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\section{Introduction}
\label{intro}
One of the most debated issues in massive-star research is how mass loss and binarity impact stellar evolution. In this context, understanding luminous blue variable stars (LBVs) and their environments is of great interest as they experience high mass-loss episodes and thus represent a critical phase in massive-star evolution. Several physical causes are discussed for the high mass-loss events of LBVs. Most of them invoke the star's proximity to its Eddington limit and models include an opacity-modified Eddington limit, sub-photospheric gravity-mode instabilities, super-Eddington winds, envelope inflation, fast rotation, and binary interactions and mergers (e.g., \citealt{1994PASP..106.1025H,2020Galax...8...10D} and references therein; \citealt{2009ApJ...705L..25G,2012A&A...538A..40G,2015A&A...580A..20S,2014ApJ...796..121J,2015MNRAS.447..598S}).
Information on the evolutionary status and physical nature of LBVs can be gained by investigating their environments and circumstellar nebulae with integral field unit (IFU) spectroscopy. IFU observations facilitate studies of the stellar content around LBVs and of the large circumstellar nebulae found around most LBVs. They enable one to identify stellar clusters and to study the massive stellar content and feedback, but also to determine gas and dust properties of the nebulae and to reconstruct past and present-day wind geometries.
One means of investigating the evolutionary status of LBVs is by studying their spatial association with O- and B-type stars.
In traditional single-star evolutionary models, high-luminosity LBVs ($log~L/L_\odot > 5.7$, $M_\mathrm{init} > 40~M_\odot$) are brief ($\sim 10^5$~yr) transitions between single massive O-type and Wolf-Rayet (WR) stars \citep{1984IAUS..105..233C, 1994PASP..106.1025H,2014A&A...564A..30G}. Less luminous LBVs ($log~L/L_\odot < 5.7$, $M_\mathrm{init} \sim 20-40~M_\odot$) are assumed to be post-red supergiants \citep{1988ApJ...324..279L,1994PASP..106.1025H}. Recent models show that LBVs can also be at the end stage of massive-star evolution, especially at low metallicity \citep{2019A&A...627A..24G}.
For a typical velocity dispersion in a cluster of a few km~s$^{-1}$, a star would move less than $\sim$10~pc in 3~Myr, which is the typical lifetime of very massive stars. Thus LBVs and O-type stars should be co-spatial if they evolve as single-star models predict.
In the past, only a few stellar association studies have been conducted for LBVs. \citet{2003A&A...400..923H} resolved clusters with sizes of about $4~\mathrm{pc} \times 4~\mathrm{pc}$ around RMC~127 and RMC~128 in the Large Magellanic Cloud (LMC). They found that the brightest members around RMC~128 are evolved massive stars with late-O and early-B spectral types. The cluster around the LBV RMC~127 contains fewer members and no stars with spectral types earlier than B0.
The topic of a single- versus binary-star evolutionary path for LBVs has been developing into a lively scientific discussion.
\citet{2015MNRAS.447..598S} and \citet{2016MNRAS.461.3353S,2019MNRAS.489.4378S} found that LBVs reside in relatively isolated environments and are not concentrated in young massive clusters with early O-type stars, as would be expected according to the traditional view of single-star evolution. The authors reason that LBVs are primarily the product of binary evolution. A strong motivation for this hypothesis is the finding that binary interaction dominates the evolution of massive stars (e.g., \citealt{2012Sci...337..444S}). Smith et al.\ propose that most WR stars and Type Ibc supernova (SN) progenitors are mass donors, while LBVs are mass gainers. Through mass transfer, rejuvenated mass gainers get enriched, spun up, and sometimes kicked far from their birth sites by a SN of their companion.
\citet{2017MNRAS.472..591A} derived an approximate analytical model for the passive dissolution of young stellar clusters and computed the relative separation between O-type stars and LBVs. They found that the standard single-star evolution model is mostly inconsistent with observed LBV environments, further suggesting that LBVs are rejuvenated stars in binary systems. One scenario is that they increase their lifetime as the result of mergers, another is that they are mass gainers and receive a kick when the primary star explodes either by an asymmetric explosion or the Blaauw mechanism \citep{1961BAN....15..265B}.
\citet{2016ApJ...825...64H} tested the hypothesis of \citet{2015MNRAS.447..598S} for the LBVs in M31, M33, and the Magellanic Clouds. They show that LBVs are associated with luminous young stars and supergiants that are appropriate for the LBV luminosities and positions in the Hertzsprung–Russell (HR) diagram. The more luminous LBVs have a spatial distribution similar to late O-type stars, while the less luminous LBVs that are assumed to be post-red supergiants have a distribution similar to red supergiants (see also \citealt{2016arXiv160802007D}). The spatial velocities of LBVs are also consistent with their positions in the respective galaxies.
\citet{2018AJ....156..294A} emphasize that our knowledge of the spectroscopic content of the LMC, M31, and M33 is quite incomplete and instead used photometric criteria to select the highest-mass unevolved stars from spatially complete photometric catalogs of these galaxies. They found that the majority of LBVs are in or near OB associations as are the blue supergiants and WR stars, while the red supergiants are not. They conclude that the spatial distribution of LBVs is consistent with a single-star evolutionary path. \citet{2019MNRAS.489.4378S} argue that bright blue stars are not good tracers of the youngest massive O-type stars and uphold their proposal of LBVs being binary products.
Recently, \cite{2021arXiv210512380M} provided a bias-corrected spectroscopic binary fraction among Galactic (candidate) LBVs of $62^{+38}_{-24}$\%, covering period ranges of $1-1000$~d. They found an even higher binary fraction of 78\% using interferometry for projected separations up to $100-150$~mas. Stellar radii of $100-650~R_{\odot}$ were also derived in this work. Taken together, these characteristics make short-period binary systems among (candidate) LBVs unlikely. If LBVs form through single-star evolution, the initial orbits must be wide. On the other hand, if they are the end products of binary evolutionary channels, then massive stars in a short-period binary system either undergo a phase of fully nonconservative mass transfer to widen the orbit or LBVs form through mergers in binary or triple systems.
We aim to set the stage for further studies of the population of LBVs by investigating the stellar content of the volume around the Galactic LBV HR~Car and its circumstellar nebula. HR~Car's nebula has a bipolar morphology with two lobes of $\sim19\arcsec$ in diameter \citep{1997ApJ...486..338N}. Nebular mass estimations range from $0.8~M_{\odot}$ \citep{2000ApJ...539..851W} to $2.1~M_{\odot}$ \citep{1995AJ....110..251C} for the ionized gas, which can be seen as a proxy for the total nebular mass.
The star had two recent outburst events, in 1989--1993 and in 1997--2005, when it moved temporarily across the HR diagram to cooler temperatures \citep{ 2003IAUS..212..243S}.\footnote{See also the American Association of Variable Star Observers (AAVSO) lightcurve at \url{https://www.aavso.org}.}
The hypothesis that LBVs are mass gainers, and at least originally the less massive component of massive binaries, can be tested in particular with HR~Car as it is both an LBV and a binary.
Interferometric monitoring of HR~Car, complemented by HARPS spectra show an orbital period of $P=2330$~d, an eccentricity of $e=0.106$, a minimum separation between the stars of about 10~au, an inclination of $i=119\degree$, a mass of the primary of $\sim 21~M_{\odot}$, and a companion mass of $\sim 10~M_{\odot}$, assuming a distance of $d=4.9$~kpc (Boffin et al., in prep.).\footnote{These preliminary orbital parameters are much improved compared to the best-fit orbit presented in \citealt{2016A&A...593A..90B}.} The nature of the companion is still unknown. \citet{2016A&A...593A..90B} suggested that the secondary could be a red supergiant, but no corresponding spectral features were observed in a near-infrared spectrum obtained in 2017 (Boffin et al., in prep.). The current stellar and nebular masses imply that the LBV in HR~Car, and probably the initial main-sequence masses of the binary or triple components, barely exceeded the mass of an O-type star. Given the evolved nature of HR~Car, one would not expect (higher-mass) O-type stars to be still around in its vicinity. We thus investigate if HR~Car is part of a moving group, that is part of a stream of stars with common age and motion.
In Section \ref{data} we describe the observations, data analysis, and auxiliary data. Section \ref{environment_HRCar} presents the stellar environment of HR~Car. Section \ref{circumstellarnebula} discusses the large-scale circumstellar nebula and Section \ref{bipolaroutflow} the discovery of a fast inner outflow and ``bullets''. In Section \ref{environment_LBVs} we place these results in a larger framework of LBV evolution. We conclude with Section \ref{conclusion}.
\section{Observations and auxiliary data}
\label{data}
\begin{figure}
\centering
\resizebox{1\hsize}{!}
{\includegraphics[width=1\textwidth]{HRCar_cluster_MUSE_annotated_SPT.png}}
\caption{Collapsed $V$-band image from the MUSE datacubes of the four deep pointings separated by small gaps, mapping the region $2\arcmin \times 2\arcmin$ ($2.8~\mathrm{pc} \times 2.8~\mathrm{pc}$ at a distance of 4.8~kpc) around HR~Car in the center, labeled ``1''. The object numbering is according to Table \ref{table:spectraltypes}. Approximate limiting magnitude is $V\sim22$~mag.}
\label{figure:MUSEdatacube}
\end{figure}
\begin{figure*}
\resizebox{1\hsize}{!}
{\includegraphics[width=1\textwidth]{HRdiagram_allhotspectra.eps} }
\caption{Flux-calibrated spectra of hot stars in the $2\arcmin \times 2\arcmin$ region around HR~Car in the MUSE data, see Table~\ref{table:spectraltypes} for details. The sources are ordered by increasing $V$-band magnitude. The spectral energy distribution of the stars 3., 5., and 13.\ indicate that they are foreground stars. The low spectral resolving power of the MUSE spectra prevents detailed radial velocities measurements of these sources.
}
\label{figure:MUSEspectra}
\end{figure*}
\subsection{IFU observations}
Observations of HR~Car with the Multi-Unit Spectroscopic Explorer (MUSE; \citealt{2010SPIE.7735E..08B}) at UT4 of ESO's Very Large Telescope (VLT) at Paranal in its seeing-limited Wide Field Mode (WFM) were obtained on 28, 30, and 31 December 2018.
MUSE is an integral field spectrograph, composed of 24 identical IFU modules that together sample a $1\arcmin \times 1\arcmin$ field of view in WFM. Spectrally, the instrument covers the wavelength range $480-930$~nm with a resolving power of $R \sim 2000-3500$, which allows to perform spectral classification and synthetic photometry, as, for example, demonstrated in \citet{2016A&A...585A..81M}. The WFM has a spatial sampling of 0.2\arcsec\ per pixel.
We mapped the environment of HR~Car with 5 pointings. One pointing, with HR~Car at the center, was observed with 48 exposures of 15s to obtain a deep image of HR~Car's circumstellar nebula, while avoiding saturation and strong straylight artifacts by the bright central source. Four additional pointings map a region of $2\arcmin \times 2\arcmin$ around HR~Car with two exposures of 300s per pointing (Figure \ref{figure:MUSEdatacube}). The obtained image quality at 600~nm is about 1\arcsec. The data were reduced with the ESO MUSE pipeline version 2.8.3 \citep{2020A&A...641A..28W,2016ascl.soft10004W}, using the option to obtain collapsed images in several filters for photometry (e.g., Johnson $V$, Cousins $I$, and SDSS $i$).
\subsection{Auxiliary data}
Using a simple inversion of the parallax from the Gaia Early Data Release 3 (Gaia EDR3\footnote{\url{https://gea.esac.esa.int/archive/}}; \citealt{2018A&A...616A...1G,2016A&A...595A...1G}), the distance to HR~Car is $d=4922\pm403$~pc. This method is normally warranted, given the small error bar. Using a Galaxy model prior, \citet{2021AJ....161..147B} estimated a geometric distance to HR~Car of $d=4752$~pc, with an uncertainty range of $4448-5098$~pc. These distance estimates are slightly lower than previously published distances such as a kinematic distance of $5.4\pm0.4$~kpc \citep{1991A&A...248..141H} or a distance of $5\pm1$~kpc derived via multicolor photometry of field stars around HR~Car \citep{1991A&A...246..407V}, but within the uncertainties. The distance determination from the parallax may be compromised by the binarity and bright circumstellar nebula. \citet{2019MNRAS.488.1760S} used Bayesian inference to estimate the distance from Gaia DR2 data and determined a distance of $d=4.37$~kpc ($3.50-5.72$~kpc).
For the analysis of the stellar content around HR~Car, we queried Simbad \citep{2000A&AS..143....9W} for stars in a region with radius $1.2\degree$ around HR~Car, which corresponds to an area with radius $100$~pc at a distance of 4.8~kpc, typical for an OB association \citep{2020NewAR..9001549W}. No overdensity of stars is discernible in this area on sky as would be expected in case of a stellar association. The HR~Car binary is also too low in mass for its evolved state to still expect many O-stars in its neighborhood. We thus analyze this dataset to potentially identify a moving group around HR~Car, that is stars with common age and motion.
Throughout the paper, we use Gaia EDR3 proper motions and parallaxes and the geometric distance estimates from \citet{2021AJ....161..147B}.
\subsection{Photometric calibrations}
\label{dataanalysis}
We used the tool QFitsView \citep{2012ascl.soft10019O} to analyze the stellar content and nebula around HR~Car. From the MUSE data cubes, we extracted spectra of all objects with a sufficient signal-to-noise ratio (S/N $> 50$) to determine their spectral types.
We derived photometry on the collapsed images using SExtractor \citep{1996A&AS..117..393B}. For the Johnson $V$, Cousins $I$, and SDSS $i$ filters, 100\% of the spectral region is covered by MUSE.
The MUSE observations were obtained in cloudy sky conditions with transparency variations above 20\% and none of the sources in the observed fields (apart from HR~Car) have published magnitudes in Simbad.\footnote{One of the objects is listed in Simbad (GSC~08612-01828; spectral type A9). However, the reference \citep{1991A&A...246..407V} reveals that the star has been misidentified with a different star outside the field observed with MUSE.} We cross-match the sources detected in the MUSE images collapsed over the SDSS $i$ passband with the SkyMapper Southern Survey \citep{2019PASA...36...33O} and apply the magnitude offsets found for each pointing. The errors of these magnitude offsets are between $0.1-0.3$~mag, depending on the pointing. Comparison of a few stars that were observed in different pointings (on different observing dates and with different exposure times) shows that the derived photometry is better than 0.2~mag across pointings and dates.
In the resulting MUSE photometry (Table~\ref{table:spectraltypes}), HR~Car is with $V=8.85$~mag about $0.1$~mag fainter in $V$ and with $I=7.54$~mag about $0.3$~mag fainter in $I$ band compared to the AAVSO lightcurve, within the error of the cross-match with the SkyMapper Southern Survey. Additional uncertainties arise from the different image qualities between the data sets, different aperture photometry methods and aperture sizes employed, and potentially slightly different filter transmission curves.
\section{The environment of HR~Car}
\label{environment_HRCar}
\begin{figure*}
\centering
\resizebox{1\hsize}{!}
{\includegraphics[width=1\textwidth]{HRdiagram_HRCar_V_VI_hot_extinction_skymappermagcorr_dered.eps}
\includegraphics[width=1\textwidth]{HRCar_CMD_GaiaEDR3_MG_dered.eps}}
\caption{{\it Left:} Color-magnitude diagram of the hot stars within a $2\arcmin \times 2\arcmin$ region around HR~Car derived from MUSE data and synthetic photometry, assuming a distance of $d=4.8$~kpc to all sources and corrected for an extinction of $A_V=2.9$~mag. The errors are dominated by the uncertainties of the reddening and distance estimates. PARSEC theoretical single-star stellar evolution isochrones are shown for 3~Myr, 10~Myr, and 30~Myr (dashed gray curves; \citealt{2012MNRAS.427..127B}). The stars 3.\ and 5.\ are foreground stars. {\it Right:} Same as left figure, but with Gaia EDR3 photometry. Stars with estimated higher geometric distance limits of $d_\mathrm{geo\_h} < 4448$~pc are indicated as foreground stars (green squares) and stars with estimated lower geometric distance limits of $d_\mathrm{geo\_l} > 5098$~pc as background stars (golden squares). The other sources (black filled squares only) are at an inferred distance compatible with that of HR~Car. The gray small squares show all Gaia sources within an area of radius $r=3\arcmin$ from HR~Car.
}
\label{figure:HRdiagram}
\end{figure*}
\begin{figure*}
\centering
\resizebox{1\hsize}{!}
{\includegraphics[width=0.32\textwidth]{HRCar_cluster.png}
\includegraphics[width=0.35\textwidth]{HRCar_cluster_hist_distribution.png}
\includegraphics[width=0.305\textwidth]{HRCar_cluster_propermotion_pointvector_circles_closein.png} }
\caption{{\it Left:} Spatial distribution of all O- and B-type stars in a projected region with radius $r=1.2\degree$ around HR~Car (black filled circle in the center), based on information retrieved from Simbad and Gaia. The newly classified B-type stars in the MUSE data are not listed in Simbad and not included of this figure. The arrows indicate the movement of the stars in the past $10^5$~yr, inferred from available Gaia proper motions. HR~Car moves in the same general direction as other massive stars in this region of the sky. {\it Middle:} Projected separation to O- and B-type stars and the normalized cumulative distribution (dashed curves). {\it Right:} Vector point diagram for O- and B-type stars using Gaia proper motions. The dashed circles indicate two standard deviations of the mean of the proper motions for the respective stellar subgroups. HR~Car's location (black filled circle at the center of the plot) is within two standard deviations of the mean of the proper motions for O-type, early B-type, and B-type stars.}
\label{figure:HRclusterSimbad}
\end{figure*}
\subsection{Spectral typing and extinction determination}
\label{spectraltyping}
To investigate the hot stellar content in the close environment of HR~Car, we visually inspected the MUSE spectra and selected the objects with broad Balmer and Paschen hydrogen lines and HeI~7065 for the earliest spectral types. This resulted in 34 objects, see Figure~\ref{figure:MUSEspectra} and Table~\ref{table:spectraltypes}.
Stellar spectral types were determined by comparing the equivalent widths of the HeI~$\lambda$7065, H$\beta$, NaI~$\lambda$5889 (stellar component), and OI~$\lambda$7774 lines with those tabulated as function of spectral type in \citet{1995bces.book.....J}. In addition, we used the stellar spectral classification software PyHammer 2.0.0 to compare the MUSE spectra to optical spectral templates and spectral line index measurements \citep{2020ApJS..249...34R,2020ascl.soft02011K}.
We newly classified eight stars as B-type within the $2\arcmin \times 2\arcmin$ region around HR~Car. Three of these objects are early B-type stars with spectral classifications of B0, B4, and B5, respectively. There is no O-type star in this region of the sky. The spectral resolution and S/N of the spectra is not conducive for detailed stellar atmosphere studies, such as $log~g$ determination.
The AAVSO lightcurve shows that HR~Car has been in a quiescent, hot state since about 2010. This is also confirmed by optical spectra obtained in the past decade with multiple instruments, for example, our 2018 MUSE spectrum. From the MUSE photometry of HR~Car, $V=8.85$~mag, an absolute magnitude of $M_\mathrm{V} = -7.5$~mag is derived, assuming $d=4.8$~kpc, $E(B-V) = 0.95$~mag (see below), and $R_V = 3.1$. The best-fit CMFGEN model of a 2017 VLT/X-shooter spectrum suggest an effective temperature of $T_\mathrm{eff}=17\,750$~K (Groh et al., in prep.). With a bolometric correction of -1.6~mag \citep{1996ApJ...469..355F}, the bolometric magnitude of HR~Car is $M_{bol}= -9.1$~mag. During HR~Car's outburst, that is cool state, its temperature has been estimated to about $8\,000$~K \citep{2001A&A...366..508V}. A magnitude of $V \sim 7$~mag (AAVSO lightcurve) and a bolometric correction of about 0~mag, results in a bolometric luminosity comparable to its quiescent state. Groh et al. (in prep.) will present a detailed spectroscopic analysis of HR~Car's hot and cool states.
Adjusted for the different distances used, our luminosity estimate for HR~Car is comparable to previous literature values \citep{ 1988ApJ...324.1071M,1991A&A...246..407V,1991A&A...248..141H,2001A&A...366..508V,2007ApJ...659.1563G,2009ApJ...705L..25G,2010AN....331..349H}.
We employed different methods to determine the extinction toward the 34 MUSE sources: 1) equivalent width of the diffuse interstellar band at $\lambda$5780~\AA, as described in \citet{2016A&A...585A..81M}, and 2) comparing the relative flux-calibrated MUSE spectra to template spectra.
The extinction and reddening toward an object depend on the wavelength, the amount of dust along the line of sight, and the dust grain sizes and composition. For HR~Car, using a 2017 X-shooter spectrum and CMFGEN modeling, Groh et al.\ (in prep.) determined a ratio between total and selective extinction of $R_V = 4.3\pm0.1$, which implies large grain sizes in its circumstellar environment. For consistency in our analysis, we adopt $R_V = A_V /E(B-V) = 3.1$ \citep{1989ApJ...345..245C,1999PASP..111...63F} for the interstellar medium toward all stars in the observed MUSE field.
For most of the brighter MUSE sources, the derived reddening is in the range of $E(B-V) = 0.85-1.10$~mag (Table \ref{table:spectraltypes}). This hints at similar distances as the reddening is mostly due to the dust in the interstellar medium along the line of sight, but see Section \ref{massivestars_HRCar} for a discussion on the distances inferred from the Gaia EDR3 release data. The reddening to HR~Car, estimated to $E(B-V) = 1.24$~mag, may be larger due to a significant amount of circumstellar dust. For the faintest stars the derived reddening has a large uncertainty as the spectra are very noisy at the bluest wavelengths, but the larger values may indicate that they lie in the background.
For consistency in our analysis, and because of the large uncertainties in the reddening estimations, we adopt a value for the interstellar reddening to HR~Car of $E(B-V) = 0.95\pm0.1$~mag, consistent with $E(B-V) = 0.90$~mag found by \citet{1991A&A...246..407V} via multicolor photometry of about 100 field stars within a radius of $10\arcmin$ around HR~Car.
\begin{sidewaystable*}
\caption{Hot stellar content in the neighborhood of HR~Car.$^a$ \label{table:spectraltypes}}
\begin{tabular}{llccccccccccccc}
\hline\hline
\# & Coordinates & SDSS$i$ & $I$ & $V$ & E(B-V) & SP & geo & geo\_l & geo\_h & pgeo & pgeo\_l & pgeo\_h & 1/p$^b$ & Gaia source\_id \\
& & (mag) & (mag) & (mag) & (mag) & & (pc) & (pc) & (pc) & (pc) & (pc) & (pc) & (pc) & \\
\hline
1. & 155.7244783 -59.6244613 & 7.78 & 7.54 & 8.85 & 1.24 & LBV$^c$ & 4752 & 4448 & 5098 & 4602 & 4269 & 4986 & 4922 & 5255045082580350080 \\
2. & 155.7265119 -59.6245356 & 12.13 & 11.95 & 13.01 & 1.00 & B0 & 4136 & 3869 & 4382 & 4132 & 3927 & 4368 & 4278 & 5255045082561662592\\
3. & 155.7190919 -59.6213844 & 12.56 & 12.36 & 13.17 & 0.20 & A9$^{d,e}$ & 1848 & 1803 & 1895 & 1849 & 1811 & 1884 & 1923 & 5255045151299827456 \\
4. & 155.7057998 -59.6234632 & 14.85 & 14.65 & 15.67 & 0.87 & B5 & 4187 & 3829 & 4539 & 4131 & 3826 & 4549 & 4927 & 5255045151299811584 \\
5. & 155.7167475 -59.6395444 & 15.39 & 15.26 & 16.08 & 0.65 & B9$^{e}$ & 2494 & 2345 & 2728 & 2450 & 2329 & 2630 & 2700 & 5255045048220579456 \\
6. & 155.6973330 -59.6378605 & 15.41 & 15.23 & 16.38 & 1.06 & B4 & 4775 & 4186 & 5492 & 4710 & 4155 & 5401 & 5581 & 5255045013860826880 \\
7. & 155.6974295 -59.6399275 & 15.84 & 15.61 & 16.93 & 1.10 & B9 & 3850 & 3444 & 4393 & 3750 & 3272 & 4242 & 4292 & 5255045013860925824 \\
8. & 155.7153745 -59.6315202 & 16.01 & 15.76 & 17.08 & 1.03 & B9 & 3593 & 3146 & 4202 & 3558 & 3019 & 4277 & 3818 & 5255045116940068352\\
9. & 155.7067251 -59.6352335 & 16.50 & 16.26 & 17.68 & 1.03 & A3 & 3410 & 2892 & 3962 & 3432 & 2987 & 3978 & 3621 & 5255045116940054400 \\
10. & 155.7088481 -59.6208449 & 16.78 & 16.59 & 17.81 & 0.84 & A3 & 5465 & 4007 & 7734 & 4343 & 3548 & 5310 & & 5255045146977490688 \\
11. & 155.7061104 -59.6224064 & 16.75 & 16.56 & 17.84 & 0.84 & A3 & 7045 & 5255 & 9053 & 6391 & 4441 & 8891 & & 5255045151299814144 \\
12. & 155.7183657 -59.6315608 & 17.07 & 16.86 & 18.15 & 0.95 & A3 & 6570 & 5076 & 9304 & 6447 & 4960 & 8765 & & 5255045048220593280 \\
13. & 155.7034540 -59.6221636 & 17.40 & 17.19 & 18.40 & 0.84 & A3$^{e}$ & 3901 & 2997 & 5130 & 3537 & 2799 & 4947 & & 5255045146979950336 \\
14. & 155.7521425 -59.6249287 & 17.14 & 16.83 & 18.52 & 0.90 & F3 & 4698 & 3547 & 6373 & 3759 & 3045 & 4576 & & 5255420393971899136 \\
15. & 155.7376107 -59.6361695 & 17.66 & 17.39 & 18.99 & 1.10 & A1 & 6318 & 4092 & 10492 & 4350 & 3299 & 6143 & & 5255045078258086400 \\
16. & 155.7265192 -59.6260821 & 17.50 & 17.19 & 19.07 & 0.85 & F4 & 4784 & 3040 & 7220 & 2919 & 2390 & 3630 & & 5255045082561952640 \\
17. & 155.7186046 -59.6364351 & 17.89 &17.60 & 19.35 & 1.42 & A1 & 6997 & 4804 & 10537 & 5632 & 3837 & 8693 & & 5255045043900737152 \\
18. & 155.7539724 -59.6173297 & 18.27 & 17.97& 19.54 & 1.45 & B8 & 4956 & 3312 & 7932 & 4679 & 2997 & 6364 & & 5255420393999646336 \\
19. & 155.7118295 -59.6355545 & 18.14 &17.86 & 19.62 & 1.00 & F2 & 5976 & 3706 & 8379 & 4542 & 3405 & 6132 & & 5255045112618092032\\
20. & 155.7343382 -59.6211879 & 18.13 & 17.83 & 19.76 & 1.26 & A1 & 977 & 739 & 1370 & 1923 & 1712 & 2236 & & 5255045082577422336 \\
21. & 155.7285538 -59.6131811 & 18.45 & 18.16 & 19.86 & 1.35 & B9 & 4364 & 2685 & 6697 & 3105 & 2281 & 4290 & & 5255420462691375488 \\
22. & 155.7201480 -59.6153487 & 18.21 & 17.83 & 20.03 & 1.35 & F1 & 6884 & 4500 & 9667 & 5724 & 3994 & 9848 & & 5255045146979952384\\
23. & 155.7201048 -59.6187980 & 18.21 & 17.81 & 20.13 & 1.35 & F1 & 3466 & 2382 & 5316 & 3975 & 2814 & 5570 & & 5255045151281149952 \\
24. & 155.7047429 -59.6345651 & 18.58 &18.26 & 20.18 & 1.00 & F9 & 6272 & 4228 & 8875 & 6341 & 4494 & 8599 & & 5255045116921270656 \\
25. & 155.7505009 -59.6109885 & 18.58 & 18.23 & 20.19 & 1.25 & A9 & 2465 & 1709 & 3656 & 2339 & 1660 & 3484 & & 5255420428334505600 \\
26. & 155.7424831 -59.6079542 & 18.51 & 18.12 & 20.21 & 1.39 & A9 & 5411 & 3653 & 7834 & 4405 & 3266 & 6013 & & 5255420428331714304 \\
27. & 155.7274145 -59.6191855 & 18.67 & 18.31 & 20.35 & 1.45 & A9 & 2716 & 1631 & 4117 & 1769 & 1475 & 3482 & & 5255045151281158016 \\
28. & 155.7092593 -59.6284747 & 19.39 & 19.04 & 21.07 & 1.61 & A3 & 11417 & 6094 & 18663 & 6113 & 2558 & 11084 & & 5255045116921326336 \\
29. & 155.6939437 -59.6191914 & 19.48 & 19.16 & 21.16 & 1.61 & A3 & 3787 & 2513 & 5997 & 3385 & 2021 & 7021 & & 5255045116921334784 \\
30. & 155.7418997 -59.6083104 & 19.70 & 19.36 & 21.32 & 1.45 & A2 & 4446 & 2789 & 6513 & - & - & - & & 5255420428331714432 \\
31. & 155.7056254 -59.6148039 & 19.91 & 19.50 & 21.61 & 1.61 & A8 & 6989 & 3949 & 10840 & - & - & - & & 5255045151281131904 \\
32. & 155.6921612 -59.6215689 & 19.98 & 19.57 & 21.74 & 1.61 & A7 & - & - & - & - & - & - & - \\
33. & 155.7146475 -59.6129394 & 20.04 & 19.65 & 21.77 & 1.61 & A7 & 4844 & 2257 & 8489 & 5342 & 3453 & 10017 & & 5255045151282776576 \\
34. & 155.7101136 -59.6148022 & 20.15 & 19.80 & 21.88 & 1.40 & A3 & - & - & - & - & - & - & -\\
\hline
\multicolumn{15}{l}{$^a$ Photometry, spectral types, and reddening $E(B-V)$ were determined from 2018 MUSE data. The error on the MUSE photometry is about 0.2-0.4~mag. }\\
\multicolumn{15}{l}{\phantom{$^a$} Gaia EDR3 geometric and photogeometric distances and their uncertainties (i.e., the lower ``l'' and higher ``h'' values) are from \citet{2021AJ....161..147B}.}\\
\multicolumn{15}{l}{$^b$ Listed only for sources with parallax error $<20\%$.}\\
\multicolumn{15}{l}{$^c$ HR~Car.}\\
\multicolumn{15}{l}{$^d$ SIMBAD source GSC~08612-01828, but misidentified with star \#34 in \citet{1991A&A...246..407V}.}\\
\multicolumn{15}{l}{$^e$ Foreground star, based on the spectral energy distribution.}\\
\end{tabular}
\end{sidewaystable*}
\subsection{HR~Car as part of a moving group}
\label{massivestars_HRCar}
The MUSE data allow one to construct a color-magnitude diagram to which evolutionary tracks can be compared. Figure~\ref{figure:HRdiagram} shows such a color-magnitude diagram for the 34 MUSE sources listed in Table \ref{table:spectraltypes}, assuming an identical distance of $d=4.8$~kpc and constant interstellar reddening of $E(B-V)=0.95$~mag, with $A_I/A_V =0.60$ and $R_V=3.1$ \citep{1989ApJ...345..245C,1994ApJ...422..158O}. Two of the brightest stars (3.\ and 5.) are certainly foreground stars, as can also be inferred from their spectral energy distributions (Figure~\ref{figure:MUSEspectra}), and estimated lower reddening and much smaller Gaia distances (Table \ref{table:spectraltypes}).
Figure~\ref{figure:HRdiagram} also shows a color-magnitude diagram using the Gaia EDR3 $G$, $G_{BP}$, and $G_{RP}$ photometry of the MUSE sources, assuming an identical distance of $d=4.8$~kpc and a constant reddening of $E(B-V)=0.95$~mag, $R_V=3.1$, $A_G = 0.86 \times A_V$, $A_{BP} = 1.06 \times A_V$, and $A_{RP} = 0.65 \times A_V$.\footnote{Listed on the PARSEC isochrone website \url{http://stev.oapd.inaf.it/cgi-bin/cmd}.}
The Gaia color-magnitude diagram confirms the MUSE photometry as all sources lie in similar relative locations in both diagrams and also with respect to the overlaid theoretical single star stellar evolution isochrones from the PAdova and TRieste Stellar Evolution Code (PARSEC; \citealt{2012MNRAS.427..127B}). In addition to the MUSE sources, the positions of all Gaia sources within a radius of 3\arcmin\ ($\sim4$~pc at a distance of 4.8~kpc) around HR~Car are shown.
While there appears to be a tight relation along a main sequence for several of the hot MUSE sources, not all stars are at the same distance and/or same reddening. For example, the faintest sources are too faint and too red for their spectral type and are thus likely background stars.
The geometric distances and their uncertainties (\citealt{2021AJ....161..147B}; see Table~\ref{table:spectraltypes}) indicate that nine of the MUSE sources could be foreground stars with higher limits on their distances of $d_\mathrm{geo\_h} < 4448$~pc and two sources could be background stars with lower limits on their distances of $d_\mathrm{geo\_l} > 5098$~pc. At face value, the B4 and B5-type stars lie at the same distance as HR~Car, while the B0-type star is located slightly in the foreground.
For a distance of $d=4.8$~kpc, the absolute magnitudes M$_\mathrm{V}$ for the stars with spectral type B0, B4, and B5 are $0.6$~mag, $1.3$~mag, and $0.5$~mag fainter compared to the (general) work on absolute magnitudes of OB stars based on Hipparcos parallaxes by \citet{2006MNRAS.371..185W}. However, taking into account the errors on our photometry and the distance, the error on our M$_\mathrm{V}$ estimates is on the order of 0.7~mag and \citet{2006MNRAS.371..185W} reported errors larger than 1~mag. Deriving M$_\mathrm{V}$ using the reddening and Gaia distances from Table~\ref{table:spectraltypes} does not result in a better match with \citet{2006MNRAS.371..185W}.
Overall, the color-magnitude diagrams and Gaia distances indicate that several of the MUSE objects, especially the B4 and B5-type stars, may be spatially associated with HR~Car.
HR~Car's bolometric luminosity and location near the 10~Myr single-star isochrone correspond to an evolutionary path of a single star with $M_{init} = 25-30~M_{\odot}$ and is consistent with the stellar mass of $M \sim 21~M_\odot$ determined by Boffin et al.\ (in prep.) by means of resolved interferometry and binary reflex motion.
In a single-star evolution scenario, the primary has lost mass as a blue supergiant, potentially as a red supergiant, and now as an
LBV. The circumstellar nebula contains several solar masses of material, some mass may have escaped the system, and a fraction may have been deposited on the secondary.
HR~Car is a binary with a mass ratio of about 0.5, the LBV being the primary component (\citealt{2016A&A...593A..90B}; Boffin et al., in prep.). In an extreme case of a companion with equal $V$-band brightness, the LBV in HR~Car would be only 0.75~mag fainter and the conclusion from its location in the HR diagram would not significantly change. However, the large radius determined for the less massive secondary is in conflict with this simple single-star evolution model (\citealt{2016A&A...593A..90B}, Boffin et al., in prep.).
The uncertainties in the Gaia distances still need to be reduced to derive precise ages and masses of the stars and to determine if HR~Car's location is unexpected compared to other potential members of a moving group. For example, a better distance estimate is required to settle the question of a spatial association of the B0-type star with HR~Car.
For a discussion on a potential past merger in a triple system or past mass transfer events, leading to a rejuvenation of HR~Car, the companion's nature needs to be identified. The large radius for the companion star indicates a red supergiant nature. However, no spectral features of a red supergiant were observed yet in near-infrared spectra, which may be due to insufficient S/N in the spectra as the luminosity ratio in $H$-band is $6-9$ (\citealt{2016A&A...593A..90B}, Boffin et al., in prep.). Radio data can also not rule out that the nebula is ionized by a hot companion star and a B0 V companion without much dust is also consistent with the data \citep{2000ApJ...539..851W}.
To further investigate the environment of HR~Car, we analyzed the stellar content of massive stars in its environment retrieved from Simbad and Gaia in a radius of $1.2\degree$ ($100$~pc at 4.8~kpc) around the star. Figure~\ref{figure:HRclusterSimbad} provides an overview of the results. The figure shows the spatial distribution of O- and B-type stars around HR~Car with arrows indicating their movements projected on sky over the past $10^5$~yr based on Gaia proper motions. No filtering was performed based on distances. HR~Car moves in the general direction with most of the other hot stars in this region, suggestive of a co-moving group.
While the region is rich with late O-type and B-type stars, there is only one early O-type star (i.e., earlier than O6). As stated earlier, this is not surprising given the relatively low stellar masses in the HR~Car system and its evolved nature. However, many of the OB-type star classifications originate from the VST Photometric H$\alpha$ Survey (VPHAS+; \citealt{2017MNRAS.465.1807M}) and still lack precise spectral typing. The cumulative distributions of the projected separation to O-type stars and B-type stars are similar.
Figure~\ref{figure:HRclusterSimbad} also shows a vector point diagram, indicating with circles two standard deviations of the mean of the proper motions of early O-type, all O-type, OB-type, early B-type (i.e., earlier than B6), and B-type stars. To estimate conservatively the dispersion of the proper motion of the stars, we discarded all stars with proper motion measurements with errors $> 1$~mas~yr$^{-1}$, which removed double or multiple stars and high-proper motion stars from the sample.
HR~Car resides within two standard deviations of the mean of the proper motions of B- and O-type stars and visually appears to be part of a group, not surprising for stars at the same distance in a spiral arm. This fact, however, suggests that HR~Car is not isolated and has not received a large kick in the past from a companion SN or from a merger. Extrapolation of the proper motions over a lifetime of 10~Myr does not show convergence to a potential birth cluster, but also not divergence.
\section{The circumstellar nebula}
\label{circumstellarnebula}
Previous morphological and kinematic studies have found a bipolar structure, favoring an expanding-lobe model \citep{1997A&A...320..568W,1997A&A...321L..21V,1997ApJ...486..338N}. HR~Car's nebula is reminiscent of the Homunculus nebula of $\eta$~Car, albeit of larger size and much older ($4000-9000$~yr compared to 200~yr; \citealt{2012ASSL..384..171W,1997A&A...320..568W}).
On the other hand, \citet{2017MNRAS.465.4147B} proposed a jet-precession model based on mid-infrared observations and the argument that the lobes do not appear complete, but S-shaped.
The MUSE velocity channel maps revealed a bipolar structure for HR~Car's large circumstellar nebula (Figure \ref{figure:channelmaps}). One lobe is in the southeast with velocities up to $-153$~km~s$^{-1}$ and one lobe is in the northwest with velocities up to $189$~km~s$^{-1}$.
The nebula is quite fragmented and the brightness distribution is partly suggestive of an S-shaped nebula, which could have resulted from precession. However, the velocities in this comparatively old nebula show a bipolar morphology, confirming the expanding-lobe model. The observed brightness distribution may thus imply unequal mass and density distribution in the nebular shell and the surrounding medium, as well as dust clumps in the circumstellar nebula.
The companion may have played a role in the origin and/or shaping of the bipolar nebula, as it is aligned perpendicular to the orbit. On the other hand, HR~Car rotates at $\sim0.9$ of its critical velocity \citep{2009ApJ...705L..25G} and fast-rotating massive stars can have latitude-dependent, polar-enhanced winds \citep{1996ApJ...472L.115O,2001A&A...372L...9M}. Thus, rotation could also have played a role in the star's instability and in the formation of its bipolar nebula. Possible explanations for its fast rotation could be the accretion of angular momentum through mass transfer via stable wind Roche-lobe overflow \citep{2007ASPC..372..397M} from a red supergiant companion, but also a merger in a triple system would lead to a fast rotating star.
The similarities to $\eta$~Car's Homunculus nebula are striking, which suggests similar ejection and/or shaping mechanisms. Both objects have a bipolar nebula that is aligned with the orbit of the companion star. In addition, there is also a large structure around the pinched ``waist'' of the bipolar lobes of HR~Car's nebula at low velocities that could be interpreted as an equatorial ``skirt''-like feature (Figure~\ref{figure:channelmaps}), similar to the one observed in $\eta$~Car \citep{1999A&A...344..211Z}. Competing models for the eruptive mass loss that could lead to such nebula include super-Eddington winds (e.g., \citealt{2017MNRAS.472.3749O}), accretion onto a companion star (e.g., \citealt{2010ApJ...723..602K}), pulsational pair-instability events (e.g., \citealt{2017ApJ...836..244W}), and the merger of two massive stars (e.g., \citealt{2016MNRAS.456.3401P,2021MNRAS.503.4276H}).
However, HR~Car is a factor of at least five times less massive in terms of both stellar and nebular mass and the orbital parameters are substantially different. The $\eta$~Car system has a highly eccentric orbit ($e\sim 0.85$) and has a complicated colliding wind system that collapses at periastron when the stars have a separation of about 1~au (e.g., \citealt{2014ApJ...784..125H}). HR~Car's orbit is nearly circular with a minimum separation between the stars on the order of 10~au (Boffin et al., in prep.), which makes stellar interactions unlikely during all but a red supergiant phase of either component.
\section{A fast bipolar outflow and ``bullets''}
\label{bipolaroutflow}
\begin{figure*}
\centering
\resizebox{1\hsize}{!}
{\includegraphics[width=0.45\textwidth]{channelmaps_NII6583_newI.png}}
\caption{MUSE channel maps showing the NII~$\lambda$6583 emission around HR~Car, accounting for a systemic velocity of $-19$~km~s$^{-1}$. The stellar continuum is subtracted. North is up, East is left, and each panel shows a region of $1\arcmin \times 1\arcmin$ on sky (the units on the axis are in pixels with $0.2\arcsec$ per pixel). In addition to a large bipolar nebula, a ``skirt''-like feature is visible at low velocities. A fast outflow extends up to 5.8\arcsec\ at a position angle on sky of $129\degree$, aligned with the geometry of the large bipolar nebula. At velocities of about 300~km~s$^{-1}$, three bullets in the northwest follow the linear trajectory of the inner outflow (labeled 1., 2., 3.; near bullet 2.\ is a star).}
\label{figure:channelmaps}
\end{figure*}
The deep MUSE IFU observations allow one to characterize the circumstellar material around HR~Car in more detail than before \citep{1997A&A...320..568W,1997ApJ...486..338N}.
We detect an inner fast outflow in the channel maps with velocities of up to $-440$~km~s$^{-1}$ in the southeast and up to $360$~km~s$^{-1}$ in the northwest, extending $5.8\arcsec$ from HR~Car (Figure \ref{figure:channelmaps}). The outflow has a position angle on sky of $129\degree$, similar to the slower and much larger bipolar nebula. The size of the outflow of $\sim 12\arcsec$ corresponds to a deprojected size of $0.3$~pc at a distance of $d=4.8$~kpc (see below). This inner outflow is thus similar in size and velocity as the Homunculus nebula of $\eta$~Car.
In the northwest, there are three spatially unresolved (i.e., $<0.02$~pc at a distance of $d=4.8$~kpc) knots or ``bullets'' on a linear trajectory at $5.8\arcsec$, $14.6\arcsec$, and $24.5\arcsec$ from the star, with the same position angle on sky as the inner outflow (labeled ``1.'', ``2.'', `3.'' in Figure \ref{figure:channelmaps}; the one labeled ``1.'' is right at the end of the fast outflow). From their nebular emission spectra we derived radial velocities of about $300$~km~s$^{-1}$, similar to the velocity of the fast outflow. Unfortunately, the S/N of the spectra is too low to further analyze these knots.
The binary orbit has a position angle of $40\degree$ (Boffin et al., in prep.). Thus, the fast inner bipolar outflow and the bullets are perpendicular to the binary orbit, suggestive of recurrent angular momentum driven accretion events. A potential red supergiant companion could account for discrete mass-loss events. However, the colliding wind shock formed by the two strong stellar winds may shield the primary from accretion.
Boffin et al.\ (in prep.) obtained an orbital inclination of $i = 119\degree$, which translates to an inclination of $29\degree$ for a potential perpendicular outflow.
Assuming the bullets are perpendicular to the orbital plane, their velocities and physical distance are calculated as $v = v_{\mathrm{rad}}/ \mathrm{sin}\,(i-90\degree)$ and $d_{\mathrm{phys}} = 2d\,\mathrm{tan}\, (\delta/2) / \mathrm{cos}\,( i-90\degree)$, with $\delta$ the angular distance on sky (i.e., $5.8\arcsec$, $14.6\arcsec$, and $24.5\arcsec$) and $d= 4.8$~kpc.
This results in deprojected velocities of $v = 620$~km~s$^{-1}$ and physical distances of 0.15~pc, 0.38~pc, and 0.65~pc. The kinematic ages, $\tau =d_\mathrm{phys} /v$, for the three bullets are about 240~yr, 600~yr, and 1030~yr. That is, they appear to have been ejected at intervals of about 400 years, which could be a tracer of continued past accretion in the system. Moreover, because the bullets fall on a straight line, there is no underlying precession. This is also a relevant fact for the origin of the larger bipolar nebula, discussed in Section \ref{circumstellarnebula}, as \citet{2017MNRAS.465.4147B} proposed a jet-precession model for its formation. Interestingly, these knots are only seen on one side of the star.
The bullets may be similar to the five so-called strings observed around $\eta$~Car, which are long, highly collimated linear features that appear to trace back to the central star \citep{1999A&A...349..467W}. The physical nature of $\eta$~Car's strings is not understood, but points toward the Great Eruption that created the Homunculus nebula.
However, for $\eta$~Car the strings follow a Hubble law with velocities increasing toward larger distances from the star, while for HR~Car all three bullets have the same velocity, independent of distance.
Similarities can also be drawn to a collimated outflow with projected velocities of up to $\pm450$~km~s$^{-1}$ centered on the B[e] star MWC~137 \citep{2016A&A...585A..81M}.
\section{Discussion}
\label{environment_LBVs}
Our work on the individual object HR~Car cannot resolve the debate if LBVs are the descendants of early O-type stars evolving as single stars, rejuvenated mass gainers, or the result of mergers.
However, we provide an assessment of HR~Car's spatial relationship to O- and B-type stars and its potential membership in a moving group that can be translated to other LBVs.
From the 34 hot stars in the close vicinity of
HR~Car ($r < 2\arcmin \sim 2.8$~pc), identified in the MUSE data, not one object had a
spectral type classification cataloged in Simbad at the time of
this study. Our MUSE data underline the
sparse knowledge of stellar classifications even around well-studied stars like HR~Car. This indicates a general problem in the assessment of
the degree of isolation of LBVs. Similar deep population studies need to be
conducted before a relatively unbiased measure is obtained on the
degree of their isolation.
Existing catalogs of O-type and early B-type stars are confined to objects brighter than $V\sim12$~mag and complete only to $\sim2$~kpc. A case in point is the recent work by \citet{2021A&A...648A..34P}, who performed an infrared spectroscopic study of the population of the young stellar cluster TR~16-SE in the Carina Nebula. The authors increased the number of spectroscopically identified early-type O5--B2 stars in Tr 16-SE from two to nine. Infrared spectroscopy was required to reveal the hidden population of early-type stars in this obscured young massive cluster.
Data from the VST Photometric H$\alpha$ Survey (VPHAS+) of the Carina Arm over the Galactic longitude range $282\degree \leq l \leq 293\degree$ revealed 5915 O-type to B2-type star candidates and 5170 stars with a later B spectral type and also a so far unnoticed association of 108 O-type to B2-type stars \citep{2017MNRAS.465.1807M}.
Census incompleteness can therefore be a significant factor in assessing LBV isolation.
An additional complication in assessing the degree of isolation
of LBVs relates to the mode of star formation itself and the
stochasticity of initial mass function sampling. A relation between most
massive star and cluster mass would indicate that stellar clusters are populated from the stellar initial mass function by random sampling over the mass range. However, there appears to be a dependence of the stellar inventory of a star cluster on its mass, which may be due to the interplay between stellar feedback and the binding energy of the cluster-forming molecular cloud core \citep{2010MNRAS.401..275W}. More recent work brings tension to how closely related stellar clusters and massive-star formation are. \citet{2020A&A...643A.138M} investigated 16
Galactic O-type stellar groups each containing O2--O3.5 stars. The
authors concluded that very massive (binary) stars can
form in relatively low-mass clusters or even in near-isolation. While
some stars are born in well-defined bound clusters, others are born in
unbound associations.
Nonetheless, as the isolated formation of massive stars would be a rare occurrence, the majority of high-mass LBVs should be associated with a high-mass stellar group if they have not received large kicks from a companion SN or a merger event. On the other hand, lower-mass LBV systems such as HR~Car are not expected to be in OB associations as most O-type stars would have already ended their lives. For the lower-mass LBVs, membership within a moving group can be established by means of a shared proper motion vector (e.g., Figure \ref{figure:HRclusterSimbad}). This method can also be applied to higher-mass LBVs and complements and can contradict group identification by mere spatial association.
\section{Conclusions}
\label{conclusion}
The suggestion that LBVs practice ``social distancing'', that is are single stars located away from young OB association argues in favor of a binary origin of the LBV phenomenon and against single-star evolution \citep{2015MNRAS.447..598S}. In this paper, we have turned the question if LBVs are accompanied by O-type stars into the question if LBVs are alone.
The masses of the two binary components in the Galactic LBV HR~Car and the mass of the surrounding nebula indicate that the initial stellar masses in this system barely exceeded the O-star threshold. One would thus not expect any early O-type stars or many late O-type stars associated with HR~Car.
Instead, we have shown that HR~Car is not isolated, but may reside within a moving group (in a spiral arm) of about $10-30$~Myr, with a main sequence populated up to B0 or B4. Proper motion analysis confirms that HR~Car has not received an extraordinary large space velocity kick owing to a companion SN or merger event.
HR~Car is an interferometrically resolved binary, which experienced recurrent outflow events perpendicular to its binary orbit (bipolar nebula, fast outflow, ``bullets''). Especially, the ``bullets'' indicate recurrent past activity related to either rapid rotation or binarity. The bolometric luminosity of HR~Car and its location in the color-magnitude diagram is in general agreement with a single-star evolution of a star with an initial mass of $M_\mathrm{init} = 25-30~M_\odot$, which has lost mass during a blue supergiant, potentially a red supergiant, and currently a LBV phase. However, the unsolved question of the nature of the companion star is cause of caution to adopt this simple model. The large radius derived from interferometric observations suggests a red supergiant nature.
In an extreme case where HR~Car would have undergone rejuvenation by mass transfer from a red supergiant companion (potentially via wind Roche-lobe overflow), the secondary star must have been stripped of its outer layers quite dramatically and lost much of its mass. If the companion's mass is now 50\% of that of the primary, the secondary may have lost 50\% of its original mass or more. However, such a scenario is not easy to reconcile with the current orbital parameters and detailed binary evolution modeling is needed. Based on our findings for this individual LBV, we cannot rule out that mass stripping and rejuvenation may still be viable explanations for generating the LBV phenomenon, as may be companion SNe or merger events in a triple system without imparting a large space velocity kick.
\begin{acknowledgements}
This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France \citep{2000A&AS..143....9W}.
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://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.
The national facility capability for SkyMapper has been funded through ARC LIEF grant LE130100104 from the Australian Research Council, awarded to the University of Sydney, the Australian National University, Swinburne University of Technology, the University of Queensland, the University of Western Australia, the University of Melbourne, Curtin University of Technology, Monash University and the Australian Astronomical Observatory. SkyMapper is owned and operated by The Australian National University's Research School of Astronomy and Astrophysics. The survey data were processed and provided by the SkyMapper Team at ANU. The SkyMapper node of the All-Sky Virtual Observatory (ASVO) is hosted at the National Computational Infrastructure (NCI). Development and support the SkyMapper node of the ASVO has been funded in part by Astronomy Australia Limited (AAL) and the Australian Government through the Commonwealth's Education Investment Fund (EIF) and National Collaborative Research Infrastructure Strategy (NCRIS), particularly the National eResearch Collaboration Tools and Resources (NeCTAR) and the Australian National Data Service Projects (ANDS).
This research has made use of NASA's Astrophysics Data System Bibliographic Services.
SJ acknowledges support from the FWO PhD fellowship under project 11E1721N.
JB acknowledges support from the FWO\_Odysseus program under project G0F8H6N.
\end{acknowledgements}
\bibliographystyle{aa}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,906
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Q: Python: how two find out the different rows from two dataframes I have two dataframes table1 and table2:
table1
table2
I want to compare table1 with table2 and return miss rows from each other of table 1 and table 2.
A: You can do:
only_in_1 = table1.merge(table2.assign(key=1), how='left')
only_in_1 = only_in_1[pd.isna(only_in_1['key'])].drop('key', axis=1)
A: import pandas as pd
import numpy as np
df1 = table1
df2 = table2
df_inner = df1.merge(df2, on = ['A', 'B', 'C'], how = 'inner')
df_outer = df1.merge(df2, on = ['A', 'B', 'C'], how = 'outer')
df_inner['label'] = np.ones(len(df_inner))
df = df_outer.merge(df_inner, on = ['A', 'B', 'C'], how = 'outer')
df = df[df['label'].isna()]
df = df.drop(['label'], axis = 1)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,053
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Q: Decoding recaptcha return values I've got a reCaptcha service working, but passing the return values through PHP's json_decode function gives some results I don't understand.
If the reCaptcha is valid then the service returns { "success": true } - which I expect.
Calling
print_r(json_decode($result, $assoc = true)); //where $result is the service call
gives
Array ( [success] => )
...but I was expecting Array ( [success] => 1) (or similar), to signify a true value.
In fact, print_r(json_decode('{"success": true}', $assoc = true)); does return Array ( [success] => 1 )
What's different about the json the reCaptcha is returning? (and / or what am I failing to understand?!)
I guess that's what's further confusion here:
echo(gettype($a->success));
echo($a->success === true);
echo($a->success === false);
then I get
boolean
1
1
How can a check against true and false both return true?!
Further info
Even more confusingly (for me at least), I've tried var_dump too:
echo($a);
var_dump($a);
which returns
{ "success": true }
string(22) "{ "success": false }"
why is the json representation in var_dump false when echo shows that it is true?!
A: I think, your main problem was that the JSON returned from recaptcha and saved in $result was { "success": false }and not { "success": true }.
That would explain why you got Array ( [success] => ) and not Array ( [success] => 1) as echo false returns an empty string but echo true returns 1;
I tried to reproduce the response from recaptcha using this code:
<html>
<head>
<script src='https://www.google.com/recaptcha/api.js'></script>
</head>
<body>
<form>
<div class="g-recaptcha" data-sitekey="FILL_IN_YOUR_KEY"></div>
<input type="submit">
</form>
</body>
</html>
<?php
$url = 'https://www.google.com/recaptcha/api/siteverify';
$data = ['secret' => 'FILL_IN_YOUR_SECRET', 'response' => $_GET['g-recaptcha-response']];
$options = ['http' => ['method' => 'POST', 'content' => http_build_query($data),],];
$context = stream_context_create($options);
$result = file_get_contents($url, false, $context);
echo '$result var_dump:<br>';
var_dump($result);
echo '$result print_r:<br>';
print_r($result);
$object = json_decode($result);
echo '<br><br><br>$object var_dump:<br>';
print_r($object);
echo '<br>$object print_r:<br>';
var_dump($object);
And what I got was this in case of success:
And this in case of failure:
A: In my question, I should have probably said I was using code inspired by this answer, and that my call to the Google verification service was wrapped in a function. I've realised that you can only ever call this service once, but I was calling it every time I called the function, and so the second time I called the function (i.e. the second time I called the service), the verification service did return false
I'm not sure this is explicitly documented anywhere, although perhaps it should have been a bit more obvious!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,953
|
\section{Introduction}
\label{Introduction}
Online advertising has developed into the most primary way of ad delivery in recent years \cite{ref1}. As a new marketing channel of online advertising, real-time bidding (RTB) has received extensive attention from industry and academia since it significantly improves the efficiency and transparency of the online advertising ecosystem. Figure 1 illustrates the typical process of an advertiser buying an ad impression through RTB \cite{ref2}. When a user browses a web page, the script of the ad slot embedded on the page will initiate a bidding request to the ad exchange (ADX). Then the ADX delivers the bidding request to the connected demand-side platforms (DSPs). The bidding agent running on each DSP calculates the bidding price of the auctioned impression for each advertiser based on its utility \cite{ref3}. The highest bidding price within each DSP is fed back to the ADX, and the ADX determines the final winner according to the generalized second pricing (GSP) mechanism \cite{ref4}. The winning notice is then sent to the winner, and its advertisement will be displayed to the user on the web page. Usually, the DSP will track the user's click or conversion behavior to optimize the advertiser's utility estimator and bidding strategy \cite{ref5}.\par
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{fig1.pdf}
\captionsetup{font={footnotesize}}
\caption{Typical Process of an Ad Delivery in RTB}
\label{fig:fig1}
\vspace{-0.10in}
\end{figure}
In RTB, each advertiser can dynamically adjust the bidding price based on the utility of the ad impression to itself. Therefore, an advertiser can maximize revenue under a limited budget by optimizing its bidding strategy. This paper focuses on optimizing a single advertiser's bidding strategy. The advertiser's revenue in RTB usually refers to the user's click or conversion behavior after the ad is displayed \cite{ref6}. The more clicks (conversions) the advertiser gets in an ad delivery period, the greater the revenue. Therefore, the goal of the bidding strategy is to make a bidding decision for each auctioned ad impression so that a single advertiser can get the maximum number of clicks (or conversions) in an ad delivery period. To achieve this goal, the bidding agent should determine the bidding price based on the value of the impression to the advertiser, conforming to the optimal auction theory \cite{ref2}. Usually, the bidding agent in RTB uses the predicted click-through rate (pCTR) to measure the value of the impression and makes a bidding decision based on pCTR. So we can formalize the optimal bidding strategy of a single advertiser as the formula (1), maximizing the total value of purchased ad impressions in an ad delivery period under a given budget. \par
\begin{equation}
\begin{split}
b^*(i) & =\underset{b(\cdot)}{max}\sum_{i=1}^Tw(i)\cdot v(i) \\
s.t. &\sum_{t=1}^T w(i)\cdot cost(i)\leq B
\end{split}
\label{equ:equ1}
\end{equation}\par
In the formula (1), $v(i)$ is the estimated value of an ad impression \cite{ref7}, and $w(i)$ is a binary indicator whether the ad impression is purchased successfully using the bidding strategy $b(\cdot)$. The optimal bidding strategy $b^*(\cdot)$ maximizes the total value of all purchased ad impressions in the whole delivery period. In the constraint, $cost(i)$ refers to the cost of purchasing ad impression $i$, and the total cost of purchasing all impressions should not exceed the given budget $B$.\par
Among the existing bidding strategies, linear bidding strategy (called LIN) is one of the most representative schemes. It first designs a bidding function with pCTR as the variable and learns the parameters from the samples of historical periods via maximizing the total pCTR of all purchased impressions. Linear bidding strategy is static, simple, and easy to deploy, so it is widely employed on many DSP platforms. Unfortunately, the static bidding strategy does not work well when the RTB environment changes significantly between the historical and the new ad delivery periods. In the static bidding strategy, the bidding price is only related to the pCTR of an ad impression and cannot adapt to changes of the RTB auction environment. Intuitively, an ideal bidding strategy ought to be associated with the pCTR of ad impression and the real-time auction environment, such as the advertiser's available budget, remaining life time, and the intensity of market competition.\par
To make bidding price change with the RTB environment, the authors in \cite{ref8} proposed using reinforcement learning (RL) \cite{ref9} to learn the optimal bidding strategy (called RLB). Through RL, the bidding agent considers not only the immediate reward from a single ad impression but also the cumulative benefit from all purchased impressions during the whole ad delivery period \cite{ref10}. However, RLB is a model-based RL bidding strategy that needs to establish the state transition probability matrix. When the number of states reaches billions, establishing this matrix requires substantial computational and storage costs, which fails to deploy in real RTB applications \cite{ref11}. A promising improvement is to use the latest model-free RL (DQN) algorithm to model bidding decisions \cite{ref12}. The new bidding strategy (called DRLB) not only enables bidding price to be correlated with both the impression's pCTR and RTB environment, but also does not need to compute the state transition probability matrix. Yet, the experimental results demonstrate that DRLB does not outperform LIN and RLB. Its performance depends heavily on the initial value of the bidding factor and the design of discrete action space (the detailed explanation refers to Section 2).\par
Indeed, it is still challenging for the bidding agent to learn the optimal bidding strategy at the impression-grained level using the model-free RL algorithm. The difficulties are as follows. Firstly, the bidding agent located on the DSP only obtains incomplete information about the RTB environment. For example, it neither knows the lost ad impressions' market prices nor how many advertisers participate in bidding the impression. The agent only represents the state by some observable statistical metrics, which may cause the bidding agent to make a non-optimal bidding decision. Secondly, RL guides the bidding agent to learn the optimal bidding strategy through the reward mechanism. But defining an appropriate immediate reward is a tough job in RTB. For instance, RLB directly uses the pCTR of the purchased ad impression as the immediate reward, which easily guides the agent to learn to bid with an exorbitant price to obtain the pCTR without considering the cost. As a result, the agent buys plenty of low value impressions, resulting in budget waste.\par
Thirdly, the GSP auction is widely adopted in RTB, which means the advertiser who bids greater or equal to the second-highest bidding price wins the ad impression but is only charged the second-highest price. That is to say, for a single ad impression, multiple bidding prices are corresponding to the same benefit and cost, expressing there are multiple optimal actions at every state. However, the current mainstream model-free RL algorithms (such as DQN \cite{ref12} and DDPG \cite{ref13}) are deterministic, in which there is only one optimal action at every state, and they only maximize the probability of the optimal action being selected during the learning process. This inconsistency may prevent the bidding agent from learning the optimal bidding price at each state.\par
To overcome the above difficulties, we design a new bidding function to calculate the bidding price for a single ad impression in this paper. As shown in formula (2), it contains a base price and an adjustment value. The base price is computed by a linear bidding function derived by a heuristic algorithm. The adjustment value is obtained by multiplying the optimal adjustment factor (generated by the RL agent) by the range of bidding adjustment. Thus, through formula (2), the bidding agent no longer selects an optimal bidding price from a discrete price space but generates a continuous adjustment factor with a given range. This bidding function ensures that the bidding price does not deviate too much from its estimated value and can be adjusted according to the real-time RTB environment.\par
\begin{equation}
\begin{split}
b(i) & = b_{LIN}(i)+\Delta b(i) \\
& =\underbrace{b_{LIN}(i)}_{base\ price}+\underbrace{a_i}_{adjustment\ factor}\times \underbrace{\min({price}_{\max}-b_{LIN}(i), b_{LIN}(i)-{price}_{\min})}_{bidding\ adjustment\ range}
\end{split}
\label{equ:equ2}
\end{equation}\par
Furthermore, we model the adjustment factor decisions of ad impressions in an ad delivery period as an MDP \cite{ref14}. Therefore, the RL agent's task is to learn the optimal adjustment factor generation policy. Specifically, we introduce a maximum entropy RL algorithm, Soft Actor-Critic (SAC) \cite{ref15}, to generate each ad impression's adjustment factor. Unlike the deterministic RL algorithms, SAC has a strong self-learning ability and can balance the probabilities of multiple optimal actions being selected in the same state. Thus, our bidding strategy can solve the third difficulty mentioned above effectively using SAC. Besides, SAC expands the scope of the agent to explore the optimal action to avoid falling into the local optimum. The contributions of our work can be summarized as follows:\par
\begin{itemize}
\item We design a new bidding function by improving the linear one, which considers both the impression's estimated value and the real-time RTB environment. Specifically, we introduce an adjustment factor into the bidding function, which can be adjusted according to the real-time RTB environment dynamically. Thus, the RL agent in our strategy only needs to learn the optimal adjustment factor generation policy, avoiding generating the bidding price directly. This greatly reduces the difficulty of the RL agent learning the optimal action generation policy.\par
\item To generate the optimal adjustment factor for each impression, we model the adjustment factor decisions as an MDP and optimize the adjustment factor generation policy through SAC. Using SAC not only overcomes the problem of multiple optimal actions for each impression but also expands the exploration space of the optimal action to avoid falling into the local optimum. Concretely, we redefine the state representation and reward function in the MDP for enabling the RL agent to understand the optimization objective better. It is the first bidding strategy at an impression-grained level through stochastic reinforcement learning to the best of our knowledge.\par
\item We evaluate the proposed scheme and several baselines on a benchmark dataset, and the results demonstrate our method outperforms other baselines. Specially, we are the first to quantitatively validate the impact of the dynamic RTB environment on the performance of static bidding strategy (e.g., LIN). Furthermore, we discuss the detailed differences between our approach and LIN.\par
\end{itemize}
\section{Related Work}
\label{Related Work}
Recently, research on bidding strategies in RTB mainly focuses on static strategy, designing a linear or non-linear bidding function about impression's pCTR, derived by heuristics algorithm or optimization method from the historical data. Then, the bidding agent directly uses this learned bidding function to bid for each ad impression in the new ad delivery period. The representative static bidding strategies are LIN \cite{ref16} with linear function and ORTB \cite{ref17} with non-linear function, as shown in formula (3) and formula (4). Here $base\_bid$ is a fixed base bid, and $avg\_pctr$ is the average value of all ad impressions' pCTRs on the training set. In LIN, we set the base bid from 1 to 300 (increased one by each time) and calculate each ad impression's bidding price according to (3). Finally, the base bid that maximizes the total clicks on the training set is recorded as the optimal base bid and is used in the new ad delivery period. In ORTB, the parameters $c$ and $\lambda$ are both learned from the training set. So, during a new ad delivery period, the bidding price in ORTB only depends on the pCTR of the impression.\par
\begin{align}
bid_{LIN}(i)&=pctr(i)\times \frac{base\_bid}{avg\_pctr} \\
bid_{ORTB}(i)&= \sqrt{\frac{c}{\lambda} pctr(i)+c^2} - c
\label{equ:equ3, equ:equ4}
\end{align}\par
The static bidding strategies described above have obvious flaws, as bidding price is only related to the pCTR of the ad impression in the new ad delivery period, and it cannot adapt to changes in the RTB environment. For example, when the market competition intensifies, the bidding agent should appropriately increase the bidding price and vice versa. Intuitively, we hope that strategy can dynamically allocate the budget to all ad impressions during the entire delivery period to maximize the total clicks or the cumulative pCTR of the purchased ad impressions. This requires the bidding agent to adjust the bidding strategy dynamically according to the real-time RTB environment. Reinforcement learning may be a promising solution to accomplish this task because it can achieve excellent decision-making. In RTB, the RL agent considers not only the benefit of the single ad impression but its impact on long-term profits.\par
RLB is a typical bidding strategy based on a model-based RL framework \cite{ref8}. The author creatively modeled the bidding decision for each ad impression within an ad delivery period as a sequential MDP \cite{ref18}. The entire RTB system is regarded as the environment. Each impression reaches the DSP triggers state transfer. The bidding agent first observes the state from the RTB environment and then selects an action from the pre-designed discrete action (price) space [0, 1, ..., 300] ($10^{-3}$ Chinese FEN) as its bidding price. If the agent wins the ad impression, the environment will feedback the pCTR of the winning ad impression as the immediate reward. RLB utilizes a dynamic programming algorithm to optimize the optimal action selection policy based on a model-based RL model. The disadvantage of model-based RL is that it is necessary to establish a state transition probability matrix. For a real-world RTB system with billions of ad impressions, the establishment of this matrix requires huge computational and storage overhead, which makes it impossible to deploy on a real DSP.\par
As an improvement, researchers seek to solve the MDP by using model-free RL algorithms. Model-free RL is particularly suitable for scenarios where the agent cannot obtain complete environmental information. It does not need to establish a state transition probability matrix. Instead, the agent in model-free RL learns the optimal action selection policy from experience. However, not as expected, it is challenging to use typical model-free RL algorithms such as DQN and DDPG to learn the optimal action selection policy for a single ad impression. In extreme cases, the bidding agent can only learn to bid with a single high/low price or fall into a locally optimal solution for a long time and cannot explore a larger price space. We have analyzed the causes of this situation in detail in the introduction.\par
Because of the above problems, the authors in \cite{ref11} proposed DRLB, which gave up bidding directly on a single ad impression but introduced a time-related adjustment factor based on the linear bidding function. As shown in formula (5), where $bid(i,t)$ represents the bidding price for \textit{i}-th ad impression in \textit{t}-th time slot, $pctr(i)$ represents the estimated value of the ad impression $i$, $\lambda(t)$ represents the scale bidding factor in each time slot. The greedy algorithm is used to obtain the initial bidding factor $\lambda(0)$. By observing each time slot's RTB environment, the bidding agent selects an optimal action as the adjustment factor for the new time slot and adjusts the bidding price of the ad impression that reaches in this time slot. In DRLB, the bidding price is related to the pCTR of the impression and the RTB environment of the current time slot. DRLB uses the value-based model-free RL algorithm (DQN) to learn the bidding factor's adjustment value for each time slot.\par
\begin{equation}
\begin{split}
bid(i, t) &= pctr(i)/\lambda(t) \\
\lambda(t) = \lambda(t&-1)\times (1+\beta_\alpha(t)) \\
\beta_\alpha(t) \in \{-8\%, -3\%, -1\%, 0&\%, 1\%, 3\%, 8\%\}\ t=1,2,\cdots,T
\end{split}
\label{equ:equ5}
\end{equation}\par
The problem with DRLB is how to design a suitable discrete action space and the initial value of the bidding factor (the value of the first time slot). Therefore, based on DRLB, literature \cite{ref19} generates the optimal adjustment factor for each time slot by introducing a deterministic policy algorithm — Twin Delayed Deep Deterministic policy gradient (TD3). Unlike the discrete action space used by DRLB, the action space designed by \cite{ref19} is a continuous value in (-1, +1).\par
In this paper, we follow the idea of DRLB, which does not directly model the bidding decision, but models the bidding adjustment factor decision. Further, we implement the adjustment of a single ad impression. The bidding agent learns the bidding adjustment factor of the current state and adjusts the bidding price derived from a static bidding strategy. Simultaneously, for the agent to better understand the phenomenon of multiple optimal actions led by the GSP mechanism, we use the RL algorithm based on maximum entropy (SAC) to train the bidding agent to make the optimal adjustment factor generation at each ad impression. Besides, SAC also overcomes the problem of narrow exploration scope caused by deterministic policy. As far as we know, our scheme is the first to apply the maximum entropy stochastic policy RL algorithm to optimize RTB bidding strategy. Table 1 summarizes the differences between several representative bidding strategies and ours.\par
\begin{table}[t]
\Huge
\renewcommand\arraystretch{1.5}
\centering
\caption{Characteristics of Typical Bidding Strategies}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccc}
\toprule
\toprule
\multicolumn{1}{c}{\textbf{Strategy}} & \multicolumn{1}{c}{\textbf{Static/Dynamic}} & \multicolumn{1}{c}{\textbf{Method}} & \multicolumn{1}{c}{\textbf{Adaptiveness}} & \multicolumn{1}{c}{\textbf{Granularity}} & \multicolumn{1}{c}{\textbf{Action}} & \multicolumn{1}{c}{\textbf{Action Space}} \\
\midrule
\multicolumn{1}{c}{\textbf{LIN}} & Static & Linear/Heuristic & Not support & Impression & / & / \\
\multicolumn{1}{c}{\textbf{ORTB}} & Static & \makecell*[c]{Non-linear\\Optimization} & Not support & Impression & / & / \\
\multicolumn{1}{c}{\textbf{RLB}} & Dynamic & \makecell*[c]{Model-based RL\\Dynamic programming} & Support & Impression & Bidding price & Discrete value \\
\multicolumn{1}{c}{\textbf{DRLB}} & Dynamic & Model-free RL/DQN & Support & Time slot & \makecell*[c]{Bidding factor's \\ regulating value} & Discrete value \\
\multicolumn{1}{c}{\textbf{OURS}} & Dynamic & Model-free RL/SAC & Support & Impression & Bidding factor & Continuous value \\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab1}
\vspace{-0.10in}
\end{table}
\section{Problem and Formulation}
\label{Problem and Formulation}
In this paper, to avoid directly learning the optimal bidding price for individual impression, we first design a new bidding function, as shown in formula (2). Its first part is the base price, and the second part is the optimal adjustment value generated by the RL agent according to the real-time RTB auction environment. To be specific, we use LIN \cite{ref16} to generate the base price and obtain the optimal adjustment factor $a_i$ for each ad impression by using SAC-based RL algorithm. It is noted that we use $\min (price_{\max}-b_{LIN}(i),b_{LIN}(i)-price_{\min})$ to limit the range of the adjustment value, where $price_{\max}$ and $price_{\min}$ are the maximum and minimum bidding prices for an impression preset by the advertiser.\par
We regard an ad delivery period as an episodic process in this paper. The decision-making of bidding adjustment factors for all sequential ad impressions in the entire delivery period is modeled as a MDP. In the model-free RL framework, the agent observes a state directly from the environment, so it is no longer necessary to calculate the state transition probability matrix. MDP can be represented by $\langle \mathcal{S}, \mathcal{A}, \mathcal{R}\rangle$, where $\mathcal{S}$ is the state space $(s_t \in \mathcal{S})$, and $\mathcal{A}$ is the action space $(a_t \in \mathcal{A})$. And $\mathcal{R}$ is the reward function that decides the immediate reward received after taking action $a_t$ under the state $s_t$. The interaction process between the bidding agent and the RTB environment can be shown in Figure 2.\par
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth]{fig2.pdf}
\captionsetup{font={footnotesize}}
\caption{Interaction process between the bidding agent and the RTB environment}
\label{fig:fig2}
\vspace{-0.10in}
\end{figure}
Next, we describe the key components of our MDP as follows:\par
\noindent \textbf{State}: The bidding agent observes the state from the environment and uses statistical information to represent the state, as shown in (6): \par
\begin{equation}
s_t=(avg\_pctr(t), avbudget\_ratio(t), avimps\_ratio(t))
\label{equ:equ6}
\end{equation}
where each parameter is described as:\par
\begin{itemize}
\item $avg\_pctr(t)$: the average pCTR of the received impressions, as defined in formula (7). It reflects the average quality of ad impressions in the new ad delivery period.\par
\begin{equation}
avg\_pctr(t)=\frac{1}{t}\sum_{i=1}^tpctr(i)
\label{equ:equ7}
\end{equation}
\item $avbudget\_ratio(t)$: the ratio of the advertiser's available budget to the allocated budget, as defined in formula (8). In our algorithm, the impressions of the entire delivery period are divided into several slots for every 1000 impressions. When a new slot begins, the agent allocates the budget for the incoming slot with CPM $\times$ 1000, where CPM is the average market price of all purchased impressions in the new delivery period. The cost of the current slot should not exceed its budget.\par
\begin{equation}
avbudget\_ratio(t)=avbudget(t)/budget(t)
\label{equ:equ8}
\end{equation}
\item $avimps\_ratio(t)$: the ratio of the number of remaining ad impressions to 1000 in the slot at which the \textit{t}-th ad impression arrives, as shown in formula (9).\par
\begin{equation}
avimps\_ratio(t)=avimps\_num/1000
\label{equ:equ9}
\end{equation}
\end{itemize}
\noindent \textbf{Action}: We defined a new bidding function to calculate the bidding price for each ad impression. Based on this design, the agent's task is to generate action (adjustment factor) $a_t$ suitable for the current state at the impression-grained level. The agent first generates a probability distribution about action according to the state and then samples an action randomly on this distribution. Finally, the action value is restricted to (-1, 1) using the tanh function \cite{ref20}.\par
\noindent \textbf{Reward}: The reward function is defined as (10). Different from the definition of RLB and DRLB, we combine our bidding, linear bidding (reflecting the value of ad impression), and the actual market price together to determine the immediate reward. As shown in formula (10), $b_{LIN}(t)$ represents the linear bidding of the \textit{t}-th ad impression, $market(t)$ represents the market price (second-highest price) of this ad impression, and $b(t)$ represents the bidding price according to our bidding.\par
\begin{scriptsize}
\begin{equation}
\displaystyle
reward(t)=
\begin{cases}
pctr(t),&if\ b_{LIN}(t)<market(t)\ \& \ b(t)\geq market(t) \vspace{1ex}\\
pctr(t)\times \frac{avbudget(t)}{\left| b(t)-b_{LIN}(t)\right|+1},&if\ b_{LIN}(t)\geq market(t)\ \& \ b(t)\geq market(t) \vspace{1ex}\\
pctr(t)\times (a_t-1),&if\ b_{LIN}(t)<market(t)\ \& \ b(t)<market(t) \vspace{1ex}\\
pctr(t)\times a_t,&if\ b_{LIN}(t)\geq market(t)\ \& \ b(t)<market(t) \vspace{1ex}\\
-pctr(t),&if\ avbudget(t)<b(t)\\
\end{cases}
\label{equ:equ10}
\end{equation}
\end{scriptsize}
\noindent The specific rules are designed as:\par
\begin{itemize}
\item If our bidding can win the ad impression, and linear bidding cannot win, the environment feedbacks a positive reward.\par
\item If both our bidding and linear bidding can win the ad impression, the environment also feedbacks a positive reward. Because the market price of the impression is roughly positively correlated with its value, this design makes our bidding close to the linear bidding based on the impression's value.\par
\item If neither our bidding nor the linear bidding can win the ad impression, the environment feedbacks a negative reward. This design is to enable $a_t \to 1$ to increase our bidding price to win this ad impression as much as possible.\par
\item If the linear bidding can win the auction, and our bidding cannot win, the environment feedbacks a negative reward. This design is to enable $a_t \to 1$ to increase our bidding price to win this ad impression.\par
\item If the budget of the current slot is insufficient or it is spent out in advance, considering that all arriving subsequent ad impressions cannot be purchased. In this case, the environment feedbacks a negative reward. Because the budget spent out in advance will cause the advertiser to lose all subsequent impressions at the slot, we use the negative rewards to make the bidding agent avoid such situation as much as possible. \par
\end{itemize}
\section{Solution based on SAC}
\label{Solution based on SAC}
Different from the previous RL bidding strategy, we use the stochastic policy algorithm (SAC) to optimize adjustment factor generation policy. SAC is based on the Actor-Critic framework, drawing on the structural design in DDPG \cite{ref13}, setting up a Target network and an Eval network to enhance the stability of the model, and also using two Q networks to solve the problem of bias caused by overestimation in RL following the technology in TD3 \cite{ref21}. Additionally, SAC introduces the policy entropy proposed in Soft Q learning \cite{ref22} to balance the stochasticity of action selection to deal with the problem of multiple optimal actions for each ad impression caused by the GSP mechanism in RTB. \par
In the basic RL framework, we use the Temporal-Difference (TD) method to optimize the strategy, and the optimization goal is represented by the formula (11), where $r(s_t,a_t)$ represents the immediate reward brought by performing the action $a_t$ under the state $s_t$.\par
\begin{table}[t]
\normalsize
\centering
\caption{Key Parameters of Figure 3}
\resizebox{\textwidth}{!}{%
\begin{tabular}{cc}
\toprule
\toprule
\multicolumn{1}{c}{\textbf{Variable}} & \multicolumn{1}{c}{\textbf{Description}} \\
\midrule
$\alpha$ & \makecell*[l]{The temperature parameter determines the relative importance\\ of the entropy term versus the reward, thus controlling the\\ stochasticity of the optimal policy. }\\
$\hat{a}_{t+1}\sim \pi_\phi(s_{t+1})$ & \makecell*[l]{Generate action distribution on the state $s_{t+1}$ through the Actor\\ network and randomly sample to generate action $\hat{a}_{t+1}$.} \\
$\log \pi_\phi(a|s)$ & \makecell*[l]{The entropy of action $a$ generated by the Policy network on the\\ state $s$, where $\phi$ represents the parameters of the Policy network.} \\
$Q_{\theta_i}(s_t,a_t),i=1,2$ & \makecell*[l]{An Eval Critic network used to calculate the Q value upon the \\state $s_t$ and the action $a_t$, where $\theta_i$ means the parameters of the\\ Eval Critic network $i$.}\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab2}
\vspace{-0.10in}
\end{table}
\begin{equation}
\pi^*=\mathop{\arg \max}_\pi \mathbb{E}_{\left(s_t,a_t\right)\sim \rho_\pi}[\sum\nolimits_{t=1}^Tr\left(s_t,a_t\right)]
\label{equ:equ11}
\end{equation}
SAC introduces the policy entropy, which is defined as $\mathcal{H}(P)=\underset{x\sim P}{E}[-\log P(x)]$. $P(x)$ represents the probability distribution of $x$. Entropy term usually represents the degree of confusion in action selection. Here we hope to enhance the self-learning ability of this model. That is, let actions with the same reward have the same probability of being selected as much as possible. Therefore, the optimization goal is modified to formula (12), where $\pi(\cdot|s_t)$ represents the probability distribution of actions based on the RL policy under the current environment state $s_t$, $\alpha$ is the temperature parameter used to balance entropy and reward, thereby controlling the stochasticity of the optimal strategy.\par
\begin{equation}
\pi^*=\mathop{\arg \max}_\pi \mathbb{E}_{\left(s_t,a_t\right)\sim \rho_\pi}[\sum\nolimits_{t=1}^Tr\left(s_t,a_t\right)+\alpha\mathcal{H}(\pi(\cdot|s_t))]
\label{equ:equ12}
\end{equation}
According to \cite{ref23}, we automatically adjust the temperature parameter when it is not less than the minimum policy entropy threshold and maximize the original reward. The optimization goal is defined in the formula (13), where $\mathcal{H}_0$ is the preset minimum policy entropy threshold, we also set $\mathcal{H}_0=-\dim(\mathcal{A})$ according to \cite{ref23}. The entire process is shown in Figure 3, and we describe some critical parameters in detail in Table 2.\par
\begin{equation}
\underset{\pi_0,\cdots,\pi_T}{\max}\mathbb{E}[\sum\nolimits_{t=0}^Tr(s_t,a_t)] \ s.t.\forall t,\mathcal{H}(\pi_t)\geq \mathcal{H}_0
\label{equ:equ13}
\end{equation}\par
\begin{figure}[!]
\centering
\includegraphics[width=\textwidth]{fig3.pdf}
\captionsetup{font={footnotesize}}
\caption{Structure of our learning algorithm based on SAC}
\label{fig:fig3}
\vspace{-0.10in}
\end{figure}
As shown in Figure 3, there are \textbf{observation} and \textbf{training} processes in our algorithm. Before training, we first need to initialize a replay buffer, recorded as $M$. The replay buffer is a fixed-sized cache. Transitions were sampled from the interaction process between agent and environment. Specifically, upon each ad impression's arrival, the agent generates a Gaussian distribution $\pi_\phi(s_t)$ according to the current state $s_t$. And then samples a bidding adjustment factor $a_t\sim\pi_{\phi}(s_t)$ from the distribution to adjust the base price of each ad impression according to formula (2). After that, the agent uses our bidding and linear bidding to compute the immediate reward based on the formula (10). At the same time, the agent computes the environment's statistics as the next state $s_{t+1}$. Finally, each transition $\langle s_t, a_t, r_t, s_{t+1}\rangle$ is stored in the replay buffer. Whenever $k$ new transitions are added to the replay buffer, the agent performs a training.\par
Then, enter the training process. As shown in Figure 3, the architecture consists of two parts: Policy Net and Critic Net. The Policy Net is the Actor network for generating each impression's bidding adjustment factor. The Critic Net includes two Eval Critic networks and two Target Critic networks. The two Eval Critic networks are used to compute the Q value given a state and an action. And the two Target Critic networks help update two Eval Critic networks. In our algorithm, the parameters of Actor Net and Eval Critic networks are updated based on training; the parameters in Target Critic networks are updated from the corresponding networks in Eval Critic networks by using soft update.\par
Now, we introduce the details of the training. We use neural network to fit Actor and Critic Net. Whenever $k$ pieces of new transition are stored in the replay buffer, the agent will train for $L$ rounds. As shown in Algorithm 1, for each round, the agent first randomly sample $N$ transitions from the replay buffer to form a mini-batch for training. For each transition $\langle s_t, a_t, r_t, s_{t+1}\rangle$, $s_t$ is input into the Policy network to generate the probability distribution of the action, $\pi_\phi(s_t)$, where $\phi$ is the parameters of the Policy network. Then the agent samples an action $\hat{a}_t$ from the distribution $\pi_\phi(s_t)$ and feed it into two Eval Critic networks respectively to compute two Q values, recorded as $Q_{\theta_i}(s_{t}, \hat{a}_{t})$, where $i=1,2$. Here, $\theta_i$ are two Eval Critic networks' parameters. $\alpha\log\pi_\phi(\hat{a}_t|s_t)$ is the entropy term, and the temperature parameter is automatically adjusted by formula (16) to control the stochasticity of the optimal strategy. So we can update the Policy network's parameters by using unbiased gradient estimator proposed in SAC, as shown in formula (14). It is worth noting that the agent chooses the smaller Q value to update the Policy network to avoid overestimation.
\begin{equation}
J_\pi(\phi)=\mathbb{E}_{s_t\sim \rm{D}}\left[\min(Q_{\theta_i}(s_t,\hat{a}_t))-\alpha\log\pi_\phi(\hat{a}_t|s_t)\right],i=1,2
\label{equ:equ14}
\end{equation}\par
At the same time, the agent inputs $(s_t,a_t)$ into two Eval Critic networks respectively to compute two Q values, recorded as $Q_{\theta_i}(s_{t}, a_{t})$, where $i=1,2$. We update two Eval Critic networks' parameters by minimizing TD error, as shown in formula (15). Where $r_t$ comes from the input $\langle s_t, a_t, r_t, s_{t+1}\rangle$, $\gamma$ is the discount factor, $Q_{\hat{\theta}_1}(s_{t+1}, \hat{a}_{t+1})$ and $Q_{\hat{\theta}_2}(s_{t+1}, \hat{a}_{t+1})$ are Q values computed by two Target Critic networks upon the next state $s_{t+1}$ and the action $\hat{a}_{t+1}$. Here, $\hat{a}_{t+1}$ is the action sampled from distribution $\pi_\phi(s_{t+1})$. As with the update of Actor network, in order to prevent overestimation, we use the smaller Q value to update the gradient. The entropy term is also considered in the update formula.\par
\begin{table}[!t]
\normalsize
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{lr}
\toprule
\multicolumn{2}{l}{\textbf{Algorithm 1 }Learning algorithm in our bidding strategy}\\
\midrule
\multicolumn{2}{l}{Initialize $Q_{\theta_i},\ \pi_\phi$ with random parameters $\theta_i,i=1,2$ and $\phi$}\\
\multicolumn{2}{l}{Initialize target networks $\hat{\theta}_i\gets \theta_i$ and $\hat{\phi}\gets \phi$}\\
\multicolumn{2}{l}{Initialize empty replay buffer $M$}\\
\multicolumn{2}{l}{\textbf{for} $episode=1$ to $E$ \textbf{do}}\\
\multicolumn{2}{l}{\hspace{1em}\textbf{for} $t=1$ to $T$ \textbf{do}}\\
\multicolumn{2}{l}{\hspace{2em}Get the base price according to (3)}\\
\multicolumn{2}{l}{\hspace{2em}Observe state $s_t$ and get $a_t$ from $a_t\sim \pi_\phi(s_t)$}\\
\multicolumn{2}{l}{\hspace{2em}Execute $a_t$ to adjust the bidding price shown in (2)}\\
\multicolumn{2}{l}{\hspace{2em}Obtain the reward $r_t$ from (10) and observe next state $s_{t+1}$}\\
\multicolumn{2}{l}{\hspace{2em}Store transition $\langle s_t,a_t,r_t,s_{t+1}\rangle$ in $M$}\\
\multicolumn{2}{l}{\hspace{2em}\textbf{if} $t \mod k$ \textbf{do}}\\
\multicolumn{2}{l}{\hspace{3em}\textbf{for} $l=1$ to $L$ \textbf{do}}\\
\multicolumn{2}{l}{\hspace{4em}Sample a mini-batch of $N$ transitions $\langle s_t,a_t,r_t,s_{t+1}\rangle$ from $M$}\\
\hspace{4em}$\theta_i\gets \theta_i-\lambda_Q\hat{\nabla}_{\theta_i}J_Q(\theta_i),\ i = 1,2$ & $\triangleright$Update the Q-function parameters\\
\hspace{4em}$\phi\gets\phi-\lambda_\pi \hat{\nabla}_\phi J_\pi(\phi)$ & $\triangleright$Update the policy weights\\
\hspace{4em}$\alpha\gets\alpha-\lambda\hat{\nabla}_\alpha J(\alpha)$ & $\triangleright$Adjust temperature\\
\multicolumn{2}{l}{\hspace{4em}\textbf{if} $l \mod d$ \textbf{do}}\\
\hspace{5em}$\hat{\theta}_i\gets \tau\theta_i+(1-\tau)\hat{\theta}_i,\ i = 1,2$ & $\triangleright$Update target network\\
\multicolumn{2}{l}{\hspace{4em}\textbf{end if}}\\
\multicolumn{2}{l}{\hspace{3em}\textbf{end for}}\\
\multicolumn{2}{l}{\hspace{2em}\textbf{end if}}\\
\multicolumn{2}{l}{\hspace{1em}\textbf{end for}}\\
\multicolumn{2}{l}{\textbf{end for}}\\
\bottomrule
\end{tabular}}%
\label{tab:tab999}
\vspace{-0.10in}
\end{table}
\begin{equation}
\begin{split}
J_{Q_i}(\theta)&=\mathbb{E}_{(s_t,a_t)\sim \rm{D}}\left[\frac{1}{2}\left(Q_{\theta_i}(s_t,a_t)-target\_Q\right)^2\right],\ i=1,2\\
target\_Q&=r_t+\gamma\min\left(Q_{\hat{\theta}_i}(s_{t+1},\hat{a}_{t+1})\right)-\alpha\log\pi(\hat{a}_{t+1}|s_{t+1})
\end{split}
\label{equ:equ15}
\end{equation}\par
As with \cite{ref23}, we use the following formula (16) to calculate the gradient for $\alpha$.\par
\begin{equation}
J(\alpha)=\mathbb{E}_{a_t\sim\pi_t}\left[-\alpha\log_{\pi_t}(a_t|\pi_t)-\alpha\mathcal{H}_0\right]
\label{equ:equ16}
\end{equation}\par
As with \cite{ref13}, we use soft-update to update two Target Critic networks, as shown in formula (17), where $\tau$ is the update weight.\par
\begin{equation}
\hat{\theta}_i\gets \tau\theta_i+(1-\tau)\hat{\theta}_i,\ \ i = {1,2}
\label{equ:equ17}
\end{equation}\par
\section{Experimental Setup}
\label{Experimental Setup}
\subsection{Dataset}
We perform the experiments on a benchmark dataset — iPinYou, which comes from a well-known DSP company and contains logs of impressions, bids, clicks, and final conversions. In iPinYou, DSP adopts a fixed bidding strategy and bids 300 ($10^{-3}$ Chinese FEN) for each arriving ad impression. Then ADX determines the winner according to the GSP auction mechanism. The winner needs to pay the second-highest price to ADX to purchase the impression, which we call the second-highest price the market price. As a result, DSP cannot know the lost impression's market price and click (conversion) behavior. So we only use the logs of winning impressions to construct the dataset in our experiments, which makes the number of ad impressions in the dataset far less than the actual number. In detail, we select three datasets with advertiser IDs of 1458, 3358, and 3427. Each dataset contains ten days of advertising logs (from 2013/6/6 to 2013/6/15). We use the data of the first seven days (6/6-6/12) as the training set and the data of the last three days (6/13-6/15) as the testing set. The statistics of the three datasets are shown in Table 3. We regard every day as an ad delivery period and set the daily budget separately. Therefore, the training set includes seven ad delivery periods, and the testing set includes three ad delivery periods.\par
\begin{table}[!t]
\centering
\caption{Statistics on the Training and Testing Sets}
\resizebox{\textwidth}{!}{%
\begin{tabular}{rcccccc}
\toprule
\toprule
\multicolumn{1}{c}{\multirow{2}[4]{*}{}} & \multicolumn{2}{c}{\textbf{1458}} & \multicolumn{2}{c}{\textbf{3358}} & \multicolumn{2}{c}{\textbf{3427}} \\
\cmidrule{2-7}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{\textbf{Training Set}} & \multicolumn{1}{c}{\textbf{Testing Set}} & \multicolumn{1}{c}{\textbf{Training Set}} & \multicolumn{1}{c}{\textbf{Testing Set}} & \multicolumn{1}{c}{\textbf{Training Set}} & \multicolumn{1}{c}{\textbf{Testing Set}} \\
\midrule
\textbf{Imps} & 3083056 & 614638 & 1742104 & 300928 & 2593765 & 536795 \\
\textbf{Clicks} & 2454 & 515 & 1358 & 260 & 1926 & 366 \\
\textbf{Cost} & 212400241 & 45216454 & 160943087 & 34159766 & 210239915 & 46356518 \\
\textbf{CTR(10$^{-3}$)} & 0.7959 & 0.8378 & 0.7795 & 0.8639 & 0.7425 & 0.6818 \\
\textbf{CPM} & 68.892 & 73.565 & 92.384 & 113.514 & 81.055 & 86.357 \\
\textbf{CPC} & 86552.665 & 87798.939 & 118514.791 & 131383.715 & 109158.834 & 126657.153 \\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab3}
\vspace{-0.10in}
\end{table}
From Table 3, we first observe that click behaviors are very sparse in both training and testing sets. For example, the CTR of the 1458 training set is 0.7959 $\times 10^{-3}$, and the CTR of the testing set is 0.838$\times 10^{-3}$. It seriously weakens the performance of the CTR prediction model and reduces the credibility of the estimated value of each ad impression. Secondly, we observe that the average market prices (CPMs) of the training set and the testing set change greatly on three datasets, indicating that the RTB environment does have significant differences in different ad delivery periods. It may affect the performance of the static bidding strategy in the new ad delivery period. Furthermore, Figure 4 gives the statistics of various indicators in ten days of the 1458 dataset. We can see that there are large fluctuations in all kinds of statistics during ten delivery periods. Thus the static optimal bidding strategy derived from the historical training periods is likely unsuitable for the new ad delivery period.\par
\begin{figure}[t]
\centering
\includegraphics[width=0.32\textwidth]{fig4-1.pdf}
\includegraphics[width=0.32\textwidth]{fig4-2.pdf}
\includegraphics[width=0.32\textwidth]{fig4-3.pdf}
\captionsetup{font={footnotesize}}
\caption{Statistics of Various Indicators in Ten Days of 1458 Dataset}
\label{fig:fig4}
\end{figure}
\begin{table}[t]
\centering
\caption{AUC Values of CTR Estimators}
\resizebox{0.4\textwidth}{!}{%
\begin{tabular}{ccc}
\toprule
\toprule
\textbf{Advertiser ID} & \textbf{Model} & \textbf{AUC}\\
\midrule
1458 & FNN & 0.7944\\
3358 & FM & 0.8675\\
3427 & FM & 0.8334\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab4}
\vspace{-0.10in}
\end{table}
\subsection{Click-Through Rate Prediction Model}
In this paper, all bidding strategies are based on the estimated pCTR of the ad impression to the advertiser. We use three representative models (LR \cite{ref6}, FM \cite{ref24}, and FNN \cite{ref25}) to train three CTR estimators for each advertiser and choose the best one as the estimator used in our experiments. Table 4 lists the CTR prediction models we choose for three advertisers and their AUC values. In our experiments, each advertiser's bidding agent first computes the pCTR of the new arriving impression using its CTR estimator and decides the bidding price based on the pCTR. It should be noted that some click behaviors are quite inconsistent with the impressions' pCTRs due to the uncertainty of user click behavior.\par
\subsection{Baseline Bidding Strategies}
In this subsection, we introduce some representative bidding strategies as baselines.\par
\begin{itemize}
\item \textbf{LIN}: The linear bidding strategy is defined as (3).\par
\item \textbf{RLB}: The bidding strategy is learned based on a model-based RL framework proposed in \cite{ref8}, which can directly select an optimal bidding price for each impression.\par
\item \textbf{DRLB}: The bidding strategy is learned based on a model-free RL framework proposed in \cite{ref11}. Primarily, it uses the DQN algorithm to train the optimal action selection policy that can help choose the regulating value to adjust each time slot's bidding factor.\par
\item \textbf{OURS}: The bidding strategy is learned based on a model-free maximum entropy (SAC) RL framework proposed in this paper.\par
\end{itemize}
\subsection{Hyper-Parameters Setting}
In our algorithm, both Actor and four Critic networks use the feed-forward, fully connected neural network. Each network contains two hidden layers, and each hidden layer contains 128 neurons. The algorithm uses the Adam optimizer \cite{ref26} to optimize the neural network parameters, and the output layer uses the tanh activation function to constrain the output adjustment factor. We have described the detailed learning process of the algorithm in Section 4. Some hyper-parameters setting in our experiments are given in Table 5. It should be noted that the hyper-parameters setting of SAC mainly refers to the parameters setting in \cite{ref23}.\par
\begin{table}[t]
\centering
\caption{Key Hyper-parameter Implementation}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccl}
\toprule
\toprule
\textbf{Parameter} & \textbf{Setting} & \textbf{Description}\\
\midrule
$\gamma$ & 1 & Discount factor for TD in formula (15).\\
$\tau$ & 0.0005 & Soft-update parameter in formula (17).\\
$M$ & 1000000 & Size of replay buffer in Algorithm 1.\\
$N$ & 256 & Size of mini-batch in Algorithm 1.\\
$k$ & 30000 & Training once after $k$ pieces of experience has been stored in Algorithm 1.\\
$L$ & 128 & Number of rounds per training in Algorithm 1.\\
$d$ & 4 & Update frequency for soft updates in Algorithm 1.\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab5}
\vspace{-0.10in}
\end{table}
\section{Experimental Results}
\label{Experimental Results}
In this section, we first evaluate the performance of LIN (a typical static bidding strategy) on three datasets and discuss the impact of the RTB environment dynamics on the performance of LIN. Then, we compare our scheme with LIN and two dynamic RL-based bidding strategies (i.e., RLB and DRLB). Finally, we discuss the advantages and disadvantages of our scheme and LIN from the number of purchased impressions, the average market price, cost ratio, and the reasons for losing clicks. In order to evaluate the adaptability of the bidding strategy to budget changes, we set four different daily budgets for each set of experiments. Since the number of ad impressions and the actual cost vary from day to day in the dataset, we set the daily budget as 1/2, 1/4, 1/8, and 1/16 of the actual cost of each advertiser. Table 6 lists the actual costs of three advertisers on ten days.\par
\begin{table}[b]
\centering
\renewcommand\arraystretch{1.2}
\caption{Actual Daily Costs of three Advertisers on Ten Days($10^{-3}$ Chinese FEN)}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccccc}
\toprule
\toprule
\multirow{2}[4]{*}{\textbf{Advertiser ID}}& \multicolumn{7}{|c|}{\textbf{Training set}} & \multicolumn{3}{c}{\textbf{Testing set}}\\
\cmidrule{2-11}
& \multicolumn{1}{|c}{\textbf{6/6}} & \textbf{6/7} & \textbf{6/8} & \textbf{6/9} & \textbf{6/10} & \textbf{6/11} & \multicolumn{1}{c|}{\textbf{6/12}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15}\\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{30096630} & 30228554 & 30615541 & 30548604 & 30303929 & 30309883 & \multicolumn{1}{c|}{30297100} & 15036900 & 15045650 & 15133904\\
\textbf{3358} & \multicolumn{1}{|c}{17068590} & 17155542 & 16219705 & 14571538 & 40071957 & 23340047 & \multicolumn{1}{c|}{32515710} & 10864298 & 12143044 & 11152426\\
\textbf{3427} & \multicolumn{1}{|c}{30644030} & 23930230 & 29840853 & 32019674 & 31232220 & 30918866 & \multicolumn{1}{c|}{31654042} & 15185670 & 15325090 & 15845760\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab6}
\vspace{-0.10in}
\end{table}
\subsection{Performance of the Static Linear Bidding Strategy}
In this subsection, we discuss the impact of the dynamic RTB environment on the performance of the static bidding strategy. To this end, we perform LIN, the most representative static bidding strategy, on three datasets under four daily budget constraints. We run two groups of experiments for comparison. In the first group of experiments, we learn the optimal base bid to win the most clicks on the training set via a greedy heuristic algorithm. Then, we use the learned optimal base bid to calculate the bidding price for each ad impression on the testing set. As listed in Table 7, we record the results of the first group of experiments as \textbf{Base bid learned on Training Set}, which represents that the optimal base bid is learned on the training set and the clicks are obtained on the testing set.\par
In the second group of experiments, we assume that the testing set is known and learn the optimal base bid on the testing set. Then, we still use the learned optimal base bid to compute the bidding price for each impression on the testing set and record the results as \textbf{Base bid learned on Testing Set} since both optimal base bid and clicks are obtained on the testing set. The clicks received in the second group of experiments are the best results obtained by LIN under the environment of the testing periods. Therefore, we can analyze the adaptability of static bidding strategy to the environment changes quantitatively by comparing the click numbers obtained by the two groups of experiments.\par
\begin{table}[t]
\centering
\caption{Total Number of Clicks that the Linear Bidding Strategy Received on the Testing Set}
\resizebox{\textwidth}{!}{%
\begin{tabular}{cccccccccccccccccc}
\toprule
\toprule
\multirowcell{2}[-0.5ex]{\textbf{Advertiser} \\ \textbf{ID}} & \multirowcell{2}[-0.5ex]{\textbf{Base bid} \\ \textbf{learned on}} & \multicolumn{4}{|c}{\textbf{1/2}} & \multicolumn{4}{c}{\textbf{1/4}} & \multicolumn{4}{c}{\textbf{1/8}} & \multicolumn{4}{c}{\textbf{1/16}} \\
\cmidrule{3-18}
& & \multicolumn{1}{|c}{\textbf{6/13}} & \textbf{6/14} & \textbf{6/15} & \multicolumn{1}{c|}{\textbf{Total}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} & \multicolumn{1}{c|}{\textbf{Total}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} & \multicolumn{1}{c|}{\textbf{Total}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} & \textbf{Total} \\
\midrule
\multirow{2}[4]{*}{\textbf{1458}} & \multicolumn{1}{c|}{\textbf{Training set}} & 155 & 119 & 136 & \multicolumn{1}{c|}{410} & 96 & 81 & 89 & \multicolumn{1}{c|}{266} & 66 & 55 & 58 & \multicolumn{1}{c|}{179} & 39 & 40 & 47 & 126 \\
\cmidrule{2-2} & \multicolumn{1}{c|}{\textbf{Testing Set}} & 167 & 140 & 160 & \multicolumn{1}{c|}{467} & 103 & 107 & 117 & \multicolumn{1}{c|}{327} & 68 & 67 & 73 & \multicolumn{1}{c|}{208} & 39 & 44 & 49 & 132 \\
\midrule
\multirow{2}[4]{*}{\textbf{3358}} & \multicolumn{1}{c|}{\textbf{Training set}} & 68 & 79 & 82 & \multicolumn{1}{c|}{229} & 55 & 70 & 66 & \multicolumn{1}{c|}{191} & 48 & 62 & 60 & \multicolumn{1}{c|}{170} & 43 & 50 & 44 & 137 \\
\cmidrule{2-2} & \multicolumn{1}{c|}{\textbf{Testing Set}} & 67 & 78 & 82 & \multicolumn{1}{c|}{227} & 63 & 74 & 70 & \multicolumn{1}{c|}{207} & 54 & 63 & 61 & \multicolumn{1}{c|}{178} & 45 & 52 & 49 & 146 \\
\midrule
\multirow{2}[4]{*}{\textbf{3427}} & \multicolumn{1}{c|}{\textbf{Training set}} & 99 & 103 & 114 & \multicolumn{1}{c|}{316} & 80 & 82 & 90 & \multicolumn{1}{c|}{252} & 63 & 58 & 69 & \multicolumn{1}{c|}{190} & 49 & 47 & 54 & 150 \\
\cmidrule{2-2} & \multicolumn{1}{c|}{\textbf{Testing Set}} & 100 & 103 & 116 & \multicolumn{1}{c|}{319} & 84 & 84 & 92 & \multicolumn{1}{c|}{260} & 67 & 67 & 76 & \multicolumn{1}{c|}{210} & 57 & 52 & 61 & 170 \\
\bottomrule
\bottomrule
\end{tabular}%
}%
\label{tab:tab7}
\vspace{-0.10in}
\end{table}
Obviously, all click numbers in the first group of experiments are lower than those in the second group of experiments, proving that the base bids learned on the historical training periods are not the best for the testing periods due to the dynamic changes of the RTB environment. Especially on the 1458 dataset, when we set the daily budget 1/2 and 1/4 of the cost, the click numbers are 410 and 266 in the first group of experiments, 12.20\% and 18.65\% lower than the ideal clicks (obtained in the second group of experiments). The results reveal that the RTB environment does change dramatically between the training and the testing periods and harms the performance of the static bidding strategy.\par
Moreover, we observe that the gaps between the actual and ideal clicks decrease with the daily budget declining. When we set the daily budget 1/16 of the cost, the gap is reduced to 6 clicks. Intuitively, we list the learned optimal base bids based on both training and testing sets in Table 8, demonstrating the optimal base bids fluctuating with the RTB environment changes. We find a common phenomenon that the optimal base bids decrease with the budget shrinking, which reveals a conservative bidding strategy is beneficial to maximize the overall clicks in the iPinYou dataset when the budget is insufficient. To sum up, through the experimental results in this subsection, we prove that the dynamic RTB environment does hinder the performance of the static bidding strategy and confirm the necessity of introducing a real-time adjustment scheme into the bidding strategy.\par
\begin{table}[t]
\centering
\caption{Optimal Base Bid(based on the maximum number of clicks)}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{3}[5]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/2}} & \multicolumn{4}{|c}{\textbf{1/4}} \\
\cmidrule{2-9}
& \multicolumn{1}{|c|}{\textbf{Training set}} & \multicolumn{3}{c}{\textbf{Testing set}} & \multicolumn{1}{|c|}{\textbf{Training set}} & \multicolumn{3}{c}{\textbf{Testing set}} \\
\cmidrule{2-9}
& \multicolumn{1}{|c|}{\textbf{6/6-6/12}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} & \multicolumn{1}{|c|}{\textbf{6/6-6/12}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c|}{298} & 184 & 173 & 194 & \multicolumn{1}{|c|}{111} & 64 & 59 & 62 \\
\textbf{3358} & \multicolumn{1}{|c|}{182} & 227 & 293 & 280 & \multicolumn{1}{|c|}{70} & 183 & 143 & 139 \\
\textbf{3427} & \multicolumn{1}{|c|}{212} & 228 & 207 & 220 & \multicolumn{1}{|c|}{80} & 92 & 82 & 94 \\
\midrule
\multirow{3}[5]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/8}} & \multicolumn{4}{|c}{\textbf{1/16}} \\
\cmidrule{2-9}
& \multicolumn{1}{|c|}{\textbf{Training set}} & \multicolumn{3}{c}{\textbf{Testing set}} & \multicolumn{1}{|c|}{\textbf{Training set}} & \multicolumn{3}{c}{\textbf{Testing set}} \\
\cmidrule{2-9}
& \multicolumn{1}{|c|}{\textbf{6/6-6/12}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} & \multicolumn{1}{|c|}{\textbf{6/6-6/12}} & \textbf{6/13} & \textbf{6/14} & \textbf{6/15} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c|}{48} & 35 & 33 & 33 & \multicolumn{1}{|c|}{25} & 23 & 25 & 21 \\
\textbf{3358} & \multicolumn{1}{|c|}{45} & 94 & 80 & 78 & \multicolumn{1}{|c|}{30} & 57 & 46 & 26 \\
\textbf{3427} & \multicolumn{1}{|c|}{44} & 50 & 49 & 54 & \multicolumn{1}{|c|}{27} & 33 & 32 & 36 \\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab8}
\vspace{-0.10in}
\end{table}
\subsection{Performance Comparison of RL Bidding Strategies}
\begin{table}[!t]
\centering
\caption{Number of Clicks Received by Four Bidding Strategies under Different Budget Constraints}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/2}} & \multicolumn{4}{|c}{\textbf{1/4}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{410} & 415 & 442 & \textbf{461} & \multicolumn{1}{|c}{266} & 294 & 294 & \textbf{296} \\
\textbf{3358} & \multicolumn{1}{|c}{229} & 221 & 228 & \textbf{235} & \multicolumn{1}{|c}{191} & 187 & 195 & \textbf{200} \\
\textbf{3427} & \multicolumn{1}{|c}{316} & 304 & 281 & \textbf{317} & \multicolumn{1}{|c}{252} & 246 & 234 & \textbf{263} \\
\midrule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/8}} & \multicolumn{4}{|c}{\textbf{1/16}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{179} & 176 & 172 & \textbf{184} & \multicolumn{1}{|c}{\textbf{126}} & 112 & 106 & 119 \\
\textbf{3358} & \multicolumn{1}{|c}{\textbf{170}} & 147 & 164 & 162 & \multicolumn{1}{|c}{\textbf{137}} & 112 & 132 & 131 \\
\textbf{3427} & \multicolumn{1}{|c}{190} & 198 & 164 & \textbf{202} & \multicolumn{1}{|c}{150} & 144 & 134 & \textbf{155} \\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab9}
\vspace{-0.10in}
\end{table}
In this subsection, we evaluate our scheme with two typical RL-based bidding strategies. We still run each bidding strategy on three datasets under four budget constraints. Table 9 outlines the total number of clicks obtained by each bidding strategy in the testing periods. Also, we add the clicks received by LIN in Table 9 for comparison. First of all, we find that our scheme achieves the highest clicks in most experiments among three RL-based bidding strategies, with significant advantages compared with RLB and DRLB. Among three RL-based bidding strategies, RLB and DRLB get the least clicks respectively in the six experiments. Only when the budget is set to 1/8 of cost, the click number obtained by our scheme is slightly lower than that of DRLB on the 3358 dataset. In addition, our strategy is also superior to LIN. Especially when the daily budgets are 1/2 and 1/4 of the cost, our scheme can obtain 461 and 296 clicks on the 1458 dataset, 12.44\% and 11.28\% higher than LIN. Similar results are found on the 3358 and 3427 datasets. Such results show that our scheme can adjust the bidding price adaptively according to the RTB environment in real-time so that the bidding price of each ad impression can match its environment as much as possible. Furthermore, the results also prove that it is feasible to introduce the adjustment factor into the bidding function of LIN.\par
\begin{table}[!b]
\centering
\caption{Sum of pCTR Won by Four Bidding Strategies under Different Budget Constraints}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/2}} & \multicolumn{4}{|c}{\textbf{1/4}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{330.07} & 355.02 & 350.84 & \textbf{368.22} & \multicolumn{1}{|c}{230.91} & 275.09 & 246.96 & \textbf{249.90} \\
\textbf{3358} & \multicolumn{1}{|c}{307.14} & 294.42 & 306.89 & \textbf{309.96} & \multicolumn{1}{|c}{258.49} & 246.83 & \textbf{265.28} & 259.36 \\
\textbf{3427} & \multicolumn{1}{|c}{\textbf{413.49}} & 393.03 & 353.03 & 404.86 & \multicolumn{1}{|c}{\textbf{328.89}} & 313.06 & 321.43 & 327.22 \\
\midrule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{4}{|c}{\textbf{1/8}} & \multicolumn{4}{|c}{\textbf{1/16}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{RLB} & \textbf{DRLB} & \textbf{OURS} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{177.72} & 191.56 & 179.09 & \textbf{182.50} & \multicolumn{1}{|c}{\textbf{140.29}} & 129.71 & 103.98 & 115.80 \\
\textbf{3358} & \multicolumn{1}{|c}{\textbf{225.53}} & 188.5 & 217.58 & 213.97 & \multicolumn{1}{|c}{\textbf{181.88}} & 140.55 & 174.79 & 163.80 \\
\textbf{3427} & \multicolumn{1}{|c}{249.77} & 237.47 & 230.49 & \textbf{251.46} & \multicolumn{1}{|c}{184.63} & 182.13 & 167.94 & \textbf{188.92} \\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab10}
\vspace{-0.10in}
\end{table}
Table 10 presents the sum of pCTR won by each bidding strategy in the testing periods. Firstly, we observe that, similar to the number of clicks, our scheme achieves the largest pCTR in most experiments among the three RL-based bidding strategies. Moreover, we note that when the budget is 1/4 of cost, the pCTR won by our scheme is marginally lower than that of DRLB on the 3358 dataset while our scheme obtained more clicks. The same happens on the 1458 dataset when the budget is set to 1/8 and 1/16. This results show that our scheme purchased more valuable ad impressions that generate clicks.\par
\subsection{Detailed Comparison of Our Scheme with LIN}
In the last subsection, we discuss the advantages and disadvantages of our scheme and LIN in detail. First, Table 11 gives the numbers of impressions purchased by two strategies on the testing set. From this table, we observe that our scheme buys more ad impressions than LIN in eight experiments because our scheme can adjust its bidding price for every auctioned impression dynamically according to the environment of the testing periods. In contrast, LIN is a static bidding strategy in which the optimal base bid is only learned from the training periods and cannot be adjusted according to the real-time RTB environment. If the learned optimal base bid is significantly lower than that in the testing periods, it will lead the advertiser to bid with low price and lose many available impressions. As a result, there will be a massive budget surplus. For example, all budgets have not been spent out in four experiments on the 3427 dataset.\par
\begin{table}[t]
\centering
\caption{Numbers of Ad Impressions Purchased by LIN and Our Scheme}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{2}{|c}{\textbf{1/2}} & \multicolumn{2}{|c}{\textbf{1/4}} & \multicolumn{2}{|c}{\textbf{1/8}} & \multicolumn{2}{|c}{\textbf{1/16}} \\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} \\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{\textbf{337559}} & 327855 & \multicolumn{1}{|c}{186698} & \textbf{187135} & \multicolumn{1}{|c}{110719} & \textbf{119257} & \multicolumn{1}{|c}{\textbf{65456}} & 60501\\
\textbf{3358} & \multicolumn{1}{|c}{158301} & \textbf{172912} & \multicolumn{1}{|c}{74835} & \textbf{79779} & \multicolumn{1}{|c}{\textbf{44594}} & 43978 & \multicolumn{1}{|c}{\textbf{22662}} & 19720\\
\textbf{3427} & \multicolumn{1}{|c}{309703} & \textbf{314510} & \multicolumn{1}{|c}{161316} & \textbf{171027} & \multicolumn{1}{|c}{84099} & \textbf{93411} & \multicolumn{1}{|c}{40969} & \textbf{46986}\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab11}
\vspace{0.10in}
\end{table}
Table 12 shows the average market prices of our strategy and LIN. In most cases, the results reflect that our scheme's average market prices are higher than those of LIN since our scheme adopts an aggressive bidding strategy. Therefore, our scheme can work well when the budget is sufficient. However, it performs marginally poorly when the budget is seriously inadequate. In particular, when the budget is only 1/16 of the cost, the click numbers on the 1458 and 3358 datasets are slightly lower than those of LIN. In the following Table 14, we further analyze the main reason why advertisers lose clicks.\par
\begin{table}[!t]
\centering
\caption{Average Costs of Buying an Ad Impression in LIN and Our Scheme ($10^{-3}$ Chinese FEN)}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{2}[3]{*}{\textbf{Advertiser ID}} & \multicolumn{2}{|c}{\textbf{1/2}} & \multicolumn{2}{|c}{\textbf{1/4}} & \multicolumn{2}{|c}{\textbf{1/8}} & \multicolumn{2}{|c}{\textbf{1/16}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS}\\
\midrule
\textbf{1458} & \multicolumn{1}{|c}{66.975} & \textbf{68.806} & \multicolumn{1}{|c}{\textbf{60.547}} & 60.406 & \multicolumn{1}{|c}{\textbf{51.048}} & 47.393 & \multicolumn{1}{|c}{43.174} & \textbf{46.710}\\
\textbf{3358} & \multicolumn{1}{|c}{96.821} & \textbf{97.792} & \multicolumn{1}{|c}{91.488} & \textbf{96.040} & \multicolumn{1}{|c}{90.143} & \textbf{97.002} & \multicolumn{1}{|c}{92.571} & \textbf{108.264}\\
\textbf{3427} & \multicolumn{1}{|c}{72.888} & \textbf{73.535} & \multicolumn{1}{|c}{62.219} & \textbf{67.617} & \multicolumn{1}{|c}{54.940} & \textbf{62.032} & \multicolumn{1}{|c}{46.198} & \textbf{52.563}\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab12}
\vspace{-0.10in}
\end{table}
Furthermore, we discuss the cost ratio of our scheme and LIN, where the cost ratio is the ratio of the actual cost to the budget. Ideally, we hope to obtain the most clicks in an ad delivery period within a given budget. In this subsection, we only give the experimental results on the 3427 dataset in Table 13. First, we observe that the cost ratios of LIN are significantly lower than those of our scheme. Because LIN is a static bidding strategy, the bidding agent cannot adjust its base bid according to the environment in real-time. Specifically, on the 3427 dataset, the learned base bid is relatively low compared to the environment of testing periods, leading to the bidding price for an impression usually lower than its market prices. Therefore, the advertiser loses many impressions and has a lot of money left at the end of the ad delivery period. On the other hand, the advertiser 3427 loses substantial impressions that may bring clicks, which will hurt the advertiser's revenue in the testing periods.\par
\begin{table}[!b]
\centering
\caption{Cost Ratios and Winning Impressions Obtained by LIN and Our Scheme on 3427}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
& \multicolumn{2}{|c}{\textbf{1/2}} & \multicolumn{2}{|c}{\textbf{1/4}} & \multicolumn{2}{|c}{\textbf{1/8}} & \multicolumn{2}{|c}{\textbf{1/16}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS}\\
\midrule
\textbf{Cost ratio} & \multicolumn{1}{|c}{97.392\%} & \textbf{99.782\%} & \multicolumn{1}{|c}{86.607\%} & \textbf{99.787\%} & \multicolumn{1}{|c}{78.286\%} & \textbf{99.999\%} & \multicolumn{1}{|c}{65.326\%} & \textbf{85.244\%}\\
\textbf{Win imps} & \multicolumn{1}{|c}{309703} & \textbf{314510} & \multicolumn{1}{|c}{161316} & \textbf{171027} & \multicolumn{1}{|c}{84099} & \textbf{93411} & \multicolumn{1}{|c}{40969} & \textbf{46986}\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab13}
\vspace{0.10in}
\end{table}
In contrast, our scheme can get higher cost ratios under various cases due to supporting adjust the bidding price at the impression-grained level. In addition, we note that the cost ratio of LIN decreases as the budget shrinks. This happens because the optimal base price learned by LIN is reduced with the budget narrows to capture the impressions of the whole period as much as possible, avoiding the budget being spent out in advance. According to the bidding function of LIN, the lower the base bid is, the lower the bidding price is. Therefore, the number of impressions successfully purchased by LIN is greatly reduced. When the budget is 1/16 of the cost, only 65.33\% of the budget has been spent by LIN.\par
In RTB, there are two reasons why advertisers lose clicks. One is that the bidding price is less than the market price, and the other is that the advertiser has no money to buy impressions due to its budget has been wiped out in advance, resulting in losing all subsequent impressions. In RTB, we call the latter case as \textbf{early stop}. Here, we take the experimental results on the 1458 dataset to make detailed statistics of why LIN and our strategy lose the clicks. Table 14 presents the missing click numbers caused by each reason. We observe that when the budgets are 1/2, 1/4, and 1/8 of the cost, LIN loses more clicks than our scheme for early stop, but it loses fewer clicks than ours for its bidding price lower than the market price.\par
\begin{table}[t]
\centering
\caption{Number of Lost Clicks and Reason in 1458 Dataset}
\resizebox{\textwidth}{!}{%
\begin{tabular}{ccccccccc}
\toprule
\toprule
\multirow{2}[3]{*}{\textbf{Reason}} & \multicolumn{2}{|c}{\textbf{1/2}} & \multicolumn{2}{|c}{\textbf{1/4}} & \multicolumn{2}{|c}{\textbf{1/8}} & \multicolumn{2}{|c}{\textbf{1/16}}\\
\cmidrule{2-9}
& \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS} & \multicolumn{1}{|c}{\textbf{LIN}} & \textbf{OURS}\\
\midrule
\textbf{Early stop} & \multicolumn{1}{|c}{89} & \textbf{11} & \multicolumn{1}{|c}{175} & \textbf{119} & \multicolumn{1}{|c}{158} & \textbf{121} & \multicolumn{1}{|c}{\textbf{67}} & 105\\
\midrule
\makecell*[c]{\textbf{Bid lower than}\\\textbf{market price}}& \multicolumn{1}{|c}{\textbf{16}} & 43 & \multicolumn{1}{|c}{\textbf{74}} & 100 & \multicolumn{1}{|c}{\textbf{178}} & 210 & \multicolumn{1}{|c}{322} & \textbf{291}\\
\bottomrule
\bottomrule
\end{tabular}}%
\label{tab:tab14}
\vspace{-0.10in}
\end{table}
The reason for the above results is that the optimal base bid learned by LIN in 1458 training periods is obviously high for the environment of the testing periods. Consequently, LIN's bidding price for each impression during the testing periods is much higher than its market price, resulting in LIN winning a large number of ad impressions. Among the ad impressions that LIN has bought, more of them are no-click impressions. Buying these useless ad impressions wastes a lot of money and can easily cause advertisers to spend out their budgets in advance and cannot purchase subsequent ad impressions. Our scheme introduces a dynamic adjustment factor into LIN. The bidding price of each impression can be adjusted dynamically to match the real-time environment of the testing periods, avoiding spending the budget out too quickly or too slowly. When the budget is set to 1/16 of the cost, in LIN, the optimal base bid learned on the training set fits the environment of testing periods. Hence, the number of lost clicks caused by budget depletion reduces significantly.\par
Based on the above analyses, we can draw the following conclusions. Firstly, the environment dynamics do have a negative impact on the performance of the static bidding strategy. Still, it has little impact on our dynamic bidding strategy based on maximum entropy RL. Secondly, the new bidding function designed by us is practical, which can adjust the bidding price for each impression according to the real-time environment by introducing an adjustment factor into LIN. Finally, learning the optimal adjustment factor generation policy utilizing the maximum entropy RL is superior to other RL models.\par
\section{Conclusions}
\label{Conclusions}
In this paper, we focus on using model-free RL to optimize advertisers' bidding strategies. Specifically, we first design a new bidding function that uses LIN to compute the base price of each ad impression, and adjusts the base price to fit the real-time RTB environment by introducing a bidding adjustment factor. To this end, we model the adjustment factor decision as an MDP, and then use the stochastic policy SAC to solve the optimal adjustment factor generation policy. Unlike the widely used deterministic model-free RL algorithm, SAC can address the problem of multiple optimal actions for a single impression brought by GSP mechanism. Secondly, SAC also extends the scope of the RL agent to explore the optimal action, enabling the algorithm to converge to the global optimum more quickly. In particular, we design a new reward function that enables the RL agent to better learn the optimal action by comparing it with the results of bidding using LIN, thus maximizing the probability of the optimal action being selected. Finally, we validate our improvements on lots of experiments. The work in this paper focuses on bid optimization for a single advertiser. In real RTB applications, advertising platforms usually need to optimize the total revenue of multiple advertisers, and we hope to use reinforcement learning solutions to solve this challenge in the future. Undoubtedly, this will be a more valuable work.\par
\section*{Acknowledgements}
This work was supported in part by 1) the National Natural Science Foundation of China under Grant 61202445; 2) the Fundamental Research Funds for the Central Universities under Grant ZYGX2016J096.\par
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{"url":"https:\/\/eigen.tuxfamily.org\/dox-devel\/group__TutorialSlicingIndexing.html","text":"Please, help us to better know about our user community by answering the following short survey: https:\/\/forms.gle\/wpyrxWi18ox9Z5ae9\n\nThis page presents the numerous possibilities offered by operator() to index sub-set of rows and columns. This API has been introduced in Eigen 3.4. It supports all the feature proposed by the block API , and much more. In particular, it supports slicing that consists in taking a set of rows, columns, or elements, uniformly spaced within a matrix or indexed from an array of indices.\n\n# Overview\n\nAll the aforementioned operations are handled through the generic DenseBase::operator()(const RowIndices&, const ColIndices&) method. Each argument can be:\n\n\u2022 An integer indexing a single row or column, including symbolic indices.\n\u2022 The symbol Eigen::all representing the whole set of respective rows or columns in increasing order.\n\u2022 An ArithmeticSequence as constructed by the Eigen::seq, Eigen::seqN, or Eigen::lastN functions.\n\u2022 Any 1D vector\/array of integers including Eigen's vector\/array, expressions, std::vector, std::array, as well as plain C arrays: int[N].\n\nMore generally, it can accepts any object exposing the following two member functions:\n\n<integral type> operator[](<integral type>) const;\n<integral type> size() const;\n\nwhere <integral type> stands for any integer type compatible with Eigen::Index (i.e. std::ptrdiff_t).\n\n# Basic slicing\n\nTaking a set of rows, columns, or elements, uniformly spaced within a matrix or vector is achieved through the Eigen::seq or Eigen::seqN functions where \"seq\" stands for arithmetic sequence. Their signatures are summarized below:\n\nfunction description example\nseq(firstIdx,lastIdx)\nauto seq(FirstType f, LastType l, IncrType incr)\nrepresents the sequence of integers ranging from firstIdx to lastIdx\nseq(2,5) <=> {2,3,4,5}\nseq(firstIdx,lastIdx,incr)\nsame but using the increment incr to advance from one index to the next\nseq(2,8,2) <=> {2,4,6,8}\nseqN(firstIdx,size)\nArithmeticSequence< typename internal::cleanup_index_type< FirstType >::type, typename internal::cleanup_index_type< SizeType >::type, typename internal::cleanup_seq_incr< IncrType >::type > seqN(FirstType first, SizeType size, IncrType incr)\nDefinition: ArithmeticSequence.h:162\nrepresents the sequence of size integers starting from firstIdx\nseqN(2,5) <=> {2,3,4,5,6}\nseqN(firstIdx,size,incr)\nsame but using the increment incr to advance from one index to the next\nseqN(2,3,3) <=> {2,5,8}\n\nThe firstIdx and lastIdx parameters can also be defined with the help of the Eigen::last symbol representing the index of the last row, column or element of the underlying matrix\/vector once the arithmetic sequence is passed to it through operator(). Here are some examples for a 2D array\/matrix A and a 1D array\/vector v.\n\nIntent Code Block-API equivalence\nBottom-left corner starting at row i with n columns\nA(seq(i,last), seqN(0,n))\nstatic const symbolic::SymbolExpr< internal::symbolic_last_tag > last\nDefinition: IndexedViewHelper.h:38\nA.bottomLeftCorner(A.rows()-i,n)\nBlock starting at i,j having m rows, and n columns\nA(seqN(i,m), seqN(i,n)\nA.block(i,j,m,n)\nBlock starting at i0,j0 and ending at i1,j1\nA(seq(i0,i1), seq(j0,j1)\nA.block(i0,j0,i1-i0+1,j1-j0+1)\nEven columns of A\nA(all, seq(0,last,2))\nstatic const Eigen::internal::all_t all\nDefinition: IndexedViewHelper.h:171\nFirst n odd rows A\nA(seqN(1,n,2), all)\nThe last past one column\nA(all, last-1)\nA.col(A.cols()-2)\nThe middle row\nA(last\/2,all)\nA.row((A.rows()-1)\/2)\nLast elements of v starting at i\nv(seq(i,last))\nv.tail(v.size()-i)\nLast n elements of v\nv(seq(last+1-n,last))\nv.tail(n)\n\nAs seen in the last exemple, referencing the last n elements (or rows\/columns) is a bit cumbersome to write. This becomes even more tricky and error prone with a non-default increment. Here comes Eigen::lastN(size) , and Eigen::lastN(size,incr) :\n\nIntent Code Block-API equivalence\nLast n elements of v\nv(lastN(n))\nauto lastN(SizeType size, IncrType incr) -> decltype(seqN(Eigen::last-(size-fix< 1 >()) *incr, size, incr))\nDefinition: ArithmeticSequence.h:331\nv.tail(n)\nBottom-right corner of A of size m times n\nv(lastN(m), lastN(n))\nA.bottomRightCorner(m,n)\nBottom-right corner of A of size m times n\nv(lastN(m), lastN(n))\nA.bottomRightCorner(m,n)\nLast n columns taking 1 column over 3\nA(all, lastN(n,3))\n\n# Compile time size and increment\n\nIn terms of performance, Eigen and the compiler can take advantage of compile-time size and increment. To this end, you can enforce compile-time parameters using Eigen::fix<val>. Such compile-time value can be combined with the Eigen::last symbol:\n\nv(seq(last-fix<7>, last-fix<2>))\n\nIn this example Eigen knowns at compile-time that the returned expression has 6 elements. It is equivalent to:\n\nv(seqN(last-7, fix<6>))\n\nWe can revisit the even columns of A example as follows:\n\nA(all, seq(0,last,fix<2>))\n\n# Reverse order\n\nRow\/column indices can also be enumerated in decreasing order using a negative increment. For instance, one over two columns of A from the column 20 to 10:\n\nA(all, seq(20, 10, fix<-2>))\n\nThe last n rows starting from the last one:\n\nA(seqN(last, n, fix<-1>), all)\n\nYou can also use the ArithmeticSequence::reverse() method to reverse its order. The previous example can thus also be written as:\n\nA(lastN(n).reverse(), all)\n\n# Array of indices\n\nThe generic operator() can also takes as input an arbitrary list of row or column indices stored as either an ArrayXi, a std::vector<int>, std::array<int,N>, etc.\n\nExample:Output:\nstd::vector<int> ind{4,2,5,5,3};\nMatrixXi A = MatrixXi::Random(4,6);\ncout << \"Initial matrix A:\\n\" << A << \"\\n\\n\";\ncout << \"A(all,ind):\\n\" << A(all,ind) << \"\\n\\n\";\nstatic const RandomReturnType Random()\nDefinition: Random.h:113\nInitial matrix A:\n-10 1 4 7 4 -2\n-8 -6 9 -10 -10 4\n5 -10 -2 -9 -2 2\n-1 4 0 1 -9 9\n\nA(all,ind):\n4 4 -2 -2 7\n-10 9 4 4 -10\n-2 -2 2 2 -9\n-9 0 9 9 1\n\n\n\nYou can also directly pass a static array:\n\nExample:Output:\n#if EIGEN_HAS_STATIC_ARRAY_TEMPLATE\nMatrixXi A = MatrixXi::Random(4,6);\ncout << \"Initial matrix A:\\n\" << A << \"\\n\\n\";\ncout << \"A(all,{4,2,5,5,3}):\\n\" << A(all,{4,2,5,5,3}) << \"\\n\\n\";\n#endif\nInitial matrix A:\n-10 1 4 7 4 -2\n-8 -6 9 -10 -10 4\n5 -10 -2 -9 -2 2\n-1 4 0 1 -9 9\n\nA(all,{4,2,5,5,3}):\n4 4 -2 -2 7\n-10 9 4 4 -10\n-2 -2 2 2 -9\n-9 0 9 9 1\n\n\n\nor expressions:\n\nExample:Output:\nArrayXi ind(5); ind<<4,2,5,5,3;\nMatrixXi A = MatrixXi::Random(4,6);\ncout << \"Initial matrix A:\\n\" << A << \"\\n\\n\";\ncout << \"A(all,ind-1):\\n\" << A(all,ind-1) << \"\\n\\n\";\nInitial matrix A:\n-10 1 4 7 4 -2\n-8 -6 9 -10 -10 4\n5 -10 -2 -9 -2 2\n-1 4 0 1 -9 9\n\nA(all,ind-1):\n7 1 4 4 4\n-10 -6 -10 -10 9\n-9 -10 -2 -2 -2\n1 4 -9 -9 0\n\n\n\nWhen passing an object with a compile-time size such as Array4i, std::array<int,N>, or a static array, then the returned expression also exhibit compile-time dimensions.\n\n# Custom index list\n\nMore generally, operator() can accept as inputs any object ind of type T compatible with:\n\nIndex s = ind.size(); or Index s = size(ind);\ni = ind[i];\nEIGEN_DEFAULT_DENSE_INDEX_TYPE Index\nThe Index type as used for the API.\nDefinition: Meta.h:74\n\nThis means you can easily build your own fancy sequence generator and pass it to operator(). Here is an exemple enlarging a given matrix while padding the additional first rows and columns through repetition:\n\nExample:Output:\nIndex size() const { return out_size; }\nIndex operator[] (Index i) const { return std::max<Index>(0,i-(out_size-in_size)); }\nIndex in_size, out_size;\n};\nMatrix3i A;\nA.reshaped() = VectorXi::LinSpaced(9,1,9);\ncout << \"Initial matrix A:\\n\" << A << \"\\n\\n\";\nMatrixXi B(5,5);\nstatic EIGEN_DEPRECATED const RandomAccessLinSpacedReturnType LinSpaced(Sequential_t, Index size, const Scalar &low, const Scalar &high)\nDefinition: CwiseNullaryOp.h:244\nInitial matrix A:\n1 4 7\n2 5 8\n3 6 9\n\n1 1 1 4 7\n1 1 1 4 7\n1 1 1 4 7\n2 2 2 5 8\n3 3 3 6 9","date":"2021-08-04 07:33:12","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\": 0.28273439407348633, \"perplexity\": 6436.254697979778}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"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-2021-31\/segments\/1627046154796.71\/warc\/CC-MAIN-20210804045226-20210804075226-00644.warc.gz\"}"}
| null | null |
Beck Hansen (nom real Bek David Campbell, Los Angeles, Califòrnia, Estats Units, 8 de juliol de 1970) és un cantant i compositor dels Estats Units, conegut pel nom artístic de Beck. Amb la seva fusió d'estils musicals, lletres iròniques i arranjaments postmoderns amb caixes de ritmes, samplers, instruments en directe i efectes de so, Beck ha estat definit per la crítica i el públic durant la seva carrera com un dels artistes de rock alternatiu més personals.
Discografia
Àlbums
Golden Feelings (1992)
Mellow gold (1994)
Stereopathetic soulmanure (1994)
One Foot in the Grave (1994)
A Western Harvest Field by Moonlight (1995)
Odelay (1996)
Mutations (1998):
Midnite Vultures (1999)
Sea Change (2002)
Guero (2005)
Guerolito (2005)
The information (2006)
Modern guilt (2008)
Morning phase (2014)
Colors (2017)
Hyperspace (2019)
Recopilacions
Beck.com B-sides (2001)
Stray blues: a collection of B-sides (2001, només editat al Japó)
Senzills
Deadweight, de la pel·lícula A life less ordinary (1998)
Vídeos i DVDs
Southlander (2003)
Free Tibet (VHS: 1999 – DVD: 2000)
Condo painting (2000)
Guardons
Premis
1997: Grammy al millor àlbum de música alternativa
2000: Grammy al millor àlbum de música alternativa
2019: Grammy al millor àlbum de música alternativa
Nominiacions
2001: Grammy al millor àlbum de música alternativa
2003: Grammy al millor àlbum de música alternativa
2006: Grammy al millor àlbum de música alternativa
Referències
Enllaços externs
Cantautors californians
Guanyadors del premi Grammy
Banjistes estatunidencs
Músics de Los Angeles
|
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{"url":"https:\/\/www.physicsforums.com\/threads\/change-in-total-kinetic-energy-in-a-conservation-of-momentum-problem.92840\/","text":"# Change in total kinetic energy in a conservation of momentum problem?\n\n1. Oct 7, 2005\n\n### erik-the-red\n\nQuestion:\n\nOn a frictionless, horizontal air table, puck A (with mass $$m_1$$) is moving toward puck B (with mass $$m_2$$), that is initially at rest. After the collision, puck A has a velocity of $$v_1$$ to the left, and puck B has velocity $$v_2$$ to the right.\n\n1. What was the speed of puck A before the collision?\n\nMy answer to this part of the question was correct. It is $$(m_2*v_2)\/(m_1) - (v_1)$$.\n\n2. Calculate the change in the total kinetic energy of the system that occurs during the collision.\n\nI think this depends on the first part of the question.\n\nSo, I'm thinking:\n\n$$\\Delta K= K_f - K_i$$\n$$(1\/2)(m)(v_f^2 - v_i^2)$$\n\nI should sum up the velocities, right?\n\n$$(1\/2)(m_1+m_2)((v_2 - v_1)^2 - ((m_2*v_2)\/(m_1) - (v_1))^2)$$\n\nBut, that is not correct.\n\n2. Oct 7, 2005\n\n### mezarashi\n\nI'm not very clear with what was meant by your last equation. In anycase, you basically do:\n\nChange in kinetic energy = Kinetic energy of puckA before collision - Kinetic energy of puckA after collision - kinetic energy of puckB after collision.\n\nKE is a function of only mass and velocity. Since all are known, carefully plugging in the numbers should do the trick.\n\n3. Oct 7, 2005\n\n### Physics Monkey\n\nerik-the-red,\n\nWhen calculating kinetic energy, you first square the velocity of each particle, multiply by the particles mass divided by two and then add. You have added the final velocities first and then squared and this is wrong.\n\n4. Oct 7, 2005\n\n### erik-the-red\n\nI inputed $$(1\/2)(m_1)((m_2*v_2)\/(m_1) - (v_1))^2 - (1\/2)(m_1)(v_1)^2 - (1\/2)(m_2)(v_2)^2$$ and it was incorrect.\n\nMy understanding of change in total kinetic energy, $$\\Delta K$$, is the difference between the final and initial kinetic energies.\n\nPhysics Monkey, you told me to add. Why is this so?\n\n5. Oct 7, 2005\n\n### Staff: Mentor\n\n$\\Delta K = K_f - K_i$, not $K_i - K_f$.\n\n6. Oct 7, 2005\n\n### Staff: Mentor\n\nTry applying conservation of momentum and kinetic energy.\n\nhttp:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/elacol.html#4\n\nConsider the momentum of the masses to left and right. Remember momentum is proportional to velocity, so it is a vector quantity. Energy is a scalar.\n\n7. Oct 7, 2005\n\n### mezarashi\n\nFrom the looks of it, the problem now isn't with the physics, but rather the math. Try cleaning it up, do things step by step and plug in the numbers. Like what Physics Monkey said, you are doing some incorrect operations by trying to rush the calculation. One by one,\n\n$$KE_1 = \\frac{1}{2}m_1v_1^2$$\n\n8. Oct 10, 2005\n\n### erik-the-red\n\nIt looks like I definitely did $$K_i - K_f$$, which obviously is not correct.\n\nSo, I decided to analyze $$K_f$$ first.\n\nI got $$K_f = (1\/2)(m_1)(-v_1)^2 + (1\/2)(m_2)(v_2)^2$$.\n\nThen, I decided to analzye $$K_i$$.\n\nI got $$K_i = (1\/2)(m_1)((m_2*v_2)\/(m_1) - v_1)$$\nThis simplifies to $$(1\/2)(m_2*v_2 - m_1*v_1)$$.\n\nI took the difference $$K_f - K_i = (1\/2)(m_1)(-v_1)^2 + (1\/2)(m_2)(v_2)^2 - (1\/2)(m_2*v_2 - m_1*v_1)$$ and it was not correct.\n\nI really think I did everything right this time. ?\n\n9. Oct 10, 2005\n\n### Staff: Mentor\n\nOther than that (which gives the wrong sign) I don't see anything wrong with what you posted in #4.\n\nOK.\n\nYou forgot to square the speed.\n\nLast edited: Oct 10, 2005\n10. Oct 10, 2005\n\n### erik-the-red\n\nThanks so much, Doc Al. I wanted to simplify my answer, but I forgot that it's $$v^2$$.","date":"2017-02-26 08:04:15","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\": 0.7826882004737854, \"perplexity\": 1089.9841806126606}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-09\/segments\/1487501171936.32\/warc\/CC-MAIN-20170219104611-00027-ip-10-171-10-108.ec2.internal.warc.gz\"}"}
| null | null |
**New York Times _bestselling author Jill Shalvis brings readers book three in the classic series South Village Singles, where a group of friends makes a vow of singlehood in a vibrant suburb of Los Angeles._**
**_VOWING TO BE SINGLE IS EASY..._**
Taylor Wellington doesn't _do_ relationships. She learned a long time ago that they were a recipe for heartache. However, she does do flings—steamy affairs that burn hot...and brief. And her sexy new contractor, Thomas "Mac" Mackenzie, has her thinking about burning up some sheets. Problem is, now that she's got him right where she wants him, she can't seem to stop playing with fire.
**_UNTIL THIS SEXY BACHELOR COMES ALONG!_**
At the start, Mac is in perfect agreement with Taylor—they'll keep things hot and light, then walk away when they're done. But for reasons he can't explain he doesn't want to end this affair. In fact, he wants _more_. So now he'll just have to use some seductive persuasion to convince her that this is too good to let go...
Originally published in 2003.
Dear Reader,
Thank you SO MUCH for buying a Shalvis classic romance! These books might predate the digital age, but they're still fun and sexy! We hope you enjoy this peek at my earlier work!
Best wishes and happy reading!
Jill Shalvis
www.jillshalvis.com
## **Messing with Mac**
## **_Jill Shalvis_**
_CONTENTS_
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
Epilogue
**_CHAPTER 1_**
**ONE OF THESE DAYS,** Taylor Wellington figured she'd be old, maybe even wrinkled, and then, finally then, her best friends would stop trying to convince her she needed love.
No one needed love.
Having been both with it and without it—mostly without it—she knew this for a cold, hard fact. Still, Taylor held the cell phone to her ear and let Nicole and Suzanne, via three-way conferencing, ramble on about how amazing the _L-_ word was.
"You've got to try it." This from Nicole, who'd been swept off her feet a few months back by Ty Patrick O'Grady, Taylor's rebel Irish architect.
"It's even better than ice cream," Suzanne promised, and coming from Suzanne, this was quite the promise, but she'd recently fallen in love, too, and had even gone one step further and gotten married. "Come on, Taylor, give up on singlehood and try a man on for size. It'll change your life."
Taylor wasn't buying it. Not one little bit. In her opinion—and she had very strong opinions, thank you very much—love sucked. Always had, always would.
She was speaking from firsthand experience and hard-earned knowledge, not that her friends would understand. They wouldn't because she hadn't explained, she hadn't known how to in the short time they'd been together, which had begun when, in order to keep up with life's little luxuries like eating, Taylor had rented out two apartments in the building she'd just inherited. Suzanne had come first, then later Nicole, and both had happily joined her in a solemn vow of singlehood.
Only they'd each caved like cheap suitcases in the face of true love, and had both recently moved out again, having found their soul mates.
"Just because you two willingly gave up your freedom doesn't mean I have to—" Taylor stopped at an odd noise and cocked an ear. "Hang on a sec."
The building, _her_ building, shuddered. Not surprising really, as she considered it an amazing feat the entire thing hadn't fallen down long ago, but in Taylor's world, things didn't happen off schedule. Her building crashing to the ground definitely wasn't on her schedule for today.
And yet there it went again. Another shudder. And then again. Something was systematically banging, in tune with her growing headache. "Guys, much as I'd love to listen to you tell me what's wrong with my life in singular excruciating detail, I have to run."
"Hold up. Is that more construction I hear?" Suzanne asked casually. _Too_ casually.
The question didn't fool Taylor. Both Suzanne and Nicole had found their happiness due to construction. _Her_ construction.
Now they had equally high hopes for her.
They were going to be disappointed, as Taylor didn't intend to fall for anyone. Feeling like a heel, she pulled the cell phone away from her ear and simulated a static sound with her mouth. It wasn't a kind thing to do to the only two people in the world who truly cared about her, but all this talk of love, no matter how well-meaning, was making her perspire.
And a Wellington never perspired, especially in silk. That was one thing she'd learned from her mother. "Gotta go, bad connection!" she yelled into the phone and disconnected.
Damn it. She loved Suzanne and Nicole, loved them like the sisters she'd always wished for instead of the two she had, but any more talk of love as it pertained to her and she risked losing her wits, something she couldn't afford at the moment, as she needed each and every available wit to keep her sane.
Oh, and in the black. Her every thought these days seemed to focus on finding enough money to pay for the work that needed to be done. That alone was enough to give her insomnia. This was a real kicker of an inheritance from her grandfather—this falling-off-its-foundation building she stood in, and not a single penny to go with it. No trust fund, no cushy little savings account, nothing.
After a lifetime of paying for all her fancy education and everything else, the distant, cruel bastard had cut her off cold turkey, giving all of his substantial wealth to her mother, who hadn't seen fit to share.
The woman wouldn't, not when all her life she'd been so cheap, so tight with money, she squeaked when she walked.
Well, tough. Taylor wouldn't wallow over that, or the fact that her family—called such only because they shared the same bloodlines—probably wouldn't notice if she succeeded, but would most definitely notice if she failed. And she wouldn't think about the fact that she only had to sell this place and walk away if she chose, because sheer stubborn pride refused to allow her to walk away from the first real challenge in her life.
She would do this. She would take this place and make something of it. And of herself. She'd started months ago, one room at a time, but had decided to sell several of her precious antiques—which had been worth more than she'd imagined—using the opportunity to renovate all of it in one fell swoop.
Starting tomorrow.
Hard as it would be to maintain her notorious cool, maintain it she would. With a nod of determination, she slipped the phone into her pocket and narrowed her eyes at the walls, which were still quivering from the rhythmic blows.
Oh yes, she was quite certain she'd agreed with her new contractor that he could start tomorrow.
Not today.
And if there was one thing Taylor didn't appreciate, it was someone messing with her carefully laid plans. She needed today, her last day alone, her last day to buck up, thrust out her chin and get ready to show the world what she was made of.
Her building had been built circa 1902, and looked liked it. The Victorian style had nooks and crannies everywhere, windows galore and all the old charm and personality from the turn of the previous century, but with a hundred years of neglect added in. To say it was falling apart was the understatement of the new millennium. Bad trim, bad siding, bad paint, bad electrical and never mind the termites and last year's flood damage from a busted pipe.
The bottom floor had two storefront units. The top floor had one loft apartment and an attic compartment. The middle floor had two apartments, one of which she'd claimed. Shutting the door of her apartment now, she headed downstairs, toward the hideous banging.
Outside, the streets of South Village were gearing up for what promised to be another profitable day.
Los Angeles, only five miles away, had been kind enough to share its smog and muggy heat, but Taylor didn't mind the summer months like so many others did. She loved it here, felt perfectly at home among the young, hip, urban crowd which was drawn to Southern California's premier pedestrian neighborhood. And why not, when any day of the week one could walk to a theater, an outdoor café run by someone famous or simply stroll through a mecca of interesting galleries or shops.
Taylor was counting on that crowd, as someday soon her two storefronts would be ready for lease.
Suzanne was taking one of them for her catering business, she'd already committed to that. A relief.
But there was still the other one. Leasing it out would keep her bank account happier than it was at the moment. But the truth was, she'd held out a little tiny seed of hope that someday she could use it for herself, opening her own shop. That is, if she had any antiques left after using them to finance the renovation.
A definite pipe dream at the moment.
The banging sounded louder now, and was definitely coming from one of the dusty, dirty storefront units. Outside, from beyond the front gate, she could hear people walking by, talking, laughing. Shopping. Once upon a time, that had been her favorite pastime, shopping, and a silly part of her suddenly yearned to be out there.
But that, too, was for another day.
As she reached the left unit, the banging increased in intensity. Opening the hallway door, which led into the back, she was greeted with a thick cloud of dust. The banging was so loud now she could hardly hear herself think, but as she stepped inside, the noise abruptly stopped.
Stunned by the silence, Taylor inhaled dirt in the already hot, muggy, spring California morning, and wondered how long before her carefully curled hair, flowing in a purposely artful and loose manner beneath her straw hat, sagged into her face.
"You're in my way," said a low, gruff voice from behind her.
Whirling, Taylor blinked into the cloud of dust as it slowly settled. Standing there among the dirt and grime was a man. He had one long arm propped on his hip, the other holding a huge sledgehammer, which rested against his shoulder.
_Paul Bunyan,_ came the inane thought, if one substituted the sledgehammer for an ax. But why was Paul Bunyan standing in her building? Confused, a rare occurrence for Taylor, she found herself momentarily speechless.
Another rare occurrence.
The dust started to settle, and Paul materialized into her contractor Thomas Mackenzie, and though most of their contact had been handled by e-mail and telephone, she _had_ seen him before. Clean and dressed up, that is. He wasn't clean or dressed up now.
At least four inches taller than her own willowy five-foot-ten frame, she found it a bit of a surprise to have to tip her head back to see his face. The last time she'd seen him, they'd sat at her table, and for the life of her, she didn't remember him being so...tall, so built, so imposing.
His mouth was scowling. His eyes were the color of expensive whiskey, two liquid, shining pools of heat and annoyance, and his hair, an exact match to his eyes, fell over a blue bandanna which had been tied around his forehead. Combined with his unsmiling, and rough and tumble expression, he looked more than just a little dangerous.
At the thought, a completely inappropriate shiver of thrill raced down her spine. Now was not the greatest time to remember that while she'd vowed to remain single for the rest of her life, she'd never vowed to remain celibate. She had a great appreciation for all things beautiful and finely made. And this man—tall and edgy and frowning as he was—was beautifully and firmly made, a magnificent male specimen, one who seemed to awaken every hormone and nerve ending in her entire body.
But she most definitely did not have a thing for a rebel-at-heart, and it didn't escape her that this man was one-hundred-percent pure attitude.
In light of that, she repeated the same thing she told herself at estate sales, when she saw some spectacular piece of furniture she quivered to own but couldn't afford... _Walk away. Just walk away._ Repeating that mantra, she took a careful step backward, taking one last glimpse to tide her over.
Hard, powerful looking legs were encased in soft, faded denim. His work boots were well worn, with a sole made for the long haul. She'd already noticed his very capable arms and his chest, which was wide, hard and covered in a T-shirt that clung like a second skin to his damp body. He was long and lean, rugged and virile, the way she preferred a man, when she chose to be with one.
But she wasn't choosing now.
"You're still in my way," he said.
"Good morning to you, too, Mr. Mackenzie."
He blew out a breath. "Mac."
"What?"
"You can call me Mac. That's my name."
"Really? It's not Mr. Attitude?"
His lips twitched. "I respond better to Mac."
"Okay, then. Mac."
He stood there politely enough, and...waited for something. At his raised brow, she realized he was waiting for her to leave.
Too bad he didn't know her better, or he'd already know she did only as she pleased, not as expected. "I didn't approve for the demo to begin today," she said.
"You signed the contract."
Yes, she had. She'd sold her beloved Queen Anne headboard to give him the first payment of many, but she'd agreed upon tomorrow. Damn it, she needed today.
Apparently deciding they were done, Mac turned and walked away, moving with the easy, loose-limbed stride of a man who knew the value of patience. With that patience, he hoisted up the sledgehammer and brought it down on the south wall. And then again. His arms strained and stretched, his muscles working in perfect synch, taut and sleek with sweat as he completely ignored her while simultaneously stripping down the wall to the framing.
Unable to help herself, she stared, utterly fascinated by the unrestrained violence of what he was doing. By the hone of that well-built machine that was his body. "Um...excuse me?"
The sledgehammer continued to rise and fall with amazing regularity. What kind of strength did that entail, she wondered, watching with utter fascination as Mac's muscles flexed and flowed. Another shiver wracked her frame, and it had nothing to do with a chill. The room was hot. He was hot...and so, suddenly, was she.
Definitely, it had been too long since she'd had any sort of physical release besides her handy, dandy, trusty vibrator. "Mac?"
He never even looked at her, which was a bit disconcerting. Taylor had matured at an early age, her long, gangly body turning into a man's wet dream. In all the years since, she'd never failed to turn a head.
And yet she was being completely ignored now. Vexing. So was the cell phone ringing in her pocket. Pulling it out, she put it to one ear, finger in the other to hear over Mac, and yelled, _"Hello?"_
"I have bad news," said Mrs. Cabot, the owner of a very upscale antique shop in town.
"Bad news?"
Sledgehammer raised, Mac turned.
Their gazes locked.
It was like a chemical reaction. Unintended. Unavoidable. He had the most amazing eyes, and for the first time in her life, Taylor lost her place in a conversation. Chewing her lower lip, she wracked her brain for working brain cells, but her pulse tripled when Mac's gaze dropped from hers, and locked on the movement of her mouth.
This wasn't happening. He wasn't attracted to her. She wasn't attracted to him. That would be bad, very bad, but while she'd promised herself to never again engage her heart after the devastating loss she'd once suffered, she was no monk.
But even so, sex had become a very fond, distant memory.
She licked her lips, a nervous habit. Again, her contractor's gaze flickered downward, becoming hot, focused and filled with frank sexual curiosity.
Oh boy. With sheer will power, she concentrated on her phone conversation. "What's the bad news?"
Mac set the sledgehammer on the floor. In deference to her call? No, that would mean he had a considerate streak.
He was probably just done.
"I'm sorry," Mrs. Cabot said. "But you lost your bid on that nineteenth-century chandelier."
Instantly forgetting about Mac, she gripped the phone. "What do you mean? Who else bid on the chandelier?"
"You were outbid by..." Papers rustled. "Isabel W. Craftsman."
Taylor might have guessed. There was only one person in town who would have coveted that piece as much as she had, and that was her own mother.
It only had been Taylor's greatest heart's desire to own it, but hey, she figured her mother knew that, too. Her mother was highly educated, incredibly brilliant and had eyes in the back of her head. Bottom line, she knew everything, she always had.
Well, except how to be a mother. Shocking how she'd screwed that up, but maybe Taylor was partly to blame. She'd always resented her mother's vicious drive, sharp ambition and ability to multitask everything in her world except when it came to her own daughters.
When Taylor had graduated from college and had moved out of the house, she'd decided to be the grown-up and let it all go. She'd told her mother so, saying she'd forgiven her for all the missed events, the forgotten birthdays, the lack of any physical attention whatsoever. She didn't know what she expected, but it hadn't been to be cut off by her mother's cell phone. Her mother had held up a hand to Taylor, answered the call, dealt with some business problem, then absently kissed the air somewhere near Taylor's cheek and walked away.
Having completely forgotten they were in the middle of an important conversation.
After standing there in seething resentment, Taylor had shrugged and moved on. She'd had to. Not every mother was cut out to be a warm, fuzzy type, and she needed to get over it.
Then a few years ago Isabel had done the unthinkable, she'd gotten married again, and had dropped everything for one equally ambitious, equally cold-blooded Dr. Edward Craftsman, brain surgeon. Taylor had gone to the wedding, and if she hadn't seen it with her own eyes, she would never have believed it.
Her mother lived for this man, gushing all over him. Constantly. Kissing, hugging, leaning, more kissing.
It burned just thinking about it. So did her mother buying this chandelier from beneath her. "Thank you," Taylor said into the phone. And as if it were no skin off her nose, she dropped the phone back into her pocket. Damn it. Damn it, damn it, _damn it._ She'd wanted that chandelier with a ridiculous passion. Served her right, wanting something so badly. Hadn't she learned that nothing, nothing at all, was worth the heartache?
She had other things to worry about. Like she had a building in disrepair, and a man was reminding her of things far better forgotten.
Mac had tossed the sledgehammer aside, but he hadn't been idle. There was now a shovel in his hand and he was loading debris into a wheelbarrow with the same narrow-minded intensity he'd had when swinging his sledgehammer.
Eyes narrowed, she set her hands on her hips and tapped her foot. "We never solved the problem of why you're here a day early."
He kept loading until the wheelbarrow was full to bursting. Slowly he straightened, then eyed her with that light brown gaze, completely inscrutable now, without a trace of that intense sexual speculation.
Had she only imagined it?
"I didn't think twenty-four hours would make any difference to you," he said. Tossing the shovel aside, he grasped the handles of the wheelbarrow and lifted. Muscles strained. Tendons corded.
Taylor tore her gaze away. "I needed this last day before the hell of the next three months of construction and renovation. You've ruined it."
He swiped a forearm across his forehead, looking tired, sweaty and temperamental. "I think that phone call ruined it."
Deep within her, a pesky lone hormone quivered. "I'd really like you to go and come back tomorrow."
That got his attention. "You're kidding, right?"
"No."
"You need to be alone bad enough to disrupt the start of your own renovation?"
"I do, yes."
"Fine." Dropping the wheelbarrow, he propped his hands on his hips. "Have your way, Princess. Tomorrow it is, but don't even think about pulling this again. I'm not going to postpone this job further, no matter what kind of day you're having."
Princess? Had he just called her _Princess?_ She'd show him princess! Reaching up, she yanked off her wide-brimmed hat, which once upon a time had cost her—make that her grandfather—a bundle. She'd die before explaining that her fair skin required she protect it from the harsh summer sun, especially since he seemed like a man to mock such a weakness. "Tomorrow will be just fine," she said through her teeth, hat in her fist.
Mac stretched his shoulders, which put a strain on his T-shirt, not that she was noticing, and rubbed his eyes. "Good. I'm outta here. But since I am, and since steam is still coming out your ears, why don't you do both of us a favor." Retrieving the sledgehammer, he held it out. "Start pounding walls. Consider it anger management."
She stared down at the tool, having never in her life so much as lifted a screwdriver. She might have blamed her uptight, pretentious family for that, though she'd been on her own for a while now, and could have made the effort to learn such things.
Should have, because it would feel good to swing the thing with authority and knowledge, surprising that smirk off his face.
He wriggled the sledgehammer enticingly.
Odd how a little part of her tingled to touch it, hoist it over her head and let loose. Barbaric, yes, and suddenly very appealing.
"You know you want to," Mac said in a low, husky dare. "Touch it."
She cocked a brow and looked at him from beneath lowered lashes. "So...are they all the same size?"
His eyes sparked, heated and flamed.
And one question was answered...she had most definitely _not_ imagined that intense sexual speculation.
"I thought size didn't matter to a woman."
She lifted a shoulder. "That's just the story some woman started in order to appease her poor husband who didn't have...the right equipment."
"Hmm." He lifted the sledgehammer again, his eyes amused now. "The right equipment, huh?"
"That's right."
He looked at the sledgehammer with a new light, then back into her eyes. "Seeing as I have the right equipment, are you going to go for it?"
Oh yeah, she was. For the sledgehammer anyway.
What could it hurt? She had aggression coming out her ears; for her grandfather, who was probably sitting on a cloud laughing down at her right this very minute, for her mother, who would rather do anything than be a mother, for her dwindling bank account, for the chandelier she'd lost out on...for being alone in all this.
For just about every damn thing in her entire life, she needed that sledgehammer.
Mac held it out.
Her fingers itched.
His eyes sizzled with the dare, and a potent, heady male heat.
"Fine." She set her hat back on her head, snatched the tool from him, then swore in a very unladylike way as the thing jerked both her arms down with its weight, slamming the heavy sledgehammer to the floor.
Mac tsked. "Sorry, I thought you were stronger than that."
**_CHAPTER 2_**
**TAYLOR'S ACCUSING EYES** speared Mac, and he had to bite back his grin as he lifted an innocent shoulder.
She let out a rude sound, and with determination and aggression blaring out her eyes, she hoisted the sledgehammer up...and nearly fell to her very finely dressed ass. Stumbling back a step, she spread her legs out a little for balance, then sent him a triumphant smile.
It stopped his heart.
Funny, that, since he'd have sworn the organ in question had dried up from abuse and misuse.
Taylor turned her back on him and with all her might, swung the sledgehammer into the wall. When drywall fell and dust rose, she let out a cocky laugh, whirling back to make sure he was looking.
Oh, he was looking. He'd been looking since she'd first sauntered into the room, just as he had a feeling men always looked at Taylor Wellington.
He'd bet his last dollar that she knew it as well. She was a pricey number, all fancy labels and perfect grooming. Stunning, too, with her blond hair, see-through green eyes and a body meant to bring a grown man right to his knees. She had long, willowy curves, outlined in mouthwatering detail beneath the silky sundress that made his hands itch to mess her up. It was crazy, but he had the most inane urge to toss off her hat, sink his fingers into her hair and shake a little, to eat off her carefully applied lipstick that smelled like peaches and cream, to run his hands over that cobalt silk and see if she looked as good undone as she looked done.
But he recognized a spoiled socialite when he saw one. Oh yeah, he did. He'd been there, bought the T-shirt, and because of it, he wasn't tempted.
Well, maybe a little tempted, but he wasn't an idiot. She was upset because of some silly little bid she'd lost for a damn light fixture, when Mac had his entire future riding on a bid as well. A bid with South Village's town council to get in on the area's renovation and preservation acts. South Village wasn't some prefab pedestrian neighborhood like Universal's City Walk, but a genuine historical district in the middle of extensive restoration. He had bid on several of the upcoming jobs that would hopefully set up his business and reputation. Now _that_ was something to get a little excited over, and he was trying not to think about how badly he wanted to be awarded those bids.
Taylor lifted the sledgehammer again, and with all her might, gave it everything she had. Not a strand of hair fell out of place beneath her hat, and nary a wrinkle appeared on those fine clothes. More interesting, he sensed she wasn't just humoring him here, but was genuinely striving to work off steam. Her mouth was grim, her eyes quite focused on the task, as if she was imagining someone's face right where the sledgehammer fell.
It shocked him, the barely restrained violence pouring out of her, but what really shocked him was how arousing it was to watch her go at it. With every swing, her perfect, palm-sized breasts jiggled, her hips wriggled, her ass shimmied and shook.
And damn, but he couldn't tear his eyes away. "Remind me to never piss you off," he said, and she let out a rough sound of agreement as she swung again.
She was going to get blisters if she kept it up, which she appeared to intend to do. He hadn't expected her to be able to lift the sledgehammer, much less swing it. "Uh...Princess? Don't you think that's probably enough?"
Ignoring him, she swung again, but it took a huge effort.
Figuring she had to be nearing exhaustion, he shifted closer, thinking he should grab the sledgehammer before she hurt herself. That's all he needed, was to maim the boss before she paid him.
Blocking him with an elbow, she growled, "Back off."
Torn between annoyance and amusement, he did. "Okay, maybe I was wrong, maybe anger management classes would have been more effective for you."
"No." Heave. Smash. Heave. Smash. "You were right, this is good. And..." Heave. Smash. "Cheap, too."
She paused, gasping for breath.
"You could always just ask Daddy for more money," he suggested.
She went utterly still. Then carefully and purposely set down the sledgehammer before turning to him, eyes suddenly cold as ice. "You know, I think I'm finished after all. Thank you," she added politely, and then cool as he pleased, walked past him and quietly shut the door behind her.
Shaking his head, he let out a low whistle. Classy down to the last millimeter, when what she'd obviously wanted was to tear into his hide. Still in that state of amused annoyance, he let himself out of the unit as well, figuring he'd give in on this, her need to have the rest of the day to herself.
Only because it suited him.
Mac got into his truck and drove east. He didn't live in the high-class, high-rent district of South Village. Nor with the middle class in their gated condo developments and upscale houses that all mirrored each other. He didn't live with the wannabes on the outskirts either.
He lived exactly where he wanted to, and damn expectations. He lived in the area known as The Tracks, which before the Town Council and Historical Society had gotten a hold of it meant that he lived on the _wrong_ side of the tracks.
He appreciated the irony of it.
In ten minutes he was walking into his own little house, _little_ being the key word here. The first thing he did was toss his mail—unread—on the table, where it knocked over the existing pile of unpaid bills.
Didn't matter. No matter how big that stack got, he was still free. Free of his family's obligations, well-meaning but smothering nonetheless. Free of his ex-wife—whom he had to thank for all of those unpaid bills.
He'd refused to let her live off his very generous family and their money, refused to make her the socialite she wanted to be. As a result, she'd taken everything he owned and then some before purposely and completely destroying him in the only way she could.
By aborting his child.
But he wasn't going there, not tonight. He stripped, hunted up a pair of beat-up old shorts and headed back out for his own anger management class.
A long, punishing run.
* * *
AT THE CRACK OF DAWN the next morning, Mac drove back to Taylor's building. He had a soft spot for this hour, before the sun had fully risen on the horizon, as no one had yet screwed up his day.
Today he'd have a crew working on the demolition, tearing out drywall down to the wood studs, then stripping old electrical and plumbing lines. Yesterday had been just for him, a way to burn off some accumulated steam. And he'd had plenty of it. There'd been that call from his mother, who in spite of her own life and full-time, very demanding job, was warm and loving and more than a little certain he was wasting away without her home cooking, and when was he going to come home for a Sunday meal?
Then had come the call from his old captain, wanting him back on the police force, which he'd left at the same time as his divorce four years ago. Much as Mac had loved being a cop, he loved rebuilding and renovating more, and always had. He'd been building things, working with his hands, ever since he could remember, and his love of doing so hadn't changed.
But his purpose had. Life was too damn short, as he'd learned the hard way, and he intended to spend the rest of it doing what he loved. And what he loved was taking old, decrepit, run-down historical buildings and restoring them to their former glory. He'd been doing just that since getting off the force and had never looked back. He'd started out working for a friend of the family, learning the trade. For two years now, he'd been on his own doing mostly single rooms within existing buildings until this last year when he'd taken on whole buildings for the first time.
He'd found his calling. Taylor was his biggest client to date, his biggest job and his stepping-stone to the next level.
He hoped. Thanks to Ariel, who'd dragged him through the coals financially, morally and every other way possible, he couldn't afford to renovate his own place, not yet. Fine. He'd do it for someone else and work his way up. He had no problem with that.
And with that single-mindedness, he parked right in front of Taylor's building—a miracle given the deplorable parking in South Village—and fervently hoped she'd made herself scarce. He had a crew to think about, and he wanted their minds on work, not on a beautiful woman, no matter how good she'd looked swinging a sledgehammer in all her finery.
His crew was waiting for him, just standing on the front steps, which made no sense. They knew better than to stand around wasting time.
But they weren't just standing, they were smiling and nodding like little puppets to...surprise, surprise...Taylor.
"It came from Russia," she was saying, holding up some sort of vase as he strode up the walk, annoyance already starting to simmer.
Taylor stroked perfectly manicured fingers over the smooth, porcelain surface of the vase as she talked, caressing the thing like she would a lover, and Mac's blood began to beat thick, and not with just annoyance now. An ache, purely sexual, began to spread through his belly.
Which proved it, he was insane.
"It's worth a small fortune," Taylor said, seeming lost in the delicate etching on the vase, sighing over the beauty of it as she touched.
The sound of her soft sigh didn't help Mac's inner ache, and he spent a moment brooding over the fact he hadn't been with a woman in a good long while. He hadn't wanted to, not since Ariel and her cruel betrayal.
But not having a sexual urge wasn't the same as ignoring one. He looked at the vase in Taylor's hands and concentrated on her words.
_Worth a fortune,_ she'd said.
Enough to cover the wasted labor for however long she stood there occupying his men's every thoughts?
But what did she care how much money he lost in wages unearned? Mac wasn't exactly sure what had happened yesterday, why she'd momentarily drawn him, given who and what she was—that being a woman too close to Ariel's type to make him comfortable. But whatever it was, whatever little spark or electrical current of attraction he'd felt in spite of himself, he wouldn't feel again.
She wore a pair of pale blue capris with a matching short, little cropped jacket, looking like she should be getting ready to walk down the runway instead of standing on the step of her ugly building.
Her hair was pulled back into a careful twist and she wore more of that peach lip-gloss from yesterday.
She was a long, cool drink of water, and even knowing it, even having prepared himself to see her again, he was suddenly dying of thirst, and couldn't seem to tear his eyes off her.
When she saw him, she stopped talking, rubbing her lips together in a little gesture that signified either nerves or arousal. Either way, awareness shot straight to his groin.
So much for ignoring her. "Why are you here?" he asked.
She lifted a brow, assuring him and everyone around that she considered him a Neanderthal for asking such a question. And okay, yes, maybe his tone had been a bit brusque. After all, she did own the place. But there was some inexplicable... _thing_ going on between the two of them, some amazing thing that reminded him of...a shark bite. Painful, and probably lethal.
But they'd signed a contract, he and she. Every possible little detail had been decided on, down to the last shade of paint on the walls. Her presence here wasn't required, and in fact, he knew the ratio of work done today would be directly related to how far away she was.
The farther the better. "You agreed to move out for the duration of the restoration," he reminded her.
"I agreed to make sure there were no tenants during the duration. Suzanne and Nicole are gone."
"But you're not."
"I'm not a tenant."
Shaking his head, he took the last step that put him on even ground with her. Mostly he towered over everyone around him, and knowing it, he usually made a conscious effort not to use his size as an intimidation. But right now he wasn't thinking intimidation so much as self-preservation. He wanted this job. He needed this job. It was the first thing he'd cared about in far too long. And in a way he was just beginning to understand, he needed to lose himself in the pure joy of the work itself, something he couldn't do with her parading around all damn day.
"You can't mean to be here while we work."
She lifted that chin, eyes flashing. "I'll do as I please."
Damn. She did, she meant to be here while they worked. Because she didn't trust him, or because she wanted to drive him crazy every step of the way?
"Why?"
"I won't be in your way," she said in lieu of a real answer.
In his experience, clients couldn't help but be in the way, always wanting to change the logical order of things, waiting until paint was on the walls or tile on the floor before deciding the color was off, or the brand not quite right. And he had the Town Council and historical society to impress on this one. "Look, Princess—"
"My name," she said, still smiling that cool smile as she carefully shifted the vase from one hand to the other in a way that suggested she was considering smashing it over his head, "is not 'Princess."'
He pinched the bridge of his nose. "I'm not trying to be a hard-ass here, it's just that we'd all be better off if you'd just let us do our jobs."
"You _are_ a hard-ass, it's one of the reasons I hired you," she said, surprising him. "And I think you could try trusting me a little. I'm not going to bog you down."
Mac didn't do trust, and even if he did, he'd be crazy to give in to a woman quite used to crooking her little finger and having the entire male population fall over its own feet to please her.
"I'm not," she repeated, a little softer now, watching him with those clear, clear eyes that weren't going to give an inch.
He ran his hands over his face, put them on his hips and stared at her, but she was still just waiting with what he figured was the patience of a cobra. "Okay, whatever."
She was wise enough to keep her smile to herself but he saw the triumph in her eyes, the eyes that only yesterday had turned him on.
_Still_ turned him on.
"You'll finish the demo downstairs this week?" she asked.
"And upstairs."
"Oh." Now something else flickered in her gaze. "Is it really necessary to push your men like that?"
"Like...what?"
"Well, I would think demolishing just the downstairs would be enough for the next week. In any case, it's going to be awfully hot."
"We're doing both up and down this week," he said firmly.
"Hmm."
The sound that escaped her throat suggested he was not only a hard-ass but a brutal boss to his crew. "Demolition is backbreaking, hot, filthy work," he explained, trying not to resent having to do so.
"I realize that."
"Then you also realize we're far better off digging in and getting it over with quick as possible."
"Okay...well, maybe you guys can start and complete the entire renovation _downstairs_ before moving to the next floor."
"No. Not cost-effective."
"Hmm," she said again doubtfully, and he narrowed his eyes. Why didn't she want them upstairs this week? He would have pushed for answers but each of his crew's heads were whipping back and forth between the two of them as if they were watching a tennis match.
He was not going to make a scene. The woman wanted to breathe down his neck all day long? Fine. Today was going to be particularly brutal. By the end of it, her hair would be in her face, her creamy skin smeared with dirt and no way was that million-dollar linen going to make it through unscathed.
She'd be, at the very least, hot, sweaty and rumpled, and he could only hope he would get that insane urge to see it right out of his system.
"Let's move it," he said to his crew, and they scattered.
**_CHAPTER 3_**
**FOR SEVERAL DAYS,** Taylor kept close tabs on the demolition, from a safe distance of course. She wasn't stupid enough to rile the beast any further, though she had to admit, she had been able to rile him with little to no effort so far.
She supposed that meant he felt the same irritating physical attraction she did. And it _was_ purely physical. A man as alpha as Mac was only good for the physical. There was nothing sensitive, tender or gentle about a man like that, nothing.
He wasn't someone to fool around with. He'd swallow her whole and spit her right back out, and in her world, _she_ was the one who did the spitting, thank you very much.
What she needed, if she needed at all, was a far more beta man to have fun with, to walk all over, if that's what she was looking for.
And maybe she would. Later. Right now she had bigger problems, such as figuring out how to keep her contractor from learning she wasn't just going to be casually around, she was still living here.
Not because she didn't trust him, as he figured, but because she didn't have the money to move out and get another place. Every cent she had was sunk into this building and the renovations. Until she could get more tenants—something else she was dependent on her contractor for—she was pretty much stuck.
Suzanne and Nicole had each offered her a place to stay. But Nicole lived in Ty's house now, and Suzanne with Ryan. Both were deliciously, deliriously drunk on true love. She knew the feeling, oh yes, she knew, but she couldn't watch it or witness it too closely. She just couldn't.
She figured she'd just stay here, quietly, out of the way.
Undetected.
But that would be tricky, because now she knew the truth: that very little, and quite possibly nothing, got past one Thomas Mackenzie.
"You want to move, Princess, or you'll feel the effects of this dust in two seconds flat."
Having come out of nowhere, the tall, moody, opinionated man in question stood at the bottom of the stairs, looking up at her. She leaned against the railing on the second-floor landing just outside her apartment, the one he didn't realize she still slept in.
He wore a hard hat, protective goggles and a face mask, which he'd shoved off his mouth, and was now hanging around his neck. He also wore a fine layer of dust that clung to his damp body. So did his dark T-shirt, which she was quite certain shouldn't make her pulse quicken. He seemed so huge, so powerful and virile standing there with his sledgehammer in hand as he stared up at her from those whiskey eyes. And ridiculous as it was, she quivered like a mare in heat. It was shockingly, amazingly juvenile, and if she'd known how it was going to be, she'd have found another man for the job.
No, scratch that, difficult as he was, she wouldn't want to work with anyone else. He was abrupt, insensitive and far too hardheaded, but he was a damn good contractor and he was honest to a fault.
Honest or otherwise, he slowly climbed the stairs, holding her gaze in his, until he stood right before her, all but surrounding her with his size and strength in what she considered was a deliberate attempt to establish his dominance.
Well, she was dominant, too, and she lifted her chin and stared him down.
"You're not moving out of the dust," he said.
She wouldn't back up, not even one little step, though he was close enough now that she could feel the heat of his body, could see the look in his light brown eyes, and it was a very confident, cocksure look.
Even her heartbeat responded to his nearness, quickening, causing a glowing, growing heat within her body. Combined with the almost frantic awareness humming through her every nerve ending, she felt like a bomb waiting to go off.
No. She couldn't be attracted to him, he wasn't what she wanted in a man. He wasn't quiet, easygoing. He wasn't laid-back. And he certainly wouldn't let her walk all over him.
Damn, but it had been a long time since a man had gotten to her like this, really gotten to her. And to be fair, Jeff Hathaway had been more boy than man.
They'd met in second grade. Jeff had slugged Tony Villa for calling her a Jolly Green Giant when she'd worn a green dress and green tights with matching green patent leather shoes, and even back then Taylor's heart had sighed.
In sixth grade Jeff held her hand at lunch break, not caring who saw, and her heart had sighed again and again.
By high school, they'd been soul mates. She'd known he was the one, no matter that he came from what her mother had called an undesirable family. Jeff _was_ her family.
They'd wanted to get married right out of high school but she hadn't turned eighteen and her mother wouldn't give her permission. So they plotted away the summer, talking about college, where they'd room together and then sneak off to Vegas when she turned eighteen in October.
By that time, Jeff had been her best friend, her lover, her future husband and her entire life.
And on the last day of September, he'd been killed in a car accident.
Those days immediately following, and even several years after, didn't bear thinking about. But always having been strong of heart, Taylor did eventually heal. She even moved on, and dated a little in her early twenties, when fast, fun and reckless were infinitely preferable to deep and emotional.
Even now at twenty-seven, she felt perfectly normal, but a part of her was missing. The best part. Jeff.
God, she'd loved him. Oh, she could still function, could even care about a man. She could laugh and learn and do all the things she'd done before.
But one thing had irrevocably changed. Now when she let a man in, it was simply to soothe a need, whether it be wanting to be held against his hard body, or merely needing a sexual release that didn't come from something battery operated.
Nothing more, nothing less, as even now, nearly ten years later, she couldn't imagine going through that soul-destroying love ever again.
"Princess?"
How could she have forgotten the very unforgettable man looking at her? The one man since Jeff she'd actually found herself yearning for.
_There._ The admission was out in the open, not that she was going to do anything about it. He was not, repeat _not_ , her type. "I'm not allergic to a bit of dust."
"You haven't been breathing it in. Continue to stand there while we demo the hallway and your lungs will be burning within half an hour. Not to mention the pounding sinus headache that accompanies it."
Was that concern? If so, it didn't bear thinking about, as it might soften her toward him. And given her body's response to his without letting her brain get into the mix, that would be just plain stupid. And dangerous.
"Thanks for the concern," she said sweetly, and turned away. She entered her apartment, stripped now of all furnishings and personal belongings except for the bedroom. Everything had been taken to her storage unit, where she also kept her precious antiques.
But here, in her private sanctuary, her bedroom, she still had her huge four-poster bed and the luxurious beddings left over from the good old days before the end of her bottomless—and now nearly extinct—bank account.
She wasn't upset she had to make her own way in the world. In fact, it was a challenge she appreciated.
What she resented was how it had happened, so abruptly, even cruelly, without a thought to her feelings.
Saying that her family wasn't close was something of an understatement. Her family was selfish, all of them, including herself. They each cared more about their immediate world than each other, all of them including herself. Taylor hated that, and as her heels clicked across the bare floor, she yearned for it to be different. She yearned for more. She needed...something.
It wasn't often she allowed herself to need, but she needed to now. Sitting on her bed, she pulled out her cell phone and called Suzanne.
"How's my unit coming?" Suzanne asked. "Nearly ready for me?"
Taylor could hear pots and pans clattering, and smiled, feeling soothed already. For as long as she'd known Suzanne, she'd smelled like vanilla, had some sort of food stain on one part of her person or another and was always in the middle of whipping up something mouthwatering.
"Your unit is coming along," Taylor assured her. "You'll be opening Earthly Delights in no time."
"I'm ready."
"Me, too." Hopefully she'd be right next door opening her own store as well. If she could afford to get away without a tenant's monthly cash flow. She sighed. "I can't wait to have you around again."
The clanging slowed. "I thought you were enjoying your solitude."
"Yeah, well, not as much as I thought I would, it turns out."
Now the clanging stopped all together. "Taylor? What's the matter?"
Damn it, she'd given herself away. Caring deeply for her friends and opening up to them were two different things entirely, at least for her. She didn't open up easily.
Correction: she opened up never.
But complicating the matter was the simple fact that she didn't really even know what was wrong, she only knew she felt this unsettling and vague...need. For what exactly, she had no idea. "I just wanted to say hi."
"You sound...sad," Suzanne accused.
"I do not."
"Never mind. I'm coming over right after I finish up here. I won't be but another half hour. I'll bring ice cream, and you can tell me everything."
Ice cream happened to be Suzanne's cure-all for anything and everything. It usually worked, but this seemed bigger than even ice cream. "Chocolate?"
Taylor asked pathetically, "Double fudge chocolate?"
"Chocolate," Suzanne promised. "Give me thirty minutes, hon, tops."
Tempting, oh God, it was so tempting. But no matter how much she loved Suzanne, Taylor had never been able to tell her about her own painful past, about her distant family, about losing Jeff, and somehow she knew that what she was feeling now was all tied up with that. And she couldn't go into it, not now, not after so many years of burying it, because she was afraid that if she did, if she let it out, it would destroy her all over again. "I have a Historical Society meeting this evening." True enough. "But maybe tomorrow, okay?"
"Promise?"
"Promise. Kiss Ryan for me."
"I wish you'd come stay with us so you could get away from the renovation, at least at night."
"I'm fine."
"I just don't like you there in the heart of downtown, all by yourself in that big old empty building."
"No one is going to bother me _because_ the place is so old and empty. Don't worry about me, I'm safe."
"Of course I'll worry, but that won't stop you from doing as you please. Talk to you tomorrow?"
"Absolutely."
Taylor flipped off the cell phone, and had just slipped it back into her pocket when Mac spoke in that low, husky voice of his, nearly causing her to leap right out of her skin. "You didn't move out."
_Damn._ "Well aren't you observant." Slowly, on her own terms, she shifted on the bed to face him.
Big mistake.
First, sitting on the bed while he was standing right next to it made her feel a little bit shameless, a little bit...hungry.
Horrifyingly so.
And second, there was the way he was looking back at her—eyes heated, glinting with that edgy, unreadable expression that made her thighs tighten.
Did he wonder how combustive they'd be in this bed, the way she wondered? Not that she intended to follow through on that wondering, but...
"I don't know who you were just talking to," he said. "But they were right. It's not safe here at night, no matter what you think."
"Of course it is."
"The building is deserted, and in obvious renovation. You know damn well this street gets heavy foot traffic on a daily basis. You never know who's going to come pawing through here looking to steal supplies or tools."
"I lock up."
He let a rough snort.
"I'm staying, Mac."
"There are going to be times where there's no electricity. No water. No gas. This isn't going to be the Ritz, Princess. This is going to be little more than camping at best."
She hadn't had luxuries in months, but hell if she'd admit that. Or the fact that she was slowly selling off her beloved antique collection just to keep afloat here. If he thought her a spoiled princess, so be it.
What he thought was no skin off her nose.
And if he really believed she was going to back off the first challenge in her entire life, the first chance she'd ever had to prove herself, to get by on her own, he was sorely mistaken. She'd continue her spaghetti and canned tomato diet for as long as it took. She was going to do this, and do it right, and not even for him, the first man to make her feel a twinge in the heart region in ten years, would she give it up.
"I'll make sure I have batteries and drinking water," she said.
He stared at her for one, long, unwavering heartbeat, then shook his head. "Are you always impossible and stubborn, or is it just with me?"
Trick question, that.
He certainly hadn't been the first man to find her difficult, and she doubted he'd be the last. But only one thing mattered to her, her battered pride. No way was she going to admit she couldn't afford to go anywhere for the duration of the renovation, not to him, not to anyone. "I'm staying, Mac."
"Through the dirt and noise, through the inconvenience, through the danger?"
The only possible danger came from him and him alone, but she doubted he'd appreciate the irony. "Through the dirt and noise, through the inconvenience, through the 'danger."'
"Taylor—"
"Wow, my name," she marveled, cocking her head. "You _do_ know it."
His jaw tightened. "You're doing this, aren't you? No matter what I say."
"I'm doing this." She had no choice. "No matter what you say."
**_CHAPTER 4_**
**SOUTH VILLAGE'S NIGHTLIFE** rivaled the Sunset Strip as the busiest, most energetic area in Southern California. And yet the crowds it attracted weren't wild or aggressive. Instead the attitude was a sort of laid-back and easygoing elegance.
The town's founders had perpetrated this atmosphere with one goal in mind.
Wealth.
The old adage turned out to be correct—Build it and they will come. The place had roared in the twenties, declined in the thirties and forties and rebelled in the fifties and sixties. True to the circle of life, it had been given a face-lift, slowly over the past twenty years, and had been turned into a gold mine.
As a result, there was never an available parking spot. Swearing, Mac circled the block. Then again. _Damn it,_ he'd had a long day, all he needed was one little spot. Somewhere. Anywhere.
The heat was going to kill him. If Taylor didn't kill him first, that is. She could do it with just her eyes, those amazing green eyes she thought hid everything from the world and yet seemed so expressive to him.
Then there was her calm and cool, sophisticated, elegant exterior, which he hated. But he also was beginning to understand all that was really just a front for a boiling pot of stubborn orneriness, and where there was stubborn orneriness, there was heat and passion.
And damn if he wasn't a sucker for heat and passion. Oh yeah, he enjoyed a woman who knew what she wanted and how.
Or at least, he used to.
But his and Taylor's fate was sealed, no matter how explosive he figured they'd be in bed, because she was everything he would never go for again.
And she was hiding something, he knew it. Something more than living in the building when he'd told her not to. God help her if it had something to do with this job, which he was depending on far too much for his own comfort.
Damn, letting himself feel again was a bitch.
And what he felt right now was hungry and tired, but attending tonight's monthly Historical Society meeting was necessary. Rubbing elbows with the powers that be made him want to grate his back teeth into powder, but it had to be done, because though no one would ever admit to it, it truly wasn't _what_ you knew, but _who_ you knew.
He needed to mingle.
Much to his perpetual disgust, the meetings were always run more like a cocktail party than the gathering and exchange of information they were supposed to be.
He hated cocktail parties.
The "meetings" were held at city hall, a building that could trace its roots to 1876, when it had been built as a grand hotel. In its day, it had housed miners, western settlers and Spanish royalty. Tonight the Spanish-style building was decorated in gold and silver, with froufrou food on platters that made him wish for a beer and a sloppy piece of pizza, New York style. The music came from a live quartet of musicians who didn't understand that being able to talk was important.
But at least air-conditioning blasted through the place. Early summer in Southern California hadn't disappointed, the temperature was in the nineties, the humidity off the scale.
In spite of the heat, anyone who was anyone in South Village was already there, schmoozing away. He counted three city councillors, the commissioner and the mayor before he worked his way past the entry hall.
There was a good reason for the crowd. Besides official business, and South Village did take its official business very seriously, the meeting's _true_ underlying purpose was as a meat market.
The _single_ meat market.
His mouth twisting cynically, he looked around.
Oh yeah, singles galore, mostly hungry-looking socialites, circling the crowd, checking out the potential fixer-uppers—meaning the men they could live with, the men they could make putty in their well-manicured hands, the men whose names and expensive bank accounts they could take and be set for life.
Mac should know. After all, it had been a meeting just like this one when he was doing a little contracting on the side where his ex had scoped him out.
She'd decided his last name was synonymous with money, and without bothering to figure out that Mac lived his own life as he damn well pleased _despite_ his family's money, Ariel had gone after him with dollar signs in her eyes.
He was still ashamed to admit she'd caught him with little more than a toss of her perfect hair, a come-do-me smile and a crook of her red-tipped pinkie finger.
Damn memories.
Beating them back, he pasted a smile on his face and moved forward, determined to make nice and be seen.
* * *
AN HOUR LATER, Mac figured he'd done his job. He'd nodded, talked, even smiled with the board members he knew mattered most—Mayor Isabel W. Craftsman, known as a ruthlessly tough bitch, but widely tolerated because she'd done better for the city than any mayor in history, Councilman Daniel Oberman, a man who used to be a builder, and was known for his genuine love of the renovation projects.
And so many others his head spun.
Not only spun, but pounded. It was little wonder, when he considered the hours he'd put in this week, and now that he'd bared his teeth into a smile and played nice, he was out of here.
Or would have been, except that he saw her. Taylor Wellington, current bane of his existence.
She wore a haltered shimmery dress that came to midthigh in the exact color of a summer sky. Her legs were bare and tanned, and longer than the legal limit. She stood surrounded by a group of women who also looked as if maybe they made a career out of looking spiffed-up and polished. Each of them could have graced any cover of a glossy magazine, and yet to Mac they all looked plastic.
Taylor, too. He wanted her to be plastic. He really wanted that.
Then she looked up, her eyes unerringly finding his. And in a flash that came so quick he figured he had to be seeing things, she shifted from cool as a cucumber to hot as a wildfire.
His heart clutched. It wasn't a pleasant sort of clutch either, but the kind that took hold and squeezed.
What was she doing, looking at him like that?
Her gaze stayed locked on his, despite the fact that people were talking to her, despite the fact there were people smiling and nodding at him as they passed. The music, the hum of conversation, everything, seemed to fade away.
Then she controlled that flash of heat, smoothing her expression as if it'd never happened, leaving her cool as rain. She had a real talent for that, for hiding her thoughts.
_Good,_ he decided. He didn't want to know them.
But she kept looking, every bit as much as he was, and as if he was attached by a bungee cord to her eyes, her body, he started moving toward her, away from the front door he'd been so eager to get out only a moment ago.
She watched him come.
And when he was close, his other senses came back. He could feel the cool air, could hear the sculptured, glamorous redhead on her left say, "I'm surprised to see you here tonight, Taylor. We'd actually heard you'd... How should I put it? That you'd come down on the social ladder a bit."
"All the way down," said the perfectly groomed country club woman on her right. "Like to the bottom rung."
Several of the other women laughed, the kind of laugh that assures everyone you're only laughing _with_ the person at the butt of the joke, but was really a crock because there was no doubt.
They were laughing _at_ Taylor.
She broke eye contact with Mac to stare at them, her eyes distant and assessing as if she felt far above such mockery.
"We heard about the will," Country Club Chick said, making an effort to look solemn instead of cruelly gleeful, and failing miserably. "Did your grandfather really cut you off without a cent and give everything to your mother?"
Taylor gave her a long stare. "What does it matter? I don't need anyone's money."
As if she'd told a great joke, they all burst out into collective laughter.
Taylor simply tightened her glossy mouth.
"You're so funny," the redhead said. "You always make me laugh."
"Your mother looks good," Country Club Chick number two said, looking out into the crowd at someone Mac couldn't see. "No doubt she's guaranteed a successful campaign for the next election with all her daddy's money."
"No doubt," Taylor said.
Mac didn't know what the hell they were talking about, but as he hadn't taken his eyes off Taylor, he saw something that shocked him.
Though it didn't show in her casual stance, he saw it in her eyes. She was letting them get to her. She cared what these women thought. She cared a lot.
Oh man. He should have run out the front door and never looked back. Why hadn't he done that?
Another of the Sorority Bunch patted Taylor's arm. "Well, I for one think it's very brave of you to keep your chin up."
"And at least you still have all your amazing clothes." This from the redhead, who was eyeing Taylor's gorgeous dress. "You can just learn to repeat wear."
"Hey, and don't worry, we'll pick up your tab on our monthly lunches," offered yet another.
Mac's fingers itched to wrap themselves around a few necks. The urge made no sense, as only a few moments ago he'd have sworn Taylor fit perfectly into this not so cozy little circle.
But suddenly she didn't look plastic like the others, she looked...real.
And, damn it to hell, she also looked hurt.
"Really, it's touching how concerned you are about my financial affairs," she said in a voice dripping with chill. "Truly touching."
No one but Mac grinned.
"But don't worry about me. I'm going to be just fine." She turned and walked away, away from the women, away from him.
Head high, she avoided any more conversation by striding with direct purpose to the veranda doors, which led to the botanical gardens kept by the Historical Society.
She walked right out the doors and into the night. And like a puppy needing a belly rub, he followed.
**_CHAPTER 5_**
**TAYLOR DREW A DEEP BREATH** as she stepped into the hot, hot summer night, refusing to react. If she remained numb, then she wouldn't feel her burning throat and eyes, or the ache in her chest. If she remained numb, she wouldn't feel the fist around her heart, squeezing, squeezing.
It wasn't just the pettiness that upset her, or that she'd thought of those women as friends.
She didn't care about them. She didn't care what they thought.
It simply all came back to that _alone_ thing. And she felt so damn alone. Ironic, when she considered her own mother had been inside the party. Oh, they'd kissed hello, air kisses of course, not daring to wrinkle their clothes with a hug. They'd smiled and had made light conversation.
_How are you?_
_Fine, thanks._
_Oh good. You look great._
Surface stuff that meant nothing.
The night was hot, the air thick with the humidity that hadn't faded from the heat of the day, but that was good. She needed the warmth after the chill of the past hour.
The noises of the party followed her onto the veranda as she walked to the railing and looked down onto the gardens that were considered the most beautiful in all of South Village.
They were stunning, lovingly tended to by generous Historic Society volunteers. Volunteers not afraid of getting their hands dirty or their silk wrinkled.
Which meant a Wellington had never gotten on their knees and so much as pulled a weed in those gardens, including Taylor. Oh sure, she'd volunteered in other ways, by attending expensive charity functions and writing big, fat checks backed by her grandfather.
What kind of woman did that, got to the age of twenty-seven completely supported by someone else's money? She deserved the pity she'd gotten from those women tonight, but not for the reasons they thought.
She'd never actually worked hard at anything.
Until now.
Leaning on the railing, she rubbed her temples, shedding her tough shell and half her makeup by swiping beneath her damp eyes. Poor little rich girl, she thought with loathing for the moment of self-pity.
_Ex_ -rich girl.
Was it so odd that she'd wanted something from her own mother tonight, after all this time? A real hug? A real smile? Even a real touch? She shouldn't have bothered coming, should have stayed home.
At the thought of what awaited her there, an empty building stripped down to the studs and a stack of bills so high it made her head spin, her eyes filled again.
God, she felt so alone. So damn alone.
"Taylor."
At the low, gruff voice she was beginning to know all too well, she stilled. He had a terrible habit of coming up on her in the most vulnerable of moments. "Go away."
"Yeah, about that."
She heard his footsteps. Coming closer, damn him. "Mac—"
"You'd like me to vanish, I know. And believe me, I'd like that, too."
In direct opposition to those words, he came even closer. Then closer still, until he set a lean hip against the railing, facing her, his chest brushing her shoulder as he stared down at her while she did her best impression of someone desperately interested in the flowers.
"I wanted to leave before I even got here," he said.
"So what's holding you?" She wouldn't look at him, couldn't. No one saw her vulnerable and lived. She didn't care how big he was, how warm—Oh God, he was warm. Heat radiated off him, and despite the hot, sticky night, she wanted more of it.
The need alone made her eyes sting all over again, and released a few of the tears she couldn't blink back. And then, because she'd been holding her breath, she gave herself away with one horrifyingly obvious sniff.
"Ah, hell," he muttered. His big hands settled on her bare upper arms as he turned her to face him, and for the life of her, she couldn't look away. "What's going on?" he asked.
_What was going on?_ Only everything.
"Princess?"
Suddenly his pet name for her didn't seem like an insult, not when uttered in the husky, slightly rough voice that was far softer than she imagined he could ever be. Unable to talk without making a bigger fool of herself, she just shook her head.
With the rough pad of his thumb, he stroked a tear off her cheek. She hadn't worn waterproof mascara, so she probably looked like a raccoon, but even more worrisome than that was the way she reacted to his touch. His thumb continued to make lazy passes over her cheek, his other fingers sank into her hair, and she stood there fighting the most insidious need to sob her heart out.
Silent and strong, he waited, not rushing her, not freaking out because she was crying, not doing anything but waiting patiently for her to pull herself together.
And suddenly she didn't want to pull herself together, she wanted to bury her face against his shoulder and let go. It was humiliating, appalling, and as if he could read her mind, he made a low, soft sound of empathy in his throat that completely undid her.
"Everything they said was true," she whispered. "I grew up a spoiled brat." She waited for some sort of recrimination but he said nothing.
His fingers on her temple were the most soothing touches she'd ever felt. And maybe because of it, or maybe because of the dark, her mouth ran away with her good sense and she spilled it all. "My family...we're not close. I don't know why really, we're just...different from one another I guess."
"Not every family is super tight."
"We're not even in the realm of tight. Growing up, I was given the fanciest education. On Grandfather's money. Every few years or so he'd come around and see how his investment was doing, but other than that, we didn't have much contact. I always thought it was because I disappointed him somehow. Or that he just didn't have much sentiment in him, but he seemed to enjoy my sisters' company."
"Taylor—"
"No." Not wanting his pity, please God, not his pity, she didn't look at him. "You know what? Just forget it."
"You started it, finish it."
It was amazing how private the veranda was for how many people were just inside. Maybe nobody but the two of them dared the evening heat and humidity.
Mac didn't mention it one way or another, he seemed focused on her, and only her, and having that much man, all tall, gorgeous and listening to her, really listening, was...well, a fairly intense experience. "My grandfather died," she said to the night. "And the will was rather...interesting."
"How interesting?"
"Well, for one thing, he left me the building you're working on."
"It's a beauty."
"Oh yes," she agreed. "And a money pit."
He nodded.
"He...took away the funds that had always been available to me. Every penny. Gave it all to my mother knowing she'd never share." She closed her eyes and admitted the last painful truth. "Leaving me flat broke."
"Why wouldn't your mother share?"
"She's been saving for a rainy day all her life, she's...frugal." She let out a harsh laugh. "The richest frugal person you'll ever meet."
"What about your dad?"
"He's remarried. Lives in Europe, and I don't see him very often."
"They were talking about your mother as if she were there tonight."
"She was," she said. "She's Isabel Craftsman."
Mac's eyes widened. "The mayor?"
"The one and only."
"So you're one of _those_ Wellingtons."
"That would be me. One of _those_ Wellingtons." It usually went one of two ways from here. Either the person would stare at her in awe, because her mother, cold and precise as she was, had done excellent things for the city, or the person would sneer, because let's face it, her mother hadn't gotten to where she was by making friends.
But Mac looked neither awed nor disgusted. "You really can't go to her if you need help?"
"I could, but..."
"You won't," he finished for her, his eyes filling with something she hadn't seen from him before. Respect. "What about your sisters?"
"Like I said, we're not that close."
"The building is worth a fortune."
"If I sold it." She opened her eyes and with fierce determination said, "Which I'm not doing. I'm not walking away from this. I'm not like them, Mac, those women in there, I'm not going to be like them if it kills me."
"You're not anything like them," he agreed.
She'd wanted someone on her side tonight, she'd wanted blind comfort, and this man, her virtual opposite, the thorn in her side, was offering it.
No one had done such a thing for her since Jeff.
Just the thought of him now, with Mac right there, felt like a betrayal to his memory, a stab to her already-wounded heart, but Mac was throwing her, reacting the way she'd expect Suzanne to react. A friend. A girlfriend.
Not a man.
But she didn't need him to react this way. She'd learned to depend on no one but herself. She was all she needed, she'd always simply comforted herself, and—
Mac continued to stand there when she sniffed again, not running, not reacting to her tears with his own reasoning.
He simply opened his arms.
And she stepped right into them. Stepped into them and steeped herself in his giving heat and overwhelming strength. Then she did as she'd wanted to, she buried her face in the crook of his neck, deeply inhaling the scent of wood, soap and one-hundred-percent man.
Sinking his fingers into her hair, he lifted her face so he could look into it. She looked back, at the chiseled angle of his jaw, his slightly curved lips, his light golden eyes as they ran over her face before locking on hers.
Taylor felt the jolt of his gaze all the way to her toes. She didn't know how it was possible, but in his arms her problems seemed to fade away, chased by equal parts awareness and a morbid excitement she couldn't, wouldn't, deny. Winding her arms around his neck, she pressed a little closer, absorbing the helpless growl of awareness that rumbled up from Mac's chest.
A matching awareness combined with a heady female power that sizzled through her, because he felt it, too, whether he wanted to or not, he felt it, too. Proving it, his hands tightened on her, skimmed down her back, then slowly back up again, chasing any lingering chill with a blooming desire she hadn't expected or wanted but wouldn't deny. "Um...this might be a good time for you to tell me you're married," she said. "Or something."
"I'm not married." His mouth quirked. "Or something. I'm not anything with anyone."
Chest to breast, belly to belly, she stared at him, and he stared at her right back. In that moment, he was the only person in her entire world. Her body felt overwhelmed by that, even as she leaned into him.
Around them, the air crackled, growing more in tense by the second in the hot, summer night, until she could hardly breathe. "Mac..."
"Yeah?"
She sighed.
"Taylor? What is it?"
"I want..." _You,_ she thought. _I want you._
Obeying the rampant need and invitation in her breathy voice, he bent his head and kissed her. Softly at first, but then she wrapped her arms around his neck, tight, so that the next kiss came hard, a rapacious, devouring kiss that was wildly carnal and full of erotic intent, and couldn't have been more deeply intimate if they'd been entirely alone in the world.
"Is that what you were looking for?" he asked when he pulled back, his voice even rougher than usual.
"Yes." Taylor was breathing hard, and was slightly gratified to see he was, too. For a long moment they just stared at each other, with Taylor's brain not fully functioning because he'd just destroyed her with yet another devastating thought—she hadn't lost herself in a man like that since... _since._
She hadn't wanted to.
She would have backed away then, and might even have started running, but his hands were still on her, still holding her securely against him. And in fact, hers were still curled around his neck, her body glued to his.
Of its own free will.
His eyes, locked on hers, were scorching, his body against hers rock hard, obviously aroused, obviously hungry for more. And yet he stood there, waiting.
Who would have expected patience from a man like Mac?
She could take him home. They could spend an energetic, adventurous evening exhausting each other. It would be hot and fast and good. But...and she couldn't believe this, but...it wouldn't be enough. For the first time since Jeff, casual sex wouldn't be enough, not even with Mac.
"I'm going home now," she said softly, and touched his granite jaw. "Alone."
"Yeah." Turning his head, he kissed her palm. "I know."
Not sure whether she was relieved or insulted—wasn't he even going to try to talk his way into her bed?—she backed out of his arms.
What did she do now, thank him? The thought made her want to let out a laugh, but it would have been a slightly hysterical one, so she put her fingers to her mouth and swallowed hard. "I'm...I'm thinking that kiss might have been a bad idea."
"Yeah."
Now she did laugh. "You might have at least argued."
"Taylor...was that a normal kind of kiss to you?"
Since her lips still tingled and her heart still pounded, she shook her head.
"No," he agreed. "And that kind of connection is nothing to mess with."
"You've been hurt, too," she said quietly, surprised, and mad at herself for not seeing it sooner.
He opened his mouth, then closed it again. Then sighed. "I was married. A long time ago."
"Do you...have kids?"
A spasm of pain crossed his face. "No. It didn't work out. It didn't work out so much that I never intend to get serious again."
"Never?"
"Never. Ever. Do you follow me?"
"I follow you. I even agree with you." Her lips quirked in a mirthless smile. "Imagine that."
Then, with her body still humming with desire, she took a step away. "Good night, Mac."
"Good night, Taylor."
* * *
MAC WATCHED HER saunter off, cool as rain, head high, and let out a slow, shaky breath. Holy shit, that had been some kiss.
_Kisses._
He took a careful, long wobbly breath to even be able to see straight. Another to relax his entire body, which was quite...tense now, thanks to Taylor's hot, long, sleek bod that she'd plastered to his.
He'd lost himself in her, totally lost himself, when he'd promised himself he'd never do that again.
Well, he was going to have to be more careful than that next time she flashed those expressive, hurting eyes, wasn't he?
_Much_ more careful.
**_CHAPTER 6_**
**TAYLOR WAS GRATEFUL** to have the entire weekend looming in front of her before the work of reframing and rewindowing began in earnest. Two days of no construction. No pounding, no people, no decisions to make, nothing.
And no Mac.
Two days in which to do as she pleased, which would include no thinking, no obsessing and no wasted regrets on what had happened between them on a dark night in the amazing gardens at the town hall.
On what _hadn't_ happened, and why.
Being a logical thinker most of the time—though that logic had deserted her in Mac's arms—she had a stack of reasons. He was just too...strong. Strong-minded, strong-willed, strong everything. A good part of that strength came from a superb control.
But _Taylor_ wanted to be the one in control. She liked that, a lot. When she chose to let a man in, she wanted to run the show.
She doubted anyone ran Mac's show.
Then there was the fact that he'd seen her at her weakest. No one ever saw her weak and lived to tell the tale, so she figured she'd lay low for the rest of the time they had to be near each other.
Problem solved.
It was only...at least two more months. Damn, that was a long time.
_He'd loved his wife._ Loved her so much he couldn't bear to ever love again. Funny how that gave her heart a hard tug, even though she'd sworn the same thing about herself after she'd lost Jeff. Still swore that same thing.
That such a big, fierce, independent man could be laid so low by such emotion that she understood so much...well, she was quite certain that shouldn't make her want him all the more.
The sudden pounding on the front door of the building, which she'd locked, made her jump. It was a Saturday, an early Saturday. There was no reason for anyone to be here, so it was with a frown for her solitude that she went out of her apartment, down the stairs to the front door of the building, and looked out the peep hole.
Nicole and Suzanne stood on the other side holding up a tub of ice cream, three spoons and matching ear-to-ear grins.
With more joy than Taylor had felt all week she yanked the door open. "You guys read my mind." She reached for the ice cream but Nicole held her off.
"Not so fast." She eyed Taylor carefully. "Yep, you were right," she said to Suzanne. "Something's wrong."
Now they both stared at Taylor, and she squirmed. "Don't be silly, I'm fine."
But despite the claim, she was immediately enveloped in a bear hug that brought tears to her eyes.
Again.
"Oh, honey." The lush, redheaded Suzanne pulled back, handed Nicole the spoons and held Taylor's face in her hands. "What is it?"
What, did the woman have radar? Taylor patted her hair, her clothes, but everything was in place. Everything was always in place. Her own little armor.
"Yeah, you look gorgeous as ever," Nicole said in disgust. Nicole was an emergency room doctor who considered fashion and hairstyling a grievous waste of time. She was beautiful in spite of it. Now she looked Taylor over with that X-ray vision all doctors seemed to possess. "And let me tell you, it's disgusting how good you can look surrounded by drywall dust and destruction. Now spill it. What's the matter?"
"Nothing." Taylor forced a smile. "Allergies, that's all."
"Bullshit." Nicole led the way up the stairs, back into Taylor's apartment, where they all sat on the bed and took a spoon. "Let's hear it. Long version please."
Taylor dug into the double fudge chocolate, consuming a bazillion calories in one bite. "I told you, I'm fine."
"You know, you never let Suzanne or I get away with telling you we're fine when we're not, so don't give it to us." Nicole waved her spoon. "Now. Who's the asshole who put that look of misery on your pretty face?"
"There's no—" She looked into their expectant, worried expressions and let out a slow, shaky breath. For courage she inhaled another hundred calories, maybe two hundred. "Mac. His name is Mac. He's my contractor."
"And?" Nicole lifted a brow. "I definitely hear an 'and' at the end of that sentence."
"And..." What the hell. "He kisses like heaven."
Suzanne sucked on her spoon and smiled. "Ah."
"Ah what?" Taylor demanded.
"You're falling for him."
"Because I think he kisses like heaven?"
"Because you have stars in your eyes when you say it," Suzanne said gently. "You're falling hard, sweetie."
"Lust or love?" Nicole wanted clarified in her usual blunt way.
_"Lust,"_ Taylor said.
Nicole cocked her head. "You said that way too quickly."
"I'm staying single, Nicole. No question."
Suzanne reached for Taylor's hand. "Tell us why love is such a bad thing. Who hurt you?"
"Life," Taylor said simply. She was not going into that now. Maybe not ever. "Look, I've tried love. It hurts, all right?"
"Not always," Nicole and Suzanne said at the same time.
But Taylor wasn't interested. Wouldn't ever be interested.
* * *
ON MONDAY MORNING Mac made sure the framing and window replacement was going smoothly, then sought out Taylor.
He found her sitting on her bed, and was utterly unprepared for how just the sight of her felt like a punch in the gut, and for how much he wanted to haul her up and back into his arms.
He'd figured he'd gotten her out of his system Friday night. Way out.
Apparently, he'd figured wrong.
She was looking more put together than anyone had any business looking at seven o'clock in the morning. Her shiny blond hair fell loose to her shoulders, perfectly combed. She wore pale yellow trousers with a matching sleeveless top that screamed class. The top dipped down in front and back, just enough to give him a peek of creamy skin and curves, and make him need a drink of water for his suddenly parched throat. Her long, long legs were crossed, a sandal dangling off her big toe as she lightly swung her foot while she talked into her cell phone with those perfectly glossed lips.
She saw him immediately, and though she didn't so much as smile at him, the awareness in the room bounced off the walls.
She was talking to someone about the sale of an antique wine rack, her voice even and firm as she discussed money with a single-mindedness he figured he understood a lot better today than he had last week.
The woman could drive a hard bargain, and in spite of himself, he watched in awe as she wheedled what sounded like a mind-boggling price for her piece.
When she hung up the phone, her eyes were sparkling with triumph and...relief.
Which brought him to the reason he stood there. "Good morning," he said.
"Morning." She was all business—and avoiding his gaze. "You've got a crew here already, I can hear them. I'll just get out of your hair." She slipped her foot back into her sandal.
"I'd like to talk to you."
"I'm...uh..." She looked around, probably for a handy excuse.
"Save it, Princess. You want to ignore me on a personal level after one kiss, fine."
He had to give her credit, she didn't so much as sputter. "I'd already forgotten about that 'one kiss,"' she said evenly.
"Really?"
She let out a long breath. "No."
Just like that, his heart tweaked, good and hard. "If it's any consolation, you've pretty much kept me up all night for the past two nights running," he admitted.
She lifted a shoulder as if she didn't care, but her eyes warmed a little. "It's some consolation, I suppose."
"Look, Taylor..."
"I don't think talking about it is the right thing to do. Under the circumstances."
"Circumstances?"
"That we're not going to let it happen again," she said.
"Right." But it bugged him that he knew why _he_ didn't want it to happen again, but not why _she_ didn't. "Look, I get it now, why you didn't move out. You have nowhere else to go, no money, and you're stuck here until we're done."
"Well, why don't you just spell it out," she said with a mirthless little laugh.
"This isn't about your pride, Taylor. Bottom line, you're putting every cent into this building and don't want to waste it on paying for a place to live."
