsky scandal, which in 1998 revealed that Clinton was having an affair with a White House intern. But apparently, 16 years has not been enough time for Paul to let it go.
"I think he's a bad role model for the workplace, for women's rights, for all of that, and I think really they ought to be a little embarrassed to be associated or being seen with him," Paul said on Fox News.
We know we can take Paul's word for it because he is a noted feminist scholar who has declared that "the war on women" is already over. And this isn't the first time he's piled on Clinton — he has called Clinton a "sexual predator" on occasion, and has accused the American media of moving past the scandal. So if anyone has the moral authority to call someone "a bad role model for women's right," it's Rand Paul.
Image credit: Fox News
While it's understandable that Paul is lashing out at a Democrat surrogate who is campaigning against his colleague, his attempts to discredit the former president also serve a dual purpose: to lay the groundwork to campaign against Hillary Clinton, who is still the presumptive Democrat nominee for the 2016 presidential race. Paul himself is currently a popular choice for the GOP.
Bill was the last Democrat president to carry a chunk of the South, and he may put his folksy surrogate charm to use if and when his wife runs for office. He is also a big fundraiser for the Democratic Party (he raised $700,000 for Grimes at a single event on Tuesday).
And that's exactly what Paul is afraid of — the senator has gone as far to suggest that Hillary should return campaign funds that Bill raised.
Hillary has not yet definitely announced plans to run for office again, but the Republican Party is already pulling out all the stops to prevent it from happening. Though Sen. Ted Cruz (R-Texas) declined to follow Paul's attack, he has gone after Hillary for having a "consistently wrong record when it comes to foreign policy, when it comes to domestic policy." (It's worth noting that in the same interview, Cruz fields questions about his own presidential ambitions.)
In the meantime, the GOP has put up this hilarious petition to "stop the Clintons from taking back the White House:
Image credit: GOP
But the GOP's attempts to date the Clintons in the same category as Austin Powers and the Macarena may not working. More than 80% of Democrats want Hillary to run, and virtually every prediction poll has Hillary trumping potential Republican nominees.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,145
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