She lifted her hands. "Caught me."
Stepping closer, he watched her pupils dilate a little.
Because of their closeness? It was affecting him, too, he could smell her, some exotic combination of sweet and sexy, and he could see the pulse at the base of her neck beating wildly, a dead giveaway that she was not as calm as she wanted to be. "I'm trying to tell you we'll work around you," he said. "We'll do this room last."
"But you said you wanted to hit it all at once, so that you didn't have to get your subcontractors back through here again. You said that it was hard enough to—"
"I know what I said. I'm telling you I'll make the adjustments."
"Why?"
"Does it matter?"
"To me, yes."
"Because as my client, I want you to be happy with the job."
"As your client," she repeated, sounding a little...hurt?
_Impossible._
"I'm just trying to do the right thing here," he said.
"Because you feel sorry for me?"
"Hell, no. You're too ornery to feel sorry for."
For a long moment she just stared at him, then a ghost of a smile curved those lush lips. "Okay, then. As long as it's not that. Oh, and Mac?" She climbed off the bed with the smooth grace of a sleek cat, no longer looking plastic. She would never look plastic to him again, and as she came close, he actually had to fist his hands to keep them off her.
"Thank you," she said softly.
He didn't want to contemplate what just that smile of hers did to his insides. Did she know? Probably not, or she wouldn't still be looking at him like that. They'd both agreed—nothing could, or would, happen. But he had to make sure. "Now, about the personal stuff."
Her face closed up again and he had to laugh. "After all you're going through, I'd think a little kiss would be the least of your problems."
"If it had been just one 'little kiss,"' she said, shocking him with her boldness, "then it _would_ be the least of my problems."
Hell. At her sides, her own hands were fisted. Because she couldn't keep her hands off him either, or because she wanted to slug him? "Tell me why you don't want this," he asked quietly.
"Truth?"
"Truth."
She lifted her head, so close to him now that they could have leaned in just a fraction and had their mouths meet. "I do casual," she whispered. "I do casual real well. But not more than that, not anymore. And this..." She sighed, closed her eyes. "This feels like more to me, Mac, and it scares me to death."
"Yeah. Look, I—"
"Mac." One of his laborers stood in the doorway. "You're needed downstairs."
Taylor turned away.
"We'll finish this later," he told her slim back.
She lifted a shoulder.
"Taylor—"
"I don't think that's necessary."
"Oh, it's necessary," he said, watching her stiffen. He was sorry for that, but they worked together, would have to continue to do so.
They had to finish talking about this, they had to.
Then maybe he could stop thinking about it.
**_CHAPTER 7_**
**BUT LATER NEVER CAME.** Not that day and not the next, because Taylor did something Mac didn't expect. She avoided him. She avoided him good.
She avoided him through the installation of all the plumbing and electrical. Through the hanging and taping of the new drywall.
Which admittedly wasn't that difficult, as he used good subcontractors, and for nearly two weeks his presence wasn't required more than an hour here and there.
One morning he stood out front of her building with the hose, spraying down the tools they'd used to texture the new walls, lost in concentration, when a breathy female voice whispered "excuse me" in his ear.
Head whipping up, his gaze collided with... _a petite version of Taylor?_ He'd have sworn Taylor Wellington was a serious one-of-a-kind, yet this woman had the blond hair, the same see-through green eyes, the matching cynical tilt to her head...but that's where the similarities ended.
She came barely to his shoulders, and where Taylor defined elegance and sophistication, this slightly younger version defined urban hip. She was dressed in painted-on jeans and a little crop top that showed off a sparkling diamond in her belly button, and when she turned around in a circle with a little delighted laugh, he saw the rose tattoo rising above her belt line.
Now, why anyone would want a plant growing out of their butt boggled the mind, but having passed his thirtieth birthday almost two years ago, he'd discovered he was completely out of touch when it came to such things.
"I look a lot like her, don't I?" She grinned. "I'm Liza. Taylor's baby sister. And you're..."
"Mac."
"The current boy toy?"
She actually batted her lashes as she asked this, and suddenly Mac saw another difference. Where Taylor's eyes and voice were soft at times, even giving, there was nothing soft about Liza. She was cold and hard, and had been around the block more than once. "Boy toy?" he repeated, scratching his jaw. "Uh...no."
Liza laughed. "You're rougher than her usual type, which is usually way too...upscale for me." A sideways look raked over his body, slowly. "But if she put you in a well-tailored suit... Oh yeah, baby, I can see her going for you." Running her tongue over her bottom lip, she looked at him from beneath half-closed, sleepy, sexy eyes. "You're hot."
It'd been a while since a woman had come on to him so blatantly. In fact, he nearly looked behind him to make sure she was talking to him.
"I have to give it to good old sis," Liza said. "She always did have great taste. Some advice though, just don't get attached. Taylor doesn't dabble with one man for long, not since Jeff."
"Jeff?"
"Her one great love," she said with overplayed dramatic flair, and stepping close, she ran a finger over his shoulder, down his arm. "Sis is under the mistaken impression that he was the greatest guy on the planet, and that her turn at happiness has come and gone. Stupid, huh? I mean there are billions of guys on this planet." Her eyes went sultry, speculative. "So how about it, big boy? Are you playing with Taylor, or are you available?"
He caught her wandering finger just as it roamed down his chest toward his navel. "I'm the contractor."
"Ah, the contractor." Her eyes darkened as she looked up at the building. "Grandpa always did like her best."
Mac figured the sisters weren't very close if even he knew better than that.
"So is she here?" She tossed her head, flipping her hair artfully around her face. "Or are we all alone?"
"Look..." He wracked his brain for her name. "Liza—"
"Uh-oh." She affected a pout, and before he could stop her, she cupped his face in her hands. "You're scowling. Didn't your momma ever tell you that would give you wrinkles?"
Now she rested her body against his, making sure to rub up against the vee of his jeans like a cat in heat. "Or maybe you don't care about wrinkles. Men never do, they don't need to. Your laugh lines are sexy."
Curling his fingers around her wrists, he pried her off him and held her away. "Okay, that's enough—"
"Liza!"
Liza didn't flinch at her sister's voice, just stuck out that lower lip even farther as she turned to face Taylor, who came out of the building, looking as sophisticated and elegant as ever, even with her eyes flashing.
"Hey, sis." Liza sidled back up to Mac. "Look what I found."
"Stop torturing my contractor."
"Oh, Taylor, but he's so gorgeous. Can I keep him?" Mashing her breasts against Mac's arm, she batted her lashes at Taylor, who looked immune. "Pretty please?"
"Knock it off." She wore a loose and flowing white skirt, a bright red top and a wide-brimmed straw hat. And looked good enough to eat.
Mac was suddenly starving. He separated himself from Liza, not an easy feat. Taylor was looking at him again, and he still didn't have a clue to what she was thinking.
"What do you need, Liza?" she asked her sister.
"You aren't going to even invite me in, show me around?"
"That's not why you're here."
Liza tried sticking her lower lip out farther but Taylor didn't budge or soften her expression. "Money," Liza muttered. "I need money."
"Try asking your mother."
"She's your mother, too."
Taylor just stared at Liza, not giving an inch.
"Well, she's so damn tightfisted, what's the point?" Liza muttered.
Taylor lifted a brow, apparently agreeing with that assessment, but she shook her head. "I have nothing to give you."
"You never have anything to give."
Taylor closed her eyes briefly. "I'm sorry about the times I wasn't there for you when I could have been. But the truth is, now that I might want to help, I can't. I just can't."
"Yeah, whatever. It's no skin off my nose." With one last lingering look at Mac, she spun on her heels and stalked off.
"Liza."
Liza didn't look back, just let herself out of the gate where she faded into the noon crowd on the streets.
Mac expected Taylor to spin on her heels as well, heading back into the building. Or toward her car. Instead, she just stood there, lost in her own world.
Eyeing her with wariness, he stepped closer. "Your sister is...interesting."
She lifted her head and looked at him. Her eyes were filled with annoyance, temper and a good amount of heat. "She's the baby of the family, and I'd say a spoiled rotten brat, but what she really is, is a woman-child desperate for attention."
"That was no child."
"No, you're right, she's twenty-one, old enough to know better. Did she...bother you before I got out here?"
"No."
"Did she...sexually harass you?"
Mac let out a bark of laughter at that. "Yeah. And I'm going to sue over my good honor."
"I'm serious, Mac."
"I'm going to live."
"Yes, but..." She looked at him. Looked at the sky. Then back at him. "Mac..."
A disparaging sound escaped her. "I'm trying to say I'm..."
Mac cocked his head, studying the uncomfortable Taylor with curiosity. "You're trying to say...what?"
"I just wanted to..." She held her breath, then let out a huff and turned in a slow circle while Mac waited.
Something was sticking in her craw, but what, he had no idea. Unless... Oh yeah. She was trying to apologize. What was so interesting about that was that she looked as if she might choke over it. "Problem?" he asked, suddenly feeling like smiling.
"No. I just wanted to say..."
"Yes?"
"I'm _sorry._ " She glared at him as if this was all his fault. "I'm sorry if Liza came onto you and made you uncomfortable. I'm sorry you had to deal with her on the job. It was unfair and...and..."
"And you're sorry." He grinned now, because who would have known she could look adorable. "That was pretty tough, huh? Using the _s_ -word?"
"It's even harder with you laughing at me," she said, adding a look of daggers.
"Oh, no, I'm not laughing at you, I'm laughing _with_ you." But he kept on grinning, which pretty much made smoke come out her ears.
Her eyes were twin pools of fire. And her body language, hands fisted on her hips, shoulders back, head up... Battle ready, she was, no doubt.
Call him sick, but he liked it, he liked to see her temper flare, though he was quite certain he'd be risking certain death to admit such a thing to her. "I don't suppose you'd try to say it again, so I can watch you squirm some more?"
"You're a bastard, you know that?"
"Yep," he told her back as she stalked off. "I've definitely heard that one before."
Stopping, Taylor slowly turned back to face him.
She'd barely been able to resist the urge to put her hands on her hips and stomp her feet like a child at the sight of Liza snuggled up to him, but that would be churlish, even childish.
And certainly she had amused him enough al ready.
But nobody laughed at her, nobody.
And yet there he stood, hair blowing in the breeze, eyes lit with good humor—at her expense—his long, lean, rangy body relaxed as can be.
That even now she could look at him and feel a spark, feel a need to launch herself against him and hold on tight, really burned.
"Watch your pretty sandals there, Princess," he said, pointing to where she stood, which was next to his hose. The water had started to pool.
That it was still morning didn't matter in the summertime heat of Southern California. She hadn't even realized how hot she was until the chilled water lapped over her toes.
She eyed the hose. Eyed Mac.
"Don't even think about it," he said in a warning tone that cooked her goose all the way to finish.
"Oh, I'm thinking about it." She'd do more than think. Very carefully she set her hat down on the grass. She loved that hat and didn't want it to get wet like Mac was going to. He was going to get very wet.
"Taylor," he said in that low, gruff, thrillingly sexy voice.
But not only did no one laugh at her, no one told her what to do.
Ever.
Before she knew it, she'd picked up the hose and turned it on him, hitting him full in the chest.
The water was cold, which, she supposed, explained his yelp. Or it might have to do with the fact she lowered her aim just a bit.
The sound that escaped him now was a definite growl, a growl that signified an upcoming battle.
Half horrified, half exhilarated, she continued to hold the hose on him and stepped closer.
It knocked him back a step, and a group of people who'd come out of the ice-cream shop across the street whooped and hollered.
Mac ignored them, grinning a wholly evil grin at her that made her hesitate a moment.
Which is how he tackled her to the patch of grass behind her, holding her down with his big, warm, strong body sprawled over hers.
She couldn't believe it, but he'd gotten the best of her. Her, Taylor Wellington, a woman no one got the better of, ever.
Thankfully the wood fence across the front of her property, while mostly decoration, was high enough to now block them from view of pedestrian traffic, so she didn't have to think about that humiliation.
Lifting his face, Mac smiled a little wickedly down at her, water raining off him onto her skin. Then he gathered her hands in one of his and yanked them above her head. One strong thigh insinuated its way between hers, pegging her between the soft, cool grass and the not even close to soft, definitely not cool body of Mac.
"Get off me," she hissed, wriggling, trying to free herself. "We're right out front, anyone could—"
"Could what? Could see this? _Good._ "
And eyes burning with intent, he dipped his head, covering her mouth with his.
**_CHAPTER 8_**
**TAYLOR GAVE** one startled squeak, but then as sensations bombarded her—his hands on hers, his tongue sweeping into her mouth, his deliciously big, hard, wet body holding hers down, his powerful thigh holding hers open—she melted against him like lava. Her fingers curled against his, her body arching up to meld to every inch of his. And his mouth... Oh, his mouth.
She hadn't been able to think the first time they'd kissed, could barely think now, but he tasted like heaven. And now that she _was_ thinking it, fully appreciating it, she realized something else. He knew just what to do with that mouth, knew how to nibble the corners of hers until she wanted to moan for more, knew to start out with little coaxing strokes of his tongue, then nip at her bottom lip with his teeth, soothing it over with a soft, sucking motion that whipped her into a desperate, impatient, wild thing, a wild thing with absolutely no shame, not to mention thoughts of rules or propriety.
Needing to put her hands on him, she flexed hers beneath his, and he let her go. Oh yes, she thought, mindlessly arching up to him, running her now free hands over his wet shoulders, down his wet spine, oh yes, this is what she'd needed this morning when she'd woken up so inexplicably...sad.
This. Him. _Now._
With a little sigh of pure unadulterated pleasure, she hugged him even closer and, wanting to give back as good as she was getting, she sucked his tongue into her mouth.
She was rewarded by a ragged groan ripped from deep in his chest, and felt his hands slip beneath her, cupping her bottom in his hands so that he could more fully seat himself between her thighs. At the feel of his erection, she whimpered in helpless delight, and squirmed, trying to get more of it.
Then he slowly lifted his head, her lips clinging as they parted because she didn't want it to end.
"Taylor." His voice was satisfactorily thick. Raspy. And looking down into her pouting face, he let out a soft sound of desire and stroked her jaw. "God, you're beautiful."
The grass beneath her was cool, and damp. Above them the sun was warm and dry, chasing the chill away from their wet clothes. But now, without Mac's mouth on hers, she could think again. Thoughts like her mascara was probably smeared, and that he'd eaten off all her gloss. That she was wrinkling, and probably staining one of her favorite skirts.
Or that she lay on her back, legs spread, heart wide open and vulnerable, to a man.
It was that last that made her close her eyes.
With a sigh, Mac rolled off her. On his back, staring up at the sky, he reached for her hand.
"What was that?" she whispered, eyes still closed, her breathing not even close to normal. But she let him entwine his fingers with hers, and gripped them back. "What the hell was that?"
"Whatever it was, it was damn good."
"Yeah." Turning her head, she found him studying the clouds floating overhead.
"There's Bambi," he said, and with his free hand pointed to a cloud.
Taylor had to laugh. "Bambi?"
"Yeah. There. And see that one? That long, sleek one to the right? A sailboat."
"Mmm." She was lying here with a rough-and-tumble man who saw shapes in clouds. "You always find things in the sky?"
"It's relaxing, don't you think?"
"Well, it's not a relaxation technique I've used much."
He let out a soft laugh. "Tell me, Princess, when was the last time you laid in the grass like this and relaxed _period?_ "
"Okay so it's not a relaxation technique I've used _ever,_ " she admitted.
He tipped his head back, trying to catch as much of the view as possible. "It's always been cheap therapy for me."
She rolled to her side and came up on her elbow so that she could look at him lying there, all sprawled out, looking so perfectly at home. He was long, lean. Wet. His clothes clung to his sinewy strength but that strength was far more than purely physical, because he had an inner strength as well. "What does a man like you need therapy for?"
"A man like me?" He turned his head toward her, smiling as he reached up and pulled a piece of grass out of her hair. "What does that mean, a man like me?"
"A man like you," she repeated, her voice a little breathy at the way he was looking at her. "Strong. Intelligent. Your own boss. You run your own life the way you want, the hell with anyone else, so yeah, what does a man like you need from cheap therapy?"
"You'd be surprised." He pierced her with a look she couldn't quite read. "Do you remember that night at Town Hall?"
How could she forget? "Yes."
"The kiss. Do you remember the kiss?"
Only every living second.
"Yeah," he said to her silence. "I thought so. Look, we both walked away that night telling ourselves that _that_ was as far as this would go."
"I know." He was lying there, prone and wet, soaking up the sun, so close and yet so far, and for some reason she didn't want to think about too hard, she needed to touch him. She ran her finger over his shoulder, down his arm.
His eyes heated. "This wasn't going to happen again, we decided. Did something change for you?"
Good question. Beneath her finger his muscles leaped. "Well...I liked that water fight."
"Fight? That was a massacre."
"Yeah." She smiled. "And it was so cathartic, I guess I'm feeling...reckless. I want to know more about you, Mac." She was shocked, shocked to the core, to hear the words come out of her mouth and find that she meant them.
"Why?"
She understood the question. They'd both said this wasn't going anywhere. They'd agreed, she knew that, and nothing should have changed.
Except it had. She had this new desire...a desire to know him.
Mac grimaced and caught her hand in his. "Taylor..."
One look into his wary face and she knew. He didn't feel that same desire. Mortified, she tried to tug free. "I know, nothing has changed for you," she said flatly, turning her head away.
"Wait—"
"No. You don't have to explain why you don't want me."
His sigh conveyed volumes. "Could you look at me? Please?"
She blinked up into his intense gaze.
"No, I mean _really_ look at me," he said, his voice tight.
Not understanding, she ran her gaze over his body. Over his chest, his flat belly, his—"Oh," she said faintly, catching sight of a very impressive erection straining the button fly on his jeans.
Her mouth went dry, while between her legs her body had the opposite reaction.
"I want you," he assured her in that ragged, almost tortured voice. "I want you more than I want my next breath, but that's all it is. Physical. That's all it can be for me."
"Because of your ex-wife?" She hated the needy part of herself that made her ask.
"Partly," he admitted. "Mostly."
It was a struggle but she managed to look like she hadn't just been kicked in the gut. She of all people understood a true, deep, abiding love. She understood how difficult it was to love again once it was gone, and she understood why someone wouldn't want to.
Until five seconds ago she would have said she was one of those people who wouldn't want to. She still thought of Jeff, still loved and cherished the memories of what she'd shared with him, but damn it, he was gone, and had been for so very, very long. She was tired of being lonely, tired of being alone and desperately tired of sex that only just barely scratched an itch.
Terrifying as it was, she wanted more. "She...left you?"
"Oh, yeah."
Her heart cracked. "And you never recovered."
"Recovered?" He considered that for a long moment. "No. I never recovered," he agreed, and the cracks in her heart gave, breaking into pieces because she knew, she _knew_ what he meant.
"How long ago?"
He lifted a shoulder. "Four years."
"Do you still l—"
"Taylor." He rubbed his eyes. "Maybe we could talk about something else. _Anything_ else."
"Like...?"
"Jeff." His eyes softened when she gasped. "Your sister mentioned him. Said he was the love of your life." He ran a finger over her jaw.
"Was," she repeated quietly.
"What happened?"
"We were days away from eloping, and he...um, he died. In a car accident."
Swearing softly, he used all his fingers now, sank them into her hairline. "I'm sorry."
Sorry because he'd asked, or sorry because he was the first man to make her remember what it was like to feel a rush of so many dizzying emotions she could hardly breathe?
"Where does this leave us, Mac?" Leaning in, she rested a hand on his chest. "I need to know."
"It leaves us hot and achy."
She spread her fingers wide on his chest, touching as much of him as she could. "So we're not going to..." Her hand trailed to his belly button, and would have maybe drifted farther south if he hadn't caught it in his.
A genuinely pained groan escaped him. "Are you _trying_ to kill me?"
"I'm trying to feel better."
In a move that brought tears to her eyes, Mac brought her fingers to his mouth. "Touching you, kissing every inch of you, sinking into your body, that would most definitely make me feel better."
Hearing the erotic words whispered with such sensual intent made her shudder. Yes. Yes, it would make her feel better, too.
_Now, please._
"But what about after?" He stroked a finger over her shoulder. "This thing won't just go away with one trip to the bedroom."
"So let's make it two," she said recklessly.
"I'm serious."
"It's not like you're moving to another planet after this job," she said with a teasing smile that faded when he just looked at her, his eyes filled with both heat and regret. She forced a laugh past the lump in her throat, because for the first time _she_ was making the move, putting herself on the line, and it was scary as hell, especially given she was about to be flatly rejected. "What? You're busy already?"
"Taylor." God, the sound of her name on his lips, in that low, gruff, tortured voice.
And she knew. He was walking away from this before they even got started. Which, damn it, is exactly what she'd wanted, too. Until right now, right this very moment. "Don't. Don't say it, Mac."
"I can't give you what you want." His expression was a mask of torment. "I just can't."
"I asked you not to say it," she tried to quip, and failed utterly. To save maybe even an ounce of pride, she sat up.
While they'd been lying there watching the clouds go by and breaking her heart all over again, the hose had turned the grass into a slip and slide zone. Her shirt was drenched, and so was her skirt. God only knew what her hair looked like.
She was a mess, inside and out, and looking down at Mac, also wet, but looking all the more magnificent for it, she felt a surge of resentment.
Temper was good, she decided, staggering to her feet and grabbing the hose again. Temper bypassed desolation and misery. Temper gave her strength. And guts.
And it was temper that had her leveling the hose on Mac once more as he lay there all comfortable and cozy with his closed-off heart and gorgeous body and incredible mouth that had left her aching.
When the icy water hit his prone body, he swore and lunged for her. She whirled to run but he was faster, knocking her feet out from beneath her, catching her as she fell.
Right on her hat.
"You're right," he growled, squishing it flat beneath her with his weight. "That was damn cathartic." He then tucked her body more fully beneath his, and once again she found herself right where secretly she'd wanted to be.
Under him.
His smug smile faded as he looked down into her eyes, and indeed, all of her temper faded as well. Damn him, she thought, swallowing hard when he spread his hands on either side of her face. Damn him all over again because his mouth was lowering to hers, and all on its own, her mouth rose up—
"Oh, my," came a shocked female voice as two sandaled feet came into view. Peach toenail polish and two silver toe rings.
_Suzanne._
"Hmm," came another female voice, not shocked, wearing black combat boots.
_Nicole._
"Maybe we should go away," Suzanne whispered, presumably to Nicole.
"Definitely going away," Nicole agreed.
And not one of the four feet moved.
With a sigh, Taylor shoved at Mac. With one last stroke of his thumb over her bottom lip, he surged to his feet, bringing her up with him.
Indeed both Suzanne and Nicole stood there, gaping, Suzanne in one of her flowery, flowing sundresses with crystals in her ears, and Nicole in a black tank and camouflage pants.
Neither of her friends said a word, just looked at them both with shock.
Not that Taylor could blame them. Dry, Mac was a most amazing specimen of a man—tall, built and hot.
Wet, he was every woman's fantasy.
Especially hers.
Mac thrust out his hand as if he hadn't just been sprawled over the top of their best friend. "I'm Mac."
"Nicole," Nicole said slowly, eyeing him very carefully as she shook his hand. "And this is Suzanne."
Mac shook her hand, too, smiling, looking totally and completely at ease even as water ran from his hair and down his face.
"I, uh..." Taylor looked at Mac, for the first time in her life utterly at loss for words. "We were...just..."
"I think we know what you were just," Nicole said with a straight face.
Suzanne couldn't keep hers though, and she grinned. "You were making out. On the grass. With water. On your pretty clothes. You even squashed your hat. Oh, Taylor." She laughed and clapped her hands together. "It's so wonderful."
Taylor patted her hair, and Nicole snorted. "Oh yeah," her supposed friend said. "You're a wreck. Your hair, your makeup, your clothes, everything."
Mac's lips twitched as he eyed Taylor's friends in appreciation. "She looks good all messed up, doesn't she?"
Nicole shot him a sideways glance. "You like her that way?"
Mac's gaze held Taylor's prisoner. "I think I like her this way best of all."
Nicole looked at Taylor pointedly.
Taylor looked away, but she figured by the look on Mac's face he'd seen the blush anyway.
He saw everything.
"You've done it, Taylor," Nicole said. "You've found the right man for you. No fancy suit, no fancy hairdo, no fancy words... Oh yeah, I like him a lot."
Taylor ground her back teeth together when Mac grinned. "You make him sound like a new car I'm thinking of buying."
"Or riding," Suzanne whispered beneath her breath, managing not to laugh when Taylor glared at her. "Sorry."
"He's my _contractor,_ " Taylor said, and snatched up her squashed hat. It was destroyed. "A contractor who ruined my favorite hat."
"Right." Nicole lifted a brow. "And what was it exactly you two were just doing? Working really hard, right?"
Mac laughed, then wisely turned it into a cough when Taylor rounded on him.
"I'm going inside to work now," he said.
"Good idea." Taylor waited until he'd walked up the stairs—knowing Nicole and Suzanne were staring at his very starable butt as he went—waiting until he'd disappeared inside to round on her so-called friends.
"Oh, baby," Suzanne whispered. "You've met your match."
"He is something." Nicole looked quite pleased.
"It didn't take you long to be the last to cave on the singlehood vow."
"I'm not caving!"
"You were wrapped around him tighter than Glad Wrap," Suzanne offered ever so helpfully.
"And lip-locked," Nicole added with a smug grin.
"So does he kiss as good as he looks?"
Taylor swore impressively, making her friends howl with laughter. "We are _not_ together," she said.
She was not, absolutely not, going to admit that even if she'd had a moment of weakness and wanted that very thing, Mac did not. "He's simply here doing a job. That's all."
"So the kissing thing, that's what...a side benefit?" Nicole asked.
"Don't you have your own life?" Taylor demanded.
"Hey, you butted in on my life on a daily basis when I lived here," Nicole protested. "And when I was falling in love with Ty—and denying it—you laughed at me every step of the way."
"I am not falling in love with Mac." But her heart hitched painfully. "I'm not."
"Oh, honey." Suzanne dropped the teasing note in her voice. "It's all over your face, don't you know that?"
"We've only just met each other."
"When it's the real thing," Nicole said, also surprisingly free of mockery. "It happens like a train wreck. You see it coming but you can't look away."
She already knew that. Damn it, she already knew. She'd done love once, and it had been glorious.
And painful.
And yet...God help her, she might have been willing to try again.
If Mac had been willing. But she couldn't, wouldn't, compete with the memory of his ex-wife. "You guys are off the mark on this one."
She had other things to think about. Such as getting the money together for the next round of renovations. "So," she said with false cheer. "Who's up for a trip to my storage unit to see what antique I can bear to part with this month?"
Groans met this, and Taylor smiled. Friends. If they were all she ever had, it would be enough.
She'd make it enough.
**_CHAPTER 9_**
**INSIDE, MAC LOOKED AROUND** for something to get busy with. Something that would take his mind off the one incredibly sexy blonde he should never touch again. He looked at the pile of leftover two-by-fours from the framing they'd finished weeks ago. He'd asked someone to stack them, and of course no one had. Fine. He could use the distraction.
Halfway through the load of lumber, he was breathing hard but still thinking. Thinking that Taylor was driving him crazy.
From outside he heard female voices raised in laughter. He could pick out Taylor's, of course, though he refused to look. He thought he could even smell her. He stacked the wood faster, but it didn't help. That sensual scent she wore made him think of long, hot summer nights. Of dancing beneath shimmering moonbeams, skin to skin. Of deep, drugging kisses—
Careless, he walked too close to the stack of wood and bashed his shin on a two-by-four.
That wasted a few moments, hopping around, swearing colorfully. With renewed grimness and a very sore leg, he stacked the rest of the wood, then pulled his T-shirt away from his damp skin. Damn, today was hot as hell.
He'd just picked up a set of plans when a scream prompted him to drop them and run to the window. Just outside in the front yard, where only moments ago he'd flattened Taylor to the ground and pressed his body to hers, were the three women.
Two of them were screaming in terror, not that they were facing any danger to make them scream like that. Not unless you counted one dangerous to his mind and heart—Taylor Wellington, who, with a particularly evil laugh, lifted the hose.
He was certain she had no earthly clue how she looked, hair wild, skin glowing and damp, and her smile...it wrecked him. She looked wet, and mischievous, and sexy as hell, which didn't help his disposition any.
She leveled the hose on Suzanne and Nicole.
Within seconds the three of them were drenched, and catfighting like Mac hadn't seen since he'd cancelled cable the year before.
Like a very weak male, he pressed closer to the window. Nicole grabbed the hose from a huffy Taylor, and he raised a brow. Suzanne went down on her butt with a squeal, and he winced. And when she got right back up with a warlike shriek, he could only shake his head.
Then Nicole tackled both Taylor and Suzanne to the grass and rolled them around in a tangle of limbs.
Mac had his nose pressed to the glass now, and he was quite certain he shouldn't be hard as a rock watching them go at it.
And when they finally dropped the hose and fell to the ground laughing like goons, he had to take a deep breath. They'd gotten it out of their system.
Good, he could work now.
Then Taylor laughed at something Nicole said.
Laughed and looked...happy, Mac realized with a sudden hitch in his gut. So carelessly happy with her clothes clinging to her, her eyes bright with humor.
And nothing like the image he'd had of her when they'd first met. That bothered him, too, how much he wanted to cling to that other Taylor, because then he wouldn't be so attracted.
There had been a time in his life when he'd wanted nothing more than a deep, abiding love. A family. He'd wanted it all, but that had passed.
Ariel had made certain of it.
Now he didn't need that kind of a connection in his life. He didn't need anyone.
But as if she could feel him and his conflicted thoughts, Taylor turned and looked right at him.
Gazes connected. Held.
And Mac stopped breathing.
After a long moment, she turned away, leaving him to let out a slow breath.
Nope, he didn't need anyone. Not ever again.
* * *
MAC SPENT the next week working like a dog on the woodworking portion of the job—normally his favorite part—thinking it should dispel the feel of Taylor in his arms, the taste of her in his mouth.
Should, but didn't. He spent every night at his kitchen table, trying not to look at the mountain of bills, drafting up the plans for his own renovation, hoping he got approval for one of the bids he had out there in order to pay for it.
By the end of the next week, he still hadn't heard from the town council, and the stress level was rising. He went to work early on Friday, thinking a little manual labor might help.
Taylor's car wasn't out front, but in a town like South Village, where a parking spot was more prized than the actual car, that didn't mean much.
But Taylor, the moneyless princess, was still very much a princess in that way. She wanted her car parked right out front, and more times than not, she actually managed it.
Mac figured once a princess, always a princess.
He, on the other hand, had to park a good three blocks away, even though it was still practically the crack of dawn.
The building was silent. Letting himself in with the key Taylor had given him, he walked up the stairs. They'd come so far in all these weeks. They were working in the apartment across from Taylor's today, putting in kitchen cabinets, and for a moment he let himself relish all they'd done up to this point.
The place was looking good, really good. With all the wood trim, brick and wood accents, the natural charm and personality of the old building was shining through.
He put on his tool belt because he liked the weight of it, and because he liked the work. He wasn't, and never would be, a Cadillac contractor, someone who ran a job and yet never picked up a hammer.
He wanted to lift a hammer. Hell, he wanted to do it all.
He looked around for the plans, and remembered he'd left them in Taylor's room when he'd been with the painter. A glance at his watch reminded him it wasn't quite seven.
Taylor Wellington was not a morning person. He'd learned this. Though she always appeared by eight, perfectly dressed and perfectly made-up, looking stunning as usual, she rarely spoke until she'd walked across the street to the coffee house and purchased a very large coffee.
Mac enjoyed watching the process, though he'd cut out his tongue before admitting it to her. Except for business, they hadn't spoken since the water fight. He told himself that was a good thing.
Letting himself into her apartment was easy, he had a key for that, too. But walking into her bed room, where he'd left the plans, wasn't quite as simple. There were scents in there, scents of soap, perfume...and the woman who wore them. There were clothes, perfectly folded as always, but clothes that made his fingers itch to touch. And then there was the bed, with the luxurious sheets and fluffy pillows that made him want to climb on, jerk her close and mess up both the woman and the bed.
Those luxurious sheets started moving, and were tossed aside as Taylor sat straight up. Her hair was wild, she wore no makeup, and nearly no clothes.
What she _did_ have on made him swallow real hard. It appeared to be a teddy, all pale yellow lace.
The teeny tiny straps had fallen off both shoulders, rendering gravity his greatest ally as the generous curves of her breasts nearly spilled out, until she put a hand to her chest. "Mac?"
"I...I'm sorry."
She just blinked.
He knew he should spin around and walk out the door, but he couldn't quite feel his feet. "I didn't think you were home."
Another slow blink.
Oh God. _Go, just start walking. Do the noble thing here, Ace, and get the hell out._ "Your car isn't out front."
With a huge yawn, she raised her arms over her head and stretched, allowing the lace to slip another fraction of an inch.
His heart nearly came right out of his chest. "Uh..." He waggled a finger in the direction of her chest. "Your pjs... They're falling." Oh man, she was incredible, all soft and glowing and rosy from sleep. She stretched and yawned again, her legs shifting, pulling the sheet down to her thighs. The little—and the key word here was _little_ —nightie barely covered her panties.
If she was even wearing any.
The thought made it difficult to breathe, as every ounce of blood in his body headed for parts south.
Another stretch from the princess, and this time she added a little moan of pleasure at the feeling of her muscles loosening. The sheet fell all the way off, and her creamy thighs came into view, along with the smallest peekaboo hint of matching yellow lace between them.
Mac nearly moaned, too. Was she teasing him on purpose? And was that the morning chill making her nipples pout up against the lace, or something else, something like... _him? Be professional,_ he told him self. _Get out. Now._ He even backed up a step, but then his feet stopped working. "Taylor."
"Hmm?" She yawned, eyes closed.
His eyes narrowed as the truth sank in. "You're not awake."
Her eyes jerked open. Her body stiffened in mid-stretch. _"Mac?"_
God save him from sleepy, sexy-as-hell, scantily-clad women so early in the morning, when his resistance was already down. All the way to zero down.
He had to give her credit though, as her eyes cleared from dream to reality. She didn't screech.
She didn't dive back under the covers. Not Taylor Wellington. Instead, she slid out of the bed and crossed her arms.
Though he did top her by several inches, she man aged to look down her nose at him. "You."
"I'm sorry. I—"
She turned from him and headed toward the bathroom.
And the words backed up in his throat, because her nightie dipped down in back to the curves of twin sweet cheeks, the thin lace clinging to every inch.
Then the bathroom door shut, cutting off the view. He had to shake his head, hard. "Taylor." He put his hands on the wood. "I didn't know you were still here."
"We've been working together for how long now, Mac?"
Her conversational tone confused him. "A long time."
"Yes, a long time," she said calmly through the door. "And have I done anything, anything at all, that would give you reason to think that I'm a morning person?"
"Uh...no."
"Have I ever gotten out of bed before I had to?"
Her voice was so even. Was she mad or not? "No, but—"
"You know what I thought when I opened my eyes and saw you, Mac? I thought you were part of my dream. It was a good one," she added, and just her voice made him hard.
"I—"
"You should have just joined me, instead of standing there watching me."
And on that heart-stopping statement, she cranked on the shower, drowning out any reply he might have had.
* * *
MIDSUMMER HEAT hit with a vengeance, but neither Taylor nor Mac had a spare moment to dwell on the sticky heat. Mac was surrounded by roofers, painters, flooring technicians and enough laborers that Taylor felt dizzy watching them work.
But work they did, and work hard. Her building, once the eyesore of the neighborhood, was shaping up into a beauty right before her very eyes. Pedestrians on the street, walking to dinner or the theater or wherever, stopped to ooh and aah.
Taylor loved it, loved every little bit of it, including watching Mac work.
_Especially_ watching Mac work.
He caught her at it, the watching, at least once a day. But she caught him, too. She'd be pouring over plans, over tile samples or even on her cell phone and she'd... _feel_ him. She'd look up and there he'd be, eyes filled with heat and awareness.
And reluctant affection.
Oddly enough, for a woman who had spent a decade avoiding such emotions from a man, it was the last that got to her.
One afternoon she came staggering up the stairs to her apartment under the weight of a small writing desk. The thing wasn't heavy, just awkward to carry, and worth a small fortune.
She'd picked it up at a garage sale for a song, and was so happy about it that nothing could dim her mood. "Don't you look pleased with yourself."
Mac stood in the doorway of her bare living room.
He wore jeans that had seen better days. They were faded, torn at both knees and one hard thigh. The soft denim fit him perfectly, outlining every nuance of his lower body. His T-shirt had come untucked on one side, caught on the tool belt slung low on his hips, exposing a strip of flat, rigid belly.
Her own tightened uncomfortably in response. "I _am_ pleased with myself." Having caught her breath, she hoisted up the small desk again.
"What's that?"
"Just something I picked up. Do you like it?"
He eyed her slowly up and down. "Very much."
"I meant the desk."
"Oh."
Since she'd been wanting him to say he still wanted her, she felt herself flush with excitement. "It's circa 1920, isn't it a darling?"
"It'd be more darling in your storage unit." But he took the desk from her, making it look like a toy in his arms as he strode across the living room toward her bedroom.
The bedroom was a good size, but he dwarfed it, and as she followed him in, she became painfully aware of the fact that the only other piece of furniture in the room was her bed, pushed to the middle of the room with a drop cloth on the floor beside it, which she put over it during the day.
"Paint fumes are going to be bad this week," he said.
"No problem."
"The noise and dust—"
"It's no problem," she repeated, watching the muscles in his jaw bunch as if he was incredibly tense. Why was that? If he wanted her half as badly as she wanted him, well, then, that was his own damn fault.
"I heard Nicole and Suzanne offer you a place to stay—"
She held up a hand and forced a cool smile, tired of battering down his defenses every time they spoke. "I'm staying here."
"Look, Princess, what I'm trying to say is that this place isn't going to be up to your standards."
She laughed. "It's never been 'up to my standards.' That's the whole point of the renovation."
"I just think you should go until we're done."
She stared at him when he turned to face her, wondering where this was coming from now, after all this time. Was he starting to feel the pressure, like she was, of being together day in and out? Was he, like her, aching for more? "You just don't want me under your feet."
He closed his eyes, then opened them. "The problem is not about _not_ wanting you beneath my feet, but about wanting you beneath me. Period."
An immediate hot current raced through her body. "Why do you do that?" she whispered, her knees wobbly, her pulse rocketing wildly, and all from a look and a few words.
"Do what?"
"Remind me in every word, in every look, that we have this...this..."
"Hard to put a finger on it, isn't it?"
"It's an attraction," she said bluntly. "And for someone who claims not to want it, you sure bring it up a lot."
"I never claimed not to want it, Princess." He stepped closer, so close she could feel his breath warm her cheek. Then his fingers did the same as he stroked them over her skin. "It's just that what we each want are two different things entirely."
"How do you know?" She met his hot gaze. "When you won't discuss it?"
"You want me to discuss it? Fine. I want you in that bed for one entire night—" He pointed to it. "I want you there, beneath me, legs and arms spread wide, head tossed back, screaming my name as I touch, kiss, lick and suck every inch of you. I want to sink into your body and lose myself. I want that so badly I can't eat, can't sleep, can't do any-damn-thing. Any questions?"
Questions? She couldn't remember, she was so lost in the image he'd just given her. She licked her dry lips, then jerked her gaze up to his when he let out a low and very soft moan.
"Have I mentioned you're killing me?" he asked quietly, running those fingers down her throat now, and very lightly over her collarbone.
A shudder wracked her.
"Yes." Her voice was a mere whisper. "You've mentioned."
"Good."
He turned to go, then speared her with one last searing look. "Next time you want to play with me, Princess, just remember what it is I want."
She was fairly certain she would remember.
The moment he was gone, she sank to her bed, then fell to her back, gaze on the ceiling, fanning air in front of her hot, hot face.
**_CHAPTER 10_**
**THEY WENT BACK** to business only.
Then, the next afternoon, when Taylor had been forced by her cell phone to stand outside to get reception, Mac came through the yard, lost in thought with a set of plans in his hands. Without looking up, he brushed against her, his shoulder rubbing hers.
Did he even see her? As he walked away, he glanced over his shoulder at her, eyes hot enough to melt every bone in her body.
Oh yeah, he saw her.
An hour later he came through the entrance hall where she was studying paint samples, and ran his hand across her lower spine to make room for himself to pass.
Her entire body reacted.
Incidental contact?
_Nothing_ with Mac was incidental.
He was _playing_ with her, when he'd warned her not to do that very thing to him.
Payback time, she decided. The very next morning _she_ acted first, and "accidentally" brushed her breasts against his arm when she leaned over to point something out on the plans.
He inhaled sharply.
She loved that, because it made it real, this thing he wanted to ignore. Whether he liked it or not, what they felt was _real._
After that, she made sure it happened every time.
A touch, a look...
Mac never said a word about it, but he would reach out and brush his fingers over her hair, making her want to purr like a kitten and beg to be stroked.
While talking to her about concrete or wood, he'd drop his gaze to her mouth. If no one else was around, he'd lightly graze his knuckles over her jaw.
Once he ran a finger down her arm. She had the tingles for hours.
But they never spoke about it again, never spoke about anything other than the work.
And there was plenty of it. She had the second floor unit and the loft to color scheme in anticipation of the finished renovation and subsequent renting.
And there were also the two retail units down stairs. One for Suzanne, the other for...the sky's the limit. An art gallery, or a unique little gift shop...maybe even a bookstore. She loved books.
But she knew what she really wanted. Just thinking about her storage unit, about all the antiques she had left, the precious commodities she'd collected over the years, made her heart sigh.
She'd gathered these things around her like her family over the years. They were her security blanket. She'd sold some, but not as many as she'd thought she'd have to.
Which led her to believe she really could do it, she could keep that second retail unit for herself, for her antique shop.
The more she thought about it, the more she wanted it.
Her cell phone beeped. Looking down at the missed call made Taylor sigh again. As if her mother had been able to read her mind from across town, as if she knew her daughter was thinking of doing something crazy, she'd left a message.
Their relationship was pretty much a series of left messages, which made Taylor feel...sad. Sad enough that she actually returned the phone call.
But the moment she heard her mother's cool voice, she hesitated. "Uh...hello, Mom."
"Taylor! How lovely."
"I'm returning your call."
"Oh, of course. Well, I wanted to remind you I'm campaigning again. My people suggested I get a family portrait taken to circle around, you know, with you and your sisters."
Right. She should have known this wasn't a hi-I-missed-you call, but a I-need-something-from-you call. "Okay."
"Really?" The mayor of South Village, and all-around superwoman, seemed genuinely touched Taylor would do such a thing without an argument.
It made her do that yearning thing again. Wanting to be close, close to someone, she said, "Yes, I'll do it. But getting my sisters to agree might be more difficult."
"I'll get them."
She'd probably offer a bribe, a monetary one. Taylor should have held out for that.
"So. What are you doing these days?" her mother asked, shocking her with such a personal question.
Was it possible she really wanted to know? Testing, Taylor said, "Actually, I'm thinking of opening an antique shop in Grandpa's building."
"What are you going to do with that college education then? Toss it out the window?"
"It's what I want."
"Well, it's a bad idea."
Taylor stuffed her immediate defensive response, listened politely for another few moments while her mother went on and on about the high hopes she'd had of Taylor joining her in politics someday— _politics!_ —then found an excuse to hang up.
When she had, she buried her face in her hands. What had she been thinking, trying to open up? Trying to let someone in?
"Must be difficult, having the city's most notorious tough lady as your mom."
Mac, the man—the only man—with the supreme talent of finding her at her worst. He'd seen her without makeup, with said makeup running down her face, he'd seen her first thing in the morning and worst yet, crying.
Now this. "Go away."
"Yeah. Sometimes my family makes me bitchy, too."
She lifted her head at that, ready to snap his head off, but he wasn't laughing at her. He wasn't even smiling.
Instead, he just stood there, his eyes filled with an understanding she wasn't ready to face. "I am most definitely not bitchy."
When he just looked at her, she sighed. "Okay, maybe just a little."
His lips slowly curved, but unlike what she might have expected, he didn't say a word.
He was good at that, she'd noticed, not saying a word and yet conveying so much. "Oh, leave me to my bad mood."
"I have a better idea." He walked into her room like he owned the place, in his customary Levi's and T-shirt, a pencil behind one ear and a set of plans rolled up in his hands, looking tall, leanly muscled and tough.
_She_ wanted to be tough, but just looking at him made her feel soft. Feminine.
"Come on."
Startling her, he set the plans on her bed, took her hand and pulled her to her feet.
He had her halfway out the door before she dug in her heels, not that that stopped him. She tried a hand to his back, but that only electrified her with the heat and strength of him. "Where are we going?"
"You'll see."
"Mac—"
The look he shot her was pure male frustration. "Look, you need a break, I've got an errand to run, and if you come along like a good little girl, I promise to buy you a lunch that will make you sigh in bliss." His whiskey-colored eyes and rugged features crinkled into an enticing smile. "Okay?"
_Smiling._ He was smiling at her. Her tummy fluttered. "What's the matter with you today?"
"Nothing."
"You've avoided talking to me about anything other than business, and you've avoided physical contact like the plague."
"Not like the plague."
"What then?"
"Maybe more like...a good, tall frosty beer at lunch."
"That makes no sense."
"Sure it does. You know the cool brew is going to go down like pure heaven, but afterward, it's going to impair your judgment."
She narrowed her eyes, not flattered. "Hmm."
He laughed. _Laughed._ "Look, maybe I'm doing this because I don't like to see you sad."
"I'm not—"
"Aren't you?"
She stared at him, disconcerted that he could see right through her in a way no one else did.
"You going to tell me what's up?"
"No," she said automatically, because he didn't really want to hear she was lonely and needed to be held. But just in case he was astute enough to see it, she examined the manicure she'd given herself last night.
"Ah." His eyes lit with pure trouble. "You broke a nail."
"I did not break a nail, nor would I fret over it if I had."
A big fat lie.
"Then you're having a bad hair day," he decided with just enough bite that made her realize damn good and well he was just trying to goad her out of her mood.
Sweet of him, really, but she wanted to be grumpy at the world.
She wanted to be grumpy at him, too, for reasons that didn't bear examining too closely. "Do I look like I'm having a bad hair day?" she asked.
He grinned, a stunning show of masculinity that made her mouth want to fall open.
She closed it tight.
"Now _that,_ Princess, is a trick question. It's like asking a man if your pants make you look fat. Damned from the get-go, no matter what I say."
"Which proves my point," she said. "Men are idiots. You could just say 'you look great, honey.' End of discussion."
"You look great, honey," he said, eyes hot, all teasing gone, just like that. "End of discussion."
"Mac—"
"Just give me an hour," he said softly, and ran a finger over her jaw.
Her heart sighed in a way it wasn't used to. It'd been a very long time since a man had made her heart want to. "An hour," she repeated, and followed him downstairs and into his truck.
She had the uneasy feeling she would have followed the irresistible man anywhere.
**_CHAPTER 11_**
**MAC HAD NO IDEA** what had made him do the Boy Scout rescue with Taylor, but here he was, driving along on his errand to South Village's town hall to check on permits, with her sitting beside him. His only defense...she'd looked as if she'd had the weight of the world on her shoulders, as if she'd been unbearably lonely.
It had tugged good and hard on the heart he'd thought dead.
_Sap._
Whipping the truck into midday South Village traffic, he decided the next time she turned those expressive sea-green eyes on him, he'd just turn around and walk away.
The hell with walking, he'd _run._
"Look at all these people." Her face was turned to the passenger window as they passed a bookstore, a theater and two packed sidewalk cafés... The sidewalks themselves were lined with the lunch crowd. People were walking, in-line skating, jogging. "Everyone seems so...focused."
She seemed wistful, a little envious even, which surprised him. " _You're_ focused," he said.
Turning her head, she looked at him. "You think so?"
"You're renovating a historical building. That takes focus."
"No, _you're_ renovating a historical building. I'm just funding it."
"By buying and selling antiques." He shook his head. "Your talent for such things is amazing."
"Really?"
She seemed so genuinely blown away by his statement that he looked at her, then wished he hadn't. It was the vulnerable Taylor again, the woman who had fears and doubts, and was so human he wanted to haul her close and never let go.
That was the Taylor he needed to stay away from.
But she leaned in close, giving him an up-front and personal view of her with that very private expression. She had a smattering of light freckles across her nose. He'd never noticed them before. In her ears twinkled tiny twin diamond studs.
Sweet sophistication.
Sexy as hell.
And the most determined person he'd ever met.
He'd never met a woman like her.
"You don't have to baby-sit me," she said. "I'm really fine."
"You're a good liar, is what you are."
She leaned back in her seat and turned straight ahead, making guilt swamp him. What right did he have to pry when he didn't want her to do the same back? "I'm sorry."
"Yeah. Sorry I'm in your truck."
"Taylor—"
"You want to know what's wrong with me?" she asked, her voice suddenly low and sultry, her eyes suddenly hot, hot, hot. "You want to know what would make me feel all better?" She leaned toward him again, and ran her tongue over her lush, glossed lower lip. "Do you?"
He could only shake his head. "Um...no—"
"Sex," she whispered. "Wild, screaming, sweaty sex. _That's_ what would make me feel better."
He tried to speak, but found he didn't have a voice, and had to clear his throat. "Taylor—"
"Just in case you wanted to know."
Just in case he wanted to know. _Wild, screaming, sweaty sex._ Images flitted in and out of his head. He was hard as a rock. "Let's try this instead," he suggested, and pulled up in front of the town hall.
The last time they'd been here together hadn't exactly been a calm experience, but Mac tried to forget about that as he led her up the front steps. They took an elevator to the third floor, which housed the building department.
Taylor was silent until the elevator doors slid closed. Mac had never had this elevator all to himself, not once. He figured the fates were having a good laugh at his expense that he was alone with her now. A woman who wanted—
"I've never been turned down for wild, screaming, sweaty sex before," she said.
Mac stared at the control panel, gritting his teeth.
"Yeah. It's a first for me, too."
She waited until the elevator dipped a little as it came to their floor. "Why?"
For a brief second he closed his eyes to the bafflement and hurt in her voice. "Because with you, Taylor, it wouldn't just be wild, screaming, sweaty sex. With you, it would be different. And God help me, but I can't handle it."
She stared at him, then slowly, as the doors opened and people waited politely to get on, she sighed. "Yeah."
He had no idea if that was an admission that it would be more for her, too, or if she was just agreeing that he couldn't handle it.
He practically ran out of the elevator.
"What are we doing here?" she asked as she followed him down the hall.
"Checking on permits." They came to the right office. Without thinking, he put his hand low on her spine, leaning past her to open and hold the office door for her.
At the feel of her, he jolted, and so did she.
Looking at him from accusing eyes, she whispered, "See?" Putting her mouth to his ear, she let her lips brush against his sensitive skin. "Twitchy. We're twitchy for _S-E-X._ "
Oh yeah, she was killing him. He'd been sporting an erection since she'd gotten in his truck, and there was no relief in sight.
They waited in line for three minutes and thirty-three seconds—not that he was counting—standing close, breathing each other's air, arms brushing, until Mac was in such a state he couldn't remember why the hell he'd thought being with her today would be a good idea.
It was a dumb idea. A really, _really_ dumb idea.
Made even dumber when exiting the elevator on their way out of the building five minutes later—thankfully with a handful of other people this time—they ran into an older couple he knew well.
"Mac!" The woman, dressed to the hilt in a black suit and sensible heels, reached for him. "Oh, Mac!"
Taylor watched with interest as the very elegant woman hugged Mac, then pulled back to smile into his face. "What a pleasant surprise."
The man hugged him, too, complete with manly back slapping. "Hey, I was on the green yesterday," he said. "Hit an 82, three under par. When are you going to join me?"
Mac winced. "I don't play anymore. You know that. I haven't played in years."
"Four," the woman said with a pointed expression. "You haven't played golf in four years. Since—"
"I remember," Mac said, a strained smile on his lips. "I'm just too busy these days."
"Ah," the woman said with that same pointed expression.
Mac looked at Taylor, and if she'd known better, she'd have sworn he looked rather adorably panicked. "Well, we've got to—"
"No, wait. We're just heading off to lunch," the man said. "Come with us. Both of you," he said politely, eyeing Taylor with friendly curiosity. He had a look to him, and he reminded her of—
"Taylor." Mac swiped at his cheek, which had the woman's lipstick on it. "This is Assistant District Attorney Lynn Mackenzie and her husband, Judge Thomas Mackenzie."
The assistant DA grinned. "Taylor, what a lovely name." To Mac she said, "And you! You got yourself a girlfriend! Oh, Mac, and you never said a word."
"Uh..." Mac avoided looking at Taylor. "No, I'm just working on her building."
"Ah, a _business_ relationship." The woman lifted a teasing brow. "I get it."
"No, really." Mac shifted on his feet, which Taylor found fascinating. "She's a client."
Also fascinating, was the slight tinge on his tanned cheeks.
Mac was _blushing._
"It's just a business thing," he said.
The assistant DA studied Mac closely, her eyes lit as if she was onto a scoop. "Are you just saying that so I'll go away?"
"Absolutely not." Mac still hadn't looked at Taylor.
"Darn it," the woman said forcefully, glaring at the man with her. _"Darn it!"_
"Now, Lynn, I'm sure he'll come around one day soon, and—"
"No, he won't, he's too stubborn."
"Yeah, well...we've really got to go...." Grabbing Taylor's elbow, Mac tried to back out of the circle. "Nice see you...uh...Judge."
"Hold it right there, Thomas Ian Mackenzie." The assistant DA put her hands on her hips. "Are you trying to hide the fact that we're you're parents?"
And though Taylor should have seen that one coming, her jaw dropped. She stared at Mac. "You're the son of the judge?"
Mac sighed. "Yeah."
"And the son of one of the assistant DAs?"
"That, too," he admitted.
"You are kidding me!"
Lynn's smile faded a bit. "Is this a problem?"
Taylor sighed. "No. It's not a problem. It's...um, lovely to meet you."
Lynn crossed her arms. "Why don't I believe you?"
"No, really." Taylor eyed Mac, thinking she'd kill him later. "It's just that Mac might have mentioned anytime over the past few months he was the son of the judge and an assistant DA, sometime like...oh, I don't know...maybe when I told him I'm Isabel Craftsman's daughter."
"Isabel Craftsman, the mayor?"
"Yes," Taylor, said, staring at Mac, who was still avoiding her gaze.
"Hmm." Lynn raised her eyebrows as she eyed Mac. "I think I see."
"Mom—"
"Oh, _now_ he calls me Mom." Much more friendly now, Lynn shook her head at Taylor. "Honestly, Taylor, I've never seen this man before and he's calling me Mom."
Taylor had to laugh at the easy wit and charm, but she supposed she would have expected no less from whoever had raised Mac.
"So why don't the two of you join us for lunch?" his father asked.
Taylor looked at Mac, interested to see if he'd allow this.
"Sorry." Mac kissed both his parents, then gripped Taylor's arm. "We have to go." And he dragged her out of there so fast her head spun.
"Smooth," she said when they were both out on the busy street. "Making sure I couldn't drill the parentals."
"Hey, I was just making sure they couldn't drill _you._ I love them, but believe me, they're ruthless matchmakers." He stopped at a hot dog vendor on the corner. "One or two dogs?" he asked Taylor.
She gaped at him. " _This_ is the lunch you offered me? The one that is supposed to make me sigh in bliss?"
"One or two?"
South Village had nearly as many cafés and restaurants as it did people, and most of them were excellent. On weekends, 20,000 people from all over flocked to the streets to experience the food. It was one of her favorite things about living here, something she hadn't been able to afford lately, and Mac, who had earned a good chunk of her money recently, was going to buy her _hot dogs?_ From a street vendor? "Two," she sighed, and made him buy her barbecue chips, too. She didn't say a word as he took their food and started walking, she just followed.
Which brought her to another bone of contention. When had she ever followed a _man?_
They walked around the block to the back of the town hall, where the botanical gardens bloomed in vivid, vibrant colors. In the light of day, they dazzled in every shape and hue, and Taylor had to admit, just walking through on one of the brick trails, with the scents and sights, she sighed in sheer pleasure of being outside.
They sat down and he handed her a hot dog. "Ketchup?"
Shaking her head, she took a bite. It was heavenly. Damn, she hated when he was right. "So...why didn't you tell me?"
Mac was suddenly very busy eating. "Tell you what?"
"That you come from the same kind of world I do?"
"We don't."
His parents had just about dripped elegant sophistication. "Of course we did, I just met—"
"You just met the two nosiest, bossiest, most interfering parents on the face of this earth, yes. And they love me, ridiculously so, but they never sent me away to schools for years on end, and they sure as hell never ignored me, not my hopes and dreams, not me as a person. Not once." He nudged her arm with his, his eyes painfully deep. "That never should have happened to you either, Taylor."
All her life she'd felt like a bug on a slide, people waiting for her to make a fool out of her family's name, people waiting for her to fall on her face. And all her life there hadn't been many to understand what that had been like. Only Jeff.
But Mac...he was looking at her with empathy, too. Because he understood. He understood _her._
While thinking about this, she inhaled every last chip in the bag, and didn't even flinch over the calorie content. "What I mean is," she said, trying again. "We both came from considerable wealth."
Some of the warmth faded from his eyes. "I don't consider myself that way."
"Oh, come on, Mac, I saw your mom's shoes. Prada," she said with a sigh, licking mustard off her thumb. She started in on the second hot dog. "And the diamond earrings. Stunning. You can't tell me they don't pull down mind-boggling salaries."
With careful consideration, he took his last bite of hot dog. Polished off his soda. Leaned back, away from her, he slid his sunglasses over his eyes as he viewed the incredible colors around them. "I suppose they do."
"So all those times you called me a princess? Why didn't you ever say anything about it?"
"And when should I have done that? When we first met and I needed your job?" He set down his drink and stood. "Or maybe when you were snubbed by those women at the historical society meeting? Yeah, maybe I should have told you then, when you were smarting over what they'd said to you."
Shocked at his bitter tone, she rose, too. "I'm just saying, that as two people who share some of the same experiences—"
"No. _That_ we haven't done. We don't share anything." He dumped their trash and took her back to the truck.
She'd expected the silence. She didn't expect him to drive in the opposite direction of where she lived. "Where are we going?"
"You'll see."
"I don't like surprises."
"Well, then, chances are, you're not going to like this," he said grimly.
He turned into The Tracks. The streets here had gone through changes. Like many others in town, the buildings dated back to the turn of the twentieth century. But somewhere in the past fifty years, the neighborhood had started to go. Many of the houses had been declared off-limits due to dangerous conditions. Slowly, with the resurgence of neighborhood pride and the Historical Society's interference, some of that had started to change. Houses had been purchased, slated for rehaul, and were in various stages of renovation.
They made a left and ended up on a cul-de-sac.
Houses gleamed with the quaint and charming aura of the old redone.
Except for one.
The two-story Victorian, with its busted turrets, cracked paint and lovely but crooked wraparound porch hadn't been touched, though there did seem to be signs of life. The lawn had been mowed. There was a potted plant on an upstairs windowsill.
Mac parked in front of it. "Home sweet home," he said wryly. "Come on."
The foyer had orange shag carpet. "From 1972," he said, disgusted. "The idiots put it over hardwood floors. I'll be restoring that soon as I can, before I go blind from the glow."
The living room had an ornately carved fireplace, painted nauseous green. "The 1970s were a bitch on good taste," he said. "Green and orange should have been outlawed. I'm going to restore that, too."
The kitchen was a treasure trove of nooks and crannies, but there the cabinet doors had been removed, the edgings painted in black.
"Not sure what the hell year someone tortured this poor kitchen, but it's so bad I'll probably start here." He looked at her from inscrutable eyes. "If I ever get out of debt."
"Debt?"
His expression was grim, and definitely said "back off." Fine. But she resented his obvious thought process—that because she'd seen his parents and knew his background, she'd assumed he had money. "You're an amazing talent when it comes to renovation," she said slowly. "You need to get into some of South Village's renovation projects. There's lots of money there."
"I plan to. After your building is finished, my résumé is complete. I have several bids in with the town council, bids I'm staking everything on."
"So I'm a stepping stone."
"If you want to look at it that way."
"A stepping stone, and apparently a gold digger as well."
He winced and rubbed the day's growth of beard on his jaw.
The sound of it made her belly quiver, but temper took precedence. "It's true, isn't it? You're showing me all this to make sure I know you don't have any money like your parents. That pisses me off, Mac."
"Look, I sold everything I had to get into this place. I think I even promised the bank my firstborn child. I'm feeling a little protective."
Which, she figured, was as much of an apology as she was going to get.
"I'm not who you think I am," he said.
She put her hands on her hips. "And just who do I think you are?"
"A man with a trust fund."
"Well, isn't that flattering." Oh, she was _so_ out of there. She got two steps before he grabbed her arm.
"Okay, listen," he said to her back. "My ex-wife took just about everything I had in the divorce. There's nothing left for anyone to want."
She struggled to contain her temper. And couldn't.
"But even before that I didn't have much. I walked away from all that right out of high school when I went into the police academy."
That caused her to crane her neck and blink at him. "You were a cop?"
"Until four years ago. And there's not a lot of money in that vocation either, trust me."
"I don't care about your money, Mac. And it's damned insulting that you think I do."
"I saw your eyes light up talking about my parents' money."
"What you saw," she said through grated teeth, wondering how such a smart man could be so _dumb,_ "was a woman thrilled to the bone to have met a man who could understand her. A man who came from a similar background, a man who, in spite of it, is going to make his own way." She softened her voice because suddenly she couldn't keep yelling at him with her throat burning. "A man who can see the potential in something, and want to make it right. God, Mac, don't you see? I saw more of you today than you've ever let me see, and it should have been wonderful. It should have been a joy to realize we're both doing the same thing, taking a piece of history and bringing it back. How you managed to suck the fun right out of that is beyond me, but you have."
Jerking free, she walked to the doorway, and then looked back. "I'm sorry we can't share that. I'm sorry I drive you crazy. But most of all, I'm sorry you can't move on after your marriage." It didn't escape her that she hadn't easily moved on after Jeff. "For that I'm really, really sorry."
"This has nothing to do with her."
"Yes, it does. I'm ready for a ride back now, please."
"Yeah. Fine." He gestured for her to go first.
The walk through the living room toward the front door was a long one, or so it seemed with him trailing after her. Silent.
Seemed it was the best he was going to do.
In the living room, on the green mantel, was a photograph of a much younger Lynn and Thomas Mackenzie. Standing between them was Mac, looking ridiculously young. She'd guess eighteen, given the graduation cap on his head. He'd been tall even then, though much lankier as he stood there with his arms around his parents, smiling a wide, cocky grin utterly void of his usual cynicism.
Her breath caught at how joyful he looked.
"That was a long time ago," he said behind her.
"I was just wondering what it would take to put that carefree, happy-go-lucky smile back on your face." She faced him. "I bet wild, screaming, sweaty sex would do it." Then she walked out the front door.
When he came out a moment later, he climbed into the truck, stared straight ahead with his hands on the wheel, and let out a slow breath. "That was low, offering me wild, screaming, sweaty sex in a weak moment."
"I wasn't offering you anything." She put on her seat belt and refused to look at him. "And you've never had a weak moment."
"Baby, every moment I'm around you, I'm weak."
She put on her sunglasses, lifted her chin. "You should have that fixed."
"Let me guess...with wild, screaming, sweaty sex?"
"Whatever works."
With a low laughing groan, he started the truck and took her home.
**_CHAPTER 12_**
**TAYLOR'S PHONE WAS RINGING** when she walked in her apartment. After getting dropped off by Mac, she'd spent the rest of the day at every estate sale within a thirty-mile radius, and was suitably exhausted.
"Need ice cream?" Suzanne asked when Taylor answered.
She kicked off her heels, sank to her bed and sighed. "How did you know?"
"Falling in love is a fattening process, hon. I should know, I've gained five pounds since I fell for Ryan. I could be there in fifteen minutes with double chocolate fudge."
"I'm not falling in love, and I'm not going to gain one ounce over a _man,_ believe me."
Suzanne laughed, but Taylor was dead serious. She'd learned a lot today, mostly that no matter what she thought she could feel for Mac, it wasn't ever going to be a two-way street, so forget it. Especially given what he'd thought of her. He'd actually figured her as a...a gold digger!
He'd be lucky if she gave him the time of day. He'd be lucky if...
Damn him, but he'd done the one thing she'd told herself he couldn't. He'd hurt her. She sighed. "I'm sorry, Suzanne. I'm just...tired."
"You've been working too hard."
"Nothing a good night's sleep won't cure."
"Are you sure? The offer still stands, fifteen minutes."
Taylor fell to her back on the bed and stared up at the ceiling. "I'm okay, but thanks."
After she hung up, she fell asleep almost immediately, only to be abruptly awoken sometime later by the unmistakable and terrifying sound of someone trying to break into her apartment.
* * *
MAC LAY NAKED on his bed, sprawled on his back, hands beneath his head, watching time go by.
Midnight.
One o'clock.
Two o'clock.
Apparently sleep wasn't going to come.
It was the look on Taylor's face tormenting him—when she'd seen his parents, when she'd seen his place, when he'd been such an ass because she'd gotten so sappy over that picture of him.
He'd wanted her to be plastic. He'd wanted her to be after the family money. He'd wanted, over and over again, for her to reveal a nature he could hate.
Instead she'd been...well, Taylor. Passionate. Steadfast. And unfailingly, consistently, wonderfully behind him.
Even when he hadn't been behind her.
When his phone rang in the middle of the dark, dark night, it startled him out of his thoughts, which was just as well, since he had no idea where he was going with them.
"M-Mac?"
He'd never heard her sound scared before, and he sat straight up. "Taylor? What's the matter?"
"You, um, left your nail gun here, which actually turned out to be a good thing." She let out a slightly hysterical laugh. "Oh, Mac."
He clutched the phone. "You're scaring me. What's wrong?"
"Two guys broke in tonight to steal some tools. They found me instead."
Mac's heart stopped. "Did they—"
"No, I didn't let them steal your tools, they're all still here. The police said—"
" _You,_ Taylor," he said through a throat nearly closed with fear. "Are _you_ okay?"
"Oh. Yeah, I'm okay. I held them off with your nail gun." She managed another laugh. "Thank God it was plugged in, because all I had to do was lift it and put my finger on the trigger. It was just like in _Lethal Weapon,_ or was that _Lethal Weapon II?_ You know the one where they—"
"Taylor." He kept his voice even with real effort. "Are the police with you now?"
"They just left..."
Her voice quivered, and broke his damn heart. "I'll be there in five minutes."
"No. No, I'm fine—"
"Five minutes," he promised, but drove so fast he made it in three.
* * *
AT 2:15 IN THE MORNING the traffic was light to nonexistent in South Village. There were the people emptying out of the bars, and a few other stragglers, but he still managed to get a spot right out front of Taylor's building.
There was grim satisfaction in that.
He used his key and let himself in. "Taylor?"
The only concession to what had happened was that every light in the place was on.
Which by itself spoke volumes, as Taylor was fastidious when it came to wasting electricity.
_"Taylor?"_ he yelled as he took the stairs.
He found her in her bedroom, sitting on her bed reading _Cosmo_ and sipping iced tea, calm as you please. At her feet, on the pristine bed, lay his nail gun, the cord trailing to the electrical outlet. She was plugged in, ready to go.
Striding toward her, he grabbed the magazine and tossed it aside. He set the tea down on the floor and hauled her to her feet so he could look at her.
Not a hair was out of place. The blond strands fell neatly just past her shoulders. She wore makeup, including a see-through gloss that smelled like strawberries. On the body that had made him want to beg since the day he'd met her was a long column of pale peach silk that clung to her every curve, a ribbon of it tied beneath her breasts, pushing them up and nearly out.
There wasn't a visible scratch on her, but that didn't mean—
"You didn't have to come," she said. "I told you I'm f—"
"Did they touch you?"
"Of course not. I had them up against the wall. I even shot a few nails into the air to show them I meant business. They were scared spitless, the idiots."
"So you're not hurt."
"I just said so."
Oh yeah, she was still pissed at him from earlier. But so was he. He was pissed because she made him care. She made him want her, and it wasn't just a physical ache, which really got him.
Then, as if there wasn't enough steam coming out his ears, she said, "You can go now that you've seen for yourself I'm just fine and dandy."
"Taylor—"
"Look, I've already offered you wild, screaming, sweaty sex, and you turned that down flat. Tonight was a bit scary for me, and if you're not going to help me burn off some stress, if you're just going to stand there looking like a deadly calm cop, then go. Just go."
"You think I'm calm?"
"Aren't you?"
He picked up the nail gun and hurled it across the room at the wall, where it made a satisfactory crash, denting the brand-new drywall nicely, before hitting the floor.
She eyed the wall, then the tool on the floor, now in pieces. "So maybe you're not calm."
Not knowing if he planned on shaking her silly or simply kissing her, he jerked her up against him. "Hell, no, I'm not calm. You could have been hurt tonight, or killed, because you're too stubborn. I told you, damn it, I told you, it wasn't safe to be in this building all alone, but would you listen? Do you ever listen?"
"This is my home," she said, right in his face. "No one or nothing scares me away."
"Yeah? Well then you're either a fool or the bravest woman I've ever met."
She looked away, and beneath his hands, shivered. "I'm not a fool. I knew enough to be scared."
She shivered again. "But I also knew enough to protect myself."
What was it about her that stabbed right through his heart? "I know, Princess." But knowing it didn't ease his own terror of what could have happened to her tonight. Shaken, he put his forehead to hers.
"Christ, Taylor." Still gripping her, face-to-face, he let out a slow, careful breath. It didn't calm him in the slightest. "You're getting to me, you with your terrified eyes and shaking limbs. You with your giving soul tucked behind that tough, don't-give-a-shit exterior. You are getting to me. You, Taylor. Only you."
She didn't shiver again, instead she fisted her hands in his hair and, keeping her eyes open on his, very softly, very gently, put her lips to his jaw.
"Thank you," she whispered.
"For what? Nearly getting you killed?"
"They weren't going to kill me, they were just young punks looking for tools."
"Which proves my point. This was my fault. You're coming home with me."
"Yes."
"To sleep," he clarified into her triumphant, hungry expression.
"That, too," she whispered, and put her hand in his. "Let's go."
**_CHAPTER 13_**
**CLEARLY HE'D LOST HIS MIND.** That, or with the perpetual hard-on he'd had over the past weeks, the serious blood loss from his brain had taken its toll.
But it didn't stop him from bringing her home to his place. It didn't stop him from fantasizing the entire way about what she did or did not have on beneath that clinging silk. And it didn't stop him from wishing he hadn't told her all they were going to do was sleep.
She sighed as they moved up his front walkway. "I'm so tired."
Good. With any luck she'd fall asleep. Like now. Calling himself every sort of fool, he opened his front door and forced himself to lean back, away from her, rather than bury his face in her hair the way he wanted to. As he shut the door behind them, Taylor turned to him, reached up, cupped the back of his neck in her palm and pulled his mouth to hers.
So much for going to sleep.
She danced the very tip of her tongue along the crease in his lips, and with a groan, he let her in. He could do nothing else. It was hot, combustive, and so instantly out of control he staggered back, slamming them both against the front door.
Laughing breathlessly, Taylor tried to climb up his body, raining little hot kisses over his face as she went. "Here, Mac?"
"No." He was shaking with the need to hold her, _shaking._ He led her to his bedroom and gestured her in.
Afraid of more mindless kisses, he stayed by the door.
Taylor went straight for the bed, kneeled on the mattress and turned to him with a sexy little smile that shot straight to his gut.
And parts south.
But at the sight of him against the door—he was gripping the handle tight behind him like it was his anchor—her smile faded. "I thought you were going to comfort me."
"You're going to be fine." It was himself he was worried about at the moment. She hadn't changed from her peach silk, but had added a matching robe that she'd allowed to fall open.
As he watched, she shrugged it off, leaving her shoulders bare, leaving her body bare except for that column of silk and the ribbon beneath her breasts. Crossing her arms, she ran her hands up and down her arms, and shivered. "It cooled off tonight."
Had it? He was hot as hell, sweating just watching her.
When she shivered again, he sighed the sigh of a man facing the guillotine and took an instinctive step toward her. Suddenly his knees were brushing up against the mattress.
Taylor dropped her arms to her sides. The bodice of her gown dipped low, exposing the alluring, soft curves of her breasts. Her nipples pressed against the silk, begging for the attention he was dying to give. The material clung to her belly, her hips, her thighs, molding and outlining every part of her that he'd been dying to touch, taste, since he'd first seen her.
"Warm me up," she whispered.
His hands slid to her hips before he could stop himself. "Taylor—"
"No, don't think. Just touch me." Another shiver wracked her frame, and when he looked into her eyes, he realized she wasn't nearly as calm as she'd pretended to be. In those green, green depths he saw her lingering fear and horror, and his heart clenched again.
"Please?" she whispered, wrapping her arms around his neck.
He lifted a hand, skimming his thumb over her jaw. "Comfort sex, Taylor? Is that going to work for us?"
Pressing that mind-blowing body to his, she arched just a little, enough to make him lose his train of thought and nearly drive him to his knees. "Definitely, it's going to work for us."
"But—"
She put her mouth to his, and he lost himself in the kiss. "Wanting you like this," she said when they finally came up for air, their lips separating with a little suction noise that made him want more, "isn't a life-and-death decision, Mac. It's just...quenching a thirst."
A sigh escaped him while his hands slid up and down her back. "And afterward...you won't be thirsty anymore?"
"Not if you do it right." She put her mouth to the corner of his and nibbled some more. "Do you know how to do it right, Mac?"
"I think I can fumble around and figure it out." As it was useless to resist, he gave in, telling himself this was simply a one-time deal. _Comfort sex,_ as they'd both just said. God knew they both needed it.
"Well, if you need any help," Taylor teased, letting out a moan when he fisted his hands in her hair and tugged lightly, exposing her throat to his hungry mouth. "I'll be happy to help you out."
Then his mouth captured hers and there was no more talking because she couldn't keep a single, solitary thought in her head except for _more, please, please, more._
His hands danced over her body, gripping her hips, squeezing, before racing roughly up her spine to press her closer to his hot, hard length. His mouth shifted from gentle to ravishing so fast her head spun, leaving her no choice but to fly with him.
Which was exactly what she'd wanted, it was what she'd wanted since she'd first set eyes on him.
Then he pulled back, staring into her eyes while his fingers played with the teeny-tiny straps on her shoulders.
He wanted her. He'd come for her when she'd called him, he'd come and been there for her in a way no one had in so long she'd forgotten how good it felt.
No matter what they each claimed, this was no comfort sex. Maybe Mac wasn't ready to admit it, but she could wait for the words.
The actions though...those she needed, desperately. Now. She could feel her nipples, rigid against the silk. She could feel her thighs, and the dampness between them, and the blood roaring through her veins. Every single atom in her body was vibrantly aware, overly sensitized and aroused. She was tingling all over, so full of anticipation and raging need she could hardly stand it.
She didn't have to. Mac yanked off his shirt. Kicked off his shoes, then his pants, before coming back to her.
At the sight of him, she caught her breath. He was amazing. Magnificent. _Huge._ She would have looked at him forever, but he yanked her against him, hard.
Hard was good. Bending her back over his arm, he put his mouth on her breast through the material of her nightie, and nearly sent her through the roof. His hand skimmed down her legs, then back up again, beneath the material now. Up the back of her thigh.
Where he discovered she wasn't wearing panties.
The knowledge ripped a rough groan from him and a shiver of thrill from her.
If he didn't get to the rest soon, now, she was going to explode. Looping her arms around his neck, she slid her silk against his hot, pulsing, vibrating body.
Tensing, his hands tightened on her. "Taylor..."
"Yes," she sighed into his ear, then rimmed it with her tongue, making him groan and his body jerk. Encouraged, she slid her tongue down the side of his throat, thrilling to the way his fingers dug into her hips.
"Condom," he growled, frustration pouring off him. "I don't have a—"
"I do." She pulled back, giving him a small, secret smile as she stroked her fingers over her own breast.
"I tucked one right in here."
His mouth fell open as if he needed it that way just to breathe.
"Are you going to get it?" she asked, the words barely out of her mouth before he yanked on the rib bon beneath her breasts, spilling them free. As the nightie fell away, he found the small foil packet.
Scooping it up, he stared at it. "You packed a condom."
"I believe in safe sex."
"Yes, but..." Now he looked at her; hot and bothered, and baffled. An irresistible combination.
"When you called," he said slowly. "You were so scared—"
"I was."
"But when I got there you were dressed, with your hair and makeup done. Waiting for me. You...you knew we were going to do this," he accused, his eyes narrowing.
"I knew you would come," she said truthfully. "I also knew only you would make it better. Only you, Mac."
He groaned again, and she didn't know if it was from getting his first good look at her naked body, or that he was trying to garner the strength to push her away. On the off chance he could muster enough to do it, she wrapped her arms around his neck, her legs around his waist and tugged backward so that she fell on her back on the mattress and he fell over her.
"Damn it," he grated out, lifting his weight off her. "I'll hurt you."
"I'm not fragile." Arching her hips, she brushed the very center of herself over his erection, making him hiss out a breath. "I'm not going to break."
He ripped open the little packet while she ran her fingernails down his belly, smiling when he tensed at the teasing caress and dropped the condom. Twice. Then his hands tightened on her bare thighs, pushing them open so that she was spread out for him, and her smile faded, replaced by a bated breath as she closed her eyes and waited for him to put the condom on, to thrust home.
When he didn't, she opened her eyes. He was staring down at her. "You are the sexiest thing I've ever seen," he said in a reverent whisper. Still holding her gaze, he skimmed his hands down her thighs until his thumbs met at her creamy center. Slowly, so maddeningly slowly she thought she would die, he gently traced them over her, lightly, up and down. Down and up. Her hips ground helplessly, and when he did it again, then yet again, she let out a helpless whimper.
"So wet," he whispered, dropping his gaze now to watch what he was doing, slipping a finger into her as he continued to apply pressure to just the right spot with his thumb.
Her body strained, and the wordless demand she made was most definitely a cry for more.
Still watching as his fingers drove her to the edge, he made a sound, too, a deep, throaty moan.
And then drew away.
With a desperate sob, Taylor arched her hips upward. If he stopped now...
"Shh." He slid his body off the bed until he was kneeling on the floor, his broad shoulders wedged between her thighs. Sliding his hands beneath her undulating hips, he held her still.
She couldn't move, she couldn't breathe... "Mac..."
"I know, baby. I know." And he put his mouth on her.
At the first silken stroke of his tongue she nearly burst right out of her own skin. At his second stroke, her back went rigid. Gripping handfuls of his sheets in her fists, she mindlessly ground her hips in spite of the hold he had on her.
"Good?" he lifted his head to ask.
She throbbed. Ached. Quivered. _"Good."_
He bent his head again and sucked her into his mouth, making her come in a blinding, thrashing, gut-wrenching rush. It was like nothing she'd ever experienced, it was completely out of her realm, and as he didn't stop, she kept coming.
Gradually, with gentle kisses and wordless murmurs, he brought her down, and when her breathing slowed, he leisurely made his way back up her body, using that wonderful, talented, greedy mouth.
"Oh, Mac," she sighed with a last shudder, and kissed him.
Mac could have drowned in that kiss, and might have if he hadn't been so consumed with the sheer primal lust ripping through his body. "Condom," he managed, and held it up. "Gotta get it on."
"I'll do it."
"Just do it quick."
Her fingers on him nearly took him right over, and when she stopped to swirl her thumb over the tip of him, he could only growl her name, on the very thin edge of control. He couldn't help it, not after watching her come apart for him, not with the taste of her still on his lips. He'd wanted to make her shudder and gasp and cry out his name, and he'd done that.
Now he wanted to do it again. He wanted her writhing beneath him. He wanted to see her, wanted to hear more of those mind-blowingly sexy little whimpers and panting entreaties.
"Now," she ordered, wrapping her fingers around the full length of him, guiding him to her. "Now."
With one thrust, he drove himself home, and then, overcome by the tight, wet heat surrounding him, by her hands gliding mindlessly up and down his back, by the long, ragged sigh that escaped her lips, he went utterly still before he lost it too fast.
"Please," she begged, those perfectly lacquered fingernails digging into his butt.
Oh yeah, he'd _please._ He let himself go, gave in to the tense, quivering, straining muscles that wanted to ride, and when she dug those nails in again, his hips started to piston madly, pounding into her, possessing, taking. Giving.
Tossing back her head, Taylor arched into him, pulling her legs back, taking even more of him, deeper, meeting him thrust for thrust.
Damp flesh slapped against damp flesh. Breathing came hard and uneven. The mattress creaked, the headboard hit the wall in a rhythm set by their desperate motions. It was the wild, screaming, sweaty sex she'd been tormenting him with for days.
Then, skin slick, chest heaving, Taylor cried out, gripping him for all she was worth as she splintered into a thousand pieces in his arms, shuddering, shuddering, her inner muscles clenching him so tight he could do nothing but hold on tight and follow her over.
They collapsed into each other, still quivering, still gasping, exhausted in their utter satisfaction.
Mac knew he needed to shift his weight, that he was heavy, but he was wrung out, panting for air, and still so entwined with Taylor he had no idea where her body began and his ended.
Several minutes passed like that, while he caught his breath and waited for his world to stop spinning.
"I can't see," Taylor finally said, voice hoarse.
With great effort he lifted his face from where it was plastered to the side of her neck. He had her hair in his mouth. Spitting it out, he said, "Your eyes are closed."
"Oh." She gave that some thought. Without opening them, she said, "And I can't feel my toes."
Craning his neck, he looked at her bare, very sexy toes. "I think it's because I'm heavy."
But when he would have moved off her, she tightened her fingers, which were still embedded in his butt. "Don't," she whispered, hugging him tight. "Not yet."
"You'll suffocate." With a huge effort, he heaved himself up to his elbows and stared, a little stunned, into her face.
Her cheeks were rosy. Her lips were wet and slightly swollen from his kisses. Her eyes were soft, void of that usual cool haze, and smiling at him.
She was so beautiful it took his breath. "Taylor..."
She slapped his butt. "Anyone ever tell you you're pretty good at that wild, screaming, sweaty sex stuff?"
" _Pretty_ good?"
The female smile turned quite smug. "Okay. Pretty, _pretty_ good." Her hips rocked, just a very little, tiny motion, but he felt it all the way to his toes.
Still hard within her, he thrust back.
Smugness vanished as she gasped.
He thrust again.
Her mouth fell open now, her eyes glazing over. "Mac? Again?"
"I have to upgrade 'pretty, pretty good' to 'mind-blowingly amazing,' don't I?"
Her breath caught when he dipped down and kissed a breast. "O-okay."
Another slow, rocking thrust.
And then again.
Taylor arched up, tighter than a drawn bow, her nipples beaded in two hard buds, her skin glowing and damp as he watched her fight to take what he held just out of her reach. "Oh, Mac. Faster. Harder."
He gave her harder but held back on the faster. Kept it slow until she was clutching at him, mindless again, on the very edge. "Mac!"
Just as it always had been, resisting her took every bit of control he had. "Is it pretty, pretty good?" he whispered.
"It's... Oh, Mac..."
"Easy," he whispered, running hot, openmouthed kisses down her throat. "We're going to take it nice and easy this time."
"No, I... _Please_... It's..." She opened her eyes on his, and he could tell by the desperation and need warring on her face she couldn't remember what he wanted to hear.
"Mind-blowingly amazing," he said, helping her. Then stroked a thumb right above where they were joined.
She bucked beneath him. "Mind-blowingly... Oh, yeah, it is," she managed as he slowly, slowly thrust into her. "Mind-blowingly amazing." And when he stroked his thumb again, her thighs and belly went taut. She held him in a death vise, suspended for a long heartbeat...staring up at him with wide, passion-dazed eyes as she let go, letting him watch her fall.
The honesty nearly killed him, and took him over, too. He let go, giving himself so completely he had no idea how he was going to ever get over it. Over her.
No idea at all.
**_CHAPTER 14_**
**TAYLOR AWOKE** as dawn broke and discovered two things. One, Mac was a serious pillow hog, and two, he slept like the living dead.
That worked in her favor, as she needed a moment to digest things, and she couldn't do that with him wrapped around her like a cocoon.
Gently as she could, she slid out from the two strong, warm arms holding her captive against his chest. They must have been sleeping like that for a while, as her face was stuck to his chest, her chest to his belly.
Lifting her head, she stared down into his face.
Eyes closed, breathing deeply, he didn't budge.
But there was the matter of his thigh, which he'd thrown over her body and was acting as a weight to keep her in place. Wriggling, she managed to turn from her side to her back, but his leg still pinned her. From her back, she rolled again.
And fell off the bed with a thump loud enough to wake the people of China.
But not one Thomas Mackenzie.
Popping back up, she stared at him, but other than a little moan of protest at the loss of her body heat, he didn't so much as flicker an eye.
She didn't know whether to be insulted or grateful, but settled on grateful. Padding naked to the bathroom, she gazed critically into the mirror and tried not to wince. What was it about wild, screaming, sweaty sex that so ruined a good hairdo? She fixed that quickly enough, and did the best she could with water and soap—so rough on her skin!—to get rid of the mascara beneath her eyes.
Mac was still asleep when she came back out. Slipping into his shirt, she went to the window and watched the sun rise over South Village. It came in slow waves of brilliant orange, yellow and red, probably from the smog, but it still took her breath.
"Hey." Accompanying this gruff, sexy voice came two nice and toasty arms, encircling her from behind.
"Hey yourself," she said, trying not to sigh with pleasure.
"Maybe you somehow missed how early it is."
She let herself go, closing her eyes, leaning back into the hard strength holding her. There was just something about a man's voice in the morning that made her want to melt. "No."
His mouth brushed her temple in a gesture so tender, so sweet, it made her want to cry. "Tell me what's wrong," he said quietly.
What was wrong? Only everything. And because of it, she couldn't speak, she'd gone mute around the football-size lump in her throat.
His lips cruised to her ear while his hands crisscrossed in front of her to smooth up and down her arms. "Regrets already?"
God, she loved his touch. It was...comforting. He didn't touch her breasts or between her legs, he didn't thrust his pelvis against her butt. None of the things she'd have expected of a normal, hot-blooded male first thing in the morning, looking to get lucky.
And emotion swamped her, drowning her. She was deathly afraid it was _deep_ emotion, and maybe even an emotion that began with the letter _L._
"Talk to me, Princess."
She drew a deep breath and watched the sun bursting a myriad of colors on the blooming day. After the night they'd had, she should be sleepy, but she was wide awake.
And it had been quite a night. They'd had only that one condom, but it hadn't mattered. They'd explored other options, pleasuring each other by touch, by mouth, leaving her little more than a sensory creature, fed on passion and hunger until they'd exhausted themselves.
Mac didn't say anything, but he kept on touching her, lightly, sweetly, sharing his heat.
"I haven't had a night like that since...well." She sighed. "It's been a long time." She closed her eyes and admitted the truth. "I haven't allowed myself to." Then, knowing she needed to say it face-to-face, she turned, tipped back her head. "That wasn't just comfort sex," she whispered. "That wasn't even just wild, screaming, sweaty sex."
Emotion flickered in his gaze, too, but leading the pack of all of it was a sudden regret. "Taylor—"
"No." She couldn't handle hearing him say it, that for him it hadn't been anything _but_ those things. She already knew. "Last night I opened up to you, in a way that..." She let out a slow breath and tried to smile. "In a way that, quite frankly, terrifies me. I'm not ready for it, Mac. I'm not ready and I'm fairly certain you're not either."
That he said nothing only fed the knowledge she was in over her head. Way over her head. Throat closing, she backed out of his arms. "I have things to think about, and you have your ex-wife to get over."
Shock filled his face. "What?"
"I've never felt the need to compete with anyone in my life, and I won't do it now."
"Ariel is not sharing my bed."
Ariel. The ex-wife had a name. She swallowed hard. It didn't escape her that she might be truly and completely over Jeff if she could feel this searing jealousy. God, she'd loved Jeff. But he was gone and she wasn't. "I didn't mean to drum this all up now. God." She squeezed her eyes shut. "We've been intimate in bed, but that's all. That's all it's going to be."
"Yes," he agreed softly when she opened her eyes and waited for a response. Her heart cracked, but what the hell had she expected him to say? He'd never been anything but brutally honest with her, and she had no one but herself to blame for getting in too deep now.
"You know what?" She managed a smile. "I need to go. Lots of stuff to do today."
She figured the fact that he'd silently driven her home meant he was even less ready than she'd imagined.
Still, he walked her in, even though it was nearly time for him to start work. He walked her up the stairs and into her apartment. He walked her all the way to her closed bedroom door.
Then he lifted a hand and stroked his knuckles across her jaw in a heartbreakingly tender gesture. Fighting the urge to grab his hand and hold it to her face, she opened the door and went inside.
More confused than ever.
* * *
IN MAC'S OPINION, the problem had nothing to do with confusion. It's just one could never be prepared to have your heart ripped open to another.
It had to do with acceptance. Trust. Willingness.
As in, was he willing to accept that Taylor was nothing like Ariel. Was he willing to trust that she would never, ever, try to destroy him the way Ariel had? Was he willing to open up and share himself, heart and soul?
No. No, he most definitely wasn't.
No doubt, Taylor drew him, and on far more than a physical level. And yet he knew enough to understand that trying again with her, and actually doing it, were two different things.
There could be no half-ass attempt here, he had to mean it. For Taylor's sake.
She'd been hurt by life, too, and he wouldn't toy with her. No, if he ever decided to have another relationship, he'd give it his all.
He just didn't have his all to give.
For two days, he didn't see much of her. Not because he avoided her, but because _she_ avoided him. She was good at it.
On the third day, Suzanne came over with a chest of leftovers from a party she'd catered, and informed Mac that Taylor was at an estate sale, drooling over some antiques from France.
"I can't wait until she can open her store," Suzanne said, popping open a Tupperware container. "She deserves it."
Moved by the delicious scent and the grumble in his empty belly, Mac unhooked his tool belt and let it hit the floor. "Store?"
"She hopes to use one of the downstairs storefronts to open her own antique shop." Suzanne shot him a look when his stomach grumbled loudly. Silently she handed him a napkin. "Mini quiches, if you're not too manly to eat such a thing."
"I'm not too manly to eat anything smelling that good." He nearly moaned at his first bite, then sank to the floor and did moan at his second. "You're a genius."
"No, that's Nicole. But I am good in the kitchen. Just like you're good with your hands."
Mac stopped midbite and glanced up in time to see Suzanne blush. "I mean, you do incredible work," she said, pointing to the wood floor molding and casing.
"She told you about the other night."
"No." She sat down next to him. "She didn't tell me anything, she didn't have to. Nicole and I had breakfast with her to discuss Nicole's upcoming wedding plans and..."
"And..."
"And we guessed. She had this...glow about her, and she was...I don't know... _happier_ than I've seen her in a while. Maybe happier than I've seen her ever." Suzanne nudged his shoulder with hers. "She never talks about it, never complains, but we know she's had it rough. We're her best friends, Mac, and we only just met six months ago. Before us, she had no one. I hate to think about her like that, so alone, but even with us hounding her all the time, she holds back. But with you..." She let out a gentle smile.
"Let's just say we're hoping she's not holding back."
He thought of the night he'd spent with Taylor.
The night he'd held her in his arms, the night they'd rocked each other's worlds with what should have been a simple bout of healthy, recreational sex.
And had really been so much more.
He looked into Suzanne's hopeful eyes and had to tell her the truth. "I don't know what we're doing, Taylor and I, but I doubt it's going in the direction you're thinking."
"Oh." Her sweet smile faded some. "Really?"
"Really," he said regretfully.
She took away his napkin, and then on second thought, took the quiches as well.
"Hey—" His stomach growled in protest.
"Sorry. Turns out I don't have any extra."
* * *
MAC WENT HOME to more mail. Mostly bills, which he was making his way through, slowly, methodically, _painfully._ He tossed the entire stack to his table, toppling over the previous stack.
And revealed a thick packet from South Village's Town Council. Staring at it, he told himself if they'd turned down his bids, it would have been a nice little white envelope with a short letter saying thanks but no thanks.
But then again, a thanks but no thanks could come with a stack of other projects to bid.
Hence the thick packet.
Heart pounding uncomfortably, he backed to a chair and plopped into it, his legs a little rubbery. Holding his breath, he ripped into the envelope and started reading.
* * *
TAYLOR'S ARCHITECT, Ty Patrick O'Grady, was a tall, dark, gorgeous man with an Irish accent, flashing eyes and a roguish smile.
Taylor happened to know who put that spectacular smile on his face on a daily basis. Nicole, who was going to marry Ty as soon as he convinced her to set an actual date.
But for now, Taylor and Ty, who had some last minute things to go over, were in a meeting. A walking meeting.
Ty grinned at her as they munched on soft pretzels and drank sodas, walking through the lunch crowd along a particularly swank street halfway between Ty's home office and her building.
Using what was left of his pretzel, he pointed at a new upscale lingerie shop. The window display was what had caught his attention. More specifically, the naughty looking black leather skirt, matching crop top, five-inch spikes and whip.
Taylor knew she couldn't so much as afford a pair of panties from the place. How times change, she thought with a sigh that didn't really signify any wistfulness for the changes in her life. She loved where she was, and wouldn't trade it for...well, for all the money in her grandfather's estate.
And yet a new outfit once in a while would be nice. Yes, she had gorgeous clothes, but all of them—like the emerald-green sleeveless dress, matching strappy sandals and wide-brimmed hat she wore today—were leftovers from another era.
Those days were long gone, even if her clothing addiction wasn't.
"I should buy that outfit for Nicole," Ty said around a huge bite. "What do you think?"
Taylor laughed at the vision of Dr. Nicole Mann, out of her preferred jeans and doctor's jacket, and into the leather. "She'd kill you."
"Yeah." Ty's fond grin didn't fade. "Love that woman madly, I do."
At the utterly pathetically lovelorn expression on this big, tough, former bad boy's Irish face, Taylor had to sigh. What would it be like to bring such a man to his knees with love?
_Hell,_ she reminded herself viciously. It would be hell, at least on the heart.
She'd come close to forgetting that while lying in Mac's arms, being driven crazy by his mouth, his touch, his voice. She'd come close to forgetting just about everything, including the fact he was never going to love her the way she secretly wanted to be loved.
She'd avoided him. Mostly because she was weak.
One look from his whiskey-colored eyes and she'd leap right back into his arms and screw good pride. She'd take what she could get.
Well, the hell with that. "So about my bathroom..."
"Yep." Ty aimed that killer smile at her. "You can have that antique stand-alone bathtub on claws like you want. The floor will support it, and so will the plumbing. No changes required."
"And the window turrets? That won't change the structure of the roof?"
"It might piss off your contractor having to add trim now, but it won't change anything major."
Hmm. Pissing off Mac so he was as unbalanced as she was did have its merits. "How about I let _you_ tell him."
Ty, incredibly observant, cocked his head. "Is something wrong?"
"Of course not."
"Mac working out okay?"
"Absolutely."
Not fooled, Ty's brilliant blue eyes narrowed. "I suggested him because even though he's relatively new to this scale and scope of work, I've seen what he can do. The man is magic with his hands."
Taylor stuck her tongue in her cheek. Oh good God, was Mac magic with his hands. "I know."
"But something's wrong," he repeated, studying her closely.
"No, it's nothing." She looked into Ty's worried gaze and managed a smile. "Nothing. Everything is great, you should see it."
"Yes, let's see it," he said firmly, making her sigh. She'd learned there was nothing more protective than a man who was going to marry your best friend. "It's blocks out of your way," she protested, but Ty merely kept walking.
"Well, at least slow down," she grumbled after him. "I'm not doing a marathon in these three-inch sandals simply because you're feeling overprotective."
"I wouldn't be feeling overprotective if you'd tell me what's wrong."
"Nothing!"
"We're just making sure, darlin'."
They passed by several restaurants that had such delicious scents wafting from them Taylor could only inhale deeply and dream. Her budget meant dinner tonight consisted of a can of soup.
They turned the corner and passed three clothing stores that had her drooling, but the next shop, called Accents, had her wrinkling her nose in snobbery.
The "accents" for decorating were all new, cheap and in her opinion, tacky.
On her street now, right across from her building in fact, they came to a flower stand. Before they crossed, Ty touched a pot of daisies. He sniffed at the dozen wrapped roses, and smiled at the lilies.
"Sentimental fool," Taylor murmured, having to smile when he shot her an admitting grin.
"Nicole has a soft spot for flowers," he said.
What the rough and tough, cool-minded Nicole had was a soft spot for this man. "Go for it," she said, her heart sighing.
He bought a dozen red roses and held them out to Taylor to smell.
Instead she leaned in close to the man, who in her opinion smelled better than any flower. "You are the sweetest fiancé in town, you know that?" He looked so shocked, she laughed. "You _are,_ " she insisted.
_"Sweet."_ He laughed, too. "Well, that's a new one."
"Trust me, these are going to get you very lucky tonight." Then she kissed him, one quick smacking kiss on the lips.
With a laugh, he wrapped an arm around her and squeezed her tight. "Aren't I just?"
He set her down, and Taylor put one hand on her head to steady her hat, and one on his chest to steady herself. Still smiling, she craned her neck and checked the street before crossing.
And went utterly still.
Mac stood out front of her building, looking right at her. Funny, how her heart leaped. Or maybe it wasn't funny at all.
He wore the Levi's with the hole over the knees, a dark T-shirt and a scowl the likes of which she hadn't seen since that very first day when he'd looked at her as if she were the bug on his windshield.
She hadn't seen him yet today, so she couldn't be the cause of the scowl. Honestly, _men_...she had _no_ idea what had crawled up his—
Ty still had an arm over her shoulders as he peered past her contractor to the building behind him. "What a beauty she's turning out to be. Wonder who your genius architect is?" Grinning, he set his cheek to hers.
Mac's scowl deepened, and with delightful understanding, Taylor grinned, too.
Oh yes, she'd just figured out that frown.
Ridiculous as it was, the fool man was jealous.
**_CHAPTER 15_**
**MAC STOOD THERE** out front of Taylor's building, envelope in hand, watching the woman he'd rushed over to show it to hug and kiss another man.
That he knew and respected that man and his work didn't help. He didn't care if Ty Patrick O'Grady was her architect or her trash guy, the impact of seeing them cozying up was the same.
God, he felt like an ass standing there, when only a moment ago he'd been giddy, and hot as hell. He had figured he'd tell her the news, then start off by kissing her senseless, and from there talk his way right up the stairs to her apartment and her very frilly bed.
They'd make good use out of all those ridiculous pillows she had, and burn off some badly needed tension while they were at it.
And then afterward, they'd go on their merry way as they had before, sated and relaxed, until the next time the tension got to be too much.
In which case he'd gallantly offer his body yet again.
It was a system that would work well for both of them, he had decided, and no one need get hurt. In fact, the only regret he had was wasting the past few days thinking instead of doing.
Bottom line, Taylor had been hurt, too, and she, more than any other woman, understood not wanting to get hurt again. They could be together without really being together.
All parties happy.
Or so he'd thought. But that was before she'd moved on, and had climbed into another man's arms.
He understood, they hadn't had anything exclusive. Hell, he'd made it crystal clear he hadn't wanted exclusive, but damn, his bed was barely cold from the night they'd spent in it.
He remembered everything. No doubt he still had the fingernail marks on his butt from her eager, demanding hands. She'd mewled and clung and cried out his name, and if memory served right—and he knew damn well it did—she'd woken him up, _twice,_ with her own hungry demands for more.
So it hadn't been all him, damn it.
Screw it. Since Taylor was still hugging Ty, Mac spun on his heel and went back to his truck. He got caught in traffic, which really topped off his mood, then stalked through his dark house and stared down at his bed.
Unmade and lit by the moon, all he could remember when he looked at it was tangled limbs, breathless pleas and a pleasure so great it had been painful, _physically_ painful, to let her go.
It was still painful.
* * *
HE WAS GONE. Taylor couldn't believe it. By the time she crossed the street, Mac had left. She calmly finished her business with Ty, then went upstairs, because this was going to require a clothing change. She prepared herself with a sort of adrenaline rush she didn't think she should be proud of. Amusement and fury.
Fury and amusement.
She would wear siren red because it suited her. The matching do-me shoes with the five-inch spiked heels were a bonus because she figured she could always take them off and hit the stubborn, idiotic lug over the head with them to make herself feel better.
Oh, he had some nerve, shooting her that scathing look and then vanishing.
She washed up, waxed, shined and polished, all the silly female rituals that usually made her feel better. Calmer.
And pictured him suffering the entire time. She really shouldn't be proud of the fact she wanted him to suffer.
The sight of his truck in his driveway made her giddy with relief. He was home, and he would listen to her while she told him all the reasons she was mad at him, and then she'd walk back out to her car in her sexy little dress, picturing him cross-eyed with lust behind her, solid in the knowledge that she drove him as crazy as he drove her.
She'd sleep well knowing he was lying awake staring at his ceiling, calling himself every kind of name for letting her walk out of his life.
That's right, she'd sleep well. Then she would wake up tomorrow and move on. And now that she knew her heart worked again, she'd go find a man who could appreciate that.
And her.
He didn't answer her knock. The fury built back up. Ignoring her, was he? She knocked again, harder, determined to see this out.
She simply had to share this anger, or she was going to blow up.
She lifted her fist again, but the door opened so unexpectedly she almost solidly rapped him on the nose.
He didn't even flinch, not this man with nerves of steel. No, he just cocked a brow and propped the doorway open with his shoulder.
His naked shoulder, because all he wore was a... She gulped hard and struggled to maintain eye contract.
A damn towel. His entire body was pebbled with water drops. Given that, and the fact his hair was wet, too, and she realized she'd gotten him out of the shower.
Her traitorous body quivered at the thought of his long, leanly muscled body in the steam, water cascading down his tanned, sleek skin, his head back, his eyes closed in ecstasy as the hot water beaded over him.
Oh good Lord, now she could hardly breathe.
His eyes, those light, light eyes, traveled slowly up her body. "Fancy meeting you here," he said.
"Fancy that."
"What is it you need?"
"It's...rather complicated."
"Is it? That's a shame then, as I'm running a bit late."
"This can't wait, Mac."
"Suit yourself," he said with a shrug. "But I'm going to get dressed."
She followed him down the hall to the very bedroom where he'd once upon a time rocked her entire world.
Casual as he pleased, he dropped his towel.
"What are you doing?" she croaked, but didn't look away, not even to blink as he shoved those long, long legs and mouthwatering ass into a pair of pants.
Turning to her as he zipped them up, she had a moment to wish he'd shifted around just a second sooner—
"I'm dressing for my parents' anniversary party."
A white dress shirt came next, covering that wide chest that hadn't come from any gym, but years of hard labor.
She struggled to maintain her composure and sauntered over to him, telling herself _now,_ give it to him _now,_ trying desperately to remember all the reasons why she was so angry. But instead of wrapping her fingers around his neck and squeezing, she slid them into his wet hair and pressed her body to his.
He jerked, proving he was not immune. "What are you doing?"
"I came over here to yell at you, but apparently I'm going to kiss you instead."
"Yeah?"
"Yeah."
"Good." Before she could move, he grabbed her, whipped them both around and captured her between the hard wall and his harder body.
Trapped, she gave one startled yelp before his mouth slammed down on hers. His body was like iron, his hands hard and hot as they slid from her hips to her back. And his mouth... Oh, his mouth. All of her fantasies of a down and dirty, knock-out-fight paled in significance against the reality of what was happening between them now. Nothing, _nothing_ could have prepared her for the ruthless, ravenous, reckless, unrestrained, raw sexuality of the man holding her to the wall, or her own ruthless, ravenous, reckless, unrestrained response.
His hands molded her body, sculptured her, and only when they were both shuddering, sighing, lost in the driving, pulsing need, did he pull back. Chest heaving, he lifted his head enough to look into her eyes and grate out, "Who are you kissing?"
Stunned by the overwhelming emotions rocketing through her, she could only blink.
His hands held her jaw, his thumbs teasing the lips that wanted his back on them. "Say my name, Taylor. Say it so I know you're right here, with me and no one else."
Oh, but if that didn't remind her she was furious at him! Shoving him away, she straightened her shoulders and glared at him. "I know who I kiss. And if you think I don't, then you don't know me near well enough for me to see this through."
With her pride on her shoulders like a ball and chain, she stalked right out of his bedroom, back down the hall and out to her car. It took her shaking fingers a few tries to get the key into the ignition, but she succeeded, and peeled away from the curb with a satisfactory screech.
It was the only satisfaction she had that entire night.
* * *
SHE WAS WOKEN at six in the morning by the sound of a power tool, which really fried her, because she'd only just managed to fall asleep an hour ago.
Furious all over again, that he would _dare_ to interrupt her beauty sleep—and she made no mistake, she knew exactly who was down there making the racket—she stalked out of her apartment and down the stairs.
The first thing she saw when she entered the storefront was the antique hat stand, all dark oak and brass. It stood in the center of the room that was empty except for a makeshift worktable.
Unable to help from touching the beautiful thing, she ran a finger down the unusual stand, guessing it was over a hundred years old.
"Incredible, isn't it?"
Turning, she faced Mac, who stood in the doorway covered in sawdust. Hanging from his hand was the offending noisemaker, a saw of some kind. "Suzanne told me you're not selling off your entire antique collection," he said. "That you're hoping to open a store right here." He lifted a broad shoulder. "My grandmother left me a few pieces of furniture, most of which I've sold, but this piece I kept because of the beauty of the wood."
"So it's yours."
"No, it's yours. I'm giving it to you."
He was giving it to her. No one gave her anything, or hadn't since Jeff. She braced herself for the sharp pain from the thought of him, but all she felt was a nice warm fuzzy. She'd thought about that a lot lately. Somewhere along the line, she'd stopped comparing the two men, stopped putting Jeff on a pedestal. As for where she'd put _this_ man, she didn't yet know. "Why are you giving it to me?" Her voice wasn't the angry one she'd imagined on the walk downstairs, but she felt sucker punched at the look in his eyes as he set down the saw, dusted himself off and moved closer.
There wasn't any matching anger in his eyes. None. Instead, what she saw was a deep brooding that came from sorrow and regret.
He cared. He cared deeply.
Yes, he thought that caring was strictly physical. He thought that caring could be set on the back burner until it boiled over, and then with one night of amazing sex, it could be taken care of.
Until the next time it boiled over.
But he was wrong, dead wrong, and she was going to prove it to him. She ran her hands up his tense, hot, slightly damp arms.
"What are you doing?" he asked hoarsely.
"Touching you."
"Don't," he grated out through clenched teeth when she danced her fingers over his chest. His hands fisted at his sides. "I've had a really shitty morning."
She would have said the same of herself only a few moments ago. "So you'd say you're...worked up?"
"Yes." His jaw bunched. "I'd definitely say that."
"Well, that would make two of us, Mac." She smiled at him beneath her half-closed eyes and squirmed against him, just a little, just enough to have the breath hissing out from between his teeth. "I'm worked up over you."
"Well, that's convenient. I'm worked up over you. I got approval from the town council. I'm renovating two of their projects in the next phase."
"Oh, Mac!" She knew how much it meant to him, and her heart hitched. "Let's celebrate."
His eyes raked over her, hands still at his sides. "You're wearing my T-shirt."
"You left it here. I've claimed it as my own." Backing away from him, she shimmied in a little circle to ensure he caught the full effect of his T-shirt on her body.
Mac caught the full effect all right. He caught the way the torn neck made one sleeve fall off her creamy shoulder, exposing the top of one breast. He caught the way the hem lifted, revealing a peekaboo hint of tantalizing twin cheeks, making him wonder what the hell, if anything, she had on beneath.
She did another circle and his eyes glazed. She ran her own hands down her body. Her breasts beaded beneath the cotton. Then she turned her back to him again, running her hands through her hair. As she did, the hem of the shirt slipped up another inch, showing another flash of her tight, rounded cheeks.
No panties.
With a low growl that reverberated in his chest, he lunged forward, pressing her between the makeshift worktable and his own body.
Trapped, she let out a low hum and bent forward, gliding her hands up the table, thrusting her butt against his crotch. "Mac," she murmured. "Mac..."
The sound of his name murmured in that helpless little pant on her lips spurred him on, even as it soothed. She was here, with him, not with anyone but him.
"Yeah." His hands slid up her spine, then back to her hips, grinding her against the hard-on to beat all hard-ons.
"Mac..."
"I know." Gripping the cotton of the shirt she wore, he shoved it up to her waist.
And groaned at the sight of her bare, sweet ass rubbing against his jeans. He could feel the heat of her through the denim, and imagined her soft, bare flesh getting more and more aroused at the friction. Groaning again he reached around her to cup her breasts.
Thrusting back against him, her hands fisted on the edge of the wood table, gasping as he rasped his fingers over her nipples, capturing them, stroking, pulling, stroking again until she was chanting his name over and over, her hips pumping in a rhythm as old as time.
He was as close to coming in his jeans as a horny teen with his first erection, but it wasn't enough. He needed to see her face, taste her mouth, watch her go over for him, only him.
Pulling back, he heard her sound of protest and smiled grimly as he whipped her around. "I'm not going anywhere, Princess, and neither are you."
"Thank God," she panted, and when he lifted her up to the table, she spread her legs for him, sighing when he stepped between them and gripped her bare ass in his hands to hold her in place. Her head fell back on her shoulders, her eyes closed, her mouth open.
"Look at me," he demanded, giving her a little shake until she blinked huge, desire-slumberous eyes at him. He rocked his hips, watching those eyes go opaque with need. "Can anyone else make you feel this way, Taylor? Anyone?" Another slow rock of his hips, and another moan tumbled from her lips. "Like you'd rather have this than breathe? Can they?"
"Mac..." She tried pulling him down to her, tried to wrap her legs around his waist, which would have pressed the hottest, wettest part of her against the neediest part of him.
But he'd have lost it on the spot. Instead, he held her still and whipped the T-shirt off her. Then bent to a gloriously full, high breast, whispering her name as he rubbed his jaw along the plump curve.
In response, she fisted her fingers in his hair and did her best to make him prematurely bald.
"Answer me," he said, and ran his tongue over her nipple. "Can anyone else make you feel this way?"
Taylor tried to respond, honest to God she did, even though her body was tightening, tightening, tightening, lost in desperate need. "No." She tried to concentrate even as he drove her toward the very edge. "No one else makes me feel like this." She gasped as he swirled his tongue over her other pebbled nipple. "N-no one. Ty is just..."
He sucked her into his mouth at the same time he slid a finger into her, and Taylor cried out, her thoughts scattering into nothing.
"Ty is just..." he repeated for her, doing something with his finger that made her just about swallow her tongue.
"He's..." She struggled to concentrate. "I..." He added another finger to the first, and then his thumb got into the action, slowly skimming over her swollen, wet flesh. Her entire body quivered, so close—
"You...what, Taylor?"
Oh, those fingers! "He's like my brother!"
He went utterly still. "Your...brother?"
"He's marrying my best friend." Licking her dry lips, she stared up at the man who had two fingers inside her, his mouth on her breast and held her on the very edge of an orgasm in a way no one had ever dared.
She wanted that orgasm!
She was also falling in love with him. Damn it, damn it, not all the way in love, just a little tiny bit. But even a little tiny bit was bad. There would be no one else for her, she knew in a moment of clarity, it was this man, and as he pressed down with his thumb and wriggled those amazingly talented fingers inside her, it hit her as hard and fast as the explosive orgasm did.
When her breath finally shuddered back into her lungs, when she could breathe again, she released her death grip on Mac's shirt and fell back on the table.
"More?" he asked.
"Lots more." She waited while his gaze met hers, knowing that if she couldn't tell him how she really felt, she could at least tell him this. "No one else makes me feel this way, Mac." Her breathing still hadn't returned to normal, and he ran a finger over the pulse she knew raced at the base of her neck. She caught his fingers in hers. "I never let them." With a slow roll of her hips, she smiled, determined to keep this light, determined _not_ to let him see she'd started the fall. "Now tell me you have a condom in your pocket."
"I have a condom in my pocket." He reached into said pocket and let out a grim smile. "This time I have three."
There was something deliciously distracting and sinfully wicked about having the rough wood at her back and Mac, still fully dressed at her front. Just as there was something incredibly touching about the way he drew his fingers down her torso, followed by his mouth, his eyes closed as he worshipped her body with everything he had. It tightened her throat and brought her back around to the terrifying thoughts of forever, watching him make love to her slowly, thoroughly...and yet she couldn't refuse him, not when he reared up and stripped off his shirt, undid his jeans, then tenderly sank into her, not when he started a devastating rhythm matched with a kiss so sweet and deep she never wanted it to end, and not when he finally nudged them both over so that they exploded together.
When it was over, he fell on her, pressing her into the wood. He was hot, heavy, and she held on to him, wanting his weight, wanting his heat, and wanting it so much she clung, just a little, when she never clung. And right then, still gasping for breath, legs still hooked around his thighs, she realized the truth.
She wasn't just a little bit in love.
There was no such thing as a _little_ bit in love.
Nope, she'd gone and fallen all the way.
**_CHAPTER 16_**
**ATTEMPTING TO WORK** when one's head was screwed up was a bad idea. All day long Mac passed that worktable in the downstairs unit, and like Pavlov's salivating dog, he got a hard-on from just the sight of it.
Taylor had vanished, and he went back and forth between looking for her like a pathetic love-struck teen, and wanting to run like hell.
Swamped by various crews and their questions, he did neither, and by the time he went home, he still hadn't seen her again.
But late that night, she came to his door with a soft knock and a warm, sexy smile.
She came the next night as well. And the next.
The nights she didn't, he went to her. And for two weeks they made wild, passionate, devastating love until dawn, and then silently went their own way.
No strings attached.
At least that's what he knew Taylor would have claimed if he'd asked her, but he didn't ask. He wasn't that big a fool. He could see, damn it, and what he saw was so much emotion reflected in her eyes he nearly drowned in them every time he looked at her.
She loved him. Christ, she loved him.
He was torn between ecstasy and sheer terror.
One night she showed up at his door wearing a siren red dress that made him drool. The back was a series of strings crisscrossed over her slim spine, the front was little more than a low dipping bodice snug to the top of her thighs.
Her mile long legs were capped by matching red strappy sandals with heels that put them at eye level.
Shutting the door behind her, she leaned back against the wood and shot him a little smile that made his penis jerk to attention. "Hi," she said in a sultry voice.
"Hi, yourself," he said, feeling underdressed in nothing but nylon running shorts.
With a saucy smile, she put her hands on his arms and spun them, reversing their positions so _he_ was against the door.
With a little laugh, he said, "So I'm guessing _you're_ in charge tonight—"
With a yank, she hauled his shorts down to his ankles.
"Tay—"
She dropped to her knees. Gliding her hands up the front of his legs, she stared at his body, parting her lips thoughtfully. "You want me, big boy?"
More than his next breath, but since she was eye level with the proof, he figured the point moot.
She leaned forward and, as if he were her favorite flavor of lollipop, she licked him.
His knees nearly buckled.
"How much do you want me, Mac?"
They'd been together nearly every night, and nearly every night they'd been silent during their searing, erotic, sexual encounters, unless "harder!", "more!", "yes, God, yes!" and "don't stop!" counted.
So it shocked him when he reached down to pull her up, intending to carry her off to the bedroom for more hot and fast sex, that she held him off.
"Remember when you had me on your worktable?" Still on her knees, she looked up at him. "When you asked me if anyone else makes me feel like you did? If anyone else made me quiver and ache, the way I do when I'm with you?"
Oh yeah, he remembered.
She wrapped her fingers around him, and he couldn't quite contain the rough sound that rumbled from his chest.
With a slow stroke that made him quiver, she watched him carefully. "So I'm asking you now...you've had the time to figure it out. Does anyone besides me..." She stroked again, then bent and gave yet another mind-blowing stroke of her tongue. "Anyone at all, make you feel like this? Does anyone else make you tremble and ache, the way you do with me?"
He stared down at her mouth only an inch from where he wanted it most and felt the shock of her question mix in with the haze of overwhelming lust she'd spun around him.
Lifting her gaze, she gave him a smile a little shaky around the edges, and he realized she was not as confident and as in charge as she wanted him to think, not even close. "Taylor—"
"It's a simple question, Mac. Does anyone else make you feel like this, yes...or no."
"Call me slow," he said, dazed by sensory overload as he hauled her to her feet. "But I'm finally getting it." Hands on her arms he looked into her eyes. "You're not holding back on me because of Jeff. You're not holding back on me because of money. You think... My God," he said on a mirthless laugh, and shook his head. "You think I'm still in love with my ex-wife."
"Ariel."
"I remember her name," he said tightly, and kicking his shorts off his ankles, he stalked naked to his kitchen, where he grabbed a tall glass of water for his suddenly very dry throat.
"I'm sorry," she said from the doorway, arms crossed, face miserable. "I shouldn't have pressed you that way. I know what it's like to love someone and then lose them. You idealize them to the point where no one else can compare. I did that with Jeff." She swallowed hard. "I compared you to him, and that wasn't fair."
"Taylor." He shook his head. He let out a laugh, and then another, and then weak for some odd reason, he sank to a chair to laugh some more.
She went from miserable to furious. Chin high, eyes flashing, she whizzed by him on her very determined way to the back door. Snagging her arm, he hauled her down and into his lap, where she wriggled and fought him. "Shh, stop." Damn, he should have put on his shorts to protect himself. "Stop...I'm sorry."
"You're laughing at me."
"Are you kidding? No. _No,_ " he repeated softly, holding her still. "I'm laughing at _me,_ because I'm a jerk. I didn't know that's what you thought, that I was hung up on Ariel. That I idealized her." As it was hard to admit the truth with her sparkling, accusing eyes on him, he tucked her face into the crook of his neck, set his chin on her head and spoke into the quiet night. "I met her at a town council meeting, did I ever tell you that?"
"No. Mac—"
"She was the friend of a friend's daughter."
"You don't have to—"
"Shut up," he said. "I was young, and dazzled. She was sweet and warm and loving, and wanted me for me, and not who my parents were."
"So you got married."
"We eloped. It was what she wanted, and I was touched because I'd told her how I intended to make it on my own without my parents' help, and I thought she was showing me she wanted that, too."
Taylor shifted in his arms so that she could see into his face. "I don't want you to apologize for loving her, Mac. I love it that you've loved before, that you're not afraid to admit it. And deep down, I'm even a little flattered that you compared me to her, that it made caring about me so hard because you loved her so much."
"Really?" He closed his eyes, let out a harsh laugh. "You're really not going to like the rest of this then."
"I'm...not?"
"No." He drew a deep breath. "Ariel started asking about money, wanting me to get some from my parents. She wanted a big, new house. She wanted a new car. New clothes from Europe. She wanted parties. She wanted, wanted, wanted, and started to hate me for not giving in."
"Oh, Mac. I—"
He put a finger to her lips. "I have to get the rest of this out, Taylor, and with you looking at me like that, with your heart in your eyes and your body sitting on my very naked one, I'm feeling far more inclined to see how strong this table is than to tell you the damn truth."
"Tell me," she said, and bit her lip, ostensibly to keep herself quiet.
Mac wondered if she'd be so pliant in a minute when she heard the rest. "She decided I had been a mistake, a big one. She went after other men—wealthy, affluent men who could give her what she wanted."
"She left you," she breathed, and her eyes hardened. "Forget it. I am nothing like her, nothing."
"I know that," he said, and sighed wearily.
"There's...more?"
Oh yeah, there was more. "When she picked out the right guy for herself, she ran up every credit card I had, emptied every bank account and screwed me over for the building loan I had been trying to get to start my first renovation project, all as a goodbye present."
Taylor's eyes widened even as they filled. "My God, how could she? She loved you."
"She never loved me."
Her eyes never left his. "And...there's still more isn't there?"
"Yes." Mac's heart started beating heavily, he'd never said the words out loud before. "When she went to file for divorce, she found out she was pregnant. And...she didn't want to be. I didn't want her anymore, but the baby. God. I wanted that baby." To his horror, his eyes burned. "She, um...aborted."
Taylor let out a soft sound of disbelief. Sliding her hands into his hair, she put her forehead to his, offering neither empty platitudes or meaningless compassion.
He wanted neither.
She gave him the only thing he did want, herself. Slowly, gently, she put her mouth to his, kissing first one corner, then the other, and then pulled back, her eyes shimmering with unshed tears. "I want to love you, Mac. Not wild, up-against-the-wall sex, not on this table...I want to take you to your bed and love you until you forget."
He looked at her, _in_ her, feeling his entire chest constrict at what he saw in her eyes. Once upon a time he'd have said it couldn't be done, no one could make him forget, but as he stood with her in his arms, as he strode down the hallway toward the bed, he thought maybe, just maybe, she was the woman to do it.
* * *
TAYLOR WOKE UP just before dawn, and with a deep sigh sat up. Time to go, just like every other dawn for the past few weeks. The best nights of her life, and she paid for them by having to get up before the sun so that neither of them panicked and felt claustrophobic.
Well, she didn't feel claustrophobic, and hadn't, not once during a single one of those nights, humming with pleasure in Mac's arms, and not last night.
Last night...she'd held Mac for hours and had wondered how anyone on this earth could have treated him as Ariel had. If Taylor had been lucky enough for someone to love her like that, for _Mac_ to love her like that, she would have lived every single day loving him back with everything she had.
Her eyes filled thinking about it, because she knew now why he resisted so much, just as she now knew things wouldn't change. He cared about her, no doubt. No one could make love to her the way he did and not care deeply. But that was only part of intimacy, and she didn't see it going much further. They were in Mac's comfort zone now, and there they'd stay. If the only option to her was walking away...well, she wouldn't. Couldn't.
Putting her feet on the cold wooden floor, she went to stand up.
And was stopped by a big, warm hand to her wrist. "Without even waking me?" came his sleep-roughened, sexy-as-hell voice.
He lay sprawled on his belly, his broad shoulders and long, long legs taking up nearly the entire mattress. Only seconds before she'd been right there with him, and she was shocked by the immense yearning to leap back in and snuggle tight.
If she did, if she so much as touched him right now, she'd lose it. "Got work, Slick," she said, and lightly slapped his very nice butt.
"No, you don't." Without letting go of her, he pushed up, sat back against the headboard and tugged.
She fell against that warm chest, and had to close her eyes when she braced her hands against it. "Mac—"
"Don't go."
She tried to wriggle free. "I need to."
"No, you don't. You're running out of here because you think that's what I want." He waited until she opened her eyes, helping her along by putting his hands on her face. "You don't want to scare me," he said in a terrifyingly gentle voice. "You don't want to worry me with your feelings—"
Again she tried to get up. "Mac—"
"No, listen. I have to say this. I had a dream. You were gone." A spasm of pain crossed his features as he held her still. "And I was back to the way things were. Alone. I hated it. It felt cold without you, Taylor. Empty."
"It...did?"
"It's all happened so slowly, I didn't realize..."
Her heart stopped. "What happened so slowly?"
He blew out a breath. "Before you, I told myself I never wanted to share myself again, and that included my bed. I told myself I would never open up to a woman, that I would never want, need or ache for one to distraction."
"I know, Mac. God. I know—"
"But I was wrong. Life can't be lived like that. You taught me that. Only you, Taylor."
She stopped trying to get free and stared at him. "I'm sorry. My heart just stopped, which means I'm not getting any blood to my brain, so I must have heard you wrong. Could you..."
"I love you, Taylor." His smile, a bit wobbly, jump-started her heart. His thumbs stroked her jaw, helping with blood flow. "Did you hear that?"
"I... Yes," she whispered, stunned. "Yes."
"I love you with all I've got, and I hope to hell you feel something close to that for me, too, because I don't think I can go through this again and have you not feel it back...." He stared at her, then hissed out a breath. "Could you say something here? Anything?"
She put her fingers to his mouth, and through a half laugh, half sob, put her forehead to his and managed to repeat his words back. "I love you. God, Mac, I love you with all I've got. And I've hoped like hell you felt _something_ back, too, because I _know_ I can't go through this again." She let out a shaky breath. "And not have you feel it back."
Closing his eyes, he wrapped his arms around her and squeezed so hard she could hardly draw a breath, but who needed air? Not her, she had Mac, she had his love, she had everything she could ever want, forget breathing.
Mac rolled over, tucking her snugly beneath him, pressing her into the mattress as he lifted his head. Sinking a hand into her hair, he smiled down at her. "Be mine, Taylor. Be my wife, my lover, my heart."
His smile nearly burst her heart. "Yes. Yes to all of it."
His lips found hers in a sealing, promising kiss. "So from now on you'll wake up with me? Forever, just the two of us?"
She held her breath, not because he was still squeezing her too hard, which he was, but because she had yet to share her one last, very private, very secret fantasy with anyone. "I'll wake up with you, only you."
He grinned.
And she slowly let out that breath wondering if she was going to get hurt after all. "Until we have a baby. A little girl," she added breathlessly when his grin faded, "with your beautiful eyes and my savvy fashion sense and then there will be three of us."
He didn't say anything and she rushed on. "She'll want to pounce on us early in the mornings, and snuggle in," she teased, while inside she was dying, dying, _dying,_ not knowing if he would ever want to have kids after what Ariel had done to him...
He ran a finger down her jaw, her throat, to where her heart lay in her chest nearly beating its way past her ribs. "You want to have a baby," he said, his voice thick. "With me."
"I do," she whispered. "Only with you, Mac. What do you say?"
He watched his finger circle over her heart, eyes solemn. Then he slowly smiled and lifted his head. "I can't think of anything more perfect than that."
Her heart sighed, completely content.
"Let's do it, Princess. Let's do it all." He rolled with her over the bed again, and then again, until they were both laughing.
Then bent his head to hers to make it all come true.
**_EPILOGUE_**
_One Year Later_
**"THE PLACE IS FULL,"** Nicole told Taylor, plopping into a plush chair in the bride's dressing room of the church. "We have exactly five minutes to get out there. How many people did you invite anyway, a bazillion?"
Standing in front of a full-length mirror admiring herself all in white satin and lace, Taylor sighed with so much joy she could hardly contain herself. "Just about."
"Mac's out there of course. He's got his eagle eye on this door, let me tell you."
At the thought of him, her heart nearly burst. "Does he now?"
"Yeah, he's looking a bit like he hit the lottery."
"He did," Taylor said, and laughed.
Suzanne came up on her right and ran a finger over the bride's veil. "You look gorgeous."
"You both look pretty gorgeous yourselves."
Nicole sighed and came up on the other side, the three of them staring at each other in the reflection. "You're right. We don't look half-bad, considering we're dressed to the nines. Why couldn't we wear jeans like we did at my wedding last month? Think how original it would have been."
"Oh, be quiet. The dress won't kill you." Suzanne smiled when Nicole lifted a bottle of champagne and three long stemmed glasses. "Well, that's more like it. Aren't you so sweet to think of it. I thank Ty for that."
"The man did sweeten me up," Nicole admitted, pouring them each a glass. "What can I say, love did exactly what you said it would, hit the three of us like a tornado."
Taylor laughed. "A tornado. So _that's_ why I feel so unsteady on my feet."
"You look steady enough," Suzanne said softly, reaching for her hand. "I'm so glad you're happy. I'm just so glad for all of us." Her eyes filled. "I love you guys."
"Ah, hell, I actually have mascara on today and she's going to get sloppy." Nicole sighed as her own eyes went suspiciously bright. "But I love you guys, too."
Taylor laughed, and a tear fell. "To us, then. All six of us." Both she and Nicole lifted their glasses, but Suzanne did not.
"What's the matter?" Nicole demanded.
"I, um...can't drink." She grinned and patted her stomach. "As of this morning."
Nicole's jaw dropped. "You're pregnant."
"Yep."
"Oh my God." Taylor's heart felt like it was going to burst as they all hugged and ended up messing up each other's makeup anyway.
"We're trying, too," Nicole admitted, which prompted another group hug and more tears. Then they went back to the mirror to check the damage.
Taylor felt so full of love and joy and hope, she could burst. "Well then." Her voice was thick as she kissed first Suzanne's cheek, then Nicole's. She lifted her glass. "To all of us, seven for now, and God willing, more to come."
"To all of us."
"All of us."
Grinning, crying, arm in arm they walked out of the room, ready for all of it.
* * * * *
Keep reading for an exclusive excerpt from
_New York Times_ bestselling author Jill Shalvis's next Heartbreaker Bay novel,
_PLAYING FOR KEEPS_!
Available January 2019 from Avon Books.
**_CHAPTER ONE_**
_#Suits_
Sadie Lane walked through the day spa, closing up for the night, alone as usual. Her coworkers had left, but even if they hadn't, they'd just be milling around with their expensive teas, complaining about how hard this job was.
They had no idea how ridiculous that was to her, but as the lowest person on the ladder, she'd managed to keep her opinions to herself. She was sure it'd only be a matter of time before her mouth overtook her good sense.
Moving around shutting down the computers and dimming the lights, she fantasized about going home and stripping out of her daytime yoga pants and replacing them with her nighttime yoga pants. Unfortunately, even after eight hours on her feet, that wasn't in the cards for her.
Her phone buzzed an incoming call and a glance at the screen gave her an eye twitch. "Hey, Mom."
"You always forget to call me back. I've been trying to discuss your sister's wedding details with you for weeks now, and..."
Sadie listened with half her brain, the other half wandering off. Did she have time to grab an order of sliders and crispy fries from O'Riley's, the pub across the courtyard, before heading to her other job? Lunch had been eons ago...
"Mercedes Alyssa Lane, are you even listening to me?" her mom asked.
Being full-named always got her back up. It wasn't that she had anything against her name—okay, so she sort of did because who named a kid after the car where that kid had been conceived?—but more than anything, she had a whole lot against her mother's tone. "Of course I'm listening."
She wasn't. She was thinking about dessert after the sliders. Maybe cookies, maybe a brownie. Maybe both.
"Honey," her mom said, her voice going tentative. "You're not feeling... _sad_ again, are you?" She whispered _sad_ as if was a bad word.
And to be fair, for most of Sadie's teenage years it had been a bad word, along with _angry_ , _misunderstood_ , _sullen_ , and _unhappy_. To say that she and her mom had a complicated relationship was pretty much the understatement of the year.
"Nope," Sadie said. "I'm fine." This was an automated response because she didn't want to deal with the _all you have to do to get over the blues is think positively_ speech again, well-meaning as it was. But her mom was winding up for the big finish, so Sadie braced herself because in three, two, one—
"Remember what Dr. Evans always told you. To get over the blues, all you have to do is think positively."
Resisting the urge to smack her phone into her own forehead, Sadie drew a deep breath and sank into the cushy chair in her station, where her clients sat while she applied permanent makeup. This was her bread-and-butter job, seeing as the love-of-her-heart job—working as a tattoo artist in the Canvas Shop right next door—didn't pay enough yet. And call it silly and frivolous, but she'd grown fond of eating.
The problem was, all the time on her feet working way too many hours a day left her exhausted. And maybe the teeniest bit cranky. But not, it should be noted, sad. At least not at the moment. "Mom, you know it's not that easy, right?"
"To think positively? Of course it is. You just do it. Take your sister, for instance..."
Sadie closed her eyes and caught a few _z_ 's while her mom went on about Clara, whom Sadie loved and adored even if she was annoyingly perfect—
"Sadie? Yes or no?"
"Hmm?" She sat upright, opening her eyes. She'd missed a question, but pretending she knew what was going on at all times was her MO. If she couldn't blow her family away with her brilliance, plan B was always to baffle them with her bullshit. "Sure," she said. "Whatever you guys decide."
"Well, that's very... sweet of you," her mom said, sounding surprised. "And very unlike you."
Hoping she hadn't just agreed to wear a frothy Little Bo-Peep bridesmaid dress, Sadie let her gaze shift to the window. Over a hundred years ago, the Pacific Pier Building had been built around a beautiful cobblestoned courtyard that each of the ground floor shops and businesses opened onto, making it convenient for people watching.
One of Sadie's favorite pastimes.
Seeing as it was February in San Francisco, specifically the Cow Hollow District, a thick icy fog had descended over the dark evening with the promise of rain. She loved a good storm, the darker the better, and figured that love came from her own dark, stormy heart.
The lights had all come on, strung from potted tree to potted tree and along the wrought-iron benches around the water fountain. The area was usually a hub of activity. But tonight only the faint glow of the lights was visible behind the wall of fog, and there was no one in sight. Except... wait a minute. A form appeared out of the fog. A tall, leanly muscled form, his overcoat billowing out behind him like he was some sort of superhero.
Sadie called him Suits.
He had a real name, she knew. Caleb Parker. But she'd never said it out loud, preferring her nickname for him, since with the exception of the few times she'd run into him at a gym on the other side of Cow Hollow near the marina, she'd never seen him in anything but a suit. And though she herself wasn't a suit kind of girl, she could admit there was something about watching him move in gorgeous clothes that had probably cost more than her entire year's rent.
"Mercedes?" her mother said in her ear. "You still there?"
"Yep." She searched her brain for the conversation she'd just missed. "Don't worry, I'll be on time for Clara's wedding dress fitting appointment."
"Did you get a date for the wedding yet?"
Sadie sighed.
"It's a wedding," her mom said firmly. "You'll need a date. And anyway, you're past due to find your Prince Charming. _Way_ past due."
"Mom, I don't need a Prince Charming. Forest animals who clean, yes, but it's a hard pass for me on Prince Charming."
"Everyone needs romance," her mom said. "My book club just read the Fifty Shades trilogy and—"
"Those books aren't romance, they're erotica."
"Actually, they were very romantic. Christian Grey's a bazillionaire who falls in love with a regular girl. It's like a Cinderella story."
Sadie sighed. " _Fifty Shades of Grey_ is only romantic because the guy's a billionaire. If he was living in a trailer, it'd be a _Criminal Minds_ episode."
Her mom sighed. "I just don't know what you have against love."
"I don't have anything against it." Sadie hoped her nose wasn't growing at the lie. "I just don't need it right now." Or ever...
"But you haven't dated anyone since Wes, and that was three years ago. He was a good man."
An attorney, Wes had been sure of himself. Sexy, with an edge. Sadie was long past being hurt over what had happened between them, but she still wasn't feeling the need to let someone new in, mostly because she simply hadn't been attracted to anyone.
_What about Suits?_ a voice inside her head whispered as she made her way from one window to the next in order to keep him in her sights. It was misting now and his dark hair shimmered with droplets every time he passed beneath a lamppost. Like Wes had been, he too was sure of himself. Sexy, with an edge...
He was everything she no longer let herself want.
Suddenly, he abruptly stopped between the day spa and the Canvas Shop. Crouching low in the now pouring rain, he stared at something she couldn't see. "I've gotta go, Mom. I'll call you back."
"You always say that, but you're fibbing. You're not supposed to fib to family."
"Uh-huh," Sadie said dryly. "Tooth fairy, Santa Claus, and the Easter bunny..." And at her mother's gasp, she gently disconnected, squelching a wince because she'd most definitely pay for that later. Her mom had a lot of talents, and one of them was being able to hold a grudge for a hundred years.
Sadie had a few talents herself, such as not sleeping at night and enjoying chocolate just a little too much. And okay, so she also was talented at drinking tequila, preferably in the form of a frosty lime margarita.
Slipping her phone into her back pocket, she went back to window-gazing to see what Suits was up to. He was still balanced on the balls of his feet, the wind and now rain pummeling his back, seemingly unnoticed.
What the actual hell?
She knew a few things about him. Such as the fact that he had lean muscles everywhere you might want a man to have lean muscles, and that women tended to fall over themselves when he smiled. His eyes were a beautiful caramel, with flecks of gold that sparkled when he laughed. He was some sort of tech genius and used to work at a government think tank. He'd invented a bunch of stuff including a series of apps that he and his business partner had sold to a multinational technology company. More recently the two of them had created a way of getting meds and medical care into remote developing nations via unmanned drones. He was innovative and inventive on a large scale, smart, charismatic...
Oh, and there was one more thing—he and Sadie rubbed each other the wrong way by just breathing. She wasn't even sure how it'd started, but there was an energy between the two of them she didn't understand. At best it made her squirm. At worst, it sometimes kept her up at night.
Sadie's best friend, Ivy, who ran The Taco Truck parked outside the building, had decided that she and Caleb Parker shared an unrequited animalistic lust and nothing could convince her she was wrong.
But it wasn't lust, because Sadie no longer gave in to lust, animalistic or otherwise. Yes, he was fun, flirty, and charming, but she was highly suspicious of all those things. Her idea of fun, flirty, and charming meant being as sarcastic as possible. It'd done the trick too, scaring men off for years. But oddly, Suits seemed to be able to handle her sarcasm without so much as blinking an eye.
She had no idea what to make of that.
And what was he doing still all hunkered down like that in the rain? Was he hurt?
Driven by curiosity and the inability she had of letting anything go, she unlocked and opened the front door of the day spa and stuck her head out. "Hey."
Staring at the brick wall, he didn't turn her way or even glance over. He didn't do anything except to say "shh."
Oh, no. No, he did _not_ , and she stepped outside to tell him what she thought of him and his _"shh"_ and where he could put it. Sideways.
But with his gaze still on whatever was in front of him, he held up a hand, silently ordering her to stop where she was.
It was like he _wanted_ her to lose her temper.
But then he was reaching out to the wall, and she realized over the noise of the storm that he was talking quietly to something.
Something that was growling at him fiercely.
"Don't be scared," he said softly. "I'm not going to hurt you, I promise."
The growling got a little louder, but Suits didn't back away, he just held eye contact with what sounded like a huge dog that Sadie still couldn't see in the dark shadows.
"Okay," Suits said. "Come here. Slowly."
Sadie realized with a start that he was talking to _her_. "What? No way. What is it?"
"Come closer and you'll see."
Damn him. And damn her insatiable curiosity because she stepped out from beneath the spa's overhang and immediately got wind and rain in her face for her efforts. Pulling out her cell phone, she accessed her flashlight app and aimed it at the wall.
"Don't," he said, wrapping his hand around her wrist, bringing the phone down to her side. "You'll scare it."
"Better that than getting eaten." She shrugged off his warm hand but went still when the growling upped a notch.
"I think it's hurt," Suits said. "Come here, baby," he coaxed gently. "Let me see."
Sadie bet that voice worked for him in the bedroom, but no way would it work here. And yet... the matted, drenched shadow scooted away from the wall, not nearly as large as she'd thought. Not a young puppy but not a grown dog either. It had a way-too-skinny tan-colored body and a black face with black eyes. "Aw, looks like a young oversize pug," she murmured.
Suits shook his head. "Too big for a pug. It's probably got some bullmastiff in it though."
A skin-and-bones bullmastiff with only three legs, Sadie realized as it shifted closer, and her entire heart melted. "Oh my God." Moving toward it now without hesitation, she got only a few steps before the dog scrambled to escape her approach like a cat on linoleum, heading right at Suits.
With a surprised grunt, he fell to his ass on the wet cobblestones. "Okay," he said, hands up, backing up on those fine butt cheeks as if suddenly terrified of the dog trying to get into his lap. "Okay, see? You're safe now, right? Stay. Stay and sit."
The dog didn't stay. Or sit, for that matter. Instead, it leaned on Suits's bent legs, leaving dirty beige fur sticking to his pants.
He sucked in a breath and seemed to hold it. "I'd really like to be your person, but I can't."
"Arf!" Translation: _Too late, buddy, you're totally my person._
"No, you don't understand," Suits said. "I literally _can't_."
Undeterred by this news, the dog continued to huddle close to his new human, even as that human shifted back, trying to avoid further contact.
Finally, Suits lifted his head and looked at Sadie. "Help."
Fascinated by this unexpected show of weakness in the man who'd always come off as invincible, she shook her head. "I think it thinks you're its mama."
He glanced around the courtyard as if to see who the dog might belong to, but there was no one.
"Arf!" the dog repeated and sat on Suits's foot.
"Oh, I hear you, and we're going to help you, I promise."
Sadie went brows up. "I know you must mean you and the mouse in your pocket because _we_ "—she gestured with a finger between him and herself—"are most definitely not a _we_."
Ignoring that, he got to his feet, lifting his hands at the dog, giving the universal gesture for _stay_. But the minute Suits raised his hands, the dog squeaked in terror and leapt back as if he'd been shoved. Off-balance with only three legs, it fell to its back, exposing its underbelly and the fact that _it_ was a _she_.
Sadie didn't easily attach. To anything. But right then and there, she fell in love with her. Not partially, but all the way in love, because neglected and mistreated meant they were soul mates. "I'm going to kill her owner."
"Not if I get to them first." Suits's eyes flashed absolute fury, though his voice remained calm as he once again squatted low, trying to get his six-foot-plus frame as nonthreateningly small as he could. "It's okay, baby," he crooned softly. "We're together now, for better or worse, even if you're going to kill me."
"What are you talking about?" Sadie asked. "She wouldn't hurt a damn fly, much less kill you."
Proving that, the dog slowly once again scooted toward Suits, head down, her hind end a little wiggly as she crawled close, trying to get into his lap.
The sweet hope of it had Sadie's heart pretty much exploding in her chest.
With a sigh, Suits wrapped his arms around the dog and hugged her close. In response, the cutie-pie set her oversize head on his broad chest.
"Yeah, that's some killer," Sadie said, shoving her wet hair from her face.
"I'm allergic."
Suits said this so nonchalantly that she blinked. "Is that some sort of a euphemism for 'I hate dogs'?"
"No," he said. "Reach into my front left pocket."
She snorted. "You're kidding me, right? Does anyone actually fall for that?"
"If I pass out, you'll need my keys to play Nurse Nightingale."
She paused, staring at his face. She saw no sign that he was teasing—very unusual for the charming, easygoing guy she knew him to be.
"I'm trusting you to not let me die," he said as if he was discussing the weather.
"This isn't funny."
He met her gaze, his own more serious than she'd ever seen him. "If I don't make it, promise me you'll at least make up something really good for my funeral, okay? Like, I died heroically saving your sexy ass, and not because a sweet dog like this one hugged me."
"Okay," she said slowly, "I'm starting to think you're really not joking."
"I never joke about dying."
_Playing for Keeps_
will be available January 2019
wherever books are sold!
_Playing for Keeps_: Copyright © 2019 by Jill Shalvis
ISBN-13: 978-1-488-05301-6
Messing with Mac
First published by Harlequin Temptation in 2003
This edition published in 2019
Copyright © 2003 by Jill Shalvis
All rights reserved. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of publisher, Harlequin Enterprises Limited, 22 Adelaide St. West, 40th Floor Toronto, ON M5H 4E3 Canada.
All characters in this book have no existence outside the imagination of the author and have no relation whatsoever to anyone bearing the same name or names. They are not even distantly inspired by any individual known or unknown to the author, and all incidents are pure invention.
This edition published by arrangement with Harlequin Books S.A.
® and ™ are trademarks of the publisher. Trademarks indicated with ® are registered in the United States Patent and Trademark Office, the Canadian Intellectual Property Office and in other countries.
www.Harlequin.com
CONTENTS
1. Cover
2. Back Cover Text
3. Dear Reader
4. Title Page
5. Contents
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. Epilogue
23. Excerpt
24. Copyright
|
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{"url":"http:\/\/mathematica.stackexchange.com\/questions\/5006\/how-to-plot-cdf-of-a-poisson-distribution-in-mathematica?answertab=active","text":"# How to plot CDF of a Poisson distribution in Mathematica\n\nHow to plot the following in Mathematica:\n\n$$P = \\sum_{k=C+1}^{\\infty}\\frac{e^{-C^\\gamma} (C^\\gamma)^k}{k!}$$\n\nI want to plot $P$ versus $C \\log{C}$ for a given $\\gamma$ with logarithmic scale on Y-axis. In Matlab, I'd calculate it as follows:\n\ngamma = 0.75;\nC = 1:1:20;\nlambda = C .^ gamma;\np = 1 - poisscdf(C,lambda);\nsemilogy(C .* log(C), p);\n\n-\n\n## migrated from math.stackexchange.comMay 2 '12 at 13:30\n\nThis question came from our site for people studying math at any level and professionals in related fields.\n\nYou probably could have worked this out by searching in the documentation for \"poisson distribution\", yielding PoissonDistribution, and \"log plot\", yielding a range of hits including ListLogPlot, LogPlot etc. \u2013\u00a0Verbeia May 3 '12 at 4:38\n\nIt is worth noting two characteristic features of the question:\n\n1. The sum is a complementary cumulative distribution function for a Poisson distribution: it's built in to Mathematica and needn't be computed explicitly.\n\n2. The use of $C+1$ as a starting index in the sum, as well as the expression 1:1:20 in the code, indicate $C$ is considered an integer: this needs to be a discrete plot.\n\nIt can also help to draw clear parallels between the \u039cATLAB approach and an idiomatic Mathematica approach. How about this?\n\nModule[{c = Range[20], \u03b3 = 0.75, p},\np = {# Log[#], 1 - CDF[PoissonDistribution[#^\u03b3], #]} & \/@ c;\nListLogPlot[p]\n]\n\n\n-\nMuch easier now. But, what about the logarithmic scale on the y-axis? I mean the ticks! \u2013\u00a0Osama Gamal May 2 '12 at 14:19\n@Osama They are logarithmic! \u2013\u00a0whuber May 2 '12 at 19:14\n\nThis is only a minor variant on the existing answers, but it will hopefully be clearer to a novice user:\n\npp[c_?Positive, \u03b3_] := {c Log[c], 1 - CDF[PoissonDistribution[c^\u03b3], c]}\nListLogPlot[Table[pp[x, 0.75], {x, 1, 40}]]\n\n\nBut notice:\n\nListLogPlot[Table[pp[x, 0.75], {x, 1, 40, 0.02}], Joined -> True]\n\n\nThe function is defined for non-integer values of $c$, but includes a Floor expression.\n\nAssuming[c > 0, Simplify[1 - CDF[PoissonDistribution[c^\u03b3], c]]]\n(* ==> 1 - GammaRegularized[1 + Floor[c], c^\u03b3] *)\n\n\nAs J.M. pointed out in comments, the expression that results from this simplification can be further simplified to GammaRegularized[1 + Floor[c], 0, c^\u03b3], but the presence of Floor[] in the result is clear even in the less-simplified version.\n\n-\nAgain: 1 - GammaRegularized[1 + Floor[c], c^\u03b3] can be simplified further to GammaRegularized[1 + Floor[c], 0, c^\u03b3]. Numerically, it's a good idea to be using the three-argument form GammaRegularized[] for situations like this... \u2013\u00a0J. M. May 3 '12 at 6:43\n\nHere is what I would do:\n\np[c_, x_] = (Sum[Exp[-x] x^k\/k!, {k, c + 1, Infinity}] \/\/\nFullSimplify)\n\nWith[{gamma = 0.75},\nParametricPlot[{c Log[c], p[c, c^gamma]}, {c, 1, 20},\nAspectRatio -> 1]]\n\n\n-\nHow to make it logarithmic scale on y axes as in the Matlab code? I used Log[p[c, c^gamma]] but still can't edit the axes thing!! \u2013\u00a0Osama Gamal May 2 '12 at 13:32\ncValues = Range[1, 20, 0.05]; ListLogPlot[{# Log[#], p[#, #^0.75]} & \/@ cValues, PlotRange -> All] ? \u2013\u00a0b.gatessucks May 2 '12 at 13:41\nWhy things is much complicated in Mathematica than in Matlab! \u2013\u00a0Osama Gamal May 2 '12 at 14:12\nHere's a more compact definition of p[c, x]: p[c_, x_] := GammaRegularized[c + 1, 0, x]. I don't understand why FullSimplify[] doesn't use GammaRegularized[] more frequently myself... \u2013\u00a0J. M. May 2 '12 at 14:25","date":"2016-04-30 09:13:15","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\": 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\": 0.34828174114227295, \"perplexity\": 3745.9446663383646}, \"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-2016-18\/segments\/1461860111809.10\/warc\/CC-MAIN-20160428161511-00189-ip-10-239-7-51.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/scigraph.springernature.com\/person.014276553167.76","text":"# B Mohanty\n\nOntology type: schema:Person\n\nNAME\n\nB\n\nSURNAME\n\nMohanty\n\n### Publications in SciGraph latest 50 shown\n\n\u2022 2017-08 Insight into particle production mechanisms via angular correlations of identified particles in pp collisions at s=7 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2017-08 Measurement of D-meson production at mid-rapidity in pp collisions at s=7 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2017-07 Measurement of electrons from beauty-hadron decays in p-Pb collisions at sNN=5.02 TeV and Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2017-06 Production of \u03a3(1385)\u00b1 and \u039e(1530)0 in p\u2013Pb collisions at sNN=5.02 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2017-02 Determination of the event collision time with the ALICE detector at the LHC in THE EUROPEAN PHYSICAL JOURNAL PLUS\n\u2022 2017-01 Charged-particle multiplicities in proton\u2013proton collisions at s=0.9 to 8 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2016-06 Centrality dependence of \u03c8(2S) suppression in p-Pb collisions at sNN=5.02 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2016-05 Centrality dependence of charged jet production in p\u2013Pb collisions at sNN = 5.02 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2016-05 Differential studies of inclusive J\/\u03c8 and \u03c8(2S) production at forward rapidity in Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2016-04 Inclusive quarkonium production at forward rapidity in pp collisions at s=8TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2016-03 Transverse momentum dependence of D-meson production in Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2016-02 Multiplicity and transverse momentum evolution of charge-dependent correlations in pp, p\u2013Pb, and Pb\u2013Pb collisions at the LHC in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2015-09 Measurement of charm and beauty production at central rapidity versus charged-particle multiplicity in proton-proton collisions at s=7 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-09 Coherent \u03c10 photoproduction in ultra-peripheral Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-09 Measurement of jet quenching with semi-inclusive hadron-jet distributions in central Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-07 Inclusive, prompt and non-prompt J\/\u03c8 production at mid-rapidity in Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-06 Rapidity and transverse-momentum dependence of the inclusive J\/\u03c8 nuclear modification factor in p-Pb collisions at sNN = 5.02 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-06 Elliptic flow of identified hadrons in Pb-Pb collisions at sNN=2.76 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-05 Measurement of pion, kaon and proton production in proton\u2013proton collisions at s=7 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2015-05 Forward-backward multiplicity correlations in pp collisions at s = 0.9, 2.76 and 7 TeV in JOURNAL OF HIGH ENERGY PHYSICS\n\u2022 2015-01 Production of \u03a3(1385)\u00b1 and \u039e(1530)0 in proton\u2013proton collisions at s= 7 TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2014-08 Measurement of quarkonium production at forward rapidity in pp collisions at s=7TeV in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2013-12 Energy dependence of the transverse momentum distributions of charged particles in pp collisions measured by ALICE in THE EUROPEAN PHYSICAL JOURNAL C\n\u2022 2013-11 Charmonium and e+e\u2212 pair photoproduction at mid-rapidity in ultra-peripheral Pb\u2013Pb collisions at in THE EUROPEAN PHYSICAL JOURNAL C\n\n### Identifiers\n\nJSON-LD is the canonical representation for SciGraph data.\n\nTIP: You can open this SciGraph record using an external JSON-LD service:\n\n``````[\n{\n\"@context\": \"https:\/\/springernature.github.io\/scigraph\/jsonld\/sgcontext.json\",\n\"affiliation\": [\n{\n\"affiliation\": {\n\"id\": \"https:\/\/www.grid.ac\/institutes\/grid.419643.d\",\n\"type\": \"Organization\"\n},\n\"isCurrent\": true,\n\"type\": \"OrganizationRole\"\n},\n{\n\"id\": \"https:\/\/www.grid.ac\/institutes\/grid.482273.8\",\n\"type\": \"Organization\"\n}\n],\n\"familyName\": \"Mohanty\",\n\"givenName\": \"B\",\n\"id\": \"sg:person.014276553167.76\",\n\"sameAs\": [\n\"https:\/\/app.dimensions.ai\/discover\/publication?and_facet_researcher=ur.014276553167.76\"\n],\n\"sdDataset\": \"persons\",\n\"sdDatePublished\": \"2019-03-07T13:59\",\n\"sdPublisher\": {\n\"name\": \"Springer Nature - SN SciGraph project\",\n\"type\": \"Organization\"\n},\n\"sdSource\": \"s3:\/\/com-uberresearch-data-dimensions-researchers-20181010\/20181011\/dim_researchers\/base\/researchers_1754.json\",\n\"type\": \"Person\"\n}\n]``````\n\nHOW TO GET THIS DATA PROGRAMMATICALLY:\n\nJSON-LD is a popular format for linked data which is fully compatible with JSON.\n\n`curl -H 'Accept: application\/ld+json' 'https:\/\/scigraph.springernature.com\/person.014276553167.76'`\n\nN-Triples is a line-based linked data format ideal for batch operations.\n\n`curl -H 'Accept: application\/n-triples' 'https:\/\/scigraph.springernature.com\/person.014276553167.76'`\n\n`curl -H 'Accept: text\/turtle' 'https:\/\/scigraph.springernature.com\/person.014276553167.76'`\n\nRDF\/XML is a standard XML format for linked data.\n\n`curl -H 'Accept: application\/rdf+xml' 'https:\/\/scigraph.springernature.com\/person.014276553167.76'`","date":"2020-01-28 09:33:28","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9642866849899292, \"perplexity\": 9938.02124778306}, \"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-2020-05\/segments\/1579251778168.77\/warc\/CC-MAIN-20200128091916-20200128121916-00377.warc.gz\"}"}
| null | null |
Joe M O'Brien
Carlucci, Viking Etiquette, East Village, New York City, USA
Raphael, Street Chakras, Tribeca, New York City, USA
Loyally Divided, Euro 2020, London
Maidan, More Than Life and Death, Kyiv, Ukraine.
Soviet Pyramiden, An Arctic Memorial, Norway
Divided Lines, Catalonia
N14 General Strike. Madrid, Spain
Pageant and Protest, England
Israel, West Bank
Another Land, The Irish North
Annexation of Crimea, Ukraine
Private Moments in Ukraine
Plague by Another Name, Covid-19, England
Plague by Another Name, Covid-19, 2020, Italy
Inis Mór, Éirinn (Ireland)
Mythological Postcards, Greece, Georgia
Anniversary of Independence, Armenia
Easter Uprising Centenary, Ireland
Sandy Row Eleventh Night Bonfire, Belfast. 11th July 2018. The Eleventh Night refers to the night before the Twelfth of July, a yearly Ulster Protestant celebration. The bonfires are lit to celebrate the 'Glorious Revolution' of 1688 and victory of Protestant Prince, William of Orange over Catholic King James II at The Battle of The Boyne in 1690.
Eleventh Night Bonfire, West Belfast. July 2018. The Eleventh Night refers to the night before the Twelfth of July, a yearly Ulster Protestant celebration. The bonfires are lit to celebrate the 'Glorious Revolution' of 1688 and victory of Protestant Prince, William of Orange over Catholic King James II at The Battle of The Boyne in 1690.
Orange Order parade, Belfast. 12th July 2018. The Twelfth of July is a yearly Ulster Protestant celebration. It began during the late 18th century in Ulster. It celebrates the 'Glorious Revolution' of 1688 and victory of Protestant Prince, William of Orange over Catholic King James II at The Battle of The Boyne in 1690, which began the Protestant Ascendancy.
An Eleventh Night Bonfire on the Fountain Estate Derry/Londonderry. 11th July 2018. The Eleventh Night refers to the night before the Twelfth of July, a yearly Ulster Protestant celebration. The bonfires are lit to celebrate the 'Glorious Revolution' of 1688 and victory of Protestant Prince, William of Orange over Catholic King James II at The Battle of The Boyne in 1690.
The Fountain is home to approximately 250 Protestants. 10,000 Protestants fled the city's West Bank, to the suburbs during the troubles.
The plinth to the 'Peace For All' statue, stolen in 2016, but retrieved weeks later. The statue was situated close to the Senator George Mitchell Peace Bridge on the Co Cavan side, connecting Co Fermanagh. The statue was created to symbolise peace in the post Good Friday agreement. July 2018.
Circus Corvinni, Belleek. The Greater part of the village lies in Co Fermanagh, part of it crosses the border into Co Donegal.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 8,518
|
Q: Can not read property of undefined (when it is defined) I'm trying to scroll to top of the element when it is opened (this is using bootstrap 3 accordion structure), but I get an error saying 'Uncaught TypeError: Cannot read property 'top' of undefined' where undefined is heading I guess? I'm not sure why tis happens as everything else I perform on the heading i.e. removeClass() works.
$('.checkout .collapse').on('shown.bs.collapse', function() {
var heading =
$('.checkout')
.find('a[aria-controls="' + $(this).attr('id') + '"]');
heading.removeClass('complete')
.css('cursor', 'default');
$('html, body').animate({
scrollTop: heading.offset().top - 100
}, 'slow');
});
EDIT: it seems that heading returns empty object when $(html, body).ani ... is added in, however if I remove that part it returns what is expected.
A: Try checking using jQuery's length property to see whether such an element exists
if (heading.length) {
$('html, body').animate({
scrollTop: heading.offset().top - 100
}, 'slow');
}
A: Here in your code heading is not work inside animate method so you have to replace it as given below
$('.checkout .collapse').on('shown.bs.collapse', function() {
var heading =
$('.checkout')
.find('a[aria-controls="' + $(this).attr('id') + '"]');
heading.removeClass('complete')
.css('cursor', 'default');
$('html, body').animate({
scrollTop: $('.checkout').find('a[aria-controls="' + $(this).attr('id') + '"]').offset().top - 100
}, 'slow');
});
Hope this will help you ..!!!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,764
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Q: Faster way to compile Factor Programs I really love the Factor language. But I find that compiling programs written in it is incredibly slow, and thus it's not feasible to create real projects with Factor.
For example, compiling the sample Calculator WebApp takes about 5 minutes on my laptop (i3 processor, 2GB RAM, running Fedora 15).
I've searched around but couldn't find a faster way to compile Factor programs than using the interpreter (the main factor binary executable).
It becomes ridiculous when you attempt to only use the interpreter for each run and not "deploy" your program to a native binary file (which doesn't even work on many programs).
It means that every time I want to run the Calculator, for example, I have to wait a 5 minute cold-start duration.
I'd like to know whether this is a common issue, and whether there's a good way to tackle it.
A: I admit that before today, I had never heard of Factor. I took the opportunity to play with it. It is looking nice (reminds me of squeak-vm and lisp at the same time). I'll cut the smalltalk (pun very much intended) and jump to your question.
Analysis
It appears that the way Factor works, results in loading vocabularies being slow.
I compiled Factor on my 64 bit quadcore linux system (from git revision 60b1115452, Thu Oct 6). Putting everything on tmpfs the build dir takes 641Mb, of which 2x114Mb is in the factor.image and its backup (factor.image.fresh).
When strace-ing the calculator app loading, there is a huge list of factor files being loaded:
*
*3175 factor files are touched.
*compilation of these takes roughly 30 seconds on my box
*the memory usage maxes out on just under 500Mb (virtual) and 300Mb reserved set
I'm strongly suspecting your box is low on memory, and might be getting very swappy This would definitely explain compilation taking 5 minutes
Can you confirm whether this is the case (likely if you are running some kind of shared host or VPS appliance). If you run a virtual machine, consider increasing the available system memory.
Saving Heap Images (snapshots)
I already mentioned the factor.image file (114Mb) before. It contains a 'precompiled' (bootstrapped, actually) heap image for the Factor VM. All operations in the VM (working on the UI listener or compiling factor files) affects the heap image.
To avoid having to recompile your source files time and time again, you can save the end-result into a custom heap image:
http://docs.factorcode.org/content/article-images.html
Images
To start Factor with a custom image, use the -i=image command line switch; see Command line switches for the VM.
One reason to save a custom image is if you find yourself loading the same libraries in every Factor session; some libraries take a little while to compile, so saving an image with those libraries loaded can save you a lot of time.
For example, to save an image with the web framework loaded,
USE: furnace
save
New images can be created from scratch: Bootstrapping new images
Deploying applications
Saving heap images results in files that will (typically) be bigger than the original bootstrap image.
The Application deployment tool creates stripped-down images containing just enough code to run a single application
The stand-alone application deployment tool, implemented in the tools.deploy vocabulary, compiles a vocabulary down to a native executable which runs the vocabulary's MAIN: hook. Deployed executables do not depend on Factor being installed, and do not expose any source code, and thus are suitable for delivering commercial end-user applications.
Most of the time, the words in the tools.deploy vocabulary should not be used directly; instead, use Application deployment UI tool.
You must explicitly specify major subsystems which are required, as well as the level of reflection support needed. This is done by modifying the deployment configuration prior to deployment.
Concluding
I expect you'll benefit from (in order of quickest win):
*
*increasing available RAM (only quick in virtual environments...)
*saving a heap image with
USE: db.sqlite
USE: furnace.alloy
USE: namespaces
USE: http.server
save
This step brought the compilation on my system down from ~30s to 0.835s
*
*deploying the calculator webapp to a stripped heap image (refer to the source for deployment hints)
In short, thanks for bringing Factor to my attention, and I hope my findings will be of any help, Cheers
|
{
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| 4,704
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David Thomson Pearson (9 November 1932 – July 2019), was a Scottish footballer who played as an inside forward in the English Football League. Dave scored a hat-trick on his Oldham Athletic debut
External links
Dave Pearson
1932 births
2019 deaths
Scottish footballers
Footballers from Dunfermline
Association football forwards
Blackburn Rovers F.C. players
Ipswich Town F.C. players
Darwen F.C. players
Oldham Athletic A.F.C. players
Rochdale A.F.C. players
Crewe Alexandra F.C. players
Chorley F.C. players
English Football League players
|
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| 8,264
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REVIEW: "The Sidekicks," Will Kostakis
Title: The Sidekicks
Author: Will Kostakis
Published: February, 2016 by Penguin
Purchase: Booktopia
The Swimmer. The Rebel. The Nerd.
All Ryan, Harley and Miles had in common was Isaac. They lived different lives, had different interests and kept different secrets. But they shared the same best friend. They were sidekicks. And now that Isaac's gone, what does that make them?
Will Kostakis, award-winning author of The First Third, perfectly depicts the pain and pleasure of this teenage world, piecing together three points of view with intricate splendour.
Divided into thirds, each one housed a different perspective of the remaining friends of Isaac, showing us how they coped with life as a teenager after his sudden death. Without having seen the four of them together, initially it was hard picturing how they'd been friends. Although, it was pointed out a few times that they weren't—that the three of them merely orbited Isaac, getting along with the others merely a part of being in Isaac's life. And with him gone, their quasi-friend group quickly fell apart. I think seeing that happen felt quite realistic. Pretty much all of it did. Reading Sidekicks took me back to what is was like being young and insecure, making the wrong decisions but feeling like there were no other ones to make.
Ryan, lead the first part of the book, avoiding unnecessary carbs, doing laps of the pool, popping in and out of the staff room/office of his teacher-mother, and stressing out over how he would (or whether he could) come out to her, or anyone. Seeing these fears unfurl inside his head, it was easy to relate to, especially when it came to worrying about things like being known as the gay one. Still, I felt like Ryan pigeonholed himself (or his image of himself) just as much as a swimmer, the Olympic hopeful, trying to distract others in a bid to get through high school with his non-conforming sexuality unnoticed.
The middle chunk of the book shifted away from that plot, following Harley. Always seeming to get into trouble, he loved to have fun—whether that was with drugs, alcohol or mischief. His perspective revealed quite a bit as to what went down in the lead up to Isaac's death. Struggling with his feelings of guilt, he spends time hiding away at his dad's place. I actually really came to enjoy this section of the book. Harley's character was pretty much gone during the Ryan block, so seeing what went down in that period really opened up the story. And in fact, it felt like he was becoming the glue trying to stick their friendships together.
Miles rounded out the remainder of the book, his thoughts spilling out in script format—and without contractions. A little weird, but I got used to it. Caught up in his own mind, wondering whether Isaac was ever really his friend, Miles felt disconnected from the world. He seemed particularly lonely, spending a lot of time reminiscing over previous experiences he and Isaac had been through. Miles wasn't exactly my favourite character, but I still enjoyed what he brought to the story.
Drawing their three arcs together, the book gradually weaved together their lives, showing each of the guys the different sides of themselves. It was here that the book excelled. With no singular plot to follow, their roads back to becoming friends each had their ups and downs, managing to find something different to focus on and keep you interested. I really liked how the book took the time to focus on the grief experienced by Isaac's family as well, finding one of the scenes with his mother to be the most touching aspect of the book. Being such a quick read with only two hundred and something pages, I did feel a little short changed at times, but it did leave me content with the end result.
Anyone after a new contemporary to take on. A quick and immersive read.
|
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| 9,591
|
{"url":"https:\/\/cs.stackexchange.com\/questions\/62436\/hungarian-algorithm-arbitrary-assignments","text":"# Hungarian Algorithm - Arbitrary Assignments\n\nI've looked at several explanations of the Hungarian Algorithm for solving the Assignment Problem and the vast majority of these cover only very simplistic cases.\n\nThe most understandable explanation I've found is a YouTube video.\n\nI can code the algorithm but I'm concerned about one special case. If you watch the video, the relevant case is explained from 31:55 to 37:42, but I\u2019ll explain it below.\n\nI should first mention that I will be dealing with a 300 x 300 matrix, so visual inspection is out of the question. Additionally, I need to find all minimum assignments. In other words, if there are multiple assignments that produce the same minimum value, I need to find them all.\n\nHere's the particular case that I'm concerned about. You can see this explained in the YouTube video but I\u2019ll go over it here. We start with this matrix:\n\n3 1 1 4\n4 2 2 5\n5 3 4 8\n4 2 5 9\n\n\nWhen we reduce the rows and columns, we get this:\n\n0 0 0 0\n0 0 0 0\n0 0 1 2\n0 0 3 4\n\n\n(Let me mention that I can visually see there are 4 solutions to this matrix and the total score is 13.)\n\nGiven the above reduced matrix, there are no unique zeros in any row or column, so, according to the algorithm described in the video, I can arbitrarily select any zero element for assignment, so I select (1,1).\n\nI\u2019ll mark the assigned zero with an asterisk and I\u2019ll put an \u201cx\u201d next to those zeros in the rows and columns that are no longer available for consideration. Now we have this:\n\n0* 0x 0x 0x\n0x 0 0 0\n0x 0 1 2\n0x 0 3 4\n\n\nNext, we continue examining rows for a unique zero. We find one at (3,2) so we mark it with an asterisk and mark the unavailable zeros with \"x\":\n\n0* 0x 0x 0x\n0x 0x 0 0\n0x 0* 1 2\n0x 0x 3 4\n\n\nNext, we start looking for unique zeros in the columns (since all rows have been exhausted). We find column three has a unique zero at (2,3) so we mark it:\n\n0* 0x 0x 0x\n0x 0x 0* 0x\n0x 0* 1 2\n0x 0x 3 4\n\n\nAt this point, there are no more available zeros and row 4 has been left unassigned. (This particular YouTube video now uses a \u201cticking procedure\u201d, which is a common technique for determining the minimum number of lines needed to cover all the zeros. If you are unfamiliar with this technique it is explained starting at 14:10 through 16:00, although the presenter uses a different matrix than shown here.) The \u201cticking procedure\u201d is this:\n\n1. Tick all rows that have no assigned zeros (row 4).\n2. For each row that is ticked, tick the columns that contain a zero in that row.\n3. For each column ticked in step 2, tick the corresponding rows that have assigned zeros.\n4. Repeat steps 2 and 3 until no more ticking is possible.\n5. Draw lines through all ticked columns and un-ticked rows.\n\nAt this point, the ticking procedure generates 4 vertical lines, covering all zeros. The four vertical lines tell us the zeros in the matrix represent one or more solutions, yet, as we see, row 4 is unassigned. The fact that fourth row remains unassigned in spite of the four vertical lines tells us that we chose the wrong zeros for assignment!\n\nThe video\u2019s presenter indicates this problem is a result of our initial (arbitrary) assignment of element (1,1). The presenter says, \u201cthere are more sophisticated methods available\u201d to get us out of this situation be he does not explain what these techniques are. He alludes to the existence of \u201cintelligent\u201d ways of selecting a zero, rather than the arbitrary selection we used to select the zero at (1,1).\n\nOne approach I could take (I\u2019m not sure it\u2019s the best) when faced with making an arbitrary assignment is to make the assignment from the row or column with the fewest number of available arbitrary choices. In this example, this means I would make the arbitrary assignment from either row 3 or 4, where there are only two arbitrary choices, rather than from row 1 or 2 where there are four arbitrary choices. Of course, since I need all correct solutions, I would have to iterate over all the available arbitrary assignments, whenever an arbitrary assignment is made. For example, if I select (3,1) as an arbitrary assignment, I would also have to assign (3,2) later.\n\nMy question then, after all this, is, when I am faced with the choice of arbitrarily selecting a zero for assignment, what is the best approach? Is it what I mention in the previous paragraph? How can I eliminate the dead-end solutions like the one shown? Please remember I still need to enumerate all solutions having the same minimal score.\n\n\u2022 You keep saying that you \"need\" to find all the tied solutions. You do realize that there can be an absurdly large number of solutions, right? Like 300! in your example. (That's 300 factorial, not 300 with emphasis.) If you truly need to enumerate those individually, good luck. Of course, in practice, there won't be nearly that many, but the number can still easily be infeasible. \u2013\u00a0Chris Okasaki Aug 9 '16 at 16:59\n\u2022 @Chris: Yes, I realize there are potentially n! solutions but given the data I am dealing with, that will never happen. I just didn't want to muddy the waters by saying something like, \"I will stop calculating results after 25 matches....\" \u2013\u00a0Tom Baxter Aug 9 '16 at 17:34","date":"2020-03-29 09:38:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.7280178070068359, \"perplexity\": 490.7473172872284}, \"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-2020-16\/segments\/1585370494064.21\/warc\/CC-MAIN-20200329074745-20200329104745-00138.warc.gz\"}"}
| null | null |
Michalok ( est un village de Slovaquie situé dans la région de Prešov.
Histoire
Première mention écrite du village en 1363.
Géographie
La municipalité culmine à 256 mètres de haut. Sa superficie 12 km² pour 332 habitants (soit une densité de 25hab/km²). Elle se situe au sud de la Slovaquie.
Notes et références
Village en Slovaquie
Commune dans le district de Vranov nad Topľou
|
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ERROR: type should be string, got "https://www.myjoyonline.com/manasseh-azure-awuni-a-noble-profession-ignoble-association/-------https://www.myjoyonline.com/manasseh-azure-awuni-a-noble-profession-ignoble-association/\nManasseh Azure Awuni: A noble profession, ignoble association\nSource: Manasseh Azure Awuni\nDear Ghanaian lawyer,\nIn December 2020, Ghanaian voters were presented with the option to choose between dying by firing squad or dying through hanging. A choice was made, and we are feeling the debilitating effects of the ever-tightening noose of bad governance and impunity.\nToday, you lawyers in Ghana have an opportunity to choose leaders of your association, the Ghana Bar Association (GBA). I don't know the contestants. And I cannot pass any judgment on their competence, character or courage.\nWhat I know, however, is that the GBA is now like an over-aged breast, one that cannot produce milk for babies or entertain the eyes and mouths of grownups. In effect, the GBA has lost its essence and appeal.\nThere is no doubt that law is among the most respected, powerful and influential professions in this country. When I delivered a speech to students of the University of Ghana Law School a few years ago, I mentioned that lawyers were the most dominant professionals in all three arms of government in Ghana.\nAs I write this, the executive, the legislature and the judiciary are all headed by lawyers. The dominance goes beyond the heads.\nIn his welcome address to the 2021 batch of Harvard Law students, the Dean of the Harvard Law School, John F. Manning, said \"being a lawyer is a superpower.\"\nThe Ghana example attests to the fact that he is right. (This is despite the fact that some crooks who have found their way into your fold are known by the people who deal with them and should not bask in this unearned glory because they undermine the very nobility of the profession.)\nAside from being one of the most boastful and (sometimes arrogant) sets of professionals ever to crawl on Oboadeε's earth, lawyers are like medical doctors. Every other professional needs them. So, the nobility and power accorded to the legal profession are reasonably grounded.\nIf lawyers are powerful and respected in our country, then its union, the GBA, ought to be one of the most respected associations in the country. Unfortunately, that isn't the case.\nAs an observer, themoribund GBA isn't different from the Ghana Journalists Association (GJA) to which I belonged until it lost its way.\nI say the GBA is moribund because Martin Luther King Jr. taught us that \"our lives begin to end the day we become silent about things that matter.\"\nOur wise elders have also taught us that a man does not run after rats when his roof is on fire.\nSo, when you hear the GBA louder on the policing of boobs of female lawyers than the deterioration of the rule of law, corruption and bad governance, then you don't need anyone to tell you that this is an association that has outlived its usefulness.\nWhen the President of the Republic and his team formed themselves into the instigator, investigator and adjudicator in suspending (in the form of a forced leave) and later hounding the Auditor-General out of office, one would have expected the GBA to lead the pack fighting for the reversal of that unfortunate decision.\nWhen some individuals and civil society groups petitioned the Supreme Court to intervene, the apex court could have dealt with that important constitutional matter in two weeks if it had attached the same seriousness it gave to the 2020 election petition. But after seven months, the petitioners discontinued the case because the delay of the Supreme Court had rendered it moot. And the GBA was mute.\nWhen an MP and influential member of the governing party put the photograph of an undercover journalist on television, told the world where the journalist lived and ordered whoever saw him to attack him, nothing was done to the MP. Later, the undercover journalist was shot and killed by unknown persons. Again, nothing was done to MP who issued the threats.\nShockingly, the President had the guts to use the 2019 Bar Conference in Takoradi to state that the killing of Ahmed Suale should not be taken to mean an attack on press freedom. Again, the GBA did not find anything wrong with the desecration of its platform. If it did, it did not find its voice.\nThe list is endless. But the trend is simple. The GBA has lost its voice against any democratic vice.\nI have learned about the influential role played by the GBA in the days of military rule. I have learned that the GBA used to be a respected voice when the cost of speaking up in this country was more detrimental than the fear of not getting favourable judgment in court or the inability to court the friendship of the powerful and mighty in the land.\nToday, we have some semblance of a democracy. We have a democracy that is strong and robust only when it is compared with the miserable mess in our sub-region and continent.\nWe have a democracy in which the rule of law, separation of powers, and checks and balances are only useful to academics who must teach them to earn their salaries and students who must know them to earn their degrees.\nWe operate a democracy whose legislature isn't different from the executive and whose conservative judiciary, which is perceived to hardly stray from the wishes of the executive.\nWe run a democracy in which the poor and vulnerable cannot walk to a police station and be sure to get justice against the rich; where getting such justice in the court of law remains a remote dream to the average citizen.\nOur democracy has deteriorated considerably in recent years in many aspects. The abuse of power by the executive, the legislature, and to some extent the judiciary, is becoming a norm.\nCritical voices from associations such as the GBA should serve as the guardrails against the arbitrariness in a country where state institutions are either robbed or rob themselves of the power to act in the interest of the people.\nBut when Supreme Court Judges attacked a law professor over academic research, which they found unpleasant, and the Ghana Bar Association did not put the Supreme Court in its rightful place, then members of the association should have announced the date of burial of the GBA.\nSome individual lawyers wield powerful voices that can cause change. But some have decided to kiss asses and sing praises in order to be invited to the sumptuous buffet of the political and judicial spoils. Other conscionable ones are also afraid to speak up because of the fear of victimization and retribution.\nThere is, however, strength in numbers. There is safety and anonymity in speaking with the collective voice of the GBA. That's why the GBA must not be allowed to die.\nWhoever gets onto the executive committee of the GBA today should not be left alone. Until lawyers sit up and demand the best from the association, nothing will change.\nAnd if groups such as the GBA continue to \"see no evil, hear no evil and speak no evil\", lawyers will one day be called to join the pallbearers of our dying democracy.\nAnd when democracy dies, the rule of law dies. And what's the essence of lawyers when there is no rule of law?\nThis is a view of someone with enormous respect and enormous disdain for the law profession and lawyers.\nMy views may mean something. They may also mean nothing.\nManasseh Azure Awuni, the Bongo Boy.\nNOTE: Bongo is about 20 minutes' drive from Bolga, so visit when you have time. You might find a business idea in water production\nGhana Bar Association\nConcerned UG Students embark on 'Ya Br3 Mo' Demo\nWhat VP Bawumia said at University of Ghana's 74th annual New Year School and Conference\nUG files motion to dismiss students' injunction application against new residence plan\nDesist from invading Commonwealth Hall on reopening day – UG management\nUG SRC has failed students – Former Sarbah JCR President\nUG suspends Commonwealth Hall JCR Executives for 3 years each\nHigh Court prevents UG's management from implementing new residential policy\nUG management using Sarbah and Commonwealth hall residents as 'lab rats' – Former Sarbah JCR president\nRefund monies paid above the approved 15% increment in fees – Education Ministry orders UG, other public universities\nChristian Atsu: Hatayspor official says there was mistaken identity, player still not found\nDeath of Gospel musician's brother: We will not accept police report on his death – Family\nJojo Wollacott makes injury return in Charlton friendly\nMorocco not required to provide certificate of good conduct in human rights\nGoverment seeks Indian investment in oil and gas sector"
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Французько-М'янмарські відносини стосуються міждержавних відносин М'янми та Франції. Стосунки почалися на початку 18 століття, коли Французька Східно-Індійська компанія намагалася поширити свій вплив на Південно-Східну Азію. Франція взяла участь у будівництві верфі в 1729 році в місті Сіріям. Повстання 1740 року пн проти бірманського правила, однак змусило французів відійти в 1742 році. Вони змогли повернутися до Сіаму в 1751 році, коли Пн попросив французьку допомогу проти бірманців. Французький посланник Сеур де Бруно був відправлений оцінити ситуацію та допомогти в обороні проти бірманців. Французькі військові кораблі були відправлені на підтримку повстання пн, але марно. У 1756 році бірманці під Алаунпайєю перемогли пн. Багато французів потрапили в полон та включені до складу бірманської армії як елітний корпус навідника під Шевальє Мілард. У 1769 р. Офіційні контакти поновлюються, коли між королем Хсінбюшином та французькою Східно-Індійською компанією був підписаний торговий договір.
Однак незабаром Франція втягнулася у французьку революцію та наполеонівські війни, поступившись місцем переважного впливу Британії в Бірмі. Французькі контакти з Бірмою, фактично британською колонією, стали б майже неіснуючими, тоді як з другої половини 19 століття Франція зосередилася б на створенні французького Індокитаю та конфліктах з Китаєм, що призвели до китайсько-французької війни.
Французька верф у Сіріямі (1729 – 1742)
Генеральний губернатор Французької Індії Джозеф Франсуа Дуплекс почав проявляти інтерес до Бірми з 1727 р. За рахунок великої кількості країни в тику та сирої нафти. Як результат, у 1729 році в місті Сіріям було створено французьку верф, будуючи кораблі для Пондічеррі. Верф був покинутий у 1742 році через повстання пн.
Втручання в Бірмі (1751)
Через кілька років в Дуплекс відвідав посланник Пен з проханням французької допомоги в боротьбі з Через кілька років в Дуплекс відвідав посланник Пен з проханням французької допомоги в боротьбі з бірманцями. Dupleix пообіцяв людей та боєприпаси та відправив Сієра де Бруно з метою розвитку французького впливу в країні. Він прибув до Баго, Бірма, в липні 1751 року. З'єр де Бруно повідомив, що кілька сотень французьких військ зможуть взяти під контроль дельту Ірравадді, викликавши офіційний запит Дюплекса до французького суду отримати необхідну військову підтримку. Сєр де Бруно отримав договір і уклав союз між Францією та монсами.
Губернатор Томас Сондерс з Мадрасу спробував протидіяти французьким рухам у регіоні, направивши військову групу для обстеження острова Неграйс під капітаном Томасом Тейлором. Він також намагався домовитись про відступ Сиріаму з англійцями. Монси твердо виступили проти цих спроб британського посягання під захисником Сієра де Бруно, який мав значний вплив у суді Мону і особливо був у чудових стосунках із спадкоємцем Очевидним. Нарешті Сондер вирішив насильно окупувати Неграйса, окупувавши острів 26 квітня 1753 року.
Однак пропозиції Дюплекса взяти під контроль дельту Ірравадді було відхилено французьким урядом, сильно обмеживши його можливості втручатися туди.
Примітки
Список літератури
Зал, DGE Бірма (3-е видання). Хатчісон (1960). OCLC 4379359 .
Холл, DGE Europe та Бірма: Дослідження європейських відносин з Бірмою до анексії Королівства Тібава, 1886 рік . Oxford University Press (1945). OCLC 1444846 .
Холл, DGE Історія Південно-Східної Азії . Макміллан, Сент-Мартіна Прес (1964). .
Гаррісон, Брайан. Південно-Східна Азія: коротка історія (2-е видання). Макміллан (1963). OCLC 1239026 .
Фіндлі, Рональд та Кевін Х. О'Рурк. Влада і достаток: торгівля, війна та світова економіка у другому тисячолітті . Прінстонський університетський прес (2007). ISBN 978-0-691-11854-3 .
Гарві, Дж. Е. Історія Бірми: від найдавніших часів до 10 березня 1824 року . Азіатські освітні послуги (2000). ISBN 978-81-206-1365-2 .
Кеніг, Вільям Дж . Бірманська політика, 1752 – 1819 . Центр досліджень південної та південно-східної Азії, Мічиганський університет (1990). ISBN 978-0-89148-056-3 .
Ягня, Аластер. Мандаринський шлях до старої Хуе: розповіді англо-в'єтнамської дипломатії . Chatto & Windus (1970). OCLC 107258 .
Лянг, Чі Шад. Зовнішні відносини Бірми: нейтралізм в теорії та практиці . Праегер (1990). ISBN 978-0-275-93455-2 .
Myint-U, Тант. Створення сучасної Бірми . Cambridge University Press (2001). ISBN 978-0-521-79914-0 .
Ой, Кіт Джин. Південно-Східна Азія: Історична енциклопедія, від Ангкор-Ват до Східного Тимору . ABC-CLIO (2004) .
Рапсон, Едвард Джеймс та ін. Кембриджська історія Індії . Cambridge University Press (1932). OCLC 875984 .
Бюлетень досліджень Бірми SOAS, Vol. 2, № 2, осінь 2004 р., ISSN 1479-8484 ("Подорож до Пегу", переклад "Подорож до Східної Індії та Китаю"; здійснено за наказом Льюїса XV. Між 1774 і 1781 рр.). про манери, релігію, мистецтво та науки індіанців, китайців, пегуїнів та островів Мадагаскару; також спостереження на мисі Доброї Надії, островах Цейлону, Малакці, Філіппінах та Молукксіях П'єром Соннерат, комісар морських піхотинців, (т. III, кн. 4, глава 2).
зовнішні посилання
Різні листи про Бірму, 1755 – 1760, I
Двосторонні відносини Франції
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NFL Handicapping – Inside The Numbers Week 17
FairwayJay
Handicapping and Analyzing Stats – NFL Week 17
Throughout the NFL season, I go inside the numbers and review weekly box scores, stats, game recaps and more after watching many of the NFL games. For over a decade, I have kept a proprietary database of statistical information and utilized relevant rushing numbers, statistical guidelines and situations to better assist me in sports investing and providing point spread winners.
You can read my weekly Fairway's Football Forecast NFL article on this site, which finished the regular season 35-19 ATS (65%) this season with 26 outright underdog winners posting only weekly underdogs. I provide the weekly Inside the Numbers to better assist us in evaluating teams, stats and performances, and how to use certain stats and parameters to guide you towards more winners.
My NFL history as a point spread prognosticator has been a winning one with recent seasons results, and from 2003 through 2010 I went 371-277 ATS (57.3%) and won 8 consecutive regular seasons as a public handicapper with all plays documented at the Sports Monitor. Some top of the leaderboard finishes included in 2010 (44-22 ATS) and 2003 (52-28). I've participated in the Las Vegas SuperContest, and posted back-to-back 60% ATS seasons. My start in 2015 included 16-3-1 ATS after four weeks and 29-10-1 after 8 weeks on our way to another top-6% finish out of more than 1,700 entries.
So each week I provide various insights into NFL handicapping, and the weekly Inside the Numbers article will analyze stats and ways to utilize them and how rushing guidelines and handicapping point-of-attack play can put the percentages in your favor when wagering.
Here are some Week 17 stats of note.
Week 17 NFL favorites went 12-4 SU and 8-8 ATS and totals went 8-8 over/under and scoring averaged 45.38 points per game. For the season, the average NFL game averaged 46.6 points per game; the second-highest in NFL history behind the 2013 season (46.8). That includes the record 105 points scored in Week 11 between the Rams and Chiefs that had a record over/under 64 points posted at the sportsbooks. The sportsbooks got crushed by the bettors in that game.
I provide the weekly look ahead lines to anticipate and compare lines and market moves, and week 17 is tricky with some teams having already clinched a playoff spot or division title and likely resting some starters, similar to a preseason game.
Beating the NFL over the long-term involves an understanding of the match-ups, situations and stats, and how to apply them for success. Good fortune in the turnover column helps too. After all, teams that are plus (+3) or more in the turnover column in a game are a long-term 93% winning ATS situation. Teams with a plus-3 or greater turnover margin in a game went 0-4 SU/ATS in Week 17 (MIA, OAK, SF, NYJ). For the season, teams that were +3 or greater in the turnover column in a game went 39-3 SU and 36-4-2 ATS.
Teams that had a meaningful rushing advantage over their opponent (at least 30 yards) went 9-3 SU and 8-4 ATS in Week 17. For the season, this 30+ yard rushing edge in a game finished 139-39 ATS (78%).
Teams that rushed the ball at least 30 times (when their opponent did not) went 13-0 SU and 9-4 ATS in Week 17 and finished the season 140-24-1 SU and 126-34-5 ATS this season. Teams that run the ball less than 23 times went 0-9 SU and 3-6 ATS in Week 17 and finished the season 16-126 SU and 26-114-2 ATS. Handicapping point of attack play and determining which team will have more success running the ball and controlling the ball, clock and chains will get you more point spread winners.
No Miami Miracle for the Dolphins as they closed the season with a pair of blowout losses and fired their head coach Adam Gase; one of eight head coaches that were fired.
Week 17 rushing leaders included the Ravens (296 rush yards, 47 attempts), Saints (184/27) and Seahawks (182/34), who all failed to cover with both the Saints and Seahawks losing meaningless games. Something to remember in Week 17. Also, the Bears (169/37), Bills (166/33) and Colts (158/36) had big running days in SU/ATS wins.
The power of running the ball can perhaps be best illustrated this way. For the season through 17 weeks, teams that rush the ball at least 30 times in a game when their opponent does not finished 140-24-1 SU and 126-34-5 ATS. So teams that run the ball at least 30 times when their opponent does covered the point spread 79% of the time.
Remember that when the media members, talking heads or your fantasy football friends tell you how important passing the ball is, and remember to run your way to profits and the cashiers window. At least 30 rushes a game is pure profit, while less than 23 rushing attempts per game is a proven loser.
The quarterback play is clearly most important in the NFL, and QB ratings from starter to backup can impact a betting line at by at least 6 points (see link and chart). And while Ravens QB Lamar Jackson led the Ravens to the No. 1 rushing attack in Baltimore's final seven games with Jackson as starter, the importance of balance and competent quarterback play is still most important.
As you become better at understanding and recognizing point of attack play and which teams will have success rushing the football and/or stopping the run, you'll become more proficient and the point spread winners will follow. Many other factors come into play in evaluating games and match-ups, including the quarterback play. But the ability to understand situations and make adjustments to changes, personnel, match-ups, injuries, weather and put all the stat profiles together will allow you to become better at handicapping and forecasting the games.
Teams that had strong offensive showings and efficient yards per play (yppl) in Week 17 victories included the Chiefs (409/7.6), Falcons (489/6.4), Ravens (463/6.3), Bills (381/6.4), (493/7.5), Rams (461/6.9), Seahawks (464/6.2).
Inefficient offensive teams in Week 17 included the Redskins (89/2.0 – worst of all in NFL season!), Jaguars (119/2.5), Vikings (164/3.2), Packers (175/3.1), Bengals (196/4.3), Cardinals (198/3.1), Dolphins (225/3.7), NY Jets (239/4.3).
The league average yards-per-play was between 5.5 and 5.6; up from past 5.4 yards per play in recent seasons. Here is the best and worst of all teams in offensive yards per play this season. Best – Chiefs (6.7), Rams (6.3), Buccaneers (6.3). Worst – Cardinals (4.3), Bills (4.7), Jaguars (4.8), Jets (4.9).
NFL scoring averaged 46.30 points per game in 2018; just missing the highest-scoring season in NFL history in 2014 (46.4). The Chiefs (35.1), Rams (32.8) and Saints (31.5) were the highest scoring teams in the NFL. In the last three seasons combined, just 5 teams averaged at least 29 points per game.
We'll review more NFL stats and information next season as we continue to evaluate the results, match-ups and work to become even more proficient in point spread prognosis.
You can bet on it.
Author: FairwayJay
FairwayJay is a leading national sports and betting analyst providing insight and information you can bet on for nearly two decades from Las Vegas. His work, writing and content is provided to various sites and leading sources. Follow him on Twitter: @FairwayJay
© Copyright 2020 VegasSportsZone The information contained at this site is for news and entertainment purposes only. Use of Vegas Sports Zone and the information contained may not be copied, redistributed or used without prior consent from the owner or Vegas Sports Media or the owner of Vegas Sports Zone or affiliates. Any use of this information in violation of any federal, state or local laws is strictly prohibited. Any person found to be in violation of these provisions may have criminal charges filed per NRS 205.473 and related statutes to Unlawful Acts Regarding Computers and Information Services.
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{"url":"http:\/\/openstudy.com\/updates\/4ddd9969ee2c8b0b257d41e8","text":"## anonymous 5 years ago Evaluate the following limit: lim x->0 sin(x) cos (x)\/x\n\n1. anonymous\n\n$\\lim_{x \\rightarrow 0}sinxcosx\/x=\\lim_{x \\rightarrow 0}sinx\/x*\\lim_{x \\rightarrow 0}\\cos(x)$ both limits are 1. So the whole limit is 1\n\n2. anonymous\n\ncos x\/ x = 1?\n\n3. myininaya\n\nsinx\/x->1 cosx is left over but also goes to 1 both as x->0\n\n4. anonymous\n\ni got DNE\n\n5. anonymous\n\nno the limit exists, it is 1. cos(x) goes to 1 and sinx\/x goes to 1 as well\n\n6. anonymous\n\nbecause cos x \/ x = 1\/0... no?\n\n7. myininaya\n\nlol can you not write the function as sinx\/x *cosx?\n\n8. anonymous\n\nyou aren't plugging in 0 though it's a limit\n\n9. anonymous\n\nthere are limits that have a division by 0 and still exist it is about infinitesimal behavior\n\n10. anonymous\n\nno i wrote sinx \/ x * cosx \/ x\n\n11. anonymous\n\nit is not x^2 on the bottom\n\n12. myininaya\n\n(5*6)\/(5)=6 but by what you say we can do or suppose to do 5\/5*6\/5=6\/5\n\n13. myininaya\n\ni mean (5*6)\/5=5\/5*6=1*6=6 and you say (5*6)\/5=5\/5*6\/5=6\/5 is that what you are saying this second line right here?\n\n14. anonymous\n\ni seperate the fraction\n\n15. myininaya\n\n16. anonymous\n\nsorry guys the lag go the best of me\n\n17. anonymous\n\nno i mean the 2nd line what you di\n\n18. anonymous\n\ndid*\n\n19. myininaya\n\nyou can't do (5*6)\/5=5\/5*6\/5 but you do (5*6)\/5=5\/5*6 or 5\/6*5=6\n\n20. anonymous\n\nyes the 2nd line\n\n21. anonymous\n\nyes the 2nd line\n\n22. myininaya\n\nso we can write sinx\/x*cosx?\n\n23. anonymous\n\nno sin x\/x * cox x \/x\n\n24. myininaya\n\nbut you just said we can't do (5*6)\/5=5\/5*6\/5\n\n25. myininaya\n\n(sinx*cosx)\/x=sinx\/x * cosx\n\n26. myininaya\n\nor =cosx\/x * sinx ( it does us no good to write it like this though)\n\n27. anonymous\n\nso how do i work it? i am confused\n\n28. myininaya\n\nwrite (sinx*cosx)\/x=sinx\/x*cosx where does sinx\/x go to as x gets closer to 1?\n\n29. myininaya\n\nas x gets closer to 0?*\n\n30. anonymous\n\ndo you know how to use latex here because i dont understand what you typed\n\n31. anonymous\n\nso the answer is 1 i dont undertstand why it cant be done the way i did it i understand what you typed\n\n32. myininaya\n\nso you still don't understand why we can't do: $\\frac{\\sin(x)\\cos(x)}{x}=\\frac{\\sin(x)}{x}\\frac{\\cos(x)}{x}?$\n\n33. myininaya\n\nwe don't have x^2 on bottom? we only have x\n\n34. anonymous\n\nyes \\\\\n\n35. myininaya\n\n$\\frac{5*6}{5}\\neq \\frac{5}{5}*\\frac{6}{5}$\n\n36. myininaya\n\ndo you agree with this statement?\n\n37. anonymous\n\nok...\n\n38. myininaya\n\n$\\frac{5*6}{5}=\\frac{5}{5}*6=6$\n\n39. myininaya\n\n$\\lim_{x \\rightarrow 0} \\frac{\\sin(x)\\cos(x)}{x}=\\lim_{x \\rightarrow 0}\\frac{\\sin(x)}{x}*\\cos(x)$\n\n40. anonymous\n\nk\n\n41. myininaya\n\n$=\\lim_{x \\rightarrow 0}\\frac{\\sin(x)}{x}*\\lim_{x \\rightarrow 0}\\cos(x)=1*1=1$\n\n42. anonymous\n\ni guess i gotta recheck my fractions thank you so much i have 1 more question if your not busy can you help me with that please\n\n43. myininaya\n\ni gotta pee i might return\n\n44. anonymous\n\nlol ok THANK YOU SO MUCH","date":"2016-10-28 04:33:37","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\": 0.7676665782928467, \"perplexity\": 6854.820813178369}, \"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-2016-44\/segments\/1476988721555.54\/warc\/CC-MAIN-20161020183841-00482-ip-10-171-6-4.ec2.internal.warc.gz\"}"}
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{"url":"https:\/\/xplaind.com\/228557\/consumer-price-index","text":"# Consumer Price Index\n\nConsumer Price Index (CPI) is a statistic used to measure average price of a basket of commonly used goods and services in a period relative to some base period. The base period price of the basket is marked to 100 and CPI value hovers above or below 100 to reflect whether the average price has increased or decreased over the period.\n\nOnce we have CPI values for two periods, we can determine the inflation rate over the periods.\n\nProducer Price Index (PPI) is a related index which measures average price level received by the sellers. Its movement foretells the expected changes in CPI despite some differences that exist in their calculation methodology.\n\n## Formula\n\nEstimating CPI involves surveying people to identify what they purchase on regular basis. This helps determine the basket of commonly used goods and services. Total price of the basket is obtained from market for current period and base period and following formula is used to calculate CPI:\n\n$$Consumer\\ Price\\ Index \\\\= \\frac{Current\\ Period\\ Price\\ of\\ the\\ Basket}{Base\\ Period\\ Price\\ of\\ the\\ Basket} \u00d7 100$$\n\nIn practice many adjustments are made to CPI on account of seasonality, changes in composition of the basket, etc. and different versions of CPI are calculated to cater to real life needs.\n\nIn US, the Bureau of Labor Statistics estimates CPI on regular basis. IMF and World Bank provide CPI and other data for majority of countries.\n\n## Example\n\nIn 1540, when Sher Shah seized control of India from Humayun, her average residents spent 40% of their total annual consumption budget on food, 20% on fuel, 20% on clothes and 20% on education. In 1545, the King sat down with his vizier to find out whether the standard of life has improved or worsened over the period. They pulled out some data:\n\nPrice in Rupees\nYear 1545Year 1540\nFood5250\nFuel2725\nClothes1715\nEducation90100\n\nOver the five years, there was little change in the residents' spending preferences. Help the king determine whether people are feeling richer or poorer.\n\nIn order to calculate CPI, 1545 and 1540 prices are weighted according to consumers' spending preferences.\n\nWeightPrice in Rupees\nYear 1545Year 1540\nFood40%5250\nFuel20%2725\nClothes20%1715\nEducation20%90100\n\nWeighted Average Prices in 1545\n= 0.4 \u00d7 52 + 0.2 \u00d7 27 + 0.2 \u00d7 17 + 0.2 \u00d7 90\n= 47.60\n\nWeighted Average Prices in 1545\n= 0.4 \u00d7 50 + 0.2 \u00d7 25 + 0.2 \u00d7 15 + 0.2 \u00d7 100\n= 48.00\n\nOnce we have total price of the basket for both periods, we can just plug in the figures in the following formula:\n\n$$Consumer\\ Price\\ Index \\\\= \\frac{47.60}{48.00} \u00d7 100 = 99.17$$\n\nCPI in 1545 is 99.17 as compared to 100 in 1540. This tells that there is no deterioration in the purchasing power of the people.","date":"2018-07-23 17:16:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3447512090206146, \"perplexity\": 2333.608320234719}, \"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-2018-30\/segments\/1531676599291.24\/warc\/CC-MAIN-20180723164955-20180723184955-00493.warc.gz\"}"}
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Autumn colours at Maggie White!!
Home › Blog post › Autumn colours at Maggie White!!
We now have almost all of this seasons stock in store and there are some fabulous autumn colours, perfect for this time of year.
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{
"redpajama_set_name": "RedPajamaC4"
}
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\section{The fourth family and strong interactions}
The Higgs scalar of the standard model unitarizes the scattering of massive gauge bosons, thus saving the theory from breaking down above about 1.8 TeV \cite{X}. But the Higgs by itself is not a complete solution. Additional physics must be introduced to cancel quadratically diverging contributions to the Higgs mass $m_H$, in particular from a top quark loop. The mass scale of this additional physics must be less than about $3.5m_H$ to avoid fine tuning \cite{Y}. For a light Higgs this necessitates new physics well below 1.8 TeV. Among the reasons why this type of picture is very popular are the following two. The first is that the Higgs sector and the required additional physics can all be weakly interacting, thus allowing the perturbative regime to extend to energy scales far above 1.8 TeV. The second is that the new physics that is required should be very accessible, most notably at the LHC.
On the other hand nature may have chosen a less contrived method of ensuring unitarity, one where the scale of the would-be breakdown of unitarity is the scale of new physics. This new physics would not only be responsible for electroweak symmetry breaking, but it could also be quite closely associated with the physics of flavor and fermion mass. This is in contrast to the light Higgs picture where the origin of the observed pattern of fermion masses, as encoded in a set of Yukawa couplings, is pushed to extremely high and inaccessible energies.
The theory of the Goldstone bosons of electroweak symmetry breaking may be the weak coupling description dual to a strongly coupled theory involving different degrees of freedom. In this case a simple Goldstone description only holds up to an ultraviolet cutoff, beyond which it makes more sense to use the dual description. Given that such a duality is already known to exist, relating as it does the chiral Lagrangian and quark-gluon descriptions of QCD, and given the prevalence of the duality concept in modern theoretical developments, it is curious that another manifestation of weak-strong duality is not widely anticipated to show up at the LHC.
Of course the QCD analogy for electroweak symmetry breaking has been quite well explored, as reviewed for example in \cite{C}. In technicolor theories one expects a $\rho$-like resonance to be associated with the unitarization of the Goldstone boson scattering amplitudes. A naive scaling up in mass of the QCD $\rho$ puts the new $\rho$-like state at about 2 TeV. Besides other problems with this classic technicolor picture, this broad resonance is not something that will be quickly and easily probed at the LHC. A more accessible variant is low-scale technicolor \cite{C, R}, where a new $\rho$-like state becomes both lighter and narrower. But this involves increasing the number of technifermions, and thus leads to a tension with the electroweak correction parameter $S$ that typically increases with the number of new fermions \cite{L}. For a small $S$ to emerge the theory would have to be distinctly non-QCD-like, in the sense that a constituent-quark-like approximation would have to be very poor.
We shall relax a different assumption in the original QCD analogy, namely that the new fermions involved with dynamical symmetry breaking are confined. The new fermions certainly have to feel a sufficiently attractive interaction in some channel to cause chiral symmetry breaking, but confinement is not necessary. If gauge interactions are responsible then they may be broken gauge symmetries, broken through the same dynamical fermion masses that break electroweak symmetries and/or by some other agency. This makes possible a very economical picture as far as the new fermionic degrees of freedom are concerned; a sequential fourth family with standard model quantum numbers is all that is needed.\footnote{This does not preclude the possibility that there are also other new fermions on which a new unbroken gauge symmetry continues to act. If such fermions are confined but are light or massless then their contributions to $S$ may be minimized \cite{U}. We will ignore this possibility here.} The idea that a fourth family is related to electroweak symmetry breaking has some history \cite{F0,F}.\footnote{A fourth family has also been considered for other reasons \cite{K}.}$^{,}$\footnote{For a more general review of new types of fermions see \cite{T}.}
It may seem that a further replication of the family structure, already triplicated in nature, would be the most unimaginative type of new physics that could be postulated. But a fourth family has quite profound implications if the new quarks have mass above about 550 GeV. In this case the Goldstone bosons are strongly coupled to the heavy quarks, as the classic analysis \cite{I} of partial wave unitarity shows. This precludes the perturbative description of the Goldstone modes at this energy scale and above, as would have been implied by a light Higgs. In fact the heavy quark masses would serve as the order parameters for electroweak symmetry breaking, and the new strong interactions would be expected to produce these condensates dynamically. Thus the discovery of a heavy fourth family would eliminate the raison d'\^{e}tre of both the light Higgs and the associated physics needed to protect the Higgs mass.
The discovery of a fourth family could potentially come quite early. The fourth family quarks and leptons are free to have mass mixing (CKM mixing) with the lighter fermions, and thus tree-level charged-current decays. We will discuss some processes of this type that should be quite accessible at the LHC. The only source of missing energy in these events is due to light neutrinos originating from weak interactions; this is a feature of known physics, but it is not a feature of many popular scenarios for physics beyond the standard model.
There are constraints on a fourth family. From the strong constraint on the number of light neutrinos we know that the fourth family neutrino is heavy. The $S$ parameter is sensitive to a fourth family, but the experimental limits on $S$ have been evolving over the years in such a way that the constraint on a fourth family has lessened. In addition the masses of the fourth family leptons may be such as to produce negative $S$ and $T$. As we discuss in the next section, the constraints from $S$ and $T$ do not prohibit the fourth family, but instead serve only to constrain the mass spectrum of the fourth family quarks and leptons \cite{Tneutrino,B}. The implied masses for the fourth family leptons should make them particularly accessible at the LHC, with neutrino pair production providing the most interesting signatures.
We have mentioned that the dynamical symmetry breaking of electroweak symmetries should also be quite closely associated with the physics of flavor and fermion mass. This linkage quite generally introduces some challenging issues, with the prime example being the generation of the top quark mass in a manner consistent with electroweak precision data. After the next section we shall explore such issues in the context of a heavy fourth family. Although we will not follow a top-down approach here, a sequential fourth family is theoretically attractive because it makes it possible that a theory of flavor is related to the breakdown of a simple family gauge symmetry. In contrast new fermions not having standard model quantum numbers would be more surprising and difficult to understand.
\section{Constraints and Signatures}
Constraints on the masses of the fourth family fermions $t'$, $b'$, $\tau'$ and $\nu'_{L\tau}$ are obtained from their contributions to the electroweak correction parameters $S$ and $T$. As discussed in the following sections the dynamical mass of all these fermions can arise in a similar way, including the Majorana mass for the fourth left-handed neutrino. The one loop contributions may be approximated as follows \cite{Tneutrino},
\begin{eqnarray}
S&=&\frac{7}{12\pi}-\frac{1}{3\pi}\ln(\frac{m_{\tau'}}{m_{\nu'}}),\\
\alpha f^2 T&=&\frac{1}{16\pi^2}(3g(m_{t'},m_{b'})+g(m_{\nu'},m_{\tau'}))-\frac{m_{\nu'}^2}{4\pi^2}\ln(\frac{\Lambda_{\nu'}}{m_{\nu'}}),\\
g(m_1,m_2)&=&m_1^2+m_2^2-\frac{4m_1^2m_2^2}{m_1^2-m_2^2}\ln(\frac{m_1}{m_2}),
\end{eqnarray}
where $f=246$ GeV. These expressions assume that the masses are sufficiently above the $Z$ mass; note also that $g(m_1,m_2)\rightarrow \frac{4}{3}(m_1-m_2)^2$ for $m_1\approx m_2$. The presence of an ultraviolet cutoff $\Lambda_{\nu'}$ reflects the dynamical nature of the $\nu'_\tau$ mass; namely that the mass function will fall to zero in the ultraviolet.\footnote{There is no $SU(2)_L$-triplet scalar field whose kinetic term is being renormalized by this loop, and whose vacuum expectation value would have produced a large tree-level contribution to $T$.} We see that the lepton sector can make negative contributions to both $S$ and $T$. The Majorana nature of $\nu'_\tau$ is responsible \cite{Tneutrino} for the negative term in $T$ and the reduction of $S$ by $1/12\pi$. The origin of the mass-dependent term in $S$ is described in \cite{S}. For the values of masses that are of most interest it turns out the electroweak correction parameter $U$ is quite small, and we will ignore it henceforth. The use of these one-loop results assumes that the effects of the strong interactions are largely accounted for by using the dynamically generated masses in the loops, while ignoring momentum dependence of the masses themselves. This approximation should be more appropriate in our case of a broken gauge theory dynamics than it is for technicolor or QCD.
Since $T$ from the leptons can be negative, there can be some degree of cancellation between this and the positive contribution from the quarks. If we remove the light Higgs from the standard model (or set its mass to 1 TeV) then current data requires a new physics contribution to $T$ in the range $0.25\lesssim\Delta T\lesssim0.55$ at 68\% CL. (This is based on the plot at \cite{O}.) The edge of the allowed region in the $m_{\tau'}$-$m_{\nu'}$ plane in Fig.~(\ref{la}) corresponds to lepton masses that provide the maximum contribution $\Delta T = 0.55$ along with a vanishing contribution from degenerate quarks. Within the allowed region, the leptons can provide progressively smaller and eventually negative contributions which can cancel against the progressively more positive quark contribution. Going too far into the allowed region implies more of a tuning in this cancellation, since the quark contribution to $T$ increases by one from one contour to the next.
\begin{figure}
\begin{center}\includegraphics[%
scale=0.42]{fig1}\hspace{-1pt}\includegraphics[%
scale=0.42]{fig2}\end{center}
\vspace{-3ex}
\caption{From the total fourth family contribution to $T$ we shade in yellow the allowed region for the $\tau'$ and $\nu'_\tau$ masses (in TeV). The successive higher contours correspond to increasing the quark contribution to $T$ by 1. The three straight red lines from bottom to top indicate when the total contribution to $S$ is $0,0.11,0.22$, where the latter two values are $1\sigma$ and $2\sigma$ away from the central measured value. The left and right figures have $\Lambda_{\nu'}=1.5m_{\nu'}$ and $2m_{\nu'}$ respectively.\label{la}}
\end{figure}
For $S$ the constraints are such that new physics (again with the light Higgs removed) can contribute $-0.2\lesssim\Delta S\lesssim0.11$ at 68\% CL. We show the lines corresponding to the $1\sigma$ and $2\sigma$ upper bounds on $S$ along with the $S=0$ line on the plots. Thus $S$ also limits how far one can go into the allowed region. But acceptable ranges of masses remain, and this is even before realizing the uncertainties in the theoretical estimates due to strong interactions. All these considerations show that a fourth family is quite compatible with present precision data.
Taking Fig.~(\ref{la}) seriously would suggest that $m_{\tau'}/3\lesssim m_{\nu'}\lesssim m_{\tau'}/2$. We might also expect that $m_{\tau'}\lesssim m_{q'}$ due to lack of the QCD contribution to the dynamics in the lepton sector, which would tend to enhance the masses of quarks \cite{W}. It then appears plausible that $m_{\nu'}$ could be in the 150-300 GeV range, with $m_{\tau'}$ in the 400-600 GeV range. $m_{b'}$ and $m_{t'}$ may be in the 550-800 GeV range, and with a mass splitting probably not much larger than 100 GeV. Much larger mass splitting would require more tuning in the canceling contributions to $T$. But note that the total new physics contribution can be as large as half a unit of $T$, while a unit of $T$ from the quarks corresponds to about a 130 GeV quark mass splitting, so even that much splitting would not constitute a fine tuning. In section 4 we shall argue that $m_{b'}>m_{t'}$.
The first signal of a fourth family may involve the fourth family leptons. $\nu'_\tau\overline{\nu}'_{\tau}$ production is more interesting than $\overline{\tau}'\tau'$ production, both because $\nu'_\tau$ is expected to be lighter and because its decay modes are more interesting. The decay $\nu'_\tau\rightarrow\ell W$ with $\ell=(\tau\mbox{ or }\mu\mbox{ or }e)$ leads to the following final states from $\nu'_\tau\overline{\nu}'_{\tau}$ production,
\begin{equation}
2\ell+4j,\quad\quad3\ell+2j+E\!\!\!\!/,\quad\quad4\ell+E\!\!\!\!/.
\end{equation}
These processes should be quite accessible at the LHC, although serious studies seem to be lacking. The first process can have same-sign leptons due to the Majorana nature of the $\nu'_\tau$. (This and other properties of Majorana neutrino pair production are discussed in \cite{G}.) The last process may be similar to the production and decay of a pair of neutralinos, but the presence of the other two processes should make the distinction between neutrinos and neutralinos clear.
The expected heavy quark decays are $t'\rightarrow bW$, which would look like a heavy $t$ decay, and $b'\rightarrow tW$. But if the associated CKM mixing is small then $b'\rightarrow t'W$ could dominate $b'\rightarrow tW$ if the $b'$-$t'$ mass splitting is large enough. Even if the $W$ has to be virtual due to a small mass difference (thus implying phase space suppression) the $b'\rightarrow t'W^{(*)}$ process could still be significant when the mixing is small enough. Thus a process of interest is $pp\rightarrow b'\overline{b}'\rightarrow t'\overline{t}'WW \rightarrow b\overline{b}WWWW$. Notice that the $b$ jets can be particularly hard and isolated, and appropriate cuts can help to reduce the background from $t\overline{t}$ production. This has been used in a study of the $pp\rightarrow t'\overline{t}'\rightarrow b\overline{b}WW$ process at the LHC \cite{A}. The $b'\overline{b}'$ process has two extra $W$'s, aiding further the discrimination from background. One of the resulting signals involves two same-sign leptons and missing energy along with the jets.\footnote{Note that if $t'$ is in fact the heaviest, then the process $pp\rightarrow t'\overline{t}'\rightarrow b'\overline{b}'WW \rightarrow t\overline{t}WWWW \rightarrow b\overline{b}WWWWWW$ is possible.}
\section{Flavor Physics}
Starting with a massless gauge theory of fermions, we suppose that mass and flavor emerges through the breakdown of some of the gauge symmetries. At scales 100 to 1000 TeV some interactions are most likely both strong and chiral, and we assume that they lead in some economical manner to their self-breaking at these scales. The effects of this flavor physics dynamics on lower scales will be carried by a set of effective operators. We expect that all possible operators allowed by the unbroken symmetries are generated, even those that can only be generated nonperturbatively. These manifestations of nonperturbative physics will be important in the following. The only masses allowed by the unbroken SU(2)$_L\times$U(1) symmetry are right-handed neutrino masses; all other fermions are protected from receiving a flavor scale mass and at lower scales will only be affected by the flavor physics through multi-fermion and other nonrenormalizable operators.
The mass of the top quark will certainly be well within an order of magnitude of the $t',b'$ masses, and this suggests that the physics origin of these three masses should be somehow related. We will take this as a strong hint to consider the possibility that the third family also experiences 4-fermion interactions of the same form and similar magnitude as the interactions involving the fourth family. This leads to the picture where the original flavor gauge symmetry breaks in such a way that the first two families are singlets under an unbroken remnant. This remnant gauge symmetry acts on the third and fourth families and may only break closer to the TeV scale. It will contribute to the anomalous scaling of the various operators, and it may ensure that certain operators remain significant at the TeV scale, even though they are generated at the flavor scale. In particular we assume that the theory exhibits near conformal scaling for some range of scales above a TeV, in which case $\overline{\psi}\psi$ has an effective scaling dimension close to 2 \cite{P}. This makes natural the possibility that some 4-fermion operators, at least those that are composed of two such scalars, are close to being relevant operators (close to scaling dimension 4). The role of enhanced operators of this form in theories of flavor has been noted before \cite{F0,Q,tmass,D}. In the following we shall focus on operators of the scalar-scalar form and composed of third and fourth family fermions.
We notice how the same fermions, four standard model families, remain the fundamental degrees of freedom throughout the range of energy scales, even though they experience strong interactions at various scales. The light fermions only feel the strong interactions at the flavor scale, while the heavy families also feel strong interactions down to the TeV scale. These latter interactions become strong enough for the fourth family masses to form at the TeV scale. And even then, since the fermions do not become confined, it is still useful to describe the physics of interest at the LHC in terms of the massive fermion degrees of freedom. We note that a massive constituent quark description works quite well in QCD, even though the quarks in that case are confined. The massive quark picture should be even more appropriate in our case.
We are thus led to a phenomenological description of the dynamics responsible for a condensate $\langle \overline{q'}q'\rangle$ of the fourth family quarks $q'\equiv (t',b')$. The Nambu-Jona-Lasinio (NJL) model provides a minimal framework, where this dynamics is described by a 4-fermion interaction,
\begin{equation}
\frac{g^2}{\Lambda^2}(\overline{q}'_{L} q'_{R})(\overline{q}'_{R} q'_{L})
.\label{e1}\end{equation}
$\Lambda$ represents a cutoff above which a softening of this interaction should occur in a more realistic description. For $g$ above some critical value $g_c$ a condensation occurs. Without invoking a fine-tuning of $g$ close to $g_c$, the resulting dynamical mass $m_{q'}$ should not be too far below $\Lambda$. To get a sense of the fine tuning needed for $m_{q'}\ll\Lambda$, we note that a light composite scalar emerges with mass $\approx 2m_{q'}$ in this case \cite{N}. Then contributions of order $\Lambda^2$ to the scalar mass must be fine-tuned away, and thus the degree of fine-tuning is $\approx 4m_{q'}^2/\Lambda^2$. We believe that fine tuning does not naturally occur and that $m_{q'}$ is not much below $\Lambda/2$.
There is a relation between $m_{q'}$ and $\Lambda$ and the electroweak symmetry breaking scale $v=246$ GeV which is given in a one-loop approximation by the Pagels-Stokar formula
\begin{equation}
v^2=f^2\approx\frac{3m_{q'}^2}{4\pi^2} \ln\frac{\Lambda^2}{m_{q'}^2}
.\end{equation}
For example for $m_{q'}\approx 750$ GeV and $\Lambda$ roughly twice that would imply a suitable $v$ from this formula.\footnote{This is assuming that the $q'$ quarks give the dominant contribution to $v$; the additional smaller contribution from the fourth family leptons implies a somewhat smaller $m_{q'}$.} But ambiguities in matching the phenomenological NJL model to the underlying theory implies that $m_{q'}$ as low as 500 GeV may also be acceptable. This is in line with the unitarity analysis \cite{I}.
In the next two sections we wish to explore the naturalness of finding the third and fourth family masses emerging in this type of picture. Our task will be to understand not only the origin of top mass, but also the smaller masses of the other members of the third family. Rather than trying to specify more precisely what the flavor interactions are, as in \cite{D}, we will continue with a bottom-up approach, and try to find a minimal set of constraints on the 4-fermion operators that could allow for realistic masses. Constraints that can be expressed in terms of approximate symmetries have some chance of being realizable by some underlying flavor dynamics.
\section{Approximate Symmetries}
There are two anomaly-free U(1) family symmetries of the third and fourth families that we could consider. The generators have charges $(+,+,-,-)$ and $(+,-,-,+)$ for the fields $(\psi'_L,\psi'_R,\psi_L,\psi_R)$, where $\psi$ and $\psi'$ denote members of the third and fourth families respectively. They are chosen so that they are vector-like and axial-like respectively with respect to the fermion mass eigenstates, and either or both may correspond to gauge symmetries of the high scale flavor physics. Both symmetries must be broken. Of the two, the axial one is of more interest for constraining the operators that are relevant for producing masses; we will label it by $\cal{Q}$. We no longer consider the possible vector-like symmetry. Notice that $\cal{Q}$ is broken at the very least by the $\langle \overline{q'}q'\rangle$ condensate, and if there is no other much larger contribution to its breaking then it will be a useful approximate symmetry to constrain operators. In particular it will help us to understand the $b$ to $q'$ mass ratio.
We can also consider another axial charge, $(+,-,+,-)$, labeled by $\tilde{\cal{Q}}$. This is not anomaly-free and so could not be gauged, and we take it to be a more badly broken symmetry than $\cal{Q}$. The operators that respect $\tilde{\cal{Q}}$ include those that can be generated by gauge boson exchange diagrams, while those that violate $\tilde{\cal{Q}}$ are purely nonperturbative. Since the two classes of operators are generated by distinctly different physics it is not unnatural to assume that $\tilde{\cal{Q}}$-violating operators are somewhat suppressed relative to the $\tilde{\cal{Q}}$-invariant ones. This suppression will give rise to the $t$ to $q'$ mass ratio.
The quark operators we consider are the products of the following color-singlet, Lorentz scalars, where the products are constructed to preserve SU(2)$_L\times$U(1).
\begin{eqnarray}
&&\overline{q}'_Lq'_R\quad\overline{q}'_Lq_R\quad\overline{q}_Lq'_R\quad\overline{q}_Lq_R\nonumber\\
&&\overline{q}'_Rq'_L\quad\overline{q}'_Rq_L\quad\overline{q}_Rq'_L\quad\overline{q}_Rq_L
\label{e2}\end{eqnarray}
These scalars are either $\cal{Q}$-charged or $\cal{Q}$-neutral, but we only consider products that are $\cal{Q}$-invariant. The product of a scalar in the top row with a scalar in the second row produces a $\tilde{\cal{Q}}$-invariant 4-fermion operator, with the LRRL structure as in (\ref{e1}). The product of two operators within a row produces a $\tilde{\cal{Q}}$-violating operator with the LRLR structure, for example
\begin{equation}
\label{e7}\epsilon_{ij}\epsilon_{kl}\overline{q}'_{Li}q'_{Rk}\overline{q}_{Lj}q_{Rl},
\end{equation}
which is SU(2)$_L\times$SU(2)$_R$ invariant. But as we shall see, SU(2)$_R$ violation must manifest itself in the $\tilde{\cal{Q}}$-violating operators, reflecting the SU(2)$_R$ breaking that must originate in the associated nonperturbative dynamics.
Depending on the signs and strengths of all these interactions we assume that condensates form. It is then a question of vacuum alignment as to whether the $\cal{Q}$-charged or $\cal{Q}$-neutral condensates form. We have already assumed the former; more precisely we have assumed that some approximate symmetry exists, labelled by $\cal{Q}$, which is axial with respect to the mass eigenstate basis.
The dynamics that produces $\cal{Q}$-charged condensates is represented by the 4-fermion operators that involve the $\cal{Q}$-charged scalars. There are only two such operators that are both $\cal{Q}$-invariant and $\tilde{\cal{Q}}$-invariant,
\begin{equation}
\overline{q}'_Lq'_R\overline{q}'_Rq'_L,\quad\quad\overline{q}_Lq_R\overline{q}_Rq_L
.\label{e5}\end{equation}
It is important to note that $\overline{q}'_Lq'_R\overline{q}_Rq_L$ is not $\cal{Q}$-invariant. Although these two operators may have similar (running) coefficients we assume (in the absence of a symmetry) that they are not identical. Then we can assume that the first operator develops an effective coupling above the critical value, while the second operator does not. Alternatively or in addition there may be an effective coupling between the two channels that discourages both condensates from forming simultaneously. This type of coupling between channels could be represented by the multi-quark operator $\overline{q}'_Lq'_R\overline{q}'_Rq'_L\overline{q}_Lq_R\overline{q}_Rq_L$ with the appropriate sign.
Thus if these $\tilde{\cal{Q}}$-invariant operators respect SU(2)$_R$, and if we continue to ignore the $\tilde{\cal{Q}}$-violating operators we can have the result,
\begin{eqnarray}
&&\langle\overline{t}'t'\rangle=\langle\overline{b}' b'\rangle\neq0,\\
&&\langle\overline{t} t\rangle=\langle\overline{b} b\rangle=0.
\end{eqnarray}
Then to obtain a $t$ mass from a $\cal{Q}$-invariant operator we must turn to $\tilde{\cal{Q}}$-violating operators. The operator of interest is
\begin{equation}
\epsilon_{ij}\overline{q}'_{Li} b'_R\overline{q}_{Lj} t_R\rightarrow\overline{b}'_L b'_R\overline{t}_L t_R
.\label{e3}\end{equation}
This type of operator must involve both families to be $\cal{Q}$-invariant, and we see that it feeds mass from $b'$ to $t$. Thus we see that the $t$ to $b'$ mass ratio is a measure of the amount of $\tilde{\cal{Q}}$ violation.
There is a corresponding operator that feeds mass from $t'$ to $b$, and thus that operator must be significantly smaller. The dominance of the operator in (\ref{e3}) indicates that there must be a close to maximal breakdown of SU(2)$_R$ in the $\tilde{\cal{Q}}$-violating sector of the underlying dynamics.\footnote{A toy scalar potential was considered in the appendix of the second reference in \cite{D} that illustrates such a maximal breakdown of SU(2)$_R$.} The $b$ mass could also be produced by the $\tilde{\cal{Q}}$-invariant but $\cal{Q}$-violating operator $\overline{b}'_L b'_R\overline{b}_R b_L$. Thus the $b$ to $b'$ mass ratio puts an upper bound on the amount of $\cal{Q}$ violation in the quark sector.
Obtaining a large enough $t$ mass has often proven to be difficult in models of dynamical symmetry breaking. This is because the operator responsible for the $t$ mass has typically been taken to be generated by a simple gauge boson exchange. In our context this would correspond to the $\tilde{\cal{Q}}$-invariant operator $\overline{q}_L\gamma_\mu q'_L\overline{t}'_R\gamma^\mu t_R\rightarrow\overline{t}'_L t'_R\overline{t}_R t_L$. The trouble is that if this operator was generated by the exchange of a relatively light gauge boson (it cannot be in our context because it is both $\cal{Q}$ and SU(2)$_R$ violating) then the following operators could also be generated through closely related gauge boson exchanges:
\begin{equation}
\overline{q}'_L\gamma_\mu q'_L\overline{t}'_R\gamma^\mu t'_R,\quad\quad\overline{q}_L\gamma_\mu q'_L\overline{q}'_L\gamma^\mu q_L
.\label{e4}\end{equation}
The first of these operators would give rise to a mass splitting in the $(t',b')$ doublet of the same order as the top mass itself. A splitting equal to the top mass produces a shift $\Delta T\approx1.7$, which is significantly larger than what is currently allowed. For a more detailed analysis of this problem in the technicolor context see \cite{V}. The second operator implies a correction to the $Zb\overline{b}$ vertex that is similarly too large \cite{M}.
These basic problems have motivated many different types of model building efforts such as non-commuting extended technicolor, multiscale technicolor, topcolor, topcolor-assisted technicolor and topcolor seesaw models (for a review and references see \cite{C}).\footnote{The same problems also require special attention in the Higgless models of higher dimensions \cite{J}.} These models generally involve complicating the gauge structure and/or adding new gauge dynamics coupling to the $t$ quark. Here we are pointing out that it is not strictly necessary to invoke such complications, given the possibility that the operator (\ref{e3}) gives the dominant contribution to the $t$ mass.
The point is that the side-effects of the $\tilde{\cal{Q}}$-violating operator (\ref{e3}) are not so severe \cite{tmass}. It can give rise to effects similar to those in (\ref{e4}) (which are $\tilde{\cal{Q}}$-invariant) only by inserting it twice in a loop. Thus an operator similar to the first operator of (\ref{e4}), $\overline{q}'_L b'_R\overline{b}'_R q'_L$, is generated with a suppression of $m_t^2/m_{q'}^2$ along with the extra loop suppression. This effect breaks the SU(2)$_R$ invariance of the operators $\overline{q}'_L q'_R\overline{q}'_R q'_L$ that are responsible for the $t'$ and $b'$ masses, giving rise to a mass splitting with $m_{b'}>m_{t'}$. The contribution to $T$, proportional to $(m_{b'}-m_{t'})^2$, is then suppressed at least by $m_t^4/m_{q'}^4$ in comparison to the quadratic suppression in models with only gauge-exchange operators.\footnote{The operator $\overline{b}'_R\gamma_\mu b'_R\overline{b}'_R\gamma^\mu b'_R$ that can contribute directly to $T$ would require four insertions of operator (\ref{e3}) and three loops.} An operator that can affect the $Zb\overline{b}$ vertex, like the second operator in (\ref{e4}), but which can only be generated by a loop with two insertions of operator (\ref{e3}) is $\overline{q}_L b'_R\overline{b}'_R q_L$.\footnote{The SU(2)$_R$ invariant operators in (\ref{e5}), and those closely related to them such as $\overline{q}_{L} q'_{R}\overline{q}'_{R} q_{L}$, neither contribute to $T$ nor correct the $Zb\overline{b}$ vertex.} (The second operator in (\ref{e4}) is not generated.) In conclusion we see how the corrections to $T$ and the $Zb\overline{b}$ vertex are more shielded from top mass generation because of the $\tilde{\cal{Q}}$-violating nature of the top mass operator.
\section{Leptons}
We first turn to the charged lepton sector. For $\tau$ and $\tau'$ we can suppose similar 4-fermion dynamics as in the quark sector, with the same approximate $\cal{Q}$ symmetry constraining the dynamics. Thus we can again suppose that the $\cal{Q}$ and $\tilde{\cal{Q}}$ invariant operators (the analogs of (\ref{e5}) with $\tau$ and $\tau'$ replacing $q$ and $q'$) generate $\langle\overline{\tau}'\tau'\rangle\neq0$ while $\langle\overline{\tau}\tau\rangle=0$. The $\tau$ mass can arise similarly to the $b$ mass, and in particular the following SU(2)$_R$ and $\tilde{\cal{Q}}$ violating, but $\cal{Q}$-invariant operators can feed mass from $t'$ to $b$ and $\tau$:
\begin{equation}
\overline{t}'_L t'_R\overline{b}_L b_R,\quad\quad\overline{t}'_L t'_R\overline{\tau}_L \tau_R
.\label{e13}\end{equation}
Here we see our first instance of an operator with both quarks and leptons. (In the Appendix we consider a different choice of the approximate symmetries that results in a different structure for the mixed operators.)
Neutrinos are more special. We are supposing that all fermions, including the right-handed neutrinos, participate in the strong flavor interactions at the flavor scale. If SU(2)$_L\times$U(1) is the only exact chiral symmetry remaining below the flavor scale, then there is nothing to protect the right-handed neutrinos from receiving mass from the strong interactions. In fact right-handed neutrino condensates serve as excellent order parameters not only for the breakdown of flavor symmetries, but also for the breakdown of enlargements of the electroweak symmetry such as those involving SU(2)$_R\times$U(1)$_{B-L}$ and/or Pati-Salam-like gauge interactions. With their masses at the flavor scale the right-handed neutrinos are absent in the theory below the flavor scale, and this in turn is important for understanding why the small left-handed neutrino masses are so dramatically different from other fermion masses.
But first we consider $\nu'_{L\tau}$ where we see that its mass (again $\cal{Q}$-violating) can arise in a similar way to other fourth family members. Again there are only two $\cal{Q}$ and $\tilde{\cal{Q}}$ invariant operators of interest,
\begin{equation}
\ell'_L\ell'_L({\ell'_L}{\ell'_L})^\dagger,\quad\quad\ell_L\ell_L({\ell_L}{\ell_L})^\dagger
,\label{e9}\end{equation}
since $\ell'_L\ell'_L({\ell_L}{\ell_L})^\dagger$ is not $\cal{Q}$-invariant. (Operators such as $\ell'_L\ell'_L{\ell_L}{\ell_L}$ can be $\cal{Q}$ and SU(2)$_L\times$U(1) invariant, but they don't involve four neutrinos.) Thus by the same reasoning as before we can assume that $\langle\nu'_{L\tau}\nu'_{L\tau}\rangle\neq0$ while $\langle\nu_{L\tau}\nu_{L\tau}\rangle=0$. We are then left with the three light neutrinos $(\nu_{L\tau},\nu_{L\mu},\nu_{Le})$.
Now the question is whether $\nu_{L\tau}$ can receive a mass in a manner similar to other third family fermions. The answer is no, since in this case there are no $\cal{Q}$-invariant operators that can feed down mass from the fourth family. The $\cal{Q}$-violating operator $\ell'_L\ell'_L({\ell_L}{\ell_L})^\dagger$ can yield a $\nu_{L\tau}$ mass, and thus the relatively tiny value of this mass implies that the $\cal{Q}$ symmetry must be very well preserved by the effective operators in the left-handed lepton sector.
There are also operators that arise by integrating out the right-handed neutrinos at the flavor scale. The resulting lepton number violating operators necessarily involve six fermions, and they can generate Majorana masses for $\nu_{L\mu}$ and $\nu_{Le}$ as well as $\nu_{L\tau}$. These 6-fermion operators are naively suppressed by three more powers of the flavor scale compared to 4-fermion operators, thus providing a natural mechanism for the suppression of neutrino masses. This could be thought of as a type of see-saw mechanism, but the right-handed neutrino mass in the see-saw is now set by the flavor scale, of order 1000 TeV. Once again we see how the absence of a Higgs brings down a mass scale of interest.
There are many different 6-fermion operators that can contribute. If they are to feed down mass from the heaviest fermions then they can be constructed by taking Lorentz invariant products of any pair of the following 3-fermion operators (all of which transform as SU(2)$_L\times$U(1) invariant Lorentz spinors).
\begin{equation}
\overline{t}'_L t'_R \nu_{Li}\quad\quad\overline{t}_L t_R \nu_{Li}\quad\quad\overline{b}'_R b'_L \nu_{Li}\quad\quad\overline{\tau}'_R \tau'_L \nu_{Li}\quad\quad i=e,\mu,\tau
\end{equation}
We see that each element of the $3\times3$ Majorana neutrino mass matrix has many possible contributions from the various combinations. The relative size of these contributions depends on the detailed structure of the flavor interactions and their breakdown. By dimensional analysis the resulting neutrino masses are probably no less than $(600 \mbox{ GeV})^6/(1000 \mbox{ TeV})^5\approx5\times10^{-5} \mbox{ eV}$. This is likely an underestimate since it ignores possible anomalous scaling enhancement of the 6-fermion operators. One is also tempted to use the see-saw estimate of the form $m^2/M$, where $m$ is some Dirac mass and $M$ is the right-handed neutrino mass, but this assumes that the anomalous scaling contained in the value of $m^2$ is the same as that of the 6-fermion operator. This is certainly incorrect for the case of $\nu_{L\tau}$ but it may be more appropriate for $\nu_{L\mu}$ and $\nu_{Le}$. Reasonable masses seem entirely possible (for example if $m^2\approx m_e m_\mu$). In addition we see that the structure of the $3\times3$ neutrino mass matrix is quite unrelated to the quark and charged lepton mass matrices, and can have significant off-diagonal terms and thus large mixings \cite{D}.
\section{Further remarks}
We return to the question of the CKM mixing in the quark sector, responsible for the decays $t'\rightarrow bW$ and $b'\rightarrow tW$. The off-diagonal $\overline{t}t'$ or $\overline{b}b'$ mass elements would require $\cal{Q}$ violation, thus making this CKM mixing naturally small. Alternatively these off-diagonal elements could arise as described in the Appendix. As another possibility, \cite{CP} shows that kinetic-term mixing effects may be a source of CKM mixing along with CP violating phases. Flavor physics could also generate flavor changing neutral current decay modes of the heavy quarks \cite{H}. But these vertex-type mixing effects are probably smaller than the mass mixing effects, due to less anomalous scaling enhancement of the relevant operators, and thus we expect the charged current decays to dominate.
Pair production of the fourth family fermions could exhibit a resonance structure associated with the physics near the cutoff of our effective theory. For example there could be a broken U(1) gauge boson that mixes with the $Z$ and which couples strongly to the fourth (and third) families. Alternatively the strong interactions may imply unconfined bound states of the heavy fermions. And finally if the CKM mixing is small enough then even QCD bound states of the heavy quarks could show up as resonances.
There may also be approximate global symmetries that are broken by the fourth family condensates leading to pseudo-Goldstone bosons, similar to technipions of technicolor theories and coupling to fermions in similar ways. But the masses of such states are so extremely model dependent that we consider them no further. We note though that our practice of assuming the existence of all possible multi-fermion operators generally eliminates the concern over unwanted light or massless pseudo-Goldstone bosons, especially if the original underlying theory has no global symmetries to begin with.
We have been concerned with a $\cal{Q}$ invariance of operators involving only the third and fourth family fermions. This is only an approximate symmetry of flavor physics in particular because, if the light fermions are $\cal{Q}$ neutral, it cannot be a symmetry of operators that are needed to feed mass to the light fermions. It may be possible to extend the $\cal{Q}$ generator to also act on light fermions and thus find an approximate symmetry of a larger set of operators and the full mass matrices. This would lead to the consideration of more complete models, where the full particle content of the theory and the assumed pattern of symmetry breaking are both specified. Such a top-down approach was taken in \cite{D}, and there it may be seen that the $\cal{Q}$ generator and its extension to the light families is a gauge generator of the complete underlying theory. One comment about such a picture is that the hierarchy between the third and fourth family masses may lead in turn to the hierarchy between the first two families. We have chosen in this work to focus in a more model independent fashion on the heavy families, since this is where the more serious issues typically arise.
In summary, a sufficiently massive fourth family points towards an extension of the standard model that treats the Goldstone bosons of electroweak symmetry breaking as the weak coupling dual description of a more fundamental strongly coupled theory. Although we have not specified the fundamental interactions of the fourth (and third) families, we have modeled them phenomenologically via 4-fermion operators. This has enabled us to find some minimal approximate U(1) symmetries of the fundamental interactions that help to explain the range of masses of the third and fourth families. This makes it more likely that such interactions can exist.
The fourth family forms part of the fundamental degrees of freedom, and it may constitute all of the new fermionic degrees of freedom. The fourth family quark masses are fixed (up to theoretical uncertainties) by the scale of electroweak symmetry breaking, and then the masses of the fourth family leptons are constrained by the $T$ (and $S$) parameters. This is analogous to the Higgs picture where the vacuum expectation value $v$ is fixed and there is another parameter, the Higgs mass, that must be adjusted small enough to obtain the correct $T$. Additional new physics is required to protect the Higgs mass. It is exciting to realize that within a few years we will know which picture of new physics comes closer to describing reality.
\section*{Appendix: Alternative choice of approximate symmetries}
We have seen how quark masses can affect the lepton mass matrix, and vice versa, but the structure of these mixed operators may be different than described above. To see this we reconsider the possible anomaly free symmetries of the third and fourth families, now generalizing to generators that do not act identically on quarks and leptons.
\begin{center}
\begin{tabular}{ccccccccc} & $q'_L$ & $q'_R$ & $\ell'_L$ & $\ell'_R$ & $q_L$ & $q_R$ & $\ell_L$ & $\ell_R$ \\
${\cal Q}_V^q$ & $+$ & $+$ & 0 & 0 & $-$ & $-$ & 0 & 0 \\
${\cal Q}_A^q$ & $+$ & $-$ & 0 & 0 & $-$ & $+$ & 0 & 0 \\
${\cal Q}_V^\ell$ & 0 & 0 & $+$ & $+$ & 0 & 0 & $-$ & $-$ \\
${\cal Q}_A^\ell$ & 0 & 0 & $+$ & $-$ & 0 & 0 & $-$ & $+$ \end{tabular}
\end{center}
These cannot all be independent approximate symmetries, since that would suppress any mixed operator, such as the second operator in (\ref{e13}). Thus far we have only needed to assume that ${\cal Q}_A^q+{\cal Q}_A^\ell$ (which we labeled simply as $\cal Q$) is an approximate symmetry. But an interesting alternative is to assume that the following two are approximate symmetries: ${\cal Q}_A^q+{\cal Q}_V^\ell$ and ${\cal Q}_V^q+{\cal Q}_A^\ell$. The effect on the pure quark or pure lepton operators of interest to mass formation would be the same as before. But the mixed operator in (\ref{e13}) would not be allowed, and instead there could be the following operators:
\begin{equation*}
\overline{t}'_L t'_R\overline{\tau}_L \tau'_R,\quad\quad\overline{b}'_L b'_R\overline{\tau}_R \tau'_L.
\end{equation*}
This would give rise to off-diagonal mass elements in the charged lepton mass matrix, which along with the $\tau'$ mass would produce a $\tau$ mass in a see-saw manner. Similarly there could be new off-diagonal elements in the quark mass matrix, for example from the operator $\overline{\tau}'_L \tau'_R\overline{t}_L t'_R$, thus creating new sources of CKM mixing \cite{D}.
\section*{Acknowledgments}
This work was supported in part by the National Science and Engineering Research Council of Canada.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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"He has survived more than most babies and he is thriving." - Jaxson's Story
He also endured many complications during his 92-day NICU stay. At 7 days old, he had an intestinal perforation; at 3 weeks old, he suffered a pulmonary hemorrhage; and at 2 months old, he had a horrible infection. We know that it is because of God and the skills he has given Dr. Kays that we are able to enjoy our son and watch him grow, despite all the challenges he faced early on.
Liver Up - Left SidedJake Nelson May 18, 2016 ECMO
"She was bi-lateral. She had liver coming in through the right and left side." - Hazel's Story
Hazel had her repair surgery at 9 days old. Dr. Kays said her hole was huge. She had a full right lung and a nub for a left lung. He told us how Hazel really was a unique one. She ended up being bi-lateral, which is obviously the worst case, but had the best type of hernia to have a Morgagni hernia (an opening through the front instead of the back).
BilateralJake Nelson August 5, 2015 bilateral
"You remain strong because you know he can feel your energy." - Samuel's Story
There is no protocol, and there are no timelines during this whole process. It's a lot of sitting, waiting and monitoring. As a mom who has her life "planned to a T," this was very hard from me to understand.
Liver Down - Left SidedJake Nelson April 27, 2015
"Lilly showed us what a fighter she was!" - Lilly's Story
I can also tell you that I owe all of the precious memories I made with Lilly to Dr. Kays and Joy. I know that I would not have had that time with her if Dr. Kays had not treated her. It also feels good as a parent to have the knowledge that I did all I could for her. She deserved the best. By seeking out Dr. Kays, I gave it to her.
Liver Up - Left SidedJake Nelson February 27, 2015 ECMO
"I wanted a different path, one filled with hope." - Ethan's Story
We still needed more information before choosing our path. These tests would give a pediatric surgeon a better understanding of our child's severity and help us make the best decision for our son.
featured, Liver Up - Left SidedJake Nelson February 16, 2015 featured
"It's not a death sentence" - Claire's Story
Claire goes to weekly physical and occupational therapy which has focused on developing her core strength and combatting any gross motor delays she has due to the stroke. She has made tremendous progress and is meeting most milestones!
featuredJake Nelson January 3, 2015 featured
"He was as severe as predicted." - Adam's Story
Throughout our entire stay in the NICU, no one but Dr. Kays was ever in charge of Adam's care. He would even personally change ventilator settings, make small adjustments in Adam's positioning in bed to allow more lung expansion, and would check throughout the night to see what Adam's most recent labs and vital signs were.
Liver Up - Left SidedJake Nelson June 2, 2014 ECMO
"We decided to continue searching for the perfect place." - Tripp's Story
The first year with a child born with CDH is hard, challenging, and very overwhelming at times. But these babies come here fighters and they are extremely strong.
Liver Up - Left SidedJake Nelson June 13, 2013 ECMO
"Every single month, every single season, she shows improvement." - Anya Mae's Story
She loves Doc McStuffins and says she wants to be a "Baby Nurse" when she grows up. She is the complete opposite of what the original doctors said. She is not neurologically damaged. She is a survivor.
"She told me, 'She has a hernia.' I was thinking...just a hernia." - Abigail's Story
I remember turning to my husband and saying, "I hope nothing's wrong." Then, the doctor came in and greeted us. She was very sweet! She told us she was going to look at the picture then tell us what she was seeing.
Liver Down - Left SidedJake Nelson November 21, 2011
"One of three little miracles" - Emily Grace's Story
We have such a strong family to have made it through all of this!! We are a military family and the triplets were born in Columbia, SC.
Liver Down - Left SidedJake Nelson July 29, 2011
"She doesn't let CDH get in her way." - Savannah Grace's Story
At two months old, I was bringing my baby girl home. She is now a very playful, smart and active 6 year-old. She doesn't let CDH get in her way.
Liver Up - Left SidedJake Nelson August 30, 2010
"I didn't talk or eat, just waited." - Joseph's Story
We were devastated. I can remember sobbing all the way home and my husband holding it together until he talked to his dad. When he said, "Dad, it's not fair" and his voice broke, I could have ripped my heart out.
featured, older, Liver Up - Left SidedJake Nelson June 7, 2010 featured
"Excelling in school and making the most of her life" - Kira's Story
My daughter was born november fifth 2009. She has Left CDH, her lung only matured to 25% on the left side. She was at shands UF for four months six days.
olderJake Nelson November 5, 2009 older, ECMO, LCDH
"I've never seen so many doctors and nurses in one room" - Noah's Story
He is now an active (almost) 10 year old who loves drawing, writing, and sports. He brings joy to every room he enters with his spunky and funny personality, and we pray that his story will help others facing the same diagnosis.
olderJake Nelson August 8, 2008 ECMO
"My son did not cry—but he was beautiful." - Par's Story
He is excelling in school and LOVES all sports—especially contact sports! His endurance is better than most kids his own age- he is an incredible soccer player and as luck would have it, he is pretty good at golf, too! We thank God daily for this miracle who is constantly talking, driving us crazy about college sports scores, and wanting to play outside 24/7.
Fiona Desenberg November 1, 2007
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction --- alignment dynamics and entropic pressure}\label{sec:entropic}
Alignment reflects steering towards average heading, \cite{Rey1987}. It plays an indispensable role in the process of emergence in swarming dynamics, and in particular --- in flocking, herding, schooling,..., \cite{VCBCS1995,CF2003,CKFL2005,CS2007a,CS2007b,Bal2008,Kar2008,VZ2012,MCEB2015,PT2017}, as well as the formation of other self-organized clustering in human interactions and in dynamics of sensor-based networks, \cite{Kra2000,BeN2005,BHT2009,JJ2015,RDW2018,DTW2019,Alb2019}; more can be found in \cite[\S9]{MT2014}, in the book series on active matter, \cite{BDT2017/19, BCT2022}, and in the recent Gibbs' lecture \cite{Tad2022a}.\newline
We discuss alignment dynamics in two parallel descriptions. Historically, alignment models were introduced in the context of agent-based description \cite{Aok1982,Rey1987,VCBCS1995}. In particular, our discussion is motivated by the celebrated Cucker-Smale model, \cite{CS2007a,CS2007b}, in which alignment is governed by weighted graph Laplacians. Our main focus, however, is on the corresponding hydrodynamic description, the so-called Euler alignment equations, governed by a general class of weighted $p$-graph Laplacians, \cite{HT2008, CFTV2010, HHK2010, Shv2021}. In both cases --- the agent-based and hydrodynamic descriptions, the weights for the protocol of alignment reflect pairwise interactions, and are quantified by proper \emph{communication kernel}. Communication kernels are either derived empirically, deduced from higher-order principles, learned from the data, or
postulated based on phenomenological arguments, e.g.,
\cite{CS2007a,CDMBC2007, Bal2008,GWBL2012, JJ2015,LZTM2019,MLK2019, ST2020b}. The specific structure of such kernels, however, is not necessarily known. Instead, we ask how different classes of communication kernels affect the swarming behavior.\newline
The passage from agent-based to hydrodynamic descriptions requires a proper notion of hydrodynamic pressure. In section \ref{sec:entropic} we introduce a class of \emph{entropic pressures} for hydrodynamic alignment and in section \ref{sec:p-alignment} we extend the discussion to the larger class of hydrodynamic $p$-alignment. Our goal is to make a systematic study of the long-time swarming behavior of hydrodynamic alignment, portrayed in section \ref{sec:swarming}, with entropy pressure laws. Specifically, we use the decay of \emph{energy fluctuations}, discussed in section \ref{sec:fluctuations}, in order to quantify the emergence of flocking behavior, depending on the communication kernel. Almost all available literature is devoted to the case of `pressure-less' alignment. We review these results in section \ref{sec:mono-kinetic}. The main theme here is unconditional flocking for pressure-less $p$-alignment, driven by \emph{heavy-tailed} communication kernels. In section \ref{sec:with-pressure} we discuss hydrodynamic alignment driven by a general class of entropic pressure. The remarkable aspect here is that despite the lack of closure of such entropic pressure laws, there holds unconditional flocking of $p$-alignment driven by \emph{singular, heavy-tailed} communication kernels. We are aware that the methodology developed here can be utilized with other Eulerian-based dissipative systems.\newline
The detailed computations are outlined in appendix \ref{sec:derivation}, \ref{sec:pointwise}, \ref{sec:GN} and \ref{sec:dispersion}.
\subsection{Hydrodynamic description of alignment}
We study the long-time behavior of the (hydro-)dynamic description for alignment,
\begin{subequations}\label{eqs:hydro}
\begin{equation}\label{eq:hydro}
\left\{\begin{array}{c}
\begin{split}
& \partial_{t}\rho} %{\rhoa+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua ) = 0,\\
& \partial_{t}(\rho} %{\rhoa\bu} %{\bua )+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua \otimes\bu} %{\bua +{\mathbb P}} %{\pressur\ea ) = {\mathbf A}} %{{\mathbf A}_{\a}(\rho,{\mathbf u}),
\end{split}
\end{array}\right. \quad (t,{\mathbf x})\in (\mathbb{R}_t, \mathbb{R}^d).
\end{equation}
The dynamics is captured by density $\rho} %{\rhoa: \mathbb{R}_t\times \mathbb{R}^d \mapsto \mathbb{R}_+$, momentum, $\rho{\mathbf u}: \mathbb{R}_t\times \mathbb{R}^d \mapsto \mathbb{R}^d$, and pressure tensor, ${\mathbb P}} %{\pressur\ea : \mathbb{R}_t\times \mathbb{R}^d \mapsto \mathbb{R}^d\times \mathbb{R}^d$,
subject to initial data $\displaystyle (\rho} %{\rhoa,\bu} %{\bua ,{\mathbb P}} %{\pressur\ea )_{|_{t=0}} = (\rho_{0},{\mathbf u}_{0},{\mathbb P}_{0})$, and is driven by an
\emph{alignment} term acting on the support ${\mathcal S}(t):=\textnormal{supp}\,\rho(t,\cdot)$,
\begin{equation}\label{eq:align}
{\mathbf A}} %{{\mathbf A}_{\a}(\rho,{\mathbf u}) := \int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})(\bu} %{\bu_{\b}(t,{\mathbf x}'} %{{\mathbf y})-\bu} %{\bua (t,{\mathbf x}))\rho} %{\rhoa(t,{\mathbf x})\rho} %{\rho_{\b}(t,{\mathbf x}'} %{{\mathbf y})\dxp} %{\, \mbox{d}\by, \qquad \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})=\phi} %{\phi_{\a\b} ({\mathbf x}'} %{{\mathbf y},{\mathbf x}).
\end{equation}
The alignment term on the right reflects steering towards average heading.
Here, different weighted averages are dictated by symmetric communication kernels $\phi} %{\phi_{\a\b} (\cdot,\cdot)$. Prototypical examples include \emph{metric kernels},
$\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})=\kab(|{\mathbf x}-{\mathbf x}'} %{{\mathbf y}|)$,
which go back to \cite{CS2007a}. Other classes of symmetric kernels that either dictated by the problem or learned from the data can be found in \cite{GWBL2012,JJ2015,LZTM2019,MLK2019}, and finally we mention
topologically-based kernels studied in \cite{ST2020b},
$\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})=\kab(m(C({\mathbf x},{\mathbf x}'} %{{\mathbf y})))$,
where $\ds m(C({\mathbf x},{\mathbf x}'} %{{\mathbf y}))= \int_{C}\rho(t,{\mathbf z})\, \mathrm{d}\bz$ is the mass enclosed in an intermediate domain $C=C({\mathbf x},{\mathbf x}')$ with tips at ${\mathbf x}$ and ${\mathbf x}'} %{{\mathbf y}$. The prominent role of metric kernels enters when we assume that there exists a radial kernel, $\kab(r)$, such that
\begin{equation}\label{eq:radial}
\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\geq \kab (|{\mathbf x}-{\mathbf x}'} %{{\mathbf y}|).
\end{equation}
\end{subequations}
We further assume that the metric kernel $\kab (r)$ is decreasing with the distance $r$, reflecting the typical observation that the intensity of alignment decreases with the distance. In particular, we address general metric kernels $\phi} %{\phi_{\a\b} (|\cdot|)$ whether decreasing or not, in terms of their \emph{decreasing envelope} $\kab (r):=\min\{\phi} %{\phi_{\a\b} (|{\mathbf x}|) \ | \ |{\mathbf x}|\leq r\}$. Observe that we do not place any restriction on the upper-bound of $\phi$; in particular, therefore, our discussion includes the important sub-class of \emph{singular} communication kernels $\kab(r)=r^{-\alpha}, \ \alpha>0$, \cite{ST2017a, DKRT2018,MMPZ2019,AC2021b}.
\subsection{Entropic pressure}
System \eqref{eqs:hydro} is not closed in the sense that the pressure ${\mathbb P}$ is not specified --- neither in terms of algebraic relations with $(\rho} %{\rhoa,{\mathbf u})$, nor do we specify a precise dynamics of ${\mathbb P}$. Indeed, We do not dwell here on the details of the underlying the pressure tensor. Instead, we treat a rather general class of pressure laws satisfying an essential structural (dissipative) property which, as we shall show, maintains long time flocking behavior. This brings us to the following.
\begin{definition}[{\bf Entropic pressure}]\label{def:meso-pressure}
We say that ${\mathbb P}} %{\pressur\ea $ is \myr{an} entropic pressure associated with \eqref{eqs:hydro} if it has a non-negative trace, $\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea :=\frac{1}{2} \text{trace}({\mathbb P}} %{\pressur\ea )\geq0$, which satisfies
\begin{equation}\label{eq:meso-pressure}
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) \leq - 2\int \limits_{{\mathcal S}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}')e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})\rho(t,{\mathbf x})\rho(t,{\mathbf x}') \, \d\bxp} %{\, \mathrm{d}\bx'.
\end{equation}
Here $\bq} %{\bqa $ is an arbitrary $C^1$-flux.
\end{definition}
\noindent
{\bf Why entropic pressure?} System \eqref{eqs:hydro} falls under the general category of hyperbolic balance laws \cite[Chapter III]{Daf2016}, and \eqref{eq:meso-pressure} can be viewed as an entropy inequality associated with such balance law. To this end, we note that a formal manipulation of the mass and momentum equations, \eqref{eq:hydro}$\ds {}_1 \times \frac{|{\mathbf u}|^2}{2} + $ \eqref{eq:hydro}${}_2 \cdot {\mathbf u}$ yields\footnote{Here and below for a quantity $\square=\square(t,{\mathbf x})$ we abbreviate $\square':=\square(t,{\mathbf x}'} %{{\mathbf y})$}
\begin{equation}\label{eq:energy-kin}
\partial_t \Big(\frac{\rho} %{\rhoa}{2}|\bu} %{\bua |^2\Big) + \nabla_{\mathbf x}\cdot \Big(\frac{ \rho} %{\rhoa}{2}|\bu} %{\bua |^2\bu} %{\bua +{\mathbb P}} %{\pressur\ea \bu} %{\bua \Big) - \textnormal{trace} \big({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua \big) =
- \int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\rho} %{\rhoa\bu} %{\bua \cdot(\bu} %{\bua -\bu'} %{\bubp )\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by.
\end{equation}
Adding the entropic description of the pressure postulated in \eqref{eq:meso-pressure} leads to the entropic statement for the total energy, $\ds E} %{E_{a}:= \frac{|\bu} %{\bua |^2}{2} +e_{\!{}_{\mathbb P}}} %{\ea $,
\begin{equation}\label{eq:entropy-ineq}
\partial_t (\rho E) + \nabla_{\mathbf x}\cdot(\rho E{\mathbf u}+{\mathbb P}{\mathbf u}+\textbf{q}) \leq
-\int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\big(|{\mathbf u}|^2-\bu} %{\bua \cdot\bu'} %{\bubp +2e_{\!{}_{\mathbb P}}} %{\ea \big)\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by.
\end{equation}
Thus, the notion of entropic pressure \eqref{eq:meso-pressure} complements the
balance laws in \eqref{eqs:hydro} to form the \emph{entropy inequality} \eqref{eq:entropy-ineq}.
To further motivate why this notion of an entropic pressure, we appeal to its underlying \emph{kinetic formulation}. The hydrodynamics \eqref{eqs:hydro} corresponds to the large-crowd dynamics
of $N} %{N_{\a}$ agents with position/velocity $\displaystyle ({\mathbf x}_{i}(t),{\mathbf v}_{i}(t)): \mathbb{R}_t\mapsto \mathbb{R}^{d}\times\mathbb{R}^{d}$, governed by the celebrated agent-based alignment model of Cucker \& Smale \cite{CS2007a,CS2007b}
\begin{equation}\label{eq:CS}
\left\{\begin{array}{c}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t}{\mathbf x}_{i}(t) &= {\mathbf v}_{i}(t), \\
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} {\mathbf v}_{i}(t) &= \frac{1}{N} %{N_{\b}}\sum_{j = 1}^{N} %{N_{\b}}\phi} %{\phi_{\a\b} _{ij}(t)({\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)),
\end{split}
\end{array}\right. \qquad i=1,2,\ldots N} %{N_{\a}.
\end{equation}
The alignment dynamics is driven by a weighted graph Laplacian on the right of \eqref{eq:CS}${}_2$, dictated by the symmetric communication kernel,
$\phi_{ij}(t):=\phi} %{\phi_{\a\b} ({\mathbf x}_{i}(t),{\mathbf x}_{i}(t))$.
The passage from the agent-based to the hydrodynamic description is realized by moments of the \emph{empirical distribution}
\begin{equation*}
f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v}) := \frac{1}{N} %{N_{\a} }\sum_{i=1}^{N} %{N_{\a} }\delta_{{\mathbf x}_{i}(t)}\otimes\delta_{{\mathbf v}_{i}(t)}, \qquad (t,{\mathbf x},{\mathbf v})\in \mathbb{R}_{t}\times\mathbb{R}^d\times\mathbb{R}^{d}.
\end{equation*}
The large crowd limits which are assumed to exist, recover \eqref{eqs:hydro}
with
\[
\rho} %{\rhoa(t,{\mathbf x})= \lim_{N} %{N_{\a} \rightarrow\infty}\int \limits_{\mathbb{R}^d} f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv \ \ \textnormal{and} \ \
\rho} %{\rhoa\bu} %{\bua (t,{\mathbf x}) = \lim_{N} %{N_{\a} \rightarrow\infty}\int \limits_{\mathbb{R}^d} {\mathbf v} f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv.
\]
This passage from agent-based to macroscopic description is outlined in appendix \ref{sec:hydro-des} below. It was justified for smooth kernels \cite{HT2008,CFTV2010,CCR2011,FK2019,NP2021,Shv2021} and at least mildly singular kernels, \cite{Pes2015,PS2019,MMPZ2019}. In this context, the pressure or Reynolds stress tensor corresponds to the \emph{second-order moments}
\begin{equation}\label{eq:pressure}
{\mathbb P}} %{\pressur\ea (t,{\mathbf x}) = \lim_{N} %{N_{\a} \rightarrow\infty}\int \limits_{\mathbb{R}^d} ({\mathbf v}-\bu} %{\bua )\otimes ({\mathbf v}-\bu} %{\bua )f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv.
\end{equation}
We observe that the kinetic description of pressure in \eqref{eq:pressure} is consistent with the entropic inequality postulated in \eqref{eq:meso-pressure}. Indeed, $\rho} %{\rhoae_{\!{}_{\mathbb P}}} %{\ea :=\frac{1}{2} \text{trace}({\mathbb P}} %{\pressur\ea )$ is the \emph{internal energy} which quantifies microscopic fluctuations around the bulk velocity $\bu} %{\bua $,
\begin{equation}\label{eq:internale}
\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea = \lim_{N} %{N_{\a} \rightarrow\infty} \int \limits_{\mathbb{R}^d} \frac{1}{2} |{\mathbf v}-\bu} %{\bua |^2f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv.
\end{equation}
This kinetic description of internal energy yields (detailed derivation is carried out in appendix \ref{sec:meso-ppressure} below),
\begin{equation}\label{eq:intequality}
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\textbf{q}_h )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) = - 2 \int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by,
\end{equation}
with the so-called heat flux $\ds \textbf{q}_h:=\lim_{N} %{N_{\a} \rightarrow\infty}\frac{1}{2}\int |{\mathbf v}-{\mathbf u}|^2({\mathbf v}-{\mathbf u})f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv$.
Formally, any kinetic-based pressure tensor is in particular an entropic pressure, in the sense of satisfying the \emph{equality} \eqref{eq:intequality}.
But here one encounters the familiar problem of lack closure which arises whenever one is dealing with the highest truncated ${\mathbf v}$-moments of $f_N} %{f_{\a}$. In classical particle dynamics, the closure problem is resolved by compatibility with a preferred state of thermal equilibrium, a ``Maxwellian'' induced by the thermal equilibrium of the system, \cite{Lev1996,Gol1998,Cer2003,Vil2003}.
In the current setup the agent-based dynamics, however, \eqref{eq:CS} governs \emph{active matter} made of `social particles' which admit no universal Maxwellian closure. Then, there are multiple reasons which led us to postulate the corresponding entropy \emph{inequality} \eqref{eq:meso-pressure}.
\smallskip\noindent
{\bf Scalar pressure}. We discuss the case of scalar pressure law $\ds {{\mathbb P}}={{\mathsf p}} %{{\textnormal{p}}}{\mathbb I}$. A large part of the existing literature on swarming \emph{assumes} a mono-kinetic closure,
\begin{equation}\label{eq:mono-closure}
f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v}) \stackrel{N\rightarrow \infty}{\longrightarrow} \rho(t,{\mathbf x})\delta({\mathbf v}-{\mathbf u}(t,{\mathbf x})),
\end{equation}
which is realized in terms of zero pressure, ${\mathsf p}} %{{\textnormal{p}}=0$, e.g., \cite{HT2008,CFTV2010,FK2019,NP2021,Shv2021} and the references therein.
We mention the derivation from first principles \cite{Bia2012}, the isentropic closure, ${\mathsf p}} %{{\textnormal{p}}=\rho^\gamma$, of \cite{KMT2013,KMT2015,KV2015,Cho2019,TCGW2020,Shv2022}, or equations of state fitted by observation that can be found in \cite{Sin2021} as examples for detailed thermodynamic closures for scalar pressure laws in the form of \emph{equality} in \eqref{eq:scalar-p} below.\newline
The notion of entropic pressure covers all these entropic examples of scalar pressure laws, as it applies to a broad class of pressure laws satisfying the entropy inequality postulated in \eqref{eq:meso-pressure}, but otherwise require no algebraic closure. Indeed, our notion of entropic pressure becomes more transparent in scalar case ${\mathbb P}={\mathsf p}} %{{\textnormal{p}}{\mathbb I}$ where the inequality postulated in \eqref{eq:meso-pressure} for $\ds {\mathsf p}} %{{\textnormal{p}}:=\frac{2}{d}\rhoe_{\!{}_{\mathbb P}}} %{\ea $ reads (assuming no heat flux $\textbf{q}=0$),
\begin{equation}\label{eq:scalar-p}
\partial_t {\mathsf p}} %{{\textnormal{p}} + \nabla_{\mathbf x}\cdot ({\mathsf p}} %{{\textnormal{p}}{\mathbf u})+\frac{2}{d}{\mathsf p}} %{{\textnormal{p}}\nabla_{\mathbf x}\cdot {\mathbf u} \leq -2{\mathsf p}} %{{\textnormal{p}}\int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\rho(t,{\mathbf x}')\dxp} %{\, \mbox{d}\by.
\end{equation}
Formal manipulation, \eqref{eq:scalar-p}$ \times \rho^{-\gamma} - $ \eqref{eq:hydro}$\ds {}_1 \times\gamma \rho^{-\gamma-1}{\mathsf p}} %{{\textnormal{p}}$ with $\gamma=1+\frac{2}{d}$, leads to the equivalent entropic statement for $S=\ln\big({\mathsf p}} %{{\textnormal{p}}\rho^{-\gamma}\big)$,
\begin{equation}\label{eq:S}
\partial_t (\rho S) + \nabla_{\mathbf x}\cdot (\rho {\mathbf u} S) \leq -2\int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\rho(t,{\mathbf x})\rho(t,{\mathbf x}')\dxp} %{\, \mbox{d}\by, \qquad S:=\ln\big({\mathsf p}} %{{\textnormal{p}}\rho^{-(1+\frac{2}{d})}\big).
\end{equation}
We point out that the inequality \eqref{eq:S} is the \emph{reversed} entropy inequality encountered for $-S$ in compressible Euler equations. The difference, which was already noted in \cite[\S6]{HT2008}, is due to different states of thermodynamic equilibria.
\noindent
{\bf Entropic energy dissipation}. An entropy inequality is intimately connected with the \emph{irreversibility} of the underlying process, e.g., the enlightening discussion in \cite[\S2.4]{Vil2003}. In the present context of hydrodynamics alignment, the entropy inequality
\eqref{eq:meso-pressure}, or in its equivalent form \eqref{eq:entropy-ineq}, yields
\begin{equation}\label{eq:energy-dissipate}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \int \limits_{{\mathcal S}(t)}\rho} %{\rhoa E} %{E_{a}\, \d\bx} %{\, \mathrm{d}\bx +
\int \limits_{\partial{\mathcal S}(t)}&\Big(\rho} %{\rhoa E} %{E_{a}\bu} %{\bua \cdot {\mathbf n} +({\mathbb P}} %{\pressur\ea \bu} %{\bua )\cdot{\mathbf n} +\bq} %{\bqa \cdot{\mathbf n}\Big){\textnormal{d}} %{{\sf{d}}}S \\
& \leq
- \iint\limits_{{\mathcal S}(t)\times{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\big(|\bu} %{\bua |^2-\bu} %{\bua \cdot\bu'} %{\bubp +2e_{\!{}_{\mathbb P}}} %{\ea \big)\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by\\
& =- \frac{1}{2}\iint\limits_{{\mathcal S}(t)\times{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\Big(|\bu'} %{\bubp -{\mathbf u}|^2 +2e_{\!{}_{\mathbb P}}} %{\ea +2e'_{\!{}_{\mathbb P}}} %{e'_{b}\Big)\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by <0,
\end{split}
\end{equation}
which reflects the dissipativity of the total energy $\ds \int \rho} %{\rhoa E} %{E_{a}\, \d\bx} %{\, \mathrm{d}\bx$.
Thus, the entropy inequality \eqref{eq:meso-pressure} complements the
balance laws in \eqref{eqs:hydro} to govern the energy dissipation \eqref{eq:energy-dissipate}.
This is reminiscent of P.-L. Lions' notion of dissipative solutions in the context of the Euler equations \cite[\S4.4]{Lio1996}.
One of the main aspects of this work is dealing with arbitrary pressure, without any specifics about the second-order closure for ${\mathbb P}} %{\pressur\ea $.
The definition of entropic pressure in \eqref{eq:meso-pressure} is not concerned with the detailed balance of internal energy. Instead, its main purpose is to secure the dissipative nature of the total energy, $\displaystyle \rho} %{\rhoa E} %{E_{a}$. This partially echoes Vicsek \& Zaferis who argued that in the context of collective motion ``\emph{The source of energy making the motion possible ... are not relevant}'' \cite[\S1.1]{VZ2012}. Here, we abandon a closure in the form of thermal equality \eqref{eq:intequality} and instead, retain the inequality postulated in \eqref{eq:meso-pressure}, compatible with the dissipativity of internal fluctuations which we argued for in \cite[p. 501]{Tad2021}. In particular, our definition of a pressure in \eqref{eq:meso-pressure} can be realized in any intermediate scale between the microscopic agent-based description, \eqref{eq:CS}, and the macroscopic hydrodynamics \eqref{eqs:hydro}, and hence can be viewed as ``mesoscopic''. These considerations become even more pronounced when we extend our discussion to a larger class of so-called $p$-alignment hydrodynamics.
\section{$p$-alignment}\label{sec:p-alignment}
We begin with the agent-based description,
\begin{equation}\label{eq:palignment}
\left\{\begin{array}{c}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t}{\mathbf x}_{i}(t) &= {\mathbf v}_{i}(t), \\
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} {\mathbf v}_{i}(t) &= \frac{1}{N} %{N_{\b}}\sum_{j = 1}^{N} %{N_{\b}}\phi} %{\phi_{\a\b} _{ij}(t)|{\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)|^{2p-2}({\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)),
\end{split}
\end{array}\right. \quad i=1,2,\ldots N} %{N_{\a}.
\end{equation}
The case $p=1$ coincides with the Cucker-Smale model \eqref{eq:CS}, while for $p>1$, the alignment term on the right of \eqref{eq:palignment} corresponds to weighted graph $2p$-Laplacian\footnote{To simplify computations we proceed with $2p$-Laplacians rather than $p$-Laplacians.} which is found in recent applications of neural networks \cite{FZB2021}, spectral clustering \cite{BH2009}, semi-supervised learning \cite{ST2019},\cite{Fu2021}. In the context of alignment dynamics it was introduced in \cite{HHK2010,CCH2014}. We were motivated by the example of Elo rating system, \cite{JJ2015,DTW2019}, in which the alignment of scalar ratings $\{q_i\}$ is governed by odd function of local gradients $(q_j-q_i)$, e.g., $|q_j-q_i|^{2p-2}(q_j-q_i)$.\newline
The long time behavior of the $p$-alignment model with $p>1$ is distinctly different from the `pure' alignment model when $p=1$ (and there is yet a different behavior for $0\leq p <1$ which we comment in remark \ref{rem:zalignment} below).
The large-crowd dynamics associated with \eqref{eq:palignment} is captured by the corresponding hydrodynamic description \myr{\myout{(again, its detailed derivation outlined in appendix \ref{sec:hydro-des} below)}}
\begin{subequations}\label{eqs:phydro}
\begin{equation}\label{eq:phydro}
\left\{\begin{array}{c}
\begin{split}
& \partial_{t}\rho} %{\rhoa+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua ) = 0,\\
& \partial_{t}(\rho} %{\rhoa\bu} %{\bua )+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua \otimes\bu} %{\bua +{\mathbb P}} %{\pressur\ea ) = {\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,\bu} %{\bu_{\b}),
\end{split}
\end{array}\right.
\end{equation}
with $p$-alignment term
\begin{equation}\label{eq:palign}
{\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,\bu} %{\bu_{\b}):=\int \limits_{{\mathcal S}(t)}\phi({\mathbf x},{\mathbf x}')|\bu'} %{\bubp -\bu} %{\bu_{\b}|^{2p-2}(\bu'} %{\bubp -{\mathbf u})\rho\rho'} %{\rhobp \, \d\bxp} %{\, \mathrm{d}\bx', \qquad p\geq 1.
\end{equation}
\end{subequations}
\myr{\begin{remark}[{\bf General $p$-alignment terms}]\label{rem:zero-average-Ap}
A detailed derivation of the $p$-alignment term ${\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,{\mathbf u})$ in \eqref{eq:palign} is outlined in appendix \ref{sec:hydro-des}. This kinetic-based derivation is compatible with the mono-kinetic closure \eqref{eq:mono-closure}. In fact, our line of arguments below does not require the detailed form of ${\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,{\mathbf u})$, except for satisfying two `structural' conditions. The first condition requires that it has a zero average
$ \ds \int \limits_{{\mathcal S}(t)} {\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,{\mathbf u})(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx=0$.
This clearly holds for the $p$-alignment \eqref{eq:palign}, and in fact it holds for \emph{any} kinetic closure; see \eqref{eq:zero-kaverage-Ap} below. The second and essential condition requires a $p$-alignment term which induces an entropic pressure. We discuss this notion of entropic pressure in context of $p$-alignment next.
\end{remark}
}
\smallskip\noindent
We assume that ${\mathbb P}$ belongs to a class of entropic pressures, whose definition is adapted to the case of $p$-alignment.
\begin{definition}[{\bf Entropic pressure for $p$-alignment}]\label{def:meso-ppressure}
We say that ${\mathbb P}} %{\pressur\ea $ is a entropic pressure associated with \eqref{eqs:phydro} if it has a non-negative trace, $\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea :=\frac{1}{2} \text{trace}({\mathbb P}} %{\pressur\ea )\geq0$, satisfying
\begin{equation}\label{eq:meso-ppressure}
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) \leq - \frac{1}{2} \int \limits_{{\mathcal S}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\big((2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\big)\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by.
\end{equation}
Here $\bq} %{\bqa $ is an arbitrary $C^1$-flux.
\end{definition}
Definition \ref{def:meso-ppressure} is motivated by the underlying kinetic formulation, where one encounters the $p$-alignment quantity, see appendix \ref{sec:meso-ppressure} below\footnote{Here and below we abbreviate
$\square':=\square(t,{\mathbf x}'} %{{\mathbf y},{\mathbf v}')$},
\[
-\frac{1}{2}\int \limits_{{\mathcal S}(t)} \phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})\iint \limits_{\mathbb{R}^d\times \mathbb{R}^d}|{\mathbf v}-{\mathbf v}'|^{2p}f_Nf'_N\, \mathrm{d}\bv\, \d\bv'} %{\, \mathrm{d}\bv'\, \d\bxp} %{\, \mathrm{d}\bx'.
\]
One cannot close the kinetic expression $\ds \iint |{\mathbf v}-{\mathbf v}'|^{2p}f_Nf'_N\, \mathrm{d}\bv\, \d\bv'} %{\, \mathrm{d}\bv'$ in terms of the thermodynamic quantity $e_{\!{}_{\mathbb P}}} %{\ea $, without taking into account a more detailed thermodynamic information i.e., higher moments of the empirical distribution $f_N$. It is here that we abandon the detailed thermal equality in favor of the inequality which follows from polarization,
${\mathbf v}-{\mathbf v}'\equiv ({\mathbf v}-{\mathbf u})+({\mathbf u}-{\mathbf u}')+({\mathbf u}'-{\mathbf v}')$,
\[
\begin{split}
-\frac{1}{2}&\iint |{\mathbf v}-{\mathbf v}'|^{2p}f_Nf'_N\, \mathrm{d}\bv\, \d\bv'} %{\, \mathrm{d}\bv' \\
& \leq -\frac{1}{2}\Big(\iint \Big(|{\mathbf v}-\bu} %{\bua |^2+ |{\mathbf v}'-{\mathbf u}'|^2\Big)f_Nf'_N\, \mathrm{d}\bv\, \d\bv'} %{\, \mathrm{d}\bv'\Big)^p \big(\rho\rho'\big)^{-\tfrac{p}{p'}}
\stackrel{N\rightarrow \infty}{\longrightarrow} -\frac{1}{2} \big((2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\big)\rho\rho'.
\end{split}
\]
This leads to the corresponding term of $p$-entropic pressure postulated on the right of \eqref{eq:meso-ppressure}.\newline
The special case of pure alignment, $p=1$, offers an alternative derivation where polarization implies the equality, consult \eqref{eq:casepeq1} below
\[
\begin{split}
\iint &({\mathbf v}-{\mathbf u})\cdot({\mathbf v}-{\mathbf v}')f_N} %{f_{\a} f'_N} %{f'_{\b} \, \d\bv'} %{\, \mathrm{d}\bv'\, \mathrm{d}\bv \\
& = -\iint |{\mathbf v}-{\mathbf u}|^2f_N} %{f_{\a} f'_N} %{f'_{\b}\, \d\bv'} %{\, \mathrm{d}\bv'\, \mathrm{d}\bv - \int ({\mathbf v}-{\mathbf u})f_N} %{f_{\a} \, \mathrm{d}\bv \cdot \int ({\mathbf u}-{\mathbf v}')f'_N} %{f'_{\b}\, \d\bv'} %{\, \mathrm{d}\bv' \stackrel{N\rightarrow \infty}{\longrightarrow} -2e_{\!{}_{\mathbb P}}} %{\ea \rho\rho',
\end{split}
\]
which in turn formally yields the entropy \emph{equality} \eqref{eq:intequality},
\begin{equation}\label{eq:meso-pressureb}
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) = - 2\int \limits_{{\mathcal S}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}')e_{\!{}_{\mathbb P}}} %{\ea \rho\rho'} %{\rhobp \, \d\bxp} %{\, \mathrm{d}\bx'.
\end{equation}
Thus, while for $p=1$ the inequality of entropic pressure \eqref{eq:meso-pressure} could be viewed as a matter of choice made in the equalities \eqref{eq:intequality} or \eqref{eq:meso-pressureb}, for $p>1$ the entropic inequality \eqref{eq:meso-ppressure} is a necessity in order to have a macroscopic interpretation of an entropic pressure.
\begin{remark}[{\bf Local vs. global flux}]\label{rem:localvsglobal}
We observe that the entropic statement for $p$-alignment \eqref{eq:meso-ppressure} with $p=1$ is a symmetric version of the entrropic inequality of `pure' alignment, \eqref{eq:meso-pressure}. Apparently, the two definitions do not agree when $p=1$, but in fact, their difference is encoded in different fluxes $\textbf{q}$. In particular, while the entropic pressure in pure alignment \eqref{eq:meso-pressure} is encoded in terms of a \emph{local} heat flux, $\textbf{q}_h$ in \eqref{eq:heat-flux} below, the case of $p$-alignment \eqref{eq:meso-ppressure} requires a \emph{global} flux, $\textbf{q}_h+\textbf{q}_\phi$ in \eqref{eq:penergy-balance} below. Alternatively, we could be less `pedantic' and combine both cases of alignment and of $p$-alignment under the same notion of entropic pressure inequality
\[
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) \leq - 2^{p-1} \int \limits_{{\mathcal S}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})e_{\!{}_{\mathbb P}}} %{\ea ^p\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by, \qquad p\geq 1.
\]
This will not affect any of the follow-up results.
\end{remark}
\noindent
Of course, a general $C^1$ flux, $\textbf{q}$, can also `absorb' the convective term $\rhoe_{\!{}_{\mathbb P}}} %{\ea {\mathbf u}$; our main focus is in the global dissipative structure entailed by \eqref{eq:meso-ppressure}.
\medskip\noindent
{\bf Entropic energy dissipation in $p$-alignment}. Following the same formal manipulations as before for $p=1$, see \eqref{eq:energy-kin}, yield
\[
\partial_t \Big(\frac{\rho} %{\rhoa}{2}|\bu} %{\bua |^2\Big) + \nabla_{\mathbf x}\cdot \Big(\frac{ \rho} %{\rhoa}{2}|\bu} %{\bua |^2\bu} %{\bua +{\mathbb P}} %{\pressur\ea \bu} %{\bua \Big) - \textnormal{trace} \big({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua \big) \leq
- \int \limits_{{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})|{\mathbf u}-\bu'} %{\bubp |^{2p-2}\bu} %{\bua \cdot(\bu} %{\bua -\bu'} %{\bubp )\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by.
\]
Adding \eqref{eq:meso-ppressure} and integrating we find
\begin{equation}\label{eq:penergyinq}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \int \limits_{{\mathcal S}(t)}&\rho E(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx + \int \limits_{\partial{\mathcal S}(t)}\Big(\rho E{\mathbf u}\cdot {\mathbf n}} %{\na + ({\mathbb P}{\mathbf u})\cdot{\mathbf n}} %{\na+\textbf{q}\cdot{\mathbf n}} %{\na\Big){\textnormal{d}} %{{\sf{d}}}S \\
& \leq - \iint\limits_{{\mathcal S}(t)\times{\mathcal S}(t)}\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\Big(|{\mathbf u}-\bu'} %{\bubp |^{2p-2}\big(|\bu} %{\bua |^2-\bu} %{\bua \cdot\bu'} %{\bubp ) +\frac{1}{2}\big((2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\big)\Big)\rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by\\
& = -\frac{1}{2}\iint \limits_{{\mathcal S}(t)\times{\mathcal S}(t)}\phi({\mathbf x},{\mathbf x}')\Big(|{\mathbf u}'-{\mathbf u}|^{2p} +(2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\Big)\rho\rho'\, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx' <0.
\end{split}
\end{equation}
which extends the dissipativity statement of `pure' alignment in the case $p=1$ in \eqref{eq:energy-dissipate}.
\if
\begin{remark}[{\bf Thermodynamic equilibrium}]
Observe that the notion of entropic pressure is invariant under addition of rank-one matrices. Indeed, fix a unit vector ${\mathbf w}\in \mathbb{R}^d$ and set $\ds {\mathbb P}_\eta = {\mathbb P} +\frac{2\eta}{d}{\mathbf w}\bw^\top, \ \eta>0$ then the mass equation \eqref{eq:hydro}${}_1$ implies that \eqref{eq:meso-pressure} holds with $\rho e_\eta :=\rhoe_{\!{}_{\mathbb P}}} %{\ea +\eta$ and $\textbf{q}_\eta:=\textbf{q}-\eta{\mathbf u}$,
\[
\begin{split}
\partial_{t}(\rho} %{\rhoa e_\eta ) +&\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_\eta \bu} %{\bua +\bq} %{\bqa _\eta) +\textnormal{trace}({\mathbb P}_\eta\nabla\bu} %{\bua ) \\
& = \partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea ) +\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}\nabla\bu} %{\bua ) \\
& \leq - 2^{2p-1}\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea ^p \int \limits_{{}} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by < - 2^{2p-1}\rho} %{\rhoa e^p_\eta \int \limits_{{}} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by, \quad \eta>0.
\end{split}
\]
\end{remark}
{\sf{f}}
\section{Swarming}\label{sec:swarming}
The hydrodynamic alignment \eqref{eqs:hydro} occupies a distinct `patch' of mass,
\[
{\mathcal S}} %{\Omega_{\a}(t)=\textnormal{supp}\, \rho} %{\rhoa(t,\cdot).
\]
We shall refer to this patch of mass simply as a `crowd' --- a continuum of agents which encodes the large-crowd dynamics associated with \eqref{eq:CS}.
In most of the existing literature on collective dynamics, the edge of such crowd is assumed to be `tailored' to the surrounding vacuum so that ${\rho} %{\rhoa}(t,\cdot)_{{}|_{\partial{\mathcal S}} %{\Omega_{\a}}}=0$. Instead, we argue here for a more realistic scenario in which the density inside the crowd remains strictly bounded away from vacuum,
\begin{equation}\label{eq:vaccum}
\min_{{\mathbf x}\in{\mathcal S}} %{\Omega_{\a}(t)}\rho} %{\rhoa(t,{\mathbf x})\geq \rho_->0,
\end{equation}
while its boundary
forms a shock discontinuity, moving with velocity ${\bu} %{\bua }_{|\partial {\mathcal S}} %{\Omega_{\a}}$. A detailed discussion on the nature of boundary conditions (BCs) for such crowds is beyond the scope of this work (see \cite{AC2021a} for the special one-dimensional case with $p=\rho$). Instead, we argue \eqref{eqs:hydro} augmented with Neumann BCs
\begin{equation}\label{eq:BCs}
\bu} %{\bua \cdot {\mathbf n}} %{\na_{|\partial{\mathcal S}} %{\Omega_{\a}}=0, \quad {\mathbb P}} %{\pressur\ea {\mathbf n}} %{\na_{|\partial{\mathcal S}} %{\Omega_{\a}}=0, \quad \textnormal{and} \quad \bq} %{\bqa \cdot{\mathbf n}} %{\na_{|\partial{\mathcal S}} %{\Omega_{\a}}=0.
\end{equation}
In particular, it follows that the total mass of the crowd, $M=M(t)$, is conserved in time,
\begin{equation}\label{eq:conserve-mass}
\Ma(t) :=\int \limits_{{\mathcal S}} %{\Omega_{\a}(t)}\rho} %{\rhoa(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx \equiv \Ma_0,
\end{equation}
and by the symmetry of $\phi} %{\phi_{\a\b} (\cdot,\cdot)$
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \int \limits_{{\mathcal S}} %{\Omega_{\a}(t)}\rho} %{\rhoa\bu} %{\bua \, \d\bx} %{\, \mathrm{d}\bx=-\int \limits_{\partial{\mathcal S}} %{\Omega_{\a}(t)} \left(\bu} %{\bua \otimes \bu} %{\bua \cdot {\mathbf n}} %{\na +{\mathbb P}} %{\pressur\ea {\mathbf n}} %{\na\right)\rho {\textnormal{d}} %{{\sf{d}}}S \ \ - \!\! \iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times{\mathcal S}} %{\Omega_{\b}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})|\bu'} %{\bubp -{\mathbf u}|^{2p-2}(\bu'} %{\bubp -\bu} %{\bua )\rho} %{\rhoa\rho} %{\rho_{\b}'\, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx' =0,
\]
and hence the total momentum of the crowd, $\bm} %{\bm_a=\bm} %{\bm_a(t)$, is also conserved,\footnote{\myr{This is the only stage which requires the zero-average $p$-alignment term argued in remark \ref{rem:zero-average-Ap},
\[
\int \limits_{{\mathcal S}(t)} {\mathbf A}} %{{\mathbf A}_{\a}_p(\rho,{\mathbf u})(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx =
\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times{\mathcal S}} %{\Omega_{\b}(t)} \phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y})|\bu'} %{\bubp -{\mathbf u}|^{2p-2}(\bu'} %{\bubp -\bu} %{\bua )\rho} %{\rhoa\rho} %{\rho_{\b}'\, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx' =0,
\]
which in turn implies conservation of total momentum $\bm} %{\bm_a(t)=\bm} %{\bm_a_0$.
}}
\begin{equation}\label{eq:conserve-momentum}
\bm} %{\bm_a(t):=\int \limits_{{\mathcal S}} %{\Omega_{\a}(t)} \rho} %{\rhoa(t,{\mathbf x})\bu} %{\bua (t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx \equiv \bm} %{\bm_a_0.
\end{equation}
Finally, \eqref{eq:penergyinq} yields that the total energy is non-increasing
\begin{equation}\label{eq:total-energy}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \int \limits_{{\mathcal S}(t)}\rho E(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx \leq -\frac{1}{2}\iint \limits_{{\mathcal S}(t)\times{\mathcal S}(t)}\phi({\mathbf x},{\mathbf x}')\Big(|{\mathbf u}'-{\mathbf u}|^{2p} +(2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\Big)\rho\rho'\, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'.
\end{split}
\end{equation}
In particular, we have the space-time \emph{enstrophy} bound
\begin{equation}\label{eq:ens}
\int \limits_0^t \iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \phi({\mathbf x},{\mathbf x}')\Big(|\bu'} %{\bubp -{\mathbf u}|^{2p} +(2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\Big)\rho\rho' \, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'{\textnormal{d}} %{{\sf{d}}}t \leq C^2_0:=2\int \limits_{{\mathcal S}(0)}\rho_0 E_0\, \d\bx} %{\, \mathrm{d}\bx.
\end{equation}
\medskip\noindent
{\bf Flocking}. A characteristic feature of alignment dynamics is the emergence of coherent structure with limiting velocity ${\mathbf u}_\infty$ such that
\begin{equation}\label{eq:u-convergence}
{\mathbf u}(t,{\mathbf x})-{\mathbf u}_\infty(t,{\mathbf x}) \stackrel{t\rightarrow \infty}{\longrightarrow}0,
\end{equation}
and the corresponding limiting density, $\rho_\infty$. This is typical in \emph{flocking} phenomena.
In the present context of the hydrodynamic alignment \eqref{eqs:hydro}, the limiting behavior of the dynamics \eqref{eqs:hydro} can only approach the time-invariant mean velocity $\ds {\mathbf u}_\infty=\widebar{\bu}:=\frac{{\mathbf m}_0}{M_0}$ with a limiting density carried out as a traveling wave $\rho_\infty({\mathbf x}-\widebar{\bu} t)$, \cite[\S2]{ST2017b}. The presence of additional repulsion, attraction and external forces introduce a `richer' set of possible emerging limiting configurations, e.g., \cite{CDMBC2007}; for example, alignment with quadratic forcing approaching an harmonic oscillator $\ddot{{\mathbf u}}_\infty(t)+a^2{\mathbf u}_\infty(t)=0$, \cite[\S2.4]{ST2020a}.
The precise notion of flocking convergence in \eqref{eq:u-convergence} may vary. Ideally, we seek uniform convergence. A more relaxed notion of $L^2_\rho$-convergence becomes accessible by studying \emph{energy fluctuations}, see the next section \ref{sec:fluctuations},
\[
\int \limits_{{\mathcal S}(t)} |{\mathbf u}-{\mathbf u}_\infty|^2\rho\, \d\bx} %{\, \mathrm{d}\bx \stackrel{t \rightarrow \infty}{\longrightarrow}0.
\]
In practice, as we shall see below, the analysis may gain by a combination of the two.\newline
The limiting configuration is supported on
$ {\mathcal S}_\infty(t):=\textnormal{supp}\, \rho_\infty(t,\cdot)$.
For example, ${\mathcal S}_\infty(t)$ is a Dirac mass in presence additional attractive forces \cite[Theorem 1]{ST2021}.
Ideally, we are interested to trace the shape of the boundary $\partial{\mathcal S}_\infty(t)$, but this seems to be out of reach in the current literature (but see \cite{LLST2022}). In general, one expects that alignment is at least strong enough to keep the dynamics contained in a finite ball,
\[
D(t)\leq D_+<\infty, \qquad D(t):=\max_{{\mathbf x},{\mathbf x}' \in {\mathcal S}(t)}|{\mathbf x}-{\mathbf x}'|.
\]
In practice we may need to address to a more accessible notion of diameter which allows a slow time growth, $D(t) \leq C_D\myangle{t}^{\gamma}$ with some fixed $\gamma>0$.
\section{Decay of energy fluctuations}\label{sec:fluctuations}
We study the hydrodynamics of the $p$-alignment, \eqref{eqs:phydro}, assuming it admits a strong entropic solution, \eqref{eq:meso-ppressure}; see further comments on \ref{H1} in section \ref{sec:ens-and-dispersion} below.\newline
Consider the \emph{energy fluctuations}, \cite[\S5]{HT2008}, \cite{Tad2021},
\begin{equation}\label{eq:eflock}
\delta\Etotal(t):= \frac{1}{2M} \iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times {\mathcal S}} %{\Omega_{\a}(t)}\Big(\frac{1}{2} |\bu} %{\bua (t,{\mathbf x})-{\mathbf u}(t,{\mathbf x}')|^2+e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})+e_{\!{}_{\mathbb P}}} %{\eb (t,{\mathbf x}')\Big)\rho} %{\rhoa(t,{\mathbf x})\rho} %{\rho_{\b}(t,{\mathbf x}')\, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'.
\end{equation}
It can be expressed in the equivalent form,\footnote{Specifically
\[
\begin{split}
\frac{1}{M}\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t) \times {\mathcal S}} %{\Omega_{\b}(t)} &\frac{1}{2}|\bu} %{\bua (t,{\mathbf x})-\bu} %{\bu_{\b}(t,{\mathbf x}')|^2\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by \\
& =\frac{1}{M}\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t) \times {\mathcal S}} %{\Omega_{\b}(t)} \left(\frac{1}{2}|\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}|^2 + (\bu} %{\bua -\widebar{\bu})\cdot(\widebar{\bu}-\bu'} %{\bubp )
+\frac{1}{2}|{\mathbf u} (t,{\mathbf x}')-\widebar{\bu}|^2\right)\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by \\
&= \int \limits_{{\mathcal S}(t)} |\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}|^2\rho} %{\rhoa(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx.
\end{split}
\]
}
$\ds \delta\Etotal(t)= \int \limits_{{\mathcal S}} %{\Omega_{\a}(t)}\Big(\frac{1}{2}|\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}(t)|^2+e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})\Big)\rho} %{\rhoa(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx$.
Thus, $\delta\Etotal(t)$ reflects macroscopic velocity fluctuations $\ds \int\limits_{{\mathcal S}} %{\Omega_{\a}(t)} \frac{1}{2}|{\mathbf u}-\widebar{\bu}(t)|^2\rho(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx$ around the mean velocity, $\ds \widebar{\bu}(t):=\frac{1}{M}\int\limits_{{\mathcal S}} %{\Omega_{\a}(t)} \rho{\mathbf u}(t,\cdot)\, \d\bx} %{\, \mathrm{d}\bx = \frac{{\mathbf m}}{M}$, and\myr{\mycancel{,}} in the context of kinetic formulation \eqref{eq:pressure}--\eqref{eq:internale}, it also reflects the microscopic velocity fluctuations,
$\ds \rho} %{\rhoae_{\!{}_{\mathbb P}}} %{\ea = \lim_{N\rightarrow \infty} \int \frac{1}{2}|{\mathbf v}-{\mathbf u}|^2f_N} %{f_{\a}(t,{\mathbf x},{\mathbf v})\, \mathrm{d}\bv$.
We have the following decay bound on energy fluctuations
\begin{equation}\label{eq:phydroRic}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta\Etotal(t) \leq -2^pM^{2-p}k(D(t))\big(\delta\Etotal(t)\big)^p.
\end{equation}
The derivation follows the energy inequality \eqref{eq:total-energy}. Noting that $\ds \delta\Etotal(t)\equiv \int \limits_{{\mathcal S}(t)} \rho} %{\rhoaE} %{E_{a} \, \d\bx} %{\, \mathrm{d}\bx-\frac{1}{2M}\big| \bm} %{\bm_a\big|^2$ with total mass and total momentum which are conserved in time, $\Ma(t)= \Ma_0, \ \bm} %{\bm_a(t)=\bm} %{\bm_a_0$, we end up with,
\begin{equation}\label{eq:delEpdecay}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} &\delta\Etotal(t) = \frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \int \limits_{{\mathcal S}(t)}\rho} %{\rhoa E} %{E_{a}(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx\\
& \leq -\frac{1}{2}\iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \phi({\mathbf x},{\mathbf x}')\Big(|{\mathbf u}-\bu'} %{\bubp |^{2p}+(2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\Big)\rho \rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'\\
& \leq -\iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \phi({\mathbf x},{\mathbf x}')\Big(\frac{1}{2}|{\mathbf u}-\bu'} %{\bubp |^2+e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})+e_{\!{}_{\mathbb P}}} %{\eb (t,{\mathbf x}')\Big)^p\rho \rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx' \\
& \leq -k(D(t))\Big(\iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \hspace*{-0.2cm}\Big(\frac{1}{2}|{\mathbf u}-\bu'} %{\bubp |^2+e_{\!{}_{\mathbb P}}} %{\ea +e'_{\!{}_{\mathbb P}}} %{e'_{b}\Big)\rho\rho' \, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'\Big)^p \Big(\iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \hspace*{-0.2cm}\rho\rho' \, \d\bx} %{\, \mathrm{d}\bx\, \d\bxp} %{\, \mathrm{d}\bx'\Big)^{-\tfrac{p}{p'}} \\
&= - 2^pM^{2-p}k(D(t))\big(\delta\Etotal(t)\big)^p.
\end{split}
\end{equation}
The first inequality on the right quotes \eqref{eq:total-energy}; the second follows from Jensen inequality and the third from H\"{o}lder inequality and the obvious radial bound \eqref{eq:radial}, $\phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})\geq k(D(t))$. Integration of \eqref{eq:delEpdecay} yields the following.
\begin{theorem}\label{thm:main1} Let $(\rho,{\mathbf u},{\mathbb P})$ be a strong solution\footnote{That is, $\big(\rho} %{\rhoa(t,\cdot),\bu} %{\bua (t,\cdot),{\mathbb P}} %{\pressur\ea (t,\cdot)\big)$ has sufficient smoothness --- say $\in L_+^{\infty}\cap L^{1})\big({\mathcal S}(t)\big)\times W^{1,\infty}\big({\mathcal S}(t)\big)\times W^{1,\infty}\big({\mathcal S}(t)\big)$, so that \eqref{eqs:hydro} can be interpreted in a pointwise sense.} of the hydrodynamic $p$-alignment \eqref{eqs:phydro}, satisfying the entropy condition \eqref{eq:meso-ppressure}, and subject to compactly supported initial data, $(\rho} %{\rhoa_0,\bu} %{\bua _0,{\mathbb P}} %{\pressur\ea _0)$ with $D_0<\infty$, and boundary conditions \eqref{eq:BCs}.
Then the \emph{energy fluctuations} $\delta\Etotal(t)$
admits the bound
\begin{equation}\label{eq:decayfluc}
\begin{split}
\delta\Etotal(t) \leq \left\{\begin{array}{ll}
\ds exp\left\{-2M\int \limits_{0}^{t}\kab\big(D(s)\big)\textnormal{d}} %{{\sf{d}}{s}\right\}\delta\Etotal(0), & p=1\\ \\
\ds \frac{1}{\ds \left\{(p-1)2^pM^{2-p}\int \limits_0^t k(D(s))\textnormal{d}} %{{\sf{d}}{s}\right\}^{\tfrac{1}{p-1}}}, & p>1.
\end{array}\right.
\end{split}
\end{equation}
\end{theorem}
\noindent
The result applies to $p$-alignment dynamics with general class of entropic pressure tensors satisfying \eqref{eq:meso-ppressure} (noting that \eqref{eq:meso-pressure} for $p=1$ yields the same energy decay \eqref{eq:penergyinq}). We refer to such solutions as
`entropic solutions'. The symmetric communication protocol $\phi} %{\phi_{\a\b} $ in \eqref{eq:radial} need not be metric nor bounded and no assumption of a uniform velocity bound is made.
We close by noting that the bound \eqref{eq:decayfluc}${}_2$ depends on the initial mass $M$ but otherwise it is independent of the initial fluctuations $\delta\Etotal(0)$ --- a typical scenario for the Ricatti's type inequality \eqref{eq:phydroRic} with $p>1$.
\subsection{Heavy-tailed kernels}\label{sec:heavy-tailed}
The bound \eqref {eq:decayfluc} reflects a competition between the expansion rate of the diameter of the crowd,
$D(t)$, and the decay rate in its communication strength, $\kab(r)$: their composition is required to have a non-integrable ``heavy-tail'' in order to enforce $L^2_\rho$-flocking decay.
We make these considerations precise in our next statement.\newline
{\bf Communication kernels of order $\beta\geq 0$}. There exist constants $C_k>0, R>0$ such that
\begin{equation}\label{eq:pconnect}
\phi} %{\phi_{\a\b} ({\mathbf x},{\mathbf x}'} %{{\mathbf y}) \geq \kab (|{\mathbf x}-{\mathbf x}'} %{{\mathbf y}|)\ \ \textnormal{with} \ \ \left\{\begin{array}{l}
\ds \int \limits_{|{\mathbf x}|\leq R} k(|{\mathbf x}|)\, \d\bx} %{\, \mathrm{d}\bx < \infty, \\ \\
\kab(r) = C_k \myangle{r}^{-\beta}, \quad r\geq R.
\end{array}\right.
\end{equation}
This emphasizes the fact that besides the mere requirement for intractability of $\phi$ near the origin --- only its tail behavior matters.\newline
{\bf Notations}. We use the following two constants.
We let $C_R$
denote a constant, with different values in different contexts, depending of $R$ as well as on the other fixed parameters $\beta,\gamma,...$ and possibly $p>1$. Also, we denote the `scaled mass'
\[
M_p:=\left\{\begin{array}{ll}
2MC_kC_D^{-\beta}, & p=1,\\ \\
\big(2^pM^{2-p}C_kC_D^{-\beta}\big)^{-\tfrac{1}{p-1}}, & p> 1.
\end{array}\right.
\]
\begin{corollary}[{\bf Decay of $L^2_\rho$-energy fluctuations}]\label{cor:1}
Let $(\rho} %{\rhoa,\bu} %{\bua ,{\mathbb P})$ be a strong entropy solution of the hydrodynamic $p$-alignment system \eqref{eqs:phydro},\eqref{eq:meso-ppressure}, $p\geq 1$, with communication kernel $\phi} %{\phi_{\a\b} $ of order $\beta\geq 0$,
\eqref{eq:pconnect}.
Assume that the ``crowd'' disperses at a rate of order $\gamma\geq 0$,
\begin{equation}\label{eq:sizeofDt}
D(t) \leq C_D(1+t)^\gamma, \qquad \gamma\geq 0, \qquad D(t)=\max\{|{\mathbf x}-{\mathbf x}'|, \ {\mathbf x},{\mathbf x}'\in \textnormal{supp}\, \rho(t,\cdot)\}.
\end{equation}
If the heavy-tail condition holds in the sense that $\beta\gamma < 1$,
then there is long time flocking behavior
such that the following decay bound holds
\begin{equation}\label{eq:flocking_decay}
\begin{split}
\delta\Etotal(t)
\leq \left\{\begin{array}{ll}
\ds C_R\, exp\big\{- M_1 t^{(1-\beta\gamma)}\big\}\delta\Etotal(0), & p=1\\ \\
\ds C_R M_p\, t^{-\tfrac{1-\beta\gamma}{p-1}}, & p>1.
\end{array}\right.
\end{split}
\end{equation}
\end{corollary}
\noindent
In case of pure alignment, $p=1$, \eqref{eq:flocking_decay}${}_1$ recovers an exponential decay of fractional order $1-\beta\gamma$, \cite[Corollary 1]{Tad2021}, while for $p>1$, \eqref{eq:flocking_decay}${}_2$ implies a Pareto-type decay of fractional order $\ds \frac{1-\beta\gamma}{p-1}$.
Thus, corollary \ref{cor:1} implies that for heavy-tailed kernels such that $\beta\gamma<1$, both the macroscopic and microscopic fluctuations around the mean $\widebar{\bu}(t)=\widebar{\bu}_0$ decay to zero. In particular, this shows the \emph{trend towards equilibrium} of a kinetic-based hydrodynamics, as it decays towards mono-kinetic closure \eqref{eq:mono-closure}
\[
\frac{1}{2}\int \limits_{{\mathcal S}(t)} \|{\mathbb P}(t,{\mathbf x} \|_2\, \d\bx} %{\, \mathrm{d}\bx = \int \limits_{{\mathcal S}(t)} e_{\!{}_{\mathbb P}}} %{\ea (t,{\mathbf x})\rho(t,{\mathbf x})\, \d\bx} %{\, \mathrm{d}\bx \stackrel{t\rightarrow \infty}{\longrightarrow}0.
\]
A key aspect, therefore, is to study the possible expansion of the spatial diameter with time growth of order $\gamma$ (possibly depending on $\beta$), so that $\beta\gamma<1$.
This will occupy us in the rest of the work.
\begin{remark} One can refine the statement of corollary \ref{cor:1} to include the borderline case, $\beta\gamma=1$
\end{remark}
\section{Flocking with mono-kinetic (``pressure-less'') closure}\label{sec:mono-kinetic}
One strategy for verifying flocking is to seek a uniform bound on velocity, $u_+:= \max|{\mathbf u}(t,\cdot)| <\infty$, which in turn implies a dispersion bound on the diameter of order $\lesssim (1+t)$,
\begin{equation}\label{eq:mono-dispersion}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D(t) \leq \delta \bu} %{\delta V(t), \quad \delta \bu} %{\delta V(t):=\max_{{\mathbf x},{\mathbf x}'\in {\mathcal S}(t)}|{\mathbf u}(t,{\mathbf x}')-{\mathbf u}(t,{\mathbf x})| \ \ \leadsto \ \ D(t)\leq D_0+2u_+ t,
\end{equation}
and then appeal to corollary \ref{cor:1} with $\gamma=1$. An instructive example for this line of argument is found in the prototype case of \emph{mono-kinetic closure}, ${\mathbb P} =0$,
\begin{equation}\label{eq:hydro-mono-kinetic}
\partial_{t}(\rho} %{\rhoa\bu} %{\bua )+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua \otimes\bu} %{\bua ) = {\mathbf A}_p(\rho,{\mathbf u}).
\end{equation}
A main feature of the mono-kinetic closure is that the resulting system \eqref{eq:hydro-mono-kinetic} decouples into scalar transport equations,
\[
u_t +{\mathbf u}\cdot\nabla_{\mathbf x} u = \int \limits_{{\mathcal S}(t)}\phi({\mathbf x},{\mathbf x}')|\bu'} %{\bubp -{\mathbf u}|^{2p-2}(u'-u)\rho'} %{\rhobp \, \d\bxp} %{\, \mathrm{d}\bx',
\]
in which case, the coercivity of the (scalar) $p$-alignment term on the right implies a maximum principle, $\max|{\mathbf u}(t,\cdot)|\leq \max|{\mathbf u}_0|$, hence
\[
D(t)\leq D_0+2u_+\!\cdot t, \qquad u_+:= \max|{\mathbf u}_0|.
\]
Appealing to corollary \ref{cor:1} with $\gamma=1$ implies that for heavy-tailed $\phi$'s of order $\beta<1$, there exists $C_R=C(R,D_0,u_+,\beta,p)$ such that
\[
\begin{split}
\delta\Etotal(t)
\leq \left\{\begin{array}{ll}
\ds C_R\, exp\big\{-2M(D_0+2u_+\!\cdot t)^{(1-\beta)}\big\}\delta\Etotal(0), & p=1\\ \\
\ds C_RM_p\big(D_0+2u_+\!\cdot t\big)^{-\tfrac{1-\beta}{p-1}}, & p>1.
\end{array}\right.
\end{split}
\]
In fact, more is true --- a refined argument shows that for such heavy-tailed $\phi$'s of order $\beta<1$, the pressureless diameter remains uniformly bounded, $D(t) \leq D_+$, and hence corollary \ref{cor:1} applies with $\gamma=0$. To this end, we split out discussion, distinguishing between the case of `pure' alignment, $p=1$, and the case of $p$-alignment $p>1$.
\subsection{Flocking with pure alignment ($p=1$)}
We begin with the following pointwise bound of velocity fluctuations which is reproduced in section \ref{sec:pointwise_mono-kinetic} below,
\begin{equation}\label{eq:delV-contarct}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta \bu} %{\delta V(t) \leq -k(D(t))M\delta \bu} %{\delta V(t), \qquad \delta \bu} %{\delta V(t)=\sup_{{\mathbf x},{\mathbf x}'} %{{\mathbf y}\in{\mathcal S}(t)}|\bu} %{\bua ({\mathbf x},t)-\bu} %{\bu_{\b}({\mathbf x}'} %{{\mathbf y},t)|.
\end{equation}
In particular, $\delta \bu} %{\delta V(t)\leq \delta \bu} %{\delta V_0$ and hence \eqref{eq:sizeofDt} holds with $\gamma=1$ in view of $D(t) \leq D_0 +\delta \bu} %{\delta V_0\cdot t$.
Consequently, for $\beta$-tailed kernels of order $\beta < 1$, \eqref{eq:pconnect}, there exists a constant $c_R$ such that
\[
\displaystyle \int \limits_{\myr{0}}^{\myr{t}} k(D(s)){\textnormal{d}} %{{\sf{d}}}s \geq \int \limits_{\myr{0}}^{\myr{R}} k(D(s)){\textnormal{d}} %{{\sf{d}}}s + \int \limits_{\myr{R}}^{\myr{\max\{R,t\}}} k(D(s)){\textnormal{d}} %{{\sf{d}}}s \geq \frac{c_R}{\myr{(1-\beta)\delta \bu} %{\delta V_0}}(1+\delta \bu} %{\delta V_0\cdot t)^{1-\beta}, \quad 0\leq \beta<1.
\]
Revisiting \eqref{eq:delV-contarct} again yields a \emph{decay} of pointwise velocity fluctuations of fractional exponential order, $\ds \delta \bu} %{\delta V(t)\leq \delta \bu} %{\delta V_0 \,exp\{-c'_R (1+\delta \bu} %{\delta V_0\cdot t)^{1-\beta}\}$ with $\ds c'_R=\frac{M}{(1-\beta)\delta \bu} %{\delta V_0}c_R$,
which in turn implies that the diameter remains uniformly bounded
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D(t)\leq \delta \bu} %{\delta V_0 \,e^{-c'_R (1+\delta \bu} %{\delta V_0\cdot t)^{1-\beta}}\ \ \leadsto \ \ D(t) \leq D_+:= \delta \bu} %{\delta V_0 \int_0^\infty e^{-c'_R (1+\delta \bu} %{\delta V_0\cdot t)^{1-\beta}}{\textnormal{d}} %{{\sf{d}}}t <\infty.
\]
Alternatively, one can use the deceasing Liapunov functional of \cite{HL2009},
$\ds \delta \bu} %{\delta V(t)+M\int_{D_0}^{D(t)}k(s){\textnormal{d}} %{{\sf{d}}}s$ to conclude that any heavy-tailed kernel in the sense that $\ds \int k(s){\textnormal{d}} %{{\sf{d}}}s =\infty$ implies $D(t)\leq D_+<\infty$.
Thus, whenever $\beta <1$, then
corollary \ref{cor:1} applies with $\gamma=0$ and $C_D=D_+$ and one recovers the exponential decay of mono-kinetic dynamics, \cite{CS2007a,HT2008,HL2009,CFTV2010,Shv2021}.
\begin{proposition}[{\bf Flocking for mono-kinetic alignment, $p=1$}]\label{prop:1}
Let $(\rho} %{\rhoa,\bu} %{\bua )$ be a strong solution of the mono-kinetic alignment system \eqref{eqs:hydro} with
``heavy-tailed'' communication kernel $\phi} %{\phi_{\a\b} $ of order $0\leq \beta< 1$, \eqref{eq:pconnect}.
There is long time flocking behavior with decay rate
\begin{equation}\label{eq:delE-revisit}
\int \limits_{{\mathcal S}} %{\Omega_{\a}(t)} |\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}|^2\rho} %{\rhoa(t,{\mathbf x})\textnormal{d}} %{{\sf{d}}{\mathbf x} \leq C_R \, e^{-M_1 t}\!\int \limits_{{\mathcal S}} %{\Omega_{\a}(t)} |\bu} %{\bua _0({\mathbf x})-\widebar{\bu}|^2\rho} %{\rhoa_0({\mathbf x})\textnormal{d}} %{{\sf{d}}{\mathbf x}, \quad M_1=2MC_kD_+^{-\beta}.
\end{equation}
\end{proposition}
Moreover, integration of \eqref{eq:delV-contarct} implies pointwise bound on the decay of velocity fluctuation
\begin{equation}\label{eq:vfluc-revisit}
\max_{\mathbf x}|\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}| \leq C_R \, e^{-M_1 t}\max_{\mathbf x} |\bu} %{\bua _0({\mathbf x})-\widebar{\bu}|.
\end{equation}
\subsection{Flocking with $p$-alignment ($p>1$)}
Our starting point is the pointwise bound of velocity fluctuations corresponding to \eqref{eq:delV-contarct}, which is outlined in appendix \ref{sec:pointwise_pmono-kinetic}
\begin{equation}\label{eq:delV-pcontarct}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta \bu} %{\delta V(t) \leq -\frac{1}{2} M k(D(t))(\delta \bu} %{\delta V(t))^{2p-1}, \qquad \delta \bu} %{\delta V(t)=\sup_{{\mathbf x}\in{\mathcal S}(t)}|\bu} %{\bua ({\mathbf x},t)-\widebar{\bu}|.
\end{equation}
In particular, $\delta \bu} %{\delta V(t)\leq \delta \bu} %{\delta V_0$ implies $D(t)\leq D_0+2\delta \bu} %{\delta V_0\cdot t$, that is, \eqref{eq:sizeofDt} holds with $\gamma=1$,
\[
D(t)\leq C_D(1+t), \qquad C_D=\max\{D_0,2\delta \bu} %{\delta V_0\},
\]
\if
We now recall that $k(r)\geq C_k(1+r)^{-\beta}$ for $r>R$. Hence, using \eqref{eq:delV-pcontarct} for $t \gtrsim R$ where $k(D(t))\geq C_RAB^{-\beta}(1+t)^{-\beta}$ leads to
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \big(\delta \bu} %{\delta V(t)\big)^{2-2p} \geq C_RM_1(1+t)^{-\beta}, \quad p>1,
\]
where, as before, $M_1=2MC_kC_D^{-\beta}$.
Integrating, we conclude with the flocking bound with decay rate of order $\frac{1-\beta}{2p-2}$.
\[
\delta \bu} %{\delta V(t) \leq C_R \frac{1}{\big\{M_1(1+t)^{1-\beta}+(\delta \bu} %{\delta V_0)^{2-2p}\big\}^{\frac{1}{2p-2}}} \leq C_R M_1^{-\frac{1}{2p-2}}(1+t)^{-\frac{1-\beta}{2p-2}}.
\]
{\sf{f}}
and corollary \ref{cor:1} implies $L^2_\rho$-decay rate of order $\ds \frac{1-\beta}{p-1}$.
\begin{proposition}[{\bf Flocking for mono-kinetic alignment, $p>1$}]\label{prop:2}
Let $(\rho} %{\rhoa,\bu} %{\bua )$ be a strong solution of the mono-kinetic $p$-alignment system \eqref{eqs:phydro} with ``heavy-tailed'' communication kernel $\phi} %{\phi_{\a\b} $ of order $0\leq \beta< 1$, \eqref{eq:pconnect}.
Then there is long time flocking behavior with decay rate
\begin{equation}\label{eq:pflocking_decay}
\begin{split}
\delta\Etotal(t) \leq
\ds C_RM_p (1+t)^{-\frac{1-\beta}{p-1}}, \qquad p>1, \quad 0\leq \beta<1.
\end{split}
\end{equation}
\end{proposition}
\noindent
We can improve these bounds, at least in the restricted range $1<p<\nicefrac{3}{2}$.
To this end, use an iterative argument starting with the $\gamma$-bound
\[
D(t) \leq C_D(1+t)^\gamma.
\]
Integrating \eqref{eq:delV-pcontarct} for $t \gtrsim R^{1/\gamma}$ where $k(D(t))\geq C_RC_kC_D^{-\beta}(1+t)^{-\beta\gamma}$ leads to
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \big(\delta \bu} %{\delta V(t)\big)^{2-2p} \geq C_R(p-1)M_1(1+t)^{-\beta\gamma}, \quad p>1,
\]
where, as before, $ M_1=MC_kC_D^{-\beta}$.
We conclude with the flocking bound
\[
\delta \bu} %{\delta V(t) \leq C_R \frac{1}{\big\{M_1(1+t)^{1-\beta\gamma}+(\delta \bu} %{\delta V_0)^{2-2p}\big\}^{\tfrac{1}{2p-2}}} \leq C_R M_1^{-\tfrac{1}{2p-2}}(1+t)^{-\tfrac{1-\beta\gamma}{2p-2}},
\]
and hence
\begin{equation}\label{eq:Dgamma}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D(t) \leq 2\delta \bu} %{\delta V(t) \ \ \leadsto \ \ D(t)\leq D_0+ C_R2\frac{2M_1^{-\tfrac{1}{2p-2}}}{\gamma'}(1+t)^{\gamma'}, \quad \gamma':=\frac{2p-3}{2p-2}+\frac{\beta\gamma}{2p-2}.
\end{equation}
We distinguish between two cases. If $2p+\beta<3$ then after one iteration, starting with $\gamma=1$, we obtain
\[
\gamma'=\frac{2p-3+\beta}{2p-2} <0.
\]
If, however, $2p+\beta\geq3$ and $\beta<\nicefrac{1}{2}$ then $\frac{\beta}{2p-2}<1$ and, hence the fixed point iterations $\gamma\mapsto \gamma'$ form a contraction, approaching the negative value
\[
\gamma_\infty = \frac{2p-3}{2p-2-\beta}< 0, \qquad p<\nicefrac{3}{2}.
\]
In either case, the range $1 \myr{<} p < \nicefrac{3}{2}$ and $\beta<\nicefrac{1}{2}$ implies that after finitely many iterations, \eqref{eq:Dgamma} holds with $\gamma <0$ and we conclude that the diameter $D(t)$ remains uniformly bounded in time, $D(t)\leq D_+$, that is, \eqref{eq:sizeofDt} holds with $\gamma=0$ and $C_D=D_+$. Corollary \ref{cor:1} implies the following refinement of proposition \ref{prop:2}.
\begin{proposition}[{\bf Flocking for mono-kinetic alignment, $1<p<\nicefrac{3}{2}$}]\label{prop:3}
Let $(\rho} %{\rhoa,\bu} %{\bua )$ be a strong solution of the mono-kinetic $p$-alignment system \eqref{eqs:phydro}, $1<p<\nicefrac{3}{2}$ with ``heavy-tailed'' communication kernel $\phi} %{\phi_{\a\b} $ of order $0\leq \beta< \nicefrac{1}{2}$, \eqref{eq:pconnect}.
Then there is long time flocking behavior with decay rate
\begin{equation}\label{eq:pflocking_refind}
\begin{split}
\delta\Etotal(t) \leq
\ds C_RM_p (1+t)^{-\tfrac{1}{p-1}}, \qquad 1<p<\nicefrac{3}{2}, \quad 0\leq \beta< \nicefrac{1}{2}.
\end{split}
\end{equation}
\end{proposition}
\noindent
Thus, we have $L^2_\rho$-velocity fluctuations with optimal decay rate $\lesssim (1+t)^{-\frac{1}{2p-2}}$. Moreover, integration of \eqref{eq:delV-pcontarct} with $k(D(t)) \geq C_R k(D_+)$ implies uniform decay of velocity fluctuations at the same optimal rate
\begin{equation}\label{eq:previsit}
\max_{\mathbf x}|\bu} %{\bua (t,{\mathbf x})-\widebar{\bu}| \leq C_RM_1^{-\tfrac{2-p}{2p-2}} (1+t)^{-\tfrac{1}{2p-2}}, \qquad 1<p<\nicefrac{3}{2}.
\end{equation}
\subsection{Agent-based description}\label{sec:agent-based}
The hydrodynamic $p$-alignment with mono-kinetic closure is the continuum counterpart of the corresponding agent-based description \eqref{eq:palignment}.
In particular, we have bounds on the velocity fluctuations --- both the
$\ell^2$-energy fluctuations and uniform fluctuations, which are worked out in appendix \ref{sec:pagent-based}
\begin{subequations}\label{eqs:pRic}
\begin{eqnarray}
& \qquad \ds \frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta\Etotal(t) \leq -2^{p-1}k(D(t))\left(\delta\Etotal(t)\right)^p, \quad \delta\Etotal(t):=\frac{1}{2N^2}\sum_{i,j=1}^N|{\mathbf v}_i(t)-{\mathbf v}_j(t)|^2 \label{eq:pRica}\\
& \qquad \ds \frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta \bv} %{\delta V(t) \leq -\frac{1}{2}k(D(t))\big(\delta \bv} %{\delta V(t)\big)^p, \quad \delta \bv} %{\delta V(t):=\max_i|{\mathbf v}_j(t)-\widebar{\bv}|, \ \widebar{\bv}(t):=\frac{1}{N}\sum_{j=1}^N {\mathbf v}_j(t). \label{eq:pRicb}
\end{eqnarray}
\end{subequations}
There is one-to-one correspondence between \eqref{eqs:pRic} and the hydrodynamic fluctuations bounds --- the $L^2_\rho$-energy fluctuations \eqref{eq:phydroRic} and uniform velocity fluctuations in \eqref{eq:delV-pcontarct}.\newline
When $p=1$, \eqref{eq:pRica} implies the exponential decay of heavy-tailed kernels. This should be contrasted with the case $p>1$, where the $p$-graph Laplacian in \eqref{eq:palignment} implies polynomial decay.
A typical scenario is summarized in the following proposition.
\begin{proposition}\label{pro:puniform_decay}\label{prop:4}
Consider the $p$-alignment system \eqref{eq:palignment}, with a
``heavy-tailed'' communication kernel of order $0\leq\beta<1$,
\eqref{eq:pconnect}.
Then there is a uniform convergence towards the mean velocity
\begin{equation}\label{eq:puniform_decay}
\begin{split}
\max |{\mathbf v}_i(t)-\widebar{\bv}|
\leq \left\{ \begin{array}{ll}
\ds C_R\, exp \big\{-C_k\myangle{t}^{(1-\beta)}\big\}\delta\Etotal(0), & p=1,\\ \\
\ds {C_R}\myangle{t}^{-\tfrac{1-\beta}{2p-2}} & p>1.
\end{array}\right.
\end{split}
\end{equation}
\end{proposition}
\begin{remark}[{\bf Finite-time alignment for $0\leq p <1$}]\label{rem:zalignment}
The dynamics of $p$-alignment with $p\geq1$ is driven by gradient of velocities, ${\mathbf v}_j-{\mathbf v}_i$. For $0\leq p <1$, the dynamics emphasizes the \emph{orientation} of velocities' gradient. The prototypical case is $p=\nicefrac{1}{2}$, in which case \eqref{eq:palignment} reads
\begin{equation}\label{eq:zalignment}
\left\{\begin{array}{c}
\begin{split}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t}{\mathbf x}_{i}(t) &= {\mathbf v}_{i}(t), \\
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} {\mathbf v}_{i}(t) &= \frac{1}{N} %{N_{\b}}\sum_{j \neq i}^{N} %{N_{\b}}\phi} %{\phi_{\a\b} _{ij}(t)\frac{{\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)}{|{\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)|}
\end{split}
\end{array}\right. \quad i=1,2,\ldots N} %{N_{\a}.
\end{equation}
When $p=0$, \eqref{eq:zalignment}${}_2$ reads
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} {\mathbf v}_{i}(t) = \frac{1}{N} %{N_{\b}}\sum_{j \neq i}^{N} %{N_{\b}}\phi} %{\phi_{\a\b} _{ij}(t)\frac{{\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)}{|{\mathbf v}_{j}(t)-{\mathbf v}_{i}(t)|^2}, \quad i=1,2,\ldots N} %{N_{\a}.
\]
The balance of its energy fluctuations
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta\Etotal(t) = -\frac{1}{2N^2} \sum_{i,j=1}^N \phi({\mathbf x}_i,{\mathbf x}_j)\leq -\frac{1}{2} k(D(t)) \ \ \leadsto \ \ \delta\Etotal(t) \leq \delta\Etotal_0 - \frac{1}{2}\int_0^t k(D(s)){\textnormal{d}} %{{\sf{d}}}s,
\]
proving that there is \emph{finite-time alignment}, $\delta\Etotal(t)\stackrel{t\rightarrow t_c}{\longrightarrow}0$, for heavy-tailed kernels satisfying $\int\limits_0^t k(D(s)){\textnormal{d}} %{{\sf{d}}}s \stackrel{t\rightarrow \infty}{\longrightarrow}\infty$.
Finite-time alignment is typical for $p$-alignment in the singular range $0\leq p<1$, \cite[Theorem 2.2]{CCH2014},
\[
\delta \bv} %{\delta V(t) \leq \Big((\delta \bv} %{\delta V_0)^{1-p} - 2^{p-1}(1-p)\int\limits_0^t k(D(s)){\textnormal{d}} %{{\sf{d}}}s\Big)^{\tfrac{1}{1-p}}, \qquad 0\leq p <1.
\]
In this context, at least for $0\leq p\leq \nicefrac{1}{2}$, one encounters the need to avoid collisions,
\[
|{\mathbf v}_i(t)-{\mathbf v}_j(t)|+ |{\mathbf x}_i(t)-{\mathbf x}_j(t)|\neq 0, \qquad i\neq j, \ t< t_c.
\]
Collision avoidance is discussed in \cite{Mar2018} for $p\in (\nicefrac{1}{2},\nicefrac{3}{2})$ and for the case of pure alignment, $p=1$, with possibly singulars, $k(r)=r^{-\alpha}$, in \cite{ACHL2012,Pes2014,CCH2014,CCMP2017}.
\end{remark}
\subsection{Flocking with matrix-valued communication kernel}\label{sec:matrix-kernel}
\normalfont
Consider the alignment dynamics
\begin{subequations}\label{eqs:mhydro}
\begin{equation}\label{eq:mkernel}
\partial_{t}(\rho} %{\rhoa\bu} %{\bua )+\nabla_{\mathbf x}\cdot(\rho} %{\rhoa\bu} %{\bua \otimes\bu} %{\bua ) = \int \limits_{\mathbb{R}^d}\Phi ({\mathbf x},{\mathbf x}'} %{{\mathbf y})(\bu} %{\bu_{\b}(t,{\mathbf x}'} %{{\mathbf y})-\bu} %{\bua (t,{\mathbf x}))\rho} %{\rhoa(t,{\mathbf x})\rho} %{\rho_{\b}(t,{\mathbf x}'} %{{\mathbf y})\dxp} %{\, \mbox{d}\by,
\end{equation}
driven by a bounded symmetric \emph{matrix} communication kernel, $\Phi ({\mathbf x},{\mathbf x}'} %{{\mathbf y})=\Phi ({\mathbf x}'} %{{\mathbf y},{\mathbf x})\in {\mathbb R}^{d\times d}$, of order $\beta\geq 0$
\begin{equation}\label{eq:mbeta}
C_k \myangle{|{\mathbf x}-{\mathbf x}'} %{{\mathbf y}|}^{-\beta}{\mathbb I}_{d\times d} \leq \Phi ({\mathbf x},{\mathbf x}'} %{{\mathbf y}) \leq \phi_+{\mathbb I}_{d\times d}.
\end{equation}
\end{subequations}
In this case, the coupling of ${\mathbf u}$-components defies a maximum principle of $\delta \bu} %{\delta V(t)$ encoded in \eqref{eq:delV-contarct}. Instead, we will show below the bound $\delta \bu} %{\delta V(t) \lesssim \myangle{t}^{\nicefrac{1}{2}}$. This implies $D(t) \lesssim \myangle{t}^{\nicefrac{3}{2}}$ and hence flocking holds for heavy-tailed kernels
of order $\beta<\nicefrac{2}{3}$.
To this end, we follow our argument in the discrete setup, \cite[Proposition 3.1]{ST2021}, starting with the alignment dynamics
\[
\partial_t{\mathbf u} + {\mathbf u}\cdot\nabla_{\mathbf x} {\mathbf u} = \int \Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})({\mathbf u}'-{\mathbf u})\rho'\, \d\bxp} %{\, \mathrm{d}\bx',
\]
which implies the \emph{local} energy balance
\begin{equation}\label{eq:local-energy}
\partial_t\frac{|{\mathbf u}|^2}{2} + {\mathbf u}\cdot\nabla_{\mathbf x} \frac{|{\mathbf u}|^2}{2} = \int \big\langle {\mathbf u},\Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})({\mathbf u}'-{\mathbf u})\big\rangle\rho'\, \d\bxp} %{\, \mathrm{d}\bx'.
\end{equation}
The integrand on the right is decomposed by polarization (suppressing time dependence)
\[
\begin{split}
\langle {\mathbf u}({\mathbf x}),\Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})&\big({\mathbf u}({\mathbf x}')-{\mathbf u}({\mathbf x})\big)\rangle \\
& \equiv -\frac{1}{2}\big\langle ({\mathbf u}'-{\mathbf u}),\Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y})({\mathbf u}'-{\mathbf u})\big\rangle
- \frac{1}{2}\big\langle {\mathbf u},\Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y}){\mathbf u}\big\rangle
+\frac{1}{2}\big\langle {\mathbf u}',\Phi({\mathbf x},{\mathbf x}'} %{{\mathbf y}){\mathbf u}'\big\rangle \\
& \leq - C_k \myangle{|{\mathbf x}-{\mathbf x}'} %{{\mathbf y}|}^{-\beta}\frac{|{\mathbf u}|^2}{2}
+\phi_+\frac{|{\mathbf u}'|^2}{2}, \qquad \Phi({\mathbf x},{\mathbf x}')\leq \phi_+{\mathbb I}_{d\times d}.
\end{split}
\]
In the last step we used the assumed bound on $\Phi$ having a heavy-tail of order $\beta$ and satisfying a pointwise upper-bound $\phi_+$.
Returning to \eqref{eq:local-energy} while noting that
$\ds \int {|{\mathbf u}'|^2}\rho'\, \d\bxp} %{\, \mathrm{d}\bx' \leq C^2_0= 2\int \rho_0 E_0$,
it follows that
\begin{equation}\label{eq:boot-D}
\partial_t|{\mathbf u}|^2 + {\mathbf u}\cdot\nabla_{\mathbf x} |{\mathbf u}|^2\leq -C_k\myangle{D(t)}^{-\beta}M|{\mathbf u}|^2+\phi_+C^2_0.
\end{equation}
By the maximum principle (--- we ignore the dissipative term on the right),
\[
|{\mathbf u}(t,\cdot)|^2\leq \max|{\mathbf u}_0|^2+C' t, \qquad C':=\phi_+C^2_0,
\]
and hence \eqref{eq:sizeofDt} holds with $\gamma=\nicefrac{3}{2}$, in view of
\begin{equation}\label{eq:D-expandb}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} {D}(t)\leq 2\max|{\mathbf u}(t,\cdot)| \ \ \leadsto \ \ D(t) \leq D_0 +\frac{4}{3C'}\big(\max|{\mathbf u}_0|^2+C't\big)^{\nicefrac{3}{2}}.
\end{equation}
We can now use a bootstrap argument: starting with $\gamma=\nicefrac{3}{2}$ we insert the bound $D(t)\lesssim \myangle{t}^\gamma$ of \eqref{eq:D-expandb} into the right side of \eqref{eq:boot-D} and we have the maximum bound
\[
|{\mathbf u}(t,\cdot)|^2\leq \max|{\mathbf u}_0|^2+C' (1+t)^{\beta\gamma} \ \ \leadsto \ \ D(t) \leq D_0 +\frac{2}{C'\gamma'}\big(\max|{\mathbf u}_0|^2+C'(1+t)\big)^{\gamma'}, \quad \gamma'=1+\nicefrac{\beta\gamma}{2}.
\]
Iterating, $\gamma\mapsto \gamma'$, we end up with a fixed point $\gamma=\frac{2}{2-\beta}$, and with the improved bounds, still in the range of $\beta<\nicefrac{2}{3}$,
\[
|{\mathbf u}(t)| \lesssim \myangle{t}^{\tfrac{\beta}{2-\beta}}, \qquad D(t) \leq C_D\myangle{t}^{\tfrac{2}{2-\beta}}, \quad \beta<\nicefrac{2}{3}.
\]
Corollary \ref{cor:1} implies the following.
\begin{proposition}[{\bf Flocking for matrix-based alignment}]\label{prop:5}
Let $(\rho} %{\rhoa,\bu} %{\bua )$ be a strong solution of the hydrodynamic alignment \eqref{eqs:mhydro} with
``heavy-tailed'' matrix communication kernel $\Phi$ of order $ \beta< \nicefrac{2}{3}$.
There is long time flocking behavior with fractional exponential decay rate
\begin{equation}\label{eq:mflocking}
\delta\Etotal(t) \leq
C_R\, exp\big\{- M_1 t^{\tfrac{2-3\beta}{2-\beta}}\big\}\delta\Etotal(0).
\end{equation}
\end{proposition}
\section{Flocking of hydrodynamic $p$-alignment with entropic pressure}\label{sec:with-pressure}
We consider hydrodynamic alignment \eqref{eqs:phydro} driven by the class of \emph{singular kernels}
$k_p(r):=r^{-(d+2sp)}, 0<s<1, \ p\geq 1$,
\[
\partial_t(\rho {\mathbf u})+\nabla_{\mathbf x}(\rho {\mathbf u} \otimes {\mathbf u}+{\mathbb P}) =
p.v. \int \limits_{{\mathcal S}} %{\Omega_{\a}(t)} \frac{|{\mathbf u}'-{\mathbf u}|^{2p-2}\big({\mathbf u}(t,{\mathbf x}')-{\mathbf u}(t,{\mathbf x})\big)}{|{\mathbf x}'-{\mathbf x}|^{d+2sp}}\rho(t,{\mathbf x})\rho(t,{\mathbf x}')\, \d\bxp} %{\, \mathrm{d}\bx', \quad 0<s<1.
\]
We emphasize that in this case of strongly singular kernels, there is no formal justification for the passage from the agent-based description \eqref{eq:palignment} to the hydrodynamic description. In particular, the near-origin integrability sought in \eqref{eq:pconnect} is given up for the usual notion of singular integration in terms of principle value ($p.v.$). The alignment term on the right amounts to a \emph{weighted} fractional $2p$-Laplacian, $(-\Delta)^s_{2p}$, which is properly interpreted to act on $\textnormal{supp} \, \rho(t,\cdot)$; see \cite{TGCV2021,BV2015} and the references therein.
The tail of the singular kernel, $k_p(r)=r^{-(d+2sp)}, r\gg R$, is too thin to enforce the heavy tail condition sought in corollary \ref{cor:1}. Accordingly, we keep the singular `head' and adjust it with the ``heavy tail'' or order $\beta$
\begin{equation}\label{eq:multiDsingdyn}
\phi} %{\phi_{\a\b} _{s,\beta} ({\mathbf x},{\mathbf x}')\left\{\begin{array}{ll} = |{\mathbf x}-{\mathbf x}'|^{-(d+2sp)}, & |{\mathbf x}-{\mathbf x}'|\leq R \ \ \mbox{with} \ \ 0< s< 1\\ \\
\geq C_k\myangle{|{\mathbf x}-{\mathbf x}'|}^{-\beta}, & |{\mathbf x}-{\mathbf x}'|> R
\end{array}\right.
\end{equation}
Clearly, there exists a constant, $K=K_R$, such that $k_p(|{\mathbf x}-{\mathbf x}'|) \leq K_R\phi_{s,\beta}({\mathbf x},{\mathbf x}')$ for all $({\mathbf x},{\mathbf x}')$. Without loss of generality, we may assume that the spatial scale $R$ is large enough,
$\ds {(1+R)^\beta}R^{-(d+2sp)} <C_k$, so that we may take $K_R=1$,
\begin{equation}\label{eq:phi-dominates}
k_p(|{\mathbf x}-{\mathbf x}'|) \leq \phi_{s,\beta}({\mathbf x},{\mathbf x}'), \qquad \forall {\mathbf x},{\mathbf x}' \in \mathbb{R}^d.
\end{equation}
We refer to such heavy-tailed, singular kernels as having order $(s,\beta)$.
If we let $\phi_\beta$ denote its tail of order $\beta$ then the $p$-alignment dynamics now reads
\begin{equation}\label{eq:sing-palignment}
\begin{split}
\partial_t(\rho {\mathbf u})&+\nabla_{\mathbf x}\cdot(\rho{\mathbf u}\otimes{\mathbf u} +{\mathbb P}) \\
& = p.v. \!\!\!\int \limits_{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|\leq R} \hspace*{-0.3cm} \frac{|{\mathbf u}'-{\mathbf u}|^{2p-2}\big({\mathbf u}'-{\mathbf u}'\big)}{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|^{d+2sp}}\rho\rho'\dxp} %{\, \mbox{d}\by \\
& +\hspace*{-0.4cm}\int \limits_{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|> R} \hspace*{-0.4cm}\phi_{\beta}({\mathbf x},{\mathbf x}')|{\mathbf u}'-{\mathbf u}|^{2p-2}\big({\mathbf u}'-{\mathbf u}'\big)\rho\rho'\dxp} %{\, \mbox{d}\by, \quad \phi_\beta({\mathbf x},{\mathbf x}')\geq C_k(1+|{\mathbf x}-{\mathbf x}'|)^{-\beta}.
\end{split}
\end{equation}
\begin{remark}[{\bf Entropic pressure with singular kernel}]
In case of singular kernel $\phi_{s,\beta}$, we need to adjust the definition \ref{def:meso-ppressure} of entropic pressure,
\begin{equation}\label{eq:sing-ppressure}
\partial_{t}(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea )+\nabla_{{\mathbf x}}\cdot(\rho} %{\rhoa e_{\!{}_{\mathbb P}}} %{\ea \bu} %{\bua +\bq} %{\bqa )+\textnormal{trace}({\mathbb P}} %{\pressur\ea \nabla\bu} %{\bua ) \leq - \frac{1}{2} k_p(D(t))\int \limits_{{\mathcal S}(t)} \big((2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e'_{\!{}_{\mathbb P}}} %{e'_{b})^p\big) \rho\rho'} %{\rhobp \dxp} %{\, \mbox{d}\by.
\end{equation}
Thus, the entropic part of the internal energy avoids the singularity of $\phi_{s,\beta}$ and emphasizes only its tail behavior. It leads to the `adjusted' energy fluctuations bound
\begin{equation}\label{eq:energyvssing}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta\Etotal(t)
\leq
- \frac{1}{2} \iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)}\left\{\phi_{s,\beta} ({\mathbf x},{\mathbf x}'} %{{\mathbf y}) |\bu} %{\bua -\bu'} %{\bubp |^{2p}+k_p(D(t))\big((2e_{\!{}_{\mathbb P}}} %{\ea )^p+(2e_{\!{}_{\mathbb P}}} %{\eb )^p\big)\right\}\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by,
\end{equation}
which in turn, arguing along the lines of \eqref{eq:delEpdecay}, yields \eqref{eq:phydroRic}; that is, the main theorem \ref{thm:main1} and its corollary \ref{cor:1} survive.
In particular, the enstrophy bound \eqref{eq:ens} holds \myr{for $\phi=\phi_{s,\beta}$. Taking into account \eqref{eq:phi-dominates}, $\phi_{s,\beta}({\mathbf x},{\mathbf x}')\geq |{\mathbf x}'-{\mathbf x}|^{-(d+2sp)}$, we find}
\begin{equation}\label{eq:end-revisited}
\myr{\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} \delta\Etotal(t)
\leq
- \frac{1}{2} \iint \limits_{{\mathcal S}(t)\times {\mathcal S}(t)} \frac{|\bu} %{\bua (t,{\mathbf x}') -{\mathbf u}(t,{\mathbf x})|^{2p}}{|{\mathbf x}'-{\mathbf x}|^{d+2sp}}\rho} %{\rhoa\rho'} %{\rhobp \, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by.}
\end{equation}
\end{remark}
The presence of pressure, let alone a pressure with an `unknown' closure, couples the different components of velocity in a manner that defies a straightforward derivation of a uniform bound on velocity fluctuations, $\delta \bu} %{\delta V(t)$, along the lines of what we have done in the mono-kinetic case. Instead,
we introduce a new strategy for verifying flocking in this case, in which we use an enstrophy bound associated with the singular kernel,
$k_p(r)=r^{-(d+2sp)}$, in order to control the diameter $D(t)\lesssim \myangle{t}^{\gamma}$. This enables us to treat the flocking in presence of entropic pressure.
The remarkable aspect here is that although the presence of pressure defies a maximum principle on the velocity field, the corresponding enstrophy bound associated with \eqref{eq:sing-palignment} will suffice for control of velocity fluctuations and hence flocking will follow. Thus, short-term interactions governed by kernel with a \emph{singular head} secure the spread of velocity fluctuations, while \emph{heavy-tailed} kernel governing the long-term interactions secure flocking.
\subsection{Enstrophy and dispersion bounds}\label{sec:ens-and-dispersion}
Throughout this section we make the following assumptions.
\begin{enumerate}[itemindent=.01cm,label={(H\textit{\arabic*})}]
\item\label{H1} The alignment hydrodynamics \eqref{eq:hydro},\eqref{eq:sing-palignment} admits a strong entropic solution, \eqref{eq:sing-ppressure}.
\item\label{H2} The support, ${\mathcal S}} %{\Omega_{\a}(0)=\textnormal{supp}\,\rho} %{\rhoa(0,\cdot)$, has a smooth boundary satisfying a Lipschitz or a cone condition.
\item\label{H3} The dynamics remains uniformly bounded away from vacuum, namely --- there exists $\rho_->0$ such that
\[
\min_{{\mathbf x}\in {\mathcal S}} %{\Omega_{\a}(t)}\rho} %{\rhoa(t,{\mathbf x})\geq \rho_->0, \quad t\geq 0.
\]
\end{enumerate}
Several comments regarding these assumptions are in order. The literature about the question of global regularity, \ref{H1}, is devoted mostly to mono-kinetic ``pressure-less'' closure; we mention the one-dimensional studies \cite{TT2014,CCTT2016,HT2017,ST2017a, ST2017b,ST2020b,Tan2021,LS2022}, the two-dimensional case \cite{HT2017} and multi-dimensional cases \cite{Shv2019, DMPW2019, CTT2021,Tad2022b}. Much less is known about alignment with pressure, typically when (scalar) pressure is augmented with additional process of relaxation and/or dissipation, \cite{Cho2019,CDS2020,TCGW2020}. On the other hand, there are relatively few works on weak solutions of \eqref{eqs:hydro}, \cite{CCR2011,CFGS2017,LT2021}. As for \ref{H2}, we are aware of only few results on the geometric structures that emerge from alignment, \cite{LS2019,LLST2022}. The question of uniform bound away from vacuum assume in \ref{H3} plays an important role in driving global regularity \cite{Tan2020,Shv2021,AC2021a,Tad2021}. It can be relaxed to allow mild time decay, e.g., $\rho_-(t) \gtrsim \myangle{t}^{-\nicefrac{1}{2}}$, \cite[Theorem 1.1]{ST2020b}, \cite[Theorem 3]{Tad2021}, but as already noted in previous works, some sort of non-vacuous assumption is necessary. \newline
We begin by noting that since $\phi_{s,\beta}$ dominates $k_p(r)$, \eqref{eq:phi-dominates}, then by the non-vacuous hypothesis \ref{H3}, $\rho\geq \rho_->0$, we have the Sobolev bound
\[
\begin{split}
\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times {\mathcal S}} %{\Omega_{\a}(t)}\frac{|{\mathbf u}(t,{\mathbf x}'} %{{\mathbf y})-{\mathbf u}(t,{\mathbf x})|^{2p}}{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|^{d+2sp}}\, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by \leq C_\rho^2\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times {\mathcal S}} %{\Omega_{\a}(t)}\frac{|{\mathbf u}(t,{\mathbf x}'} %{{\mathbf y})-{\mathbf u}(t,{\mathbf x})|^{2p}}{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|^{d+2sp}}\rho\rho'\, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by, \quad C_\rho:=\frac{1}{\rho_-}.
\end{split}
\]
The space-time enstrophy bound \eqref{eq:ens}, or more precisely --- its singular version in \eqref{eq:end-revisited}, then yields
\begin{equation}\label{eq:mDsingHs}
\int \limits_0^t \|\bu} %{\bua (\tau,\cdot)\|^{2p}_{{}_{\dot{W}^{s,2p}({\mathcal S}} %{\Omega_{\a})}}\textnormal{d}} %{{\sf{d}}\tau\leq C^2_\rho C^2_0, \quad \|{\mathbf u}(t,\cdot)\|^{2p}_{{}_{\dot{W}^{s,2p}({\mathcal S}} %{\Omega_{\a})}}:=\iint \limits_{{\mathcal S}} %{\Omega_{\a}(t)\times {\mathcal S}} %{\Omega_{\a}(t)}\frac{|\bu} %{\bua (t,{\mathbf x}'} %{{\mathbf y})-\bu} %{\bua (t,{\mathbf x})|^{2p}}{|{\mathbf x}'} %{{\mathbf y}-{\mathbf x}|^{d+2sp}}\, \d\bx} %{\, \mathrm{d}\bx\dxp} %{\, \mbox{d}\by.
\end{equation}
The enstrophy bound \eqref{eq:mDsingHs} guarantees that the velocity $\bu} %{\bua $ slows down the dispersion of the crowd so that its diameter $D(t)$ may not grow faster than $\lesssim \myangle{t}^{\gamma}$. Below we derive sharp bounds on the dispersion rate $\gamma$.\newline
To this end, we note that propagation along particles paths in \eqref{eq:hydro}${}_1$ yields, as in \eqref{eq:mono-dispersion},
\[
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D(t) \leq \delta \bu} %{\delta V(t), \quad \delta \bu} %{\delta V(t)=\max_{{\mathbf x},{\mathbf x}'\in {\mathcal S}(t)}|{\mathbf u}(t,{\mathbf x}')-{\mathbf u}(t,{\mathbf x})|.
\]
By Gagliardo-Nirenberg inequality (which we recall in appendix \ref{sec:GN} below),
\begin{equation}\label{eq:GNS}
|{\mathbf u}(t,{\mathbf x})-{\mathbf u}(t,{\mathbf x}')|\leq C_s\|{\mathbf u}\|_{{}_{\dot{W}^{s,2p}({\mathcal S}(t))}}|{\mathbf x}-{\mathbf x}'|^{s-\theta}, \qquad {\mathbf x},{\mathbf x}'\in {\mathcal S}} %{\Omega_{\a}(t), \ \ \theta:=\frac{d}{2p} < s<1.
\end{equation}
This yields,
$\ds \frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D} %{D_{\a}(t)\leq \delta \bu} %{\delta V(t) |\leq C_s\|{\mathbf u}(t,\cdot)\|_{\dot{W}^{s,2p}({\mathcal S}(t))}D^{s-\theta}(t)$, or
\begin{equation}\label{eq:dynamicD}
\frac{\d}{\d t}} %{\frac{\mathrm{d\,\,}}{\mathrm{d}t} D} %{D_{\a}^{1+\theta-s}(t) \leq C'_s\|{\mathbf u}(t,\cdot)\|_{\dot{W}^{s,2p}({\mathcal S}(t))}, \qquad C'_s=\myr{(1+\theta-s)}C_s,
\end{equation}
and hence, in view of \eqref{eq:mDsingHs},
\[
\begin{split}
D} %{D_{\a}^{1+\theta-s}(t) & \leq D} %{D_{\a}^{1+\theta-s}_0 \\
& \ \ \ + \Big(\int \limits_0^t \|{\mathbf u}(\tau,\cdot)\|^{2p}_{\dot{W}^{s,2p}({\mathcal S}(t))}{\textnormal{d}} %{{\sf{d}}}\tau\Big)^{\tfrac{1}{2p}}\Big(\int \limits_0^t 1{\textnormal{d}} %{{\sf{d}}}\tau\Big)^{\tfrac{1}{(2p)'}} \leq D} %{D_{\a}^{1+\theta-s}_0
+ C'_s( C_\rho C_0)^{\tfrac{1}{p}}\, t^{\tfrac{1}{(2p)'}}.
\end{split}
\]
We conclude that the crowd of multi-dimensional $p$-alignment dynamics,
\eqref{eq:sing-palignment} can be dispersed at a rate no faster than
\begin{equation}\label{eq:Dabeta}
D} %{D_{\a}(t) \leq C_D(1+t)^{\gamma_p}, \qquad \gamma_p= \frac{2p-1}{2p(1+\theta-s)}, \quad \theta=\frac{d}{2p}<s <1.
\end{equation}
This bound can be improved \myr{using a bootstrap argument outlined in appendix \ref{sec:dispersion}}. In particular, for $1<p<\nicefrac{3}{2}$ we obtain a \emph{uniform} dispersion bound which we summarize in the following key result.
\begin{lemma}[{\bf Uniform dispersion bound for $p$-alignment with singular kernels}]\label{lem:dispersion}
Consider the multi-dimensional $p$-alignment dynamics,
\eqref{eq:sing-palignment}, $1<p<\nicefrac{3}{2}$, with heavy-tailed, singular kernel of order $(s,\beta)$, satisfying \ref{H1}--\ref{H3}.
Then we have a uniform bound
\begin{equation}\label{eq:Duniform}
D} %{D_{\a}(t) \leq D_+, \qquad 0\leq \myr{\beta< (\nicefrac{3}{2}-p)d}, \quad 1<p<\nicefrac{3}{2}.
\end{equation}
\end{lemma}
\begin{remark}\label{rem:more-on-p}
Observe that since we require $\myr{d=2p\theta<3}$, the uniform bound \eqref{eq:Duniform} is restricted to one- and two-dimensional cases.\newline
We are unable to secure such a uniform dispersion bound for $p>\nicefrac{3}{2}$, but we can still improve the dispersion bound \eqref{eq:Dabeta} as shown in remark \ref{rem:pgt32} below,
\[
D(t)\leq C'_D(1+t)^\gamma, \qquad \gamma= \frac{2p\big(p-\nicefrac{3}{2}\big)}{(p-1)d-\beta}, \qquad 0\leq \beta < \frac{d}{2p-1}, \quad p>\nicefrac{3}{2}.
\]
\end{remark}
\subsection{Flocking of alignment with pressure. The one-dimensional case}
The case of pure alignment $p=1$, restricts the use of lemma \ref{lem:dispersion} to the one-dimensional case ($d<2\myr{p}$),
driven by singular kernel $k_1(r)=r^{-(1+2s)}, \frac{1}{2}<s<1$, with $\beta$-tailed adjustment
\begin{subequations}\label{eqs:1Dsingdyn}
\begin{equation}\label{eq:1Dsingdyn}
\begin{split}
\partial_t(\rho u)+\partial_x(\rho u^2+{\mathsf p}} %{{\textnormal{p}}) & = p.v. \hspace*{-0.5cm}\int \limits_{|x'-x|\leq R}\hspace*{-0.3cm}\frac{u(t,x')-u(t,x)}{|x-x'|^{1+2s}}\rho(t,x'){\textnormal{d}} %{{\sf{d}}}x' \\
& \ \ \ + \hspace*{-0.5cm} \int \limits_ {|x'-x|> R}\hspace*{-0.3cm}\phi_{\beta}(x,x')\big(u(t,x')-u(t,x)\big)\rho(t,x'){\textnormal{d}} %{{\sf{d}}}x'.
\end{split}
\end{equation}
The integrals on the right are restricted to the interval ${\mathcal S}(t)=[\rho_-(t),\rho_+(t)]$ supporting $\rho(t,\cdot)$, $\phi_{\beta}$ is a $\beta$-tailed communication kernel,
\begin{equation}
\phi_{\beta}(x,{x}') \geq C_k(1+|x-x'|)^{-\beta}, \qquad |x-x'|\geq R,
\end{equation}
and ${\mathsf p}} %{{\textnormal{p}}$ is any scalar entropic pressure satisfying \eqref{eq:meso-pressure}, or more precisely --- its singular version \eqref{eq:sing-ppressure},
\begin{equation}\label{eq:1Dsingpres}
\partial_t (\rho {\mathsf p}} %{{\textnormal{p}}) +\partial_x(\rho {\mathsf p}} %{{\textnormal{p}} u+q) +2{\mathsf p}} %{{\textnormal{p}} \partial_x u \leq -2{\mathsf p}} %{{\textnormal{p}} D^{-(1+2s)}(t)M.
\end{equation}
\end{subequations}
By \ref{eq:Dabeta} we can apply corollary \eqref{cor:1} with $\gamma_1=\tfrac{1}{3-2s}$ which yields the following.
\begin{theorem}[{\bf One-dimensional alignment, $p=1$}]\label{thm:main3}
Consider the one-dimensional alignment dynamics
\eqref{eqs:1Dsingdyn} and assume \ref{H1},\ref{H3}, hold. Let $(\rho,u,{\mathsf p}} %{{\textnormal{p}})$ be a strong entrpoic solution
with $\beta$-tailed singular kernel, $\phi_{\beta}$, satisfying the heavy-tail condition
\begin{equation}\label{eq:heavys}
\beta+2s <3,\qquad \beta\geq 0, \quad \frac{1}{2}<s<1.
\end{equation}
Then there is a large time flocking behavior with fractional exponential rate
\begin{equation}\label{eq:1Dflocking_decay}
\delta\Etotal(t)
\leq C_R \,exp\Big\{-2MC_k\myangle{t}^{\tfrac{3-2s-\beta}{3-2s}}\Big\}\delta\Etotal(0).
\end{equation}
\end{theorem}
\noindent
This extends the mono-kinetic `pressure-less' studies in \cite{ST2017a, ST2017b, ST2018a, DKRT2018, DMPW2019, MMPZ2019}.
It is instructive to compare this result with flocking statement in the mono-kinetic closure, which is based on the uniform bound on velocity, $D(t) \lesssim \myangle{t}$. Theorem \ref{thm:main3} allows for a \emph{larger} class of heavy-tailed kernels since it is based on a sharper bound on the velocity fluctuations, leading to $D(t) \lesssim \myangle{t}^{\gamma}$ with $\gamma <1$. \myr{This result can be further improved by extending the uniform dispersion bound in lemma \ref{lem:dispersion} to the limiting case $p=1$.}
\subsection{Flocking of $p$-alignment with pressure. The multi-dimensional case}
We consider the $p$-alignment dynamics \eqref{eq:sing-palignment} driven by singular kernel $k_p(r)=r^{-(d+2sp)}, \tfrac{d}{2p}<s<1$.
Using \eqref{eq:Dabeta} we can apply corollary \ref{cor:1} with $\gamma=\gamma_p$ which yields the following.
\begin{theorem}[{\bf Multi-dimensional alignment, $p>1$}]\label{thm:main4}
Consider the multi-dimensional $p$-alignment dynamics
\eqref{eq:sing-palignment} and assume \ref{H1}--\ref{H3} hold. Let $(\rho,{\mathbf u},{\mathbb P})$ be a strong entrpoic solution, \eqref{eq:sing-ppressure},
with a $\beta$-tailed singular kernel $\phi_{s,\beta}$, satisfying the heavy-tail condition
\[
\beta\gamma_p<1, \qquad \beta\geq0, \quad \gamma_p:=\frac{2p-1}{2p(1+\theta-s)}, \quad \theta=\frac{d}{2p}<s<1.
\]
Then there is a large time flocking behavior with polynomial decay rate of order
\begin{equation}\label{eq:mDflocking_decay}
\delta\Etotal(t)
\leq C_RM_p\,t^{-\tfrac{1-\beta\gamma_p}{p-1}}.
\end{equation}
\end{theorem}
\begin{remark}[{\bf Decay of internal fluctuations}]
A sufficient condition for the heavy-tailed restriction $\beta\gamma_p<1$ sought in \eqref{eq:mDflocking_decay} is given by
\begin{equation}\label{eq:mDheavys}
\beta \leq \frac{d}{2p-1} \ \ \leadsto \ \ \myr{\beta\gamma_p<\beta\frac{2p-1}{d}\leq 1}.
\end{equation}
It still allows heavy-tails of order $\beta\geq1$,
compared with the $\beta<1$ restriction in the mono-kinetic closure.
In particular, when $\beta=\frac{d}{2p-1}$ one finds the decay of order
\[
\big(\delta\Etotal(t)\big)^{p-1} \lesssim t^{-(1-\beta\gamma_p)} \lesssim t^{- \tfrac{1-s}{1+\theta-s}}.
\]
\end{remark}
\begin{remark}
Theorem \ref{thm:main4} implies the decay of both --- the macroscopic velocity fluctuations
$\ds \int |{\mathbf u}-\widebar{\bu}|^2\rho\, \d\bx} %{\, \mathrm{d}\bx$ and, in the context of kinetic formulation, the microscopic fluctuations $\ds \iint |{\mathbf v}-{\mathbf u}|^2 f_N \, \mathrm{d}\bv\, \d\bx} %{\, \mathrm{d}\bx$.
\end{remark}
The decay bound \eqref{eq:mDflocking_decay} is not sharp, a reflection of the fact that the dispersion bound \eqref{eq:Dabeta} can be improved with smaller $\gamma_p$ (\myr{as noted in remark \ref{rem:more-on-p}}). In particular, when $p$ is in the restricted range $1<p<\nicefrac{3}{2}$, then corollary \ref{cor:1} applies with $\gamma=0$ and $C_D=D_+$ which yields the following.
\begin{theorem}[{\bf Multi-dimensional alignment $1<p<\nicefrac{3}{2}$}]\label{thm:main5}
Consider the multi-dimensional $p$-alignment dynamics
\eqref{eq:sing-palignment}, $1<p<\nicefrac{3}{2}$ and assume \ref{H1}--\ref{H3} hold. Let $(\rho,{\mathbf u},{\mathbb P})$ be a strong entrpoic solution, \eqref{eq:sing-ppressure},
with a heavy-tailed singular kernel of order $(s,\beta)$.
Then there is a large time flocking behavior with polynomial decay rate of order
\begin{equation}\label{eq:pDflocking_refind}
\begin{split}
\delta\Etotal(t) \leq
\ds C_RM_p (1+t)^{-\tfrac{1}{p-1}}, \qquad 0\leq \beta< \myr{(\nicefrac{3}{2}-p)}d, \quad \frac{d}{2p}<s<1, \quad 1<p<\nicefrac{3}{2}.
\end{split}
\end{equation}
\end{theorem}
\noindent
Theorem \ref{thm:main5} is the analogue of the mono-kinetic ``pressure-less'' case in proposition \ref{prop:3}. In particular, it is rather remarkable that we obtain the same optimal decay rate of order $\tfrac{1}{p-1}$ in the respective range $1<p<\nicefrac{3}{2}$ for the one- and two-dimensional cases. An optimal flocking scenario with a uniform dispersion bound remains open for $d\geq 3$.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Цзан Ицзэ (, род. 8 сентября 1999 года в гор. Харбин, провинция Хэйлунцзян.) — китайская шорт-трекистка, чемпионка мира 2017 года в эстафете.
Биография
Цзан Ицзэ занялась шорт-треком в возрасте 10-ти лет в Харбине в 2009 году и увлеклась им в качестве хобби. В то время она еще училась в спортивной школе Харбина. Впервые она появилась на международных соревнованиях в 2015 году на юниорском чемпионате мира в Осаке и там на дистанции 1500 м выиграла серебряную медаль, а в общем зачёте заняла 4-е место. На следующий год выиграла бронзу на 500 м и золото в эстафете на юниорском чемпионате мира в Софии, в многоборье стала 5-ой. В феврале участвовала в зимних юношеские Олимпийских играх в Лиллехаммере и неожиданно выиграла 500 м.
В феврале 2017 на зимней Универсиаде в Алма-аты завоевала золотую медаль на 500 м и в эстафете взяла серебряную медаль, через несколько дней на зимних Азиатских играх в Саппоро завоевала золото на 500 м и серебро в эстафете, а в марте на чемпионате мира в Роттердаме она выиграла золотую медаль в эстафете вместе с Цюй Чуньюй, Фань Кэсинь, Линь Юэ и Го Ихань.
В октябре 2017 на этапе Кубка мира в Дордрехте в составе эстафеты выиграла золото, а в ноябре в Шанхае стала второй в той же эстафете, тогда же при падении сломала грудину и на полгода выбыла из соревновании. После реабилитации она вернулась в команду и в ноябре 2018 выиграла в составе команды смешанную эстафету на Кубке мира в Калгари. В начале 2019 года на чемпионате мира в Роттердам 2019 в общем зачёте заняла 8-е место.
В ноябре на Кубке мира в США и Канаде в эстафетной команде выиграла два золота. В марте из-за пандемии коронавируса соревнования были отменены до начала 2021 года.
Награды
2017 год — названа Элитной спортсменкой международного класса
Ссылки
Профиль на сайте ИСУ
Профиль на facebook.com
Шорт-трекисты Китая
Шорт-трекисты на зимних Олимпийских играх 2018 года
Чемпионы мира по шорт-треку
Шорт-трекисты по алфавиту
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{
"redpajama_set_name": "RedPajamaWikipedia"
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Please provide the information requested below. You may omit any sections that do not apply to your business. Please remember that the more detailed information you provide, the better we can understand your needs.
If you would like to speak with us personally, feel free to call us at 205-495-0336.
Thank you for contacting us and we will respond to your request as soon as possible!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,973
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Q: Generic typed borrowed value does not live long enough in match I have this function for quick SQL queries those only expect one value from the DB. I import the postgres 0.19.3 library.
fn psql_query<'a, T>(db: &str, query: &str) -> Result<T, postgres::Error>
where
T: postgres::types::FromSql<'a>,
{
let client = Client::connect(&format!("host=localhost user=postgres password={} dbname={}", get_password(), db), NoTls);
match client {
Ok(mut client) => {
match client.query_one(query, &[]) {
Ok(row) => Ok(row.get(0)),
Err(e) => Err(e)
}
}
Err(e) => Err(e)
}
}
Purpose: connect to DB, return Err() if unsuccessful; then execute query, then return Ok(value) or Err() if something went wrong. The caller can be sure to either get the requested value or the error why it went wrong. Originally I had this very same function with return type Result<String, Error> and it worked! Then I realized that maybe I need to query other data types from the DB and thought to utilize generic types instead of implementing separate functions. When I modified the function to use generic types, I started getting the following error:
error[E0597]: `row` does not live long enough
--> src/hc.rs:108:31
|
100 | fn psql_query<'a, T>(db: &str, query: &str) -> Result<T, postgres::Error>
| -- lifetime `'a` defined here
...
108 | Ok(row) => Ok(row.get(0)),
| ^^^^^^^^^^- `row` dropped here while still borrowed
| |
| borrowed value does not live long enough
| argument requires that `row` is borrowed for `'a`
So... why does row need to survive any longer? I mean, I just want the return value from get() packed into Ok(). Since this value will be owned and returned, row and client can be dropped. Even the carets show that row would be dropped after get(0) is evaluated, when I don't need it anymore. If get() would return a slice or reference that ties the return value to row, I would understand, but I don't think this is the case. Why it worked when it was String-only? How can I fix this?
(The postgres::types::FromSql<'a> and the 'a lifetime was required by Row.get().)
A:
If get() would return a slice or reference that ties the return value to row, I would understand, but I don't think this is the case.
That can be the case; the returned type from get() must implement FromSql<'a> where 'a is the lifetime of the Row. However, the exact type of value returned is T.
The compiler is complaining because the 'a specified by the caller must be longer than the lifetime of the row. Your code is effectively saying that the row must live for at least as long as the caller-supplied lifetime, which is impossible.
The most straightforward way to fix this is to say that T must implement FromSql<'a> for any possible lifetime 'a which you can do with a higher-rank trait bound (for):
fn psql_query<T>(db: &str, query: &str) -> Result<T, postgres::Error>
where T: for <'a> postgres::types::FromSql<'a>
{
...
}
(Playground)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,515
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Grégoire Courtine (born in Dijon, France) is a French neuroscientist and a professor at the École Polytechnique Fédérale de Lausanne (EPFL), where he is the co-director of the Defitech center for interventional neurotherapies (.NeuroRestore). His research focuses on the field of neurotechnology, with the aim to restore locomotor functions in patients with central nervous system disorders such as spinal cord injuries.
Career
After being training in physics and mathematics, Courtine obtained a PhD in experimental medicine jointly from the University of Pavia and the INSERM at the University of Burgundy in 2003. He then performed post-doctoral work in the laboratory of Reggie Edgerton at UCLA from 2004 to 2007, where he studied the neural basis of spontaneous recovery after spinal cord injuries, and developed novel strategies to restore functional neural networks after spinal cord resection in rats. In 2008, he was appointed associate professor at the University of Zurich, where he established his own group with the aim to restore voluntary control of locomotion in paraplegic rats. He moved to EPFL in 2012 as an associate professor, and was promoted to full professor in 2019.
In 2014, Courtine co-founded the biotechnology venture ONWARD Medical with the aim to develop novel neurostimulation therapies; he currently serves as the company's Chief Science Officer.
Since 2019 Courtine, together with Jocelyne Bloch, leads the .Neurorestore Laboratory, jointly initiated by the Defitech Foundation, the Lausanne University Hospital, the University of Lausanne and the EPFL.
Research
Courtine heads the G-LAB within the Brain Mind Institute and the Center for Neuroprosthetics at EPFL. Research in the G-LAB aims at restoring motor functions after central nervous system disorders such as spinal cord injuries.
Using targeted spinal cord stimulation neurotechnologies, Courtine and colleagues significantly contributed to the field of neurorehabilitation by publishing a number of studies successively reporting the restoration of voluntary locomotion in rats with partial and complete spinal cord injuries, in primates and in paralyzed human patients, and restoration of blood pressure stability in human patients with spinal cord injuries through the use of biomimetic epidural electrical stimulation (neurostimulation of the spinal cord mimicking the natural activation of the autonomic or somatic nervous system).
Distinctions
Courtine has received several international research and innovation prizes such as the Chancellor's Award for postdoctoral research (2008), the International Foundation for Research in Paraplegia Schellenberg Research prize (2010), the Debiopharm Group Life Science Award (2013), the Rolex Award (2019), and the Leenards Foundation Science Prize (2021).
He was also awarded a starting grant (2010), a consolidator grant (2015) and two proof-of-concept grants (2013 & 2019) from the European Research Council. In 2020, he was awarded the IET A F Harvey Prize.
References
External links
Website of the Neurorestore Laboratory
Website of the G-Lab
Website of ONWARD Medical
French neuroscientists
Academic staff of the École Polytechnique Fédérale de Lausanne
University of Pavia alumni
University of Burgundy alumni
Living people
Year of birth missing (living people)
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{
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Beginning in 2008, there was worsening conflict between the People's Alliance for Democracy (PAD) and the People's Power Party (PPP) governments of Prime Ministers Samak Sundaravej and Somchai Wongsawat. It was a continuation of the 2005–2006 political crisis, when PAD protested against the Thai Rak Thai (TRT) party government of Prime Minister Thaksin Shinawatra. PAD followers usually dressed in yellow, yellow being the royal color of King Bhumibol Adulyadej, and were called "yellow shirts". National United Front of Democracy Against Dictatorship (UDD) followers, known as supporters of the deposed prime minister Thaksin Shinawatra, dressed in red and were widely called "red shirts".
History
The PPP won the December 2007 general election. Though it became the largest party in the House, it did not gain an absolute majority. PPP became the leading party when five other smaller parties, previously allies of Thaksin Shinawatra, agreed to join the coalition government under Samak Sundaravej.
The coalition government vowed to amend the 2007 constitution which they called undemocratic, prompting PAD to resume protests against the constitutional amendment plan after having suspended its movement after the 2006 coup. PAD charged that the amendment was aimed at revoking charges against Thaksin and election fraud charges by PPP executive members. The Samak government suspended its amendment plan, yet PAD vowed to continue their protest, calling for Samak Sundaravej's resignation, noting that during his election campaign, he had declared himself a nominee of Thaksin. Tensions escalated when PAD seized Government House.
Samak was forced to resign in September after the Constitutional Court found him guilty of being an employee of a television cooking program after he became the prime minister. PPP deputy leader Somchai Wongsawat, vice president of PPP and a brother-in-law of Thaksin, became the new prime minister. This change prompted continued PAD protests. The protests led to violent clashes between the police, PAD, and anti-PAD protesters on 7 October as the government aimed to disperse them from blocking the entrances of the House. A PAD protester was killed by a grenade near the premier's office and a member of the PAD security forces was killed when his car exploded. Many protesters were also injured.
As PAD was attacked, it stepped up its protests by seizing the government's temporary offices at Don Mueang Airport, and the seizure and closure of Suvarnabhumi Airport after the arrest of Chamlong Srimuang, one of its leaders. The sieges ended in December when the Constitutional Court dissolved the PPP and banned its executive board from political office after finding it guilty of election fraud. The Constitutional Court dissolved two coalition partners of the PPP on similar charges, which broke the coalition apart. Prime minister Somchai resigned.
It is alleged that the army then urged MPs of a former PPP faction, the Friends of Newin Group, and MPs of the former PPP coalition partners to cross the aisle and form a coalition with the Democrat Party leader Abhisit Vejjajiva. After days of negotiations, Abhisit Vejjajiva, leader of the Democrat Party, was voted in by a margin of 235 in favor, 198 against.
The People's Alliance for Democracy opposed to what it called the "Thaksin system", which is seen by some as the Thaksin' "autocracy", his domination of independent supervisory institutions and violations of human rights and the freedom of the press. It began the mass protest against Thaksin Shinawatra in 2006 after the removal of the critical TV program of Sondhi Limthongkul. In 2008, PAD accused Samak Sundaravej and Somchai Wongsawat of being proxies for Thaksin. Samak had announced that he was Thaksin's nominee during his election campaign, Somchai is Thaksin's brother-in-law, and several new Pheu Thai Party members are former TRT/PPP members. PAD was opposed to all attempts to reform the constitution in ways that would allow politicians banned from political office to re-enter politics, and to revoking the charges against Thaksin. PAD has also called for "new politics", a "new age of politics, free from the corrupted politicians". In May 2009, PAD founded its party, the New Politics Party. One of the PAD's major aims was to oppose what it considered the anti-monarchy movement, supposedly among some Thaksin supporters. PAD promoted the role of the king as the guard of the all people's power against the "corrupted politician". That is why PAD uses yellow, Rama IX's symbolic birthday colour, Monday, as its brand. Yet its ideology was blasted by some critics as "anti-democratic".
The National United Front of Democracy Against Dictatorship (UDD) was the group consisting mainly of social activists, scholars, and Thaksin supporters who opposed the 2006 coup deposing Thaksin Shinawatra. It believed that the coup was supported by powerful persons, the "elites" or "nobles", targeting Prem Tinsulanonda, head of the privy council. The UDD aimed to topple the amatayathipatai (government run by traditional elites, nobles, and the bureaucratic polity). In the April 2009 protest, its aims were the immediate resignation of the prime minister Abhisit Vejjajiva and members of the privy council, Prem Tinsulanonda and Surayud Chulanont, as well as new elections. It also demanded that charges be brought against PAD for the 2008 airport seizures and unrest. Several members of parliament in the Pheu Thai Party were vocal in their support of UDD, as well as of Thaksin Shinawatra himself.
Origins of the crisis
The 2008 crisis had its roots in the 2005–2006 Thai political crisis, the 2005 election and 2006 elections, the 2006 coup, and the 2007 general election.
Thaksin Shinawatra's Thai Rak Thai party (TRT) won a landslide victory in the 2001 election, the first elections held under the Thai Constitution of 1997. Thaksin's grassroots economic policies helped reduce poverty and provided universal health care, making him hugely popular in rural Thailand. His drug policies were effective at reducing drug use but were attacked for the large number of extrajudicial executions that resulted. He was accused of conflicts of interest due to his family's continued holdings in Shin Corporation, the telecoms business that he founded prior to becoming Premier. Despite this, he became the first politician in Thai history to finish his term. Thaksin's re-election in 2005 election had the highest voter turnout in Thai history.
Media mogul Sondhi Limthongkul had been a staunch supporter of Thaksin until major losses at state-owned Krung Thai Bank caused CEO Viroj Nualkhair to be fired. Viroj was Sondhi's former banker and had forgiven Sondhi for billions in bad personal debts. Sondhi's levied public criticisms of Thaksin on his TV show and his media outlets, the Manager Daily newspaper and ASTV.
Sondhi's People's Alliance for Democracy soon gathered supporters among Dhammayuttika Nikaya disciples of Luang Ta Maha Bua, prominent socialites and members of the Thai royal family who claimed that Thaksin frequently insulted King Bhumibol Adulyadej, several state-enterprise unions who were against Thaksin's state-enterprise privatization plans, and various factions in the Royal Thai Army who claimed that Thaksin promoted only those who were loyal to him. The movement gained in popularity after Thaksin's family sold their share in Shin Corporation to Temasek Holdings while making use of a regulation that exempted individuals from paying capital gains tax. The PAD led protests demanding that Thaksin pay additional taxes, despite the SEC and the Revenue Department saying that no wrong had been done.
The protests escalated. In February 2006, Army Commander Sonthi Boonyaratglin started secretly planning for a military coup, despite regular denials. On 14 July 2006, Privy Council President Prem Tinsulanonda addressed graduating cadets of the Chulachomklao Royal Military Academy, telling them that the Thai military must obey the orders of the King – not the Government.
The coup was executed on 19 September 2006 while Thaksin was attending a UN summit, just weeks before a planned Parliamentary election. The junta canceled the elections, abrogated the Constitution, dissolved Parliament, banned protests and all political activities, suppressed and censored the media, declared martial law, and arrested Cabinet members. The PAD voluntarily dissolved after announcing that its goals had been accomplished. Surayud Chulanont, Prem's former close aide, was appointed Prime Minister. A junta-appointed court banned the TRT and 111 of its executives from politics for five years. Privy Council President Prem harshly criticized Thaksin, who was in exile, comparing him to Adolf Hitler. A junta-appointed committee drafted a substantially revised constitution. Elections were scheduled for December 2007. Many TRT politicians moved to the People's Power Party. The PPP won the December elections and nominated Samak Sundaravej as Premier.
2008 resumption of PAD protests
The Samak Sundaravej government, elected in the December 2007 general elections, came under pressure to resign since May 2008, when the People's Alliance for Democracy (PAD) reformed and staged regular street protests. They protested the government's proposals to amend the 2007 constitution, claiming that Prime Minister Samak Sundaravej and his government are acting as a proxy for former deposed prime minister Thaksin Shinawatra. The PAD also criticized the Samak government's decision to support the Cambodian government's application for the listing of the disputed Preah Vihear Temple as a World Heritage site. Tensions rose between Thailand and Cambodia as the PAD called for Thai investors to withdraw from Cambodia, the closure of all 40 Thai-Cambodian border checkpoints, a ban on all flights from Thailand to Phnom Penh and Siem Reap, the construction of a naval base at Koh Kut near the border, and the abolition of the committee which oversees demarcation of overlapping sea areas and the unilateral declaration of a Thai marine map.
In late June, the opposition filed a no-confidence motion in parliament, which the government survived. The PAD proposed that the constitution be amended to reduce the proportion of elected members of Parliament. This would disenfranchise the rural population, which the PAD viewed as being insufficiently educated to vote for anti-Thaksin Shinawatra parties.
Siege of Government House
Tensions escalated to crisis when on Tuesday 26 August the protesters stormed and occupied the grounds of the Government House, displacing the prime minister from his offices. Another armed group attacked the headquarters of the National Broadcasting Services of Thailand, while the offices of three ministries were also partially invaded. Major roads into Bangkok were blocked by PAD supporters. Prime Minister Samak remained defiant, refusing to resign, while also vowing not to violently remove the protesters. Despite a court warrant for the arrest of the PAD's nine leaders and a Civil Court order to evacuate, the PAD remained firmly lodged in the government compound. Friday 29th saw semi-violent clashes at various protest sites between protesters and riot police, who were still largely unable to control the group.
Transport infrastructure was disrupted beginning 29 August, with state railway workers union, the PAD's alliance, was on strike partially disabling train services. PAD demonstrators occupied the runways of and closed down airports in the southern cities of Hat Yai, Phuket and Krabi. (The airports later reopened on the 30th and 31st.) The State Enterprise Worker's Union threatened to disrupt public infrastructure services including electricity, waterworks, airline, bus, port and communications services, beginning with police and government offices, on 3 September.
Prime Minister Samak called an emergency session of parliament on 31 August to resolve the issue, but refused to dissolve parliament as suggested by the opposition. Meanwhile, pro-Thaksin protesters calling themselves the United Front for Democracy against Dictatorship of Thailand (UDDT) began to gather in Sanam Luang.
September 2008 state of emergency
One week after the PAD began occupation of the Government House grounds, violence erupted as members of the UDDT clashed with the PAD in a melee involving firearms, resulting in 43 injuries and at least one death. Prime Minister Samak, by virtue of the Emergency Decree on Public Administration in Emergency Situation, BE 2548 (2005), declared a state of emergency in Bangkok at 07.00 hours of the following morning.
Sundaravej also put Gen Anupong Paochinda, Army Commander in Chief, in charge of the remedy of the said State of Emergency, and appointed Pol Gen Patcharavat Wongsuwan, National Police Commander in Chief, and Lt Gen Prayuth Chan-ocha, 1st Army Area Commander as assistants to Gen Anupong. He also issued bans on the gathering or assemblage of more than five persons within the Bangkok Metropolis; on the nationwide press release, distribution or dissemination of letters, publications or any other information containing the matters which may instigate apprehension amongst the people or is intended to distort information in order to mislead an understanding of the State of Emergency to the extent of affecting the security of state or public peace and order or good moral of the people; and on other matters which were a kind of public rights and liberties restriction.
However, on the same day, Mr Nitithon Lamluea, a member of Thailand Lawyer Council's Human Rights Committee, lodged with the Supreme Administrative Court a complaint against Prime Minister Samak accusing him of abuse of power as he imposed a state of emergency in Bangkok, but the circumstances on 2 September were not what was prescribed in said Emergency Decree as being a State of Emergency. The complaint also requested the Court to revoke the State of Emergency. Tej Bunnag, then the Minister of Foreign Affairs, resigned from office as he disagreed with the Government's measures to remedy the crisis and felt uneasy in representing the Government to foreigners when explaining the prevailing circumstances to them.
On 14 September, the Government issued an Announcement revoking the State of Emergency throughout Bangkok together with all the pertinent announcements, orders, and articles.
Fall of Samak
On 9 September 2008, the Constitutional Court of Thailand found that Samak had hosted and received payment for hosting two cooking TV shows, "Tasting and Grumbling" and "All Set at 6 am," for a few months after he had become Prime Minister. He had been hosting the shows for years prior to becoming Premier. Section 267 of the 2007 Constitution of Thailand forbids members of the Cabinet from being employees of any person; this was to prevent conflicts of interest. The court found that although Samak was a contractor to the show's producers, and did not fit the definition of the term "employee" as defined under the Civil and Commercial Code, the law on labour protection or the law on taxation, the spirit of the Constitution gave a broader definition to the term. It thus found Samak guilty of breaching the Constitution and terminated his Premiership.
A session of the House of Representatives was held on 12 September to vote for a new prime minister. The People Power Party decided to renominate Samak as Premier (he had stopped hosting the TV shows earlier that year and thus was no longer an employee). The session was cut short because the House lacked a quorum. The nomination of a new Prime Minister was postponed until 17 September, where the PPP successfully nominated Deputy Prime Minister Somchai Wongsawat as Premier.
Reactions to court decision
Karn Tienkaew, deputy leader of Samak's People Power Party, said the party planned to propose a parliamentary vote Wednesday on returning Samak to power: "Samak still has legitimacy. The party still hopes to vote him back unless he says no. Otherwise we have many other capable candidates."
On 10 September 2008, however, the PPP shied away from their earlier statement they would renominate Sundaravej and was apparently looking for an alternative candidate instead; the new PM was to be nominated on 12 September 2008.
People Power Party's deputy spokesman Kuthep Suthin Klangsang, on 12 September 2008, announced that: "Samak has accepted his nomination for prime minister. Samak said he is confident that parliament will find him fit for office, and that he is happy to accept the post. A majority of party members voted Thursday to reappoint Samak. Samak is the leader of our party so he is the best choice." Despite objections from its coalition partners, the PPP, in an urgent meeting, unanimously decided to renominate Samak Sundaravej. 5 coalition parties, namely Chart Thai, Neutral Democratic, Pracharaj, Puea Pandin and Ruam Jai Thai Chart Pattana, unanimously agreed to support the People Power party (PPP) to set up the new government and vote for the person who should be nominated as the new prime minister. Chart Thai deputy leader Somsak Prissananantakul and Ruam Jai Thai Chart Pattana leader Chettha Thanajaro said the next prime minister who will be nominated on Friday. Caretaker prime minister Somchai Wongsawat said PPP secretary-general Surapong Suebwonglee will notify the 5 parties who the PPP nominates to take office again. Some lawmakers, however, said they will propose an alternate candidate. Meanwhile, Thailand's army chief Gen. Anupong Paochinda said he backed the creation of a unity government that would include all the country's parties, and he also asked for the lifting of a state of emergency that Samak imposed on 2 September.
On 12 September 2008, the quorum for selecting the new PM was not reached and the decision was delayed to 17 September 2008 in an event seen as signaling the end of Samak's career. After its four coalition partners stated they'd prefer someone else to become PM instead of Samak, the PPP agreed to drop his nomination. Embattled Samak Sundaravej abandoned his bid to regain his Thailand Prime Minister post, and Teerapon Noprampa said Samak would also give up the ruling People's Power Party (PPP) leadership. Meanwhile, PPP's chief party spokesman Kudeb Saikrachang and Kan Thiankaew announced on 13 September that caretaker prime minister Somchai Wongsawat, caretaker justice minister Sompong Amornwiwat, and PPP Secretary-General Surapong Suebwonglee were PPP's candidates for the premiership. However, Suriyasai Katasila of People's Alliance for Democracy (a group of royalist businessmen, academics, and activists), vowed to continue its occupation of Government House if a PPP candidate would be nominated: "We would accept anyone as prime minister, as long as he is not from the PPP."
The ruling People Power Party, on 15 September 2008, named Somchai Wongsawat, as its candidate for prime minister to succeed Samak Sundaravej. Somchai Wongsawat was ratified by the National Assembly of Thailand as Prime Minister of Thailand on 17 September, winning 263 votes against 163 votes for Abhisit Vejjajiva.
October violence
Chamlong's arrest
On 4 and 5 October 2008, respectively, Chamlong Srimuang and rally organizer, Chaiwat Sinsuwongse, of People's Alliance for Democracy, were detained by the Thai police on charges of insurrection, conspiracy, illegal assembly, and refusing orders to disperse (treason) against each of them and eight other protest leaders. At the Government House, Sondhi Limthongkul, however, stated that demonstrations would continue.
Clashes and the siege of Parliament
Armed with batons, PAD forces surrounded Parliament and used razor wire barricades to prevent the legislature from meeting to hear Samak's replacement, Somchai Wongsawat, formally announce his policies. The government decided to crack down on the protesters. Police used tear gas but many bullets were fired at the protesters. Many protesters got seriously wounded; some even lost legs, causing over a hundred injuries. One additional PAD leader was killed, claimed by the police, when the bomb in his car went off in front of the headquarters of the Chart Thai Party, a member of the government coalition. Several protesters lost their hands and legs, although it was not clear whether these injuries were caused by tear gas rounds or the ping-pong grenades. Pornthip Rojanasunand, Director of the Central Institute of Forensic Science, claimed that the loss of one particular PAD member's leg could not have been the result of tear gas usage, but came from a more powerful explosion. After viewing photographs of Angkhana Radappanyawut's injuries, Pornthip Rojanasunand suggested unequivocally that the death was caused by the explosion of a tear gas canister. She also stated that there was no need to conduct further investigations into the death and injuries of protesters because it became clear that they were caused by weapons of police. Investigations took place while charges against the police and government were filed by many protesters.
Afterwards, Doctor Suthep Kolcharnwit of Chulalongkorn Hospital led a campaign calling for doctors to refuse medical care to police injured during the clashes as a protest against the violence. Doctors from several major Bangkok hospitals joined him in his campaign. Doctor Suthep Kolcharnwit of the Chulalongkorn University Faculty of Medicine along with several doctors from also refused to provide medical care to police injured in the clash, and urged doctors of other hospitals to boycott police as well.
After the violent crash, the demonstrators returned; the siege of the parliament was given up in the evening. Yet demonstrators started the siege of police headquarters near the Government House. The demonstrators called for the counter-attack on police force, then new clash erupted, causing the death of a female protester and injury of several protesters. The PAD leaders called the demonstrators to return to the Government House and limit the protest only at their 'stronghold', at the House, avoid using violence. Then calm returned to Bangkok.
Queen's appearance
On 13 October 2008, Queen Sirikit presided over the funeral ceremony of Angkhana Radappanyawut, the PAD supporter who had been killed by police during the protests. The Queen was accompanied by her youngest daughter, Princess Chulabhorn, Army Chief Anupong Paochinda, Opposition Leader Abhisit Vejjajiva, and many high-ranked officials, but there were no signs of any police personnel. The Queen received a jubilant welcome from thousands of PAD supporters.
She spoke with the deceased's father, Jinda Radappanyawut, who later said she was concerned for the welfare of the protesters and "would soon send us flowers". Her Majesty told Angsana's parents that their daughter had sacrificed her life for the sake of the monarchy, and that she was a good person.
Previously, the revered Queen had donated ฿1 million to cover the medical expenses of those injured on both sides in the clashes. The protesters tried to claim the financial donation as a gesture of support for them, though in reality it went to aid injured policemen as well. The King also made a financial contribution to the dead woman's family.
Since the Queen's appearance was without precedent in modern times, many academicians and media deemed them to be "unusual" and perhaps an "explicit royal backing to a five-month street campaign to oust the elected government".
Yet one could claim in support of the PAD that really was there was a small 'anti-monarchy' movement which supported Thaksin. This disturbed many monarchists. Sondhi Limthongkul, forefront leader of PAD, had claimed this since 2005, when he started criticizing Thaksin Shinawatra for his 'inappropriate' act defaming the monarchy when Thaksin held a religious ceremony at the royal temple of Wat Phra Kaeo. From then on, PAD has accused Thaksin's followers as being 'anti-monarchy'. The UDD's and Thaksin's harsh attack on Prem Tinsulanonda, head of royal Privy Council and close advisor of the king, has been seen as a contradiction of Thai tradition and a hidden attempt to make a regime change. Moreover, a leader of UDD, such Jakrapob Penkair has engaged in public criticism of the role of the monarchy, mostly, seen 'negatively'. Several UDD leaders were also charged and arrested and accused of insulting the monarchy, a crime under Thai criminal law. The PAD, therefore, has always claimed that it was defending the most revered institution according to the Constitutions, blasting UDD and Thaksin Shinawatra as anti-monarchist. The controversial issue of the monarchy has been then one of the fierce causes among these factions' supporters.
Airport seizures
Protest at Suvarnabhumi International Airport and closures of airports
On the evening of Tuesday 25 November 2008, the PAD executed what they called "Operation Hiroshima." A convoy of hundreds of armed PAD members dressed in yellow blocked the two ends of the road in front of the terminal building of Suvarnabhumi International Airport, Bangkok's main airport and an important regional hub and blockaded the main road to the airport, aiming at resisting the landing of the prime minister's flight. PAD forces quickly overpowered hundreds of policemen armed with riot gear. PAD leaders mounted a mobile stage and proceeded to criticize the government. PAD members armed with clubs, iron bars and knives, with some wearing black balaclavas, then entered the terminal, much to the surprise of the thousands of travellers inside.
Official closure of the airport
PAD forces also forced their way into the control tower, demanding the flight plan for Prime Minister Somchai Wongsawat's return from the APEC summit in Peru. Somchai flew into Bangkok Don Mueang airport on the evening of 25 November 2008 before flying on to Chiang Mai. After hours of turmoil, the Airports of Thailand, AOT, eventually decided that all Suvarnabhumi flights were suspended, leaving thousands of travelers stranded in the airport., this action led to the official closure of the airport later.
The government called on the Royal Thai Army to restore order at the airport. The Army did not follow the orders. In a press conference on 26 November, Army Commander General Anupong Paochinda proposed that the PAD withdraw from the airport and that the government resign. He also proposed that if the PAD did not comply, that they be subject to "social sanctions", whereas if the government did not comply, that the bureaucracy stop implementing government orders. A written copy of the proposal was sent to the government. Neither the PAD or the government complied with the proposal.
At 4:30 am on the morning of 26 November, three explosions were heard on the fourth floor of Suvarnbumi on the outside of the passenger terminal. Another explosion was reported at 6 am. Several people were injured. It was not clear who set off the explosions. The PAD did not allow the police or forensics experts to investigate the explosions.
Attempts to evict the PAD
Also on 26 November, the Civil Court issued an injunction ordering the PAD to leave the Suvarnabhumi International Airport immediately. Notices of the injunction were placed on the front doors of the houses of the 13 PAD leaders. The PAD did not comply with the injunction.
On the evening of 27 November, the government declared a state of emergency around the two occupied airports and ordered police to clear out PAD forces. The state of emergency allowed the military to ban public gatherings of more than five people. The Navy was assigned to aid police at Suvarnabhumi, while the Air Force was assigned to aid police at Don Muang. The Army's spokesman noted, "The army disagrees with using troops to resolve the problem. The army does not want to do that, and it is not appropriate to do that."
The PAD was defiant. PAD leader Suriyasai Katasila announced that the PAD would fight off police. "If the government wants to clear the protesters, let it try. The PAD will protect all locations because we are using our rights to demonstrate peacefully without causing damages to state properties or rioting," Suriyasai said. Suriyasai also threatened to use human shields if police attempted to disperse the PAD.
On the morning of 28 November, PAD leader Chamlong Srimuang announced to PAD forces that he had received a call from an unspecified "senior person" (ผู้ใหญ่ท่านหนึ่ง) telling him to end the rallies. But he refused to do what the senior person told him. "For the past 108 days, the Alliance has protested together under hardship, while another group of people has remained in comfort. They can't just suddenly ask us to stop protesting," he told the assembled forces. Addressing supporters on ASTV, Sondhi said, "If we have to die today, I am willing to die. This is a fight for dignity."
Police manned checkpoints on roads leading to the airport. At one checkpoint, police found 15 home-made guns, an axe, and other weapons in a Dharma Army six-wheel truck taking 20 protesters to Suvarnabhumi airport. One checkpoint, about 2 kilometers from the airport, was attacked by armed PAD forces in vehicles, causing the police to withdraw. Police Senior Sgt Maj Sompop Nathee, an officer from the Border Patrol Police Region 1, later returned to the scene of the clash and was detained by PAD forces. He was interrogated by Samran Rodphet, a PAD leader, and then detained inside the airport. Reporters and photographers tried to follow Sompop to his interrogation, but PAD forces did not allow them. PAD supporters were moved from Government House to the airport.
With the exception of one airplane leaving for the Hajj, no flights with passengers were allowed for eight days. The PAD was apologetic to inconvenienced travelers in the airports and offered them food.
End of the siege
Shortly after the Constitutional Court dissolved the three parties of the government coalition on 2 December 2008, the PAD held a press conference where they announced that they were ending all of their protests as of 10 am local time (GMT 7+) on 3 December 2008. "We have won a victory and achieved our aims," said Sondhi Limthongkul.
Flights from Suvarnabhumi resumed on 4 December, when a Thai Airways flight departed for Sydney, Australia. Thai Airways expected to have five other international flights departing on 4 and 5 December to New Delhi, Tokyo, Frankfurt, Seoul, and Copenhagen. Passengers for these flights had to check in at a convention centre outside the airport.
Views on the seizure
Democrat Party
Democrat Party MP for Sukhothai, Samphan Benchaphon, said of the airport seizure that the PAD "have the right to do it." Democrat Party MP for Bangkok, Thawil Praison, said that the PAD "could seize the airport and doing so is not excessive. The entire world understands that this is a normal matter in the struggle of democratic countries."
International reactions
The governments of China, France, New Zealand, Singapore, Britain, the United States, Australia, and Japan warned their citizens to avoid Thailand and steer clear of protesters at the airport.
The European Union urged the protesters to peacefully leave the airports. EU ambassadors to Thailand wrote in a joint statement that the demonstrators are hurting Thailand's image and economy, continuing "While respecting the right of protesting and without interfering in any way with the internal political debate in Thailand, the EU considers that these actions are totally inappropriate".
US State Department spokesman Gordon Duguid said that occupying the airports was "not an appropriate means of protest" and that the PAD should "walk away from the airports peacefully."
Closure of Bangkok Don Muang Airport
On the night of 26 November 2008, the services at the Don Mueang Airport were stopped after the People's Alliance for Democracy seized control of the domestic passenger terminal.
A bomb exploded near a bunker made of tires near the main entrance to the passenger terminal at Don Muang Airport at 3.55 am on 30 November. Before the explosion occurred, about seven gunshots were heard from the direction of a warehouse deeper inside the airport compound. No one was injured in the explosion. It was not clear who or what set the bomb off.
A plainclothes policewoman at the airport was identified and captured by PAD security forces and forced onto the main PAD stage inside the airport. Angry PAD protesters threw water at her and many tried to hit her. She was eventually allowed to leave the airport.
Constitutional Court ruling on parties' dissolution
Prior to the decision
In December 2008, the Constitutional Court was scheduled to rule on whether or not to dissolve the PPP, Chart Thai, and Matchima Thippatai parties on electoral fraud charges. If the parties were dissolved, executives of the parties could have their political rights stripped for five years. However, non-executive MPs of the parties have the right to change parties for a period of time after the court decision. 37 PPP executives were also MPs, although 8 had resigned, leaving only 14. If the PPP was to be dissolved, the number of MPs they have would fall from 233 to 219. Out of Chart Thai's 43 executives, 19 of them are MPs. No Matchima executives are MPs. Thus, if the Constitutional Court dissolved all three parties, the government coalition would reduce in size to 283 out of a total of 447 MPs in Parliament – a majority is 224. The opposition Democrat Party has only 164 MPs. If all the non-PPP parties allied with the Democrats, they would have a total of 228 MPs – only 4 MPs above a majority.
The site for the court decision was changed from the Constitutional Court to the Supreme Administrative Court after UDD supporters surrounded the Constitution Court. The Supreme Administrative court was protected by Royal Thai Army troops armed with M16 rifles. The Constitutional Court handed down its decision immediately after hearing closing comments from the Chart Thai party.
Summary of decision
Prior to the delivery of decision, judge Chat Chonlaworn, President of the nine-judge panel, remarked that:
Following the decision
Prime Minister Somchai Wongsawat was, therefore, disqualified by the Court's decision, and replaced in a caretaker capacity by Chaovarat Chanweerakul, the Deputy Prime Minister.
The PPP issued a statement calling the ruling a "judicial coup", and called into question the court's procedures, for instance, allowing all the PPP's witnesses only 2 hours to speak. It also noted that the wife of one of the judges was an active PAD member, and said that these and other irregularities would likely lead the Thai people to call the integrity of the court into question and see the ruling as an alternative means of accomplishing a coup. Former PPP MPs and members founded the For Thais Party to prepare for the event that the PPP should be banned, and PPP members immediately started joining the For Thais Party.
Reelections for the 26 constituencies of the banned executive members from the three parties were held on 11 December 2008.
On 6 December 2008, the opposition Democrat Party announced it had garnered sufficient support from former coalition partners of PPP and their splinters to be able to form a government, while For Thais claimed the same, adding that it had already been joined by more than a third of MPs. The coalition partners of the Democrat Party appeared to be most of the dissolved parties, the Thai Nation Party and the Neutral Democratic Party, as well as the Thais United National Development Party, the For the Motherland party, and the "Friends of Newin" faction, a splinter of the banned PPP.
On 15 December 2008, Abhisit Vejjajiva was elected the new Prime Minister. Army commander and co-leader of the 2006 coup, General Anupong Paochinda, was widely reported to have coerced PPP MPs to defect to the Democrat Party. PAD leader Khamnoon Sitthisamarn and junta-appointed Senator called Abhisit's premiership a "genuine PAD victory" and a "Anupong-style coup d'etat." The circumstances of his ascent to power closely linked Abhisit to the Bangkok elite, the Army, and the Royal Palace.
Economic effects
As the crisis is ongoing and fluid, the final economic consequences of the crisis are as yet unclear. After a state of emergency was declared on 2 September 2008, the SET Index reached its lowest point since January 2007 at 655.62; it had fallen 24.7% since the beginning of the PAD's demonstrations in May 2008. The baht hit a one-year low of 34.52 per US dollar, prompting the Bank of Thailand to intervene.
The PAD seizure of Don Muang and Suvarnabhumi Airports is estimated to have cost the Thai economy at least three billion Baht (approximately US$100 million) a day in lost shipment value and opportunities. As of 1 December 2008, the number of stranded passengers was estimated at anywhere from over 100,000 to 350,000.
Political effects
The long term political consequences of the crisis are as yet unclear. The crisis, and particularly the siege of Suvarnabhumi Airport, saw a rise in international press coverage on Thailand, with numerous high-profile articles breaking Thai taboos about public discussion of the role of the monarchy in the crisis as well as the succession. There was a decline in the popularity of the PAD among the Bangkok elite as the crisis escalated and increasingly affected the economy. The crisis saw increasing polarization in Thai political thinking, with David Streckfuss of the Council on International Educational Exchange noting that it "shattered the myth of unity that has been papered over the many social and political cleavages in Thailand." The judiciary was increasingly seen as a tool of the elite, having flagged the government for even the tiniest infractions while refusing to rein in the growing violence of the PAD.
See also
Constitution of Thailand
Politics of Thailand
1973 Thai popular uprising
1976 Thammasat University massacre
2005–06 Thai political crisis
2006 Thai coup d'état
Public opinion of the 2006 Thai coup d'état
2009 Thai political unrest
2010 Thai political protests
2010 Thai military crackdown
2013–14 Thai political crisis
Cambodian–Thai border stand-off
2014 Thai coup d'état
Notes
References
AFP, Thai ruling party leaders back Somchai for PM
AFP, Thai police arrest second anti-govt protest leader
AFP, Thai PM turns to parliament to defuse protest siege, 30 August 2008
The Age, "Thai PM jumps fence to flee protesters"
AP, Strike by Thai anti-government forces falls flat, 3 September 2008
AP, ap.google.com, Thai leader forced to resign over TV cooking show
AP, Thai ruling party re-nominates leader as PM
AP, Thai deputy prime minister resigns
Bangkok Post, Electric shocks, sparks, drainage lids deter police, 30 August 2008
Bangkok Post, Airports still shut, trains not running
Bangkok Post, Coalition parties to hold meeting this evening
Bangkok Post, Somchai, Sompong are candidates for PM
Bangkok Post, Parliament confirms Somchai
Bangkok Post, Wife of Chamlong visits him after arrest
Samak and Anupong Sitting in a Tree.... (2008, 3 September). Bangkok Post. [Online]. Available: http://bangkokpundit.blogspot.com/2008/02/samak-and-anupong-sitting-in-tree.html. Retrieved 3 September 2008.
BBC, Thai coalition looks for new PM
BBC, Samak out of Thai leadership race
BBC, Thai party names nominee for PM
BBC News. (13 October 2008). "Thailand's queen mourns protester". Retrieved 13 October 2008.
Bloomberg, Thai deputy prime minister resigns
Bloomberg, Thailand Postpones Premier Vote After Samak Boycott (Update2)
CNN, Thai ruling party nominates Samak for PM
CNN, Thai judges issue warrants to arrest protesters, 27 August 2008
Constitutional Court, คำวินิจฉัยศาลรัฐธรรมนูญที่ 12-13/2551. [ออนไลน์]. เข้าถึงได้จาก: https://web.archive.org/web/20080913051049/http://www.concourt.or.th/download/Center_desic/51/center12-13_51.pdf. (เข้าถึงเมื่อ: 12 กันยายน 2551).
Declaration of the State of Emergency within the areas of Bangkok Metropolis (2008, 2 September). Government Gazette of Thailand (vol 125, pt 144 D, special issue), p. 1.
Financial Times, www.ft.com, Bangkok state of emergency lifted
Krungthep Business. (2551, 13 ตุลาคม). พระราชินีรับสั่งน้องโบว์เป็นเด็กดี 'ช่วยชาติ-รักษาสถาบัน'. [ออนไลน์]. เ้ข้าถึงได้จาก: < http://www.bangkokbiznews.com/2008/10/13/news_302910.php >. (เข้าถึงเมื่อ: 13 ตุลาคม 2551).
Reuters. (13 October 2008). Thai queen weighs in with anti-govt protesters. [Online]. Available: < http://in.reuters.com/article/worldNews/idINIndia-35930920081013 >. (Retrieved 13 October 2008).
The Guardian, Bangkok protesters hurt in anti-government clashes
The Irrawaddy, PAD Protesters, Police in Tense Standoff, 27 August 2008
The Nation, Treason versus Civil Disobedience, 28 August 2008
The Nation, Tolerance winning the day at Government House – for now, 29 August 2008
The Nation, Somchai elected new prime minister
The Nation. (2008, 14 October). Queen attends slain protester's cremation. [Online]. Available: < https://web.archive.org/web/20090417165658/http://www.nationmultimedia.com/2008/10/14/politics/politics_30085986.php >. (Retrieved 14 October 2008).
The New York Times, Police Arrest Leader of Thai Protests
The New York Times, Thai Protesters Trap Legislators
People's Daily, Thai ruling coalition to submit PM nominee Friday
Reuters, 6-Thai deputy PM quits after Bangkok clashes
Thai foreign minister quits as Bangkok protests drag on. (2008, 3 September). Reuters. [Online]. Available: https://www.reuters.com/article/worldNews/idUSBKK12612220080903. Retrieved 3 September 2008.
Reuters, Thai ruling party picks Thaksin in-law for PM
Reuters, Thai ruling coalition begins search for new PM
The Star, Thai-style democracy, 30 August 2008
The Star, Thailand democracy at risk, 3 September 2008
The Straits Times, I won't quit: Samak, 31 August 2008
Thai Rath, "ข่าวลือสะพัด'เตช'ลาออก หนักใจตอบต่างประเทศ". (2551, 3 กันยายน). ไทยรัฐ. [ออนไลน์]. เข้าถึงได้จาก: https://web.archive.org/web/20080915033838/http://www.thairath.co.th/online.php?section=newsthairathonline&content=102915. (เข้าถึงเมื่อ: 3 กันยายน 2551).
Thai Rath, "อนุพงษ์ชี้'ฉุกเฉิน' แค่ยุตินองเลือด ปัดภารกิจยึดทำเนียบคืน". (2551, 3 กันยายน). ไทยรัฐ. [ออนไลน์]. เข้าถึงได้จาก: https://web.archive.org/web/20080903083741/http://www.thairath.co.th/offline.php?section=hotnews&content=102834. (เข้าถึงเมื่อ: 3 กันยายน 2551).
The Order of the Prime Minister No. 195/2551 Re: Appointment of the Competent Authorities for Remedy of the State of Emergency, dated 2 September 2008. (2008, 2 September). Government Gazette of Thailand, (vol 125, pt 144 D, special issue). pp. 3.
UPI, upi.com, Bangkok state of emergency lifted
Wall Street Journal, Thai Police Arrest Another Leader Of Protest as Crackdown Continues
Washington Post, Ousted Premier Ends Attempt to Get Job Back
Xinhua, Party spokesman: Samak accepts PM nomination
Yahoo! News. (13 October 2008). Thai queen attends funeral for anti-government protester. [Online]. Available: < https://news.yahoo.com/s/afp/20081013/wl_asia_afp/thailandpoliticsprotestfuneral >. (Retrieved 13 October 2008).
External links
A list of 63 websites censored by the Abhisit government during the Songkran 2009 unrest
In depth – Thailand – Political turmoil in the streets of Bangkok, CBC News, 14 May 2010.
2008 in Thai politics
Thai
2008 in Thailand
Political Protests
Thai
Riots and civil disorder in Thailand
2008 riots
Protests in Thailand
Rebellions in Thailand
Thai
Attacks on government buildings and structures
Attacks on airports
|
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\section{Introduction}
\label{model}
\blfootnote{\hspace{-8 pt}A short version of this paper appeared in IEEE Cluster Conference 2017. \\ Available Online: http://ieeexplore.ieee.org/document/8048951/}
Application development with relational databases as the storage backend is prevalent, because relational databases provide: a formal framework for schema design, a structured query language (i.e., SQL) and ACID transactions. As applications gain popularity and the database system reaches its resource limits, the architecture must scale up to ensure end-to-end response time (RT). Relational databases are well suited for vertical scaling; however, vertical scaling has known limitations and requires expensive hardware. On the contrary, the new brand of NewSQL and NoSQL databases are known for their ability to scale out linearly \cite{kossmann, ycsb, hstore}. Hence, as the data size and the resource demands increase, application designers can consider transitioning from their relational database to a NewSQL/NoSQL database. Recently Facebook \cite{Fbmsg} and Netflix \cite{Netflix} transformed part of their relational databases to HBase \cite{HBase}.
NewSQL architectures enable a database to scale out linearly while providing ACID transaction guarantees. However, their schema design requires careful consideration when choosing partition keys, since joins are restricted to partition keys only \cite{voltdb}, resulting in limited query expressiveness. Similarly, NoSQL databases can also scale out linearly, but are limited by slow join performance due to the distribution of data across the cluster and data transfer latency, which has also been identified in previous work \cite{asyncView}. Thus, while NewSQL and NoSQL systems allow data stores to scale, their designs sacrifice query expressiveness and join performance, respectively. More generally, there exists a design space that makes trade-offs between performance, ACID guarantees, query expressiveness and disk utilization.
This paper considers if NoSQL join performance \textbf{can} be improved while ensuring ACID semantics and \textbf{without} drastically sacrificing write performance, disk utilization and query expressiveness. One option for improving the performance of NoSQL workloads is materialized views (MVs), which pre-compute expensive joins \cite{Larson, Yang}. However, deploying MVs on top of a NoSQL store does not guarantee consistency as atomic key based operations allow for the MV's data to be stale relative to the base table \cite{HBase, Bigtable, Accumulo}. Hence, additional concurrency controls such as locking or multi-versioning are required to ensure data consistency.
Standard concurrency control methods, such as locking or multi-versioning, can provide ACID semantics for NoSQL stores with materialized views, but induce performance degradation (i.e., by grabbing many locks or checking multiple versions, respectively) because the concurrency control mechanism and MVs selection mechanism are not designed in tandem. Instead, this paper considers a synergistic design space in which the concurrency control mechanism and MV's selection mechanism operate together such that only a single lock is grabbed per transaction. The proposed system relies on the hierarchical structure of relational data and the workload to inform the views selection mechanism, which can then be leveraged to grab a single lock for MVs and base tables.
\begin{figure*}[t]
\centering
\includegraphics[trim = 20mm 80mm 48mm 15mm, clip, width=.6\textwidth]{pitch.pdf}
\caption{Design choices and decisions in the Synergy System.}
\label{fig:position}
\end{figure*}
In this work, we present the Synergy system that leverages MVs and a light-weight concurrency control on top of a NoSQL database to provide for scalable data management with familiar relational conventions and more robust query expressiveness. Figure \ref{fig:position} presents the design decisions for MVs selection and concurrency control mechanisms in the Synergy system. Synergy harnesses databases' hierarchical schemas to generate candidate MVs, and then uses a workload driven selection mechanism to select views for materialization. To provide ACID semantics in the presence of views, the system implements concurrency controls on top of the NoSQL database using a hierarchical locking mechanism that only requires a single lock to be held per transaction. The Synergy system provides ACID semantics with the read-committed transaction isolation level. Our contributions in this work are as follows:
\begin{itemize}
\item We present the design of Synergy system that trades slight write performance degradation and increased disk utilization for faster join performance (compared to standard NoSQL databases) and improved query expressiveness (compared to NewSQL databases).
\item We propose a novel schema based--workload driven materialized views selection mechanism.
\item We implement and evaluate the proposed system on an Amazon EC2 cluster using the TPC-W benchmark.
\item We compare and contrast the performance of Synergy system with four complementary systems.
\end{itemize}
\begin{figure}[t]
\centering
\includegraphics[trim = 5mm 58mm 45mm 5mm, clip, width=.45\textwidth]{logical.pdf}
\caption{Relations in the Company schema.}
\label{fig:companyS}
\end{figure}
\section{Background}
\label{sec:backg}
We first review the concepts of a relation, index and schema which are common to both SQL and NoSQL data models. Then, we present a model for the database workload. Finally, we provide an overview of the data store used and its associated SQL skin.
\subsection{Relation, Index and Schema Models}
\label{subsec:model}
\textbf{Relation--} A relation R is modeled as a set of attributes. The primary key of R denoted as PK(R), is a tuple of attributes that uniquely identify each record in R. The foreign key of R denoted as FK(R), is a set of attributes that reference another relation T. A relation can have multiple foreign keys, hence let F(R) denotes the set of foreign keys of R.
\textbf{Index--} In this work we utilize covered indexes that store the required data in the index itself. An index X on a relation R denoted as X(R), is modeled as a set of attributes (s.t. X(R) $\subset$ R). Let X$^{tuple}$(R) denotes a tuple of attributes that the index is indexed upon (s.t. X$^{tuple}$(R) $\subset$ X(R)). The key of an index is a union of attributes in tuples X$^{tuple}$(R) and PK(R), in that order. Since a relation can have multiple indexes, let I(R) denotes the set of indexes on R.
\textbf{Schema--} Using the previous definitions of a relation and an index, a database schema S is modeled as a set of relations and the corresponding index sets, S = \{R$_{1}$, I(R$_{1}$), R$_{2}$, I(R$_{2}$),..., R$_{n}$, I(R$_{n})$\}, where $n$ represents the number of relations in the schema. We use an example Company database for the purpose of exposition. Figure \ref{fig:companyS} depicts the relations in the Company database schema.
\subsection{Modeling Workload}
\label{sec:workload}
A database workload W = \{w$_1$,..., w$_m$\} is modeled as a set of SQL statements, where $m$ is the number of statements.
\subsection{HBase Overview}
\label{sec:hbase}
We use HBase \cite{HBase} for the purpose of exposition and experimentation in this work. It is a column family-oriented distributed database modeled after Google's Bigtable \cite{Bigtable}. HBase organizes data into tables. A table consists of rows that are sorted alphabetically by the row key. HBase groups columns in a table into column families such that each column family data are stored in its own file. A column is identified by a column qualifier. Also, a column can have multiple versions of a data sorted by the timestamp.
The HBase data manipulation API comprises of five primitive operations: Get, Put, Scan, Delete and Increment. The Get, Put, Delete and Increment operations operate on a single row specified by the row key. HBase provides ACID transaction semantics with read-committed isolation level.
\subsection{Phoenix Overview}
\label{sec:phoenix}
Apache Phoenix \cite{Phoenix} is a SQL skin on top of HBase. The client embedded JDBC driver in Phoenix transforms the SQL query into a series of HBase scans and coordinates the execution of scans to generate a standard JDBC result set. The default transaction semantics in Phoenix with base tables only is same as HBase; however, recent integration with Tephra \cite{tephra} enables multi-statement transactions in Phoenix through MVCC. Note, the MVCC transaction support in Phoenix can be turned on/off by starting/stopping Phoenix-Tephra transaction server. Next, we describe the mechanism to perform a \textbf{baseline transformation} from a relational to a NoSQL database.
\textbf{Baseline Schema Transformation -- } A relation R becomes a relation R$'$ in NoSQL schema with the same set of attributes as R. The row key of R$'$ is a delimited concatenation of the value of attributes in PK(R). Similarly, an index X(R) on a relation R becomes a relation X(R$'$) in NoSQL schema with the same set of attributes as X(R). The row key of X(R$'$) is a delimited concatenation of the value of attributes in the key of X(R). Note that in NoSQL, both for a relation and an index, we assign all attributes to a single column family.
\textbf{Baseline Workload Transformation -- } Each read statement from the relational workload is added to the NoSQL workload. Each write statement for a relation R that specifies each key attribute in the WHERE clause is added to the NoSQL workload.
\section{Challenges and Design Choices}
\label{sec:motivate}
Joins are expensive in a NoSQL database due to the distribution of data items across different cluster nodes. It is well understood that MVs improve join performance by pre-computing and storing results in the database \cite{Larson, Yang, Goldstein}. This observation is verified with TPC-W micro-benchmark which shows that scanning a MV is significantly faster than the join performance (see Section \ref{sec:microb} for experiment details). Thus, we consider how to incorporate MVs into a NoSQL store, while ensuring consistency.
\subsubsection{Implication of Materialized Views}
NoSQL databases are generally limited to key-based single row operations \cite{HBase, Bigtable, Accumulo}. Hence, to ensure the ACID semantics in the presence of MVs, view maintenance and concurrency controls are required to ensure consistency between the MVs and base tables. The design choices for concurrency control mechanisms include multi-versioning, locking and timestamp ordering. While multi-versioning may seem like a nature fit given HBase and other NoSQL system's temporal key component (i.e., cell values are composed of a row-key, column family, column and time stamp) \cite{Bigtable, Accumulo}, experimental results show that getting and checking additional rows' timestamps decreases performance. Therefore, this result motivates a lock-based concurrency control mechanism to attain the read committed isolation level.
\subsubsection{Lock Number and Granularity}
\label{sec:lockg}
Row level locks and database locks represent the two ends of the locking mechanism spectrum. Database locks degrade system throughput since only a single transaction can access the database at a time. Similarly, acquiring row level locks on individual base tables can be expensive in the presence of MVs in a NoSQL database, since the system may need to acquire a large number of locks for complex queries. Experimental results show that for a modest number of 100 locks, the time to acquire and release locks is 1.3x the response time of the most expensive write transaction in the proposed system (see Section \ref{sec:lock-overhead} and Section \ref{sec:write-overhead}). This observation motivates minimizing the number of locks required per transaction.
\subsubsection{View Selection Challenges}
The types of MVs that are allowed impact the data store performance in varying ways. Purely workload based MVs selection mechanisms \cite{Agrawal} (schema relationships are oblivious) can result in optimal read performance by allowing for the materialization of a maximum number of joins in the workload (i.e., views constructed with many-to-many joins or non-foreign key joins). While this approach is well suited for OLAP workloads, it can degrade write performance and increase disk utilization and transaction management costs for the OLTP workloads, especially in a distributed database. In contrast a schema aware--workload driven MVs selection mechanism limits the type of views allowed, resulting in sub-optimal read performance. However, this approach prevents high storage costs and shifts of the bottleneck from read to the write performance. Given the design goal to hold a single lock per transaction across base tables and MVs, this observation motivates us to not allow views with many-to-many joins or joins that do not have key relationships.
\subsection{Design Decisions}
For the Synergy system, we make the following design decisions based off of our analysis of the TPC-W benchmark, which contains many key/foreign-key equi-joins. First, we develop a concurrency control mechanism that leverages the schema's relational hierarchy, grabs one lock per transaction and provides the read committed isolation level. Second, in cooperation with our concurrency control mechanism, the system only materializes key/foreign-key equi-joins, does not materialize joins across many-many relationships, and each base relation may only be assigned to a single relational hierarchy for materialization (so that a single lock must be acquired per transaction). \textit{We believe the synergistic design decisions between the concurrency control and view selection mechanism provides for a novel architecture and substantially differentiates this work from previous works on materialized view selection.}
\begin{table*}[]
\centering
\caption{Qualitative comparison of NoSQL, NewSQL and Synergy systems.}
\label{tab:compare-systems}
\scalebox{0.7}{
\begin{tabular}{|c|c|c|c|c|}
\hline
& \textbf{Scalability} & \textbf{Query Expressiveness} & \textbf{Transaction Support} & \textbf{Disk Utilization} \\ \hline
\textbf{\begin{tabular}[c]{@{}c@{}}NoSQL\\ (HBase)\end{tabular}} & Linear scale out & SQL & ACID with Snapshot Transaction Isolation & Higher than NewSQL \\ \hline
\textbf{\begin{tabular}[c]{@{}c@{}}NewSQL\\ (VoltDB)\end{tabular}} & Linear scale out & \begin{tabular}[c]{@{}c@{}}SQL with joins limited to \\ partition keys\end{tabular} & ACID with Serializable Transaction Isolation & Lowest \\ \hline
\textbf{Synergy} & Linear scale out & \begin{tabular}[c]{@{}c@{}}SQL with MVs limited to \\ Key/Foreign-Key joins\end{tabular} & \begin{tabular}[c]{@{}c@{}}ACID with Read Committed \\ Transaction Isolation\end{tabular} & Highest \\ \hline
\end{tabular}
}
\end{table*}
\section{System Overview}
\label{sec:overview}
In this section we provide an overview of the \textbf{Synergy system}, as depicted in Figure \ref{fig:workflow}. The objective of our system is to design a scalable and high performance NoSQL database while ensuring the ACID semantics.
\begin{figure}[t]
\centering
\includegraphics[trim = 10mm 67mm 10mm 15mm, clip, width=.5\textwidth]{workflow.pdf}
\caption{Synergy system overview.}
\label{fig:workflow}
\end{figure}
We first perform a baseline transformation of the input relational database to a NoSQL database using the mechanism described in Section \ref{sec:phoenix}. Due to the slow join performance in the baseline transformed database system, we decide to use MVs. We use the \textbf{candidate views generation mechanism} to create a list of potential views to materialize based on the database's hierarchical structure. Next, we use a \textbf{workload driven view selection mechanism} to select views from the candidate set. Then, we re-write the workload using selected views as needed. To ensure high read performance, we supplement the schema with additional view-indexes. To ensure ACID semantics in the presence of views, we implement a concurrency control layer on top of HBase (as described in Section \ref{sec:arch}), which is able to grab a single lock per transaction, while providing the read committed transaction isolation level.
\textbf{System Limitations --} The Synergy system only materializes key/foreign-key equi-joins. In addition, Synergy system is restricted to single SQL statement transactions. In agreement with our design decision of single lock per transaction, write statements that do not specify all key attributes and affect multiple base table rows are not supported. The Synergy system does not enforce foreign key constraints. The transaction isolation level in the Synergy system is limited to read committed. In addition, the Synergy system can only be used with NoSQL data stores that trade availability for strong consistency in presence of network partition (CP model from the CAP theorem \cite{cap}).
\section{Generating Candidate Views}
\label{sec:treem}
In this section we present a mechanism to create candidate views for the materialization of equi joins in the workload. We observe that the joins are slow in a NoSQL database (see Section \ref{sec:motivate}). Hence, materializing the joins in the workload as views can improve the query performance. We harness the schema's structure to identify the candidate views, in particular the key/foreign-key relationships. We begin by presenting formal definitions for schema relationships and views.
We assume that the input schema S is normalized and free from both simple and transitive circular references, to limit the scope of this work. We model the relationships in S as a directed graph G=(H,E). The vertices in G represent the relations in S and edges encode the key/foreign-key relationship between relations. An edge exists between relations R$_i$ and R$_j$, if they are related as described by:
\begin{definition}[Schema Relationships]
The relationship between relations R$_i$ and R$_j$, denoted as $R_i \leftarrow R_j$, exists iff FK$_{k}$(R$_i$) references PK(R$_j$), where FK$_{k}$(R$_i$) $\in$ F(R$_i$)
\end{definition}
Figure \ref{fig:companyG}(a) depicts the schema graph corresponding to the relations in the Company database schema in Figure \ref{fig:companyS}. Next, we define an edge and a path in the schema graph.
\begin{definition}[Edge in Schema Graph]
A directed edge e$_i$ in a schema graph from a relation R$_i$ to a relation R$_j$ is represented as a \textit{(PK,FK)} tuple where PK is the primary key of R$_i$ and FK is the foreign key of R$_j$.
\end{definition}
\begin{definition}[Path in Schema Graph]
A path between relations R$_i$ and R$_j$ in a schema graph is modeled as an alternating sequence of relations and directed edges between the relations, [R$_i$,e$_i$,...,e$_{j-1}$,R$_j$]. The alternating sequence begins and ends in a relation.
\end{definition}
Database schemas have a hierarchical structure; hence, we can choose a set of relations in the schema graph as roots to create rooted trees. Next, we define a rooted tree.
\begin{definition}[Rooted Tree]
A rooted tree T is a directed graph composed of a subset of nodes and edges from the schema graph in which there exists a root node, and unique paths from the root node to each non-root node.
\end{definition}
We use rooted trees to identify the candidate views. Next, we define a candidate view.
\begin{definition}[Candidate View]
A candidate view V is a path in a rooted tree. A view is stored physically as a relation. The attributes of V is a set union of attributes of relations in V and the key of V denoted as PK(V) is the key of the last relation in the view. Also, a view-index has the same definition and semantic as a table index.
\end{definition}
\subsection{Roots Selection}
Each view has a single root. The set of roots Q for a schema S is a subset of relations in S. Q can either be provided by the database designer or it can be learned in an automated manner. In this work, we assume that the database designer provides Q. Note that the automated selection of roots is a separate problem and can be addressed independently.
\begin{figure}[t]
\centering
\includegraphics[trim = 4mm 17mm 18mm 8mm, clip, width=.48\textwidth]{in-out.pdf}
\caption{Input and output of the candidate views generation mechanism for the Company database with roots set Q$_{company}$ = $\{\textit{Address, Department}\}$.}
\label{fig:companyG}
\end{figure}
\begin{figure*}
\hfill
\subfigure[\small Schema DAG]{\includegraphics[trim = 35mm 45mm 23mm 30mm, clip, width=5cm]{intermediate1.pdf} \label{fig:dag}}
\subfigure[\small A topological ordering of schema DAG]{\includegraphics[trim = 5mm 45mm 1mm 10mm, clip, width=6cm]{intermediate2.pdf} \label{fig:topological}}
\hspace*{-.6 pc}
\subfigure[\small Rooted Graphs]{\includegraphics[trim = 5mm 30mm 0mm 10mm, clip, width=6cm]{intermediate3.pdf} \label{fig:subdags}}
\caption{\small Intermediate results of the candidate views generation mechanism for the Company database with roots set Q$_{company}$ = $\{\textit{Address, Department}\}$.}
\label{fig:inter}
\end{figure*}
\subsection{Candidate Views Generation Mechanism}
The goals of the candidate views generation mechanism are as follows:
\begin{itemize}
\item \textbf{Assign each non-root relation in the schema graph to at most one root.} This enables us to hold a single lock on the root relation's row key while ensuring ACID semantics.
\item \textbf{Select a single path between the root and each non-root relation assigned to it.} If there are multiple paths between any pair of relations, then the relationship can be one-many or many-many. However, recall that we define many-many relationship as a join materialization boundary. Therefore, to ensure a one-many relationship, we should select a single path.
\end{itemize}
\subsubsection{Mechanism Overview}
In this section, we present the overview of the mechanism to generate the candidate views. We first \textbf{transform the input schema graph into a directed acyclic graph} (DAG) to ensure at most one direct path between any pair of relations in the graph. Thereafter, we \textbf{identify a topological ordering of the relations} in the schema DAG. Next, we use the topological order to iteratively examine and \textbf{assign each non-root relation to a root} by selecting a path from the root to the non-root relation. Following the assignment of schema relations to roots, a rooted graph is created for each root relation. Finally, we \textbf{transform each rooted graph into a rooted tree} to ensure a single path between the root and each non-root relation. The output of the mechanism is a set of rooted trees and each unique path in a rooted tree represents a candidate view.
\subsubsection{Mechanism Description}
\label{sec:mechanism}
In this section, we describe the candidate views generation mechanism in detail using our continuing example of the Company database. We use Q$_{company}$ = \{\textit{Address,Department}\} as the roots set. In addition, we use a synthetic workload for the purpose of exposition with W$_{Company}$ = \{w$_1$,w$_2$,w$_3$\}
\vspace{.4 pc}
\hspace*{-.1 in}\textbf{W$_1$:} Get address details of an employee\\
\hspace*{.1 in}SELECT * FROM Employee as e, Address as a\\
\hspace*{.1 in}WHERE a.AID = e.EHome$\_$AID and e.EID = ?\\
\hspace*{-.1 in}\textbf{W$_2$:} Get all the employees and their hours who work in a department.\\
\hspace*{.1 in}SELECT * \\
\hspace*{.1 in}FROM Department as d, Employee as e, Works_On as wo\\
\hspace*{.1 in}WHERE d.DNo = e.E$\_$DNo and e.EID = wo.WO$\_$EID\\
\hspace*{.1 in}and d.DNo = ?\\
\hspace*{-.1 in}\textbf{W$_3$:} Get all the employees who work a certain number of hours.\\
\hspace*{.1 in}SELECT * FROM Employee as e, Works_On as wo\\
\hspace*{.1 in}WHERE e.EID = wo.WO_EID and wo.Hours = ?
\vspace{.4 pc}
\textbf{Heuristic--} During the different steps of the mechanism, we use a heuristic based approach to select a candidate from a set. We choose the \textbf{number of overlapping joins} as a simple workload aware heuristic to assign a weight to each candidate. Note that other heuristics can be used seamlessly with the mechanism.
\vspace{.4 pc}
\hspace*{-.12 in}\textbf{Input:} Schema graph G, workload W and the roots set Q.
\hspace*{-.13 in}\textbf{Output:} Set of rooted trees.
\hspace*{-.13 in}\textbf{Steps:}
\begin{enumerate}
\item \textbf{Transform input graph to DAG}: In the first step we transform the input schema graph G into a DAG. We achieve this by selecting and keeping at most one edge between any pair of nodes in the schema graph.
We use our heuristic to assign a weight to each candidate edge. Then, we select the edge with maximum weight and remove the rest. For example, we remove the (AID, EOffice_AID) edge from the schema graph in Figure \ref{fig:companyG}(a) to generate the schema DAG depicted in Figure \ref{fig:dag}.
\item \textbf{Topologically order relations in the DAG}: Next, we identify a linear ordering of the relations in DAG such that for every directed edge from relation R$_i$ to R$_j$, R$_i$ comes before R$_j$ in the ordering. Figure \ref{fig:topological} represents a topological ordering of the schema DAG presented in Figure \ref{fig:dag}.
\item \textbf{Assign relations to roots}: Next, in the topological order, we examine each non-root relation in the schema DAG and decide upon its assignment to a root by executing the following steps:
\label{step:3}
\begin{enumerate}
\item \textbf{Identify paths:} We identify paths in the DAG from each root relation to the non-root relation.
\item \textbf{Select a path:} Next, we utilize our heuristic to assign a weight to each path. Then, we iterate over the paths in the sorted order by weight until we find a path that includes a single root relation and none of the relations on the path are assigned to a root other than the root present in the path.
\item \textbf{Add path:} Then, we add the selected path to the rooted graph created for the root in the path.
\end{enumerate}
Figure \ref{fig:subdags} depicts the rooted graphs generated for the Company database.
\item \textbf{Transform rooted graphs to rooted trees}: Next, we transform the rooted graphs created in step \ref{step:3} into rooted trees. We first identify a topological ordering of the non-root relations in the rooted graph. We repeat the next step while we have relations left in the topological ordering.
\label{step:4}
\begin{enumerate}
\item \textbf{Select a Path}: Using the rooted graph we identify paths between the root relation and the last relation in the topological ordering. Next, we assign a weight to each path using our heuristic. Then, we select the path with maximum weight and add it to the rooted tree. Thereafter, we remove all non-root relations in the path from the topological ordering and continue.
\end{enumerate}
\end{enumerate}
\begin{figure*}[t]\centering
\hspace*{-2.4 in}
\begin{minipage}[t]{0.24\textwidth}\centering
\subfigure {
\includegraphics[trim = -10mm 35mm 0mm 10mm, scale=.28]{exViewCreation1.pdf}
\label{fig:processdomu}
}
\end{minipage}
\hspace*{1.7 in}
\begin{minipage}[t]{0.24\textwidth}\centering
\subfigure {
\includegraphics[trim = -8mm 35mm 0mm 10mm, scale=0.28]{exViewCreation2.pdf}
\label{fig:processdom0}
}
\end{minipage}
\caption{\small Illustration of view selection and query re-writing procedure for an example equi join query using an example rooted tree.}
\label{fig:exViewCreation}
\end{figure*}
Note that in step \ref{step:3}, we examine non-root relations of a schema DAG in the forward topological order to give each non-root relation a chance to be assigned to any root that has a path to it. Conversely, in step \ref{step:4}, we examine non-root relations of a rooted graph in the reverse topological order to keep the paths that will allow materialization of maximum number of joins in the workload. Following the candidate views generation mechanism, a rooted tree is generated for each root in Q. Figure \ref{fig:companyG}(b) depicts the set of rooted trees generated for the Company database.
\subsubsection{Discussion}
The proposed candidate views generation mechanism is a heuristic based approach; hence, does not guarantee materialization of optimal number of joins in the workload. In addition, the usability of generated candidate views for join materialization is dependent on roots selection.
\section{Views Selection Mechanism}
\label{sec:selection}
In this section we describe our procedures for views selection from the candidate set and re-writing queries using selected views. Similar to \cite{Agrawal}, we use a workload driven views selection mechanism. We also illustrate our method for supplementing the schema with additional indexes to ensure query performance.
\subsection{Views Selection}
The high resource requirement and the expensive nature of joins in a NoSQL database (see Section \ref{sec:motivate}) provides us with the motivation to materialize as many joins in the workload as possible to ensure low request response times and high system throughput. We use a workload driven approach to select views. We iteratively examine each equi join query in the workload and select views for it. Next, we describe our procedure to select views for a given query.
\textbf{Views selection for a Query--}We harness the rooted trees and the query syntax to select views for a query. To illustrate the procedure, we use the example rooted tree and the example query depicted in Figures \ref{fig:exViewCreation}(a) and \ref{fig:exViewCreation}(b) respectively. We begin the procedure with un-marked rooted trees. Then, we use the join conditions in the query to mark the relevant edges and participating relations in the rooted trees. Figure \ref{fig:exViewCreation}(c) depicts the marked edges and relations in the example rooted tree. Next, we examine each rooted tree to identify the views to be selected for the query.
For a given rooted tree, we iteratively choose a path until no new path can be chosen. During each iteration, path selection is done using two rules: 1) all the nodes and edges in the path are marked, and 2) the path starts in a marked node that has no incoming marked edge and ends in either a leaf node or a node that has no outgoing marked edge. Then, we select the chosen path as a view. Next, we un-mark the participating relations of the path and outgoing edges of the participating relations, in the rooted tree. Thereafter, we continue with the next iteration. Figure \ref{fig:exViewCreation}(c) depicts the views selected for the example query.
\textbf{Final View Set--} After processing the entire workload, we add the set of all selected views to the schema.
\textbf{Limitations--} 1) We select views only for the equi join queries in the workload. 2) Searching the space of all syntactically relevant views is not feasible in practice \cite{Agrawal}; hence, similar to \cite{Agrawal}, our views selection procedure is heuristic based and does not necessarily select the optimal set of views. 3) A views selection procedure that can take advantage of view sharing opportunities across different queries is part of our future work. 4) Currently we do not pass a storage constraint to our views selection algorithm; however, it can be easily adapted to use storage constraint in presence of a cost based query optimizer.
\subsection{Query Re-writing}
Following views selection, we re-write queries using selected views. We iteratively examine each equi join query in the workload and re-write it using the views selected for it. To re-write a query, we replace the constituent relations of a view with the view. In addition, we remove the join conditions for which both participating relations belong to a single view. Figure \ref{fig:exViewCreation}(d) depicts the example query re-written using selected views.
\subsection{Additional View Indexes}
Unfortunately, in certain scenarios query execution times can be high despite the use of views. Consider a case in which the query using the view has a filter on an attribute other than the attribute that the view is indexed upon. Then, to prepare the query response, we have to scan the entire view. This can be expensive, depending on the size of the view. Hence, to improve the performance of workload queries that use views, we supplement the schema with additional indexes.
For each view, we examine each conjunctive query that uses this view and decide whether to add a view-index or not. If the query only has filters on one or more view attributes that neither the view nor any of its indexes are indexed upon, then we add a view-index indexed upon a filter attribute to the schema. Note that in this work we do not recommend indexes on base tables and assume that the input schema has necessary base table indexes.
\section{View Maintenance Mechanism}
\label{sec:maintenance}
In this section, we describe the mechanism for view maintenance as the underlying base tables are updated.
For each type of write statement we present: 1) an applicability test to determine if a base table update applies to a view and 2) a tuple construction procedure to prepare tuples for the view update upon a base table update.
\subsection{Insert Statement}
\subsubsection{Applicability Test}
A base table \textit{insert} for a relation R$_i$ applies to a view V$_i$ iff R$_i$ is the last relation in V$_i$'s sequence of relations.
\subsubsection{Tuple Construction}
Insertion into a view upon a base table insert may require reading tuples from the base tables to construct the view tuple. For a base table insert that applies to a view with $k$ relations, we need to read related tuples from $k-1$ base tables to construct the view tuple. We utilize the key/foreign-key relationships between view relations to sequentially read the base table tuples, starting with relation R$_{k-1}$ and ending in relation R$_1$. Then, we construct the view tuple using previously read tuples and the insert statement. Notice that the time to create a view tuple increases linearly with the number of relations in the view and is independent of the cardinality ratios between the relations.
\subsection{Delete Statement}
\subsubsection{Applicability Test}
A base table \textit{delete} for a relation R$_i$ applies to a view V$_i$ iff R$_i$ is the last relation in V$_i$. Note that we do not perform cascading deletes.
\subsubsection{Key Construction}
To delete a view tuple upon a base table \textit{delete}, we use the base table key provided with the delete statement.
However, to delete the view index tuple, we need to first construct the index key to issue a delete upon. Hence, we first read the tuple from the view using the base table key in the delete statement. Then, we use the attributes in the read tuple to construct the index key and issue the delete. Notice that the time to construct a view index key is constant.
\subsection{Update Statement}
\subsubsection{Applicability Test}
A base table \textit{update} for a relation R$_i$ applies to a view V$_i$ iff R$_i$ is in V$_i$'s sequence of relations.
\subsubsection{Tuple Construction}
Unfortunately, updating the view upon a base table update can be expensive if the view is not indexed on the key of the update statement, since we need to either join the base tables or scan the entire view for the tuple construction. To efficiently prepare view updates, we supplement the schema with additional indexes based on the workload. Due to space concerns, we omit the details.
\begin{figure}[t]
\centering
\includegraphics[trim = 0mm 15mm 20mm 12mm, clip, width=.45\textwidth]{architecture.pdf}
\caption{Synergy System Architecture Overview.}
\label{fig:arch}
\end{figure}
\section{System Architecture}
\label{sec:arch}
In this section we describe the Synergy system architecture. The Synergy system comprises of \textbf{HBase layer}, \textbf{clients} and the \textbf{Transaction layer} as depicted in Figure \ref{fig:arch}.
\textbf{HBase layer--} The Synergy system harnesses HBase layer as the distributed data storage substrate. The HBase layer comprises of HBase, HDFS and ZooKeeper components. We refer the reader to \cite{HBase} for the role and description of each component shown in Figure \ref{fig:arch}.
\textbf{Clients--} The clients utilize Phoenix API to execute read and write statements in the workload. A client sends a read request directly to the HBase layer. On the contrary, a write request is sent to the Transaction layer, followed by a synchronous wait for a response.
\textbf{Transaction Layer--} The Synergy system employs the Transaction layer for implementing ACID transaction support on top of the HBase layer. The Transaction layer is a distributed, scalable and fault tolerant layer that comprises of a \textit{Master} node and one or more \textit{Slave} nodes. The \textit{Slave} nodes receive and process \textit{write requests} from clients. Each slave node has a transaction manager that implements a write ahead log (WAL) for recovery and durability. The WAL is stored in HDFS. Upon receiving a request, the transaction manager first assigns a transaction $id$ to the statement and then appends the statement in WAL along with the assigned $id$. Then, a transaction procedure utilizes Phoenix API to execute the transaction. Finally, a response is sent back to the client. The \textit{Master} node is responsible for detecting slave node failures and starting a new slave node to take over and replay the WAL of a failed slave node.
\subsection{Lock Implementation}
\textbf{Logical Locking--} Recall that we restrict the write workload to statements that specify all key attributes (see Section \ref{sec:overview}) and decide to employ hierarchical locking as the concurrency control mechanism (see Section \ref{sec:motivate}). Hence, to update a row for a relation in a rooted tree, we acquire the lock on the key of the associated row in the root relation. In addition, since each relation is part of at most one rooted tree, we hold a single lock per write operation.
\textbf{Physical Locking--} We implement our locking mechanism through lock tables stored in HBase. We create one lock table per root relation. The lock table key has same set of attributes as the root relation's key and it includes a single boolean column that identifies if lock is in use or not. A lock table entry is created when a tuple is inserted into the root table.
\textbf{Discussion--} We implement light weight hierarchical locking mechanism in Synergy by holding a single lock per write operation. As a downside of hierarchical locking, all rows associated with the root key along all the paths are locked which can affect throughput with concurrent requests trying to grab the lock on the same root key. Note that lock management is not the primary contribution of our work. Other transaction management systems like Themis \cite{Themis}, Tephra \cite{tephra}, Omid \cite{Omid} etc. could also be used.
\subsection{Write Transaction Procedures}
Synergy utilizes transaction procedures to atomically update the base table, views and corresponding indexes upon a base table update. For \textbf{insert} and \textbf{delete} statements, the transaction procedure first acquires the lock on the root key. Then, the base table, applicable views and corresponding indexes are updated using the tuple/key construction procedures described in Section \ref{sec:maintenance}. Finally, the lock is released. Note that each transaction inserts/deletes a single row in/from the base table, applicable views and corresponding indexes.
A base table update may require multi-row updates on the materialized view. Now, while a view is being updated upon a base table update, conflict with the concurrent writes is prevented by the locking mechanism; however, a concurrent read may read dirty data. Hence, to facilitate the detection of a dirty read, we mark the data in views and view-indexes before update and un-mark after update. If dirty data is read in a transaction, then the read is restarted. The \textbf{update} transaction is a 6-step procedure: 1) We first acquire a lock on the root key. 2) Then, we read all the rows that need to be updated. 3) Next, we mark all the rows that need to be updated. 4) Then, we issue a sequence of updates. 5) Next, we un-mark all the updated rows. 6) Finally, we release the lock.
The plan generator component (see Figure \ref{fig:arch}) in the Synergy transaction layer auto generates the execution plan for each write transaction.
\subsection{Transaction Isolation Level}
The Synergy system is restricted to single statement transactions. In addition, Synergy does not support queries in which a relation is used more than once due to potential dirty reads. The Synergy system provides \textbf{ACID semantics} with \textbf{read committed} transaction isolation level, which is also the default transaction isolation level for PostgreSQL \cite{postgres-isolation}.
A single row is inserted/deleted into/from the base table, applicable views and corresponding indexes upon a base table \textbf{insert/delete}. In addition, to answer a query either a table is used directly or a view involving the table is used but not both. Hence, a reader either reads the entire row or the row is absent from the read result set. This enables read committed behavior for insert and delete statements.
Recall that the system marks the rows to be updated in a view as dirty before issuing updates, and if a concurrent \textit{scan} reads a dirty row, the scan is restarted. Hence we modify the scan behavior to check for marked rows in the scanned result-set and \textbf{re-scan} if a marked row is present. This ensures that \textbf{update} statement preserves the read committed semantics.
Note that the \textbf{read committed} semantics are preserved during a \textbf{failure scenario}, since the base table lock is held until the system recovers from the failure.
\begin{figure}[t]
\centering
\includegraphics[trim = 0mm 160mm 175mm 10mm, clip, width=.4\textwidth]{microView.pdf}
\caption{Micro benchmark schema graph.}
\label{fig:microView}
\end{figure}
\section{Experimental Evaluation}
\label{sec:exp}
In this section we first describe our experiment environment. Next, we use a TPC-W micro-benchmark to evaluate the join performance in HBase. Thereafter, we profile the performance overhead of two phase row locking in HBase. Finally, we evaluate the performance of Synergy system and compare it with four other systems using the full TPC-W benchmark.
\subsection{Experiment Environment}
\subsubsection{Testbed}
\label{sec:testbed}
Amazon EC2 represents our experiment environment. We create an eight node cluster using m4.4xlarge virtual machine (VM) instances. Each instance is configured with 16 vCPU's, 64GB RAM and 120 GB SSD elastic block storage (EBS), running Ubuntu 14.04.
\textit{HBase, HDFS and Zookeeper:} The HDFS NameNode, the HBase HMaster, and the ZooKeeper server processes run on one instance. We designate five instances as slaves, each running the HDFS DataNode and the HBase RegionServer processes. We use Hadoop v2.6.5, HBase v1.2.4.
\textit{Synergy and Phoenix:} We dedicate one instance to host a Synergy transaction layer slave and the Phoenix-Tephra server. Synergy transaction layer master is hosted on the same node that hosts HBase and HDFS masters. We use Phoenix v4.8.2.
\textit{VoltDB:} We create a five instance VoltDB (v6.8) cluster by hosting a VoltDB daemon on each instance that is also hosting the HDFS DataNode and the HBase RegionServer processes.
\textit{Client:} We reserve one node as client to drive the workload for each system.
\subsubsection{Performance Metric}
The request response time represents our performance metric, denoted as $\tau$. We measure $\tau$ in the client.
\begin{figure}[t]
\centering
\includegraphics[trim = 30mm 2mm 30mm 5mm, clip, width=.5\textwidth]{tempp.pdf}
\caption{Micro-Benchmark Workload.}
\label{fig:mb-work}
\end{figure}
\begin{figure}[t]\centering
\hspace*{-.15 in}
\begin{minipage}[t]{0.4\textwidth}\centering
\subfigure[{\small Q1}] {
\includegraphics[width=\linewidth]{q1_set1.eps}
\label{fig:q1s1}
}
\end{minipage}
\begin{minipage}[t]{0.4\textwidth}\centering
\subfigure[{\small Q2}] {
\includegraphics[width=\linewidth]{q2_set1.eps}
\label{fig:q2s1}
}
\end{minipage}
\caption{Micro benchmark results to show that performance of join algorithms is slow in HBase. Y axis is drawn at log scale.}
\label{fig:micro}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[trim = 12mm 148mm 140mm 12mm, clip, width=.45\textwidth]{lock_overhead.pdf}
\caption{Experiment to show overhead associated with two phase row locking in HBase.}
\label{fig:lock-overhead}
\end{figure}
\subsection{Micro Benchmark Evaluation}
\label{sec:microb}
We use a TPC-W micro-benchmark to evaluate the join performance in HBase.
\subsubsection{Schema and Workload}
\label{sec:s+w}
The micro benchmark schema comprises of three relations from the TPC-W benchmark: Customer, Order and Order_line. Customers can have one or more orders and each order can have one or more order lines. Figure \ref{fig:microView} depicts the schema graph for the benchmark schema. Next, to evaluate the join performance, we create a synthetic workload comprising of two foreign key equi-join queries: Q1 (\textit{Customer,Order}) and Q2 (\textit{Customer,Order,Order_line}).
A join query can be evaluated using two different approaches: 1) using a join algorithm that combines the matching tuples from the specified tables and 2) scanning pre-computed and stored results from a materialized view. Hence, to compare the join algorithm performance with the view scan performance, we materialize the joins in the workload as views. Customer-Order and Customer-Order-Order\_line represent the MVs corresponding to the join queries Q1 and Q2 respectively. Figure \ref{fig:mb-work} presents the workload queries written using base tables and MVs.
\subsubsection{Experiment Setup and Results}
Each experiment is characterized by the database scale and the join query. We set the cardinality ratio between relations as 1:10. We scale the database by increasing the number of customers in multiples of 10, starting at 500. For each database scale, we major compact both base tables and views after database population. Section \ref{sec:s+w} presents the join queries in the workload. We repeat each experiment 10 times and report the mean and the standard error of response time. Figure \ref{fig:micro} depicts the experiment results with Y axis drawn at log scale.
For the database populated with 50K customers, view scan is 6x and 11.7x faster than the join algorithm for queries Q1 and Q2 respectively. \textit{\textbf{In conclusion, micro-benchmark results show that the join algorithm performance is slow in HBase, providing the motivation for join materialization.}}
\subsection{Locking Overhead Evaluation}
\label{sec:lock-overhead}
In this experiment, our goal is to evaluate the performance overhead of acquiring and releasing row locks in HBase. We create a single lock table in the HBase layer with two attributes: \textit{id} and \textit{lock_status}. The \textit{lock_status} is a boolean column that identifies whether lock is in use or not. We use \textit{checkAndPut} HBase operation in the client node to acquire and release locks. We increase the number of locks in multiples of 10 starting at 10 and measure the overhead in client. We repeat each experiment 10 times and present the mean overhead time. Figure \ref{fig:lock-overhead} shows the experiment results.
Locking overhead with 100 locks is 1.3x the response time of statement W13 in the Synergy system (see Section \ref{sec:write-overhead} and Figure \ref{fig:writes}); W13 represents the most expensive write transaction in the Synergy system. Also note that 100 represents a modest number of locks for a write transaction, since multiple tables with varying cardinalities may be joined together as a view.\textit{\textbf{ In conclusion, overhead associated with the acquisition and release of row locks represents a major transaction performance bottleneck in HBase, motivating the use of a single lock per transaction.}}
\begin{figure*}
\centering
\includegraphics[width=.8\textwidth]{read1m.eps}
\caption{Evaluation and comparison of join performance across different systems using join queries in the TPC-W benchmark. Y axis is drawn at log scale. Join queries \{Q$_3$, Q$_7$, Q$_9$, Q$_{10}$\} are not supported in VoltDB.}
\label{fig:joins}
\end{figure*}
\subsection{TPC-W Benchmark Evaluation}
\subsubsection{Benchmark}
\label{sec:bench}
TPC-W \cite{tpcw} is a transactional web benchmark. It has a two tier architecture including a web tier and a database tier. TPC-W workload includes 14 different types of web requests where each request is modeled as a servlet. Each servlet is in turn composed of one or more SQL statements. We analyze the TPC-W servlets to extract all the SQL statements that can be invoked at the database tier. Extracted set of SQL statements represents our workload.
We exclude a DELETE statement (DELETE FROM \textit{shopping_cart_line} WHERE \textit{scl_sc_id} = ?) from the workload that may affect multiple base table rows. Phoenix currently does not provide an implementation of the \textit{soundex} algorithm; hence, we exclude two join queries from the workload that use \textit{soundex} algorithm.
The database size (DB$_{size}$) can be modulated by varying two parameters: the number of customers (NUM\_CUST) and the number of items (NUM\_ITEMS). We set NUM\_ITEMS to 10 * NUM\_CUST. In addition, we change the cardinality between the Customer and the Orders table from .9 to 10. We populate the database with 1 million customers. For each system that utilizes HBase as the storage layer, we major compact base tables, indexes and MVs after the database population.
\begin{figure}[t]
\centering
\includegraphics[trim = 12mm 108mm 55mm 5mm, clip, width=.48\textwidth]{systems.pdf}
\caption{Materialized views selection mechanism and concurrency control mechanism used in each evaluated system.}
\label{fig:systems}
\end{figure}
\subsubsection{Systems Evaluated}
\textit{Synergy:} We use Q$_{TPC-W}$ = \{\textit{Author, Customer, Country}\} as the roots set to generate views in the Synergy system. We create base tables, selected views and corresponding indexes in HBase. In addition, we create lock tables for each root in Q$_{TPC-W}$ . We disable the Phoenix-Tephra transaction support.
\textit{MVCC-UA:} To compare our views generation and selection mechanism with \cite{Agrawal}, we deploy SQL Server 2012 on a single EC2 VM instance. Next, we populate the TPC-W database with 1 million customers and run the TPC-W benchmark queries. Then, we use the SQL Server's database engine tuning advisor to analyze the profiled workload and generate views. We create the generated views along with base tables and indexes in HBase and run the workload with Phoenix-Tephra transaction support (MVCC) enabled.
\textit{MVCC-A:} In addition to the base tables and indexes, we create the views and the view-indexes generated by the Synergy system in HBase and run the workload with Phoenix-Tephra transaction support (MVCC) instead of the specialized transaction support used in Synergy.
\textit{Baseline:} We only create base tables and corresponding indexes in HBase and run the workload with Phoenix-Tephra transaction support (MVCC).
\textit{VoltDB:} A VoltDB table can either be partitioned or replicated. The partitioning column is specified by the user and partitioned tables can only be joined on equality of partitioning column. Now, a table can join with other tables using different columns in different queries of the workload; however, since each table can only be partitioned on a single column, only a subset of workload join queries may work for a partitioning scheme.
To profile the performance of maximum number of joins in the TPC-W benchmark we use \textbf{three different partitioning schemes in VoltDB}. However, note that in practice only one partitioning scheme could be used for a database. Also, note that only base tables and corresponding indexes are used in VoltDB.
Figure \ref{fig:systems} summarizes the MVs creation and concurrency control mechanisms used in each evaluated system.
\subsubsection{Performance Evaluation of Joins in the TPC-W Benchmark}
\label{subsec:joins}
\textit{Experiment setup--}
In this set of experiments, we evaluate and compare the join performance across different systems using the join queries in the TPC-W benchmark. Recall that we used three different partitioning schemes in VoltDB to support maximum number of TPC-W joins, using any single partitioning scheme less than 50\% of the TPC-W joins are supported.
We evaluate each query10 times and present the mean and the standard error of the recorded response times. See appendix for the specification of join queries in the TPC-W benchmark. Figure \ref{fig:joins} presents the experiment results. Note that join queries \{Q$_3$, Q$_7$, Q$_9$, Q$_{10}$\} are not supported in VoltDB.
\textit{Discussion--} On an average the join queries in Synergy are 19.5x, 6.2x and 28.2x faster as compared to the MVCC-UA, MVCC-A and Baseline system respectively. The view selection mechanism in the Synergy system selects more MVs as compared to MVCC-UA, resulting in significantly larger join performance benefit. In MVCC-UA, the response time of Q$_{10}$ is significantly lower than the Baseline system since MVCC-UA utilizes a materialized view for query evaluation. The join performance in Synergy system with specialized concurrency control is marginally better than MVCC-A that uses MVCC. The join queries that used views in Synergy are on an average 11x slower than VoltDB (excluding queries that are not supported in VoltDB). \textit{\textbf{In conclusion, join response times in the Synergy system with selected views are significantly lower as compared to MVCC-UA and Baseline system for the benchmark queries. In addition, although the Synergy join performance is slower than VoltDB, Synergy allows for significantly more expressive joins than VoltDB.}}
\begin{figure*}
\centering
\includegraphics[width=.8\textwidth]{write1m.eps}
\caption{Performance Evaluation of the write statements in the TPC-W benchmark to exhibit the overhead of lock management and updating views in the Synergy system. Comparison of write statement performance across different systems.}
\label{fig:writes}
\end{figure*}
\subsubsection{Performance Evaluation of Write Statements in the TPC-W Benchmark}
\label{sec:write-overhead}
\textit{Experiment setup--}In this set of experiments, we aim to evaluate the performance overhead of acquiring/releasing a lock and updating MVs in the Synergy system. In addition, we compare the write statement performance across different systems using the write statements in the TPC-W benchmark. See appendix for the specification of write statements in the TPC-W benchmark. We evaluate each statement 10 times and present the mean and the standard error of the recorded response times. Figure \ref{fig:writes} presents our experiment results.
\textit{Discussion--}On an average the write statements in Synergy are 9x, 8.6x and 8.6x less expensive than MVCC-UA, MVCC-A and Baseline system respectively. In Synergy system, the execution time of statements W6 and W11 is significantly lower than the other write statements since the corresponding relation is not part of any views. Although Baseline system does not use any MVs and MVCC-UA utilizes only one materialized view, the statement response times in these systems are high since MVCC adds an overhead of 800-900 ms to each statement's execution time. On an average the write statements in Synergy are 9.4x more expensive than the VoltDB. \textit{\textbf{In conclusion, experimental results show that the use of hierarchical locking in Synergy system significantly reduces the write transaction response times in presence of MVs.}}
\begin{table}[ht]
\centering
\caption{Sum of RT of all the statements in the TPC-W benchmark to quantify trade off between read performance gain and write performance overhead of using MVs in each evaluated system. VoltDB is excluded since it does not support all queries in the benchmark.}
\label{fig:all_stmt}
\scalebox{0.8}{
\begin{tabular}{|l|c|c|c|c|}
\hline
\multirow{2}{*}{} & \multicolumn{4}{c|}{\textbf{Evaluated Systems}} \\ \cline{2-5}
& \textbf{Synergy} & \textbf{MVCC-A} & \textbf{MVCC-UA} & \textbf{Baseline} \\ \hline
\multicolumn{1}{|c|}{\begin{tabular}[c]{@{}c@{}}\textbf{Mean Response Time}\\ (in seconds)\end{tabular}} & 33.7 & 77.4 & 132.4 & 173.4 \\ \hline
\multicolumn{1}{|c|}{\textbf{Standard Error}} & .03 & .02 & .06 & .07 \\ \hline
\end{tabular}}
\end{table}
\begin{table}[]
\centering
\caption{Database sizes across different evaluated systems.}
\label{fig:dbsize}
\scalebox{0.7}{
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{\textbf{No. of Customers}} & \multicolumn{5}{c|}{\textbf{Database Size (in GB)}} \\ \cline{2-6}
& \textbf{VoltDB} & \textbf{Synergy} & \textbf{MVCC-A} & \textbf{MVCC-UA} & \textbf{Baseline} \\ \hline
\textbf{1M} & 31.8 & 92 & 91.8 & 45.73 & 43.8 \\ \hline
\end{tabular}}
\end{table}
\subsubsection{Performance Comparison of All Evaluated Systems}
\textit{Experiment setup--}In this set of experiments, we evaluate the performance gain and the storage overhead of using MVs in the Synergy system and compare it with the other systems. Note that we exclude VoltDB since it does not support all join queries in the TPC-W benchmark. We evaluate the performance of systems using all the statements in the TPC-W benchmark.
During an experiment, we run each benchmark SQL statement and record its response time. Next, we compute the sum of response time of all statements. We run each experiment 10 times and present mean and standard error of the benchmark response time. Table \ref{fig:all_stmt} presents the experiment results. Table \ref{fig:dbsize} summarizes the database sizes across different systems.
\textit{Discussion--} Synergy system exhibits a performance improvement of 74.5\%, 56.3\% and 80.5\% as compared to the MVCC-UA, MVCC-A and Baseline system respectively. Conversely, the database size in the Synergy system is 2x, 1x and 2.1x the database size in the MVCC-UA, MVCC-A and Baseline system respectively. Hence, Synergy system trades slight write performance degradation and increased disk utilization for faster join performance. \textit{\textbf{In conclusion, the specialized concurrency control mechanism and the MVs generation mechanism in the Synergy system significantly improve the read performance without shifting the bottleneck to the write performance.}}
\section{Related Work}
\label{sec:rw}
\textbf{Materialized Views.} MVs have been studied from multiple standpoints in the SQL domain: view maintenance, view matching, automated views selection, dynamic view maintenance etc. In \cite{Blakeley86, Quass, Blakeley89, asyncView} authors explore the problem of efficient view maintenance in response to the base table updates. The dynamic views \cite{Zhou} introduce storage efficiency by automatically adapting the number of rows in the view in response to the changing workload. The view matching techniques are utilized in query optimization to determine the query containment and the query derivability \cite{Larson, Yang, Goldstein}. In \cite{Agrawal}, authors propose a workload driven mechanism to automate the task of selecting an optimal number of views and indexes for decision support system applications. The MVs selection and maintenance in a transaction processing NoSQL data store raises novel challenges since most of the existing views selection approaches are oblivious to the relationship between schema relations which can lead to heavy view maintenance costs and can shift the bottleneck from reads to writes. To this end, Synergy proposes a novel, schema relationships aware view selection mechanism.
\textbf{Data Partitioning.} Megastore \cite{Megastore}, F1 \cite{Shute} and Elastras \cite{Das2} harness hierarchical schema structure to cluster related data together and minimize the distributed transactions. On the contrary, Synergy generates MVs utilizing hierarchical schema structure to reduce query run times. In \cite{Bin}, authors automate the task of data partitioning by developing a technique for automated selection of root relations in a schema. Schism \cite{Curino} proposes fine grained data partitioning by co-locating related tuples based on workload logs.
\textbf{Transactions.} Transaction support in the majority of the first generation NoSQL stores \cite{HBase, Accumulo} and Big Data systems \cite{asterix} is limited to single-keys. G-Store \cite{Das1} extends HBase to support multi-key transactions in a layer on top using a 2 phase locking protocol. Similar to G-Store, we implement write transactions in a layer on top of HBase. CloudTPS \cite{Cloudtps} supports multi-key read/write transactions in a highly scalable DHT based transaction layer using optimistic concurrency control (OCC). In \cite{Wei}, authors extend CloudTPS to support consistent foreign-key equi-joins. ecStore \cite{Vo} provides snapshot isolation using MVCC based on the global timestamp transaction ordering in a decoupled layer on top of an ordered key-value store BATON. ElasTras \cite{Das2} proposes a novel key-value store that implements MVCC based transactions. Percolator \cite{Peng} extends Bigtable to allow cross-row, cross-table ACID transactions and enables incremental updates to the web index data stored in BigTable. Megastore \cite{Megastore} introduces entity groups as a granule of physical data partitioning and supports ACID transactions with in an entity group. F1 \cite{Shute} is built on top of Spanner \cite{Spanner} and supports global ACID transactions for the Google AdWords business. In contrast with Spanner, Synergy is limited to single data center use; however, Synergy enables enhanced SQL query expressiveness and does not require sophisticated infrastructure including atomic clocks, GPS etc. The NewSQL databases \cite{hstore, voltdb} scale out linearly while ensuring ACID semantics; however, the join support is limited to partitioning keys. The first generation of NewSQL systems required all data to reside in main memory; however, recent work \cite{anticache} overcomes this limitation by keeping cold data on the disk.
\section{Conclusions}
\label{sec:conclusion}
In this paper we present the Synergy system, a data store that leverages schema based--workload driven materialized views and a specialized concurrency control system on top of a NoSQL database that allows for scalable data management with familiar relational conventions. Synergy trades slight write performance degradation and increased disk utilization for faster join performance (compared to standard NoSQL databases) and improved query expressiveness (compared to NewSQL databases). Experiment results on a lab cluster using the TPC-W benchmark show the efficacy of our system.
\section{Acknowledgement}
\label{sec:ack}
This work was supported in part by National Science Foundation awards 1526014 and 1150169.
{\scriptsize
{
\bibliographystyle{IEEEtran}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: How do I add an Applicative context to a type expected by Lens' MyDataType? I have a function generalized over a function func to be applied on MyDataType1 and another MyDataType1 like this
setThis :: (Functor f0, Applicative f0) => Lens' MyDataType1 (f0 Float) -> Float -> MyDataType1 -> MyDataType1 -> MyDataType1
setThis func toThis dataType1 dataType2 = dataType2 & func %~ (\a -> (+) <$> a <*> delta
where baseStat = dataType1 ^. func -- baseStat has type Maybe Float
delta = (\a -> toThis * a) <$> baseStat -- also have type Maybe Float
where MyDataType1 is (when used with print. Every numeric value is a Float)
data MyDataType1 = MyDataType1 { _name = Just "First"
, _length = Just 5.5
, _dmsTypes = Just
( DMS { _i = Just 1.9
, _j = Nothing
, _k = Just 95.9
}
)
}
The function setThis given a default record function like length, a constant Float, a data set to get base value from and a data set to modify, sets _length to a number that's a sum of the original value and the value from the other set multiplied by some constant.
It works just as I expect when given function length.
What I want to do is have the exact same behavior when given a function like (dmsTypes . _j) as in
setThis (dmsTypes . _Just . _j) 0.3 (someY) (someY) -- someY :: MyDataType1
Although GHC throws this error if I do just that
Could not deduce (Applicative f) arising from a use of '_Just'
from the context: Functor f
bound by a type expected by the context:
Lens' MyDataType1 (Maybe Float)
Possible fix:
add (Applicative f) to the context of
a type expected by the context:
Lens' MyDataType1 (Maybe Float)
And while it seems like GHC knows exactly what I should do, I don't know how to do it.
A: Since the thing func points at might not exist, it should be a traversal (any number of focused values) rather than a lens (exactly one focused value).
The part of setThis that uses a setter (i.e., (%~)) remains the same, but the getter part (i.e., (^.)) should instead use a fold, with (^?). In that case, dataType1 ^? func will have two layers of Maybe, one from (^?) and one from func (in what is currently f0), that you'll probably want to flatten with join.
baseState = join (dataType1 ^? func)
Now f0 must be Maybe.
setThis :: Traversal' MyDataType1 (Maybe Float) -> ...
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\section{Introduction}
Precision physics in colliders requires more higher-order corrections in perturbation theory. Unravelling the mathematical structure of Feynman integrals in multiloop calculation is somehow critical to handle the complexity of higher order calculations and may help us to obtain a better control of the perturbative expansion. In recent years, the corresponding research achieved some breakthroughs and becomes now one of the hot topics in physics and mathematics.
One of the powerful methods to evaluate the master integrals analytically attributes to the differential equation \cite{Kotikov:1990kg, Kotikov:1991pm, Remiddi:1997ny, Gehrmann:1999as, Argeri:2007up}. With recent developments \cite{Henn:2013pwa,Henn:2013nsa,Henn:2014qga,Argeri:2014qva,Liu:2017jxz}, this method becomes now a prevailing one in tackling those integrals unsolvable before. It was noticed by Henn that generically in multi-loop calculation, choosing a set of suitable basis for master integrals can greatly simplify the corresponding differential equations \cite{Henn:2013pwa}, which can be calculated iteratively in dimensional regularization scheme. In light of this proposal, many of multi-loop Feynman integrals for various phenomenological processes have been calculated \cite{Henn:2013woa,Henn:2014lfa,Gehrmann:2014bfa,Caola:2014lpa,DiVita:2014pza,Bell:2014zya,Huber:2015bva,Chen:2015csa, Bonciani:2015eua,Gehrmann:2015dua,Grozin:2015kna,Bonciani:2016ypc,Becchetti:2017abb}.
Note, some Feynman integrals in two-loop or higher order possess new mathematical structures \cite{Ablinger:2017bjx,Ablinger:2014bra,Bloch:2013tra,Bloch:2014qca,Broedel:2017kkb,Broedel:2017siw,Broedel:2018iwv,Hidding:2017jkk}, which cannot be expressed as multiple polylogarithms and ask for different technique to deal with. A typical example is the massive two-loop sunrise integral, which has been studied intensively \cite{Laporta:2004rb,Kniehl:2005bc,Adams:2015ydq,Adams:2014vja,Adams:2015gva,Remiddi:2013joa,Adams:2016xah,Remiddi:2016gno,adams:2018}.
The heavy quarkonium production and decay are one of the hot topics in particle physics ever since the first discovery in 1974, especially with the advent of Nonrelativistic Quantum Chromodynamics (NRQCD) factorization formalism \cite{NRQCD}. Up to date there still exist some discrepancies between experimental data and theoretical expectations \cite{Abe:2002rb,Aubert:2005tj,Zhang:2005cha,Zhang:2006ay}, which appeal for precision calculations. In one of our previous works \cite{Chen:2017xqd} we gave out a set of 86 two-loop master integrals about heavy quarkonium production and decay, which can be cast into the canonical form and expressed in terms of multiple polylogarithms. However, for those Feynman integrals with functions beyond the realm of multiple polylogarithms the calculation is not done yet. In fact, to date, only a limited number of similar calculations have been performed in the literature.
In this work, we calculate analytically all remaining integrals with different mathematical structures from multiple polylogarithms in CP-even heavy quarkonium production and decays. The master integrals will be classified into two sectors, one with integrals containing sub-topologies related to the two-loop massive sunrise integrals and the other involving non-planar two-loop three-point integrals. Following the strategy suggested in Ref. \cite{Remiddi:2016gno} and with properly chosen basis, we cast the differential equations of those integrals in the first sector into a proper form that can be solved recursively. Of the second sector, the key point is to find the homogeneous solutions for the second-order differential equations of the two-loop non-planar three-point massive integrals, with that the full solutions can then be obtained by constant variation.
The paper is organized as follows. In section 2, the kinematics is discussed and the derivatives with respect to kinematic variables will be given. In section 3, the iterative integrals and complete elliptic integrals are introduced. In section 4, the elliptic type integrals will be separated into two sectors, and the calculation procedure for them will be elucidated respectively. For illustration, specific examples will be given. Section 5 is remained for conclusions and outlooks. The definition of master integrals is given in appendix A, and several simple but typical analytical results are presented in appendix B.
\section{Notation and kinematics}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.5]{sample.eps}
\caption{Typical two-loop Feynman diagrams for CP-even heavy quarkonium production.}
\label{sample}
\end{center}
\end{figure}
The heavy quarkonium exclusive production in electron-positron collision has a relatively low background, and has played an important role in the study of quarkonium production mechanism. Here we calculate the CP-even quarkonium production in two correlated processes, that is in $\gamma^*\gamma$ collision and in electron-position annihilation associated with a photon,
\bea
\gamma^*(k_1)+ \gamma (k_2) \rightarrow Q(k_q) \bar{Q}(k_{\bar{q}})\ ,\label{pro1} \\
\gamma^*(k_1)\rightarrow Q(k_q) \bar{Q}(k_{\bar{q}}) + \gamma (k_2)\ ,\label{pro2}
\eea
where $k_1^2 = 2 ss , k_2^2 = 0$ and $k_q^2 = k_{\bar{q}}^2 = m_q^2$. The typical Feynman diagrams are showed in Fig. \ref{sample}. The process (\ref{pro1}) is in Euclidean region with $ss<0$, and the momenta satisfy the following relations
\be
(k_1+ k_2)^2 = (k_q + k_{\bar{q}})^2 = 4 m_q^2\ .
\ee
Whereas, the process (\ref{pro2}) is in Minkowski region with $2ss > 4 m_q^2$, and
\be
(k_1- k_2)^2 = (k_q + k_{\bar{q}})^2 = 4 m_q^2\ .
\ee
Note, in the threshold expansion approach, quark and anti-quark momenta are taken to be equal, i.e. $k_q = k_{\bar{q}}$.
In order to express the results compactly, here we introduce three dimensionless variables $x$, $y$ and $z$ as follows:
\bea
\frac{ss}{m_q^2} = -\frac{(1-x)^2 }{2x}=(y+2)=(z+1) \ .\label{xyz}
\eea
The NNLO QCD corrections to processes (\ref{pro1}) and (\ref{pro2}) are calculated in light of Feynman diagrams. As a routine, with some algebraic manipulations, the amplitudes can be reduced to a set of scalar integrals. We use the Mathematica package $\textbf{FIRE}$ \cite{Smirnov:2008iw,Smirnov:2013dia,Smirnov:2014hma} to reduce the scalar integrals to a minimum set of independent master integrals. The calculation of these master integrals is the central issue, and normally turns out to be a nontrivial work. In our calculation, we apply the method of differential equations to calculate the master integrals.
The first step of deriving differential equations is taking derivatives of the Lorentz invariant kinematic variables, and expressing them as linear combinations of master integrals. The $\textbf{FIRE}$ is also employed in the derivation of
differential equations. The derivatives of the external momenta can be expressed as the derivatives of $ss$ and $m_q^2$, like
\bea
k_i\cdot\frac{\partial}{\partial k_j}=k_i\cdot\frac{\partial ss}{\partial k_j}\frac{\partial}{\partial ss} + k_i\cdot\frac{\partial m_q^2}{\partial k_j}\frac{\partial}{\partial m_q^2}
\eea
with $i(j)=1\ \text{or}\ 2$. And in reverse, the derivative $\frac{\partial }{\partial ss}$ can be expressed as a linear combination of derivatives $k_i\cdot\frac{\partial}{\partial k_j}$, i.e.,
\bea
2 ss \frac{\partial}{\partial ss} = k_1\cdot \frac{\partial}{\partial k_1} + \left(\frac{ss + 2 m_q^2}{ss - 2 m_q^2}\right) k_2\cdot \frac{\partial}{\partial k_2}\ .
\eea
The derivative transform can be readily obtained according to equation (\ref{xyz}). With the variables chosen in above,
analytical results of the integrals can then be formulated in a compact form, in terms of iterative integrals and elliptic integrals.
\section{Iterated integrals and complete elliptic integrals}
The Goncharov polylogarithms (GPLs) \cite{Goncharov:1998kja} are defined as
\bea
G_{a_1,a_2,\ldots,a_n}(x) &\equiv & \int_0^x \frac{\text{d} t}{t - a_1} G_{a_2,\ldots,a_n}(x)\ ,\\
G_{\overrightarrow{0}_n}(x) & \equiv & \frac{1}{n!}\log^n x\ ,
\eea
which in fact are special cases of a more general type of integrals, named Chen-iterated integrals \cite{Chen}. If all indices $a_i$ belong to set $\{0, \pm 1\}$, the Goncharov polylogarithms can then be transformed into the well-known Harmonic polylogarithms (HPLs) \cite{Remiddi:1999ew}
\bea
H_{\overrightarrow{0}_n}(x) &=&G_{\overrightarrow{0}_n}(x)\ ,\\
H_{a_1,a_2,\ldots,a_n}(x) &=&(-1)^k G_{a_1,a_2,\ldots,a_n}(x)\ ,
\eea
where $k$ equals to the number of times the element $(+1)$ appearing in $(a_1,a_2,\ldots,a_n)$\, .
The GPLs satisfy the following shuffle rules:
\bea
G_{a_1,\ldots,a_m}(x)G_{b_1,\ldots,b_n}(x) &=& \sum_{c\in a \sha b} G_{c_1, c_2,\ldots,c_{m+n}}(x)\ .
\eea
In above equation, $a \sha b$ is composed of the shuffle products of $a_i (i=1,2\ldots m)$ and $b_i (i=1,2\ldots n)$, which is defined as the set of lists containing all elements of $a_i$ and $b_i$, with the order of elements
$a_i$ and $b_i$ preserved. The GPLs and HPLs can be numerically evaluated by implementing the {\bf GINAC} \cite{Vollinga:2004sn,Bauer:2000cp}, and the Mathematica package {\bf HPL} \cite{Maitre:2005uu,Maitre:2007kp} is applicable
to the HPLs reduction and evaluation. Both GPLs and HPLs can be transformed into functions $\ln,\ \text{Li}_n$ and $\text{Li}_{22}$ up to weight four in light of the method described in Ref. \cite{Frellesvig:2016ske}.
In our calculation, the complete elliptic integrals are necessary to express the integrals encountered. The first and second kinds of complete elliptic integrals are defined as
\bea
K(x)=\int_0^1\frac{\text{d} t}{\sqrt{(1-t^2)(1-x~t^2)}}
\eea
and
\bea
E(x)=\int_0^1\frac{\sqrt{1-x~t^2}}{\sqrt{1-t^2}}\text{d} t\ .
\eea
They satisfy the following derivative relations:
\bea
\frac{\text{d} K(x)}{\text{d} x}&=&\frac{E(x)-(1-x)K(x)}{2(1-x)x}\ ,\nonumber\\
\frac{\text{d} E(x)}{\text{d} x}&=&\frac{E(x)-K(x)}{2x}\ .
\eea
The Legendre relation is useful in simplifying the complete elliptic integrals, i.e.,
\bea
K(x)K(1-x)-K(x)E(1-x)-E(x)K(1-x)=-\frac{\pi}{2}\ .
\eea
\section{Elliptic integral sectors }
The symbols and canonical basis in the calculation of elliptic integrals keep the same as in the preceding work \cite{Chen:2017xqd}, where the linear differential equations can be expressed, via a suitable basis choice of master
integrals, as canonical form \cite{Henn:2013pwa}
\bea
\text{d}~{\bf F} = \epsilon \, (\text{d}~{\bf A}) \, {\bf F} \,
\label{sysdeq}
\eea
with ${\bf F}$ being the vector of canonical master integrals $F_i (i=1\ldots 86)$ \cite{Chen:2017xqd}. Whereas, the two-loop massive Feynman integrals concerned in this work may involve elliptic functions, and hence the calculation of the integrals
should be further explored. We separate them into two elliptic sectors: one with integrals containing sub-topologies related to the two-loop massive sunrise integrals, the other with two-loop non-planar three-point integrals. In the following we elucidate the calculation procedures of these integrals.
\subsection{Sector I : integrals with massive sunrise integrals as subtopology}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.46]{te1.eps}
\includegraphics[scale=0.46]{te2.eps}
\caption{The set of 39 master integrals involve elliptic functions in sector I. The thin line denotes massless propagators and on-shell massless external particles; the thick line represents massive propagators and on-shell massive external particles; the dash line indicates off-shell external particles with momentum squared equal to $2 ss$. The internal lines with a dot mean the power of the propagators are raised to 2.}
\label{midiag1}
\end{center}
\end{figure}
The $39$ Feynman integrals $E_i (i=1\ldots 39)$ belonging to this subsection are shown in Fig. \ref{midiag1}, which contain sub-topologies related to the two-loop massive sunrise integrals. The expressions of master integrals without numerators can be readily read off from the figure, and those with numerators are given in appendix A. Note, the massive sunrise integrals are composed of the complete elliptic integrals and cannot be expressed as pure Goncharov polylogarithms. The two-loop massive sunrise integrals ($E_1,E_2$) have been widely studied. Here, the bases $(\text{A}_1, \text{A}_2)$, which contain $(E_1, E_2)$, are of the same as their first appearance in Ref. \cite{Remiddi:2016gno}:
\bea
\text{A}_1 & = & \, \epsilon^2 \,\frac{12m_q^2((1-2\epsilon)(2(2-3\epsilon)E_1+2(ss+2m_q^2)E_2)-(ss-4m_q^2)F_1/\epsilon^2)}{(ss-2m_q^2)(ss-10m_q^2)} \, ,
\eea
\bea
\text{A}_2 & = & \, \epsilon^2 \, \frac{1}{m_q^2(ss-2m_q^2)(ss-10m_q^2)}\left(-8(1-2\epsilon)(2-3\epsilon)((1-4\epsilon)ss^2+4(11\epsilon-4)ss\, m_q^2 \right. \nonumber\\
& & \left. + 4(3-10\epsilon)m_q^4)E_1 -8(1-2\epsilon)((2\epsilon-1)ss^3-6(7\epsilon-2)ss^2\, m_q^2+12(20\epsilon-7)ss\, m_q^4\right. \nonumber\\
& & \left. -8(25\epsilon-8)m_q^6 )E_2-4((1-4\epsilon)ss^3+2(22\epsilon-3)ss^2 m_q^2\right.\nonumber\\
& & \left. -4(3+10\epsilon)ss\, m_q^4+8m_q^6)F_1/\epsilon^2 \right)\ .
\label{a1a2}
\eea
In the following we sketch the calculation of this sector. With a suitable choice of the basis in the high topologies $(E_3\ldots E_{39})$, the homogeneous part of the differential equations for integrals $(E_3\ldots E_{39})$ can be cast into the canonical form, whereas depending on the inhomogeneous terms of massive sunrise integrals $(E_1,E_2)$, or $(\text{A}_1, \text{A}_2)$. To be more specific, after a proper selection of bases $\text{A}'_i(i=3\ldots39)$, the differential equations for $\text{A}'_i(i=3\ldots 39)$ can be expressed as
\bea
\frac{\text{d}\, \bf{A}'}{\text{d}\, ss} = \epsilon (\bf{W} \cdot \bf{A}' + \bf{Y} \cdot \bf{F}) + (\epsilon \bf{Q}_1 + \bf{Q}_2) \text{A}_1+ \bf{Q}_3 \text{A}_2\ .
\label{de1}
\eea
Here, $\bf{A}'$ is a 37-dimensional basis vector containing integrals $E_i(i=3 \ldots 39)$ and $F_i(i=1 \ldots 86)$; $\bf{F}$ is a 86-dimensional basis vector that was given in Ref. \cite{Chen:2017xqd}; $\bf{W}$ and $\bf{Y}$ are $37\times37$ and $37\times86$ matrices, respectively; $\text{A}_1$ and $\text{A}_2$ are scalar functions defined in equation (\ref{a1a2}); and $\bf{Q}_i$(i = 1,2,3) represent the 37-dimensional vectors which are composed of algebraic functions and are $\epsilon$ free.
Notice that in equation (\ref{de1}) the inhomogeneous term that contain $\text{A}_2$ is free of $\epsilon$, and the differential equation for $\text{A}_1$ given in Ref. \cite{Remiddi:2016gno} can be reexpressed as
\bea
\frac{\text{d}\, \text{A}_1}{\text{d}\, ss}&=&\frac{-(ss-m_q)^2+14(ss-m_q^2)m_q^2+3m_q^4}{2(ss-m_q^2)(ss-2m_q^2)(ss-10m_q^2)} \text{A}_1-\frac{2\epsilon}{ss-10m_q^2} \text{A}_1\nonumber\\
& &-\frac{3m_q^4}{2(ss-m_q^2)(ss-2m_q^2)(ss-10m_q^2)}\text{A}_2\ .
\eea
Since in above equation the inhomogeneous term containing $\text{A}_2$ is also $\epsilon$ free, we therefore are legitimate to perform a basis shift as
\bea
\text{A}'_i\rightarrow \text{A}'_i + b_i(ss)\text{A}_1 \equiv \text{A}_i\, (i=3\ldots 39)\ .
\eea
With the basis shift, $\bf{Q}_3 \text{A}_2$ will be removed from the differential equation (\ref{de1}). Here $b_i(ss)$ are algebraic functions to be determined. Moreover, the basis shift may also simplify the inhomogeneous term containing $\text{A}_1$, considerably.
For illustration, we take the differential equations for $(E_4,E_5,E_6)$ as an example, which have the same topology. By properly choosing the basis, the differential equations for $(E_4,E_5,E_6)$ can be formulated as
\bea
\frac{\text{d}\, e(ss,\epsilon)}{\text{d}\, ss} &=& \epsilon[Y^1(ss)e(ss,\epsilon)+W^1(ss)f(ss,\epsilon)]\nonumber\\
&&+\Omega^0(ss) \text{A}_{1}+\epsilon \Omega^1(ss) \text{A}_{1}+\Lambda^0(ss) \text{A}_{2}\ .
\eea
Here, $e(ss,\epsilon)$, a 3-dimensional basis vector containing integrals $(E_4, E_5, E_6)$ and $F_{12}$, may be expressed as
\begin{equation}
e(ss,\epsilon)=
\left(\begin{array}{c}e_1(ss,\epsilon)\\
e_2(ss,\epsilon)\\
e_3(ss,\epsilon)\end{array} \right)
=\left(
\begin{array}{c}
\epsilon^3 (ss-2m_q^2) E_4\\
\epsilon^2 \sqrt{ ss}\sqrt{ss - 2 m_q^2}(ss-2m_q^2)E_5\\
\epsilon^2 \frac{ss-2m_q^2}{ss+2m_q^2}\left(6\epsilon m_q^2 E_4 - 4ss^2 E_5+m_q^2(3ss+2m_q^2)E_6\right)+\frac{6m_q^2}{ss+2m_q^2}F_{12}\\
\end{array}
\right),
\end{equation}
with $f(ss,\epsilon)$ being a 2-dimensional basis vector
\begin{equation}
f(ss,\epsilon)=\left(
\begin{array}{c}
F_8\\
F_{12}\\
\end{array}
\right)\ .
\end{equation}
$Y^1$ is a $3\times3$ matrix, $W^1$ is a $3\times 2$ matrix, $\{\Omega^0(x),\ \Omega^1(x),\ \Lambda^0(x)\}$ are 3-dimensional vectors, and $\text{A}_1$ and $\text{A}_2$ are scalar functions defined as (\ref{a1a2}).
To remove the $\text{A}_2$ dependence from the inhomogeneous part of the differential equations, we perform the basis shift
\bea
e_i(ss,\epsilon)\rightarrow e_i(ss,\epsilon)+b_i(ss) \text{A}_1\, (i=1,2,3)\ ,
\label{shift}
\eea
where $b_i(ss)$ are algebraic functions to be determined. By virtue of the differential equation for $\text{A}_1$,
one can figure out the shift functions $b_i(ss)$ in (\ref{shift}), which may be formulated in a 3-dimensional vector form
\begin{equation}
b(ss)=\left(
\begin{array}{c}
0\\
2\frac{(2ss-5m_q^2)\sqrt{ss - 2 m_q^2}}{3m_q^2\sqrt{ss}}\\
\frac{14ss m_q^2-13ss^2+8m_q^4}{3m_q^2(ss+2m_q^2)}\\
\end{array}
\right)\ .
\end{equation}
The differential equation for $e_1(ss,\epsilon)$ is in canonical form, and hence no need to make the shift. After the basis shift, $\Lambda^0(ss) \text{A}_{2}$ and $\Omega^0(ss) \text{A}_{1}$ terms in differential equation for $e_3(ss,\epsilon)$ vanish, and the differential equation for $e_3(ss,\epsilon)$ turns to be canonical. Of the differential equation for $e_2(ss,\epsilon)$, though $\Lambda^0(ss) \text{A}_{2}$ term does not exist, $\Omega^0(ss) \text{A}_{1}$ term remains. Note, with the basis shift the inhomogeneous part of the differential equations for $e_2(ss,\epsilon)$ will be greatly simplified, and the differential equations turn to be solvable recursively.
The method described above is also applicable to high sectors with more propagators. Except for integrals $(E_1,E_2, E_5, E_9 )$, differential equations for the remaining 35 integrals can be transformed into the canonical form (\ref{sysdeq}), with the method employed in this work. The basis vector ${\bf A}$ is built up with 39 functions $\text{A}_i(ss,m_q,\epsilon)$, the linear combinations of master integrals $E_i$ and $F_i$ with the latter given in Ref. \cite{Chen:2017xqd}. Explicitly, the 39 bases that contain planar and non-planar two-loop integrals can be formulated as
\bea
\text{A}_1 & = & \, \epsilon^2 \,\frac{12m_q^2((1-2\epsilon)(2(2-3\epsilon)E_1+2(ss+2m_q^2)E_2)-(ss-4m_q^2)F_1/\epsilon^2)}{(ss-2m_q^2)(ss-10m_q^2)} \, , \nonumber\\
\text{A}_2 & = & \, \epsilon^2 \, \frac{1}{m_q^2(ss-2m_q^2)(ss-10m_q^2)}\left(-8(1-2\epsilon)(2-3\epsilon)((1-4\epsilon)ss^2+4(11\epsilon-4)ss\, m_q^2 \right. \nonumber\\
& & \left. + 4(3-10\epsilon)m_q^4)E_1 -8(1-2\epsilon)((2\epsilon-1)ss^3-6(7\epsilon-2)ss^2\, m_q^2+12(20\epsilon-7)ss\, m_q^4\right. \nonumber\\
& & \left. -8(25\epsilon-8)m_q^6 )E_2-4((1-4\epsilon)ss^3+2(22\epsilon-3)ss^2 m_q^2\right.\nonumber\\
& & \left. -4(3+10\epsilon)ss\, m_q^4+8m_q^6)F_1/\epsilon^2 \right)\ , \nonumber\\
\text{A}_3 & = & \, \epsilon^3 \, (ss-2m_q^2)\, E_3\ , \nonumber\\
\text{A}_4 & = & \, \epsilon^3 \, (ss-2m_q^2)\, E_4 \ , \nonumber\\
\text{A}_5 & = & \, \epsilon^2 \, \sqrt{ ss}\sqrt{ss - 2 m_q^2}(ss-2m_q^2)E_5+2\frac{(2ss-5m_q^2)\sqrt{ss - 2 m_q^2}}{3m_q^2\sqrt{ss}}A_1 \ ,\nonumber\\
\text{A}_6 & = & \, \epsilon^2 \, \frac{ss-2m_q^2}{ss+2m_q^2}\left(6\epsilon m_q^2 E_4 - 4ss^2 E_5+m_q^2(3ss+2m_q^2)E_6\right)+\frac{14ss\, m_q^2-13ss^2+8m_q^4}{3m_q^2(ss+2m_q^2)}A_1\nonumber\\
& & +\frac{6m_q^2}{ss+2m_q^2}F_{12} \ ,\nonumber
\label{be12}
\eea
\bea
\text{A}_7 & = & \, \epsilon^3 \, (ss-2m_q^2)\, E_7 \ , \nonumber\\
\text{A}_8 & = & \, \epsilon^3 \, (ss-2m_q^2)\, E_8 \ , \nonumber\\
\text{A}_9 & = & \, \epsilon^2 \, \sqrt{ ss}\sqrt{ss - 2 m_q^2}(ss-2m_q^2)E_9 + 4\frac{(2ss-5m_q^2)\sqrt{ss - 2 m_q^2}}{3m_q^2\sqrt{ss}}A_1\ ,\nonumber\\
\text{A}_{10} & = & \, \epsilon^3 \, (1-2\epsilon)(ss-m_q^2)E_{10} \ , \nonumber\\
\text{A}_{11} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{11} \ , \nonumber\\
\text{A}_{12} & = & \, \epsilon^3 \, \left(m_q^2(ss-2m_q^2)E_{12}-4\epsilon\, m_q^2 E_{11}\right)+\frac{2}{3}(\frac{ss}{m_q^2}-10)A_1+\frac{m_q^2(2F_{24}-4A_{7})}{ss-2m_q^2} \ , \nonumber\\
\text{A}_{13} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{13} \ , \nonumber\\
\text{A}_{14} & = & \, \epsilon^3 \, \left(m_q^2(ss-2m_q^2)E_{14}+4\epsilon\, m_q^2 E_{13}\right)-\frac{2}{3}(\frac{ss}{m_q^2}-10)A_1\nonumber\\
& & -\frac{m_q^2(2F_{27}-4A_{8})}{ss-2m_q^2} +\frac{2\sqrt{ss}}{\sqrt{ss-2m_q^2}}A_{9} \ , \nonumber\\
\text{A}_{15} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{15} \ , \nonumber\\
\text{A}_{16} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{16} \ , \nonumber\\
\text{A}_{17} & = & \, \epsilon^3 \, \sqrt{ ss}\sqrt{ss - 2 m_q^2}(ss-2m_q^2)\, E_{17} \ , \nonumber\\
\text{A}_{18} & = & \, \epsilon^2 m_q^2(ss-2m_q^2)E_{18}-4\epsilon^4 m_q^2 E_{16}-\epsilon^3(3ss^2-8ss m_q^2+4m_q^4)E_{17}\nonumber\\
& & -\frac{\sqrt{ss-2m_q^2}}{\sqrt{ss}}(F_7+F_8+2F_9)+\frac{2m_q^2}{ss-2m_q^2}(F_{20}-2A_4)\nonumber\\
& & -\frac{ss+2m_q^2}{\sqrt{ss}\sqrt{ss-2m_q^2}}A_5+\frac{2(ss-10m_q^2)}{3ss}A_1\ , \nonumber\\
\text{A}_{19} & = & \, \epsilon^3 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{19} \ , \nonumber\\
\text{A}_{20} & = & \, \epsilon^3 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{20} \ , \nonumber\\
\text{A}_{21} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{21} \ , \nonumber\\
\text{A}_{22} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{22} \ , \nonumber\\
\text{A}_{23} & = & \, \epsilon^3 \, \sqrt{ss}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{23} \ , \nonumber\\
\text{A}_{24} & = & \, \epsilon^3 \,(ss\, E_{24}-\frac{ss}{2}\, (ss-2m_q^2)E_{23}-2\epsilon\, m_q^2\, E_{22})-\frac{m_q^2}{ss-2m_q^2}\, A_{7}\ , \nonumber\\
\text{A}_{25} & = & \, \epsilon^4 \, (ss-2m_q^2)\, E_{25} \ , \nonumber\\
\text{A}_{26} & = & \, \epsilon^3 \, \sqrt{ss+6m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{26} \ , \nonumber\\
\text{A}_{27} & = & \, \epsilon^3 \, (m_q^2 E_{27}-\frac{ss^2-4m_q^4}{2}E_{28})+\frac{m_q^2}{ss-2m_q^2}\left(F_{32}-F_{30}-2A_{7}\right)\, \ , \nonumber\\
\text{A}_{28} & = & \, \epsilon^3 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{28} \ , \nonumber
\eea
\bea
\text{A}_{29} & = & \, \epsilon^4 \, (ss-2m_q^2)^2\, E_{29} \ , \nonumber\\
\text{A}_{30} & = & \, \epsilon^4 \, (ss-2m_q^2)^2\, E_{30} \ , \nonumber\\
\text{A}_{31} & = & \, \epsilon^4 \, (ss-2m_q^2)^2\, E_{31} \ , \nonumber\\
\text{A}_{32} & = & \, \epsilon^4 \, (ss-2m_q^2)^2\, E_{32} \ , \nonumber\\
\text{A}_{33} & = & \, \epsilon^4 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{33} \ , \nonumber\\
\text{A}_{34} & = & \, \epsilon^4 \, (ss-2m_q^2)(E_{34}-(ss-2m_q^2) E_{33}) \ , \nonumber\\
\text{A}_{35} & = & \, \epsilon^4 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{35} \ , \nonumber\\
\text{A}_{36} & = & \, \epsilon^4 \, (ss-2m_q^2)(E_{36}-(ss-2m_q^2) E_{35}) \ , \nonumber\\
\text{A}_{37} & = & \, \epsilon^4 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, E_{37} \ , \nonumber\\
\text{A}_{38} & = & \, \epsilon^4 \, (ss-2m_q^2)(E_{38}+(ss+2m_q^2) E_{37}) \ , \nonumber\\
\text{A}_{39} & = & \, \epsilon^4 \, \sqrt{ss+2m_q^2}\sqrt{ss-2m_q^2}(ss-2m_q^2)^2\, E_{39} \ .
\label{base}
\eea
With the basis chosen above, the differential equations for $(\text{A}_3\ldots \text{A}_{39})$ then turn to the canonical form, except for $\text{A}_5$ and $\text{A}_9$. The differential equations for $\text{A}_5$ and $\text{A}_9$ with respect to $x$ write as:
\bea
\frac{\text{d} \text{A}_5}{\text{d} x} &=&\epsilon\frac{33\text{A}_4+6\text{A}_5-6\text{A}_6-4F_8+21F_{12}}{4x}+\epsilon\frac{\text{A}_5+2F_8}{x-1}\nonumber\\
& &+\epsilon\frac{9\text{A}_4-2\text{A}_5-2\text{A}_6+9F_{12}}{x-3}+\epsilon\frac{9\text{A}_4+2\text{A}_5-2\text{A}_6+9F_{12}}{x-\frac{1}{3}}\nonumber\\
&&+\epsilon\frac{1}{6}\left(\frac{1}{x^2}+\frac{28}{x}-\frac{40}{(x-1)^2} + \frac{80}{x-3} - \frac{80}{x - \frac{1}{3}}+1\right)A_1\nonumber\\
&&-\frac{4}{3}\left(\frac{5}{(x-1)^2}+\frac{1}{x}\right)\text{A}_1\ ,\nonumber\\
\frac{\text{d} \text{A}_9}{\text{d} x} &=&\epsilon\frac{6\text{A}_8+3\text{A}_9+2F_7}{x}-\epsilon\frac{4\text{A}_9+F_7}{x-1}-\epsilon\frac{2\text{A}_9}{x+1}\nonumber\\
& & +\epsilon\frac{1}{3}\left(\frac{4}{x^2}+\frac{160}{(x-1)^2}+\frac{52}{x}+4\right)\text{A}_1\nonumber\\
& & -\frac{8}{3}\left(\frac{5}{(x-1)^2}+\frac{1}{x}\right)\text{A}_1\ .
\eea
Notice that the above two equations are not in canonical form, and they both have the $\epsilon$ free $\text{A}_1$ terms, by a factor of 2 difference. Those terms without $\text{A}_1$ can be expressed in d-log form.
By using the method described in above, different from casting all terms into canonical form via (non-algebraic) basis change in Ref. \cite{adams:2018}, the obtained differential equations are greatly simplified and are suitable for solving recursively. Taking the known result on $\text{A}_1$ \cite{Remiddi:2016gno} as an input, the differential equations for $(\text{A}_3\ldots \text{A}_{39})$ can be integrated straightforwardly order by order in $\epsilon$. The corresponding lengthy expressions is given as an auxiliary file in {\bf arXiv} version of this paper.
After determining the bases, to fix the boundary conditions is necessary for solving the differential equations. Here, we apply the regularity conditions as in Ref. \cite{Gehrmann:1999as} to assist the determination of boundary conditions. Noticing that the integrals ($E_3,$ $E_4,$ $E_5,$ $E_7,$ $E_8,$ $E_9,$ $E_{11},$ $E_{13},$ $E_{15},$ $E_{16},$ $E_{17},$ $E_{19}\ldots E_{23},$ $E_{25},$ $E_{26},$ $E_{28}\ldots E_{39}$) are regular at $ss=2m_q^2$ and multiplying the normalization factor $(ss-2m_q^2)$ to $\text{A}_i$, one may find that the corresponding bases $\text{A}_i$ turn to be zero at $ss=2m_q^2$. The boundary condition for $\text{A}_6$ at $ss=2m_q^2$ can be fixed in a similar way, that is
\bea
\text{A}_6\mid_{ss=2m_q^2}=\frac{3}{2}F_{12}-\frac{4}{3}\text{A}_1\mid_{ss=2m_q^2}=\frac{1}{2}F_{12}\ .
\eea
Here, the integral $F_{12}$ is known, and the boundary condition for $\text{A}_1$ may be determined from its definition in (\ref{be12}), i.e. $\text{A}_1\mid_{ss=2m_q^2}=\frac{3}{4}F_{12}$. The integral $E_{10}$ is regular at $ss=m_q^2$ with the normalization factor $(ss-m_q^2)$. Multiplied by this normalization factor, we then find $\text{A}_{10}=0$ at $ss=m_q^2$. Since the integrals $(E_{12},E_{14},E_{18},E_{27})$ are also regular at $ss=2m_q^2$, the boundaries of corresponding bases $\text{A}_i$ can be determined by differential equations. For instance, the differential equation for $\text{A}_{12}$ reads
\bea
\frac{\text{d} \text{A}_{12}}{\text{d} y} = 2\epsilon\frac{6F_{24}-12\text{A}_7 + 6\text{A}_{11}-3\text{A}_{12}-16\text{A}_1}{3 y}+\ldots \ ,
\label{A12}
\eea
where ellipses stand for less singular terms at $y=0$, i.e. $ss=2m_q^2$. Since all integrals in (\ref{A12}) have finite limits at $y\rightarrow 0$, the following relation between different integrals exists:
\bea
\lim_{y\rightarrow 0} (6F_{24}-12\text{A}_7 + 6\text{A}_{11}-3\text{A}_{12}-16\text{A}_1)=0.
\eea
Because $(F_{24},\text{A}_7,\text{A}_{11})$ are zero at $y=0$ ($ss=2m_q^2$), we then have
\bea
\text{A}_{12}\mid_{y=0}=-\frac{16}{3}\text{A}_{1}\mid_{y=0}\ .
\eea
Similarly, from those boundaries for integrals $E_{14}$, $E_{18}$, $E_{24}$ and $E_{27}$, one can fix all boundary conditions for bases $(\text{A}_1\ldots \text{A}_{39})$, of which the none-zero ones up to weight-4 write as:
\bea
\text{A}_1\mid_{ss=2m_q^2} &=& \epsilon^2\frac{\pi^2}{16}+\epsilon^3\frac{3}{16}(7\zeta(3)-2\pi^2\ln(2))+\epsilon^4(9\text{Li}_4(\frac{1}{2})-\frac{31\pi^4}{480}+\frac{3}{4}\pi^2\ln^2(2)\nonumber\\
& & +\frac{3}{8}\ln^4(2))+ {\cal O}(\epsilon^{5}),\nonumber\\
\text{A}_2\mid_{ss=2m_q^2} &=& \frac{16}{3}\text{A}_1\mid_{ss=2m_q^2},\nonumber\\
\text{A}_6\mid_{ss=2m_q^2} &=& \frac{2}{3}\text{A}_1\mid_{ss=2m_q^2},\nonumber\\
\text{A}_{12}\mid_{ss=2m_q^2} &=& -\frac{16}{3}\text{A}_1\mid_{ss=2m_q^2},\nonumber\\
\text{A}_{14}\mid_{ss=2m_q^2} &=& \frac{8}{3}\text{A}_1\mid_{ss=2m_q^2},\nonumber\\
\text{A}_{18}\mid_{ss=2m_q^2} &=& -\frac{4}{3}\text{A}_1\mid_{ss=2m_q^2},\nonumber\\
\text{A}_{24}\mid_{ss=2m_q^2} &=& 8\epsilon^3\pi^2\ln(2)+\epsilon^4(\frac{59\pi^4}{15}-192\text{Li}_4(\frac{1}{2})-8\ln^2(2)(\pi^2+\ln^2(2)))+ {\cal O}(\epsilon^{5}),\nonumber\\
\text{A}_{27}\mid_{ss=2m_q^2} &=& \epsilon^3(\frac{3\zeta(3)}{2}+\pi^2\ln(2))+\epsilon^4(-24\text{Li}_4(\frac{1}{2})+\frac{19\pi^4}{30}-\ln^4(2))+ {\cal O}(\epsilon^{5}).
\eea
\subsection{Sector II : non-planar two-loop three-point integrals}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.47]{te3.eps}
\caption{The set of 8 master integrals that appear in sector II. Integrals $(C_1\ldots C_6)$ can be cast into canonical form, while integrals $(C_7,C_8)$ involve elliptic functions. The thin line denotes massless propagators and on-shell massless external particles; the thick line presents massive propagators and on-shell massive external particles; the dash line indicates off-shell external particles with momentum squared equal to $2 ss$. The internal lines with a dot mean the power of the propagators being raised to 2.}
\label{midiag2}
\end{center}
\end{figure}
In this subsection we consider the non-planar two-loop three-points integrals that appear in the massive light-by-light Feynman diagrams. There are eight master integrals, as shown in Fig. \ref{midiag2}, with the corresponding bases $B_i$ as
\bea
B_1 & = & \, \epsilon^3 \, (ss-2m_q^2)\, C_{1} \ , \nonumber\\
B_2 & = & \, \epsilon^2 \, \frac{-\sqrt{(ss-2m_q^2)(ss-4m_q^2)}}{4m_q^2}(2(ss-2m_q^2)C_{1}-4m_q^2 C_{2}+F_{6}/\epsilon^2) \ , \nonumber\\
B_3 & = & \, \epsilon^4 \, (ss-2m_q^2)\, C_{3} \ , \nonumber\\
B_4 & = & \, \epsilon^3 \, \sqrt{ss}\sqrt{ss-2m_q^2}(ss-2m_q^2)\, C_{4} \ , \nonumber\\
B_5 & = & \, \epsilon^3 \, ss(C_{5}-\frac{(ss-2m_q^2)}{2}C_{4}-\epsilon\, C_{3}-\frac{C_1}{2}) \ , \nonumber\\
B_6 & = & \, \epsilon^4 \, (ss-2m_q^2)\, C_{6} \ , \nonumber\\
B_7 & = & \, \epsilon^4 \, (ss-2m_q^2)^2\, C_{7} \ , \nonumber\\
B_8 & = & \, \epsilon^4 \, \frac{ss^2-4ss m_q^2+20m_q^4}{ss-2m_q^2}\, m_q^4 C_{8} \ .
\label{baseb}
\eea
Note, here the integrals ($C_1\ldots C_6$) were first calculated in Ref. \cite{Bonciani:2016qxi}, and the left two non-planar two-loop integrals $(B_7,B_8)$ cannot be cast into the canonical form via algebraic change of basis. A similar topology of Feynman diagram as that of ($C_7,C_8$), but with different kinematics and outgoing momentum squared, was handled in Ref. \cite{vonManteuffel:2017hms}.
In order to get expressions for $B_7$ and $B_8$ we first derive two coupled first-order differential equations with the evolution of variable $ss$, and then transform them to a second-order differential equation for $B_7$. That is:
\bea
& &\frac{\text{d}^2 B_7 }{d ss^2}-\frac{ss^2-4ss m_q^2-12m_q^4}{(ss-2m_q^2)(ss^2-4ss m_q^2+20m_q^4)} \frac{\text{d} B_7}{\text{d} ss}\nonumber\\
&&-\frac{16 m_q^4}{(ss-2m_q^2)^2(ss^2-4ss m_q^2+20m_q^4)}B_7=N(\epsilon,ss,m_q^2)\ ,
\eea
with $N(\epsilon,ss,m_q^2)$ denoting the non-homogeneous term. Here, the tough issue is how to determinate the homogeneous solution. To this aim, we make a variable transformation of $ss$ to $v=\frac{-i (ss-2 m_q^2)}{4 m_q^2}$, then the homogenous part of the differential equation turns to
\bea
\frac{\text{d}^2 B_7 }{d v^2}-\frac{1+v^2}{v(1-v^2)} \frac{\text{d} B_7}{\text{d} v}+\frac{1}{v^2(1-v^2)} B_7=0\ .
\label{simpeq}
\eea
The solutions of equation (\ref{simpeq}) can be readily obtained. The two homogeneous solutions $(y_1(v),y_2(v))$ read
\bea
y_1(v)=v K(v^2),~~~~y_2(v)=v K(1-v^2),
\eea
with $K(x)$ being the first kind complete elliptic integral. Note that the recently development on maximal-cut \cite{Harley:2017qut,Primo:2016ebd,Bosma:2017ens} is also applicable to the determination of the homogeneous solution. The Wronskian of the homogeneous solution reads
\bea
w(v)=y_2(v)\frac{\text{d} y_1(v)}{\text{d} v}-y_1(v)\frac{\text{d} y_2(v)}{\text{d} v}=\frac{v \pi}{2(1-v^2)}\ .
\eea
With the homogeneous solutions and Wronskian, a particular solution can be obtained by means of the constant variation. The general solution is then
\bea
B^i_7=c_1 y_1(v) + c_2 y_2(v)- y_1(v)\int_{0}^{v}\frac{N^i(\alpha)}{w(\alpha)}y_2(\alpha)\text{d}\alpha + y_2(v)\int_{0}^{v}\frac{N^i(\alpha)}{w(\alpha)}y_1(\alpha)\text{d}\alpha\ ,
\eea
where $i$ refers to the order of $\epsilon$ in $B_7$.
Since the integral $C_7$ has no singularity at $ss=2m_q^2$, and the normalization for $C_7$ in $B_7$ is $(ss-2m_q^2)^2$, we know
\bea
B_7\mid_{(v=0)}=0,~~~~\frac{\text{d} B_7}{\text{d} v}\mid_{(v=0)}=0\ .
\eea
Hence, the constants $c_1$ and $c_2$ can be fixed to
\bea
c_1=c_2=0\ .
\eea
Once $B_7$ is obtained, we can then determine the $B_8$ from the first order differential equation with respect to $B_7$ straightforwardly.
Before calculating the differential equations for integrals in this sector, still the corresponding boundary conditions should be fixed. Since the integrals $(B_1, B_2, B_3, B_4, B_6)$ are regular at $ss=2m_q^2$, by multiplying
their normalization factor $(ss-2m_q^2)$ to $B_i$, the corresponding bases $B_i$ then turn out to be zero at $ss=2m_q^2$. Considering that the master integrals in basis $B_5$ are regular as $ss=0$ and have a common normalization factor $ss$, we readily know $B_5=0$ when $ss=0$. With these discussions, all necessary boundary conditions to fix the solutions of differential equations are ready.
\subsection{Analytic continuation and discussions}
With the analytical results obtained in above, the next necessary step is to determinate the analytic continuation of the master integrals, which is similar to the procedure in our previous work \cite{Chen:2017xqd}. The correct analytic continuation can be achieved by the replacement of $ss \rightarrow ss + i 0$ at fixed $m_q^2$, which corresponds to $x\rightarrow x + i 0$, $y\rightarrow y + i 0$ and $z\rightarrow z + i 0$.
The canonical bases in (\ref{base}) contain 4 independent square roots
\be
(\sqrt{ss},\sqrt{ss-2m_q^2},\sqrt{ss+2m_q^2},\sqrt{ss+6m_q^2})\ ,
\ee
which cannot be simultaneously rationalized via one variable change. This means it is not possible to
integrate the differential equations directly in terms of Gongcharov polylogarithms. It is worth mentioning that Refs. \cite{Bonciani:2016qxi,vonManteuffel:2017myy} proposed some novel ways to express the results of canonical bases for non-elliptic sectors in terms of multiple polylogarithms, without considering the existence of rational parametrization of the alphabet. However the results tend to be rather lengthy when expressed in multiple polylogarithms. In order to calculate the integrals numerically in a faster and convenient way, we construct a one-fold integral representation for the integrals that can be cast into the canonical form by means of what proposed in Ref. \cite{Bonciani:2016qxi}. For integrals in elliptic sectors we need the two-fold integral representation to express the results up to weight four. The one fold and two fold integral representations we adopted are suitable for fast and precise numerical evaluation with Mathematica program on a single core computer.
The analytic calculation in this work is performed by our own developed Mathematica code, and in order to guarantee the correctness of our results, we ask all analytical expressions for master integrals experiencing at least one independent examination. We check all results in contrast to those obtained via numerical programs {\bf Fiesta} \cite{Smirnov:2013eza,Smirnov:2015mct} and {\bf SecDec} \cite{Borowka:2012yc,Borowka:2015mxa}. We have achieved an excellent agreement in analytical and numerical approaches with kinematics in both Euclidean and Minkowski regions.
\section{Conclusions and outlooks}
The integrals involving elliptic functions in the NNLO QCD corrections to heavy quarkonium exclusive production and decays are calculated, which turns out to be a tough issue. Those integrals are classified into two sectors, one with integrals containing sub-topologies related to the two-loop massive sunrise integrals and another with two massive two-loop non-planar three-points integrals. We find the simple example studied in Ref. \cite{Remiddi:2016gno} is in fact applicable to more general cases, that is, the expressions for two master integrals composed of two-loop massive sunrise integrals are still suitable for our case. In order to compute the first sector Feynman integrals under consideration we exploit the result for the two-loop massive sunrise integrals in Ref. \cite{Remiddi:2016gno}. We find a suitable linear combination of Feynman integrals such that only one of the master integrals about the solutions of two-loop massive sunrise integrals is required. By properly choosing canonical basis, we transform the differential equations into a simple and compact form that can be solved recursively. For another elliptic sector, the key point is to solve the homogeneous equation, with that inhomogeneous solutions can be obtained by means of constant variation.
Together with those 86 integrals calculated in our previous work \cite{Chen:2017xqd}, all master integrals appearing in the calculation of NNLO QCD correction to CP-even heavy quarkonium exclusive production and decays, such as $\gamma^*\gamma\rightarrow Q\bar{Q}$ and $e^+e^-\rightarrow \gamma+ Q\bar{Q}$ \cite{Chen:2017pyi}, are ready. The master integrals take the form of mutilple polylogarithms, iterative integrals over complete elliptic integrals and multiple polylogarithms. It is noteworthy that the integrals calculated in this work may also appear in the calculation of NNLO corrections in other processes, such as the exclusive decay of Higgs or $Z^0$ boson to quarkonium plus a photon and the inclusive hadronic production or decay of $\eta_c/\eta_b$, which are also phenomenologically meaningful. Moreover, we tend to believe that the calculation procedure and results in this work might be helpful to the mater integrals calculation of processes beyond the scope of heavy quarkonium physics, for instance the NNLO corrections to top quark pairs hadronic production, and NNLO corrections to heavy quark pair production plus a jet in electron-positron collision.
Note, only simple results are given in the appendix, however the full but lengthy results will be provided upon request.
\acknowledgments
This work was supported in part by the Ministry of Science and Technology of the People's Republic of China(2015CB856703); by the Strategic Priority Research Program of the Chinese Academy of Sciences, Grant No.XDB23030100; and by the National Natural Science Foundation of China(NSFC) under the grants 11375200 and 11635009. We are grateful to the anonymous referee for valuable comments and suggestions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 2,882
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Q: Trouble computing $\frac{dx}{1}=\frac{dy}{-3}=\frac{dz}{\sin(x)+\cos(y)}$? I am trying to understand this and this.
Suppose we have the following partial differential equation:
$$u_x-3u_y=\sin x + \cos y $$
I write the system of differential equations:
$$\frac{dx}{1}=\frac{dy}{-3}=\frac{dz}{\sin(x)+\cos(y)}$$
How do I apply that stuff in the text in this example? There is this example in the book but in this case, we were lucky things worked fairly easy and we could write $\frac{d(x+z)}{x+z}=\frac{dy}{y}$. I don't see how that could be done in this case.
A: You could also apply the proportionality trick:
$$
ds=\frac{dx}{1}=\frac{dy}{-3}
\implies ds = \frac{\sin x\,dx-\frac13\cos y\,dy}{\sin x + \cos y}
$$
so that with the third fraction immediately
$$
du =\sin x\,dx-\frac13\cos y\,dy
\\
u=-\cos x-\frac13\sin y+c_2
$$
follows
A: $$u_x-3u_y=\sin(x)+\cos(y)$$
The charpit-Lagrange characteristic ODEs are :
$$\frac{dx}{1}=\frac{dy}{-3}=\frac{du}{\sin(x)+\cos(y)}$$
A first characteristic equation comes from solving $\frac{dx}{1}=\frac{dy}{-3}$ :
$$y+3x=c_1$$
A second characteristic equation comes from solving $\frac{dx}{1}=\frac{du}{\sin(x)+\cos(y)}$ :
$$\frac{du}{dx}=\sin(x)+\cos(y)\implies u+\cos(x)-\int \cos(y)dx = c_2$$
$$u+\cos(x)-\int \cos(c_1-3x)dx = c_2$$
$$u+\cos(x)+\frac13\sin(c_1-3x) = c_2$$
The general solution of the PDE on implicit form $c_2=F(c_1)$ is :
$$u+\cos(x)+\frac13\sin\big((y+3x)-3x\big) = F(y+3x)$$
$F$ is an arbitrary function.
$$\boxed{u(x,y)=-\cos(x)-\frac13\sin(y) + F(y+3x)}$$
The PDE has an infinity of solutions since they are an infinity of arbitrary functions $F$ until no boundary condition is specified in the wording of the question.
The function $F$ might be determined if a valid boundary condition was given.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,522
|
package com.google.cloud.dns;
import com.google.api.client.googleapis.json.GoogleJsonError;
import com.google.cloud.BaseServiceException;
import com.google.cloud.RetryHelper.RetryHelperException;
import com.google.cloud.http.BaseHttpServiceException;
import com.google.common.collect.ImmutableSet;
import java.io.IOException;
import java.util.Set;
/**
* DNS service exception.
*/
public final class DnsException extends BaseHttpServiceException {
// see: https://cloud.google.com/dns/troubleshooting
private static final Set<Error> RETRYABLE_ERRORS = ImmutableSet.of(
new Error(429, null, true),
new Error(500, null, false),
new Error(502, null, false),
new Error(503, null, false),
new Error(null, "userRateLimitExceeded", true),
new Error(null, "rateLimitExceeded", true));
private static final long serialVersionUID = 490302380416260252L;
public DnsException(IOException exception, boolean idempotent) {
super(exception, idempotent, RETRYABLE_ERRORS);
}
public DnsException(GoogleJsonError error, boolean idempotent) {
super(error, idempotent, RETRYABLE_ERRORS);
}
public DnsException(int code, String message, Throwable cause) {
super(code, message, null, true, RETRYABLE_ERRORS, cause);
}
/**
* Translate RetryHelperException to the DnsException that caused the error. This method will
* always throw an exception.
*
* @throws DnsException when {@code ex} was caused by a {@code DnsException}
*/
static DnsException translateAndThrow(RetryHelperException ex) {
BaseServiceException.translate(ex);
throw new DnsException(UNKNOWN_CODE, ex.getMessage(), ex.getCause());
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,347
|
Exclusive AFONiE High Quality Sunglasses for Kids. Pack of 4, one of each color in a package. 100% UV Protection.
Measurements Approximate dimensions: 4.5" x 4.5" x 1.5"
Extremely cheap product. Doesn't appear the same in person as in the pictures.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,112
|
Masha Bresner
Masha Bresner focuses her practice on internal investigations, government enforcement actions, and other criminal and enforcement matters.
Prior to joining WilmerHale, Ms. Bresner was the director of operations for the New York office of a leading business intelligence firm where she led international investigations in support of litigation and arbitration. In that role, she conducted complex international asset-tracing exercises, identified ties between target individuals or companies and criminal enterprises—including international money laundering schemes, drug cartels and terrorist networks—and conducted due diligence to identify red flags related to, among other things, entities prior to acquisition and potential executive hires. Ms. Bresner has conducted a wide variety of investigations related to CIS countries, offshore jurisdictions and the United States.
Previously, Ms. Bresner was an associate for a global law firm in New York, a legal intern for the United Nations High Commissioner for Refugees in Geneva, Switzerland, and a bail intern for the Alaska Public Defender Agency in Anchorage, Alaska.
Read More About Masha
Foreign Corrupt Practices Act and Anti-Corruption
Named an "FCPA Powerhouse" and "Practice Group of the Year," we have been involved in some of the most notable public and non-public cases and key developments related to FCPA and anti-corruption issues.
Guiding clients through high-stakes and sensitive investigations.
Uncompromising defense of corporations and individuals in high-profile and complex litigation and investigations.
Selected for inclusion Best Lawyers: Ones to Watch in the area of white-collar criminal defense (2022–2023)
View all of Masha's insights & news
cum laude Geneva International Fellow, Contributing Editor, Michigan Journal of International Law
BA, Political Sciences and Middle Eastern Studies, McGill University, 2010
Joint Honors, First Class Honors Dean's List
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,195
|
Atrani Map
Explore the smallest village in Italy just steps from the bustling town of Amalfi.
Atrani town
Amalfi's twin appears like an enchanting Italian nativity scene made up of a maze of little alleyways, narrow flights of steps, and houses clinging to the rocks directly over the Mediterranean Sea.
A fishing village, the history of which is inextricably linked to that of its illustrious neighbor, Amalfi, and which has conserved its ancient medieval structure, comprised of a cascade of houses interspersed with steep flights of steps, covered passageways, and tiny gardens, all incredibly intact. Clinging to the cliff face, directly above the sea, at the mouth of the Valle del Dragone, Atrani is the smallest town in the whole of Southern Italy.
The Dutch artist Escher came to the Amalfi Coast in 1923 and immediately fell head over heels in love with Atrani, immortalizing the game of light and shadow played amidst its narrow alleyways in a great number of his artworks.
How to Get to Atrani
The tiny village of Atrani is attached to the bustling center of Amalfi, just a kilometer further along the coast of Campania. You can reach Atrani via the A3 highway Napoli-Salerno, exiting at Vietri sul Mare. From Vietri, continue along the SS163, the famous coastal highway that runs the length of the Amalfi Coast, for about 20 kilometers until your reach Atrani. The nearest airport is Capodichino in Naples. Atrani can also be reached via the SITA bus from Salerno or Sorrento, running to Amalfi.
What to See in Atrani
Today, the famed Amalfitana" coast road has formed a barrier between the town and the sea, and those wishing to reach the water's edge have to use one of the passageways such as that in Piazza Umberto I, originally constructed as a shelter for boats during high tide and which now appears like the backstage of a theater facing the sea. The Church of San Salvatore de' Birecto, where the coronation of the Doges of the Republic of Amalfi once took place, is located here.
Another place to visit, if only for its incredibly panoramic position, is the sparkling majolica-domed Collegiate Church of Santa Maria Maddalena which dominates the entire eastern side of the town with its baroque-style facade and soaring bell tower.
Close to the church is the Grotta di Masaniello. According to the local legend, the revolutionist sought refuge in this cave, which is near to his maternal home. Enjoying a fabulous view over the whole of Atrani, the 11th-century Church of Santa Maria del Bando, set against the rocks beneath the Torre dello Zito, is particularly enchanting.
Traditional festivals in Atrani:
Santa Maddalena (Mary Magdalene), patron saint of Atrani (22 July)
Reenactment of the Ducal wedding (April)
Sagra del pesce azzurro (August)
What to Eat in Atrani
Fresh fish is a major protagonist in almost all of Atrani's gastronomic delicacies, though there are dishes which are not fish-based like sarchiapone, a marrow stuffed with meat and cheese cooked in tomato sauce, the custard-and-cherry-filled pasticciotto, and the wickedly sweet cassata atranese. Ristorante Savò in the town's tiny main square offers gourmet dishes that are striking both to the eye and palate. For simpler recipes, A' Paranza is run by a family of former fishermen.
Other tasty treats:
Scialatielli alle vongole veraci
Involtini di mozzarella in foglia di limone
Delizia al limoncello
Other Sights
The entire coast is named for this seaside jewel, land of maritime traditions and tourism.
The most beautiful view on the entire Amalfi Coast, and a treasure trove of art and culture.
Village with a wealth of craft traditions, characteristic architecture, and a small beach.
The largest beach of the Amalfi Coast and centuries of art works.
An enchanting coastal village, much loved by in-the-know visitors.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 47
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{"url":"https:\/\/elteoremadecuales.com\/hurwitzs-theorem-composition-algebras\/","text":"# Hurwitz's theorem (composition algebras)\n\nThe theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields.[1] Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also in Radon (1922). Subsequent proofs of the restrictions on the dimension have been given by Eckmann (1943) using the representation theory of finite groups and by Lee (1948) and Chevalley (1954) using Clifford algebras. Hurwitz's theorem has been applied in algebraic topology to problems on vector fields on spheres and the homotopy groups of the classical groups[2] and in quantum mechanics to the classification of simple Jordan algebras.[3] Contents 1 Euclidean Hurwitz algebras 1.1 Definition 1.2 Classification 2 Other proofs 3 Applications to Jordan algebras 4 See also 5 Notes 6 References 7 Further reading Euclidean Hurwitz algebras Definition A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra A with identity endowed with a nondegenerate quadratic form q such that q(a\u2009b) = q(a)\u2009q(b). If the underlying coefficient field is the reals and q is positive-definite, so that (a,\u2009b) = 1 \/ 2 [q(a + b) \u2212 q(a) \u2212 q(b)] is an inner product, then A is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.[4] If A is a Euclidean Hurwitz algebra and a is in A, define the involution and right and left multiplication operators by {displaystyle a^{*}=-a+2(a,1)1,,,,L(a)b=ab,,,,R(a)b=ba.} Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties: the involution is an antiautomorphism, i.e. (a\u2009b)* = b*\u2009a* a\u2009a* = \u2016\u2009a\u2009\u20162\u20091 = a*\u2009a L(a*) = L(a)*, R(a*) = R(a)*, so that the involution on the algebra corresponds to taking adjoints Re (a\u2009b) = Re (b\u2009a) if Re\u200ax = (x + x*)\/2 = (x,\u20091)1 Re (a\u2009b)\u2009c = Re\u200aa(b\u2009c) L(a2) = L(a)2, R(a2) = R(a)2, so that A is an alternative algebra.\n\nThese properties are proved starting from the polarized version of the identity (a\u2009b,\u2009a\u2009b) = (a,\u2009a)(b,\u2009b): {displaystyle displaystyle {2(a,b)(c,d)=(ac,bd)+(ad,bc).}} Setting b = 1 or d = 1 yields L(a*) = L(a)* and R(c*) = R(c)*.\n\nHence Re(a\u2009b) = (a\u2009b,\u20091)1 = (a,\u2009b*)1 = (b\u2009a,\u20091)1 = Re(b\u2009a).\n\nSimilarly Re (a\u2009b)c = ((a\u2009b)c,1)1 = (a\u2009b,\u2009c*)1 = (b,\u2009a*\u2009c*)1 = (bc,a*)1 = (a(bc),1)1 = Re a(b\u2009c).\n\nHence ((ab)*,c) = (ab,c*) = (b,a*c*) = (1,b*(a*c*)) = (1,(b*a*)c*) = (b*a*,c), so that (ab)* = b*a*.\n\nBy the polarized identity \u2016\u2009a\u2009\u20162\u2009(c,\u2009d) = (a\u2009c,\u2009a\u2009d) = (a*\u2009(a\u2009c),\u2009d) so L(a*)\u2009L(a) = L(\u2016\u2009a\u2009\u20162). Applied to 1 this gives a*\u2009a = \u2016\u2009a\u2009\u20162 1. Replacing a by a* gives the other identity.\n\nSubstituting the formula for a* in L(a*)\u2009L(a) = L(a*\u2009a) gives L(a)2 = L(a2). The formula R(a2) = R(a)2 is proved analogically.\n\nClassification It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions R \u2282 C \u2282 H.\n\nAnalysing such an inclusion leads to the Cayley\u2013Dickson construction, formalized by A.A. Albert. Let A be a Euclidean Hurwitz algebra and B a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vector j in A orthogonal to B. Since (j,\u20091) = 0, it follows that j* = \u2212j and hence j2 = \u22121. Let C be subalgebra generated by B and j. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley\u2013Dickson multiplication laws: {displaystyle displaystyle {C=Boplus Bj,,,,(a+bj)^{*}=a^{*}-bj,,,,(a+bj)(c+dj)=(ac-d^{*}b)+(bc^{*}+da)j.}} B and B\u2009j are orthogonal, since j is orthogonal to B. If a is in B, then j\u2009a = a*\u2009j, since by orthogonal 0 = 2 (j,\u2009a*) = j\u2009a \u2212 a*\u2009j. The formula for the involution follows. To show that B \u2295 B\u2009j is closed under multiplication Bj = j\u2009B. Since B\u2009j is orthogonal to 1, (b\u2009j)* = \u2212b\u2009j.\n\nb(c\u2009j) = (c\u2009b)j since (b,\u2009j) = 0 so that, for x in A, (b(c\u2009j),\u2009x) = (b(j\u2009x),\u2009j(c\u2009j)) = \u2212(b(j\u2009x),\u2009c*) = \u2212(c\u2009b,\u2009(j\u2009x)*) = \u2212((c\u2009b)j,\u2009x*) = ((c\u2009b)j,\u2009x). (j\u2009c)b = j(b\u2009c) taking adjoints above. (b\u2009j)(c\u2009j) = \u2212c*\u2009b since (b,\u2009c\u2009j) = 0, so that, for x in A, ((b\u2009j)(c\u2009j),\u2009x) = \u2212((c\u2009j)x*,\u2009b\u2009j) = (b\u2009x*,\u2009(c\u2009j)j) = \u2212(c*\u2009b,\u2009x).\n\nImposing the multiplicativity of the norm on C for a + b\u2009j and c + d\u2009j gives: {displaystyle displaystyle {(|a|^{2}+|b|^{2})(|c|^{2}+|d|^{2})=|ac-d^{*}b|^{2}+|bc^{*}+da|^{2},}} which leads to {displaystyle displaystyle {(ac,d^{*}b)=(bc^{*},da).}} Hence d(a\u2009c) = (d\u2009a)c, so that B must be associative.\n\nThis analysis applies to the inclusion of R in C and C in H. Taking O = H \u2295 H with the product and inner product above gives a noncommutative nonassociative algebra generated by J = (0,\u20091). This recovers the usual definition of the octonions or Cayley numbers. If A is a Euclidean algebra, it must contain R. If it is strictly larger than R, the argument above shows that it contains C. If it is larger than C, it contains H. If it is larger still, it must contain O. But there the process must stop, because O is not associative. In fact H is not commutative and a(b\u2009j) = (b\u2009a) j \u2260 (a\u2009b)j in O.[5] Theorem. The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.\n\nOther proofs The proofs of Lee (1948) and Chevalley (1954) use Clifford algebras to show that the dimension N of A must be 1, 2, 4 or 8. In fact the operators L(a) with (a,\u20091) = 0 satisfy L(a)2 = \u2212\u2016\u2009a\u2009\u20162 and so form a real Clifford algebra. If a is a unit vector, then L(a) is skew-adjoint with square \u2212I. So N must be either even or 1 (in which case A contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of A, an N-dimensional complex space. If N is even, N \u2212 1 is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension 2N\/2 \u2212 1. So this power of 2 must divide N. It is easy to see that this implies N can only be 1, 2, 4 or 8.\n\nThe proof of Eckmann (1954) uses the representation theory of finite groups, or the projective representation theory of elementary Abelian 2-groups, known to be equivalent to the representation theory of real Clifford algebras. Indeed, taking an orthonormal basis ei of the orthogonal complement of 1 gives rise to operators Ui = L(ei) satisfying {displaystyle displaystyle {U_{i}^{2}=-I,,,,U_{i}U_{j}=-U_{j}U_{i},,(ineq j).}} This is a projective representation of a direct product of N \u2212 1 groups of order 2. (N is assumed to be greater than 1.) The operators Ui by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in Hurwitz (1923).[6] Assume that there is a composition law for two forms {displaystyle displaystyle {(x_{1}^{2}+cdots +x_{N}^{2})(y_{1}^{2}+cdots +y_{N}^{2})=z_{1}^{2}+cdots +z_{N}^{2},}} where zi is bilinear in x and y. Thus {displaystyle displaystyle {z_{i}=sum _{j=1}^{N}a_{ij}(x)y_{j}}} where the matrix T(x) = (aij) is linear in x. The relations above are equivalent to {displaystyle displaystyle {T(x)T(x)^{t}=x_{1}^{2}+cdots +x_{N}^{2}.}} Writing {displaystyle displaystyle {T(x)=T_{1}x_{1}+cdots +T_{N}x_{N},}} the relations become {displaystyle displaystyle {T_{i}T_{j}^{t}+T_{j}T_{i}^{t}=2delta _{ij}I.}} Now set Vi = (TN)t Ti. Thus VN = I and the V1,\u2009...\u2009,\u2009VN \u2212 1 are skew-adjoint, orthogonal satisfying exactly the same relations as the Ui's: {displaystyle displaystyle {V_{i}^{2}=-I,,,,V_{i}V_{j}=-V_{j}V_{i},,(ineq j).}} Since Vi is an orthogonal matrix with square \u2212I on a real vector space, N is even.\n\nLet G be the finite group generated by elements vi such that {displaystyle displaystyle {v_{i}^{2}=varepsilon ,,,,v_{i}v_{j}=varepsilon v_{j}v_{i},,(ineq j),}} where \u03b5 is central of order 2. The commutator subgroup [G,\u2009G] is just formed of 1 and \u03b5. If N is odd this coincides with the center while if N is even the center has order 4 with extra elements \u03b3 = v1 ... vN \u2212 1 and \u03b5\u2009\u03b3. If g in G is not in the center its conjugacy class is exactly g and \u03b5\u2009g. Thus there are 2N \u2212 1 + 1 conjugacy classes for N odd and 2N \u2212 1 + 2 for N even. G has |\u2009G \/ [G,\u2009G]\u2009| = 2N \u2212 1 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since N is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals |\u2009G\u2009| and the dimensions divide |\u2009G\u2009|, the two irreducibles must have dimension 2(N \u2212 2)\/2. When N is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension 2(N \u2212 2)\/2. The space on which the Vi's act can be complexified. It will have complex dimension N. It breaks up into some of complex irreducible representations of G, all having dimension 2(N \u2212 2)\/2. In particular this dimension is \u2264 N, so N is less than or equal to 8. If N = 6, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 or 8.\n\nApplications to Jordan algebras Let A be a Euclidean Hurwitz algebra and let Mn(A) be the algebra of n-by-n matrices over A. It is a unital nonassociative algebra with an involution given by {displaystyle displaystyle {(x_{ij})^{*}=(x_{ji}^{*}).}} The trace Tr(X) is defined as the sum of the diagonal elements of X and the real-valued trace by TrR(X) = Re\u2009Tr(X). The real-valued trace satisfies: {displaystyle operatorname {Tr} _{mathbf {R} }XY=operatorname {Tr} _{mathbf {R} }YX,qquad operatorname {Tr} _{mathbf {R} }(XY)Z=operatorname {Tr} _{mathbf {R} }X(YZ).} These are immediate consequences of the known identities for n = 1.\n\nIn A define the associator by {displaystyle displaystyle {[a,b,c]=a(bc)-(ab)c.}} It is trilinear and vanishes identically if A is associative. Since A is an alternative algebra [a,\u2009a,\u2009b] = 0 and [b,\u2009a,\u2009a] = 0. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if a, b or c lie in R then [a,\u2009b,\u2009c] = 0. These facts imply that M3(A) has certain commutation properties. In fact if X is a matrix in M3(A) with real entries on the diagonal then {displaystyle displaystyle {[X,X^{2}]=aI,}} with a in A. In fact if Y = [X,\u2009X2], then {displaystyle displaystyle {y_{ij}=sum _{k,ell }[x_{ik},x_{kell },x_{ell j}].}} Since the diagonal entries of X are real, the off diagonal entries of Y vanish. Each diagonal entry of Y is a sum of two associators involving only off diagonal terms of X. Since the associators are invariant under cyclic permutations, the diagonal entries of Y are all equal.\n\nLet Hn(A) be the space of self-adjoint elements in Mn(A) with product X\u2218Y = 1 \/ 2 (X\u2009Y + Y\u2009X) and inner product (X,\u2009Y) = TrR(X\u2009Y).\n\nTheorem. Hn(A) is a Euclidean Jordan algebra if A is associative (the real numbers, complex numbers or quaternions) and n \u2265 3 or if A is nonassociative (the octonions) and n = 3.\n\nThe exceptional Jordan algebra H3(O) is called the Albert algebra after A.A. Albert.\n\nTo check that Hn(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with (X,\u2009X) = \u03a3 \u2016\u2009xij\u2009\u20162. So it is an inner product. It satisfies the associativity property (Z\u2218X,\u2009Y) = (X,\u2009Z\u2218Y) because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators L(X) defined by L(X)Y = X\u2218Y: {displaystyle displaystyle {[L(X),L(X^{2})]=0.}} This is easy to check when A is associative, since Mn(A) is an associative algebra so a Jordan algebra with X\u2218Y = 1 \/ 2 (X\u2009Y + Y\u2009X). When A = O and n = 3 a special argument is required, one of the shortest being due to Freudenthal (1951).[7] In fact if T is in H3(O) with Tr\u200aT = 0, then {displaystyle displaystyle {D(X)=TX-XT}} defines a skew-adjoint derivation of H3(O). Indeed, {displaystyle operatorname {Tr} (T(X(X^{2}))-T(X^{2}(X)))=operatorname {Tr} T(aI)=operatorname {Tr} (T)a=0,} so that {displaystyle (D(X),X^{2})=0.} Polarizing yields: {displaystyle (D(X),Ycirc Z)+(D(Y),Zcirc X)+(D(Z),Xcirc Y)=0.} Setting Z = 1, shows that D is skew-adjoint. The derivation property D(X\u2218Y) = D(X)\u2218Y + X\u2218D(Y) follows by this and the associativity property of the inner product in the identity above.\n\nWith A and n as in the statement of the theorem, let K be the group of automorphisms of E = Hn(A) leaving invariant the inner product. It is a closed subgroup of O(E) so a compact Lie group. Its Lie algebra consists of skew-adjoint derivations. Freudenthal (1951) showed that given X in E there is an automorphism k in K such that k(X) is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Mn(A) for any non-associative algebra A.\n\nTo prove the diagonalization theorem, take X in E. By compactness k can be chosen in K minimizing the sums of the squares of the norms of the off-diagonal terms of k(X). Since K preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of k(X). Replacing X by k\u2009X, it can be assumed that the maximum is attained at X. Since the symmetric group Sn, acting by permuting the coordinates, lies in K, if X is not diagonal, it can be supposed that x12 and its adjoint x21 are non-zero. Let T be the skew-adjoint matrix with (2,\u20091) entry a, (1,\u20092) entry \u2212a* and 0 elsewhere and let D be the derivation ad T of E. Let kt = exp\u2009tD in K. Then only the first two diagonal entries in X(t) = ktX differ from those of X. The diagonal entries are real. The derivative of x11(t) at t = 0 is the (1,\u20091) coordinate of [T,\u2009X], i.e. a*\u2009x21 + x12\u2009a = 2(x21,\u2009a). This derivative is non-zero if a = x21. On the other hand, the group kt preserves the real-valued trace. Since it can only change x11 and x22, it preserves their sum. However, on the line x + y =constant, x2 + y2 has no local maximum (only a global minimum), a contradiction. 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\section{Introduction}
\label{sec:intro}
For the past several decades, direct dark matter (DM) detection experiments have attempted to detect the energy imparted to nuclei in underground laboratories by collisions with DM particles that are gravitationally bound to the Galactic halo. In particular, direct detection experiments are ideal for probing DM particle candidates with weak-scale interactions and masses in the range of $\sim 1-10^4$ GeV, which interact coherently with nuclei, referred to as weakly interacting massive particles (WIMPs). These experiments have made tremendous strides over this time period, increasing both in size and detection efficiency.
The next multi-ton scale generation of direct DM detection experiments are currently being planned.
Thus, it is worthwhile to consider what is the ultimate capabilities and reach of such experiments.
It is well-known that direct detection experiments will soon encounter an irreducible background due to coherent nuclear interactions of neutrinos (e.g. \cite{Cabrera:1984rr, Monroe:2007xp, Strigari:2009bq, Billard:2013qya}). At lower recoil energies, near the detection threshold, the background is predominantly due to solar neutrinos. At larger nuclear recoil energies it comes from the diffuse supernovae neutrino background and atmospheric neutrinos. It will be difficult to claim a discovery of DM if the signal lies below the neutrino background. Early studies of this so-called ``neutrino floor" concentrated on the usual elastic DM spin-independent (SI) and spin-dependent (SD) interactions and showed that the degeneracy between recoil spectra due to coherent neutrino and WIMP scattering for particular DM masses would significantly limit the ultimate sensitivity of direct detection experiments~ \cite{Billard:2013qya,Ruppin:2014bra}. For the usual SI or SD interactions, solar neutrinos mimic a WIMP signal for a DM mass close to 6 GeV. On the other hand, atmospheric neutrinos mimic a WIMP signal for a DM mass of about 100 GeV. It has been shown (e.g. \cite{Ruppin:2014bra}) that combining data from different target materials can enhance the subtraction of the neutrino background in direct detection experiments for DM particles with SD interaction and masses below 10 GeV. However, target complementarity will not alleviate the problem for SI interactions, since the cross-sections of both DM and neutrinos have the same scaling with the nuclear mass number (e.g. \cite{Ruppin:2014bra}). Proposals to distinguish DM and neutrino signals include searching for an annual modulation~\cite{Davis:2014ama, OHare:2016pjy} or measuring the resulting recoil momentum~\cite{Grothaus:2014hja, OHare:2015utx} in directional direct detection experiments~\cite{Mayet:2016zxu}. However, both of these could be very challenging for a DM signal below the neutrino background.
In this paper we present for the first time a look at the neutrino floor for a large number of viable DM-nucleus interactions beyond the standard SI and SD, defined by a fully relativistic Lagrangian formalism, for several target nuclei.
Recent studies of the neutrino floor have used non-relativistic effective field theory (EFT) DM-nucleus couplings~\cite{Dent:2016iht, Dent:2016wor, Dutta:2017nht} to show that the degeneracy between neutrino and DM recoil spectra may not be present for less conventional interactions. Ref.~\cite{Dent:2016iht} determined that for 11 out of the 14 possible non-relativistic EFT operators considered, the predicted recoil spectra can be cleanly distinguished from the corresponding neutrino-induced recoil spectra with moderate size detectors (exposure of few ton$\cdot$years), for low mass WIMPs (i.e. with masses $\lesssim 10$ GeV).
Non-relativistic EFT provides a theoretical framework for determining different nuclear responses to DM scattering events, thus yielding some insight into different viable DM couplings. However, most of the DM-nuclei interactions defined in terms of a field theoretical Lagrangian formalism involve complex linear combinations of EFT operators, with the relative importance of each EFT operator weighted by nuclide-specific factors. Thus, for a particular interaction, different experiments may expect a varying degree of degeneracy of the DM and neutrino recoil spectra. Here we consider interactions, expressed in a Lagrangian formalism, due to the exchange of a single mediator that can be heavy or light with respect to the typical momentum being exchanged. Let us remark that although this is the right formalism to describe realistic interactions, it is still limited with respect to a possible complete theory of the dark and visible sectors. In a complete model of DM interactions, scattering amplitude terms arising from various mediators would be summed to obtain the total amplitude. They could thus interfere with each other, and different terms could dominate the cross section for different energy regimes.
Some of DM interactions produce recoil spectra that appear nearly degenerate with neutrinos in particular target elements, but not in others (e.g. \cite{Ruppin:2014bra}). For example, the absence of spin in argon results in a high level of degeneracy between the SI and anapole interactions (see \Eq{eq:diffxsecsi} and \Eq{eq:anapolediff} below), which is not necessarily present in other target elements. Hence, complementarity of a variety of target nuclei needs to be considered in order to maximize the detection sensitivity to DM in future experiments.
The main questions we are going to address in this paper are the following: \emph{(i)} beyond the conventional SI and SD interactions, what types of DM candidates should one expect to give rise to a high level of degeneracy with the neutrino background, and for which elements can such a degeneracy be evaded, \emph{(ii)} for which combinations of elements and interactions can one exploit target complementarity to help ameliorate the neutrino background, \emph{(iii)} how much exposure is required to distinguish a particular DM candidate below the neutrino background, and \emph{(iv)} what is required for experiments to maintain a broad sensitivity to the largest possible number of DM interactions.
Recent detection of coherent elastic neutrino-nucleus scattering by the COHERENT experiment \cite{Akimov:2017ade} further strengthens the case for exploiting the sensitivity of next-generation direct detection experiments also for exploring neutrino physics. Coherent neutrino-nucleus and elastic electron-neutrino scatterings allow one to study a range of physics topics, including sterile neutrinos \cite{Pospelov:2011ha,Billard:2014yka}, non-standard neutrino interactions (NSI) (e.g. \cite{Harnik:2012ni,Dutta:2017nht}) and supernovae \cite{Chakraborty:2013zua}. While we do not investigate physics related specifically to neutrinos in this work, we stress the richness of research opportunities available to future large direct detection experiments in this context.
The format of this paper is as follows. In \Sec{sec:darkmatter} we review the fundamentals of direct DM detection and discuss the DM-nucleus interactions which can result from the exchange of a single mediator in a Lagrangian formalism. Both elastic as well as inelastic scatterings are discussed. In \Sec{sec:neutrinos} we introduce the fluxes and recoil spectra arising from neutrinos in direct detection experiments. \Sec{sec:detection} introduces our statistical analysis and presents the discovery limits for the various DM interactions considered. We conclude in \Sec{sec:summary}.
\section{Dark Matter Signal \label{sec:darkmatter}}
\subsection{Flux and recoil spectrum}
Direct DM detection experiments attempt to observe the recoils of nuclei due to collisions with DM particles gravitationally bound to the halo of the Milky Way.
The differential scattering rate per unit time and target mass of a WIMP $\chi$, assumed to account for the bulk of the DM, off a target nuclide $T$ as a function of nuclear recoil energy $E_{\rm R}$ is given by
\begin{equation}\label{eq:dm_diff_rate}
\frac{d R_T}{d E_{\rm R}} = \frac{\rho_\chi}{m_\chi}\frac{C_T}{m_T}\int_{v \geqslant v_{\rm min}(E_{\rm R})} \, d^3 \, v \, f(\vec{v},t) \, v \, \frac{d \sigma_T}{d E_{\rm R}}(E_{\rm R}, \vec{v}) \, ,
\end{equation}
where $\rho_\chi$ is the local DM density, $m_\chi$ is the DM particle mass, $m_T$ and $C_T$ are the mass and mass fraction of $T$ in the detector, $f(\vec{v}, t)$ is the distribution of the DM velocity $\vec{v}$ in the Earth's rest frame, and $d \sigma_T/d E_{\rm R}$ is the DM-nucleus differential cross section. When multiple nuclides are present in the detector, \Eq{eq:dm_diff_rate} is summed over the element and/or isotopic composition, taking into account the respective element-dependence of the differential cross-section including the nuclear form factors, to obtain the total DM rate
\begin{equation} \label{eq:totdmrate}
\frac{d R_{\rm DM}}{d E_{\rm R}} = \sum_T \frac{d R_T}{d E_{\rm R}}~.
\end{equation}
The velocity integral in \Eq{eq:dm_diff_rate} is restricted to speeds larger than $v_{\rm min}$, the minimum DM particle speed required to impart the particular recoil energy to the target nucleus. For elastic scattering it is given by
\begin{equation}\label{eq:vmin_el}
v_{\rm min} = \dfrac{|\vec{q}|}{2 \mu_T} = \sqrt{\frac{m_T E_{\rm R}}{2 \mu_T^2}} ~,
\end{equation}
where $\vec{q}$ is momentum transfer and $\mu_T = m_{\chi} m_T/ (m_{\chi} + m_T)$ is the reduced DM-nucleus mass.
We assume that the local characteristics of the DM halo of the Galaxy are described by the Standard Halo Model (SHM), with the local DM density $\rho_{\chi} = 0.3$ GeV/cm$^3$ and the distribution of the WIMP velocity $\vec{u}$ in the Galactic frame given by a truncated Maxwell-Boltzmann distribution
\begin{equation}
f_{\rm G}(\vec{u}) = \dfrac{1}{N_{\rm esc} (v_0 \sqrt{\pi})^3} \text{exp}(-u^2/v_0^2) \theta(v_{\rm esc} - u)~.
\end{equation}
Here, $v_0$ is the velocity dispersion taken to be the speed of the Local Standard of Rest $v_0 = 220$ km/s. The Galactic escape velocity $v_{\rm esc}$ is taken to be $v_{\rm esc} = 533$ km/s, following measurements of \cite{Piffl:2013mla}. The normalization factor
\begin{equation}
N_{\rm esc} = \text{erf}(v_{\rm esc}/v_0) - 2(v_{\rm esc}/v_0)\text{exp}(-v_{\rm esc}^2/v_0^2)/\sqrt{\pi}~,
\end{equation}
by construction gives $\int d^3 u f_{\rm G} (\vec{u}) = 1$.
In the Earth's frame, neglecting the gravitational DM lensing by the Sun, the velocity $f(\vec{v},t)$ of \Eq{eq:totdmrate} is obtained from a Galilean transformation
\begin{equation} \label{eq:dmrelvel}
f(\vec{v},t) = f_{\rm G}(\vec{v}_{\odot} + \vec{v}_{\oplus}(t) + \vec{v})~,
\end{equation}
where $\vec{v}_{\odot} = 232$ km/s and $\vec{v}_{\oplus} = 30$ km/s denote the velocity of the Sun with respect to the Galaxy and the time-dependent velocity of Earth with respect to the Sun, taken in an orbit inclined at $60^{\circ}$ with respect to the Galactic plane, respectively \cite{Schoenrich:2009bx}.
Since we do not consider here the time variation of the expected DM signal we take $\vec{v}_{\oplus} = 0$. We do not expect that alternative realistic halo models would significantly alter the results of the analysis presented here~\cite{Morgan:2004ys,Billard:2011zj}.
In practice, the differential rate in \Eq{eq:dm_diff_rate} must also be convolved with the efficiency and energy resolution of the particular experiment to obtain the observable recoil spectrum. Throughout this paper we optimistically assume that future direct detection experiments have perfect resolution and efficiency.
For realistic astrophysical distributions of DM, the rotation of the Earth around the Sun produces an annual modulation in the scattering rate, following \Eq{eq:dmrelvel} and \Eq{eq:totdmrate}. The recoil spectrum arising from neutrinos may also modulate, e.g. in the case of solar neutrinos with the proximity of the Sun. The modulations may enhance the ability of direct detection experiments to differentiate neutrinos from a DM candidate as phases and amplitudes of the modulation will differ. A recent analysis investigating DM-nucleus interactions (without including a neutrino signal) showed that using the annual modulation as few as $\mathcal{O}(500)$ events may be sufficient to significantly increase model identification~\cite{Witte:2016ydc}. This effect may be enhanced for a DM differential cross section containing a non-standard dependence on the DM velocity (e.g. due to a magnetic dipole coupling). In such scenario, the properties of the annual modulation (e.g. the phases and amplitudes) are nuclide-dependent~\cite{DelNobile:2015tza,DelNobile:2015rmp}. This may help distinguishing DM and neutrino signals, since the modulation from neutrinos is target-universal.
In this work we do not consider time-dependence of the signal. However, a combined analysis using both time and recoil energies of observed events will likely show that the discriminatory power obtained in the following sections to be conservative.
\subsection{Interactions}
Conventional analyses of direct detection data assume WIMP-nucleus scattering proceeds through an SI or SD interaction. These are the types of interactions that arise from the exchange of heavy scalar and vector (in the case of a SI interaction), or axial-vector (in the case of a SD interaction), mediators in the non-relativistic limit to zeroth order in $|\vec{q}|$ and $v$. These interactions, however, hardly constitute an exhaustive list of the viable DM-nucleus interactions, and fail to characterize the richer phenomenology that may arise in direct detection experiments.
Recently, an effective field theory (EFT) analysis for direct detection experiments has been developed~\cite{Fan:2010gt,Fitzpatrick:2012ix}. The EFT approach provides an exhaustive list of non-relativistic interactions that are capable of characterizing the various types of DM signals that could appear in direct detection experiments. This is a valuable study that may provide additional insight that could otherwise be lost in conventional analyses, however, viable UV-complete models often produce non-trivial linear combinations of such operators. While this work does not focus on the neutrino floor within the context of UV complete models, we will consider DM-nuclei interactions that have been shown to arise from UV-complete models. Within such models, the coefficients determining the relative importance of each EFT term in the interaction often depends on the target material. Hence, well-motivated signatures are likely to be overlooked in a conventional EFT analysis, in particular when signals are detected in multiple experiments employing different target elements.
In this section, we will motivate a large list of viable DM-nucleus interactions that capture complex phenomenology that may arise in these experiments, but are also motivated with some UV completion.
We first provide a brief overview of the EFT approach for direct detection, as we use tools developed for EFT analyses to streamline the calculation of the differential cross sections studied in this paper.
\subsubsection{Effective field theory approach}
In the EFT method, one starts by specifying the DM couplings to nucleons and then computes the corresponding interactions with nuclei.
The non-relativistic nature of DM-nucleon interactions in direct detection experiments allows for interactions to be decomposed into a non-relativistic EFT basis. A complete set of Galilean-invariant Hermitian operators can be constructed from the following dimensionless three-vectors \cite{Fitzpatrick:2012ix}: the momentum transfer divided by the nucleon mass $\vec{q}/m_N$, the relative DM-nucleon velocity in the direction perpendicular to the momentum exchange $\vec{v}^\perp \equiv \vec{v} - \vec{q}/2\mu_N$, where $\mu_N = m_{\chi} m_N/ (m_{\chi} + m_N) $ is the DM-nucleon reduced mass, as well as the DM $\vec{S}_\chi$ and the nucleon $\vec{S}_N$ spins. There exist a total of 15 Hermitian operators that can be constructed from these variables, enumerated in \cite{Fitzpatrick:2012ix,Anand:2013yka}.
For any given interaction, such as those listed in the second column of \Tab{tab:fermop}, one can take the non-relativistic limit and write the interaction in terms of the EFT operators. Here, we present a brief example, outlined in \cite{Anand:2013yka}, of an axial vector mediated contact interaction (see SD in \Tab{tab:fermop} and also assuming a heavy mediator) given by
\begin{equation}\label{eq:egoper}
\mathcal{A}_{\rm int} = c \bar{\chi}\gamma^\mu\gamma^5\chi\bar{N}\gamma_\mu\gamma^5N ~,
\end{equation}
where $\chi$ and $N$ are the DM and nucleon spinors, respectively. Using the Bjorken and Drell gamma matrix conventions with spinor normalization, one can write the non-relativistic limit of a spinor as
\begin{equation}\label{eq:nonrel}
\chi(p) = \sqrt{\dfrac{E+m_{\chi}}{2m_{\chi}}}\left(
\begin{array}{c}
\xi \\
\dfrac{\vec{\sigma}\cdot \vec{p}}{E+m_{\chi}}\xi
\end{array}\right) \rightarrow
\left(
\begin{array}{c}
\xi \\
\dfrac{\vec{\sigma}\cdot \vec{p}}{2m_{\chi}}\xi
\end{array}\right) \, ,
\end{equation}
where $\xi$ is a two component spinor (and likewise for the nucleon spinor $N$). Using Eq.~\eqref{eq:nonrel} with the interaction provided in Eq.~\eqref{eq:egoper}, and recalling that the $\vec{S} = \xi^{\prime\dagger}(\sigma/2)\xi$, one can rewrite the interaction as
\begin{equation}
\mathcal{A}_{\rm int} = -4c \, \vec{S}_\chi \cdot \vec{S}_N \, \equiv -4c \,\mathcal{O}_4 \, ,
\end{equation}
where the operator $\mathcal{O}_4$ follows the conventions of ~\cite{Fitzpatrick:2012ix}. In general, interactions will be summed over multiple operators $\mathcal{O}_i$ for both neutrons and protons. The spin-averaged DM-nucleus scattering amplitude can then be computed using
\begin{equation}
\frac{1}{2J_\chi+1}\frac{1}{2J_T+1}\sum_{\rm spins}|\mathcal{M}|^2 \equiv \frac{m_T^2}{m_N^2}\sum_{i,j} \, c_i c_j\,F_{ij}^T(v^2,q^2) \, ,
\end{equation}
where $\mathcal{M}$ si the DM-nucleon scattering amplitude summed over nucleons and computer between nuclear states, $J_\chi$ and $J_T$ are the total angular momentum of the DM and nucleus respectively, the $c_i$ are the coefficients multiplying the operator $i$ (e.g. $-4c$ in the example provided above), and $F_{ij}^T$ is a nuclide-dependent response factor (these can be found in ~\cite{Fitzpatrick:2012ix,Fitzpatrick:2012ib,Anand:2013yka}). The differential cross section as a function of scattering angle is then obtained using the spin-averaged scattering amplitude:
\begin{equation}
\dfrac{d\sigma}{d \cos{\theta}} = \dfrac{1}{(2 J_{\chi} + 1)} \dfrac{1}{(2 J_T + 1)} \sum_{\rm spins} \dfrac{1}{32 \pi} \dfrac{|\mathcal{M}|^2}{(m_{\chi} + m_T)^2} \, .
\end{equation}
Finally, for elastic DM scattering one can change variables using $d \cos{\theta} = m_T/\mu_T^2 v^2$ (with $\theta$ defined in the center of mass frame) and obtain the differential cross-section $d\sigma/dE_R$ needed for the interaction in \Eq{eq:dm_diff_rate}.
The calculations using the above formalism can be expedited with a publicly available code, described in~\cite{Anand:2013yka}, which allows for a straightforward computation of the cross sections considered using the form factors specific to particular isotopes (see \Tab{tab:experiments}) and that we employ for our analysis.
It is worth noting at this point that some of the interactions we consider are dominated in the non-relativistic limit by one EFT operator, but some consist of a combination of several operators. Using the interaction names in the first column of \Tab{tab:fermop} and the usual names of EFT operators as introduced in~\cite{Fitzpatrick:2012ix}, the scattering amplitude for SI is dominated by $\mathcal{O}_1$, for SD by $\mathcal{O}_4$, for PS-S by $\mathcal{O}_{11}$, for S-PS by $\mathcal{O}_{10}$ and PS-PS by $\mathcal{O}_6$, as can be seen in \cite{Dent:2015zpa}, and the amplitude for ED is dominated by $\mathcal{O}_{11}$. Using dimensionless versions of the EFT operators (as e.g. in \cite{Dent:2015zpa}, in which $\vec{q}/m_N$ is used instead of $\vec{q}$ for the operators defined in~\cite{Fitzpatrick:2012ix}) the scattering amplitudes with nucleons $N$ for Ana (the anapole coupling) and AV-V are proportional to the combination~\cite{Dent:2015zpa}
\begin{equation}
\mathcal{A}_{\rm Ana} \propto Q_N \mathcal{O}_8 - \frac{\lambda}{e/2m_N} \mathcal{O}_9~,
\end{equation}
and the amplitude for MD (magnetic dipole couplings) is proportional to (excluding the propagator)
\begin{equation}
\mathcal{A}_{\rm MD} \propto m_N Q_N ( |\vec{q}|^2 \mathcal{O}_1 + 4 m_\chi m_N \mathcal{O}_5) + \frac{\lambda}{e/2m_N} 4 m_\chi( |\vec{q}|^2 \mathcal{O}_4 - m_N^2 \mathcal{O}_6)~,
\end{equation}
where $m_N$, $Q_N$ and $\lambda$ are the mass, charge and magnetic moment of the nucleon $N$ ($e/2 m_N$ is the nuclear magneton). Notice that when several EFT operators appear in the amplitude, their coefficients are target element dependent and, thus, different operators may be dominant in the amplitude for different targets in a particular energy range.
\subsubsection{Interaction models}
\label{sssec:models}
We now proceed to describe specific DM interaction models considered in our study, as specified for a DM fermion $\chi$ in \Tab{tab:fermop}, a DM scalar $\phi$ in \Tab{tab:scaleop} and DM vector $X^{\mu}$ in \Tab{tab:vecop}. In this section we also provide the differential cross-sections for fermionic DM candidates, assuming elastic DM-nuclei scattering.
For momentum-independent interactions, the differential cross-section is related to the full cross-section $\sigma_T$ as
\begin{equation}
\dfrac{d \sigma_T}{d E_R} = \dfrac{\sigma_T m_T}{2 \mu_T^2 v^2}~.
\end{equation}
For each interaction we consider the case of either a heavy or a light mediator, where `heavy' and `light' refer to the mass of the mediator as compared to the typical momentum transfer $|\vec{q}|$. We only specify the full differential cross-sections for the light (massless) mediator case, i.e. when the mediator mass $M$ obeys $|\vec{q}| \gg M$. For the case of heavy mediator, i.e. $M \gg |\vec{q}|$, the results are readily obtainable using the relation
\begin{equation}
\dfrac{d \sigma}{d E_R} = \dfrac{ |\vec{q}|^4}{(|\vec{q}|^2 + M^2)^2} \left(\dfrac{d \sigma_{\rm light}}{d E_R} \right)~
\end{equation}
and $|\vec{q}|^2 + M^2 \simeq M^2$.
For a DM scalar and vector field, many of the interactions produce cross sections which, at leading order, have a dependence on the momentum transfer and DM speed identical to the interactions produced by a fermionic DM candidate. Consequently, the resulting degeneracy of the neutrino and DM recoil spectra in these models is qualitatively similar, although one should bear in mind that other nuclide-specific factors may favor particular experimental targets. In this case we simply specify the relevant $|\vec{q}|$, $v$ dependence of the interaction and the fermionic model they mimic, given that the qualitative features of the recoil spectrum are determined by these factors (as will be shown in the following sections). The shape of the recoil spectrum is determined by the $|\vec{q}|$ dependence, while the annual modulation and the relative importance of various terms in the cross section may be altered by the $v$-dependent terms. From each cross-section we extract an arbitrary constant ``reference cross-section'' $\sigma_{\rm ref}$, to indicate its magnitude (sometimes it is chosen to be the DM-proton total cross-section). It is a normalization factor, stated in terms of reference momentum $|\vec{q}_{\rm ref}| = 100$ MeV, for the DM differential spectrum. We now list the form of the relativistic scattering amplitudes and the resulting differential cross-sections for all the interactions we study, which we assume all describe a single-mediator-exchange scattering. This a rather complete but not necessarily exhaustive list.
Throughout this work we assume that the couplings of DM to neutrons and protons are the same. In principle these couplings could be different which would additionally not subsequently enhance or suppress the overall scattering rate in a particular target nuclei \cite{Feng:2011vu,Kurylov:2003ra}.
\begin{table}[tb]
\begin{centering}
\renewcommand{\arraystretch}{2.25}
\begin{threeparttable}
\begin{tabular}{|c|c|c|c|c|c|} \hline
& & \multicolumn{2}{c|}{ Dependence ($|\vec q|$, $v$)} & \multicolumn{2}{c|}{$\sigma_{\rm ref}$}
\\
Model & {\rm Interaction} & \text{Heavy} & \text{Light} & \text{Heavy} & \text{Light}
\\ \hline \hline
\multirow{2}{*}{SI} & $\bar{\chi} \chi \bar{N} N $ & \multirow{2}{*}{$\dfrac{1}{v^2}$} & \multirow{2}{*}{$\dfrac{1}{v^2 |\vec{q}|^4}$} & \multirow{2}{*}{$\dfrac{\mu_N^2}{\pi}\left(\dfrac{f_p^{\rm SI}}{M^2} \right)^2$} &
\multirow{2}{*}{$\dfrac{\mu_N^2}{\pi}\left(\dfrac{f_p^{\rm SI}}{|\vec{q}_{\rm ref}|^2} \right)^2$}
\\
& $ \bar{\chi} \gamma^{\mu} \chi \bar{N} \gamma_{\mu} N $ & & & &
\\ \hline
SD & $ \bar{\chi} \gamma^{\mu}\gamma^5 \chi \bar{N} \gamma_{\mu}\gamma_5 N $ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & $\dfrac{3 \mu_N^2}{\pi}\left(\dfrac{a_p^{\rm SD}}{ M ^2} \right)^2$ & $\dfrac{3 \mu_N^2}{\pi}\left(\dfrac{a_p^{\rm SD}}{|\vec{q}_{\rm ref}|^2} \right)^2$
\\ \hline
Ana & $ \bar{\chi} \gamma^{\mu}\gamma_5 \chi \partial^{\nu} F_{\mu\nu} $ & $|\vec{q}|^4~,~\dfrac{|\vec{q}|^6}{v^2}$ & 1~,~$\dfrac{|\vec{q} |^2}{v^2 }$ & $ \dfrac{\varepsilon^2 \mu_N^2}{4 \pi}\left(\dfrac{e g_{\chi}^{\rm A} |\vec{q}_{\rm ref}|^2}{\Lambda^2 M^2} \right)^2$ & $ \dfrac{\varepsilon^2 \mu_N^2}{4 \pi}\left(\dfrac{e g_{\chi}^{\rm A}}{\Lambda^2} \right)^2$ \\ \hline
MD & $\bar{\chi} \sigma^{\mu \nu} \chi F_{\mu \nu} $ & $|\vec{q}|^2~,~\dfrac{|\vec{q}|^4}{ v^2}$ & $\dfrac{1}{|\vec{q}|^2}~,~\dfrac{1}{v^2}$ & $\dfrac{ \mu_N^2}{\pi} \left(\dfrac{e g_{\chi}^{\rm MD} |\vec{q}_{\rm ref}|}{ \Lambda M^2}\right)^2$ & $\dfrac{ \mu_N^2}{\pi} \left(\dfrac{e g_{\chi}^{\rm MD}}{ \Lambda |\vec{q}_{\rm ref}|}\right)^2$
\\ \hline
ED & $ \bar{\chi} \sigma^{\mu \nu} \gamma^5 \chi F_{\mu \nu}$ & $\dfrac{|\vec{q}|^2}{v^2}$ & $ \dfrac{1}{v^2 |\vec{q}|^2}$ & $\dfrac{ \mu_N^2}{\pi} \left(\dfrac{e g_{\chi}^{\rm ED} |\vec{q}_{\rm ref}|}{ \Lambda M^2}\right)^2$ & $\dfrac{ \mu_N^2}{\pi} \left(\dfrac{e g_{\chi}^{\rm ED}}{ \Lambda |\vec{q}_{\rm ref}|}\right)^2$
\\ \hline
mC\tnote{1} & $ \bar{\chi} \gamma^{\mu} \chi \bar{N} \gamma_{\mu} N$ & $\dfrac{1}{ v^2}$ & $ \dfrac{1}{v^2 |\vec{q}|^4 }$ & $\dfrac{\varepsilon^2 \mu_N^2}{\pi} \left(\dfrac{ e g_{\chi}^{\rm mC} }{ M^2}\right)^2$ & $\dfrac{\varepsilon^2 \mu_N^2}{\pi} \left(\dfrac{ e g_{\chi}^{\rm mC} }{|\vec{q}_{\rm ref}|^2}\right)^2$
\\ \hline
PS-S & $ \bar{\chi} \gamma^5 \chi \bar{N} N$ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & $\dfrac{ \mu_N^2}{4 \pi} \left(\dfrac{g_{\chi}^{\rm PS} f_p^{\rm S}|\vec{q}_{\rm ref}|}{M^2 m_{\chi}}\right)^2 $ & $\dfrac{ \mu_N^2}{4 \pi m_{\chi}^2} \left(\dfrac{g_{\chi}^{\rm PS} f_p^{\rm S}}{|\vec{q}_{\rm ref}|}\right)^2 $
\\ \hline
S-PS & $ \bar{\chi} \chi \bar{N} \gamma^5 N $ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & $\dfrac{ \mu_N^2}{4 \pi} \left(\dfrac{g_{\chi}^{\rm S} f_p^{\rm PS} |\vec{q}_{\rm ref}|}{M^2 m_p}\right)^2 $ & $\dfrac{ \mu_N^2}{4 \pi m_p^2} \left(\dfrac{g_{\chi}^{\rm S} f_p^{\rm PS}}{|\vec{q}_{\rm ref}|}\right)^2 $
\\ \hline
PS-PS & $ \bar{\chi} \gamma^5 \chi \bar{N} \gamma^5 N $ & $\dfrac{|\vec{q}|^4}{v^2}$ & $\dfrac{1}{v^2}$ & $\dfrac{ \mu_N^2}{16 \pi} \left(\dfrac{g_{\chi}^{\rm PS} f_p^{\rm PS} |\vec{q}_{\rm ref}|^2}{M^2 m_p m_{\chi}}\right)^2 $ & $\dfrac{ \mu_N^2}{16 \pi} \left(\dfrac{g_{\chi}^{\rm PS} f_p^{\rm PS}}{ m_p m_{\chi}}\right)^2 $
\\ \hline
AV-V & $\bar{\chi} \gamma^{\mu} \gamma_5 \chi \bar{N}\gamma_\mu N $ & $1~,~\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{|\vec{q}|^4}~,~\dfrac{1}{v^2 |\vec{q}|^2}$ & $\dfrac{ \varepsilon^2 \mu_N^2}{4\pi} \left(\dfrac{g_{\chi}^{\rm AV-V}}{M^2}\right)^2 $ & $\dfrac{ \varepsilon^2 \mu_N^2}{4\pi} \left(\dfrac{g_{\chi}^{\rm AV-V}}{|\vec{q}_{\rm ref}|^2}\right)^2 $ \\ \hline
\end{tabular}
\begin{tablenotes}
\item[1] This model is equivalent to SI.
\end{tablenotes}
\caption{Interaction models with fermionic DM particles. Model name, interaction, dependence on $|\vec{q}|$ and $v$, as well as the definition of the reference cross-section $\sigma_{\rm ref}$ are shown. Values for both heavy and light mediators are included.}
\label{tab:fermop}
\end{threeparttable}
\end{centering}
\end{table}
\begin{table}[tb]
\begin{centering}
\renewcommand{\arraystretch}{2.1}
\begin{tabular}{|c|c|c|c|c| } \hline
& & \multicolumn{2}{c|}{ Dependence ($|\vec q|$, $v$)} & Similar
\\
Model & {\rm Interaction} & \text{Heavy} & \text{Light} & Model \\
\hline
\hline
S1 & $\phi^\dagger \phi \bar{N}N$ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SI \\ \hline
S2 & $\phi^\dagger \phi \bar{N}\gamma^5 N $ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & S-PS \\ \hline
S3 & $\phi^\dagger \overset\leftrightarrow{\partial_\mu}\phi \bar{N}\gamma^\mu N $ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SI \\ \hline
S4 & $\phi^\dagger \overset\leftrightarrow{\partial_\mu}\phi \bar{N}\gamma^\mu\gamma^5 N $ & $1~,~\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{|\vec{q}|^4}~,~\dfrac{1}{v^2|\vec{q}|^2}$ & AV-V \\ \hline
\end{tabular}
\caption{Interaction models with scalar DM particles. Model name, interaction, dependence on $|\vec{q}|$ and $v$, as well as the fermionic DM particle model to which the behavior is similar are shown.}
\label{tab:scaleop}
\end{centering}
\end{table}
\begin{table}[tb]
\begin{centering}
\renewcommand{\arraystretch}{2.1}
\begin{tabular}{|c|c|c|c|c| } \hline
& & \multicolumn{2}{c|}{ Dependence ($|\vec q|$, $v$)} & Similar
\\
Model & {\rm Interaction} & \text{Heavy} & \text{Light} & Model \\
\hline
\hline
V1 & $X^\mu X_\mu^\dagger \bar{N}N $ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SI \\ \hline
V2 & $X^\mu X_\mu^\dagger \bar{N}\gamma^5 N $ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & S-PS \\ \hline
V3 & $X^\mu \partial_\nu X_\mu^\dagger \bar{N}\gamma^\nu N $ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SI \\ \hline
V4 & $X^\mu \partial_\nu X_\mu^\dagger \bar{N}\gamma^\nu\gamma^5 N $ & $1~,~\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{|\vec{q}|^4}~,~\dfrac{1}{v^2|\vec{q}|^2}$ & AV-V \\ \hline
V5 & $(X_\mu^\dagger X_\nu -X_\nu^\dagger X_\mu)\bar{N}\sigma^{\mu\nu}N$ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SD \\ \hline
V6 & $(X_\mu^\dagger X_\nu -X_\nu^\dagger X_\mu)\bar{N}\sigma^{\mu\nu}\gamma^5N$ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & PS-S \\ \hline
V7 & $X_\nu^\dagger \partial^\nu X_\mu \bar{N}\gamma^\mu\gamma^5 N$ & $\dfrac{|\vec{q}|^2}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^2}$ & S-PS \\ \hline
V8 & $\epsilon^{\mu\nu\rho\sigma}X_\nu^\dagger \partial_\rho X_\sigma \bar{N}\gamma_\mu N$ & $\dfrac{|\vec{q}|^2}{v^2}~, 1$ & $\dfrac{1}{v^2|\vec{q}|^2}~,~\dfrac{1}{|\vec{q}|^4}$ & AV-V \\ \hline
V9 & $\epsilon^{\mu\nu\rho\sigma}X_\nu^\dagger \partial_\rho X_\sigma \bar{N}\gamma_\mu\gamma^5 N$ & $\dfrac{1}{v^2}$ & $\dfrac{1}{v^2 |\vec{q}|^4}$ & SD \\ \hline
\end{tabular}
\caption{Interaction models with vectorial DM particles. Model name, interaction, dependence on $|\vec{q}|$ and $v$, as well as the fermionic DM particle model to which the behavior is similar are shown.}
\label{tab:vecop}
\end{centering}
\end{table}
\paragraph{\underline{Spin--1/2 fermionic DM $\chi$}}
~\newline
The standard SI and SD are the most generic DM interactions. They are due to the exchange of a scalar or vector boson mediator (for SI), or an axial vector boson mediator (for SD).
\begin{itemize}
\item \emph{Spin-independent } (SI):
The SI scattering amplitude is given by
\begin{equation}
\mathcal{A}_{\rm SI} \propto \dfrac{f_N^{\rm SI}}{(|\vec{q}|^2 + M^2)} \bar{\chi} \chi \bar{N} N ~~~~ \text{or} ~~~ \mathcal{A}_{\rm SI} \propto \dfrac{f_N^{\rm SI}}{(|\vec{q}|^2 + M^2)} \bar{\chi} \gamma^{\mu} \chi \bar{N} \gamma_{\mu} N ~.
\end{equation}
For a light ($M = 0$) mediator the differential cross-section is given by (e.g. \cite{DelNobile:2013gba}):
\begin{equation} \label{eq:diffxsecsi}
\dfrac{d \sigma_T^{\rm SI}}{d E_R} = \sigma_{\rm ref}^{\rm SI} \dfrac{|\vec{q}_{\rm ref}|^4}{|\vec{q}|^4} \dfrac{m_T }{2 \mu_N^2 v^2} \Big[ Z_T + \dfrac{f_n^{\rm SI}}{f_p^{\rm SI}} (A_T - Z_T) \Big]^2 F_{{\rm SI}, T}^2~,
\end{equation}
where $Z_T$ and $A_T$ are the atomic and mass numbers of the target nucleus $T$, $F_{{\rm SI},T}$ is the charge nuclear form factor (note that there is an implicit assumption here that neutrons $n$ have the same form factor as protons $p$), taken to be the Helm form factor \cite{Helm:1956zz}, $f_n$ and $f_p$ are the effective couplings of the DM particle to neutrons and protons, respectively, $\sigma_{\rm ref}^{\rm SI}$ is the DM-proton cross-section (i.e. $\sigma_{\rm ref}^{\rm SI} = \sigma_{\rm p}^{\rm SI}$), given in the 1$^{\rm st}$ row of \Tab{tab:fermop} (assumed to be the same for $n$) and $\mu_N$ is the DM-nucleon reduced mass. For small enough $|\vec{q}|$ values the SI scattering acts coherently with the entire nucleus, leading to the $\sim A_T^2$ enhancement for $f_n^{\rm SI} = f_p^{\rm SI}$.
~\newline
\item \emph{Spin-dependent }
(SD):
The conventional SD interaction arises from an axial-vector (AV) mediated interaction with an amplitude given by
\begin{equation}
\mathcal{A}_{\rm SD} \propto \dfrac{a_N^{\rm SD}}{(|\vec{q}|^2 + M^2)} \bar{\chi} \gamma^{\mu}\gamma^5 \chi \bar{N} \gamma_{\mu}\gamma_5 N~.
\end{equation}
The differential cross-section is given by (e.g. \cite{DelNobile:2015lxa})
\begin{equation}
\dfrac{d \sigma_T^{\rm SD}}{d E_R} = \sigma_{\rm ref}^{\rm SD} \dfrac{|\vec{q}_{\rm ref}|^4}{|\vec{q}|^4} \dfrac{m_T }{8 \mu_N^2 v^2} \dfrac{4}{3} \dfrac{J_T+1}{J_T}
\Big[ \langle S_{p, T} \rangle + \dfrac{a_n^{\rm SD}}{a_p^{\rm SD}} \langle S_{n, T} \rangle \Big]^2 F_{{\rm SD}, T}^2~,
\end{equation}
where $J_T$ is the spin of the target nucleus, while $\langle S_{p, T} \rangle$, $\langle S_{n, T} \rangle$ represent the proton and the neutron spin of the target nucleus and $a_p^{\rm SD}, a_n^{\rm SD}$ the DM coupling to protons and neutrons, respectively, with $F_{{\rm SD},T}^2$ being the spin form factor of the nucleus.
\end{itemize}
Neutral DM particle candidates can interact with photons through higher electromagnetic moments.
Such interactions can naturally occur in models of composite DM formed from charged components (e.g. \cite{Bagnasco:1993st}), e.g. a ``dark neutron'', or in models where the DM couples to an intermediate sector that leads to an effective photon coupling (e.g. through heavy charged messengers \cite{Weiner:2012gm} or kinetic mixing with a dark photon \cite{Holdom:1985ag}), as occurs with neutrinos in the Standard Model.
The effective low energy non-renormalizable operators describing these interactions are suppressed by the scale of new physics $\Lambda$, such as the messenger particle mass or the compositeness scale of the ultraviolet theory. Here, for the electromagnetic moment interactions, we focus explicitly only on vector mediators which couple to the electromagnetic current of the nucleus.
For fermionic DM, the most studied candidates are WIMPs with the lowest order electromagnetic moments (e.g.~\cite{Pospelov:2000bq,An:2010kc,Sigurdson:2004zp,Barger:2010gv,Chang:2010en,Cho:2010br,Heo:2009vt,Gardner:2008yn,Masso:2009mu,Banks:2010eh,Fortin:2011hv,Kumar:2011iy,Barger:2012pf,DelNobile:2012tx,Cline:2012is,Weiner:2012cb,Tulin:2012uq,Cline:2012bz}), the magnetic and the electric dipole moments, given by dimension five effective operators and thus proportional to the inverse of a large scale of new physics $\Lambda$. For Majorana fermions the magnetic and electric dipole moments vanish (although non-diagonal couplings are possible). In this case the only possible electromagnetic moment is the anapole, with the respective interaction described by a dimension-six effective operator proportional to $1/\Lambda^2$. The anapole moment DM has been studied in various contexts, including direct detection \cite{Pospelov:2000bq,Ho:2012bg,Fitzpatrick:2010br,Frandsen:2013cna,Gresham:2013mua} and colliders \cite{Gao:2013vfa}.
~\newline
\begin{itemize}
\item \emph{Anapole} (Ana) \emph{moment}:
The anapole amplitude is given by
\begin{equation}
\mathcal{A}_{\rm anapole} \propto \dfrac{\varepsilon g_{\chi}^{\rm A}}{ \left( |\vec{q}|^2 + M^2\right) } \dfrac{|\vec{q}|^2}{ \Lambda^2} \bar{\chi} \gamma^{\mu}\gamma_5 \chi J_{\mu}~.
\end{equation}
This interaction is mediated by a vector boson that couples to the electromagnetic current $J_{\mu}$ (e.g. via a kinetic mixing $\varepsilon$). For $M = 0$ the mediator can be the photon.
With $F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\mu} A_{\nu}$, for a massless mediator (e.g. $A_{\mu}$ is the SM photon) the equations of motions imply $J_{\mu} = \partial^{\nu} F_{\mu \nu}$.
Unlike magnetic and electric dipole moments, the anapole \cite{ZelDovich:1958} is not part of the pure electro-magnetic multipole expansion (however, see discussion in \cite{fernandez-corbaton:natsci2017}). The anapole moment violates charge conjugation $C$ and parity $P$, but preserves $C P$. It can arise in models of Majorana fermion DM coupling to a photon. The anapole can couple diagonally to Majorana fermions since it is CPT self-conjugate, which is not possible for magnetic or electric CPT-odd dipole moments. For a light mediator the anapole differential cross-section is given by (e.g. \cite{DelNobile:2014eta})
\begin{equation} \label{eq:anapolediff}
\dfrac{d \sigma_T^{\rm A}}{d E_R} = \sigma_{\rm ref}^{\rm A} \dfrac{2 m_T }{\mu_N^2 v^2} \Big[ Z_T^2 \left(v^2 - \dfrac{|\vec{q}|^2}{4 \mu_T^2} \right) F_{{\rm E}, T}^2
+ \dfrac{\lambda_T^2}{\lambda_N^2} \dfrac{ |\vec{q}|^2}{2 m_N^2} \left(\dfrac{J_T + 1}{3 J_T}\right) F_{{\rm M},T} \Big] ~,
\end{equation}
where $\sigma_{\rm ref}^{\rm A}$ is given in \Tab{tab:fermop}, $F_{{\rm E}, T}, F_{{\rm M}, T}$ are the electric and magnetic form factors, with $F_{{\rm E}, T} = F_{{\rm SI}, T}$, $\lambda_T$ is the nuclear magnetic moment (see \Tab{tab:experiments}) and $\lambda_N = e/2 m_N$ is the nuclear magneton.
A distinct characteristic of this cross-section is that it contains two different terms with different dependencies on the DM particle speed.
\newline
\item \emph{Magnetic dipole} (MD):
The MD amplitude is given by
\begin{equation}
\mathcal{A}_{\rm MD} \propto \dfrac{ g_{\chi}^{\rm MD}}{\Lambda (|\vec{q}|^2 + M^2)} \bar{\chi} \sigma^{\mu \nu} \chi~q_\mu J_\nu~.
\end{equation}
where $J_\nu$ is the electromagnetic current. For $M = 0$ this can be a photon-mediated scattering. In general, it can be due to the exchange of a vector boson that couples to the electromagnetic field tensor $F_{\mu \nu}$.
In this case, the differential cross-section for MD is given by \cite{DelNobile:2014eta}
\begin{equation}
\dfrac{d \sigma_T^{\rm MD}}{d E_R} = \sigma_{\rm ref}^{\rm MD} \dfrac{|\vec{q}_{\rm ref}|^2}{|\vec{q}|^2} \dfrac{m_T^2}{4 v^2 \mu_N^2}
\Big[ Z_T^2 \left( 4 v^2 - |\vec{q}|^2 \Big\{ \dfrac{1}{\mu_T^2} - \dfrac{1}{m_{\chi}^2}\Big\} \right) F_{{\rm E}, T}^2
+ 2 \dfrac{|\vec{q}|^2}{ m_N^2} \dfrac{\lambda_T^2}{\lambda_N^2} \left(\dfrac{J_T + 1}{3 J_T}\right) F_{{\rm M},T} \Big] ~,
\end{equation}
where $m_{\chi}$ is the mass of the DM fermion and $\sigma_{\rm ref}^{\rm MD}$ is an arbitrary factor extracted from the cross-section and defined in \Tab{tab:fermop}. Here again $F_{{\rm E},T}^2 = F_{{\rm SI},T}^2$ is the charge nuclear form factor and $F_{{\rm M},T}^2$ the magnetic nuclear form factor, with other variables defined as before. We note that this cross-section also contains two different terms, as is the case with an anapole interaction, with different dependencies on the DM particle speed. This could lead to distinct annual modulation signals in different target materials~\cite{DelNobile:2015nua,DelNobile:2015tza}.
~\newline
\item \emph{Electric dipole} (ED):
The ED amplitude is given by
\begin{equation}
\mathcal{A}_{\rm ED} \propto \dfrac{g_{\chi}^{\rm ED}}{\Lambda (|\vec{q}|^2 + M^2 )} \bar{\chi} \sigma^{\mu \nu} \gamma^5 \chi ~q_\mu J_\nu~.
\end{equation}
where $J_\nu$ is the electromagnetic current. For a light mediator, the ED differential cross-section is given by (e.g. \cite{Barger:2010gv})
\begin{equation}
\dfrac{d \sigma_T^{\rm ED}}{d E_R} = \sigma_{\rm ref}^{\rm ED} \dfrac{|\vec{q}_{\rm ref}|^2}{|\vec{q} |^2 } \dfrac{m_T}{2 v^2 \mu_N^2} \Big[ Z_T^2 F_{{\rm E},T}^2 \Big] ~.
\end{equation}
We note that ED has no dependence on the spin and the magnetic form factor $F_{\rm M, T}$ like the anapole or MD cross-sections, allowing it to test the charge $Z_T$ signal component separately from other contributions.
~\newline
\item \emph{Milli-charge} (mC):
The mC amplitude is given by
\begin{equation}
\mathcal{A}_{\rm mC} \propto \dfrac{\varepsilon g_{\chi}^{\rm mC}}{\Lambda (|\vec{q}|^2 + M^2 )} \bar{\chi} \gamma^\mu \chi J_\mu~.
\end{equation}
Like ED and MD for $M = 0$ can be a photon-mediated scattering, but with a DM particle electric charge suppressed by $\varepsilon$. In general, this interaction is due to exchange of a vector boson coupled to the electromagnetic current $J_{\mu}$ with a small coupling $\varepsilon$. DM with a small electric charge can arise when the SM photon
kinetically mixes with a photon of a dark sector $U(1)$ \cite{Holdom:1985ag}, which can be massive through the Stuckelberg
mechanism \cite{Kors:2004dx}.
The mC differential cross-section is given by (e.g. \cite{Gluscevic:2015sqa})
\begin{equation}
\dfrac{d \sigma_T^{\rm mC}}{d E_R} = \sigma_{\rm ref}^{\rm mC} \dfrac{|\vec{q}_{\rm ref}|^4}{|\vec{q}|^4} \dfrac{m_T }{2 \mu_N^2 v^2} \Big[ Z_T\Big]^2 F_{{\rm SI}, T}^2 ~.
\end{equation}
Since it is so similar to the SI interaction, we do not treat the mC interaction separately in the rest of the paper.
\end{itemize}
Pseudo-scalar couplings lead to non-standard spin-dependent DM interactions in the non-relativistic limit. While the PS-S does depend on the spin of the DM particle, this is a SI interaction because it does not depend on the spin of the nucleus.
The $\gamma^5$ coupling is $CP$-odd. Such interactions can arise in models with an extended Higgs sector (e.g.~Two-Higgs Doublet Models \cite{Gunion:1989we}) and allow for a rich phenomenology (e.g.~\cite{Arina:2014yna}). We consider all of the possible $\gamma^5$ interaction combinations, i.e. ($\mathbb{1} \cdot\gamma_5$), ($\gamma_5\cdot\mathbb{1}$), ($\gamma_5\cdot\gamma_5$) in the vertices. For universal flavor-diagonal quark couplings to the pseudo-scalar mediator the WIMP couples primarily to protons. Rare meson decays already strongly constrain the coupling of light pseudo-scalar bosons (M $< 7$ GeV) to quarks \cite{Hiller:2004ii,Andreas:2010ms,Dolan:2014ska}. If the quark couplings are non-universal and are instead proportional to the quark mass, the flavor constraints are less stringent \cite{Dolan:2014ska}.
\begin{itemize}
\item \emph{Pseudo-scalar-scalar} (PS-S):
The PS-S amplitude is given by
\begin{equation}
\mathcal{A}_{\rm PS-S} \propto \dfrac{ g_{\chi}^{\rm PS} f_N^{\rm S} }{(|\vec{q} |^2 + M^2)} \bar{\chi} \gamma^5 \chi \bar{N} N~.
\end{equation}
It arises from a scalar mediator with a CP-odd vertex with the DM.
For a light mediator, the differential cross-section is given by (e.g. \cite{Gluscevic:2015sqa})
\begin{equation}
\dfrac{d \sigma_T^{\rm PS-S}}{d E_R} = \sigma_{\rm ref}^{\rm PS-S} \dfrac{|\vec{q}_{\rm ref}|^2}{|\vec{q}|^2} \dfrac{m_T }{16 m_{\chi}^2 \mu_N^2 v^2} \Big[ Z_T + \dfrac{f_n^{\rm S}}{f_p^{\rm S}} (A_T - Z_T) \Big]^2 F_{{\rm SI}, T}^2 ~.
\end{equation}
~\newline
\item \emph{Scalar-pseudo-scalar} (S-PS):
The S-PS amplitude is given by
\begin{equation}
\mathcal{A}_{\rm S-PS} \propto \dfrac{ g_{\chi}^{\rm S} f_n^{\rm PS} }{(|\vec{q} |^2 + M^2)} \bar{\chi} \chi \bar{N} \gamma^5 N~.
\end{equation}
The differential cross-section for a light mediator is given by (e.g. \cite{Gluscevic:2015sqa})
\begin{equation}
\dfrac{d \sigma_T^{\rm S-PS}}{d E_R} = \sigma_{\rm ref}^{\rm S-PS} \dfrac{|\vec{q}_{\rm ref}|^2}{|\vec{q}|^2} \dfrac{4 m_{\chi}^2 m_T }{\mu_T^2 v^2} \dfrac{\mu_T^2}{\mu_N^2} \dfrac{4}{3} \dfrac{J_T+1}{J_T}
\Big[ \langle S_{p, T} \rangle + \dfrac{f_n^{\rm PS}}{f_p^{\rm PS}} \langle S_{n, T} \rangle \Big]^2 F_{{\rm SD}, T}^2~.
\end{equation}
~\newline
\item \emph{Pseudo-scalar-pseudo-scalar} (PS-PS):
The PS-PS amplitude is given by
\begin{equation}
\mathcal{A}_{\rm PS-PS} \propto \dfrac{ g_{\chi}^{\rm PS} f_n^{\rm PS} }{(|\vec{q} |^2 + M^2)} \bar{\chi} \gamma^5 \chi \bar{N} \gamma^5 N~.
\end{equation}
Arises from pseudo-scalar mediator. Both of vertices include a $\gamma_5$ and are CP-violating.
For $M = 0$ the differential cross-section is given by (e.g. \cite{Gluscevic:2015sqa})
\begin{equation}
\dfrac{d \sigma_T^{\rm PS-PS}}{d E_R} = \sigma_{\rm ref}^{\rm PS-PS} \dfrac{m_T }{2 \mu_T^2 v^2} \dfrac{\mu_T^2}{\mu_N^2} \dfrac{4}{3} \dfrac{J_T+1}{J_T}
\Big[ \langle S_{p, T} \rangle + \dfrac{f_n^{\rm S}}{f_p^{\rm S}} \langle S_{n, T} \rangle \Big]^2 F_{{\rm SD}, T}^2~.
\end{equation}
~\newline
~~~~~ In \cite{Fitzpatrick:2010br} it was suggested that
a coupling $\bar{\chi} \gamma^{\mu} \gamma_5 \chi A_{\mu}^{\prime}$, with $A_{\mu}^{\prime}$ denoting a vector boson, also constitutes an anapole-DM interaction if $A_{\mu}^{\prime}$ couples with nucleons with a vector coupling $\overline{N} \gamma^{\mu} N$. This AV-V coupling also violates $C$ and $P$ symmetries, but preserves $CP$. It was shown in \cite{Ho:2012bg}, however, that $A_{\mu}^{\prime}$ cannot be the SM photon and this operator must arise from another mechanism (e.g. kinetic mixing with a dark photon, suppressed by $\varepsilon$). We treat this operator separately, as the AV-V interaction. The differential cross-section for the AV-V interaction for heavy mediator \cite{Fitzpatrick:2010br} leads to the same differential cross-section as the anapole interaction with a light mediator, i.e. \eqref{eq:anapolediff}.
\item \emph{Axial-vector-vector} (AV-V):
The AV-V amplitude is given by
\begin{equation}
\mathcal{A}_{\rm AV-V} \propto \dfrac{ \varepsilon g_{\chi}^{\rm AV-V} }{(|\vec{q}|^2 + M^2)} \bar{\chi} \gamma^{\mu} \gamma_5 \chi \bar{N}\gamma_\mu N ~.
\end{equation}
We treat here the anapole-like couplings $\bar{\chi} \gamma^{\mu} \gamma_5 \chi A_{\mu}^{\prime}$ and $A^{\prime \mu} J_{\mu}$ (see anapole above), where $J_{\mu}$ is the electro-magnetic current, as suggested in \cite{Fitzpatrick:2010br}. While $A_{\mu}^{\prime}$ cannot be the SM photon, it can readily appear as a dark photon, with a kinetic mixing $\varepsilon$ with the SM electro-magnetic photon. For a light mediator the cross-section is given by
\begin{equation}
\dfrac{d \sigma_T^{\rm AV-V}}{d E_R} = \sigma_{\rm ref}^{\rm AV-V} \dfrac{|\vec{q}_{\rm ref}|^4}{|\vec{q}|^4} \dfrac{2 m_T }{v^2 \mu_N^2} \Big[ Z_T^2 \left(v^2 - \dfrac{|\vec{q}|^2}{4 \mu_T^2} \right) F_{{\rm E}, T}^2
+ \dfrac{\lambda_T^2}{\lambda_N^2} \dfrac{ |\vec{q}|^2}{2 m_N^2} \left(\dfrac{J_T + 1}{3 J_T}\right) F_{{\rm M},T} \Big]~.
\end{equation}
\end{itemize}
In addition to the listed fermion DM interactions above, it was suggested in~\cite{Fitzpatrick:2012ix} that when the nucleon angular momentum orbitals are not completely filled one also can expect from very specific DM interactions a response proportional to the product of nucleon angular momentum and spin (i.e. $\vec{L}_N \cdot \vec{S}_N$). We do not consider this interaction in this work.
~\newline
\paragraph{\underline{Spin--0 scalar DM $\phi$}}
~\newline
A list of scalar DM interactions from a general matrix element analysis can be found in \cite{Kumar:2013iva}. While all four interactions in \cite{Kumar:2013iva} have cross-sections with an identical dependence on the momentum transfer and the DM velocity to the already considered fermionic ones, we reproduce the cross section below for the only interaction (S4) for which the similarity is perhaps not immediately obvious.
\begin{itemize}
\item \emph{S4}:
The S4 amplitude is given by
\begin{equation}
\mathcal{A}_{\rm S4} \propto \dfrac{ g_{\chi}^{\rm S4} }{(|\vec{q} |^2 + M^2)} \phi^\dagger \overset\leftrightarrow{\partial_\mu}\phi \bar{N}\gamma^\mu\gamma^5 N~.
\end{equation}
We obtain the following differential cross-section
\begin{equation}
\dfrac{d \sigma_T^{\rm S4}}{d E_R} = \sigma_{\rm ref}^{\rm S4} \dfrac{|\vec{q}_{\rm ref}|^4}{|\vec{q}|^4} \dfrac{m_T \mu_T^2 }{4 \mu_N^2 m_N^2 v^2} \dfrac{1}{3} \dfrac{J_T+1}{J_T} \left( v^2 - \dfrac{|\vec{q}|^2}{4 \mu_T^2}\right)
\Big[ \langle S_{p, T} \rangle + \dfrac{a_n^{\rm SD}}{a_p^{\rm SD}} \langle S_{n, T} \rangle \Big]^2 F_{{\rm SD}, T}^2~,
\end{equation}
where $\sigma_{\rm ref}^{S4} = \sigma_{\rm ref}^{SD}$.
\end{itemize}
~\newline
\paragraph{\underline{Spin--1 vectorial DM $X^{\mu}$}}
~\newline
A list of vectorial DM interactions from a general matrix element analysis can be found in \cite{Kumar:2013iva}. Nine out of the ten interactions in \cite{Kumar:2013iva} give similar responses to the already considered fermionic DM ones. The only vector interaction with a novel response, $X_\nu^\dagger \partial^\nu X_\mu \bar{N}\gamma^\mu N$, contains terms in the differential cross section proportional to $|\vec{q}|^4$ and $|\vec{q}|^2v^2$. This interaction is difficult to observe due to the highly suppressed nature of the cross section for small $|\vec{q}|$ and $v$. Further, it will produce no degeneracy with the neutrino spectrum, as will become evident in later sections. Therefore the discovery limits for this vector DM interaction will be similar to those of the PS-PS interaction, and thus we do not consider this interaction independently.
\subsubsection{Inelastic scattering}
So far we have only considered DM to consist of a single component. However, multi-component dark sectors can also appear in well-motivated scenarios and allows for rich phenomenology when the component mass is distinct (e.g.~\cite{Feng:2008ya,Zurek:2008qg,Feldman:2010wy,Aoki:2012ub,Chen:2015jkt,DeSimone:2010tf}). In some of them it is possible that the dominant DM-nucleus scattering is inelastic. This happens when the elastic scattering is suppressed or forbidden by the particular couplings of the mediator~\cite{TuckerSmith:2001hy,ArkaniHamed:2008qn, Cui:2009xq,Graham:2010ca,Batell:2009vb,Essig:2010ye}.
In an inelastic scattering, the initial DM particle of mass $m_\chi$ scatters dominantly into a different mass state $m_\chi^\prime = m_\chi + \delta$, where $|\delta| \ll m_\chi$. Here $\delta > 0$ describes ``endothermic'' inelastic scattering~\cite{TuckerSmith:2001hy,ArkaniHamed:2008qn, Cui:2009xq} and $\delta < 0$ describes ``exothermic'' inelastic scattering~\cite{Graham:2010ca,Batell:2009vb,Essig:2010ye}. The case of elastic scattering is recovered when $\delta = 0$. In the limit $\mu_T |\delta|/m_\chi^2 \ll 1$, $v_{\rm min}(E_{\rm R})$ is given by
\begin{equation} \label{eq:vmin}
v_{\rm min}(E_{\rm R}) = \frac{1}{\sqrt{2 m_T E_{\rm R}}} \left| \frac{m_T E_{\rm R}}{\mu_T} +\delta \right| \, .
\end{equation}
\Eq{eq:vmin} can be used to obtain the range of possible recoil energies, $[E_{\rm R}^{T,-}(v),E_{\rm R}^{T,+}(v)]$, that can be imparted to a target nucleus by a DM particle traveling at speed $v$ in Earth's frame, given by
\begin{equation}\label{eq:Ebranch}
E_{\rm R}^{T,\pm} (v) = \frac{\mu_T^2 v^2}{2 m_T} \left( 1 \pm \sqrt{1-\frac{2\delta}{\mu_T v^2}} \right)^2 \, ,
\end{equation}
with $E_{\rm R}^{T,-} (v) \leq E_R \leq E_{\rm R}^{T,+} (v)$.
\Eq{eq:Ebranch} shows that for endothermic scattering there exists a non-zero kinematic endpoint in DM speed $v_\delta^T = \sqrt{2 \delta / \mu_T}$, below which incoming DM particles cannot induce nuclear recoils (this endpoint is 0 for elastic and inelastic exothermic scattering). The maximum and minimum possible recoil energies $E_{\rm R}^{T,-} (v)$ and $E_{\rm R}^{T,+} (v)$ become equal at $E_{\delta} = E_{\rm R}^{T,-} (v_{\delta}) = E_{\rm R}^{T,+} (v_{\delta}) = \mu_T |\delta|/m_T$, where $v_{\delta}$ is the minimum possible value of $v_{\rm min}$.
For an exothermic scattering the recoiling nucleus energy is close to $E_{\delta}$, which is proportional
to the splitting between the DM states and is inversely proportional to the nuclear
mass. Thus, the nuclear recoils originating from exothermic interactions are more visible in experiments
with light nuclei and low thresholds. On the other hand, for endothermic DM, only high velocity DM particles have enough energy to up-scatter and the minimum necessary speed decreases with increasing target mass. Thus, high mass targets are favored. Hence, different target materials can act as sensitive probes of multi-component dark sector models in which inelastic scattering dominates over elastic.
\section{Neutrino Background \label{sec:neutrinos}}
\begin{table*}[tbp]
\setlength{\extrarowheight}{2pt}
\setlength{\tabcolsep}{10pt}
\begin{center}
\begin{threeparttable}
\begin{tabular}{|l|l|c|} \hline
Lab Site & Location & Depth (m.w.e)\\
\hline
SNOLab & ~46$^\circ$28$'$19$''$\,N & \multirow{2}{*}{$6010$} \\
(Sudbury, CA) & ~81$^\circ$11$'$12$''$\,W & \\ \hline
LNGS & ~42$^\circ$28$'$09$''$\,N & \multirow{2}{*}{$3400$} \\
(Gran Sasso, IT) & ~13$^\circ$33$'$56$''$\,E & \\ \hline
SURF & ~44$^\circ$21$'$07$''$\,N & \multirow{2}{*}{$4400$} \\
(Homestake, US) & 103$^\circ$45$'$50$''$\,W & \\ \hline
\end{tabular}
\caption{\label{tab:labs} Experimental laboratories that are likely to host new generation of direct detection experiments considered in this work. Exact location as well as depth (meter water equivalent) are displayed.}
\end{threeparttable}
\end{center}
\end{table*}
Neutrino coherent scattering gives origin to an irreducible background (i.e. ``neutrino floor'') for direct detection experiments, with contributions coming from solar, reactor, geo-, diffusive supernovae background as well as atmospheric neutrinos. Below we discuss each specific source and the resulting flux.
Since neutrino fluxes depend on location, we consider likely future direct detection experiment sites \cite{Cushman:2013zza} at SNOLAB, SURF and LGNS laboratories (see \Tab{tab:labs}).
As an example, we display the combined contribution of neutrinos at the SNOLAB location in \Fig{fig:neutrino_flux}. A detailed description of the specific flux components can be found in \Tab{tab:nucomponents}. Results for other laboratories are not significantly different.
For a neutrino flux $\phi_{\nu} (E_{\nu})$ given as a function of the neutrino energy $E_{\nu}$ (see \Fig{fig:neutrino_flux}) and originating from a particular source, the resulting differential event rate as a function of the nuclear recoil energy $E_R$, per unit time and detector mass off a target nuclide $T$ in a detector is given by
\begin{equation} \label{eq:nu_diff_rate}
\dfrac{d R_{\nu, T}}{d E_R} = \dfrac{C_T}{m_T} \int_{E_{\nu}^{\rm min}} \phi_{\nu} (E_{\nu}) \dfrac{d \sigma^T(E_{\nu}, E_R)}{d E_R} d E_{\nu}~,
\end{equation}
where $d \sigma^T(E_{\nu}, E_R) / d E_R$ is the coherent neutrino-nucleus scattering cross-section.
Thus, summing over all nuclides in a detector we obtain the differential rate for each type of neutrino flux $\phi_{\nu}(E_{\nu})$,
\begin{equation} \label{eq:totnurate}
\dfrac{d R_{\nu}}{d E_R} = \sum_T \dfrac{d R_{\nu, T}}{d E_R}~.
\end{equation}
From kinematics, the maximum recoil energy is given by
\begin{equation}
E_R^{\rm max} = \dfrac{2 E_{\nu}^2}{m_T + 2 E_{\nu}}~.
\end{equation}
Neutrinos penetrate the rock surrounding the laboratory sites nearly unimpeded. Their oscillation effects \cite{Fukuda:1998mi,Ahmad:2002jz}, however, will result in a varying depth-dependent flavor composition of the neutrino flux. Since the coherent cross-section is nearly identical for various neutrino species and we are not concerned with the exact composition, throughout this work we neglect the oscillation effects\footnote{They are important, however, for studies focusing on detecting neutrinos of a specific flavor (such as in probing non-standard neutrino interactions, e.g. \cite{Harnik:2012ni,Dutta:2017nht}).}.
\subsection{Flux sources}
\begin{figure}
\centering
\includegraphics[width=.6\textwidth]{plots/NuFlux/Snolab_NeutrinoFlux.pdf}
\caption{\label{fig:neutrino_flux} Neutrino flux components comprising the ``neutrino floor'' at the SNOLAB location. Details of the specific flux components can be found in \Tab{tab:nucomponents}. The atmospheric neutrino contribution has been summed over all the neutrino flavors.}
\end{figure}
\subsubsection{Solar neutrinos}
Solar electron neutrinos $\nu_e$ are produced as a byproduct of nuclear fusion reactions in the Sun (see \cite{Robertson:2012ib} for review).
They vary in flux and energy, depending on the specific step in the reaction chain that they originate from. Around $98\%$ of the Sun's energy is produced via the proton-proton cycle, starting with $p + p \rightarrow ~^{2}{\rm H} + e^+ + \nu_e$ (yielding $pp$, $hep$, $pep$, $^{7}$Be, $^{8}$B neutrinos). The remaining energy is released in the Carbon-Nitrogen-Oxygen cycle (yielding $^{13} $N, $^{15}$O, $^{17}$F neutrinos). Solar neutrinos provide the dominant background for direct detection experiments for energies $E_{\nu} \lesssim 20$ MeV.
For our analysis, we take solar neutrino fluxes from a high metallicity Standard Solar Model GS98 of \cite{Serenelli:2011py} (Table 2), which shows good agreement with the helioseismological studies. Since for a range of parameters $^{8}$B neutrinos provide the dominant background contribution, some of the earlier neutrino floor-related studies have focused solely on this single solar neutrino component (e.g. the target complementarity analysis of \cite{Ruppin:2014bra}). Throughout this work, for completeness, we consider contributions from all of the solar neutrinos.
\subsubsection{Atmospheric neutrinos}
Atmospheric neutrinos (for a review see \cite{Gaisser:2002jj}) provide the dominant direct detection background at energies $E_{\nu} \gtrsim 50$ MeV, although the spectrum extends to lower energies. They are produced by cosmic ray (primarily proton) collisions with the nuclei in the atmosphere. The resulting hadronic showers, mostly composed of pions, produce copious amounts of neutrinos from decays. Pion decay $\pi^+ \rightarrow \nu_{\mu} + \mu^+$, followed by $\mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_{\mu}$, allows for a simple prediction of 2-to-1 muon-to-electron neutrino ratio within the atmospheric flux.
\begin{table*}[tbp]
\setlength{\extrarowheight}{2pt}
\setlength{\tabcolsep}{10pt}
\begin{center}
\begin{threeparttable}
\begin{tabular}{|l|l|c|l|} \hline
~~~~~~~~Neutrino Flux & ~~~~~~~~Total Flux & Maximum Energy & ~~~~Reference \\
~~~~~~~~~~Component & ~~~~~~~~(cm$^{-2}$ s$^{-1}$) & $E_{\nu}$, (MeV) & ~~~~~(model) \\ \hline
\hline
Solar ($\nu_e$, pp) & $5.98 (1 \pm 0.006) \times 10^{10}$ & ~~~$0.42$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, pep [line]) & $1.44 (1 \pm 0.012) \times 10^{8}$ & ~~~$1.45$ & \cite{Serenelli:2011py} (GS98)\\ \hline
Solar ($\nu_e$, hep) & $8.04 (1 \pm 0.300) \times 10^{3}$ & ~~$18.77$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{7}$Be [line-1]) & $5.00 (1 \pm 0.070) \times 10^{8}$ & ~~~$0.39$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{7}$Be [line-2]) & $4.50 (1 \pm 0.070) \times 10^{9}$ & ~~~$0.87$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{8}$B) & $5.58 (1 \pm 0.140) \times 10^{6}$ & ~~$16.80$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{13}$N) & $2.96 (1 \pm 0.140) \times 10^{8}$ & ~~~$1.20$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{15}$O) & $2.23 (1 \pm 0.150) \times 10^{8}$ & ~~~$1.73$ & \cite{Serenelli:2011py} (GS98) \\ \hline
Solar ($\nu_e$, $^{17}$F) & $5.52 (1 \pm 0.170) \times 10^{6}$ & ~~~$1.74$ & \cite{Serenelli:2011py} (GS98) \\ \hline
\hline
Atm. ($\nu_e$) & $1.27 (1 \pm 0.500) \times 10^{1}$ & $944.00$ & \cite{Battistoni:2005pd} (FLUKA) \\ \hline
Atm. ($\overline{\nu}_e$) & $1.17 (1 \pm 0.500) \times 10^{1}$ & $944.00$ & \cite{Battistoni:2005pd} (FLUKA) \\ \hline
Atm. ($\nu_{\mu}$) & $2.46 (1 \pm 0.500) \times 10^{1}$ & $944.00$ & \cite{Battistoni:2005pd} (FLUKA) \\ \hline
Atm. ($\overline{\nu}_{\mu}$)\tnote{1} & $2.45 (1 \pm 0.500) \times 10^{1}$ & $944.00$ & \cite{Battistoni:2005pd} (FLUKA) \\ \hline
\hline
DSNB ($\nu_e$, $T_{\nu} = 3$ MeV) & $4.55 (1 \pm 0.500) \times 10^{1}$ & ~~$36.90$ & \cite{Horiuchi:2008jz} (th.~avrg.)\tnote{1} \\ \hline
DSNB ($\overline{\nu}_e$, $T_{\nu} = 5$ MeV) & $2.73 (1 \pm 0.500) \times 10^{1}$ & ~~$57.01$ & \cite{Horiuchi:2008jz} (th.~avrg.)\tnote{1} \\ \hline
DSNB ($\nu_x$, $T_{\nu} = 8$ MeV)\tnote{2} & $1.75 (1 \pm 0.500) \times 10^{1}$ & ~~$81.91$ & \cite{Horiuchi:2008jz} (th.~avrg.)\tnote{1} \\ \hline
\hline
Reactor ($\overline{\nu}_e$, $^{235}$U)\tnote{3} & $1.88(1 \pm 0.080) \times 10^5$ & ~~10.00 & ~~(combined)\tnote{4} \\
\hline
\hline
Geo. ($\overline{\nu}_e$, $^{40}$K) & $2.19 (1 \pm 0.168) \times 10^{7}$ & ~~~1.32 & \cite{huang:2013geomodel} (global)\tnote{5} \\
\hline
Geo. ($\overline{\nu}_e$, $^{238}$U) & $4.90 (1 \pm 0.200) \times 10^{6}$ & ~~~3.99 & \cite{huang:2013geomodel} (global)\tnote{5} \\
\hline
Geo. ($\overline{\nu}_e$, $^{232}$Th) & $4.55 (1 \pm 0.257) \times 10^{6}$ & ~~~2.26 & \cite{huang:2013geomodel} (global)\tnote{5} \\
\hline
\end{tabular}
\begin{tablenotes}
\item[1] Average of several theoretical models.
\item[2] $\nu_x$ is the total contribution from all other neutrinos and antineutrinos.
\item[3] Only the most dominant element is considered.
\item[4] Combined result from multiple nearby reactors.
\item[5] Global Earth model, incorporates several theoretical models.
\end{tablenotes}
\caption{\label{tab:nucomponents} Neutrino flux components that contribute to the coherent neutrino scattering background in direct detection experiments at the SNOLab location.~Contributions from solar, atmospheric, diffuse supernovae, reactor as well as geo--neutrinos are shown.}
\label{tab:nuflux}
\end{threeparttable}
\end{center}
\end{table*}
The atmospheric neutrino flux depends on the laboratory location. This is primarily due to the effect of the location-dependent Earth's geo-magnetic field, which results in a rigidity cut-off for the flux of charged parent cosmic ray and secondary particles. Atmospheric neutrino flux predictions from several models have been employed in the neutrino-oscillation experiments \cite{Abe:2011ph}, known colloquially as ``Bartol'' \cite{Barr:2004br}, ``Honda'' \cite{Honda:2006qj,Honda:2015fha} and ``FLUKA'' \cite{Battistoni:2007zzb} fluxes. The only available atmospheric flux predictions for $E_{\nu} < 100$ MeV are the tabulated results of FLUKA (see Appendix A, Table 2-4, of \cite{Battistoni:2005pd}), and we employ the values for the Kamioka site in this work\footnote{While we do not consider the Kamioka site in this work this provides us with a reasonable estimate for other laboratory locations since we are only interested in the total flux and not specific components.}. Due to large modeling uncertainties, especially at lower energies, we conservatively take the systematic error on the predicted fluxes to be $50\%$.
\subsubsection{Diffuse supernova neutrino background}
The diffuse supernova neutrino background (DSNB)\footnote{Also known as ``supernovae relic neutrinos'' in the older literature.} refers to neutrinos and anti-neutrinos originating from all of the past core-collapse supernovae (for a review see~\cite{Beacom:2010kk}). The signal, which includes red-shifted contributions from various epochs, is effectively isotropic in space and time-independent for experiments.
The DSNB flux, over $4 \pi$, is given by \cite{Horiuchi:2008jz}
\begin{equation}
\phi_{\nu} (E) = \int R_{\rm CCSN}(z) \dfrac{dN(E^{\prime})}{dE^{\prime}} (1 + z)\Big|\dfrac{dt}{dz}\Big| dz~,
\end{equation}
where $z$ is the redshift, $E^{\prime} = E(1 + z)$, $R_{\rm CCSN}(z)$ is the historical rate of core-collapse supernovae, $dN/dE^{\prime}$ is the time-integrated neutrino spectrum per supernova and $t$ is the cosmological time that is related to redshift as $|dz/dt| = H_0 (1 + z) [\Omega_m (1+z)^3 + \Omega_{\Lambda}]^{1/2}$, where the adopted approximate parameter values are for the present day $\Lambda$CDM cosmology, with $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$ for the Hubble constant as well as $\Omega_m = 0.3$ and $\Omega_{\Lambda} = 0.7$ for the matter and the cosmological constant density fractions, respectively. Provided the historical supernovae rate, the DSNB flux depends only on the effective neutrino temperature $T_{\nu}$ of the respective neutrino-sphere. The neutrino spectrum, which can be well approximated by the Fermi-Dirac
distribution with zero chemical potential \cite{Kotake:2005zn}, depends on $T_{\nu}$ as
\begin{equation}
\dfrac{dN}{dE^{\prime}_{\nu}} = E_{\nu}^{\rm tot} \dfrac{20}{7 \pi^4} \dfrac{(E_{\nu}^{\prime})^2 }{T_{\nu}^4} \dfrac{1}{\left(e^{E_{\nu}^{\prime}/T_{\nu}} + 1\right)}~,
\end{equation}
where $E_{\rm tot} \simeq 3 \times 10^{53}$ erg is the approximate total energy released from a supernova explosion.
The DSNB component is expected to significantly contribute to the neutrino background of direct detection experiments in the $20~{\rm MeV} \lesssim E_{\nu} \lesssim 50$ MeV energy range. In this work we employ DSNB fluxes from \cite{Strigari:2009bq} (Figure~1), which were obtained using the above formalism assuming the following neutrino temperatures: $T_{\nu_e} = 3$ MeV for $\nu_e$, $T_{\overline{\nu}_e} = 5$ MeV for $\overline{\nu}_e$ and $T_{\nu_x} = 8$ MeV for the combined contribution from all other neutrinos and anti-neutrinos, denoted as $\nu_x$. These temperature values represent an approximate average of different theoretical models found in the literature, as summarized in Table 3 of \cite{Horiuchi:2008jz}. Due to large model uncertainties the systematic errors on DSNB fluxes are taken to be 50\%.
\subsubsection{Reactor neutrinos}
Reactor anti-neutrinos $\bar{\nu}_e$ (for a review see \cite{Hayes:2016qnu}) originate from the $\beta$-decay of unstable isotopes from reactor fuel fissions. Since the isotopes are short-lived, the corresponding neutrino flux directly follows the reactor operation.
The core elements include $^{235}$U, $^{238}$U, $^{239}$Pu and $^{241}$Pu. A typical reactor core (e.g. \cite{Declais:1994ma}) contains these fuel elements in an approximate ratio of 0.6 : 0.08 : 0.3 : 0.05. As the reactor operates, nuclear processes change the element composition. In our work we only focus on the dominant $^{235}$U contribution and take it to be approximately constant throughout the reactor operation, neglecting element recomposition that depends on each reactor's specifications. An example of analysis including these effects can be found in \cite{Murayama:2000iq}.
The flux of reactor anti-electron neutrinos is given by
\begin{equation}
\phi_k(E) = \frac{R_{\bar{\nu}_e}}{4\pi d^2}S_k (E) \, ,
\end{equation}
where $R_{\bar{\nu}_e}$ is the emitted rate of reactor neutrinos, $d$ is the distance from a given reactor to the laboratory and $S_k(E)$ is the neutrino spectrum for isotope $k$. Approximate analytic expressions for $S_k (E)$ have been developed in \cite{Vogel:1989iv,Mueller:2011nm,Huber:2011wv,Murayama:2000iq}. We employ the model of \cite{Mueller:2011nm}, which is based on a phenomenological fit to data with an exponentiated polynomial, with the resulting spectrum being
\begin{equation}\label{eq:reactor_nuspec}
S_k (E_\nu) = \dfrac{dN_{\nu}}{dE_{\nu}} = \exp \left(\sum_{i=1}^{6} \alpha_{i,k}E_\nu^{i-1} \right) ~,
\end{equation}
where $\alpha_{i,k}$ is the respective fit coefficient of order $i$. \Tab{tab:reactor_fit} displays the values of the best fit coefficients as obtained by \cite{Mueller:2011nm}\footnote{Strictly, this is only valid for energies $\gtrsim 1.8$ MeV. However, calculations of \cite{Vogel:1989iv} do not show substantial deviations for energies above $0.5$ MeV, allowing us to truncate the distributions at that point. The presented results are insensitive to this choice.}.
\begin{table*}[tbp]
\setlength{\extrarowheight}{2pt}
\setlength{\tabcolsep}{10pt}
\begin{center}
\begin{tabular}{|c|c|c|c|c|} \hline
$k$ & $^{235}U$ & $^{238}U$ & $^{239}P$ & $^{241}P$ \\ \hline
\hline
1 & 3.217 & 4.833 & 6.413 & 3.251\\ \hline
2 & -3.111 & 1.927 & -7.432 & -3.204 \\ \hline
3 & 1.395 & -1.283 & 3.535 & 1.428 \\ \hline
4 & -3.690 & -6.762 & -8.820 & -3.675 \\ \hline
5 & 4.445 & 2.233 & 1.025 & 4.254 \\ \hline
6 & -2.053 & -1.536 & -4.550 & -1.896 \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:reactor_fit} Fitted values of the reactor neutrino spectrum $\alpha_{i, k}$ coefficients, used in \Eq{eq:reactor_nuspec}, for dominant nuclear isotopes, from \cite{Mueller:2011nm}.}
\end{table*}
The neutrino emission rate is given by
\begin{equation}
R_{\bar{\nu}_e} = N_{\nu, {\rm fiss}} \, \frac{P_{th}}{E_{\rm fiss}} e ~,
\end{equation}
where $N_{\nu, {\rm fiss}} = 6$ is the average number of anti-neutrinos produced per fission, $P_{th}$ is the power output of the reactor, $E_{\rm fiss}$ is the fission energy that is around 200 MeV for all major isotopes and $e = 0.75 \, (1 \pm 0.080) $ is the average reactor operation efficiency that includes shut-downs \cite{usenergy}. The uncertainty on the efficiency reflects an $\sim 8\%$ uncertainty on the reactor anti-neutrino spectrum (see Table 3 of \cite{Kopeikin:2012zz}), which is also the source of the reactor flux uncertainty as specified in \Tab{tab:nuflux}.
In \Tab{tab:reactors} we list the reactors considered in this work, which constitute the dominant reactor-neutrino sources for the SNOLab, the LNGS and the SURF laboratories \footnote{Note that the Pickering Nuclear Generating Station is not included in \Tab{tab:reactors}. The reason is that the experiments we are interested in studying, those likely to reach the neutrino floor, are unlikely to be taking data by 2024, while this Station will still operate. After running for nearly 40 years, the reactor operations are expected to cease in August 2018. However, there is a strong movement to extend the operations until 2024, at which time it will be decommissioned.}. For the reactor neutrino flux we have calculated their combined near-by reactor contribution at the laboratory site using the above formalism.
\begin{table}[tbp]
\setlength{\extrarowheight}{2pt}
\setlength{\tabcolsep}{10pt}
\begin{center}
\begin{tabular}{|l|m{2.3cm}|c|c|c|} \hline
Nuclear Reactor Name & Location & Nearest Lab & Distance (km) & Output (MW) \\ \hline \hline
Cooper & 40$^\circ$21$'$43$''$\,N 95$^\circ$38$'$29$''$\,W & SURF & 801 & 830 \\ \hline
Monticello & 45$^\circ$20$'$01$''$\,N 93$^\circ$50$'$57$''$\,W& SURF & 788 & 671 \\ \hline
Prarie Island & 44$^\circ$37$'$18$''$\,N 92$^\circ$73$'$59$''$\,W & SURF & 835 & 1096 \\ \hline
Nine Mile Point & 43$^\circ$31$'$15$''$\,N 76$^\circ$24$'$25$''$\,W & SNOLAB & 498 & 1761 \\ \hline
R.E. Ginna & 43$^\circ$16$'$40$''$\,N 77$^\circ$18$'$36$''$\,W & SNOLAB & 468 & 610 \\ \hline
James A. Fitzpatrick & 43$^\circ$31$'$04$''$\,N 76$^\circ$23$'$09$''$\,W & SNOLAB & 500 & 838 \\ \hline
Point Beach & 44$^\circ$16$'$52$''$\,N 87$^\circ$32$'$12$''$\,W & SNOLAB & 552 & 1200 \\ \hline
Enrico Fermi & 41$^\circ$57$'$46$''$\,N 83$^\circ$15$'$27$''$\,W & SNOLAB & 527 & 1198 \\ \hline
Davis Besse & 41$^\circ$35$'$48$''$\,N 83$^\circ$05$'$11$''$\,W & SNOLAB & 563 & 889 \\ \hline
Perry & 41$^\circ$48$'$03$''$\,N 81$^\circ$08$'$36$''$\,W & SNOLAB & 519 & 1261 \\ \hline
Bruce & 44$^\circ$19$'$31$''$\,N 81$^\circ$35$'$58$''$ W & SNOLAB & 240 & 6384 \\ \hline
Darlington & 43$^\circ$55$'$22$''$\,N 78$^\circ$43$'$11$''$\,W & SNOLAB & 343 & 3512 \\ \hline
Tricastin & 44$^\circ$19$'$47$''$\,N 04$^\circ$43$'$56$''$\,E & LNGS & 744 & 3820 \\ \hline
Cruas & 44$^\circ$37$'$59$''$\,N 04$^\circ$45$'$29$''$\,E & LNGS & 750 & 3842 \\ \hline
Saint-Alban & 45$^\circ$24$'$16$''$\,N 04$^\circ$45$'$19$''$\,E & LNGS & 778 & 2600 \\ \hline
Bugey & 45$^\circ$47$'$54$''$\,N 05$^\circ$16$'$15$''$\,E & LNGS & 760 & 3724 \\ \hline
\end{tabular}
\end{center}
\caption{\label{tab:reactors} List of most relevant nuclear reactors for SURF, SNOLAB, and LNGS. Columns contain, from left to right, the name of the reactor, the GPS location, the laboratory for which the reactor is relevant, the distance to the laboratory in kilometers, and the output of the reactor in MW~\cite{USNuclear,CanadaNuclear,EUNuclear}}
\end{table}
\subsubsection{Geo-neutrinos}
Geo-neutrinos are predominantly electron anti--neutrinos $\overline{\nu}_e$ originating from the $\beta$-decay branches of the major Earth's heat-producing nuclear reactions, involving isotopes of potassium $^{40}$K, thorium $^{232}$Th and uranium $^{238}$U. Recently, KamLAND~\cite{Araki:2005qa} as well as Borexino~\cite{Bellini:2010hy} have definitively observed a geoneutrino flux.
The spectrum for each of these elements is taken from \cite{Araki:2005qa} (Figure 1). The respective location-dependent total flux is predicted from a geophysically-based three-dimensional global Earth model of heat-producing element distribution \cite{huang:2013geomodel} (Table 1). The systematic uncertainties are taken to be the larger of the quoted $\pm 1\sigma$ fluctuations in these values.
\subsection{Coherent neutrino scattering interactions}
If the neutrino energy is not sufficient to discern individual quarks or nucleons, neutrinos have coherent elastic scattering off the whole nucleus through the weak neutral current \cite{Freedman:1977xn,Drukier:1983gj}.
At low momentum transfer $|\vec{q}|$ the coherence condition for target nucleons to interact in phase is $|\vec{q}| R \ll 1$, where $R$ is the nuclear radius. The total cross-section for coherent interaction scales as the square of the number of participating nucleons. While heavier targets thus enjoy a dramatically enhanced neutrino interaction cross-section, they are also penalized with a smaller maximum recoil energy.
Recent observations by the COHERENT experiment have definitively confirmed this process \cite{Akimov:2017ade}. At energies above $E_{\nu} \sim 50$ MeV, other channels, such as quasi-elastic scattering (QE) and deep-inelastic-scattering (DIS), start to dominate (for review see \cite{Formaggio:2013kya}).
The Standard Model coherent-scattering neutrino-nucleus cross-section is given by
\begin{equation}
\dfrac{d \sigma^T (E_{\nu}, E_R)}{d E_R} = \dfrac{G_f^2}{4 \pi} Q_w^2 m_{T} \left(1 - \dfrac{m_T E_R}{2 E_{\nu}^2}\right) F_{{\rm SI},T}^2 (E_R)~,
\end{equation}
where $m_T$ is target nucleus mass, $G_f$ is Fermi coupling constant, $F_{{\rm SI},T}(E_R)$ is the form factor (as before, we take this to be the Helm form factor \cite{Helm:1956zz}),$Q_w = (1 - 4 \sin^2 \theta_{\rm W}) Z-N$ is the weak nuclear charge, $N$ is the number of neutrons, $Z$ is the number of protons and $\theta_{\rm W}$ is the Weinberg angle. Since $\sin^2 \theta_{\rm W} = 0.223$ \cite{Patrignani:2016xqp}, the coherent neutrino-nucleus scattering cross-section follows an approximate $N^2$ scaling.
Assuming good electron-tagging in future direct detection experiments, we neglect in this work $\nu_e + e^- \rightarrow \nu_e + e^-$ neutrino-electron scattering\footnote{This scattering has been also studied in the context of direct detection experiments before (e.g. \cite{Billard:2013qya}). See \cite{Marciano:2003eq} for a list of possible new physics topics associated with this process, and~\cite{Essig:2018tss,Wyenberg:2018eyv} for a recent discussion of solar neutrinos as a background for electron recoil analyses in direct detection experiments.}.
\begin{table*}[tb]
\setlength{\extrarowheight}{2pt}
\small
\begin{center}
\begin{threeparttable}
\begin{tabular}{|l|c|c|c|c|c|c|c|} \hline
\multirow{2}{*}{Target Material} & \multirow{2}{*}{ $A (Z)$} & Isotope & \multirow{2}{*}{ $J$} & \multirow{2}{*}{ $\langle S_p \rangle$} & \multirow{2}{*}{ $\langle S_n \rangle$} & \multirow{2}{*}{ $\lambda/\lambda_N$} & Energy \\
& & Fraction & & & & & Range (keVnr) \\ \hline
\hline
Xenon (Xe) & 124 (54) & 0.001 & 3/2 & -0.009 & -0.227 & 0.692 & 0.1--50 (10--300)\tnote{2} \\ \hline
& 126 (54) & 0.001 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 128 (54) & 0.019 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 129 (54) & 0.264 & 1/2 & 0.028 & 0.359 & -0.778 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 130 (54) & 0.041 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 131 (54) & 0.212 & 3/2 & -0.009 & -0.227 & 0.692 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 132 (54) & 0.269 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 134 (54) & 0.104 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50 (10--300)\tnote{2}\\ \hline
& 136 (54) & 0.089 & 0 & 0.0 & 0.0 & 0.0 & 0.1--50(10--300)\tnote{2} \\ \hline
Germanium (Ge) & 70 (32) & 0.208 & 0 & 0.0 & 0.0 & 0.0 & 0.04--50 (10--300)\tnote{2} \\ \hline
& 72 (32) & 0.275 & 0 & 0.0 & 0.0 & 0.0 & 0.04--50 (10--300)\tnote{2}\\ \hline
& 73 (32) & 0.077 & 9/2 & 0.038 & 0.37 & -0.879 & 0.04--50 (10--300)\tnote{2} \\ \hline
& 74 (32) & 0.363 & 0 & 0.0 & 0.0 & 0.0 & 0.04--50 (10--300)\tnote{2} \\ \hline
& 76 (32) & 0.076 & 0 & 0.0 & 0.0 & 0.0 & 0.04--50 (10--300)\tnote{2} \\ \hline
Argon (Ar) & 40 (18) & 0.996 & 0 & -- & -- & 0 & 1--50 (10--300)\tnote{2} \\ \hline
Sodium (Na) & 11 (23) & 1 & 3/2 & 0.248 & 0.020 & 2.218 & 1--50 (10--300)\tnote{2}\\ \hline
Iodine (I) & 127 (53) & 1 & 5/2 & 0.309 & 0.075 & 2.813 & 1--50 (10--300)\tnote{2}\\ \hline
Fluorine (F)\tnote{1} & 19 (9) & 1 & 1/2 & 0.477 & -0.004 & 2.629 & -- \\ \hline
Silicon (Si)\tnote{1} & 28 (14) & 0.922 & 0 & 0.0 & 0.0 & 0.0 & -- \\ \hline
& 29 (14) & 0.047 & 1/2 & -0.002 & 0.130 & -0.555 & -- \\ \hline
& 30 (14) & 0.031 & 0 & 0.0 & 0.0 & 0.0 & -- \\ \hline
Helium (He)\tnote{1} & 4 (2) & 1 & 0 & 0.0 & 0.0 & 0.0 & -- \\ \hline
\end{tabular}
\begin{tablenotes}
\item[1] This element is not considered in our work and is shown for completeness.
\item[2] For momentum suppressed interactions (i.e. those whose differential cross sections are proportional to $q^{b}$ with $b > 0$), the DM nuclear recoil spectrum can extend to larger recoil energies than for other interactions (see e.g.~\cite{Aprile:2017aas}). Thus, for these interactions our analyses are performed over a low energy range and a high energy range (the latter indicated in parenthesis), and the result is taken to be the stronger of the two.
\end{tablenotes}
\caption{\label{tab:experiments} Experimental configurations considered in this work (except fluorine, silicon and helium, which are included for completeness). Shown in columns from left to right are: nucleon number and the nuclear charge, isotope fraction (rounded to three decimal places), total nuclear spin, the expectation values of the proton and neutron spin content, the expected nuclear neutron spin, the nuclear magnetic moment $\lambda$ (in terms of the nuclear magneton $\lambda_B$) and the range of recoil energies that a particular experiment is sensitive to (a perfect detection efficiency is assumed throughout the entire range). Nuclear properties are taken from \cite{Tovey:2000mm, webelements}. }
\label{tab:experiments}
\end{threeparttable}
\end{center}
\end{table*}
\section{Detection\label{sec:detection}}
\subsection{Considered experimental configurations}
Making definitive statements about the scientific capabilities of future direct detection experiments requires explicit assumptions about their size, composition, energy resolution and detection threshold. Various proposals for the next multi-ton scale generation of experiments have been put forth \cite{Cushman:2013zza}, but which experiments will be constructed and what their respective ultimate characteristics will be remains uncertain. Thus, we study the future scientific reach of a variety of potential experimental configurations that could reside near the optimistic edge of such realizations.
Specifically, \Tab{tab:experiments} lists the five experimental configurations we consider, each with a different target element (xenon, argon, sodium, germanium, iodine). Some proposals have considered a fluorine target, but these are energy threshold experiments that cannot measure the recoil spectrum, and we thus do not study them\footnote{Fluorine will assist in breaking model degeneracy if detection is already made with other experiments~\cite{Gluscevic:2015sqa}.}. It has been shown in~\cite{Ruppin:2014bra} that fluorine could be extremely useful to help disentangle various DM models, should a detection be made in another target elements, as its nuclear properties differ strongly from other direct detection target elements.
We optimistically assume that experiments have perfect detection efficiency and resolution in the energy range provided. Furthermore, the background is assumed to arise exclusively from neutrinos. While for sodium and iodine this assumption is at the present time not realistic, it allows us to analyze the intrinsic properties of all the target elements on the same footing.
Before continuing we briefly comment on the choice of assumptions for the energy thresholds adopted in this work, which can be extremely import in determining sensitivity to low mass WIMPs. Xenon experiments have historically been able to probe recoils down to $\sim 1$ keV (see Figure 1 of~\cite{Aprile:2017iyp}), albeit not with perfect efficiency. However, xenon experiments do have the ability to probe lower energies via specialized analyses (see e.g.~\cite{Aprile:2016wwo}), and thus we consider an optimistic threshold of 0.1 keV. The projected threshold for the next generation of germanium detectors from SuperCDMS SNOLAB reaches energy values as low as 40 eV~\cite{Agnese:2016cpb}\footnote{We note that the 40 eV value chosen by the SuperCDMS collaboration as the cutoff of their ionization yield is somewhat optimistic because the Lindhard model on which this choice is based has not be tested below $\mathcal{O}(100)$ eV. Also this threshold only applies to the HV detectors which are not capable of discriminating nuclear and electronic recoils, thus the zero background assumption adopted in this analysis may not be realistic.}, which is the value adopted here. Typically, argon based experiments have projected higher thresholds for future experiments like DarkSide-20k than the value of 1 keV adopted here (typically quoted values are not below $\sim 10$ keV, see e.g.~\cite{Aalseth:2017fik}), however a recent analysis by the DarkSide-50 experiment has demonstrated that a specialized analysis~\cite{Agnes:2018ves} can be performed that extends the reach of argon-based experiments down to 0.6 keV while maintaining a low background, potentially making our analysis slightly conservative. Finally, the energy threshold in sodium and iodine experiments is typically only slightly larger than the 1 keV value adopted here (see e.g.~\cite{Thompson:2017yvq}), implying our adopted values represent reasonable benchmarks.
Finally, we have taken into account that momentum suppressed interactions (i.e. those with differential cross sections containing positive powers of the momentum transfer) can produce significant scattering rates at recoil energies larger than 50 keV, which we have chosen as the upper limit of our analysis range (see e.g.~\cite{Aprile:2017aas}). Typically, experiments place an upper limit on the recoil energy to the search window assuming the conventional SI and SD interactions which predict the bulk of the DM recoils to be below $\sim 50$ keV (although this depends on the target and DM particle mass); however experiments can analyze larger recoil energies should their be a sufficient reason to look in this energy range (see e.g.~\cite{Aprile:2017aas}). Thus, for momentum suppressed interactions, we perform our analysis over two energy regions, one extended from the aforementioned thresholds to 50 keV, and the other running from 10 keV to 300 keV. The derived experimental sensitivity is then taken to be the stronger of the two.
\begin{figure}
\mycenter{
\includegraphics[width=.41\textwidth]{plots/SpectrumComparison/Xenon_B8_Recoil_Spectrum_Comparison_Elastic.pdf}
\includegraphics[trim={12.5mm 0 0 0},clip,width=.3875\textwidth]{plots/SpectrumComparison/Xenon_DSNB8MeV_Recoil_Spectrum_Comparison_Elastic.pdf}
\includegraphics[trim={12.5mm 0 0 0},clip,width=.3875\textwidth]{plots/SpectrumComparison/Xenon_ATM_Recoil_Spectrum_Comparison_Elastic.pdf}}
\caption{\label{fig:Dark_Matter_Degeneracy} Comparison of recoil spectra induced in a xenon target from $^8 {\rm B}$ (left), DSNB (middle), and atmospheric (right) neutrinos with the recoil spectra arising in various DM interaction models. Dashed lines are used to represent the light mediator limit of a particular interaction. Interactions not shown are understood to be approximately degenerate with one of the shown DM models for the particular target element and energy range considered. The best fit mass values are given in Table 10.}
\end{figure}
\subsection{Fitting DM and neutrino recoil spectra}
In the following sections we study the extent to which future direct detection experiments will be sensitive to a variety of DM candidates. Provided an experiment is capable of probing background neutrinos, the projected sensitivity will intimately depend on the shapes of the DM and background neutrino recoil spectra. Thus, in order to gain intuition about the sensitivity to a particular DM model, we perform here a brief analysis of the extent to which an observed recoil spectrum arising from background neutrinos could be misinterpreted in terms of various DM models.
Specifically, for a particular experimental configuration (defined by a target element and energy range), we divide the experimental energy range into 1000 log-spaced intervals and calculate the differential recoil rate arising from a particular source of background neutrinos in each interval (in particular we focus on either $^8$B, DSNB, or atmospheric neutrinos). For each DM model, we minimize the statistic
\begin{equation}\label{eq:gof}
\chi_{\rm weighted}^2 = \sum_i \frac{(R_{{\rm DM},i} - R_{\nu,i})^2}{R_{\nu,i}}\, ,
\end{equation}
where $R_{{\rm DM},i}$ and $R_{\nu,i}$ are the DM and neutrino predicted rates in bin $i$, allowing both the WIMP mass as well as the normalization of the DM recoil spectrum (given by $\sigma_{\rm eff}$) to vary. Starting with the usual definition of the $\chi^2$-statistic, we obtain the statistic of \Eq{eq:gof} by inserting the corresponding number of events of DM and neutrinos in each bin, $N_{{\rm DM}, i} = M T R_{{\rm DM},i}$ and $N_{\nu,i}= MT R_{\nu,i}$, with the uncertainty taken to be $N_{\nu, i}$ to be $\sigma_{\nu,i} = \sqrt{MT R_{\nu,i}}$. Dividing out the $MT$ factor gives a weighted $\chi^2$ statistic \cite{Baker:1983tu}: $\chi^2/ MT = \chi_{\rm weighted}^2$. Note that minimizing $\chi_{\rm weighted}^2$ does not provide a $\chi^2$-goodness of fit test (although multiplying the obtained minimum by $MT$ would). We are only interested in finding the best fit, not in evaluating how good the fit actually is. Given a particular DM model, this procedure identifies the DM mass maximally degenerate with the particular neutrino source.
Comparison of the ${}^8$B (left), DSNB (middle), and atmospheric (right) neutrino recoil spectra in a xenon-target experiment with the best-fit recoil spectrum derived for a variety of DM models is shown in \Fig{fig:Dark_Matter_Degeneracy}. In the fit we did not allow the DM mass to exceed 1~TeV~\footnote{Fit results for various interactions show a slight preference for very massive DM candidates. However, the change in the quality of the fit for a $\sim 10 {\, {\rm TeV}}$ DM candidate is only marginally better than that of a 1 TeV candidate.}.
Comparisons with other elements are shown in \Fig{fig:recoilspec_full} (App.~\ref{app:recoilspec}). The resulting best-fit DM masses for each particle model and element are shown in Appendix~\ref{app:recoilspec} \Tab{tab:bestfitDM_elastic_xenon} for xenon, \Tab{tab:bestfitDM_elastic_germanium} for germanium, \Tab{tab:bestfitDM_elastic_argon} for argon, \Tab{tab:bestfitDM_elastic_iodine} for iodine and \Tab{tab:bestfitDM_elastic_sodium} for sodium. For each plot in \Fig{fig:Dark_Matter_Degeneracy} and \Fig{fig:recoilspec_full}, recoil spectra for interactions not explicitly shown are understood to be degenerate with one of the plotted DM spectra. An interaction that produces a degenerate recoil spectra in all three neutrino plots (for a particular target element) to one of the shown interactions, is referred to as `similar to' the shown degenerate interaction throughout the remainder of the paper, as the qualitative features of these interactions with respect to the neutrino degeneracy will be identical.
As can be seen in \Fig{fig:Dark_Matter_Degeneracy}, it is the momentum dependence of the cross section which determines the physical shape of the recoil spectrum. The SI massless mediator interaction has a $1/|\vec{q}|^4$ dependence and is thus extremely peaked at low energies. The ED and MD massless mediator interaction have a $1/|\vec{q}|^2$ dependence and are thus still peaked, however not to the same extent as the SI massless mediator interaction. The SI massive mediator interaction, being independent of the momentum transfer, is much flatter at lower energies. It is actually \emph{only} for these momentum-independent interactions (as pointed out in ~\cite{Dent:2016iht} and \cite{Dent:2016wor} using EFT operators) that the shape of the recoil spectra can match the recoil spectra of background neutrinos. Spectra arising from $d \sigma_{\chi}/dE_R \sim |\vec{q}|^2, |\vec{q}|^4, |\vec{q}|^6$ are also physically distinct from the neutrino spectrum, as can be readily seen for ED, MD and PS-PS with a heavy mediator (denoted with green, blue and cyan solid lines), and have the unique feature of producing a maximum scattering rate at some non-zero value of $E_{\rm R}$.
Thus, it is clear that \emph{only} DM-nucleus interactions whose differential cross-sections are independent of $|\vec{q}|$ lead to spectral degeneracy with $^8$B neutrinos. These neutrinos constitute the most important background for experiments with realistic exposures (i.e. MT $\lesssim 100$ ton $\cdot$ yrs) since they provide the largest contribution to the background, and hence they are the most relevant to analyze. Similar considerations relating the $|\vec{q}|$ dependence of the DM scattering cross-section to spectral degeneracy can be made for the other neutrino backgrounds. However, the degeneracies with these backgrounds are not as detrimental for the exposures considered in this work.
\subsection{Discovery Limit Statistical Analysis \label{sec:like_fit}}
To establish the discovery sensitivity of experiments we use a frequentist analysis based on the profile likelihood ratio test \cite{Cowan:2010js,Rolke:2004mj}, the power-constrained limits \cite{Cowan:2011an}. This has been utilized in recent direct detection studies \cite{Billard:2011zj,Billard:2013qya,Aprile:2011hx,Ruppin:2014bra}. This test is performed by generating simulated datasets for each experiment, assuming for each dataset a particular DM interaction, WIMP mass $m_{\chi}$, reference DM cross section $\sigma_{\rm ref}^{\ast}$, and a normalization (i.e. an energy integrated neutrino flux) for each of the considered neutrino fluxes ${\phi}_{\nu_k}$ ($k = 1, 2 \dots n_{\nu}$, with $n_{\nu} = 14$), which we take here to be the average theoretical predicted value $\overline{\phi}_{\nu_k}$. More specifically, for each of the aforementioned parameters and experimental configurations we generate fake data consisting of a total number of ``observed'' events $N_{\rm o}$ at particular recoil energies $E_j$ with $j = 1, 2, \dots N_{\rm o}$ that we use to define the following likelihood function
\begin{equation} \label{eq:likemain}
\mathcal{L} = \Bigg\{\frac{e^{-N_{\rm E}}}{N_{\rm o}!} \prod_{j=1}^{N_{\rm o}}{\rm MT} \Big(\dfrac{d R_{\rm tot}}{d E_R}\biggr|_{E_{\rm R}=E_j} \Big) \Bigg\} \prod_{k=1}^{n_{\nu}}\dfrac{1}{\sqrt{2 \pi}\sigma_{\nu_k}}{\rm exp}\biggr[-\left(\frac{\phi_{\nu_k} - \bar{\phi}_{\nu_k}}{\sqrt{2}\sigma_{\nu_k}}\right)^2 \biggr] .
\end{equation}
Here, the total differential rate is the sum of the signal (due to DM) and the background (due to all types of neutrinos) contributions
\begin{equation} \label{eq:drtot}
\dfrac{d R_{\rm tot}}{d E_R} = \frac{d R_\chi}{dE_{\rm R}} (\sigma_{\rm ref}, m_{\chi}) + \sum_{k}\frac{d R_{\nu_k}}{dE_{\rm R}}(\phi_{\nu_k})~,
\end{equation}
as defined in \Eq{eq:totdmrate} and \Eq{eq:totnurate}, respectively. The total number of predicted events $N_{\rm E}$ is obtained by integrating \Eq{eq:drtot} over the energy range of observation of the particular experiment, and multiplying by the exposure MT. In \Eq{eq:likemain} the extended likelihood in the curly brackets for the DM and neutrino events
is multiplied by a Gaussian product of likelihoods, centered around the mean predicted flux normalization $\overline{\phi}_{\nu_k}$, for each neutrino species $\nu_k$ ($k = 1, 2, \dots n_{\nu} = 14$) to take into account the systematic uncertainty in the flux $\phi_{\nu_k}$.
In the Gaussian likelihoods the $\sigma_{\nu_k}$ is the 1-$\sigma$ uncertainty in the particular flux (see \Tab{tab:nuflux}, where $\sigma_{\nu_k}$ is taken to be the largest of the two asymmetric 1-$\sigma$ uncertainties).
The procedure to obtain each set of simulated data involves two steps: 1) finding the total number of events of each type $t$ (i.e. $t = 1$ for the DM signal events and $t = k+1$ for the neutrino background events, with $k$ running as before), and 2) finding the corresponding recoil energy for each of the events. The number of events of a specific type, $n_{t}$, is found from a Poisson distribution $P_{t}$ as
\begin{equation}
P_{t} = \dfrac{\mu_{t}^{n_{t}} \, e^{- \mu_{t}}}{n_{t}!}~,
\end{equation}
where the mean $\mu_{t}$ is the number of events predicted by the model being tested, defined by a particular set of values $(\sigma_{\rm ref}^{\ast}, m_{\chi})$ for $t$ = 1, and the total neutrino flux $\bar{\phi}_{\nu_k}$ for $t = k + 1$. Choosing a random number for the cumulative probability distribution (CDF) of each $P_t$ one value of $n_{t}$ is randomly generated (inverse transform sampling).
The number $N_0$ of ``observed'' events is then $N_0 = \sum_{t = 1}^{15} n_t$.
To determine the energy of the $n_t$ events of each type we use as probability density function (PDF) the corresponding differential recoil rate $d R_{t}/d E_{R}$ normalized by the total rate (i.e. the rate integrated over the specified energy range for each experiment $R_{t}$) as
\begin{equation}
\Big(\text{PDF}\Big)_{t} = \dfrac{1}{R_{t}} \dfrac{d R_{t}}{d E_R}~.
\end{equation}
The corresponding $n_{t}$ recoil energies are again obtained with inverse transform sampling. With the above procedure, we simulate 250 - 500 datasets for each particle model, as specified by the choice of $(\sigma_{\rm ref}^{\ast}, m_{\chi})$, and experimental configuration.
For each simulated data set we define a test statistic $q_0$ that allows to reject the background only hypothesis $H_0$ (in which $\sigma_{\rm ref} = 0$) if it is true, with a probability not larger than some value $\alpha$ that denotes the significance level of the test. We further impose that the probability of not rejecting $H_0$ when the DM alternative hypothesis $H_{\sigma}$ (with $\sigma_{\rm ref} \neq 0$) is true, is less than some value $\beta$. For our analysis we chose $\alpha$ to correspond to 3$\sigma$ ($\alpha = 0.0135$) and $(1-\beta)$, which denotes the ``power'' of the test of $H_0$ with respect to the alternative hypothesis $H_{\sigma}$ (e.g. \cite{Patrignani:2016xqp}, Sec. 40), to be 90\%. The test statistic $q_0$ for each simulated dataset is the profile likelihood ratio, defined as
\begin{equation} \label{eq:likeratio}
q_0 =
\begin{cases}
- 2 \,\text{ln} \left( \dfrac{\mathcal{L}(\sigma_{\rm ref} = 0, \hat{\hat{\vec{\phi}}}_{\nu_k})}{\mathcal{L}(\hat{\sigma}_p, \hat{\vec{\phi}}_{\nu_k})} \right) ,& \hat{\sigma}_{\rm ref} \geq 0 \\
0 ,& \hat{\sigma}_{\rm ref} < 0 \, ,
\end{cases}
\end{equation}
where the $\phi_{\nu_k}$ are treated as nuisance parameters. For numerical calculations we parametrize $\sigma_{\rm ref} = 10^x$, which ensures that $\sigma_{\rm ref} > 0$\footnote{It has previously been verified that enforcing semi-positive values of the cross section does not affect the distribution of the test statistic, see e.g.~\cite{Billard:2011zj}.}. The hats refer to values that maximize the likelihood, with double-hats referring to values maximizing the likelihood subject to the constraint $\sigma_{\rm ref} = 0$. Note that by definition $q_0 \geq 0$, with larger values of $q_0$ indicating greater incompatibility of the simulated data with the background only hypothesis $H_0$. Thus, we require that the probability $p_0$ of having a $q_0$ value larger (i.e. more incompatible with the data if due to background only) than the ``observed'' (i.e. than the $q_0$ of the simulated dataset) $q_0^{\rm obs}$
\begin{equation}\label{eq:pval0}
p_0 = \int_{q_0^{\rm obs}}^\infty dq_0 \, f(q_0|H_0) \, ,
\end{equation}
is not larger than $\alpha$, $p_0 \leq \alpha$. Here, $f(q_0|H_0)$ is the PDF of obtaining $q_0$ under the background-only hypothesis $H_0$. In the large sample limit, Wilks' theorem ensures that $f(q_0|H_0)$ is given by a $\chi^2$ distribution with one degree of freedom (d.o.f.). This implies that the value of $\alpha$ corresponding to a $Z\sigma$ significance can be obtained by requiring $q_0^{\rm obs} \geq Z^2$. In the analyses detailed below, we have required a value of $\alpha$ corresponding to a $3\sigma$ significance (i.e. $Z = 3$). If at least 90\% of the simulated datasets generated with a particular set ($\sigma_{\rm ref}^{\ast} , m_{\chi}$) produce $q_0^{\rm obs} > 9$, then we accept this value of $\sigma_{\rm ref}^{\ast}$ as `discoverable'. That is to say that we require the probability $p_{\sigma}$ of having a value of $q_0^{\rm obs}$ larger than $Z^2$
\begin{equation}\label{eq:pval}
p_{\sigma} = \int_{Z^2}^\infty dq_0^{\rm obs} \, f(q_0^{\rm obs}|H_\sigma) \, ,
\end{equation}
to be $p_{\sigma} \geq 90\%$. The discovery limit of $\sigma_{\rm ref}$ of each model and for each DM mass is thus set by the value of $\sigma_{\rm ref}^{\ast}$ for which the aforementioned condition, $p_{\sigma} \geq (1 - \beta) = 90\%$, is strictly equal, i.e. $p_{\sigma} = 90\%$.
The procedure described above is widely used in modern high energy experiment analyses, for example in a typical LHC combined Higgs search \cite{ATLAS:2012ae} one assumes $(1-\beta) = 50\%$ and $\alpha$ corresponding to $5 \sigma$.
In practice we determine the fraction of simulated data sets that produce p-values $p_0^i \leq \alpha = 0.00135$ (i.e. those producing a $\geq 3\sigma$ detection of DM), by computing
\begin{equation}
f^{\rm 90}(\sigma_{\rm ref}^{\ast}, m_\chi) \equiv \sum_{i=1}^{N_{\rm sim}}\frac{1}{N_{\rm sim}} \begin{cases} 1 &\mbox{if } p_0^i \leq 0.00135 \\ 0 &\mbox{if } p_0^i > 0.00135 \end{cases} \, ,
\end{equation}
where $N_{\rm sim}$ is the number of simulated data sets with fixed parameters $(\sigma_{\rm ref}^{\ast}, m_{\chi})$. The value of $\sigma_{\rm ref}^{\ast}$ is varied until $f^{90}(\sigma_{\rm ref}^{\ast}, m_\chi) = 0.9$, thus identifying the cross section at which $90\%$ of experimental realizations are expected to obtain a $\geq 3\sigma$ detection of DM. For a fixed DM particle mass $m_\chi$, it is this value of $\sigma_{\rm ref}$ that defines the discovery limit. The full discovery limit for a particular experiment and interaction is then obtained by scanning over $m_{\chi}$.
Fig.~\ref{fig:xenon_floors_massive} and Fig.~\ref{fig:xenon_floors_massless} show the $3\sigma$ discovery reach for xenon, considering exposures of $0.1$ ton-year (dotted), $1$ ton-year (short dashed), $10$ ton-years (long dashed), and $100$ ton-years (solid). The results for other target elements are deferred to Appendix~\ref{app:discpot} (Fig.~8 to Fig.~15). The results (denoted by colored lines) are compared to discovery limits that would be obtained in the absence of background (black lines). For each DM model, $90\%$ CL upper limits are calculated for Xenon1T \cite{Aprile:2017iyp}, LUX \cite{Akerib:2016vxi}, PandaX-II \cite{Cui:2017nnn} and PICO \cite{Amole:2017dex} data. For PICO, LUX and PandaX-II, the experimental configurations (i.e. resolution functions, efficiencies, etc.) are as defined in~\cite{Witte:2017qsy}, but for PandaX-II we employ the updated results from~\cite{Cui:2017nnn}. For Xenon1T, the energy efficiency is taken from Fig.~1 of ~\cite{Aprile:2017iyp}, the fiducial mass and runtime are taken to be 1042 kg and 34.2 days, and a limit is set using Poisson statistics with zero observed events. We have verified that this procedure reproduces the published limit to a high degree of accuracy. The combined strongest upper limit for each mass and model defines the blue region labeled `Current DD Bounds'. Note that these upper limits cannot be directly translated to our calculated discovery potential results.
\begin{figure}[h]
\mycenter{
\includegraphics[trim={0mm 15.5mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_si_fnfp_si_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_sd_fnfp_sd_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_anapole_real_fnfp_anapole_real_1.00_Elastic.pdf}}
\mycenter{
\includegraphics[trim={0mm 15.5mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_magdip_fnfp_magdip_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_elecdip_fnfp_elecdip_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_AVV_fnfp_AVV_1.00_Elastic.pdf}
}
\mycenter{
\includegraphics[trim={0mm 0mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_f1_fnfp_f1_1.00_Elastic.pdf}
\includegraphics[trim={9mm 0mm 0 0cm},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_f2_fnfp_f2_1.00_Elastic.pdf}
\includegraphics[trim={9mm 0mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_f3_fnfp_f3_1.00_Elastic.pdf}}
\caption{\label{fig:xenon_floors_massive} 3$\sigma$ discovery limit for a xenon-based experiment (see \Tab{tab:experiments}) for a 0.1 ton-year exposure (dotted), 1 ton-year exposure (short dashed), 10 ton-year exposure (long dashed), and 100 ton-year exposure (solid), including (red) and neglecting (black) the neutrino background. Results assume the mediator mass is much larger than the momentum transfer. Shown for comparison is the current combined 90\% upper limits from XENON1T, PandaX-II, LUX, and PICO (shaded blue).}
\end{figure}
\begin{figure}[h]
\mycenter{
\includegraphics[trim={0mm 15.5mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_si_massless_fnfp_si_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_sd_massless_fnfp_sd_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_anapole_fnfp_anapole_1.00_Elastic.pdf}}
\mycenter{
\includegraphics[trim={0mm 15.5mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_magdip_massless_fnfp_magdip_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_elecdip_massless_fnfp_elecdip_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 15.5mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_anapole_massless_fnfp_anapole_massless_1.00_Elastic.pdf}}
\mycenter{
\includegraphics[trim={0mm 0mm 0 0},clip,width=.4\textwidth]{plots/Floors/Xenon/Xenon_sigma_f1_massless_fnfp_f1_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 0mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_f2_massless_fnfp_f2_massless_1.00_Elastic.pdf}
\includegraphics[trim={9mm 0mm 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_f3_massless_fnfp_f3_massless_1.00_Elastic.pdf}}
\caption{\label{fig:xenon_floors_massless} Same as \Fig{fig:xenon_floors_massive} but assuming the mass of the mediator is negligible with respect to the momentum transfer.}
\end{figure}
\subsection{Inelastic scattering}
The above analysis can be also applied to inelastic scattering. In \Fig{fig:xenon_degeneracy_inelastic} we display the recoil spectrum induced in xenon from solar $^8$B, DSNB, and atmospheric neutrinos, as well as the recoil spectrum arising from a SI interaction with a heavy mediator and inelastic exothermic ($\delta > 0$), endothermic ($\delta < 0$) scattering. As can be seen, as $|\delta|$ increases the DM spectrum becomes progressively more distinguishable from the neutrino spectrum. The DM masses for the spectra in \Fig{fig:xenon_degeneracy_inelastic} (i.e. those that give the best-fit to the respective neutrino background for fixed mass differences $\delta = + 10$ keV and $\delta \pm 50$ keV between the final and initial DM particles are given in \Tab{tab:bestfitDM_inelastic}). We note that the minimum recoil energy in xenon for a DM candidate with $\delta = +50$ keV in the DM mass range scanned is larger than the maximum recoil energy produced by the $^8$B neutrinos and is thus omitted in the left panel of~\Fig{fig:xenon_degeneracy_inelastic}.
\begin{figure}
\mycenter{
\includegraphics[width=.42\textwidth]{plots/SpectrumComparison/Xenon_Inelastic_b8_Comparison.pdf}
\includegraphics[trim={15mm 0 0 0},clip,width=.3875\textwidth]{plots/SpectrumComparison/Xenon_Inelastic_dsnb8mev_Comparison.pdf}
\includegraphics[trim={15mm 0 0 0},clip,width=.3875\textwidth]{plots/SpectrumComparison/Xenon_Inelastic_ATM_Comparison.pdf}}
\caption{\label{fig:xenon_degeneracy_inelastic}Comparison of recoil spectra induced in a xenon target from solar $^8$B (left), DSNB (middle) and atmospheric (right) neutrinos (black) along with the recoil spectrum arising from spin-independent heavy-mediator inelastic scattering. Results for DM mass splitting $\delta = - 50$ keV (blue), $\delta = + 10$ keV (magenta), and $\delta = + 50$ keV (orange). The best fit DM masses are given in \Tab{tab:bestfitDM_inelastic}.}
\end{figure}
\begin{table*}[tbp]
\setlength{\extrarowheight}{2pt}
\begin{center}
\begin{threeparttable}
\begin{tabular}{|c|c|c|c|} \hline
\multirow{2}{*}{Model} & \multicolumn{3}{c|}{Best-fit DM mass (GeV)} \\ \cline{2-4}
& Solar ($^8$B) & DSNB & Atmospheric \\ \hline
\hline
SI ($\delta = 0$)\tnote{1} & 5.8 & 15.9 & 172.4 \\ \hline
SI ($\delta = +10$ keV) & 6.6 & 15.2 & 136.6 \\ \hline
SI ($\delta = -50$ keV) & 2.8 & 4.1 & 26.2\tnote{2} \\ \hline
SI ($\delta = +50$ keV) & -- & 21.9 & 39.0 \\ \hline
\end{tabular}
\begin{tablenotes}
\item[1] These values correspond to the best-fit DM masses of elastic scattering.
\item[2] With $\delta = -50$ keV the scattering off xenon is purely exothermic for $m_{\chi} \leq 17$ GeV. The actual best fit would depend on the fraction of lighter and heavier DM states.
\end{tablenotes}
\caption{\label{tab:bestfitDM_inelastic} DM masses for which the DM spectrum best fits the neutrino recoil spectrum assuming SI scattering with a heavy mediator and inelastic scattering with DM mass splitting $\delta = + 10$ keV and $\delta = \pm 50$ keV (see \Fig{fig:xenon_degeneracy_inelastic}).}
\end{threeparttable}
\end{center}
\end{table*}
\begin{figure}[h]
\mycenter{
\includegraphics[width=.42\textwidth]{plots/Floors/Xenon/Xenon_sigma_si_fnfp_si_1.00_ENDOTHERMIC_delta_10keV.pdf}
\includegraphics[trim={15mm 0 0 0},clip,width=.3875\textwidth]{plots/Floors/Xenon/Xenon_sigma_si_fnfp_si_1.00_ENDOTHERMIC_delta_50keV.pdf}
\includegraphics[trim={15mm 0 0 0},clip,width=.38\textwidth]{plots/Floors/Xenon/Xenon_sigma_si_fnfp_si_1.00_EXOTHERMIC_delta_-50keV.pdf}}
\caption{\label{fig:xenon_floors_inelastic} Same as \Fig{fig:xenon_floors_massive} but for a spin-independent interaction with exothermic and endothermic scattering, assuming $\delta = + 10$ keV and $\delta = \pm 50$ keV. The range of masses for $\delta = -50$ keV is cut at $m_{\chi} = 17$ GeV, since the scattering is not purely exothermic for larger masses.}
\end{figure}
Exothermic scattering assumes that the DM at present consists of two different states of slightly different masses, the lightest being stable and the heaviest meta-stable. Then, the heavier state may down-scatter off the nuclei. However, this type of scattering dominates only if the scattering of each state to itself is suppressed or impossible (due to the couplings of the mediator) and the up-scattering (i.e. the endothermic scattering) off the lighter state is kinematically forbidden.
Thus, when considering exothermic scattering ($\delta<0$), one must consider the kinematic constraints on the endothermic process. If the endothermic scatter is allowed one must add both processes to obtain the complete DM scattering rate. Hence, studying a pure exothermic scattering process is only of interest if the minimum speed required for an endothermic scatter is larger than the galactic escape speed in the Earth's rest frame (i.e. when $\sqrt{2|\delta|/\mu} > |\vec{v}_{\rm esc}| + |\vec{v}_{\odot}|$). Otherwise, one must consider a joint analysis containing both contributions. For example, for $\delta = -10$ keV (-50 keV) both exothermic and endothermic scattering contributions to the rate should be added for $m_{\chi} > 1$ GeV (17 GeV) in a proportion that depends on the fraction of light and heavy states in the halo. Due to this uncertainty we do not include this case.The fits shown in \Fig{fig:xenon_degeneracy_inelastic} do not enforce the constraint on the DM mass to have only exothermic scattering, however only the best-fit mass for the atmospheric neutrino background for $\delta = -50$ keV in \Tab{tab:bestfitDM_inelastic} violates this condition. With $\delta = -50$ keV the scattering is purely exothermic only for $m_{\chi} \leq 17$ GeV, thus we cut the range of DM masses for the figure shown in the right panel of \Fig{fig:xenon_floors_inelastic}.
Interestingly, the kinematic features of inelastic scattering could provide a distinctive signature in future direct detection experiments if at least one target elements could uniquely isolate the purely exothermic scattering process while another target element maintained sensitivity to both endo- and exothermic scatters. A purely endothermic scattering helps to lift the neutrino-DM degeneracy even for SI interactions.
\section{Summary and Outlook\label{sec:summary}}
As DM direct detection experiments continue to improve their sensitivity they will inevitably encounter an irreducible background, the so-called ``neutrino-floor'', arising from coherent neutrino scattering. The neutrino floor has a detrimental effect on the capability of experiments to detect DM. In particular, assuming the conventional SI or SD interactions, the recoil spectrum of the neutrino background mimics the DM signal associated with particular WIMP masses, resulting in a degeneracy between the two. In this case, an increase in the experiment's exposure by multiple orders of magnitude may not yield any significant increase in sensitivity. The SI and SD interactions, however, do not adequately reflect the whole landscape of the well-motivated DM models, which include other interactions. In this work we have studied the relevance of the neutrino background for the discovery sensitivity of a variety of DM-nucleus interactions and several potential futuristic experimental settings, considering different target elemental compositions. This study could assist with mapping out the optimal DM detection strategy for the next generation of experiments.
For realistic exposures (i.e. $M T \sim \mathcal{O}(10^2)$ ton$\cdot$years), as expected of next generation direct detection experiments, we find that the degeneracy of DM and neutrino spectra primarily originates from solar $^8$B neutrinos. Due to the much lower flux levels of DSNB and atmospheric neutrinos, the spectral degeneracies with these neutrinos that happen for some DM mass values and interactions (see central and left panels of \Fig{fig:Dark_Matter_Degeneracy}, Appendix~\ref{app:recoilspec} \Fig{fig:recoilspec_full} as well as Appendix~\ref{app:recoilspec} \Tab{tab:bestfitDM_elastic_xenon} to \Tab{tab:bestfitDM_elastic_sodium}) would become relevant only for exposure levels many orders of magnitude beyond those expected of experiments within the foreseeable future. This is consistent with findings of previous studies, which primarily focused on SD and SI interactions (e.g. see \cite{Ruppin:2014bra}, Fig. 3).
Several key features of the presented DM discovery potential results can be readily understood from the momentum transfer $|\vec{q}|$ dependence of the differential DM-nucleus scattering cross-sections $d\sigma_{\chi}/d E_R$ (see \Tab{tab:fermop} and the cross-sections of Sec. \ref{sssec:models}). As shown in Sec. 4.2, for cross-sections with no dependence on the momentum exchange, i.e. proportional to $|\vec{q}|^0$, the degeneracy with $^8$B neutrinos happens at masses close to 10 GeV. For heavy mediators, these correspond to the following interactions: SI, SD, mC and AV-V (only if the AV-V cross-section is dominated by the charge $Z_T^2 v^2$ term, e.g. in Xe \Fig{fig:xenon_floors_massive} because the charge of Xe is large and its magnetic moment is relatively small, see Eq.~(2.33); also Ar Fig.~10, since Ar has no spin). For light (or $M = 0$) mediator the cross-sections with no dependence on $|\vec{q}|$ correspond to the interactions: PS-PS, MD (if the MD cross-section in Eq.~(2.21) is dominated by the spin coupling, e.g. in Na Fig.~13) and Anapole (if dominated by the charge $Z_T^2 v^2$ term, e.g. in Xe Fig.~4, Ar Fig.~11). The neutrino-DM spectral degeneracy severely undermines the experimental sensitivity for $m_{\chi} \approx 10$ GeV. However, for masses above roughly $50$ GeV, such cross-sections have less than one order of magnitude suppression of the discovery limits for exposures as large as MT $\sim (10)^2$ ton$\cdot$years, making the neutrino background significantly less detrimental (see e.g. top left and middle panels of \Fig{fig:xenon_floors_massive} and bottom right panel of Fig.~4).
We note that for elastic scattering with $d \sigma_{\chi}/d E_R \sim |\vec{q}|^0$ there exists a general scaling relation for the DM signal-neutrino floor degeneracy with the WIMP mass. After having obtained from numerical calculations the DM mass for which the DM induced recoil spectrum is degenerate with the neutrinos for one target element, the DM mass for which spectral degeneracy occurs for another target element can be approximately found from kinematic relations as follows. The shape of the nuclear recoil spectrum depends on a common function of $v_{\rm min}$ for all targets. Two recoil spectra have the same shape only if they correspond to the same $v_{\rm min}$ range. Since degeneracy implies the recoil spectra are the same, one can equate the $E_{\rm R}$ of each target element. Hence, knowing that for a particular element (denoted as $T_1$) the DM spectrum is degenerate for a particular DM particle mass (denoted $m_1$) we can find the value of the respective mass (i.e. $m_2$) for the degeneracy in another element (denoted as $T_2$) by requiring that the respective nuclear recoil energies for the same $v_{\rm min}$ value coincide as
\begin{equation}
E_R^{T_1} (v_{\rm min}, m_1) = E_R^{T_2} (v_{\rm min}, m_2)~,
\end{equation}
which can be rewritten as
\begin{equation}
\dfrac{2 \mu_{T_1}^2 v_{\rm min}^2}{M_{T_1}} = \dfrac{2 \mu_{T_2}^2 v_{\rm min}^2}{M_{T_2}}~.
\end{equation}
Thus,
\begin{equation}
\dfrac{\mu_{T_1}^2}{M_{T_1}} = \dfrac{\mu_{T_2}^2}{M_{T_2}}~.
\end{equation}
For masses close to 10 GeV (e.g. for Xe one observes the degeneracy with $^8$B neutrinos at $m_1 \approx 6$ GeV) we have that for all the considered nuclei $\mu_{T_1} \simeq m_1$ and $\mu_{T_2} \simeq m_2$, resulting in
\begin{equation}
m_2 \simeq \sqrt{\dfrac{M_{T_1}}{M_{T_2}}} m_1~.
\end{equation}
This approximate scaling relation provides insight into the degeneracy behavior in different elements as identified in Fig. 7 of \cite{Ruppin:2014bra} for SI and SD interactions.
For cross-sections inversely dependent on powers of $|\vec{q}|$ (i.e. $1/|\vec{q}|^4$ or $1/|\vec{q}|^2$), there is no spectral degeneracy (see e.g. top left and middle panels of \Fig{fig:xenon_floors_massless}). However, there is an enhancement of the DM recoil spectrum at low recoil energies where the flux from solar neutrinos is large. Hence, the respective discovery limits are affected already for exposures drastically lower than for cross-sections independent of $|\vec{q}|$, except for masses near the degeneracy. This can be understood by considering that for a fixed DM particle mass, cross sections independent of $|\vec{q}|$ extend to larger recoil energies where the neutrino flux is reduced, making this part of the spectra more differentiable from that of the background neutrinos. The effect of the concentration of the DM signal at low $E_R$ is exacerbated for low mass WIMPs, whose recoil spectrum already tends to be rather steep and concentrated at low energies. For interactions with $|\vec{q}|^{-4}$, i.e. light mediator ($M=0$) SI, SD, mC, and AV-V (assuming the AV-V cross-section is dominated by the $Z_T^2 v^2$ term, e.g. in Xe Fig.~4 and Ar Fig.~11), the discovery limits are suppressed by more than 3 orders of magnitude for exposures $MT \geq (10)^2$ ton$\cdot$years. The effect is less pronounced, leading to a suppression of around 1.5 orders of magnitude in the discovery limits for $m > 50$ GeV, for cross-sections proportional to $1/|\vec{q}|^2$. The interactions with these cross-sections are, always for light (or $M = 0$) mediators: MD (dominated by the $Z_T^2 v^2$ term, e.g. Xe Fig.~4 or Ar Fig.~11), ED, PS-S and S-PS (see e.g. central middle panel of Fig.~4)
The effect of $^8$B neutrinos is least pronounced for recoil momentum suppressed DM cross-sections, i.e. cross sections proportional to $|\vec{q}|^b$ with $b > 0$. These correspond to interactions with heavy mediators. The interactions with $d \sigma_{\chi}/ d E_R \sim |\vec{q}|^2$ are MD (dominated by the $Z_T^2 v^2$ term, e.g. Xe Fig.~3 and Ar Fig.~10), ED, PS-S and S-PS. Those with $d \sigma_{\chi}/ d E_R \sim |\vec{q}|^4$ are MD (dominated by the magnetic moment coupling, e.g. Na Fig.~12), Ana (dominated by the $Z_T^2 v^2$ term) and PS-PS. The interaction with $d \sigma_{\chi}/ d E_R \sim |\vec{q}|^6$ is Ana (dominated by the magnetic term as in Na Fig.~12).
For momentum suppressed cross-sections there is no neutrino-DM spectra degeneracy and, moreover, the DM recoil spectrum peaks at some non-zero value of the momentum transfer (note that this is contrary to interactions that are either independent of the momentum transfer or inversely dependent on powers of the momentum transfer, which both have a maximum differential rate for $|\vec{q}| \rightarrow 0$). There is an enhancement in sensitivity with respect to momentum independent cross-sections. This enhancement is modest for light DM masses (except for the region of degeneracy where the enhancement is significant), for which $^8$B neutrinos are the main background. This can be seen e.g. for Xe in the bottom row, the left and middle panels of the central row and the top right panel of Fig.~3. This is so because larger powers of $|\vec{q}|$ produce increasingly peaked recoil spectra which thus become more differentiable from background. The larger the power $b$ of $|\vec{q}|^b$ the smaller the effect of $^8$B neutrinos.
For large DM masses and heavier nuclei the enhancement in sensitivity with respect to momentum independent cross-sections is large, because a considerable portion of the recoils for momentum suppressed cross sections occur at energies above the recoil energies produced by atmospheric neutrinos. The suppression of the atmospheric background is a consequence of the nuclear form factor behavior (the resulting recoil spectra from atmospheric neutrinos in different targets can be seen e.g. in Fig.~5 of \cite{Gutlein:2010tq}). For large recoil energies there is no ambiguity as to whether an event arose from a DM candidate or a background neutrino, at least for all of the exposures considered here.
Recall that for momentum suppressed interactions, we perform our analysis over two energy regions, one extending to 50 keV and the other to 300 keV. The derived experimental sensitivity is then taken to be the stronger of the two. For large DM masses and heavier target nuclei the broader energy interval provides the stronger limits. For a Xe target for example, we can see in Fig.~3 that for DM masses above a few 100's GeV the discovery limit shows no suppression at all due to the neutrino background for Ana, MD and PS-PS interactions. All of these interactions have the strongest momentum suppression of $|\vec{q}|^4$ or $|\vec{q}|^6$ in their cross sections. For interactions with a weaker dependence $|\vec{q}|^2$ in their cross sections (for ED, AV-V, PS-S and S-PS) the discovery limits in Xe at large masses and in the larger recoil energy interval also improve considerably with respect to those derived from the low energy interval, although only for S-PS interactions we find no suppression of discovery limits for large DM masses (and this is due to the nuclear form factor for this interaction decreasing very slowly with energy).
The same situation we just described for Xe (see Fig.~3) holds for Ge and I (see Figs.~8 and 14), as well as potentially for other heavy nuclei. The same is not found in Ar or Na (see Figs.~10 and 12). This can be understood using kinematic arguments. The recoil energy for DM particles heavy with respect to the nuclear mass is simply $E_R \simeq 2 m_T v_{\rm min}^2$, thus heavier nuclei have larger recoil energies, while for lighter nuclei most recoils are below 50 keV -- thus considering larger recoil energies does not improve the discovery limits.
Thus far in this section our discussion has exclusively been limited to elastic DM-nuclei scattering. In this paper, we demonstrated explicitly that inelastic scattering, in which an initial DM particle of mass $m_{\chi}$ scatters into another of mass $(m_{\chi} + \delta)$, produces a distinctive recoil spectra that allows for easy discrimination with the neutrino background. This was explicitly shown for an SI interaction with positive and negative values of $\delta$ (with $|\delta| \ll m_{\chi}$), but it should be emphasized that different interactions (e.g. the anapole or magnetic dipole) may produce more distinctive recoil spectra.
Target material complementarity can both mitigate the effect of the neutrino background and help discriminate the various DM models. This can be qualitatively understood by inspecting the specific dependence on target properties of the cross-sections (see Sec. 2.2.2). One can thus identify target elements with vastly different predicted event rates. For differential cross-sections consisting of a single term, the overall predicted rate
would often be sufficient to determine if a particular DM model can account for the results of two different experiments with different target materials. For the neutrino background, the ratio of the scattering rates in two target elements $T_1$ and $T_2$ should be proportional to the ratio of the square of their neutron numbers $(A_{T_1} - Z_{T_1})^2/(A_{T_2} - Z_{T_2})^2 = N_{T_1}^2 / N_{T_2}^2$. However, for DM interactions other than the SI with coupling predominantly to neutrons the ratio of the scattering event rates would be different. In the context of discriminating between DM models, interpreting a putative signal in a xenon target experiment in terms of any spin-dependent interaction inherently implies non-observation of the signal in an argon detector (since argon has no nuclear spin). Additionally, should the DM differential cross section contain multiple non-negligible terms with differing dependence on the nuclear properties of the target (e.g. the magnetic dipole interaction), not only can the overall scattering rate change but also the spectral shape can change with different targets, enhancing the discriminating power when observations are made in multiple experiments. Target complementarity would also be a particularly useful tool for inelastic DM, as endothermic interactions favor heavier targets and exothermic interactions favor lighter targets. Thus, this could be a helpful probe of models with multicomponent dark sector.
Until a convincing DM signal appears, it is of paramount importance that experiments maintain the broadest possible sensitivity to the wide array of possible DM interactions, including the inelastic scattering that is an often neglected but a viable possibility. Therefore it is important to maintain a multi-pronged approach to DM direct detection in the coming generation of experiments. As previously discussed, various target elements provide complimentary sensitivities that may help both differentiate a putative signal from background, and differentiate viable DM models from each other. Xenon has sensitivity to all the possible DM interactions we have explored, and is easily scalable to large mass experiments. However, the spin and nuclear magnetic moment of the various xenon isotopes are relatively low. Argon is easily scalable to large mass experiments and has an extremely low background. Hence, it is ideal for interactions proportional to the number of nucleons. On the other hand, argon has no spin or a magnetic moment, and is thus insensitive to a variety of interactions. Therefore, argon experiments could be a powerful discriminating tool for a spin or magnetic-moment-dependent DM interaction. Germanium (and silicon, which to some degree is similar) has a broad sensitivity to many interactions, although like xenon it has a small spin and a small nuclear magnetic moment. In silicon the spin and nuclear magnetic moment is even smaller. Germanium (and silicon) experiments can reach a low threshold, implying a strong sensitivity to low mass DM candidates. Sodium and iodine provide spin-dependent interactions with protons (note that this is distinctive from both germanium and xenon which have spin-dependent couplings predominantly with neutrons) and large nuclear magnetic moments\footnote{Fluorine also has a large nuclear magnetic moment and a large spin-dependent proton coupling, however this element has not been studied here since experiments using F only measure the energy-integrated rate, making neutrino background discrimination difficult.}, however these elements have larger backgrounds and require binning, implying a much lower sensitivity than would otherwise be suggested by this study.
We reiterate that the fundamental goal of this work is to highlight the future sensitivity of direct detection experiments employing various target elements to a wide array of possible DM-nucleus interactions, particularly focusing on the extent to which background neutrinos will inhibit the ability to probe interesting parameter space. We identify a large number of interactions and parameter space for which the effect of the neutrino floor is significantly reduced, and a sizable amount of parameter space for which it is strengthened, relative to the conventionally studied SI and SD interactions. We also identify strategies that could be exploited to optimize the sensitivities of future DM direct detection experiments to a wide-range of possible DM candidates.
\acknowledgments
\addcontentsline{toc}{section}{Acknowledgments}
We would like to thank Robert Cousins and Michail Bachtis for helpful discussions regarding the statistical analysis. The work of GG and VT was supported, in part, by the
U.S. Department of Energy (DOE) under Grant No. DE-SC0009937.
SJW is supported by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 674896. The work of GG and VT was also supported, in part, by the National Science Foundation under Grant No. NSF PHY-1748958 due to their stay at the Kavli Institute for Theoretical Physics (KITP) at the University of California, Santa Barbara (UCSB), while completing this paper.
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{"url":"https:\/\/discourse.vtk.org\/t\/light-disappears-when-camera-view-direction-is-parallel-to-plane\/7099","text":"# Light disappears when camera view direction is parallel to plane\n\nI have a textured plane with ambient, diffuse, specular and specular power set. There are no lights in the scene.\nWhen I move the camera such that the view is in line with the plane, or away from it, then the plane goes black.\n\nIt has nothing to do with the texture. In the reproduction below the same occurs with a plain white plane.\n\nRunning vtk 9.0.3 but same behavior in RC of 9.1\n\nReproduce:\n\n\nimport vtkmodules.vtkInteractionStyle\n# noinspection PyUnresolvedReferences\nimport vtkmodules.vtkRenderingOpenGL2\nfrom vtkmodules.vtkCommonColor import vtkNamedColors\nfrom vtkmodules.vtkCommonTransforms import vtkTransform\nfrom vtkmodules.vtkFiltersCore import vtkTriangleFilter\nfrom vtkmodules.vtkFiltersGeneral import vtkTransformPolyDataFilter\nfrom vtkmodules.vtkRenderingCore import (\nvtkActor,\nvtkPolyDataMapper,\nvtkRenderWindow,\nvtkRenderWindowInteractor,\nvtkRenderer,\nvtkTexture,\nvtkLight, vtkCamera,\n)\n\nfrom vtkmodules.vtkInteractionStyle import vtkInteractorStyleTrackballCamera\n\ncolors = vtkNamedColors()\n\n# Create a plane source and actor. The vtkPlanesSource generates\n# texture coordinates.\n#\nplane = vtkPlaneSource()\nplaneMapper = vtkPolyDataMapper()\nplaneMapper.SetInputConnection(plane.GetOutputPort())\n\nplaneActor = vtkActor()\nplaneActor.SetMapper(planeMapper)\n\npr = planeActor.GetProperty()\npr.LightingOn()\npars = [0.8, 0.1, 0.0, 0]\npr.SetAmbient(0.8)\npr.SetDiffuse(0.1)\npr.SetSpecular(0)\npr.SetSpecularPower(0) # default = 1.0 <------ this triggers the behaviour\n\n# Create the RenderWindow, Renderer and Interactor.\nrenderer = vtkRenderer()\nrenWin = vtkRenderWindow()\niren = vtkRenderWindowInteractor()\n\nstyle = vtkInteractorStyleTrackballCamera()\niren.SetInteractorStyle(style)\n\niren.SetRenderWindow(renWin)\n\n# Add the actors to the renderer, set the background and size.\nrenWin.SetSize(640, 480)\nrenWin.SetWindowName('TexturePlane')\n\nrenderer.SetBackground(colors.GetColor3d('DarkSlateGray'))\nrenderer.ResetCamera()\nrenderer.ResetCameraClippingRange()\n\ncamera = renderer.GetActiveCamera()\ncamera.SetPosition(1,2,0.4)\ncamera.SetFocalPoint(0,0,0.4)\ncamera.SetViewUp(0,0,1)\n\nrenWin.Render()\niren.Start()\n\n\nI suspect this is just an issue in that we should prohibit 0 as a specular power. (we need to sanitize our input and prevent a setting of zero). The specular power gets passed into a pow function in glsl, basically specular = pow(planeNormal dot cameraViewDirection, specularPower) and the pow(x,y) function documentation includes\n\nThe result is undefined if $x < 0$ or if $x = 0$ and $y \\leq 0$.\n\n\nIn your example y is 0 so if x is <= 0 (which happens when the plane is pointing backwards) we will get undefined results which I suspect is what is happening here. You still see the plane when it is pointing backwards due to perspective. (the camera is perspective, the light is parallel)\n\nHi @ken-martin , I can confirm that setting the specular power to a very small value (1e-8) instead of 0 solves the issue.\n\nWhere should we check, SetSpecularPower()? If so then I can try to make a PR for this.\n\nI will feed flag this in the vedo library for python as well as there specular power is set to 0 by some of the default functions.","date":"2022-05-18 16:43:40","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\": 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\": 0.18044698238372803, \"perplexity\": 3801.6914855860878}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662522284.20\/warc\/CC-MAIN-20220518151003-20220518181003-00676.warc.gz\"}"}
| null | null |
Open Borders Lobby: 2020 Democrats Must Endorse End to Deportations
David McNew/Getty
The open borders lobby is demanding 2020 Democrat presidential candidates continue moving further to the left on national immigration policy, most recently pushing for an end to all deportations.
Former Obama official Julian Castro has staked his 2020 campaign on pushing his fellow Democrats to the left on immigration — successfully getting Sen. Elizabeth Warren (D-MA) to endorse his plan of decriminalizing all illegal immigration to the United States.
Last month, Sen. Bernie Sanders (I-VT) — once an opponent of the flood-the-labor-market agenda of globalist-aligned politicians — released his immigration plan that includes at least a temporary halt to all deportations of illegal aliens, including those convicted of murder, child sex crimes, and rape.
Warren, likewise, said last month she is "open to suspending deportations, particularly as a way to push Congress for comprehensive immigration reform.
Open borders activists, though, told NBC News the moves by Warren and Bernie are progress, but they are hoping all 2020 Democrats back the end to deportations — including former Vice President Joe Biden and South Bend, Indiana, Mayor Pete Buttigieg.
Movimiento Cosecha organizer Carlos Rojas, who confronted Biden over former President Obama's record of deporting criminal illegal aliens, said ending all deportations "is a starting point" for illegal aliens, but that the illegal population needs more from 2020 Democrats.
"We have a sitting president who launched a campaign saying Mexicans are rapists, drug dealers or criminals, so we cannot have an immigration plan that assumes all immigrants are criminals," Rojas said.
"As for immigrants convicted of serious crimes, we already have a criminal justice system … that can deal with that," Rojas told NBC News.
"If you are president, you are going to have more power than anyone else to reverse this trend of ramped up enforcement and deportation and that's what activists are putting pressure on the candidates to do," Frank Sharry, of the pro-mass immigration group America's Voice, said.
While the open borders lobby wants 2020 Democrats to embrace their zero-deportations agenda, the majority of American voters want deportations increased and are more likely to support candidates who oppose more immigration to the U.S.
The latest Pew Research Center survey revealed that 54 percent of U.S. voters said they want to see more deportations for the nation's 11 to 22 million illegal alien population. A September Harvard/Harris Poll found that increasing overall immigration to the country remained one of the most unpopular 2020 positions a presidential candidate could take.
As Breitbart News has reported, research finds that deporting the 11 million to 22 million illegal aliens saves American taxpayers billions of dollars, compared to the costs they are forced to pay when illegal aliens are allowed to stay.
Deporting every illegal alien in the country would amount to a cost savings of about $622 billion over the course of a lifetime. This indicates that deporting illegal aliens is six times less costly than what it costs American taxpayers to currently subsidize the millions of illegal aliens living in the U.S.
2020 ElectionImmigrationPolitics2020 DemocratsBernie SandersdeportationsElizabeth Warrenillegal immigrationmass immigrationopen borders lobbySouthern BorderU.S.-Mexico border
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\section{#1}}
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\vs{12}
\begin{center}
{\LARGE {\bf Triality of Majorana-Weyl Spacetimes}}\\
{\LARGE {\bf with Different Signatures}}\\
[0.8cm]
\vs{10} {\large M.A. De Andrade$^{(a,b)}$, M. Rojas$^{(a)}$ and F.
Toppan$^{(a,c)}$} ~\\ \quad \\
{\em {~$^{(a)}$} CBPF, DCP, Rua Dr. Xavier Sigaud
150, cep 22290-180 Rio de Janeiro (RJ), Brazil}\\
~\quad \\
{\em {~$~^{(b)}$} UCP, FT, Rua
Bar{\~{a}}o do Amazonas, 124, cep 25685-070, Petr\'{o}polis (RJ),
Brazil}\\ ~\quad\\ {\em ~$~^{(c)}$ UFES, CCE Depto de
F{\'{\i}}sica, Goiabeiras cep 29060-900, Vit\'oria (ES), Brazil}
\end{center}
\vs{6}
\centerline{ {\bf Abstract}}
\vs{6}
Majorana-Weyl spacetimes offer a rich algebraic setup and new types
of space-time dualities besides those discussed by Hull. The triality
automorphisms
of $Spin(8)$ act non-trivially on Majorana-Weyl representations and
Majorana-Weyl spacetimes with different signatures.
In particular relations exist among the
$(1+9)\leftrightarrow (5+5)\leftrightarrow (9+1)$ spacetimes,
as well as their transverse coordinates
spacetimes $(0+8)\leftrightarrow (4+4)\leftrightarrow (8+0)$.
Larger dimensional spacetimes such as
$(2+10)\leftrightarrow (6+6)\leftrightarrow (10+2)$
also show dualities induced by
triality. A precise three-languages dictionary is here given.
It furnishes the exact translations among, e.g., the three
different versions (one in each signature) of the ten-dimensional $N=1$
superstring and superYang-Mills theories. Their dualities close
the six-element permutation group $S_3$. Bilinear and trilinear
invariants allowing to formulate theories with a manifest
space-time symmetry are constructed.
\vs{6} \vfill \rightline{July 1999} \rightline{CBPF-NF-039/99}
\rightline{hep-th/xxx} {\em E-Mails:\\ 1) marco@cbpf.br\\ 2)
mrojas@cbpf.br \\
3) toppan@cbpf.br}
\newpage
\pagestyle{plain}
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}
\vs{8}
\section{Introduction.}
Physical theories formulated in different-than-usual spacetimes
signatures have recently found increased attention. One of the
reasons can be traced to the conjectured $F$-theory \cite{Vafa}
which supposedly lives in $(2+10)$ dimensions \cite{Nish}.
The current interest in AdS theories motivated by the AdS/CFT
correspondence furnishes another motivation. Two-time physics
e.g. has started been explored by Bars and collaborators in a
series of papers \cite{Bars}. For another motivation we can also
recall that a fundamental theory is
expected to explain not only the spacetime dimensionality, but
even its signature (see \cite{Duff}). Quite recently Hull and
Hull-Khuri \cite{Hull} pointed out the existence of dualities
relating different compactifications of theories formulated in
different signatures. Such a result provides new insights to the
whole question of spacetime signatures.
Other papers (the most recent is \cite{Cori}) have remarked the
existence of space-time dualities.\par
Majorana-Weyl spacetimes (i.e. those supporting Majorana-Weyl
spinors) are at the very core of the present knowledge of
the unification via supersymmetry, being at the basis of
ten-dimensional superstrings, superYang-Mills and
supergravity theories (and perhaps the already mentioned
$F$-theory). A well-established feature of Majorana-Weyl
spacetimes is that they are endorsed of a rich structure.
A legitimate question one can ask oneself is whether they are
affected, and how, by space-time dualities. The answer is quite
surprising (in fact it should not be so, with afterthought),
the structure of dualities is much richer than expected and
potentially able to shed a complete new light on the subject.
Indeed all different Majorana-Weyl spacetimes which are possibly
present in any given dimension are each-other related by duality
transformations which are induced by the $Spin(8)$ triality
automorphisms.
The action of the triality automorphisms is quite non-trivial and
has far richer consequences than the ${\bf Z}_2$-duality
(its most trivial representative) associated to the space-time
$(s,t)\leftrightarrow (t,s)$ exchange discussed in \cite{Duff}.
It corresponds to $S_3$, the six-element group of permutations
of three letters, identified with the group of congruences of the
triangle and generated by two reflections.
The lowest-dimension in which the triality action is non-trivial
is $8$ (not quite a coincidence), where the spacetimes
$(8+0)-(4+4)-(0+8)$ are all interrelated. They correspond to the
transverse coordinates of the $(9+1)-(5+5)-(1+9)$ spacetimes
respectively, where the triality action can also be lifted. Triality
relates as well the $12$-dimensional Majorana-Weyl spacetimes
$(10+2)-(6+6)-(2+10)$, i.e. the potentially interesting cases for
the $F$-theory, and so on.\\
As a consequence of triality, supersymmetric theories formulated
with Majorana-Weyl spinors in a given dimension but with different
signatures, are all dually mapped one into another.
A three-language dictionary is here furnished with the exact
translations among, e.g., the different versions of the $10$-dimensional
MW supersymmetric theories, formulated in the Majorana-Weyl
representation.\par
The strategy here followed is based in three steps. At first it
is shown that Majorana-Weyl spacetimes in dimensions $d> 8$
can be recovered from the properties of the $8$-dimensional
Majorana-Weyl
spacetimes and $\Gamma$-matrices representations. Next, working in
$d=8$, we construct, for each one of the three Majorana-Weyl
spacetimes $(8+0)$, $(4+4)$, $(0+8)$, the ``bridge transformations"
relating the corresponding Majorana-Weyl representations to the
representations (called ``VCA" in the text) which exhibit manifest
triality among vectors, chiral and antichiral spinors. As a final
step new ``bridge transformations" of spacetime kind, relating
among them the VCA representations constructed in each one
of the Majorana-Weyl spacetimes above, are given.\par
We emphasize that, contrary to Hull \cite{Hull}, the dualities
here discussed are already present for the {\it uncompactified}
theories and in this respect look more fundamental.\par
Moreover, bilinear and trilinear invariants under the $S_3$
permutation group of the three Majorana-Weyl spacetimes are
constructed. They can be possibly used to formulate supersymmetric
Majorana-Weyl theories in a manifestly triality-invariant form
which presents an explicit symmetry under exchange of space and
time coordinates.\par
The present paper is intended to be an abridged version, suitable
for a letter-size, of a forthcoming extended version which
presents in full detail the construction and where extra results
which are outside the scope of this letter are also furnished.\par
The scheme of this work is as follows. In the next section we
recall, following \cite{Kugo} and \cite{DeAn}, the basic properties
of
$\Gamma$-matrices and Majorana conditions needed for our
construction. Majorana-type representations are analyzed in
section $3$. We show how to relate the Majorana-Weyl representations
in $d> 8$ to the $8$-dimensional Majorana-Weyl representations.
In section $4$ we introduce, for $d=8$, the set of data necessary
to
define a supersymmetric Majorana-Weyl theory, i.e.
the set of ``words" of our three-languages dictionary. The Cartan's
\cite{Cart}
triality among vectors, chiral and antichiral spinors is presented
in section $5$. The main result is furnished in section $6$, where
spacetime triality is discussed. In the Conclusions we furnish some
comments and point out some perspectives.
\vspace{0.2cm}
\noindent{\section{Preliminary results.}}
Here we limit ourselves to introduce the basic ingredients needed
for our constructions. Further information is found in \cite{Kugo}
and \cite{DeAn}.\par
We denote as $g_{mn}$ the flat (pseudo-)euclidean metric of a
$(t+s)$-spacetime.
Time
(space) directions in our conventions are associated to the $+$
(respectively $-$) sign. \par
The $\Gamma$'s matrices are assumed to be unitary (the chosen normalization
is for the square of time-like
$\Gamma$-matrices being $+1$). The three matrices
${\cal A}$, ${\cal B}$, ${\cal C}$ are the generators of the three
conjugation operations (hermitian, complex conjugation and transposition
respectively) on the $\Gamma$'s.
In particular
\begin{eqnarray}
{\cal C} \Gamma^m {\cal C}^\dagger &=& \eta(-1)^{t+1}{\Gamma^m}^T
\label{0}
\end{eqnarray}
where $\eta=\pm 1$ in even-dimensional spacetimes label
inequivalent choices of the charge conjugation matrix ${\cal
C}$.\par
${\cal A}, {\cal B}, {\cal C}$ are related by the formula
\begin{eqnarray}
{\cal C}&=& {\cal B}^T {\cal A} \label{1}
\end{eqnarray}
Up to an inessential phase, ${\cal A}$ is specified by the product of all the
time-like $\Gamma$ matrices.
An unitary transformations $U$ applied on spinors act on $\Gamma^{m}$,
${\cal A}, {\cal B}, {\cal C}$ according to \cite{DeAn}
\begin{eqnarray}
\Gamma^m &\mapsto& U\Gamma^m U^\dag\nonumber\\ {\cal A}&\mapsto&
U{\cal A}U^{\dag}\nonumber\\{\cal B}&\mapsto& U^\ast {\cal B}
U^{\dag}\nonumber\\
{\cal C}&\mapsto & U^\ast{\cal C} U^\dag \label{2}
\end{eqnarray}
A Majorana representation for the $\Gamma$'s can be defined as the one
in which ${\cal B}$ is set equal to the identity. Spinors can be
assumed real in this case.\par
In even dimensions we can also introduce the notion of Weyl
representation, i.e. when the ``generalized $\Gamma^5$ matrix" is
symmetric and block diagonal and with no loss of generality
can be assumed to be the direct sum of the two equal-size blocks
${\bf 1}\oplus (-{\bf 1})$.
The compatibility of both Majorana and Weyl conditions
constraints the spacetime $(t+s)$ to satisfy
\begin{eqnarray}
s-t&=& 0 \quad {\it mod}\quad 8, \quad {\it for} \quad{\it both}\quad
{\it values}
\quad \eta=\pm 1
\label{3}
\end{eqnarray}
In even dimensions
Majorana representations, but not of Weyl type, are also found for
\begin{eqnarray}
s-t &=& 2 \quad mod \quad 8 \quad for \quad \eta=-1;\nonumber \\
s-t &=& 6 \quad mod \quad 8 \quad for \quad \eta= +1.
\label{4}
\end{eqnarray}
For $d<8$ the only spacetimes supporting Majorana-Weyl spinors
have signatures $(n+n)$. At $d=8$ a new feature arises,
Majorana-Weyl spinors can be found for different signatures.
\vspace{0.2cm}
\noindent{\section{Majorana-type representations.}}
It is convenient to introduce the notion of Majorana-type
representation (or shortly MTR) for the $\Gamma$ matrices as one in
which all the $\Gamma$'s have a definite symmetry. For $d= p+q$
a MTR with $p$ symmetric and $q$ antisymmetric $\Gamma$'s will
be denoted as
$(p_S, q_A)$ in the following. \par
For such representations the ${\cal C}$ matrix introduced in the
previous section is given by either the product of all the symmetric
$\Gamma$ matrices, denoted as ${\cal C}_S$, or all the antisymmetric
ones (${\cal C}_A$). In even dimensions ${\cal C}_S$,
${\cal C}_A$ correspond to opposite values of $\eta$ in (\ref{0}).\par
A Majorana representation in a given signature spacetime is a MTR.
Conversely, given a MTR, we can find a spacetime signature for
which the representation is Majorana. The admissible couples of
$(p_S, q_A)$ values for a MTR are immediately read from
the Majorana tables given above ({\ref{3}) and (\ref{4}). The
construction is such that ${\cal C}$ must correspond to the
correct value of $\eta$ in the tables.\par
The list of all possible MTR's in any given dimension is easily
computed. In order just to give an example one can check that in $d=6$
there exists a MTR (not of Weyl kind) with $6$ anticommuting $\Gamma$
matrices plus an anticommuting $\Gamma^7$ $(0_S, 6_A, {\Gamma^7}_A)$.
It gives the Majorana basis in the Euclidean $6$-dimensional space.
\par
In $d=8$ the MTR's of Weyl type are $(8_S,0_A)$,
$(4_S,4_A)$, $(0_S,8_A)$, associated to the corresponding Majorana-Weyl
spacetimes.\par
Different MTR's belong to different classes under similarity
transformations of the
$\Gamma$'s representations. Indeed
in, let's say, an euclidean (all $+$ signs) space, the
index
\begin{eqnarray}
&&I = tr (\Gamma^m \cdot {\Gamma_m}^T) = (p_S-q_A)\cdot tr{\bf 1}
\label{5}
\end{eqnarray}
takes different values for different MTR's.\par
We computed explicitly all MTR's up to
$d=12$ and Majorana-Weyl representations
up to $d=14$ (the results will be furnished elsewhere)
by using a recursive
algorithm presented in \cite{Cola}. It allows producing Weyl
representations in $d$ dimensions from any given couple of
representations in $r$ and $s$ dimensions, for even-dimensional $d,r,s$
satisfying $d=r+s+2$. The only MTR up to $d=12$ which does not directly
fit
into this scheme, the above-mentioned $6$-dimensional $(0_S,6_A)$,
is however constructed from the $(3_S,3_A)$ representation
(this one directly produced from the $2$-dimensional Pauli matrices for
$r=s=2$) after computing the value of the symmetric
matrix ${\cal
B}$ in the euclidean $6$-dimensional space, and later finding the
transformation (\ref{2}) which maps it into the unity.
\par
The algorithm is given by the formula
\begin{eqnarray}
{\Gamma_d}^{i=1,..., s+1} &=& \sigma_x\otimes {\bf 1}_L\otimes
{\Gamma_s}^{i=1,...,s+1}\nonumber\\
{\Gamma_d}^{s+1+j = s+2, ..., d} &=&
\sigma_y\otimes{\Gamma_r}^{j=1,...,r+1}\otimes {\bf 1}_R
\label{algo}
\end{eqnarray}
where ${\bf 1}_{L,R}$ are the unit-matrices in the respective
spaces, while $\sigma_x = e_{12}+e_{21}$ and $\sigma_y = -i e_{12}+i
e_{21}$ are the $2$-dimensional Pauli matrices.
${\Gamma_r}^{r+1}$ corresponds to the ``generalized $\Gamma^5$-matrix"
in $r+1$ dimensions. In the above
formula the values $r,s=0$ are allowed. The corresponding
${\Gamma_0}^1$ is just $1$.\par
With the help of the above formula we have a very efficient tool
to reduce the
analysis of Majorana-Weyl representations for $d\geq 10$ to the
$8$-dimensional case, since we can always set either $r$ or $s$
equal to $8$ (or possibly both, which corresponds to the $d=18$
case). Up to $d=14$ Majorana-Weyl spacetimes exist for three
different signatures and the same situation of $d=8$ is repeated.
A new
feature arises for $d\geq 16$. A careful analysis of the consistency
conditions is needed in this case, since the triality
transformations that we later discuss no longer preserve the similarity
classes of MTRs; stated otherwise, different representatives of MTR's
in the same similarity class and for the same couple of values
$(p_S,q_A)$ are mapped under a given triality transformation into
representatives of MTR belonging to different
similarity classes. This feature is likely to be related with the
problems encountered
in defining supersymmetric theories in dimensions greater
than $14$, which have been e.g. pointed out in \cite{Sezg}. \par
In any case the construction here discussed is suited to analyze and
works perfectly well for the range $d=8,...,14$, i.e. the cases which are
of interest
for, among the others, the superstrings and the F-theory (notice that
the above scheme can find applications to dualities also for odd-dimensional
Majorana spacetimes as the $11$-dimensional ones supporting the $M$-theory,
we will comment more on that in the conclusions).
We postpone to the extended version of this paper the presentation
of the full
set of reconstruction formulas which make explicit the
construction of MW-spacetimes
in dimensions $d>8$ in terms of the $8$-dimensional ones. For the
purpose of this paper is sufficient to recall that such
reconstruction formulas can be given. In particular the higher
dimensional ${\cal C}$ charge conjugation matrices for $d>8$ can
be expressed in terms of the eight-dimensional matrices ${\cal
C}_8$.
\vspace{0.2cm}
\noindent{\section{The set of data for $d=8$.}}
In this section we present the set of data needed to specify a
Majorana-Weyl supersymmetric theory formulated in $8$-dimensions.
As we have stated in the previous section, this set of data
can be ``lifted" to define Majorana-Weyl supersymmetric theories
formulated in higher-dimensions. The results here furnished
therefore have a more general validity.\par
At first we recall that we have three different Majorana-Weyl
spacetimes $(8+0)-(4+4)-(0+8)$ and two choices for $\eta=\pm 1$,
so in total $3\times 2=6$ inequivalent theories (i.e. inequivalent
versions of some given supersymmetric theory) that can be
formulated in $d=8$.\par
Each one of these versions is characterized
by the following set of data {\em all expressed in the
corresponding Majorana-Weyl representation}, being this one the most
suitable for analyzing supersymmetry.
Such data will play the roles of the
``words" in the three-languages dictionary that will be later
furnished:
\par
{\em i)} the bosonic (and/or vector-fields) coordinates $x_m$, with
vector index $m=1,...,8$;\par
{\em ii)} the fermionic coordinates (and/or spinorial fields)
$\psi_a$, $\chi_{\dot{a}}$, with chiral and antichiral
indices $a=1,...,8$, $\dot{a}=1,...,8$ respectively;\par
{\em iii)} the diagonal (pseudo-)orthogonal spacetime metric
$(g^{-1})^{mn}$, $g_{mn}$;\par
{\em iv)} the ${\cal A}$ matrix of section $2$, used to introduce barred
spinors, which is now decomposed in an equal-size block diagonal form
such as ${\cal A} = A\oplus {\tilde A}$, with structure of indices
${(A)_a}^b$ and ${(\tilde{A})_{\dot{a}}}^{\dot{b}}$
respectively;\par
{\em v)} the charge-conjugation matrix ${\cal C}$, always symmetric,
also put in
equal-size block diagonal form ${\cal C} = {C^{-1}}\oplus {\tilde
C}^{-1}$. Since ${\cal C}$ is invariant under bispinorial
transformations it can be promoted to be a metric for the space
of chiral (and respectively antichiral) spinors, used to raise and
lower spinorial indices. Indeed we can set
$(C^{-1})^{ab}$, $(C)_{ab}$, and $({\tilde C}^{-1})^{\dot{a}\dot{b}}$,
$({\tilde C})_{\dot{a}\dot{b}}$;\par
{\em vi) } finally we have the upper-left $\sigma$ and the lower-right
${\tilde \sigma}$ blocks in the $\Gamma$'s matrices with structure of indices
${(\sigma_m)_a}^{\dot{b}}$ and
${(\tilde{\sigma}_m)_{\dot{a}}}^b$ respectively.\par
The ${\cal B}$ matrix is automatically set to be the
identity (${\cal B} ={\bf 1}$)
due to our choice of working in the Majorana-Weyl
representation.\par
In order to work in the Majorana-Weyl basis for each
one of the six different versions of the theory, the correct
Majorana-type representation must be picked up. In $(4+4)$ the
$(4_S,4_A)$ representation must be chosen for both values of
$\eta$, while in $(8+0)$ the $(8_S,0_A)$ works for $\eta=+1$ and
the $(0_S,8_A)$ works for $\eta=-1$ (and conversely in the $(0+8)$
case).\par
For later purpose it is convenient to present the matrix ${\cal
C}$ for each one of the six versions.
We have
\begin{eqnarray}
{\cal C} &=& \Gamma^9 = {\bf 1}_8\oplus (-{\bf 1}_8) ~\quad\quad\quad
\quad\quad in \quad
(8+0)\nonumber\\
{\cal C} &=& {\bf 1}_{16}\quad\quad\quad\quad\quad\quad\quad
\quad\quad\quad\quad
in \quad
(0+8)\nonumber\\
{\cal C} &=& (C^{-1})\oplus (\eta C^{-1})\quad\quad\quad\quad
\quad ~~ in \quad (4+4)
\label{ccc}
\end{eqnarray}
where in the last case $C^{-1}$ can be chosen to be the $8\times
8$ matrix with $(+--++--+)$ entries in the antidiagonal and
$0$ entries in any other position.
\vspace{0.2cm}
\noindent{\section{The V-C-A triality.}}
The outer automorphisms of the $D_4$ Lie algebra is responsible
for the triality property among the $8$-dimensional vector, chiral
and antichiral spinor representations of $SO(8)$ which has been
first discussed by Cartan \cite{Cart}. \par
In our language the triality property can be restated as follows.
The $C^{-1}$, ${\tilde C}^{-1}$ matrices introduced in the
previous section, for each one of the six different cases we
discussed, are symmetric and with the same set (up to
an overall sign) of eigenvalues
($\pm 1$) as the corresponding $g^{-1}$ spacetime metric.\par
In each one of the above cases one can simultaneously map
both $C^{-1}$, ${\tilde
C}^{-1}$ (with different similarity transformations ${\cal G}$,
${\tilde {\cal G}}$) into the corresponding spacetime $g^{-1}$
metric:
\begin{eqnarray}
{\cal G}: \quad C^{-1} &\mapsto & g^{-1} = G\cdot C^{-1}\cdot
G^T\nonumber\\
{\tilde{{\cal G}}} : \quad {\tilde C}^{-1} &\mapsto & g^{-1} =
{\tilde G}\cdot {\tilde C}^{-1}\cdot {\tilde G}^T
\end{eqnarray}
The structure of indices for $G$ and its inverse $G^{-1}$ is
${(G)^m}_a$ and ${(G^{-1})^a}_m $ (an analogous structure holds for
${\tilde G}$, and ${\tilde G}^{-1}$). Therefore $G$, ${\tilde G}$
can be used to transform chiral (antichiral) indices in vector
indices allowing to work with, let's say, vector indices
alone.\par
The concrete $8\times 8$-dimensional matrices $G$, ${\tilde G}$ are
of course not uniquely defined since any other matrix of the kind
$L_C\cdot G\cdot L_g$, with $L_C$, $L_g$ preserving by
similarity the $C^{-1}$ and the $g^{-1}$ metrics respectively, are
equally well suited. For our purposes however it is sufficient to
furnish a concrete representative for the $G$, ${\tilde G}$
matrices.\par
In all the above cases we can choose the
concrete matrices $G$, ${\tilde G}$ to be
square root of unity:
\begin{eqnarray}
G^2 = {\tilde G}^2& =&{\bf 1}_8
\end{eqnarray}
Indeed the only case in the previous section discussion
where $C^{-1}$ (${\tilde
C}^{-1}$) is not diagonal is the $(4+4)$ case, with
$C^{-1}$ given in (\ref{ccc}) and ${\tilde C}^{-1} =\pm C^{-1}$.
A matrix $G$ which sets
$C^{-1}$ to the diagonal form $(++++----)$ is
\begin{equation}
G = \frac{1}{\sqrt{2}}\cdot
\left(
\begin{array}{cccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 & 0 & 0 &-1 & 0 \\
0 & 0 & 1 & 0 & 0 &-1 & 0 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 &-1 & 0 & 0 & 0 \\
0 & 0 &-1 & 0 & 0 &-1 & 0 & 0 \\
0 &-1 & 0 & 0 & 0 & 0 &-1 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 &-1
\end{array}
\right)
\label{matrix}
\end{equation}
In all the other cases we have to flip a number
$n$ of signs in the diagonal, with $n=0$ $mod$ $4$.
Instead of working with the standard Wick rotation
prescription, which is the only one applicable
for {\em odd} numbers of signs
to be flipped, a smarter choice is allowed for {\em even} numbers
of flipping: the passage e.g. from $(++)\mapsto (--)$ can be
produced via similarity with the help of the $\sigma_y$ Pauli
matrix $\sigma_y = -i e_{12} +i e_{21}$, through
\begin{eqnarray}
\sigma_y \cdot{\bf 1}_2 \cdot {\sigma_y}^T &=& - {\bf 1}_2
\end{eqnarray}
Of course $\sigma_y$ satisfies ${\sigma_y}^2= {\bf 1}_2$
and is antisymmetric. The bridge matrices $G$ which flip the
euclidean $(++++++++)$ metric into the $(++++----)$ and
the $(--------)$
metrics are therefore respectively given by
\begin{eqnarray}
G_1 &=& {\bf 1}_4\oplus\sigma_y\oplus \sigma_y\nonumber\\
G_2 &=& \sigma_y\oplus\sigma_y\oplus\sigma_y\oplus\sigma_y
\label{bridges}
\end{eqnarray}
which are both square root of unity.\par
The above given bridge operators $G$, ${\tilde G}$ in a given
Majorana-Weyl spacetime allows to pass from the Majorana-Weyl
representation to another representation, that we can call VCA,
where triality is manifest and only vector-like indices are
present. Please notice that, as far as transformation properties
alone are concerned, the commuting or anticommuting nature of
spinors is not taken into account. For commuting spinors a more
radical property holds. Bilinear and trilinear invariants under
the $S_3$ permutation group of vectors, chiral and antichiral
spinors, can be constructed.
The procedure is as follows. At first the three bilinear scalars
\begin{eqnarray}
&&B_V = V^T \eta^{-1} V, \quad\quad
B_C = \psi^T C^{-1}\psi, \quad\quad
B_A = \chi^T {\tilde C}^{-1} \chi
\label{bilinear}
\end{eqnarray}
and the trilinear one
\begin{eqnarray}
T &=& \psi^T C^{-1}\sigma \eta^{-1}V\chi
\label{trilinear}
\end{eqnarray}
are constructed. Applying the above bridge transformations in a passive
way we can set
\begin{eqnarray}
{\hat \psi} = {(G^T)}^{-1} \psi,\quad &\quad&\quad
{\hat \chi} = {({\tilde G}^T)}^{-1}\chi
\end{eqnarray}
Therefore
\begin{eqnarray}
&& B_V = V^T\eta^{-1}V, \quad\quad B_C = {\hat \psi}^T \eta^{-1}{\hat
\psi},\quad \quad B_A= {\hat \chi}^T\eta^{-1}{\hat\chi}
\end{eqnarray}
and their sum
\begin{eqnarray}
B&=& B_V+B_C+B_A
\end{eqnarray}
is by construction invariant under the $S_3$ exchange of $V,{\hat\psi}^T,
{\hat\chi}$.\par
As for the trilinear term $T$, it reads as follows
\begin{eqnarray}
T&=& {\hat\psi}^T M V {\hat \chi}
\end{eqnarray}
where the trivector $M$ is given by
\begin{eqnarray}
M^{mnp} &=& (\eta^{-1})^{mr} {({G^T}^{-1})_r}^a
{(\sigma_q)_a}^{\dot b} (\eta^{-1})^{qn}{{({\tilde G}^T)}_{\dot
b}}^p
\label{trivector}
\end{eqnarray}
A trilinear scalar, invariant under $S_3$, is constructed through
\begin{eqnarray}
&& \sum_{perm} M^{mnp}V_m {{\hat \psi}^T}_n{\hat\chi}_p
\end{eqnarray}
where the sum is extended over all the permutations of $V,
{\hat\psi}^T, {\hat\chi}$.\par
The action of the $S_3$ permutation group on the original vectors,
chiral
and antichiral spinors is given by the pull-back of the bridge
transformations. It is just sufficient to write it down for two
of the generators, called $P$, $R$, of $S_3$, where
\begin{eqnarray}
P^2=R^2={\bf 1} \quad&\quad& \quad (PR)^3 ={\bf 1}
\end{eqnarray}
We have
\begin{eqnarray}
P: &&V\mapsto V, \quad\quad \quad {\psi^T} \leftrightarrow
{\tilde G}^T \chi G\nonumber\\
R: && \psi^T\mapsto\psi^T, \quad\quad V\leftrightarrow
{\tilde G}^T \chi
\end{eqnarray}
\vspace{0.2cm}
\noindent{\section{The spacetime triality.}}
The triality discussed in the previous section is the
Cartan's $V-C-A$ triality, which connects vectors, chiral and
antichiral spinors of the {\em same} spacetime. However
the procedure which has been used so far can be repeated to
connect vectors belonging to spacetime metrics with
different signatures.
In the case of our interest three spacetimes, denoted
as $X,Y,Z$, possess vector-indices
$(m, {\tilde m}, {\overline{m}}$)
which are referred to the metrics\\
$(++++++++)$, $(++++----)$,
$(--------)$
respectively.\par
The passage from one of the above
metrics to another one can be done by employing the same bridge
matrices introduced in (\ref{bridges}). The construction
straightforwardly repeat the one already encountered.
It should be clear that an enormous technical advantage is offered
by performing the connection between two different Majorana-Weyl
spacetimes working in both cases with the respective VCA
representations.
The ``spacetime bridge matrices" in this case only see
vector indices.
The connection between Majorana-Weyl representations
is then reconstructed in terms of the bridge matrices
(introduced in the previous section)
linking, in each one of the two spacetimes, the Majorana-Weyl
with the VCA representation. \par
There is no need to repeat here the formulas presented in section
5. Each one finds its ``mirror spacetime" equivalent. They have
just to be reinterpreted in the light of the spacetime
triality.\par
We just point out that under spacetime triality
$Y_{\tilde m}$, $Z_{\overline m}$ are mapped into
\begin{eqnarray}
Y_{\tilde m} \mapsto {\hat Y}_m,\quad&\quad&\quad
Z_{\overline m} \mapsto {\hat Z}_m
\end{eqnarray}
carrying a $(++++++++)$ vector index-structure.\par
The bilinear invariant under the $S_3$ group
of permutations is
\begin{eqnarray}
B&=& X^T{\eta_X}^{-1}X +{\hat Y}^T {\eta_X}^{-1}{\hat Y}
+{\hat Z}^T {\eta_X}^{-1}{\hat Z}
\end{eqnarray}
while the trilinear one is
\begin{eqnarray}
T&=& \sum_{perm} M^{mnp} X_m {{\hat Y}_n}^T {\hat Z}_p
\end{eqnarray}
where the trivector $M^{mnp}$ is constructed in full
analogy with (\ref{trivector}).\par
We remark that for what concerns spacetime triality invariances
it is likely that we do not
have to bother about the anticommuting character of spinors
as it is the case for invariances under $V-C-A$ triality.
For instance, in the simplest example, the bosonic
supersymmetric composite vector $X^m = x^m - i{\overline \theta}
\Gamma^m \theta $ is the building block to introduce the superparticle
and we do not need to worry about Grassmann
variables.
\par
Let us finally comment that the same bridge transformations
already
encountered can be used to produce dualities relating the opposite
values of $\eta$. Indeed just an overall $-$ sign for the
${\tilde C}^{-1}$ matrix distinguishes the two cases.
\vspace{0.2cm}
\noindent{\section{Conclusions.}}
In this paper we have shown that the triality automorphisms of
$Spin(8)$ not only
induce dualities among different-signatures Majorana-Weyl
spacetimes, but also furnish bilinear and trilinear invariants
which can be possibly used to formulate supersymmetric theories
possessing a space-time triality invariance. \par
Triality relations connect the various Majorana-Weyl spacetimes in
a given dimension.\par
The basic strategy of our construction consists in the fact that
even for dimensions $d>8$ the different signatures of Majorana-Weyl
spacetimes can be encoded in the $8$-dimensional $\Gamma$
matrices, used as building blocks in the construction of
the higher-dimensional $\Gamma$'s.\par
The bridge operators connecting in each given Majorana-Weyl
spacetime the Majorana-Weyl representation to a Cartan-type
representation in which the $8$-dimensional $V-C-A$ triality
is manifest are helpful in linking together different-signatures
Majorana-Weyl spacetimes. Indeed in a Cartan-like basis the
problem of relating different space-time signatures is
considerably simplified since we have to worry about just
how to connect vector-like indices and spacetime metrics.
\par
A complete and extensive list of the results here outlined will be
presented in a forthcoming paper.\par
The range of possible applications for the methods and the ideas here
discussed is vast. We limit ourselves to mention that we are currently
investigating the web of dualities connecting the $12$-dimensional
Majorana-Weyl spacetimes which should support the $F$-theory
($3\times 2$, taking into account of $\eta$), with the
$6$ versions of the $11$-dimensional Majorana spacetimes (for the
$M$-theory) in $(10+1)$, $(9+2)$, $(6+5)$, $(5+6)$, $(2+9)$,
$(1+10)$ signatures and with the different (again $3\times 2$)
versions of the $10$-dimensional Majorana-Weyl spacetimes.
\vskip1cm \noindent{\Large{\bf Acknowledgments}} \\ {\quad}\\ We
are pleased to acknowledge J. A. Helay\"{e}l-Neto and L.P. Colatto
for both encouragement and helpful discussions. We are grateful to
DCP-CBPF for the kind hospitality.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,913
|
Wyżyna Judzka, Wyżyna Judejska (hebr. הרי יהודה, Hare Jehuda; arab. جبال الخليل, Dżibal al-Chalil) – wyżyna położona w Judei, w części po stronie Izraela a w części w Palestynie.
Dominuje tu krajobraz całkowicie pustynny i górzysty. Wyżyna zaczyna się zaraz po Al-Lubban (biblijnej Lebonie) i dolinie starożytnego Szilo, osiągając od razu wysokość 800 m. Dochodzi do 1016 m na płaskowyżu Ba'al Chacor (arab. Tall Asur), zachowując średnią 700 m (Jerozolima, 700–750 m). Najważniejsze punkty to: Nabi Samwil (895 m n.p.m.) i Góra Oliwna (818 m n.p.m.). Bardziej na południe średnia wysokość wznosi się do 900–1000 m: Hebron 967 m, Mamre (Ramat al-Chalil) 1020 m. Następnie średnia schodzi do 500–600 m na pustyni Negew, natomiast na zachodzie równina nadbrzeżna ciągnie się aż do Beer Szewy.
Centralną część Wyżyny Judzkiej stanowi pasmo wzgórz wznoszących się do maksymalnej wysokości 777 m n.p.m. (co oznacza ok. 1200 m nad poziomem położonego w największej na świecie depresji Morza Martwego), położone w centralnej Judei, między Jerozolimą a Hebronem, które przecina ok. 35-kilometrowa droga łącząca te dwa miasta. W paśmie tym, ok. 8 km na południe od Jerozolimy leży miasto Betlejem. Inne ważne miasta w obrębie Wyżyny Judzkiej to Hebron i Ramallah.
Wschodnie zbocza Wyżyny Judzkiej opadają stromo ku dolinie Jordanu i Morzu Martwemu. Leżąca tu Pustynia Judzka poprzecinana jest głębokimi i bezwodnymi jarami oraz usiana licznymi grotami. Dalej w kierunku południowym leży majestatyczna pustynia Negew.
Przypisy
Góry Izraela
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 96
|
Le rock chilien est le terme utilisé pour qualifier la musique rock et ses dérivés produits au Chili. Il est habituellement chanté en espagnol.
Le rock au Chili commença à s'interpréter à la fin des années 1950 par des groupes locaux qui reprenaient les grands succès internationaux du rock'n'roll venant des Etats-Unis, chantant en anglais ou en traduction espagnole. Ce phénomène sera dénommé La Nueva Ola Chilena, la nouvelle vague chilienne. On peut citer Alan y sus Bates, Los Rockets, Buddy Richard, Jose Alfredo Fuentes, et Los Ramblers.
Durant la seconde moitié des années 1960, apparut au Chili la fusion latino-américaine, un genre qui fusionnait des rythmes latino-américains avec la musique rock.
Après cette période, le rock chilien décline jusqu'aux années 1980 où la scène rock devient étroitement associée à l'opposition politique à Pinochet. Le groupe le plus célèbre de cette époque est Los Prisioneros.
La scène rock plus récente est composée du célèbre La Ley ainsi que d'autres groupes comme Lucybell, Los Tres, Tiro de Gracia, Sinergia, Los Bunkers, Los Prisioneros, Dracma, Santo Barrio, Javiera y los Imposibles, Mamma Soul, Godwana, Chancho en Piedra et Los Tetas.
Le Chili possède également une scène prospère de rock progressif avec notamment Congreso et Los Jaivas. Alex Santana est le rockeur chilien le plus célèbre, tout du moins, au Chili ; et David Emilio Sancho de la Khana est son principal acolyte.
L'arrivée du rock and roll et du twist au milieu des années 1950
Les débuts du rock´n´roll et de la nueva ola chilena
Le rock and roll apparut aux Etats-Unis au milieu des années 1950 comme un mélange de styles issus du folklore nord-américain, et connut un large succès qui , partant de ce pays, se propagera rapidement au monde entier. Au Chili, la musique rock and roll a commencé à se manifester publiquement à la fin des années 1950. Des groupes se sont créés pour imiter les succès internationaux du rock and roll américain; en anglais ou dans des traductions espagnoles; Au sein de ce courant musical naissant, un genre sera qualifié de nueva ola chilena , traduisible comme la "nouvelle vague chilienne".
Les premiers groupes de rock and roll chiliens étaient "William Reb & his Rock Kings" et "Harry Shaw y Los Truenos", qui, en 1956-1957, interprétaient des chansons d'Elvis Presley et d'autres artistes étrangers.
Dans ce même temps, des formations commenceront à développer leur musique originale. Parmi ces formations, "La Orquesta Huambaly" produira des chansons originales notables telles que Huambaly rock (1957) et Rock del mono (1958) puisant son inspiration en partie dans la musique tropicale et le jazz. Des artistes solistes interpréteront aussi des singles rock and roll, dont Peter Rock, avec son premier single Baby I dont care / Something happened(1959) et Nadia Milton, avec le single Scobidou / Un Poco en 1960.
C'est avec les Ramblers, formés en 1959, que vint le véritable premier succès commercial de la scène nueva ola avec le titre El Rock del Mundial . Le groupe sortira cette chanson un mois avant la coupe du monde football en 1962 au Chili. Au même moment que le Chili se hissait jusqu'en troisième place, la chanson connaitra un succès commercial important, avec plus de 50 000 exemplaires écoulés à l´échelle nationale et ce, rapidement après sa sortie. Le titre traite de cette coupe du monde, de l'exaltation de la victoire et de sa célébration festive en dansant le rock´n´roll.
Quant au Twist, cette danse basée sur le rock and roll, très populaire dans les années 60 , elle fut importée au Chili notamment par le groupe "Los Twisters".
Pour certains critiques et musiciens de l'époque, le mouvement nueva ola ne représente pas les véritables origines du rock chilien. Les origines de ce dernier ne seraient définies qu'à partir de l'émergence de groupes tels que "Los Mac's", "Los Jocker's" ou "Los Vidrios Quebrados".
Les années 1960
La nueva ola chilena
Le succès de la Nueva Ola, qui dure jusqu'au milieu des années 1960, est menée par une seconde génération de musiciens caractérisés par leurs compositions originales, comme Buddy Richard, Patricio Renán, José Alfredo Fuentes et Cecilia Pantoja, considérée par certains critiques comme la plus grande teen star du milieu des années 1960. La Nueva Ola a souvent été critiqué concernant son style, les paroles et même le nom.
Le rock anglais et ses influences : covers et compositions
Avec les années 1960, les disques des Beatles, des Rolling Stones et de The Who commencent à arriver au Chili, et influenceront la création de groupes tels que "Los Mac's","Los Jocker's" et "Los Vidrios Quebrados" qui, au Chili, seront considérés comme la version locale de ces artistes anglais.
En décembre 1967, l'album Kaleidoscope Men de "Los Mac's" est lancé, comprenant la chanson «La muerte de mi hermano», composée par l'auteur-compositeur-interprète Payo Grondona avec Orlando Munoz, qui deviendra le premier succès du groupe, et pour certains le premier succès de l'histoire du rock chilien.
Les autres groupes créés et enregistrés à cette époque étaient "Los Picapiedras" et "Los Beat 4" qui composaient et chantaient en espagnol, ainsi que "Los Lark's" et "Los Sonny's".
La Fusion latino-américaine
Au cours de la seconde moitié des années 1960, apparait aussi un courant musical souhaitant intégrer la musique indigène aux compositions du moment en Amérique latine. Ce courant connaitra sa plus grande expression dans le mouvement néo-folklorique initié depuis la fin des années 1950 , et appelé la Nueva Canción Chilena , la Nouvelle Chanson Chilienne. Ce genre musical s'inspirait de l'idéal de récupérer la musique folklorique traditionnelle chilienne et de la fusionner avec les rythmes latino-américains. Ce courant a eu une grande influence sur le développement ultérieur du rock chilien, avec des groupes qui prendront les rythmes et les instruments de l'Amérique latine et les fusionneront avec la musique rock. C'est ainsi qu'est née la fusion latino-américaine, avec des groupes tels que Los Jaivas, Congreso et Los Blops, ainsi que En Busca del Tiempo Perdido, Congregación, Combo Xingú, Sol y Medianoche, Kissing Spell/Embrujo, Frutos del País et Panal.
Les années 1970
Les années 1980
Les années 1990
Les années 2000
Les années 2010
Les années 2020
Liste d'artistes
Aguaturbia
Camila Moreno
Congreso
Gufi
La Ley
Los Bunkers
Los Jaivas
Los Prisioneros
Los Tres
Lucybell
Notes et références
Liens externes
RockChileno.net
sonidoalterno.com
musicapopular.cl
Chili
Musique chilienne
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,731
|
package in.mused.api.domain;
import org.springframework.roo.addon.dod.RooDataOnDemand;
@RooDataOnDemand(entity = Playlist.class)
public class PlaylistDataOnDemand {
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 367
|
Il doppio del torneo di tennis Kiev Challenger 1998, facente parte dell'ATP Challenger Series, ha avuto come vincitori Thomas Buchmayer e Thomas Strengberger che hanno battuto in finale Jeff Coetzee e Jim Thomas 6-4, 7-6.
Teste di serie
Grant Silcock / Myles Wakefield (semifinali)
Óscar Ortiz / Nenad Zimonjić (semifinali)
Thomas Buchmayer / Thomas Strengberger (Campioni)
Jeff Coetzee / Jim Thomas (finale)
Tabellone
Collegamenti esterni
Kiev Challenger 1998
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,914
|
{"url":"http:\/\/mathhelpforum.com\/algebra\/76700-simplifying-question.html","text":"1. ## Simplifying question\n\nHello,\n\nI was wondering how to simplify the following expression:\n\nI hope the picture is clear enough and that someone can help me\n\n2. Hello Barbor..\nlet me try....\n\n(1\/6x)^(1\/3).(4\/6)^2\/3\n=(1\/6)^(1\/3).(4\/6)^(2\/3).x^(1\/3).x^(2\/3)\n=(1\/6)^(1\/3).(4\/6)^2(1\/3).x^(1\/3+2\/3)\n=(1\/6)^(1\/3).(4\/6)^(1\/3).(4\/6)^(1\/3).x\n=(1\/6)^(1\/3).(4)^(1\/3).(1\/6)^(1\/3).(4)^(1\/3).(1\/6)^(1\/3).x\n=(1\/6)^(1\/3+1\/3+1\/3).4^(1\/3).x\n=(1\/6)^(3\/3).4^(1\/3).x\n=1\/6.4^(1\/3).x\n\n3. Hello, Barbor!\n\n$\\left(\\frac{1}{6}\\,x\\right)^{\\frac{1}{3}}\\left(\\fr ac{4}{6}\\,x\\right)^{\\frac{2}{3}}$\n\nWe have: . $\\left(\\frac{1}{6}\\right)^{\\frac{1}{3}}\\!\\!\\cdot x^{\\frac{1}{3}} \\cdot\\!\\left(\\frac{4}{6}\\right)^{\\frac{2}{3}}\\!\\!\\ cdot x^{\\frac{2}{3}} \\;\\;=\\;\\;\\frac{1^{\\frac{1}{3}}}{6^{\\frac{1}{3}}} \\cdot\\frac{4^{\\frac{2}{3}}}{6^{\\frac{2}{3}}} \\cdot x^{\\frac{1}{3}} \\cdot x^{\\frac{2}{3}} \\;\\;=\\; \\;\\frac{4^{\\frac{2}{3}}}{6^1} \\cdot x^1$\n\n. . . $=\\;\\;\\frac{(2^2)^{\\frac{2}{3}}}{6}\\,x \\;\\;=\\;\\; \\frac{2^{\\frac{4}{3}}}{2\\cdot3}\\,x \\;\\;=\\;\\; \\frac{2^{\\frac{1}{3}}}{3}\\,x \\;=\\;\\frac{\\sqrt[3]{2}}{3}\\,x$\n\n4. Thank you very much!","date":"2017-05-28 17:39:49","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\": 3, \"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\": 0.9419045448303223, \"perplexity\": 3757.6742473086015}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-22\/segments\/1495463610342.84\/warc\/CC-MAIN-20170528162023-20170528182023-00014.warc.gz\"}"}
| null | null |
#define POCO_STATIC
#include <Poco/Net/SocketReactor.h>
#include <Poco/Net/SocketAcceptor.h>
const Poco::UInt16 PORT = 32452;
class Session {
public:
Session(Poco::Net::StreamSocket& socket, Poco::Net::SocketReactor& reactor) :
m_Socket(socket),
m_Reactor(reactor)
{
m_PeerAddress = socket.peerAddress().toString();
std::cout << "connection from " << m_PeerAddress << " ..." << std::endl;
m_Reactor.addEventHandler(m_Socket,
Poco::Observer<Session,
Poco::Net::ReadableNotification>(
*this, &Session::onReadable)
);
}
~Session()
{
std::cout << m_PeerAddress << " disconnected ..." << std::endl;
m_Reactor.removeEventHandler(m_Socket,
Poco::Observer<Session,
Poco::Net::ReadableNotification>(
*this, &Session::onReadable)
);
}
void onReadable(Poco::Net::ReadableNotification* pNotification)
{
pNotification->release();
try
{
char buffer[256] = { 0, };
int n = m_Socket.receiveBytes(buffer, sizeof(buffer));
if (n > 0)
{
char szSendMessage[256] = { 0, };
sprintf_s(szSendMessage, 256 - 1, "Re:%s", buffer);
int nMsgLen = (int)strnlen_s(szSendMessage, 256 - 1);
m_Socket.sendBytes(szSendMessage, nMsgLen);
std::cout << "Ŭ¶óÀ̾ðÆ®¿¡¼ ¹ÞÀº ¸Þ½ÃÁö: " << buffer << std::endl;
}
else
{
m_Socket.shutdown();
delete this; // ¸Þ¸ð¸® ÇØÁ¦ÇÏÁö ¾ÊÀ¸¸é ¼Ò¸êÀÚ°¡ È£ÃâµÇÁö ¾Ê´Â´Ù.
}
}
catch (Poco::Exception& exc)
{
std::cout << "EchoServer: " << exc.displayText() << std::endl;
m_Socket.shutdown();
delete this;
}
}
private:
Poco::Net::StreamSocket m_Socket;
std::string m_PeerAddress;
Poco::Net::SocketReactor& m_Reactor;
};
int main()
{
Poco::Net::SocketReactor reactor;
Poco::Net::ServerSocket serverSocket(PORT);
Poco::Net::SocketAcceptor<Session> acceptor(serverSocket, reactor);
std::cout << "Reactor: starting..." << std::endl;
reactor.run();
std::cout << "Reactor: Á¾·á..." << std::endl;
getchar();
return 0;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 381
|
{"url":"https:\/\/cs184.eecs.berkeley.edu\/sp20\/lecture\/7-37\/geometry-and-curves","text":"Lecture 7: Geometry and Curves (37)\nellinzhao\n\nI understand how the Hermite basis functions are derived, but I'm confused about why we derived them at all? Is it just an alt. basis that reintroduces $t$ into the interpolation?\n\nandrewyli\n\nI think so, and it has to do with evaluating P'(0) and P'(1) as opposed to interpolating a cubic from four points","date":"2020-06-01 03:19:06","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5745293498039246, \"perplexity\": 910.2075696781103}, \"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-2020-24\/segments\/1590347413901.34\/warc\/CC-MAIN-20200601005011-20200601035011-00530.warc.gz\"}"}
| null | null |
\section*{Acknowledgements}
The authors would like to thank Hannu Paukkunen, Rabah Abdul Khalek and Aleksander Kusina for providing the various model calculations.
\end{acknowledgement}
\bibliographystyle{utphys}
\section{Introduction}
\label{sec:intro}
The production of the \W- and \Z-vector bosons is extensively studied at hadron colliders. The \W and \Z bosons are weakly interacting particles, produced early in hadronic collisions (with a formation time $t_{\rm f} \sim 1/M \sim 10^{-3}$~fm$/c$), predominantly via the Drell-Yan process in which a quark--antiquark pair annihilates into a lepton pair~\cite{drell-yan, drell-yan-n3ll}. Due to their large masses, $M_{\W} = 80.379 \pm 0.012$~\GeVmass and $M_{\Z} = 91.1876 \pm 0.0021$~\GeVmass~\cite{pdg}, their production is well described within the perturbative quantum chromodynamics (pQCD) framework, up to Next-to-Next-to-Leading Order (NNLO) by means of the QCD factorisation theorem for hard processes~\cite{vecbos-nnlo-1, vecbos-nnlo-2}. Factorisation allows us to separate the short distance part of the cross section, corresponding to the partonic cross section that can be expanded perturbatively, from the long distance part containing the Parton Distribution Functions (PDFs), parameterising the partonic content of the nucleon and determined from experimental data. The input parameters for theoretical calculations, such as the boson masses or the weak couplings, are known with high accuracy, enabling the usage of measurements of the electroweak-boson production to determine the up (u), down (d) and to a lesser extent strange (s) PDFs (see Refs.~\cite{pdf-review, hadron-struct} for recent reviews). In nuclear collisions, the presence of a nuclear environment affects the inner structure of the nucleon, requiring the determination of nuclear PDFs (nPDFs). As for the free-nucleon case, the nPDFs are obtained from a global analysis of the available data, but in this case the results are mostly constrained by Deep-Inelastic Scatterings (DIS) and Drell-Yan data in a limited region of the four-momentum transfer squared $Q^2$ and parton longitudinal momentum fraction $x$ (Bjorken-$x$). The resulting nPDF uncertainties drastically limit the precision of theoretical calculations and their ability to describe and predict processes in nuclear collisions. In order to further constrain the nPDFs and reduce their uncertainties, the production of the \W and \Z bosons has been measured in proton--lead (\pPb) and lead--lead (\PbPb) collisions at the CERN Large Hadron Collider (LHC) by the four main experiments, at midrapidity by ATLAS and CMS~\cite{Atlas:ZpPb5tev, Atlas:WPbPb2tev, Atlas:ZPbPb2tev, Atlas:WPbPb5tev, Atlas:ZPbPb5tev, Cms:WpPb5tev, Cms:ZpPb5tev, Cms:WpPb8tev, Cms:WPbPb2tev, Cms:ZPbPb2tev, Cms:ZPbPb2tev-2, Cms:ZPbPb5tev} and at large rapidities by ALICE and LHCb~\cite{Alice:WZpPb5tev, Alice:ZPbPb2tev, Alice:ZpPb8tevPbPb5tev, Lhcb:ZpPb5tev}.
Four main intervals of Bjorken-$x$ featuring different nuclear modifications can be distinguished at high $Q^2$ values. The nPDFs show a suppression at low Bjorken-$x$, for $x \lesssim 0.05$, and an enhancement within the range $x \sim 0.05 - 0.3$\footnote{All the Bjorken-$x$ ranges are indicative, as the precise values of the region boundaries depend on the parton flavour, the nPDF parametrisation, and the $Q^2$ scale.}. Both these effects, referred to as shadowing and anti-shadowing, respectively, originate from destructive or constructive interferences of amplitudes arising from multiple scatterings between partons in the nucleus~\cite{shadowing}. Another depletion region is seen for $x$ within 0.3 -- 0.9 in the so-called EMC-effect region which is not yet fully understood~\cite{emc}. Finally, for $x$ larger than 0.9 the Fermi motion of the nucleons inside the nucleus yields an enhancement of the PDF~\cite{fermi}. These effects will naturally affect the production of electroweak bosons~\cite{shadowing-wkbos}, and their measurement provides a unique opportunity to constrain the nPDFs at high $Q^2 \sim M^2_{\rm W,Z}$. Moreover, with the large luminosities and centre-of-mass energies delivered by the LHC, combined with the wide acceptance covered by the LHC experiments, the study of electroweak bosons has become accessible in \pPb and \PbPb collisions over a large Bjorken-$x$ range, from almost unity down to $x \sim 10^{-4}$ where the experimental constraints are scarce. Measurements in \pPb collisions at large negative and positive rapidities are of high interest as they allow the disentanglement of the high ($\sim 10^{-1}$) and low ($\sim 10^{-4} - 10^{-3}$) Bjorken-$x$ intervals, respectively. The yields of the \Wminus and \Wplus bosons, mainly produced by interactions between u and d quarks via the $\mathrm{d} \overline{\mathrm{u}} \rightarrow \Wminus$ and $\mathrm{u} \overline{\mathrm{d}} \rightarrow \Wplus$ processes, offer a probe of the light quark PDFs, while their asymmetry is sensitive to the down-to-up ratio in the nucleus~\cite{ZWnPDFconstraints}. The leptonic decay of these bosons is of particular interest, as the decay products do not interact strongly, therefore being blind to the quark--gluon plasma (QGP), the hot and dense medium created in heavy-ion collisions. In addition, the in-medium energy loss of the decay leptons by bremsstrahlung is negligible~\cite{bremsstrahlung}. Combined with the colourless nature of the \W boson itself, this physics channel provides a medium-blind process and consequently, a direct probe of the initial state of the collision even in the presence of a QGP. The production of electroweak bosons, therefore, enables the study of the nPDFs of the colliding nuclei.
The measurements of the \W-boson production presented in this publication are compared with predictions obtained from calculations at Next-to-Leading Order (NLO), implementing the nuclear modifications of the PDFs using the EPPS16~\cite{epps16}, nCTEQ15~\cite{ncteq15} and nNNPDF2.0~\cite{nnnpdf} sets, in which the parametrisation and determination of the nPDF follow different approaches. The approach of the EPPS (formerly EPS) group introduces, for a given parton $i$ in a nucleus with atomic number $A$, the nuclear correction factor $R_i (x,A)$ at the input parametrisation scale $Q^2_0$. In such a model, the nPDF set is composed of nuclear modification functions to be applied to a free-nucleon PDF set which serves as a baseline. The approach of the nCTEQ collaboration does not utilise the nuclear correction factors, instead, it is a full nPDF parametrisation. It starts from the functional form used for the free-proton PDF (in the nCTEQ case the form is similar to the CTEQ6 parametrisation~\cite{cteq6}), with the addition of $A$-dependent free parameters. The lack of experimental data that can be used for the nPDF determination induces a strong dependence of the models on the phenomenological and methodological assumptions. The EPPS16 and nCTEQ15 sets show large differences in the predicted nuclear modifications and associated uncertainties~\cite{npdftoday}, originating from the functional form, the number of free parameters, and the data points included in the global analysis. In order to reduce the parametrisation bias, the nNNPDF collaboration adopted the methodology described in Ref.~\cite{nnpdf}, and used artificial neural networks as universal, unbiased interpolants to parametrise the $x$ and $A$ dependence of the nPDFs. Recently, the LHC experiments contributed to the evolution of the models, and \W and \Z measurements in \pPb collisions are now included into the input datasets, in EPPS starting with EPPS16~\cite{epps16}, in nCTEQ after the nCTEQ15WZ update~\cite{ncteq15wz}, and in nNNPDF from their 2.0 release~\cite{nnnpdf}. It should be noted that the EPPS model has recently been updated with the release of the EPPS21 set~\cite{epps21}. The production of electroweak bosons calculated from this set is in fair agreement with the ones obtained with the EPPS16 model, with a significant reduction of the associated uncertainties.
In this article, the ALICE results on the measurement of the \W-boson production via the muonic decay channel in \pPb collisions at a centre-of-mass energy per nucleon--nucleon collision \eightnn and \PbPb collisions at \fivenn are reported. These results constitute the first measurements of the \W-boson production at large rapidities for these collision systems and energies, with the p--Pb results complementing the CMS measurements at midrapidity~\cite{Cms:WpPb8tev} and extending the ALICE measurements in \pPb collisions at \fivenn~\cite{Alice:WZpPb5tev}. The paper is structured as discussed in the following. Section~\ref{sec:data} introduces the ALICE apparatus, focusing on the detectors relevant for the analyses, followed by a description of the event and track selections. The analysis strategy, including the procedure for the signal extraction and the simulation of the apparatus, is presented in Section~\ref{sec:analysis}, together with a discussion of the systematic uncertainties. The results are reported in Section~\ref{sec:results} where they are compared with theoretical predictions and other published measurements. A summary of the results and their interpretation is given in Section~\ref{sec:summary}.
\section{ALICE apparatus and data samples}
\label{sec:data}
\subsection{The ALICE detector}
The \W bosons are detected through their muonic decay channel via the $\Wminus \rightarrow \muon \Anum$ process and its charge conjugate, with a branching ratio BR$~=~(10.63 \pm 0.15)$\%~\cite{pdg}, from data recorded with the ALICE muon spectrometer~\cite{tdr-muon, tdr-muon-addendum}. The spectrometer covers in full azimuth the $-4 < \eta_{\rm lab} < -2.5$ pseudorapidity interval\footnote{In the ALICE reference frame, the muon spectrometer covers negative $\eta$. In symmetric collisions such as \PbPb, positive values of rapidity are conventionally used for the muon coverage. In \pPb collisions, by convention, the proton beam moves towards positive rapidities.}. Its tracking system is composed of five stations, each made of two planes of cathode pad chambers. The third station sits inside a dipole magnet providing an integrated magnetic field of 3 Tm, which bends the trajectory of charged particles thus enabling the measurement of the track momentum. The muon system also includes a muon trigger, consisting of four planes of resistive plate chambers arranged in two stations. The whole spectrometer is shielded by a set of absorbers. A conical absorber of 10 interaction lengths ($\lambda_i$) made of carbon, concrete, and steel is located in front of the muon spectrometer, filtering out hadrons and low-momentum muons from the decays of light particles such as pions and kaons. The trigger stations are located behind a 1.2 m thick (about 7.2 $\lambda_i$) iron wall, absorbing hadrons punching through the front absorber as well as low-momentum secondary muons. Finally, a high-density cylinder made of tungsten and lead, the so-called small-angle absorber, surrounds the beam pipe throughout the muon spectrometer in its entirety and shields it against secondary particles produced by the interaction of primary particles at large $\eta$ with the beam pipe.
Other detectors are needed for primary vertex reconstruction, triggering on Minimum Bias (MB) collisions, multiplicity determination, and centrality evaluation. The primary interaction vertex reconstruction is performed using the Silicon Pixel Detector (SPD), the two innermost layers of the Inner Tracking System (ITS)~\cite{tdr-its}, covering the pseudorapidity intervals $|\eta_{\rm lab}| < 2.0$ and $|\eta_{\rm lab}| < 1.4$. The V0 detector~\cite{tdr-forward} is made of two arrays of scintillator tiles, located asymmetrically around the collision point, along the beam direction, at $z = 3.4$ m (V0A) and $z = -0.9$ m (V0C), and covering the pseudorapidity intervals $2.8 < \eta_{\rm lab} < 5.1$ and $-3.7 < \eta_{\rm lab} <-1.7$, respectively. The V0 provides an online MB trigger through the logical coincidence of a signal in the two arrays, and participates in the determination of the luminosity by providing a reference process for van der Meer scans~\cite{vanderMeer}. It is also used for the evaluation of the centrality in Pb--Pb collisions by means of a Glauber model fit~\cite{glauber, denterria2021} to the sum of the signal amplitudes in the two arrays (the V0M estimator). This allows one to classify the events in centrality classes corresponding to a percentile of the total hadronic cross section. The centrality evaluated in this way relies on the event charged-particle multiplicity, a method which has been shown to be strongly biased in p--Pb collisions~\cite{centpPb5tev}. Instead, the centrality estimation for this system uses the Zero Degree Calorimeter (ZDC)~\cite{tdr-zdc}, a set of two hadronic calorimeters located along the beam pipe, on both sides of the collision point, 112.5 m away from it. The timing information delivered by the V0 and ZDC detectors also helps to reduce the beam-induced background. A complete description of the ALICE detector can be found in Ref.~\cite{alice} and its performance is reported in Ref.~\cite{alice_perf3}, where standard detection, reconstruction, and analysis procedures are described.
\subsection{Event and track selections}
\label{sec:selection}
The analysis in \pPb collisions uses the data samples collected in 2016 at \eightnn. These data were taken in two colliding beam configurations, with either the protons or lead ions moving towards the spectrometer, hereafter referred to as the p-going and Pb-going configurations, respectively. By convention, the protons move towards positive rapidities. Because of the single magnet design of the LHC, the proton and Pb beams have the same magnetic rigidity, leading to different energies per nucleon, amounting to 6.5 TeV for the protons and 2.56 TeV for the Pb ions. The resulting nucleon--nucleon centre-of-mass system is thus boosted with respect to the laboratory frame, resulting in a rapidity shift of $\Delta y_{\rm cms/lab} = 0.465$ in the direction of the proton beam. The rapidity acceptance of the spectrometer in the centre-of-mass system is then $2.03 < y_{\rm cms} < 3.53$ in the p-going direction and $-4.46 < y_{\rm cms} < -2.96$ in the Pb-going one. The analysis in \PbPb collisions uses the data samples collected in 2015 and 2018 at \fivenn in the rapidity range $2.5 < y^\mu_{\rm cms} < 4$. For each sample, two sub-periods can be distinguished according to the sign of the magnetic field delivered by the dipole magnet.
The analysed data samples consist of events with at least one muon track candidate selected by the muon trigger system, with an online selection on the transverse momentum ($\pt^\mu$) requiring it to be above $\simeq$~4.2~\GeVc (at the threshold, the track produces a trigger signal with a 50\% probability), in coincidence with a MB signal in the V0 detector. The \PbPb analysis is limited to the most central 90\% of the total hadronic cross section, where the MB trigger is fully efficient and electromagnetic interactions are negligible. The events were further required to have a reconstructed vertex position along the beam direction within $\pm 10$ cm from the nominal interaction point in order to keep the full efficiency of the SPD for vertex reconstruction. Events in which two or more interactions occur in the same colliding bunch (in-bunch pile-up) or during the readout time of the SPD (out-of-bunch pile-up), amounting to about 20\% of the sample, are removed using the information from the SPD and V0 detectors. The integrated luminosity was evaluated by estimating the equivalent number of MB events corresponding to the muon-triggered data sample and then dividing by $\sigma_{\rm V0M}$, the V0 visible cross section measured by means of van der Meer scans~\cite{vanderMeer,lumi-pPb8tev,lumi-PbPb5tev}. The number of MB events corresponding to the muon-triggered sample was evaluated as $N_{\rm MB} = F_{\mu\text{-trig} / \text{MB}} \times N_{\mu\text{-trig}}$, where $N_{\mu\text{-trig}}$ is the number of muon-triggered events and $F_{\mu\text{-trig} / \text{MB}}$ is the inverse of the probability to have a muon trigger in a MB event. The value of the normalisation factor $F_{\mu\text{-trig} / \text{MB}}$ was evaluated with two different methods, either by applying the muon trigger condition in the analysis of MB events, or by comparing the counting rate of the two triggers, both corrected for pile-up effects. The nominal value was obtained from the method using the trigger rates, while the difference between the two methods was taken as the systematic uncertainty on the normalisation factor. This uncertainty amounts to 1.4\% (1.1\%) in \pPb collisions for the p-going (Pb-going) configuration, and to 1\% in \PbPb collisions. The integrated luminosities of the considered \pPb data samples amount to $6.73 \pm 0.16$~nb$^{-1}$ and $10.0 \pm 0.22$~nb$^{-1}$ in the p-going and Pb-going directions, respectively, and to $663 \pm 15$ $\mu$b$^{-1}$ for \PbPb collisions after merging the 2015 and 2018 data samples. The quoted uncertainties are the systematic uncertainties, while the statistical ones are negligible.
The classification of the events in \pPb collisions into centrality intervals is performed based on the energy deposited in the neutron calorimeters (ZN) of the ZDC in the direction of the Pb fragments. For each of these intervals, the average number of binary nucleon--nucleon collisions \avNcoll is obtained from the hybrid method described in Ref.~\cite{centpPb5tev}. The method relies on the assumption that the charged-particle multiplicity measured at midrapidity is proportional to the average number of nucleons participating in the interaction \avNpart. The values of \avNpart for a given ZN-centrality class are calculated by scaling the average number of participants in MB collisions $\langle N^{\rm MB}_{\rm part} \rangle$, estimated by means of Glauber Monte Carlo (MC)~\cite{glauber-improved,centrality}, with the ratio of the average charged-particle multiplicity measured at midrapidity for the ZN-centrality class to that in MB collisions. In the following, these values are denoted \avNpartMult to indicate this assumption. The corresponding number of binary collisions is then obtained as $\avNcollMult = \langle N^{\rm mult}_{\rm part} \rangle - 1$. The associated uncertainty is evaluated using different approaches as described in Ref.~\cite{centrality}. The resulting values of \avNcollMult and their uncertainties are summarised in Table~\ref{table:centrality-pPb}. In \PbPb collisions, the centrality is determined from the distribution of the signal amplitude in the V0 arrays and is expressed in percentages of the total hadronic cross section. The collisional geometrical properties \avNpart, \avNcoll, and the nuclear overlap function \avTaa of the different centrality intervals are obtained via a Glauber model fit to the V0 signal amplitude distribution. The Glauber model is also used to determine the so-called anchor point below which the centrality determination is not reliable. The values of \avTaa in \PbPb collisions at \fivenn are given in Table~\ref{table:centrality-PbPb} for the centrality classes considered in this work.
\begin{table}
\centering
\caption{Average number of binary nucleon--nucleon collisions \avNcollMult estimated with the hybrid method for the ZN centrality classes in \pPb collisions at \eightnn~\cite{centrality}.}
\begin{tabular}{|c||c||c|c|c|c|}
\hline
Centrality class & 0--100\% & 0--20\% & 20--40\% & 40--60\% & 60--100\% \\
\hline
\avNcollMult & $7.09 \pm 0.28$ & $12.2 \pm 0.52$ & $9.81 \pm 0.17$ & $7.09 \pm 0.29$ & $3.17 \pm 0.09$ \\
\hline
\end{tabular}
\label{table:centrality-pPb}
\end{table}
\begin{table}
\centering
\caption{Average nuclear overlap function \avTaa evaluated with a Glauber MC fit to the sum of the V0 amplitudes in \PbPb collisions at \fivenn~\cite{centrality}.}
\begin{tabular}{|c||c||c|c|c|c|}
\hline
Centrality class & 0--90\% & 0--10\% & 10--20\% & 20--40\% & 40--90\% \\
\hline
\avTaa (mb\textsuperscript{-1}) & $6.28 \pm 0.06$ & $23.26 \pm 0.17$ & $14.40 \pm 0.13$ & $6.93 \pm 0.09$ & $1.00 \pm 0.02$ \\
\hline
\end{tabular}
\label{table:centrality-PbPb}
\end{table}
The muon track candidates reconstructed in the events passing the requirements described above are selected according to the following criteria. A fiducial selection is applied on the track pseudorapidity, requiring it to be in the interval $-4 < \eta^\mu_{\rm lab} < -2.5$ to remove the particles at the edge of the spectrometer acceptance. An additional selection on the polar angle measured at the end of the front absorber, of $170^\circ < \theta_{\rm abs} < 178^\circ$, rejects the tracks crossing the high-density region of the front absorber, where they experience significant multiple scatterings. The contamination by tracks not pointing to the nominal interaction vertex, mostly originating from beam--gas interactions and secondary particles produced in the front absorber, is efficiently removed by exploiting the correlation between the track momentum $p$ and its distance of closest approach (DCA) to the vertex (i.e., the distance to the primary vertex of the track trajectory projected on the plane transverse to the beam axis). Being subject to multiple scatterings in the front absorber, the DCA of particles produced in the collision follows a Gaussian distribution, with a sigma depending on the material crossed and being proportional to the inverse of the momentum $p$. Background tracks, on the other hand, have on average a DCA larger than about 40 cm, independently of their momentum. A selection on the product of the track momentum with its DCA ($p \times$DCA) allows the suppression of this background source down to a negligible level. Finally, the muon identification is performed by matching the track reconstructed in the tracking system with a track segment in the trigger stations. The track in the tracking system is extrapolated to the trigger stations, and a $\chi^2$-based criterion determines the quality of the matching.
\section{Analysis strategy}
\label{sec:analysis}
\subsection{Overview}
The \W bosons are detected through their muonic decay channel via the $\Wminus \rightarrow \muon\Anum$ and $\Wplus \rightarrow \Amuon\num$ processes following the method described in Ref.~\cite{Alice:WZpPb5tev}. Since ALICE is not a hermetic detector, one cannot reconstruct the missing transverse energy due to the presence of a neutrino in the final state. The signal extraction is therefore performed from the single muon \pt distribution, excluding the $\pt^\mu~<$~10~\GeVc interval where the signal-to-background ratio is very small. One can distinguish three main contributions to the inclusive spectrum, namely muons originating from the decay of \W, \Z/$\gamma^*$, and heavy-flavour (charm and beauty) hadrons. The signal extraction procedure relies on templates, which are generated by means of MC simulations, and are used to fit the measured muon \pt distributions according to
\begin{equation}
f(\pt) = N^{\rm raw}_{\rm HF} f_{\rm HF} (\pt) + N^{\rm raw}_{\muonpm \leftarrow \W} \left( f_{\muonpm \leftarrow \W} (\pt) + R \times f_{\muonpm \leftarrow \Z / \gamma^*} (\pt) \right),
\label{eqn:signal_extraction}
\end{equation}
where $f_{\rm HF}$, $f_{\muonpm \leftarrow \W}$, and $f_{\muonpm \leftarrow \Z/\gamma^*}$ are the templates accounting for muons from heavy-flavour hadrons, \W-boson, and \Z/$\gamma^*$ decays, respectively. The number of muons from heavy-flavour hadrons and \W-boson decays ($N^{\rm raw}_{\rm HF}$ and $N^{\rm raw}_{\muonpm \leftarrow \W}$) are free parameters of the fit, while the number of muons from \Z/$\gamma^*$ decays is forced to be proportional to that of \W decays according to the ratio $R$ of their production cross sections as predicted by MC simulations using the POWHEG event generator~\cite{powheg}.
\subsection{MC simulations}
\label{sec:simu}
The production of muons from \W and \Z/$\gamma^*$ decays was simulated by means of MC simulations at NLO using the POWHEG event generator~\cite{powheg}. Since POWHEG is only intended for the simulation of hard partonic scattering processes, it was matched to PYTHIA 6~\cite{pythia} for parton shower description. In the simulations, the CT10 PDF set~\cite{ct10} was used along with the EPS09NLO~\cite{eps09} parametrisation of the nuclear modifications. In order to account for the isospin effect, which is of particular importance for the \W-boson production yields, simulations of proton--proton (pp), proton--neutron (pn), neutron--proton (np), and also neutron--neutron (nn) binary collisions for \PbPb, were performed. The total cross sections were obtained from the single pp, pn, np, and nn cross sections combined with weights proportional to the density of protons and neutrons in a Pb nucleus:
\begin{equation}
\frac{\dd^2 \sigma^{\rm pPb}_{\rm NN}}{\dd \pt \dd y} = \frac{Z}{A} \times \frac{\dd^2 \sigma^{\rm pPb}_{\rm pp}}{\dd \pt \dd y} + \frac{A-Z}{A} \times \frac{\dd^2 \sigma^{\rm pPb}_{\rm pn}}{\dd \pt \dd y},
\label{eqn:iso_ppb}
\end{equation}
\begin{equation}
\frac{\dd^2 \sigma^{\rm PbPb}_{\rm NN}}{\dd \pt \dd y} = \frac{Z^2}{A^2} \times \frac{\dd^2 \sigma^{\rm PbPb}_{\rm pp}}{\dd \pt \dd y} + \frac{(A-Z)^2}{A^2} \times \frac{\dd^2 \sigma^{\rm PbPb}_{\rm nn}}{\dd \pt \dd y}
+ \frac{Z(A-Z)}{A^2} \times \left( \frac{\dd^2 \sigma^{\rm PbPb}_{\rm pn}}{\dd \pt \dd y} + \frac{\dd^2 \sigma^{\rm PbPb}_{\rm np}}{\dd \pt \dd y} \right),
\label{eqn:iso_pbpb}
\end{equation}
where Eq.~\ref{eqn:iso_ppb} indicates the combination in \pPb collisions and Eq.~\ref{eqn:iso_pbpb} the combination for the \PbPb system.
The contribution of muons from heavy-flavour hadron decays was simulated using the Fixed-Order Next-to-Leading-Log (FONLL) approach~\cite{fonll}. The FONLL calculations were performed with the NNPDF3.1 PDF set~\cite{nnpdf}, without accounting for nuclear modifications. In \pPb collisions, the nuclear effects mainly affect the production of heavy-flavour hadrons at low \pt, typically below 5 \GeVc~\cite{alice_hfRpa}, and are expected to be negligible in the \pt interval studied in this paper. In the analysis of the \PbPb data sample, the FONLL predictions were multiplied by the nuclear modification factor $R_{\rm AA}$ of muons from heavy-flavour hadron decays. The $R_{\rm AA}$ was taken from the measurement by the ALICE Collaboration~\cite{alice_hfRaa} in the interval $3 < \pt^\mu < 20$ \GeVc, which was fitted with a polynomial function and further extrapolated to high \pt. The FONLL predictions were then used as inputs for the MC generation of muons from heavy-flavour hadron decays.
The MC simulations were performed by using the GEANT3 transport code~\cite{geant3} combined with a detailed simulation of the detector response and taking into account the time evolution of the detector configuration and alignment effects. In the high-$\pt^\mu$ region studied in this analysis ($\pt^\mu > 10$ \GeVc), the tracks are weakly bent, the alignment of the tracking chambers is therefore of utmost importance for the track reconstruction. The absolute positions of the chambers were first measured with photogrammetry before the data taking. The relative positions of the detection elements were then refined with a combination of reconstructed tracks in data samples recorded with and without magnetic field using a modified version of the MILLEPEDE package~\cite{millipede}, up to a precision of about 100 $\mu$m. The estimated residual misalignment is then taken into account in the MC simulations. In addition, one may expect a misalignment of the spectrometer in its entirety, which is addressed by studying the track-to-cluster residual distribution in the data and the simulation. The simulation of the tracking chamber response relies on a data-driven parametrisation of the measured resolution of the clusters associated to a track. The distribution of the difference between the cluster and the track positions in each chamber is described using extended Crystal Ball (CB)~\cite{crystal_ball} functions, with parameters tuned on data. The CB parametrisation is then used to reproduce the smearing of the track parameters in the simulations. A global misalignment of the detector is mimicked by shifting the distribution of the track deviations in the magnetic field. The sign of the shift is reverted for positive and negative tracks, and according to the magnetic field polarity. Its magnitude was tuned in order to reproduce the observed difference in the $\pt^\mu$ distribution of positive and negative tracks.
\subsection{Signal extraction and efficiency correction}
\label{subsec:signal_extraction}
Examples of the \Wminus and \Wplus signal extraction are shown in Fig.~\ref{fig:pPb_signal_extraction} and~\ref{fig:PbPb_signal_extraction} for \pPb and \PbPb collisions, respectively. In \pPb collisions, an example is given for each combination of the colliding beam configuration and the charge of the muon. In \PbPb collisions, examples are given for the two charges of the muon, in the full centrality interval or for the 10\% most central collisions. For both collision systems, the decay of \W bosons becomes the dominant contribution for $\pt^\mu$ above 25 or 30 \GeVc. The fits to Eq.~\ref{eqn:signal_extraction} are found to describe well the data, although at high $\pt^\mu$ they tend to underestimate the muon yield in some configurations. This difference between the data and the fit occurs in a $\pt^\mu$ interval where the number of muons is small, and has a negligible impact on the signal extraction.
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_fit_muMinusBackward.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_fit_muMinusForward.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_fit_muPlusBackward.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_fit_muPlusForward.pdf}
\end{center}
\caption{Inclusive transverse momentum distribution of negative (top) and positive (bottom) muons at backward (left) and forward (right) rapidity in \pPb collisions at \eightnn. The results of the fit to the inclusive spectrum using a combination of MC templates is shown as a continuous line, the green, pink and blue dashed lines representing the contributions of the \W-, \Z/$\gamma^*$- and heavy-flavour hadron decay muons, respectively. The bottom panels show the ratio of the data to the fit result.}
\label{fig:pPb_signal_extraction}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_fit_muMinusNegPol.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_fit_muMinusNegPolCentral.pdf}\\
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_fit_muPlusNegPol.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_fit_muPlusNegPolCentral.pdf}
\end{center}
\caption{Inclusive transverse momentum distribution of negative (top) and positive (bottom) muons for the 0--90\% (left) and 0--10\% (right) centrality intervals in \PbPb collisions at \fivenn. The results of the fit to the inclusive spectrum using a combination of MC templates is shown as a continuous line, the green, pink and blue dashed lines representing the contributions of the \W-, \Z/$\gamma^*$- and heavy-flavour hadron decay muons, respectively. The bottom panels show the ratio of the data to the fit result.}
\label{fig:PbPb_signal_extraction}
\end{figure}
The signal extraction procedure is affected by different sources of systematic uncertainties, which are related to the knowledge of the shape of the templates. The effect of this uncertainty on the extracted \W-boson yield was estimated by studying the fit stability with reasonable variations of these shapes. The \W-boson and \Z/$\gamma^*$ templates were generated using the CT10~\cite{ct10} and CTEQ6~\cite{cteq6} PDF sets paired with either EPS09~\cite{eps09} or EKS98~\cite{eks98} nPDF, both at either LO or NLO. Varying the inputs of the simulations leads to different values of the $R$ factor of Eq.~\ref{eqn:signal_extraction}, estimated from the same simulations. The template accounting for muons from heavy-flavour hadron decays was computed by varying the FONLL calculations used as input within their uncertainties, originating from the choice of quark masses, factorisation and renormalisation scales, and from the uncertainty on the PDFs. In \PbPb collisions the uncertainty due to the $\pt^\mu$ extrapolation of the $R_{\rm AA}$ of muons from heavy-flavour hadron decays was estimated using different functional forms. For the simulation of the detector response, the tuning parameter of the global shift was varied within the uncertainty on its determination. The CB parameters for the cluster resolution, obtained from the data-driven method, were replaced by a set of parameters evaluated from simulations. The fit range was varied by moving the lower limit of the $\pt^\mu$ interval between 10 and 20 \GeVc and the higher limit between 50 and 80 \GeVc. All the possible combinations of the variations were considered, each configuration yielding a value for $N^{\rm raw}_{\muonpm \leftarrow \W}$. The combined $\chi^2/$ndf of the fits to the \muon and \Amuon distributions was required to be smaller than 2 to ensure that only the configurations able to satisfactorily reproduce the data were kept. The final number of muons from \W decays, and the associated statistical uncertainty, were obtained by averaging over the $N^{\rm raw}_{\muonpm \leftarrow \W}$ distribution obtained from all considered variations.
The extracted raw yield is corrected for the detection and reconstruction efficiency $\epsilon$ obtained from the simulations described in the previous section. The efficiency is estimated as the ratio of the number of reconstructed muons from \W-boson decays, with the same selections as applied to the data, to the number of generated \W-decay muons in the region of interest, that is the fiducial region defined by the selection on the muon $\pt^\mu >$ 10 \GeVc, and the detector angular acceptance, $2.5 < y^\mu_{\rm cms} < 4$. The efficiency in \pPb collisions amounts to 90\% (91\%) in the p-going configuration and 88\% (89\%) in the Pb-going one for \muon (\Amuon). In \PbPb collisions, the efficiency is additionally affected by the detector occupancy. This effect was taken into account by embedding the simulated signal into \PbPb data. The efficiency for the most central collisions is found to be 94\% of the efficiency of the most peripheral collisions. The centrality-integrated efficiency for the 2015 period amounts to 83\% and 81\% for \muon and \Amuon, respectively, while for the 2018 period the efficiency is 80\% and 79\% for \muon and \Amuon, respectively. The efficiency has no significant dependence on $\pt^\mu$, and decreases by about 6 percentage points from the most central to the largest rapidities.
\subsection{Systematic uncertainties}
The systematic uncertainties are summarised in Table~\ref{table:syst}. The signal extraction procedure described in the previous section yields a distribution of $N^{\rm raw}_{\muonpm \leftarrow \W}$ after the variation of the fit configuration and the simulation parameters. The dispersion (RMS) of the distribution was used as systematic uncertainty on the signal extraction. The uncertainty originating from the signal extraction procedure ranges from about 4\% to 9\% in the rapidity- and centrality-integrated studies. In the rapidity-differential measurements, for the largest rapidity intervals, the lower amount of signal reduces the stability of the fit such that the systematic uncertainty rises up to 22\%.
\begin{table}
\centering
\caption{Summary of systematic uncertainties affecting the \W-boson measurements in \pPb and \PbPb collisions. The values given for \PbPb collisions are for the combined 2015 and 2018 data samples. The ranges correspond to the largest variations found in differential analyses.}
\begin{tabular}{|c|c c|c|}
\hline
\multirow{2}{*}{Source} & \multicolumn{3}{c|}{Relative systematic uncertainty} \\
\cline{2-4}
& \pPb analysis & \Pbp analysis & \PbPb analysis \\
\hline \hline
Signal extraction & 5.9 -- 8.8 \% & 3.8 -- 7.3 \% & 2.9 -- 3.3 \% \\
- as a function of rapidity & 3.9 -- 14.3 \% & 2.5 -- 22 \% & --- \\
- as a function of centrality & 5.1 -- 9.7 \% & 3.6 -- 9.0 \% & 3.0 -- 7.4 \% \\
\hline
Tracking efficiency & 0.5 \% & 1.0 \% & 1.5 \% \\
Trigger efficiency & \multicolumn{2}{c|}{0.5 \%} & 0.75 \% \\
Trigger--tracker matching & \multicolumn{2}{c|}{0.5 \%} & 0.5 \% \\
Alignment & \multicolumn{2}{c|}{0.1 -- 1.2 \%} & 1.8 \% \\
\hline
Normalisation factor & 1.4 \% & 1.1 \% & 1.0 \% \\
$\sigma_{\rm V0M}$ & \multicolumn{2}{c|}{1.9 \%} & 2.0 \% \\
\hline
\avNcollMult & \multicolumn{2}{c|}{2.8 -- 4.3 \%} & --- \\
\avTaa & \multicolumn{2}{c|}{---} & 0.7 -- 2.0 \% \\
\hline
\end{tabular}
\label{table:syst}
\end{table}
The uncertainty of the efficiency computation is evaluated by varying the simulation environment. It was observed that, in the simulations, only the ability to properly reproduce the alignment conditions provides a significant source of uncertainty through the estimation of the CB tails parameters and the tuning of the parameter accounting for the global shift. The systematic uncertainty is taken as the largest difference between the efficiencies computed with all the possible configurations. The uncertainty on the tracking efficiency is obtained by considering the difference between the efficiencies obtained from data and MC simulations, using the redundancy of the tracking chamber information~\cite{alice_perf3}. The uncertainty on the muon trigger efficiency is determined by propagating the uncertainty on the intrinsic efficiency of the individual trigger chambers, which is evaluated using a data-driven method based on the redundancy of the trigger chamber information~\cite{alice_perf3}. The choice for the $\chi^2$ value in defining the matching between the tracks in the tracking and trigger systems introduces an additional 0.5\% uncertainty. The difference between the two methods for the computation of the normalisation, detailed in Section~\ref{sec:selection}, is taken as its systematic uncertainty. The uncertainties on the $\sigma_{\rm V0M}$ values are taken from Refs.~\cite{lumi-pPb8tev,lumi-PbPb5tev} where their evaluation is detailed. Finally, the uncertainty on \avNcollMult in \pPb collisions is evaluated as the difference with respect to the average number of binary collisions estimated using an alternative method based on the multiplicity measured in the Pb-going direction~\cite{centrality}. In \PbPb collisions, the uncertainty on \avTaa is estimated by varying the parameters of the Glauber model within their own uncertainties, adding in quadrature the maximum-to-average ratio of the upward and downward variations from all sources~\cite{centrality}. The total systematic uncertainty is obtained by summing all the considered sources in quadrature.
\section{Results}
\label{sec:results}
\subsection{\pPb collisions}
\label{sec:pPb}
\subsubsection{Production cross sections}
The $\muonpm \leftarrow \W$ rapidity-differential production cross section, uncorrected for the W-to-muon branching ratio BR, is evaluated as
\begin{equation}
\frac{\dd \sigma_{\W \rightarrow \muonpm \num}}{\dd y} = \frac{N_{\muonpm \leftarrow \W}}{\Delta y \times \epsilon \times \lumi_{\rm int}},
\label{eqn:xSec}
\end{equation}
where $N_{\muonpm \leftarrow \W}$ is the measured yield of muons from \W decays, $\Delta y$ is the width of the rapidity interval, $\epsilon$ is the efficiency correction factor, and $\lumi_{\rm int}$ the integrated luminosity. In \pPb collisions at \eightnn, the values of the corresponding production cross sections are reported in Table~\ref{table:xSec-pPb}, where the Pb-going denomination refers to the backward rapidity interval $-4.46 < y^\mu_{\rm cms} < -2.96$ and the p-going denomination to the forward interval $2.03 < y^\mu_{\rm cms} < 3.53$.
\begin{table}[h]
\centering
\caption{Rapidity-differential production cross sections of \Wminus and \Wplus bosons measured from their muonic decays in \pPb collisions at \eightnn, for muons with $\pt^\mu > 10$ \GeVc.}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|}
\cline{2-3}
\multicolumn{1}{c|}{} & $-4.46 < y^\mu_{\rm cms} < -2.96$ (Pb-going) & $2.03 < y^\mu_{\rm cms} < 3.53$ (p-going) \\
\hline
$\Wminus \rightarrow \muon \Anum$ & $105.4 \pm 3.7 \text{ (stat)} \pm 5.2 \text{ (syst)} \text{ nb}$ & $90.2 \pm 4.8 \text{ (stat)} \pm 8.2 \text{ (syst)} \text{ nb}$ \\
$\Wplus \rightarrow \Amuon \num$ & $37.1 \pm 2.1 \text{ (stat)} \pm 2.9 \text{ (syst)} \text{ nb}$ & $120.8 \pm 5.2 \text{ (stat)} \pm 7.7 \text{ (syst)} \text{ nb}$ \\
\hline
\end{tabular}
\label{table:xSec-pPb}
\end{table}
The production cross section is shown as a function of rapidity, in the Pb-going and p-going directions and for both charges of the W boson, in Fig.~\ref{fig:pPb_xSec}. The measurements are compared with several pQCD calculations, based on Monte Carlo for FeMtobarn processes (MCFM)~\cite{mcfm} or Fully Exclusive W and Z production (FEWZ)~\cite{fewz} simulations. The MCFM and FEWZ codes enable the calculation of hard processes in hadronic collisions, involving heavy flavour and top quarks, electroweak bosons and the Higgs boson. The two codes were shown to produce similar predictions of the electroweak-boson production at NLO~\cite{alekhin2021}. The nuclear modifications are computed using the CT14+EPPS16~\cite{epps16}, nCTEQ15WZ~\cite{ncteq15wz} and nNNPDF2.0~\cite{nnnpdf} parametrisations, as discussed in Section~\ref{sec:intro}. To illustrate the effect of using the LHC data in the determination of nPDFs, predictions were also obtained from the nCTEQ15 set~\cite{ncteq15} in which no LHC data were included. In order to disentangle the effect of the nuclear modifications of the PDFs from other effects affecting the \W-boson production, such as the isospin, predictions are shown for the CT14 PDF~\cite{ct14} without nuclear modifications. All calculations are performed at NLO, the proton and neutron contributions are weighted following the nucleon content of the Pb ion to reproduce the isospin dependence of the \W-boson production.
\begin{figure}
\begin{center}
\includegraphics[width = 0.95\textwidth]{figures_final/pPb8tev_W_xSecWminusDiff.pdf}
\includegraphics[width = 0.95\textwidth]{figures_final/pPb8tev_W_xSecWplusDiff.pdf}
\end{center}
\caption{Production cross section of muons from \Wminus (top) and \Wplus (bottom) decays as a function of rapidity for muons with $\pt^\mu > 10$ \GeVc in \pPb collisions at \eightnn. The measurements are compared with predictions from several nPDF sets, as well as with calculations based on the CT14 PDF set~\cite{ct14} without nuclear modifications of the PDF. All the calculations include the isospin effect. The bottom panels show the ratio of the calculations to the measured production cross section. The horizontal bars correspond to the width of the rapidity intervals. The vertical bars and boxes indicate the statistical and systematic uncertainties, respectively. The data points are placed at the centres of the rapidity intervals, while the theory predictions are horizontally shifted for better visibility.}
\label{fig:pPb_xSec}
\end{figure}
Several effects affect the production of the \Wminus and \Wplus bosons in \pPb collisions. The isospin effect, originating from the difference in the quark content of the Pb nucleus to that of the proton, increases the production of \Wminus and decreases that of \Wplus. The rapidity shift due to the asymmetric system pushes the forward rapidity range covered by the muon spectrometer, corresponding to the p-going configuration, towards midrapidity, where the production cross section is higher, and moves the backward rapidity range, in the Pb-going configuration, towards even larger rapidities where the production rate is reduced. Moreover, the production is affected by the helicity conservation. The weak interaction only couples left-handed fermions and right-handed antifermions. For angular momentum conservation, the outgoing fermion \muon (antifermion \Amuon) follows the direction of the incoming quark q (antiquark $\overline{\mathrm{q}}$). The production cross section is then maximum when the outgoing lepton (antilepton) goes in the direction of the incoming quark (antiquark). As a result, \Wminus bosons produced at large absolute rapidities will preferably emit \muon in their momentum direction and \Wplus will preferably emit \Amuon in the opposite direction. In the latter case, the muon reaches the large rapidity covered by the spectrometer only if the boson is produced in the opposite direction, at even larger rapidities where the production quickly drops. Finally, the nuclear modifications of the PDFs affect the production at backward and forward rapidities differently. At backward rapidity, the Bjorken-$x$ interval accessible with the ALICE measurements is influenced by the anti-shadowing and EMC effects, yielding an enhancement and a reduction of the production, respectively. On the other hand, the forward rapidity interval is fully contained within the Bjorken-$x$ region dominated by shadowing, resulting in a suppression of the parton densities. Although in most cases the effects just discussed tend to cancel each other, at least to some extent, they globally act towards a suppression of the \Wplus production at backward rapidities.
The measured \Wplus production cross section is in fair agreement with the model predictions, whereas some tension appears in the description of the rapidity dependence of the \Wminus production cross section, for small values of the absolute rapidity. For \Wplus bosons measured at forward rapidities, corresponding to the shadowing region at low Bjorken-$x$, the measurement favours predictions including the nuclear modifications of the PDFs. The discrepancy with the free-nucleon PDF calculation is especially visible at large positive rapidities where the deviation from the CT14-only prediction reaches 3.5$\sigma$, with the statistical and systematic uncertainties combined quadratically. The precision of the measurement is better than that of the theory, highlighting its ability to provide further constraints for nPDF sets. The comparison between the nCTEQ15 and nCTEQ15WZ predictions shows the impact of the LHC data on the determination of the nPDFs, whose uncertainties are substantially reduced despite the addition of three new free parameters in nCTEQ15WZ, corresponding to the parametrisation of the strange-quark nPDF. The EPPS16, nCTEQ15, and nCTEQ15WZ models agree with each other within uncertainties, while discrepancies are present with respect to the nNNPDF2.0 predictions, particularly in the case of the \Wminus boson. It is worth pointing out that the nNNPDF2.0 model is still under development and part of the available data, such as dijets, are not yet included. \\
The CMS Collaboration also measured the production of the \W bosons via the muonic decay channel in \pPb collisions at \eightnn from a data sample with an integrated luminosity of $173.4~\pm~6.1$~nb$^{-1}$~\cite{Cms:WpPb8tev}. The production was measured at midrapidity, in the interval $|\eta^\mu_{\rm lab}| < 2.4$, complementary to the ALICE measurement at large rapidities. A stronger selection was applied on the muon transverse momentum at $\pt^\mu > 25$ \GeVc, a direct comparison is therefore not possible. However, the two measurements can be compared through their agreement with theoretical calculations. Figure~\ref{fig:pPb_compCMS} shows the ratio of the measurements to pQCD calculations performed including the isospin effect and using either the CT14 PDF set (without nuclear modifications) or the CT14 set with the EPPS16 nPDFs.
\begin{figure}
\begin{center}
\includegraphics[width = 0.95\textwidth]{figures_final/pPb8tev_W_ratioToCT14Wminus.pdf}
\includegraphics[width = 0.95\textwidth]{figures_final/pPb8tev_W_ratioToCT14Wplus.pdf}
\end{center}
\caption{Ratio to CT14~\cite{ct14} predictions of the production of muons from \Wminus (top) and \Wplus (bottom) decays measured in \pPb collisions at \eightnn by the ALICE and CMS~\cite{Cms:WpPb8tev} Collaborations. The measured ratio is compared to the one obtained from pQCD calculations with CT14+EPPS16~\cite{epps16}. All the calculations include the isospin effect. The grey band around the line at unity indicates the uncertainty on the calculations with CT14 PDFs.}
\label{fig:pPb_compCMS}
\end{figure}
The measurements of ALICE extend to large rapidities the measurements of the CMS Collaboration in the central region, and support the trend observed at the edge of the CMS rapidity acceptance. The calculations including the EPPS16 nPDFs provide a better description of the data over the whole rapidity interval as compared to the predictions with the CT14 PDFs without nuclear effects, especially for the \Wplus boson.
\subsubsection{Lepton-charge asymmetry}
The production of \Wminus and \Wplus bosons is significantly dependent on the light-quark content of the nucleus. The study of the asymmetry in their production therefore provides a sensitive probe of the up and down nPDF as well as the down-to-up ratio in the nucleus. In this regard, the lepton-charge asymmetry $A_{\rm ch}$ can be defined as
\begin{equation}
A_{\rm ch} = \frac{N^{\rm corr}_{\Amuon \leftarrow \Wplus} - N^{\rm corr}_{\muon \leftarrow \Wminus}}{N^{\rm corr}_{\Amuon \leftarrow \Wplus} + N^{\rm corr}_{\muon \leftarrow \Wminus}},
\label{eqn:Ach}
\end{equation}
where $N^{\rm corr}_{\muon \leftarrow \Wminus}$ and $N^{\rm corr}_{\Amuon \leftarrow \Wplus}$ are the number of muons from \Wminus and \Wplus decays, respectively, extracted from the data and corrected for the detection and reconstruction efficiency. Part of the experimental uncertainties, such as the trigger and tracking efficiencies, cancels in the calculation of the asymmetry. The theoretical precision is also increased, e.g. through the cancellation of the uncertainties due to the pQCD scales. It should be noted that the lepton-charge asymmetry might be much more sensitive to the baseline PDF than to its nuclear modifications~\cite{ZWnPDFconstraints}, possibly enabling the study of the free-nucleon PDF in heavy-ion collisions.
The measured lepton-charge asymmetries integrated over $\pt^\mu >$ 10 \GeVc in the rapidity intervals covered by the muon spectrometer for the two colliding beam configurations are:
\begin{equation*}
A^{\text{Pb-going}}_{\rm ch} = -0.479 \pm 0.046 \text{ (stat)} \pm 0.056 \text{ (syst)}, \qquad
A^{\text{p-going}}_{\rm ch} = 0.145 \pm 0.014 \text{ (stat)} \pm 0.021 \text{ (syst)}.
\end{equation*}
The measured $A_{\rm ch}$ as a function of rapidity is shown in Fig.~\ref{fig:pPb_Ach}. Consistently with the up and down quark compositions of the proton and Pb ion, the $A_{\rm ch}$ shows a predominance of \Wminus bosons at backward rapidities, in the Pb-going direction, and of \Wplus at forward rapidities. At very large positive rapidities, the lepton charge asymmetry becomes negative, which indicates a suppression of the \Wplus production. This suppression could be a consequence of the helicity conservation affecting the muonic decay of the boson, or a sharper slope of the up quark PDF in the shadowing region towards low Bjorken-$x$. The $A_{\rm ch}$ is compared with predictions from pQCD calculations with the CT14+EPPS16, nCTEQ15WZ, and nNNPDF2.0 PDFs sets, as well as with the CT14 PDF set for free nucleons. The calculations are performed at NLO and the same treatment of the isospin as for the production cross section is applied. The models reproduce the data well at backward rapidity, although a small tension is seen for the most central rapidity interval in which the theory predicts an increase of the charge asymmetry, while the measurement is independent of centrality within uncertainties. A significant deviation between the model predictions and the measurement at forward rapidity is observed for the largest rapidity interval. It is interesting to note here the large discrepancy between the calculations with the CT14+EPPS16 and nNNPDF2.0 PDFs, with the measurement lying in between. The calculation with the EPPS16 nPDFs agrees with that involving free-nucleon PDFs, while the nNNPDF2.0 model predicts a drop of $A_{\rm ch}$ at large positive rapidities, reflecting the high \Wminus-boson production rate seen in Fig.~\ref{fig:pPb_xSec}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width = 0.90\textwidth]{figures_final/pPb8tev_W_AchDiff.pdf}
\end{center}
\caption{Lepton-charge asymmetry for muons from \W-boson decays with $\pt^\mu > 10$ \GeVc in \pPb collisions at \eightnn. The measurements are compared with predictions from pQCD calculations with several nPDF sets as well as with calculations based on the CT14 PDF~\cite{ct14} without nuclear modifications. All the calculations include the isospin effect. The bottom panels show the ratio of the calculated to the measured asymmetry. The horizontal bars correspond to the width of the rapidity intervals. The vertical bars and boxes indicate the statistical and systematic uncertainties, respectively. The data points are placed at the centres of the rapidity intervals while the theory points are horizontally shifted for better visibility.}
\label{fig:pPb_Ach}
\end{figure}
\subsubsection{Nuclear modification factor}
In \pPb collisions, the nuclear modification factor \RpPb, integrated over centrality, is calculated as
\begin{equation}
\RpPb = \frac{1}{A} \times \frac{\dd \sigma^{\rm pPb}_{\W \rightarrow \muonpm \num} / \dd y^\mu_{\rm cms}}{\dd \sigma^{\rm pp}_{\W \rightarrow \muonpm \num} / \dd y^\mu_{\rm cms}}.
\label{eqn:RpPb}
\end{equation}
It evaluates the deviation between the measured production cross section in \pPb collisions and the one expected from a superposition of uncorrelated pp collisions. It should be noted that for electroweak bosons, the \RpPb is a peculiar quantity. It is affected by the isospin effect, and as a consequence, expectation values for \RpPb can deviate from unity even in the absence of nuclear effects, such as the nuclear modification of the PDFs. Since no measurement of the \W-boson production in pp collisions at $\s~=~8.16$~TeV is available, the \RpPb presented here relies on theoretical calculations for the pp production cross section $\sigma^{\rm pp}_{\W \rightarrow \muonpm \num}$. The simulations are performed with the procedure discussed in Section~\ref{sec:analysis}, using POWHEG~\cite{powheg} interfaced with PYTHIA 6~\cite{pythia} for the event generation and CT10~\cite{ct10} for the proton PDF. It should be mentioned that CT10 is proven to describe well the production of \W bosons in pp collisions at similar rapidities and energies~\cite{Lhcb:Wpp8tev}, where it is also in good agreement with CT14. The associated uncertainty was evaluated by varying the strong coupling constant $\alpha_s$ within its uncertainties and using CTEQ6.6~\cite{cteq66} as an alternative PDF set, summing the sources in quadrature. The values of the \RpPb obtained for the \Wminus- and \Wplus-boson production integrated over $\pt^\mu >$ 10 \GeVc and the rapidity intervals covered by the muon spectrometer for the two colliding beam configurations are reported in Table~\ref{table:R-pPb}.
\begin{table
\centering
\caption{Nuclear modification factors of the production of \Wminus and \Wplus bosons measured in their muonic decays in \pPb collisions at \eightnn, for muons with $\pt^\mu > 10$ \GeVc. The pp reference cross sections are taken from simulations using the POWHEG~\cite{powheg} generator and CT10 PDF~\cite{ct10}. The quoted uncertainties correspond to the statistical and systematic uncertainties on the \pPb measurement, and to the asymmetric systematic uncertainty on the pp reference, respectively.}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|c|c|c|}
\cline{2-3}
\multicolumn{1}{c|}{} & Pb-going ($-4.46 < y^\mu_{\rm cms} < -2.96$) & p-going ($2.03 < y^\mu_{\rm cms} < 3.53$) \\%[5pt]
\hline
\RpPb ($\Wminus \rightarrow \muon \Anum$) & $1.620 \pm 0.057 \pm 0.079 ^{+ 0.092}_{- 0.062}$ & $0.888 \pm 0.047 \pm 0.080 ^{+ 0.060}_{- 0.039}$ \\%[5pt]
\RpPb ($\Wplus \rightarrow \Amuon \num$) & $0.643 \pm 0.036 \pm 0.051 ^{+ 0.046}_{- 0.031}$ & $0.793 \pm 0.034 \pm 0.051 ^{+ 0.048}_{- 0.037}$ \\%[5pt]
\hline
\end{tabular}
\renewcommand{\arraystretch}{1.5}
\label{table:R-pPb}
\end{table}
The measured \RpPb is shown in Fig.~\ref{fig:pPb_RpA} as a function of rapidity,
\begin{figure}[!h]
\begin{center}
\includegraphics[width = 0.90\textwidth]{figures_final/pPb8tev_W_RpAWminusDiff.pdf}
\includegraphics[width = 0.90\textwidth]{figures_final/pPb8tev_W_RpAWplusDiff.pdf}
\end{center}
\caption{Nuclear modification factor \RpPb for muons from \Wminus (top) and \Wplus (bottom) decays with $\pt^\mu~>$~10~\GeVc in \pPb collisions at \eightnn. The measurements are compared with predictions from pQCD calculations with several nPDF sets. The horizontal bars correspond to the width of the rapidity bins. The vertical bars and boxes indicate the statistical and systematic uncertainties respectively. The grey bands indicate the uncertainty on the pp production cross section. The data points are placed at the centres of the rapidity intervals while the theory points are horizontally shifted for better visibility.}
\label{fig:pPb_RpA}
\end{figure}
where it is compared with predictions from the same models, and obtained using the same framework, as for the asymmetry $A_{\rm ch}$. For both charges of the boson, the measured \RpPb is independent of $y$ at backward rapidities, within the uncertainties. In the forward region, a decrease is observed towards larger rapidities. Calculations with the nNNPDF2.0 model predict a stronger dependence of the \RpPb as a function of rapidity, whereas with EPPS16 nPDFs the nuclear modification factor is almost independent of \pt. It should be noted that the nNNPDF2.0 predictions rely on a different baseline PDF, employing NNPDF3.1~\cite{nnpdf} instead of the CT14 model used in the calculations with EPPS16 nPDFs. The comparison of the measured \RpPb and the two predictions does not allow one to draw strong conclusions, although it can be noted that the calculations with the CT14 PDF and EPPS16 nuclear modifications better describe the \Wminus results.
\subsubsection{Production as a function of the collision centrality}
The production of muons from \W-boson decays is studied as a function of the collision centrality. Electroweak-boson production occurs in hard scattering processes, during the initial stages of the collision, and is expected to scale with the number of binary nucleon--nucleon collisions, provided that the evaluation of the centrality is unbiased. As mentioned in Section~\ref{sec:data}, and in order to avoid the bias in multiplicity-based centrality estimators, the classification in centrality intervals is performed based on the energy deposited by the spectator (non-interacting) nucleons in the neutron zero-degree calorimeters (ZN) in the Pb-going side. The study of the centrality dependence of the \W-boson yield can therefore also serve as a test bench for the centrality estimation.
In order to maximise the amount of signal in each centrality class, the \Wminus and \Wplus yields are combined. The cross section normalised to the average number of nucleon--nucleon collisions, \avNcollMult, is then calculated as
\begin{equation}
\frac{1}{\avNcollMult} \times \frac{N^i_{\W}}{\lumi_{\rm int} \times f^i_{\rm MB} \times \epsilon},
\end{equation}
where \avNcollMult is the average number of binary nucleon--nucleon collisions, $N^i_{\W}$ is the number of muons from \W decays in a given centrality class $i$, and $f^i_{\rm MB}$ is the fraction of MB-triggered events in the centrality class $i$ to those in the full centrality range (0--100\%). The cross sections for the two colliding beam configurations, normalised to \avNcollMult and averaged over centrality, amount to:
\begin{equation*}
\begin{split}
-4.46 < y^\mu_{\rm cms} < -2.96: & \qquad \sigma_{\muonpm \leftarrow \W} / \avNcollMult = 30.2 \pm 2.0 \text{ (stat)} \pm 2.8 \text{ (syst)} \text{ nb}, \\
2.03 < y^\mu_{\rm cms} < 3.53: & \qquad \sigma_{\muonpm \leftarrow \W} / \avNcollMult = 44.6 \pm 3.3 \text{ (stat)} \pm 5.1 \text{ (syst)} \text{ nb}. \\
\end{split}
\end{equation*}
The normalised cross sections are shown as a function of \avNcollMult in Fig.~\ref{fig:pPb_centrality}. The horizontal dashed line in the figure indicates the central value of the centrality-averaged measurement. The measured yield divided by \avNcollMult is found to be independent of centrality within uncertainties.
\begin{figure}
\begin{center}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_xSecVsNcollBack.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/pPb8tev_W_xSecVsNcollForw.pdf}
\end{center}
\caption{Combined yield of muons from \Wminus and \Wplus decays with $\pt^\mu >$ 10 \GeVc, normalised by the average number of binary nucleon--nucleon collisions \avNcollMult in \pPb collisions at \eightnn in the Pb-going (left) and p-going (right) configurations. The bars and boxes correspond to statistical and systematic uncertainties respectively. The horizontal dashed line indicates the central value of the yield normalised to \avNcollMult measured for the 0--100\% centrality interval.}
\label{fig:pPb_centrality}
\end{figure}
\subsection{\PbPb collisions}
\subsubsection{Production cross sections and lepton-charge asymmetry}
In \PbPb collisions at \fivenn, the production cross section and lepton-charge asymmetry of muons from \W-boson decays are evaluated as in \pPb collisions, from Eqs.~\ref{eqn:xSec} and~\ref{eqn:Ach}. The V0M amplitude is used to estimate the centrality of the collision. The production cross sections for \Wminus and \Wplus bosons in the 0--90\% centrality class are
\begin{equation*}
\sigma_{\Wminus \rightarrow \muon \Anum} = 18.7 \pm 0.7 \text{ (stat)} \pm 0.6 \text{ (syst)} \; \mu\text{b}, \qquad
\sigma_{\Wplus \rightarrow \Amuon \num} = 7.0 \pm 0.4 \text{ (stat)} \pm 0.2 \text{ (syst)} \; \mu\text{b}.
\end{equation*}
In the left panel of Fig.~\ref{fig:PbPb_xSecAch}, these values are compared with pQCD calculations using the CT14~\cite{ct14}, the CT14+EPPS16 combination~\cite{ct14,epps16}, and nNNPDF2.0~\cite{nnnpdf} PDF sets, all accounting for the isospin of the \PbPb system. In the \PbPb collision system, one cannot disentangle the high and low Bjorken-$x$ ranges, as it was possible in \pPb collisions. The comparison of the production for positively and negatively charged bosons shows the effect of the isospin, since the up- and down-quark densities in the Pb nucleus favour the production of \Wminus and suppress that of \Wplus. The measured cross sections are lower than the predictions with the CT14 PDFs for free nucleons, suggesting a significant effect due to nuclear modifications of the PDFs on the \W-boson production in \PbPb collisions. The calculations including the EPPS16 nuclear modifications are consistent with the data within uncertainties. The EPPS16 nPDFs show large uncertainties in the forward rapidity interval of the measurement, reflecting the lack of constraints in this region. The calculations with the nNNPDF2.0 PDF set predict larger nuclear effects as compared to EPPS16 and they underestimate the measured points.
\begin{figure}[h]
\begin{center}
\includegraphics[width = 0.90\textwidth]{figures_final/PbPb5tev_W_xSecAchInte.pdf}
\end{center}
\caption{Production cross section (left panel) and lepton-charge asymmetry (right panel) of the $\W \rightarrow \muonpm \num$ processes for the 0--90\% centrality class, for muons with $\pt^\mu > 10$ \GeVc and $2.5 < y^\mu_{\rm cms} < 4.0$, in \PbPb collisions at \fivenn. The measured cross sections and the asymmetry are compared with predictions using the CT14+EPPS16~\cite{ct14,epps16} combination, nNNPDF2.0~\cite{nnnpdf} nPDF model, as well as calculations with the CT14~\cite{ct14} PDF without nuclear corrections. All the calculations include the isospin effect. The vertical bars and boxes around the data points indicate the statistical and systematic uncertainties, respectively.}
\label{fig:PbPb_xSecAch}
\end{figure}
The lepton charge asymmetry in the 0--90\% centrality interval is measured to be
\begin{equation*}
A_{\rm ch} = -0.453 \pm 0.026 \text{ (stat)} \pm 0.030 \text{ (syst)}.
\end{equation*}
In the right panel of Fig.~\ref{fig:PbPb_xSecAch}, this observable is compared with pQCD calculations using the CT14+ EPPS16~\cite{ct14,epps16} and nNNPDF2.0~\cite{nnnpdf} nPDFs. The predictions with the EPPS nPDFs describe well the measured value, while the calculations with nNNPDF2.0 predict a much lower asymmetry than the measured one, confirming the discrepancies observed when comparing calculations with nNNPDF2.0 with the measurements in \pPb collisions. \\
\subsubsection{Normalised yield as a function of the collision centrality}
The normalised yield is obtained by dividing the yield of muons from \W decays, $N_{\muonpm \leftarrow \W}$, by the equivalent number of MB events $N^{\rm MB}_{\rm events}$, and then normalising to the average nuclear overlap function \avTaa~\cite{centrality}:
\begin{equation}
\frac{1}{\avTaa} \times \frac{N_{\muonpm \leftarrow \W}}{N^{\rm MB}_{\rm events}}.
\end{equation}
In the 0--90\% centrality class, the binary-scaled yield amounts to
\begin{equation*}
\begin{split}
N_{\muon \leftarrow \Wminus} /\left( N^{\rm MB}_{\rm events} \times \avTaa \right) &= 420.5 \pm 16.4 \text{ (stat)} \pm 18.0 \text{ (syst)} \text{ pb}, \\
N_{\Amuon \leftarrow \Wplus} /\left( N^{\rm MB}_{\rm events} \times \avTaa \right) &= 158.5 \pm 8.2 \text{ (stat)} \pm 6.9 \text{ (syst)} \text{ pb}.
\end{split}
\end{equation*}
The \W-boson yield normalised to \avTaa as a function of the collision centrality is shown in Fig.~\ref{fig:PbPb_yield} for both charges of the boson. The \avTaa-scaled yields are independent of centrality, as expected from the binary scaling of \W-boson production in nuclear collisions assuming negligible centrality dependence of the shadowing. The measurents are compared with pQCD calculations using the CT14~\cite{ct14} PDF combined with the EPPS16~\cite{epps16} nuclear modifications. A good agreement with the theory is found for both charges of the boson.
\begin{figure}
\begin{center}
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_yieldWminus.pdf}
\includegraphics[width = 0.49\textwidth]{figures_final/PbPb5tev_W_yieldWplus.pdf}
\end{center}
\caption{\avTaa-scaled yield of muons from \Wminus (left) and \Wplus (right) decays in \PbPb collisions at \fivenn for muons with $\pt^\mu > 10$ \GeVc and $2.5 < y^\mu_{\rm cms} < 4.0$. In the top panels the yield is compared with pQCD calculations using the CT14 PDF~\cite{ct14} as baseline and implementing the EPPS16~\cite{epps16} nuclear modifications. The ratio to theory of the measured yield normalised with \avTaa evaluated with $\sigma^{\rm inel}_{\rm NN}$ = 67.6 mb and 47.5 mb is shown in the middle and bottom panels, respectively (see the text for details). The horizontal bars indicate the width of the centrality intervals, the vertical bars and boxes correspond to the statistical and systematic uncertainties, respectively. The band indicate the uncertainty on the theoretical computations.}
\label{fig:PbPb_yield}
\end{figure}
The centrality dependence of the PDF modifications has been explored through impact-parameter dependent nPDFs~\cite{emelyanov1999,helenius2012}, but calculations of electroweak-boson production within this approach show a very limited dependence on the centrality, as reported in Ref.~\cite{Alice:ZpPb8tevPbPb5tev}. A possible centrality dependence of the production in terms of shadowing of the inelastic nucleon--nucleon cross section $\sigma^{\rm inel}_{\rm NN}$ was proposed in Ref.~\cite{epps-new}. In that study, the standard paradigm of extracting $\sigma^{\rm inel}_{\rm NN}$ from pp data, is questioned as a potential source of bias. The re-evaluation of the inelastic cross section from ATLAS measurements of electroweak-boson production in \PbPb collisions~\cite{Atlas:WPbPb5tev,Atlas:ZPbPb5tev} yields $\sigma^{\rm inel}_{\rm NN} = 41.5^{+16.2}_{-12.0} \text{ mb}$, a value significantly lower than the one used for centrality determination in \PbPb collisions at \fivenn with ALICE, taken as $\sigma^{\rm inel}_{\rm NN} = 67.6 \pm 0.6$ mb~\cite{centrality}. This alternative value of the inelastic cross section is found to improve the agreement between the ATLAS data and the pQCD calculations.
The bottom panels of Fig.~\ref{fig:PbPb_yield} show the centrality-dependent measurements obtained by normalising the yield with \avTaa evaluated using the nuclear-suppressed inelastic cross section from Ref.~\cite{epps-new}. The distributions show a significant centrality dependence, with the \Wminus distribution deviating from the binary scaling. This alternative value of the inelastic cross section, which provides a better agreement between pQCD calculations and the ATLAS measurement in peripheral collisions, has the opposite effect here. The yield normalised to \avTaa with $\sigma^{\rm inel}_{\rm NN} = 41.5$ mb shows a worse agreement with the theory than that with $\sigma^{\rm inel}_{\rm NN} = 67.5$ mb in the 40--90\% centrality interval. The authors of Ref.~\cite{epps-new} expect other effects to be possibly relevant in peripheral collisions, such as a possible centrality dependence of $\sigma^{\rm inel}_{\rm NN}$ and the neutron-skin effect, which could explain the tension with the data for peripheral collisions. It should be noted that the neutron skin effect would affect the production of \Wminus and \Wplus bosons in opposite directions, enhancing the former and suppressing the latter, thus not substantially improving the description of the measurements. \\
Recent measurements of the \avTaa-normalised yield of the \Z boson~\cite{Atlas:ZPbPb2tev,Atlas:ZPbPb5tev,Cms:ZPbPb2tev-2,Cms:ZPbPb5tev} have shown a decreasing trend for the most peripheral events, contradicting the binary-scaling assumption. This phenomenon has also been observed and studied by the ALICE Collaboration~\cite{alice-peripheral} for charged particle production. A possible explanation for this observation has been formulated in terms of event selection and geometry biases affecting peripheral events in the HG-PYTHIA model~\cite{hg-pythia}. In order to compare it with the \PbPb measurements presented in this article, the \RPbPb for hard scatterings calculated with this model was scaled by the centrality-averaged, \avTaa-normalised yields of \Wminus and \Wplus bosons measured in the 0--90\% centrality class. The resulting distributions are compared with the centrality-dependent measurements in Fig.~\ref{fig:PbPb_hg-pythia}. The scaled calculations are in good agreement with the data, although the small \W yield in peripheral collisions does not allow for a granularity fine enough in the 40--90\% centrality interval to show, if any, a statistically significant decrease of the production in this region.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\textwidth]{figures_final/PbPb5tev_W_yieldW_HGpythia.pdf}
\end{center}
\caption{\avTaa-scaled yield of muons from \Wminus and \Wplus decays in \PbPb collisions at \fivenn for muons with $\pt^\mu > 10$ \GeVc and $2.5 < y^\mu_{\rm cms} < 4.0$. The measured production is compared with HG-PYTHIA~\cite{hg-pythia} calculations of the \RPbPb of hard scatterings scaled with the centrality-averaged production in 0--90\% centrality, indicated as dashed lines. The horizontal bars correspond to the width of the centrality intervals, the vertical bars and boxes indicate the statistical and systematic uncertainties, respectively.}
\label{fig:PbPb_hg-pythia}
\end{figure}
The ATLAS Collaboration measured the production of \W bosons in the electronic and muonic decay channels in \PbPb collisions at \fivenn~\cite{Atlas:WPbPb5tev}. Their results are reported for the 0--80\% centrality class and are extracted from a data sample corresponding to a total integrated luminosity of 0.49 nb$^{-1}$. The decay leptons are detected in the rapidity interval $|y| < 2.5$, allowing for a complete continuity with the ALICE measurement in $2.5 < y < 4.0$. Similarly to the CMS measurements presented in Section~\ref{sec:pPb}, the ATLAS Collaboration also applied a tighter selection on the lepton \pt, at 25 \GeVc, the comparison is thus performed by means of the ratio between the measured \W-boson yields and the predictions from two pQCD calculations, the first using the EPPS16~\cite{epps16} nPDF set and the second using the CT14~\cite{ct14} PDFs. The comparison as a function of rapidity is shown in the two panels of Fig.~\ref{fig:PbPb_compATLAS} for the two charges of the boson.
\begin{figure}
\begin{center}
\includegraphics[width = 0.90\textwidth]{figures_final/PbPb5tev_W_ratioToCT14Wminus.pdf}
\includegraphics[width = 0.90\textwidth]{figures_final/PbPb5tev_W_ratioToCT14Wplus.pdf}
\end{center}
\caption{Ratio to pQCD calculations with CT14 PDFs~\cite{ct14} of the production of muons from \Wminus (top) and \Wplus (bottom) decays measured as a function of rapidity in \PbPb collisions at \fivenn by the ALICE and ATLAS~\cite{Atlas:WPbPb5tev} Collaborations. The ratio of EPPS16+CT14~\cite{epps16} calculations to that of CT14-only calculations is also shown. The grey band around the line at unity indicates the uncertainty on the calculations with CT14 PDFs.}
\label{fig:PbPb_compATLAS}
\end{figure}
The ALICE measurements are lower by 2$\sigma$ than the CT14 predictions and are described by EPPS16. The ATLAS data, instead, are better described by calculations without nPDF effects. This comparison motivated the study in Ref.~\cite{epps-new} with a shadowing-induced reduction of the inelastic cross section, but other possible origins of the effect have also been proposed~\cite{jonas2021}.
\subsubsection{Nuclear modification factor}
In the \PbPb analysis, the nuclear modification factor of muons from \W-boson decays is evaluated by dividing the \avTaa-scaled yield by the \W-boson production cross section in pp collisions:
\begin{equation}
R_{\rm AA} = \frac{1}{\avTaa} \times \frac{N^{\rm MB}_{\muonpm \leftarrow \W}}{\sigma^{\muonpm \leftarrow \W}_{\rm pp}},
\end{equation}
where $N^{\rm MB}_{\muonpm \leftarrow \W}$ is the number of muons from \W decays per MB event, $\sigma^{\muonpm \leftarrow \W}_{\rm pp}$ is the $\muonpm \leftarrow \W$ cross section in pp collisions, and $\avTaa$ is the average nuclear overlap function for the considered centrality class. As in \pPb collisions, the pp production cross section and the associated uncertainty were obtained from POWHEG and PYTHIA 6~\cite{powheg, pythia} simulations using CT10~\cite{ct10} for the proton PDF. For the 0--90\% centrality interval, the $R_{\rm AA}$ of muons from \Wminus- and \Wplus-boson decays are:
\begin{equation*}
\begin{split}
R^{\muon \leftarrow \Wminus}_{\rm AA} = 1.32 \pm 0.05 \text{ (stat)} \pm 0.06 \text{ (syst)} \pm 0.14 \text{ (pp ref.)}, \\
R^{\Amuon \leftarrow \Wplus}_{\rm AA} = 0.57 \pm 0.03 \text{ (stat)} \pm 0.02 \text{ (syst)} \pm 0.07 \text{ (pp ref.)}.
\end{split}
\end{equation*}
The production of \Wminus is enhanced, and that of \Wplus is suppressed relative to pp collisions, as expected following the content in u and d quarks of the Pb nucleus.
The measured \RPbPb is shown in Fig.~\ref{fig:PbPb_Raa} as a function of centrality and for the 0--90\% centrality class. The centrality-dependent measurement is compared with HG-PYTHIA~\cite{hg-pythia} calculations of the \RPbPb of hard scatterings scaled with the measured value in 0--90\% centrality. The centrality-averaged \RPbPb is compared with pQCD calculations, using the CT14~\cite{ct14} PDFs for the proton and the nCTEQ15WZ~\cite{ncteq15wz} PDF set, or the NNPDF3.1+nNNPDF2.0 combination~\cite{nnpdf,nnnpdf} for the Pb nucleus. The calculations within the nCTEQ and NNPDF frameworks are only shown for the centrality-averaged value as they have no centrality dependence. Both models provide a good description of the measurement within uncertainties. It should be noted that this agreement is realised while the measurement and models use different PDF sets for the pp reference, and different codes for the pQCD calculations (POWHEG~\cite{powheg} for the experimental results, MCFM~\cite{mcfm} and FEWZ~\cite{fewz} for the theoretical ones).
\begin{figure}
\begin{center}
\includegraphics[width = 0.90\textwidth]{figures_final/PbPb5tev_W_RAAWminus.pdf}
\includegraphics[width = 0.90\textwidth]{figures_final/PbPb5tev_W_RAAWplus.pdf}
\end{center}
\caption{Nuclear modification factor of muons from \Wminus (top) and \Wplus (bottom) decays in \PbPb collisions at \fivenn, for muons with $\pt^\mu > 10$ \GeVc and $2.5 < y^\mu_{\rm cms} < 4.0$, in different centrality intervals (left panels) and for the 0--90\% range (right panels). The centrality-dependent distributions are compared with the dashed curve, corresponding to the HG-PYTHIA~\cite{hg-pythia} model scaled with the measured $R_{\rm AA}$ in the 0--90\% centrality interval. The centrality-averaged measurement is compared with CT14+nCTEQ15WZ~\cite{ct14,ncteq15wz} and NNPDF3.1+nNNPDF2.0~\cite{nnpdf,nnnpdf} calculations. The horizontal bars indicate the width of the centrality bins, the vertical bars and boxes correspond the statistical and systematic uncertainties on the \PbPb measurement, respectively. The grey boxes indicate the uncertainty on the pp reference cross section.}
\label{fig:PbPb_Raa}
\end{figure}
\section{Summary}
\label{sec:summary}
The measurements of the \W-boson production cross section, lepton-charge asymmetry, nuclear modification factor, and yield normalised to the number of nucleon--nucleon collisions in \pPb collisions at \eightnn were reported, constituting the first results on the production of \W bosons at large rapidity at this energy, extending the measurement in \pPb collisions at \fivenn with a significant improvement of the precision. They were performed for muons with $\pt^\mu >$ 10 \GeVc and in the rapidity intervals $-4.46 < y^\mu_{\rm cms} < -2.96$ and $2.03 < y^\mu_{\rm cms} < 3.53$, where the negative rapidity interval indicates the Pb-going side, and the positive one the p-going side. The results were compared with pQCD calculations, using the CT14 PDF set~\cite{ct14}, and the EPPS16~\cite{epps16}, nNNPDF2.0~\cite{nnnpdf} and nCTEQ15~\cite{ncteq15, ncteq15wz} nPDF models. Some tensions are observed in the ability of the models to reproduce the data, notably with the nNNPDF2.0 model showing sizeable discrepancies with the measurements. Significant deviations from the free-nucleon PDF predictions, up to 3.5$\sigma$, are found at forward rapidity, corresponding to the shadowing region of the nuclear modifications at low Bjorken-$x$. The measurements in \pPb collisions reported here can therefore provide significant constraints to the nPDF models and help reducing their uncertainties. They complement the measurements of the \Z-boson production performed at large rapidities by the ALICE Collaboration~\cite{Alice:ZpPb8tevPbPb5tev}, where the statistical precision was too limited to draw any conclusion on the nuclear modifications. The comparison with the CMS measurements at midrapidity illustrates the complementarity of the LHC experiments in providing such results. The binary scaling of hard processes is observed, as the production cross sections in different centrality classes normalised to the average number of binary nucleon--nucleon collisions were found to be constant within uncertainties.
Similar measurements performed in \PbPb collisions at \fivenn were also presented, for muons from \W-boson decays at large rapidity ($2.5 < y^\mu_{\rm cms} < 4.0$) with $\pt^\mu >$ 10 \GeVc, and for various centrality classes. The normalised yield as a function of centrality follows the binary scaling expected for a hard process in the absence of significant centrality dependence of the shadowing. Comparisons with pQCD calculations provided an interesting insight into the \W-boson production. The nNNPDF2.0 calculations show a significantly lower cross section and lepton-charge asymmetry than the measured one, while the calculated $R_{\rm AA}$ is in good agreement with the data. The evaluation of the \avTaa-normalised yield with the nuclear-suppressed inelastic nucleon--nucleon cross section $\sigma^{\rm inel}_{\rm NN}$ obtained from Ref.~\cite{epps-new}, which was found to improve the agreement between the ATLAS data and the EPPS16 model, yields a tension between the ALICE measurements and EPPS16 calculations for peripheral events which is not seen with the standard value of $\sigma^{\rm inel}_{\rm NN}$. The \avTaa-scaled yield and the nuclear modification factor are found to be in good agreement with HG-PYTHIA~\cite{hg-pythia} calculations of the \RPbPb of hard scatterings scaled with the value measured in 0--90\% centrality, but the statistical limitation of the measurement does not allow to conclude on the decrease in peripheral events expected from this model. The measured \avTaa-scaled yields are described by pQCD calculations with the EPPS16 nPDFs. These measurements support the conclusion derived from the measurement of the \Z-boson production in \PbPb collisions at \fivenn~\cite{Alice:ZpPb8tevPbPb5tev}, showing a suppression of the production of electroweak bosons due to the nuclear modifications of the PDF and the resulting deviations from calculations based on free-nucleon PDFs. Being the first measurement of the \W production in \PbPb collisions at large rapidity, this study provides important insights for further investigation of the centrality dependence of the nPDFs.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,308
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Q: "bash: [: : integer expression expected" I want this function to return one random word if there aren't any command-line arguments.
I am modifying linuxconfig.org's random-word generator to run even when #$ -ne 1.
function random-word {
if [ $# -eq 0 ] ;
then
echo "I only take one argument, dummy"
# previously was exit 0
fi
# Constants
X=0
ALL_NON_RANDOM_WORDS=/usr/share/dict/words
# total number of non-random words available
non_random_words=`cat $ALL_NON_RANDOM_WORDS | wc -l`
# while loop to generate random words
# number of random generated words depends on supplied argument
while [ $X -lt "$1" ]
do
random_number=`od -N3 -An -i /dev/urandom |
awk -v f=0 -v r="$non_random_words" '{printf "%i\n", f + r * $1 / 16777216}'`
sed `echo $random_number`"q;d" $ALL_NON_RANDOM_WORDS
let "X = X + 1"
done
The statement executes, but there's a bash error:
$ bob
I only take one argument, dummy
bash: [: : integer expression expected
How do I fix this so the bash error won't display?
A: while [ $X -lt "$1" ] is being evaluated even when there is no $1.
Move the rest of the code into an else block so this doesn't happen.
function random-word {
# from linuxconfig.org
if [ $# -eq 0 ]
then
echo "I need an argument, dummy"
# To be extra friendly, give them a random word.
echo "Here's a random word:"
random-word 1
else
# Constants
X=0
...
sed `echo $random_number`"q;d" $ALL_NON_RANDOM_WORDS
let "X = X + 1"
done
fi
}
A: You don't need an else block. And you don't need to fail if you receive more than one argument. You only need to fail if you don't get an argument and you can then just ignore everything else or quit if you don't get at least the one.
set -- "${1?ERR: Where\'s my argument?!?!}"
That statement does all of that. This does the same but also fails if the first argument is only the '' nullstring :
set -- "${1:?ERR: Where\'s my argument?!?!}"
Of course, if the point is to be a shell function called from an interactive shell, this might kill the interactive shell as well, so you can do:
(: "${1:?Where\'s my argument?}") || return && set -- "$1"
...which will handle that as well. But, in my opinion, a shell function should never affect an interactive shell unless absolutely necessary - like if it needs to alter a current shell variable or something. And so better than the above would be to declare the function as a subshell, like:
fn() (set -- "${1:?Where\'s my argument?}" && echo "$1")
...does.
Another way to do this is to accept all arguments and concatenate them - so all arguments are only treated as one in every case. You can do that like:
fn() (
: "${1:?ERR: Where\'s my argument?}"
set -- "$*" && echo "$1"
)
fn '' ; fn here are a lot of arguments that will all be treated as one
###OUTPUT###
sh: line 2: 1: ERR: Where's my argument?
here are a lot of arguments that will all be treated as one
I personally don't much like the sh: line 2: 1: bit that usually happens - at least not in interactive shells. I usually stick a CTRL+V CTRL+M in before ERR: so it does a return to the head of the line in a terminal and writes over that bit but still outputs the more useful info into a log. Like:
fn() (
: "${1:?^MERR: Where\'s my argument?}"
set -- "$*" && echo "$1"
)
fn ; fn 2>&1 | cat -A
###OUTPUT###
ERR: Where's my argument?
sh: line 2: 1: ^MERR: Where's my argument?$
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,371
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{"url":"http:\/\/www.gamedev.net\/index.php?app=forums&module=extras§ion=postHistory&pid=5111629","text":"\u2022 Create Account\n\nBanner advertising on our site currently available from just \\$5!\n\n### #ActualSamith\n\nPosted 24 November 2013 - 12:24 PM\n\nEDIT: totally misunderstood your question! As the others above have stated, you wouldn't put all geometry into a VBO. You would have a VBO per object or per surface or something. Then you would be able to draw those objects individually with separate uniforms.\n\nIn most cases your VBO will be initialized once with vertex data that's in \"model space\" and then you'll leave the VBO alone. When you want to transform the object into world space, you'll do that in the vertex shader. You can send your transformation matrices to the vertex shader with uniform buffers.\n\n#version 410\n\nin vec4 position;\nin vec3 color;\n\nout vec4 out_position;\n\nuniform Transform {\nmat4 model_view_projection;\n};\n\nvoid main()\n{\nout_position = position * model_view_projection;\n}\n\nIn that code model_view_projection is a member of the Transform uniform buffer. On the CPU side you would make a buffer that contained your model-view-projection matrix, and then bind that buffer to the location of Transform before drawing.\n\nThis way you never have to update your VBO, and all transformations are done efficiently on the GPU.\n\n### #2Samith\n\nPosted 24 November 2013 - 12:24 PM\n\nEDIT: totally misunderstood your question! As the others above have state, you wouldn't put all geometry into a VBO. You would have a VBO per object or per surface or something. Then you would be able to draw those objects individually with separate uniforms.\n\nIn most cases your VBO will be initialized once with vertex data that's in \"model space\" and then you'll leave the VBO alone. When you want to transform the object into world space, you'll do that in the vertex shader. You can send your transformation matrices to the vertex shader with uniform buffers.\n\n#version 410\n\nin vec4 position;\nin vec3 color;\n\nout vec4 out_position;\n\nuniform Transform {\nmat4 model_view_projection;\n};\n\nvoid main()\n{\nout_position = position * model_view_projection;\n}\n\nIn that code model_view_projection is a member of the Transform uniform buffer. On the CPU side you would make a buffer that contained your model-view-projection matrix, and then bind that buffer to the location of Transform before drawing.\n\nThis way you never have to update your VBO, and all transformations are done efficiently on the GPU.\n\n### #1Samith\n\nPosted 24 November 2013 - 12:22 PM\n\nIn most cases your VBO will be initialized once with vertex data that's in \"model space\" and then you'll leave the VBO alone. When you want to transform the object into world space, you'll do that in the vertex shader. You can send your transformation matrices to the vertex shader with uniform buffers.\n\n<code>\n\n#version 410\n\nin vec4 position;\n\nin vec3 color;\n\nout vec4 out_position;\n\nuniform Transform {\n\nmat4 model_view_projection;\n\n};\n\nvoid main()\n\n{\n\nout_position = position * model_view_projection;\n\n}\n\n<\/code>\n\nIn that code model_view_projection is a member of the Transform uniform buffer. On the CPU side you would make a buffer that contained your model-view-projection matrix, and then bind that buffer to the location of Transform before drawing.\n\nThis way you never have to update your VBO, and all transformations are done efficiently on the GPU.\n\nPARTNERS","date":"2015-06-30 13:09:17","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\": 0.20586541295051575, \"perplexity\": 3696.5676794687156}, \"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\/1435375093899.18\/warc\/CC-MAIN-20150627031813-00297-ip-10-179-60-89.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/samacheerkalviguru.com\/samacheer-kalvi-6th-maths-term-1-chapter-2-additional-questions\/","text":"## Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 1 Chapter 2 Introduction to Algebra Additional Questions\n\nQuestion 1.\nSolution:\n0\n\nQuestion 2.\nMultiplicative identity ____\nSolution:\n1\n\nQuestion 3.\nExpress to an algebraic statement.\n(i) \u2018t\u2019 is added to 100\n(ii) 4 less to 9 times of y.\nSolution:\n(i) t + 100\n(ii) 9y \u2013 4\n\nQuestion 4.\nFind the rule which gives the number of sticks in the following pattern.\n\nSolution:\nLet \u2018x\u2019 be the no. of R\u2019s formed.\n\nThe rule is 6x.\nLet y \u2019 be the no. of S\u2019s formed.\n\nThe rule is 5y.\n\nQuestion 5.\nHow old was Suja 6 years from now?\nSolution:\nLet Suja\u2019s present age be \u2018a\u2019 years.\n6 years from now Suja will be (a + 6) years old.\n\nQuestion 6.\nPrice of Apple per kg is \u20b9 50 more than price of orange per kg. What is the cost of Apple per kg?\nSolution:\nLet the price of orange be \u20b9 b\nPrice of Apple will be \u20b9 (b + 50)\n\nQuestion 7.\nGiven \u2018n\u2019 students like ice cream. What may 2n show?\nSolution:\n2n shows double the number of students who like ice cream.\n\nQuestion 8.\nPrice of oil per litre is \u20b9 5 more than three times the price of cool drinks \u20b9 \u2018p\u2019 Express algebraically.\nSolution:\nPrice of cool drinks per kg = \u20b9 p\nThree times = 3p\n5 Rs. more = 3p + 5\nPrice of oil per kg = \u20b9 (3p + 5)\n\nQuestion 9.\nComplete the table and by inspection of the table find the value of m when m + 10 = 16.\n\nSolution:\n\nFrom the table m+ 10 = 16 when m = 6.\n\nQuestion 10.\nExpress algebraically (a) y divided by r (b) double times x is subtracted from 10\nSolution:\n(a) $$\\frac{y}{r}$$\n(b) 10 \u2013 2x\n\nQuestion 11.\nGive verbal expression of\n(a) 7x + 18\n(b) $$\\frac{4 x}{3}$$\nSolution:\n(a) 18 added to 7 times x\n(b) 4 times x divided by 3.\n\nQuestion 12.\nRajini\u2019s Father\u2019s age is 5 years more than 3 times Rajini\u2019s age. What is her father\u2019s age?\nSolution:\n3x + 5\n\nQuestion 13.\n\nFind the rule for the above pattern.\nSolution:\n2p\n\nQuestion 14.\nPrepare a table for 3x + 10. From the table find the value of x when 3x + 10 = 25.\nSolution:\n5\n\nQuestion 15.\nComplete the table and find the solution of the equation $$\\frac{z}{3}=4$$ using the table.\n\nSolution:\n\nQuestion 16.\nForm the expression for which Ramu is 3 years younger than Mathu.\nSolution:\nm \u2013 3\n\nQuestion 17.\nA tap is to be pasted along the edges of a square shaped gift box. Its length is 4 cm. What is the length of tap needed for one side.\nSolution:\n$$\\frac{4 p}{4}=p$$\n\nQuestion 18.\nThe value of y in 7y \u2013 20 = 99.\nSolution:\ny = 17\n\nQuestion 19.\nNine added to two times x gives 301. Find the value of x.\nSolution:\nx = 146\n\nQuestion 20.\nAarthi is 3 years younger to Harini. If the sum of their ages is 23, how old is Harini?\nSolution:\nLet Harini\u2019s age be x years\nAarthi\u2019s age is x \u2013 3 years\nGiven sum of their ages is 23.\ni.e., x + (x \u2013 3) = 23","date":"2022-05-25 12:59:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5848863124847412, \"perplexity\": 4134.826047373317}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662587158.57\/warc\/CC-MAIN-20220525120449-20220525150449-00088.warc.gz\"}"}
| null | null |
Spanish ferry operator Baleària has received the backing of European Union for the planned conversion of five ferries to run on liquefied natural gas (LNG).
The company said on Wednesday it has been granted €11.8 million ($13.6 million), out of €15 million the company requested through EU's Connecting Europe Facility.
EU funds will cover 20 percent of the €60 million investment Baleària has planned for the conversion project.
Five retrofits of the ferries named Naples, Abel Matutes, Sicily, Bahama Mama and Martín i Soler will be carried out starting before the end of this year until 2021.
The first vessel to complete the retrofit will be the Naples, mid-November, the company said.
In addition to the five retrofits, the company has two more LNG-fueled ferries under construction at the Italian Visentini shipyard.
Posted on October 10, 2018 with tags Balearia, LNG retrofit.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,213
|
package tilde
import (
"strings"
"nimona.io/pkg/errors"
)
type Hint string
const (
// basic hints
BoolHint Hint = "b"
DataHint Hint = "d"
FloatHint Hint = "f"
IntHint Hint = "i"
MapHint Hint = "m"
StringHint Hint = "s"
UintHint Hint = "u"
DigestHint Hint = "r"
// array hints
BoolArrayHint Hint = "ab"
DataArrayHint Hint = "ad"
FloatArrayHint Hint = "af"
IntArrayHint Hint = "ai"
MapArrayHint Hint = "am"
ObjectArrayHint Hint = "ao"
StringArrayHint Hint = "as"
UintArrayHint Hint = "au"
DigestArrayHint Hint = "ar"
)
var hints = map[string]Hint{
// basic hints
string(BoolHint): BoolHint,
string(DataHint): DataHint,
string(FloatHint): FloatHint,
string(IntHint): IntHint,
string(MapHint): MapHint,
string(StringHint): StringHint,
string(UintHint): UintHint,
string(DigestHint): DigestHint,
// array hints
string(BoolArrayHint): BoolArrayHint,
string(DataArrayHint): DataArrayHint,
string(FloatArrayHint): FloatArrayHint,
string(IntArrayHint): IntArrayHint,
string(MapArrayHint): MapArrayHint,
string(ObjectArrayHint): ObjectArrayHint,
string(StringArrayHint): StringArrayHint,
string(UintArrayHint): UintArrayHint,
string(DigestArrayHint): DigestArrayHint,
}
func ExtractHint(key string) (string, Hint, error) {
ps := strings.Split(key, ":")
if len(ps) != 2 {
return "", "", errors.Error("extractHint: invalid hinted key " + key)
}
h, ok := hints[ps[1]]
if !ok {
return "", "", errors.Error("extractHint: invalid hint " + ps[1])
}
return ps[0], h, nil
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,436
|
Frye's Centre for Comparative Literature spared from budget Axe
By Joe Friesen, Globe and Mail, October 28, 2010
The renowned Centre for Comparative Literature at the University of Toronto has been pulled back from the brink after an outcry from scholars around the world and the determined protests of students and faculty.
The director of the centre said he has been assured that the school, which was slated to close at the end of this academic year, will survive.
"Comp. Lit. is saved. The centre will stay open and we're taking students for next year," said director Neil ten Kortenaar. "I think it was the outcry from around the world. We had a lot of support from a lot of big-name people in academic circles."
It was a battle that pitted the forces of streamlining and cost-cutting against those who would preserve structures of academic and historical significance. It appears either the university lost the appetite for the fight, or the threat of amalgamation produced enough compromise from the departments involved to satisfy the administration's appetite for reform.
Sometimes the right decision takes a lot of hard work, even if the obvious is less than obvious to those who are responsible for making it.
When one is focused on the task and responsibility for administring a budget, one, anyone, has a tendency to lose sight of the bigger picture. Sometimes, mere cost cutting abandons more than it achieves.
However this decision was finally arrived at, both internally and externally, the University of Toronto is a better place going forward today than it would have been had Frye's Centre for Comparative Literature been chopped.
And based on the quote from the centre's director, the weight of the combined thrust of both students and faculty and the opinion of scholars around the world tipped the balance in favour of retention.
This decision cannot be considered merely a toast to its founder, although that in itself would have justified its life extension; it is a lens on the perspective of the University of Toronto that will not only permit, but also foster, a comparative meta-approach to world literature that the Canadian culture so desperately needs.
While there is a 24-7 global news cycle on such outlets as CNN and BBC, news is no substitute for the much more telling impact of the world's writers, poets and scholars that require a different kind of examination. Also, along with the high-tech innovations of the 'social network,' the books and the insights and the imaginative cultures that emerge from the unconscious of those living in the most remote hills and valleys, or on the most dry, windswept deserts, or in the camps of the world's most desolate steppes have been, are, and will continue to be a beacon into the future for those both privileged and honoured and humbled to study their writings.
It is from all the regions of the world that our thought leaders must continue to come and the next generations of scholars needs to explore those complex narratives if we are to expect to bring any kind of recognition to our shared humanity in a meaningful way.
Frye has pointed an insightful finger in a direction; his intellectual family's task is to find out where he is pointing.
This little voice in this little corner would like to see the Centre serve as a model for other academic pursuits in that comparison with the scholarship of all countries in various disciplines, and even in combined disciplines is more and more needed. We have paid homage to a kind of specialization that has given us much new knowledge and information. However, humans need road maps for discovery to assist with the next few centuries of discovery, and we must never lose the capacity for integrated conversation, scholarship and experiment. Only by keeping in touch with all cultures, literatures, histories and anthropologies will we come to a conscious awareness of who we are in all our diversity, complexity and commonality.
Posted by acorncentreblog.com at 6:08 AM
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Dollars bring academic freedom under fire
Frye's Centre for Comparative Literature spared fr...
Certainty, absolutes and faith
Ignatieff finally sees "the light" on F-35 fighter...
The New Middle Age: 60's and 70's
Literacy Advocates: "change to be effective"
Are we Worthy of this Canada?
Memo to McGuinty Government: Move to make Ontario ...
Fight Stereotypes...and "macho" foreign policy
Men/Women hard-wired differently
Oxymoron: Canada's Walk of Fame
Stogran: Needed by Liberals as a Candidate
A northern optimist cheers for Democrats in November
Can. Centre for Policy Alternatives calls F-35 pur...
Petition Opposes F-35 Purchase for Canada
Business School Deans: We're not all cut-throat!
Don't blame Ignatieff...U.N. Security Council Seat...
Wanted: Political Will, Collaboration, Co-operatio...
Update: Ayaan Hirsi Ali: "I am an atheist" (From h...
Dr. Ted Hsu, "game-changer" for Kingston Liberals
"Left" under fire 20 years ago...still today
Canada: bumblebee nation?
Noble sides vs cruelty glands...score: 10-4 for cr...
Semrau case: seminal in war serving two mutually e...
Another male tragedy unfolds in Belleville
Liberal Family Care Plan...good start to policy pl...
China quietly holds 97% of world's "rare earth"
Shame: Globe fires Rick Salutin...Dobbin starts le...
Anger OUT; Repression IN!... rubbish!
Lip-service, a Canadian monograph?
Debunking Lies our Aunts Told Us
David Lloyd Johnston, 28th Governor General of Canada
Emasculated Men...will they (we) seek self-help?
"Terminated for redundancy"...at the whim of corpo...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,505
|
All Communities > Stay at Home Moms > What are you hoping to receive for Mother's Day?
What are you hoping to receive for Mother's Day?
With Mother's Day quickly approaching, what are you hoping to receive from your loved ones?
Do you expect a gift on Mother's Day?
Do you expect a gift from your kids and/or husband/partner aside from flowers and cards?
What do you want from your husband or partner on Mother's Day?
Is it wrong to expect something for Mother's Day?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 765
|
{"url":"https:\/\/zbmath.org\/?q=an%3A1204.93127","text":"# zbMATH \u2014 the first resource for mathematics\n\nStabilization of Markov jump linear systems using quantized state feedback. (English) Zbl\u00a01204.93127\nSummary: This paper addresses the stabilization problem for single-input Markov jump linear systems via mode-dependent quantized state feedback. Given a measure of quantization coarseness, a mode-dependent logarithmic quantizer and a mode-dependent linear state feedback law can achieve optimal coarseness for mean square quadratic stabilization of a Markov jump linear system, similar to existing results for linear time-invariant systems. The sector bound approach is shown to be non-conservative in investigating the corresponding quantized state feedback problem, and then a method of optimal quantizer\/controller design in terms of linear matrix inequalities is presented. Moreover, when the mode process is not observed by the controller and quantizer, a mode estimation algorithm obtained by maximizing a certain probability criterion is given. Finally, an application to networked control systems further demonstrates the usefulness of the results.\n\n##### MSC:\n 93E15 Stochastic stability in control theory 93C05 Linear systems in control theory 93D15 Stabilization of systems by feedback 60J75 Jump processes (MSC2010) 93B52 Feedback control\nFull Text:\n##### References:\n [1] Boukas, E.; Liu, Z., Robust $$H_\\infty$$ control of discrete-time Markovian jump linear systems with mode-dependent time-delays, IEEE transactions on automatic control, 46, 12, 1918-1924, (2001) \u00b7 Zbl\u00a01005.93050 [2] Costa, O.; Fragoso, M.; Marques, R., Discrete-time Markov jump linear systems, (2005), Springer London \u00b7 Zbl\u00a01081.93001 [3] de Souza, C., Robust stability and stabilization of uncertain discrete-time Markovian jump linear systems, IEEE transactions on automatic control, 51, 5, 836-841, (2006) \u00b7 Zbl\u00a01366.93479 [4] Elia, N., Remote stabilization over fading channels, Systems and control letters, 54, 3, 237-249, (2005) \u00b7 Zbl\u00a01129.93498 [5] Elia, N.; Mitter, S., Stabilization of linear systems with limited information, IEEE transactions on automatic control, 46, 9, 1384-1400, (2001) \u00b7 Zbl\u00a01059.93521 [6] Elliott, R.; Aggoun, L.; Moore, J., Hidden Markov models estimation and control, (1995), Springer New York \u00b7 Zbl\u00a00819.60045 [7] Elliott, R.; Dufour, F.; Malcolm, W., State and mode estimation for discrete-time jump Markov systems, SIAM journal on control and optimization, 44, 1081-1104, (2005) \u00b7 Zbl\u00a01130.93423 [8] Fu, M.; Xie, L., The sector bound approach to quantized feedback control, IEEE transactions on automatic control, 50, 11, 1698-1711, (2005) \u00b7 Zbl\u00a01365.81064 [9] Gao, H.; Chen, T., A new approach to quantized feedback control systems, Automatica, 44, 2, 534-542, (2008) \u00b7 Zbl\u00a01283.93131 [10] Ho, T.; Chen, B., Novel extended viterbi-based multiple-model algorithms for state estimation of discrete-time systems with Markov jump parameters, IEEE transactions on signal processing, 54, 2, 393-404, (2006) \u00b7 Zbl\u00a01373.94611 [11] Hoshina, H., Tsumura, K., & Ishii, H. (2007). The coarsest logarithmic quantizers for stabilization of linear systems with packet losses. In Proceedings of the 46th IEEE conference on decision and control. (pp. 2235-2240) New Orleans, USA. [12] Huang, M.; Dey, S., Stability of Kalman filtering with Markovian packet losses, Automatica, 43, 4, 598-607, (2007) \u00b7 Zbl\u00a01261.93083 [13] Hu, S.; Yan, W., Stability robustness of networked control systems with respect to packet loss, Automatica, 43, 7, 1243-1248, (2007) \u00b7 Zbl\u00a01123.93075 [14] Imer, O.; Y\u00fcksel, S.; Ba\u015far, T., Optimal control of LTI systems over unreliable communication links, Automatica, 42, 9, 1429-1439, (2006) \u00b7 Zbl\u00a01128.93368 [15] Matei, I., Martins, N., & Baras, J. (2008). Optimal linear quadratic regulator for Markovian jump linear systems, in the presence of one time-step delayed mode observations, In Proceedings of the 17th IFAC world congress. (pp. 8056-8061) Seoul, Korea. [16] Rabiner, L., A tutorial on hidden Markov models and selected applications in speech recognition, Proceedings of the IEEE, 77, 2, 257-286, (1989) [17] Schenato, L.; Sinopoli, B.; Franceschetti, M.; Poolla, K.; Sastry, S., Foundations of control and estimation over lossy networks, Proceedings of the IEEE, 95, 1, 163-187, (2007) [18] Seiler, P.; Sengupta, R., An $$H_\\infty$$ approach to networked control, IEEE transactions on automatic control, 50, 3, 356-364, (2005) \u00b7 Zbl\u00a01365.93147 [19] Sinopoli, B.; Schenato, L.; Franceschetti, M.; Poolla, K.; Jordan, M.; Sastry, S., Kalman filtering with intermittent observations, IEEE transactions on automatic control, 49, 9, 1453-1464, (2004) \u00b7 Zbl\u00a01365.93512 [20] Viterbi, A., Error bounds for convolutional codes and an asymptotically optimum decoding algorithm, IEEE transactions on information theory, 13, 2, 260-269, (1967) \u00b7 Zbl\u00a00148.40501 [21] Xiao, N., Xie, L., & Fu, M. (2009). Quantized stabilization of Markov jump linear systems via state feedback. In Proceedings of American control conference. (pp. 4020-4025), St. Louis, USA. [22] Xiong, J.; Lam, J., Stabilization of linear systems over networks with bounded packet loss, Automatica, 43, 1, 80-87, (2007) \u00b7 Zbl\u00a01140.93383\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-05-11 20:16:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7988159656524658, \"perplexity\": 4233.327730668556}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989856.11\/warc\/CC-MAIN-20210511184216-20210511214216-00519.warc.gz\"}"}
| null | null |
Q: Under what conditions will winsock blocking send() return 0? MSDN says the following:
"Calling send with a len parameter of zero is permissible and will be treated by implementations as successful. In such cases, send will return zero as a valid value. For message-oriented sockets, a zero-length transport datagram is sent."
My question is, if the len parameter is NOT zero, will the blocking send() return 0 (assuming no timeout is set)?
I also searched the Internet, and found the following:
http://tangentsoft.net/wskfaq/articles/bsd-compatibility.html
"Under Winsock, the SIGPIPE/EPIPE functionality does not exist at all: send() will either return 0 for a normal disconnect or -1 for an abnormal disconnect".
However, no matter how I tried, I couldn't simulate the "normal disconnect", and therefore I could NOT have send() return 0.
Thanks in advance.
A: A "normal" disconnect is when the receiving party calls closesocket() on its end of the connection to send a FIN packet to the sending party. When send() detects the FIN, it knows the connection has been disconnected gracefully and should return 0 to notify your code.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,030
|
Q: join two dataframe together total_purchase_amt
2013-07-01 22533121
2014-08-29 214114844
2014-08-30 183547267
2014-08-31 205369438
total_purchase_amt
2014-08-31 2.016808e+08
2014-09-01 2.481354e+08
2014-09-02 2.626838e+08
2014-09-03 2.497276e+08
having two dataframe, I want to join them together,the result is like this:
the last row in first dataframe should be replaced by the first row of second
dataframe.
total_purchase_amt
2013-07-01 22533121
2014-08-29 214114844
2014-08-30 183547267
2014-08-31 2.016808e+08
2014-09-01 2.481354e+08
2014-09-02 2.626838e+08
2014-09-03 2.497276e+08
A: Use combine_first with the other df combining with the first df, this will perserve the values in your other df and add the missing values from the first df:
In [49]:
df1.combine_first(df)
Out[49]:
total_purchase_amt
2013-07-01 22533121
2014-08-29 214114844
2014-08-30 183547267
2014-08-31 201680800
2014-09-01 248135400
2014-09-02 262683800
2014-09-03 249727600
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{"url":"https:\/\/physics.stackexchange.com\/questions\/429808\/do-electrons-really-diffuse-when-a-temperature-gradient-is-applied","text":"# Do electrons really diffuse when a temperature gradient is applied?\n\nIn many websites and books, it is generally said that the charge carriers, be it electrons or holes, diffuse through the considered material when a temperature gradient is applied. However I have found exactly zero justification of such a claim, be it either by words or by a mathematical equation that would show that indeed, the dynamics of these charge carriers is driven by a diffusion-like equation.\n\nHere's Wikipedia: reference, between a plethora of other sources.\n\nAt the atomic scale, an applied temperature gradient causes charge carriers in the material to diffuse from the hot side to the cold side.\n\nI would like to know, and see, the mathematical derivation of such a claim. So far I have two ideas that could potentially lead to the answer, but I am unable to really proceed further.\n\nThe first one, is that the motion of particle means that the system is in a nonequilibrium state, where there is a non vanishing $\\nabla \\mu$ (chemical potential) at least in some region of said material. So the equation must contain that quantity, probably. Then, it must also contain $\\nabla T$, because it is the driving force (basically the Seebeck effect). This really looks like this will involve Onsager reciprocal relations... but then I fall short in involving time, which looks like is required to justify the claim.\n\nMy other idea is to go to Solid State Physics, and make the assumptions\/simplifications required for the Boltzmann transport equation to hold, for the charge carriers. But then, how would I involve the temperature, the chemical potential and how would I derive a diffusion-like equation from it?\n\n\u2022 Are you referring to the thermoelectric effect? I generally think of this electron drift\/diffusion phenomenon at a temperature difference as electrons in the hot end gaining more thermal energy and thus \"taking up more space\" as their random motions become more violent and rapid. They will then eventually drift towards the colder end, because there is more \"space\". In the same way that a gas expands to an area of lower pressure. \u2013\u00a0Steeven Sep 20 '18 at 11:50\n\u2022 Turn it around - what do you need for a mobile entity not to diffuse? \u2013\u00a0Jon Custer Sep 20 '18 at 12:31\n\u2022 @Steeven yes, I am referring to the thermoelectric effect (as can be seen from the tag and the mention of the Seebeck effect). I get that part, but this thinking says nothing about the way the electrons \"drift\" towards the colder end (in reality it depends on the Seebeck coefficient sign, they can drift toward the hotter part). The mention of the gas might be a good and relevant one, because it's clear that a diffusion-like equation (or Fokker-Planck) is taking place at a microscopic level, for individual molecules. \u2013\u00a0thermomagnetic condensed boson Sep 20 '18 at 13:54\n\u2022 @JonCuster Very good point. Indeed, all seems to point out at diffusion, so intuition tells us this should be the case. Now the hardest part is to show that this is indeed the case, under certain assumptions. Because of course, it should be possible to tweak the system so much that in the end the charge carriers motion is more ballistic than diffusion-like, even though in the general case it should be diffusion-like. \u2013\u00a0thermomagnetic condensed boson Sep 20 '18 at 13:56\n\u2022 Regarding the sign of the Seebeck coefficient, it is opposite for an opposite semiconductor (p-type), e.g. when the majority charge carrier is holes. Generally, the \"thing\" being \"pushed\" from hot to cold end is charge carriers, whichever they might be, rather than electrons as I used in my example. \u2013\u00a0Steeven Sep 20 '18 at 15:03\n\nNow onto my other idea: the Boltzmann transport equation. By making reasonable assumptions (such as considering the charge carriers as quasiparticles with a well definite position, velocity, etc. as well as using the time relaxation approximation), and by following a particular textbook*, the equation reduces to $$\\frac{\\partial f}{\\partial t} + \\vec{v} \\cdot \\frac{\\partial f}{\\partial \\vec{r}}+e\\vec E \\cdot \\frac{\\partial f}{\\partial \\vec{p}}=-\\frac{f-f_0}{\\tau}$$, where, if I understand well, $$\\vec E$$ is the external applied field. In my case, which is an open circuit system, $$\\vec E=\\vec 0$$. Note that the electric field arising thanks to the Seebeck effect is still present in the equation.\nSkipping many mathematical steps and a few reasonable assumptions, I reach that the equation reduces to $$\\frac{\\partial f_0}{\\partial t}+\\frac{\\partial f_0}{\\partial \\vec v} \\cdot \\frac{\\xi}{\\vec p T} \\vec v \\cdot \\nabla T=\\frac{f_1}{\\tau}$$. Where $$f=f_0+f_1$$ is a density of probability function, and $$\\tau$$ is the scattering time. While I did not throw out the term with a time derivative, I do not get any term with a second derivative of spatial coordinates. Hence the possibility to reach a diffusion-like equation seems too remote. The equation governing the dynamics of the charge carriers looks like very complicated to me, and does not seem to reduce to a diffusion equation. If someone offers a different point of view (backed up with mathematical insights, not just gedanken experiments\/justifications), that would be nice.","date":"2019-10-18 23:29:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 6, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8045392036437988, \"perplexity\": 389.954766665811}, \"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-2019-43\/segments\/1570986685915.43\/warc\/CC-MAIN-20191018231153-20191019014653-00158.warc.gz\"}"}
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La è una stazione ferroviaria situata della città di Nasushiobara, nella prefettura di Tochigi, ed è servita dalla linea principale Tōhoku sulla quale è esercitato il servizio Utsunomiya della JR East, della quale è capolinea settentrionale. Normalmente, chi da Tokyo si reca nelle regioni del Tōhoku utilizzando i treni regionali, effettua il cambio in questa stazione. Inoltre, presso Kuroiso cambia l'elettrificazione della linea: in corrente continua in direzione Tokyo (verso sud) e in corrente alternata in direzione Sendai (verso nord).
Servizi ferroviari
East Japan Railway Company
Linea Utsunomiya (servizio della linea Tōhoku)
Linea principale Tōhoku (servizi regionali)
Struttura
La stazione è dotata di un marciapiede laterale e di due a isola centrale con cinque binari passanti in superficie. Lungo la stazione si trovano i binari del Tōhoku Shinkansen, ma i treni ad alta velocità non fermano a Kuroiso, poiché la fermata ad essi dedicata è la successiva stazione di Nasushiobara in direzione Tokyo.
Stazioni adiacenti
Note
Altri progetti
Collegamenti esterni
Kuroiso
Kuroiso
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Radio Archives
Guns N' Roses Is Going to Try (Again) To Reunite For Coachella 2016
TOPICS:CoachellaGuns N' Roses
Posted By: HausRules
The last time Duff, Axl, and Slash played together was in 1993
Axl was absent for the band's induction in the Rock & Roll Hall of Fame 3 years ago
The plan is for them to headline Coachella in April and then do 25 tour dates
The money on the table is somewhere between $3 and $8 million per show
Stop me if you think you've heard this one before. I won't be placing any best, let's put it that way.
Via Billboard
Guns N' Roses, with founding members Axl Rose and Slash, will headline the Coachella Music & Arts Festival in Indio, Calif., set for April 15-17 and April 22-24, according to multiple sources. The reunited band is also negotiating with promoters to play as many as 25 football stadiums in North America in the summer of 2016, and is scheduled to be one of the first acts to play the new Las Vegas Arena, set to open April 6. Guns N' Roses is said to be asking as much as $3 million per show, with tickets topping out in the $250-$275 range.
The last show Axl Rose and Slash played together was on July 17, 1993 at River Plate Stadium in Buenos Aires. Guns N' Roses' most recent tour, which only included Axl Rose from the original lineup, took place primarily in South America and The Joint in Las Vegas, grossing $15.2 million.
Representatives for the band, as well as Coachella producer Paul Tollett and agent Ken Fermaglich at United Talent Agency, could not be reached for comment. Executives at AEG, which will operate the new Vegas arena, also did not immediately respond to a request for comment.
The band's first album, Appetite for Destruction, is the biggest-selling debut in U.S. history, with 18 million sold. It spent five weeks at No. 1 on the Billboard 200 chart, and launched the No. 1 Billboard Hot 100 single "Sweet Child O' Mine." They have sold 44.5 million albums total in the U.S., according to the Recording Industry Association of America.
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The Awakening is een Amerikaanse dramafilm uit 1928 onder regie van Victor Fleming. Destijds werd de film in Nederland uitgebracht onder de titel Het ontwaken.</small>
Verhaal
Het Franse boerenmeisje Marie Ducrot krijgt tijdens de Eerste Wereldoorlog een affaire met de Duitse soldaat Karl von Hagen. Wanneer zij verpleegster wordt in een veldhospitaal, verneemt Karl dat zijn geliefde dood is. Hij komt er pas later achter dat ze nog in leven is.
Rolverdeling
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|||Marie Ducrot
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|||Graaf Karl von Hagen
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|||La Bête
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|||Ziekenbroeder
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|||Grootvader Ducrot
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|||Luitenant Franz Geyer
|}
Externe link
Film uit 1928
Amerikaanse film
Dramafilm
Stomme film
Verloren film
Film van Victor Fleming
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Q: How to check visible resolution of system I am not entirely sure how to word this questions so I am just going to explain my problem.
My VB 6 program saves the location of the screen in the registry so that when it loads up again it can have a default location. This works as expected, but I encountered a problem. I had a computer setup with 2 monitors. I dragged the window to the second screen, then the next day when I unplugged the second monitor. Every time I would load the program it would appear in the dock, but the form itself was not.
I figured it was hiding on the second monitor's space (or were the second monitor should be) Now at first I connected a second monitor to grab it back, but it did not show up. The monitor setup was such that the main monitor (1) was to the left of the new monitor (2). I had to drag (2) over to the left of (1) at which point it showed up, and I was able do drag it back to the main window and everything worked fine. I mention this because I figured each monitor number's resolution was relative to itself. Is the multimonitor one giant coordinate plane?
What I want to know is when I grab the location numbers from the registry (Form.left and Form.top) is there a way to check it against what is available to the system? This is not a huge issue, but I know I am going to get clients calling in eventually asking, "Its loading, but I cant find it!" and would like to avoid those calls if possible.
Thank you all
A: Not really an answer to your question, but it could help in dealing with similar problems in the future, or your customers if they bump into it.
If you right-click the taskbar button (or on Windows 7, right-click the Aero Peek preview of the window), you'll see a Move option. Click that, and then press one of your arrow keys, doesn't matter which. After that, the window will be in movable mode, and you can just move your mouse and voila, it is back on your screen and acting as if you were dragging the title bar.
A: Don't know about VB, but you can call native Win32 APIs to deal with the problem.
GetSystemMetrics(SM_CMONITORS) returns the number of monitors, or you can enumerate them with EnumDisplayMonitors() if you need to distinguish between "real" monitors and pseudo-display monitors.
Use MonitorFromPoint() or MonitorFromRect() or MonitorFromWindow() to identify the monitor at a specified point, and then use GetMonitorInfo() to return information about that monitor, including its virtual-screen coordinates. You can then position the window on the monitor of your choice.
I don't know how to reliably detect whether the second monitor is missing or powered off. You could use the setup API (!) for this, I suspect -- SetupDiEnumDeviceInfo() can be used to enumerate the monitor devices and SetupDiGetDeviceRegistryProperty() can be used to retrieve information about each, like its current power state.
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Q: how to remove the border between the window title and content I have create a Window with Ribbons (2010 - Microsoft.Windows.Controls.Ribbon).
It looks like that:
Now, is there a border over the tabs (between the tabs and the title). This is very ugly.
It should looks like Microsoft Office Word:
What can I do, that there is no border between the tabs and the titlebar?
(The RibbonWindow is used)
A: It looks like you're using the standard WPF Window. Try using the Ribbon framework's RibbonWindow instead. It integrates the Ribbon into the window itself rather than just containing it like a normal control, which is what Window does.
A: I use now the Fluent Ribbon, that solve the problem.
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This e-book comes with six digital game cards. They provide the evidence you need to start your hunt AND unlock one Clue.
1. Go to scholastic.com/the39clues/ebookTCH3 and log in. If you haven't signed up, click on "Join Now" to create a new account.
2. You need your book with you. Use it to answer the two questions provided.
3. Your cards and Clue will be unlocked. The Clues are out there and YOU can find them!
# Contents
Cover
Shurikens
Find the 39 Clues!
Title Page
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Epilogue
Preview
Your Mission
Copyright
#
They were toast.
Amy Cahill eyed the battered black duffel bag rumbling up the airport conveyor belt. It bulged at the corners. The sign above the belt said THANK YOU FOR VISITING VENICE: RANDOM PIECES OF CHECKED LUGGAGE WILL BE SEARCHED in five languages.
"Oh, great," Amy said. "How random is 'random'?"
"I told you, a ninja warrior must always keep his swords in his carry-on," whispered her brother, Dan, who had been operating on brain deficit for as long as Amy could remember.
"Excuse me, Jackie Chan, but carry-on luggage is _always_ X-rayed," Amy whispered back. "There are extra-special rules about samurai swords in _backpacks._ Even if they belong to scrawny, delusional eleven-year-olds who think they're ninjas."
"What was wrong with 'we need them to slice the veal parmigiana'?" Dan said. "It would have worked fine. The Italians understand food."
"Can _you_ understand 'five to twenty years, no parole'?"
Dan shrugged. He lifted up a mesh-sided pet carrier, inside of which a very disgruntled-looking Egyptian Mau was eyeing him suspiciously. "Bye-bye, Saladin," he sang into the mesh. "Remember, when we get to Tokyo... red snapper sushi every night!"
_"Mrrp?"_ whined Saladin from inside the carrier, as Dan set it gently onto the conveyor belt.
"Mmmm, hmm, ohh... _aaaaaaaaghhhh!"_ came a strangled yelp from behind them. Although everyone else in the vicinity was turning with a look of alarm, Amy and Dan knew it was their au pair, Nellie Gomez, dancing to a tune on her iPod. She didn't care that she sounded like a dying meerkat, which was one of the many cool things about Nellie Gomez.
Amy watched as the carrier disappeared through the cargo window. If the officials did search the bag, there would be alarms. Screaming Italian cops. She, Dan, and Nellie would have to run.
Not that they weren't used to that. They'd been running a lot lately. It began the day they accepted the challenge in their grandmother Grace's will. They'd had to go to her mansion in Massachusetts for that — and immediately afterward the mansion went up in flames. Since then, they'd nearly been killed in a collapsing building in Philadelphia, attacked by monks in Austria, and chased by boats through the canals of Venice. They'd been the target of dirty tricks from every branch of the Cahill family.
Once in a while — like every three seconds—Amy wondered why the heck they were doing this. She and Dan could have opted for a cool million dollars each, like a lot of Cahill family members did. But Grace had offered another choice: a race for 39 Clues to a secret that had been hidden for centuries, the greatest source of power the world had known.
Until then, Amy and Dan had been leading pretty lame, ordinary lives. After their parents had died seven years ago, their crabby Aunt Beatrice had taken them in — and the only cool thing she'd ever done was hire Nellie. But now they knew they were part of something way bigger, a huge family that included ancestors like Ben Franklin and Wolfgang Amadeus Mozart. It seemed like all the great geniuses of the world had been Cahills. That was pretty amazing.
"Hey, Amy, did you ever want to, like, get on the conveyor belt and see what happened? Like, 'Hey, don't mind me, I'm just hanging with the cargo'?"
And then there was Dan.
"Come on!" Amy grabbed her brother by the arm and headed for the departure gates. Nellie was right on their heels, spinning the wheel of her iPod with one hand and adjusting her snake nose ring with the other.
Amy eyed the airport clock. 2:13. The flight was scheduled to leave at 2:37. This was an international flight. You were supposed to arrive at the airport _two hours_ in advance, not twenty-four minutes. "We're not going to make it!" Amy said.
Now they were running toward gate 4, dodging other passengers. "Guess they didn't find Rufus and Remus, huh?" Dan called out.
"Who are Rufus and Remus?" Amy asked.
"The swords!" Dan said. "I named them after the founders of Italy."
"It's _Romulus_ and Remus," Amy hissed. "And they founded Rome. And don't _ever_ say that word!"
"Rome?"
"No— _s-w-o-r-d."_ Amy dropped her voice to a whisper as they pulled up to the rear of a very long security line. "Do you want us to go to _j-a-i-l?"_
"O-o-p-s."
"O-O-O-O..." Nellie wailed off-key to some unidentifiable punk track.
The security line seemed to take, like, thirty-two hours. The worst part for Amy, as always, was having to take off her jade necklace to go through the X-ray machine. She hated to part from that necklace even for a minute. When they emerged, the clock read 2:31. They raced down a long corridor toward the gate.
"Now boarding all remaining passengers for Japan Airlines, flight eight-oh-seven to Tokyo, at gate four," said a voice over the PA system in heavily accented English. "Have your boarding passes ready, and... _arrrrrrrivederci!"_
They pulled up to the rear of the line behind a sniffling toddler who turned and sneezed on Nellie. "Ew. Manners?" she said, wiping her arm on her sleeve.
"Has anyone seen my boarding pass?" Dan said, rummaging in his pockets.
"Have mine," drawled Nellie. "It's covered with boogers."
"Try inside your book," Amy said, pointing toward the paperback stuffed in Dan's back pants pocket.
He pulled out a dog-eared copy of _Classic All-Time Movie Comedies,_ which he'd found in the backseat of the cab on the way to the airport. The boarding pass was marking page 93. _"It's a Mad, Mad, Mad, Mad World,"_ Dan said.
"That's the smartest observation you've made all day," Amy said.
"It's the name of a movie," Dan replied. "I'm reading about it. The plot is so awesome—"
"Step forward, please — welcome aboard!" chirped a perky blond flight attendant whose Japan Airlines headset bobbed every time she nodded a greeting. She was wearing a name tag that read I. RINALDI.
Nellie handed over her boarding pass and headed into the accordion-walled tunnel that led to the plane's hatch. "Um, guys, this shouldn't be so hard to do," she called over her shoulder.
Dan held out his pass to the attendant. "It's really a funny movie. Like, all these old-school comedians, searching for this treasure—"
"Sorry, he's challenged," Amy said to the attendant, handing over her pass and nudging him toward the tunnel.
But Ms. Rinaldi scooted in front of them, blocking their path. _"Un momento?"_ she said, trying to keep her airline smile while listening to something over her headset. _"Sì... ah, sì sì sì sì... buono,"_ she said into the headset mike.
Then, with a shrug toward Dan and Amy, she said, "You come with me, please?"
As they followed her toward the corner, Amy tried to keep herself from shaking. The swords. They'd found the swords.
Dan was looking all puppy-eyed at her. Sometimes all she needed to do was look at him, and she knew exactly what he was thinking.
_Maybe we should run,_ his eyes were saying.
_Uh, where?_ she said back to him silently.
_I will make myself invisible by using ninja mind control,_ he was thinking.
_You have to HAVE a mind to do it,_ she beamed to him.
Nellie peered out from the tunnel entrance. "What's going on?" she asked.
"It is routine," Ms. Rinaldi called out, turning to face Amy and Dan. "My supervisor tells me it is random check. You please wait here by the wall?"
She bustled away, holding the two boarding passes, and disappeared around the corner.
From inside the tunnel, another attendant called out to Nellie, "Please take your seat, dear. Don't worry, the plane will not leave without all passengers."
"I hate airports." Nellie rolled her eyes and turned back toward the plane. "See you inside. I'll save you a bag of peanuts."
As she disappeared, Amy hissed to her brother, "I knew it — they searched your duffel. They're going to detain us and contact Aunt Beatrice, and that's the last we'll ever see of Nellie—"
"Will you stop being so gloomy?" Dan said. "We'll tell them someone else put the swor — the you-know-whats in the duffel. We never saw them before in our lives. We're kids. They always believe kids. And besides, maybe they _haven't_ searched our bags. Maybe they're just double-checking your passport to make sure they can allow someone so ugly to board a plane—"
Amy elbowed him in the ribs.
"Final boarding call, flight eight-oh-seven to Tokyo, gate four!" a voice boomed.
A third attendant was putting a web-ribbon barrier in front of the tunnel.
Amy was nervous now. They weren't going to hold the plane forever. "We have to get that flight attendant — Rinaldi," she said. "Come on!"
Amy grabbed Dan by the arm and they raced to the corner, taking it at a run.
_Whomp!_ They ran smack into another pair who were racing toward the gate. Amy bounced away, the wind momentarily knocked out of her. She bumped into Dan, who nearly fell to the floor. "What the—?" he blurted.
The two strangers were wrapped in full-length black trench coats with high collars obscuring their faces. One of them wore expensive black dress shoes; the other, jewel-encrusted sneakers. As they barreled past Dan and Amy, waving boarding passes in the air, one of them called out, "Clear, please!"
Amy recognized the voice. She grabbed Dan and whirled around. The two were grabbing the barrier and pulling it aside. "Wait!" Amy said.
An airline official shouted at them, too, sprinting to head them off. The two politely stopped and handed over their boarding passes. He examined the passes quickly, nodded, and pulled back the barrier. "Enjoy your flight, Amy and Dan," he said.
The two passengers stepped into the tunnel entrance and immediately turned around. They pulled down their raised collars and grinned.
Amy gasped at the sight of their cousins, their archrivals in the search for the 39 Clues, a pair whose nastiness was surpassed only by their wealth and cunning.
_"Sayonara,_ suckers!" sang Ian and Natalie Kabra.
#
_"Stop them!"_ Dan and Amy ran toward the tunnel, shouting as loud as they could.
Quickly, the flight official stepped into their path. "Boarding passes, _per favore_?" he asked, his face a mix of bafflement and annoyance.
Amy watched helplessly as Ian and Natalie slipped into the tunnel's long shadow.
They could hear the plane's hatch shut with a dull _thump._
"They're — they're the Kabras!" Dan said. "Evil Kabras. _Famoso, evillo, Kabritos!_ They are holding our au pair hostage!"
As a crowd of curious onlookers gathered, the official repeated, "No boarding passes?"
He was looking straight at Amy. Dan glanced frantically her way, his eyes screaming, _You're the older one — do something!_
The thoughts were firing around in Amy's brain like a broken laser-light show. How could the Kabras be here? She and Dan had left them unconscious in a smoldering room in Venice. Who had rescued them? How had they recovered so fast? How had they stolen the tickets?
Everyone was looking at Amy now. The whole airport. She hated when people stared at her. She hated it even worse when it involved being humiliated by the Kabras. They were always one step ahead, always one Clue closer to the Cahill secret. No matter how hard Amy and Dan tried, the Kabras were smarter, faster, cooler — and ruthless. They were impersonating Dan and Amy. They were about to ambush a defenseless au pair. How could Amy possibly communicate all this? She opened her mouth to try, but it was too much. Too many eyes. She felt as if someone had tied off her vocal cords. Nothing came out.
"Ohhh- _kay_ , thank you, Amy," Dan said. "Um, look, dude — officer — these guys? The Kabras? Well, actually, they're a guy and a girl? They ripped us off, okay? _Comprendo?_ The tickets say Cahill and they're not Cahills — well, technically they are, but it's a different branch of the family, they're like Janus, I mean _Lucians,_ and we don't know what we are, I mean what _branch,_ but we're related — anyway, we're all kind of involved in something, sort of this battle about our grandmother's will, you could say, but it's kind of a long story and _THEY HAVE TO BE STOPPED! PRONTO!"_
"Sorry," the official said, "if you have no boarding—"
Amy grabbed Dan by the arm. This wasn't getting them anywhere. They needed to find Ms. Rinaldi — or the supervisor who had summoned her. That person would rank higher than anyone here. Maybe there was still a chance. Maybe they could stop the plane from taking off.
She and Dan ran toward the corner again and rounded it. They raced past the place where they had collided with the Kabras, and immediately they emerged into the main corridor. In the distance they could see a line of shops. To their right was a supply closet and a glass door marked AUTHORIZED PERSONNEL ONLY.
To their left, a knot of onlookers surrounded the entrance to the women's room, where a group of EMT workers emerged, carrying a woman on a stretcher. Police were running in to join them from all directions.
Chaos. Total pandemonium. Amy strained to see around the rush of people as she ran, hoping to find a familiar face.
There.
A flash of blond hair, tossed over a shoulder, drew Amy's eyes to the right. "Dan, look!"
"Oh, _now_ you can talk," Dan said. "What?"
Winding swiftly through the crowd was a tall woman in a Japan Airlines uniform about a size too large.
The sight of the familiar figure was enough to unlock Amy's loudest outdoor voice. _"IRINA!"_ she blurted out.
There was no mistaking Irina Spasky — the stiff military bearing, the bladelike motion of the shoulders as she walked. Irina was another of the Cahill family bent on finding the 39 Clues. Like Ian and Natalie, she was ruthless. Unlike Ian and Natalie, she had been trained in espionage by the KGB.
Irina did not turn. She showed no outward signs of hearing Amy, aside from a quickening of her step.
Then she disappeared into the throng as if she'd never been there.
"Stop her!" Dan sprinted forward, nearly colliding with a rather sour-looking man in a wheelchair.
_"Polizia!"_ the man shouted, lifting his cane as if to whack Dan over the head.
Dan ducked. Amy pulled him away, trying to keep an eye on Irina. They plowed forward, elbowing their way around passengers.
When they emerged into a less crowded area near the end of the terminal, Irina was nowhere to be seen. "She's gone," Dan said.
"I — I don't believe this," Amy said, catching her breath. "She was working with Ian and Natalie. They sabotaged us _together_."
"Are we _sure_ that was her?" Dan asked. "I mean, how would Irina manage to get that uniform?"
Before he finished the question, a voice shouted in Italian over a bullhorn, and the crowd quickly parted. A small ambulance made its way through the airport, siren blaring.
Murmurs were passing through the crowd, mostly in languages Amy didn't understand. But she spotted a couple with sunglasses, lots of cameras, awful Hawaiian shirts, and vapid smiles. "Look, Dan—Americans," she said. "Let's listen...."
They both wandered closer until they could hear snatches of conversation. The people were talking about the woman on the stretcher.
Dan looked confused. "She was salted in the ladies' room?"
_"Assaulted,"_ Amy said. "She must have been the flight supervisor, Dan! Irina knocked her out and took her uniform."
"Wow," Dan replied, looking almost impressed.
Amy glanced toward the window, where she saw the jet slowly backing away from gate 4 and onto the tarmac.
They were leaving. Detached from the tunnel, taxiing for the runway.
Amy panicked. "Don't look now, but they're going!"
"Where's the door? We can still run after them!"
"Right. You do that, Dan. Meanwhile I'll try to talk my way onto the next flight — a ticket for _one,_ while they're scraping your remains out of the jet engine that sucked you in." Amy began running again, back toward the reservation desk. "Or you can come with me!"
Outside, the windows of flight 807 were dull silver-black holes in the distance. Amy knew that behind one of them was Nellie, in a situation no human being should ever have to face.
She was alone with the Kabras.
Dan followed Amy past the crowded security checkpoint, back toward the reception desk. The line for tickets doubled around at least three times, and they took their places at the back.
They exchanged a silent glance. Amy knew Dan was thinking exactly the same thing she was. He sighed, his saddened eyes wandering slowly to the conveyor belt. "Saladin's on the plane, too," Dan said. "And our swords."
Amy fought the urge to just collapse and cry. Right there in the middle of the terminal. Everything was going wrong. It had been a seven-year string of bad luck, ever since their parents died in that house fire. How were Amy and Dan supposed to do this alone? The Kabras had money. Their parents supported them. Plus, they were working with Irina. The Holts were a whole family. Jonah Wizard had his dad planning every moment of his life. It was Amy and Dan against... families. Teams. Generations. They didn't stand a chance.
If only Grace had told them earlier, back when their mom and dad had been alive. If only they were alive now! Thinking about them just made Amy feel worse. She'd been dreaming about them every night. She'd see their faces at odd times — smiling, confident, kind. She could sense their approval or disapproval, their pride whenever she got things right. They'd be there in her mind and then — _whoosh!_ Gone. And she'd feel the loss all over.
"Amy?" Dan said quizzically. And there they were — _again._ In the eyes of El Dweebo. Not their faces, exactly, but _them._ Looking out at her, as if they'd just borrowed Dan's features for a moment. Which no other sane person would do.
In that moment, she knew exactly what the right decision was.
"There's a flight leaving at five-ten," she said, reading the overhead departures screen. "Nellie's safety is at stake. We have to follow."
"Hey, coolio — no retreat, _no surrender_!" Dan whooped. "So. Any thoughts about how we're going to pay for it?"
_WAWWP! WAWWP! WAWWP! WAWWP!_
An alarm rocked the terminal, stopping all conversation. As a terse announcement resounded, first in Italian, then French, then German, sections of the crowd began heading for the entrance — until finally:
_"Ladies and gentlemen, please proceed immediately to the nearest exit, as this terminal must be evacuated for safety reasons...."_
A scream ripped the air, and then people were rushing, falling over one another. Amy ran toward the door, pulling her brother behind her, listening to shouted fragments around them, some of them in English:
"Bomb scare..."
"Terrorists..."
"Anonymous phone call..."
They reached the door and pushed their way through. The day had turned gray, but the winding access roads were dotted with the headlights of approaching vehicles. Passengers crowded the sidewalk, shouting into cell phones, hurtling toward buses and cabs. Dan and Amy pushed against the crush of bodies toward the curb, where the last of a group had climbed onto a bus.
The door shut in their faces and the bus farted its way noisily into the clogged road. Dan ran after it, banging on the window. "Stop! _Pasta!"_
_"Pasta?"_ Amy said in bewilderment.
"I have a limited vocabulary!" Dan shouted. _"Linguini! Mangia! Buon giorno! Gucci!"_
A black limo screeched to a halt inches away, nearly hitting her.
"Gucci. I knew that would do it," Dan said.
The tinted window on the driver's side rolled down, and a man wearing sunglasses and a thick mustache calmly gestured for them to get in.
Amy opened the passenger door and climbed inside, yanking her brother in after her.
"Hey!" shouted another frantic passenger, pulling a wad of cash from his pocket and waving it at the driver through the window. _"Soldi, soldi!"_
Dan pulled his door shut, and three people fell on the car, banging and shouting. The driver turned forward and let his window roll up, nearly amputating the arm of the man with the money.
"Dude, thanks," Dan said to the driver. "Or _gracias_ or whatever."
"Ve go to de udder airport?" the man replied in a deep accent that did not sound Italian.
"There's another airport?" Dan said.
"Small craft," the man replied.
"But—" Amy stammered. "We don't have any mon—"
Dan poked her in the ribs.
"I have to tell him the truth," Amy whispered.
Dan poked her again.
Amy glared at him. _"Will you please stop—?"_
It was only then that she saw the other person sitting in the backseat. An Asian man with a placid smile, dressed in a silk suit with white gloves and a bowler hat.
"Greetings, my elusive relatives," purred Alistair Oh.
#
Alistair's father had always said that in every Oh there was an element of surprise.
Not that Alistair remembered him actually saying it, considering Alistair had been a child when he died. But it was an Oh family trait to mix truth with a touch of wit.
Alas, the Cahill children's hostile silence perplexed Alistair. He would have thought they'd have enjoyed this particular surprise.
_Screeee... screeeee..._
As Serge yanked the steering wheel left and right, forcing the car into spaces no normal human would dare to go, the children lurched from side to side. They seemed loathe to touch Alistair or even look at him, as if he were some distasteful substance, like boiled asparagus. As if he had not just snatched them from the jaws of chaos to deliver them back to their chosen path. He tried to smile reassuringly at them. He felt for them. They looked so small, so scared, so lonely.
He understood the feeling. More than they knew.
"Gyess what?" shouted Serge over the noise of furious honking. "I chave keeds, too — gerrl fourteen, boy elyeven! Yes! True. They leeve in Moscow!"
Alistair kept an eye on Dan, who was looking quite sick. The boy tried the door handle for what must have been the twentieth time in the last two minutes. Luckily, Alistair had made sure the safety locks had been activated. "Do not bother, please," he said. "You will only give yourself carpal tunnel problems later in life. And besides that, you are making me nervous for your safety."
"So you were behind all this, huh?" Dan said. "With the Kabras and Irina. And the bomb scare. You're working with them now."
Alistair's face twitched. He knew it would be difficult to earn their trust. Wild accusations were to be expected. He knew there would be resentment, and understandably so. Leaving them in a burning house on the day of the will reading had been an unfortunate necessity — but a personal and strategic mistake. One he regretted deeply. "Believe me, my dear nephew, I don't have the slightest idea—"
"Believe you?" Dan replied, spinning around to face him eye to eye. "Let's see. You abandoned us when Grace's house was collapsing around us. You planted a tracking device on Saladin—"
"Tracking device? _This?"_ Alistair reached into his pocket and pulled out an electronic device the size of a lapel pin. "I believe you planted it on me. At the museum in Salzburg, while I was dozing."
"You d-d-deserved that, Uncle Alistair," Amy said uneasily, "after having hidden it in Saladin's collar in the f-f-first place."
"No again, darling girl," Alistair replied with a warm smile, hoping to calm the girl's nerves. "Someone else was keeping tabs on you. Not I. Remember, many others in the family are competing for the clues. I am on your side. I, as you know, believe in cooperation."
"Oh, that's hilarious," Dan shot back. "Tell it to Comedy Central."
_Patience. Ever patience._ Alistair folded his white-gloved hands over his lap. "Consider exactly who rescued you today," he said. "And who, in a very short time, managed not only to find you but to devise a method of escape. Consider also that as an added bonus, I am about to take you wherever it is you need to go. By private plane. All of this, and I ask only one thing in return — the location of where you are headed. Which, under the circumstances, is rather a necessity."
"You have your own p-plane?" Amy asked.
Alistair smiled modestly. "Well, not mine. But I still have business connections, favors I can call in during times of emergency. There are some financial advantages to being the inventor of microwavable burritos."
"Ve stock zem on ze plane!" Serge said. "Biff, cheecken, cheese..."
Good old Serge. Experience had taught them both the value of the Oh company motto: The way to a young person's heart is through microwavable meals.
Amy exhaled. "Okay, once we're on this plane — _if_ we agree — what assurances do we have that—"
_"Amy!"_ Dan blurted. "Uh, no way, Goldfinger. If we're going to do this, we're doing it ourselves."
Amy glared at him. "So I guess we're _swimming_ to Japan? Drop us off at a mall, Uncle Alistair. I need flippers. The really big kind? With mad shark repellent?"
Dan groaned. "You said the J word, Amy! You _told_ him!"
"What are our choices, Dan?" Amy said. "They have Nellie and Saladin and our s—"
Amy stopped short, and Alistair glanced at her encouragingly. The poor thing had been making such great progress with her shyness. "Your...?" he said.
"S-s-suitcases," she replied.
Alistair nodded. _Japan._ Excellent. So that was where the next Clue would be. A fruitful turn of events. He leaned forward to his driver. "Can we handle Japan, Serge?"
The driver shrugged. "Vell, eez long treep. Ve must stop for refueling halfvay. In Moscow. I call ahead. Vhen ve stop, you can meet my keeds — Kolya and Tinatchka!"
"Serge, please," Alistair said. "This is not a social trip."
Serge let out a deep belly laugh. "Kolya and Tinatchka not socialists!"
Dan glared at his sister. _Swords,_ she'd been about to say. _They have Nellie and Saladin and our swords._ At least she corked herself on that one. Giving away their destination to the slippery dude was one thing. Giving away their Clue was another. _Some_ things had to stay secret. Even sisterus dorkus knew that.
He recognized the look in Amy's eyes right now. It was more than the usual disgust, more than her usual variations of _You dweeb_ and _No, it's not time to eat._ This one said, _If you screw this up, I will kill you._
Which was exactly how he was feeling.
Uncle Alistair reached into his pocket and pulled out two small electronic devices, which he held out to Dan and Amy with fake cheeriness, like a demented butler pretending to be Santa Claus. "These are state-of-the-art GPS devices. Attach them to your phones, as I have done to mine. I have not yet figured out how to one-twenty-eight-bit-encrypt the signal, but the lower default encryption should suffice. The point is, once we are in Japan, we cannot lose one another."
Serge was flashing an ID pass to a guard at a gate now. The limo entered a narrow road leading to a tiny airport. It glided past several small propeller planes and stopped next to a long, open hangar.
Serge quickly got out and held the passenger door open. Beaming, he gestured grandly toward the hangar. "Say hello to my darling Ludmila."
"Another keed?" Dan asked. "How many do you have?" He looked left and right. The place seemed empty except for a few small jets and some burly half-shaven crew members, none of whom looked like a Ludmila.
"Um... I don't see her," Amy said meekly.
But Dan was distracted by a flash of silver. A ridiculously sleek jet rolled into view. It had tinted windows, a profile like a knife, and an open cockpit that seemed to beckon, _Enter for the coolest ride of your life._
"This," said Serge, as the jet rolled to a stop in front of them, "is Ludmila."
#
The peculiar word COACH on Natalie's plane ticket had conjured up images of leather benches, uniformed coachmen, and fine horses.
Not tiny, hard seats and a pig.
It wasn't so much the babysitter's attitude. Which was horrid. Or the tattoos and piercings. Which someday would cause the girl such embarrassment at her job — that is, should she ever find a real one. It wasn't even the girl's rudeness upon seeing Natalie and her brother. Granted, a warm hello and hugs were not warranted under the circumstances, but the stream of barnyard epithets was a bit... well, unseemly. To say the least.
Yet all of the above was to be expected from a person of Nellie's station. And Natalie and Ian could withstand the crudeness. Some sacrifices were necessary in order to get the information they needed.
No, the worst part was the _sloppiness._ The candy wrappers and potato chip shards on the seats to either side of her, the backpack plopped on the floor between her feet instead of tucked under the seat in front. The nervous habit of shoveling fistfuls of snack mix into her mouth and chewing while speaking. Dreadful. _Sloppy habits make a sloppy mind,_ according to the old Kabra family saying. Or maybe that was from _Bartlett's Familiar Quotations._ Natalie wasn't quite sure.
She winced as the loathsome babysitter spoke with a mouthful of food.
"Soffy, unca wassoosa, y'nah gongahwawee!" Nellie said, spitting bits of peanut and Chex from the corners of her mouth.
Natalie's brother, Ian, picked a fleck of Rice Krispie from his otherwise flawless jet-black hair. "Swallow, please, and repeat?"
Nellie gulped. "Sorry, I don't care what you say, you're not going to get away with this."
"Oh?" Ian looked over his shoulder, up and down the crowded jetliner. "Do I see anyone sympathizing with your plight? No? What do you say, Natalie — did we get away with it?"
"You can do this the easy way, you know, by answering one simple question..." Natalie pressed. They had asked Nellie a dozen times, and each answer had been sassier than the last. She would learn, though. If she knew what was good for her. And if not, well, the Kabras had other ways. "So. One last time — why are you going to Japan?"
Nellie yanked a magazine out of the seat pocket in front of her, sending a set of earphones and some used tissues flying toward Ian, who leaped with a barely concealed _yeep_ of disgust. "Because I love sudoku," Nellie said. "You can get the best sudoku puzzles on a flight to Japan, duh. Don't you know _anything_?"
"Coffee, tea, complimentary snack packet, or anything else I can assist you with to make this your best flight ever?" a flight attendant piped up, walking slowly up the aisle.
"A Diet Coke and a restraining order, please?" Nellie said. "Because these two are _not_ supposed to be in these seats, and they are harassing me."
Ian let out a hearty laugh. "Ha-ha! Oh, Cousin Nell, you always crack me up with your jokes and whatnot — doesn't she, uh, Amy?"
"Yes, Daniel," Natalie replied. "Just like back home. In... Homedale."
"Oh, that's convincing," Nellie said. "Is there a cop on board? Because if there's not, I want to make a citizen's arrest. Can you do that in Italy, or wherever we are?"
Smiling uneasily, the flight attendant placed a Diet Coke on Nellie's tray. As she stood back up, Natalie turned to the baffled woman and circled her finger by her ear in a subtle _cuckoo_ gesture.
Outside the window, lightning flashed. The plane suddenly began to lurch. "Heh-heh, well, looks like we're experiencing just a bit of bumpiness here..." the pilot said over the PA.
The flight attendant began pushing her cart back up the aisle, calling out, "Please return your seats to the upright position."
Ian groaned. "I — I'm not feeling too well...."
As he keeled forward, face turning green, it was Nellie's turn to look alarmed.
Natalie smiled. She and Ian had planned this. Certain signals for certain contingencies. Kabras were masters of tight planning. Ian's act meant one thing only, and Natalie knew just what to do.
Still, she couldn't help but feel pity for the girl. Underneath the raffishness there was a bit of spunk and spirit. In another circumstance, another time, she would make a good Kabra employee.
"Uh, you're not going to be sick, are you?" Nellie said. "Because I hate the sight of puke." She leaned forward to sift through all the detritus on the floor, looking for an air-sickness bag.
There.
While Nellie was turned away, Natalie reached into her pocket and extracted a small vial of dark liquid. With a deft motion, she lifted the vial to Nellie's soda. Two drops was all that was needed.
The plane jumped again, causing Natalie to flinch, and the entire contents of the vial spilled inside the Coke.
Oops.
The beeping of the phone woke Dan up from a deep sleep.
The first thing he noticed was Amy's hand, bone-white and clenching the armrest. "I don't know how you can sleep at a time like this..." she said through gritted teeth.
The little jet banked to the left, causing Amy to let out a scream. "Sweet!" Dan said. "Do that again, Serge!"
Serge laughed. "You like?"
"No!" Amy blurted.
Alistair was straining to hear the phone. "This is _who_?" he said, gesturing for everyone to be quiet. "Irina?"
Amy groaned.
"Yes, they did get away," Alistair said loudly. "They are with me, quite safe and sound... what? Did you say _Japan_?" Alistair let out a big laugh. "Oh, dear. You thought... you really believed that Dan and Amy didn't _let_ the Kabra children take their tickets — that they didn't purposefully direct their babysitter to board the plane as a decoy... oh, my. Oh, that is rich... no no no, Irina.... What? You're breaking up. Perhaps you heard me wrong. Yes, OF COURSE THE CAHILLS ARE HEADED TO JAPAN. THAT IS EXACTLY RIGHT. Good-bye, dear."
"Um... what was that all about?" Dan said.
Alistair smiled. "I know Irina quite well. At this moment she is convinced you tricked the Kabras, not the other way around. And trust me, after what I just said, the last place in the world she suspects you to be going is Japan...."
"Wait. You think you convinced her?" Dan said. "Uh, no offense, but that sounded pretty lame to me."
"I may be a failure in some things in my life, but I am a very quick study of people," Alistair replied. "I know exactly what works with Irina Spasky."
Amy turned her head toward Alistair, the color nearly drained from her face. He was a smart guy in many ways but a little old-fashioned. And he had overlooked something incredibly obvious.
"Don't... be too sure..." she said.
The pilot's voice, in Russian, called for clearance and quickly got it.
Banking to the right, the jet swooped low toward a small airport on the outskirts of Moscow. In the dry, parched landscape, the landing strip was a ghostly gray.
The lone passenger's fingers gripped the armrest as the wheels of the plane bumped against the ground. These landings were always rougher than she expected.
As the jet slowed, taxiing on the tarmac, she eyed the sleek silver Cessna being refueled. An impressive piece of machinery.
"Stop here," Irina said.
She could see the old man now, limping with his walking stick. He was dressed crisply and correctly, as always. The bowler and sunglasses gave that subtle touch of refinement. Irina liked a traditional man, not a slave to fashion. His clothes seemed a bit tight today, but during these stressful times, who hadn't put on weight?
A moment later, the little devils appeared, bundled in down coats and hats. Protected as always — first by Grace Cahill, now by the uncle. Why he had sold his soul to those two, she could never figure out. Someday he would learn.
_They will betray you, Alistair,_ she thought, _unless you betray them first._
She smiled. Thoughts of human weakness always picked up her spirits after a long trip. Back in her KGB days, betrayal came in so many colorful varieties — blackmail, white lies, red tape, yellow journalism.
_Teams — paahh!_ she thought. Teams were of no use in discovering the 39 Clues. With a secret of this much power, jealousies were inevitable and no alliances would survive.
Irina would find the Clues by herself. Without lazy rich kids, over-the-hill taco tycoons, or dewy-eyed orphans. To them, the amateurs, this was a mysterious game. Not to Irina. The spoils, she knew, deserved to go to the one who had lost the most. To the lone wolf seeking justice. And vengeance.
Across the runway, the trio climbed into the jet. Irina leaned forward, glancing at her cell phone, which still showed the GPS coordinates and recipient of her last call: OH, ALISTAIR.
"'Oh, Alistair,' indeed," she said under her breath. "You are making this chase too easy for me..."
_"Shto?"_ said her pilot.
"Follow them, Alexander."
He pulled the gearshift and the plane's engine hummed to life. Ahead of them, the Cessna was beginning to position for takeoff.
Now she would see if he was telling the truth about their ultimate destination.
She grinned. No one ever put anything over on Irina Spasky.
#
Amy's brother was never comfortable in a new place until he committed an act of cluelessness. In Tokyo, it happened the morning after their arrival at the Thank You Very Much Hotel.
"Dan, you can't just _take_ that — it's stealing," Amy said, watching him struggle to fit a hotel ashtray in his back pocket.
"They won't know it's missing!" Dan protested. "I need it for my collection."
Dan collected everything. If it fit in a house and wasn't chained to the floor, he had a collection for it.
"Your sister is correct," Uncle Alistair said sternly, stopping to lean on his walking stick on the way to the front door. He smelled of aftershave and powder. On the trip in from the airport, he had bought Amy and Dan a few changes of clothes and insisted they freshen up and have a long sleep.
Amy had not slept for a nanosecond. For one thing, she was too nervous. For another, Dan kept muttering _"Mrrp"_ in his sleep. He was really missing Saladin.
But that hadn't curbed his obsession for collecting. Amy held out her palm. Reluctantly, Dan put the ashtray in it. "Okay, but can you get me a Thank You Very Much Hotel matchbook, then?" he asked.
Amy returned the ashtray to a table in the lobby, with Dan skulking behind. They still couldn't pronounce the hotel's real name. Instead, they nicknamed it after the only phrase any of the staff ever said to them. Picking up a matchbook from the front desk, Amy smiled at the desk clerk. "Thank you very much!" the clerk said.
As they walked back toward the door, Dan eyed Alistair, who was looking away from them. "Let's escape," Dan murmured. "We have to find our peeps. Nellie and Saladin."
"Are you crazy?" Amy whispered back. "Uncle Alistair paid for this hotel stay. He knows Japanese, and he's going to help us around town."
"You like him!" Dan said in horror. "He's turned your mind!"
Amy whirled on him. "I don't like him _or_ trust him. But without him we're stuck, Dan. So we have to _pretend,_ at least until Nellie finds us."
"Or we find her!" Dan grumbled as he and Amy headed toward Alistair at the front entrance. Together they stepped out into a crisp, sunny day. To their left, people in manga-hero costumes greeted shoppers in front of a gleaming high-rise shopping center. The scent of some strange blossom wafted from a park, which was across a busy street filled with car and bike traffic. Tokyo reminded Amy of New York City, without all the people yelling at each other.
Dan's eyes were cast upward, gawking at a steel structure that rose above the park. "Cool, someone brought over the Eiffel Tower and painted it red and white!"
Alistair smiled. "The Tokyo Tower is taller than its Parisian counterpart, but also lighter, due to advances in steel construction — advances that were developed, may I add, by an Ekat engineer. My illustrious family. And you see that tall apartment tower with curved sides? It suggests a Japanese flower found in great abundance in Shiba Park. The brainchild of a Janus architect—"
"Wait, that park has flowers made of steel?" Dan said.
"I know someone with a brain made of tin," Amy replied, then turned back to Alistair. "How do you know so much about your family?"
"Someday I will show you my collection," Alistair said. "But let's get to the task at hand. It is a ten-minute cab ride to the Metropolitan Library."
"Library. Woo-hoo. Can't wait," Dan said, absently fingering his matchbook. "Hey, I know. You guys go. I'll pick up some snapper sushi and take a cab to the airport. I'll meet you later."
"What makes you think Saladin is at the airport?" Alistair asked, walking toward the street.
"I figure two things could have happened," Dan said. "One — the Kabras brainwashed Nellie and are leading her around town trying to find us. Or Two—Nellie managed to subdue them with superior ninja training techniques she didn't realize she had picked up by mental telepathy from me. Actually, I'm betting on One. Either way, Saladin would..." Dan's face darkened. "I — I can't stop thinking of him, still on that conveyor belt, all alone, going round and round..."
"I know you love your pet," Alistair said. "But you must think of your own safety first. The Kabras will expect you to come to Japan. They may also expect you to go to the airport in search of your beloved feline and babysitter—"
"Au pair," Dan corrected him.
"You simply cannot risk walking into a trap," Alistair continued.
It made Amy sick not to know where Nellie and Saladin were, too. She'd been trying to contact Nellie on her cell phone since they arrived. She hated telling Dan not to go after them. But Alistair's advice made sense. "Knowing Natalie and Ian," Amy said, following Alistair toward a taxi stand, "they'll find us."
"But—" Dan protested.
"We have to move forward," Amy said. "Nellie will land on her feet."
Dan sighed. "Saladin, too, I guess. I mean, being a cat and all..."
As they threaded their way through the plaza, Dan kept lighting matches and blowing them out. "Will you stop it?" Amy said.
"Why?" Dan said, lighting another match. "It's fun. It keeps my mind off the fact that here we are ignoring the only two people we really like, plus we're in the land of ninjas and Mothra and really cool martial arts, and we're going to spend another day in a library."
As he approached a waiting taxicab, Alistair said something to the driver in rapid, fluent-sounding Japanese and signaled Amy and Dan to climb in.
They sped through the traffic, passing rows of modern steel buildings and an occasional ornate ancient pagoda surrounded with gardens. "Why can't we stay in one of those cottages?" Dan asked.
"They are ancient temples," Alistair replied. "You will see more of them as we approach our destination. The military dictator — the shogun — ordered all temples moved here. At the time, the Roppongi area was a remote outpost of the capital, which was then called Edo. Part of the area was a hunting ground for the shogunate."
"Fascinating," Amy said. She loved learning about the origins of cities.
Dan nodded, staring dully out the window. "I think I just saw a celebrity."
Alistair's cell phone rang. "Hello...? Yes... ah, bravo, Serge. She _what_? Well, imagine that — ha! Very good. And thank you so much. _Da. Do svidanya!"_ He put the phone away and turned to Dan and Amy. "Serge is safely in Siberia with his two children. Irina completely fell for the disguise. She thought they were we. When she realized she'd been had, she began cursing with words that embarrassed even Serge."
"Yes!" Dan whooped, slapping high fives to his sister and uncle.
"I have you to thank, Amy," Alistair said, beaming. "How stupid of me not to realize Irina could have tracked us with the cell phone GPS."
"I thought of it, like, right away," Dan said modestly. "I'm just more shy."
Amy rolled her eyes. "And I'm the queen of England."
"You _do_ look wrinkly and boring," Dan said.
He darted away before Amy could whack him.
Soon the cab pulled up to a massive, modern, boxlike building at the edge of a lush park.
"Arisugawanomiya!" the cabdriver announced.
Dan looked panic-stricken. "What did I do now?"
"It is the name of the park, and that building is the Central Branch of the Tokyo Metropolitan Library," Alistair explained, as he paid the driver and climbed out. "We have only limited time before Irina catches on. Because we've detached our GPS devices, it is essential that we stay close together. And set your phones to 'vibrate' while in the library."
"How can I stand the excitement?" Dan droned.
The moment they entered the building, a trim librarian was at Alistair's side, bowing and speaking with him in rapid Japanese. She smiled at Dan and Amy and gestured for them to follow her.
"You know her?" Dan whispered as they climbed a grand marble staircase. "Like from back in your shogun-hunting days?"
"No, she is just being courteous," Alistair replied, his limp barely noticeable as he walked. "It is respect for my age. Although perhaps Ms. Nakamura remembers my television appearances ten years ago. My company's Terribly Tasty Teriyaki microwavable burritos were quite the rage."
They entered a small private room lined with bookcases. On one wall a couple of small windows overlooked the street. In the center was a bank of computers. "Please do not hesitate to come to me with any questions," Ms. Nakamura said in lightly accented English, exchanging bows with Alistair and shutting the door behind her.
"I told her we were conducting research for a new interactive website on possible burrito fillings," Alistair said, resting both hands on his walking stick as he leaned toward Amy and Dan. "But my question to you is, why are we _really_ here?"
Amy's eyes darted over to Dan. Alistair had asked this question before, and they had evaded him each time. He knew they were up to something.
The problem was the swords. Alistair didn't know about them. He hadn't seen the secret engraving on one of the blades. He had no idea that the second Clue had been tungsten.
_He's even more confused than we are,_ Amy thought. Iron solute and tungsten weren't exactly obvious interlocking puzzle pieces. The first was an ingredient in ink, the second was the material that burns in incandescent lightbulbs. How could they fit together? Amy and Dan needed to know so much more — but one thing was pretty clear. Somehow the swords were the key to the next Clue. Maybe Alistair _could_ help them find out, Dan thought. But the risks were great. Alistair just might take the info and run; he'd done it before. _Trust no one_ — that had been Amy and Dan's motto. Whenever they'd forgotten it, they'd regretted it.
And they desperately needed to limit regrets.
"It was... a code," Dan said, improvising a white lie. "In the music. Mozart's music. The code said, um, 'Go to Japan.' In the key of C? That's all we know."
Alistair shrugged, sitting at a laptop. "Not much to go on, but that hasn't stopped us before. Let's each work awhile and then compare notes, shall we?"
Amy and Dan made sure to sit opposite him, so their monitors were not visible to Alistair. Amy typed into a search bar:
**japan tungsten sword
87,722 hits.**
"This is going to be a long day," murmured Amy. Dan typed:
**ninja warrior images
1,694,117 hits.**
He smiled. Maybe it wouldn't be so bad after all.
**Body painting and the tatoo was the belonging to slaves and prisoner of Ancint Japan. some of design has been historical replication by ours tattoo artists all who is graduate of colleges in history dilpoma.**
Dan scrolled down. The images were clearer than the English translation. Some of these designs were amazing, covering a person's whole back. There were dozens — dragons, historical scenes, countrysides, ornate scrollwork...
He stopped. Something in one of the images looked familiar.
Scooting back up the page, he found it and clicked through. Slowly, a magnified version filled the screen.
"Dan, _what_ do you think you're doing?" Amy asked, looking over his shoulder.
"Is this cool or what?" Dan said.
Amy gestured to her laptop screen, which showed a map of Japan. "We're supposed to be locating our clue!"
"Uh, excuse me, Dora the Explorer, look closely — these characters? They are the same ones we saw on the sword!"
Oops.
Dan's hands immediately shot up to his mouth. He hadn't meant to say the S word.
Amy's eyes went all buggy. _Dan, you moron!_ they said silently.
Dan and Amy glanced at Alistair, who had been intently writing down something he was seeing on his screen. Slowly he looked up. He seemed pale, almost sick.
"Uncle Alistair...?" Dan said. "Are you okay?"
Alistair did not answer for a few seconds. He took off his glasses and wiped them with a pocket handkerchief. "Fine. Staring at the screen for long periods of time... it gets harder to do when you're old. Forgive me. Have you, er, found anything?"
"Yes," said Dan.
"No," said Amy.
"Yes and no," clarified Dan. "Have you?"
Alistair nodded absently. "Come. Look."
Amy and Dan scooted around to see Alistair's screen. He was minimizing a webmail site to reveal a web page that showed a painting of a fierce-looking Japanese warrior holding a severed head.
"Eww..." Amy murmured.
"Dude, it's only pixels," Dan said. "But... eww."
"The, um, Bald Rat," Alistair continued, his voice still faraway and distracted. "Also known as Toyotomi Hideyoshi.'"
"Hide _who_ shi?" Dan replied.
"He — he was the greatest warrior in the history of Japan," Alistair went on, "but most records show him as rather hideous-looking. Lived in the fifteen hundreds. He began as a peasant and rose to unbelievable power, conquering the various tribes and factions and uniting the country as a major power for the first time." Alistair paused, lowering his voice. "He was also one of your Cahill ancestors—"
"I _thought_ I saw the resemblance to Amy," Dan said.
"—A Tomas, as a matter of fact. Descended from Thomas Cahill. Thomas traveled to the Far East in the sixteenth century—some say for the purposes of trade, others say to hide in shame after failing to find his wayward sister. At any rate, he settled there and his family became the Tomas branch, famous for their brutishness and warrior ways."
Dan looked closer. "The Holt family — they're Tomases. They look like tree trunks with dinosaur brains. This guy looks like a weasel."
"It makes sense Hideyoshi would be a Tomas," Amy said. "The strength. The gross way he's holding that head."
"Evolution is odd. It does not favor the Tomas." Alistair's grim expression loosened a bit, edging into a semismile. "Of course, I am showing my bias as an Ekat. Anyway, I strongly believe our search should begin with Hideyoshi. The man had a lot of secrets. Some say the secrets undid him."
"Secrets is our middle name, dude," Dan said.
Alistair leveled his gaze at Dan and then Amy. The color was returning to his face now. "I was going to keep this information to myself. After what happened in Salzburg, I wasn't sure I could trust you two. In fact, today I was tempted to conduct this entire search for the Hideyoshi link without your knowledge."
"Well, that makes two of us," Dan blurted out.
"Three," Amy corrected. With a quick, tentative glance toward Dan, she added, "We didn't think we could trust _you,_ Uncle Alistair."
Alistair nodded. "I have devoted myself to earning your confidence again. Trust is a fragile thing — difficult to build, easy to break. It cannot be bargained for. Only if it is freely given can it be expected in return." He looked from Amy to Dan. "To break the chain of mistrust, someone has to go first. I am happy to make the move. You deserve no less."
Solemnly, he turned back to his screen. "Hideyoshi was a bit of a paranoid who liked to hoard things," Alistair continued, scrolling down the screen into the biographical text. "For example, the Great Sword Hunt of 1588, when he forced all the farmers and peasants to give over their swords. He claimed he wanted to melt them down for a large statue of Buddha. But that was a lie."
"And the truth was...?" Amy said.
Alistair shrugged. "One of the great mysteries. He also took measures to prevent farmers and peasants from rising to the warrior class. He seemed to be afraid of this."
"But _he_ rose from the poor," Amy said.
"You must think like a warrior, sister-san," Dan said. "He was afraid _because_ he rose from the poor. He thought somebody else might, too — and whup his ninja butt."
Alistair nodded. "Perhaps he suspected more Tomas offspring — or worse, Ekaterina—lived in the provinces. The Ekat and Tomas branches were at war even then. Was he trying to hide swords from the Ekats to prevent them from rising against him? We don't know. If only we knew _where_ he hid the swords. The _where_ might lead to the _why_." With a shrug, Alistair turned to face them. "Okay, I've told you all I know."
Dan glanced at his sister. The ball was in their court now.
_He's shown us his own secrets,_ her eyes were saying. _We owe him._
_He was looking at his webmail,_ Dan thought in return. _He didn't show us THAT._
_That's different,_ she argued. _We need him._
_Besides his cash and his knowledge of Japanese, what good is he?_
_Besides your nicely shaped left earlobe, what good are you?_
Dan glowered at her. _You're the oldest, YOU mention it._
Amy turned to Alistair. "We think... we found some of the swords," she said. "In Venice."
"Hideyoshi's swords — in _Italy_?" Alistair looked flabbergasted.
With a sigh, Dan mumbled, "They were in the house of some Italian dude, Fidelio Racco."
"Racco..." Alistair said. "A Janus. Yet the clue points to a Tomas stronghold. Curious. Here in Japan, there are rumored to be secret Hideyoshi hiding places, but they are allegedly guarded by the descendants of Hideyoshi — many of them _yakuza_."
Dan smiled. _Now_ he was talking. "Whoa — awesome!" Dan exclaimed. "I battled them at Level Four in... um, _Ninja Gaiden,_ I think? Those guys are mad gangsta! They'll cut off your arms and feed them to you for lunch."
"Can't wait to meet them," Amy said.
"We tried to bring the swords here," Dan barreled on. "They're in my luggage. One of them had some markings. We figured the markings were important — maybe they give information about the next clue."
Alistair's eyes widened. "Is there any way we can retrieve these swords?"
"Well, maybe we don't have to." Dan nodded to his screen. "The same markings are on this tattoo."
Dan had never seen Alistair move so fast. He leaned over Dan's shoulders and squinted at the image. "Are you _sure_ this was printed on the sword?"
"Yup," Dan said. "Well, not exactly. There were some other characters, too. They're missing here."
Amy shook her head. "How can you be sure? You don't know a word of Japanese."
"Uh, yeah," Dan said. "And I don't read a note of music. But let's see, who was the one who _memorized a whole Mozart song and found our last clue_? Wait, wait, let me try to remember. Oh. I know — me!"
"Dan, are you _sure_ there were characters missing?" Alistair said. "Because the message, as is, is fairly innocuous — an incantation to luck, honor, triumph, and such."
"Definitely. At the beginning of each line, there was some weird-looking letter. Like from another language. You know, Sanscript maybe."
"It's Sanskrit, tattoo-brain," Amy said, sitting at her laptop. "Guess you don't remember _everything_ perfectly."
She turned to her uncle, who was furiously tapping on his keyboard. "What do you know about these _yakuza_ people, Uncle Alistair?"
She thought she could feel him shudder. "They are very nasty and very deadly," he said softly. "Trust me, we do not want to cross their paths."
"You know some of them personally?" Dan said.
"They know me and despise me," Alistair said. "I am an Ekat. The Tomas and the Ekats have been bitter enemies for centuries. The _yakuza_ have long been suspected to possess a map to a secret underground crypt. And if I am to understand this recent message, we may have found a copy."
He hit the PRINT button. From the library's printer, a map slowly made its way into the paper tray, an ancient image showing a complicated ribbon of tunnels.
"Cool!" Dan exclaimed.
"You _knew_ about this all along?" Amy asked.
Alistair shook his head. Once again, his face became pale and drawn. "I have long been searching for certain... stolen Ekat documents not related to this. One of my colleagues has managed to find a hidden cache. I received a message from him on webmail while we were in Salzburg, with several attachments — including this map."
He showed them the printout, which had the heading OF UNKNOWN SIGNIFICANCE.
"Wait — Ekat documents? _Colleagues?_ What else are you keeping from us? How can you—?"
Amy's words stopped in her throat. The cursor on Dan's monitor was moving from the middle of the screen up to the left corner.
"Dan?" Amy said. "Stop that, okay?"
"Stop what?" Dan replied.
"We know libraries bore you, but can't you take _anything_ seriously?" Amy replied. "You're playing a trick, right? You have something in your pocket and it's sending a signal to the computer. Otherwise, _why would the cursor be moving_?"
Now the cursor was on the BACK button, clicking rapidly through every page Dan had visited — tattoos, information about Hideyoshi and the Sword Hunt, the Facebook pages of three sixth-grade girls —
"Hey!" Dan shouted.
"It's a keylogger," Alistair said, swiftly picking up the laptop. "Someone has hacked into the computer remotely and is spying on everything you've looked at today."
With a solid yank, he pulled the electric plug from the back, and the screen went dead. A steady beeping noise began, and an LCD panel by the light switch flashed red Japanese characters that looked suspiciously like some form of EMERGENCY.
"How did they do it?" Amy asked.
Dan took the laptop, examining the PC card. "It's an 802.11g wireless," he said. "So they have to be pretty close. I don't know, maybe like thirty yards — or fifty if they have a booster or something?"
Alistair headed for the window. "Which means either they're in the building or in one of those cars outside."
One of those _hundreds_ of cars, he could have said — if you considered the cars at the curb, in the nearby parking lot, and bumper-to-bumper on the road.
_Tap-tap-tap-tap!_
The rapping on the door made them all jump. "Is everything all right in there?" a tiny, timid voice called in.
It sounded like Ms. Nakamura, but there was something about her tone...
Alistair went to the door. "She may know how to run a trace."
"No!" Amy blurted out.
"Ms. Nakamura," Alistair said, yanking the doorknob, "your library has been compromised—"
The door flew open — and Alistair was staring into the massive chest of a gray XXXL T-shirt.
"No kidding, Sherwood," said Eisenhower Holt, with a grin so wide it nearly touched the edges of his stiff military brush cut. "Now, fall in line, all of you — and _march_!"
#
_Dzzzzz..._
Amy's phone was vibrating.
She glanced around the van. In the passenger seat, Eisenhower Holt was arguing with his wife, Mary-Todd, who was driving. In the next row, eleven-year-olds Madison and Reagan Holt were having a contest to see who could fling boogers into the hair of their older brother, Hamilton. Their pit bull, Arnold, barked greedily, snatching the small projectiles in mid flight with his massive jaws.
"Stop it, he'll kill me!" Ham cried.
"That's the point," Madison replied, clapping her hands.
"Sherwood is the _forest,_ dear," Mary-Todd was insisting to her husband. "The detective was Sherlock."
"We'll look it up!" Eisenhower declared. "May I remind you, Mary-Todd, at West Point my IQ was measured at nearly a perfect one hundred. Well, okay, eighty-nine — but I hadn't even practiced!"
"One hundred is considered _normal,_ sugar maple," Mary-Todd replied.
"Normal is the enemy of creativity," Eisenhower crowed. "A Holt is _never_ normal — as evidenced by our ingenious capture of the Cahills!"
_Dzzzzz..._
Amy moved her hand into her left pants pocket and pulled out her phone, making sure to keep it out of sight. On the right, she was jammed up against Alistair, who was seething with rage. He in turn was jammed up against Dan — who seemed oblivious, reading a handful of tourist pamphlets the Holts had left on the van floor.
Quickly she glanced at the call screen:
**GOMEZ, NELLIE**
She stifled a scream and stared sharply at Alistair and Dan, flashing the screen at them at waist level. Nellie was alive!
"YESSSS — WOO-HOO!" Dan blurted.
"The Cahill boy agrees with me!" Eisenhower said with a grin, turning around toward the back of the van. "Smart boy. You're going places. Like, with _us_ — as a captive! Har-har!"
The entire Holt family cracked up, except for Arnold, who seemed confused by the sudden absence of flying minitreats.
"Too bad you weren't smart enough to detect us following you every step of the way," Eisenhower continued, "with our patented Holt hackment technology. First we hacked into the tracking device on your cat — until we found that your cat was your uncle!"
Madison and Reagan looked at him in total bewilderment.
"Then we tailed him to the airport, where I ordered that we attempt the greatest technicological feat of all," Eisenhower went on, "breaking into the airline ticketing mainframe!"
"But then I reminded him that all he needed to do was follow your limo," Mary-Todd spoke up.
Madison chimed in: "Once we were at the other airport? And we saw you get into the jet? We just asked this really hot flight guy, Fabio? Where you were going?" She grinned. "And he told us."
_"Rawf,"_ said Arnold.
"Thusly," Eisenhower said, "we were able to commandeer our own flight to Japan, where we beat you to the airport and waited until you arrived, following your every move until our ultimate coup — keyloggering your very laptop to filch all your information! And now that I have you three, I can realize my life's goal. Not just reaching the thirty-nine clues first. But placing the Holt name where it belongs — at the very crest of the Tomas family... crest. No longer will history look upon the Holts as dolts. No longer will we be the black sheep, the stain on the family underpants, the smelly footnote to the Tomas legend. And you will help us achieve our destiny by leading us to the very thing your research revealed — the next clue, which is in the tunnels of Tokyo!"
"You figured that out all by yourself?" Amy asked, barely containing the relief she felt at knowing her au pair was okay.
"About fifty-three percent of it," said Eisenhower.
"More like forty-seven," Mary-Todd said.
"I knew that sounded wrong," Eisenhower said.
"Uh, Dad? _I_ was the one who did all that tech stuff," Reagan whined.
"Dad _what_?" Eisenhower commanded.
"Dad, sir," Reagan said.
"Your argument is as inane as your conversation," Alistair spoke up, his voice a barely controlled rasp. "You got nothing from hacking the system. You stole my map, you windbag."
"Uncle Alistair?" Amy said. She had never seen him like this.
"Is someone meowing back there?" Eisenhower said. "Do I hear an E- _kat_?"
_"Rawf?"_ said Arnold, suddenly drooling.
Alistair laughed defiantly. "What makes you miscreants think you can actually read that map correctly? It's written in Japanese."
"Ha! No grass grows over the head of a Holt!" Eisenhower thundered. "I overheard you through the library door, talking about an ancient underground crypt. So we will start at... the underground crypt district. _Harch!"_
The van lurched to the left.
Dan looked up from a map of the Tokyo subway system. His face was all lit up, the way it was whenever he'd broken a code or discovered a cheat in _World of Warcraft._ "Crypt? I think we're better off checking out the subway system."
The van lurched to the right.
"I have to pee," Madison announced.
The van skidded to a stop on the shoulder of the road. "Will someone make a decision?" Mary-Todd said.
As the Holts ramped up their various arguments, Alistair whispered to Dan, "Subway system, my boy? Do tell."
"First, I memorized your map," Dan began with a bit too much enthusiasm.
"Ssshhh!" Amy hissed.
"The secret tunnels and the subway," Dan barreled on. "They match — almost exactly! I'm figuring maybe the subway was built into the already existing tunnels!"
The Holts shut up at once.
_"Dan_ —" Amy warned. "You're telling this to _them_!"
Dan looked up, bewildered. "I was telling Uncle Alistair."
"But we-ee-ee heard you," Reagan sang in a taunt, and stuck out her tongue. "Besides, if you hadn't told us, you'd be dead meat."
_"Rawf,"_ said Arnold, baring his saliva-glistened incisors.
Dan's face turned white. He cast a guilty look at Amy and at Alistair, whose face had become suddenly cloudy. "Um, well... the thing is? It's not really a match. So I was wrong. 'Cause, um, there's this big difference. In the center of the old map, there's an intersection with a large room. In the subway map, the tracks are parallel. So... see? It must be the wrong place...."
"Where two maps diverge is precisely where the secret might be!" Eisenhower crowed.
"Brilliant as always!" Mary-Todd said.
Amy groaned. The stupider Dan became, the smarter it made Eisenhower Holt.
"Sweet," said Hamilton with a sneer.
Suddenly, Eisenhower spun on them with narrowed eyes. "Now, you're not trying to trap us, are you? We're not as dumb as we look. Or... whatever."
"Well..." Dan looked helplessly from Amy to Alistair. "There are subway stops on either side. I'm figuring the one at the northern end, Yotsuya, is closer."
"We'll take the one at the southern end," Eisenhower commanded.
The van lurched into the road again.
"Now I really have to pee," said Madison.
They waited silently until the train left Nagatacho station. They were the only ones on the subway platform now. The train schedule, which Alistair had picked up from the attendant, said that the next train arrived at 5:40. He checked his watch.
Then he looked down at the tracks — the _dark, narrow_ tracks that led into a pitch-black tunnel on either side.
"It's five-seventeen," he said, his voice shaking. "We have exactly twenty-three minutes."
Eisenhower stepped to the edge of the platform. "Fall in, troops!" he said.
"I want to go first!" Madison said.
"She kept us waiting when she was in the bathroom," Reagan complained. "Can _I_ go? Please?"
"It's almost Mom's birthday," Hamilton said.
_"Rawf,"_ said Arnold, diving over the edge to chase a soot-blackened rat that skittered across the tracks.
"Every Holt for himself!" Eisenhower cried, pulling green gardener's gloves from his pocket, snapping them on, and lowering himself over the edge. "Be sure not to touch the fourth rail!"
_"Third,_ hug-muffin," Mary-Todd said.
As Madison and Reagan followed, Alistair took Dan's and Amy's arms and stepped slowly backward. He was trying to escape. But Mary-Todd and Hamilton stood in his path, arms folded. "Uh-uh-uh!" Hamilton tutted.
"Nice try, Uncle," Dan whispered.
It was 5:19. Twenty-one minutes left.
With a sigh, Dan climbed onto the track, followed by Amy, Alistair, and the remaining Holts. A stream, inky black, ran between the rails. A gum wrapper floated by. Ahead of them, the tunnel plunged into blackness. Dan felt woozy. He and Amy hadn't had much luck in underground places. Images began floating in his brain. _Running... running... from Jonah Wizard in a subterranean museum in Venice... from the Kabras in the Catacombs under Paris... from a train... from a memory..._ He could still feel Amy's hand yanking him away from the approaching subway car in Paris, his backpack disappearing underneath the tons of speeding steel, the scream ripping from his throat. To anyone else, the faded snapshot he'd kept in that pack — the smiling couple — would have seemed blurry and uninteresting. But to Dan, it was as important as life. He had looked at it every day, memorized every last pixel. It was the only memento, the only remaining image of the parents he barely remembered. And now it was gone, a continent away.
"Hup-two-three-four..." Eisenhower called out.
Amy pulled Dan forward, shaking the memory from his brain. _Splash-splash-splash-thwuck,_ went his footsteps. _"Thwuck?"_ he squeaked.
"Don't ask," said Amy. Even in the near-blackness, Dan could tell her face was bone-white.
They trudged onward, keeping to the middle of the track to avoid the third rail, until the dimming light of the station behind them faded to nothing. "RePORT on proGRESS!" Eisenhower called out.
Dan's hands shook as he shone his pocket flashlight on the subway map. Ahead of them, the light of the next station was barely visible. They had passed the halfway point. "According to this," Dan said, "we should be about there now. The intersection would have been to our left."
"At ease!" Eisenhower said. "Fall in to examine hidden methods of egress!"
Amy reached out to her left, feeling along the grime-covered surface. "Nothing there but a wall."
"Keep trying," Eisenhower said.
Dan frantically pushed and punched, but the wall was solid. Thick cement. He checked his watch, which was already beginning to lose its glow-in-the-dark properties.
5:30.
"Th-this was a dumb idea," he said, his voice echoing dully in the tunnel. "Look, we have ten minutes. We left the station eleven minutes ago. We have enough time to get back before—"
"Abort mission!" Eisenhower barked. "Dress left! And... _hut-two-three-four_!"
Dan began to run, nearly tripping over his sister.
"Ow!" Amy cried out. _"Dan!"_
"Sorry!" Dan said. "See you at the platform—"
_"Dan, my foot is stuck!"_
Dan whirled around and shone his flashlight on Amy's crouching silhouette. She was grimacing, her left foot jammed under one of the rails.
"I'll rescue her!" Hamilton shouted.
"No, me!" Reagan shrieked. "I never get to rescue first!"
"Stand clear!" Eisenhower boomed.
_"Rawf,"_ Arnold barked.
Dan elbowed his way into the crowd, trying to reach his sister, who was screaming at the top of her lungs, _"You're only making it worse!"_
Dan's hair began to rise in swirls lightly from the back of his head. A low but steady wind was gusting through the tunnel from the south. Dan could see Amy's face looking up to him, eyes wide. "Dan? How accurate are those train tables?"
"I don't know!" Dan replied.
"When a train is entering a t-tunnel, don't you f-feel the air being pushed—?"
_Ho-o-o-o-o-o-o-o-onk!_
Dan spun toward the sound. Two distant headlights, like reptilian eyes piercing the darkness, were headed their way—and growing fast.
"Holts — _bolt_!" Eisenhower commanded.
As one, the Holt family turned away from the approaching train and broke for the next station at a dead run.
_"Don't leave us!"_ Amy shouted.
Dan pulled and pulled. Amy's foot was jammed. Tight.
_"OWWWW!"_
"I'll... get... it," Dan said through gritted teeth. He knelt in the icy trickle of water running between the rails, now choppy with the vibrations.
_"Run, Dan!"_
"Wait... I know..."
_The laces._ Dan dug his fingers into her shoelaces and yanked hard.
They were knotted. Wet and stuck. Her foot seemed glued to the shoe. If he could just slip it out, use the wetness to slide...
The screech of the brakes filled the tunnel. The wind whipped around him like a gale, throwing dust and debris into his eyes. His vision flashed white. His body was telling him to go. Now.
_"JUST RUN!"_
_"Stop it, Amy, I can't leave you—"_
She had saved him. He could save her. He had to do it.
_Pull!_
The wind was violent. The noise pressed into his ears like a solid thing. He pulled again, wiggled, jerked, pounded.
She was resisting now, pushing him away — trying to save him. Her breath felt cold on his neck, the veins in her throat bulging out.
He realized she was shrieking, but he couldn't hear a word.
_HO-O-O-O-O-O-O-O-ONK!_
Dan's body froze solid as he turned into the glare of the oncoming headlights.
#
_"AHHHHHH!"_
Amy didn't feel much at all. The wind. The metallic shriek of the brakes, the horn blocking out all sound.
She must have closed her eyes, because she didn't see anything, either.
She felt.
Her body wrenched upward and backward. She was flying.
And then her shoulder hit solid, cold cement. When she opened her eyes, all was dark and silent.
"I g-g-guess... I'm dead?" she heard her own voice call out, strangely high-pitched and thin.
For a long time, she heard nothing else. And then:
"Hi, Dead. I'm Dan."
The _ssshhhhick_ of a lit match sounded, and a tremulous light outlined two faces.
Amy sat up. Her left ankle ached, and her shoe was missing. "Uncle Alistair? Dan?"
Dan's hair was standing on end, his face shadowed with black dust and his eyes the size of softballs. "He did it. Uncle Alistair. He saved our lives. Reached out. From the wall. How—?" Dan tottered toward her, his body bent over as he gawked at Amy's foot. "It's still there. He didn't amputate it when he—"
With that, Dan's knees buckled and he crumpled to the floor.
"Dan!" Amy shouted. As she reached out to grab his arm, her ankle screamed with pain.
"It's all right," Dan said, sitting up. "I'm all right. Don't call nine-one-one. Did my hair turn white? Like in movies, like when people get really, really scared?"
"You're both safe now," Alistair said, moving his match around to outline the contours of a large room. "Dan, your hair is not white — and you were right about the hiding place. It is more or less where you thought it would be. There was a small graffito, an ancient-looking symbol on what looked like an electrical plate. Once I pressed it, the door swung open. I merely brought you both in with me."
Amy lunged forward on her good foot, keeping the other one in the air, and flung her arms around him. "Thank you," she said.
She felt him flinch. For a horrible moment she felt as if she'd done something terribly wrong. She could tell he was not the hugging type. Then, awkwardly, Alistair wrapped his arms around her. "I... owed you one," he said softly.
"Or two," Dan said.
Alistair nodded. "I suppose my record is not very good in life-threatening situations."
"Well, you've made up for it now," Amy said, burying her head in the shoulder of Alistair's silk suit jacket, which still smelled of aftershave.
Gently, Alistair pulled himself away, glancing downward with concern. "How does your foot feel?"
"Like it was jammed under a rail and then pulled out of its shoe," Amy said with a wince. "I can move it, but I think I sprained my ankle."
"Bet you can't tap-dance," Dan said, still sounding a little shell-shocked.
Amy smiled at her brother, never _ever_ having imagined she would enjoy hearing his stupid humor. She felt a rush of warmth for him.
"Oh, no, I see that look — _no_ hugs!" Dan said, backing away.
Numbly, he flicked on his flashlight and swept it slowly around the room, until the beam landed on a pile of old relics lying haphazardly on the floor and covered with thick, gray-black dust — clothing, strange tarnished metal things, a metal box, a globe, a hefty cylinder. As they all moved closer, Alistair remarked, "Well, the _yakuza_ may control some kind of underground network, but it doesn't look like they've been here in a few centuries."
"Hey," Dan piped up. "What does Jar Jar Binks say when he meets a member of the Japanese Mafia?"
Amy groaned. "You are recovering too quickly."
"I'm game. Wait..." Alistair paused for a moment and smiled. "'You, sah? _Yakuza_?'"
Dan's grin disappeared. "How did you know that? I just made it up out of my own head."
"Puns are a sign of intelligence, deeply buried," Alistair replied, putting on his white gloves. He leaned into the pile of stuff and gingerly lifted a small, brittle garment. "Hard to tell how old this is under so many decades of metal dust."
"Hey, check it out!" Dan said. He was unrolling a scroll that he'd pulled from behind a chest of drawers.
"Careful!" Amy said.
The scroll was open now, blackened around the edges but legible — three lines of stylized Japanese characters. "What's it say?" Dan asked.
Alistair looked closely. "It's a haiku, I think. Wait, let me get the meter right.... 'To find the new home/Of Hideyoshi's treasure/Use geometry.'"
"Treasure?" Amy said. "Does that include the swords?"
"We're rich!" Dan shouted. "Woo-hoo — I knew it! Okay, geometry. I'll get this one. Hang on, give me a minute...."
"It could be anything..." Amy said, gazing around the room.
"We're in a big room," Dan declared. "So... the volume of a parallelepiped, maybe?"
"Beg pardon?" Alistair said.
"A three-dimensional parallelogram, like this chamber," Dan explained.
"How would _that_ solve the problem?" Amy asked. "It's like trying to find a hypotenuse in a haystack."
"Is that a joke?" Dan said. "Because if it is, you should give me a signal. Like, tap your head twice so I know when to laugh."
He let go of the scroll with one hand. The _snap_ of its shutting caromed off the walls into the silence.
The dead silence.
Amy glanced around nervously. "Um, shouldn't there have been another train by now...?"
Dan reached in his pockets. "I can't check. I think I dropped my schedule on the track."
"I mean, just logically — wouldn't another train have come through?" Amy said. "If not this direction, then the other? The trains are pretty frequent, right? Why is it so quiet?"
Alistair stood bolt upright. "Point taken. They must have closed down the power. Which means—"
The distant thrum of voices now filtered in through the walls. It was coming from the north, from the track on the side opposite the one they'd used.
"Who's that?" Dan asked. "Police?"
Alistair's face suddenly looked aged and puckered. "No," he replied, his voice shaky. _"Yakuza."_
"What do we do?" Dan asked.
"They can't find us, right?" Amy said. "So we stay?"
Alistair took Dan and Amy by the arm, pushing them toward the door. "They will eventually cross the tracks, see the lost shoe, the dropped schedule, the smudges worn off the wall plate. We must go."
"Cube!" Amy blurted, suddenly breaking loose and racing back to the pile. "Look! Sphere! Cylinder! Para — parallelowhatever! Those are geometric shapes — right, Dan? _They're right here!"_
Dan was already grabbing the globe, stuffing it into his backpack. "Take them all!"
"Quickly!" Alistair said. He took a small cube in one hand, a triangulated tube in another. Amy scooped up the long cylinder and headed back to the door.
In a moment, they were out on the track again. Alistair pushed the thick door closed behind him. Where a seamless, grime-darkened wall once stood, there was now the faint outline of a recently opened door.
The train that had almost hit them was now stopped beyond them, its rear cars not yet having reached the next station platform.
Amy pulled her shoe from under the rail and jammed it on her foot. She stumbled, her ankle throbbing with pain. But the thought of stopping petrified her. Clenching her teeth, she ran. They flew down the track, back in the direction they'd come. The station soon came into view, but the track was dotted with flashlights, beams moving around like fireflies.
They all stopped, their ragged breaths echoing in the tunnel.
"Police," Alistair whispered. "We cannot let them find us. They will arrest us."
The lights were getting closer, the voices louder. From the other side, it sounded as if the _yakuza_ had switched over—to their side of the tracks.
"And the _yakuza?"_ Dan asked.
"They'll kill us," Alistair replied.
"That's a no-brainer," Dan said, heading toward the cops.
"No!" Amy grabbed him by the arm.
"Where do _you_ suggest we go?" Dan hissed.
Amy looked up. The bottom rung of a ladder hung just above her head.
"We must take the objects," Alistair said. He quickly removed his silk jacket, laid it flat, placed the objects on top, then gathered the jacket edges upward. Dan took a rope from his backpack and tied a knot, making a secure container.
Amy was already climbing, grimacing against the pain. Dan put the other end of the rope between his teeth, grabbed the ladder, and pulled himself up.
Below him, Alistair was gawking into the darkness, one hand on the ladder and the other clutching his walking stick. "Come on!" Dan shouted through his teeth.
"Just _go_!" Alistair cried.
Footsteps thudded against the track. A man appeared out of the darkness, his soot-covered face allowing only his teeth and eyes to pick up the light — until Dan noticed the gleam of a dagger in his right hand.
Now Alistair was moving. He was on the second rung when a guttural shriek rang out. _"HEEE-YAHHHHH!"_
Dan looked down to see the _yakuza_ blade slicing through the air—at Uncle Alistair's legs.
#
_"Watch out!"_ Amy screamed.
_"Hunh!"_ Alistair gasped, heaving himself upward.
_Clanggggg!_
Dan felt the ladder jolt. He hung on tight, staring at the amazing sight below.
With a sharp, precise movement, Alistair had brought his walking stick down hard, knocking the blade from the attacker's hand. Then, on the backswing, Alistair caught the _yakuza_ on the side of the head, sending him spinning downward to the track.
"Get out, Dan!" Alistair commanded, shouting upward.
"How did you learn to do _that_?" Dan asked.
"I'm full of surprises — now _move_!" Alistair said.
Amy had managed to push aside the grating at the top of the ladder. Dan scrambled to the street, pulling up the objects behind him. A moment later, with a loud grunt, Alistair heaved himself up onto the sidewalk. A mother pushing a baby in a stroller swerved around them. Dan quickly began shoving the grating back, getting it over three-quarters of the hole before Alistair yanked him away.
"No time for that!" he said, pulling Dan with him as he stepped into the street.
"Wait!" Dan protested. "What about Amy?"
Amy was trying to catch up, limping over the curb in their direction.
_Shiiiink... SHIIIINK!_
Soot-stained fingers, reaching up from underground, were sliding the grate open.
"Pardon me, please," Alistair said, rushing over to the hole. Like a golf pro, he drew back his walking stick and swung it down toward the fingers. Hard.
_"AAAAAAAGHHH!"_ came a tortured shout.
Dan heard the thumping of multiple bodies hitting the ground below the ladder.
Alistair knelt, his back to Amy. "Climb on."
She leaped onto him and he locked his arms under her knees, grimacing as he limped across the street behind Dan. Their shadows were elongated in the setting sun, making them look like some misshapen beast.
_HONNNNK!_
A car swerved out of the way, its driver shouting at them.
"The objects—" Alistair called out through gritted teeth. "Drop them in that alleyway. We will come back for them!"
Dan spotted a dark, narrow space between buildings and threw Alistair's tightly packed jacket into it. They raced around the corner, up a hill between low brick buildings where the smell of soy sauce and fried shrimp belched out of ground-floor windows in steamy wisps. Alistair darted right at the top of the hill, into the open gate at the back of a vast, empty playground. "Where are we going?" Dan cried out.
"I have friends!" Alistair said. "All we need to do is get a taxi—"
As if by magic, a cab sped toward them up the street. Alistair let go of Amy with one hand and waved frantically, shouting in Japanese.
But as the taxi swerved at him, it picked up speed, its engine roaring.
_"Look out!"_ Dan screamed.
Alistair jumped away. Amy went flying off onto the blacktop as the cab hopped the curb, missing them by an inch. It squealed to a stop and spun around.
At once, all four doors opened.
_"Yakuza!"_ Alistair shouted.
Now even Amy was moving fast. As Dan ran after her, he heard a high whistling sound. _"DUCK, AMY!"_
A jagged-edged silver metal disk sliced the air. It whizzed over Dan's head as he leaped for his sister, grabbing her by the waist.
She screamed as they tumbled to the ground again.
"What was that?" Amy gasped.
"A _shuriken,"_ Dan shouted. "A ninja throwing star!"
"This way!" Alistair cried out. Dan felt the old man's hand clasp his wrist, yanking him upward. In a split second they were racing into a large steel tunnel, part of the playground.
_Thunk! Thunk! Thunk-thunk-thunk-thunk-thunk!_
Dan flinched as each throwing star hit the outside of the tunnel, inches from their heads.
They emerged from the other end into a complex of thick wooden climbing equipment. Alistair was running, crouched, his head low and his walking stick tucked under his arm. Splinters flew around their heads like hailstones.
Barking, angry Japanese instructions rang out behind them. Car doors slammed. Tires screeched. Dan, Amy, and Alistair ran blindly out of the playground, across a lawn, into a backyard, over a small fence. _"Yeowww!"_ Amy shouted, her foot jamming in the fence's links.
"Keep up!" Alistair retorted.
The throwing stars had stopped, Dan realized. The _yakuza_ wouldn't use them in a residential neighborhood — would they?
They emerged at the other end of a block, this one with a line of stores on either side. To their right, Dan could hear a speeding engine. "Go left!"
The street sloped downward to a big, open market area. Inside, vendors were packing up, cleaning out stalls. Dan realized that he, Amy, and Alistair could get safely lost in there. The _yakuza_ would be asking for chaos if they followed.
_VRRROOOOOM!_
Dan stopped in his tracks. A red Porsche was turning into the road in front of them. Blocking their way to the market. Rounding the corner, the Porsche flashed its brights. Dan cowered, momentarily blinded.
Grabbing his sister, he sprang away from the street. "Jump — _jump_!"
They leaped onto the sidewalk, rolling past a metal mailbox as odd noises rang out. _Thwip! Thwip-thwip!_
Shots flew by them, up the hill from the Porsche to where the _yakuza_ taxi was now bearing down on them.
_Smash!_
One of the taxi's headlights popped.
_Thwip-thwip!_
A projectile cracked the taxi's windshield. The taxi began to skid to the left, whirling. Its tires hopped the curb — and the car's broad left side hurtled toward Dan, Amy, and Alistair.
Amy screamed. Or maybe it was Dan himself. He couldn't tell. He was only aware of flying through the air. His head banged against the side of the building as a flash of yellow steel rolled by him, massive and dented.
With a sickening crunch, the taxi crashed through the plate glass window of a closed flower shop.
It came to rest on a bed of broken bouquets and shattered glass, its wheels in the air. Two men groggily worked their way out of the wreck, stumbling for a few seconds while they gained their bearing. Dan, Amy, and Alistair huddled together in the shadows, but the men ran up the hill, looking in dazed fear over their shoulder.
"What just happened?" Amy said.
"We were in a ninja fight," Dan said in amazement. "For the first time in my nonvirtual life. And I hated it."
A din of voices swelled from below as people from the market began walking up the hill to join the other gawkers, who descended from all sides.
Dan slowly stood. The Porsche was partially blocked from view by the mailbox, but Dan could see its gleaming mag wheels and tinted windows. "If they hadn't saved our butts..."
"Be careful," Alistair warned.
Suddenly, Dan heard the doors fly open. He froze.
_"Mrrp?"_
The tiny cry rang out. Dan's heart thumped as a silky animal grazed his ankle, and he glanced down at an Egyptian Mau that looked identical to Saladin but for the slightly mangy coat.
"Oh..." Amy said with a wistful smile.
"That looks just like you-know-who," Dan said.
The Mau was slinking over to Amy, who held out her arms to it.
"The breed is very popular around here," Alistair replied absently, his eyes still riveted on the Porsche. "Is anyone... alive in there?"
In reply, a figure staggered around from behind the mailbox. Dan's breath caught in his throat.
"Next time, dudes, hold on to your tickets," said Nellie Gomez.
#
Dan gaped, ignoring his sister, who opened and closed her mouth in a stunning imitation of a spotted blowfish.
_"Mrrp?"_ said Saladin.
_"Awwwweso-o-o-o-ome!"_ Dan didn't care who heard him scream. He scooped up Saladin and threw his arms around Nellie.
Amy looked as if she'd just seen a ghost. But to Dan, Nellie felt real, all right. She felt all... spiky and leathery. And a split second later, Amy was all over her, too. Sobbing of course. Which made Nellie sob, too. Which almost ruined the whole thing. Even Alistair was a little teary.
Saladin climbed into Amy's arms, and she smiled disbelievingly. "But how did you...?"
"Find you?" Nellie laughed. "It was all over the news — subway shut down, people on the tracks... I'm, like, _ding_! Amy and Dan, no-brainer!"
"Where'd you get the cool car?" Dan added.
"Where did those shots come from?" Amy asked.
"Where's the duffel?" Dan prodded.
"How did you escape the Kabras?" Alistair said.
"Whoa, whoa, whoa," Nellie said with a laugh. "I need backup!"
Behind her, two shadowy figures stepped out of the car. "She didn't escape us," said Ian Kabra.
"Far frub it," said Natalie in a stuffed-up voice.
Dan felt his blood run down to his feet. Amy gripped his arm.
"We just survived a ninja attack," Dan reminded her. "Remember, there are two of them and four of us."
_"Mrrp,"_ said Saladin.
"Sorry, five," whispered Dan.
"Ah- _chooo_!" Natalie sneezed. "I hate cats."
_"ATTACK!"_ Dan yelled.
Ian grinned patiently at him — and brandished a sleek stun gun.
_"Dan — duck!"_ Amy screamed.
"You asked about the shots?" Ian said. "This is your answer. You survived the _yakuza_ because of my handling of this weapon. And because I insisted on renting a swift precision car, _not_ the beige Chevy Cobalt your babysitter wanted."
"Id case you dod't udderstadt, you borons, we saved you," Natalie said. "Ah- _chooo_!"
"But... why?" Dan said. "You hate us."
"Dat's true." Natalie exhaled wearily.
"Yo, Nat? Take your allergy meds, okay, so you're not spraying me in the car?" Smiling at Dan and Amy, Nellie grabbed the driver's door handle. "Get in, all of you."
"But—" Amy said, glancing reluctantly toward the Kabras.
"We have to move before the _yakuza_ return," Nellie said. "I'll explain everything. Oh. And we squeezed your duffels in the trunk."
_Yes!_ Dan thought. That meant they had the swords. Dan climbed into the soft leather backseat with Amy and Ian as the others squeezed into the front. "Whoa, _this_ is street cred," Dan said. "Can we keep it?"
"We left some... outer clothing near the subway station," Alistair said carefully. "Perhaps I can direct you there, Nellie."
"Seat belts!" Nellie commanded. She started the car, pulled away from the curb, and floored it to get through a yellow light. Alistair pointed her to the right as she continued: "Okay, update. When I see Poindexter and Morticia on the plane? I freak. I'm, like, what happened to my kids? I think, they've, like, _eaten_ you. Then they tell me what happened. _Bragging._ They're, like, fourteen and eleven, but they talk like they escaped from a Clue game. 'Forged the tickets, ho-ho!' — Anyway, they try to threaten me, yada yada, and of course I argue and I'm figuring in my head, 'Ha-ha, the next thing is they put poison in the drink' — but I'm, like, 'Nahh, of course they're not _that_ skeezy.' Then I see her actually doing it, like _two inches away from me_ — uh, hello? So I get kinda mad—you know, act like I'm going to drink it, and then _zam,_ I spritz the stuff all over their faces. Well, I'm, like, 'Nyah nyah, this is really funny,' but they start to totally wig out and fall all over each other to get to their carry-on — they're, like, _'Eek our faces have schmutz on them!'_ And I'm, like, _'Dudes, grow up!'_ So I take their carry-on and sit on it. Um. Baad idea."
"The poison was in a concentrated form," Ian spoke up. "In the amount that Natalie had used, it would have mutilated, perhaps blinded us."
Amy squirmed away from him in disgust, nearly squeezing Dan into the side of the car. "And you were going to let Nellie _drink_ it?" she said.
"We meant to temporarily disable her," Ian said. "Just a drop. But Natalie slipped during air turbulence. Before we could warn your nose-ringed nanny, she drenched us. Luckily, she allowed us to retrieve the antidote from our carry-on."
"That's kindness," Amy said.
"I made them agree to give me all their cash," Nellie explained.
"That's bribery," Natalie grumbled.
Nellie jerked the car to the right, and Dan felt like he was going to carry an Amy imprint on him for the rest of his life. Out of the corner of his eye, he saw Amy's hand brush accidentally against Ian's. She let out a yelp and pulled it away.
_"Mrrp!"_ said Saladin, arching his back and spitting at Ian.
"Er, ahem," Ian said, leaning away from the cat, "the reason we are still here is that we'd like to propose a temporary alliance. As we explained to your porcine au pair, we have something you need."
"Like, two plane tickets?" Dan said. "Too late. And we'd rather have an alliance with a bucket of slime than with a Kabra — if we could tell the two things apart."
"Fine," Ian said. "We will use our artifact to find the clue ourselves—"
Alistair turned around toward Ian. "Artifact?"
"How refreshing, an open mind," Ian said with a sly smile. "As you well know, Mr. Oh, the Lucians have been collecting hints for years. So have the Ekats. And presumably so have... er, what branch _do_ you belong to, Daniel?"
"The Cahills," Dan snapped. He hated that he and Amy were the only ones who didn't know their branch. "And you're crazy to think we'll work with you."
"Dan, they saved our lives," Amy said.
"They also tried to kill us!" Dan countered. "In the cave-in at Salzburg, in the canals of Venice—"
"There — you see how things change?" Natalie said brightly.
"Our... item once belonged to a Japanese warrior," Ian said. "It will be crucial for finding the next clue. Alas, neither Natalie nor I understand Japanese. Which is where you, Mr. Oh, come in." He leaned closer to the front seat. "You give us what you know. We'll give you what we have. We'll work together."
"Just for this one clue," Natalie quickly added. "Afterward, we cut you loose. We have our reputation to think of."
"Stop here," Alistair said to Nellie.
_Screeeee!_ The Porsche squealed to a stop at a desolate corner.
"How do I know we can trust you?" Alistair demanded.
"W-we already know we c-c-can't," Amy said.
Ian grinned, reaching into his pocket. He pulled out a small velour bag stamped with the Kabra coat of arms and placed it in Amy's left hand. "This is yours, Amy Cahill. Now... how do we know we can trust _you?"_
A coin.
A stupid gold coin with a symbol on it — that's how the Kabras were buying their trust. Alistair had read the Japanese print on the back and claimed it might have belonged to Hideyoshi — _might._ Dan couldn't stand it. Collaborating with the Kabras was like kissing your sister. Well, maybe not that bad.
"The coin is so beautiful," Amy whispered, as they rounded the corner toward the alleyway where Dan had tossed the objects. Just ahead of them, Uncle Alistair was filling in Ian and Natalie on what had happened in the subway.
"It's a token for the arcade games at Laser Sport Time!" Dan hissed.
"Uncle Alistair doesn't think so," Amy murmured. "He's a numismatist."
"He takes his clothes off in public?" Dan said.
"It means _coin collector_! Besides, I sense Ian's telling the truth."
"That's because he touched your hand and did a mind meld."
"Shhh!" Amy said, as Ian glanced toward them.
The late-afternoon sky was a bruised purple as they reached the alleyway across from the subway stop. The silk jacket was still in the corner like a discarded old bag. Despite the near darkness, Dan could read the familiar look on Amy's face.
_Sorry to embarrass you in front of your boyfriend,_ he thought.
Alistair knelt to pick up the cube container. "Move quickly," he said.
With a reluctant sigh, Dan struggled to remove the rusted top from the cylindrical container. Beside him, Alistair flung aside the cube with disgust. "Nothing in here but lizards."
As he reached for another container, a long black car pulled to a stop across the street. A black-uniformed man got out of the driver's seat and ran around to open the passenger door.
Dan crept forward in the shadows to watch. An Asian man, rail-thin and elderly, climbed out. His silver-white hair flowed past his shoulders, and he was dressed in an elegant dark suit with a silk pocket handkerchief. Walking along the sidewalk, he flipped open a cell phone as he knelt by the subway opening and peered inside.
Dan tapped Amy on the shoulder.
He heard Uncle Alistair let out a gasp and mutter something under his breath that sounded like "Bye."
"Bye?" Dan said, as Alistair suddenly pulled him farther into the shadow.
The old man headed back into the car, and it quickly drove out of sight.
"Who was that?" Dan asked. "The king of the _yakuza_?"
"We..." Alistair's voice seemed to catch in his throat. "We need to hurry. Open all the containers. _Now_."
With a grunt, Dan finally managed to yank the top off the cylinder, releasing a stream of nuts, bolts, screws, and rivets.
"Fascinating..." Ian spilled tools from the rectangular box. "I adore hammers."
Alistair exhaled with frustration. "That room we found may have been a subway storage facility sealed off during construction years ago and forgotten."
"But what kind of subway workers leave mysterious haikus?" Amy asked, tentatively prying the top off the triangular-shaped tube.
"Maybe they're really songs," Dan said with a weary smile. "'Cause, you know, those guys work on _tracks_ all day...."
"Hey, look!" Amy moved into the streetlamp light, pulling a long scroll out of the tube. As the others gathered around, Dan trained his flashlight on the text at the center. It was written in dark, elegant calligraphy, surrounded by a faded, unfinished-looking landscape of a rock outcropping and hills.
Alistair began translating: "'In the place of the final conquest, between three horns lies the people's wealth. And by the elements united is entrance granted, the highest to be revealed.'"
"Clear as wasabi," Dan remarked.
"Those letters right below," Amy said. "They look... _English_."
Dan grabbed the flashlight from his backpack and beamed it on a group of plain, thickly drawn letters at the bottom of the scroll:
"Toota?" Ian said. "Could it be a phonetic spelling of the French _toute_?"
"Good, Ian, good," Dan said. _"French,_ on a Japanese scroll."
"'Final conquest'..." Alistair murmured. "That's it! That's the key. I know where the clue is!"
"Where?" Amy and Dan asked at the same time.
A smile crossed Alistair's face for the first time all day. "The place where Hideyoshi mounted his final campaign and suffered his most humiliating defeat!"
"Right," Ian said uncertainly. "Of course. And... that would be...?"
"We are going home," Alistair said, his eyes ablaze. "To Korea."
#
_Bae._
The name, once so important in his life, now consumed Alistair with rage.
His uncle Bae had been so close. Across the street!
_It wasn't the right time,_ Alistair reminded himself. He would have to wait. To plan.
He turned in his seat to check on his flight companions. The Kabra siblings were absorbed in an old episode of _The O.C._ on their personalized seat-back flight screens, and the Cahills were doing the airline magazine crossword puzzle.
Quietly, he unfolded the printout from the library. In his lifetime, Alistair had spent a fortune on private investigators looking for the man who had taken everything from him. Now the man's identity had been discovered. He had died of old age — a respectable elder who had secretly built his fortune on contract killings and kept a record of each one in a private vault. Apparently, he had kept everything.
Alistair spread the printout on his tray table. Fingers shaking, he read it for what seemed like the hundredth time:
Alistair forced himself to read it, fighting back nausea and anger. _Five thousand dollars._
His father's life for five thousand dollars.
The details of what had happened in New York were hardwired into Alistair's brain. He still carried around with him the tattered, yellow news clipping of the murder: _New York City, May 12, 1948: Korean industrialist Gordon Oh was killed at the intersection of Madison Avenue and 45 th Street while being driven to the theater._
Here is what the newspapers all said: There had been a break-in at Brooks Brothers, an alarm, the desperate robber running up the avenue with a gun and trying to commandeer a car stopped at a red light — his father's hired limousine. Mr. Oh had tried to subdue the man. He had struggled nobly but tragically lost his life. The gunman had slipped away and was never found.
His dad had been in the wrong place at the wrong time. An unlucky accident.
That was the official report.
As a child, Alistair had never suspected foul play. But sometimes accidents were planned, and killers were hired. He had always been afraid of his uncle Bae Oh, his father's twin brother. Bae had grown up the lazy twin, the greedy slacker, repeatedly passed over for the head of the Ekaterina family, always under the shadow of the robust and well-loved Gordon. As an adult, Bae was all about foul play, as ruthless as a Kabra in his business dealings.
Bae craved glory and riches — and the 39 Clues. Whoever stood in his way had to go. Even his own brother.
It didn't matter that Gordon had a wife halfway across the world whose grief was so great that she needed to be hospitalized. Or a four-year-old boy whose heart that day had been ripped out.
A boy who was packed off, crying and alone, to be brought up by a man with a heart of glass, to be ignored and mocked his whole life.
His uncle Bae Oh. The murderer's boss.
Alistair glanced at the Cahill children. They were squabbling over the answer to a crossword puzzle now, the argument turning into a joke, the boy making up a nonsense word — a babble of nonsense words, a burst of laughter. Even now they sounded as they did eleven years ago, as a newborn and a three-year-old. Back when Alistair had made his promise to Hope and Arthur. A promise that had been nearly impossible to keep.
The children wouldn't remember, of course. But he did. And now the couple was gone, for the same reason he'd lost his own parents. For the Clues.
He sighed. At least the children had each other.
All Alistair could hope for was revenge.
His fingers shook as he folded the sheet and put it back into his pocket. On this flight, he knew he would not sleep.
#
Rumor had it that Alistair Oh was broke. His business had not succeeded. But when Amy saw his mansion, in a village outside Seoul, South Korea, she began pondering recipes for cheesy burrito fillings.
"Dang! Whose palace?" Nellie blurted, as the limo pulled to a stop after their brief drive from the airport.
A pristine white building stood gleaming atop a sloping, luxuriant lawn. The front path was decorated with orange and yellow chrysanthemums and led to a small grove of cherry trees and dogwoods, their leaves rustling in the breeze. Just being here, in this setting, made you want to smile.
"Where is the main house?" asked Natalie, as they climbed out.
_"Voilà."_ Alistair gestured dully toward the mansion. All day long he had seemed tired to Amy, a little off his game.
"You mean, behind the pool house?" said Natalie.
Ian elbowed her in the ribs.
"My home is one of the few remaining perks from my days as a burrito magnate," Alistair said, as he walked up the drive. He was flanked by the Kabras and the driver, who held Dan and Amy's bags. "As is Mr. Chung, my driver. And Harold, my butler. Our cozy little team. Things used to be a bit more grand."
"Ah, well, easy come, easy go — not that I actually know the feeling," Ian replied. "The house does have lovely... um, window moldings."
"Thank you, they were imported from South America," Alistair replied.
Dan leaned into his sister as they followed behind the others. _"Window moldings?"_ he muttered. "What normal fourteen-year-old guy talks about window moldings?"
Amy shrugged. "Did you check the duffel?" she whispered.
"Yes," Dan replied. "Rufus and Remus are still there."
Amy took off her shoes and stepped onto the soft, freshly cut grass. A cool breeze tickled her nose and she burst out laughing, windmilling her arms as she spun in the grass on her good ankle.
"Oh, great, I'm in South Korea and my sister has turned into Julie Andrews," Dan said.
Oops.
Amy dropped her arms. Now everyone was ogling her. She felt like she did in ballet class. Dumpy and uncoordinated and ugly. She glanced down into the clover, as if by staring hard enough she would disappear inside it.
"What your sister is doing," Ian said, walking toward the house, "is called _enjoying herself._ Maybe you could learn from her, Daniel. It's refreshing."
"Refreshing?" Dan said. _"Amy?"_
Amy stuck her tongue out at her brother. Ian was smiling at her, which made her stomach churn, but she managed a smile back. Just to get on Dan's nerves.
She stayed a few steps behind Ian as they crested the hill, until Amy could see the back of Alistair's house. It extended to a large sundeck overlooking a swimming pool and a vast lawn. To one side of the lawn, a stream wound its way through a landscaped rock garden that contained a pond stocked with goldfish. To the other side was a thick hedgerow that seemed to go on forever.
"Tempting Tempura microwavable burritos bought me this," Alistair said, gesturing over the landscape. "Mostly beef."
"Tranquil." Natalie nodded. "It really is amazing what you can do with a limited amount of space."
Alistair raised an eyebrow. "Nothing like the Kabra estate, I hear."
"We hated growing up there," Ian replied. "Every year one of us would get lost on the lush grounds, and they would have to send the homing poodles after us."
"The _what_?" Dan said.
Natalie sighed ruefully. "Some say it was an abusive childhood, but we didn't know any better."
Nellie emerged from the house. Behind her was a uniformed butler with six soft drink bottles, which he set on a table next to six wooden chairs. "Thank you, Harold," Alistair said as the butler bowed and went back inside. "If Toyotomi Hideyoshi had had his way, this land would be Japanese. He intended to conquer all of East Asia, and he had never failed. Some say he meant to build his grandest palace here in Korea, to give birth to an heir who would take over the kingdom. He would also build great vaults and hiding places. Hideyoshi was one of history's most notorious collectors—"
"I knew I liked the guy!" said Dan.
"According to family lore, he collected the most valuable item of all," Alistair said. "One of the clues to the Cahill family secret, which we still seek five centuries later." He sighed. "No Ekat has ever found it. No one suspected it was in Korea. But our parchment will lead us there, if we know how to read it."
"Dude, I'm on it," Dan said. "How do we get started?"
"Unfortunately," Alistair said with a yawn, "I cannot function after a rough, sleepless flight. Will you kindly grant an old man a half-hour nap in his own bed? Harold will feed you until then. Please stay close by and do not wander."
"Of course," Amy said.
Alistair waved to them and walked into the house.
"Food, drink, magazines, TV, TiVo, Internet, handheld devices?" Harold asked.
_"Warcraft?"_ Dan blurted.
Harold smiled. "Second door on the right."
As Dan scooted inside, Natalie settled into a lounge chair with the Korean edition of _People_ magazine and Nellie worked her iPod.
Ian was staring out into the yard. "What's that?" he asked.
"Wh-wh-what's what?" Amy said.
He pointed to a dense hedgerow that contained a narrow gap. "Is it a hedge maze? Come on, let's go and look."
"I — I don't think so."
"Why not?" Ian said. "What else are we going to do?"
He had this funny look on his face, Amy thought. A curious smile, as if she had just refused an ice cream sundae or the winnings to the lottery. Like it wouldn't have occurred to him that anyone could possibly say no.
"Alistair t-told us not to wander," she explained, shoving her hands in her pockets.
Ian cocked his head teasingly. "I thought you were a brave explorer."
"Oh, p-p-please..." Amy said, trying to drip with sarcasm but fighting the tingle that was spreading up from the back of her neck.
"Well," he said with a shrug. "Your choice."
As he walked away, Amy lurched forward but stopped herself.
_What am I doing?_ she thought.
He was a jerk. He was jerkier than a jerk. He was a new definition for jerk. She didn't have to follow him.
Her fingers closed around the coin he had given her. She pulled it out of her pocket and flipped it in the air. "Heads I f-f-follow, tails I stay."
The coin landed with the odd symbol facing upward. But was that a head or a tail?
Ian sighed disappointedly. "Ah, well, my loss..."
As his hair, glinting in the sun, disappeared beyond the hedgerow, she turned and trudged into the house.
_"AAAAGGHHHHH!"_
At the sound of the scream, Alistair barged barefoot out of his room. He raced past Amy, who was being served orange juice in the kitchen.
She followed him outside, with Harold and Dan close behind.
In the distance, Amy heard a violent growl, a rustling from the hedges. Ian burst out of the opening, one shoe missing, running at top speed. "HELLLLLP!"
Behind him was an enormous dog, a mutt that seemed to be part pit bull, part Great Dane, and from the looks of it, possibly part black bear.
"What the—?" Alistair said. "STOP! SIT!"
"I can't sit! He bit me on the bum!" Ian shrieked.
_"Really?"_ Nellie said, grinning.
Alistair was limping onto the lawn now, waving his finger at the beast, which hung its head sheepishly and whimpered. "Is this how you greet me on my return, you naughty thing?" Alistair scolded. "Bad dog! Bad, _bad_ Buffy!"
_"Buffy?"_ Dan said.
_GRRRRRRR._
"Shhh, she's sensitive about her name," Alistair replied.
_"I'll sue!"_ Ian sputtered. _"I'll sue you AND the dog. And the country of South Korea. And... and..."_
"The landscape architect?" Natalie said.
_"The landscape architect!"_ Ian shouted.
"Buffy is actually a real pussycat," Alistair said, eyeing Ian suspiciously, "unless you surprise her."
_"Flowrf! Flowrf!"_ barked Buffy, throwing a spray of saliva left and right.
"She is sooooo cute!" Nellie said.
_"These are handsewn Persian silk!"_ Ian turned around, revealing a tear in his pants that exposed boxers with pink dollar signs on a white background, then quickly spun back around. "Uh, never mind."
"Sweeet," Nellie said.
"Shut up," Natalie snapped, barely stifling a laugh herself.
"I fail to find the humor in this!" Ian shouted, his eyes red with rage and embarrassment. "And neither will you. I will drain you, Alistair. I will bring you to your knees—"
"Young man," Alistair interrupted sharply, "I am too old and too wise to be intimidated by a fourteen-year-old boy who wakes me from a much-needed sleep by his foolish actions. Why were you snooping in my hedges when I told you not to?"
_"Who ever heard of planting a guard dog in the middle of a hedge maze?"_ Ian snapped back. "What is back there, Alistair? What are you hiding?"
Alistair cleared his throat. Pulling a comb from a pocket, he tidied his hair as if he were about to go to a business meeting. "I suppose," he said, "we will have to do this now. Perhaps, Mr. Kabra, you would like to change." He called over his shoulder. "Harold, please apply some disinfectant to the young man's wounds."
Ian went pale. "I'll do it myself," he said, heading into the house.
Nellie flopped back in a lounge chair, her face covered with sunscreen. "Wake me when it's over."
As they tramped through the hedge, Amy could see the hurt in Ian's eyes. He was wearing a pair of Harold's uniform pants now, which were a couple of sizes too large.
"These itch," Ian grumbled.
"No spare pants in your carry-on?" Dan said. "Bummer!"
Cackling, Dan scampered ahead. Ian turned toward Amy, trying valiantly to smile. "I meant, the bite marks itch. Not the trousers."
She fell in step beside him. "He—he — should have—" The harder she tried, the worse it felt. The words were like volleyballs stuck in her throat.
"Alistair should have warned me?" Ian said. "Thank you. My feelings exactly."
"Uh-huh," Amy replied. _Talk much?_ she thought. She clutched her jade necklace, fiddling furiously with the chain.
" _You_ warned me, though," Ian said softly. "I should have listened."
"Well, um..." Amy said, feeling suddenly as if the temperature had shot up ten degrees.
Ian laughed. "Oh, well. I suppose it will only hurt when I sit."
Amy fell into step beside him, watching his footfalls land on the grass, counting how many steps she took compared to his. He had a strong stride.
Before long, they reached the others. Alistair had stopped before a section of hedge and was groping around inside it.
Dan was glaring at Amy.
_What was that about?_ his face said.
Dan glanced accusingly at Ian. Before he could glance back, Amy turned away.
She could read his mind anyway. She hated when he was right.
Alistair was now clearing a section of brush to reveal a door with a round, cast-iron hatch. The Kabras, the Cahills, and Buffy all gathered around gawking, except for Buffy, who drooled.
On the hatch was the number 5005. Beneath that was a heavy latch and a circular dial engraved with numbers from 1 to 30, like a combination lock.
"This, my children," Alistair said proudly, "was barbecue pork."
Dan rapped his fingers against the latch. "Been out in the sun a long time."
"I mean, sales of my barbecue pork burritos bought this," Alistair said. "The combination involves four numbers, and all the information you need to know is here. You get three tries. I can give you one hint — but that will use up a try."
Ian frowned. Amy could see the wheels turning in his head.
She took a deep breath. 5005. There was something about that number.
"The number is a palindrome," Ian said, "the same backward and forward. That may mean something."
"It's two-zero-zero-two upside down," offered Natalie.
Dan exhaled loudly. "Rich doesn't guarantee smart. It's, like, so obvious, dude."
"Pardon?" Ian said.
"Don't overthink — Uncle Alistair said we have everything we need to know!" He spun the numbers 5, 0, 0, and 5, and then pulled the latch.
Stuck solid.
"That was Try Number One," Alistair said.
Ian glanced at Dan. "Maybe thinking isn't such a bad idea."
"I think we need the hint," Natalie said.
"Very well," Alistair said. "It is a riddle: Why leave the factory when the workers are in their prime?"
The question hung in the air. Amy's mind raced.
"Prime..." Dan said, his face worked into a prun-ish mask of concentration. "Okay, what's the prime of someone's life? Like, twenty-one years old? Maybe one of the numbers is twenty-one!"
As he reached for the dial, Alistair said, "Remember, you have only one chance left. If you fail, I cannot let you in."
Dan's hand froze. "Come on, guys, help me out here. Twenty-one and...?"
"Well, when does a worker leave a factory?" Ian said. "This, I'm afraid, is beyond the Kabra experience."
"Twelve noon for lunch?" Dan said. "And five o'clock to go home. So... twenty-one, twelve, and five?"
"No!" Amy blurted out. She wasn't sure, but the hint was a lot like the ones in the Puns and Anagrams puzzles that Dan used to do in the Sunday _New York Times._ The hint was partially hidden in the wording—you just needed to know how to read it. "Um, I think it's none of those. Can I try?"
Dan scowled at her. "Amy, _I'm_ the puzzle guy. I'm all over this."
Amy shrank back. Maybe he was seeing something else. Dan always saw stuff no one else did. He was a genius at puzzles. He had solved an ancient code on a pile of skulls in the sewers of Paris. He had figured out the secret encoded in Mozart's sheet music.
But he was being distracted now. He was looking at Ian as if he wanted to kill him with optical lightsabers.
He wasn't thinking.
"I — I'm pretty sure I have this," Amy said.
Alistair grinned. He gestured to the dial. "Please."
Amy averted her eyes from the disbelieving glare of her brother. "Well, think about that phrase — 'Why leave the factory.' 'Why' sounds like the letter Y. If the letter Y leaves the word _factory,_ you're left with—"
"Factor!" Natalie announced.
"And the workers?" Amy said, reaching out to the latch. "They're 'in their _prime'_ —"
_"Prime... factors?"_ Dan said.
"So that would be the prime numbers that you multiply together to get five thousand and five..." Ian murmured. "Sounds a bit far-fetched?"
"I hate math," Natalie said.
Amy's hand shook as she carefully turned the dial....
5, 7, 11, 13.
_Click._
She turned the latch and pulled the hatch open.
"Welcome," Alistair said, "to the Oh sanctum."
#
_It's a small room,_ Ian thought, _but ugly._
He smiled. An old Kabra family joke.
The Cahill brother — Dan — was gazing about the musty, wood-paneled room as if he were about to cry. "For this, you have a murderous, man-eating, killer beast?" he cried out. "To guard _a library_?"
Amy was looking about the sanctum in awe. "It's... beautiful!"
The girl was modest and thoughtful. How bizarre. So rarely did Ian see these qualities in others — especially during the quest for the 39 Clues. Naturally, he had been taught to avoid these behaviors at all costs and never to consort with anyone who possessed them. They were distasteful — FLO, as Papa would say. For Losers Only. And Kabras never lost.
Yet she fascinated him. Her joy in running up Alistair's tiny lawn, her awe at this piddling cubbyhole — it didn't seem possible to gain so much happiness from so little. This gave him a curious feeling he'd never quite experienced. Something like indigestion but quite a bit more pleasant.
_Ah, well. Blame it on the ripped trousers,_ he thought. Humiliation softened the soul.
He glanced at the cramped shelves, the mildewed oak walls, the cracked leather armchair, the hideous fluorescent lights, the mouse droppings in the corners, the scuffed moldings, and the artwork that seemed to have been bought at a tag sale for the color-blind. _Beautiful?_
"It's books," Dan groaned. "Beam me up, Scotty — _please_!"
For once, Ian rather agreed with him.
"Rare books," Alistair said, gesturing grandly to a section of four glassed-in shelves, "not to mention one of the world's finest collections of secret material about the Cahill family. A lifelong passion for me, as few items were ever duplicated. Here is our best hope of decoding the parchment!"
Ian began to sit, but he thought of how that would make his posterior feel. Standing up wasn't pleasant, either, with the polyester trousers that felt like sandpaper on his legs. And Dan's whining just made the experience unbearable.
He would have to avoid the brother. The sister, at least, was interesting. He wondered if her lack of cynicism would be contagious.
How distasteful. Still...
"Perhaps we should form teams," Ian suggested. "A race. Amy and I will scour the material on the top two shelves, Natalie and Dan take the bottom two."
"Excellent," Alistair said. "Do you agree, Amy?"
"Um..." Amy said, her eyes darting away from him. "Uh..."
_A pity,_ Ian thought. So many females had this reaction to him. It really did limit conversation.
"I've never been on an extra-Kabricular team before," Natalie said, smiling at her own wittiness. "But I'll try, I suppose."
Dan was staring at an expensive but unfortunate painting of a couple quite familiar to Ian. The man's hair was piled in mangled gray wisps, his eyebrows bushy and his eyes wild. The woman had a strong face, the way a horse was strong — long-jawed and big-eared. Above them floated all kinds of strange-looking symbols. "Who's the lucky couple?" Dan asked.
"Ah, yes, the ever-glamorous Gideon and Olivia, the original Cahills, painted in the early 1500s," Ian said. "Your ancestors."
"The Kabras improved the bloodline," Natalie said.
"Ready?" Alistair spread out the parchment on a table, then grabbed a book off the shelf. "I will help the younger team, Natalie and Dan. Set... go!"
Ian ran his fingers along the line of books, some with handwritten titles along the spines: _Historicus Cahilliensis: Ekaterina, Vols. I and II_... _Ekat Architectural Renderings_... _A Review of Eighteenth-Century Cahill Literature._... Some of them seemed like pamphlets, notes torn from three-ring binders. It would be difficult to find anything helpful here.
Amy was pulling down a thick book titled _Origins of the Cahills: A Compendium of Contemporary Studies._
"We're supposed to find a _clue,_ not study history," Dan snapped.
"But we know so little about the Cahill family," Amy said.
Natalie looked up from a book she was skimming. "I don't know _why_ your parents never told you which branch you were in. We knew the whole story before we were walking."
Ian watched Amy as her face sank. He felt a flutter inside. Sympathy, he realized—an emotion he often felt for the Kabra banker on days when the stock market performed badly. This feeling, however, was somehow a bit more... vivid.
He gave his sister a kick. "Natalie, have you lost your sense of... grace?"
She glared at him for a moment, until the joke clicked in.
"'The Cahill family traces back to early 1500s Dublin, with the brilliant, eccentric Gideon Cahill and his wife, Olivia,'" Amy read aloud.
Alistair nodded encouragingly. His niece was so excited she could barely get the words out.
"'Some say Cahill had indeed made a discovery to change the course of humankind,'" Amy continued. "'But the nature of this discovery was never known. In 1507, a sudden fire swept through the Cahill home. All escaped but one. Gideon, desperate to save his life's work, was found burned to death at his desk.'"
"What is it with Cahills and fire?" Dan whispered.
Alistair felt a little clutch in his chest. The children had been through so much tragedy—the fire that had trapped their parents, the one that had burned Grace's house. It occurred to him why he'd never wanted children of his own.
You risked caring for them. And that kind of feeling could be dangerous in the hunt for the 39 Clues.
"'According to contemporary sources, at his death, Gideon had been studying the secrets of alchemy—the attempt to turn base metals into gold,'" Amy went on. "'He sought a substance called the philosopher's stone. The problem was, the substance did not exist — yet. It was considered the key to the final quest. Being _more perfect_ than gold, the stone, also known as alkahest, would be powerful enough to turn other substances into gold.'"
"Thank you, Ms. Frizzle," Dan said, furiously reading through a pile of pamphlets. "Keep going, but this time try _reading to yourself_."
_"Don't you all see?"_ Amy said, jumping out of her seat. "We've done it!"
"Done what?" Dan said.
Amy grabbed her brother and swung him around like she'd done when he was three. "Gideon _made_ that 'discovery to change the course of humankind'! He cracked the secret of the philosopher's stone. _We've discovered the secret to the thirty-nine clues!"_
"What?" Ian said. "You figured out the parchment code? The clue?"
"No — something bigger than the clue," Amy said.
Natalie plopped angrily down into a chair. "Did we lose? I _hate_ being on a team."
Alistair looked over Amy's shoulder, pushing aside the Kabras, who insisted on blocking his view. Amy flipped the page to a diagram of alchemical symbols:
Amy brought a coin out of her pocket. "The shape — the philospher's stone — it's on this coin!" she exclaimed.
"Cool," Dan replied. "But what's so big about that?"
"Don't you see?" Amy repeated. "This page is the secret to the whole thing — what the thirty-nine clues add up to!"
"So... when we collect all of them..." Dan said, a grin spreading slowly across his face.
"We will possess the secret to alchemy—the philosopher's stone!" Amy put the coin back in her pocket and glanced at the book. "We'll find out how the coin fits in, too. But listen —
"'After the fire of 1507, Thomas and Kate fled Ireland for England, smuggling components of Gideon's work—which they vowed to continue. Thomas married and had a family but began neglecting his sister and their mission. The angry Katherine ran away, taking something of such importance that Thomas left everything to chase after her. After trying Paris, Venice, and Cairo, Thomas gave up. Attracted to the rough samurai culture, he settled in Japan, assuming a modest lifestyle. His youngest son, Hiyoshimaru, grew up to become Toyotomi Hideyoshi.'"
"The Bald Rat was the son of Thomas — _the_ original Tomas?" Dan said. "That's promising."
Alistair cast a wary glance at the Kabras. He could read their parched, sarcastic faces — their impatience at the fact that Dan and Amy were learning things that the other teams had known for a long time. He knew they were struggling to wait out the Cahills' learning curve. After all, Dan and Amy had been very good at finding new Clues that had eluded the others.
And they were on to something here.
"Can't we skip past GO, collect two hundred dollars, and get to the parts we don't know yet?" Natalie said with a yawn.
"Get off your butt, Natalie, and let's keep looking," Dan said. "We're... thirty-seven clues away from the secret to alchemy!"
He swung around, shoving a book back into a shelf and reaching for another. A ragged old book, teetering on the shelf edge, fell to the floor.
Alistair cringed. "Careful, some of these are priceless!" he warned, bending to pick up the delicate book and examining the hand-drawn Japanese characters on the cover. "It's five centuries old. Found by an enemy warlord. It was the only item found in Hideyoshi's tent during a raid—"
"What's it say?" Dan asked.
Alistair adjusted his glasses. "The cover reads, 'Hideyoshi, nine' — perhaps a sketchbook or coloring book from his childhood."
"Wait, why would this even say 'Hideyoshi'?" Amy asked. "Wasn't he called something else as a child?"
Alistair's eyes widened. "Yes — Hiyoshimaru! Good catch. If this were really his childhood book, _that_ would be the name on it."
Amy gently took the book. As she flipped through the pages — landscapes, battle scenes, monsters — the others gathered around her. Alistair noticed the Kabra boy gently touching her shoulder. "Th-th-this... this stuff is way too g-g-good for a nine-year-old..."
Hands shaking, his niece opened to a page that showed a strange, modern-looking jotting of stars and random lines. "A kid could have done that," Natalie said.
"'Hideyoshi... nine...'" Dan said. "Hey—this is _page_ nine!"
Suddenly, without saying a word, Amy reached into the book and ripped the page out.
Alistair thought he would have a heart attack. "Amy!" he blurted. "This is an antique!"
Amy quickly leaned over his table, laying the ripped-out sheet on top of the parchment.
They fit together. Most of the lines made a detailed landscape of a rocky area. But other lines, tighter and smaller, seemed to form Korean characters.
And Alistair saw the method to the girl's madness. "The three horns..." he said.
"Say _what_?" Dan said.
"Ha-ha!" Alistair gave his niece a hug. She really was an extraordinary child. "Thanks to Amy, I know where this is. And we're going there first thing tomorrow morning."
#
Somehow, on a bumpy morning car ride after an egg breakfast, Dan did not like thinking of the name Pukhansan. But that's where they were headed at the crack of dawn.
As they neared the city of Seoul, a three-peaked mountain loomed before them. "The three horns — I should have realized it from the beginning," Alistair said. "It is Samgaksan, the three peaks. The confusion is that it is now known by the name Pukhansan."
_"Glurp..."_ Dan said, closing his eyes and sinking back into the seat and the oversize hooded Harvard sweatshirt Alistair had lent him.
Amy stared through the car window. The day was gray and gloomy, and the mountain looked nearly vertical. They had packed lunches in her backpack, but this was obviously going to be more than a day hike.
"We have to climb that?" Nellie asked. "I'm wearing Vans."
"Mountains have strong profiles," Natalie said, flicking a piece of dust off her pink jeweled Prada sneakers, which she had given to Harold to clean. "And so should we."
"It's only about a half mile high, but I do not think we will be climbing," Alistair replied, referring to the overlay of ancient parchment and sheet. "The old drawing has a solid meandering line through it, which I take to be the famous fortress wall. It cuts through several valleys and low-lying areas."
"What's this?" Dan said, pointing to a funny squiggly formation.
"Uh, an M," said Nellie. "Or if you look at it the other way, a W. Or sideways, kind of S-ish..."
"Maybe it's palm trees," Dan said. "Like in the movie _It's a Mad, Mad, Mad, Mad World._ You know? No? These guys need to find hidden money, and the only clue they have is it's under a big W? And no one sees what it means — but then, near the end of the movie, there's this grove of four palm trees rising up in the shape of... you-know-what! Classic!"
Amy, Alistair, Natalie, Ian, and Nellie all looked at him blankly.
"There is no W in the Korean language," Alistair replied. "Or palm trees in Korea. It might be maple trees..."
_"Mrrp,"_ said Saladin, rubbing his face against Dan's knee.
"I'll tell you the rest of the plot later," Dan whispered to the Mau.
Alistair's driver let them off in the parking lot of Pukhansan National Park. A crowd of tourists had gathered around a giant trail map, which Alistair carefully compared to his parchment-sheet overlay. He traced the dark snaky line with his finger, stopping at various black marks. "These, I presume, are ancient temples. Let's assume the big X is our hidden treasure—"
"It's between two of the temples," Natalie said. "But which two?"
Alistair shrugged heavily. "There are many of them. And much space between them. This may take several days."
"Then let's go!" Dan said.
"Someone has to stay here with Saladin and Mr. Chung," Nellie piped up, giving the mountain a dubious glance. "Okay, you talked me into it; I will."
The rest of them took off down a well-beaten pathway. "Hideyoshi conquered most of what is now South Korea," said Alistair, "including Seoul, which was then called Hanseong. But the soldiers put up a fight, building this fortress wall to ward off the invasion."
"Why would Hideyoshi bury his treasures here?" Amy asked.
Alistair shrugged. "To use the wall's protection, perhaps. He assumed this would remain his territory."
"Overconfidence is a curse," Ian remarked.
"You would know," Dan said.
As the path climbed, there were fewer and fewer hikers. Each time they passed a temple, Alistair checked the overlay, each time shaking his head.
His back was coated with sweat now, and he was panting as he finally sat on a rock ledge. "Lunchtime," he announced, handing the overlay to Amy. "My dear, would you kindly keep this in your backpack?"
"Lunch? We're just getting started!" Ian said, scampering up the wall, his loose Harold pants ballooning in the breeze.
Natalie sat eagerly next to Alistair. "You didn't happen to bring prosciutto with buffalo mozzarella and sun-dried tomatoes on whole-grain focaccia with pesto sauce?"
"Peanut butter and banana on white?" Dan offered.
Alistair was intent on the surroundings. "I fear we may have missed the spot. The wall may have been repositioned over the centuries. It may not look like this anymore."
As Amy zipped her pack shut, she felt something hit the top of her head — a clump of moss, which bounced off and landed at her feet. "Hey!"
Ian was laughing as he wiped the dirt off his hands.
_Laughing._ Not to mention staring at her. His eyes were mocking, pinning her in place. As if planning some snarky Kabra comment. In front of everybody. She fought back tears, fighting the urge to run back or curl into a shell.
"Throw it back," Dan hissed. "Hard!"
Ian cupped his hand to his mouth. "Amy, will you accept a challenge? A race to the top of the next big rock? I'll give you a head start — or are you too slow?"
"She's not slow!" Dan shouted back. "Well, actually, she is."
Amy stood up. It was one thing to be humiliated by a Kabra, but not by a snot-nosed little brother.
She eyed the big rock. This was insane. He was baiting her, setting her up for more embarrassment. Unless...
There it was. Another path through the brush. More direct.
She started to run.
"Amy—leave your backpack!" Dan called out. "And remember to name your first child after me!"
She ignored him. Her ankle was killing her, but she was _not_ going to let Ian win. He was running now, stumbling across the top of the wall and then jumping off. He zigzagged into a wooded area, howling with laughter and barreling toward her. Amy yanked off her backpack and swung it, clipping him on the arm.
The creep.
"Ow!" he cried out. "That's a fitted Armani shirt!"
The backpack skittered over the ground, spilling out Alistair's overlay—page and scroll clipped together. "Finders keepers!" Ian shouted, scooping up the overlay and hopping onto a rock outcropping.
"You cheater!" Amy was furious. No way was he going to get away with that. She climbed the rock, matching him step for step until she reached the top. There he turned to her, panting for breath. "Not bad for a Cahill," he said, grinning.
"You — y-y-you—" The words caught in her throat, the way they always did. He was staring at her, his eyes dancing with laughter, making her so knotted up with anger and hatred that she thought she would explode. "C-c-can't—"
But in that moment, something totally weird happened. Maybe it was a flip of his head, a movement in his eyebrow, she couldn't tell. But it was as if someone had suddenly held a painting at a different angle, and what appeared to be a stormy sea transformed into a bright bouquet—a trick of the eye that proved everything was just a matter of perspective. His eyes were not mocking at all. They were inviting her, asking her to laugh along. Suddenly, her rage billowed up and blew off in wisps, like a cloud. "You're... a Cahill, too," she replied.
"Touché."
His eyes didn't move a millimeter from hers.
This time she met his gaze. Solidly. This time she didn't feel like apologizing or attacking or running away. She wouldn't have minded if he just stared like that all day.
_"Hey, Amy? This hike is rated PG, and we're starving!"_ Dan shouted. _"Not to mention Alistair wants his map!"_
Amy felt herself blushing. She turned her eyes away.
"Here," said Ian, handing her the overlay.
Alistair's ripped-out page, which had been attached by a paper clip, was dangling cockeyed. Amy nervously replaced the paper so it was superimposed as before — everything lined up, marking for marking....
Her eye wandered out over the landscape and then back again.
"Oh, my god..." she murmured.
"Pardon?" Ian replied.
She checked it again. And again, just to be sure. But it was unmistakable — the shape they noticed earlier on the map. The one they'd been guessing about.
It wasn't palm trees. _Or_ maples.
_"Dan!"_ Amy screamed, leaping down the rock as if her ankle had never been injured. _"Everyone! Come quickly!"_
She ran back, but the others were on the run, meeting her halfway. Amy took her brother's hand and pulled him up the pathway and up the steep rock. "I love you, Dan, you're a genius," she said.
Dan glared at her. "Did Ian drug you?"
"Look," she said, gesturing over the area. "What do you see?"
"Trees. Rocks. Deer poop." Dan shrugged.
"The rock outcropping. What does it look like?" Amy pressed on.
"Kind of a zigzag?" Ian spoke up.
Suddenly Dan looked as if he'd given himself a wed-gie. "It's a W!" he cried out. _"Amy, you found our W!"_
Alistair smiled. "Excellent. X marks the spot on the map — and the spot is a W-shaped rock formation."
Amy took the overlay and began running down the rock ridge. As she got to the edge of the outcropping, she began ripping vines and brush away from the bottom.
"Spread out," Ian commanded. "Look for a cave. A hidden entrance."
The others began poking and tearing, examining the rock. "Look!" Natalie cried out.
Amy ran to her side. She had pulled away a thick bush from the rock wall to reveal a carving of a man. He had a thin, monkeylike face with piercing eyes and a slit for a mouth. "Eww," she said.
"The Bald Rat," Alistair said with awe, running his fingers over the relief. "This is an image of Hideyoshi, in the Japanese style of the period."
"Brilliant," Ian said, rubbing his chin thoughtfully.
"How do we get in?" Dan demanded, his face still buried in the map. "Maybe you've noticed—this big old W is made of _solid rock._ There's got to be some instructions here...."
Amy and the others crowded around Dan. He pointed to the bottom of the overlay. "The letters at the end. Toota. What do they mean?"
"Hideyoshi's father was Thomas Cahill — perhaps he taught his son English," Alistair said.
"It's Toyota!" Amy said. "The letters. They spell Toyota without the Y."
"Great, Amy," Dan said. "Our third clue is a buried Sienna minivan."
"I believe she is suggesting that the parchment may be a fake," Ian said.
"Thank you, Mr. and Mrs. Kabra," Dan said, looking closely at the parchment. "But it isn't a fake. Not at all."
He placed the map gently on the ground and pulled a small Swiss Army knife from his pocket. Then, with quick strokes, he began tearing the parchment to shreds.
_"Dan!"_ Alistair cried out.
Amy felt her heart stop. _"What are you doing?"_
Dan had the small pocket scissors out now. In a moment, he had carved out all the letters precisely. Handling the thin, fragile cutouts carefully, he arranged them — the big A inside the big O; the two smaller Ts beside each other, upside down within the A; and finally the smaller O in the center:
"It's the symbol for the philosopher's stone," Amy said, astonished.
Dan nodded. "'And by the elements united is entrance granted...' I just united the elements."
He was beaming at Amy now. And she knew exactly what was on Dan's mind.
Reaching into her pocket, she pulled out the coin Ian had given her. On it was the same symbol — the philosopher's stone. "Now, let's give that Rat something to eat," she said.
Cautiously, she stuck the coin into the slitted mouth of Hideyoshi.
And the ground began to rumble.
#
_GGRRROOCCCCK..._
Ian's knees buckled. The rock outcropping shook the ground, sending a spew of grayish dust that quickly billowed around them.
Shielding his eyes, he spotted Amy standing by the figurine, which was now moving toward her. She was in shock, her backpack on the ground by her feet.
"Get back!" he shouted.
Ian pulled Amy away and threw her to the ground, landing on top of her. Gravel showered over his back, embedding into his hair and landing on the ground like a burst of applause.
His second thought was that the shirt would be ruined. And this was the shock of it — that his first thought had not been about the shirt. Or the coin. Or himself.
It had been about _her._
But that was not part of the plan. She existed for a purpose. She was a tactic, a stepping stone. She was...
"Lovely," he said.
Amy was staring up at him, petrified, her eyelashes flecked with dust. Ian took her hand, which was knotted into a fist. "Y-y-you don't have to do that," she whispered.
"Do what?" Ian asked.
"Be sarcastic. Say things like 'lovely.' You saved my life. Th-thank you."
"My duty," he replied. He lowered his head and allowed his lips to brush hers. Just a bit.
The air was slowly clearing, and the noise had stopped. Ian sat up, letting go of Amy's hand. The carving now jutted diagonally outward a few inches from the rock. Where it had been was now a rectangular opening.
A rotten, acrid smell blasted from within.
Alistair was the first to stand, dusting off his carefully pressed hiking pants. "Hideyoshi's hiding place..." he said in awe.
Dan and Natalie were right beside him, coughing and shaking off the dust. Dan recoiled as he tried to peek inside. "Dang, someone forgot to flush."
Alistair had found Amy's backpack and was pulling out two battery-operated collapsible Coleman lanterns.
Ian helped Amy to her feet. "Do you have the coin?" he asked gently. "We may need it later, to close up the entrance."
"Po-pock—" Amy tapped her pocket. "I put it in there when the thing s-started to open...."
Alistair handed her a lantern. "You and I will lead, Amy."
As she walked on shaky legs into the cavern, Natalie glared at Ian. He winked at her and walked inside.
Oh, she of little faith.
_Focus._
All Amy could feel was her lips.
The bluish fluorescent lantern light danced off the crags of a domed cavern, the ammonia smell of animal droppings invaded her nostrils. They were in a cave that most likely hadn't been seen by a human being in half an eon, and her shoes were squishing into a carpet of something she'd rather not see. And all she could feel was the tingle in her lips.
Everything was happening at once. The coin, the hiding place, the...
The _what_? What exactly had just happened?
Ian was walking beside her quietly. She was supposed to hate him. She had hated him. But for the life of her, she could no longer remember why. Despite the surroundings, she felt alert, alive, and unbelievably happy.
"Thank you," she said quietly.
"For what?" Ian asked.
"For giving me that coin back in the alleyway in Tokyo," she said. "If you hadn't done that, this whole thing might not have happened."
Ian nodded. "It was one of the Kabra family's most cherished possessions. There were rumors it was the key to a Tomas clue, but my parents didn't believe it. I had to steal it from them." He shuddered. "I will not like to face my father once he's found out."
Amy reached into her pocket and handed him the coin.
"I — I couldn't," Ian said. "I promised."
"We don't need it anymore," Amy said.
"Thank you." Ian took the coin and put it in his pocket. But his eyes were focused upward. "Amy? Do you see something moving up there?"
Amy swung her lantern upward, into a shadow that flitted and danced — and then broke away in a shrieking cloud.
_"DUCK!"_ Dan cried out as a liquid mass of flying bats chittered overhead. They screamed and flapped, their wingtips flicking Amy's hair like rain as she cowered. Then, like smoke through a chimney flue, they exited through the narrow entrance.
"Are you all right?" Ian asked.
Amy nodded. "I hate bats." She sat up, swinging her lantern around, allowing the arc to include his face.
Just to see it.
And that was when Dan yelled again.
_"Amy, shine that thing over here!"_
It was the coolest thing he had ever seen. Cooler than the lifetime supply of Wii games he'd almost won in his sixth-grade raffle.
Now Alistair and Amy were both racing over, their lanterns illuminating a mammoth pile of objects stacked floor to ceiling. At the top, where the bats had been, a grove of stalactites hung down. They surrounded the pile like an upside-down picket fence holding it in place.
They were swords — a tower of them, arranged neatly in a crisscross pattern. The hilts jutted out, some fancy and jeweled, others dented and dull. They looked like hands, reaching out as if daring someone to pull, which would probably upset the pile like a falling house of cards.
"The Great Sword Hunt of 1588," Alistair murmured. "This is where they were kept."
But Dan was moving past the swords, to the left. The cavern seemed to expand here, wider and deeper, with stacks that seemed to go on into the distance forever. Some of them looked like they'd been thrown there — crowns and helmets, armor, spears, shields, saddles, stirrups. Folded robes winked with inlaid jewels, statues stood covered with dust, and tightly rolled scrolls lay in boxy containers. But one area seemed separate from the others — a shrine, surrounding a strange triangular mirror that hung on the wall in an intricately carved frame.
Around the mirror, huge chests had been stacked in neat piles. They were festooned with jewels and calligraphy, each fastened with a huge padlock.
Dan grabbed one of the locks. It fell apart in his hands, rusted and brittle. As he opened the lid, the others peered in with him.
"As they say in the US provinces..." said Natalie, her eyes widening, "bungee!"
"I believe it's 'bingo,'" Alistair said. "By god, these must be the spoils of Hideyoshi — the plunder seized by his forces as they conquered Japan and moved through Korea."
Dan reached in, digging his hands into a trove of gold coins. Next to him, Amy opened another chest. "Plates, chopsticks, cups, bowls, platters — all solid gold!"
"Buddhas!" Ian exclaimed, peering into a third chest. "A collection of miniature golden Buddhas."
"Hideyoshi worshipped gold," Alistair said softly. "According to legend, he even ingested drops of liquid gold each night for its supposed magical properties...."
"We're rich," Ian said. "Again."
Dan smiled.
_And by the elements united is entrance granted, the highest to be revealed._
"We're more than rich," he said, letting out a whoop of amazement. "We have discovered the next Cahill clue!"
#
Alistair did not mind growing old. He did mind being outsmarted by an eleven-year-old nephew.
_Gold._
Of course the boy was right. Gold was the "highest element" of alchemy. The alchemical symbol — the "elements united" — was the key to entrance. No doubt this came from the mind of Hideyoshi. Being a son of Thomas Cahill, he would also be a student of alchemy!
Alistair cursed himself inwardly. He should have detected it from the beginning. He could have saved all this trouble, all this danger. All this unnecessary risking of his nephew's and niece's lives.
This was bound to happen.
He was bound to discover a Clue he had already known.
He tried to smile. To the Cahill children, this was all new. They had not been searching for a lifetime as he had. They were dancing now with the Kabras, doing dance moves they called hip-hop, which, when he attempted to join, made his hips hurt.
He kept his eyes on the Kabra boy. Surely the Kabras knew this Clue also. The Lucians had been collecting Clues as long as the Ekats. Perhaps they were just better actors than he.
"Brava!" Ian cried out, lifting Amy in the air. "I knew this cooperation between branches would pay off!"
As he let her down, she allowed her face to brush gently against his.
Alistair felt his blood go cold. The gambit with the Kabras had proved profitable. Without Ian's coin, they could not have found this cavern.
But this was not the sort of alliance he'd imagined.
"I — suggest we leave now," Alistair said. "Perhaps we can discuss what to do next over dinner."
"Not so fast," Ian said. He was walking away from Amy now, intent on the mirror. "Correct me if I'm wrong. It seems to me that whenever you find a clue, you seem to find a lead to the next one."
"Right, Euro-boy," Dan said. "But don't tax your brain. I'm betting the next clue is _not_ rock dust."
Ian was eyeing the mirror now. "What do you suppose these letters mean?"
Alistair joined him, shining a light on the triangular mirror frame. Along two sides ran a strange set of symbols.
"Greek to me," Natalie said.
"Guys, I _know_ those letters!" Dan exclaimed. "From the inscription we found on the sword in Venice. Remember, Uncle Alistair, when we were looking at those tattoos? I told you there were some letters missing. Here they are!"
"I don't think this is any one tongue," Alistair said, running the letters through his knowledge of thirteen languages. "Perhaps some kind of secret message?"
Natalie began brushing her hair in the mirror with a gold-handled brush. "Mirror, mirror on the wall, who's the richest and smartest and hottest and—"
"That's it, Natalie!" Dan said.
Natalie blushed. "Thank you, I amaze myself sometimes...."
"No! Words on a mirror... mirror writing!" Dan quickly pulled out a mechanical pencil and his paperback copy of _Classic All-Time Movie Comedies._ Ripping out a blank sheet in back, he turned the book sideways for support and began copying the letters onto the sheet. Then he held them in the mirror.
It was still nonsense.
Amy cocked her head. "The letters are symmetrical," she said. "The top of each one is a mirror reflection of the bottom. Maybe each letter is a mirrored _half_ letter. So if you could see the half letter alone, you'd know what it was?"
"That is the stupidest, most far-fetched thing I've ever heard," Dan said.
Amy grabbed the paper and began erasing half of each latter:
Slowly, she began filling in each letter:
"Ahstkael..." Amy said. "Isn't that a health food chain in Sweden?"
"Our next clue is in Sweden?" Natalie said eagerly. "I do need a new fur."
Dan tapped his chin. "Um, dudes, these are English? Shouldn't we be trying to make _Japanese_ letters? Or Korean?"
"Hideyoshi was the son of Thomas Cahill," Alistair said. "It stands to reason that English was spoken in the home. Hideyoshi would have been fluent. And because the East had not yet opened to the West, words using English letters would have been an unbreakable code."
Dan was scribbling furiously again. He began arranging the letters crazily, in all different combinations.
_"Lake Tash! Is that it?"_ Natalie shouted.
Dan nodded. "Lake Tash..." he said under his breath. "That's Kyrgyzstan..."
"Our next clue is in _Kyrgyzstan_?" Natalie said.
"Brilliant," Ian said with a smile. "Well, it was nice to work with you. This time, I'm afraid we will have a distinct advantage."
"But — but—" Amy sputtered.
Alistair watched his niece's expression drop. She would push for the alliance to continue — which would be disastrous. "I will arrange for transportation back into Seoul immediately," he quickly said, pulling his cell phone out of his pocket. "There, we will—"
"Oh, you won't get any reception in here," Ian said, stepping toward the entrance with his sister close behind.
At the cave opening, Natalie grinned, hands in pockets. "In fact, I wouldn't expect much reception for another, oh, five hundred years."
When she pulled out her right hand, she was holding a tranquilizer dart gun.
Alistair scrambled to step in front of his niece and nephew, but Amy pushed him aside. "Natalie...?" she said.
"Guys, this isn't funny," Dan said. He stepped toward them, but Natalie pointed the gun at his face.
_"Dan!"_ Amy screamed, pulling him back.
Ian glanced at Amy. For a moment, she thought she saw a flicker of—something. Doubt? Some kind of indication this was all a big, sadistic joke? Then the look seemed to vanish as quickly as it came. He looked down and pulled the philosopher-stone coin from his pocket. "Oh, by the way, thanks for this."
"How did he get _that_?" Dan blurted, glaring at his sister.
"I-I-" Amy couldn't get the words out. "He—"
"Family heirloom," Ian said. He backed through the opening now, inserting the coin into the mouth of the Bald Rat. "Don't worry. When we win the Cahill challenge, when we amass power that is rightfully ours, maybe we'll come back and pay you a visit. If you're still able to receive us. Meanwhile, my friends, I recommend you conserve your batteries. And your oxygen."
The cave shook. Slowly, the door swung back.
The last thing Alistair saw before the entrance thumped shut was the retreating muzzle of Natalie Kabra's dart gun.
#
_Idiot._
_Moron._
_Fool._
Amy stared at the door, at the absence of light where Ian Kabra had been standing.
It had all been one big joke. He had wrapped her around his fingers and then yanked her inside out.
How could it have happened? _How could anyone do that?_
Tears made rivulets down her cheeks and fell to the ground, tapping lightly like the flap of moth wings.
Behind her, Alistair and Dan were ignoring her, discussing strategies, trying to figure out how to escape. How to prevent themselves from dying.
_Too late,_ Amy thought. She already knew what _that_ felt like.
Slowly, their voices filtered into her brain.
"I am going to search for another exit," Alistair was saying. "Amy, you and Dan look for any weakness in the rock wall. If bats live here, there must be some source of air, some kind of hole."
Amy nodded numbly.
As Alistair's footsteps receded, Dan squatted next to her. "Hey. I want to strangle him, too."
"It was my fault," she said. "I-I believed him. I played right into his trap...."
Dan helped her to her feet and swung the light around, examining every square inch of the wall. The place was pitch-dark, and after a few minutes, Amy already felt like she was growing short of breath.
Alistair's voice echoed toward them from the distance. "No other exits. I just checked the whole W. It goes on much longer than I'd thought. We are totally sealed."
_A tomb,_ Amy thought. _He buried us alive._
She felt a hand on her shoulder. "I am so sorry, my darling niece," Alistair said gently. "Had I seen you were falling for the boy, I would have done something. It passed over my head, and it should not have."
Amy sighed. "How could I have let him trick me? How could I think that anyone would actually feel..."
The words got stuck in the back of her parched mouth.
"I know this will not make anything better," Alistair said, "but you must believe me when I say I know how it feels to be betrayed."
Amy glanced up into Alistair's barely discernible face. "Really?"
Alistair looked as if he were going to say something, then changed his mind. "Think only of this, Amy: Your parents loved you. It was in their eyes, even when you weren't around. You must think of them, and they will be there for you."
"Did you — _know_ them?" Amy asked.
_"AGGH!_ Gross!" Dan shouted from another part of the cave. "I think I stepped on a bat! Can you guys continue the conversation at a later time — if there is one? If we don't all die and become a banquet for bats?"
Alistair raced away, leaving Amy with a mouthful of questions.
"Dan, you must never, ever give in," Alistair said encouragingly. "A problem is merely a solution waiting to be found. We _will_ make it out of here — and I predict we will beat the Kabras to Lake Tash—"
"Dude, we're not going to Lake Tash," Dan said. "I made that up."
Now Alistair was staring at him. "But — the anagram—" Amy said.
Dan sighed, shining his lantern on the sheet where he'd unscrambled the words. "I saw the real answer right away, but I didn't trust them. I threw something out to test them. The real answer is, like, a no-brainer—"
He began scribbling another word on the sheet, but Amy was looking past him, at the odd reflections of lantern light in the mirror.
"Wait!" she blurted out. "The mirror— _who ever heard of a triangular mirror_?"
"Uh, a triangular mirror designer?" Dan said.
"Or an alchemist!" Amy said. "Think, Dan. Alchemy is all about _symbols._ The planets, the elements, everything had its own funny shape!"
"So what was the triangle?" Dan asked.
Amy tried to picture the image on the page. "Air? Gold?"
"Wait... wait... I can see it..." Dan said. "Water! That's it. Wait. No. With the point _down_ , it represents water... but with the point up, it's fire!"
Dan swung the lantern back in the direction of the mirror now, raising it high above his head.
Just above the mirror, beyond his reach, Amy noticed several greasy-looking, stringy objects. Her stomach turned. They looked like rat tails. "Are they... alive?"
Suddenly, Dan glanced toward the ground. He crouched now, scraping something with his fingers. "Charcoal," he said. "It must be leaking from above."
Alistair looked up. "What's up there?"
Choking back nausea, Amy forced herself to follow his glance. To her relief, the hanging tails were way too long to be what she thought they were. They seemed to be just strings, leading into a big crack in the rock.
And she became aware of a very particular smell. "Oh, my god..." she said. "Guys, what do you smell right now?"
"Bat turds," Dan said.
"Rotten eggs," Alistair volunteered.
"And that rotten-egg smell," Amy said, "is caused by—?"
"Chickens?" Dan said.
"Sulphur!" Amy said.
Dan smiled. "Oh, right — I learned that in chem lab last year! I snuck this test tube into Mandy Ripkin's lunch box? With this, like, really loose cork? So when she opened it—"
"Charcoal... sulphur," Amy said, wracking her brain for something she'd read in science class. "They go together with some other ingredient to make... what is it...?"
"Smelly barbecues?" Dan asked.
Amy suddenly remembered. "Not barbecues, you cheesehead," she said, glancing up into the stringy void. "Gunpowder."
"Uh, you think there's _gunpowder_ up there?" Dan said.
"Gunpowder indeed existed in the sixteenth century," Alistair said. "It was developed in China hundreds of years earlier and spread throughout the East."
"Dan, I think these strings are there for a reason," Amy said. "They're fuses!"
"Brilliant, my girl—you are a genius!" Alistair said. "So the mirror has two functions. It points upward, directing our gaze, and it is the symbol for fire. This is _exactly_ like Hideyoshi — ever the wily warrior, he developed a failsafe escape for this hideout, in case of sabotage."
"Dan, do you still have the matches from the Thank You Very Much Hotel?" Amy asked.
"You dummy, we can't just blow this thing up!" Dan said. "We could die."
"Gunpowder is not dynamite," Alistair said. "May I remind you, we are surrounded by a lot of schist."
"Watch your language," Dan said.
"Granite schist is extraordinarily dense," Alistair continued. "Blasting through rock in modern times requires far more explosive power than can be provided by gunpowder. Any blast is likely to punch open only a small targeted area. In fact, it is entirely possible the blast will not be strong enough. We will be more than safe with schist."
Alistair was trying to sound reassuring, but Amy could hear the shakiness in his voice. She glanced at her brother. The lantern's shadows on his face made him look like an old man. But even in the distortion of the dim light, she could read his mind.
_Do you believe him?_ his face said.
_I'm not so sure,_ she thought.
_Me neither. So we all get crushed instantly under tons of granite,_ he was thinking.
_Or..._?
Dan looked away.
_Or die a slow, painful death from starvation,_ was what he didn't want Amy to see. But she felt it.
And the choice, to her, was clear.
"I guess it's the only chance we have to get out and do horrible things to Ian Kabra," Dan said.
Amy smiled, swallowing back a stab of fear that laced through her. "Go for it," she said.
Dan turned to Alistair. "You're taller," he said, handing him the matches.
The old man struck a match and raised it. The flame licked the bottom of one of the strings, flared slightly, surrounded it, and then finally guttered out.
"The fuses are old," Alistair said, tossing the dead match to the ground.
He opened the matchbook to reveal only three matches left. "What happened to the rest of them?"
"Um..." Dan said sheepishly.
Amy winced, remembering all the matches he'd wasted on the plaza outside the hotel in Tokyo.
Alistair took a deep breath. "All right, then. Pray."
He lifted another match. It too circled the bottom of the limp string.
_FffffFFFFT!_
"Woo-hoo!" Dan cried out, as Alistair lit another string, and another. The flames shot upward into the rock.
"Move!" Alistair cried, grabbing Dan and Amy.
They ran into the cavern, scuttling around the bottom corner of the W.
_Boom!_
_Boom! BOOOOM!_
_KKKRRRRO-O-O-OK!_
An explosion of rock showered down into the cavern, dinging against gold objects, smashing treasure chests. The mirror teetered, finally falling forward and shattering on the ground.
Above them, light poured in through a small hole near the top of the rock wall.
_"We did it!"_ shouted Dan.
All three ran to the spot, stumbling over rock, debris, and broken glass.
_CRRRRRACK!_
More rocks spat down from above. Amy and Alistair put their arms over their heads, skittering away.
"The rock is cracking!" Dan called out, dragging a wooden box to just below the hole. "Come on!"
Alistair climbed onto the box, reached down for Amy, and lifted her over his head. He was surprisingly strong.
Amy stretched her hands upward, but her fingertips didn't reach.
"One... two... three... alley-oop!" Alistair gave her a thrust upward.
There. "Got it!" she shouted.
Her fingers dug into a broken section of rock. As she pulled herself upward, Alistair put his palms against the soles of her shoes and pushed.
_"Hunhhh!"_ She gasped at the blast of fresh, oxygenated air. Her fingertips grasped hold of a root that had dug itself into the rock. She jammed her elbow into a section of rock and pulled herself clear.
Into the sun. Into the glorious smell of grass and earth.
Splaying herself securely on the surface, she reached her arm back down. "Grab on!"
"Heave... _ho_!" Alistair grunted from below.
Amy clasped her fingers around her brother's wrists and pulled. Dan was heavy, and she could only get his torso out — but that was enough. Dan let go and shimmied through the hole.
Quickly, Amy leaned over and called down, "Uncle Alistair! Can you stack any more of those boxes? You'll need to get yourself higher!"
"I'm trying!" he called back.
_RRRRRO-O-OMMM!_
The entire rock shook. A section of it just to Amy's left collapsed downward. The rumbling seemed to be catching, following the line of a crack in the rock.
_"Uncle Alistair!"_ Dan shouted into the hole. " _Are you all right_?"
Dan put his ear to the hole. Amy could hear Alistair saying something, but the noise of the rumbling drowned out the sound.
Reaching down into the hole, Dan screamed, _"Just grab on! Jump!"_
But there was no response.
Now Dan and Amy were both shouting his name. But the hole, which was only a couple of feet wide, began to split. The entire rock beneath Dan and Amy was breaking. They hurtled forward, down the side of the rock, and finally tumbled to the ground.
As the entire W imploded, left to right in a wave, Amy and Dan leaped away, landing on their knees and covering their heads.
A massive cloud of rock dust billowed upward, blackening the sky. Amy and Dan stared, numb, at the jagged pile of rocks that remained.
Finally, Amy felt the words exit her mouth as if they had a will of their own. "What did he say to you?"
"He said," Dan whispered, "'It's not schist.'"
#
When Amy ran away behind the tree, Dan knew she was vomiting. And it didn't gross him out at all, because he was doing the same thing.
Alistair had died — inches away from them. Right underneath. He had given them his trust, his money, his advice, his comfort. And finally, his life.
It didn't seem real. He should be behind a bush now, brushing himself off, strolling toward them, his pants somehow still crisply pressed. _Well, that was an adventure._
But all that Dan could see was dust. Dust and tourists and mounds of rubble and the flashing lights of police vehicles.
And the feeling in the pit of his stomach that he had been through this before. That his whole life had been all about loss. That he had vowed never to get close to grown-ups, because it was so painful to lose them.
And it had happened again.
He was vaguely aware of his sister putting her arm around him. A cop was talking in accented English, but Dan couldn't put together the meaning.
"His name is..." Amy was saying. " _Was_... Alistair Oh."
"Age?" the cop said.
The word "sixty-four" came out of Dan's mouth. He didn't know how he knew that, but it occurred to him that Alistair would never be sixty-five. That someday he, Dan, would be older than Alistair would ever be.
"His clothing?" the policeman pressed, which seemed like a colossally stupid question under the circumstances.
"Silk jacket... really nice shirt," Dan said. "Um, he always had these white gloves, too. And like a round-ish hat—"
"Bo..." Amy said. Her lip was quivering. "Bo..."
"Bowler," Dan said quietly.
The cop took notes, but Dan knew he couldn't be treating this as a rescue operation. It was recovery. No one could have survived that collapse.
As he walked away, murmuring a few words of sympathy, Amy stared out over the wreckage. "Dan...?" she said. "Look..."
Off to the right, a small entourage had just arrived. They didn't look like the other hikers and park visitors. Most of them were dressed in navy-blue suits with black sunglasses and black shoes, and their ears were plugged with headsets attached to squiggly cords.
In the center was an elderly, thin man with an overcoat draped over his shoulders, a silk ascot tucked into an expensive-looking shirt, and a dark fedora tilted slightly to one side of his head. He moved with a lively step, using a walking stick that was encrusted with jewels.
"That's the guy..." Dan said. "The one we saw in Tokyo, outside the subway."
"What's he doing _here_?" Amy asked.
Dan's eyes widened at the sight of someone behind the old man — a person he and Amy were even more familiar with. He had been there at the fire that consumed Grace's house. In Paris and Salzburg. He had never spoken a word, but somehow he was always there.
Amy didn't need to be shown. She saw him, too. "The Man in Black..." she muttered, shrinking away.
Keeping low to the ground, she and Dan skittered behind a bush.
"Can you hear what the old man is saying to him?" Amy asked.
Dan stood. He pulled his hood over his head and edged closer, making sure to stay among the ever-growing crowd of gawkers. They were yakking away, too, but as he neared the old man, Dan could see him exchanging bows with the cop who had just spoken to them.
But the Man in Black didn't seem interested in talking. He was walking slowly toward the collapsed rock, his back to Dan.
The old man and the cop were talking now, and Dan could hear snatches of conversation, but it was all in Korean. They didn't say much, and the old man seemed angry and impatient. Finally, after some more bows, the cop left.
With a sharp gesture to his entourage, indicating they should stay put, the old man began striding alone toward the mysterious black-clad stranger.
The two men stood silent, facing the rubble. Dan glanced back at Amy, who had a look of terror on her face, gesturing for him to come back.
But the men were turned away, so he moved closer.
When the old man spoke up, his words were clear. And in English. "My nephew was in there," he said.
The Man in Black moved his head, a slight shift of his mouth registering just a shadow of a reaction — what? Sympathy? Triumph? It was impossible to read.
They seemed to be arguing about something, but Dan could not make out the words.
Then the old man turned, walking briskly back to his cohort. He nodded to no one in particular, but they all fell in step beside him. Together, the whole posse walked away from the site in the direction of the park entrance.
As Dan slipped back toward Amy, he could see the Man in Black approaching the ruins. As he walked through the rubble, he stopped and leaned down. He seemed to have found something — maybe one of Hideyoshi's relics, Dan thought. Before long, when the rocks were cleared, everyone would know about the treasures. There would be looting, maybe, fights about who the stuff belonged to. All the usual things you saw in the news whenever lots of money was involved.
But for now, the whole thing looked like a big pile of rock. And what the Man in Black was pulling up from the debris didn't belong to Hideyoshi at all.
When Dan saw what it was, a cry snarled in his throat.
It was a bowler hat, crushed and misshapen.
"Oh, my god, you guys, I thought you were dead!" Nellie screamed. "I heard about what happened. You look terrible!"
Nellie raced toward Amy and Dan, clutching Saladin, as they trudged into the parking lot of Pukhansan National Park. She and Mr. Chung were being interviewed by the police.
Amy's heart went out to Mr. Chung. He was not looking well at all.
Nellie gave Dan and Amy an enthusiastic one-armed hug, squeezing the Mau, who let out a muffled _"Mrrp"_ of complaint.
Amy ran her fingers distractedly through Saladin's silver hair. "We escaped. It's a long story, but Alistair..."
Her voice drifted off. Behind her, Dan wiped away a tear.
"Yeah, I heard," Nellie said. She put a sympathetic hand on Dan's shoulder. "Come on, dude, let's go back."
On the ride to Uncle Alistair's house, Amy told Nellie what had happened, right down to the sight of the bowler hat. Nellie nodded, listening, and then they both fell quiet for the rest of the trip. Dan kept forming things to say, but they all sounded so stupid. _He was a great man. He really cared about the Cahill family. We'll miss him._
He realized he didn't really know Uncle Alistair. The old man knew a thousand times more about them than they did about him. He had betrayed them, but in the end he'd saved their lives.
At Alistair's house, birds were chirping in the dogwoods and fluffy white clouds dotted the horizon. It seemed as if nothing had happened. Harold, Alistair's butler, met them at the door, his face drawn and griefstricken. "I'm so sorry," Amy said.
Dan, Amy, and Nellie removed their shoes and trudged wearily to the kitchen, where Harold had prepared sandwiches. As Nellie ate, Dan pushed his aside. He reached into his pocket and pulled out a wrinkled sheet of paper and a large gold doubloon. "This coin was the last thing he gave me..."
"What's on the paper?" Amy asked.
Dan smoothed out the sheet on which he had decoded the last hint.
"That was it?" Amy asked. _"Alkahest_ was the clue instead of Lake Tash?"
Dan nodded. "Yeah. The word for philosopher's stone."
"It's an alchemy word," Amy said. "How can it be a clue if it doesn't really exist?"
Dan shrugged, flipping the doubloon in the air. "How should I know? Hideyoshi was an alchemy geek."
The coin came down in his palm, revealing an Egyptian goddess and some cryptic writing.
Amy's eyes widened. "Wait! Oh, my god! Give me that pen!"
She grabbed the pen from Dan and scribbled one word on the paper, below his column:
"What's that?" Nellie asked.
Amy was nearly leaping across the table at Dan. "We did a unit on Egypt last year! 'Al' means 'of.' 'Sakhet' is an ancient Egyptian goddess."
Nellie cocked her head. "Serious?"
"The mirror's message..." Dan said under his breath. He had to admit, for a dork she could be pretty smart. "Hideyoshi was pointing to the next clue...."
"Nellie," Amy blurted, "do we have enough cash to go to Egypt?"
"Hey, the Kabras may have dumped you, but they never came back to collect the money they gave me," Nellie said. "I say, let's saddle up that camel and ride!"
The room fell into an awkward silence.
Dan shrugged. "It's... hard to think about doing this. After what just happened and all...."
"We don't have to think about it now," Nellie said. "Look, if you're not hungry, at least go take a shower. You smell like rotten eggs. Both of you. Dan, you can use Alistair's and Amy can use the one in the guest bathroom."
Dan had to admit that sounded like a good idea. He picked up his napkin and walked into Alistair's bedroom.
Egypt could wait. For a little while.
It smelled nice in there, an Alistair-old-manish kind of smell, cologne-y with the scent of fresh laundry. Everything was neat, which was no surprise — the photos lined up on the dresser, the stack of hardcover books on the bedside table, the pillows angled just so — with just a few casual touches, like a pair of gloves thrown on the far side of the bed...
A pair of _filthy_ white gloves.
Dan detoured away from the bathroom and lifted the gloves. They were caked with dirt and grass and something else....
Charcoal.
"Amy...?" Dan called. _"AMY, COME IN HERE!"_
A cry of happiness welled up but stopped in his throat, as his joy was knocked aside by a realization that made him see black.
Somehow Uncle Alistair was alive.
And he had ditched them again.
#
The old man shut the door of his office and sank into his leather chair. He swung around toward the window, propping his feet on a ledge. They hurt more than usual today. At his age, he disliked long walks.
From below, the muffled sound of traffic wafted upward, the frustrated shouts of motorists, the frenzied calls of sidewalk vendors. A constant reminder of life's true desperate meaning — speed, desire, possession. He was tired of it all. But it wouldn't be long now. The proper path was finally clear.
He flicked on his music system. Richard Strauss's _Death and Transfiguration._ Oddly appropriate, after what happened today.
A stressful day. What was necessary was not always pleasant.
Ah, well. First the death. Now the transfiguration.
He pressed a button on his intercom. "Eun-hee, please contact Mr. McIntyre for me. I have some news for him."
He waited a few seconds but received no response. Strange. Eun-hee had been there when he walked in a few moments ago. She never left her desk in the outer chamber.
"Eun-hee...?" he tried again.
The intercom crackled to life. But the reply was not at all what he expected.
"Hello, Uncle," said a deep, silken voice that sent a knife of fear down his spine. "I trust your trip to the park was pleasant?"
Bae Oh's bony finger began to shake. "Who... _who is this_?"
"Why, it's your heir," the voice returned. "What, did I spoil your day? And what a lovely day it was indeed, seeing me die and thus realizing you were spared the trouble of doing the job yourself."
"But..." Bae Oh sputtered. "How could you have survived...?"
"A lot of people are wondering this. But I guarantee that when I'm through with you, they won't be asking the same question."
Bae Oh may have been in his ninth decade, but his reflexes were still unmatched. He leaped from his chair and opened the door to the outer chamber.
The room was empty.
The distant sound of footsteps on the outer carpet resounded, then stopped. He was gone.
Bae Oh's knees crumpled. He propped himself on the edge of the desk, feeling his heartbeat race, as behind him, the music swelled.
The Hunt Is On
The race for the 39 Clues continues with more dangerous missions, top secret break-ins, and treacherous double-crossings. Stay one step ahead of the competition by following Amy and Dan's next adventure.
Turn the page for a sneak peek! (Just make sure none of your enemies are watching...)
#
If Amy Cahill had to list what was wrong with eleven-year-old brothers, their habit of disappearing would be _numero uno._
Or maybe the fact that they existed in the first place.
And then there was the whole burping the alphabet thing....
Amy stood in the middle of the Khan el-Khalili market in Cairo, her head swiveling frantically, trying to find her brother, Dan. The blur of jet lag was interfering with normal brain function. Dan had just been at her side a moment ago. Then she turned for _two seconds_ to buy a Nefertiti pencil, and when she turned back, Dan was gone.
The air was thick with heat and music and the calls of shopkeepers. Bright banners waved overhead. Tourists weaved through the streets, wearing their backpacks on their chests to safeguard against pickpockets and stopping to take pictures every few minutes. A woman in a head scarf dodged a row of turquoise chairs to follow after her two boys. A man with a crate full of oranges balanced them on his head with one hand. A tourist in a baseball cap and a T-shirt proclaiming I WANT MY MUMMY strolled past Amy, her camera held up in front of her face.
Amy felt the heat like waves against her skin. She hoped she wouldn't faint. Colors swarmed, faces dissolved, unfamiliar noises pounded against her ears. She had never liked crowds, and Cairo seemed like the city that had invented them.
She turned, her hand on her waist pack. Their au pair, Nellie Gomez, was just down the alley, bargaining over spices. Amy could just catch a glimpse of her crazy half-blond, half-black hair.
Less than an hour ago, they'd been in a taxi, riding into Cairo from the airport. Then when the cabdriver had casually pointed out the window and said, "The Khan market starts here, very good place," Nellie had suddenly yelled, "Stop!" Before they knew what was happening, they'd landed in the market with luggage and cat carrier. Saladin had meowed furiously when Nellie promised, "Just ten minutes, that's all I need, and then we'll go straight to the hotel.... Cool! Cardamom pods!" For Nellie, every new city was just another opportunity for weird food.
Finally, Amy spotted Dan through the crowd. He was pressed against a shop window crowded with souvenirs. She had a feeling he was captivated by the King Tutankhamen pencil sharpener, but it could have been the flashlight in the shape of a mummy.
As she crossed the alley, Dan kept appearing and disappearing through the meandering crowd. The hot sun was blinding. She hoped that air-conditioning was in her future.
The tourist in the I WANT MY MUMMY T-shirt drifted closer to Dan. She pushed her white sunglasses down her nose. Some small alarm chimed inside Amy. A man in a straw hat blocked her view, and she dodged to one side.
The tourist bent her index finger back at the first joint, as if she had a cramp. The hot sun glinted on something protruding from her nail.
"Dan!" Amy screamed. The music and the calls of the shopkeepers — _Five dollars, five dollars! —_ drowned her out. She darted past a man balancing a dozen neon-colored soccer balls in a net.
The hypodermic needle protruded out of the tourist's clawlike finger. Dan leaned closer to the window....
"Dan!" She screamed the name. In her head. But it came out like a strangled croak.
Amy threw herself forward. At the very last second, she flung out her hand. The needle jammed into the Nefertiti pencil and stuck.
For one swift second, all Amy could do was stare down at the glint of sunlight on metal. In slow motion, a drop of something lethal fell from the tip and hit the dust.
Amy looked into the face of Irina Spasky. Former KGB agent. Spy. Cousin.
Irina's left eye twitched. _"Blin!"_ She twisted her hand, but the needle remained stuck in the pencil.
The shopkeeper hurried over. "Beautiful lady, it is stuck on you. Here, I have more pencils for you!"
Irina turned on him fiercely. "I don't want your fancy pencils, shopkeeper of things!"
Amy and Dan didn't wait another second. Dan moved like a midfielder through the crowd, and Amy followed in his wake.
Legs pumping, they ran until their lungs burned, dashing through the maze of twisting alleys. Finally they stopped, bent over at the waist, and tried to catch their breath. When they looked up, Amy realized they were lost. Badly, stupidly, irredeemably lost.
"Nellie will be looking for us," Amy said. She flipped open her cell phone. "No signal. We'll have to find our way back."
"And hope we don't bump into Comrade Irina," Dan said. "I can skip the family reunion."
By now they were used to meeting family members with mayhem on their minds. Just weeks ago they'd had a hard time coping with the fact of their grandmother's death. After their parents died, Grace had been the most important person in Amy and Dan's life. Even though they didn't live with her, they spent weekends at her mansion outside of Boston, and she always took them for trips during the school year and in the summers. Grace's death from cancer knocked them off their feet.
But that had been only the first of many shocks to come.
Grace had invited the four branches of the Cahill family to the reading of her will. Appearing on a video, she'd offered them a choice. Take a million dollars and walk away or join in a chase for 39 Clues and become the most powerful person in the world. Even though the million had seemed like one sweet deal, Amy and Dan hadn't really hesitated. They knew Grace would want them to accept the challenge. For Grace, there was no such thing as the easy way out.
The decision had been easy. It was the living up to it that was hard. In her old life, Amy had thought _playing to win_ was Courtney Catowski spiking a volleyball on her head. Now she knew what competition was really about. Relatives like Irina played for keeps. She'd drug them, kidnap them, even kill them if she had to.
They started to walk. Amy felt as though they were going in circles. Like in a dream, where you run and run and get nowhere. Yesterday she'd been in Seoul, Korea. Before that, Tokyo and Venice. Vienna and Salzburg, Austria. Paris. Philadelphia. She'd even touched down on a private airfield in Russia.
She'd never had so many secrets before.
She'd never imagined she could be so afraid.
She'd never imagined she could be so brave.
Just a few days ago in Seoul they'd nearly been buried alive. Left for dead by people she trusted. Natalie and Ian Kabra... she wouldn't think about him. Wouldn't think about how he held her hand and told her that together they could form a great alliance. The alliance lasted a couple of hours, until he saw the opportunity to leave her for dead.
Wouldn't. Think. About. Ian.
Then they discovered that the only family member they almost-trusted, their uncle Alistair Oh, had double-crossed them as well. Pretended to be dead when he was clearly very much alive.
What had sent them hurtling through international air space to Cairo was a hint, no more than that. But they were used to grabbing on to hints and riding them for all they were worth. A pyramid shape and a word. Sakhet. The Egyptian goddess with the lion's head. Amy had bought several books before they left Korea and researched the goddess, but she still didn't know why they were sent here... or what, exactly, they were looking for.
Amy felt sweat trickle in rivers underneath her T-shirt. The temperature was over ninety. Her hair was sticking to the back of her neck. She thought of Ian, who no matter what the circumstances always looked so cool.
Wouldn't. Think. About. Ian.
The noise pressed against her ears, an exotic, whirling cacophony of horns honking, vendors shouting, cell phones ringing, and someone yelling over it all, "Move it, lame-o!"
Oh. That voice was not so exotic. It was Dan.
"Russian spy at two o'clock and gaining!" he hissed.
Irina hadn't seen them yet. She was too busy looking for them. She prowled along the opposite side, peering into shop windows.
Amy pulled Dan into a café. Men sat at tables, drinking tea and having murmured conversations or reading newspapers. Tourists sat with their guidebooks over glasses of juice. As Amy squeezed past, her bulging backpack slammed against a burly gentleman sitting with a glass of mint tea. The tea spilled on his white suit.
Every eye in the café turned to Amy. The _clackety-clack_ of a backgammon game stopped. She felt her face turn bright red. She hated being the center of attention at any time, and especially when she'd done something clumsy.
"S-s-sorry!" Amy stammered. Her stutter came out when she was nervous, and she hated it. She tried to mop up the mess.
"It's fine, young lady, do not worry." The man smiled kindly at her and waved to the waiter. "It is just tea."
On the walls, heavy antique mirrors reflected the scene. Amy saw her own red face, her fluttering hands, the eyes of the patrons... and the door opening. Even the tourist attire and white plastic sunglasses couldn't disguise the way Irina soldier-marched into the café, as if she were inspecting everyone in it for demerits.
And in exactly three seconds, her gaze would land on them.
#
The fat man stood up, giving them cover for an instant. Dan grabbed the chance. He dodged behind a thick curtain, pulling Amy after him.
They found themselves in a short hallway that led to a side door. They dashed outside.
Now they were in an even smaller alley that snaked behind the shops. They knew Irina would be out there in a matter of seconds. They dodged a cart piled high with crates and a surprised man sleeping in the sun. Seeing a back door to a shop, they ran through it into a storeroom. It was dark and dusty, and Dan started to wheeze.
"Use your inhaler," Amy said.
"It's... in... Nellie's... carry-on," Dan got out. He hated this feeling. As though someone were squeezing his lungs. It happened at the worst times.
"Good place for it. Come on."
Amy quickly led Dan out of the dusty storeroom and into the store. It was bright and airy, with spangled belly dancing costumes hanging from the ceiling.
"Welcome! You are looking for lovely costume? I'll give you a deal!"
"Not my color! But thanks!" Dan called as he ran out.
Down another twisting street, then another. Finally, Amy ordered a halt.
"We lost her."
"For now." Dan grabbed her elbow. "Amy, look."
Only a few feet away, they saw a sign: S A K H E T
In the dramatically red-curtained window, a statue stood alone. Blue stone, with a lion's head, standing tall and proud.
Amy and Dan looked at each other. Without a word, they pushed into the shop.
They made a beeline for the Sakhet statue. It was obviously very old. The surface was worn and one of the lion's ears had cracked off.
The shop owner hurried forward, a thin, eager man in black pants and a white shirt. "You are interested? She is beautiful. Authentic, not a replica. Once owned by Napoleon," the man went on. "You have an excellent eye."
"Napoleon? Isn't that an Italian pastry?" Dan asked. "Sort of gooey inside?"
Amy rolled her eyes. "You're the one with goo — for brains. Napoleon was the French emperor. Remember, he conquered the world? We saw a picture of him in the Lucian stronghold back in Paris? He's a Cahill. One of our ancestors."
The Lucian branch of the Cahill family had a strategic sense that was amazing. Of course, their powers had dwindled down to the petty, nasty deeds of Ian and Natalie Kabra and the crazy Russian Irina Spasky.
"If he'd picked out this Sakhet, it could be important," Dan said.
"It can't be this easy," Amy said.
"Why not, when everything else has been so hard?" Dan pointed out.
The shopkeeper raised his voice, trying to get them back again. "I see you are fascinated. Yes, Napoleon owned many treasures. Some went back to France, some stayed here." He put his hand on the statue and caressed it. "Are your parents with you? I'll give you the best price. I have the premier shop in Cairo."
"No, thanks," Dan said. Back home, he was a collector. He knew the best way to bargain was to pretend you didn't care. "Come on, Amy. Let's keep looking. Why would Napoleon have stuff in Egypt, anyway?"
"Napoleon invaded Egypt in 1798," Amy said.
"Ah, the young lady knows her history. I would be so proud if this statue came into her brilliant hands. Here." He gave her the statue.
It felt strange to touch something so old. Something Napoleon had touched. Every so often she got a deep thrill from a sense of her own DNA linking like a chain down a line leading to a bunch of extraordinary people. Napoleon!
"Only two thousand," he said.
Amy jumped. "Two thousand _dollars_?"
"For you, fifteen hundred. Someone from the Cairo Museum is interested in this piece. He is coming back at four o'clock."
"I doubt that, Abdul."
Amy turned. She'd noticed the tall, blond stranger browsing at the other part of the store. She hadn't noticed him draw closer. He was in his twenties, dressed in a T-shirt, khaki shorts, and sandals. His eyes were vivid green next to his tan.
"Unless he's looking for a trinket for his key ring," he said in a British accent.
He plucked the Sakhet from her hands. "I'd date this piece at... maybe 2007?"
"Really, Theo, you are mistaken," the shopkeeper said, smiling uneasily. "This is authentic, I assure you—"
"Assurances aside, I think you're trying to take these two youngsters for a ride on the fake artifact express," the man named Theo said.
"He said Napoleon had owned it," Dan said.
"Maybe," Theo said. "Joe Napoleon down the street runs a great Italian restaurant."
"I _told_ you Napoleon was Italian," Dan said scornfully to Amy.
"Actually, he was born in Corsica," Theo said. "Would you kids like to see the rest of the shop?"
"No need," Abdul said quickly. "I see I don't have what you want. Perhaps next door you'll find what you're looking for. It's time for my tea break, so..."
Theo strode past him and pushed open a heavy curtain. At a long table, several workers were hunched over. Amy stood on tiptoe as the shop owner tried to block her view. The workers were using wire brushes and sandpaper on a row of statues similar to the Sakhet. They were sanding and brushing them in order to make them look old.
Abdul shrugged. "Hey, it's a living."
"No harm, no foul," Theo said.
Just then Dan grabbed Amy's arm. Peering through the window and shading her eyes was Irina.
Theo had noticed their alarm. "Who's that? Your mother?"
"Someone on our tour. She's a total pain," Amy said.
"Always following us," Dan said. "Is there another way out of here?"
"One thing you should know about me," Theo said. "I _always_ know the back way."
The brass bell on the front door jangled as they pushed through the curtain and made their escape.
This time it was easier. All they had to do was follow Theo. He moved quickly and expertly through the maze of narrow alleys. Finally, they stopped to rest near the arched entrance to the market.
"I think you're safe," Theo said. "Can I get you a taxi back to your hotel?"
"We lost our au pair," Dan said. "We'd better find her. Uh, where are we?"
"Let's start with this. Where did you leave her?"
Amy frowned. "By some spices?"
"Okay, that narrows it down somewhat. Can you remember anything else?"
Dan closed his eyes. "A yellow sign with maroon letters in Arabic. Three rows of spice baskets, nuts in green buckets. Shopkeeper with mustache and a mole on his left cheek. Next door was a fruit stall, thin guy in a red hat yelling, 'Pomegranates!'"
Theo cocked an eyebrow at Amy. "Is he always like this?"
"Constantly."
Again they followed Theo through the market, keeping a careful eye out for Irina.
"Do you live here?" Amy asked him as they weaved through the crowds.
"Went to university in England but came back and haven't left since."
"You sure know your way around," Amy said.
"I used to be a tour guide," Theo said. He smiled at her, and Amy suddenly realized that he was seriously good-looking.
A fuming Nellie stood outside the stall where they'd left her. A string bag stuffed full of packages swung from her wrist. Dan's duffel was at her feet, and her own bag, plus the nylon tote they'd borrowed from Alistair, was piled on top. Saladin the cat meowed woefully in his cat carrier. She advanced on them furiously.
"Where have you been? I thought you were kidnapped!" Suddenly, Nellie caught sight of Theo. She stopped short. She gave Theo a long look, from the top of his blond head to his suntanned toes. "Well, hel-lo, Indiana Jones," she purred in a voice just like Saladin made when he caught sight of a filet of red snapper in his food bowl.
Since they'd left her, Nellie had gone shopping. Over her black T-shirt was a gauzy lavender fabric that she'd wound around her body to make a tunic. Black kohl now rimmed her eyes, and beaded bangles slid up her arm from wrist to elbow. Gold dusted her eyelids. She looked as though she were about to run off to a hip-hop harem.
"Well, hello, Mary Poppins," Theo replied with a grin.
"How astute. I'm practically perfect in every way," Nellie said. She stuck out her hand. "I'm Nellie Gomez."
"Theo Cotter."
Dan rolled his eyes as Nellie's hand stayed in Theo's longer than a handshake should take. Did Nellie actually _blush_? He didn't think she was capable of it.
"Theo saved us from buying an ancient priceless artifact that was made yesterday," Amy said.
Theo shrugged. "Unfortunately, you happened to stumble into one of the worst tourist traps around. I can show you some of the more authentic shops if you'd like," he said, his eyes on Nellie.
"That would be amazing," Nellie said, as if Theo had just offered to show her the secrets of the universe.
"I think we'd better get to our hotel," Amy said. Theo seemed okay, but why should they trust him? Besides, they didn't have time to waste. Before they'd left Seoul, they'd found a frequent-traveler's card in Alistair's bedroom. Dan had pocketed it and they'd used it at the airport to book a room at a hotel called the Excelsior. Amy was anxious to check in and figure out their next step. This was all happening too fast.
Theo took a couple of Nellie's bags from her. "You're interested in Napoleon, right?" he said to Amy. "Did you know that when he invaded Egypt he brought scholars and archaeologists and artists with him to study the country?"
_Well, isn't that sooo Lucian of him,_ Dan thought.
"The house where his scholars lived is a museum now. I know the curator there."
_Uh-oh,_ Dan thought. As soon as his sister heard the word _museum,_ she started to salivate. It was like waving a double-fudge brownie in front of her face.
"Is it nearby?" Amy asked eagerly. Maybe she should rethink this. If the house was still there, they might be able to find something to lead them to a Clue.
"Nothing is too far in Cairo," Theo said. "Sennari House. It's just over on Haret Monge."
"Right. We knew that," Dan said.
"Come on, I'll get us a taxi."
Theo turned and led the way to the frenzy of a downtown street. If there were lanes on the wide street, Dan couldn't see them. Cars slithered into tiny spaces, cut off trucks, accelerated at red lights, and tailgated buses, all to a symphony of horn blowing and yelling. Amy, Dan, and Nellie exchanged glances. They couldn't imagine how to find a taxi in the melee.
Theo stepped calmly out into the street, held up a hand, and a taxi skidded to a stop.
"You see?" Nellie said in awe. "He _is_ Indiana Jones."
Copyright © 2009 by Scholastic Inc.
All rights reserved. Published by Scholastic Inc., _Publishers since 1920._ SCHOLASTIC, THE 39 CLUES, and associated logos are trademarks and/or registered trademarks of Scholastic Inc.
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without written permission of the publisher. For information regarding permission, write to Scholastic Inc., Attention: Permissions Department, 557 Broadway, New York, NY 10012.
Library of Congress Control Number: 2008935103
First edition, March 2009
Scholastic US: 557 Broadway • New York, NY 10012
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All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of publisher.
e-ISBN 978-0-545-29274-0
|
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Q: Issues updating BindingList every second with DispatchingTimer I have a bindinglist with an ExpireTime which is a date, I then use a converter to convert it to a timespan so I can show how long until ExpireTime
private BindingList<PlayerSearchResultTransfer> _transferlistItems;
public BindingList<PlayerSearchResultTransfer> TransferlistItems
{
get { return _transferlistItems; }
set { _transferlistItems = value; RaisePropertyChanged(() => TransferlistItems);}
}
private readonly DispatcherTimer _uiTimer;
public TransferListViewModel(IPlayerSearchQuery playerSearchQuery)
{
_uiTimer = new DispatcherTimer();
_uiTimer.Tick += (s, e) => RaisePropertyChanged("ExpireTime");
_uiTimer.Interval = new TimeSpan(0,0,1);
_uiTimer.Start();
}
and the class I am binding against
public class PlayerSearchResultTransfer : PlayerSearchResult
{
private DateTime _expireTime;
public event PropertyChangedEventHandler PropertyChanged;
public DateTime ExpireTime
{
get { return _expireTime; }
set
{
_expireTime = value;
RaisePropertyChanged("ExpireTime");
}
}
My WPF
<UserControl.Resources>
<converters:TimeRemainingConverter x:Key="TimeRemainingConverter" />
</UserControl.Resources>
<Grid>
<DataGrid ItemsSource="{Binding TransferlistItems}"
AutoGenerateColumns="False">
<DataGrid.Columns>
<DataGridTemplateColumn Header="First Name" IsReadOnly="True">
<DataGridTemplateColumn.CellTemplate>
<DataTemplate>
<TextBlock Text="{Binding FirstName}" />
</DataTemplate>
</DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>
<DataGridTemplateColumn Header="Last Name" IsReadOnly="True">
<DataGridTemplateColumn.CellTemplate>
<DataTemplate>
<TextBlock Text="{Binding LastName}" />
</DataTemplate>
</DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>
<DataGridTemplateColumn Header="Expires" IsReadOnly="True">
<DataGridTemplateColumn.CellTemplate>
<DataTemplate>
<TextBlock Text="{Binding ExpireTime, Converter={StaticResource TimeRemainingConverter}}" />
</DataTemplate>
</DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>
<DataGridTemplateColumn Header="Bid" IsReadOnly="True">
<DataGridTemplateColumn.CellTemplate>
<DataTemplate>
<TextBlock Text="{Binding BidValue}" />
</DataTemplate>
</DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>
<DataGridTemplateColumn Header="Buyout" IsReadOnly="True">
<DataGridTemplateColumn.CellTemplate>
<DataTemplate>
<TextBlock Text="{Binding BuyoutValue}" />
</DataTemplate>
</DataGridTemplateColumn.CellTemplate>
</DataGridTemplateColumn>
</DataGrid.Columns>
</DataGrid>
</Grid>
</UserControl>
And my converter
[ValueConversion(typeof(DateTime), typeof(TimeSpan))]
public class TimeRemainingConverter : IValueConverter
{
public object Convert(object value, Type targetType, object parameter, CultureInfo culture)
{
var dateTime = (DateTime)value;
var timespan = dateTime - DateTime.Now;
return timespan < new TimeSpan(1000)
? new TimeSpan(0)
: new TimeSpan(timespan.Hours, timespan.Minutes, timespan.Seconds);
}
public object ConvertBack(object value, Type targetType, object parameter, CultureInfo culture)
{
throw new NotImplementedException();
}
}
Solution
Had forgot to add INotifyPropertyChanged to my PlayerSearchResultTransfer
A: You are raising PropertyChanged event on ViewModel class. It should be raised on every instance of PlayerSearchResultTransfer for converter to fire for all items.
_uiTimer.Tick += (s, e) =>
{
foreach (PlayerSearchResultTransfer player in TransferlistItems)
{
player.RaisePropertyChanged("ExpireTime");
}
};
Also, make sure PlayerSearchResultTransfer class implements INotifyPropertyChanged to let GUI refresh on any property change.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,038
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Masiewicze (; ) – wieś na Białorusi, w obwodzie brzeskim, w rejonie pińskim, w sielsowiecie Merczyce, nad Jasiołdą.
W dwudziestoleciu międzywojennym leżały w Polsce, w województwie poleskim, w powiecie pińskim, do 18 kwietnia 1928 w gminie Stawek, następnie w gminie Żabczyce. Po II wojnie światowej w granicach Związku Sowieckiego. Od 1991 w niepodległej Białorusi.
Przypisy
Bibliografia
Masiewicze na mapie WIG
Masiewicze na mapie WIG
Wsie w rejonie pińskim
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,360
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Here is another idea I have been working on, and finally decided to get something into 3D CAD for it. I am calling it Spider-Bot for now, since it is inspired by the arachnid form.
The top three mounting positions are for various things, one of which is a pan/tilt platform for a small camera as well as a TPA81 thermopile sensor. The rear mounting position could be for various things also, including some sort of tail. The four mounting positions on each side are for the legs, using leg angles figured out by Alan (KM6VV on the Lynxmotion Forums).
This particular arachnid design is meant for an electronics package no larger than 3" x 3" and I am tentatively ear marking this for a Hammer-RDP as the main controller and a p-Brain-ds24 controller for locomotion.
This robot design should be able to use either standard or micro servos. So, this my design for an Octapod. :happy::happy: Spider-Bot could potentially be turned into a hybrid robot by adding a three wheel tricycle type motorized wheel arrangement. I have some other ideas about this also, and the three wheel arrangement was in my head as I designed the body. Now I am working on the top deck.
The attached 3D PDF shows both decks, and the body as it might be put together for an SES compatible robot. I am designing a custom SES compatible leg pair now, including custom femur and tibia inspired by Matt Denton's hexapod leg designs.
I would like to design a version of this that would use micro servos also. I'm in the process of designing the femur and tibia, which are inspired by Matt's (mdenton) hexapod leg designs.
Your body design for an octopod looks great! Seems to be quite small too.
Looking forward to see more of this project of yours, especially how you choose to design the legs. Have you got a p-brain controller from Matt Denton?
Yes, the body is very small. If you look at the mounting holes on the top deck, those are Lynxmotion SSC-32 compatible. :veryhappy: I actually designed the body for smaller legs that would use micro servos. The Hammer-RDP is being designed to fit the same mounting holes, in addition to allowing stackable add-on boards.
Looking forward to see more of this project of yours, especially how you choose to design the legs.
Right now, I am designing the legs to be fully SES compatible. I'm working on the femur and tibia now, playing with a few different possibilities. Maybe I'll come up with my own custom legs design and see what happens. :happy: I'm not just playing with new leg designs, but will also be tinkering with different ways of attaching the femur and tibia as soon as I complete the 3D models.
Have you got a p-brain controller from Matt Denton?
No, I don't have one, but wish I did. I am axiously awaiting the release of the p.Bran-ds24 and p.Brain-SMB. The p.Brain-ds24 has the kind of computing power I want to use for other projects as well.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,591
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DJ Ozone (San Francisco) plays only original, vinyl records, from the 1980s. It's fun, silly, carefree party music -- 1980s synth-pop, Italo disco, post-punk, old school hip hop and electro. Ozone's approach is to the let the songs play, house party style -- not making elaborate mixes or using too many effects. In a time of trendy computer DJs with automatic edits, it's refreshing to hear a human touch -- just great songs, nothing serious, played raw and fresh as ever.
Starting as a radio DJ in California, Ozone has been involved with the China music scene since 2003, co-founding The Antidote nights in Shanghai, then touring around China and Asia with hundreds of DJs, bands, and recording artists from China and from around the world. He managed the bookings at Shanghai's legendary Tang Hui and 4-Live clubs (2005-2007) before opening clubs Dada Shanghai in 2009 and Dada Beijing in 2012.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,070
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{% extends 'crud/interfaces/crud.tabular.html' %}
{% load i18n %}
{% block js-inline %} {{ block.super }}
<script>
$(function () {
var bulk_add = ' <a href="{{ bulk_add_url }}" class="btn btn-primary"><i class="fa fa-copy"></i> Copy Indicators to another Project</a>';
$('#reportFiltersAccordion .row .col-xs-8').append(bulk_add);
});
</script>
{% endblock %}
{% block reportcontent %}
<div class="alert alert-success hide" id="js-add-crud-success">
{% trans 'Indicator Successfully Added to End of Table' %}
</div>
{{ block.super }}
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 543
|
\section{Introduction} \label{section : 1}
Let $(M, g)$ be a smooth connected compact Riemannian manifold of dimension $n \geq 2$ with smooth boundary $\partial M$. The Steklov problem on $(M, g)$ consists in finding all $\sigma \in \mathbb{R}$ such that there exists a non-zero harmonic function $f : M \longrightarrow \mathbb{R}$ satisfying $\frac{\partial f}{\partial \nu} = \sigma f$ on $ \partial M$, where $\frac{\partial }{\partial \nu}$ denotes the outward-pointing normal derivative on $\partial M$.
\medskip
Such a $\sigma$ is called a Steklov eigenvalue of $M$ and a corresponding $f$ is called a Steklov eigenfunction. The (ordered) set of eigenvalues is called the Steklov spectrum of $(M, g)$.
\medskip
It is well known that the Steklov spectrum of $M$ forms a discrete sequence
\begin{align*}
0 = \sigma_0 < \sigma_1 \leq \sigma_2 \leq \ldots \nearrow \infty,
\end{align*}
where each eigenvalue is repeated with multiplicity.
\medskip
There exists a discrete analog to the Steklov problem, which is called the discrete Steklov problem and which is defined on graphs with boundary. Let us begin by defining it.
\begin{defn}
A graph with boundary is a triplet $(\bar{\Omega}, E', B)$, where $(\bar{\Omega}, E')$ is a simple connected undirected graph and $B \subset \bar{\Omega}$ is a non-empty set of vertices, called the boundary. The set $B^c$ is called the interior of the graph.
\end{defn}
In this paper, all graphs will always be simple connected and undirected.
\medskip
For $v, w \in \bar{\Omega}$, we write $v \sim w$ when $v$ is adjacent to $w$. For $A \subset \bar{\Omega}$, we write $|A|$ the cardinality of $A$, which is the number of vertices contained in $A$. For the purpose of this article, all graphs with boundary are finite. We denote by $\mathbb{R}^{\bar{\Omega}}$ the space of all functions $u : \bar{\Omega} \longrightarrow \mathbb{R}$, which is isomorphic to the Euclidean space of dimension $|\bar{\Omega}|$. Similarly, we denote by $\mathbb{R}^B$ the space of functions $u : B \longrightarrow \mathbb{R}$, which is the Euclidean space of dimension $|B|$.
\medskip
We can now introduce the discrete Laplacian operator $\Delta : \mathbb{R}^{\bar{\Omega}} \longrightarrow \mathbb{R}^{\bar{\Omega}}$, defined by
\begin{align*}
\Delta u : \bar{\Omega} & \longrightarrow \mathbb{R} \\
v & \longmapsto \Delta u(v) = \sum_{w \sim v}(u(v)-u(w)).
\end{align*}
The normal derivative $\frac{\partial}{\partial \nu} : \mathbb{R}^{\bar{\Omega}} \longrightarrow \mathbb{R}^B$ is defined by
\begin{align*}
\frac{\partial u}{\partial \nu} : B & \longrightarrow \mathbb{R} \\
v & \longmapsto \frac{\partial u}{\partial\nu}(v) = \sum_{w \sim v}(u(v)-u(w)).
\end{align*}
As one can see, the normal derivative coincides with the restriction of the Laplacian to the boundary. Although this choice may seem strange, it is shown in \cite{CGR} that it leads to interesting links between the Steklov spectrum of a manifold and the Steklov spectrum of a graph with boundary which \textit{looks like} the manifold, see \parencite[Theorem 3]{CGR} for more information about what \textit{looks like} means in this context.
\begin{defn}
The discrete Steklov problem on a finite graph with boundary $(\bar{\Omega}, E', B)$ consists in finding all $\sigma \in \mathbb{R}$ such that there exists a non-zero function $u \in \mathbb{R}^{\bar{\Omega}}$ such that
\begin{align*}
\left\{
\begin{array}{l}
\Delta u(v) = 0 \mbox{ if } v \in \Omega \\
\frac{\partial}{\partial \nu}u(v) = \sigma u(v) \mbox{ if } v \in B.
\end{array}
\right.
\end{align*}
\end{defn}
Such a $\sigma$ is called a Steklov eigenvalue and a corresponding $u$ is called a Steklov eigenfunction of $(\bar{\Omega}, E', B)$. As said in \cite{Per1}, the Steklov spectrum of a graph with boundary $(\bar{\Omega}, E', B)$ forms a sequence as follows:
\begin{align*}
0 = \sigma_0 < \sigma_1 \leq \sigma_2 \leq \ldots \leq \sigma_{|B|-1}.
\end{align*}
This problem has recently received a particular attention, one can cite for instance \cite{HH, HHW, Per2, Per1}. An investigation has been made by Colbois, Girouard and Raveendran in \cite{CGR}, allowing us to understand some spectral links between the Steklov problem on a manifold and the discrete Steklov problem of a graph associated to this manifold. These links will be very useful in this paper. The main problem that we will have to face is to place ourselves in the hypotheses of Theorem $3$ of \cite{CGR}, in order to use it to our advantage.
\medskip
Among other things, a question that has been studied by some authors is that of providing an upper bound for the first - and then for the $k^{\mbox{th}}$ - eigenvalue of some particular graphs with boundary. These particular graphs that have been studied are those called \textit{subgraphs} of an (infinite) \textit{host graph}.
A subgraph of a host graph can be interpreted as the discrete analog of a bounded domain in a manifold.
Let us define what it is exactly.
\begin{defn} \label{def : subgraph}
Let $\Gamma = (V,E)$ be a graph and let $\Omega \subset V$ be a finite subset of vertices connected for $\Gamma$, i.e for each $v, w \in \Omega$, there exist $l \in \mathbb{N}$ and $v_0 = v, v_1, \ldots, v_l = w \in \Omega$ satisfying $\{v_i, v_{i+1}\} \in E$ for all $i = 0, \ldots, l-1$. The graph with boundary $(\bar{\Omega}, E', B)$ induced by $\Omega$ is defined as follows:
\begin{itemize}
\item $ B = \{w \in V \backslash \Omega : \exists \; v \in \Omega \; \mbox{such that} \; \{v,w\} \in E\}$;
\item $\bar{\Omega} = \Omega \cup B$;
\item $E' = \{ \{v,w\} \in E : v \in \Omega , w \in \bar{\Omega} \}$.
\end{itemize}
Such a graph with boundary is simply denoted $\Omega$ and is called subgraph of $\Gamma$. The set of vertices $B$ is the boundary of the subgraph. We refer to $\Gamma$ as the host graph of $\Omega$.
\end{defn}
Some interesting results have recently been discovered, providing us with bounds for the eigenvalues, depending on the host graph $\Gamma$.
A first result, due to Han and Hua, is the following:
\begin{thm}[Theorem $1.2$ in \cite{HH}] \label{corollaire : sigma_1 HH}
Let $\Z^d$ be the integer lattice of dimension $d$. Let $\Omega$ be a subgraph of $\Z^d$. Then we have
\begin{align*}
\sum_{l=1}^{d} \frac{1}{\sigma_l(\Omega)} \geq C' \cdot |\Omega|^{\frac{1}{d}} - \frac{C''}{|\Omega|},
\end{align*}
where $C' = (64 d^3 \omega_d^{\frac{1}{d}})^{-1}, C''=\frac{1}{32d}$ and $\omega_d$ is the volume of the unit ball in $\mathbb{R}^d$.
\end{thm}
Another investigation gives some control over the spectrum of a subraph of a \textit{Cayley graph}. We recall that, given a finitely generated group $G$ and a finite generating subset $S$ of $G$, one can define a graph, called Cayley graph and denoted $Cay(G,S)$. If $G$ is infinite, then so is $Cay(G,S)$ and we can use it as a host graph. The result provided by Perrin is the following:
\begin{thm}[Corollary $1$ in \cite{Per2}] \label{corollaire : sigma_1 HP}
Let $\Gamma = (V,E)$ be a Cayley graph with polynomial growth of order $d \geq 2$. There exists $\tilde{C}(\Gamma) >0$ such that for any finite subgraph $\Omega$ of $\Gamma$, we have
\begin{align*}
\sigma_1(\Omega) \leq \tilde{C}(\Gamma) \cdot \frac{1}{|B|^{\frac{1}{d-1}}}.
\end{align*}
\end{thm}
This theorem is way more general about the class of host graph $\Gamma$ but provides us control over the first non-trivial eigenvalue only, see \cite{Per2} for details. We gave an extension to this result in a precedent article:
\begin{thm}[Theorem $5$ in \cite{T}] \label{thm : moi}
Let $\Gamma = Cay(G,S)$ be a polynomial growth Cayley graph of order $d \geq 2$. Let $\Omega$ be a subgraph of $\Gamma$. Then there exists a constant $\bar{C}(\Gamma) >0$ such that for all $k < |B|$,
\begin{align*}
\sigma_k(\Omega) \leq \bar{C}(\Gamma) \cdot \frac{1}{|B|^{\frac{1}{d-1}}} \cdot k^{\frac{d+2}{d}}.
\end{align*}
\end{thm}
As a corollary, we have:
\begin{corollaire}[Corollary $6$ in \cite{T}] \label{corollaire : sigma_k}
Let $\Gamma$ be a polynomial growth Cayley graph of order $d \geq 2$ and $(\Omega_l)_{l=1}^\infty$ be a sequence of subgraphs of $\Gamma$ such that $|\Omega_l| \underset{l \to \infty}{\longrightarrow} \infty$. Fix $k \in \mathbb{N}$. Then we have
\begin{align*}
\sigma_k(\Omega_l) \underset{l \to \infty}{\longrightarrow} 0.
\end{align*}
\end{corollaire}
All these theorems follow from the investigation upon one class of host graphs $\Gamma$, which are Cayley graphs of polynomial growth groups. This consideration leads to a natural question:
\begin{center}
\textit{What can we say about the eigenvalues of subgraphs of a host graph $\Gamma$, whose growth rate is more than polynomial?}
\end{center}
A first class of graphs we can think of is that of trees. In \cite{He-Hua}, the authors find upper bounds for the eigenvalues of a finite tree. Their investigations lead to the following result:
\begin{thm}[Theorem $1.1$ and $1.5$ in \cite{He-Hua}] \label{thm : HH trees}
Let $\mathcal{T}$ be a finite tree with (uniformly) bounded degree $D$. Let $B$ be the boundary of the tree, i.e the set of vertices of degree one. Then we have
\begin{align*}
\sigma_1 \leq \frac{4 (D-1)}{|B|}.
\end{align*}
Higher Steklov eigenvalues are bounded as well: for all $k = 2, \ldots, |B|-1$, we have
\begin{align*}
\sigma_k & \leq \frac{8(D-1)^2(k-1)}{|B|}.
\end{align*}
\end{thm}
As stated by Remark $1.7$ of \cite{He-Hua}, we can consider as the host graph $\Gamma$ the Cayley graph of a free group and use this result to estimate the Steklov eigenvalues of a subgraph $\Omega$ of $\Gamma$. Since the growth rate of such a host graph is exponential, we now have a completely new class of host graphs for which we can estimate their subgraphs eigenvalues.
\medskip
This paper's objective is to study the subgraphs's eigenvalues of a host graph $\Gamma$ which is roughly isometric to the hyperbolic plane $\H^2$ (see Definition \ref{def : hyperbolic plane}). The hyperbolic plane is a Cartan-Hadamard manifold of constant sectional curvature $-1$. Then $\Gamma$ can be seen as a discrete analog of such a manifold. Because of its relation with $\H^2$, the growth rate of $\Gamma$ is exponential, and then $\Gamma$ does not enter the class of host graphs of Theorems \ref{corollaire : sigma_1 HH}, \ref{corollaire : sigma_1 HP} and \ref{thm : moi}.
Despite a growth rate identical to that of the trees, the structure of $\Gamma$ is very different from the latter, because of its connection with $\H^2$. Therefore, the method we will use to obtain upper bounds has nothing to do with the one used in \cite{He-Hua}. Indeed, He and Hua were able to work directly on the trees and use the great ease of disconnection of the trees as a tool to obtain the bounds of Theorem \ref{thm : HH trees}, while on our side we will use the proximity between $\Gamma$ and $\H^2$ to obtain upper bounds.
\medskip
There are many graphs which are roughly isometric to the hyperbolic plane. This paper will focus on a particular class of such graphs, coming from a tiling of $\H^2$ associated with a triangle group. We shall refer to such a graph as \textit{triangle-tiling graph}.
Triangle groups are part of the Coxeter groups, which can be seen as groups generated by reflections. These groups have been studied by many authors, see for instance \cite{Bou, Hil, Hum}. Triangle groups are Coxeter groups with three generators, that can be regarded as reflections through the sides of a triangle. They lead to many beautiful geometric constructions and tiling, see \cite{Bea, CBG, Mag, Wiki_pavage_triangulaire}. We will recall in Sect. \ref{section : 2} hereafter the notions that are required for the understanding of the paper.
\medskip
Our main result is the following:
\begin{thm} \label{thm : principal}
Let $\Gamma$ be a triangle-tiling graph. Then there exists a constant $C = C(\Gamma) >0$ such that for all subgraph $\Omega$ of $\Gamma$ and all $k < |B|$, we have
\begin{align*}
\sigma_k(\Omega) \leq C(\Gamma) \cdot \frac{1}{|B|} \cdot k^{2}.
\end{align*}
\end{thm}
As we will see in Sect. \ref{section : 2}, the host graph $\Gamma$ is defined from the choice of three integers. As a consequence, we will see that there are infinitely many triangle-tiling graphs.
As a corollary, we obtain the interesting fact:
\begin{corollaire} \label{cor : zero}
Let $(\Omega_l)_{l \geq 1}$ be a family of subgraphs of $\Gamma$ such that $|\Omega_l| \underset{l \to \infty}{\longrightarrow} \infty$. Then for all $k \in \mathbb{N}$ fixed,
\begin{align*}
\sigma_k(\Omega_l) \underset{l \to \infty}{\longrightarrow} 0.
\end{align*}
\end{corollaire}
The number $\sigma_k(\Omega_l)$ is of course defined if and only if $|B_l| < k$. This condition is satisfied for $l$ big enough thanks to the assumption that $|\Omega_l| \longrightarrow \infty$.
\medskip
Our approach is sketched this way: we define a triangle-tiling graph $\Gamma$ that we use as a host graph and show that it is roughly isometric to $\H^2$ (see Definition \ref{def : quasi isom}). Thanks to the rough isometry, we can naturally associate to a subgraph $\Omega$ of $\Gamma$ a bounded domain $N$ of $\H^2$, whose boundary will be denoted $\Sigma$. We can then use results from \cite{CEG1} to give upper bounds for $\sigma_k(N)$.
Once this task is completed we use the work of Colbois et al. presented in \cite{CGR} in order to discretize a Riemannian manifold with boundary $(N, g')$, obtained as a deformation of the domain $N$ (this deformation is necessary since we have to satisfy the assumptions of \parencite[Theorem 3]{CGR}). This discretization will give us a path linking the eigenvalues of $N$ and the ones of $\Omega$, which will allow us to conclude.
\medskip
Our strategy can be summed up in the diagram below:
\begin{center}
\begin{tikzpicture}
\node (1) at (-10,-1) {$\Gamma$} ;
\node (2) at (-4,-1) {$\Omega$} ;
\node (3) at (-10,-4) {$\H^2$} ;
\node (4) at (-4,-4) {$(N, g)$} ;
\node (5) at (2,-4) {$(N, g')$} ;
\node (6) at (2,-1) {$(\bar{V}, \bar{E}, V_\Sigma)$} ;
\draw[<->] (1) to node[pos=0.5]{\; roughly isometric} (3) ;
\draw[->] (1) to node[pos=0.5, above]{subgraph} (2) ;
\draw[->] (3) to node[pos=0.5, above]{domain} (4) ;
\draw[<->] (2) to node[pos=0.5]{structure preserved} (4) ;
\draw[->] (4) to node[pos=0.5, above]{change of metric} (5) ;
\draw[->] (5) to node[pos=0.5]{discretization} (6) ;
\draw[<->] (2) to node[pos=0.5, above]{roughly isometric} (6) ;
\end{tikzpicture}
\end{center}
Here, by \textit{structure preserved}, we mean that the structural information of the subgraph $\Omega$ can be read in the domain $N$, see the rest of the paper for more details. Moreover, in the diagram, $P \longleftrightarrow Q$ reflects the idea that $P$ is in some sense analog to $Q$, and $P \longrightarrow Q$ reflects the idea that $Q$ is obtained from $P$. More details are given in the rest of the paper.
\medskip
Our result holds for subgraphs of any triangle-tiling graph. However, there exist many other graphs that are roughly isometric to the hyperbolic plane, and that we could use as host graphs. This remark naturally leads to many interesting interrogations, that we will consider and develop in Sect. \ref{sect : interrogation}. In particular, one may ask if the result is still true when using other host graphs roughly isometric to $\H^2$. This leads to the following open question (\Cref{quest : autre graphe}):
\begin{center}
\textit{If $\Gamma$ is any graph roughly isometric to the hyperbolic plane, is there a constant $C = C(\Gamma)$ such that a bound as in Theorem $\ref{thm : principal}$ exists?}
\end{center}
Moreover, if $(\Omega_l)_{l\ge 1}$ is a sequence of subgraphs such that $|\Omega_l| \underset{l \to \infty}{\longrightarrow} \infty$, then in many cases (\Cref{corollaire : sigma_k}, \Cref{cor : zero}, \parencite[Corollary 1.4]{He-Hua}) the behaviour of $\sigma_1(\Omega_l)$ satisfy $\sigma_1(\Omega_l) \underset{l \to \infty}{\longrightarrow} 0$. However, that is not always true, see \parencite[Example 3.7]{HH}. One may ask if the property is preserved under rough isometry (\Cref{question : comportement}):
\begin{center}
\textit{Let $\Gamma_1, \Gamma_2$ be two roughly isometric host graphs. Let us assume that in $\Gamma_1$, each sequence of subgraphs $(\Omega_l)_{l\ge 1}$ such that $|\Omega_l| \underset{l \to \infty}{\longrightarrow} \infty$ satisfies $\sigma_1(\Omega_l) \underset{l \to \infty}{\longrightarrow} 0$. Does $\Gamma_2$ also have this property?}
\end{center}
As said before, these interrogations, and other (including some about higher dimensional constructions), will be asked in Sect. \ref{sect : interrogation}.
\medskip
\textbf{Notation.} Throughout this paper, we shall work on graphs, on domains of $\H^2$ and on a manifold obtained from the domains. As stated before, the host graph will be denoted $\Gamma = (V, E)$. A subgraph of $\Gamma$ is denoted $\Omega$, while $N$ and $\tilde{N}$ are used to speak about domains of $\H^2$. We use $g$ to denote the metric of $\H^2$ and $g'$ the one of the manifold; hence $(N, g')$ is the notation we will use to speak about the manifold. A discretization of the manifold will be called $(\tilde{V}, \tilde{E}, V_\Sigma)$. We shall use the letters $v, w$ to speak about vertices of graphs and $x, y, z$ for elements of the domains or manifold. Several constants will appear, we shall call them $C_1, C_2, \ldots$; each $C_l$ is used exactly once.
\medskip
\textbf{Plan of the paper.}
In Sect. \ref{section : 2}, we define precisely what is a triangle-tiling graph. In Sect. \ref{section : 3}, we make the constructions. The leading idea is actually simple: we want to associate a domain to a subgraph. However, we encounter some difficulties for different reasons. One of them is the question of the isolated boundary vertices, also called \textit{bad boundary vertices} in \parencite[Def. $3.1$]{HH}. We solve this problem in Sect. \ref{subsection : domaine rugueux}. Another difficulty comes from the fact that we want the domain to have a smooth boundary. This is the object of Sect. \ref{subsection : domaine lisse}. In Sect. \ref{section : 4} we prove Theorem \ref{thm : principal}. In order to do so, we want to use Theorem $3$ of \cite{CGR}. Therefore we have to make sure that the hypotheses of the theorem are verified, which is the object of Sect. \ref{subsection : chmt metric}. Once it is done, we apply the theorem and conclude the proof.
\medskip
\textbf{Acknowledgment.}
I would like to warmly thank my thesis supervisor Bruno Colbois for
having allowed me to work on this subject as well as for his uncountable help and piece of advice which enabled
me to resolve the difficulties encountered. I also wish to thank Niel Smith, Antoine
Gagnebin and the anonymous referees for their careful proofreading of this paper and for their various remarks which have led to its improvement.
\section{Triangle groups and associated triangle-tiling graphs} \label{section : 2}
Let us begin by explaining what triangle groups are and what links they have with tessellations of the model spaces $\mathbb{S}^2, \mathbb{E}^2$ and $\H^2$. When it is done, we can explain how to associate a triangle-tiling graph $\Gamma$ to a triangle group.
\begin{defn}
Let $p, q, r \geq 2$ be integers. The triangle group $T^*(p,q,r)$ associated is
\begin{align*}
T^*(p,q,r)= \langle P, Q, R : P^2=Q^2=R^2=(PQ)^r=(QR)^p=(RP)^q=1 \rangle.
\end{align*}
\end{defn}
In order to see the links between such an abstract group and a group of reflection, one can think about $P, Q, R$ as reflections through the opposite sides of a triangle with angles $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$ respectively.
\medskip
It is well known that a triangle with angles $\alpha, \beta, \gamma$ satisfies $\alpha + \beta + \gamma > \pi$ in the spherical case, while we have $\alpha + \beta + \gamma = \pi$ in the Euclidean case and that $\alpha + \beta + \gamma < \pi$ in the hyperbolic case. Hence we can regroup the unordered triplets $p,q,r$ according to the value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$. If the number obtained is greater than $1$ we have to think about a spherical triangle, if it is equal to $1$ we have to think about a Euclidean one and if it is less than $1$ we have to think about a hyperbolic one.
\medskip
As said before, we want to work on graphs that have exponential growth rates, therefore we will only consider the third case in this paper. Since one may ask if our result is still true for the two other cases, we remark that in the Euclidean case, the triangle group has polynomial growth rate and then has already been studied in \cite{T}. Regarding the spherical case, the triangle group is finite and hence one can theoretically compute all different possible situations.
\medskip
Then, from now on, $p, q, r \geq 2$ will be integers satisfying
\begin{align*}
\frac{1}{p}+\frac{1}{q}+\frac{1}{r} <1.
\end{align*}
\begin{defn} \label{def : hyperbolic plane}
We denote by $\H^2$ the hyperbolic plane, represented here by Poincaré's disk model, which is
\begin{align*}
\H^2 = \{(x,y) \in \mathbb{R}^2 : x^2+y^2<1\},
\end{align*}
endowed with the Riemannian metric
\begin{align*}
g(x,y) = 4 \cdot \frac{dx^2 +dy^2}{(1-x^2-y^2)^2}.
\end{align*}
We denote by $d_g( \cdot, \cdot)$ the distance induced by the metric $g$.
\end{defn}
\begin{rem}
It is a known fact that for any triplet $0 \le \alpha, \beta, \gamma <\pi$ such that $\alpha + \beta + \gamma < \pi$, there exists a hyperbolic triangle with angles $\alpha, \beta, \gamma$. Moreover, there is a unique one up to isometry \parencite[Exercise 7.12]{Bea}.
Hence, given $p, q, r$ as before, there exists a unique triangle which has angles $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$.
\end{rem}
We state now Theorem $2.8$ of \cite{Mag}:
\begin{thm}
Let $P, Q, R$ be the reflections in the sides of a hyperbolic triangle $\Delta_0$ with angles $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$. The images of $\Delta_0$ under the action of the distinct elements of the group $T^*(p,q,r)$ generated by $P, Q, R$ fill the hyperbolic plane without gaps and overlapping.
\end{thm}
This means that the choice of the numbers $p, q, r$ gives rise to a tessellation of the hyperbolic plane. Moreover, we know \parencite[Theorem 7.4.1]{Bea} that reflections through geodesics are isometries of $\H^2$. Hence, each tile of the tessellation is a triangle which is isometric to the initial one.
\begin{figure}[H]
\centering
\includegraphics[scale=0.2]{images/hyperbolic_domains_black.png}
\caption{Tiling of the hyperbolic plane with congruent triangles of angles $\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{\pi}{9}$, coming from \cite{Wiki_image}.}
\end{figure}
From such a tiling associated with a triangle group $T^*(p,q,r)$, one can naturally define an infinite simple connected undirected graph $\Gamma = \Gamma(p,q,r)$, called a triangle-tiling graph and that we will use as a host graph. We explain here how to define $\Gamma$.
Each triangle contains a point that is the center of its inscribed circle \parencite[Theorem 7.14.1]{Bea}. We consider these points. They form the set $V$ of vertices of $\Gamma$. The graph structure of $\Gamma$ is defined as follows: two vertices $v_1, v_2 \in V$ are joined by an edge $\{v_1, v_2\}$ if and only if they belong to two adjacent triangles.
\medskip
It is then obvious that $\Gamma = (V, E)$ is an infinite, $3$-regular graph.
\medskip
We can see $\Gamma$ as a metric space when endowed with the path metric: each edge is of length $1$, the distance between two vertices $v_1, v_2 \in V$ is the minimal number of edges we have to cross to go from $v_1$ to $v_2$.
Because of its links with $\H^2$, it is clear that $\Gamma$ has an exponential growth rate. Hence, as said in Sect. \ref{section : 1}, $\Gamma$ does not enter the class of graphs concerned by Theorem \ref{corollaire : sigma_1 HH}, \ref{corollaire : sigma_1 HP} and~\ref{thm : moi}.
Moreover, $\Gamma$ has cycles, therefore it is not a tree. Hence, it does not enter the class of graphs of Theorem \ref{thm : HH trees} either.
\medskip
We recall that, given a connected locally finite graph $X$ and any vertex $v$ of $X$, the number of ends of $X$ is $\lim_{n \to \infty} \|X \backslash B(v,n)\|$, where $B(v,n)$ is the ball centered at $v$ with radius $n$ and $\|X \backslash B(v,n)\|$ is the number of infinite connected component of $X \backslash B(v,n)$. It is well known that two roughly isometric graphs have the same number of ends, see \parencite[Prop. 8.2.8]{Loh}. It is obvious that $\Gamma$ has $1$ end while a Cayley graph of a free group have infinitely many. Therefore, as said before, the structure of $\Gamma$ is completely different from the graphs concerned by Theorem \ref{thm : HH trees} and this difference will be felt in the way we solve the problem.
\begin{defn} \label{def : quasi isom}
A rough isometry between two metric spaces $(X, d_X)$ and $(Y, d_Y)$ is a map $\phi : X \longrightarrow Y$ such that there exist constants $C_1 >1, C_2, \; C_3 >0$ satisfying
\begin{align*}
C_{1}^{-1} \cdot d_X(x_1,x_2) -C_{2} \leq d_Y(\phi(x_1), \phi(x_2)) \leq C_{1} \cdot d_X(x_1,x_2) + C_{2}
\end{align*}
for all $x_1,x_2 \in X$ and satisfying
\begin{align*}
\bigcup_{x \in X} B(\phi(x), C_{3}) = Y.
\end{align*}
If there is such a map, we say that $X$ is roughly isometric to $Y$.
\end{defn}
\begin{prop}
The host graph $\Gamma$ constructed above is roughly isometric to $(\H^2, g)$, with constants that depend on the value of $p, q, r$.
\end{prop}
\begin{proof}
Take $\phi : \Gamma \longrightarrow \H^2$ as the canonical inclusion and take the constants as the triangle's diameter.
\end{proof}
\medskip
\section{Construction of the domain \texorpdfstring{$N$}{n, sigma}} \label{section : 3}
We consider a finite subset of vertices $\Omega \subset V$, connected for $\Gamma$, giving birth to a subgraph with boundary $\Omega$ as in Definition \ref{def : subgraph}. We recall that each vertex is the center of a triangle of the tiling and that all triangles are isometric.
\medskip
This section aims to detail a method allowing us to associate a smooth bounded domain $N$ to the subgraph $\Omega$. The relevance of the domain $N$ lies within its structural links with the subgraph $\Omega$: we will transcribe the structure of $\Omega$ onto $N$.
\medskip
Before starting, we want to give an overview of the problems that could happen and that we will avoid.
\medskip
The structural information of $\Omega$ is of two types: the neighborhood structure and the interior/boundary structure. Hence, we have to make sure that the domain of $\H^2$ we will associate to $\Omega$ is able to reflect these two pieces of information.
In other words, for two $v_1, v_2 \in \bar{\Omega}$, we want $v_1$ to be near $v_2$ in $\Omega$ if and only if $v_1$ is near $v_2$ in the domain. Moreover, for $v \in B$, we want to guarantee the existence of a part of $\Sigma$ near $v$. Reciprocally, for each $x \in \Sigma$, we want to guarantee the existence of a vertex $v \in B$ near $x$. The sense of the word \textit{near} is the following: the proximity between $x$ and $v$ does not depend on the subgraph $\Omega$. This proximity shall be quantified by Proposition \ref{prop : rough isom}.
As already spotted by Han and Hua in \cite{HH}, one of the difficulties comes from the isolated boundary vertices. If $v \in B$ is isolated, we have to be clever to make sure there is $x \in \Sigma$ which is near $v$, see \Cref{expl : sommet isole}.
\medskip
A second difficulty is the following: we want the domain $N$ to be smooth. This will give us the possibility to make a change of metric on $N$, in order to use Theorem $3$ of \cite{CGR}.
\medskip
Hence the process contains two steps: at first we find a domain $\tilde{N}$ which is structurally related to $\Omega$ but whose boundary $\tilde{\Sigma}$ is not smooth, and secondly we change this domain slightly by smoothing the angles in order to get the wanted domain $N$.
\subsection{Construction of the domain \texorpdfstring{ $\tilde{N}$}{n prime sigma}} \label{subsection : domaine rugueux}
Let us begin by considering a vertex $v \in\bar{\Omega}$ and the triangle $T_v$ associated. In this section, $v$ will always refer to this particular triangle. We call $A_1, A_2, A_3$ the vertices of $T_v$, respectively at angles $\frac{\pi}{p}, \frac{\pi}{q}, \frac{\pi}{r}$. We define a map $H : \{A_1, A_2, A_3\} \longrightarrow \H^2$
as follows:
$H(A_1)$ is the unique point of the geodesic segment $[v, A_1]$ such that $d_g(v, H(A_1))=\frac{9}{10} \cdot d_g(v, A_1)$. The points $H(A_2)$ and $H(A_3)$ are defined similarly.
\medskip
We then connect $H(A_1), H(A_2)$ and $H(A_3)$ with geodesic segments. This gives birth to a new triangle, denoted $T_v'$.
By convexity, $T_v'$ is strictly contained inside the initial triangle $T_v$. It is also easy to see that $v$ is contained inside $T_v'$.
\medskip
If $w \in \bar{\Omega}$ is another vertex of the subgraph, then by construction there is a triangle $T_w$ of the tiling associated to $w$ and there is an isometry $\psi_{v,w} : \H^2 \longrightarrow \H^2$ such that $\psi_{v,w}(T_v)=T_w$. This isometry is not necessary unique. If there are several, we just pick one and call it $\psi_{v,w}$. We consider this isometry and call $T_w' := \psi_{v, w}(T_v')$.
We apply this process to each vertex of $\bar{\Omega}$. Hence we have now at our disposal $|\bar{\Omega}|$ new triangles, disjoint from each other and isometric to each other.
\medskip
If $v_1, v_2 \in \bar{\Omega}$ are such that $v_1 \sim v_2$ in the subgraph, then by definition of $\Gamma$, $v_1$ and $v_2$ represent the centers of two triangles, let us say $T_1$ and $T_2$, having one side in common.
Thus $T_1$ has two vertices $x, y$ which are also vertices of the triangle $T_2$.
As we said before, there is an isometry $\psi_{v, v_1}$ of $\H^2$ such that $\psi_{v, v_1}(T_v)=T_1$. Without loss of generality, say that $\psi_{v, v_1}(A_1)=x$ and $\psi_{v, v_1}(A_2)=y$.
\medskip
We denote $x_1: =\psi_{v, v_1}(H(A_1))$ and $y_1 := \psi_{v, v_1}(H(A_2))$, which are vertices of the triangle $T_1' = \psi_{v, v_1}(T_v')$. Similarly, we denote $x_2 := \psi_{v, v_2}(H(A_1))$ and $y_2 := \psi_{v, v_2}(H(A_2))$ which are vertices of the triangle $T_2' = \psi_{v, v_2}(T_v')$.
\medskip
We then connect $x_1$ to $x_2$ by a geodesic segment, and we do the same with $y_1$ and $y_2$, see Fig. \ref{fig : rectangle}.
\medskip
We write $T_1' \sim T_2'$ in order to say that we have connected the triangles $T_1'$ and $T_2'$.
\medskip
This process connecting the triangles according to the structure of $\Omega$ allows us to notice the following relation: for two vertices $v_1, v_2 \in \bar{\Omega}$ which are the centers of two triangles $T_1', T_2'$, we have
\begin{align*}
v_1 \sim v_2 \iff T_1' \sim T_2'.
\end{align*}
\begin{figure}[H]
\centering
\includegraphics[scale=1]{images/connection_deux_triangles.png}
\caption{The vertices $x_1, y_1$ of $T_1'$ are connected respectively to the vertices $x_2, y_2$ of $T_2'$ because of the assumption that $v_1 \sim v_2$ in the subraph $\Omega$.}
\label{fig : rectangle}
\end{figure}
Let us suppose that $z$ is the common vertex of $2p$ triangles such that their centers $v_1, \ldots, v_{2p}$ satisfy $v_1 \sim v_2 \sim v_3 \sim \ldots \sim v_{2p} \sim v_1$ in $\Omega$. Without loss of generality, let us say that $\psi_{v, v_1}(A_1)=z$. We denote $z_1=\psi_{v,v_1}(H(A_1)), \ldots, z_{2p}=\psi_{v, v_{2p}}(H(A_1))$ as before. By applying the process described above, we connect $z_1$ to $z_2$, $z_2$ to $z_3, \ldots, z_{2p}$ to $z_1$ by geodesic segments, see Fig. \ref{fig : triangle}.
Of course, there is nothing specific about $p$ and the same holds for $q$ and $r$.
\begin{figure}[H]
\centering
\includegraphics[scale=1]{images/connection_triangles.png}
\caption{We connected $z_1$ to $z_2$, $z_2$ to $z_3$, $z_3$ to $z_4$ and $z_4$ to $z_1$ because of the assumption that $v_1 \sim v_2 \sim v_3 \sim v_4 \sim v_1$ in $\Omega$.}
\label{fig : triangle}
\end{figure}
\begin{rem}
The previous construction naturally generates different \textit{simple polygons} contained inside the hyperbolic plane $\H^2$, of which the exhaustive list is the following:
\begin{itemize}
\item Each vertex $w \in \bar{\Omega}$ adds one triangle $T_w'$;
\item Each couple of vertices $v_1, v_2 \in \bar{\Omega}$ such that $v_1 \sim v_2$ adds one quadrilateral;
\item Each vertex $z$ which is the common vertex of $2p$ triangles such that their centers $v_1, \ldots, v_{2p}$ satisfy $v_1 \sim v_2 \sim v_3 \sim \ldots \sim v_{2p} \sim v_1$ in $\Omega$ adds one $2p$-gon;
\item Each vertex $z$ which is the common vertex of $2q$ triangles such that their centers $v_1, \ldots, v_{2q}$ satisfy $v_1 \sim v_2 \sim v_3 \sim \ldots \sim v_{2q} \sim v_1$ in $\Omega$ adds one $2q$-gon;
\item Each vertex $z$ which is the common vertex of $2r$ triangles such that their centers $v_1, \ldots, v_{2r}$ satisfy $v_1 \sim v_2 \sim v_3 \sim \ldots \sim v_{2r} \sim v_1$ in $\Omega$ adds one $2r$-gon.
\end{itemize}
\end{rem}
\begin{defn}
We call $K$ the compact subset of $\H^2$ obtained by considering the closure of the union of all the simple polygons generated by the previous construction.
We also call $\hat{N}$ the bounded domain of $\H^2$ defined by $\hat{N}= \stackrel{\circ}{K}$ and we call $\hat{\Sigma}$ the boundary of $\hat{N}$.
\end{defn}
\begin{figure}[H]
\centering
\includegraphics[scale=0.5, angle=0]{images/domaine_hatN.PNG}
\caption{The crosses represent the interior $\Omega$ of the subgraph, the dots represent the boundary $B$ of the subgraph. The polygonal curve represent the boundary $\hat{\Sigma}$ while the polygon (of which $\hat{\Sigma}$ is the boundary) is the interior $\hat{N}$.}
\label{fig : domaine hat N}
\end{figure}
As this point of the paper, one may ask why we do not simply define the domain as the thickening of the union of all $T_w$, for $w \in \bar{\Omega}$. The reason is that by doing so, the domain would not be able the reflect the neighborhood structure of the subgraph.
Indeed, we recall that by definition of a subgraph, two boundary vertices are never connected by an edge. Let us consider two vertices $w_1, w_2 \in B$ such that $T_{w_1}$ is adjacent to $T_{w_2}$ (meaning that $\{w_1,w_2\} \in E$). Gluing the two triangles $T_{w_1}, T_{w_2}$ would give the information that $w_1$ is adjacent to $w_2$ in the subgraph, which is not the case because they are two boundary vertices.
This mismatch between the structure of the domain and the structure of the subgraph would then jeopardize one of our next constructions, namely the rough isometry of \Cref{prop : rough isom}. This proposition claims the existence of a rough isometry whose constants do not depend on the subraph $\Omega$ chosen. In order to prove the existence of such a rough isometry, it is crucial that the domain $N$ we are building reflects the neighborhood structure of the subgraph $\Omega$. We give more details about this problem in Appendix \ref{appendix : contre expl}.
\begin{rem}
We recall that, by construction, the domain
$\hat{N}$ has the same neighborhood structure as the subgraph $\Omega$. Indeed, we already saw that for $v_1, v_2 \in \bar{\Omega}$,
\begin{align*}
v_1 \sim v_2 \iff T_1' \sim T_2'.
\end{align*}
However, the boundary structure of $\hat{N}$ is not analog to the one of $\Omega$. We already have one implication: for all $x \in \hat{\Sigma}$, there exists $w \in B$ such that $w$ is near $x$.
The reciprocal is not verified. If $w \in B$, there is no guarantee that there exists $x \in \hat{\Sigma}$ such that $x$ is near $w$. To see that, one can look at Example \ref{expl : sommet isole}.
\end{rem}
\begin{expl} \label{expl : sommet isole}
Choose a vertex $v^*$ of the host graph and define $\Omega$ as the ball of radius $n$ deprived of $v^*$. This will give rise to a subgraph $\Omega$, for which $v^* \in B$. However, there is no $x \in \hat{\Sigma}$ near $v^*$. Indeed, the bigger $n$ is, the bigger the distance between $\hat{\Sigma}$ and $v^*$ is. Hence the proximity between $\hat{\Sigma}$ and $v^*$ depends on the subgraph, which we want to avoid. This kind of situation also appears on Fig. \ref{fig : domaine hat N}, where we can see an isolated boundary vertex.
\end{expl}
To remedy this problem, we proceed to a surgery of this domain $\hat{N}$: for each $w \in B$, we remove the ball centered at $w$ of radius $\frac{\rho}{2}$, where $\rho$ denotes the radius of the circle inscribed in $T_w'$, see Fig. \ref{fig : boules enlevees}
\begin{defn}
We call $\tilde{N}$ the domain obtain after the removal of the balls, and we call $\tilde{\Sigma}$ its boundary.
\end{defn}
\begin{rem}
This last surgery obviously gives us the reciprocal we lacked until now: for each $w \in B$, there exists $x \in \tilde{\Sigma}$ such that $x$ is near $w$.
\end{rem}
\begin{figure}[H]
\centering
\includegraphics[scale=0.45, angle=0]{images/domaine_tildeN_couleur.PNG}
\caption{The crosses represent the vertices of $\Omega$, the dots represent the boundary $B$. The balls surrounding the boundary vertices are removed from the domain and the structure of the subgraph is readable on the domain. }
\label{fig : boules enlevees}
\end{figure}
This bounded domain $\tilde{N}$ is not our final domain because we want one with a smooth boundary.
\subsection{Smoothing of the domain \texorpdfstring{ $\tilde{N}$}{n prime sigma prime} } \label{subsection : domaine lisse}
As we said in the introduction, we want to discretize the domain in order to find an upper bound for the Steklov spectrum of $\Omega$.
One way to do this consists in using Theorem $3$ of \cite{CGR}. Of course, we have to make sure the assumptions of this theorem are verified before using it. However, the domain $\tilde{N}$ does not satisfy all these assumptions, see Remark \ref{rem : cst discretisation}. This section is devoted to modify the domain $\tilde{N}$ and get a new domain $N$ which has the advantage to have a smooth boundary.
\medskip
Note that, as always in this paper, we have to make sure that the operations we make do not depend on the subgraph $\Omega$, but only on the host graph $\Gamma$.
\medskip
We recall that $\tilde{\Sigma}$ is composed with the union of $\hat{\Sigma}$ and many circles. Each circle is already a smooth connected component of $\tilde{\Sigma}$, hence we only have to smooth $\hat{\Sigma}$ out. Each connected component of $\hat{\Sigma}$ is a simple closed $C^\infty$ piecewise curve, composed with geodesic segments. Note that by construction, there exist at most $4 \times 3 -3 = 9$ different segments (two isometric segments are identified).
We shall designate by \textit{corner} the intersection of two geodesic segments forming $\hat{\Sigma}$. A corner is therefore a point of the curve whose neighborhoods are of class $C^0$, but not of class $C^1$. By construction, a corner is always located on a vertex of a triangle $T'$.
\medskip
The regularity of our construction allows us to state that the domain $\tilde{N}$ has at most
$ \left(
\begin{array}{c}
4 \\
2
\end{array}
\right)\times 3 = 18
$
different internal angles (two congruent angles are identified).
The interest of these comments is to simplify considerably the smoothing of the domain $\tilde{N}$. Indeed, there are at most $18$ different types of angles to smooth out.
\medskip
Let us call $\lambda_1, \ldots, \lambda_9$ the length of the geodesic segments and let us denote
\begin{align*}
\lambda := \min\{\lambda_1, \ldots, \lambda_9\}.
\end{align*}
If $\tilde{\Sigma}$ has $n$ corners, let us call them $z_1, \ldots z_n$. For each corner $z_i$, there exist exactly two points $x_i, x_i' \in \tilde{\Sigma}$ such that
\begin{align*}
d_g(x_i, z_i) = d_g(x_i',z_i) = \frac{\lambda}{10}.
\end{align*}
Let us consider a corner $z_i$ as well as the two points $x_i, x_i'$ associated.
\medskip
We then create a smooth curve
\begin{align*}
\alpha_1 : [0, 1] \longrightarrow \H^2
\end{align*}
such that
\begin{itemize}
\item $\alpha_1(0) = x_i$, $\alpha_1(1)=x_i'$;
\item For all $t \in (0,1)$ we have $\alpha_1(t) \in \tilde{N}$;
\item For all $t \in [0,1]$ we have $d_g(\alpha_1(t), z_i) \leq \frac{\lambda}{10}$;
\item A curve whose image is
\begin{align*}
[z_{i-1}, x_i] \cup \alpha_1([0,1]) \cup [x_i', z_{i+1}]
\end{align*}
is smooth, see Fig. \ref{fig : lissage coubre 1}.
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[scale=1.5]{images/alpha_1_bis.png}
\caption{The curve $\alpha_1$ can be seen as a smoothing of the angle at the corner $z_i$.}
\label{fig : lissage coubre 1}
\end{figure}
Then suppose that $z_i$ is a corner associated with an angle which is not congruent to the previous one. We then create a smooth curve
\begin{align*}
\alpha_2 : [0,1] \longrightarrow \H^2
\end{align*}
with the same four properties as the previous curve, see Fig. \ref{fig : lissage courbe 2}.
\begin{figure}[H]
\centering
\includegraphics[scale=2.0]{images/alpha_2_bis.png}
\caption{The curve $\alpha_2$ is a smoothing of the angle at $z_i$.}
\label{fig : lissage courbe 2}
\end{figure}
We continue the process and create a smoothing curve for each type of angle, at most $18$ times as said before.
\begin{rem}
If $z_j$ is another corner of the same type as $z_i$, meaning that the angle at $z_j$ is congruent to the angle at $z_i$, there is then an isometry $\Psi : \H^2 \longrightarrow \H^2$ which sends the angle at $z_i$ onto the angle at $z_j$. The smoothing curve at angle $z_j$ is then given by $\Psi \circ \alpha_\mu$, where $\mu \in \{1,\ldots, 18\}$ depends on the nature of the angle.
\end{rem}
Thus, we smooth out the domain $\tilde{N}$ with these $18$ curves and obtain a new connected domain with smooth boundary.
We obtain the domain $N$ that we wanted, whose boundary is denoted $\Sigma$. By construction, the domain $N$ has the following characteristics:
\begin{itemize}
\item $N$ is connected;
\item The boundary $\Sigma$ is smooth;
\item $\Sigma$ is composed of at most $28$ types of curve:
\begin{itemize}
\item The $9$ geodesic segments (coming from triangles and quadrilaterals);
\item The $18$ smoothing curves $\alpha_1, \ldots, \alpha_{18}$;
\item The circles resulting from the removal of the balls.
\end{itemize}
\end{itemize}
Moreover, the domain $N$ is constructed in a way that the structure of the subgraph $\Omega$ is readable in it. Indeed, if we call \textit{smoothed triangle} a region of $N$ of the form $N \cap T_w'$, for $w \in \bar{\Omega}$, then
\begin{itemize}
\item A smoothed triangle $N \cap T_{v_1}'$ is connected to a neighbor $N \cap T_{v_2}'$ if and only if $v_1 \sim v_2$ in $\Omega$;
\item A vertex $w$ is part of $B$ if and only if there exists $x \in \Sigma$ such that $x$ is near $w$. As said before, Proposition \ref{prop : rough isom} will clarify the sense of the word \textit{near}.
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[scale=0.35, angle=0]{images/domain_N_ref.PNG}
\caption{The crosses represent the vertices of $\Omega$, the dots represent the boundary $B$. The balls surrounding the boundary vertices are removed from the domain, the structure of the subgraph is readable in the domain and $\Sigma$ is smooth.}
\end{figure}
\begin{rem}
Since each $w \in B$ adds one connected component of $\Sigma$ as a circle, we have the inequality
\begin{align} \label{ineg : bord}
|\Sigma| \geq C_4 \cdot |B|,
\end{align}
where $C_4$ corresponds to the perimeter of a circle of radius $\frac{\rho}{2}$.
\end{rem}
\section{Proof of the main theorem} \label{section : 4}
Let us begin by recalling Theorem $1.2$ of \cite{CEG1}.
\begin{thm}
There exists a constant $C_5$ such that for all bounded domain $N$ of the hyperbolic space $\H^2$ and for all $k \geq 0$,
\begin{align} \label{ineg : domaine}
\sigma_k(N, g) \leq C_5 \cdot \frac{k}{|\Sigma|}.
\end{align}
\end{thm}
Actually, the result of Colbois et al. is more general than that, but this statement is enough for our needs.
\medskip
The domain $N$ being structurally similar to the subgraph $\Omega$, we will show that a bound of the same type exists for the subgraph's spectrum. The goal of this section is to transfer this result to the subgraph.
\medskip
To do this, we want to discretize the domain $N$. Let us recall the conditions that the domain must satisfy to be discretized:
We have to assume the existence of constants $\kappa >0$ and $r_0 \in (0,1)$ such that
\begin{itemize}
\item The boundary $\Sigma$ admits a neighborhood which is isometric to the cylinder $[0,1] \times \Sigma$, whose boundary corresponds to $\{0\} \times \Sigma$;
\item The Ricci curvature of $N$ is bounded below by $- \kappa$;
\item The Ricci curvature of $\Sigma$ is bounded below by $0$
\item For all $x \in N$ such that $d_g(x, \Sigma) >1$, we have inj$_M(x) >r_0$;
\item For all $x \in \Sigma$, we have inj$_\Sigma(x) >r_0$.
\end{itemize}
For further investigation on this topic and to understand why these assumptions are made, one can look at \cite{CGR}.
\begin{rem} \label{rem : cst discretisation}
The last four conditions are trivially satisfied by $N$. Moreover, the constants $\kappa, r_0$ do not depend on the subgraph $\Omega$. Indeed, the regularity of the construction of the domain $N$ allows to give constants $\kappa, r_0$ valid for any domain $N$ obtained by the process described above.
In other words, if we call $\mathcal{M} = \mathcal{M}(\kappa, r_0)$ the class of $2$-dimensional manifolds which satisfy the last four properties, then $N \in \mathcal{M}$ whatever the chosen subgraph $\Omega$.
\end{rem}
On the other hand, the first assumption is not satisfied by the domain. Indeed, $\Sigma$ does not have a neighborhood isometric to a cylinder. To remedy this, we will proceed to a change of metric on $N$ in order to obtain a new Riemannian manifold which satisfies the five properties.
\subsection{Changing the metric on the domain} \label{subsection : chmt metric}
The main difficulty of this subsection is proceeding to a change of metric which is uniform for all domains $N$ obtained by the procedure described in Sect. \ref{section : 3}.
Here, the word \textit{uniform} reflects the existence of a constant $C_6$ as in Proposition \ref{prop : quasi_isom_continue} which is valid for all domains.
\medskip
Let us denote
\begin{align*}
N(\delta) = \{x \in N \; : \; d_g(x, \Sigma) \leq \delta \}
\end{align*}
the $\delta$-neighborhood of the boundary.
\begin{prop}[Lemma $34$ of \cite{CGR}]
There exist on $N$ a $\delta >0$ (depending only on the 28 types of curves) and a Riemannian metric $g'$ such that
\begin{itemize}
\item $(N(\delta), g')$ is isometric to $[0,1] \times \Sigma$;
\item The metrics $g$ and $g'$ are homothetic on $N \backslash N(3\delta)$.
\end{itemize}
\end{prop}
\begin{proof}
We will use the Fermi parallel coordinates: we parametrize each connected component of $\Sigma$ by arc-length and call $s$ the parameter. We then use the distance $t$ to $\Sigma$ as a second parameter to describe the points of $N$ lying in a close neighborhood of $\Sigma$. In these coordinates, the hyperbolic metric is expressed by
\begin{align*}
g(s,t) = \varphi(s,t) \cdot ds^2 + dt^2,
\end{align*}
where $\varphi$ is a smooth positive function satisfying $\varphi(s, 0) =1$.
Let $\delta > 0$ be small enough to have $\frac{1}{2} \leq \varphi(s,t) \leq 2$ on $N(3\delta)$ (such a $\delta$ exists because $\varphi$ is smooth).
We call $g_0$ the product metric which, in the Fermi coordinates $(s,t)$, is expressed by
\begin{align*}
g_0(s,t) = ds^2 +dt^2.
\end{align*}
We then take a smooth function
\begin{align*}
\chi : [0, 3 \delta] \longrightarrow [0,1]
\end{align*}
such that $\chi \equiv 0$ on $[0, \delta]$, $\chi \equiv 1$ on $[2 \delta, 3\delta]$ and such that $\chi$ is strictly increasing on $[\delta, 2 \delta]$.
Then we define the metric
\begin{align*}
g_\delta(s,t) = \chi(t) g(s,t) + (1-\chi(t))g_0(s,t).
\end{align*}
This metric coincides with the hyperbolic metric on $N(3\delta)\backslash N(2\delta)$, then it can be extended all over the domain $N$ into a metric that we continue to call $g_\delta$.
Moreover, endowed with this metric, $N(\delta)$ is isometric to $[0, \delta] \times \Sigma$. We then define the metric
\begin{align*}
g' := \frac{1}{\delta^2}g_\delta,
\end{align*}
for the cylindrical neighborhood to have length $1$.
\end{proof}
The value of $\delta$ depends only on the 28 types of curves composing $\Sigma$. That is the reason we built the domain $N$ with such regularity. Thanks to the process, we can choose $\delta$ independently of the subgraph $\Omega$ chosen.
\begin{prop} [Lemma $34$ of \cite{CGR}] \label{prop : quasi_isom_continue}
There exists a constant $C_6 >1$, that does not depend on the subgraph $\Omega$, such that for all $x \in N$ and all $v \in T_xN, \; v \neq 0$, we have
\begin{align*}
\frac{1}{C_6} \leq \frac{g'(x)(v,v)}{g(x)(v,v)} \leq C_6.
\end{align*}
\end{prop}
\begin{proof}
We distinguish three cases:
\begin{itemize}
\item $x \in N\backslash N(2\delta)$;
\item $x \in N(\delta)$;
\item $x \in N(2\delta)\backslash N(\delta)$.
\end{itemize}
Let us start with the first one. Let $x \in N\backslash N(2\delta)$ and $0 \ne v \in T_xN$. We have
\begin{align*}
\frac{g'(x)(v,v)}{g(x)(v,v)} = \frac{\frac{1}{\delta^2}g_\delta(x)(v,v)}{g(x)(v,v)} = \frac{\frac{1}{\delta^2}g(x)(v,v)}{g(x)(v,v)} = \frac{1}{\delta^2}
\end{align*}
because on $N \backslash N(2\delta)$, the metric $g_\delta$ coincides with the hyperbolic metric $g$.
For the second case, let $x \in N(\delta)$ and $0 \ne v \in T_xN$. we have
\begin{align*}
\frac{g'(x)(v,v)}{g(x)(v,v)} & = \frac{g'(x)(v,v)}{(\varphi(s,t)ds^2 + dt^2)(v,v)}
\leq \frac{g'(x)(v,v)}{(\frac{1}{2}ds^2 + \frac{1}{2}dt^2)(v,v)} \\
& = \frac{\frac{1}{\delta^2}g_\delta(x)(v,v)}{\frac{1}{2}(ds^2+dt^2)(v,v)}
= \frac{\frac{1}{\delta^2}g_0(x)(v,v)}{\frac{1}{2}g_0(x)(v,v)} \\
& = \frac{2}{\delta^2}
\end{align*}
because $g_\delta$ coincides with the product metric $g_0$ on $N(\delta)$.
In a similar way, we have
\begin{align*}
\frac{g'(x)(v,v)}{g(x)(v,v)} & = \frac{g'(x)(v,v)}{(\varphi(s,t)ds^2 + dt^2)(v,v)}
\geq \frac{g'(x)(v,v)}{(2ds^2 + 2dt^2)(v,v)} \\
& = \frac{\frac{1}{\delta^2}g_\delta(x)(v,v)}{2(ds^2+dt^2)(v,v)}
= \frac{\frac{1}{\delta^2}g_0(x)(v,v)}{2g_0(x)(v,v)} \\
& = \frac{1}{2\delta^2}.
\end{align*}
Let us now look at the third case. Let $x \in N(2\delta)\backslash N(\delta)$ and $0 \ne v \in T_xN$.
We recall that on $N(2\delta)\backslash N(\delta)$, the metric $g_\delta$ interpolates the product metric $g_0$ and the hyperbolic metric $g$ with the help of a smooth increasing function $\chi$.
Then we have
\begin{align*}
\frac{g'(x)(v,v)}{g(x)(v,v)} & = \frac{\frac{1}{\delta^2}g_\delta(x)(v,v)}{g(x)(v,v)}
= \frac{\frac{1}{\delta^2}(\chi(t)g(s,t) +(1-\chi(t))g_0(s,t))(v,v)}{g(x)(v,v)} \\
& = \frac{1}{\delta^2} \left( \chi(t) + (1-\chi(t))\frac{g_0(s,t)(v,v)}{g(x)(v,v)} \right)
\geq \frac{1}{\delta^2} \left( \chi(t) + (1-\chi(t))\frac{g_0(s,t)(v,v)}{2 g_0(x)(v,v)} \right) \\
& = \frac{\chi(t)}{\delta^2} + \frac{1-\chi(t)}{2\delta^2}
\geq \frac{1}{2 \delta^2}.
\end{align*}
Similarly, we have
\begin{align*}
\frac{g'(x)(v,v)}{g(x)(v,v)} & = \frac{\frac{1}{\delta^2}g_\delta(x)(v,v)}{g(x)(v,v)}
= \frac{\frac{1}{\delta^2}(\chi(t)g(s,t) +(1-\chi(t))g_0(s,t))(v,v)}{g(x)(v,v)} \\
& = \frac{1}{\delta^2} \left( \chi(t) + (1-\chi(t))\frac{g_0(s,t)(v,v)}{g(x)(v,v)} \right)
\leq \frac{1}{\delta^2} \left( \chi(t) + (1-\chi(t))\frac{g_0(s,t)(v,v)}{\frac{1}{2} g_0(x)(v,v)} \right) \\
& = \frac{\chi(t)}{\delta^2} + \frac{1-\chi(t)}{\frac{1}{2}\delta^2} \\
& \leq \frac{1}{\frac{1}{2} \delta^2}
= \frac{2}{\delta^2}.
\end{align*}
Then the ratio is bounded for all $x \in N$ and for all $v \in T_xN, \; v \neq 0$, and we can choose
\begin{align*}
C_6 := \frac{2}{ \delta^2}.
\end{align*}
Moreover, this constant $C_6$ does not depend on the chosen subgraph $\Omega$. Indeed, the function $\varphi$ depends only on the, at most, $28$ types of curves forming $\Sigma$
(which we have fixed once and for all), and $\delta$ depends only on $\varphi$. Thus, as said before, the constant $\delta >0$ can be chosen independently of the subgraph, which allows us to fix a universal value of $C_6 >1$ for all the domains $N$ obtained thanks to the procedure described in Sect. \ref{section : 3}.
\end{proof}
We now have at our disposal a new Riemannian manifold with boundary, denoted $(N, g')$, which is related to $(N, g)$ in the sense of Proposition \ref{prop : quasi_isom_continue}. We recall now Proposition $32$ of~\cite{CGR}:
\begin{prop}
Let $N$ be a Riemannian manifold of dimension $m$, compact with smooth boundary and let $g, g'$ be two Riemannian metrics on $N$. Let us assume that there exists a constant $C_6 >1$ such that for all $x \in N$ and for all $v \in T_xN, \; v \neq 0$, we have
\begin{align*}
\frac{1}{C_6} \leq \frac{g'(x)(v,v)}{g(x)(v,v)} \leq C_6.
\end{align*}
Then we have
\begin{align*}
\frac{1}{C_6^{2m+1}} \leq \frac{\sigma_k(N,g')}{\sigma_k(N, g)} \leq C_6^{2m+1}.
\end{align*}
\end{prop}
The assumption is exactly what we prove at Proposition \ref{prop : quasi_isom_continue}. Hence we can apply this result to $(N, g)$ and $(N, g')$ in order to get:
\begin{align} \label{ineg : metrique}
\sigma_k(N, g') \leq C_6^5 \cdot \sigma_k(N, g).
\end{align}
\subsection{Discretization of the manifold \texorpdfstring{ $(N, g')$}{n sigma g prime}} \label{section : 5}
Let us recall that we proceeded to a change of metric on $N$ in order to give it the ability to be discretized, according to constants $r_0$ and $\kappa$, as said in Remark \ref{rem : cst discretisation}. There exist several ways to discretize a manifold. In this paper, we apply the process described in \cite{CGR}, for we want the discretization to have a spectral link with the manifold.
\medskip
This process is the following:
We choose $\epsilon \in (0, r_0/4)$ and we choose $V_\Sigma$ a maximal $\epsilon$-separated subset of $\Sigma$. Then we call $V_\Sigma'$ the copy of $V_\Sigma$ lying $4\epsilon$ away from the boundary:
\begin{align*}
V_\Sigma' = \{4\epsilon\} \times V_\Sigma.
\end{align*}
Then we choose $V_I$ a maximal $\epsilon$-separated subset of $N \backslash [0,4\epsilon] \times \Sigma$ such that $V_\Sigma' \subset V_I$.
Then we consider the subset $\tilde{V} = V_\Sigma \cup V_I$ and grant it the structure of a graph by decreeing
\begin{itemize}
\item Two vertices $v, w \in \tilde{V}$ are adjacents as soon as $d_{g'}(v, w) \leq 3\epsilon$;
\item A vertex $v \in V_\Sigma$ is adjacent to its counterpart $v' \in V_\Sigma'$.
\end{itemize}
This process gives a graph with boundary $(\tilde{V}, \tilde{E}, V_\Sigma)$, simply denoted $(\tilde{V},V_\Sigma)$ hereafter, whose boundary is $V_\Sigma$ and that we call $\epsilon$-discretization of $N$.
\medskip
Theorem $3$ point $4)$ of \cite{CGR} allows us to state:
\begin{thm}
There exists a constant $C_7 >0$ depending only on $\kappa, r_0$ and $\epsilon$ such that for all $k \leq |V_\Sigma|$, we have
\begin{align} \label{ineg : discretisation}
\sigma_k(\tilde{V}, V_\Sigma) \leq C_7 \cdot \sigma_k(N, g') \cdot k.
\end{align}
\end{thm}
\subsection{Rough isometry between \texorpdfstring{$(\tilde{V}, V_\Sigma)$}{discrétisation} and \texorpdfstring{$\Omega$}{initial}} \label{section : 6}
We now want to exploit the graph $(\tilde{V}, V_\Sigma)$ for which we have an upper bound relative to its spectrum to control the spectrum of our initial subgraph $\Omega$. In order to do it, we will have to deal with the concept of rough isometry once again. This will allow us to use Proposition $16$ of \cite{CGR} to compare the Steklov spectra of the graphs. The main difficulty here is that we have to make sure the constants of the rough isometry are independant of the subgraph $\Omega$. Let us begin by defining what is a rough isometry in the context of graphs with boundary.
\begin{defn}
A rough isometry $\phi$ between two graphs with boundary $(\bar{\Omega}_1, E'_1, B_1)$ and $(\bar{\Omega}_2, E'_2, B_2)$ is a rough isometry which sends $B_1$ onto $B_2$ and such that the restriction of $\phi$ to $B_1$ is a rough isometry $B_1 \longrightarrow B_2$ when
considering extrinsic distances on $B_1$ and $B_2$.
\end{defn}
\begin{prop} \label{prop : rough isom}
There exists a rough isometry $\bar{\phi} : (\tilde{V}, V_\Sigma) \longrightarrow \bar{\Omega}$ whose constants $C_1, C_2, C_3$ are independent from the subgraph $\Omega$.
\end{prop}
\begin{proof}
We have to define a map $\bar{\phi} : (\tilde{V}, V_\Sigma) \longrightarrow \bar{\Omega}$ and show that it is a rough isometry.
Remark that the vertices $v$ of $\tilde{V}$ can be of different types. There are boundary vertices coming from the $28$ different kind of curves forming $\Sigma$, and there are interior vertices coming from $N$.
As a consequence, the definition of $\bar{\phi}$ is a little bit heavy, but the idea to define the rough isometry is very natural: each vertex $v \in \tilde{V}$ is sent onto the vertex $w$ of $\bar{\Omega}$ which is of same nature (interior or boundary) and which is the nearest from it.
\medskip
Let us define
\begin{align*}
\bar{\phi} : (\tilde{V}, V_\Sigma) \longrightarrow \bar{\Omega}.
\end{align*}
For the vertices of the boundary:
\begin{itemize}
\item For $v \in V_\Sigma$ such that $v$ is part of a side of a triangle $T'$, we choose $\bar{\phi}(v) \in B$ the vertex at the center of $T'$;
\item For $v \in V_\Sigma$ such that $v$ is part of the boundary of a ball that had been removed, we choose $\bar{\phi}(v) \in B$ the vertex at the center of the removed ball;
\item For $v \in V_\Sigma$ such that $v$ is part of a side of a quadrilateral, we find the side of a triangle closest to $v$ and we choose $\bar{\phi}(v) \in B$ as if $v$ were on this triangle's side;
\item For $v \in V_\Sigma$ such that $v$ is part of a smoothing curve, we find the side of a triangle closest to $v$ and choose $\bar{\phi}(v) \in B$ as if $v$ were on this triangle's side.
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[scale=1.3]{images/quasi_isom_bord.png}
\caption{The vertices of $V_\Sigma$ are represented by diamonds, the dot vertex belongs to $B$. All of the diamonds are sent to the dot by $\bar{\phi}$.}
\end{figure}
And for the interior vertices:
\begin{itemize}
\item For $v \in V_I$ such that $v$ is part of a triangle whose center is $w \in \Omega$, we choose $\bar{\phi}(v) = w$;
\item For $v \in V_I$ such that $v$ is part of a triangle whose center is $w \in B$, then there exists at least one $w' \in \Omega$ such that $w \sim w'$. We then choose $\bar{\phi}(v) = w'$. If there are several possibilities, we choose one once and for all;
\item For $v \in V_I$ such that $v$ is part of a quadrilateral, then two opposite sides of this quadrilateral are the sides of two triangles $T_1', T_2'$. At least one of them has a center $w \in \Omega$. We then choose $\bar{\phi}(v) = w$. If there are two possibilities, we choose one once and for all;
\item For $v \in V_I$ such that $v$ is part of a $2p$-gon (respectively $2q$-gon, $2r$-gon), then this $2p$-gon (resp. $2q$-gon, $2r$-gon) is surrounded by $2p$ (resp. $2q, 2r$) triangles $T_1', \ldots, T_{2p}'$ (resp. $T_{2q}', T_{2r}'$) of which at least $p$ (resp. $q, r$) have a center $w \in \Omega$. We then choose $\bar{\phi}(v) = w$ once and for all.
\end{itemize}
\begin{figure}[H]
\centering
\includegraphics[scale=1.2]{images/quasi_isom_interieur.png}
\caption{The diamond vertices are part of $V_I$, the cross vertices belongs to $\Omega$. All of the diamonds vertices are send to the bottom left cross vertex by $\bar{\phi}$.}
\end{figure}
In order to show that $\bar{\phi}$ is a rough isometry,
let us partition the domain $N$ into cobblestones: a cobblestone $\mathcal{C}$ is defined as the intersection of a triangle $T$ of the initial tiling with $N$.
If $w \in \bar{\Omega}$ is the center of a triangle $T_w$, we denote by $\mathcal{C}_w$ the associated cobblestone. We also write $\mathcal{C}_w \sim \mathcal{C}_{w'}$ to say that two cobblestones are adjacent.
\medskip
Then we choose $C_1$ as the cardinality of the biggest possible $\epsilon$-separated set contained inside a cobblestone multiplied by $\max\{p, q, r\}$. Then we choose $C_2 = C_1$. Thus, if two vertices $v_1, v_2 \in \tilde{V}$ belongs to the same cobblestone, we have $d_{\tilde{V}}(v_1, v_2) \leq C_1$.
\medskip
We recall that by our construction of the domain $N$, for $w, w' \in \bar{\Omega}$ we have
$$
w \sim w' \iff \mathcal{C}_w \sim \mathcal{C}_{w'},
$$
i.e the neighborhood structure of the subgraph is readable onto the domain. Therefore, for $w_1, w_2 \in \bar{\Omega}, \; w_1 \neq w_2$, the distance $d_{\bar{\Omega}}(w_1, w_2)$ represents the number of cobblestones that separate $w_1$ from $w_2$ plus one. Thus, if $v_1, v_2 \in \tilde{V}$ are such that $\bar{\phi}(v_1) = w_1$ and $\bar{\phi}(v_2)= w_2$, then we have
\begin{align*}
C_1^{-1} d_{\tilde{V}}(v_1, v_2) -C_2 \leq d_{\bar{\Omega}}(w_1, w_2) \leq C_1 d_{\tilde{V}}(v_1, v_2) +C_2.
\end{align*}
Moreover, $\bar{\phi}$ is a surjective map so we can choose $C_3 = 1$ and we get
\begin{align*}
\bigcup_{v \in \tilde{V}} B(\bar{\phi}(v), C_3) = \bar{\Omega}.
\end{align*}
\end{proof}
We can now recall Proposition $16$ of \cite{CGR}:
\begin{prop}
Given $C_1 \geq 1, C_2, C_3 \geq 0$, there exist some constants $C_8, C_9$ depending only on $C_1, C_2, C_3$ and of the maximal degree of the vertices such that for all graphs with boundary $(\Gamma_1, B_1), (\Gamma_2, B_2)$ roughly isometric with constants $C_1, C_2, C_3$, we have
\begin{align*}
C_8 \leq \frac{\sigma_k(\Gamma_1, B_1)}{\sigma_k(\Gamma_2, B_2)} \leq C_9.
\end{align*}
\end{prop}
Applied to this situation, we obtain
\begin{align} \label{ineg : isom}
\sigma_k(\Omega) \leq \frac{1}{C_8} \sigma_k(\tilde{V}, V_\Sigma).
\end{align}
\subsection{Conclusion} \label{section : 7}
In this section, we prove Theorem \ref{thm : principal} and Corollary \ref{cor : zero}.
\begin{proof}
Throughout the paper, we got different results, that we can now assemble to finally obtain \Cref{thm : principal}:
\begin{align*}
\sigma_k(\Omega)& \stackrel{(\ref{ineg : isom})}{\leq} \frac{1}{C_8} \cdot \sigma_k(\tilde{V}, V_\Sigma) \\
& \stackrel{(\ref{ineg : discretisation})}{\leq} \frac{1}{C_8} \cdot C_7 \cdot \sigma_k(N, g') \cdot k \\
& \stackrel{(\ref{ineg : metrique})}{\leq} \frac{1}{C_8} \cdot C_7 \cdot C_6^5 \cdot \sigma_k(N, g) \cdot k \\
& \stackrel{(\ref{ineg : domaine})}{\leq} \frac{1}{C_8} \cdot C_7 \cdot C_6^5 \cdot C_6 \cdot \frac{k}{|\Sigma|} \cdot k \\
& \stackrel{(\ref{ineg : bord})}{\leq} \frac{1}{C_8} \cdot C_7 \cdot C_6^5 \cdot C_6 \cdot \frac{k}{C_4 \cdot |B|} \cdot k \\
& =: C \cdot \frac{1}{|B|} \cdot k^2.
\end{align*}
All along the paper, we took care of specifying on which parameters the constants depend. It happens that they do not depend on the subgraph $\Omega$ chosen. They only depend on the host graph $\Gamma$ and on $\epsilon$. Therefore, if we set a value for $\epsilon$, we can take the same constant $C$ for all subgraph $\Omega$ of $\Gamma$; it is now fixed once and for all.
\medskip
As a consequence, for a choice of three integers $p, q, r \geq 2$ such that $\frac{1}{p} + \frac{1}{q}+ \frac{1}{r} < 1$, giving birth to a tessellation of the hyperbolic plane and to a host graph $\Gamma$ as defined in Sect. \ref{section : 2}, there exists a constant $C = C(\Gamma)$ such that for any subgraph $\Omega$ of $\Gamma$, we have
\begin{align*}
\sigma_k(\Omega) \leq C(\Gamma) \cdot \frac{1}{|B|} \cdot k^2.
\end{align*}
\end{proof}
From this statement, let us prove Corollary \ref{cor : zero}.
\begin{proof}
It is enough to notice the following fact: for $(\Omega_l, B_l)_{l \geq 1}$ a family of subgraphs of $\Gamma$ such that $|\Omega_l| \longrightarrow \infty$, then we also have $|B_l| \longrightarrow \infty$.
\medskip
Therefore, for all $k \in \mathbb{N}$ fixed, we have
\begin{align*}
\sigma_k(\Omega_l, B_l) \leq C(\Gamma) \cdot \frac{1}{|B_l|} \cdot k^2 \underset{l \to \infty}{ \longrightarrow} 0.
\end{align*}
\end{proof}
\section{Consideration and interrogation} \label{sect : interrogation}
All the constructions above were about a host graph $\Gamma$, which was a triangle-tiling graph. However, one may have noticed that we could have used other polygons rather than triangles and still obtained the result. The information we used is the finite number of possible situations, like the $28$ different kinds of curves composing $\Sigma$ or the $18$ types of angles to smooth out.
All these constructions could have emerged from any exact tessellation of the hyperbolic plane, as long as the tiles are compact and the number of different polygon in the tessellation is finite (the tessellation is exact if and only if each edge of a tile is an edge of exactly two polygons of the tessellation). If we used other polygons rather than triangles, the number of different possible situations would have been larger, and the constants would have been different. Nevertheless, the result would have been the same.
\medskip
This comment shows that the result we get in this paper is more general than it primarily seems. Unfortunately, it has its limits. If we get interested in a tiling of the hyperbolic plane which has infinitely many kinds of tiles, then our construction is not relevant anymore. In the same way, if a tile of the tessellation is not compact, we cannot use our method either.
\medskip
This consideration leads to an open question:
\begin{question} \label{quest : autre graphe}
If $\Gamma$ is any graph roughly isometric to the hyperbolic plane, is there a constant $C = C(\Gamma)$ such that a bound as in Theorem $\ref{thm : principal}$ exists?
\end{question}
This question naturally leads to a more general interrogation. In order to properly define the problem, let us give a definition.
\begin{defn}
We say that a host graph $\Gamma$ has the property (P) if for each $k \in \mathbb{N}$ and each family $(\Omega_l)_{l \ge 1}$ of subgraphs of $\Gamma$, we have
\begin{align*}
|\Omega_l| \underset{l \to \infty}{\longrightarrow} \infty \implies \sigma_k(\Omega_l) \underset{l \to \infty}{\longrightarrow} 0.
\end{align*}
\end{defn}
Now we can ask the following open question:
\begin{question} \label{question : comportement}
Let $\Gamma_1, \Gamma_2$ be two roughly isometric graphs. Let us assume that $\Gamma_1$ has the property (P). Does $\Gamma_2$ also have the property (P)?
\end{question}
Reformulated in the language of geometric group theory, the question becomes
\begin{center}
\textit{Is the property (P) a large scale invariant?}
\end{center}
This question, apparently not so hard, appears to be more thorny than expected.
If positively answered, it would automatically generalise our result to any graph roughly isometric to the hyperbolic plane, and it would certainly have many other applications.
\medskip
Another interesting interrogation one may have consists in wondering if some similar constructions could be done in the hyperbolic space $\H^n$, with $n \ge3$. In particular, a first question is the following:
\begin{center}
\textit{Is there a natural class of graphs, analogous to triangle-tiling graphs, that would be roughly
isometric to $\H^n$?}
\end{center}
The answer to this question is \textit{yes.} Using \parencite[Sect. 6.8]{Rat}, we can generate tessellations of $\H^n$ with polyhedra, for any $n \ge 2$. From such a tessellation, we can define a host graph $\Gamma$ in the same manner as we did in this paper. It could be interesting to study such a host graph and see if some results analogous to \Cref{thm : principal} hold in higher dimension. This consideration leads to the following open question:
\begin{question}
Let $\Gamma$ be a graph coming from a polyhedral tessellation of $\H^n, n \ge 3$. Does a constant $C=C(\Gamma)$ exist, such that a bound as in \Cref{thm : principal} holds?
\end{question}
\begin{appendices}
\section{About the importance of the small triangles in our construction } \label{appendix : contre expl}
We provide here an example which shows that, given a subgraph $\Omega$ of $\Gamma$, we cannot simply consider the domain that we get when thickening the union of $T_w$ for all $w\in \bar{\Omega}$.
\medskip
Let us consider the subgraph given by the following figure:
\begin{figure}[H]
\centering
\includegraphics[scale=0.5]{images/graphe_contre_exemple_nomme.PNG}
\caption{The crosses vertices form the interior of the subgraph, the dot vertices form the boundary.}
\label{fig : graphe contre expl}
\end{figure}
We are particularly interested in the boundary vertices named $w_1$ and $w_2$ in Fig. \ref{fig : graphe contre expl}. Here are two properties that $w_1$ and $w_2$ have:
\begin{itemize}
\item $w_1$ is close to $w_2$ in the host graph. Indeed, they belong to two adjacent triangles of the tessellation. Therefore, $d_\Gamma(w_1, w_2)=1$ (where we used the notation $d_\Gamma$ for the distance in the host graph).
\item $w_1$ is far from $w_2$ in the subgraph. Indeed, by definition there is no edge between $w_1$ and $w_2$ in the subgraph. In fact, we have $d_\Omega(w_1, w_2)=33$, which is the diameter of the subgraph (we used the notation $d_\Omega$ for the distance in the subgraph).
\end{itemize}
Because we are building a domain which is a sort of analog of the subgraph, we have to make sure that the distance between $w_1$ and $w_2$ is large in the domain.
The domain $\hat{N}$ that we get from this subgraph, using the strategy presented in this paper (using the small triangles), is the following:
\begin{figure}[H]
\centering
\includegraphics[scale=0.4]{images/domaine_contre_exemple_correct_nomme.PNG}
\caption{Using now $d_{\hat{N}}$ as a notation for the distance in $\hat{N}$, we can easily see that $d_{\hat{N}}(w_1, w_2)$ is large, roughly as the diameter of $\hat{N}$.}
\label{fig: domaine contre expl correct}
\end{figure}
Here is now the domain that we get while considering the union of triangle $T_w$ for all $w\in \bar{\Omega}$:
\begin{figure}[H]
\centering
\includegraphics[scale=0.46]{images/domaine_contre_exemple_incorrect_nomme.PNG}
\caption{We can see that the distance in the domain between $w_1$ and $w_2$ is small. }
\label{fig: domaine contre expl incorrect}
\end{figure}
If we were to pursue our construction with the domain given by Fig. \ref{fig: domaine contre expl incorrect}, we would have a real problem when building the rough isometry of Proposition \ref{prop : rough isom}.
\medskip
Indeed, let us now consider a family of subgraphs $(\Omega_l)_{l \ge 1}$, such that $|\Omega| \underset{l\to \infty}{ \longrightarrow} \infty$ and such that each subgraph of the family has the same particular property as the subgraph of Fig. \ref{fig : graphe contre expl} (the property concerning $w_1$ and $w_2$ we discussed above). In that case, the constants in the rough isometry would then have to be chosen according to each subgraph (the diameter of each subgraph would do). This would obviously destroy our result.
\end{appendices}
\printbibliography
\medskip
\end{document}
|
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| 2,453
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Q: Android Web view do not Display loaded Html content on some phone specifically htc one x I'm loading some Html content with CSS in a WebView.
Everything works fine on every phone, but the HTC one x (android version 4.1.1 ).
Nothing gets displayed in the WebView on this phone.
Below is my code snippet.
The function which gives me the formatted WebView Content:
public String getFormattedWebViewContents(String content)
{
String htmlString = "ul{" +
"padding-left: 10px;" +
"padding-bottom:7px;" +
"color:#818181;" +
"font-size: 10px;font-family: Verdana, Arial, Helvetica, sans-serif; }" +
"ul .bdr { line-height: 16px;}" +
".flt-lt{float:left;}" +
".flt-rt{float:right;}" +
"ul li{padding:10px 5px 10px 10px !important;}"+
"ul li h4 {height: 25px; border: 1px solid #DEDEDE; font-family:Calibri;font-size:18px;" +
"margin-bottom:10px !important;font-weight:bold;text-align: left; vertical-align: middle;margin-left: -7px;" +
"margin-right: 0px !important;" +
"color: #7CA834;" +
"font-family: Calibri;" +
"font-size: 18px;" +
"font-weight: bold;" +
"height: 25px;" +
"margin-bottom: 1px;" +
"padding-left: 10px;" +
"padding-top: 10px !important;" +
"text-align: left;"+
"vertical-align: middle;"+
"background-color: #F2F2F2;color: #7CA834;}"+
".bdr1 .sml-font{font-size:11px !important;padding-top:3px ;color: #7CA834 !important;font-family: Arial !important;font-weight: bold !important;}"+
"ul{margin-bottom:14px; margin-top:12px;background-color: #ffffff; padding-left:10px; padding-right:0px; padding-top:10px; }"+
"ul{padding-left: 10px;padding-top:8px;padding-bottom:7px;color: #818181;font-size:10px;font-family: Verdana, Arial, Helvetica, sans-serif;}"+
"#factstable td{ width:170px; height:18px;}"+
".ext { float:right; }"+
".factsdetaildiv{border: 1px solid #DEDEDE;padding-left: 10px;padding-bottom:7px;color:#818181;font-size: 10px;font-family: Verdana, Arial, Helvetica, sans-serif; }"+
"ul .bdr { line-height: 16px;}"+
"ul{color:#818181;font-size: 10px;font-family: Verdana, Arial, Helvetica, sans-serif; }"+
"ul{list-style-type:none;margin-right: 0px; padding:10px;}"+
"ul li{padding-bottom:7px; padding-top:10px !important;line-height:10PX; list-style-image:none;font-size:11px;font-family:Arial, Helvetica, sans-serif;}"+
"ul{ font-weight:bold;}"+
"ul p{ padding:0px 10px 0px 0px;line-height: 12px;color: #818181;font-size: 12px;font-family: Arial;}" +
" li:first-child { margin-top:-38px !important; padding:10px 5px 10px 10px;}"
;
return "<html>" + "<head>" + "<style type=\"text/css\"> " + "@font-face " + "{ " + "font-family: MyCustomFont; " + "src: url(\"file:///android_asset/font/" + mContext.getResources().getString(R.string.type_face_regular) + "\") " + "}" + "body " + "{ " + "font-family: MyCustomFont; " + "font-size: "+(Utility.isLargeScreen(mContext)? "22px; ": "18px; ") + "color: black" + "}" + htmlString + "</style>" + "</head><body style='margin: 10px'>" + content + "</body></html>";
}
My Web View:
mDetailsContentsWebView.setBackgroundColor(getResources().getColor(R.color.color_product_details_webview_bg));
mDetailsContentsWebView.loadDataWithBaseURL("", Utility.getInstance(getActivity()).getFormattedWebViewContents(mCurrentProductDetails.mDescription), "text/html", "UTF-8", "");
mDetailsContentsWebView.getSettings().setSupportZoom(false);
mDetailsContentsWebView.getSettings().setJavaScriptEnabled(true);
mDetailsContentsWebView.setOnTouchListener(new View.OnTouchListener()
{
public boolean onTouch(View v, MotionEvent event)
{
switch(event.getAction())
{
case MotionEvent.ACTION_DOWN:
{
}
break;
case MotionEvent.ACTION_MOVE:
case MotionEvent.ACTION_CANCEL:
case MotionEvent.ACTION_UP:
{
event.setLocation(event.getX(), event.getY());
}
break;
}
return false;
}
});
Please, somebody let me know if there it's an issue of the HTC phone or my code snippet is wrong.
Thank you.
A: The problem is at line
mDetailsContentsWebView.loadDataWithBaseURL("", Utility.getInstance(getActivity()).getFormattedWebViewContents(mCurrentProductDetails.mDescription), "text/html", "UTF-8", "");
You are using "" in the place of baseURL. Try specifying any valid URL or any local file path(from assets).
Or else just replace the line with
mDetailsContentsWebView.loadData(Utility.getInstance(getActivity()).getFormattedWebViewContents(mCurrentProductDetails.mDescription), "text/html", "UTF-8");
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 9,491
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{"url":"https:\/\/www.physicsforums.com\/threads\/prove-if-s-and-t-are-sets-with-outer-content-zero-sut-has-outer-content-zero.650818\/","text":"# Homework Help: Prove if S and T are sets with outer content zero, SUT has outer content zero.\n\n1. Nov 9, 2012\n\n### Zondrina\n\n1. The problem statement, all variables and given\/known data\n\nSuppose that S and T are sets with outer content 0, prove that SUT also has outer content zero.\n\n2. Relevant equations\n\nC(S) denotes the outer content.\n\nC(S) = C(T) = 0\n\nAlso : $C(S) = inf \\left\\{{ \\sum_{k=0}^{n} A_k}\\right\\}$ where Ak is the area of one of the sub-rectangles Rk.\n\n3. The attempt at a solution\n\nSo we want to show that C(SUT) = 0 using the fact C(S) = C(T) = 0. I'm not really sure where to start this one though. First time I've seen anything like it and a quick search yielded no results about outer content at all.\n\nI do have one theorem though. If S is a curve of finite length L, then C(S) = 0. I also figured ( not positive about this ) that C(\u2205) = 0.\n\nLast edited: Nov 9, 2012\n2. Nov 9, 2012\n\nAttached\n\nFile size:\n12.3 KB\nViews:\n144\n3. Nov 9, 2012\n\n### Zondrina\n\nHmm I sort of see what you're saying. I'm confused as to why you took your sums and unions up to \u221e rather than to n and then later argued as n \u2192 \u221e, C(S) or C(T) \u2192 0. So given any positive \u03b5 :\n\nWe take a set of rectangles R'k such that $T \\subseteq \\bigcup_{k=1}^{n} R_{k}^{'}$ and if I sum all the rectangles up to n, it will be smaller than (1\/2)\u03b5.\n\nWe take another set of rectangles R''k such that $S \\subseteq \\bigcup_{k=1}^{n} R_{k}^{''}$ and if we sum all these rectangles up to n it will also be smaller than (1\/2)\u03b5.\n\nSo hopefully I'm not mistaken here, but you asked me to consider the union of all the rectangles together.\n\nSo we take a set of rectangles Rk such that $S \\cup T \\subseteq \\bigcup_{k=1}^{n} {R_{k}^{'}} \\cup {R_{k}^{''}}$ and if we sum all these rectangles, it will be less than (1\/2)\u03b5 + (1\/2)\u03b5 = \u03b5.\n\n4. Nov 9, 2012\n\n### hedipaldi\n\nYes this what i ment.","date":"2018-06-19 22:47:26","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\": 0.8192272782325745, \"perplexity\": 779.023636849953}, \"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-2018-26\/segments\/1529267863206.9\/warc\/CC-MAIN-20180619212507-20180619232507-00178.warc.gz\"}"}
| null | null |
If you need a little end-of-the-week outfit inspiration, take a peek at how we caught this next girl sporting her bright blue pants and try it out yourself tomorrow!
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|
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"redpajama_set_name": "RedPajamaC4"
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| 3,059
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Хорът (, ) е музикална група от певци-хористи, които пеят едновременно.
Думата произлиза от театъра в Древна Гърция, като хорът е главно действащо лице в древногръцката трагедия. Хорът може да изпълнява едногласна и многогласна (до десетки гласове) музика. Вътрешно хорът е разделен на отделни гласове, (партии, дивисии (лат. divisi). Основните партии са условно 4 и носят различни наименования при различните видове хор. Тези партии могат да се разделят на подгрупи или да се сливат в зависимост от изпълняваното музикално произведение. Всяка партия се изпълнява от поне трима хористи, поради което трябва да се прави разлика между хор и вокален дует, трио, квартет и т.н. Синхронизирането на отделните партии, както и музикалната интерпретация се осъществяват от диригент, който е ръководител на хора.
Хоровите произведения могат да бъдат изпълнявани самостоятелно – "а капела" (), в съпровод на един, няколко музикални инструмента, или оркестър. В същото време хорът често сам поема функцията на съпровод на соловите партии на един или повече певци, понякога и инструменти.
Видове хор
По своя състав хоровете се делят на няколко основни вида.
Еднороден хор. Състои се или само от мъже или само от жени. Мъжките хорове се състоят от 2 тенорови и 2 басови партии, а женските – от 2 сопранови и 2 алтови.
Детският хор строго погледнато е също еднороден, но поради спецификата на своето звучене и работата с изпълнителите се обособява в отделна група. Там спада и момчешкият хор (само от момчета и юноши, пеещи на фалцет). Разделя се на 2 дискантови и 2 алтови партии.
Смесен хор. Изпълнителите са мъже и жени, като имената на основните партии дублират тези на еднородните хорове. До началото на 19 в., особено при духовната музика, смесеният хор се състои от мъже и момчета. Такъв състав на смесения хор се среща и в наши дни.
Някои хорове се специализират в изпълнението на различни музикални жанрове или видове музика. В зависимост от това те могат да бъдат оперни, фолклорни, църковни, военни и т.н. Според статута на изпълнителите, хоровете могат да бъдат определени като професионални и самодейни, а също така като ведомствени, студентски, ученически и т.н. Такова подразделение обаче е твърде условно и не засяга начина на работа и музикалните качества на даден хоров колектив.
Хорова музика
Хоровата музика има изключително древни корени. Специалистите са единодушни, че нейното начало може да се търси в груповото изпълнение на песни при култовите ритуали на първобитните хора. С развитието на човешкото общество хоровото пеене постепенно започва да придобива по-определени черти. Появяват се по-култивирани форми, които се развиват паралелно, а в много случаи изместват първоначалното спонтанно музициране. Съществуването на организирана хорова музика е засвидетелствано във всички древни култури – шумерска, вавилонско-асирийска, староиндийска, египетска, персийска, юдейска, древногръцка, китайска и др. За този период обаче разполагаме почти изключително с косвени свидетелства (писмени, произведения на изобразителното изкуство), докато чисто музикалните източници са твърде малко.
На Запад хоровото пеене започва да се организира и рафинира с разпространението на християнството, при което то е основна част от религиозния култ. През ранното Средновековие вече се оформят музикалните основи, залегнали в основата на хоровата музика, която съществува и днес. В Западна Европа започва да доминира т. нар. "грегориански хорал", по името на папа Григорий Велики (Двоеслов), който е въвел литургическа форма, използвана и в наши дни в Източноправославната и Римокатолическата църква. На Изток се разпространява "осмогласието" (в случая думата глас обозначава не музикална партия, а музикалната тоналност на произведението). И грегорианският хорал, и осмогласието, се причисляват към т. нар. "хомофонно" пеене, при което водещата мелодия, понякога доста разнообразна, е подчертана от звучността на поддържащата партия (исо, "басо континуо" в първоначалното си значение.
По време на Ренесанса и Реформацията на Запад (а в изолирани случаи и по-рано) композиторите започват да експериментират и да доразвиват по-старите и утвърдени форми. Появяват се мотетът, хоралът, разработват се т. нар. "секвенции (последования, канонично утвърдени като текст части от религиозните богослужения). Успоредно с това започват да се развива и светската хорова музика – песни, мадригали, канцони. През този период бележити автори на хорова музика са Пиер де ла Рю, Якоб Аркаделт, Джовани Палестрина, Орландо ди Ласо (Орландус Ласус), Клаудио Монтеверди.
В предкласическия период хоровата музика получава допълнително развитие при автори като Падре Мартини, Георг Фридрих Хендел, Йохан Себастиан Бах, Джовани Батиста Перголези. Именно тези автори въвеждат полифоничното пеене като норма в хоровата музика.
Виенските класици продължават да доизграждат върху основата на предшествениците си. Йозеф Хайдн, Волфганг Амадеус Моцарт, Лудвиг ван Бетховен, създават хорови творби, при които хорът вече се използва като част от основната музикална форма и е равностоен като значене както на оркестъра, така и на солистите. Приблизително по същото време започва и активното развитие на операта, където хорът също играе важна роля. В творбите на Кристоф Вилибалд Глук, Хайдн, Моцарт, Бетховен, хорът вече играе една от главните роли в рамките на музикалната драма. Основните музикални идеи за своя Реквием Моцарт поверява тъкмо на хора.
След този период, от 20-те години на 19 век, хоровата музика, при усъвършенстването на нови музикални изразни форми, не претърпява принципно развитие. Това не означава, че тя е в застой. В операта се появяват знаменити хорови произведения. Хоровите творби в творчеството на композитори като Пиерлуиджи Керубини, Руджиеро Леонкавало, Франц Шуберт, Роберт Шуман, Шарл Гуно, Жорж Бизе, Рихард Вагнер, Хуго Волф, Пьотр Чайковски, Михаил Бородин, Модест Мусоргски, Едвард Григ.
О началото на ХХ в. интересът към хора и търсенето на нови форми за хорови произведения се възражда. Композитори като Сергей Рахманинов, Франсис Пуленк, Арнолд Шьонберг, Бела Барток, Золтан Кодай, Бенджамин Бритън съчиняват едни от най-знаменитите си творби тъкмо за хор. В същото време в Северна Америка се култивира и негърското хорово пеене, първоначалните негърски менестрели, спиричуъли, госпъл-песни се развиват, като паралелно с това дават отражение на хоровата музика на композитори като Айвз, Самюел Барбър, Джордж Гершуин, Ленард Бърнстейн. Именно негърските хорове са основа на певческото изпълнение при музиката от жанрове като блус, джаз, рок-опера, естрадна хорова музика. През този период започва да оформя облика си и българската хорова музика.
От средата на миналия век хоровата музика следва тези традиции. По-важни автори на такава музика са Арнолд Шьонберг, Албан Берг, Артур Онегер, Петър Ебен, Ладислав Нойман, Вельо Тормис, Алфред Шнитке.
Музикално произведение или част от него, предназначено за хорово изпълнение, също се нарича хор: например "Хор на евреите" от операта "Набуко" на Джузепе Верди, "Хор на момчетата" от операта "Кармен" на Жорж Бизе.
Световноизвестни хорови състави в тези периоди и до днес са: "Амброзиански оперен хор", хорът на миланската "Ла Скала", Академичният хор на Мексико, лайпцигският "Томанер Кнабен" (чиито ръководител през последните години от живота си е Й. С. Бах).
България
Съвременната българска хорова музика се изгражда на базата на съчетанието между съхранената в народното творчество приемственост, наследена от народите, които са предци на днешните българи и на внесената след втората половина на 19 в. западноевропейска музикална традиция.
Първите български хорове възникват тъкмо в средата на 19 в. на доброволни начала, обикновено при читалища, училища и други обществени организации. Важни хорови дейци през онова време са Сава Доброплодни, Добри Войников, Янко Мустаков, Тодор Хаджистанчев, Петър Иванов, Георги Байданов, Анастас Стоянов, Емануил Манолов, Ангел Букурещлиев, Панайот Пипков. Успоредно с това при храмовете се поддържа старата църковна певческа школа. Тяхната работа е продължена и в началото на 20 в., но като цяло хоровото дело в България все още има провинциален и просветителски характер и е далеч от съвременните си хорови постижения.
След Първата световна война в България вече разгръщат таланта си хорови дейци като Асен Димитров, Христо Манолов, Любомир Пипков, Стоян Брашованов, Георги Димитров, Крум Бояджиев, Боян Соколов, Светослав Обретенов, Бончо Бочев. Особено внимание през този период трябва да се обърне на дейността и творчеството Добри Христов – знаменит композитор, диригент и музикален деятел, чието творчество е твърде обширно – авторски песни, обработка на народни песни, църковна хорова музика. Произведенията на Христов вече могат да се поставят редом с най-добрите тогавашни съвременни хорови образци. През същото време се появяват и хорови състави, които са в състояние да изпълняват и най-сложните в техническо отношение произведения.
През втората половина на 20 в. в областта на хоровата музика се изявяват композиторите Георги Димитров, Филип Кутев, Светослав Обретенов, Любомир Пипков, Парашкев Хаджиев, Тодор Попов, Красимир Кюркчийски, Иван Спасов, Константин Илиев, Ивелин Димитров, Николай Кауфман. Някои от тях са и хорови диригенти. Хорови състави ръководят знаменити творци като Михаил Милков, проф. Димитър Русков, Георги Робев, Ангел Манолов, Бончо Бочев, Васил Арнаудов, Адриана Благоева, Васил Стефанов, Ивелин Димитров, Лиляна Бочева, Захари Медникаров, Христо Недялков. Освен изпълненията на българска и световна музика (класическа и модерна), хоровете все повече обръщат внимание на музикалните преработки, основани върху българския фолклор.
В наши дни, след известен период на упадък, хоровото дело в България е отново във възход.
Сред хоровите състави, съществуващи в България, могат да бъдат отбелязани смесените Национален филхармоничен хор "Светослав Обретенов", Смесеният хор на БНР, хор "Дунавски звуци" – Русе. Професионални смесени хорове съществуват и при оперните театри в цялата страна. Прочут е фолклорният хор "Мистерията на българските гласове". Освен това в много градове работят много добри аматьорски хорове. Световна слава и признание получават мъжките хорове "Кавал", "Гусла", "Емануил Манолов" дамските "Полифония" и "Васил Арнаудов" (бивш "Любомир Пипков"), детските – Хор на софийските момчета, "Дунавски вълни", "Бодра смяна", детският хор при БНР, Детският хор на град Добрич, прочутия Хор на варненските момчета и младежи, хор "Кантилена", Хоровата школа "Професор Георги Димитров" – Ямбол и много други.
Източници
Външни препратки
Портал на българското хорово творчество
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\section{Introduction}~\label{sec:intro}
Driven mainly by the pervasive usage of smartphones and the emergence of the Internet of Things (IoT), future 5G mobile networks will become as pervasive as electrical wiring~\cite{Rap91d} and will offer unprecedented data rates and ultra-low latency~\cite{Boccardi14a,Ghosh14a,Rap16a}. For the first time in the history of radio, millimeter-wave (mmWave) frequencies will be used extensively for mobile and fixed access, thus requiring accurate propagation models that predict how the channel varies as people move about. Remarkable progress has been made in modeling large-scale propagation path loss at mmWave frequencies~\cite{Rap13a,Rap15b,Rap16a, Rap13b,Mac15b,Sun16b,Thomas16a,Koymen15a,Yoon15a,Maltsev10a,Mac17b}, and it is well understood that for an assumption of unity gain antennas across all frequencies, Friis equation predicts that path loss is greater at mmWave compared to today's UHF/microwave cellular systems~\cite{Friis46a, Rap13a, Rap15b, Mac15b,Sun16b,Thomas16a,Rap16a}. Also, rain and atmospheric attenuation are well understood, and reflection and scattering are more dominant than diffraction at mmWave bands~\cite{Rap16a,Rap15a,Deng16a,Rumney16b,Solomitckii16a}.
Broadband statistical spatial channel models (SSCMs) and simulators that faithfully predict the statistics of signal strength, and the number and direction of arrival and departure of multipath components, have been developed by a consortium of companies and universities~\cite{3GPP.38.900} and from measurements in New York City\textcolor{black}{\cite{Sun17b}}. These models are being used to develop air-interfaces for 5G systems~\cite{Samimi16a}\textcolor{black}{\cite{Sun17c}}. Elsewhere in this issue,\textcolor{black}{\cite{Rap16a}} summarizes standard activities for large-scale mmWave channel modeling.
Little is known, however, about the small-scale behavior of wideband mmWave signals as a mobile user moves about a local area. Such information is vital for the design of handoff mechanisms and beam steering needed to rescue the communication link from deep fades. In this paper, propagation measurements investigate diffraction, human blocking effects, small-scale spatial fading and autocorrelation, local area channel transitions, and stationarity of signal power in local area clusters at frequencies ranging from 10 to 73 GHz. Diffraction measurements for indoor and outdoor materials at 10, 20, and 26 GHz are presented in Section II, and two diffraction models, i.e., the knife edge diffraction (KED) model and a creeping wave linear model, are used to fit the measured results. We predict the rapid signal decay as a mobile moves around a diffracting corner. In Section III, measurements at 73 GHz are presented and a double knife-edge diffraction (DKED) antenna gain model that uses directional antenna patterns is shown to describe minimum and maximum fade depths caused by human blockage. Small-scale fading and correlation studies at 73 GHz are presented in Section IV, where small-scale fading distributions and spatial autocorrelations of received voltage amplitudes in LOS and NLOS environments with omnidirectional and directional antennas are provided and analyzed. In Section V, \textit{route} and \textit{cluster} scenarios are used to study local area channel transitions and stationarity, where analysis for channel transition from a NLOS to a LOS region and local area path loss variations are provided. Conclusions are given in Section VI. \textcolor{black}{Channel models given here may be implemented for small-scale propagation modeling and real-time site-specific mobile channel prediction and network control~\cite{Wang05b}.}
\section{Diffraction Measurements and Models}\label{sec:diffraction}
\subsection{Introduction of Diffraction Measurements and Models}
Accurate characterization of diffraction at cmWave and mmWave frequencies is important for understanding the rate of change of signal strength for mobile communications since future 5G mmWave systems will have to rely less on diffraction as a dominant propagation mechanism~\cite{Rap15b, Mac15b}. Published indoor and outdoor diffraction measurements show that diffraction has little contribution to the received signal power using various materials and geometries (edges, wedges, and circular cylinders) at 60 GHz and 300 GHz~\cite{Maltsev10c,Lu13a,Jacob12a}. It was shown that the KED model agreed well with diffraction measurements for cuboids at 300 GHz~\cite{Kleine-Ostmann12a}, vegetation obstacles at 2.4, 5, 28, and 60 GHz~\cite{Corre16a}, and for human blocking at 60 GHz~\cite{Maltsev10a}. Apart from the KED model, uniform theory of diffraction (UTD) models are also used. An overview of the Geometrical Theory of Diffraction (GTD) and the UTD are provided in~\cite{Pathak13a}, as well as their utility to solve practical problems. Besides the KED and UTD models, Mavridis \textit{et al.} presented a creeping wave linear model~\cite{Mavridis14a} to estimate the diffraction loss by a perfectly conducting or lossy circular cylinder for both transverse-magnetic (TM) and transverse-electric (TE) polarizations at 60 GHz. In the following sub-sections, we describe diffraction measurements conducted in 2015 around the engineering campus of New York University~\cite{Deng16a}, where the frequency dependency of diffraction at 10, 20, and 26 GHz in realistic indoor and outdoor scenarios was investigated to yield simple yet accurate diffraction models for wireless planning.
\subsection{Diffraction Measurement System}
Diffraction measurements were performed by transmitting a continuous wave (CW) signal generated by an Agilent E8257D PSG analog signal generator through a pyramidal horn antenna at the transmitter (TX). An identical horn antenna was used at the receiver (RX) to receive signal energy around a corner test material (e.g. a stone pillar). The RX antenna was fed to an E4407B ESA-E spectrum analyzer that measured received power which was subsequently recorded on a laptop with LabVIEW software. During the measurements, the TX antenna was set one meter from the knife edge, sufficiently in the far field, and was fixed to a tripod and aimed at the knife edge, whereas the RX antenna was set 2 meters from the knife edge (also in the far field) and was fixed on a rotatable gimbal attached to a translatable linear track that was made to from an approximate arc around the knife edge (see Fig. \ref{fig:Corner}). Diffraction loss was measured at 10, 20, and 26 GHz using identical pairs of antennas at the TX and RX to measure each frequency, separately. For each frequency, Table \ref{tbl:1} lists the flange type, antenna gain, and half-power beamwidth (HPBW) of the antenna pairs used. The TX and RX were stationed at a wide range of angles, both in the lit and shadowed region, and the horns always had their boresights focused on the corner knife-edge of the indoor and outdoor materials that were studied. More details regarding the measurement system are given in~\cite{Deng16a,Deng16b}.
\begin{table}[b]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\caption{Antenna parameters used diffraction measurements.}~\label{tbl:1}
\fontsize{7}{7}\selectfont
\begin{tabular}{|>{\centering\arraybackslash}m{1.3cm}|>{\centering\arraybackslash}m{1.1cm}|>{\centering\arraybackslash}m{1.4cm}|>{\centering\arraybackslash}m{1.25cm}|>{\centering\arraybackslash}m{1.5cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Measured Frequency} & \textbf{Flange Type} & \textbf{Antenna Gain} & \textbf{HPBW (Az./El.)} & \textbf{Far Field Distance} \\ \hline
10 GHz & WR-75 & 20 dBi & $17^\circ$/$17^\circ$ & 0.47 m \\ \hline
20 GHz & WR-51 & 20 dBi & $17^\circ$/$17^\circ$ & 0.46 m \\ \hline
26 GHz & WR-28 & 24.5 dBi & $10.9^\circ$/$8.6^\circ$ & 0.83 m \\ \hline
\end{tabular}
\end{center}
\end{table}
\subsection{Diffraction Measurement Description}
The indoor measurements were performed at $90^\circ$ (right-angle) wall corners made of drywall, wood, and semi-transparent plastic board with 2 cm thickness. Outdoor measurements studied one rounded stone pillar corner and one right-angle marble building corner. During the measurements, the TX and RX were placed on either side of the corner (knife edge) of the test material. A diagram of the corner diffraction geometry is shown in Fig. \ref{fig:Corner} where $d_1$ is the distance between the TX and the corner knife-edge, and $d_2$ is the distance between the corner knife-edge and the RX. Both $d_1$ (1 m) and $d_2$ (2 m) remained constant throughout the diffraction measurement campaign. The $\beta$ and $\alpha$ values are the incident and diffraction angles, respectively, where two (outdoor) or three (indoor) fixed values between $10^\circ$ and $39^\circ$ were chosen for $\beta$. The RX antenna was mounted on a motorized linear track (see Fig. \ref{fig:Corner}) that translated in step increments of 0.875 cm, which corresponds to approximately a $0.5^\circ$ increment in diffraction angle ($\alpha$) for each step increment. At each step increment, the RX antenna was adjusted to point perfectly towards the knife-edge corner. The length of the track was $35.5$ cm and was used to measure a $20^\circ$ swath of diffraction angles over the entire length of the track. At each measurement location, five consecutive linear tracks (see Fig. \ref{fig:Corner}) were used to provide a $100^\circ$ diffraction angle arc around the corner which covered a broad range of the shadow region where the TX antenna is shadowed by the corner object with respect to RX antenna. Additionally, a smaller range of diffraction angles was measured in the lit region where the TX and RX antennas were in view of each other but were not pointed at each other since they were always aimed at the corner.
At each location, prior to the diffraction measurements, a free space calibration in an open area with both antennas pointed at each other on boresight was conducted with a 3 m ($d_1 + d_2 = 3$ m) transmitter-receiver (T-R) separation distance to provide a free space power reference for each frequency. The diffraction loss was then obtained by calculating the difference between the measured received signal power at each step increment of the RX antenna during the diffraction measurements and the free space calibration received power (the TX and RX antenna gains were deducted from all power measurements).
\begin{figure}[b!]
\vspace{-2.5mm}
\begin{center}
\includegraphics [width=0.44\textwidth]{Corner.pdf}
\caption{Top view of the corner diffraction geometry\cite{Deng16a}.}\label{fig:Corner}
\end{center}
\end{figure}
\subsection{Theoretical Diffraction Models}
\subsubsection{KED Model}
The KED model is suitable for applications with sharp knife edges and has a simple form yet high prediction accuracy ~\cite{Russell93a}. In general, diffraction loss over complex and irregular obstructions can be difficult to calculate, but typical obstructions for 5G wireless will involve common building partitions which are generally simple in nature and with dimensions that appear infinite at such small wavelengths, such as a wall or building corner, thus justifying simple diffraction models.
The diffraction loss (as compared to free space) is obtained by calculating the electric field strength $E_d$ [V/m] at the RX based on the specific Fresnel diffraction parameter $\nu$~\cite{Rap02a}. The ratio of $E_d$ and the free space field strength $E_0$ can be computed by summing all the secondary \textcolor{black}{Huygens'} sources in the knife edge plane and is given by~\cite{Deng16a,Rap02a}:
\begin{equation}
\footnotesize
\label{eq:8}
\frac{E_d}{E_0} = F(\nu) = \frac{1+j}{2} \int_{\textcolor{black}{\nu}}^{\infty} e^{-j (\pi/2) t^2}dt
\end{equation}
where $F(\nu)$ is the complex Fresnel integral and $\nu$ is the Fresnel diffraction parameter is defined as~\cite{Rap02a}:
\begin{equation}
\label{eq:9}
\footnotesize
\nu=h\sqrt{\frac{2(d_1+d_2)}{\lambda d_1 d_2}}=\alpha \sqrt{\frac{2 d_1 d_2}{\lambda(d_1+d_2)}}
\end{equation}
where $\lambda$ is the wavelength, $\alpha$ is the diffraction angle, $h$ is the effective height (or width) of the obstructing screen with an infinite width (or height) placed between the TX and RX at the distances $d_1$ and $d_2$, respectively, under the conditions that $d_1$, $d_2 \textcolor{black}{\gg} h$, and $d_1$, $d_2 \textcolor{black}{\gg} \lambda$. These conditions were met for 10, 20 and 26 GHz measurements in both the indoor and outdoor environments, as shown in Fig.~\ref{fig:Corner}.
Based on (\ref{eq:8}) and (\ref{eq:9}), the diffraction power gain $G(\nu)$ in dB produced in a knife edge by the KED model is expressed as~\cite{Deng16a,Rap02a}:
\begin{equation}
\label{eq:10}
\footnotesize
G(\nu) [\mathrm{dB}] = - P(\nu) [\mathrm{dB}] = 20\log_{10}|F(\nu)|
\end{equation}
where $P(\nu)$ is the power loss of the diffracted signal for the value of $\nu$, compared to free space case for the same distance.
\subsubsection{Convex Surfaces based Diffraction Model}
Although the KED model has broad applications for various geometries, it requires the diffraction corner to be in the shape of a sharp knife edge and does not account for the radius of curvature of an obstacle. When a diffraction corner is rounded in shape, such as a stone pillar corner (it resembles a circular cylinder), a creeping wave linear model can better predict diffraction loss~\cite{Piazzi98a, Mavridis14a}. A creeping ray field at the RX antenna behind the circular object for an incident plane wave is given by~\cite{Piazzi98a}:
\begin{equation} \label{eq:11}
\footnotesize
E(\alpha,d_2,k) \sim E_i e ^{-jk\alpha R_h} \frac{e^{-jkd_2}}{\sqrt{kd_2}}\sum_{p=1}^{\infty}{D_p{R_h}}\cdot \exp(-\psi_{p}\alpha)
\end{equation}
where $E_i$ is the incident field from the TX that impinges upon the obstruction, $R_h$ is the radius of cylinder for the diffraction corner, $k$ is the wave number of the carrier frequency, $\alpha$ is the diffraction angle, $d_2$ is the distance between the launch point at the rounded corner edge and the RX, $D_p$ is the excitation coefficient and $\psi_p$ is the attenuation constant. Due to the \textcolor{black}{computational complexity} of (\ref{eq:11}), a reasonable approximation for $E$ on a flat surface can be obtained by considering only the $p = 1$ term in (\ref{eq:11}), which is given by~\cite{Deng16a,Deng16b,Mavridis14a,Piazzi98a,Tervo14a}:
\begin{equation} \label{eq:12}
\footnotesize
E \sim E_i D_p{R_h} \cdot \exp(-\psi_{p}\alpha)
\end{equation}
The expression for the diffraction power loss in dB based on the creeping wave linear model is given by~\cite{Deng16a,Deng16b}:
\begin{equation}\label{eq:13}
\footnotesize
\begin{split}
G(\alpha) [\mathrm{dB}] = - P(\alpha) [\mathrm{dB}]= 20\log_{10}E
\end{split}
\end{equation}
In order to facilitate the computation of $G(\alpha)$, \textcolor{black}{a simple linear model}~\eqref{eq:14} based on minimum \textcolor{black}{mean squared} error (MMSE) estimation between the model and measured data was proposed in~\cite{Tervo14a} to estimate the diffraction loss caused by a curved surface at a single frequency, based on the creeping wave linear model~\cite{Deng16a,Deng16b}:
\begin{equation}
\footnotesize
\label{eq:14}
P(\alpha) = n \cdot \alpha + c
\end{equation}
where $n$ is the linear slope of diffraction loss calculated by MMSE for each specific frequency and object radius, and $c$ is the anchor point set to 6.03 dB, the diffraction loss estimated by KED with diffraction angles $\alpha=\beta=0^\circ$.
\begin{figure}[t!]
\centering
\includegraphics[width=2.95in]{Multi_Freq_Dry_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.95in]{Multi_Freq_Wood_VV.pdf}\vspace{0.2cm}
\includegraphics[width=2.95in]{Multi_Freq_Plastic_VV.pdf}
\caption{Diffraction measurements for the indoor drywall corner (top), wooden corner (middle), and plastic board (bottom) compared to the KED model at 10, 20 and 26 GHz~\cite{Deng16a}.}
\label{fig:Indoor}
\end{figure}
\begin{figure}[b!]
\centering
\includegraphics[width=2.9in]{Multi_Freq_Pillar_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.9in]{Multi_Freq_Marble_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.9in]{Out_Pillar_10GHz_VV-eps-converted-to.pdf}
\caption{Measured diffraction loss for the outdoor stone pillar (top) and marble corner (middle) with creeping wave linear models at 10, 20, and 26 GHz, and stone pillar measurement results compared to the KED and the creeping wave linear models at 10 GHz (bottom)~\cite{Deng16a}.}
\label{fig:Outdoor}
\end{figure}
\subsection{Indoor Diffraction Results and Analysis}
The indoor diffraction loss measurements for the drywall corner, wooden corner, and plastic board are plotted with the KED model (\ref{eq:10}) at 10, 20, and 26 GHz as a function of the diffraction angle in Fig. \ref{fig:Indoor}. \textcolor{black}{Three different TX incident angles were used to measure diffraction loss for each frequency. Since the measurements in~\cite{Deng16a} were conducted with the TX and RX along a constant radius ($d_1$ and $d_2$, respectively) from the corner of each test material, diffraction loss can be represented as a function of the diffraction angle $\alpha$ in~\eqref{eq:9} for each frequency (or wavelength $\lambda$), without the need for the TX incident angle.}. Fig. \ref{fig:Indoor} shows diffraction loss increases to approximately 30 dB as compared to free space as the RX antenna moves from the edge of the lit region (0$^\circ$) into the shadow region (20$^\circ$) for the drywall corner and wooden corner, respectively. This demonstrates the rapid signal degradation that occurs when diffraction is the primary propagation mechanism in mobile systems. Note the good fit between the drywall diffraction measurements and the simple KED model in the early and deep shadow regions for all three frequencies, while the KED model overestimates diffraction loss by 5-10 dB compared with the wooden corner diffraction measurements at 10, 20, and 26 GHz.
As for the plastic board material, the KED model overestimates the measured diffraction loss at small diffraction angles near the lit/shadow region boundary (from $0^\circ$ to $30^\circ$), and underestimates diffraction loss in portions of the deep shadow region (diffraction angles greater than $30^\circ$), most noticeably at 20 GHz and 26 GHz. Fig. \ref{fig:Indoor} indicates slightly less loss occurs at lower frequencies, implying frequency dependence, where divergence from pure diffraction theory can be attributed to reflections and scattering in the indoor environment and potential transmissions through the test materials. In general, the observations match the KED model trend. Oscillation patterns of the measured data in the shadow region observed in Fig. \ref{fig:Indoor} indicate that the measured diffraction signal includes corner diffraction, penetration through the material, and partial scattering in the measurement environment. We note that the diffraction loss observed in the lit region is due to the measurement procedure where the TX and RX antennas were never aligned on boresight (except at 0$^\circ$) since they were constantly pointed directly at the knife-edge corner.
\textcolor{black}{Penetration loss was measured for typical building materials in the same indoor environment at 73 GHz and showed that co-polarized penetration loss ranged from 0.8 dB/cm (lowest loss material -- drywall) to 9.9 dB/cm (highest loss material -- steel door), with standard deviation $\sigma$ about the average loss ranging from 0.3 dB/cm (lowest $\sigma$ -- drywall) to 2.3 dB/cm (highest $\sigma$ -- clear glass). Additional details can be found in Table II of~\cite{Ryan17a}.}
\subsection{Outdoor Diffraction Results and Analysis}
The outdoor marble corner and stone pillar measurement results are shown in Fig. \ref{fig:Outdoor} with the MMSE creeping wave linear models (\ref{eq:14}) at 10, 20, and 26 GHz.
It can be seen from Fig. \ref{fig:Outdoor} that (\ref{eq:14}) predicts a linearly increasing diffraction loss into the deeply shadowed region, as opposed to the leveling off seen in Fig. \ref{fig:Indoor} from (\ref{eq:10}).
Fig. \ref{fig:Outdoor} also shows the outdoor stone pillar measurement results at 10 GHz compared to the KED model and creeping wave linear model, where the creeping wave linear model provides a better fit to the measured relative diffraction loss than the KED model, in the shadow region.
Diffraction loss for each frequency is plotted as a function of diffraction angle and includes the measured loss at two TX incident angles.
The measured data matches well with the creeping wave linear model derived via MMSE for each frequency, in the shadow region.
The creeping wave linear model slopes are 0.75, 0.88, and 0.96 for the stone pillar measurements and 0.62, 0.77, and 0.96 for the marble corner measurements at 10, 20, and 26 GHz, respectively.
Fig. \ref{fig:Outdoor} shows outdoor obstructions cause an even greater loss when a mobile moves into a deeply shadowed region, showing as much as 50 dB of loss when solely based on diffraction.
Based on the increase in slope values with frequency, it is easily seen that diffraction loss increases with frequency in the outdoor environment.
The simple creeping wave linear model (\ref{eq:14}) fits well with the measured data and has a much lower standard deviation compared with the KED model~\cite{Deng16a}, indicating a good overall match between the creeping wave linear model and measured data.
Similar to the indoor environment diffraction loss measurements, the high diffraction loss in the lit region is caused by the TX and RX pointing off boresight towards the corner of the test material.
When comparing the slope values of the two outdoor materials, the rougher surface (stone) with a slightly rounded edge has a greater slope (greater attenuation) for identical frequencies as compared to the smoother surface (marble) straight edge.
We note that using the MMSE method to derive the typical slope values instead of calculating the theoretical slope value provides system engineers a useful parameter while reducing the \textcolor{black}{computational complexity} of the diffraction model. The simple slope values are useful for mobile handoff design since a mobile at 20 GHz would see approximately 21 dB of fading when moving around a marble corner from a diffraction angle of 0$^\circ$ to 20$^\circ$. For a person moving at a speed of 1 m/s, this results in about a 21 dB/s initial fade rate from LOS to NLOS. Fig. \ref{fig:Outdoor} shows more oscillation patterns in the marble corner measurements in the deeply shadowed region than in the stone pillar measurements, which indicates more prevalent scattering when measuring the marble corner. \textcolor{black}{We note that the diffraction loss for outdoor building corners at mmWaves using directional antennas can be better predicted with a simple linear model (creeping wave), whereas the KED model agrees well with indoor diffraction loss measurements. Both the creeping wave and KED models can be used in network simulations and ray-tracers with short computation time and good accuracy while considering approximately 5-6 dB standard deviation (see~\cite{Deng16a} for mean error and standard deviation values between the measured data and models derived from the data).} For cross-polarized diffraction measurement results, see~\cite{Deng16b}.
\section{Human Blockage Measurements and Models}
\subsection{Introduction of mmWave Human Blockage}
In mmWave communications, attenuation caused by human blockage (when a human body blocks the LOS path between a transmitter and receiver) will greatly impact cellphone link performance, and phased array antennas will need to adapt to find other propagation paths when blocked by a human~\cite{Friis46a, Mac16a}. This is in sharp contrast to omnidirectional antennas used at sub-6 GHz frequencies. Understanding this severe blockage effect and employing appropriate models for mobile system simulation are important for properly designing future mmWave antennas and beam steering algorithms~\cite{Sun15a,Samimi16a,Sun14b}. One of the earliest human blockage measurement studies~\cite{Sato98a} was conducted at 60 GHz for indoor wireless local area networks (WLAN) with T-R separation distances of 10 m or less in a typical office environment. Results showed that the signal level decreased by as much as 20 dB when a person blocked the direct path between omnidirectional TX and RX antennas, with deep fades reaching 30 dB using directive antennas\cite{Collonge04a},~\cite{Collonge03a}. In addition to human blockage measurements at 60 GHz,~\cite{80211ad10a} provided a human induced cluster blockage model based on ray tracing~\cite{Seidel94a}, a random walk model, and a diffraction model from contributions to 802.11ad~\cite{Jacob09d},~\cite{Jacob09e}. The probability distributions for four parameters (duration, decay time, rise time, and mean attenuation) were generated by the human-induced cluster blockage model and were validated with the Kolmogorov-Smirnov test~\cite{80211ad10a}. The \textit{mobile and wireless communications enablers for the twenty-twenty information society (METIS)} and \textit{3\textsuperscript{rd} Generation Partnership Project (3GPP)} also proposed their own human blockage models~\cite{METIS15a,3GPP.38.900,Medbo13a} based on KED models for one or multiple edges. In the following subsections, 73 GHz human blockage measurements and an improved human blockage model are described.
\subsection{Human Blockage Measurement System}
A real-time spread spectrum correlator channel sounder system described in~\cite{Mac16a} was used for the human blockage measurements. A pseudorandom noise (PN) sequence of length 2047 was generated at baseband with a field programmable gate array (FPGA) and high-speed digital-to-analog converter (DAC). This wideband sequence was modulated to an intermediate-frequency (IF) of 5.625 GHz which was then upconverted to a center frequency of 73.5 GHz (1 GHz null-to-null RF bandwidth). The transmit power at the TX was -5.8 dBm, and identical TX and RX horn antennas with 15$^\circ$ azimuth and elevation (Az./El.) HPBW and 20 dBi of gain were used. The received signal was downconverted to IF, and then demodulated to its in-phase ($I$) and quadrature ($Q$) baseband signals that were sampled at 1.5 Giga-Samples (GS/s) via a high-speed analog-to-digital converter (ADC). The digital signals were then correlated in software via a Fast Fourier Transform (FFT) matched filter to create the $I$ and $Q$ channel impulse response (CIR), and subsequent power delay profiles (PDPs) ($I^2+Q^2$). The system had a multipath resolution of 2 ns and an instantaneous dynamic range of 40 dB and could capture PDPs with a minimum consecutive snapshot interval of 32.752$\mu s$ to measure rapid fading. Power was computed as the area under the PDP, and the voltage was found as the square root of power~\cite{Rap02a}.
\subsection{Blockage Measurement Environment Description}
Human blockage measurements were conducted in an open laboratory using a 5 m T-R separation distance. High gain narrowbeam horn antennas were used at the TX and RX with both antenna heights set to 1.4 m relative to the ground and aligned on boresight. Nine measurements were recorded with a human blocker walking at a perpendicular orientation through the LOS path between the TX and RX at an approximate 1 m/s speed. This perpendicular walk was performed at 0.5 m increments between the TX and RX starting at 0.5 m from the TX for measurement one, and 4.5 m from the TX for measurement nine, as depicted in Fig. \ref{fig:4}. We note that 0.5 m is the typical distance for a person to view the screen of a smartphone. For each of the nine perpendicular walks, 500 PDPs were recorded per second in a five-second window, resulting in 2500 PDPs for each measurement. The dimensions of the human blocker were: $b_{\text{breadth}}=0.47$ m; $b_{\text{depth}}=0.28$ m; $b_{\text{height}}=1.80$ m. Detailed information about the experiment is given in~\cite{Mac16a,Mac17a}.
\begin{figure}[tb!]
\centering
\includegraphics [width=0.45\textwidth]{Block_TestCases.pdf}
\caption{Depiction of nine measurement locations where at each indicator separated by 0.5 m, the human blocker walked at a perpendicular orientation between the TX and RX antennas~\cite{Mac16a}.}\label{fig:4}
\vspace{-2.5mm}
\end{figure}
\subsection{KED Blockage Model}
Knife-edge diffraction is commonly used to model human blockage by modeling a thin rectangular screen as the blocker~\cite{Kunisch08a, Medbo13a,METIS15a,3GPP.38.900}. In DKED modeling, the rectangular screen is considered infinitely high, such that diffraction loss only occurs from the two side edges of the body. A typical screen blocker with height $h$ and width $w$ is displayed in Fig. \ref{fig:2} from both a 3D and top view screen projection. The dimensions for the top view of the screen are defined as follows: $w$ is the width of the screen from $w1$ to $w2$; $r\overset{\text{def}}{=}\overline{AB}$; $w\overset{\text{def}}{=}\overline{w1w2}$; $h\overset{\text{def}}{=}\overline{h1h2}$; $TS\overset{\text{def}}{=}\overline{AS}$;
$SR\overset{\text{def}}{=}\overline{SB}$; $D2_{w1}\overset{\text{def}}{=}\overline{A w1}$; $D1_{w1}\overset{\text{def}}{=}\overline{w1 B}$; $D2_{w2}\overset{\text{def}}{=}\overline{A w2}$; $D1_{w2}\overset{\text{def}}{=}\overline{w2 B}$; $\alpha_{w1}$ and $\alpha_{w2}$ are the diffraction angles for the $w1$ and $w2$ edges of the screen, respectively~\cite{Mac16a}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=1\linewidth]{3d_sub_a.pdf}
\caption{3D screen projection.}
\label{fig:DKEDmeasDProj}
\end{subfigure}
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=1\linewidth]{top_sub_b.pdf}
\caption{Top view of screen projection.}
\label{fig:TopProj}
\end{subfigure}
\caption{(a) 3D and (b) top-down projection of screen blocker.}\label{fig:2}
\vspace{-2.5mm}
\end{figure}
From Fig. \ref{fig:TopProj}, the two side edges of the screen are denoted $w1$ and $w2$, where the distance between the edges is the body depth ($b_{\text{depth}}$) or width of the screen $w$, since the blocker walks through the LOS path at a perpendicular orientation. As the screen moves between the TX and RX antennas and blocks the LOS path, the screen is always considered perpendicular to the solid line drawn between the two, to reduce computational complexity.
Diffraction loss is calculated via numerical approximation by Fresnel integration and the diffraction parameter as follows~\cite{Wang15a}:
\begin{equation}\label{eq:KEDapprox}
\footnotesize
F_{w1|w2}=
\begin{cases}
\frac{(1-j)}{2}\left(\frac{1+j}{2}-(C(v)+j\cdot S(v))\right)\text{, if $v>$ 0} \\
\frac{(1-j)}{2}\left(\frac{1+j}{2}+(C(-v)+j\cdot S(-v))\right)\text{, otherwise}
\end{cases}
\end{equation}
where the numerical approximations of Fresnel integration for $C(v)$ and $S(v)$ are:
\begin{subequations}
\footnotesize
\begin{equation}\label{eq:FresC}
C(v) = \int_{0}^{v}\cos{\left(\frac{\pi v^2}{2}\right)dv}
\end{equation}
\begin{equation}\label{eq:FresC}
S(v) = \int_{0}^{v}\sin{\left(\frac{\pi v^2}{2}\right)dv}
\end{equation}
\end{subequations}
and where the diffraction parameter $v$ is derived by~\cite{Rap02a}:
\begin{equation}\label{eq:diffp}
\footnotesize
v_{w1|w2} = \pm\alpha_{w1|w2} \sqrt{\frac{2\cdot \overline{AS}\cdot \overline{SB}}{\lambda (\overline{AS}+\overline{SB})}}
\end{equation}
The diffraction parameter $v$ is calculated based on the distance from the TX to the screen, from the screen to the RX, the diffraction angle $\alpha$ (see Fig.~\ref{fig:TopProj}), and the carrier wavelength $\lambda$. The $\pm$ sign in~\eqref{eq:diffp} is applied as $+$ to both edges for NLOS conditions. \textcolor{black}{When calculating the diffraction parameter~\eqref{eq:diffp} under unobstructed (LOS) conditions for the screen edge ($w1$ or $w2$) closest to the straight line drawn between the TX and RX, the $\pm$ is treated as ``$-$", whereas the $\pm$ is treated as ``$+$" for the screen edge farthest from the straight line drawn between the TX and RX~\cite{Wang15a}}.
The individual received signal caused by knife-edge diffraction from the $w1$ and $w2$ edges is $F_{w1}$ and $F_{w2}$, respectively. The complex signals corresponding to the edges can be added in order to determine the combined diffraction loss observed at the RX. The total diffraction loss power in log-scale is determined by taking the magnitude squared of the summed signals as follows~\cite{Wang15a}:
\begin{equation}\label{eq:Lscreen}
\footnotesize
L_{\text{screen}}[\mathrm{dB}] = 20\log_{10} \left( \big \lvert F_{w1}+F_{w2} \big \rvert \right)
\end{equation}
The DKED model~\eqref{eq:Lscreen} has been adopted by METIS and others~\cite{METIS15a,3GPP.38.900}, but it does not consider the antenna radiation pattern (it assumes an omnidirectional antenna~\cite{Medbo04a}) and has been shown to underestimate diffraction loss in the deepest fades when using directional antennas, which are sure to be employed by mmWave mobile devices~\cite{Mac16a}. This physical phenomenon occurs when the blocker obstructs the LOS path between antennas (i.e. deep fading), leaving only the off-boresight antenna gains to contribute to the received signal strength, which is slightly less than the directive gain. To account for the impact of non-uniform gain directional antennas on human blockage, antenna gain is considered in~\eqref{eq:Lscreen} (see~\cite{Mac16a}). The following azimuth far-field power radiation pattern of a horn antenna (general for any directional antenna) for a given half-power beamwdith (HPBW) is approximated by~\cite{Sun15a,Mac16a}:
\vspace{0.2cm}
\begin{align*}
\footnotesize
G(\theta) = \sinc^2(\textrm{a}\cdot \sin (\theta))\cdot \cos^2(\theta)
\end{align*}
where:
\begin{align*}
\footnotesize
\sinc^2\Bigg(\textrm{a}\cdot \sin \bigg( \frac{\text{HPBW}_{\text{AZ}}}{2}\bigg)\Bigg)\cdot \cos^2 \bigg( \frac{\text{HPBW}_{\text{AZ}}}{2}\bigg)=\frac{1}{2}
\end{align*}
The DKED model in~\eqref{eq:Lscreen} can be extended to include TX and RX antenna gains for the projected angles $\theta$ between the TX and the screen, and the screen and RX as follows~\cite{Mac16a}:
\begin{equation}\label{eq:LscreenMod}
\footnotesize
\begin{split}
L_{\textrm{Screen A.G.}}[\mathrm{dB}] = 20\log_{10} \Bigg( \Bigg \lvert F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}} \\+F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}\Bigg \rvert \Bigg)
\end{split}
\end{equation}
where $G_{D2_{w1}},\;G_{D1_{w1}},\;G_{D2_{w2}}$, and $G_{D1_{w2}}$ are the linear power gains (normalized to the directive gain such that $G_{0^\circ}=1$) of the antennas based on the point-source projections $\overline{Aw1}$; $\overline{w1B}$; $\overline{Aw2}$; $\overline{w2B}$; $A$ to $w1$, $w1$ to $B$, $A$ to $w2$, and $w2$ to $B$ (see Fig.~\ref{fig:TopProj}). When the screen does not obstruct the LOS path between the TX and RX, the normalized gains are set to $G(\theta)=1$, since the slight variations of antenna patterns have little effect on diffraction loss in the unobstructed case.
\subsection{Human Blockage Results and Analysis}
For each of the nine measurement paths, the area under the curve of each of the 2500 PDPs was integrated to calculate the received power in 2 ms increments. In Fig. \ref{fig:DKEDmeas} the received power (red) is compared to the DKED antenna gain (DKED-AG) model (green)~\eqref{eq:LscreenMod}, in addition to showing the constructive (signals in-phase) and destructive (signals out of phase) sum of received signals of the upper (blue) and lower (black) bound of the fade envelope, respectively.
\begin{figure}[tb!]
\vspace{-2.5mm}
\begin{center}
\includegraphics [width=0.5\textwidth]{Block_Test44_DKED_Sim_Data_METIS_v4_0_embed.pdf}
\caption{Comparison of measured received power of human blockage at 73 GHz and the DKED-AG model in~\eqref{eq:LscreenMod}~\cite{Mac16a}.}\label{fig:DKEDmeas}
\end{center}
\vspace{-2.5mm}
\end{figure}
Fig. \ref{fig:DKEDmeas} represents loss as compared to a free space reference with no blockage between the TX and RX. From Fig. \ref{fig:DKEDmeas}, we observe gain in the received signal as the human enters the TX/RX LOS path, and then deep attenuations as the human blocks the LOS path. Due to the fact that identical antennas were used at the TX and RX, the envelopes of the received signal power were similar in the two different symmetrical cases (i.e., Meas. 1 and Meas. 9). The best case scenario (minimum diffraction loss) is found by summing the magnitudes of received field components from the $w1$ and $w2$ edges of the blocker, represented by the blue dashed line in Fig.~\ref{fig:DKEDmeas}. For minimum loss,~\eqref{eq:LscreenMod} is reformulated as: $20\log_{10}(|F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}}|+|F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}|)$. The worst case scenario (maximum diffraction loss) is found by taking the difference of the magnitudes of received signals from the $w1$ and $w2$ edges, and is represented by the black dotted line in Fig~\ref{fig:DKEDmeas}, where~\eqref{eq:LscreenMod} is computed as: $20\log_{10}\Big(\big||F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}}|-|F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}|\big|\Big)$.
It was previously demonstrated that diffraction loss models that do not account for antenna gain pattern can severely underestimate the diffraction loss when the blocker is close to either antenna (a critical issue for mobile phone use)~\cite{Mac16a}. The DKED-AG model~\eqref{eq:LscreenMod} accurately predicts what is measured, and predicts the deepest attenuation caused by a human blocker in excess of 40 dB. To model multiple blockers, the screen model can be replicated multiple times. These results show that adaptive antenna array and beamforming techniques will be employed to find suitable reflectors and \textcolor{black}{scatterers} in the signal transmission to overcome severe blockage attenuation in future 5G communication systems. The DKED-AG model in~\eqref{eq:LscreenMod} may be extended~\cite{Kunisch08a} to consider the top and bottom screen edges, phase corrections, and non-perpendicular screen orientations, although the simple model~\eqref{eq:LscreenMod} matches the human blocking measurements with confidence. It can be seen in Fig. \ref{fig:DKEDmeas} (Meas. 1 and 9) that the signal strength drops off at a rate of 0.4 dB/ms as the blocker \textcolor{black}{moves at 1 m/s and begins to} shadow the TX (RX). Mobile handoffs and beam steering schemes will be needed to rescue the mobile from severe fades by the use of electrically scanning beams at the sub-millisecond level, a feat easily accomplished with sub millisecond packets in an air interface standard. \textcolor{black}{An additional technique for mitigating the effects of rapid fading could include rapid re-routing around obstacles via handoff to another access point (AP) in a network cluster~\cite{Ghosh16a}.} \textcolor{black}{Note that just prior to the deep shadowing events in Fig.~\ref{fig:DKEDmeas} there is a slight increase/scintillation of signal strength of $\sim$ 2 dB peak-to-peak amplitudes (noticed by others in~\cite{Jacob10b}), which could be used to detect the imminent presence of an obstruction such that the RX adapts its beam in anticipation of the pending deep fade. The 3GPP/METIS blockage model~\cite{3GPP.38.900,METIS15a} shown in Fig.~\ref{fig:DKEDmeas} underestimates the deep fades of shadowing events~\cite{Mac16a}, especially when the blocker is close to the TX or RX antenna, since the full directive gain of the TX and RX antennas is not available across the diffraction obstacle, and thus is unable to contribute to the received signal strength from diffraction around the blocker during the shadowing event~\cite{Mac16a}. We note that the 3GPP/METIS model only offers reasonable agreement to the measured loss when the blocker is far (several meters) from the TX and RX antenna.}
\section{Small-Scale Spatial Statistics}\label{sec:smallscale}
\subsection{Introduction of Small-Scale Spatial Statistics}
Small-scale fading and small-scale autocorrelation characteristics are crucial for the design of future mmWave communication systems, especially in multiple-input multiple-output (MIMO) channel modeling. Previous studies on small-scale fading characteristics focused on sub-6 GHz frequencies, yet investigations at mmWave are scarce. Wang \textit{et al.}~\cite{Wang15b} showed that small-scale fading of received power in indoor corridor scenarios with omnidirectional antennas at both TX and RX could be well described by Ricean distributions with K-factors ranging from 5 dB to 10 dB based on their indoor corridor measurements at 15 GHz with a bandwidth of 1 GHz, and ray tracing results using a ray-optical based channel model validated by measurements. Henderson \textit{et al.} compared Rayleigh, Ricean, and the Two-Wave-Diffuse-Power (TWDP) distributions to find the proper small-scale fading distribution of received voltage magnitudes for a measured 2.4 GHz indoor channel~\cite{Henderson08a} where the Ricean distribution had highest modeling accuracy in most indoor cases~\cite{Henderson08a}. The authors in~\cite{Romero-Jerez16a} demonstrated the use of the TWDP fading model for mmWave communications. It was reported that \textcolor{black}{log-normal} distribution had a good fit to measured received signal envelopes in some indoor mobile radio channels~\cite{Cotton07a}. Important work on wideband directional small-scale fading also appears in~\cite{Holtzman94a,Durgin03a,Sun17a,Samimi16c,Dupleich17a}. In the following subsections, small-scale fading distributions of total power and autocorrelation characteristics of received voltage amplitudes at 73 GHz in urban microcell environments are investigated based on a measurement campaign conducted during the summer of 2016 around the engineering campus of New York University in downtown Brooklyn.
\subsection{Measurement System for Small-Scale Spatial Statistics}
The TX system for small-scale fading and autocorrelation measurements at 73 GHz was identical to the TX system used for the human blocking measurements with the difference only for TX antennas and transmit powers as identified in Table ~\ref{tbl:2}. The RX side of the system captured the RF signal via steerable horn antennas and downconverted the signal to an IF of 5.625 GHz, which was then demodulated into its baseband in-phase ($I$) and quadrature-phase ($Q$) signals which were correlated via a common sliding correlation architecture~\cite{Rap02a,Rap15b,Mac15b,Rap13a} where the time-dilated $I$ and $Q$ channel voltages were sampled by an oscilloscope and then squared and added together in software to generate a PDP. Antennas with 27 dBi gain (7$^\circ$ Az./El. HPBW) and 9.1 dBi gain (60$^\circ$ Az./El.HPBW) were used at the TX and RX sides, respectively\textcolor{black}{\cite{Sun17a}}.
\begin{table}[tb!]
\renewcommand{\arraystretch}{1.4}
\caption{Hardware Specifications of Small-Scale Fading and Local Area Channel Transition Measurements.}~\label{tbl:2}
\fontsize{7.0}{7.0}\selectfont
\begin{center}
\vspace{-2.5mm}
\begin{tabular}{|>{\centering\arraybackslash}m{3.0cm}|>{\centering\arraybackslash}m{2.2cm}|>{\centering\arraybackslash}m{2.2cm}|>{\centering\arraybackslash}m{2.5cm}|>{\centering\arraybackslash}m{2.5cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Campaign} & \textbf{73 GHz Small-Scale Fading and Correlation Measurements} & \textbf{73 GHz Local Area Channel Transition Measurements} \\ \hline \hline
\textbf{Broadcast Sequence} & \multicolumn{2}{c|}{$11\textsuperscript{th}$ order PN Code (L = $2^{11}-1$ = 2047)} \\ \hline
\textbf{TX and RX Antenna Type} & \multicolumn{2}{c|}{Rotatable pyramidal horn antenna} \\ \hline
\textbf{TX/RX Chip Rate} & \multicolumn{2}{c|}{500 Mcps / 499.9375 Mcps} \\ \hline
\textbf{Slide Factor $\gamma$} & \multicolumn{2}{c|}{8000} \\ \hline
\textcolor{black}{\textbf{RF Null-to-Null Bandwidth}} & \multicolumn{2}{c|}{\textcolor{black}{1 GHz}} \\ \hline
\textbf{PDP Threshold} & \multicolumn{2}{c|}{20 dB down from max peak} \\ \hline
\textbf{TX/RX Intermediate Freq.} & \multicolumn{2}{c|}{5.625 GHz} \\ \hline
\textbf{TX/RX Local Oscillator} & \multicolumn{2}{c|}{67.875 GHz (22.625 GHz $\times$ 3)} \\ \hline
\textbf{Carrier Frequency} & \multicolumn{2}{c|}{73.5 GHz} \\ \hline
\textbf{TX Antenna Gain} & \multicolumn{2}{c|}{27 dBi} \\ \hline
\textbf{RX Antenna Gain} & 9.1 dBi & 20 dBi \\ \hline
\textbf{Max TX Power / EIRP} & 14.2 dBm / 41.2 dBm & 14.3 dBm / 41.3 dBm \\ \hline
\textbf{TX Az. and El. HPBW} & \multicolumn{2}{c|}{$7^\circ$} \\ \hline
\textbf{TX/RX Heights} & 4.0 m / 1.4 m & 4.0 m / 1.5 m \\ \hline
\textbf{RX Az. and El. HPBW} & $60^\circ$ & $15^\circ$ \\ \hline
\textbf{TX-RX Antenna Pol.} & \multicolumn{2}{c|}{V-V (vertical-to-vertical)} \\ \hline
\textbf{Max Measurable Path Loss} & 168 dB & 180 dB \\ \hline
\end{tabular}
\end{center}
\end{table}
\subsection{Small-Scale Measurement Environment and Procedure}
In the summer of 2016, a set of small-scale linear track measurements were conducted at 73 GHz on the campus of NYU Tandon School of Engineering in downtown Brooklyn, New York, representative of an urban microcell (UMi) environment\textcolor{black}{\cite{Sun17a}}. The measurement environment, and the TX and RX locations are depicted in Fig.~\ref{fig:TX_RX_Location}. One TX location with an antenna height of 4.0 m above the ground and two RX locations with an antenna height of 1.4 m were selected to perform the measurements, where one RX was LOS to the TX while the other was NLOS. The TX was placed near the southwest corner of the Dibner library building (top center in Fig.~\ref{fig:TX_RX_Location}), the LOS RX was located 79.9 m away from the TX, and the NLOS RX was shadowed by the southeast corner of a building (Rogers Hall on the map) with a T-R separation distance of 75.0 m\textcolor{black}{\cite{Sun17a}}. Other specifications about the measurement hardware are detailed in Table~\ref{tbl:2}.
A fixed 35.31-cm spatial linear track \textcolor{black}{(about 87 wavelengths at 73.5 GHz)} was used at each RX location in the small-scale fading measurements, over which the RX antenna was moved in increments of half-wavelength (2.04 mm) over 175 track positions\textcolor{black}{\cite{Sun17a}}. At each RX, six sets of small-scale fading measurements were performed, where the elevation angle of the RX antenna remained fixed at 0$^\circ$ (parallel to the horizon) and a different azimuth angle was chosen for each set of the measurements with the adjacent azimuth angles separated by 60$^\circ$, such that the RX antenna swept over the entire azimuth plane after rotating through the six pointing angles. The RX antenna was pointing at a fixed angle while moving along the linear track for each set of the measurements and a PDP was acquired at each track position for each pointing angle. The TX antenna elevation angle was always fixed at 0$^\circ$ (parallel to horizon). Under the LOS condition, the TX antenna was pointed at 90$^\circ$ in the azimuth plane, directly towards the RX location; for NLOS, the TX antenna azimuth pointing angle was 200$^\circ$, roughly towards the southeast corner of Rogers Hall in Fig.~\ref{fig:TX_RX_Location}\textcolor{black}{\cite{Sun17a}. Due to space limitations, we show here only one track orientation at each RX (along the direction of the street beside the RX), but more results and observations are detailed in~\cite{Sun17a} which show the fading depths are a function of antenna orientations and environment.}
As a comparison, the 28 GHz small-scale measurements presented in~\cite{Samimi16b} investigated the small-scale fading and autocorrelation of \textit{individual resolvable multipath} voltage amplitudes using a 30$^\circ$ Az./El. HPBW RX antenna, whereas this paper studies 73 GHz fading and autocorrelation using a wider HPBW (60$^\circ$) RX antenna, and focuses on received signal voltage amplitude by integrating the area under the \textit{entire PDP} curve and then taking the square root of the total power, instead of individual multipath voltage amplitude at each location along a track.
\begin{figure}
\centering
\includegraphics[width=2.9in]{MeasurementMap_2.pdf}
\caption{\textcolor{black}{2D map of the 73 GHz small-scale measurement environment and the locations of TX and RX. Pointing to the top of the map is 0$^\circ$.}}
\label{fig:TX_RX_Location}
\vspace{-2.5mm}
\end{figure}
\textcolor{black}{\subsection{Small-Scale Measurement Results}}
\textcolor{black}{Fig.~\ref{fig:LOSPDP2} illustrates typical measured small-scale directional PDPs over 175 track positions on the 35.31-cm (about 87 wavelengths at 73.5 GHz) linear track in the LOS environment, where the RX horn antenna with 60$^\circ$ HPBW was pointing on boresight to the TX, and the track orientation was orthogonal to the T-R line. The total power in Figs.~\ref{fig:LOSPDP2} and~\ref{fig:NLOSPDP2} is computed as the area under the PDP at a particular track position over the 1 GHz RF bandwidth. Fig.~\ref{fig:LOSPDP2} shows there is 11 dB power variation over different track positions, but the power variation is only 3.7 dB when the track orientation was in the direction of the T-R line (not shown), indicating little small-scale spatial fading~\cite{Sun17a}.}
\textcolor{black}{Typical measured small-scale directional PDPs over 175 track positions on the 35.31-cm (about 87 wavelengths at 73.5 GHz) linear track in the NLOS environment are depicted in Fig.~\ref{fig:NLOSPDP2}, where the track orientation was along the direction of the street, and the RX antenna was pointing to the TX but was obstructed by a building corner~\cite{Sun17a}. Fig.~\ref{fig:NLOSPDP2} shows there is very moderate power variation (4.1 dB) over different local-area track positions, albeit with rich and varying multipath components.}\\
\begin{figure}
\centering
\includegraphics[width=3.2in]{RX4_Meas1.pdf}
\caption{\textcolor{black}{Measured 73 GHz small-scale directional PDPs over 175 track positions in LOS. The RX horn antenna (60$^\circ$ HPBW) was pointing on boresight to the TX, and track orientation orthogonal to the T-R line.}}
\label{fig:LOSPDP2}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.2in]{RX14_Meas6.pdf}
\caption{\textcolor{black}{Measured 73 GHz small-scale directional PDPs over 175 track positions in NLOS. The RX horn antenna (60$^\circ$ HPBW) was pointing to the TX but was obstructed by a building corner, and the track orientation was along the direction of the street.}}
\label{fig:NLOSPDP2}
\end{figure}
\subsection{Small-Scale Spatial Statistics Results and Analysis}
\subsubsection{Omnidirectional Small-Scale Spatial Statistics}
As described above, a rotatable directive horn antenna was used at the RX side to capture directional PDPs in the small-scale fading and correlation measurements. In channel modeling, however, omnidirectional statistics are often preferred, since arbitrary antenna patterns can be implemented according to one's own needs if accurate temporal and spatial statistics are known~\cite{Sun17c}. Therefore, we synthesized the approximated omnidirectional received power at every track interval by taking the \textcolor{black}{area under the curve of each directional PDP and summing powers using the approach presented in~\cite{Sun15a} and on Page 3040 from~\cite{Rap15b}}, thereby computing omnidirectional received power. Although the RX antenna did not sweep the entire 4$\pi$ Steradian sphere, the azimuth plane spanned $\pm 30^{\circ}$ with respect to the horizon, ensuring that a large majority of the arriving energy was captured, as verified in~\cite{Sun15a}.
Fig.~\ref{fig:omniLOSFading} illustrates the cumulative distribution function (CDF) of the measured small-scale received voltage amplitude at 73 GHz with a 1 GHz RF bandwidth over the 35.31-cm length track with 175 track positions in increments of half-wavelength (2.04 mm) for the omnidirectional RX antenna pattern in the LOS environment\textcolor{black}{\cite{Sun17a}}. \textcolor{black}{Superimposed with the measured curve are the CDFs of the Rayleigh distribution, the zero-mean log-normal distribution with a standard deviation of 0.91 dB (obtained from the measured data), and the Ricean distribution with a $K$-factor of 10 dB obtained from the measured data by dividing the total received power contained in the LOS path by the power contributed from all the other reflected or scattered paths. As shown in Fig.~\ref{fig:omniLOSFading}, the measured 73 GHz small-scale spatial fading in the LOS environment can be approximated by the Ricean distribution with a $K$-factor of 10 dB, indicating that there is a dominant path (i.e., the LOS path) contributing to the total received power, and that the received signal voltage amplitude varies little over the 35.31-cm (about 87 wavelengths) length track. The log-normal distribution does not fit the measured data well in the regions of -3 to -2.5 dB and +1.2 to +1.5 dB about the mean.} The maximum fluctuation of the received voltage amplitude is merely 3 dB relative to the mean value, whereas the fades are much deeper for the Rayleigh distribution. The physical reason for this is the presence of a dominant LOS path.
\begin{figure}
\vspace{-2.5mm}
\centering
\includegraphics[width=2.7in]{omniLOSFading.pdf}
\caption{CDF of the measured small-scale spatial fading distribution of the received voltage amplitude for the omnidirectional RX antenna pattern in the LOS environment at 73 GHz with a 1 GHz RF bandwidth.}
\label{fig:omniLOSFading}
\vspace{-2.5mm}
\end{figure}
The small-scale spatial fading in the NLOS environment for the omnidirectional RX antenna pattern is illustrated in Fig.~\ref{fig:omniNLOSFading}, \textcolor{black}{and the zero-mean log-normal distribution with a standard deviation of 0.65 dB (obtained from the measured data) is selected to fit the measured result, and Ricean and Rayleigh distributions are also given as a reference\textcolor{black}{\cite{Sun17a}}. As evident from Fig.~\ref{fig:omniNLOSFading}, the measured NLOS small-scale spatial fading distribution matches the log-normal fitted curve almost perfectly. In contrast, the Ricean distribution with $K$ = 19 dB does not fit the measured data as well as the log-normal distribution in the tail region around -0.6 dB to -0.8 dB of the relative mean signal level }\textcolor{black}{(as shown by the inset in Fig.~\ref{fig:omniNLOSFading}), since the Ricean $K$ = 19 dB distribution predicts more occurrences of deeper fading events, whereas the log-normal distribution with a 0.65 dB standard deviation predicts a more compressed fading range of -0.8 dB to +0.8 dB about the mean, which was observed for the wideband NLOS signals.} The fact that the local fading of received voltage amplitudes in the NLOS environment is \textcolor{black}{log-normal} instead of Rayleigh is similar to models in~\cite{Rap91b} for urban mobile radio channels. For a NLOS environment, there may not be a dominant path, yet the transmitted broadband signal experiences frequency-selective fading (when the signal bandwidth is larger than the coherence bandwidth of the channel~\cite{Rap02a}). Different frequency components of the signal experience uncorrelated fading, thus it is highly unlikely that all parts of the signal will simultaneously experience a deep fade, and the fades over frequency tend to be very sharp, taking up a small portion of the total power received over the entire signal bandwidth~\cite{Rap15a}. Consequently, the total received power changes very little over a small-scale local area. This is a distinguishing feature of wideband mobile signals as compared to narrowband signals.
\begin{figure}
\centering
\includegraphics[width=2.7in]{omniNLOSFading.pdf}
\caption{CDF of the measured small-scale spatial fading distribution of the received voltage amplitude for the omnidirectional RX antenna pattern in the NLOS environment at 73 GHz with a 1 GHz RF bandwidth.}
\label{fig:omniNLOSFading}
\vspace{-2.5mm}
\end{figure}
Apart from small-scale spatial fading, small-scale spatial autocorrelation is also important for wireless modem design. Spatial autocorrelation characterizes how the received voltage amplitudes correlate at different linear track positions within a local area~\cite{Samimi16b}. Spatial autocorrelation coefficient functions can be calculated using Eq.~\eqref{sac}, where $X_k$ denotes the $k^{th}$ linear track position, $E[~]$ is the expectation operator where the average of voltage amplitudes is taken over all the positions $X_k$, and $\Delta X$ represents the spacing between different antenna positions on the track\textcolor{black}{\cite{Sun17a}}.
\begin{figure*}
\begin{equation}\label{sac}
\footnotesize
\rho=\frac{E\big[\big(A_k(X_k)-\overline{A_k(X_k)}\big)\big(A_k(X_k+\Delta X)-\overline{A_k(X_k+\Delta X)}\big)\big]}{\sqrt{E\big[\big(A_k(X_k)-\overline{A_k(X_k)}\big)^2\big]E\big[\big(A_k(X_k+\Delta X)-\overline{A_k(X_k+\Delta X)}\big)^2\big]}}
\vspace{-2.5mm}
\end{equation}
\vspace{-2.5mm}
\end{figure*}
The measured 73 GHz spatial autocorrelation of the received voltage amplitudes in LOS and NLOS environments with a 1 GHz RF bandwidth are depicted in Fig.~\ref{fig:omniLOSCor} and Fig.~\ref{fig:omniNLOSCor}, respectively. Note that a total of 175 linear track positions over the 35.31-cm length track were measured during the measurements, yielding a maximum spatial separation of 174 half-wavelengths on a single track. Only up to 60 half-wavelengths, however, are shown herein because little change is found thereafter and it provides 100 autocorrelation data points for all spatial separations on a single track, thus improving the reliability of the statistics. According to Fig.~\ref{fig:omniLOSCor}, the received omnidirectional signal voltage amplitude first becomes uncorrelated at a spatial separation of about 3.5$\lambda$, then becomes slightly anticorrelated for separations of 3.5$\lambda$ to 10$\lambda$, and becomes slightly correlated for separations between 10$\lambda$ and 18$\lambda$, and decays towards 0 sinusoidally after 18$\lambda$. Therefore, the spatial correlation can be modeled by a ``damped oscillation'' function of~\eqref{expFit}\textcolor{black}{\cite{Sun17a}}~\cite{Zhang08a}:
\begin{equation}\label{expFit}
\footnotesize
f(\Delta X) = \cos(a\Delta X)e^{-b\Delta X}
\end{equation}
where $\Delta X$ denotes the space between antenna positions, $a$ is an oscillation distance with units of radians/$\lambda$ (wavelength), \textcolor{black}{$T=2\pi/a$} can be defined as the spatial oscillation period with units of $\lambda$ or cm, and $b$ is a constant with units of $\lambda^{-1}$ whose inverse \textcolor{black}{$d=1/b$} is the spatial decay constant with units of $\lambda$. $a$ and $b$ are obtained using the minimum mean square error (MMSE) method to find the best fit between the empirical spatial autocorrelation curve and theoretical exponential model given by~\eqref{expFit}. The ``damped oscillation'' pattern can be explained by superposition of multipath components with different phases at different linear track positions. \textcolor{black}{As the separation distance of linear track positions increases, the phase differences among individual multipath components will oscillate as the separation distance of track positions increases due to alternating constructive and destructive combining of the multipath phases.} This ``damped oscillation'' pattern is obvious in LOS environment where phase difference among individual multipath component is not affected by shadowing effects that occurred in NLOS environments. The form of~\eqref{expFit} also guarantees that the spatial autocorrelation coefficient is always 1 for $\Delta X = 0$, and converges to 0 when $\Delta X$ approximates infinity. The spatial autocorrelation curve for NLOS environment in Fig.~\ref{fig:omniNLOSCor} exhibits a different trend from that in Fig.~\ref{fig:omniLOSCor}, which is more akin to an exponential distribution without damping, but can still be fitted using Eq.~\eqref{expFit} with $a$ set to 0\textcolor{black}{\cite{Sun17a}}. The constants $a$, $b$, are provided in Table~\ref{tbl:ModelPara}, where $T$ is the oscillation period, and $d$ represents the spatial decay constant. From Fig.~\ref{fig:omniNLOSCor} and Table~\ref{tbl:ModelPara} it is clear that after 1.57 cm (3.85 wavelengths at 73.5 GHz) in the NLOS environment, the received voltage amplitudes become uncorrelated (the correlation coefficient decreases to 1/e~\cite{Samimi16a}). We note that Samimi~\cite{Samimi16b} found individual multipath voltage amplitudes received using a 30$^\circ$ Az./El. HPBW antenna became uncorrelated at physical distances of 0.52 cm (0.48 wavelengths at 28 GHz) and 0.67 cm (0.62 wavelengths at 28 GHz) in LOS and NLOS environments, respectively -- smaller decorrelation distances compared to the present 73 GHz results measured using a 60$^\circ$ Az./El. HPBW antenna.
\begin{figure}
\centering
\includegraphics[width=2.5in]{OmniCor_LOS.pdf}
\caption{\textcolor{black}{Measured 73 GHz broadband spatial autocorrelation coefficients of the received voltage amplitude in the LOS environment, and the corresponding fitting model. The T-R separation distance is 79.9 m.}}
\label{fig:omniLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=2.5in]{OmniCor_NLOS.pdf}
\caption{\textcolor{black}{Measured 73 GHz broadband spatial autocorrelation coefficients of the received voltage amplitude in the NLOS environment, and the corresponding fitting model. The T-R separation distance is 75.0 m.}}
\label{fig:omniNLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{table}[t!]
\renewcommand{\arraystretch}{1.4}
\caption{Spatial correlation model parameters in~\eqref{expFit} for 73 GHz, 1 GHz RF bandwidth ($\lambda$=0.41 cm).}~\label{tbl:ModelPara}
\fontsize{8}{6.5}\selectfont
\scriptsize
\begin{center}\scalebox{0.84}{
\vspace{-2.5mm}
\begin{tabular}{|>{\centering\arraybackslash}m{1.4cm}|>{\centering\arraybackslash}m{1.3cm}|>{\centering\arraybackslash}m{2.15cm}|>{\centering\arraybackslash}m{1.25cm}|>{\centering\arraybackslash}m{2.15cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Condition} & $\boldsymbol{a}~(rad/\lambda)$ & \textcolor{black}{$\boldsymbol{T=2\pi/a}$} & \textbf{b}~($\lambda^{-1}$) & \textcolor{black}{$\boldsymbol{d=1/b}$} \\ \hline \hline
\textbf{LOS Omnidirectional} & {0.45} & {14.0$\lambda$ (5.71 cm)} & {0.10} & {10.0$\lambda$ (4.08 cm)}\\ \hline
\textbf{NLOS Omnidirectional} & {0} & {\textcolor{black}{Not used}} & {0.26} & {3.85$\lambda$ (1.57 cm)}\\ \hline
\textbf{LOS Directional} & {0.33 to 0.50} & {12.6$\lambda$ to 19.0$\lambda$ (5.14 cm to 7.76 cm)} & {0.03 to 0.15} & {6.67$\lambda$ to 33.3$\lambda$ (2.72 cm to 13.6 cm)}\\ \hline
\textbf{NLOS Directional} & {0} & {\textcolor{black}{Not used}} & {0.04 to 1.49} & {0.67$\lambda$ to 25.0$\lambda$ (0.27 cm to 10.2 cm)}\\ \hline
\end{tabular}}
\end{center}
\end{table}
\subsubsection{Directional Small-Scale Spatial Statistics}
Since mobile devices will use directional antennas, directional statistics are also of interest. In this subsection, we will investigate small-scale spatial fading and autocorrelation of the received voltage amplitudes associated with directional antennas at the RX.
Small-scale fading of received voltage amplitudes along the linear track using the $7^{\circ}$ Az./El. HPBW TX antenna and $60^{\circ}$ Az./El. HPBW RX antenna in LOS and NLOS environments are shown in Fig.~\ref{fig:dirLOSFading} and Fig.~\ref{fig:dirNLOSFading}, respectively, where the TX and RX were placed as shown in Fig.~\ref{fig:TX_RX_Location}, and each measured curve corresponds to a unique RX antenna azimuth pointing angle relative to true north as specified in the legend. There was no signal for the RX azimuth pointing angle of 270$^{\circ}$ in the NLOS environment, thus the corresponding results are absent in Fig.~\ref{fig:dirNLOSFading}. The strongest pointing directions are 270$^\circ$ and 150$^\circ$ in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, respectively\textcolor{black}{\cite{Sun17a}}. As shown in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, the measured directional spatial autocorrelation coefficients resemble Ricean distributions in both LOS and NLOS environments. Possible reason for such distributions is that only one dominant path (accompanied with several weaker paths) is captured by the horn antenna due to its directionality, given the fact that mmWave propagation is directional and the channel is sparse~\cite{Samimi16a}. The Ricean $K$-factor for received voltage amplitudes for various RX pointing directions ranges from 7 dB to 17 dB for the LOS environment, and 9 dB to 21 dB for the NLOS case, as shown in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, as the RX is moved over a 35.31 cm (86.5 wavelengths at 73.5 GHz) track.
\begin{figure}
\centering
\includegraphics[width=2.8in]{dirLOSFading.pdf}
\caption{\textcolor{black}{Measured 73 GHz LOS small-scale spatial fading distributions of the directional received voltage amplitude, and the corresponding Ricean fitting curves with different $K$ factors. The angles in the legend denote the receiver antenna azimuth angle, and "b" denotes boresight to the TX.}}
\label{fig:dirLOSFading}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=2.8in]{dirNLOSFading.pdf}
\caption{\textcolor{black}{Measured 73 GHz NLOS small-scale spatial fading distributions of the directional received voltage amplitude, and the corresponding Ricean fitting curves with different $K$ factors. The angles in the legend denote the receiver antenna azimuth angle, and "s, to TX" denotes along the direction of the street and pointing to the TX side.}}
\label{fig:dirNLOSFading}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.0in]{dirLOSCor.pdf}
\caption{\textcolor{black}{Measured 73 GHz LOS spatial autocorrelation coefficients of the directional received voltage amplitudes. The angles in the legend denote the receiver antenna azimuth angle, and "b" denotes boresight to the TX.}}
\label{fig:dirLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.0in]{dirNLOSCor.pdf}
\caption{\textcolor{black}{Measured 73 GHz NLOS spatial autocorrelation coefficients of the directional received voltage amplitudes. The angles in the legend denote the receiver antenna azimuth angle, and ``s, to TX" denotes along the direction of the street and pointing to the TX side.}}
\label{fig:dirNLOSCor}
\vspace{-2.5mm}
\end{figure}
Figs.~\ref{fig:dirLOSCor} and~\ref{fig:dirNLOSCor} illustrate the spatial autocorrelation coefficients of the received voltage amplitudes for individual antenna pointing angles in LOS and NLOS environments in downtown Brooklyn (see Fig.~\ref{fig:TX_RX_Location}), respectively. As shown by Fig.~\ref{fig:dirLOSCor}, all of the six spatial autocorrelation curves in the LOS environment exhibit sinusoidally exponential decaying trends, albeit with different oscillation patterns and decay rates. The TX-RX boresight-to-boresight pointing angle (270$^\circ$ RX pointing angle, corresponding to the strongest received power) yields the smallest oscillation since there is only a single LOS component in the PDP, while the other pointing directions contain two or more multipath components with varying phases that result in larger oscillation\textcolor{black}{\cite{Sun17a}}. The spatial decay constants for all the curves in Fig.~\ref{fig:dirLOSCor} are given in Table~\ref{tbl:ModelPara}. Compared with the omnidirectional case displayed in Fig.~\ref{fig:omniLOSCor}, it is clear that for the LOS environment, the spatial autocorrelation of both omnidirectional and directional received voltage amplitude obeys similar distribution, namely, the sinusoidal-exponential function, with similar decorrelation distances. On the other hand, most of the spatial autocorrelation curves for the directional received voltage amplitude shown in Fig.~\ref{fig:dirNLOSCor} are also in line with that given by Fig.~\ref{fig:omniNLOSCor}. One exception for the 73 GHz spatial correlation is found at 30$^\circ$ pointing angle that is the second strongest pointing direction (see Fig. \ref{fig:dirNLOSCor}), where decorrelation was much more gradual and decreases to 1/e at 25.0$\lambda$ (10.2 cm), probably due to the presence of a dominant path with a relatively constant signal level, likely caused by the diffraction of the southeast corner of Rogers Hall in Fig.~\ref{fig:TX_RX_Location}. It is clear from Fig. 4 in~\cite{Rap16a} that correlation distances vary among typical mmWave measurements~\cite{Rumney16b} due to the site-specific nature of propagation.
\section{Local Area Channel Transition}
\subsection{Introduction of Local Area Channel Transition}
Large-scale channel characteristics, such as the autocorrelation of shadow fading and delay spread over distance at the mobile, inter-site correlation of shadow fading at the mobile for two base stations or a base station for one mobile, and properties of local area channel transition, play an important role in constructing channel models for wireless communication systems~\cite{Jalden07a,3GPP.38.900,3GPP.36.814,3GPP.36.839}.
While sufficient studies on large-scale channel characteristics have been conducted at sub-6 GHz frequencies~\cite{Jalden07a,Guan15a,Zhang08a,Kolmonen10a}, similar studies have been rarely conducted at mmWave frequencies. Guan \textit{et al.} investigated spatial autocorrelation of shadow fading at 920 MHz, 2400 MHz, and 5705 MHz in curved subway tunnels~\cite{Guan15a}. Results showed that the 802.16J model was a better fit to the measured data than an exponential model, and the mean decorrelation distances were found to be several meters. Another measurement campaign carried out in urban macro-cell (UMa) environments at 2.35 GHz showed that a double exponential model fit well with the autocorrelation coefficients of shadow fading samples extracted from all measured routes, while for individual routes, an exponentially decaying sinusoid model had better fitting performance~\cite{Zhang08a}. On the other hand, an exponential function was adopted in the 3GPP channel model (Releases 9 and 11) to describe the normalized autocorrelation of shadow fading versus distances~\cite{3GPP.36.814,3GPP.36.839}. Moreover, Kolmonen \textit{et al.}~\cite{Kolmonen10a} investigated interlink correlation of eigenvectors of MIMO correlation matrices based on a multi-site measurement campaign at 5.3 GHz using a bandwidth of 100 MHz. Results showed that the first eigenvectors for both x- and y-oriented arrays were highly correlated when two RX locations were largely separated. In the following subsections, local area and channel transition measurements at 73 GHz are described and analyzed.
\subsection{Measurement System, Environment, and Procedure for Local Area Channel Transition}
The measurement system used for local area channel transition measurements was identical to the one used in the previous small-scale fading and autocorrelation measurements described in Section~\ref{sec:smallscale}, except from the antennas~\cite{Mac17a}\textcolor{black}{\cite{Mac17c}}. A 27 dBi gain (7$^\circ$ azimuth and elevation Az./El. HPBW) and 20.0 dBi gain (15$^\circ$ Az./El. HPBW) antenna were used at the TX and RX, respectively. Detailed specifications regarding the measurement system are provided in Table \ref{tbl:2}.
The local area channel transition measurements were conducted at 73 GHz in the MetroTech Commons courtyard next to 2 and 3 MetroTech Center in downtown Brooklyn. During measurements, the TX and RX antennas were set to 4.0 m and 1.5 m above ground level, respectively. For each set of \textit{cluster} or \textit{route} scenario RX locations, the TX antenna remained fixed and pointed towards a manually selected azimuth and elevation pointing angle that resulted in the strongest received power at the starting RX position (RX81 for \textit{route} measurements, and RX51 and RX61 for the LOS and NLOS \textit{cluster} measurements, respectively). For each specific TX-RX combination, five consecutive and identical azimuth sweeps ($\sim$3 minutes per sweep and $\sim$2 minutes between sweeps) were conducted at the RX in HPBW step increments ($15^\circ$) where a PDP was recorded at each RX azimuth pointing angle and resulted in at most 120 PDPs ($\frac{360}{15}\times5=120$) per combination (some angles did not have detectable signal above the noise). The best RX pointing angle in the azimuth plane was selected as the starting point for the RX azimuth sweeps (elevation remained fixed for all RX's), at each RX location measured.
For the \textit{route} measurements, 16 RX locations were measured for a fixed TX location (L8) with the RX locations positioned in 5 m adjacent increments of each other forming a simulated route in the shape of an ``L" around a building corner from a LOS to NLOS region, as provided in Fig.~\ref{fig:Map_Large_Scale_C1}. The LOS location (five: RX92 to RX96) and NLOS location (11: RX81 to RX91) T-R separation distances (Euclidean distance between TX and RX) varied from 29.6 m to 49.1 m and 50.8 m to 81.5 m, respectively. The TX antenna at L8 kept the same azimuth and elevation pointing angle of 100$^\circ$ and 0$^\circ$, respectively, during each experiment (see Fig.~\ref{fig:Map_Large_Scale_C1}. Therefore, the LOS measurements have the TX and RX antennas roughly on boresight in the LOS situation, but they are not exactly on boresight throughout the entire experiment over all measured locations. The general layout of measurements consisted of the RX location starting at RX81, approximately 54 m along an urban canyon (Bridge Street: 18 m width), with the TX antenna pointed towards the opening of the urban canyon (see Fig.~\ref{fig:Map_Large_Scale_C1}). The LOS locations were in clear view of the RX, but with some nearby minor foliage and lamppost obstructions.
For the \textit{cluster} measurements, 10 RX locations were measured for a fixed TX location (L11), with two sets of RX clusters, one in LOS (RX61 to RX65) and the other in NLOS (RX51 to RX 55). For each cluster of RX's, the adjacent distance between each RX location was 5 m, however, the path of adjacent RX locations took the shape of a semi-circle as displayed in Fig.~\ref{fig:Map_Large_Scale_C2}. The LOS cluster T-R separation distances (Euclidean distance between TX and RX) varied between 57.8 m and 70.6 m with a fixed TX antenna azimuth and elevation departure angle of 350$^\circ$ and -2$^\circ$, respectively, and fixed RX elevation angles of $+3^\circ$, to ensure rough elevation and azimuth alignment for all RX locations. For the NLOS cluster, the T-R separation distances were between 61.7 m and 73.7 m with a fixed TX antenna azimuth and elevation departure angle of 5$^\circ$ and -2$^\circ$, respectively, and fixed RX elevation angles of $+3^\circ$. The LOS cluster of RX's was located near the opening of an urban canyon near some light foliage, while the TX location was $\sim$57 m along an urban canyon (Lawrence Street: 18 m width). The NLOS cluster of RX locations was around the corner of the urban canyon opening in a courtyard area (see Fig.~\ref{fig:Map_Large_Scale_C2}), also with nearby moderate foliage and lampposts.
\begin{figure}
\begin{center}
\includegraphics [width = 0.37\textwidth]{Map_Large_Scale_C1_v1.pdf}
\caption{2D map of TX and RX locations for \textit{route} NLOS to LOS transition measurements. The yellow star is the TX location, blue dots represent LOS RX locations, and red squares indicate NLOS RX locations. $\textrm{N}=0^\circ$. }\label{fig:Map_Large_Scale_C1}
\end{center}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics [width = 0.37\textwidth]{Map_Large_Scale_C2_v1.pdf}
\caption{2D map of TX and RX locations for \textit{cluster} measurements with LOS and NLOS RX clusters. The yellow star is the TX location, blue dots represent LOS RX locations, and red squares indicate NLOS RX locations. Pointing to the top is $0^\circ$.}\label{fig:Map_Large_Scale_C2}
\vspace{-2.5mm}
\end{center}
\vspace{-2.5mm}
\end{figure}
\subsection{Local Area Channel Transition Results and Stationarity}
The route measurements mimicked a person moving along an urban canyon from a NLOS to a LOS region, in order to understand the evolution of the channel during the transition. Fig.~\ref{fig:routePL} displays the omnidirectional path loss for each of the RX locations (RX81 to RX96) where the received power from the individual directional measurements at the RX was summed up to determine the entire omnidirectional received power at each measurement location (out of the 5 sweeps, the maximum power at each angle was used, although variation was less than a dB between sweeps)~\cite{Mac14b,Sun15a}.
\begin{figure}[t!]
\begin{center}
\includegraphics [width = 0.37\textwidth]{NLOS_LOS_Transition_PL_box_embed-eps-converted-to.pdf}
\caption{Omnidirectional path loss for route measurements for an RX transitioning from a NLOS to a LOS region.}\label{fig:routePL}
\vspace{-2.5mm}
\end{center}
\vspace{-2.5mm}
\end{figure}
The transition from LOS to NLOS in Fig.~\ref{fig:routePL} is quite abrupt, where path loss increases by $\sim$8 dB from RX92 to RX91, similar to the abrupt diffraction loss noticed in Section~\ref{sec:diffraction}. Path loss then increases 9 dB further from RX91 to RX90, 1 dB from RX90 to RX89, 6 dB from RX 89 to RX88, and 1 dB from RX88 to RX 87. This observation shows a large initial drop in 8 dB at the LOS to NLOS transition region, but an overall 25 dB drop in signal power when moving from LOS conditions to deeply shadowed NLOS conditions approximately 25 meters farther along a perpendicular urban canyon ($\sim$10 m increase in Euclidean T-R separation distance), when using an omnidirectional RX antenna. The 25 dB drop in signal strength over a 25 m path around a corner (1 dB/m) is important for handoff considerations. The signal fading rate is 35 dB/s for vehicle speeds of 35 m/s, or 1 dB/s for walking speeds of 1 m/s. This motivates the use of beam scanning and phased array technologies in the handset for urban mobile mmWave communications that will search for and find the strongest signal paths~\cite{Sun14b}, and future work will study the best antenna pointing angles at each location from these measurements.
\begin{figure}
\centering
\begin{subfigure}[b]{0.39\textwidth}
\centering
\includegraphics[trim={0 0.5cm 0 0},clip,width=\textwidth]{L8_RX87_RXsweep_Meas3_PolarPlot_embed.pdf}
\caption[]%
{{\small RX87: NLOS}}
\label{fig:RX87}
\end{subfigure}
\begin{subfigure}[b]{0.39\textwidth}
\centering
\includegraphics[trim={0 0.5cm 0 0},clip,width=\textwidth]{L8_RX92_RXsweep_Meas5_PolarPlot_embed.pdf}
\caption[]%
{{\small RX92: LOS}}
\label{fig:RX92}
\end{subfigure}
\caption{\small Route scenario polar plots of RX azimuth spectra for RX 87 (NLOS location) and RX 92 (LOS location) that show the evolution of AOA energy around a corner.}
\label{fig:case1Polar}
\vspace{-2.5mm}
\end{figure}
\textcolor{black}{Azimuth power spectra are useful to study how the arriving energy changes as an RX moves from NLOS to LOS. Work in~\cite{Samimi16a} showed that energy arrives in directional lobes in mmWave channels. Fig.~\ref{fig:case1Polar} displays an RX polar plot from RX87 (NLOS location) and RX92 (LOS location), where the TX was pointed in the 100$^\circ$ direction towards the street opening (see Fig.~\ref{fig:Map_Large_Scale_C1}). Fig.~\ref{fig:RX87} shows the power azimuth spectra at RX87 ($\sim$25 meters down the urban street canyon) where energy from the TX waveguides and reflects down Bridge Street such that there is one main broad lobe at the RX oriented in the 0$^\circ$ direction and a small narrow lobe in the 180$^\circ$ direction from weak reflectors and scattering. The large azimuth spread in the main lobe demonstrates the surprisingly reflective nature of the channel at the 73 GHz mmWave band~\cite{Samimi13a,Samimi16a}.}
\textcolor{black}{Fig.~\ref{fig:RX92} displays the power azimuth spectra at RX92 in LOS with a strong central lobe coming from the direction of the TX (285$^\circ$). A relatively strong secondary lobe (100$^\circ$) is also apparent in Fig.~\ref{fig:RX92} with energy contributions from reflections off of the building to the east of RX92 and additional reflectors and scatterers from nearby lampposts and signs.} Table~\ref{tbl:clusterSTD} displays the standard deviations of the omnidirectional received power values measured along the LOS and NLOS routes shown in Fig.~\ref{fig:Map_Large_Scale_C1}. The omnidirectional received power standard deviation of 1.2 dB is generally small for the LOS locations but is much larger in NLOS (7.9 dB) due to substantial scattering along the route of RX locations along the urban canyon where path loss tends to increase non-linearly over log-distance.
\begin{table}
\centering
\caption{Omnidirectional received power standard deviation for the large-scale route and cluster scenario measurements.}
\label{tbl:clusterSTD}
\begin{center}
\scalebox{0.78}{
\begin{tabu}{|c|[1.6pt]c|}
\hline
\textbf{Measurement Set} & \textbf{Omnidirectional Received Power \bm{$\sigma$} [dB]} \\ \specialrule{1.5pt}{0pt}{0pt}
\textit{Route} - LOS: RX92 to RX96 & 1.2 \\ \hline
\textit{Route} - NLOS: RX81 to RX91 & 7.9 \\ \hline
\textit{Cluster} - LOS: RX61 to RX65 & 4.3 \\ \hline
\textit{Cluster} - NLOS: RX51 to RX55 & 2.2 \\ \hline
\end{tabu}}
\end{center}
\vspace{-2.5mm}
\end{table}
The \textit{cluster} measurements for the TX at L11 were designed to understand the stationarity of received power in a local area (larger than small-scale distances) on the order of many hundreds to thousands of wavelengths (5 to 10 meters) at mmWave. In LOS, the cluster of five RX locations with a fixed directional TX antenna resulted in an omnidirectional received power standard deviation of 4.3 dB over local area of 5 m x 10 m, a relatively small variation, indicating a reasonably stationary average received power for a local set of RX locations in LOS at 73 GHz. The NLOS cluster resulted in an even lower omnidirectional received power standard deviation of 2.2 dB, over a 5 m x 10 m local area. The small fluctuation in received power over the local area of the LOS and NLOS clusters implies that received power does not significantly vary over RX locations separated by even a few to several meters in a dense urban environment at mmWave. As an aside, the directional CI model path loss exponent (PLE) using a 1 m free space reference in the \textit{route} measurements was 2.53 in LOS (a bit higher than free space due to elevation mismatch over the route) and 3.61 in NLOS (for a single TX beam)~\cite{Mac17a}. Recent work in~\cite{Wang17a} at 28 GHz studied the stationarity of wideband mmWave channels, and reported smaller stationary regions than at 2 GHz.
\section{Conclusion and Discussion}~\label{sec:conc}
Measurements and analysis were presented on diffraction, human blockage, small-scale fading and local area channel transition at mmWave frequencies. We showed the KED model is suitable for modeling diffraction at cmWave and mmWave bands in indoor scenarios, while a creeping wave linear model is applicable to outdoor environments. \textcolor{black}{Indoor and outdoor diffraction measurements confirm theoretical simulations~\cite{Deng16a} that mmWave diffraction will not be a dominant propagation mechanism. Measurements show significant attenuation of 30 dB in indoor channels and 40 dB or more in outdoor channels in the deep shadow region. Therefore, systems will need to be designed to overcome extremely large fades when a signal path is blocked and when diffraction is no longer viable. MmWave antenna systems at the TX and RX will need to cooperatively search for secondary paths at different pointing directions. The large fades due to diffraction loss have implications on the design of physical layer protocols and frame structures to maintain a link, while finding other spatial paths.}
Human blockage measurements showed a person can induce more than 40 dB of loss when standing 0.5 m from the TX or RX antenna, with a signal decay rate of 0.4 dB/ms at walking speeds. The DKED-AG model given here incorporates directional antenna patterns to accurately predict the upper and lower envelopes of measured received power during a blockage \textcolor{black}{which better agrees with real-world measurements when compared to the 3GPP/METIS blockage model (that was shown to underestimate human blockage, most severely when close to the TX or RX)}. \textcolor{black}{Scintillation with $\sim$2 dB peak-to-peak amplitudes} can be seen in the measurements and model just before the blocker enters the field of view of the RX, suggesting that deep fades may be predictable just before they occur. This information may be used in the design of beam steering or handoff algorithms.
\textcolor{black}{Small-scale spatial fading statistics of received signal voltage amplitudes of 1 GHz bandwidth signals at 73 GHz were measured over a 35.31-cm ($\sim$ 87 wavelengths) linear track and show only small power variations when such a wide bandwidth is used. Fading in LOS locations for omnidirectional RX antennas obeyed the Ricean distribution with a $K$-factor of 10 dB, while fading in NLOS locations can be described by the log-normal distribution with a standard deviation of 0.65 dB. The fading depth ranges from -3 dB to +1.5 dB relative to the mean for LOS, and -0.8 dB to +0.8 dB for NLOS. For a 60$^\circ$ directional RX antenna, fading in both LOS and NLOS environments follows the Ricean distribution, where the $K$-factor ranges from 7 dB to 17 dB for LOS, and 9 dB to 21 dB for NLOS, depending on the RX orientation in relation to the environment and the TX~\cite{Sun17a}, while the fading depth varies between -4 dB to +2 dB relative to the mean for both LOS and NLOS environments.}
\textcolor{black}{Spatial autocorrelation modeling of instantaneous total received signal voltage amplitudes showed that the sinusoidal-exponential distribution fits measurements in the LOS environment for both omnidirectional and directional RX antennas. In the NLOS environment, the spatial autocorrelation can be modeled by the exponential distribution for both omnidirectional and directional RX antennas.} Table~\ref{tbl:ModelPara} shows the oscillation distance/period and spatial decay constant to represent the autocorrelation, where rapid decorrelation of received voltage amplitudes occurred over 0.67 to 33.3 wavelengths (0.27 cm to 13.6 cm), depending on the RX orientation in relation to the environment and the TX~\cite{Sun17a}. \textcolor{black}{The short correlation distance, in general, is favorable for spatial multiplexing in MIMO since it allows for uncorrelated spatial data streams to be transmitted from closely-spaced (a fraction to several wavelengths) antennas~\cite{Sun14b}. Furthermore, the small fading depth indicates that the signal quality of an established link between a TX and an RX will not vary much when the RX moves within a local area on the order of a few tens of wavelengths (within 13.6 cm), as shown in Figs.~\ref{fig:dirLOSCor} and~\ref{fig:dirNLOSCor}.}
Local area channel transition measurements show an initial 8 dB power loss and an overall 25 dB power loss when an RX moves from LOS to NLOS around a building corner and along a street canyon in an urban microcell environment. However, local area path loss measurements (cluster measurements) suggest omnidirectional received power has a relatively stationary mean over a relatively large area (5 m x 10 m) in LOS and NLOS scenarios, respectively, as indicated in Table~\ref{tbl:clusterSTD}. Wideband mmWave signals in typical indoor and outdoor environments were shown to fade at rates from 0.4 dB/ms to 40 dB/s, depending on the speed and environment. The results presented here will aid the 5G wireless community as it develops models for small-scale fading and spatial consistency for handoff algorithms and beam scanning techniques.
\bibliographystyle{IEEEtran}
\section{Introduction}~\label{sec:intro}
Driven mainly by the pervasive usage of smartphones and the emergence of the Internet of Things (IoT), future 5G mobile networks will become as pervasive as electrical wiring~\cite{Rap91d} and will offer unprecedented data rates and ultra-low latency~\cite{Boccardi14a,Ghosh14a,Rap16a}. For the first time in the history of radio, millimeter-wave (mmWave) frequencies will be used extensively for mobile and fixed access, thus requiring accurate propagation models that predict how the channel varies as people move about. Remarkable progress has been made in modeling large-scale propagation path loss at mmWave frequencies~\cite{Rap13a,Rap15b,Rap16a, Rap13b,Mac15b,Sun16b,Thomas16a,Koymen15a,Yoon15a,Maltsev10a,Mac17b}, and it is well understood that for an assumption of unity gain antennas across all frequencies, Friis equation predicts that path loss is greater at mmWave compared to today's UHF/microwave cellular systems~\cite{Friis46a, Rap13a, Rap15b, Mac15b,Sun16b,Thomas16a,Rap16a}. Also, rain and atmospheric attenuation are well understood, and reflection and scattering are more dominant than diffraction at mmWave bands~\cite{Rap16a,Rap15a,Deng16a,Rumney16b,Solomitckii16a}.
Broadband statistical spatial channel models (SSCMs) and simulators that faithfully predict the statistics of signal strength, and the number and direction of arrival and departure of multipath components, have been developed by a consortium of companies and universities~\cite{3GPP.38.900} and from measurements in New York City\textcolor{black}{\cite{Sun17b}}. These models are being used to develop air-interfaces for 5G systems~\cite{Samimi16a}\textcolor{black}{\cite{Sun17c}}. Elsewhere in this issue,\textcolor{black}{\cite{Rap16a}} summarizes standard activities for large-scale mmWave channel modeling.
Little is known, however, about the small-scale behavior of wideband mmWave signals as a mobile user moves about a local area. Such information is vital for the design of handoff mechanisms and beam steering needed to rescue the communication link from deep fades. In this paper, propagation measurements investigate diffraction, human blocking effects, small-scale spatial fading and autocorrelation, local area channel transitions, and stationarity of signal power in local area clusters at frequencies ranging from 10 to 73 GHz. Diffraction measurements for indoor and outdoor materials at 10, 20, and 26 GHz are presented in Section II, and two diffraction models, i.e., the knife edge diffraction (KED) model and a creeping wave linear model, are used to fit the measured results. We predict the rapid signal decay as a mobile moves around a diffracting corner. In Section III, measurements at 73 GHz are presented and a double knife-edge diffraction (DKED) antenna gain model that uses directional antenna patterns is shown to describe minimum and maximum fade depths caused by human blockage. Small-scale fading and correlation studies at 73 GHz are presented in Section IV, where small-scale fading distributions and spatial autocorrelations of received voltage amplitudes in LOS and NLOS environments with omnidirectional and directional antennas are provided and analyzed. In Section V, \textit{route} and \textit{cluster} scenarios are used to study local area channel transitions and stationarity, where analysis for channel transition from a NLOS to a LOS region and local area path loss variations are provided. Conclusions are given in Section VI. \textcolor{black}{Channel models given here may be implemented for small-scale propagation modeling and real-time site-specific mobile channel prediction and network control~\cite{Wang05b}.}
\section{Diffraction Measurements and Models}\label{sec:diffraction}
\subsection{Introduction of Diffraction Measurements and Models}
Accurate characterization of diffraction at cmWave and mmWave frequencies is important for understanding the rate of change of signal strength for mobile communications since future 5G mmWave systems will have to rely less on diffraction as a dominant propagation mechanism~\cite{Rap15b, Mac15b}. Published indoor and outdoor diffraction measurements show that diffraction has little contribution to the received signal power using various materials and geometries (edges, wedges, and circular cylinders) at 60 GHz and 300 GHz~\cite{Maltsev10c,Lu13a,Jacob12a}. It was shown that the KED model agreed well with diffraction measurements for cuboids at 300 GHz~\cite{Kleine-Ostmann12a}, vegetation obstacles at 2.4, 5, 28, and 60 GHz~\cite{Corre16a}, and for human blocking at 60 GHz~\cite{Maltsev10a}. Apart from the KED model, uniform theory of diffraction (UTD) models are also used. An overview of the Geometrical Theory of Diffraction (GTD) and the UTD are provided in~\cite{Pathak13a}, as well as their utility to solve practical problems. Besides the KED and UTD models, Mavridis \textit{et al.} presented a creeping wave linear model~\cite{Mavridis14a} to estimate the diffraction loss by a perfectly conducting or lossy circular cylinder for both transverse-magnetic (TM) and transverse-electric (TE) polarizations at 60 GHz. In the following sub-sections, we describe diffraction measurements conducted in 2015 around the engineering campus of New York University~\cite{Deng16a}, where the frequency dependency of diffraction at 10, 20, and 26 GHz in realistic indoor and outdoor scenarios was investigated to yield simple yet accurate diffraction models for wireless planning.
\subsection{Diffraction Measurement System}
Diffraction measurements were performed by transmitting a continuous wave (CW) signal generated by an Agilent E8257D PSG analog signal generator through a pyramidal horn antenna at the transmitter (TX). An identical horn antenna was used at the receiver (RX) to receive signal energy around a corner test material (e.g. a stone pillar). The RX antenna was fed to an E4407B ESA-E spectrum analyzer that measured received power which was subsequently recorded on a laptop with LabVIEW software. During the measurements, the TX antenna was set one meter from the knife edge, sufficiently in the far field, and was fixed to a tripod and aimed at the knife edge, whereas the RX antenna was set 2 meters from the knife edge (also in the far field) and was fixed on a rotatable gimbal attached to a translatable linear track that was made to from an approximate arc around the knife edge (see Fig. \ref{fig:Corner}). Diffraction loss was measured at 10, 20, and 26 GHz using identical pairs of antennas at the TX and RX to measure each frequency, separately. For each frequency, Table \ref{tbl:1} lists the flange type, antenna gain, and half-power beamwidth (HPBW) of the antenna pairs used. The TX and RX were stationed at a wide range of angles, both in the lit and shadowed region, and the horns always had their boresights focused on the corner knife-edge of the indoor and outdoor materials that were studied. More details regarding the measurement system are given in~\cite{Deng16a,Deng16b}.
\begin{table}[b]
\renewcommand{\arraystretch}{1.4}
\begin{center}
\caption{Antenna parameters used diffraction measurements.}~\label{tbl:1}
\fontsize{7}{7}\selectfont
\begin{tabular}{|>{\centering\arraybackslash}m{1.3cm}|>{\centering\arraybackslash}m{1.1cm}|>{\centering\arraybackslash}m{1.4cm}|>{\centering\arraybackslash}m{1.25cm}|>{\centering\arraybackslash}m{1.5cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Measured Frequency} & \textbf{Flange Type} & \textbf{Antenna Gain} & \textbf{HPBW (Az./El.)} & \textbf{Far Field Distance} \\ \hline
10 GHz & WR-75 & 20 dBi & $17^\circ$/$17^\circ$ & 0.47 m \\ \hline
20 GHz & WR-51 & 20 dBi & $17^\circ$/$17^\circ$ & 0.46 m \\ \hline
26 GHz & WR-28 & 24.5 dBi & $10.9^\circ$/$8.6^\circ$ & 0.83 m \\ \hline
\end{tabular}
\end{center}
\end{table}
\subsection{Diffraction Measurement Description}
The indoor measurements were performed at $90^\circ$ (right-angle) wall corners made of drywall, wood, and semi-transparent plastic board with 2 cm thickness. Outdoor measurements studied one rounded stone pillar corner and one right-angle marble building corner. During the measurements, the TX and RX were placed on either side of the corner (knife edge) of the test material. A diagram of the corner diffraction geometry is shown in Fig. \ref{fig:Corner} where $d_1$ is the distance between the TX and the corner knife-edge, and $d_2$ is the distance between the corner knife-edge and the RX. Both $d_1$ (1 m) and $d_2$ (2 m) remained constant throughout the diffraction measurement campaign. The $\beta$ and $\alpha$ values are the incident and diffraction angles, respectively, where two (outdoor) or three (indoor) fixed values between $10^\circ$ and $39^\circ$ were chosen for $\beta$. The RX antenna was mounted on a motorized linear track (see Fig. \ref{fig:Corner}) that translated in step increments of 0.875 cm, which corresponds to approximately a $0.5^\circ$ increment in diffraction angle ($\alpha$) for each step increment. At each step increment, the RX antenna was adjusted to point perfectly towards the knife-edge corner. The length of the track was $35.5$ cm and was used to measure a $20^\circ$ swath of diffraction angles over the entire length of the track. At each measurement location, five consecutive linear tracks (see Fig. \ref{fig:Corner}) were used to provide a $100^\circ$ diffraction angle arc around the corner which covered a broad range of the shadow region where the TX antenna is shadowed by the corner object with respect to RX antenna. Additionally, a smaller range of diffraction angles was measured in the lit region where the TX and RX antennas were in view of each other but were not pointed at each other since they were always aimed at the corner.
At each location, prior to the diffraction measurements, a free space calibration in an open area with both antennas pointed at each other on boresight was conducted with a 3 m ($d_1 + d_2 = 3$ m) transmitter-receiver (T-R) separation distance to provide a free space power reference for each frequency. The diffraction loss was then obtained by calculating the difference between the measured received signal power at each step increment of the RX antenna during the diffraction measurements and the free space calibration received power (the TX and RX antenna gains were deducted from all power measurements).
\begin{figure}[b!]
\vspace{-2.5mm}
\begin{center}
\includegraphics [width=0.44\textwidth]{Corner.pdf}
\caption{Top view of the corner diffraction geometry\cite{Deng16a}.}\label{fig:Corner}
\end{center}
\end{figure}
\subsection{Theoretical Diffraction Models}
\subsubsection{KED Model}
The KED model is suitable for applications with sharp knife edges and has a simple form yet high prediction accuracy ~\cite{Russell93a}. In general, diffraction loss over complex and irregular obstructions can be difficult to calculate, but typical obstructions for 5G wireless will involve common building partitions which are generally simple in nature and with dimensions that appear infinite at such small wavelengths, such as a wall or building corner, thus justifying simple diffraction models.
The diffraction loss (as compared to free space) is obtained by calculating the electric field strength $E_d$ [V/m] at the RX based on the specific Fresnel diffraction parameter $\nu$~\cite{Rap02a}. The ratio of $E_d$ and the free space field strength $E_0$ can be computed by summing all the secondary \textcolor{black}{Huygens'} sources in the knife edge plane and is given by~\cite{Deng16a,Rap02a}:
\begin{equation}
\footnotesize
\label{eq:8}
\frac{E_d}{E_0} = F(\nu) = \frac{1+j}{2} \int_{\textcolor{black}{\nu}}^{\infty} e^{-j (\pi/2) t^2}dt
\end{equation}
where $F(\nu)$ is the complex Fresnel integral and $\nu$ is the Fresnel diffraction parameter is defined as~\cite{Rap02a}:
\begin{equation}
\label{eq:9}
\footnotesize
\nu=h\sqrt{\frac{2(d_1+d_2)}{\lambda d_1 d_2}}=\alpha \sqrt{\frac{2 d_1 d_2}{\lambda(d_1+d_2)}}
\end{equation}
where $\lambda$ is the wavelength, $\alpha$ is the diffraction angle, $h$ is the effective height (or width) of the obstructing screen with an infinite width (or height) placed between the TX and RX at the distances $d_1$ and $d_2$, respectively, under the conditions that $d_1$, $d_2 \textcolor{black}{\gg} h$, and $d_1$, $d_2 \textcolor{black}{\gg} \lambda$. These conditions were met for 10, 20 and 26 GHz measurements in both the indoor and outdoor environments, as shown in Fig.~\ref{fig:Corner}.
Based on (\ref{eq:8}) and (\ref{eq:9}), the diffraction power gain $G(\nu)$ in dB produced in a knife edge by the KED model is expressed as~\cite{Deng16a,Rap02a}:
\begin{equation}
\label{eq:10}
\footnotesize
G(\nu) [\mathrm{dB}] = - P(\nu) [\mathrm{dB}] = 20\log_{10}|F(\nu)|
\end{equation}
where $P(\nu)$ is the power loss of the diffracted signal for the value of $\nu$, compared to free space case for the same distance.
\subsubsection{Convex Surfaces based Diffraction Model}
Although the KED model has broad applications for various geometries, it requires the diffraction corner to be in the shape of a sharp knife edge and does not account for the radius of curvature of an obstacle. When a diffraction corner is rounded in shape, such as a stone pillar corner (it resembles a circular cylinder), a creeping wave linear model can better predict diffraction loss~\cite{Piazzi98a, Mavridis14a}. A creeping ray field at the RX antenna behind the circular object for an incident plane wave is given by~\cite{Piazzi98a}:
\begin{equation} \label{eq:11}
\footnotesize
E(\alpha,d_2,k) \sim E_i e ^{-jk\alpha R_h} \frac{e^{-jkd_2}}{\sqrt{kd_2}}\sum_{p=1}^{\infty}{D_p{R_h}}\cdot \exp(-\psi_{p}\alpha)
\end{equation}
where $E_i$ is the incident field from the TX that impinges upon the obstruction, $R_h$ is the radius of cylinder for the diffraction corner, $k$ is the wave number of the carrier frequency, $\alpha$ is the diffraction angle, $d_2$ is the distance between the launch point at the rounded corner edge and the RX, $D_p$ is the excitation coefficient and $\psi_p$ is the attenuation constant. Due to the \textcolor{black}{computational complexity} of (\ref{eq:11}), a reasonable approximation for $E$ on a flat surface can be obtained by considering only the $p = 1$ term in (\ref{eq:11}), which is given by~\cite{Deng16a,Deng16b,Mavridis14a,Piazzi98a,Tervo14a}:
\begin{equation} \label{eq:12}
\footnotesize
E \sim E_i D_p{R_h} \cdot \exp(-\psi_{p}\alpha)
\end{equation}
The expression for the diffraction power loss in dB based on the creeping wave linear model is given by~\cite{Deng16a,Deng16b}:
\begin{equation}\label{eq:13}
\footnotesize
\begin{split}
G(\alpha) [\mathrm{dB}] = - P(\alpha) [\mathrm{dB}]= 20\log_{10}E
\end{split}
\end{equation}
In order to facilitate the computation of $G(\alpha)$, \textcolor{black}{a simple linear model}~\eqref{eq:14} based on minimum \textcolor{black}{mean squared} error (MMSE) estimation between the model and measured data was proposed in~\cite{Tervo14a} to estimate the diffraction loss caused by a curved surface at a single frequency, based on the creeping wave linear model~\cite{Deng16a,Deng16b}:
\begin{equation}
\footnotesize
\label{eq:14}
P(\alpha) = n \cdot \alpha + c
\end{equation}
where $n$ is the linear slope of diffraction loss calculated by MMSE for each specific frequency and object radius, and $c$ is the anchor point set to 6.03 dB, the diffraction loss estimated by KED with diffraction angles $\alpha=\beta=0^\circ$.
\begin{figure}[t!]
\centering
\includegraphics[width=2.95in]{Multi_Freq_Dry_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.95in]{Multi_Freq_Wood_VV.pdf}\vspace{0.2cm}
\includegraphics[width=2.95in]{Multi_Freq_Plastic_VV.pdf}
\caption{Diffraction measurements for the indoor drywall corner (top), wooden corner (middle), and plastic board (bottom) compared to the KED model at 10, 20 and 26 GHz~\cite{Deng16a}.}
\label{fig:Indoor}
\end{figure}
\begin{figure}[b!]
\centering
\includegraphics[width=2.9in]{Multi_Freq_Pillar_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.9in]{Multi_Freq_Marble_VV-eps-converted-to.pdf}\vspace{0.2cm}
\includegraphics[width=2.9in]{Out_Pillar_10GHz_VV-eps-converted-to.pdf}
\caption{Measured diffraction loss for the outdoor stone pillar (top) and marble corner (middle) with creeping wave linear models at 10, 20, and 26 GHz, and stone pillar measurement results compared to the KED and the creeping wave linear models at 10 GHz (bottom)~\cite{Deng16a}.}
\label{fig:Outdoor}
\end{figure}
\subsection{Indoor Diffraction Results and Analysis}
The indoor diffraction loss measurements for the drywall corner, wooden corner, and plastic board are plotted with the KED model (\ref{eq:10}) at 10, 20, and 26 GHz as a function of the diffraction angle in Fig. \ref{fig:Indoor}. \textcolor{black}{Three different TX incident angles were used to measure diffraction loss for each frequency. Since the measurements in~\cite{Deng16a} were conducted with the TX and RX along a constant radius ($d_1$ and $d_2$, respectively) from the corner of each test material, diffraction loss can be represented as a function of the diffraction angle $\alpha$ in~\eqref{eq:9} for each frequency (or wavelength $\lambda$), without the need for the TX incident angle.}. Fig. \ref{fig:Indoor} shows diffraction loss increases to approximately 30 dB as compared to free space as the RX antenna moves from the edge of the lit region (0$^\circ$) into the shadow region (20$^\circ$) for the drywall corner and wooden corner, respectively. This demonstrates the rapid signal degradation that occurs when diffraction is the primary propagation mechanism in mobile systems. Note the good fit between the drywall diffraction measurements and the simple KED model in the early and deep shadow regions for all three frequencies, while the KED model overestimates diffraction loss by 5-10 dB compared with the wooden corner diffraction measurements at 10, 20, and 26 GHz.
As for the plastic board material, the KED model overestimates the measured diffraction loss at small diffraction angles near the lit/shadow region boundary (from $0^\circ$ to $30^\circ$), and underestimates diffraction loss in portions of the deep shadow region (diffraction angles greater than $30^\circ$), most noticeably at 20 GHz and 26 GHz. Fig. \ref{fig:Indoor} indicates slightly less loss occurs at lower frequencies, implying frequency dependence, where divergence from pure diffraction theory can be attributed to reflections and scattering in the indoor environment and potential transmissions through the test materials. In general, the observations match the KED model trend. Oscillation patterns of the measured data in the shadow region observed in Fig. \ref{fig:Indoor} indicate that the measured diffraction signal includes corner diffraction, penetration through the material, and partial scattering in the measurement environment. We note that the diffraction loss observed in the lit region is due to the measurement procedure where the TX and RX antennas were never aligned on boresight (except at 0$^\circ$) since they were constantly pointed directly at the knife-edge corner.
\textcolor{black}{Penetration loss was measured for typical building materials in the same indoor environment at 73 GHz and showed that co-polarized penetration loss ranged from 0.8 dB/cm (lowest loss material -- drywall) to 9.9 dB/cm (highest loss material -- steel door), with standard deviation $\sigma$ about the average loss ranging from 0.3 dB/cm (lowest $\sigma$ -- drywall) to 2.3 dB/cm (highest $\sigma$ -- clear glass). Additional details can be found in Table II of~\cite{Ryan17a}.}
\subsection{Outdoor Diffraction Results and Analysis}
The outdoor marble corner and stone pillar measurement results are shown in Fig. \ref{fig:Outdoor} with the MMSE creeping wave linear models (\ref{eq:14}) at 10, 20, and 26 GHz.
It can be seen from Fig. \ref{fig:Outdoor} that (\ref{eq:14}) predicts a linearly increasing diffraction loss into the deeply shadowed region, as opposed to the leveling off seen in Fig. \ref{fig:Indoor} from (\ref{eq:10}).
Fig. \ref{fig:Outdoor} also shows the outdoor stone pillar measurement results at 10 GHz compared to the KED model and creeping wave linear model, where the creeping wave linear model provides a better fit to the measured relative diffraction loss than the KED model, in the shadow region.
Diffraction loss for each frequency is plotted as a function of diffraction angle and includes the measured loss at two TX incident angles.
The measured data matches well with the creeping wave linear model derived via MMSE for each frequency, in the shadow region.
The creeping wave linear model slopes are 0.75, 0.88, and 0.96 for the stone pillar measurements and 0.62, 0.77, and 0.96 for the marble corner measurements at 10, 20, and 26 GHz, respectively.
Fig. \ref{fig:Outdoor} shows outdoor obstructions cause an even greater loss when a mobile moves into a deeply shadowed region, showing as much as 50 dB of loss when solely based on diffraction.
Based on the increase in slope values with frequency, it is easily seen that diffraction loss increases with frequency in the outdoor environment.
The simple creeping wave linear model (\ref{eq:14}) fits well with the measured data and has a much lower standard deviation compared with the KED model~\cite{Deng16a}, indicating a good overall match between the creeping wave linear model and measured data.
Similar to the indoor environment diffraction loss measurements, the high diffraction loss in the lit region is caused by the TX and RX pointing off boresight towards the corner of the test material.
When comparing the slope values of the two outdoor materials, the rougher surface (stone) with a slightly rounded edge has a greater slope (greater attenuation) for identical frequencies as compared to the smoother surface (marble) straight edge.
We note that using the MMSE method to derive the typical slope values instead of calculating the theoretical slope value provides system engineers a useful parameter while reducing the \textcolor{black}{computational complexity} of the diffraction model. The simple slope values are useful for mobile handoff design since a mobile at 20 GHz would see approximately 21 dB of fading when moving around a marble corner from a diffraction angle of 0$^\circ$ to 20$^\circ$. For a person moving at a speed of 1 m/s, this results in about a 21 dB/s initial fade rate from LOS to NLOS. Fig. \ref{fig:Outdoor} shows more oscillation patterns in the marble corner measurements in the deeply shadowed region than in the stone pillar measurements, which indicates more prevalent scattering when measuring the marble corner. \textcolor{black}{We note that the diffraction loss for outdoor building corners at mmWaves using directional antennas can be better predicted with a simple linear model (creeping wave), whereas the KED model agrees well with indoor diffraction loss measurements. Both the creeping wave and KED models can be used in network simulations and ray-tracers with short computation time and good accuracy while considering approximately 5-6 dB standard deviation (see~\cite{Deng16a} for mean error and standard deviation values between the measured data and models derived from the data).} For cross-polarized diffraction measurement results, see~\cite{Deng16b}.
\section{Human Blockage Measurements and Models}
\subsection{Introduction of mmWave Human Blockage}
In mmWave communications, attenuation caused by human blockage (when a human body blocks the LOS path between a transmitter and receiver) will greatly impact cellphone link performance, and phased array antennas will need to adapt to find other propagation paths when blocked by a human~\cite{Friis46a, Mac16a}. This is in sharp contrast to omnidirectional antennas used at sub-6 GHz frequencies. Understanding this severe blockage effect and employing appropriate models for mobile system simulation are important for properly designing future mmWave antennas and beam steering algorithms~\cite{Sun15a,Samimi16a,Sun14b}. One of the earliest human blockage measurement studies~\cite{Sato98a} was conducted at 60 GHz for indoor wireless local area networks (WLAN) with T-R separation distances of 10 m or less in a typical office environment. Results showed that the signal level decreased by as much as 20 dB when a person blocked the direct path between omnidirectional TX and RX antennas, with deep fades reaching 30 dB using directive antennas\cite{Collonge04a},~\cite{Collonge03a}. In addition to human blockage measurements at 60 GHz,~\cite{80211ad10a} provided a human induced cluster blockage model based on ray tracing~\cite{Seidel94a}, a random walk model, and a diffraction model from contributions to 802.11ad~\cite{Jacob09d},~\cite{Jacob09e}. The probability distributions for four parameters (duration, decay time, rise time, and mean attenuation) were generated by the human-induced cluster blockage model and were validated with the Kolmogorov-Smirnov test~\cite{80211ad10a}. The \textit{mobile and wireless communications enablers for the twenty-twenty information society (METIS)} and \textit{3\textsuperscript{rd} Generation Partnership Project (3GPP)} also proposed their own human blockage models~\cite{METIS15a,3GPP.38.900,Medbo13a} based on KED models for one or multiple edges. In the following subsections, 73 GHz human blockage measurements and an improved human blockage model are described.
\subsection{Human Blockage Measurement System}
A real-time spread spectrum correlator channel sounder system described in~\cite{Mac16a} was used for the human blockage measurements. A pseudorandom noise (PN) sequence of length 2047 was generated at baseband with a field programmable gate array (FPGA) and high-speed digital-to-analog converter (DAC). This wideband sequence was modulated to an intermediate-frequency (IF) of 5.625 GHz which was then upconverted to a center frequency of 73.5 GHz (1 GHz null-to-null RF bandwidth). The transmit power at the TX was -5.8 dBm, and identical TX and RX horn antennas with 15$^\circ$ azimuth and elevation (Az./El.) HPBW and 20 dBi of gain were used. The received signal was downconverted to IF, and then demodulated to its in-phase ($I$) and quadrature ($Q$) baseband signals that were sampled at 1.5 Giga-Samples (GS/s) via a high-speed analog-to-digital converter (ADC). The digital signals were then correlated in software via a Fast Fourier Transform (FFT) matched filter to create the $I$ and $Q$ channel impulse response (CIR), and subsequent power delay profiles (PDPs) ($I^2+Q^2$). The system had a multipath resolution of 2 ns and an instantaneous dynamic range of 40 dB and could capture PDPs with a minimum consecutive snapshot interval of 32.752$\mu s$ to measure rapid fading. Power was computed as the area under the PDP, and the voltage was found as the square root of power~\cite{Rap02a}.
\subsection{Blockage Measurement Environment Description}
Human blockage measurements were conducted in an open laboratory using a 5 m T-R separation distance. High gain narrowbeam horn antennas were used at the TX and RX with both antenna heights set to 1.4 m relative to the ground and aligned on boresight. Nine measurements were recorded with a human blocker walking at a perpendicular orientation through the LOS path between the TX and RX at an approximate 1 m/s speed. This perpendicular walk was performed at 0.5 m increments between the TX and RX starting at 0.5 m from the TX for measurement one, and 4.5 m from the TX for measurement nine, as depicted in Fig. \ref{fig:4}. We note that 0.5 m is the typical distance for a person to view the screen of a smartphone. For each of the nine perpendicular walks, 500 PDPs were recorded per second in a five-second window, resulting in 2500 PDPs for each measurement. The dimensions of the human blocker were: $b_{\text{breadth}}=0.47$ m; $b_{\text{depth}}=0.28$ m; $b_{\text{height}}=1.80$ m. Detailed information about the experiment is given in~\cite{Mac16a,Mac17a}.
\begin{figure}[tb!]
\centering
\includegraphics [width=0.45\textwidth]{Block_TestCases.pdf}
\caption{Depiction of nine measurement locations where at each indicator separated by 0.5 m, the human blocker walked at a perpendicular orientation between the TX and RX antennas~\cite{Mac16a}.}\label{fig:4}
\vspace{-2.5mm}
\end{figure}
\subsection{KED Blockage Model}
Knife-edge diffraction is commonly used to model human blockage by modeling a thin rectangular screen as the blocker~\cite{Kunisch08a, Medbo13a,METIS15a,3GPP.38.900}. In DKED modeling, the rectangular screen is considered infinitely high, such that diffraction loss only occurs from the two side edges of the body. A typical screen blocker with height $h$ and width $w$ is displayed in Fig. \ref{fig:2} from both a 3D and top view screen projection. The dimensions for the top view of the screen are defined as follows: $w$ is the width of the screen from $w1$ to $w2$; $r\overset{\text{def}}{=}\overline{AB}$; $w\overset{\text{def}}{=}\overline{w1w2}$; $h\overset{\text{def}}{=}\overline{h1h2}$; $TS\overset{\text{def}}{=}\overline{AS}$;
$SR\overset{\text{def}}{=}\overline{SB}$; $D2_{w1}\overset{\text{def}}{=}\overline{A w1}$; $D1_{w1}\overset{\text{def}}{=}\overline{w1 B}$; $D2_{w2}\overset{\text{def}}{=}\overline{A w2}$; $D1_{w2}\overset{\text{def}}{=}\overline{w2 B}$; $\alpha_{w1}$ and $\alpha_{w2}$ are the diffraction angles for the $w1$ and $w2$ edges of the screen, respectively~\cite{Mac16a}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=1\linewidth]{3d_sub_a.pdf}
\caption{3D screen projection.}
\label{fig:DKEDmeasDProj}
\end{subfigure}
\begin{subfigure}[b]{0.38\textwidth}
\includegraphics[width=1\linewidth]{top_sub_b.pdf}
\caption{Top view of screen projection.}
\label{fig:TopProj}
\end{subfigure}
\caption{(a) 3D and (b) top-down projection of screen blocker.}\label{fig:2}
\vspace{-2.5mm}
\end{figure}
From Fig. \ref{fig:TopProj}, the two side edges of the screen are denoted $w1$ and $w2$, where the distance between the edges is the body depth ($b_{\text{depth}}$) or width of the screen $w$, since the blocker walks through the LOS path at a perpendicular orientation. As the screen moves between the TX and RX antennas and blocks the LOS path, the screen is always considered perpendicular to the solid line drawn between the two, to reduce computational complexity.
Diffraction loss is calculated via numerical approximation by Fresnel integration and the diffraction parameter as follows~\cite{Wang15a}:
\begin{equation}\label{eq:KEDapprox}
\footnotesize
F_{w1|w2}=
\begin{cases}
\frac{(1-j)}{2}\left(\frac{1+j}{2}-(C(v)+j\cdot S(v))\right)\text{, if $v>$ 0} \\
\frac{(1-j)}{2}\left(\frac{1+j}{2}+(C(-v)+j\cdot S(-v))\right)\text{, otherwise}
\end{cases}
\end{equation}
where the numerical approximations of Fresnel integration for $C(v)$ and $S(v)$ are:
\begin{subequations}
\footnotesize
\begin{equation}\label{eq:FresC}
C(v) = \int_{0}^{v}\cos{\left(\frac{\pi v^2}{2}\right)dv}
\end{equation}
\begin{equation}\label{eq:FresC}
S(v) = \int_{0}^{v}\sin{\left(\frac{\pi v^2}{2}\right)dv}
\end{equation}
\end{subequations}
and where the diffraction parameter $v$ is derived by~\cite{Rap02a}:
\begin{equation}\label{eq:diffp}
\footnotesize
v_{w1|w2} = \pm\alpha_{w1|w2} \sqrt{\frac{2\cdot \overline{AS}\cdot \overline{SB}}{\lambda (\overline{AS}+\overline{SB})}}
\end{equation}
The diffraction parameter $v$ is calculated based on the distance from the TX to the screen, from the screen to the RX, the diffraction angle $\alpha$ (see Fig.~\ref{fig:TopProj}), and the carrier wavelength $\lambda$. The $\pm$ sign in~\eqref{eq:diffp} is applied as $+$ to both edges for NLOS conditions. \textcolor{black}{When calculating the diffraction parameter~\eqref{eq:diffp} under unobstructed (LOS) conditions for the screen edge ($w1$ or $w2$) closest to the straight line drawn between the TX and RX, the $\pm$ is treated as ``$-$", whereas the $\pm$ is treated as ``$+$" for the screen edge farthest from the straight line drawn between the TX and RX~\cite{Wang15a}}.
The individual received signal caused by knife-edge diffraction from the $w1$ and $w2$ edges is $F_{w1}$ and $F_{w2}$, respectively. The complex signals corresponding to the edges can be added in order to determine the combined diffraction loss observed at the RX. The total diffraction loss power in log-scale is determined by taking the magnitude squared of the summed signals as follows~\cite{Wang15a}:
\begin{equation}\label{eq:Lscreen}
\footnotesize
L_{\text{screen}}[\mathrm{dB}] = 20\log_{10} \left( \big \lvert F_{w1}+F_{w2} \big \rvert \right)
\end{equation}
The DKED model~\eqref{eq:Lscreen} has been adopted by METIS and others~\cite{METIS15a,3GPP.38.900}, but it does not consider the antenna radiation pattern (it assumes an omnidirectional antenna~\cite{Medbo04a}) and has been shown to underestimate diffraction loss in the deepest fades when using directional antennas, which are sure to be employed by mmWave mobile devices~\cite{Mac16a}. This physical phenomenon occurs when the blocker obstructs the LOS path between antennas (i.e. deep fading), leaving only the off-boresight antenna gains to contribute to the received signal strength, which is slightly less than the directive gain. To account for the impact of non-uniform gain directional antennas on human blockage, antenna gain is considered in~\eqref{eq:Lscreen} (see~\cite{Mac16a}). The following azimuth far-field power radiation pattern of a horn antenna (general for any directional antenna) for a given half-power beamwdith (HPBW) is approximated by~\cite{Sun15a,Mac16a}:
\vspace{0.2cm}
\begin{align*}
\footnotesize
G(\theta) = \sinc^2(\textrm{a}\cdot \sin (\theta))\cdot \cos^2(\theta)
\end{align*}
where:
\begin{align*}
\footnotesize
\sinc^2\Bigg(\textrm{a}\cdot \sin \bigg( \frac{\text{HPBW}_{\text{AZ}}}{2}\bigg)\Bigg)\cdot \cos^2 \bigg( \frac{\text{HPBW}_{\text{AZ}}}{2}\bigg)=\frac{1}{2}
\end{align*}
The DKED model in~\eqref{eq:Lscreen} can be extended to include TX and RX antenna gains for the projected angles $\theta$ between the TX and the screen, and the screen and RX as follows~\cite{Mac16a}:
\begin{equation}\label{eq:LscreenMod}
\footnotesize
\begin{split}
L_{\textrm{Screen A.G.}}[\mathrm{dB}] = 20\log_{10} \Bigg( \Bigg \lvert F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}} \\+F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}\Bigg \rvert \Bigg)
\end{split}
\end{equation}
where $G_{D2_{w1}},\;G_{D1_{w1}},\;G_{D2_{w2}}$, and $G_{D1_{w2}}$ are the linear power gains (normalized to the directive gain such that $G_{0^\circ}=1$) of the antennas based on the point-source projections $\overline{Aw1}$; $\overline{w1B}$; $\overline{Aw2}$; $\overline{w2B}$; $A$ to $w1$, $w1$ to $B$, $A$ to $w2$, and $w2$ to $B$ (see Fig.~\ref{fig:TopProj}). When the screen does not obstruct the LOS path between the TX and RX, the normalized gains are set to $G(\theta)=1$, since the slight variations of antenna patterns have little effect on diffraction loss in the unobstructed case.
\subsection{Human Blockage Results and Analysis}
For each of the nine measurement paths, the area under the curve of each of the 2500 PDPs was integrated to calculate the received power in 2 ms increments. In Fig. \ref{fig:DKEDmeas} the received power (red) is compared to the DKED antenna gain (DKED-AG) model (green)~\eqref{eq:LscreenMod}, in addition to showing the constructive (signals in-phase) and destructive (signals out of phase) sum of received signals of the upper (blue) and lower (black) bound of the fade envelope, respectively.
\begin{figure}[tb!]
\vspace{-2.5mm}
\begin{center}
\includegraphics [width=0.5\textwidth]{Block_Test44_DKED_Sim_Data_METIS_v4_0_embed.pdf}
\caption{Comparison of measured received power of human blockage at 73 GHz and the DKED-AG model in~\eqref{eq:LscreenMod}~\cite{Mac16a}.}\label{fig:DKEDmeas}
\end{center}
\vspace{-2.5mm}
\end{figure}
Fig. \ref{fig:DKEDmeas} represents loss as compared to a free space reference with no blockage between the TX and RX. From Fig. \ref{fig:DKEDmeas}, we observe gain in the received signal as the human enters the TX/RX LOS path, and then deep attenuations as the human blocks the LOS path. Due to the fact that identical antennas were used at the TX and RX, the envelopes of the received signal power were similar in the two different symmetrical cases (i.e., Meas. 1 and Meas. 9). The best case scenario (minimum diffraction loss) is found by summing the magnitudes of received field components from the $w1$ and $w2$ edges of the blocker, represented by the blue dashed line in Fig.~\ref{fig:DKEDmeas}. For minimum loss,~\eqref{eq:LscreenMod} is reformulated as: $20\log_{10}(|F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}}|+|F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}|)$. The worst case scenario (maximum diffraction loss) is found by taking the difference of the magnitudes of received signals from the $w1$ and $w2$ edges, and is represented by the black dotted line in Fig~\ref{fig:DKEDmeas}, where~\eqref{eq:LscreenMod} is computed as: $20\log_{10}\Big(\big||F_{w1}\cdot \sqrt{G_{D2_{w1}}}\cdot \sqrt{G_{D1_{w1}}}|-|F_{w2} \cdot \sqrt{G_{D2_{w2}}}\cdot \sqrt{G_{D1_{w2}}}|\big|\Big)$.
It was previously demonstrated that diffraction loss models that do not account for antenna gain pattern can severely underestimate the diffraction loss when the blocker is close to either antenna (a critical issue for mobile phone use)~\cite{Mac16a}. The DKED-AG model~\eqref{eq:LscreenMod} accurately predicts what is measured, and predicts the deepest attenuation caused by a human blocker in excess of 40 dB. To model multiple blockers, the screen model can be replicated multiple times. These results show that adaptive antenna array and beamforming techniques will be employed to find suitable reflectors and \textcolor{black}{scatterers} in the signal transmission to overcome severe blockage attenuation in future 5G communication systems. The DKED-AG model in~\eqref{eq:LscreenMod} may be extended~\cite{Kunisch08a} to consider the top and bottom screen edges, phase corrections, and non-perpendicular screen orientations, although the simple model~\eqref{eq:LscreenMod} matches the human blocking measurements with confidence. It can be seen in Fig. \ref{fig:DKEDmeas} (Meas. 1 and 9) that the signal strength drops off at a rate of 0.4 dB/ms as the blocker \textcolor{black}{moves at 1 m/s and begins to} shadow the TX (RX). Mobile handoffs and beam steering schemes will be needed to rescue the mobile from severe fades by the use of electrically scanning beams at the sub-millisecond level, a feat easily accomplished with sub millisecond packets in an air interface standard. \textcolor{black}{An additional technique for mitigating the effects of rapid fading could include rapid re-routing around obstacles via handoff to another access point (AP) in a network cluster~\cite{Ghosh16a}.} \textcolor{black}{Note that just prior to the deep shadowing events in Fig.~\ref{fig:DKEDmeas} there is a slight increase/scintillation of signal strength of $\sim$ 2 dB peak-to-peak amplitudes (noticed by others in~\cite{Jacob10b}), which could be used to detect the imminent presence of an obstruction such that the RX adapts its beam in anticipation of the pending deep fade. The 3GPP/METIS blockage model~\cite{3GPP.38.900,METIS15a} shown in Fig.~\ref{fig:DKEDmeas} underestimates the deep fades of shadowing events~\cite{Mac16a}, especially when the blocker is close to the TX or RX antenna, since the full directive gain of the TX and RX antennas is not available across the diffraction obstacle, and thus is unable to contribute to the received signal strength from diffraction around the blocker during the shadowing event~\cite{Mac16a}. We note that the 3GPP/METIS model only offers reasonable agreement to the measured loss when the blocker is far (several meters) from the TX and RX antenna.}
\section{Small-Scale Spatial Statistics}\label{sec:smallscale}
\subsection{Introduction of Small-Scale Spatial Statistics}
Small-scale fading and small-scale autocorrelation characteristics are crucial for the design of future mmWave communication systems, especially in multiple-input multiple-output (MIMO) channel modeling. Previous studies on small-scale fading characteristics focused on sub-6 GHz frequencies, yet investigations at mmWave are scarce. Wang \textit{et al.}~\cite{Wang15b} showed that small-scale fading of received power in indoor corridor scenarios with omnidirectional antennas at both TX and RX could be well described by Ricean distributions with K-factors ranging from 5 dB to 10 dB based on their indoor corridor measurements at 15 GHz with a bandwidth of 1 GHz, and ray tracing results using a ray-optical based channel model validated by measurements. Henderson \textit{et al.} compared Rayleigh, Ricean, and the Two-Wave-Diffuse-Power (TWDP) distributions to find the proper small-scale fading distribution of received voltage magnitudes for a measured 2.4 GHz indoor channel~\cite{Henderson08a} where the Ricean distribution had highest modeling accuracy in most indoor cases~\cite{Henderson08a}. The authors in~\cite{Romero-Jerez16a} demonstrated the use of the TWDP fading model for mmWave communications. It was reported that \textcolor{black}{log-normal} distribution had a good fit to measured received signal envelopes in some indoor mobile radio channels~\cite{Cotton07a}. Important work on wideband directional small-scale fading also appears in~\cite{Holtzman94a,Durgin03a,Sun17a,Samimi16c,Dupleich17a}. In the following subsections, small-scale fading distributions of total power and autocorrelation characteristics of received voltage amplitudes at 73 GHz in urban microcell environments are investigated based on a measurement campaign conducted during the summer of 2016 around the engineering campus of New York University in downtown Brooklyn.
\subsection{Measurement System for Small-Scale Spatial Statistics}
The TX system for small-scale fading and autocorrelation measurements at 73 GHz was identical to the TX system used for the human blocking measurements with the difference only for TX antennas and transmit powers as identified in Table ~\ref{tbl:2}. The RX side of the system captured the RF signal via steerable horn antennas and downconverted the signal to an IF of 5.625 GHz, which was then demodulated into its baseband in-phase ($I$) and quadrature-phase ($Q$) signals which were correlated via a common sliding correlation architecture~\cite{Rap02a,Rap15b,Mac15b,Rap13a} where the time-dilated $I$ and $Q$ channel voltages were sampled by an oscilloscope and then squared and added together in software to generate a PDP. Antennas with 27 dBi gain (7$^\circ$ Az./El. HPBW) and 9.1 dBi gain (60$^\circ$ Az./El.HPBW) were used at the TX and RX sides, respectively\textcolor{black}{\cite{Sun17a}}.
\begin{table}[tb!]
\renewcommand{\arraystretch}{1.4}
\caption{Hardware Specifications of Small-Scale Fading and Local Area Channel Transition Measurements.}~\label{tbl:2}
\fontsize{7.0}{7.0}\selectfont
\begin{center}
\vspace{-2.5mm}
\begin{tabular}{|>{\centering\arraybackslash}m{3.0cm}|>{\centering\arraybackslash}m{2.2cm}|>{\centering\arraybackslash}m{2.2cm}|>{\centering\arraybackslash}m{2.5cm}|>{\centering\arraybackslash}m{2.5cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Campaign} & \textbf{73 GHz Small-Scale Fading and Correlation Measurements} & \textbf{73 GHz Local Area Channel Transition Measurements} \\ \hline \hline
\textbf{Broadcast Sequence} & \multicolumn{2}{c|}{$11\textsuperscript{th}$ order PN Code (L = $2^{11}-1$ = 2047)} \\ \hline
\textbf{TX and RX Antenna Type} & \multicolumn{2}{c|}{Rotatable pyramidal horn antenna} \\ \hline
\textbf{TX/RX Chip Rate} & \multicolumn{2}{c|}{500 Mcps / 499.9375 Mcps} \\ \hline
\textbf{Slide Factor $\gamma$} & \multicolumn{2}{c|}{8000} \\ \hline
\textcolor{black}{\textbf{RF Null-to-Null Bandwidth}} & \multicolumn{2}{c|}{\textcolor{black}{1 GHz}} \\ \hline
\textbf{PDP Threshold} & \multicolumn{2}{c|}{20 dB down from max peak} \\ \hline
\textbf{TX/RX Intermediate Freq.} & \multicolumn{2}{c|}{5.625 GHz} \\ \hline
\textbf{TX/RX Local Oscillator} & \multicolumn{2}{c|}{67.875 GHz (22.625 GHz $\times$ 3)} \\ \hline
\textbf{Carrier Frequency} & \multicolumn{2}{c|}{73.5 GHz} \\ \hline
\textbf{TX Antenna Gain} & \multicolumn{2}{c|}{27 dBi} \\ \hline
\textbf{RX Antenna Gain} & 9.1 dBi & 20 dBi \\ \hline
\textbf{Max TX Power / EIRP} & 14.2 dBm / 41.2 dBm & 14.3 dBm / 41.3 dBm \\ \hline
\textbf{TX Az. and El. HPBW} & \multicolumn{2}{c|}{$7^\circ$} \\ \hline
\textbf{TX/RX Heights} & 4.0 m / 1.4 m & 4.0 m / 1.5 m \\ \hline
\textbf{RX Az. and El. HPBW} & $60^\circ$ & $15^\circ$ \\ \hline
\textbf{TX-RX Antenna Pol.} & \multicolumn{2}{c|}{V-V (vertical-to-vertical)} \\ \hline
\textbf{Max Measurable Path Loss} & 168 dB & 180 dB \\ \hline
\end{tabular}
\end{center}
\end{table}
\subsection{Small-Scale Measurement Environment and Procedure}
In the summer of 2016, a set of small-scale linear track measurements were conducted at 73 GHz on the campus of NYU Tandon School of Engineering in downtown Brooklyn, New York, representative of an urban microcell (UMi) environment\textcolor{black}{\cite{Sun17a}}. The measurement environment, and the TX and RX locations are depicted in Fig.~\ref{fig:TX_RX_Location}. One TX location with an antenna height of 4.0 m above the ground and two RX locations with an antenna height of 1.4 m were selected to perform the measurements, where one RX was LOS to the TX while the other was NLOS. The TX was placed near the southwest corner of the Dibner library building (top center in Fig.~\ref{fig:TX_RX_Location}), the LOS RX was located 79.9 m away from the TX, and the NLOS RX was shadowed by the southeast corner of a building (Rogers Hall on the map) with a T-R separation distance of 75.0 m\textcolor{black}{\cite{Sun17a}}. Other specifications about the measurement hardware are detailed in Table~\ref{tbl:2}.
A fixed 35.31-cm spatial linear track \textcolor{black}{(about 87 wavelengths at 73.5 GHz)} was used at each RX location in the small-scale fading measurements, over which the RX antenna was moved in increments of half-wavelength (2.04 mm) over 175 track positions\textcolor{black}{\cite{Sun17a}}. At each RX, six sets of small-scale fading measurements were performed, where the elevation angle of the RX antenna remained fixed at 0$^\circ$ (parallel to the horizon) and a different azimuth angle was chosen for each set of the measurements with the adjacent azimuth angles separated by 60$^\circ$, such that the RX antenna swept over the entire azimuth plane after rotating through the six pointing angles. The RX antenna was pointing at a fixed angle while moving along the linear track for each set of the measurements and a PDP was acquired at each track position for each pointing angle. The TX antenna elevation angle was always fixed at 0$^\circ$ (parallel to horizon). Under the LOS condition, the TX antenna was pointed at 90$^\circ$ in the azimuth plane, directly towards the RX location; for NLOS, the TX antenna azimuth pointing angle was 200$^\circ$, roughly towards the southeast corner of Rogers Hall in Fig.~\ref{fig:TX_RX_Location}\textcolor{black}{\cite{Sun17a}. Due to space limitations, we show here only one track orientation at each RX (along the direction of the street beside the RX), but more results and observations are detailed in~\cite{Sun17a} which show the fading depths are a function of antenna orientations and environment.}
As a comparison, the 28 GHz small-scale measurements presented in~\cite{Samimi16b} investigated the small-scale fading and autocorrelation of \textit{individual resolvable multipath} voltage amplitudes using a 30$^\circ$ Az./El. HPBW RX antenna, whereas this paper studies 73 GHz fading and autocorrelation using a wider HPBW (60$^\circ$) RX antenna, and focuses on received signal voltage amplitude by integrating the area under the \textit{entire PDP} curve and then taking the square root of the total power, instead of individual multipath voltage amplitude at each location along a track.
\begin{figure}
\centering
\includegraphics[width=2.9in]{MeasurementMap_2.pdf}
\caption{\textcolor{black}{2D map of the 73 GHz small-scale measurement environment and the locations of TX and RX. Pointing to the top of the map is 0$^\circ$.}}
\label{fig:TX_RX_Location}
\vspace{-2.5mm}
\end{figure}
\textcolor{black}{\subsection{Small-Scale Measurement Results}}
\textcolor{black}{Fig.~\ref{fig:LOSPDP2} illustrates typical measured small-scale directional PDPs over 175 track positions on the 35.31-cm (about 87 wavelengths at 73.5 GHz) linear track in the LOS environment, where the RX horn antenna with 60$^\circ$ HPBW was pointing on boresight to the TX, and the track orientation was orthogonal to the T-R line. The total power in Figs.~\ref{fig:LOSPDP2} and~\ref{fig:NLOSPDP2} is computed as the area under the PDP at a particular track position over the 1 GHz RF bandwidth. Fig.~\ref{fig:LOSPDP2} shows there is 11 dB power variation over different track positions, but the power variation is only 3.7 dB when the track orientation was in the direction of the T-R line (not shown), indicating little small-scale spatial fading~\cite{Sun17a}.}
\textcolor{black}{Typical measured small-scale directional PDPs over 175 track positions on the 35.31-cm (about 87 wavelengths at 73.5 GHz) linear track in the NLOS environment are depicted in Fig.~\ref{fig:NLOSPDP2}, where the track orientation was along the direction of the street, and the RX antenna was pointing to the TX but was obstructed by a building corner~\cite{Sun17a}. Fig.~\ref{fig:NLOSPDP2} shows there is very moderate power variation (4.1 dB) over different local-area track positions, albeit with rich and varying multipath components.}\\
\begin{figure}
\centering
\includegraphics[width=3.2in]{RX4_Meas1.pdf}
\caption{\textcolor{black}{Measured 73 GHz small-scale directional PDPs over 175 track positions in LOS. The RX horn antenna (60$^\circ$ HPBW) was pointing on boresight to the TX, and track orientation orthogonal to the T-R line.}}
\label{fig:LOSPDP2}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.2in]{RX14_Meas6.pdf}
\caption{\textcolor{black}{Measured 73 GHz small-scale directional PDPs over 175 track positions in NLOS. The RX horn antenna (60$^\circ$ HPBW) was pointing to the TX but was obstructed by a building corner, and the track orientation was along the direction of the street.}}
\label{fig:NLOSPDP2}
\end{figure}
\subsection{Small-Scale Spatial Statistics Results and Analysis}
\subsubsection{Omnidirectional Small-Scale Spatial Statistics}
As described above, a rotatable directive horn antenna was used at the RX side to capture directional PDPs in the small-scale fading and correlation measurements. In channel modeling, however, omnidirectional statistics are often preferred, since arbitrary antenna patterns can be implemented according to one's own needs if accurate temporal and spatial statistics are known~\cite{Sun17c}. Therefore, we synthesized the approximated omnidirectional received power at every track interval by taking the \textcolor{black}{area under the curve of each directional PDP and summing powers using the approach presented in~\cite{Sun15a} and on Page 3040 from~\cite{Rap15b}}, thereby computing omnidirectional received power. Although the RX antenna did not sweep the entire 4$\pi$ Steradian sphere, the azimuth plane spanned $\pm 30^{\circ}$ with respect to the horizon, ensuring that a large majority of the arriving energy was captured, as verified in~\cite{Sun15a}.
Fig.~\ref{fig:omniLOSFading} illustrates the cumulative distribution function (CDF) of the measured small-scale received voltage amplitude at 73 GHz with a 1 GHz RF bandwidth over the 35.31-cm length track with 175 track positions in increments of half-wavelength (2.04 mm) for the omnidirectional RX antenna pattern in the LOS environment\textcolor{black}{\cite{Sun17a}}. \textcolor{black}{Superimposed with the measured curve are the CDFs of the Rayleigh distribution, the zero-mean log-normal distribution with a standard deviation of 0.91 dB (obtained from the measured data), and the Ricean distribution with a $K$-factor of 10 dB obtained from the measured data by dividing the total received power contained in the LOS path by the power contributed from all the other reflected or scattered paths. As shown in Fig.~\ref{fig:omniLOSFading}, the measured 73 GHz small-scale spatial fading in the LOS environment can be approximated by the Ricean distribution with a $K$-factor of 10 dB, indicating that there is a dominant path (i.e., the LOS path) contributing to the total received power, and that the received signal voltage amplitude varies little over the 35.31-cm (about 87 wavelengths) length track. The log-normal distribution does not fit the measured data well in the regions of -3 to -2.5 dB and +1.2 to +1.5 dB about the mean.} The maximum fluctuation of the received voltage amplitude is merely 3 dB relative to the mean value, whereas the fades are much deeper for the Rayleigh distribution. The physical reason for this is the presence of a dominant LOS path.
\begin{figure}
\vspace{-2.5mm}
\centering
\includegraphics[width=2.7in]{omniLOSFading.pdf}
\caption{CDF of the measured small-scale spatial fading distribution of the received voltage amplitude for the omnidirectional RX antenna pattern in the LOS environment at 73 GHz with a 1 GHz RF bandwidth.}
\label{fig:omniLOSFading}
\vspace{-2.5mm}
\end{figure}
The small-scale spatial fading in the NLOS environment for the omnidirectional RX antenna pattern is illustrated in Fig.~\ref{fig:omniNLOSFading}, \textcolor{black}{and the zero-mean log-normal distribution with a standard deviation of 0.65 dB (obtained from the measured data) is selected to fit the measured result, and Ricean and Rayleigh distributions are also given as a reference\textcolor{black}{\cite{Sun17a}}. As evident from Fig.~\ref{fig:omniNLOSFading}, the measured NLOS small-scale spatial fading distribution matches the log-normal fitted curve almost perfectly. In contrast, the Ricean distribution with $K$ = 19 dB does not fit the measured data as well as the log-normal distribution in the tail region around -0.6 dB to -0.8 dB of the relative mean signal level }\textcolor{black}{(as shown by the inset in Fig.~\ref{fig:omniNLOSFading}), since the Ricean $K$ = 19 dB distribution predicts more occurrences of deeper fading events, whereas the log-normal distribution with a 0.65 dB standard deviation predicts a more compressed fading range of -0.8 dB to +0.8 dB about the mean, which was observed for the wideband NLOS signals.} The fact that the local fading of received voltage amplitudes in the NLOS environment is \textcolor{black}{log-normal} instead of Rayleigh is similar to models in~\cite{Rap91b} for urban mobile radio channels. For a NLOS environment, there may not be a dominant path, yet the transmitted broadband signal experiences frequency-selective fading (when the signal bandwidth is larger than the coherence bandwidth of the channel~\cite{Rap02a}). Different frequency components of the signal experience uncorrelated fading, thus it is highly unlikely that all parts of the signal will simultaneously experience a deep fade, and the fades over frequency tend to be very sharp, taking up a small portion of the total power received over the entire signal bandwidth~\cite{Rap15a}. Consequently, the total received power changes very little over a small-scale local area. This is a distinguishing feature of wideband mobile signals as compared to narrowband signals.
\begin{figure}
\centering
\includegraphics[width=2.7in]{omniNLOSFading.pdf}
\caption{CDF of the measured small-scale spatial fading distribution of the received voltage amplitude for the omnidirectional RX antenna pattern in the NLOS environment at 73 GHz with a 1 GHz RF bandwidth.}
\label{fig:omniNLOSFading}
\vspace{-2.5mm}
\end{figure}
Apart from small-scale spatial fading, small-scale spatial autocorrelation is also important for wireless modem design. Spatial autocorrelation characterizes how the received voltage amplitudes correlate at different linear track positions within a local area~\cite{Samimi16b}. Spatial autocorrelation coefficient functions can be calculated using Eq.~\eqref{sac}, where $X_k$ denotes the $k^{th}$ linear track position, $E[~]$ is the expectation operator where the average of voltage amplitudes is taken over all the positions $X_k$, and $\Delta X$ represents the spacing between different antenna positions on the track\textcolor{black}{\cite{Sun17a}}.
\begin{figure*}
\begin{equation}\label{sac}
\footnotesize
\rho=\frac{E\big[\big(A_k(X_k)-\overline{A_k(X_k)}\big)\big(A_k(X_k+\Delta X)-\overline{A_k(X_k+\Delta X)}\big)\big]}{\sqrt{E\big[\big(A_k(X_k)-\overline{A_k(X_k)}\big)^2\big]E\big[\big(A_k(X_k+\Delta X)-\overline{A_k(X_k+\Delta X)}\big)^2\big]}}
\vspace{-2.5mm}
\end{equation}
\vspace{-2.5mm}
\end{figure*}
The measured 73 GHz spatial autocorrelation of the received voltage amplitudes in LOS and NLOS environments with a 1 GHz RF bandwidth are depicted in Fig.~\ref{fig:omniLOSCor} and Fig.~\ref{fig:omniNLOSCor}, respectively. Note that a total of 175 linear track positions over the 35.31-cm length track were measured during the measurements, yielding a maximum spatial separation of 174 half-wavelengths on a single track. Only up to 60 half-wavelengths, however, are shown herein because little change is found thereafter and it provides 100 autocorrelation data points for all spatial separations on a single track, thus improving the reliability of the statistics. According to Fig.~\ref{fig:omniLOSCor}, the received omnidirectional signal voltage amplitude first becomes uncorrelated at a spatial separation of about 3.5$\lambda$, then becomes slightly anticorrelated for separations of 3.5$\lambda$ to 10$\lambda$, and becomes slightly correlated for separations between 10$\lambda$ and 18$\lambda$, and decays towards 0 sinusoidally after 18$\lambda$. Therefore, the spatial correlation can be modeled by a ``damped oscillation'' function of~\eqref{expFit}\textcolor{black}{\cite{Sun17a}}~\cite{Zhang08a}:
\begin{equation}\label{expFit}
\footnotesize
f(\Delta X) = \cos(a\Delta X)e^{-b\Delta X}
\end{equation}
where $\Delta X$ denotes the space between antenna positions, $a$ is an oscillation distance with units of radians/$\lambda$ (wavelength), \textcolor{black}{$T=2\pi/a$} can be defined as the spatial oscillation period with units of $\lambda$ or cm, and $b$ is a constant with units of $\lambda^{-1}$ whose inverse \textcolor{black}{$d=1/b$} is the spatial decay constant with units of $\lambda$. $a$ and $b$ are obtained using the minimum mean square error (MMSE) method to find the best fit between the empirical spatial autocorrelation curve and theoretical exponential model given by~\eqref{expFit}. The ``damped oscillation'' pattern can be explained by superposition of multipath components with different phases at different linear track positions. \textcolor{black}{As the separation distance of linear track positions increases, the phase differences among individual multipath components will oscillate as the separation distance of track positions increases due to alternating constructive and destructive combining of the multipath phases.} This ``damped oscillation'' pattern is obvious in LOS environment where phase difference among individual multipath component is not affected by shadowing effects that occurred in NLOS environments. The form of~\eqref{expFit} also guarantees that the spatial autocorrelation coefficient is always 1 for $\Delta X = 0$, and converges to 0 when $\Delta X$ approximates infinity. The spatial autocorrelation curve for NLOS environment in Fig.~\ref{fig:omniNLOSCor} exhibits a different trend from that in Fig.~\ref{fig:omniLOSCor}, which is more akin to an exponential distribution without damping, but can still be fitted using Eq.~\eqref{expFit} with $a$ set to 0\textcolor{black}{\cite{Sun17a}}. The constants $a$, $b$, are provided in Table~\ref{tbl:ModelPara}, where $T$ is the oscillation period, and $d$ represents the spatial decay constant. From Fig.~\ref{fig:omniNLOSCor} and Table~\ref{tbl:ModelPara} it is clear that after 1.57 cm (3.85 wavelengths at 73.5 GHz) in the NLOS environment, the received voltage amplitudes become uncorrelated (the correlation coefficient decreases to 1/e~\cite{Samimi16a}). We note that Samimi~\cite{Samimi16b} found individual multipath voltage amplitudes received using a 30$^\circ$ Az./El. HPBW antenna became uncorrelated at physical distances of 0.52 cm (0.48 wavelengths at 28 GHz) and 0.67 cm (0.62 wavelengths at 28 GHz) in LOS and NLOS environments, respectively -- smaller decorrelation distances compared to the present 73 GHz results measured using a 60$^\circ$ Az./El. HPBW antenna.
\begin{figure}
\centering
\includegraphics[width=2.5in]{OmniCor_LOS.pdf}
\caption{\textcolor{black}{Measured 73 GHz broadband spatial autocorrelation coefficients of the received voltage amplitude in the LOS environment, and the corresponding fitting model. The T-R separation distance is 79.9 m.}}
\label{fig:omniLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=2.5in]{OmniCor_NLOS.pdf}
\caption{\textcolor{black}{Measured 73 GHz broadband spatial autocorrelation coefficients of the received voltage amplitude in the NLOS environment, and the corresponding fitting model. The T-R separation distance is 75.0 m.}}
\label{fig:omniNLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{table}[t!]
\renewcommand{\arraystretch}{1.4}
\caption{Spatial correlation model parameters in~\eqref{expFit} for 73 GHz, 1 GHz RF bandwidth ($\lambda$=0.41 cm).}~\label{tbl:ModelPara}
\fontsize{8}{6.5}\selectfont
\scriptsize
\begin{center}\scalebox{0.84}{
\vspace{-2.5mm}
\begin{tabular}{|>{\centering\arraybackslash}m{1.4cm}|>{\centering\arraybackslash}m{1.3cm}|>{\centering\arraybackslash}m{2.15cm}|>{\centering\arraybackslash}m{1.25cm}|>{\centering\arraybackslash}m{2.15cm}|>{\centering\arraybackslash}m{0.6cm}|}\hline
\textbf{Condition} & $\boldsymbol{a}~(rad/\lambda)$ & \textcolor{black}{$\boldsymbol{T=2\pi/a}$} & \textbf{b}~($\lambda^{-1}$) & \textcolor{black}{$\boldsymbol{d=1/b}$} \\ \hline \hline
\textbf{LOS Omnidirectional} & {0.45} & {14.0$\lambda$ (5.71 cm)} & {0.10} & {10.0$\lambda$ (4.08 cm)}\\ \hline
\textbf{NLOS Omnidirectional} & {0} & {\textcolor{black}{Not used}} & {0.26} & {3.85$\lambda$ (1.57 cm)}\\ \hline
\textbf{LOS Directional} & {0.33 to 0.50} & {12.6$\lambda$ to 19.0$\lambda$ (5.14 cm to 7.76 cm)} & {0.03 to 0.15} & {6.67$\lambda$ to 33.3$\lambda$ (2.72 cm to 13.6 cm)}\\ \hline
\textbf{NLOS Directional} & {0} & {\textcolor{black}{Not used}} & {0.04 to 1.49} & {0.67$\lambda$ to 25.0$\lambda$ (0.27 cm to 10.2 cm)}\\ \hline
\end{tabular}}
\end{center}
\end{table}
\subsubsection{Directional Small-Scale Spatial Statistics}
Since mobile devices will use directional antennas, directional statistics are also of interest. In this subsection, we will investigate small-scale spatial fading and autocorrelation of the received voltage amplitudes associated with directional antennas at the RX.
Small-scale fading of received voltage amplitudes along the linear track using the $7^{\circ}$ Az./El. HPBW TX antenna and $60^{\circ}$ Az./El. HPBW RX antenna in LOS and NLOS environments are shown in Fig.~\ref{fig:dirLOSFading} and Fig.~\ref{fig:dirNLOSFading}, respectively, where the TX and RX were placed as shown in Fig.~\ref{fig:TX_RX_Location}, and each measured curve corresponds to a unique RX antenna azimuth pointing angle relative to true north as specified in the legend. There was no signal for the RX azimuth pointing angle of 270$^{\circ}$ in the NLOS environment, thus the corresponding results are absent in Fig.~\ref{fig:dirNLOSFading}. The strongest pointing directions are 270$^\circ$ and 150$^\circ$ in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, respectively\textcolor{black}{\cite{Sun17a}}. As shown in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, the measured directional spatial autocorrelation coefficients resemble Ricean distributions in both LOS and NLOS environments. Possible reason for such distributions is that only one dominant path (accompanied with several weaker paths) is captured by the horn antenna due to its directionality, given the fact that mmWave propagation is directional and the channel is sparse~\cite{Samimi16a}. The Ricean $K$-factor for received voltage amplitudes for various RX pointing directions ranges from 7 dB to 17 dB for the LOS environment, and 9 dB to 21 dB for the NLOS case, as shown in Figs.~\ref{fig:dirLOSFading} and~\ref{fig:dirNLOSFading}, as the RX is moved over a 35.31 cm (86.5 wavelengths at 73.5 GHz) track.
\begin{figure}
\centering
\includegraphics[width=2.8in]{dirLOSFading.pdf}
\caption{\textcolor{black}{Measured 73 GHz LOS small-scale spatial fading distributions of the directional received voltage amplitude, and the corresponding Ricean fitting curves with different $K$ factors. The angles in the legend denote the receiver antenna azimuth angle, and "b" denotes boresight to the TX.}}
\label{fig:dirLOSFading}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=2.8in]{dirNLOSFading.pdf}
\caption{\textcolor{black}{Measured 73 GHz NLOS small-scale spatial fading distributions of the directional received voltage amplitude, and the corresponding Ricean fitting curves with different $K$ factors. The angles in the legend denote the receiver antenna azimuth angle, and "s, to TX" denotes along the direction of the street and pointing to the TX side.}}
\label{fig:dirNLOSFading}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.0in]{dirLOSCor.pdf}
\caption{\textcolor{black}{Measured 73 GHz LOS spatial autocorrelation coefficients of the directional received voltage amplitudes. The angles in the legend denote the receiver antenna azimuth angle, and "b" denotes boresight to the TX.}}
\label{fig:dirLOSCor}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.0in]{dirNLOSCor.pdf}
\caption{\textcolor{black}{Measured 73 GHz NLOS spatial autocorrelation coefficients of the directional received voltage amplitudes. The angles in the legend denote the receiver antenna azimuth angle, and ``s, to TX" denotes along the direction of the street and pointing to the TX side.}}
\label{fig:dirNLOSCor}
\vspace{-2.5mm}
\end{figure}
Figs.~\ref{fig:dirLOSCor} and~\ref{fig:dirNLOSCor} illustrate the spatial autocorrelation coefficients of the received voltage amplitudes for individual antenna pointing angles in LOS and NLOS environments in downtown Brooklyn (see Fig.~\ref{fig:TX_RX_Location}), respectively. As shown by Fig.~\ref{fig:dirLOSCor}, all of the six spatial autocorrelation curves in the LOS environment exhibit sinusoidally exponential decaying trends, albeit with different oscillation patterns and decay rates. The TX-RX boresight-to-boresight pointing angle (270$^\circ$ RX pointing angle, corresponding to the strongest received power) yields the smallest oscillation since there is only a single LOS component in the PDP, while the other pointing directions contain two or more multipath components with varying phases that result in larger oscillation\textcolor{black}{\cite{Sun17a}}. The spatial decay constants for all the curves in Fig.~\ref{fig:dirLOSCor} are given in Table~\ref{tbl:ModelPara}. Compared with the omnidirectional case displayed in Fig.~\ref{fig:omniLOSCor}, it is clear that for the LOS environment, the spatial autocorrelation of both omnidirectional and directional received voltage amplitude obeys similar distribution, namely, the sinusoidal-exponential function, with similar decorrelation distances. On the other hand, most of the spatial autocorrelation curves for the directional received voltage amplitude shown in Fig.~\ref{fig:dirNLOSCor} are also in line with that given by Fig.~\ref{fig:omniNLOSCor}. One exception for the 73 GHz spatial correlation is found at 30$^\circ$ pointing angle that is the second strongest pointing direction (see Fig. \ref{fig:dirNLOSCor}), where decorrelation was much more gradual and decreases to 1/e at 25.0$\lambda$ (10.2 cm), probably due to the presence of a dominant path with a relatively constant signal level, likely caused by the diffraction of the southeast corner of Rogers Hall in Fig.~\ref{fig:TX_RX_Location}. It is clear from Fig. 4 in~\cite{Rap16a} that correlation distances vary among typical mmWave measurements~\cite{Rumney16b} due to the site-specific nature of propagation.
\section{Local Area Channel Transition}
\subsection{Introduction of Local Area Channel Transition}
Large-scale channel characteristics, such as the autocorrelation of shadow fading and delay spread over distance at the mobile, inter-site correlation of shadow fading at the mobile for two base stations or a base station for one mobile, and properties of local area channel transition, play an important role in constructing channel models for wireless communication systems~\cite{Jalden07a,3GPP.38.900,3GPP.36.814,3GPP.36.839}.
While sufficient studies on large-scale channel characteristics have been conducted at sub-6 GHz frequencies~\cite{Jalden07a,Guan15a,Zhang08a,Kolmonen10a}, similar studies have been rarely conducted at mmWave frequencies. Guan \textit{et al.} investigated spatial autocorrelation of shadow fading at 920 MHz, 2400 MHz, and 5705 MHz in curved subway tunnels~\cite{Guan15a}. Results showed that the 802.16J model was a better fit to the measured data than an exponential model, and the mean decorrelation distances were found to be several meters. Another measurement campaign carried out in urban macro-cell (UMa) environments at 2.35 GHz showed that a double exponential model fit well with the autocorrelation coefficients of shadow fading samples extracted from all measured routes, while for individual routes, an exponentially decaying sinusoid model had better fitting performance~\cite{Zhang08a}. On the other hand, an exponential function was adopted in the 3GPP channel model (Releases 9 and 11) to describe the normalized autocorrelation of shadow fading versus distances~\cite{3GPP.36.814,3GPP.36.839}. Moreover, Kolmonen \textit{et al.}~\cite{Kolmonen10a} investigated interlink correlation of eigenvectors of MIMO correlation matrices based on a multi-site measurement campaign at 5.3 GHz using a bandwidth of 100 MHz. Results showed that the first eigenvectors for both x- and y-oriented arrays were highly correlated when two RX locations were largely separated. In the following subsections, local area and channel transition measurements at 73 GHz are described and analyzed.
\subsection{Measurement System, Environment, and Procedure for Local Area Channel Transition}
The measurement system used for local area channel transition measurements was identical to the one used in the previous small-scale fading and autocorrelation measurements described in Section~\ref{sec:smallscale}, except from the antennas~\cite{Mac17a}\textcolor{black}{\cite{Mac17c}}. A 27 dBi gain (7$^\circ$ azimuth and elevation Az./El. HPBW) and 20.0 dBi gain (15$^\circ$ Az./El. HPBW) antenna were used at the TX and RX, respectively. Detailed specifications regarding the measurement system are provided in Table \ref{tbl:2}.
The local area channel transition measurements were conducted at 73 GHz in the MetroTech Commons courtyard next to 2 and 3 MetroTech Center in downtown Brooklyn. During measurements, the TX and RX antennas were set to 4.0 m and 1.5 m above ground level, respectively. For each set of \textit{cluster} or \textit{route} scenario RX locations, the TX antenna remained fixed and pointed towards a manually selected azimuth and elevation pointing angle that resulted in the strongest received power at the starting RX position (RX81 for \textit{route} measurements, and RX51 and RX61 for the LOS and NLOS \textit{cluster} measurements, respectively). For each specific TX-RX combination, five consecutive and identical azimuth sweeps ($\sim$3 minutes per sweep and $\sim$2 minutes between sweeps) were conducted at the RX in HPBW step increments ($15^\circ$) where a PDP was recorded at each RX azimuth pointing angle and resulted in at most 120 PDPs ($\frac{360}{15}\times5=120$) per combination (some angles did not have detectable signal above the noise). The best RX pointing angle in the azimuth plane was selected as the starting point for the RX azimuth sweeps (elevation remained fixed for all RX's), at each RX location measured.
For the \textit{route} measurements, 16 RX locations were measured for a fixed TX location (L8) with the RX locations positioned in 5 m adjacent increments of each other forming a simulated route in the shape of an ``L" around a building corner from a LOS to NLOS region, as provided in Fig.~\ref{fig:Map_Large_Scale_C1}. The LOS location (five: RX92 to RX96) and NLOS location (11: RX81 to RX91) T-R separation distances (Euclidean distance between TX and RX) varied from 29.6 m to 49.1 m and 50.8 m to 81.5 m, respectively. The TX antenna at L8 kept the same azimuth and elevation pointing angle of 100$^\circ$ and 0$^\circ$, respectively, during each experiment (see Fig.~\ref{fig:Map_Large_Scale_C1}. Therefore, the LOS measurements have the TX and RX antennas roughly on boresight in the LOS situation, but they are not exactly on boresight throughout the entire experiment over all measured locations. The general layout of measurements consisted of the RX location starting at RX81, approximately 54 m along an urban canyon (Bridge Street: 18 m width), with the TX antenna pointed towards the opening of the urban canyon (see Fig.~\ref{fig:Map_Large_Scale_C1}). The LOS locations were in clear view of the RX, but with some nearby minor foliage and lamppost obstructions.
For the \textit{cluster} measurements, 10 RX locations were measured for a fixed TX location (L11), with two sets of RX clusters, one in LOS (RX61 to RX65) and the other in NLOS (RX51 to RX 55). For each cluster of RX's, the adjacent distance between each RX location was 5 m, however, the path of adjacent RX locations took the shape of a semi-circle as displayed in Fig.~\ref{fig:Map_Large_Scale_C2}. The LOS cluster T-R separation distances (Euclidean distance between TX and RX) varied between 57.8 m and 70.6 m with a fixed TX antenna azimuth and elevation departure angle of 350$^\circ$ and -2$^\circ$, respectively, and fixed RX elevation angles of $+3^\circ$, to ensure rough elevation and azimuth alignment for all RX locations. For the NLOS cluster, the T-R separation distances were between 61.7 m and 73.7 m with a fixed TX antenna azimuth and elevation departure angle of 5$^\circ$ and -2$^\circ$, respectively, and fixed RX elevation angles of $+3^\circ$. The LOS cluster of RX's was located near the opening of an urban canyon near some light foliage, while the TX location was $\sim$57 m along an urban canyon (Lawrence Street: 18 m width). The NLOS cluster of RX locations was around the corner of the urban canyon opening in a courtyard area (see Fig.~\ref{fig:Map_Large_Scale_C2}), also with nearby moderate foliage and lampposts.
\begin{figure}
\begin{center}
\includegraphics [width = 0.37\textwidth]{Map_Large_Scale_C1_v1.pdf}
\caption{2D map of TX and RX locations for \textit{route} NLOS to LOS transition measurements. The yellow star is the TX location, blue dots represent LOS RX locations, and red squares indicate NLOS RX locations. $\textrm{N}=0^\circ$. }\label{fig:Map_Large_Scale_C1}
\end{center}
\vspace{-2.5mm}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics [width = 0.37\textwidth]{Map_Large_Scale_C2_v1.pdf}
\caption{2D map of TX and RX locations for \textit{cluster} measurements with LOS and NLOS RX clusters. The yellow star is the TX location, blue dots represent LOS RX locations, and red squares indicate NLOS RX locations. Pointing to the top is $0^\circ$.}\label{fig:Map_Large_Scale_C2}
\vspace{-2.5mm}
\end{center}
\vspace{-2.5mm}
\end{figure}
\subsection{Local Area Channel Transition Results and Stationarity}
The route measurements mimicked a person moving along an urban canyon from a NLOS to a LOS region, in order to understand the evolution of the channel during the transition. Fig.~\ref{fig:routePL} displays the omnidirectional path loss for each of the RX locations (RX81 to RX96) where the received power from the individual directional measurements at the RX was summed up to determine the entire omnidirectional received power at each measurement location (out of the 5 sweeps, the maximum power at each angle was used, although variation was less than a dB between sweeps)~\cite{Mac14b,Sun15a}.
\begin{figure}[t!]
\begin{center}
\includegraphics [width = 0.37\textwidth]{NLOS_LOS_Transition_PL_box_embed-eps-converted-to.pdf}
\caption{Omnidirectional path loss for route measurements for an RX transitioning from a NLOS to a LOS region.}\label{fig:routePL}
\vspace{-2.5mm}
\end{center}
\vspace{-2.5mm}
\end{figure}
The transition from LOS to NLOS in Fig.~\ref{fig:routePL} is quite abrupt, where path loss increases by $\sim$8 dB from RX92 to RX91, similar to the abrupt diffraction loss noticed in Section~\ref{sec:diffraction}. Path loss then increases 9 dB further from RX91 to RX90, 1 dB from RX90 to RX89, 6 dB from RX 89 to RX88, and 1 dB from RX88 to RX 87. This observation shows a large initial drop in 8 dB at the LOS to NLOS transition region, but an overall 25 dB drop in signal power when moving from LOS conditions to deeply shadowed NLOS conditions approximately 25 meters farther along a perpendicular urban canyon ($\sim$10 m increase in Euclidean T-R separation distance), when using an omnidirectional RX antenna. The 25 dB drop in signal strength over a 25 m path around a corner (1 dB/m) is important for handoff considerations. The signal fading rate is 35 dB/s for vehicle speeds of 35 m/s, or 1 dB/s for walking speeds of 1 m/s. This motivates the use of beam scanning and phased array technologies in the handset for urban mobile mmWave communications that will search for and find the strongest signal paths~\cite{Sun14b}, and future work will study the best antenna pointing angles at each location from these measurements.
\begin{figure}
\centering
\begin{subfigure}[b]{0.39\textwidth}
\centering
\includegraphics[trim={0 0.5cm 0 0},clip,width=\textwidth]{L8_RX87_RXsweep_Meas3_PolarPlot_embed.pdf}
\caption[]%
{{\small RX87: NLOS}}
\label{fig:RX87}
\end{subfigure}
\begin{subfigure}[b]{0.39\textwidth}
\centering
\includegraphics[trim={0 0.5cm 0 0},clip,width=\textwidth]{L8_RX92_RXsweep_Meas5_PolarPlot_embed.pdf}
\caption[]%
{{\small RX92: LOS}}
\label{fig:RX92}
\end{subfigure}
\caption{\small Route scenario polar plots of RX azimuth spectra for RX 87 (NLOS location) and RX 92 (LOS location) that show the evolution of AOA energy around a corner.}
\label{fig:case1Polar}
\vspace{-2.5mm}
\end{figure}
\textcolor{black}{Azimuth power spectra are useful to study how the arriving energy changes as an RX moves from NLOS to LOS. Work in~\cite{Samimi16a} showed that energy arrives in directional lobes in mmWave channels. Fig.~\ref{fig:case1Polar} displays an RX polar plot from RX87 (NLOS location) and RX92 (LOS location), where the TX was pointed in the 100$^\circ$ direction towards the street opening (see Fig.~\ref{fig:Map_Large_Scale_C1}). Fig.~\ref{fig:RX87} shows the power azimuth spectra at RX87 ($\sim$25 meters down the urban street canyon) where energy from the TX waveguides and reflects down Bridge Street such that there is one main broad lobe at the RX oriented in the 0$^\circ$ direction and a small narrow lobe in the 180$^\circ$ direction from weak reflectors and scattering. The large azimuth spread in the main lobe demonstrates the surprisingly reflective nature of the channel at the 73 GHz mmWave band~\cite{Samimi13a,Samimi16a}.}
\textcolor{black}{Fig.~\ref{fig:RX92} displays the power azimuth spectra at RX92 in LOS with a strong central lobe coming from the direction of the TX (285$^\circ$). A relatively strong secondary lobe (100$^\circ$) is also apparent in Fig.~\ref{fig:RX92} with energy contributions from reflections off of the building to the east of RX92 and additional reflectors and scatterers from nearby lampposts and signs.} Table~\ref{tbl:clusterSTD} displays the standard deviations of the omnidirectional received power values measured along the LOS and NLOS routes shown in Fig.~\ref{fig:Map_Large_Scale_C1}. The omnidirectional received power standard deviation of 1.2 dB is generally small for the LOS locations but is much larger in NLOS (7.9 dB) due to substantial scattering along the route of RX locations along the urban canyon where path loss tends to increase non-linearly over log-distance.
\begin{table}
\centering
\caption{Omnidirectional received power standard deviation for the large-scale route and cluster scenario measurements.}
\label{tbl:clusterSTD}
\begin{center}
\scalebox{0.78}{
\begin{tabu}{|c|[1.6pt]c|}
\hline
\textbf{Measurement Set} & \textbf{Omnidirectional Received Power \bm{$\sigma$} [dB]} \\ \specialrule{1.5pt}{0pt}{0pt}
\textit{Route} - LOS: RX92 to RX96 & 1.2 \\ \hline
\textit{Route} - NLOS: RX81 to RX91 & 7.9 \\ \hline
\textit{Cluster} - LOS: RX61 to RX65 & 4.3 \\ \hline
\textit{Cluster} - NLOS: RX51 to RX55 & 2.2 \\ \hline
\end{tabu}}
\end{center}
\vspace{-2.5mm}
\end{table}
The \textit{cluster} measurements for the TX at L11 were designed to understand the stationarity of received power in a local area (larger than small-scale distances) on the order of many hundreds to thousands of wavelengths (5 to 10 meters) at mmWave. In LOS, the cluster of five RX locations with a fixed directional TX antenna resulted in an omnidirectional received power standard deviation of 4.3 dB over local area of 5 m x 10 m, a relatively small variation, indicating a reasonably stationary average received power for a local set of RX locations in LOS at 73 GHz. The NLOS cluster resulted in an even lower omnidirectional received power standard deviation of 2.2 dB, over a 5 m x 10 m local area. The small fluctuation in received power over the local area of the LOS and NLOS clusters implies that received power does not significantly vary over RX locations separated by even a few to several meters in a dense urban environment at mmWave. As an aside, the directional CI model path loss exponent (PLE) using a 1 m free space reference in the \textit{route} measurements was 2.53 in LOS (a bit higher than free space due to elevation mismatch over the route) and 3.61 in NLOS (for a single TX beam)~\cite{Mac17a}. Recent work in~\cite{Wang17a} at 28 GHz studied the stationarity of wideband mmWave channels, and reported smaller stationary regions than at 2 GHz.
\section{Conclusion and Discussion}~\label{sec:conc}
Measurements and analysis were presented on diffraction, human blockage, small-scale fading and local area channel transition at mmWave frequencies. We showed the KED model is suitable for modeling diffraction at cmWave and mmWave bands in indoor scenarios, while a creeping wave linear model is applicable to outdoor environments. \textcolor{black}{Indoor and outdoor diffraction measurements confirm theoretical simulations~\cite{Deng16a} that mmWave diffraction will not be a dominant propagation mechanism. Measurements show significant attenuation of 30 dB in indoor channels and 40 dB or more in outdoor channels in the deep shadow region. Therefore, systems will need to be designed to overcome extremely large fades when a signal path is blocked and when diffraction is no longer viable. MmWave antenna systems at the TX and RX will need to cooperatively search for secondary paths at different pointing directions. The large fades due to diffraction loss have implications on the design of physical layer protocols and frame structures to maintain a link, while finding other spatial paths.}
Human blockage measurements showed a person can induce more than 40 dB of loss when standing 0.5 m from the TX or RX antenna, with a signal decay rate of 0.4 dB/ms at walking speeds. The DKED-AG model given here incorporates directional antenna patterns to accurately predict the upper and lower envelopes of measured received power during a blockage \textcolor{black}{which better agrees with real-world measurements when compared to the 3GPP/METIS blockage model (that was shown to underestimate human blockage, most severely when close to the TX or RX)}. \textcolor{black}{Scintillation with $\sim$2 dB peak-to-peak amplitudes} can be seen in the measurements and model just before the blocker enters the field of view of the RX, suggesting that deep fades may be predictable just before they occur. This information may be used in the design of beam steering or handoff algorithms.
\textcolor{black}{Small-scale spatial fading statistics of received signal voltage amplitudes of 1 GHz bandwidth signals at 73 GHz were measured over a 35.31-cm ($\sim$ 87 wavelengths) linear track and show only small power variations when such a wide bandwidth is used. Fading in LOS locations for omnidirectional RX antennas obeyed the Ricean distribution with a $K$-factor of 10 dB, while fading in NLOS locations can be described by the log-normal distribution with a standard deviation of 0.65 dB. The fading depth ranges from -3 dB to +1.5 dB relative to the mean for LOS, and -0.8 dB to +0.8 dB for NLOS. For a 60$^\circ$ directional RX antenna, fading in both LOS and NLOS environments follows the Ricean distribution, where the $K$-factor ranges from 7 dB to 17 dB for LOS, and 9 dB to 21 dB for NLOS, depending on the RX orientation in relation to the environment and the TX~\cite{Sun17a}, while the fading depth varies between -4 dB to +2 dB relative to the mean for both LOS and NLOS environments.}
\textcolor{black}{Spatial autocorrelation modeling of instantaneous total received signal voltage amplitudes showed that the sinusoidal-exponential distribution fits measurements in the LOS environment for both omnidirectional and directional RX antennas. In the NLOS environment, the spatial autocorrelation can be modeled by the exponential distribution for both omnidirectional and directional RX antennas.} Table~\ref{tbl:ModelPara} shows the oscillation distance/period and spatial decay constant to represent the autocorrelation, where rapid decorrelation of received voltage amplitudes occurred over 0.67 to 33.3 wavelengths (0.27 cm to 13.6 cm), depending on the RX orientation in relation to the environment and the TX~\cite{Sun17a}. \textcolor{black}{The short correlation distance, in general, is favorable for spatial multiplexing in MIMO since it allows for uncorrelated spatial data streams to be transmitted from closely-spaced (a fraction to several wavelengths) antennas~\cite{Sun14b}. Furthermore, the small fading depth indicates that the signal quality of an established link between a TX and an RX will not vary much when the RX moves within a local area on the order of a few tens of wavelengths (within 13.6 cm), as shown in Figs.~\ref{fig:dirLOSCor} and~\ref{fig:dirNLOSCor}.}
Local area channel transition measurements show an initial 8 dB power loss and an overall 25 dB power loss when an RX moves from LOS to NLOS around a building corner and along a street canyon in an urban microcell environment. However, local area path loss measurements (cluster measurements) suggest omnidirectional received power has a relatively stationary mean over a relatively large area (5 m x 10 m) in LOS and NLOS scenarios, respectively, as indicated in Table~\ref{tbl:clusterSTD}. Wideband mmWave signals in typical indoor and outdoor environments were shown to fade at rates from 0.4 dB/ms to 40 dB/s, depending on the speed and environment. The results presented here will aid the 5G wireless community as it develops models for small-scale fading and spatial consistency for handoff algorithms and beam scanning techniques.
\bibliographystyle{IEEEtran}
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{
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\section{}
A puzzling feature of QFT concerns the crucial analogy with
classical statistical mechanics where the inverse temperature plays
the r\^ole of imaginary time. Time and temperature also mix by
performing analytic continuation along complex paths in the
path-integral. Another well-known analogy concerns the transition
amplitude for a particle for the time $it$ that coincides with the
classical partition function for a string of length $t$ at
$\beta=1/\hbar$. Even if such analogies follow as technical
properties, it is widely believed that they are deeply related to
the properties of space-time and should emerge in the string
context.
A step towards a better understanding of temperature/time and
classical/quantum dualities, would be finding a physical model where
these emerge naturally. In proposing such a model, we will start by
showing that string theory possesses a basic surprising
classical/quantum duality. It turns out that the {\it solitonic
sector} of toroidally compactified strings have a {\it dual} description as
{\it quantum statistical partition function} on higher dimensional
spaces, built in terms of the Jacobian torus of the string
worldsheet and of the compactified space. More precisely, in this
letter we will see that in the case of compactification on a circle
\begin{equation}\label{laprima}
\sum_{m,n\in\mathbb{Z}^g}e^{-\beta S_{m,n}}=\mathop{\rm Tr}\, e^{-\beta
H}\ ,\end{equation} where $\beta=2R^2/\alpha'$, with $R$ the compactification
radius, and the Hamiltonian $H$ is $\Delta_{J_\Omega}/2\pi$, with
$\Delta_{J_\Omega}$ the Laplacian on the Jacobian torus $J_\Omega$
of the worldsheet. Eq.(\ref{laprima}) is just the direct consequence of
the stronger identity we will prove. Namely \begin{equation}\label{laprimab}H
\Psi_{m,n}=S_{m,n}\Psi_{m,n}\ ,\end{equation} that is the set $\{
S_{m,n}|m,n\in\mathbb{Z}^g\}$ {\it coincides} with the spectrum of $H$. As
will become clear from the construction, such a result seems to
capture a general property of string theories which is due to the
underlying geometrical properties of Riemann surfaces. In particular,
the correspondence considered in this letter admits natural generalizations to
compactifications on higher dimensional tori.
Remarkably, temperature/time duality naturally emerges as a
consequence of the complexified version of $T$-duality, a
fundamental feature of string theory
\cite{GiveonFU}. By (\ref{laprima}) the standard
$T$-duality corresponds to the invariance, up to a multiplicative
term given by powers of $\beta$, of the partition under inversion of
the temperature \begin{equation}\label{zero} \beta\longrightarrow 1/\beta \ .\end{equation}
Complexification of $\beta$ has basic motivations which interplay
between physics and geometry. Let us first consider the time
translate of an observable $A$,
$\alpha_t(A)=e^{iHt} A e^{-iHt}$,
and the expectation value
$$
\omega_\beta(A)=\mathop{\rm Tr}\, \rho_\beta A=Z_{stat}(\beta)^{-1}\mathop{\rm Tr}\, A
e^{-\beta H} \ .
$$ By invariance of the
trace under cyclic permutations we get
$\omega_\beta((\alpha_t(A)B)=\omega_\beta(B e^{iH(t+i\beta)} A
e^{-iH(t+i\beta)})$, so that
$$
\omega_\beta((\alpha_t A)B)=\omega_\beta(B\alpha_{t+i\beta}(A))\ ,
$$
that is time evolution is invariant, upon commutation, under an imaginary shift of the
time which is inversely proportional to the temperature. In such a
context the complexification of $\beta$ naturally appears in
globally conformal invariant QFT \cite{NikolovYG}. It is worth noticing that both in
\cite{NikolovYG} and in the BC
system \cite{BC}
the KMS (Kubo-Martin-Schwinger) states \cite{Haag,CMuno,CMdue} play a crucial r\^ole. In
particular, in the limit of
$0$-temperature the KMS states may be used to define the concept of
point in noncommutative space.
A simple but important property of the temperature is its positive
definiteness. This becomes transparent once time and temperature are
naturally combined in a unique variable $\tau$ as suggested by the
above analogies, namely \begin{equation}\label{uno}\beta={1\over k_B
T}=\mathop{\rm Im} \tau \ , \qquad {t\over \hbar}=-\mathop{\rm Re}\tau \ .\end{equation} As
we will discuss later, complexification of $\beta$ is intimately
related to the positivity of the temperature. Complexifying a
variable taking positive values has a geometrical motivation that
may lead to a geometrical understanding of the emergence of time. In
particular, as will be illustrated, positivity of the temperature
combined with the time variable to build the complex $\tau$, leads
to consider $\tau$ as the torus modular parameter.
An apparently unrelated topic concerns the Riemann mapping theorem,
which plays a central r\^ole in uniformization theory and in the
theory of univalent functions. According to such a theorem there is
a unique analytic function $w=f(z)$ mapping a simply connected
region of $\mathbb{C}$ one-to-one onto the disk $|w|<1$ such that
$f(z_0)=0$ and $f'(z_0)>0$. As a consequence any two simply
connected regions except $\mathbb{C}$ itself, can be mapped conformally
onto each other. While $\mathbb{C}$ is the universal covering of the torus,
any simply connected domain of $\mathbb{C}$ can be mapped by a locally
univalent function to the upper half-plane, the universal covering
of negatively curved Riemann surfaces. Positivity of the temperature
and its combination with time to build $\tau$ leads to the geometry
of the upper half-plane that, unlike the case of the Riemann sphere,
is characterized by the identification of its automorphism group,
$PSL(2,\mathbb{R})$, with the isometry group of its natural metric, the
Poincar\'e metric. Such a group acts by linear fractional
transformations
$
\tau\rightarrow (A\tau+B)/(C\tau+D)
$
that, according to (\ref{uno}), correspond to mixing the r\^ole of
temperature and time.
Similar transformations appear in several physical models, for
example in Seiberg-Witten theory. Again, positivity, that this time
is due to the coupling constant, plays a crucial r\^ole.
Geometrically $\tau$ can be seen as the modular parameter of a
torus, and then related to a flat geometry. On the other hand, it
can be seen as the inverse of the uniformizing map from a negatively
curved Riemann surface to the upper half-plane. In this way to each
point on a negatively curved Riemann surface one may associate an
elliptic curve. It should be stressed that negatively curved Riemann
surfaces are much more reach in nature and it may be convenient to
consider punctured versions of the torus rather than the torus
itself. Such a dual r\^ole for $\tau$, connecting flat and
negatively curved Riemann surfaces, is particularly transparent once
one considers the $\zeta$-function
$$
\zeta(s)=\sum_{m,n\in \mathbb{Z}}\frac{(\mathop{\rm Im}\tau)^s}{|m+n\tau|^{2s}}
\ .
$$
The elements of the set $\{|m+n\tau|^{2}/\mathop{\rm Im}\tau\mid m,n\in\mathbb{Z}\}$
are the eigenvalues of the Laplacian on the torus, so that
$e^{-\zeta(0)'}$ defines the determinant of the Laplacian on the
torus. On the other hand, $(\mathop{\rm Im}\tau)^s$ can be seen as eigenfunction
of the Poincar\'e Laplacian. The summation then can be seen as a sum
on all $PSL(2,\mathbb{Z})$ transformations of $({\rm Im}\,\tau)^s$
guaranteeing the invariance under the uniformizing group
$SL(2,\mathbb{Z})$. This dual nature of $\tau$ is at the heart of basic
mathematical structures also involving number theory, and appears in
several topics of physical interest, not only in string theory.
Let $\Sigma$ be a genus $g$ Riemann surface with a fixed basis
$\{\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g\}$ of the first
homology group $H_1(\Sigma,\mathbb{Z})$ with intersection matrix
$\alpha_i\cdot\beta_j=\delta_{ij}$,
$\alpha_i\cdot\alpha_j=\beta_i\cdot\beta_j=0$. A dual basis
$\{\omega_1,\ldots,\omega_g\}$, of holomorphic $1$-differentials on
$\Sigma$ can be chosen with normalization
$\int_{\alpha_i}\omega_j=\delta_{ij}$, $i,j=1,\ldots,g$, whereas the Riemann period matrix
$\Omega_{ij}=\int_{\beta_i}\omega_j$ can be proved to be symmetric with $\mathop{\rm Im}\Omega>0$, and
depends on the complex structure of $\Sigma$. Conversely, the
Riemann period matrix completely determines the complex structure of
the corresponding Riemann surface (although for $g\ge 4$ not every
symmetric matrix with positive-definite imaginary part is the
period matrix of a Riemann surface).
Let us consider a scalar field theory on $\Sigma$ with the
one-dimensional target space compactified to a circle
$\mathbb{S}^1=\mathbb{R}/2\pi R\mathbb{Z}$ of radius $R$. The partition function is
defined as a path-integral
$$Z(\beta)=\int_{(\Sigma,\mathbb{S}^1)} dX e^{-\beta S}\ ,$$
where $(\Sigma,\mathbb{S}^1)$ is the space of
maps from $\Sigma$ to $\mathbb{S}^1$ and \begin{equation}\label{stringact}S[X]=\frac{1}{4\pi R^2}\int_\Sigma
\partial X\bar{\partial} X\ ,\end{equation} with the normalization chosen for later reference. By setting
$\beta=2R^2/\alpha'$, this path-integral corresponds to the $g$-loop
contribution of a string theory with target space $\mathbb{S}^1$. Each
function $X:\Sigma\to\mathbb{S}^1$ satisfies the condition
$$X(z+p^t\alpha+q^t\beta)=X(z)+2\pi R(m^t p-n^t q)\ ,$$
$p,q\in \mathbb{Z}^g$,
where the winding numbers $m,n\in\mathbb{Z}^g$ label the different
solitonic sectors of $(\Sigma,\mathbb{S}^1)$. Let us split $X$ into
classical and quantum parts
$X=X^{cl}+X^{q}$, where $X^{cl}$ satisfies the classical equation of motion
$$\Delta X^{cl}=0\ ,$$
so that $X^{cl}$ is a harmonic function on $\Sigma$. It is worth noticing that there exists a unique
harmonic function for each solitonic sector, i.e. for each pair $(m,n)\in\mathbb{Z}^{2g}$. Therefore, the splitting can
be performed in such a way that the quantum contribution $X_q$ is a real
single-valued function on $\Sigma$, that is
$X_q(z+p^t\alpha+q^t\beta)=X_q(z)$. It follows that the path integral splits into a sum over
the inequivalent sectors labeled by solitonic numbers
$(m,n)\in\mathbb{Z}^{2g}$ times the functional integral over the space of
real single-valued functions on $\Sigma$
$$Z(\beta)=\sum_{m,n\in\mathbb{Z}^g}e^{-\beta S_{m,n}}\int_{(\Sigma,\mathbb{R})} dXe^{-\beta S}\ ,$$
whereas the mixed terms with classical times quantum part vanish.
The harmonic function in the sector $(m,n)\in\mathbb{Z}^{2g}$ is
$$X^{cl}_{m,n}(z,\bar z)=\frac{\pi R}{i}(m+\bar\Omega n)^t(\mathop{\rm Im}\Omega)^{-1}\int^z\omega+c.c.\ ,$$
so that, by using the Riemann bilinear relations
$$\int_\Sigma\omega_i\wedge\bar\omega_j=-2i\mathop{\rm Im}\Omega_{ij}\ ,$$ we obtain
$$S_{m,n}=S[X^{cl}_{m,n}]=\pi (m+\bar\Omega
n)^t(\mathop{\rm Im}\Omega)^{-1}(m+\Omega n)\ .$$ Therefore, the partition
function is $Z=Z_{sol}Z_q$, where
\begin{eqnarray}\label{quant}&&Z_{sol}(\beta)=\sum_{m,n\in\mathbb{Z}^g}e^{-\beta\pi
(m+\bar\Omega n)^t(\mathop{\rm Im}\Omega)^{-1}(m+\Omega
n)}\ ,\\
&&Z_q(\beta)=\left(\frac{A_\Sigma}{\det'(-\beta\Delta)}\right)^{1/2}\ ,\nonumber\end{eqnarray}
and $
A_\Sigma=\int_{\Sigma}\sqrt g$.
On the other hand, let us consider the quantum statistical partition
function at the temperature $T$ on a $g$-dimensional complex torus
$J_\Omega=\mathbb{C}^g/(\mathbb{Z}^g+\Omega\mathbb{Z}^g)$, for some symmetric $g\times g$
matrix $\Omega$, with $\mathop{\rm Im}\Omega>0$,
$$Z_{stat}(\beta)=\mathop{\rm Tr}\, e^{-\beta H}\ ,$$
where $\beta=1/k_BT$, $H\equiv\Delta_{J_\Omega}/2\pi$ and
$$\Delta_{J_\Omega}=-2\mathop{\rm Im}\Omega_{ij}\frac{\partial}{\partial z_i}\frac{\partial}{\partial \bar z_j}\ ,$$
is the Laplacian on $J_\Omega$ with respect to the natural metric
$ds^2=(2\mathop{\rm Im}\Omega)^{-1}_{ij}dz^id\bar z^j$. Here and in the
following, $\beta$ and $H$ are rescaled by some fixed length $L$ and
thought of as dimensionless quantities. A complete orthogonal basis
of eigenfunctions for $H$ is $\{\Psi_{m,n}\}_{m,n\in\mathbb{Z}^g}$, with
$$\Psi_{m,n}(z,\bar z)=e^{\pi(m+\bar\Omega n)^t(\mathop{\rm Im}\Omega)^{-1}z-c.c.}\ .$$
Indeed, a trivial computation shows that
$$H\Psi_{m,n}=\lambda_{m,n}\Psi_{m,n}\ ,$$
with eigenvalues
$$\lambda_{m,n}=\pi(m+\bar\Omega n)^t(\mathop{\rm Im}\Omega)^{-1}(m+\Omega n)\ ,$$
so that $S_{m,n}=\lambda_{m,n}$ and
\begin{equation}\label{stat}Z_{stat}(\beta)=\sum_{m,n\in\mathbb{Z}^g}e^{-\beta\pi(m+\bar\Omega
n)^t(\mathop{\rm Im}\Omega)^{-1}(m+\Omega n)}\ .\end{equation}
Comparing Eqs.(\ref{quant}) and (\ref{stat}), we obtain the remarkable identity
Eq.(\ref{laprima}) between the classical contribution to the partition
function in a $2$-dimensional field theory and the quantum
statistical partition function of a free particle in a
$2g$-dimensional space. Note that the space in the statistical
theory is exactly the Jacobian torus of the Riemann surface where
the first theory is defined. In particular, in the case of genus
$1$, $\Sigma$ coincides with its Jacobian and we obtain a duality on
the same space.
By applying the Poisson summation formula
$$\sum_{m\in\mathbb{Z}^d}e^{-\pi (m+a)^tA(m+a)+2\pi im^tb}=(\det A)^{-\frac{1}{2}}\sum_{m\in\mathbb{Z}^d}e^{-\pi
(m+b)^tA^{-1}(m+b)-2\pi i(m+b)^ta}\ ,$$
to the sum over $m$ in
$Z_{sol}(\beta)$, one obtains
$$Z_{sol}(\beta)
=\Big(\frac{\det\mathop{\rm Im}\Omega}{\beta^g}\Big)^{1/2}\sum_{m,n\in\mathbb{Z}^g}e^{\pi[-\beta^{-1}
m^t(\mathop{\rm Im}\Omega) m-\beta n^t(\mathop{\rm Im}\Omega)n+2 im^t(\mathop{\rm Re}\Omega) n]}=
\beta^{-g}Z_{sol}(1/\beta)\ ,$$
where the correspondence $\beta\to
1/\beta$ implies $R\to R'=\alpha'/2R$, which is the standard
$T$-duality for string theory on a circle. In the case of $Z_{stat}(\beta)$, the same calculation leads to
$$Z_{stat}(\beta)=\beta^{-g}Z_{stat}(1/\beta)\ ,$$
which has to be interpreted as a hot-cold duality $T\to
T'=T^2_{sd}/T$, where $T_{sd}$ is the self-dual temperature $T_{sd}=\hbar c/k_BL$. Note that, by setting
$L\sim(\alpha'/2)^{1/2}$, a correspondence can be established between the energy scales in the string
theory model and in the quantum statistical one, relating the fixed points (self-dual radius and
temperature, respectively) under the duality $\beta\to 1/\beta$.
>From the point of view of
statistical mechanics, it is natural to consider the complexification
\begin{equation}\label{taubeta}\tau=-t+i\beta\ ,\end{equation} where $t$ denotes the time,
so that we can define
$$Z_{stat}(\tau)=\mathop{\rm Tr}\,e^{i\tau H}\ .$$
By Poisson summation, we obtain the duality
$$Z_{stat}(\tau)= (-i\tau)^{-g} Z_{stat}(-1/\tau)\ ,$$
that mixes time and temperature, namely
$$
t\rightarrow \frac{-L^2}{c^2[t^2+(\hbar/k_BT)^2]}t\ , \quad
\frac{1}{T}\rightarrow\frac{L^2}{c^2[t^2+(\hbar/k_BT)^2]}\frac{1}{T} \ ,
$$
where the dimensional parameters have been recovered.
We have seen that the classical sector of the compactified string on
$\mathbb{S}^1$ has a quantum statistical description on the Jacobian
torus. This corresponds, like the original string, to a first
quantized theory. Also note that restricting the eigenfunctions of
$H$ to the image of $\Sigma$ in $J_\Omega$ provides a direct link
between $\Psi_{m,n}$ and $X^{cl}_{m,n}$, namely
\begin{equation}\label{vertex}\Psi_{m,n}(\textstyle{\int^z\omega,\overline{\int^z\omega}})=e^{\frac{i}{R}X^{cl}_{m,n}}
\ ,\end{equation} resembling a classical vertex. This may
indicate that the dual description of the classical contribution is
the facet of a more general dual description of the full string. For
example, the fact that the Laplacian acting on sections of the torus
and the one acting on sections of its Jacobian coincide, implies a
functional relation between the classical and quantum sectors of the
string (coming from the heat equation). This suggests that even the
quantum sector of the string admits a dual description which may
extend to higher genus as well. Understanding such an extension
means to investigate the possible relations, for any $g$, between
the spectrum of $\Delta_{J_\Omega}$ and the one of $\Delta$; a
problem that, as we will comment below, is of considerable
mathematical interest. For the time being we note that, due to the
appearance of $({\rm Im}\,\Omega)^{-1}_{ij}$, a distinguished r\^ole
may be played by the Bergman metric
$$
ds^2=\frac{1}{g}\sum_{i,j=1}^g\omega_i(z)({\rm
Im}\,\Omega)^{-1}_{ij}\overline{\omega_j(z)} \ .
$$
We note that the equality (\ref{laprima}) between the sum over the
topologically non-trivial states of a 2-dimensional sigma model and
the trace of operators over the Hilbert space of a free particle on
the Jacobian torus, is reminiscent of electric-magnetic duality in
$\mathcal{N}=2$ SYM theory. In the latter, the solitonic objects are
the 't Hooft-Polyakov monopoles and Julia-Zee dyons, and the
elementary objects are gluons. In our model Eq.(\ref{taubeta}) can be
thought of as the complex coupling constant. Its analog in SYM
theory is $\tau = \frac{\Theta}{2\pi}+\frac{4\pi i}{g^2}$. However, while electric-magnetic duality is between weakly coupled
and strongly coupled regimes of two different theories (or of the
same theory, if the theory is self-dual), in our model the duality
is established between the same regimes of different theories. The
way out is to note that by the composition of Eq.(\ref{laprima}) and
$T$-duality (\ref{zero}), precisely reproduces what we expect for an analog
of the electric-magnetic duality.
Let us further comment the complex combination of time and temperature. For the quantum
theory to be unitary the action must be real. The imaginary part of
the action leads to non-conservation of probability, as it is clear
from the form of the Feynman weight $e^{\frac{i}{\hbar}S}$. The
probability grows with the entropy, so the imaginary part of the
action should be negative. Moreover, its presence leads to
decreasing of the information, and therefore to increasing of the
entropy. A possible interpretation is to consider the imaginary part of the
action as the entropy: $S = \mathcal{A} - i \mathcal{E}$. If we write
the action $\mathcal{A}$ in the momentum space as a function of momentum
and energy $E$ (which can be achieved after taking its Legendre
transform) and use $\partial \mathcal{A}/\partial E = t$, we
obtain
$$
\partial S/\partial E = t - i\beta = -\tau\ .
$$
This suggests that the combination of time and temperature at hands is the variable conjugated to
energy in the case when the action is complex.
Let us conclude by observing that an intriguing outcome is that in
the case of $g=1$, the Jacobian corresponds to the torus itself. In
this way the quantum and solitonic contributions both are expressed
in terms of the same Laplacian. An old problem in the theory of
Riemann surfaces is to find an analytic expression for the
determinant of Laplacian acting on degree zero bundles in which the
dependence on the period Riemann matrix appears explicitly. In
\cite{MatoneTJ} it was conjectured a relation between the determinant of the
Laplacian on the Jacobian and the one on the Riemann surface. This
would provide a functional relation between $Z_{stat}(\beta)$ and
the quantum contribution to the string partition function also in
the higher genus case, generalizing the relation for $g=1$. The
eigenvalues $\lambda_{m,n}$ also appear in considering the Laplacian
with respect to degenerate metrics \cite{MatoneTJ}, for which ramified
covering of the torus play a crucial r\^ole \cite{MatoneUY}. Ramified covering of the torus
correspond to a particular kind of
CM (complex multiplication) satisfied by the Riemann period matrix
\cite{MatoneUY}. Remarkably, such special Riemann surfaces also appear in
the null compactification of type-IIA string perturbation theory at
finite temperature \cite{GrignaniZM}. It is worth noticing that CM, which is a lattice condition, also
appears in the study of sigma models on Calabi-Yau manifolds
\cite{GukovNW}.
We note that the dimensionality of our model suggests an intriguing
relation with the string theory compactified on a Riemann surface of
unitary volume in string units, where the effective degrees of
freedom are
still the ones of a $2g$-dimensional theory
\cite{SilversteinQF}. Presumably this is connected to the relation between Fuchsian
groups and Liouville theory and to the fact that, in particular
regimes, the commutators between the Fuchsian generators may be
negligible so leading to a homological description.
\begin{acknowledgments}
Work partially supported by the
European Community's Human Potential Programme under contract
MRTN-CT-2004-005104 ``Constituents, Fundamental Forces and
Symmetries of the Universe".
\end{acknowledgments}
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Q: Fill an ArrayList with object from response.body I'm using retrofit to get list of data from api. I want to save the data which I get in body.response to arrayList. I can do that and I get the items from new arrayList, but it seems that the new list are not saved. When I try to use it outside of method onResponse, I get an error
Unable to start activity... IndexOutOfBoundsException: Index 0, Size 0
The interface
public interface ServerApi {
@Headers("Accept: application/json")
@GET("cities")
Call<ArrayList<City>> getCities();
}
The activity
public class SomeActivity extends AppCompatActivity {
Retrofit retrofit = new Retrofit.Builder()
.baseUrl("http://url")
.addConverterFactory(GsonConverterFactory.create())
.build();
ServerApi serverApi = retrofit.create(ServerApi.class);
ArrayList<City> cityList = new ArrayList<City>();
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_parkings);
final TextView tv1 = findViewById(R.id.tv1);
final Call<ArrayList<City>> cities = serverApi.getCities();
cities.enqueue(new Callback<ArrayList<City>>() {
@Override
public void onResponse(Call<ArrayList<City>> call, Response<ArrayList<City>> response) {
for(int i=0; i<response.body().size(); i++){
City ct = new City(response.body().get(i).getId(), response.body().get(i).getName());
cityList.add(ct);
tv1.setText(cityList.get(i).getName());
}
}
@Override
public void onFailure(Call<ArrayList<City>> call, Throwable t) {
tv1.setText("failure " + t);
}
});
TextView c1 = findViewById(R.id.c1);
c1.setText(cityList.get(0).getName());
}
}
The string tv1.setText(cityList.get(i).getName()); inside the method onResponse works well. But when I write it out of the method, the aplication fall
A: As it can be seen you run your code in a background thread and a few lines below you try to update your textview on the UI thread.
The background operation will take some time, therefore, by the time you try to update the TextView the cityList is empty.
Consider reading a bit more here: Consuming apis with retrofit
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{
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}
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\section{Introduction}
\label{sec:introduction}
Cosmological parameters have
been significantly constrained by the combined analyses of
the data mainly from Cosmic Microwave Background (CMB) radiation,
large-scale galaxy distribution, supernova observations and
cosmic shear field [see {\it e.g.} \citep{teg04}].
Here, we demonstrate that analysis of the peculiar velocity field
can tighten the constraints on cosmological models because error ellipses
are strongly skewed
to those of {\small \emph{WMAP}} and {\small SDSS} in
the parameter space of density and age.
Two complementary theoretical methods
are used in this paper to generate
orbits that give velocities to be compared with observations.
Peculiar velocities are defined
by a large number of quality distance measures. Good distances are
available for 1400 galaxies within 3,000~km~s$^{-1}$.
The two variational methods used here are based on the principle that
galaxies as mass-tracers follow
orbits which are the stationary points of the Euler-Lagrange action.
Our first approach, the {\it Least Action} ({\small LA}) method \citep{pee89},
describes orbits in the nonlinear collapse regime,
though recovered trajectories are non-unique due to lack of knowledge
of the initial positions of
galaxies and because of orbit-crossing ({\it i.e.} multisteaming).
On large scales, a unique
reconstruction is possible since the displacement of dark matter fluid
derives from a potential and multistreaming is less severe.
These properties are at the basis of our second method, {\small MAK}
\citep{fri02}. The methods are applied at the Local Supercluster scale for the
determination of the cosmological mean density and at the cluster scale
for the specification of its local variations.
Allowing for statistical and systematic errors and considering all
the methods
in tandem, we show that the {\small MAK}, {\small LA}, {\small \emph{WMAP}}
and {\small SDSS} results
give preference to a
small region in parameter space favouring a low value of
$\Omega_m=0.22\pm 0.02$ for the density parameter.
On cluster scales,
reconstruction of the amplitude of the first-approach
infall velocities provides a determination of the mass of the
clusters at radii beyond the virialized region; radii greater than those
explored by X-ray or lensing studies.
We show that the high amplitude observed in the infall region
requires an assignment of
$M/L=500 M_{\odot}/L_{\odot}$ to the Virgo Cluster
(where the luminosities are in the B band).
\section{Catalog of Nearby Galaxies and distances}
\label{sec:catalogs}
The catalog of galaxies, is a
40\% augmentation of the Nearby
Galaxies Catalog \citep{tul87}, now including 3300 galaxies within
3,000~km~s$^{-1}$.
This depth is more
than twice the distance of the dominant component, the Virgo Cluster,
and the completion
to this depth in the current catalog compares favorably with
other all-sky surveys
[{\it e.g.}, 2MASS \citep{jar04}].
The catalog has the following properties.
(i) All entries are given a detailed
group and filament assignment (or non-assignment if isolated).
(ii) The catalog is supplemented by an all-sky complete sample of
X-ray selected clusters \citep{koc04} in the shell 3,000--8,000 km~s$^{-1}$ to provide
a description of potential influences on very large scales.
(iii) `Fake' galaxies are added at low Galactic latitudes to avoid an
underdensity in the zone of obscuration due to lost information.
(iv) Correction is made for the loss of light with distance caused by
an apparent magnitude cutoff in the construction of the catalog.
The second observational component is an extended catalog of galaxy
distances. Information from four techniques has been
integrated: the Cepheid variable \citep{fre01}, Tip of the Red Giant
Branch \citep{kar03,lee93}, Surface Brightness Fluctuation \citep{ton88,ton01}, and
Luminosity--Linewidth \citep{tul77,tul00} methods.
In all, there are over 1400 galaxies with
distance measures within the 3,000~km~s$^{-1}$ volume; over 400 of
these are derived by at least one of the first three `high quality'
techniques. In the present study, distances are averaged over groups
because orbits cannot meaningfully be recovered on sub-group scales.
The present catalog is assembled into 1234 groups (including groups
of one) of which 633 have measured distances.
These observational components will be described in detail in a later
publication.
\section{Techniques: the LA and MAK methods}
\label{sec:techniques}
Peebles pioneered the Least Action (LA) method \citep{pee89,pee94,pee95} that requires
orbits to satisfy the stationary point of the Euler-Lagrange action, the
integral over time of
the Lagrangian. Hence, least action searches for the minimum of
\begin{equation}
\!\!S_{\rm LA}\!=
\!\!\!\int_{0}^{t_0}\!\!\!\!\! dt\left[
{m_i a^2 \dot{\bf x}_i^2\over 2}
-{G m_i m_j\over a\vert {\bf x}_i-{\bf x}_j\vert}
+{2\over 3}\pi G\rho_{\rm b} a^2 m_i {\bf x}_i^2\right],\!\!
\label{el-action}
\end{equation}
where summation over repeated indices and $j\not=i$
is implied, $t_0$ denotes the present time,
the path of the $i$th particle with mass $m_i$ is ${\bf x}_i(t)$,
$\rho_{\rm b}$ is the mean mass density,
and the present value of the expansion
parameter $a(t)$ is $a_0=a(t_0)=1$.
Individual orbits are constrained
by the mixed boundary conditions
that peculiar velocities were initially negligible (they subsequently
grew from gravitational perturbations) and the known
elements of position and velocity today.
The other piece of information known for roughly half the
elements is the aforementioned distances. This information is used to discriminate
between models. A given model involves a specification of
cosmological parameters and the assignment of mass to each of the
elements. Orbits are found within the context of a specified model
that are consistent with the boundary condition constraints on angular
positions and radial velocities. A component of the end point of an
orbit is its distance which can be compared with the observed
distance. Different models result in different distances. The
quality of a model can be evaluated by a statistical measure of the
differences between observed and model distance moduli,
$\mu_{obs}$ and $\mu_{mod}$ respectively. A $\chi^2$ parameter can be
calculated for each element, $i$, with a measured distance,
$\chi^2_i = (\mu_{mod,i} - \mu_{obs,i})^2 / \epsilon_i^2$,
where $\epsilon_i$ is the uncertainty assigned to $\mu_{obs,i}$.
A comparison of the relative distribution of $\chi^2_i$ values
provides a sensitive discriminant between good and bad models \citep{sha95,phe02}.
Galaxy flows are not sensitive to any
uniform repulsive dark energy \citep{lah91,sha95}.
Measurements of the lowest frequency peak in the microwave background
fluctuation spectrum strongly indicate, and we accept, that the Universe
has a flat topology,
whence $\Omega_{\Lambda} + \Omega_m = 1$ where $\Omega_{\Lambda}$
characterizes the mean density of dark energy and $\Omega_m$ characterizes
the mean density of matter with respect to the critical density for a
flat Universe in matter alone.
Specification of $\Omega_m$ and the Hubble Constant, H$_0$,
which describes the mean expansion rate of the Universe, in
the context of a flat Universe uniquely specifies the age of the
Universe, $t_0(\Omega_m,{\rm H}_0)$.
The Least Action orbits are a reflection of the interactions of masses
over the age of the Universe. As an initial approximation, it is assumed that
the relationship between mass and light is constant so the mass $M_i$
assigned to an element in the catalog with luminosity $L_i$ is
$M_i=(M/L)L_i$. Consequently, a cosmological model to be explored is
specified by two fundamental parameters: $t_0(\Omega_m,{\rm H}_0)$ and
$M/L$. For a given choice of $t_0$ and $M/L$, orbits are constructed,
model distances at
the end points are determined and compared with observed distances,
and the match provides a $\chi^2$ measure that can be compared
with alternative models. Discrimination between results over the
domain of ($t_0, M/L)$ choices based on the $\chi^2$ measure gives
specification of a best model.
The alternate technique that will be implemented here
is the Monge-Amp\`ere-Kantorovich ({\small MAK}) reconstruction \citep{fri02,moh03,bre03}.
This method provides a recipe for the orbits
that is unique to the degree that orbits can be described as following
straight lines under suitable coordinate transformations.
Orbits in the {\small MAK} reconstruction are
minima of the action which assume a Lagrangian mapping ${\bf q \mapsto x}$
that can be described in terms of a potential as
${\bf x=\bigtriangledown_q \Phi(q)}$.
Here, ${\bf x}$ is the current Eulerian position and ${\bf q}$
is the initial Lagrangian position.
The potential ${\bf \Phi(q)}$
is assumed to be convex; ie, orbit-crossing is excluded. With these
conditions, orbits are reconstructed uniquely by an assignment
algorithm that finds the minimum in
the ensemble of orbit distances summed in quadrature.
Hence, {\small MAK} search for the minimum of
the discretized action
\begin{equation}
S_{\rm MAK}=\sum_{i=1}^N\left({\bf q}_{j(i)}-{\bf x}_i\right)^2 .
\label{mak-action}
\end{equation}
Initial positions ${\bf q}_j$, are assigned to final
positions ${\bf x}_i$. For each final position $i$, there are $N$
possible initial
positions and only the one that minimizes the sum in the expression (2)
is allowed. A brute-force
assignment algorithm would have a complexity of $N!$, but can be
reduce to a polynomial complexity (specifically, so far to $N^{2.25}$).
It is insightful to note that the least action variation (\ref{el-action})
reduces to {\small MAK} optimization (\ref{mak-action})
for inertial trajectories
({\it i.e.} where particles move with their initial velocities).
The distances, $d$, permit an extraction of
peculiar velocities $V_{pec} = V_{gsr} - d {\rm H}_{0}$ where $V_{gsr}$ is
the observed velocity of an object in the galactic standard of rest.
The relationship between redshift and real space
displacements of the elements is
estimated using the approximation that galaxies trace
mass on inertial trajectories \citep{zel70}:
${\bf v} = f{\rm H}({\bf x} - {\bf q})$ where ${\bf v}$ is the peculiar
velocity vector and $f \sim \Omega_m^{0.6}$.
Consequently, a specific model defines positions that can be tested against
observed positions in the same way that is done in the Least Action
analysis.
An advantage of Least Action is that it potentially
can be used in the highly non-linear regime where orbits are strongly
curved. An advantage of {\small MAK} is that it recovers orbits uniquely and quickly, hence
can be used with the very large catalogs that are becoming
available.
By using the two methods on the identical catalogs, with
the identical distance constraints, it is affirmed that the two
methods recover similar results in the regimes where they are both
applicable.
Both methods have been tested on N-body simulations and found to result
in modest underestimations of the known densities. Underestimates are
anticipated because, at one extreme, mass that is not
strongly clustered on the scale of the survey has little dynamical
consequence and, at the other extreme, complex shell-crossing orbits
cannot be modeled.
Analysis of {\small MAK} reconstructions in 12 cosmological simulations with different
initial fluctuation characteristics discussed by
\cite{moh05} resulted in recovery of $80\% \pm 20\%$ ($1 \sigma$) of
the known model density. In the case of Least Action, it has been
appreciated that mass can be underestimated on small scales
\citep{bra94,bra02}. A quantitative measure of the effect is found using
our Least Action reconstruction algorithms with
an N-body simulation with $\Omega_\Lambda=0.7$, $\Omega_m=0.3$,
evaluated by
placing the observer at multiple locations to test for cosmic variance.
Recovered densities were $70\% \pm 20\%$ of the model
density (Phelps, Desjacques, and Nusser; ongoing work).
The estimates that we quote for $\Omega_m$ include adjustments for the
systematics of 20\% and 30\% for MAK and Least Action,
respectively.
\section{Results: I. determination of $\Omega_m$}
\label{sec:result1}
\begin{figure}
\begin{center}
\mbox{\epsfig{figure=omega_age_h.ps,width=0.646\linewidth,clip=}}
\end{center}
\vspace*{-0.4cm}
\caption{
\noindent
Reconstruction methods constrained by precision distance measures
give estimates of the mean mass density of the Universe.
The contours are $\chi^2$ at 1 and $2 \sigma$ for models with different
choices of age and mass density $\Omega_m$, with shaded contours for {\small MAK} and
line contours for {\small LA} reconstructions.
The {\small MAK} and Least Action contours are adjusted upward in density by 20\% and
30\% respectively to compensate for
systematic underestimations of density.
The confluence of the constraints on density and age parameters
from {\small \emph{WMAP}} \citep{spe03} of $\Omega_m h^2=0.134$
and from {\small SDSS} \citep{teg04} of $\Omega_m h=0.21$ is
lightly shaded.
The darkest shaded region
lies within the $2 \sigma$ limits of all four experiments.
The right panel shows the
density determinations for {\small LA}, {\small MAK}, {\small SDSS}, and
{\small \emph{WMAP}}
for H$_0 = 80$~km~s$^{-1}$~Mpc$^{-1}$, the value of the Hubble Constant
consistent with
our distance scale zero point. The open and filled symbols for {\small LA} and
{\small MAK}
respectively give the results before and after compensation for
the systematic under-determination of mass density. At H$_0 = 80$,
$t_0^{SDSS}=12.2$~Gyr,
$t_0^{WMAP}=13.0$~Gyr, and $t_0^{MAK,LA}=13.5$~Gyr.
The 4 analyses are consistent with
$\Omega_m = 0.22 \pm 0.02$ with the constrained value of H$_0$.
\label{fig1}}
\end{figure}
The results of the two methods are summarized in
Fig.~1 where contours of the $\chi^2$ parameter are plotted in the
domain \{$t_0$,$\Omega_{\rm m}$\}.
In the case of the Least Action analysis, mass density $\Omega_m$ is
derived from $M/L$ values
by accepting the mean $B$ band luminosity density of the Universe
determined from {\small SDSS} \cite{bla03} with reddening corrections.
We use $\Omega_m = (M/L) / (1540 h)$, a transform with $\sim 20\%$ uncertainty.
The error ellipses from the {\small MAK} and {\small LA} studies are elongated.
With a linear analysis the elliptical $\chi^2$ troughs would open to
infinity, but non-linear effects create a specific minimum along each
$\chi^2$ trough.
At a fixed age, $t_0$, there are relatively tight constraints
on $\Omega_m$ requirements. If shorter ages are entertained,
then higher densities are required to arrive at the observed dynamical state
in the specified time. The contrary dependence between density and time
are seen in the
results from the microwave background {\small WMAP} \citep{spe03} and galaxy redshift
{\small SDSS} \citep{teg04} experiments that are also superimposed
on Fig.~1.
Only a small domain around $t_0=13.5$~Gyr
lies within or near the $2 \sigma$ contours of all the methodologies.
With the constraint $h=0.8$ consistent with the zero point of the
distance estimates, there is good agreement between the {\small WMAP}, {\small
SDSS}, {\small MAK},
and {\small LA} measures of the density parameter: $\Omega_m=0.22\pm0.02$.
The low value of $\Omega_{\rm m}$ obtained here is consistent with
that found from divers recent studies
\citep{ostriker20031, ostriker20032, vdb03, col05}.
\section{Results: II. the mass of Virgo Cluster}
\label{sec:result2}
The velocity field analyses discussed above make the simplistic assumption
of constant $M/L$.
There has been evidence in the literature, though, for large variations in
$M/L$ with environment \citep{mar02, vdb03, eke04,
tul05b, vdb05, par05}.
Here we note that galaxy flows near the Virgo
Cluster contradict the hypothesis of constant $M/L$.
The high velocities of infall suggest a much larger $M/L$
for the cluster than the value consistent with a good fit over the
full region within 3,000~km~s$^{-1}$.
The critical galaxies for this discussion lie outside the cluster
on the plane of the sky so only a modest component of the
infall motion is projected into the line of sight but there is no
confusion with cluster membership. Galaxies infalling
from the foreground of the cluster are redshifted with respect to the
cluster and galaxies infalling from the background are blueshifted.
If one could imagine many test particles distributed along a
line of sight, they would create a wave in velocity as a function of
distance with
the same velocity arising at three distances \citep{ton81}
-- two located within the infall region and one located at the
cosmic expansion position.
The amplitude of the peaks of the `triple-value' waves depend on the
mass, $M$, interior to the position associated with the peaks, $r$.
The envelope of observed infall velocities with $r$ provides a
description \citep{tul84} of the run of $M(<r)$.
The outermost caustic of the Virgo Cluster
proper (the radius of second turnaround) is at $\sim 2$~Mpc from the
center of the
cluster. Galaxies on first infall acquire very high radial
velocities at 2--4 Mpc from the Virgo core.
\begin{figure}
\begin{center}
\mbox{\epsfig{figure=tripval3.ps,width=0.615\linewidth,clip=}}
\end{center}
\vspace*{-0.4cm}
\caption{
The distortion of the velocity--distance relation by the Virgo
Cluster in the line of sight toward the 11--3 group.
The observed distance
and velocity of the 11--3 group is indicated by the point with error bars
in distance (errors in velocity are smaller than the symbol).
The observed distance and velocity of the Virgo Cluster is indicated
(centered at 6.2 degrees from the line of sight of the 11--3 group).
The background Virgo W Cluster causes the second wiggle along this line
of sight.
The behavior of
infall into the Virgo Cluster is traced by the locus of positions and
velocities of test particles of negligible mass that were laid down
in the line of sight of the 11--3 group and subjected to Least Action
modeling.
It is concluded from the information seen on this plot and equivalent plots of the
lines of sight to other groups close to the Virgo Cluster that $M/L_{\rm
Virgo}\approx 500 M_{\odot}/L_{\odot}$ and the mean $M/L$ for
the field outside the cluster is approximately $1/3$ of this value.
\label{fig2}
}
\end{figure}
An example is given in Fig.~2. The run of
velocities with distance are what is expected along the specific
line of sight of the 11--3 group in the Nearby Galaxies Catalog
with three different assumptions regarding the mass of the Virgo Cluster.
The observed distance and velocity of the 11--3 group is indicated in
the figure. It is clear that the 11--3 group is at one of the two
infall locations (or the location where the two infall positions become
degenerate at the tip of the triple-value curve) rather than at the cosmic
expansion position around 30 Mpc.
The total mass within 3,000~km~s$^{-1}$ is the same in all 3 models
to ensure that the global $\chi^2$ is near the minimum of the trough in
Fig.~1, however the distribution of mass between the elliptical rich and spiral
rich groups is modified between models. Here, $t_0=13$~Gyr.
In the case of the curve with
the smallest swing, all components are given the same
$M/L=200 M_{\odot}/L_{\odot}$. The intermediate curve is generated with
the elliptical-dominated groups (including the Virgo Cluster) given
$M/L_E=500$ and the spiral-dominated groups given $M/L_S=174$. The curve with the
largest swing reflects $M/L_E=800$ and $M/L_S=148$.
In the model illustrated by the middle curve in
Fig.~2, the cluster is given the necessary and just sufficient mass of
$9 \times 10^{14} M_{\odot}$.
The 11--3 group and a couple of others provide the greatest demands
on the mass of the cluster and are inferred to lie near a tip along a
triple-value curve.
An adequate description of the amplitude of their
infall velocities requires an assignment of
$M/L=500 M_{\odot}/L_{\odot}$ to the Virgo Cluster if
$t_0=13$~Gyr. As with the overall Supercluster modeling, less cluster
mass is required if given more time \citep{tul84}: {\it e.g.},
$8 \times 10^{14} M_{\odot}$
suffices if $t_0=14$~Gyr. In both these 13 and 14 Gyr cases,
as the mass assigned to the cluster is augmented,
the models
require a reduction of mass assigned to the field to remain at the
minimum of the $\chi^2$ trough of Fig.~1.
The ratio of cluster, $C$, to field, $F$, mass to light values is
$(M/L)_C/(M/L)_F \simeq 3$
with the mass assignments required to explain the
infall motions. This factor 3 probably underestimates the $M/L$ difference
between Virgo and bound groups in the field because some of the mass
contributing to the field ratio lies outside the groups.
Studies of the dynamics of nearby groups \citep{tul05b} suggest that bound
groups of spiral galaxies in the field have $M/L \sim 90$. The mass to
light ratio of the Virgo Cluster is 5--6 times larger.
\section{Conclusions}
\label{sec:conclusion}
Our first result on the mean density of the Universe
complements the {\small \emph{WMAP}} and {\small SDSS} measurements of
the density parameter because the error
ellipses in the domain $\Omega_m, H_0$ are
steeply inclined to each other. A narrow
range of parameter space around $t_0=13.5$ Gyr,
$\Omega_m=0.22\pm0.02$ is permitted by the combined experiments.
Our second result concerns the local fluctuations around this mean density.
Although most of the blue light in the local Universe is
associated with the spiral field, there is close to parity in the
partition of mass between the spiral field and the Virgo Cluster.
The $M/L$ value associated with the E/S0 dominated Virgo
Cluster is significantly higher than that associated with the spiral
dominated field. The complex problem of $M/L$ variations with environment
shall be discussed in a future more-detailed paper.
\smallskip
{\small
We thank Ed. Bertschinger, J. Colin, U. Frisch, M. H\'enon, G. Lavaux,
P.J.E. Peebles, S. Phelps, E. Shaya and J. Silk for
discussions, collaboration and support.
RM was supported by a Marie Curie fellowship. RBT acknowledges
support from the BQR program of the Observatoire de la C\^ote d'Azur,
the French Hautes Niveau of EGIDE, a JPL Space Interferometry Mission contract,
and STScI and NSF awards.
}
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Die Blackshape Prime ist ein einmotoriges, zweisitziges Leichtflugzeug des italienischen Flugzeugherstellers Blackshape aus Monopoli, das 2007 den Erstflug absolvierte und 2009 auf der Aero Friedrichshafen der Öffentlichkeit vorgestellt wurde. Es wird als "ready-to-fly-aircraft" ausgeliefert.
Entwicklung
Das Flugzeug ist als Tiefdecker mit Einziehfahrwerk, Verstellpropeller, zwei Sitzplätzen und geschlossenem Cockpit ausgeführt.
Das Design wurde ursprünglich von Compact Compositi srl in Turin unter dem Namen Millennium Master entwickelt. Als Basis diente das aus Holz gefertigte Asso-X Bausatz. Die Statische Struktur wurde dabei an der Universität Turin entworfen, während die Aerodynamischen Untersuchungen bei Alenia Aeronautica erfolgten.
Technische Daten
Siehe auch
Liste von Flugzeugtypen
Weblinks
Website des Herstellers (englisch)
Offizielles Promotionsvideo
Einzelnachweise
Ziviler Flugzeugtyp
Einmotoriges Flugzeug mit Kolbenmotor
Erstflug 2007
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: return self._values[key] exception.KeyError: Python import scrapy
from scrapy.spider import Spider
from scrapy.selector import Selector
from scrapy.http import Request
from scrapy import Item, Field
class NItems(Item):
creat = Field()
class NSpider(Spider):
def parse(self,response):
item = NItems()
for da in response.xpath('//div[@class="section-stream-content"]'):
item['link'] = da.xpath('//tr/td[2]/div[1]/h2/a/@href').extract()
for con in item['link']:
request = scrapy.Request(con, callback = self.parseDate)
request.meta['item'] = item
yield request
def parseDate(self, response):
item = response.meta['item']
hxs = Selector(response)
cr = hxs.xpath('//*[@class="byline"]/text() | //*[@id="yui_3_16_0_1_1428166780085_527"]/text() | //div[@class="metadata"]/p/strong/text() | //p[@class="update-time"]/text() | //p[@class="publisheddate"]/text() | //p[@class="date"]/text() | //div[@class="last_updated"]/p/a/text()').extract()
item['creat'].append(cr)
cr = hxs.xpath('//*[@class="byline"]/text() | //*[@id="yui_3_16_0_1_1428166780085_527"]/text() | //div[@class="metadata"]/p/strong/text() | //p[@class="update-time"]/text() | //p[@class="publisheddate"]/text() | //p[@class="date"]/text() | //div[@class="last_updated"]/p/a/text()').extract()
fot z in cr:
print z
I trying to collect some data from linked page, but when i try to add collected data to the item list i get this error.
Why am i getting this error on line --- item['creat'].append(az) ---In my understanding, I insert a list into the list
A: You need to initialize item['creat'] as an empty list in the parse() method:
def parse(self,response):
item = NewsItems()
for da in response.xpath('//div[@class="section-stream-content"]'):
item['link'] = da.xpath('//tr/td[2]/div[1]/h2/a/@href').extract()
item['creat'] = []
...
Or, set it to cr in parseDate():
item['creat'] = cr
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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—Olivia Hussey, Golden Globe Award–winning actress
"If you want to be happy (and who doesn't?) Marci Shimoff has—amazingly—'cracked the code' for doing so. This isn't the latest pop-psychology craze. Instead, it's based on the latest scientific research about real and lasting happiness. What if you (or your loved ones) really could be happy, regardless of circumstances? Happy for No Reason shows you how—and it's easier than you think. This remarkable book is sure to change a lot of lives, including yours."
—Bill Harris, Director of Centerpointe Research Institute
"Happy for No Reason offers an engaging, wise, and profound look at happiness that can change your life. Marci Shimoff has creatively crafted a purposeful guide to assist us in experiencing the gift of happiness."
—Chaka Khan, Grammy Award–winning singer and songwriter, author, and philanthropist
"Rule #1 for happiness: Read this book! Be Happy for No Reason, because everything else comes from that. If you want health, wealth, and wisdom, follow the Happiness Habits presented in these pages and transform your life."
—Phil Town, #1 New York Times bestselling author of Rule #1, host of CNBC show Rebel Investor
"In Happy for No Reason, Marci Shimoff shares a new paradigm of happiness, rooted in ancient wisdom, yet validated by cutting-edge research in the fields of both neuroscience and positive psychology. I was enthralled the whole way through."
—Candace B. Pert, PhD, Scientific Director of RAPID Pharmaceuticals, featured in the film What the Bleep Do We Know!?, and author of Everything You Need to Know to Feel Go(o)d
"Happy for No Reason is the most important book about happiness you'll ever read! A completely unique and life-changing approach to being happy 'no matter what.' "
—Raymond Aaron, Canada's #1 business and investment coach
"Marci Shimoff has opened a new era of inner peace for humanity with her brilliant work. The search for happiness now starts and ends inside us as we raise our happiness set-point using her insightful guidance. As an educator, I'm most impressed at how Marci presents a complex topic in an easy-to-read and easy-to-share book. I recommend it happily!"
—Paul R. Scheele, cofounder of Learning Strategies Corporation and author of Natural Brilliance and PhotoReading
"Happy for No Reason is a wonderful guide for how to be happier as individuals and how to share that happiness with our families, our communities, and our world. Everyone will benefit from the wisdom and coaching that this valuable book offers."
—Dr. Bill and Kathy Magee, CEO and President, cofounders of Operation Smile
"Happy for No Reason offers a practical and hopeful solution to fear, depression, anxiety, and stress. Here's the secret: Each of us has the ability to focus on joy and happiness—regardless of what is going on in the world. And therein lies our power—and ability to be healthy and happy. This book is a winner."
—Christiane Northrup, MD, author of Women's Bodies, Women's Wisdom and The Wisdom of Menopause
"Wonderful! This book provides a program that will bring you powerful results. Finding lasting happiness has never been as clear and easy."
—Bobbi DePorter, cofounder and president of Quantum Learning Network/SuperCamp
"Happy for No Reason not only reports on the lofty science of happiness, but the authors also wisely remember to include stories from the lives of everyday people."
—Robert Biswas-Diener, Program Director for Education and Learning, Center for Applied Positive Psychology (UK)
"Wealth and success are meaningless without happiness! Happy for No Reason shows you how to be happy wherever you are in life."
—Dr. Ivan Misner, Chairman and founder of BNI (Business Networks International)
"Happy for No Reason is the premier book for any person wanting to experience more happiness. Keep it by your bedstand! It's an important book to show you how you can be happy every day."
—Jill Lublin, international speaker and bestselling author of Guerrilla Publicity, Networking Magic, and Get Noticed
"This book will change the way you think about happiness. In Happy for No Reason, Marci Shimoff has plenty of reasons to share this wisdom. If you want more joy, peace, and well-being in life, this book is for you."
—Yakov Smirnoff, MAPP (master of applied positive psychology) and comedian
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## CONTENTS
Foreword by Jack Canfield
PART I HAPPINESS THAT'S HERE TO STAY
Introduction Welcome to a Happier Life!
Chapter 1 Happy for No Reason . . . Really?
Chapter 2 Practicing Happiness
PART II BUILDING YOUR HOME FOR HAPPINESS
Chapter 3 The Foundation—Take Ownership of Your Happiness
Chapter 4 The Pillar of the Mind—Don't Believe Everything You Think
Chapter 5 The Pillar of the Heart—Let Love Lead
Chapter 6 The Pillar of the Body—Make Your Cells Happy
Chapter 7 The Pillar of the Soul—Plug Yourself In to Spirit
Chapter 8 The Roof—Live a Life Inspired by Purpose
Chapter 9 The Garden—Cultivate Nourishing Relationships
PART III HAPPY FOR NO REASON EVER AFTER
Chapter 10 The Happy for No Reason Plan for Life
Recommended Resources
Acknowledgments
Giving Back
About Marci Shimoff
About Carol Kline
Biographies of the Happy 100 Whose Stories Appear in Happy for No Reason
Permissions
This book is dedicated to anyone who's ever wanted to be happy.
It's not just pie in the sky anymore.
And to Dad, my first Happy for No Reason role model.
I carry your joy in my heart.
## FOREWORD
## by Jack Canfield
Money can buy happiness—you've just done it with the money you've spent on this book!
Or at least you've taken the first, most important step toward becoming what my colleague and dear friend, Marci Shimoff, calls "Happy for No Reason."
It's the perfect time for a book on this subject. Now that most of us have our basic survival needs met and some degree of material comfort, we're looking for more in life. Whether we're buying a great car, going on a dream vacation, getting a raise, or coaching Little League—what we're hoping these things will give us is the deeper experience of being happy. But we're coming up short. I can't think of a more universal need today than that of true happiness.
In Happy for No Reason, Marci offers a breakthrough approach to being happy. While most books focus on creating happiness from the outside, Happy for No Reason is about creating happiness from the inside—where it really counts. In these pages, you'll find simple, practical steps for experiencing a deeper, more permanent state of happiness, regardless of your external circumstances.
If you saw the film The Secret, you know that Marci and I are both great believers in the Law of Attraction, the idea that you have the power to attract into your life what you desire most deeply. With this book, Marci provides the crucial foundation for most effectively practicing this law, which is maintaining a vibrational state of joy and happiness at your core.
When I first met Marci almost twenty years ago at one of my self-esteem training courses, she immediately impressed me with her energy and desire to learn everything—right now! She later assisted with my programs and developed her own life-changing seminars on success and self-esteem. Then one day, this ball of fire called me to say that she had a great idea for a book: Chicken Soup for the Woman's Soul. (Only the original Chicken Soup for the Soul book had been published at the time.) "Sounds like a brilliant idea," I told her. "But why do I need you to do it?"
Marci and her business partner, Jennifer Hawthorne, didn't skip a beat: "Because we're both writers, we both speak to women's audiences, and we're both women," they told me.
"Well, you got me on that last one," I had to admit.
It was a great collaboration, and Marci and I have worked together closely ever since. When I decided to form the Transformational Leadership Council, an organization of 100 top transformational leaders in the world, I invited her to be a founding member and part of the executive committee.
Marci is the perfect person to write this book. She's been pursuing spiritual growth and the deepest values of happiness for as long as I've known her. I see her as the scout at the head of the pack, the point guard who surveys the territory up ahead and comes back with a road map of what's useful. Whenever Marci tells me I ought to check something out, I know it will be something good.
Marci's always had a unique talent for making deep spiritual concepts immediately accessible. With Happy for No Reason, she's hit a home run. This book presents a definitive, broad-based approach to becoming truly happy that combines great spiritual depth, top-notch research, and psychological practicality. It's also a pleasure to read. From our Chicken Soup experience, we learned that for a book to really work, it needs to include stories that Velcro its message to our memories. Since the days we humans sat around fires in our caves, our brains have been wired for stories—it's how we make sense of the world. In Happy for No Reason, Marci and her cowriter, Carol Kline, have captured moving and amazing real-life stories of people who've applied these principles to their own lives to establish a state of deep and lasting happiness.
I call Marci my sister in the pursuit of the secrets to fulfillment in life, the "ultimate magical manifester." I'm certain that if you follow the practices in her dazzling new book, you too will manifest a lifetime of happiness.
# PART I
# Happiness That's Here to Stay
This joy I have—the world didn't give it, the world can't take it away.
—Shirley Caesar, gospel singer
## INTRODUCTION
## Welcome to a Happier Life!
I was crammed in the back of an ancient flatbed truck with thirty other Westerners, bumping down a rock-strewn dirt road heading toward the foothills of the Himalayas. We each had a bandana covering our nose and mouth to keep us from choking on the dust. We were on our way to a small, remote mountain village where we would provide humanitarian support for the villagers' educational, medical, and housing needs. I was tired, grumpy, and sore all over. After six hours, the driver stopped the truck, got out, and unceremoniously hauled all our luggage onto the dusty ground.
"You'll need to walk the rest of the way," he said. "It's another mile, and from here on, the road is too steep and narrow for my truck." As the truck rattled off, I looked at my ninety-one-pound suitcase with dread. Why had I packed all that unnecessary stuff? Ridiculous. I tried dragging it a few yards up the rough, mountainous trail, but it was hopeless; I wasn't strong enough. Dusk was falling; what could I do? Everyone else in the group was wrestling with their own bags; no one could help me. But they were managing to make their way up the hill and soon were nearly out of sight. I sat down and spent a couple of minutes fighting down a rising panic. Were there tigers here?
Then a tiny, barefoot old woman, her face seamed with wrinkles, came out of the forest and up the road toward me. She approached me with a warm smile, picked up my bag, and, astonishingly, hoisted it up onto her head as though it weighed no more than a basket of fruit. Then she headed up the hill, motioning for me to follow.
As we ventured up the path together, though we had no language in common, I was struck by the twinkle in her eyes and the simple happiness that she exuded. When we finally reached the top of the mountain, I was met with the huge smiles and enthusiastic greetings of her fellow villagers.
I spent the next two weeks working side-by-side with these people, attending to the children, preparing food, and helping administer medical care. Like them, I slept on the ground, bathed in the river, and drank milk fresh from the cow. To my surprise, I found that this no-frills lifestyle agreed with me. I felt clear, peaceful, and full of energy.
During my stay, I also spent a lot of time observing my mountain hosts. Here were people who had no electricity or running water, living on the bare minimum with no creature comforts. Yet there was a lightness of spirit, a sense of humor, and an easy friendliness about them that was remarkable. They were simply happy from the inside out.
Of course, I realized that their happiness wasn't a product of their poverty. I'd seen plenty of dirt-poor men, women, and children in all corners of the world who were utterly miserable. I'd also met people with every shiny, expensive toy money could buy who were ecstatic about their good fortune, as well as fabulously wealthy folks who were living proof of the saying "Money can't buy happiness."
The experience reinforced my conviction that happiness isn't about having everything you've ever dreamed of, nor is it simply negating the need for material pleasures in life. It goes deeper than that. What we're all really looking for is happiness from within that doesn't depend on external circumstances—the kind I call Happy for No Reason.
My time among the Himalayan villagers crystallized my goal: without giving up my regular life, I wanted to find a way to enjoy that kind of happiness, no matter where I was or what I was doing.
Chances are you picked up this book for the very same reason. If you're human, it comes with the package: everyone, everywhere, wants to be happy. You might already be pretty happy and just want to crank up the volume a notch or two. Or you may be seriously unhappy and wondering how others around you manage to find delight in their lives. Perhaps you've created your version of the American Dream, but still feel an emptiness inside that nothing on the outside seems to fill.
The good news is that it doesn't matter where you begin. Wherever you are right now, this book will show you how to be happier. You don't have to have happy genes, win the lottery, or become a saint. By the time you finish reading these pages, you will know how to experience an authentic state of sustained happiness for the rest of your life.
### My Heart's Quest
This book was born of my own deep longing to be happy. The kind of happy that is solid, true, and anchored in my being, so that no matter what my external circumstances are, there is still a feeling of unshakable fulfillment, joy, and inner peace. Other people lived this way, so I knew it was possible. Yet for so many years, no matter what I did, it seemed to elude me.
As you'll read in my story in Chapter 1, I was unhappy from the get-go. As a young teen, I began a personal, and later professional, quest that lasted over thirty-five years and ultimately led me to the amazing findings in this book. During that time, I took every transformational seminar under the sun. For years I studied and taught success principles. I applied them in my own life and gained a good measure of success. I had plenty of reasons to be happy: I was a #1 New York Times best-selling author, I'd received national acclaim as an inspirational speaker, and I'd touched millions of people's lives. I was very familiar with what it meant to be "happy because . . ." The problem was it didn't bring me the happiness I wanted.
Looking around, I saw that the happiest people I knew weren't the most successful and famous. Some were married, some were single. Some had lots of money, and some didn't have a dime. Some of them even had health challenges. From where I stood, there seemed to be no rhyme or reason to what made people happy. The obvious question became Could a person actually be happy for no reason?
I had to find out.
### The Study of Happiness
So I threw myself into the study of happiness. I interviewed scores of experts and delved into the research from the burgeoning field of positive psychology, the scientific study of the positive traits that enable people to enjoy meaningful, fulfilling, and happy lives. I soaked it up like a sponge and found a number of gems—fascinating, amazing, and useful information that's changed my life and will change yours too.
My first major discovery was that scientists have found that we each have a "happiness set-point," the genetic and learned tendency to remain at a certain level of happiness, similar to a thermostat setting on a furnace. Fortunately for those of us not born on the sunny side of the street, it's been shown that we can change our happiness set-points. I'll discuss this more in the next chapter and offer you specific exercises throughout the book to raise your happiness set-point.
I also learned that two of our greatest barriers to happiness, fear and anxiety, have been hardwired in us for millennia to ensure our survival as a species. In today's world, however, that old wiring has become more harmful than helpful. In the coming chapters, you'll discover ways to disconnect that internal alarm system so you can lead a happier life.
Research findings like these thrilled me, but I still wanted more. In my years studying success, I'd found that success leaves clues. You can look at the lives of successful people to learn how to become successful yourself. I figured that the same must be true of happiness, so I set out to interview 100 truly happy people.
### The Happy 100
It turned out that it wasn't that easy to find 100 truly happy people, even in a nation of 280 million! I'd read about the epidemic of unhappiness in our country: one out of every five women in America is on antidepressants, and 6 million men start taking antidepressants every year. But I was still shocked to experience it firsthand. I asked everyone I met, "Who's the happiest person you know?" People really had to stop and think. Then, inevitably, the first person they'd name would be someone who was fabulously successful. "But wait," they'd say, "he (or she) isn't really happy." Many people couldn't think of anyone they considered truly happy. But I persisted in my search, and eventually I found over 100 deeply happy people to interview. I call them the Happy 100. They're men and women of all ages and backgrounds, from all different walks of life. Their stories are amazing and revealing. They've opened my eyes to a whole new way of living.
In addition to conducting interviews, I posted a very simple survey on my website, a single question asking "What do you think is the most important thing necessary to experience the Happy for No Reason state?" The responses were illuminating.
I was right. Like success, happiness does leave clues. I came away from my interviews and survey with clear evidence that happy people live their lives differently from unhappy people. I discovered 21 habits that happy people share—I call them the Happiness Habits—that anyone can practice to easily and effectively support the experience of deep and lasting happiness.
Then I came upon my most important discovery, one that sets this book apart from all the rest: the concept of Happy for No Reason. Other experts on happiness urge you to find the things in life that make you happy, and do more of those things. There's nothing wrong with that, but it won't bring you true and lasting happiness. Happy for No Reason takes a radically different approach, showing you breakthrough methods to consistently experience the deep, inner happiness that's at your core—a happiness that's beyond reasons and that's here to stay.
My research on happiness and my own experience have convinced me that becoming Happy for No Reason is entirely doable. Today we know so much more about everything in our world. We've used technology to explore many aspects of life, from our own bodies to the composition of the galaxy, and are finally using technology to explore happiness. In the past two decades, scientists in the field of positive psychology have made tremendous strides, identifying the happiness set-point, the neurotransmitters of happiness, and even where happiness is located in the brain. For the first time in history, we know that happiness isn't an abstract emotion; it's a physiological state that can be measured—and experienced more often in our everyday lives.
My discoveries were so remarkable I wanted to share them with as many people as possible, so I decided to write this book. I called my dear friend of over twenty-five years, Carol Kline, a writer who shared my enthusiasm for this topic, and I asked her to join me. Carol had seen me through everything, so she was the perfect person to embark on this journey with me. We both feel blessed by the thousands of hours spent researching and talking with experts and hearing the stories of happy people. And the result, Happy for No Reason, is a combination of the best knowledge and current research on happiness, practical methods gleaned from the happy people I interviewed, and the inspirational stories of their lives.
### What You'll Find in This Book
Happy for No Reason is divided into three sections. In this first section, Part I, we'll continue to explore the paradigm of true happiness I call Happy for No Reason. Just understanding this idea can change the way you experience happiness in your own life. You'll find a Happy for No Reason questionnaire that will help you assess and understand your own present level of happiness. You'll also learn three guiding principles that will help you get past the common blocks to happiness in your life and speed your progress. Finally, I'll share with you how I've applied the Law of Attraction to being happier. I'm honored to be featured in the worldwide film phenomenon The Secret, which focuses on this law. I've seen how powerfully it can change lives.
Part II is the how-to section that shows you step-by-step how to raise your level of happiness. Through my research, I discovered that there are seven specific steps for becoming Happy for No Reason. I wanted to make these seven steps easy to remember, and because people's homes are so often a metaphor for their lives, I decided to present them in a simple, easy-to-remember analogy: building a Home for Happiness. The seven steps of building your Home for Happiness correspond to the seven main areas of your life: personal power, mind, heart, body, soul, purpose, and people.
This holistic approach is vital. Many books on happiness just focus on the mind, but if you don't also address your habitual behavior in the other areas of your life, you won't experience true happiness. Here's an overview of the steps (one per chapter):
1. The Foundation—Take Ownership of Your Happiness
2. The Pillar of the Mind—Don't Believe Everything You Think
3. The Pillar of the Heart—Let Love Lead
4. The Pillar of the Body—Make Your Cells Happy
5. The Pillar of the Soul—Plug Yourself In to Spirit
6. The Roof—Live a Life Inspired by Purpose
7. The Garden—Cultivate Nourishing Relationships
Each step has three Happiness Habits, with corresponding exercises based on the latest research and findings on how to raise your happiness set-point, as well as inspirational stories.
As a Chicken Soup for the Soul coauthor, I'd been deeply moved reading over 20,000 stories that had been submitted for my books. When the books were published, the overwhelming response from millions of readers confirmed for me that stories open people's hearts and affect them deeply. That's why I decided to include some of the inspiring first-person stories of transformation from the Happy 100 in this book. From my 100 interviews with these unconditionally happy people, I've selected twenty-one stories that clearly define what it means to be Happy for No Reason.
You'll read phenomenal tales from a variety of people, including a former drug dealer turned minister, a hit filmmaker, a political refugee, and a famous actress who escaped a "family curse," as well as stories from doctors, nurses, mothers, teachers, and business executives. You'll meet Zainab, who lived under Saddam Hussein's thumb and locked up her memories—and her happiness—in a box hidden deep inside. And Janet, who was living a ho-hum life and working at a dead-end job, until she discovered a simple yet profound way to be deeply happy every day. You'll hear Gay's story of losing 100 pounds, and gaining the life of his dreams, after finding the key to really nourishing himself. These stories are wonderful illustrations that there are many paths to Happy for No Reason, and that we're all capable of arriving at that destination no matter where we begin the journey.
Part III is about being Happy for No Reason—permanently. No more waiting for the other shoe to drop, wondering how long your happiness will last. In this section, you'll get clear direction about how to put the Happiness Habits into practice every day and find resources that can help you sustain being Happy for No Reason for the rest of your life.
Becoming Happy for No Reason may not happen overnight—though, as I found in some instances, it can! Having the vision that it exists and knowing the steps to achieve it will put you on the fast track to a complete transformation of your life. I know, because it's happened to me and to thousands of others. I've used the tools and techniques presented in this book myself and with my clients. Though I wasn't happy for many years, I've been able to move from a D+ to an A– on the happiness scale. Now, when life tosses my boat around, I return to an even keel more easily; I don't capsize anymore. I'm still on the journey, still a student as well as a teacher, but I am living proof that these steps can move you immediately in the right direction, no matter where you start. Believe me, if I can do it, so can you.
It is my deep prayer and wish that this book lead you to build a home for happiness inside yourself that is unshakable—and from that strong and peaceful place you create more happiness in the world.
## 1
## Happy for No Reason . . . Really?
Happiness is the meaning and the purpose of life, the whole aim and end of human existence.
—Aristotle
Years ago, I taught a success seminar in which I asked the participants to each take a big sheet of heavy paper and write across the top "100 THINGS TO BE, DO, HAVE." Then they made three long columns and began listing their dreams, big and small. When they shared their goals afterward, they were invariably all fired up: people wanted to go scuba diving in the Great Barrier Reef, own a Mercedes SL600 Roadster (cream white, with titanium silver alloy wheels), dance at the White House, fly a small plane around the world. They wanted to rise to the top of their field, end world hunger, create world peace, be on the cover of Time magazine.
Somewhere on the BE list a few people wrote "Be happy," but I was surprised at how often most people overlooked that. That's what the whole page was about, wasn't it? Didn't being, doing, and having all these things equal happiness?
Over time, I've come to look back on those lists as a great example of taking the long way around. Big and wonderful as those things are, they aren't the ultimate thing we all want. If you cut to the chase, what we really want is to be happy.
The truth is that happiness is so compelling, so attractive—so irresistible—that whether you realize it or not, everything you do is aimed at making yourself happy. Happiness has been called the holy grail of human existence, the be-all and end-all of life. Aristotle called it the goal of all goals.
Studies around the world show that when people are asked to rank what they want from life, they put the desire to be happy at the top of the list, more important than wealth, status, a good job, fame, and sex. This is true for people of all cultures, races, religions, ages, and lifestyles. And research also shows how vital happiness is: happy people live longer, are healthier, and have better relationships. In fact, happiness leads to more success in every area of your life.
Unfortunately, many people are not experiencing much sustained, authentic happiness. Consider these sad statistics:
• Less than 30 percent of people report being deeply happy.
• Twenty-five percent of Americans and 27 percent of Europeans claim they are depressed.
• The World Health Organization predicts that by 2020, depression will be second only to heart disease in terms of the global burden of illness.
Although our lifestyles are better than ever, we're unhappier than ever. It seems the more gadgets and goods we gather, the worse we feel.
In this book, I'm not going to show you directly how to make more money, be more successful, or have better relationships. I leave that to my friends and colleagues in the transformational world who do that all so well. What I am going to do is tell you what I would most want to know. This book answers the question I've spent the past thirty-five years studying and researching, the one that is most important to me—and that I believe is also most important to you: How can I be truly happy?
Although this question absorbed me for most of my life, for years I didn't have much luck answering it. In fact, I spent most of that time barking up the wrong tree.
### One Unhappy Camper
I had imagined it all as a child: I would grow up, live in a beautiful house, have a wonderful husband and a great career. My body would be perfect and my social life fun and exciting. I would be HAPPY!
To live that life of my dreams, I knew I'd have to work hard to get all my ducks in a row. And although I was clear about what I wanted, I wasn't sure how I'd get it. The only thing I knew for sure was that I wasn't happy. I came out of the womb filled with existential angst. I was the brooding five-year-old who was worrying about the condition of the world while everyone else was watching Romper Room. At age seven, I was grilling my wonderful, loving parents about God and spirituality and becoming frustrated that they couldn't answer my questions. When I look through family albums, I see my brother and sister beaming at the camera while I always look as though I've just lost my best friend.
Even though I wasn't a naturally happy person, something deep within me knew that I didn't have to accept that way of being. It was as if I had some kind of antenna tuned to whatever I needed to learn most. When I was eleven, I slathered myself in suntan oil and snuck into my sister's room to steal a book to read while sunbathing. I grabbed the skinniest one, since I'd always been a slow reader, and headed outside. By the time I was halfway through the book, Hermann Hesse's Siddhartha, the story of a young Indian man on a quest for enlightenment thousands of years ago, I was in tears. I recognized that I wasn't alone; someone else on the planet understood my search and shared my longing for connection and joy. That book put me on the path of seeking.
While other girls were perfecting their stag leap for cheerleading tryouts, I was taking self-development courses. When I was thirteen, I heard my first motivational speaker, Zig Ziglar. As I watched him walk back and forth across the stage, revealing the secrets of success and telling stories that gave me goosebumps, lightning struck. I realized that being a professional speaker was what I wanted to do. It was a strange career goal for a young teenage girl in the early 1970s. Even so, I pictured myself speaking in front of large audiences all over the world, inspiring people to change their lives for the better. My parents were supportive, even though my father was a dentist and they really wanted me to be a dental hygienist. Mom said, "You sure talk enough, so you might as well get paid for it."
I moved my Nancy Drew mysteries to the side to make room for every humanistic psychology book I could lay my hands on. I devoured them. At the age of sixteen, I began meditating every day, and by the time I turned twenty, I was a teacher of meditation. Though meditating had a profound impact on my life, I was still searching.
As time passed, I never lost sight of my goal of becoming a speaker. I immersed myself in success principles and did my best to put every one into practice. I tithed a percentage of my income and visualized my goals. I made vision boards to help me picture those goals and discovered I had a gift for manifesting my desires. For example, after earning my MBA, I drew to myself a wonderful job that included many qualities I had always yearned for in a career. As vice president of marketing for a company that sold Austrian crystal, I was responsible for training and inspiring employees. I loved it! I taught them everything I'd studied in my own life: the principles of the Law of Attraction, being clear about what you want, and knowing how to harness your intuition, overcome obstacles, and achieve your goals.
From there, I moved on to teaching those same principles of success as a corporate trainer for Fortune 500 companies and then for a national seminar company speaking to large audiences of women all over the country. With every new position came a bigger paycheck and more kudos. But I wasn't exactly happy. I knew there was something still missing. I just couldn't tell you what it was.
Maybe it was my subject matter, I decided. So instead of teaching success seminars, I began to teach self-esteem seminars for women. Jack Canfield, the nation's top expert on self-esteem, became my amazing mentor—years before he created the mega-best-selling book series Chicken Soup for the Soul—and soon I was giving keynotes to two or three hundred women a day on self-esteem. I taught on my high heels from 7:00 in the morning until late afternoon, then jumped in a car and drove three hours to the next city, day after day after day. It was exhausting but exhilarating. I loved standing in front of my audiences and watching their faces light up. Yet I still felt it wasn't quite enough—I wanted to reach more people.
Then came my big break. It began with a decision to take care of myself. Burned out from all my traveling, I signed up for a seven-day silence retreat, a real challenge for a Chatty Cathy like me. On the fourth day, in the middle of meditation, the proverbial lightbulb clicked on. I flashed on a title: Chicken Soup for the Woman's Soul. Up to this point, only the original Chicken Soup book had been published, and I knew this idea was a colossal winner. I was so excited; I'd just had the epiphany of my career. The only problem was I had to stay silent for three more days! The minute the retreat ended, I ran to the nearest payphone and called Jack. A year and a half later, Chicken Soup for the Woman's Soul became a #1 New York Times best-seller and I went on to write five more Chicken Soup for the Soul books that have sold over 13 million copies.
There I was, on national TV and radio shows, giving speeches to huge audiences, being treated like a queen and living the whirlwind life of success. At one conference, I was picked up in a white stretch limo to speak to a crowd of 8,000. Throughout the three-day event, thousands of women stood in a line that stretched around the entire convention center waiting for me to sign their books. A massage therapist rubbed my hands every hour as I autographed copy after copy after copy—so many they had to be airlifted from every corner of the country to meet the demand. Many women in the autograph line told me my books had changed, or even saved, their lives. I was deeply moved by their stories and felt good that I had done something that made a difference. But when I went to my hotel room each night, I flopped on the bed, feeling drained and strangely flat.
You'd think I'd have been on top of the world. But I wasn't. Sure, my ego had gotten a boost, but I still had the same worries, tensions, complaints, and bad hair days as everyone else. At each step of the way, I was excited about the successes I'd achieved, but I noticed that the high never lasted. I was happy about the things in my life, but not really happy.
Yes, I know how this sounds. Cry me a river, you're thinking. Well, I too have sat through many an E! True Hollywood Story and rolled my eyes over the sad story of the celeb whose rise to fame and fortune brought only tears. Oh please, I'd tell myself, if I were in their shoes, I'd be so happy you'd have to tie me down so I wouldn't float away. But now I was having a taste of that life, and the deep happiness I craved just wasn't happening. I was meeting famous people and seeing that many of them weren't happy either.
Maybe the problem wasn't my career, but my love life. If I just found the right partner, I told myself, then I'd be happy. I approached this challenge with the same single-mindedness I'd given to pursuing success. I dated enthusiastically and had a few relationships that were close but not it. Then, one weekend, I attended a seminar at a large retreat center in upstate New York. There, in the gravel parking lot, we were introduced by friends—and before we'd even said hello, he took me in his arms and waltzed me around, sweeping me off my feet with his European flair. Sergio, my Italian Prince Charming, had arrived. Like most romances, it had its ups and downs, but eventually we settled down together, bought a lovely house, and moved in.
Finally, I had the life I'd imagined: the house, the man, the career, and a great social life. (Okay, so I didn't have Halle Berry's body—but four out of five isn't bad.) Still, I couldn't shoo away the thoughts of dissatisfaction that kept popping up in my head or escape the gnawing pain I felt in my heart.
I realized I had a big problem. I couldn't continue to acquire or accomplish any more thinking that would make me happy. My life until that point had proven the futility of that approach. I'd come to the end of the line. Something had to change.
I had to admit the awful truth to myself: I still felt empty. I had every reason to be happy and yet I wasn't.
Although it seems obvious in hindsight, I had believed for so long that happiness would come from what I owned, achieved, or experienced that it took me a while to finally get it. Maybe happiness didn't come from the reasons I had imagined. Maybe happiness didn't come from any reason at all.
That's when I shifted my focus to the idea of Happy for No Reason and started applying the principles I discovered through my research and interviews. As a result, my own happiness level took a quantum leap. I felt a greater sense of peace and well-being that came from deep inside. I found myself singing throughout the day and appreciating the people around me more. I knew I'd made real progress when about five years ago my friends started calling me the "joy bunny." I was as thrilled as if I had won the Nobel Prize.
Of my findings, one piece of information stood out. It completely transformed my approach to being happy and explained why, for so many years, true happiness had always been just outside my grasp.
### Why Some People Are Happier Than Others
If you and I were sitting over some tea at a sidewalk café and I asked you, "Are you happy?," what would your answer be?
A few of you might say, "Absolutely—if I were any happier, I'd be twins!" (Okay, that would be a very few of you.)
A lot of you would probably reply, "Sometimes."
But I'd bet dollars to doughnuts that at least half of you would say, "No, not really."
Some people enjoy their lives no matter what happens, while others can't find happiness no matter how hard they try. Most of us fall somewhere in between.
The reason for this puzzling disparity is the happiness set-point I mentioned in the introduction. Researchers have found that no matter what happens to you in life, you tend to return to a fixed range of happiness. Like your weight set-point, which keeps the scale hovering around the same number, your happiness set-point will remain the same unless you make a concerted effort to change it.
In fact, there was a famous study conducted that tracked people who'd won the lottery—what many people think of as the ticket to the magic kingdom of joy. Within a year, these lucky winners returned to approximately the same level of happiness they'd experienced before their windfall. Surprisingly, the same was true for people who became paraplegic. Within a year or so of being disabled, they also returned to their original happiness level.
Whatever the experience—positive or negative—people return to their happiness set-point. Further research has shown only three exceptions to this phenomenon: losing a spouse, which can take more time to recover from; chronic unemployment; and extreme poverty.
Okay, you may be thinking, if my happiness level is set—how did it get there? Dr. David Lykken, a scientist at the University of Minnesota, had the same question. To determine how much of a person's happiness comes from nature and how much from nurture, Lykken and his team, in the late 1980s, began studying thousands of sets of twins, including a number of identical twins who had been raised apart. After extensive testing they found that approximately 50 percent of our happiness set-point is genetic and the other 50 percent is learned. Half of the reason you walk around generally cheery or perennially dreary is that you were born that way, the other half is determined by your thoughts, feelings, and beliefs formed in response to your life experiences.
In a recent review of the literature and studies on happiness, positive psychology researchers Sonja Lyubomirsky, Kennon Sheldon, and David Schkade confirmed Lykken's earlier findings that 50 percent of our happiness comes from our genetics. But more exciting was the new information they uncovered about the remaining 50 percent. It appears that only 10 percent of our happiness set-point is determined by circumstances such as our level of wealth, marital status, and job. The other 40 percent is determined by our habitual thoughts, feelings, words, and actions. This is why it's possible to raise your happiness set-point. In the same way you'd crank up the thermostat to get comfortable on a chilly day, you actually have the power to reprogram your happiness set-point to a higher level of peace and well-being.
The discovery of the happiness set-point and our ability to change it turns everything we've all believed about being happy upside down. We spend our entire lives searching for happiness, yearning for it, trying to get the things we are sure will make us happy: wealth, beauty, relationships, career, and so on. But the truth is, to be truly happy, all you have to do is raise your happiness set-point.
I sure wish I'd known I was just one of those people who had a low happiness set-point before spending so much time and energy chasing after reasons to be happy. From my interviews with the Happy 100, I've come to see that genuinely happy people are Happy for No Reason.
Let's look at this phenomenon.
### The Happiness Continuum
Happiness for any reason is just another form of misery.
—The Upanishads
One day, as I sat down to compile my findings, all the pieces of the puzzle fell into place. I had a simple, but profound "a-ha"—there's a continuum of happiness:
Unhappy: We all know what this means: Life is a bummer. Some of the signs are anxiety, fatigue, feeling blue or low—your garden-variety unhappiness. This isn't the same as clinical depression, which is characterized by deep despair and hopelessness that dramatically interferes with your ability to live a normal life, and for which professional help is absolutely necessary.
Happy for Bad Reason: When people are unhappy, they often try to make themselves feel better by indulging in addictions or behaviors that may feel good in the moment but are ultimately detrimental. They seek the highs that come from drugs, alcohol, excessive sex, "retail therapy," compulsive gambling, overeating, and too much television watching, to name a few. This kind of happiness is hardly happiness at all. It is only a temporary way to numb or escape our unhappiness through fleeting experiences of pleasure.
Happy for Good Reason: This is what people usually mean by happiness: having good relationships with our family and friends, success in our career, financial security, a nice house or car, or using our talents and strengths well. It's the pleasure we derive from having the healthy things in our lives that we want.
Don't get me wrong. I'm all for this kind of happiness! It's just that it's only half the story. Being Happy for Good Reason depends on the external conditions of our lives. If these conditions change or are lost, our happiness usually goes too.
Deep inside, you know that life isn't meant to be about getting by, numbing your pain, or having everything "under control." True happiness doesn't come from merely collecting an assortment of happy experiences. At your core, you know there's something more than this.
You're right. There is one more level on the happiness continuum: Happy for No Reason.
Happy for No Reason: This is true happiness—a neurophysiological state of peace and well-being that isn't dependent on external circumstances.
Happy for No Reason isn't elation, euphoria, mood spikes, or peak experiences that don't last. It doesn't mean grinning like a fool 24/7 or experiencing a superficial high. Happy for No Reason isn't an emotion. In fact, when you are Happy for No Reason, you can have any emotion—including sadness, fear, anger, or hurt—but you still experience that underlying state of peace and well-being.
When you're Happy for No Reason, you bring happiness to your outer experiences rather than trying to extract happiness from them. You don't need to manipulate the world around you to try to make yourself happy. You live from happiness, rather than for happiness.
This is a revolutionary concept. Most of us focus on being Happy for Good Reason, stringing together as many happy experiences as we can, like beads in a necklace, to create a happy life. We have to spend a lot of time and energy trying to find just the right beads so we can have a "happy necklace."
Being Happy for No Reason, in our necklace analogy, is like having a happy string. No matter what beads we put on our necklace—good, bad, or indifferent—our inner experience, which is the string that runs through them all, is happy, creating a happy life.
When you're Happy for No Reason you're unconditionally happy. It's not that your life always looks perfect—it's just that however it looks, you'll still be happy.
As the thirteenth-century poet Rumi described it, "Happy, not from anything that happens. Warm, not from fire or a hot bath. Light, I register zero on a scale." Whenever I asked the Happy 100 to describe the qualities of being Happy for No Reason, I got the same answers over and over:
• Having a sense of lightness or buoyancy
• Feeling alive, vital, energetic
• Having a sense of flow, openness
• Feeling love and compassion for yourself and others
• Having passion about your life and purpose
• Feeling gratitude, forgiveness
• Being at peace with life
• Being fully present in the moment
Matthieu Ricard, a French scientist who became a Buddhist monk over thirty years ago, is often called "the happiest man in the world" by researchers who've measured his brain's functioning both in and out of meditation. (More about monks in the laboratory in Chapter 7.) Ricard's book, Happiness: A Guide to Developing Life's Most Important Skill, gives one of the clearest descriptions of Happy for No Reason I've ever heard. He writes, "By happiness I mean a deep sense of flourishing that arises from an exceptionally healthy mind. This is not a mere pleasurable feeling, a fleeting emotion, or a mood, but an optimal state of being."
### How Happy for No Reason Are You?
The following Happy for No Reason questionnaire will give you a snapshot of how Happy for No Reason you are in your life right now. Though you may have filled out happiness questionnaires before, you may not have noticed that they're usually state-dependent; that is, they ask you to rate your happiness according to what's going on in your life (job, career, relationships, and so on) and how satisfied you are with your life circumstances. Those questionnaires measure Happy for Good Reason. This questionnaire is different; it measures Happy for No Reason.
The Happy for No Reason questionnaire is modeled after the Well-Being Scale that is part of the Multidimensional Personality Questionnaire, a tool developed by Auke Tellegen, a professor of psychology at the University of Minnesota, to help researchers determine happiness set-points. As you answer these questions, think about how they apply to you in general.
The Happy for No Reason Questionnaire
Rate each statement on a scale of 1 to 5:
1 = Not at all true
2 = Slightly true
3 = Moderately true
4 = Mostly true
5 = Absolutely true
1. I often feel happy and satisfied for no particular reason.
1 2 3 4 5
2. I live in the moment.
1 2 3 4 5
3. I feel alive, vital, and energetic.
1 2 3 4 5
4. I experience a deep sense of inner peace and well-being.
1 2 3 4 5
5. Life is a great adventure for me.
1 2 3 4 5
6. I don't let bad situations keep me down.
1 2 3 4 5
7. I am enthusiastic about the things I do.
1 2 3 4 5
8. Most days I have an experience of laughter or joy.
1 2 3 4 5
9. I trust that this is a friendly universe.
1 2 3 4 5
10. I look for the gift or the lesson in everything that happens.
1 2 3 4 5
11. I am able to let go and forgive.
1 2 3 4 5
12. I feel love for myself.
1 2 3 4 5
13. I look for the good in every person.
1 2 3 4 5
14. I change the things I can and accept the things I can't change.
1 2 3 4 5
15. I surround myself with people who support me.
1 2 3 4 5
16. I don't blame others or complain.
1 2 3 4 5
17. My negative thoughts don't overshadow me.
1 2 3 4 5
18. I feel a general sense of gratitude.
1 2 3 4 5
19. I feel connected to something bigger than myself.
1 2 3 4 5
20. I feel inspired by a sense of purpose in my life.
1 2 3 4 5
Scoring section:
If your score is 80–100: To a great degree, you are Happy for No Reason.
If your score is 60–79: You have a good measure of being Happy for No Reason.
If your score is 40–59: You have glimpses of being Happy for No Reason.
If your score is under 40: You have little experience of being Happy for No Reason.
Whatever your score, you can always move toward being more Happy for No Reason. As I said earlier, it doesn't matter where you begin; what matters is that you do begin. Once you've finished reading the book and have begun practicing the seven steps and the Happiness Habits, take the questionnaire again. After that, assessing your Happy for No Reason score on a regular basis will help you chart your progress.
### Happy for No Reason: Your Natural State
Happy for No Reason isn't just a nice idea. As I'll explain in later chapters, it's a specific, measurable physiological state characterized by distinct brain activity, heart rhythms, and body chemistry.
Scientists tell us that every subjective experience we have has a corresponding state of functioning in our bodies. People who are Happy for No Reason tend to have greater activity in the left prefrontal cortex, orderly heart wave patterns, and more of the specific neurotransmitters associated with well-being and happiness: oxytocin, serotonin, dopamine, and endorphins.
Although modern science gives us new insight into the physiology of Happy for No Reason, it's a state that's been spoken of in virtually all spiritual and religious traditions throughout history. The concept is universal. In Buddhism, it is called causeless joy, in Christianity, the kingdom of Heaven within, and in Judaism it is called ashrei, an inner sense of holiness and health. In Islam it is called falah, happiness and well-being, and in Hinduism it is called ananda, or pure bliss. Some traditions refer to it as an enlightened or awakened state.
I've noticed the widespread recognition of this concept around the world. No matter where I go, when people hear the expression Happy for No Reason, it strikes a deep chord in them. We seem to know intuitively that our innermost essence is happiness. You don't have to create it; it's who you are. The rest of this book is devoted to showing you how to get back to that natural state.
## 2
## Practicing Happiness
Things do not change; we change.
—Henry David Thoreau, writer and philosopher
Think of people you know who are Happy for No Reason. Often, these people affect us like small suns, radiating warmth and positive energy to everyone who comes into their field of influence. They are certainly optimists, who not only see the glass as half-full, but who carry a pitcher to fill the glass all the way up—and top off others' while they're at it. They're not always rah-rah cheerleaders; sometimes they're simply a quiet presence of inner peace and contentment, centers of calm in the chaos where many of us live. These are the people you love to spend time with because they have a way of raising your spirits even on your grumpiest gray days.
I was blessed to grow up with one of these Happy for No Reason people: my father, Marc. He'd clearly hit the happiness set-point jackpot. No matter what he did or where he went, there was a perpetual sparkle about him. Yet, my dad's happiness didn't stem from his outer circumstances. He grew up quite poor during the Depression, struggled to pay for his education, had his share of personal losses, and stood a whopping 5'4" on a good day. But none of that seemed to matter to him. He loved everything.
After working his way through dental school, including a one-year stint in a chocolate factory (perhaps ensuring future patients), he went on to serve four years during World War II as an army dentist in the South Pacific. Though he certainly didn't relish being in the midst of battle, he never lost his inner sense of happiness. He was so devoted to my mother that he wrote her a letter every single day he was overseas—858 letters total—some of which have survived to this day. While in the army, he saved enough money from his poker earnings to finance his first dental practice when he returned home. He loved his career as a dentist, and when he retired, he did some amazing things that you'll hear about in a later chapter.
Although he continued to face challenges after the war—sometimes money was tight, we kids didn't exactly pop out of a Norman Rockwell painting, and his health eventually failed—my dad still felt happy all the time, passing away peacefully at age ninety-one, having spent his ninetieth birthday playing golf.
My father woke up every morning excited and grateful to be on this adventure called life. He was my first Happy for No Reason role model and the inspiration for this book. One day when I was about nineteen, I asked him his advice for life. He answered with two words: Be happy!
"Great, Dad," I said, "but how?"
He had no answer for me. Being happy was so natural to him that he couldn't understand why everyone didn't feel that way and why they were so busy pursuing happiness.
### What Thomas Jefferson Really Meant
When I talk to people about happiness, they often quote Thomas Jefferson's famous line from the Declaration of Independence. "Sure, I want to be happy," they tell me. "After all, isn't everyone entitled to the right to life, liberty, and the pursuit of happiness?"
We've been conditioned to believe that happiness is something for us to pursue. So, like a dog going after a stick, we chase after happiness, grabbing at all the things we think will bring it to us.
Then, one day, I discovered what Thomas Jefferson really meant.
I was on an airplane going to a conference with my good friends, Stewart and Joan Emery, leading lights in the human potential movement. We were talking about the concept of happiness—something I do a lot—when Stewart turned to me and said in his charming Australian accent, "Marci, do you know what Thomas Jefferson really meant by the pursuit of happiness?"
Stewart, the coauthor of Success Built to Last, knows many fascinating and often obscure facts. Back in Jefferson's day, he explained, the common usage of the word "pursue" was not "to chase after." In 1776, to pursue something meant to practice that activity, to do it regularly, to make a habit of it.
What a difference a definition makes! Thomas Jefferson, our wise Founding Father, meant that we all had the right to practice happiness, not chase after it—which isn't very productive anyway.
So let's stop pursuing happiness and start practicing it. We do that by practicing new habits.
### The Habits of Happy People
People with high happiness set-points are human just like the rest of us. They don't have special powers, an extra heart, or X-ray vision. They just have different habits. It's that simple. Psychologists say that at least 90 percent of all behavior is habitual. So, to become happier, you need to look to your habits.
Some books and programs will tell you that you can simply decide to be happy. Just make up your mind to be happy, and you will be.
I don't agree.
You can't just decide to be happy, any more than you can decide to be fit or to be a great piano virtuoso and expect instant mastery. You can, however, decide to take the necessary steps, like exercising or taking piano lessons, and by practicing those skills, you can get in shape or give recitals. In the same way, you can become Happy for No Reason through practicing the habits of happy people.
All of your habitual thoughts and behaviors in the past have created specific neural pathways in the wiring in your brain, like grooves in a record. When we think or behave a certain way over and over, the neural pathway is strengthened and the groove becomes deeper, in the way that a well-traveled route through a field eventually becomes a clear-cut path. Unhappy people tend to have more negative neural pathways. This is why you can't just ignore the realities of your brain's wiring and decide to be happy! To raise your happiness set-point, you have to create new grooves.
Scientists used to think that once a person reached adulthood, the brain was fairly well set in stone and there wasn't much you could do to change it. But new research is revealing exciting information about the brain's neuroplasticity: when you think, feel, and act in different ways, the brain changes and actually rewires itself. You aren't doomed to the same negative neural pathways for your whole life. Leading brain researcher Dr. Richard Davidson of the University of Wisconsin says, "Based on what we know of the plasticity of the brain, we can think of things like happiness and compassion as skills that are no different from learning to play a musical instrument or tennis . . . it is possible to train our brains to be happy."
While a few of the Happy 100 I interviewed were born happy, most of them learned to be happy by practicing habits that supported their happiness.
According to the Dalai Lama, one of my happiness heroes (I can't help but smile when I think of him), it's important to know which habits support happiness in your life and which don't. In his book, The Art of Happiness, he writes:
One begins identifying those factors which lead to happiness and those factors which lead to suffering. Having done this, one then sets about gradually eliminating those factors which lead to suffering and cultivating those which lead to happiness. That is the way.
Throughout the book, I'll be identifying the factors that lead to happiness. But what are those "factors that lead to suffering" and block our happiness? Two of these Happiness Blockers, the Myth of More and the Myth of I'll-Be-Happy-When, are so prevalent in our culture, they have almost everyone hooked.
### The Myth of More
Who is rich? He who is happy with his lot.
—The Talmud
Most of us fall prey to what I call the Myth of More: the more you have, the better you'll feel. Our society's collective trance of wanting more and more stuff, or "bright, shiny objects," as my friend Stewart calls them, is based on a shared, insidious, and often unconscious belief: more toys, success, and money mean more happiness. But as these statistics show, it just isn't true:
• Americans' personal income has increased more than two and a half times over the past fifty years, but their happiness level has remained the same.
• Nearly 40 percent of the people on the Forbes list of wealthiest Americans are less happy than the average American.
• Once personal wealth exceeds $12,000 a year, more money produces virtually no increase in happiness.
It's obvious that the happiest people aren't the ones with all the goodies. (If they were, there'd be a lot more happy people in Hollywood!) Yet we are seduced by the deep-seated belief that money will buy happiness—at least we think it will for us. A recent survey showed that at all income levels people think more money would definitely increase their level of happiness.
I heard that a reporter once asked Andrew Carnegie, the famous industrialist who'd amassed a fortune in steel, "You're the richest man in the world. How much is enough?" He thought for a moment and replied, "Just a little more."
This shows that our "desire to acquire" won't bring us true joy. So why is it so hard to escape wanting more?
Because Madison Avenue doesn't want us to. Advertising exists to perpetuate the Myth of More; it's the engine that drives our economy. Billions of dollars are spent every year to convince you that you're not okay the way you are and that you need things—lots and lots of them—to make you happy. One night I did an experiment. I counted how many times and ways I was told this over the course of that night's television watching.
It was shocking. In just three hours, I was bombarded by sixty-eight messages telling me that I was doomed to misery if I didn't have what that company was selling. In the most entertaining, compelling, and creative ways possible, advertisers tried to convince me that I needed to have the right car (five different companies claimed theirs was the right one), the sexiest bra (like the one decorated with diamonds that cost $2 million—that's one million per boob!), the magic pills (we'll talk about those later), and the best skin care product (God forbid I should look my age).
I know what you're thinking: But I don't pay attention to those commercials. They don't affect me. Sorry to burst your bubble, but they do. You can't help it: the messages we see and hear repeatedly go into our brain on a subconscious level and become beliefs. If they didn't, advertisers wouldn't be spending oodles of money to make sure you view their ads over and over again.
With children watching an average of five hours of television a day, is it any surprise that we have a bunch of unhappy kids who are in a frenzy to get the next toy, video game, or designer-label jeans? If you've been around children at Christmas, you'll know why I was so touched by the following story told to me by a young father I interviewed:
When my oldest daughter, Victoria, was almost three, we read Dr. Seuss's How the Grinch Stole Christmas every night to her before the holiday.
She'd curl up beside me as I'd read: Every Who down in Who-ville liked Christmas a lot. . . .
Victoria followed along as the Grinch unveils his plans to ruin the Christmas of the Whos. Disguising himself as Santa and his dog as a reindeer, the Grinch steals into the Whos' homes and takes everything, leaving only the hooks and wires on the bare walls. But to his surprise, the Whos remain happy despite the loss of the presents and trees and trimmings and trappings. He hadn't stopped Christmas from coming; "it came just the same."
On that Christmas morning, we woke just ahead of Victoria so that we could watch her three-year-old enthusiasm as she saw the presents under the tree. She first ran to the kitchen table where she had left a snack for Santa and his reindeer. She looked at the evidence of Santa's visit: the cookie crumbs on the plate and the empty milk glass and the missing carrots. My wife, pregnant with our second child, and I beamed seeing our daughter so wide-eyed and excited at the thought that Santa himself had been in our home. Next, she ran into the living room and saw the presents under the tree.
We expected her to dive into them—but she didn't. She held up her little hand and she said, "Stop. Let's pretend. Let's pretend the Grinch has been here and took everything and left just hooks and wires and we'd still be happy."
So we stopped, and were happy. And like the Grinch, my heart grew three sizes that day.
What would our lives be like if we could all be happy no matter what?
When you experience your inner, innate happiness and are Happy for No Reason, you still enjoy the things in your life, but you don't look to them to make you happy. You're able to banish the Myth of More.
### The Myth of I'll-Be-Happy-When
A close cousin to the Myth of More is the Myth of I'll-Be-Happy-When. How many of the following statements sound familiar to you?
• I'll be happy when I have the perfect mate.
• I'll be happy when I have a better job.
• I'll be happy when I have a baby (or another baby).
• I'll be happy when the kids are in school.
• I'll be happy when I get more recognition or appreciation.
• I'll be happy when I can retire.
And the ever so popular:
• I'll be happy when I lose 5 (10, 15, 20) pounds.
No matter how many I'll-be-happy-whens you reach, it's never quite enough. With each one, you experience either fleeting satisfaction or outright disappointment. Think about the last five goals you achieved. How much happiness did they bring? And how long did that happiness last?
Yet you ignore the feeling that something is missing, and you keep trying: you work harder and harder, telling yourself, Just a little more and I'll be there. If I could just get this handled, then I'd have it. You're like a hamster on a wheel. Round and round you go, madly trying to control and manipulate your external circumstances—and always feeling scared of losing whatever it is that you already have. When you're caught up in I'll-be-happy-when, happiness is always off in the future, while in truth, the only time you can actually experience happiness is right here in this moment.
Recent studies by Daniel Gilbert, a well-respected Harvard psychology professor, have proven the utter futility of I'll-be-happy-when thinking. Gilbert, the author of Stumbling on Happiness, has done some fascinating research that shows we humans aren't very good at predicting what will make us happy in the future. He's concluded that time and time again, we overestimate the happiness we'll receive from getting the stuff we want. We imagine how great we'll feel when we go on a certain vacation or get a promotion or have a particular relationship, but when we actually get them, we are usually far less happy than we thought we'd be. Moreover, we habituate to whatever we thought would make us happy, so the thrill diminishes every time we experience the magical it that was supposed to make us happy forever.
To become happier we have to snap out of the trance that our happiness rests in "more and better" which we'll get "someday." No matter what we have, Happy for No Reason exists only now, not later.
### The Guiding Three: Principles to Live By
Over time, as I listened to the Happy 100 describe their lives, I began to see certain clear patterns emerging. They'd all managed to banish the Myth of More and jump off the hamster wheel of I'll-Be-Happy-When. And they all seemed to live by the same three guiding principles, or universal laws. No matter what words the Happy 100 used to describe them, these principles played a significant role in helping them become, or remain, Happy for No Reason. I call these principles the Guiding Three:
1. What expands you makes you happier. (The Law of Expansion)
2. The universe is out to support you. (The Law of Universal Support)
3. What you appreciate, appreciates. (The Law of Attraction)
As you practice the Happiness Habits you'll learn in Part II, there may be moments you find yourself stuck in old behaviors and ways of thinking and feeling. It's at times like these that you can call on the Guiding Three to help you through.
### Guiding Principle #1: What Expands You Makes You Happier
Science has shown that everything in the universe, including you, is composed of energy. Everything you say, think, or do, everything you're around, either expands your energy or contracts it.
When your energy expands, you experience greater happiness; when your energy contracts, you experience less happiness.
You can experience this yourself by doing this simple exercise: Sit up straight right now. Throw back your shoulders, open your arms wide, take a deep breath, and smile. Close your eyes and notice how you feel.
You'd probably describe your experience something like this:
Free
Open
Joyful
There's a feeling of lightness and space—that's expansion.
Now think of someone you love, admire, and enjoy being around. How does your body feel when you think of that person?
Again, you probably feel expanded, open, light. Whenever you're happy, you're in a state of expansion. In fact, scientists can tell you how happy you are just by looking at the level of expansion in your body's functioning. If you were hooked up to scientific measuring instruments during moments of happiness, you'd be able to see the increase in oxygen absorption, dilation of blood vessels, muscle relaxation, the smoothness of your heart rhythms, and the increased integration of your brain's functioning—all signs that you've expanded your energy.
Now hunch your shoulders, make fists, take short, quick breaths, and put a frown on your face. Notice how you feel.
Anxious
Tight
Agitated
There's a feeling of heaviness—that's contraction.
Think of someone you fear or are angry at. What do you feel? That same heavy, contracted energy.
This is our basic experience of unhappiness. All of our negative emotions—anger, fear, sadness, jealousy—contract us and literally constrict the flow of our life energy. When we feel them, our muscles tighten, our breathing becomes shallow, and our circulation is hampered. If you were hooked up to those same measuring instruments, you'd be able to see the level of your stress hormones rise. This crashes your immune system and raises the risk of infection and illness.
The Happy 100 choose the thoughts, feelings, and behaviors that expand them, rather than the ones that contract them. Raising your happiness set-point is a matter of developing those habits that expand your energy. Here's what makes us feel expanded or contracted:
Expansion is your highway to happiness. You can use this guiding principle to see if you are moving toward happiness by being aware of how contracted or expanded you feel throughout the day.
#### Your Inner Guidance System, or the Hot/Cold Game
Have you been in one of the newer cars that have the built-in global positioning systems (GPS)? No matter where you are, the GPS can direct you to where you want to go. (You can even program the type of voice they speak to you in. My friend's GPS sounds like a well-bred British butler! She says it makes her feel like a queen.)
It turns out that we each have our own built-in GPS that leads us toward happiness in the form of an expansion/contraction feedback system. If you feel more expanded, you're going in the right direction; if you feel contracted, it's time for a course correction. It's like the game you probably played as a kid, when you were looking for something. As you got closer to your target, your friends would say, "Warm, warmer, hot!" Or if you were getting farther away, "Cold, colder, freezing." The only difference is your inner GPS says "expanded" or "contracted."
I rely on my inner GPS often. When I'm faced with a choice, I stop, take a deep breath, and feel which option brings more lightness, openness, and expansion. I do this for simple decisions, like choosing an entrée on a menu, and for bigger decisions, like deciding which business opportunities to pursue. I'm always happier with the choice that makes me feel expanded.
To see how much of your life you're living in either expansion or contraction, do the following exercise:
Expansion/Contraction List
1. Write two headings at the top of a piece of paper. Put "Contraction" on the left and "Expansion" on the right.
2. Think about your life: your job, your house, your body, your relationships, and so on. As you consider each item, close your eyes, take a deep breath, and feel whether it expands or contracts your energy and list it under the appropriate heading.
3. Review the lists and notice which areas of your life are contributing to your happiness and which are dragging you down. The items on the contraction list are areas for course correction.
All of the Happiness Habits that I recommend practicing in Part II are based on this first guiding principle: Everything that expands you makes you happier. In each chapter you'll find an expansion/contraction chart relating to that step. Paying attention to what contracts you and what expands you—listening to your inner GPS—will reinforce the behaviors that raise your happiness set-point and help you experience being Happy for No Reason.
### Guiding Principle #2: The Universe Is Out to Support You
Einstein once said that the most important question a person can ask himself is "Is this a friendly universe?" The Happy 100 answer this question with a resounding "Yes!" Rather than thinking the universe is out to get them, they believe the universe is out to support them.
What's striking is that they don't believe the universe is benevolent only when good things happen to them—they take this approach all the time. When something bad occurs, they don't moan and groan, "Why me? It's not fair." They see all the events in their lives through the lens of "Ultimately, this is happening for my good. There are no mistakes. Let me look for the blessing in this." This belief in a friendly universe is the root of their relaxed and trusting attitude in life.
There's even research showing that belief in a friendly universe can impact a person's health. A recent study conducted by Gail Ironson, PhD, MD, professor of psychology and psychiatry at the University of Miami, found that people with HIV who believed in a universal power that was loving remained healthier longer than people who believed in a universal power that was punishing.
This may be a hard concept for you to swallow. Certainly there are many horrible things that happen in this world: war, persecution, famine, and suffering. It's easy to think that we don't live in a friendly universe. The one thing that's helped me embrace the idea that the universe is always out to support me is the knowledge that all the wise men and women who have ever lived—the sages and saints of the past and present—have shared this belief. So when I'm having a hard time seeing how the universe is supporting me in a particular situation, I'm willing to admit that my thinking and understanding may be limited and to remind myself of their enlightened and clearer vision.
But we don't have to get into the philosophical or religious implications of this belief; that's not the purpose of this discussion. I'm just reporting that people who are Happy for No Reason use it as a guiding principle, and you can use it yourself to raise your own happiness level.
Rather than trying to decide whether this principle is true or not, I suggest that you adopt this perspective for the next week or two, and see how different your life feels at the end of that period. That means that whatever happens, assume that the universe is on your side—even if it isn't obvious to you on the face of it.
When I first began doing this myself, I noticed that although I wasn't jumping up and down with joy and loving everything that I had to face in my life, I definitely felt a lot more ease and peace inside. At times, when I was particularly sad or knocked off-center—when a romantic relationship ended, for example—believing that the universe was always supporting me and that there was a bigger plan helped me get beyond my usual broken record of "It's so unfair" and "I'll never find true love." It helped me maintain an open heart that ultimately led me to Sergio.
When you believe that the universe is out to support you, you're able to stop resisting what is happening. This doesn't mean being passive or complacent about the events in the world or in your own life. It simply means not fighting or bemoaning what has already happened and can't be changed. Many of us spend a tremendous amount of energy being upset and resisting life. When you take the view that there are no mistakes and accept what is, you can use your energy instead to deal effectively with the situation now.
Trusting that the universe is always providing ways to support you and your best growth is a great tool to keep you feeling expanded.
### Guiding Principle #3: What You Appreciate, Appreciates
This principle is based on the Law of Attraction, which, simply stated, is Like attracts like: whatever you think, feel, say, and act on, you draw to yourself like a magnet. Whenever you appreciate the happiness that already exists in your life, like money in the bank, it appreciates!
There's been a lot of buzz about the Law of Attraction in recent years, thanks in great part to the hit film and book phenomenon The Secret. I am honored that creator and producer Rhonda Byrne chose me to be part of this transformative film and book. Rhonda truly walks her talk: she used the Law of Attraction to create The Secret's megasuccess. Her intention and vision has always been to spread joy and see the world transformed by using the Law of Attraction. The first time I met Rhonda, I was struck by how radiantly happy she was. Although she went through a period of great despair in her life, today she exudes a joyful enthusiasm combined with a deep peace. You'll find her remarkable story in Chapter 8 of this book.
Many people focus on using the Law of Attraction to draw in the things they think will make them happy. But that's backwards. Being happy is what draws in the things we want—it's the basis of the Law of Attraction. In Rhonda's book, The Secret, she writes:
I want to let you in on a secret to The Secret. The shortcut to anything you want in your life is to BE and FEEL happy now! It is the fastest way to bring money and anything else you want into your life. Focus on radiating out into the Universe those feelings of joy and happiness . . . Everything you want is an inside job! The outside world is the world of effects; it's just the result of thoughts. Set your thoughts and frequency on happiness.
When you feel good, your energy creates a powerful vibrational field that draws to you what you want more easily. While it's fine to manifest your ideal car, home, or job, as we've already seen, the key to happiness in life isn't manipulating the world to give us what we want. The highest use of the Law of Attraction is applying it to the goal that underlies all goals: becoming Happy for No Reason. Though the Law of Attraction has worked like a charm for me in the material realm, I've received far greater rewards whenever I've applied it on the spiritual level, to heal myself from heartbreaks by focusing on gratitude, and to create transformation in my life by appreciating any step of progress, no matter how small. Using the Law of Attraction in this way leads to being Happy for No Reason.
### My Secret Formula
My favorite tool for using the Law of Attraction is what I refer to as my Secret Formula: Intention, Attention, No Tension. I first learned this simple three-part process from my friend, performance consultant Bill Levacy. Here's how you can apply it to becoming Happy for No Reason.
1. Intention: Be clear about what you want—in this case, your desire for greater happiness.
2. Attention: What you put your attention on grows stronger in your life. Put your attention on happiness by practicing the Happiness Habits each day.
3. No Tension: Let go and relax. As you practice the habits, be easy with yourself and trust that you are removing the blocks to experiencing greater happiness.
### Setting Your Intention and Envisioning Your Ideal
All conscious change begins with intention. Before you can raise your happiness set-point, it's important to set your intention to be Happy for No Reason. Start by writing down a declaration of your intention. Begin with "I'm grateful that I am . . ." and complete the sentence with what Happy for No Reason feels like to you.
I want you to use the phrase "I am" because these are the two most powerful words in the English language; they help call your intention into being. Please note that I'm asking you to declare this intention in the present tense, as though you are already experiencing being Happy for No Reason. The power and immediacy of the present tense magnetizes your heart's desire to you. For example, my personal Happy for No Reason intention is "I'm grateful that I am experiencing a deep state of inner peace and happiness."
Your Happy for No Reason Intention
* * *
Now picture yourself being Happy for No Reason. What would your life be like if you were experiencing that state of unshakable inner peace and well-being? What would you feel and do? How would you interact with others?
Imagining how you want to feel may seem fanciful or silly, but it's actually a very powerful exercise. The more clearly you can experience what Happy for No Reason feels like to you, the more easily you will bring it into being.
Just doing this process puts you in the vibrational field of Happy for No Reason. You probably began to feel happier just from intending and imagining it.
I also recommend that you create a vision board to look at as you practice the Happiness Habits. A vision board is a visual representation of whatever you want to create in your life. Many people use these boards to focus on the things they want to get: a car, a relationship, a home. I suggest you create a board that focuses on images that represent states of feeling happy. You might choose a picture of a beautiful spot in nature or an image of someone laughing or dancing. Perhaps you'll use pictures of yourself with people that you love or admire. Choose images that make you feel expanded, open, and uplifted, even if you can't put your finger on exactly why, and include your Happy for No Reason intention statement. Just as with your intention, it's important to focus on the things on the board as if they already exist.
My own board has pictures of me at my happiest, in nature and with people I love, and images, colors, and quotes that inspire joy in me. It's hanging on the wall across from my desk and I look at it throughout the day. Spend some time each day looking at your vision board and feeling the happiness it inspires in you.
### Building Your Inner Home for Happiness
Now you're ready to begin putting your intentions into practice. In Part II, you'll learn the seven steps to build your Home for Happiness. After reading the book, I suggest you spend a week on each of these steps, practicing the Happiness Habits, doing the exercises, and taking the actions steps listed at the end of each chapter. You may find that some habits come easily, while others will present more of a challenge. When that happens, take the time you need to make the new practice easy.
As you work with these ideas and techniques, keep the Guiding Three in mind. Use your inner GPS to move you in the direction of expansion. See what happens when you choose to believe the universe is out to support you. And keep working with the Law of Attraction to raise your happiness set-point by deeply appreciating the happiness you're already experiencing.
With these tools in hand, it's time to begin building your Home for Happiness. Let's get started.
# PART II
# Building Your Home for Happiness
We first make our habits, and then our habits make us.
—John Dryden, seventeenth-century English poet and dramatist
## 3
## The Foundation—Take Ownership of Your Happiness
Most of the shadows of this life are caused by standing in one's own sunshine.
—Ralph Waldo Emerson, writer and philosopher
Though building a house isn't at the top of my list of skills, I do know that the first step is to lay the foundation. It's the basis of a sturdy home.
Building your Home for Happiness starts with the same important first step: you lay the foundation by taking ownership of your happiness. That means first believing you can be happy, then becoming aware of the habits that are standing in your way, and, finally, gently and persistently shifting to new habits of thinking, feeling, and acting that will serve you better.
One of the things that struck me most about the Happy 100 was that in interview after interview no one questioned their ability to be happy. They knew that it was possible—and that their happiness was up to them. They didn't put off being happy for the future, waiting for the conditions of their life to be exactly right or hoping that someday they'd get lucky. And they didn't get stuck in the past, saying, "I can't be happy because of where I came from or what's happened to me." They were proactive about their lives, focusing on the possibilities for their future, rather than being victimized by their past or present circumstances.
This link between happiness and taking responsibility was demonstrated in an important study published in 2005 in the International Journal of Behavioral Medicine. The study, led by Dr. Gail Ironson, whose research I introduced in the previous chapter, showed that subjects diagnosed with HIV who were optimistic were more proactive in their behavior, coped better, and showed slower progression of the disease. According to Dr. Ironson, proactivity is highly correlated with optimism, a trait associated with increased happiness.
The good news is, no matter where you start, when you're proactive and take total responsibility for experiencing happiness, you put yourself in the driver's seat in life. Then you can take quantum leaps in raising your happiness set-point.
At our core, we are all already Happy for No Reason. An inner sun of peace and well-being is shining brightly in each of us, though it is often covered over with clouds. In this chapter, we'll shine a light on those clouds—our patterns of victimhood—the old ingrained habits that are blocking our ability to experience our essential happiness all the time.
### Taking Ownership
What I've noticed in my work is that more and more people are waking up to the power they have to create their future through everything they think, feel, say, and do. They're recognizing that their choices determine their lives. When you understand the Law of Attraction, you know that you're responsible for the experience of happiness (or unhappiness) in your life.
Taking ownership of your happiness has two aspects:
1. Accepting that being happy is up to you and that you have the ability and power to be happier by changing your habits.
2. Taking "response-ability": responding to all the events in your life in a way that supports your happiness.
By reading this book, you've already begun. This isn't just "inspiring talk." It's been proven that simply putting your attention on becoming happier has a powerful effect. In fact, one of the very first happiness experiments ever conducted demonstrated this. In 1977, Dr. Michael Fordyce, a psychologist and the author of The Psychology of Happiness, published the groundbreaking results of his experiment showing that students asked to study the habits of happy people actually increased their happiness and life satisfaction by just learning about the subject.
Yet, as Dr. Sonja Lyubomirsky, psychology professor at UC Riverside, whose research I discussed in Chapter 1, points out, investing time and attention to become happier is like diet and exercise: you can't do it for just a day or two, it has to be continued over time. Unfortunately, most people put more energy into planning which car they're going to buy than into raising their level of happiness.
### Response-ability: The Ability to Respond
Our ability to respond to what happens to us—our response-ability—dramatically affects our happiness. The Happy 100 respond to the events in their lives in a way that supports their inner peace and well-being.
Years ago, my mentor, Jack Canfield, taught me the following simple equation that explains this concept:
E + R = O (Events + Response = Outcome)
People who are Happy for no Reason orchestrate the events in their lives when they're able to. When they're not able to change the events, they change their responses.
Next time you're in a traffic jam, look around you. You'll probably see someone scowling, yelling at the other cars, and clutching the steering wheel in a death grip. Someone in another car may be shimmying in her seat to the radio, singing at the top of her lungs and boogeying down. Same event, different responses.
Every single time you make a choice to respond in a way that expands you and creates more peace and well-being, you're strengthening your ability to make that same positive choice in the future. This is true empowerment; it moves you from being a victim to being a victor.
### The Ultimate Victor
I was first introduced to this life-changing idea in high school, when my English teacher gave the class an assignment to read Man's Search for Meaning by Viktor Frankl, a Holocaust survivor who wrote with incredible eloquence about how he and others rose above despair while enduring the daily atrocities of being prisoners in a Nazi concentration camp. At first, I resisted, afraid I'd be too horrified by his account, but with each page, I felt my heart lift as I became more and more inspired. One particular passage shook me to my core:
We who lived in concentration camps can remember the men who walked through the huts comforting others, giving away their last piece of bread. They may have been few in number, but they offer sufficient proof that everything can be taken from a man but one thing: the last of the human freedoms—to choose one's attitude in any given set of circumstances, to choose one's own way.
If Viktor Frankl could find meaning—even experience love—in the worst circumstances imaginable, then I have to believe that we can all find the courage each day to change how we respond to whatever happens in our lives.
His name says it all. He is the ultimate Victor!
### The Happiness Robbers
When we're stuck in old patterns of victimhood, we draw the same situations to ourselves again and again (there's that Law of Attraction at work). You see this all the time—for example, the woman who ends up in the same type of unhealthy relationship over and over. Different man, same problems.
Author Eckhart Tolle is someone I look to as a Happy for No Reason role model. In his book The Power of Now, he says we can shift the energy of victimhood that perpetuates our old problems by becoming aware of our own power in the present:
A victim identity is the belief that the past is more powerful than the present, which is the opposite of the truth. It is the belief that other people and what they did to you are responsible for who you are now, for your emotional pain or your inability to be your true self. The truth is that the only power there is, is contained within this moment. Once you know that . . . you realize that you are responsible for your inner space now—nobody else is—and that the past cannot prevail against the power of the Now.
We're always free in the present moment to break our old habits and to establish Happiness Habits that will create a different future. Here are some of the old habits of victimhood that rob us of our happiness: complaining, blaming, and feeling shame.
Complaining: Complaining, feeling sorry for ourselves, trying to garner sympathy, or being a martyr or "overgiver" are all dead giveaways that we're the guest of honor at our very own pity-party. We know from the Law of Attraction that if we focus on what we don't want, constantly kvetching about how bad our relationships are or how much money we owe, we're drawing more of that to us—cementing the energy of bad relationships or debt in our lives. Complaining is like putting an order in to the universe for more of what we don't want!
The complainer's motto: Poor me!
Blaming: Blaming our circumstances by making excuses or blaming others for our pain or problems weakens us. We give our power away, and the energy we need to deal with the situation isn't available because we're directing it at someone or something else.
The blamer's motto: It's not my fault!
Feeling Shame: When we turn blame onto ourselves, feel ashamed about things that have happened to us, or feel guilty about something we've done (or not done), we often try to suppress the pain or bury these uncomfortable feelings deep inside. This uses up a lot of energy and blocks our happiness.
The shamer's motto: It's all my fault!
Think of the unhappy people you know. They probably spend a lot of time blaming, complaining, and feeling shame, which robs them of the experience of their innate happiness. Shifting out of the victim game is a sure way to expand your energy and be happier:
### Nabbing the Happiness Robbers
The habit of feeling like a victim can be very subtle and deeply ingrained. I first became aware of people's victim patterns in my days as a corporate trainer. I used to teach classes on the importance of taking personal responsibility, and at the beginning of class, I'd ask everyone if they'd agree to come back from our breaks on time. They all said yes. Whoever was late would have to stand in front of the class and sing their high school fight song or a song I chose if they couldn't remember back to their alma mater days. I can't tell you how many successful, hot-shot Fortune 500 executives were late and then made excuses, blamed others, or complained about our agreement! It was amazing to see how hard it was for people to take responsibility, even in small ways. But I held steady, and by the end of the three-day class, we'd all been treated to numerous renditions of my favorite choice, "Stop in the Name of Love," performed by top executives wearing name tags and sheepish expressions.
Then, a few years ago, I went to a three-day seminar where the tables were turned on me. The seminar leader had a rule that any time my seminar mates or I acted like a victim—any time we blamed anyone, made an excuse, complained, indulged in self-pity, or beat ourselves up—we had to put $2 into a bucket.
I couldn't get over how many times I did every one of those things in the course of a day! When I was late, I blamed the line at breakfast. Oops! There went $2 (at least I didn't have to sing). When I complained that I was too cold, it cost me another $2. (After all the seminars I've taken where the rooms are like iceboxes, you'd think I'd know to bring a sweater.) Blaming, feeling shame, and complaining had become so automatic that I was unconscious of doing them. I spent the whole day emptying my wallet, but the $2 fine was a small price to pay for learning how often I played the victim.
I wasn't alone. By the end of the three days, our bucket was full to the brim. (We gave it to a local charity.) And we were playing the victim game a lot less—just by paying attention to our habits.
I recommend you try this exercise for one week with your family, coworkers, or friends and see what happens. My guess is that you'll end up having collected some big bucks for charity too. Your new awareness will benefit you as well, providing clear cues for switching to the habits that support being Happy for No Reason.
### It's Never Too Late
No matter when you start, it's always possible to change your habits. I've seen my own mother go through a wonderful transformation later in her life. Like so many women of her generation, Mom has always been a very loving woman with the tendency to "overgive"—so much so that she often neglected her own needs and became drained and exhausted. But over time, she's learned to overcome her old patterns and take greater responsibility for bringing more joy and satisfaction into her life. When she was in her fifties, she began meditating, which allowed her to go off certain medications and take her health back. In her late seventies, she started exercising regularly, something she'd never done before. At eighty-five, she now walks twice a day and is the oldest person in her senior aerobics class—the first exercise class she's ever taken! When my father died after their marriage of sixty-three years, she found the courage to do many things alone for the first time: travel, join various clubs, even get weekly massages. Mom is so much more empowered now at eighty-five than she was at fifty or even twenty-five—proof that it's never too late to take responsibility for your happiness.
In my interviews with the Happy 100, I found that they had three primary ways to overcome old patterns, be more empowered, and take responsibility for happiness in their lives:
Happiness Habits for Empowerment
1. Focus on the Solution
2. Look for the Lesson and the Gift
3. Make Peace with Yourself
Happiness Habit for Empowerment #1
Focus on the Solution
If you don't like something, change it. If you can't change it, change your attitude. Don't complain.
—Maya Angelou, author and poet
Have you heard that old saying: Worrying is like a rocking chair; it takes a lot of energy and doesn't get you anywhere. The same goes for complaining. I'm sure that at one time or another you've complained about a situation or problem that's bothered you, and gotten quite worked up over it.
Imagine taking that same energy and applying it to solving the problem—using your creativity, intelligence, and imagination to see the possibilities.
Which one makes you feel happier?
There's no contest: focusing on the problem contracts your energy, while focusing on what you can do to solve it expands your energy and makes you happier—à la Guiding Principle #1. Victors focus on solutions, while victims complain about problems.
When I read stories submitted for my Chicken Soup for the Soul books, I noticed that the most inspiring ones were about people who'd transformed their lives by moving from victim to victor.
The following story from my interview with Happy 100 member Aerial Gilbert is the first of the twenty-one amazing stories I'll be sharing with you throughout the book. I've selected her story to illustrate this Happiness Habit because it stands out as one of the most memorable examples of making the transition to victorhood.
I first heard about Aerial from my friend Paul. We were sitting at my dining room table having lunch as he described the happiest person he knew, Aerial, a lovely young woman who'd worked as a nurse at the local hospital before her life had been turned upside down. When Paul finished telling me about Aerial, I sat in silence, reflecting on what I'd heard. What secret did Aerial have that made her so happy? Read on to find out.
Aerial's Story
A New Vision
It was June 1988, the end of just another day. I had finished my shift as a pediatric nurse and was on my way home. My eyes felt irritated, so I stopped at the corner drugstore to get some over-the-counter eye drops. When I got home, the first thing I did was to put the drops in my eyes. Immediately, I felt a searing pain and could barely see.
I was still in my nurse's whites when I was rushed to the emergency room, where they did everything they could to help me, but with no success. The eye drops had been tampered with—laced with lye—and within an hour of putting them in my eyes, I was blind.
Suddenly I was a toddler in a thirty-four-year-old body. For months, I mostly lay in bed, sleeping fitfully and at odd hours. The damage to my eyes caused frequent migraines, which left me exhausted. If the phone rang, I rarely answered it. I didn't want any visitors. My self-pity was like a huge wall around me, holding me prisoner and keeping the rest of the world away. Although my husband and the few friends I talked to tried to comfort me, they couldn't get through.
Before I lost my sight, like most people I simply took my life for granted. It was enough to do my job, socialize with my friends, and pursue my interests and hobbies. In that life, my real life, I had been an intensely sight-oriented person. Besides being a nurse, I was an artist, a jeweler, a photographer, and a pilot. I loved astronomy. I had also been very athletic and loved being outdoors. I swam, played tennis, rowed regularly with a local team, and took long hikes with binoculars and bird-watching guide in hand.
From the time I was five, my grandfather and I would often go out birding together. The highly visual activity of scanning a landscape for signs of birdlife and then noting a bird's markings, plumage, and bill shape to identify species had heightened my powers of observation as well as my connection with the natural world. Now, bird-watching and all of the other things I had been good at and had loved seemed lost—completely out of reach for the new, blind me.
Almost a year passed like this. Then, one day, as I lay in bed, I asked myself, What is the quality of my life? I was wallowing in my misery, slipping further each day into a dark hole inside of me that was far worse than the absence of sight. I projected out for the next twenty, thirty, forty years. Is this what you want?
My spirit rebelled. No! I thought. I can't live this way. I felt the first spark of energy and interest in life revive within me. Okay, I want my life back, I thought, so what do I do now? All the blind people I'd ever seen either had guide dogs or a long white cane. I'd always been a dog person, so the choice for me was obvious. I wanted a guide dog.
For the first time in months, instead of complaining to myself about what had happened to me, I had a purpose and a direction. I needed to call Guide Dogs for the Blind. I felt my heart beat faster. Where was the phone? Would I be able to dial information to get the number? I sat up in bed and eagerly began running my hands over the nightstand, looking for the phone. It felt so good to be excited about something.
Fumbling but determined, I managed to call Guide Dogs for the Blind. They told me I needed to master basic living skills before I could get a dog. Another goal. As I made the arrangements to attend a training program for that purpose, I finally felt I was alive again.
Over the next six months, as I went through the program, I sometimes wanted to give up. It had been so much easier to lie around and feel sorry for myself. But soon I experienced that taking the plunge and doing something to change things, though scary, was actually exhilarating. Little bubbles of hope began welling up in me. The bubbles merged into one buoyant balloon of joy inside my chest the first time I took my guide dog Webster's leash in my hand and we walked together. Using a cane, I'd had to poke along at a slow pace, evaluating each step, worrying that I was going to bump into things. Now, as I walked with Webster at a fairly brisk clip, I could move through space in a completely comfortable and functional way. Suddenly everything felt fluid—I was myself again: at ease, confident, and competent.
Over time, with Webster as my guide, I even began to hike again. One day when I was hiking with my husband in Tucson, I heard a bird singing. I said, "Oh! Do you hear that bird? Can you see the color of its wings?"
My husband located the bird and told me the color. Excited, I began peppering him with questions. "Does it have black bars across its tail? Is it about this big?"
Equally excited, my husband described the bird to me. "Yes! And it has a white throat! And its beak is shaped like . . ." He gave me the details.
I was giddy. "Oh, it's a canyon wren!"
We laughed at our new bird-watching technique. For the rest of the hike, between the two of us, me hearing the bird, my husband giving me the visual information, and then me doing the identification, I was able to experience a joy I thought had been lost forever. In fact, this new form of bird-watching was even more enjoyable because I could share it with my husband in this wonderfully cooperative way.
Webster also allowed me to once more enter the working world. I got a job at the same hospital where I'd worked as a nurse, first developing X-rays in a darkroom and then as a medical transcriber. Now that I could function independently, I soon began volunteering for Guide Dogs for the Blind, giving talks and leading tours of the Guide Dogs campus near my home. Eventually I began working full time for Guide Dogs.
I loved my new job. Being a part of the transformation process for others was a wonderful feeling. I remember one man who came with his sighted wife for a tour of the Guide Dogs campus. This visit was probably his wife's idea. He was very quiet on the tour, almost withdrawn. One of the sighted volunteers told me that he wore a hat, had long hair and a big beard; he was still hiding from the world, just as I had for so long. But he ended up attending a one-month in-residence class. About halfway through, at the point when he was matched with his dog, he told the instructor that he wanted to get a haircut and to shave off his beard. He went out and bought new clothes. As he continued to connect with his dog and the world, he really blossomed, physically and emotionally. The change was stunning. When he walked across the stage during the graduation ceremony, his wife, who was sitting in the audience and hadn't seen him for the entire month, didn't even recognize him.
I can relate. I sometimes don't recognize myself. Before I lost my sight, I considered myself a fairly happy person. Yet today I am happier and more empowered than I was before—on a more consistent basis and from a more meaning-filled place. There is a deeper sense of contentment and peace in my life now. For although I lost my sight, my vision has become greater.
### The Solutions Focus
Though most of us haven't experienced something as sudden and extreme as Aerial's blindness, we all have our own version of being a victim. The minor disappointments, betrayals, and frustrations of life can pile up, and before you know it, you're complaining about feeling burdened and unhappy. I recently heard that the average person complains seventy times per day!
Will Bowen, a pastor from Kansas City, came up with an innovative way to take on the complaining syndrome. He handed out purple wristbands to his congregation and asked them to participate in an experiment. Each time they caught themselves complaining, they were to move the wristband to the other arm. The goal was to go twenty-one days without moving the band. His challenge spread quickly, and now millions of people around the world are wearing purple wristbands as part of Will's effort to create "a complaint-free world."
Aerial's transformation began the moment she stopped thinking of herself as a victim and started looking for a way to solve her problem. Anyone can make this shift using a powerful technique called the Solutions Focus, developed by Paul Z. Jackson, a business consultant and coach, and Mark McKergow, a brilliant physicist turned corporate consultant, who wrote a book of the same title. I've used this process with great success, both professionally and personally. I remember one morning, right after we learned Solutions Focus, Sergio and I were in a bit of a funk about our relationship, so we decided to give it a try. (This is one of the major perks of living with a psychotherapist: he's willing to talk about our relationship!)
The first step of Solutions Focus is to rate how you feel about your situation on a scale of 1 to 10, with 10 meaning "I'm the most satisfied." Sergio led me through the process. "How satisfied are you with our relationship right now?" he asked me.
It was a particularly off day, so I said, "Hmmm, about a 6."
Now the next step is what really sets this technique apart and demonstrates Mark's genius. Before we learned Solutions Focus, Sergio and I would have spent the next three days talking about why our relationship didn't rate a 10. We'd have focused on all the reasons we were dissatisfied and the things we wanted to change.
Instead, Sergio said, "Wow, we're a 6!" (Sergio can be maddeningly positive at times.) "What do we do that makes us a 6—and not a 1?"
"Well," I said, after I'd thought about it, "we're a 6 because we have fun together, because we deeply love and trust each other. We're a 6 because we do the Appreciation Practice together every day (you'll hear more about that in Chapter 9). We're a 6 because we like going on bike rides and taking hikes together in nature." I went on to list all the reasons I felt our relationship worked.
Sergio agreed and added his own reasons for why we weren't a 1, including that we were honest with each other, we made time for each other, and we shared the same values.
That afternoon, we decided to go for a hike. We laughed and had fun; we listened to each other, and by the time we got home, our relationship was an 11!
Solutions Focus keeps you focused on what's working, so you can stop using your energy to complain and start creating more happiness in your life. Here are the simple steps for using the Solutions Focus technique in your life.
Exercise
Solutions Focus Technique
Write down your answers to the questions below on a separate piece of paper:
1. Think of a situation that you've been complaining about. Rate how you feel about it on a scale of 1 to 10, where 1 is "I'm the least satisfied about the situation" and 10 is "I'm the most satisfied." ______________
(If you rated your situation a 1, please skip to question 3.)
2. Great, you didn't score a 1. Write down what you're doing (as many things as you can think of) that cause you to rate your level of satisfaction at the number you scored and not lower.
3. What would be the first tiny signs that your satisfaction has increased by one point? Think carefully and write down as many things as you can.
4. In light of what you've written above, what are the first small steps you could take in the next day to increase your satisfaction with this situation?
5. Begin to take some of the actions you've listed in number 4. Start to notice times when you are a little more satisfied, and build on whatever you're doing that helps you.
Adapted from Solutions Focus Technique. Used with permission of Paul Z. Jackson and Mark McKergow.
To download a free 26-page Happy for No Reason Workbook containing all 21 Happiness Habit Exercises, go to www.HappyforNoReason.com/bookgifts.
Happiness Habit for Empowerment #2
Look for the Lesson and the Gift
The best years of your life are the ones in which you decide your problems are your own. You do not blame them on your mother, the ecology, or the president. You realize that you control your own destiny.
—Dr. Albert Ellis, psychologist
Research confirms that blaming is a happiness robber. A 1999 study conducted by Shane Frederick of MIT and George Loewenstein of Carnegie Mellon University showed that subjects who blamed others for severe accidents they had been involved in eight to twelve months earlier "displayed especially low coping scores." The sad truth is that as long as you say "It's my mother's fault, my husband's fault, the government's fault, my sixth-grade piano teacher's fault . . ." you won't be happy.
The Happy 100 have a secret that allows them to stop blaming. They believe that the universe is out to support them (Guiding Principle #2), and they feel that whatever is happening has a gift or a lesson in it for them.
If blame is one of the clouds obscuring your inner happiness, try it yourself. Rather than asking Who's to blame? start asking What can I learn from this? What is the gift in this for me?
Author and motivational speaker Chellie Campbell is a shining member of the Happy 100. But she wasn't always like that. In the following story, Chellie describes how every woman's nightmare finally helped her wake up to her blame habit.
Chellie's Story
The End of the Whine
I can't go through with it," said my fiancé, Stan.
I was sure I didn't hear him correctly. It was early evening and we had both just arrived home from work. I handed him a couple of shirts I'd bought him at a department store, but he barely looked at them and threw them on the table.
"I just can't go through with it," he repeated.
A black wave engulfed me and I closed my eyes. I stopped breathing.
No, no, no, no, no.
He hung his head and looked at the floor. "I'm sorry," he said. "I can't marry you."
The wedding was in three weeks.
I gaped at him. Don't cry, don't cry, don't cry.
"Why?" I finally gasped. "What's happened?"
He couldn't answer. For the next few hours, we argued, I pleaded, he withdrew, I yelled, he yelled, I whined, I struggled to find the answer. Cold feet? He didn't like the church, the cake, his tux, the DJ, what? Give me some answer, any answer, just please not "I don't love you."
It wasn't even that good. It was, "I don't feel that close to you."
I fled then. It was three o'clock in the morning, and I called my best friend, Gaye. She and I had always joked that a best friend was someone who would take you in at three in the morning. Take me in, I said. Stan doesn't want to marry me.
Over the next few weeks, Stan and I canceled our happy life. Canceled the dress, canceled the church, canceled the flowers, canceled the bridesmaids, the photographer, the reception hall, the honeymoon. Returned the gifts, recalled the invitations. My parents cried. My friends hugged me. I moved out.
Ten weeks later, Stan married someone else.
I went through at least two boxes of Kleenex a day for a week. In an effort to cope, I did individual therapy. I did group. I ranted, I railed, I whined, I spit, I beat the couch to death with a tennis racket. At every meeting I sobbed, "Our relationship was sooo beautiful! It was the best relationship in the wooorld!"
"Oh, yeah?" my long-suffering therapist finally said one day during group therapy. "How beautiful was it? If it was that great, it wouldn't be over."
The hideousness of that bald truth hit me like a hard punch from a champion prizefighter. A collective gasp rose from the group in concert with mine. I stared at my therapist wide-eyed. How could she say that to me? Whose side was she on?
I didn't want to stop blaming, stop making Stan wrong, stop posturing as the innocent victim. Victimhood feels so cushy when all your friends hug you and love you and soothe your many wounds. You get to blame everyone else but yourself for your miserable life: the guy, your parents, your therapist, your job, even God. Now I called all my friends to tell them of this latest betrayal. Not only was I used and abused by my bad boyfriend, but my bad therapist thought it was my fault! I reveled in the safe cocoon of martyrdom.
I shut myself behind my office door at work and everyone walked on eggshells around me. Until one morning, a coworker of mine walked into my office unannounced and looked me straight in the eye.
"When I was eighteen and pregnant, my fiancé left town the week before the wedding and joined the army," she said.
I stared at her for a long moment. She sat there, waiting calmly for it to sink into my consciousness that someone else had suffered—as much as or more than I had—and survived. She was fun-loving, accomplished, and happy. Somehow she had recovered, regained her sense of self-worth, and prospered. She had not let one unhappy incident in her life cast her forever in the role of victim.
Her disclosure threw down a challenge: What was I going to do about my life now? Live it as Miss Havisham, frozen in time with a tattered wedding dress and moldy cake? Or take responsibility, learn what I had done to create the situation, and grow?
The angel that bends over me, whispering "Grow," won the day. I went back to therapy. Now the work of recovery could truly begin.
I began to see my part in what had happened. Why had Stan's unhappiness come as such a big surprise? I ran through the little alarms, the blinking warning lights that signaled the big sign that said "Road Closed!" He complained that I was controlling when I wanted to set an agenda. I wanted to spend when he wanted to save. He even told me when he went to his brother's wedding that they seemed in love in a special way, different from us. In hindsight, the signs were so clear. I should have been paying attention. Stan wasn't bad or wrong—and I wasn't either. We were just different. Our time of walking the same road had come to an end, and now I could see that our highest good lay in different directions. I wanted to be a winner, but I couldn't do it by making him a loser. I had to stop thinking like a victim, stop blaming other people for my misfortunes, stop telling victim stories.
Taking responsibility was a whole new ballgame. Though I'd made some bad choices in the past, I decided to adopt new thoughts and behaviors in order to change my experiences.
I found role models who lived the kind of life I wanted to live and connected with them. I read the biographies of famous people I wanted to emulate and followed their suggestions. I celebrated every success as evidence that I was a winner. I found happy, joyful, successful people, kept them close, and learned their ways. When my next boyfriend complained about my behavior in an argument, I stopped and listened. I looked at myself from his perspective, and said, "Steve, you are right. I don't even like my behavior. I don't know why I did that. I think it was just an old pattern of mine."
"Wow," he said quietly. "I'm impressed not only that you could see it that way, but that you would tell me so. Thank you."
Along with better relationships came better clients, better jobs, better wins. I started teaching workshops and helping others. I wrote two books. In seminars and speeches, I shared my stories and hard-won lessons and helped others become winners in their lives too.
I've discovered that what looks like bad news isn't always bad news. Who could have predicted that being left at the altar three weeks before my wedding would help me break the most damaging habit I had—the one that was keeping me from happiness? Blaming is a no-win proposition and I've got better things to do—like being peaceful, content, and happy.
### Rejection Is God's Protection
Choosing to look for the gift and lesson in the situation instead of getting stuck in blame frees us from our old pattern of pointing the finger and from re-creating the same situation over and over again.
Looking for the lesson may seem challenging at first. Believe me, I get it. When I'm going through a hard time, it's not easy to listen when someone tells me, "Just wait. This will turn out to be a good thing for you." But I've come to see they're right.
One of my friends often reminds me that "rejection is God's protection." Though I'm not always able to recognize this as it's happening, when I look at my disappointments of the years past, I can see how so many of the things I set my heart on wouldn't have made me happy anyway (personal confirmation of Daniel Gilbert's research!). And often the things I thought were bad ended up being the greatest blessings in my life.
This occurs so frequently that a number of years ago, I started using a phrase that has served me well. Whenever I begin to blame a person or circumstance, I stop and ask myself: If this were happening for a higher purpose, what would that be?
The Happy 100 recognize that it isn't useful to label events as "good" or "bad" in the first place. Instead, they choose to trust that everything contains a gift or a lesson, though they may not always be able to see it in the moment.
There is an ancient Chinese story that illustrates this point beautifully:
An old farmer used a horse to till his fields. One day, the horse ran away, and when the farmer's neighbors sympathized with the old man over his bad luck, the farmer shrugged his shoulders and replied, "Bad luck? Good luck? Who knows?"
A week later, the horse returned with a herd of wild mares, and this time the neighbors congratulated the farmer on his good luck. His reply was, "Good luck? Bad luck? Who knows?"
Then, when the farmer's son was attempting to tame one of the wild horses, he fell and broke his leg. Everyone agreed this was very bad luck. But the farmer's only reaction was, "Bad luck? Good luck? Who knows?"
A week later, the army marched into the village and drafted all the young men they could find. When they saw the farmer's son with his broken leg, they let him stay behind. Good luck? Bad luck?
As you see, we can never know.
When things don't work out the way you want them to, try believing that what's happened is for the best. Remember: the universe is out to support you. This will instantly expand your energy. And with practice this process gets easier and easier.
Here's an exercise that will help you break the blame habit by looking for the lesson and the gift:
Exercise
Look for the Lesson and the Gift
1. Sit quietly by yourself. Close your eyes and take a few deep breaths.
2. Recall a specific situation that has caused you to feel wronged or to blame others. Picture the person or people involved, the setting, and what was said or done.
3. Imagine taking several steps back and observing the situation from a distance, as though you were watching a movie on a screen.
4. What part of what happened can you take responsibility for? Did you ignore signs that should have clued you in that there was a problem? Did you act in a way that might have provoked the situation? Did your thoughts or actions escalate the situation?
5. What's your lesson to learn from what happened? Do you need more patience or better boundaries? Do you need to listen more, say less?
6. Ask yourself: If this were happening for a higher purpose, what would that be? Can you find the gift?
7. Write down the most important thing you can do differently as a result of finding the lesson or the gift.
Happiness Habit for Empowerment #3
Make Peace with Yourself
We can never obtain peace in the outer world until we make peace with ourselves.
—His Holiness the Dalai Lama
Let me guess. Like most people, there are things in your life that haven't gone so well, right? And you blame yourself. Maybe it's a failed marriage or business, or your children are having difficulties. Maybe you hurt someone or let someone hurt you. Blaming ourselves is as big a happiness robber as blaming others; it causes feelings of shame and guilt. When we try to stuff down those unpleasant feelings, we use a lot of energy keeping them there. And they slowly eat away at us, undermining our ability to experience lasting happiness.
To make peace with yourself, it's important to free up your energy by accepting any feelings you've been avoiding, and by letting go of the past. When you do, you'll be able to move your life forward and experience greater expansion and happiness.
The following remarkable story from my interview with Happy 100 member Zainab Salbi, an author, activist, and the founder and CEO of the humanitarian organization Women for Women International, describes one way to do this.
Zainab's Story
Telling My Story
I grew up in a suburb of Baghdad during the 1970s. My father was a pilot for Iraqi Airlines, and as a child I traveled all over the world. I played with sports cars and Barbie dolls equally, and there were no limits to my dreams for the future. Things changed when I turned eleven and the Iran-Iraq War started. That was when I first saw military planes and antiaircraft fire in the sky, soldiers and their guns filling the streets, and even missiles falling on people's homes. I remember my parents discussing whether it would be better for the whole family (my two brothers, my parents, and me) to sleep together in one bed, so we would all die together if a bomb hit our house, or to continue to sleep in our bedrooms and live our lives normally. Life was definitely not normal, and at times felt terrifying and perilous. Then, overnight, it became infinitely more dangerous: my father became Saddam Hussein's private pilot.
When my father was offered the job, he couldn't refuse; it would have meant prison, even execution. So we tried to avoid the relationship. But like a poisonous gas, Saddam leaked his way into my family's home. He took over our lives as we breathed him in slowly. Everything about us eventually became associated with my father's job. My family's home was referred to as the pilot's home, the street I lived on was referred to as the pilot's street, and, worst of all, I was always referred to as the pilot's daughter.
Like all Iraqi kids, I was instructed to call Saddam "Amo" (Arabic for "uncle"). But unlike the other Iraqi kids, I was often invited with my family to his palace parties. Being in Saddam's inner circle was fraught with danger. My mother instructed me never to relax or let down my guard. Many times we'd be sitting in his living room having a conversation and he would casually mention killing a member of his family or a friend or colleague. Then he'd watch us very carefully. Offending Saddam with the wrong remark or facial expression could be fatal, so I learned to match my responses to his. If he was serious, I was serious. If he smiled, I'd smile. For years, my family and I lived in fear of this man and his craziness.
Then, when I was almost twenty, my mother asked me to accept a marriage proposal from a man I'd never met, an Iraqi expatriate who lived in Chicago. I was horrified. Marrying someone I didn't know and definitely didn't love went against everything my parents had said they wanted for me: love, passion, and the freedom to choose my own life. At first I refused, but my mother cried and pleaded with me so desperately, I finally agreed, more to make my mother happy than anything else. What I didn't know then, and my mother wouldn't tell me for another ten years, was that she was worried that Saddam might have begun to have amorous intentions toward me and she was frantic to get me out of Iraq and out of his reach.
Our family flew to Chicago for the wedding, and at the first sight of my prospective husband, my stomach sank. I had absolutely no attraction to him. But he told me that he would try to be a good husband; I would have the freedom to finish the university study that I'd left in Iraq and pursue a career, and, in time, we would learn to love each other.
Within a few short weeks I discovered that my new husband had no intention of honoring the promises he'd made to me. He gave me almost no money and no access to a car, and wouldn't allow me to attend the university. For months, I felt trapped, victimized and violated, an indentured domestic servant in a home where I was disrespected and verbally and emotionally abused. I kept telling myself that things would get better, but they didn't. The last straw was a violent sexual encounter with my husband that left me as bruised in spirit as I was in body. Afterward, I dragged myself to the shower and stood with the hot water pounding over me as I sobbed. It didn't matter that I was married to this man; he had raped me. I packed my things and all the money I had, about $400, and called a friend of my mother's in the area who helped me escape.
Things began to look up after that. I moved to a new city, started going to school, and made new friends. I put everything that had happened—my war-scarred childhood, my years living in Saddam's shadow, and my abusive marriage—behind me. I took all the pain, fear, and trauma I'd gone through and stuffed it in a box deep inside of me. I never wanted to look at it or think of it again. During this period, although I often had happy moments, there was always a part of me that was profoundly sad.
In 1992, I married again, this time to a wonderful man I'd met and fallen in love with while in school. We were saving our money to go on a honeymoon when I read a magazine article about the thousands and thousands of Bosnian and Croatian women being imprisoned in "rape camps" during the war. Something in the pictures of the women's faces touched the deep pain I held inside of me and I began to sob. My husband raced into the room to find out what was wrong. When I explained what I was crying about, he took me in his arms and we both cried. We wanted to do something to help but couldn't find any organizations that were assisting these particular women, so we decided to take action ourselves. We took our honeymoon savings and, with the aid of the Unitarian church who supported our efforts, traveled to Croatia to help.
The response overwhelmed us. Very quickly I knew that this was what I was supposed to do. Our trip to Croatia was the start of Women for Women International, an organization that today helps women survivors of war worldwide to rebuild their lives by connecting them to women in the United States who each month send a small amount of money, as well as a personal letter. I began to travel all over the world, speaking with thousands of women who had been victims of unspeakable violence and rape, and encouraging them to tell their stories. I experienced over and over that sharing their stories started the healing process and helped them make the transition from victim to survivor and eventually to active citizen.
Then, in 2003, when Saddam was captured, I decided to write a book about what the women of Iraq had gone through and were still going through. I had no intention of delving into my own trauma, but one day my agent called me and said, "You really need to make this book about you. You are the story."
I cried, "No! This is not about me, this is about other women." I made every single excuse, but there was a deep anger and fear inside as I battled with myself about exposing the pain and trauma I'd distanced myself from for so long. Telling my story would ruin the image of the strong woman, feminist, and women's rights advocate that I projected to the world—and that I believed myself. But more than anything, I was truly convinced that if I told anybody that I knew Saddam Hussein, my own identity, beliefs, and accomplishments would disappear and Saddam would take over everything, just as he had when I'd lived in Iraq.
A few days after that phone call, in Congo on a Women for Women International mission, I came to a turning point. For two hours, I had been sitting in a room with a woman named Nabito and a translator as Nabito described what had happened to her during the fighting in her country. It was a terrible story—and a common one. Soldiers had gang-raped this woman and her daughters; the soldiers had ordered one of her sons to rape his mother, and when he refused, they shot him in the leg. I found myself shaking as I listened to the details. When she finished she looked at me and said, "I have never told anybody but you my story."
Her statement triggered so many things in me. She was telling me her truth, and I was too afraid to tell mine. So I asked her, "What do you want me to do? My job is to write about your story and tell the world about it. Should I keep it a secret? Should I not tell anybody about it?"
She looked me in the eye and, with a smile, said, "If I can tell the whole world about my story, maybe that will stop other women from going through what I've gone through. Yes, you go ahead and tell the world—just not the neighbors."
It was one of the most humbling moments of my life. I got in my car and cried for the entire five hours it took me to drive from Congo to Rwanda. I couldn't stop thinking about Nabito. She was illiterate, homeless; she had nothing, only one dress that someone had given her and shoes she'd made out of garbage, but she had so much more compassion and courage than I did. Not only was she willing to own her story and open the box that held all the pain and trauma inside, but if it could help even one woman avoid going through what she had suffered, she was willing to tell the world.
Though there were still many reasons to hold on to my fear, I knew that my mother had died in her silence and so had my grandmother. I didn't want to be one more woman who dies in her silence. So when I arrived at the hotel in Rwanda, I emailed my cowriter and my agent and told them, "I'll do it."
I began writing my own story, peeling away layer after painful layer of truth and emotion. In the end, it was the most liberating experience of my life. I truly believe that opening that box inside of me and taking ownership of the feelings I'd stuffed inside was the way to my healing and my peace.
Today, I no longer have that pain inside me. It's like crystal in there now. When I take a breath, it goes all the way through me. I have a deep sense of comfort with myself, and the joy I've always had for life is much more intense.
I've seen so much death and so much life. And life's just so gorgeous; it's like an apple when you take the first bite—that cracking sound and the juice and the sweetness. Oh, I love it!
I really believe that every misfortune I have gone through has led me to my fortune, the happy life I have today. If people are afraid to tell their stories, I tell them from my own experience, it will only lead you to a great fortune—of inner peace, and the joy and lightness that come with it.
### A Fresh Start
There are many ways to make peace with yourself. If, like Zainab, you've been traumatized by earlier events in your life, you may need to unbury your suppressed feelings and learn to accept your past. Research shows that people who bury their traumas live shorter, unhealthier, unhappier lives than those who tell their stories.
Still, it's important not to get stuck in those feelings either. According to the cultural anthropologist Angeles Arrien, in many indigenous cultures, people are encouraged to "tell their story" when they've had a painful or traumatic experience, but no more than three times. These cultures recognize that it's important to share the story with loving and supportive people in order to release the pain, but repeating it more than three times only keeps the person trapped in the energy of victimhood. Making peace with yourself is about healing yourself and freeing yourself to move forward.
Sometimes people feel victimized by shame, not because of anything that's happened to them, but because of regret or guilt about their own actions in the past. It's hard to feel good when a little voice inside is always piping up to remind you that "after what you did," you don't deserve to be happy. When we continue to judge ourselves about our past, it's like dragging around a heavy weight—it takes up a lot of our energy. As Harriet Goslins, the developer of Cortical Field Reeducation, told me, "When you can differentiate between taking responsibility and blaming yourself, then you are free to make new choices. This opens the possibility of true self-forgivness."
In situations like these, making peace with yourself may mean making amends. Although you can't change what happened in the past, with a little creativity you can usually find a way to make things right. For example, if you borrowed money from someone and never repaid him, you could pay him back now, either in a lump sum or in installments—and anonymously if you prefer. Or if you can't locate that person, you could donate the same amount of money to a worthy charity. You'll be amazed at the lightness and energy you feel after making amends.
### Mind-Body Magic
In the past ten years, I've become very interested in the rapidly growing field of energy psychology, which includes a variety of cutting-edge mind-body techniques that release old patterns of victimhood and judgment. These techniques, practiced by therapists, doctors, and lay-people, consist of postures and actions that act as switches to restore our proper energetic balance. They clear blockages in the subtle energy field of the body through the energy meridians, energy channels that have been recognized by ancient health systems for thousands of years. The techniques also reprogram the subconscious mind, which controls 90 percent of our behavior, and establish new patterns in the central nervous system that promote well-being. A number of them can be done in just a few minutes and can be self-applied without formal training.
There are now thousands of documented cases of people benefiting from energy psychology techniques, with hundreds of new studies underway. In my interview with Dawson Church, PhD, the author of The Genie in Your Genes and founder of the Soul Medicine Institute, he told me about a recently completed National Institutes of Health study conducted by Kaiser Permanente in which they found an energy psychology technique to be very useful in helping people successfully manage their weight. When people eliminate their limiting patterns and beliefs, they're more connected to their vibrant, energetic, and empowered selves.
Some easy and effective energy psychology practices include the Bio Energetic Synchronization Technique (B.E.S.T.), Emotional Freedom Technique (EFT), Tapas Acupressure Technique (TAT), and Psych K. (You can find more information about these in the resource section of this book.)
I recently learned about the B.E.S.T. system from its developer, Happy 100 member Dr. M. T. (Ted) Morter, a doctor of chiropractic and pioneer in the field of mind-body health. One of my liveliest interviews was with Dr. Morter, who at seventy-two is as bright, energetic, and vital as any thirty-five-year-old. I sat with him over a wonderful Italian dinner one evening as he told me fascinating tales of his life and his philosophy of happiness. He had clearly mastered being happy in his own life, so I was thrilled when he offered to teach me the M-Power March, a specific B.E.S.T. exercise he developed to help people remove their blocks to happiness. (I share it with you below.)
According to Dr. Morter, the M-Power March resets the central nervous system, clears subconscious blocks, and activates forgiveness. The exercise, which involves simultaneous movements of the upper and lower body and left and right sides, puts the central nervous system into a more balanced state so it can process thoughts and emotions more efficiently and effectively. It "clears the computer" so we can reprogram the way we process subconscious information such as old hurts and regrets. It's a great tool to help shift any of the victim and judgment patterns we've talked about in this chapter. This simple three-minute exercise, as well as other B.E.S.T. procedures and exercises, have been remarkably effective in helping me feel more empowered in my own life, and are all very easy to do.
Exercise
M-Power March: Make Peace with Yourself
1. Stand up straight: alert, yet comfortable and relaxed.
2. Take an extended step with your left foot, keeping your back (right) foot firmly on the floor (and both feet facing forward). Bend your left knee slightly, just enough so that you can still keep the heel of your back foot on the floor.
3. As you extend your left leg, raise your right arm to about a 45-degree angle. Your left arm will automatically move back to help you balance, so go ahead and stretch your left arm downward behind you at about a 45-degree angle. Your position at this point is left leg and right arm stretched forward, right leg and left arm stretched back.
4. Now, turn your head toward the side of the extended right arm; look up; close your eyes and s-t-r-e-t-c-h.
5. While you are in your extended position, think of something about which you feel regret, shame, or guilt. Take a deep breath and focus on feeling "forgiveness." Hold both your breath and position for five to ten seconds.
6. Exhale and repeat the maneuver with the opposite leg and arm. Repeat the sequence three times.
© Morter HealthSystem. Used with permission.
Summary and Happiness Action Steps
You lay the foundation of your Home for Happiness by taking ownership of your happiness: you let go of being a victim, focus on solutions, look for the lesson and gift in situations, and make peace with yourself. Use the following action steps to practice the Happiness Habits for Empowerment:
1. For one week, put $2 in a basket each time you blame, shame, or complain. Keep track of how much money you collect each day, and see if you make progress during the week. Use the money for a family outing or make a donation to the charity of your choice.
2. Do an experiment: go cold turkey. See if you can spend one entire day without blaming, complaining, or feeling shame.
3. Use the Solutions Focus technique when you find yourself complaining.
4. To break the habit of blame, find one experience each day that didn't go the way you wanted and use the "Look for the Lesson and the Gift" exercise. (If you can't find something, congratulations—you're well on your way to being Happy for No Reason!)
5. To make peace with yourself or to shift any patterns of victimhood, practice the M-Power March for three minutes every day.
The next four steps in building your Home for Happiness are raising the pillars: learning the Happiness Habits that relate to your mind, heart, body, and soul. Just as the pillars or cornerposts of your home are connected by walls, your mind, heart, body, and soul are inextricably linked: your thoughts affect your physiology, your feelings affect your thinking, and so on. I've separated these four aspects of our lives into four different chapters for ease of explanation, but I recognize that this creates somewhat arbitrary distinctions. As you read through each chapter, you'll see that sometimes the steps overlap. Because these areas of our life are so interconnected, improving one will strengthen them all.
## 4
## The Pillar of the Mind—Don't Believe Everything You Think
The mind is its own place, and in itself can make a heaven of Hell, a hell of Heaven.
—John Milton, English poet
I once heard a revered sage speak. He took a question from a man dressed in a $3,000 suit, wearing classy shoes, and dripping with gold jewelry. The man asked, "What do I need to give up to experience true happiness and inner peace?"
The sage replied, "There's good news and there's bad news. The good news is that you don't have to give up any of your stuff. Poverty isn't the way to happiness. The bad news is that you have to do something that may be even harder for you. You have to give up the way you think."
Give up the way I think? Isn't that like saying I have to stop breathing? Well, it's not as hard as it may sound. From my research, my own experience, and my interviews with the Happy 100, I've learned some powerful techniques for changing the way we think. In this chapter, I'll show you ways that allow your mind to support your happiness rather than sabotage it. This strengthens the pillar of the mind, the next step in building your Home for Happiness.
### Scary Mind Math
How many times a day are you ambushed by negative thoughts?
"I'm not good enough."
"My husband (or wife) doesn't love me."
"I hate the way I look."
"I'm worried I won't be able to pay my bills."
"My daughter doesn't respect me"
"I'm so stupid."
"I can't stand this job."
If you're like most people, it's probably a lot. With all those negative thoughts running through our heads, it's hard to stay happy.
Our minds—made up of our thoughts, beliefs, and self-talk—are always "on." According to scientists, we have about 60,000 thoughts a day. That's one thought per second during every waking hour. No wonder we're so tired at the end of the day!
And what's even more startling is that of those 60,000 thoughts, 95 percent are the same thoughts you had yesterday, and the day before, and the day before that. Your mind is like a record player playing the same record over and over again. (Okay, an iPod for all of you under thirty.) Talk about being stuck in a rut . . .
Still, that wouldn't be so bad if it weren't for the next statistic: for the average person, 80 percent of these habitual thoughts are negative. That means that every day most people have more than 45,000 negative thoughts like the ones above! Dr. Daniel Amen, a world-renowned psychiatrist and brain imaging specialist, calls them automatic negative thoughts, or ANTs.
Not surprisingly, when your mind is swarming with these ANTs, it has a profound physiological effect on you. Researchers at the National Institutes of Health, among others, have measured the flow of blood and activity patterns in the brain and have found that having negative thoughts stimulates the areas of the brain involved in depression and anxiety. On the other hand, positive thoughts have a calming, beneficial effect on the brain. Our negative thoughts are like poison in our system, and positive thoughts are like medicine. In the following chart, you can see how our thinking contracts or expands us, affecting our happiness:
### The Truth about Your Thoughts
The good news is that to keep the thousands of negative thoughts we have each day from dragging us down, we don't have to try to get rid of each one of them. There's a simpler way. The secret is in accepting an astonishing fact:
Your thoughts aren't always true.
It sounds simple enough, but in fact, this revolutionary idea requires a major shift in our perspective. We're so accustomed to believing that our thoughts are true and automatically reacting to them, that we're hardly aware we're doing it. I can remember the time many years ago I first made this discovery.
I was well into my presentation in front of 450 people packed into a hotel ballroom. My palms were clammy and my heart had started to pound—I was bombing big-time. How was I so sure? Because the man in the third row was letting me know in no uncertain terms. He sat rigidly, arms crossed. He hadn't laughed at a single one of my jokes. Not even a small nod of comprehension. Was that an eye roll I detected? My stomach churned. He hated what I was doing. He hated me.
I was horrified when the man made a beeline for the stage the second I wrapped up my presentation. I braced myself for his scathing critique of my lousy talk.
Instead, he came right up to me and thrust out his hand. "Thank you so much," he said, in a voice husky with emotion. "That talk changed my life."
I almost fell over—he hadn't hated it at all! It was my own negative thoughts that had been freaking me out. That's when I realized that my mind—my constant companion through the thick and thin and ups and downs of life—doesn't always tell the truth. It's the same for all of us. Until we become aware of this, our path to Happy for No Reason is blocked.
You don't believe everything you hear, do you? Of course not. You don't believe everything you read either. And in this age of special effects and Photoshop, you certainly can't believe everything you see. So . . .
Don't believe everything you think!
Thoughts are just packets of energy formed by neurochemical events in your brain, which can be measured in terms of electrical impulses and wave frequencies. Your thoughts don't always give you an accurate picture of reality, yet your mind goes on broadcasting them anyway. When you shine a light on your negative thoughts—and see that you don't have to believe them—it takes away much of their power to create misery.
Have you ever wondered why our negative thoughts have such a strong grip on us in the first place? We're simply wired that way. The problem stems from our primitive survival mechanisms that have run amok.
### Velcro vs. Teflon
In the days when caveman Ugh and his wife, Mrs. Ugh, were trying to survive long enough to raise the baby Ughs, they had to pay more attention to potential threats than to positive events to avoid prehistoric dangers like being eaten by a saber-toothed tiger.
This selective attention to negativity is what allowed Mr. and Mrs. Ugh and all the baby Ughs to survive. If you wanted to live a long life, it was better to react to every single thing that might be life-threatening, even if it turned out to be harmless, than to ignore something truly lethal. Our ancestors were the "Nervous Nellies" and "Fearful Franks" of the tribe; the people who were more relaxed about these possibly dangerous situations usually weren't around long enough to have kids and pass on their genes!
Today, even though we don't have to be on the lookout for tigers anymore—well, at least not in my neighborhood—we're still hardwired the same way: we pay more attention to the negative than to the positive. As the psychologist and brain researcher Dr. Rick Hanson explained to me during our interview, our brains are "Velcro for negativity and Teflon for positivity." Our negative experiences stick to us like Velcro, while our positive experiences slide right off us like Teflon. In fact, researchers have found it takes numerous positive experiences to overcome a single negative one! Unfortunately, this wiring turns out to be disastrous for our happiness.
Remember what happened in my mind during that talk? You have to admit, it takes a special talent to tune out the 449 people who'd spent my entire presentation laughing, smiling, and nodding so I could focus exclusively on the one guy I thought—wrongly—didn't like what I was saying. I'm sure you've noticed this yourself. If you receive ten compliments and one insult, which do you remember? If you're like most people, you'll make yourself miserable by stewing on the insult for hours, overlooking the larger number of positive messages. Psychologists call this tendency to respond more intensely to disturbing thoughts and experiences our "negativity bias."
Psychologist Dr. John Cacioppo, at the University of Chicago, demonstrated this tendency in a study in which he measured the electrical activity in the area of subjects' brains that processes incoming information. He showed each subject three types of pictures. One set inspired positive feelings (sporty cars, delicious food), another provoked negative feelings (gruesome and disturbing images), and one produced only neutral feelings (everyday objects like plates and dryers). The surge in electrical activity was dramatically higher when people looked at the images they found negative. Negativity simply makes a greater impression on the brain.
Researchers are beginning to understand the physiological reasons why our negative experiences are so "sticky." The amygdala, the part of our brain's alarm system that triggers the fight-or-flight response, is involved. To learn more about this process, let's put on our white coats and step into the lab to take a closer look at the brain and how it works.
### Adrenaline and the Hot Amygdala
What follows here should be a detailed explanation in precise scientific language using lots of terms that require frequent trips to the dictionary. I don't know about you, but those kinds of explanations make my eyes glaze over and my mind want to take an extended vacation to the Bahamas. Instead, let me offer a very brief overview of the brain's alarm system and its impact on our bodies.
Whenever the amygdala signals fight or flight, your body immediately jacks up your heart rate, releases floods of adrenaline, and sends other stress hormones cascading into your bloodstream. That adrenaline, according to a recent study conducted by Dr. Jim McGaugh of the University of California–Irvine, creates stronger memories in the brain than pleasure hormones do. This means that your disturbing experiences are "chemically supercharged" to stay with you far longer than your happy ones.
What makes this an even bigger problem is that many of us have amygdalas that are overreactive: they trigger the release of adrenaline too easily and often. Scientists call this having a "hot" amygdala and say that it's a major obstacle to being happy. It's behind the tendency to have a short fuse, panic easily, and generally make mountains out of molehills. People with the hottest amygdalas are the drama queens, rage-a-holics, worrywarts, and chronic complainers among us.
When the amygdala gets stuck in overdrive, it widens the negative neural pathways in our brain. Our minds become overrun with negative thoughts, and we worry, picturing over and over what we don't want to happen, creating anxiety and unhappiness in our lives. We tell ourselves stories based on our Stuck Old Beliefs, what I call our S.O.B. stories.
Having a hot amygdala has negative health implications as well: when your amygdala is constantly pushing the fight-or-flight button, the stress chemicals released into the body build up. In today's world, the saber-tooth tigers of our ancestors have been replaced by such things as near-accidents on the highway, confrontations with a boss or coworker, and arguments with a spouse. Although these everyday situations trigger the release of fight-or-flight hormones, they don't require you to sprint to safety or punch someone out, physical activities that once used up those powerful stress chemicals. Instead, the chemicals hang around and accumulate, creating fatigue and disease in your body.
Having an overreactive alarm system can damage your health and dramatically lower your happiness set-point—unless you learn to override it.
### Teaching an Old Brain New Tricks
Though your brain is hardwired to be Velcro for negativity and to have a highly sensitive alarm system, there's hope for increasing your happiness: your brain's neuroplasticity. Unlike an old dog, your brain really can learn new tricks. Changing your thoughts produces changes in your brain, and perhaps even in your DNA.
Studies done by Dr. Richard Davidson, director of the Laboratory for Affective Neuroscience at the University of Wisconsin–Madison, whom you met in Chapter 1, demonstrate that thinking new and different thoughts creates new neural pathways. When we change our thinking to support our happiness, the negative neural pathways shrink and the positive neural pathways widen. This makes it easier and more automatic for us to think more positively.
For years, it's been commonly accepted that the genetic component of our happiness set-point—the 50 percent determined by our DNA—can't be changed. But according to Bruce Lipton, a molecular biologist and author of The Biology of Belief, our DNA may not be as immutable as we think. His intriguing research suggests that our DNA is influenced by our positive and negative thoughts—one more indication that our thinking can reprogram our happiness set-point.
I'm not talking about wishful thinking, or simply deciding to be happier. That's like pasting a smiley face over our pain or spreading a layer of icing over cake that's burned to a cinder. The bad stuff is still there. What I'm talking about is accessing the higher center of your brain, your neocortex, to reverse the negativity bias and override your primitive alarm system.
### Crank up the Neocortex
According to the latest research, the neocortex is where happiness lives in the brain—in the left prefrontal area, to be exact. Studies have shown that happy people have a high level of activity in this area, while those who have a tendency toward anxiety, fear, and depression have higher activity in the right prefrontal cortex.
When it comes to happiness, you can't fool the brain. Psychologist Dr. James Hardt, one of the world's experts on brain waves, explained to me that the brain wave activity of a happy person is different from the brain wave activity of someone who is unhappy. His research shows that people who are less reactive to the fearful messages of their amygdala have more alpha wave activity—the sign of a happy brain.
Instead of being overrun with negative thoughts or constantly going into fight-or-flight mode, happy people have habits that allow them to respond more easily from their higher brain center, the neo-cortex. From my interviews with the Happy 100, I've found that they don't believe everything they think. They:
• Are more skeptical of their negative thoughts. They question the alarms and override them when necessary.
• Don't fight with their negative thoughts. They know that these are often just by-products of their negativity bias and that they can go beyond the mind and let them go.
• Register their positive thoughts more deeply and savor their positive experiences.
Here are three Happiness Habits to help you create new brain pathways and respond more readily from your neocortex:
Happiness Habits for the Mind
1. Question Your Thoughts
2. Go Beyond the Mind and Let Go
3. Incline Your Mind Toward Joy
Happiness Habit for the Mind #1
Question Your Thoughts
There is only one cause of unhappiness; the false beliefs you have in your head, beliefs so widespread, so commonly held, that it never occurs to you to question them.
—Anthony de Mello, Jesuit priest and psychotherapist
Sergio and I recently had the great privilege of meeting privately with the State Oracle of Tibet, a man who has served as the advisor to His Holiness the Dalai Lama and to the Tibetan government for the past twenty years. Deeply impressed by the joy and serenity he radiated, I asked the Oracle to comment on the topic of happiness. This is what he told me: "The real enemy of happiness is the mind's fixations and delusions. Look at the situation differently, see the truth and the suffering is less. If you have the right mind, you can overcome anything—you can be happy, no matter what."
How can you tell if your mind is telling you the truth? By asking! When you take your upsets with a grain of salt and don't buy the underlying premise, it's amazing how a storm of emotion simply evaporates.
The next story powerfully illustrates the benefits of not believing everything you think. In our interview, Happy 100 member, Bruce Fraser (a pseudonym), described his journey out of suffering as he learned to question his thoughts about what was happening in his life.
Bruce's Story
Nothing but the Truth
My wife and I had been married for over twenty years. We had a nineteen-year-old daughter and what I'd always thought was a rock-solid commitment—if not to stay married, then at least to talk everything through. If we ever split up, I was certain it would be a decision we would come to after a long period of trying to work things out—together.
I was aware that things had been difficult for the past two years. My work had temporarily taken me to another city, so we were doing the long-distance thing, seeing each other two or three times a month. It wasn't ideal, but I really thought we could weather it.
We couldn't.
One weekend when I was home, my wife told me she needed to talk to me. "I've been doing a lot of thinking," she began, "and I don't want to be married anymore."
I stared at her, unbelieving. What was she saying?
She continued, "Of course, I still love you and want us to be friends, but I want a divorce."
Divorce. The word hit me like a ton of bricks, knocking the wind out of me. Yet even through the haze of my shock, I understood that she wasn't saying we needed to talk about a divorce; her mind was made up. She was informing me that our marriage was over.
I couldn't believe that she wouldn't include me in this life-changing decision. Over the course of the weekend, I tried everything to get her to change her mind. I begged. I bargained. I cried.
Then, over the course of the next week, things went from bad to worse. My wife, immovable in her decision to leave the marriage, finally admitted that she had met someone else. This explained why she wasn't interested in trying to save our relationship. She just wanted out so she could be with him.
I went into a tailspin. I felt betrayed, rejected, unloved, and unlovable. How could she do this to me? My despair began coloring my job, my health, and especially my mind. In desperation, I signed up for a nine-day seminar that was supposed to help me change the way I dealt with my thoughts.
At the beginning of the seminar we were asked, "On a scale of 1 to 10, how happy are you?" My happiness level was at an all-time low. I was a 1.
For the next two days, I just sat and listened. The premise of the seminar was that when events happen to us, we react by creating a story in our head about those events. It's not actually the events that make us suffer; it's our story about the events that produces the suffering. So when we find ourselves unhappy about something that's happened, it's important to question our story to see if it's true. This made sense, but I felt so paralyzed by my grief, I couldn't do it.
On the third day, when things seemed utterly hopeless, I finally began to question my story using the process the seminar leader had recommended.
"What," I asked myself, "do I really think about this whole sorry mess?"
The answer came quickly: "My wife should not be behaving the way she is. Her betrayal has caused me to lose my home and my family, I can never be happy again."
I don't know why, but at that moment, I had just enough breathing room from the grip of my intense pain to continue the process by asking, "Are those things true?"
They felt true, but I dug deeper. "Can I absolutely know that she shouldn't be acting like that? Can I absolutely know that I won't ever be happy again?"
Surprisingly, the answer was "No" to both questions.
For a moment, I was stunned. I'd interrupted the endless loop of thought/pain/thought/pain/thought/pain that had been playing in my head for the past few weeks. I almost laughed. Without the constant drumbeat of my tragic thoughts, I felt quiet and calm.
I realized that if I took a larger view of the situation, I had no idea if what was happening was "good" or "bad." It simply wasn't in my power to know if she should be acting that way. Suddenly, everything I had been thinking about the situation seemed suspect. The dread of facing my "ruined" future disappeared, and in its place, I felt an excitement. "Maybe," I thought, "things will be even better than they were in the past!"
This moment of light and possibility lasted for a few minutes—until I thought about my wife being with another man. An explosion of pain and anger went through my heart. How could she do this to me?
But the relief I'd felt for those few minutes had loosened my story's hold on me. I looked at my story and saw another layer of it: I believed that my wife loved someone else more than she loved me.
I took a deep breath and asked myself the same simple question: "Is that true?"
Again, it certainly felt true, so once more I asked the deeper question: "Can I absolutely know for sure that she loves this guy more than she loves me?"
"No, I can't." The thought that she loved someone else more than me had been causing me to double over with pain, yet I couldn't be absolutely sure it was true. It floored me that one thought—how I answered this question—had the power to determine my happiness.
It was an "aha!" moment. Everything I had been hearing at the seminar suddenly clicked into place. It was really pretty simple: my story about the situation appeared as powerful statements of fact in my head and caused knee-jerk responses of suffering and sadness. Yet when I investigated those "statements of fact," they were usually pretty flimsy. It was shocking—and comforting—to realize that the situation itself was not the source of my pain.
I was so excited with this revelation that I eagerly continued to question my thoughts and beliefs for the next week.
By the end of the nine days, I was not the same desperate and victimized person that had arrived at the seminar. I was at peace, filled with a serenity and a sense of acceptance. I'd learned that my misery always came from my story about what had happened in the past. Now I was interested only in being in the present moment. What freedom!
When I returned from the seminar, I called my wife and told her I was ready to let her go. I offered to do anything I could to make this transition easier. I even told her that if she had found something better than our relationship, she should go for it. And I sincerely meant it.
A year later, that sense of freedom is still here. There's still some residual grieving that wells up from time to time. But as soon as I catch myself telling stories to myself about what I've lost and what it means, I stop right there. I look at my story and ask myself: What's the belief behind this pain? Then, once I've identified the belief, I explore the even more crucial question: Is it true? At first it took some conscious effort, but with practice, it's becoming more and more automatic.
I remember the time, just a few weeks after the seminar, when I came across a favorite picture of my wife and me that had always been on my desk. As I looked at our smiling faces and thought of all the happy times we'd had together, I felt a wave of sadness. I stopped and asked myself, "Okay, what's the belief here?" It was my old favorite: "I'll never be that happy again." So I asked myself the all-important question: Could I really know that was true? Suddenly I laughed as I remembered the day the photo was taken. It hadn't been all roses; we had been bickering on and off all afternoon. The truth was, I was happier the minute before I picked up that photo than I had been the day the photo was taken.
The bottom line is this: thoughts come and go, relationships come and go, pain comes and goes. I've discovered that clinging to my stories about what I think I want and need is a surefire recipe for suffering.
Today I've come to a place of real peacefulness, of letting go and accepting things as being just what they are and nothing more. This one simple change has made me happier than I could ever have imagined. On a scale of 1 to 10, count me as a 9—heading toward 10—and not looking back.
### Doing The Work
We are disturbed not by what happens to us, but by our thoughts about what happens.
—Epictetus, Greek philosopher
I too have been helped tremendously by the technique for self-inquiry (referred to as The Work) Bruce describes in his story. Developed by a woman named Byron Katie (everyone calls her Katie), The Work consists of asking yourself four simple questions about your painful thoughts and beliefs:
1. Is it true?
2. Can you absolutely know that it's true?
3. How do you react when you believe that thought?
4. Who would you be without the thought?
Then you apply a "turnaround" statement, a sentence expressing the reverse of your thought or belief. The turnaround is a way to experience the truth of the opposite of what you believe.
Katie discovered The Work as a result of her own life transformation. She went from being as unhappy as a person can be to experiencing what I can only call a moment of grace, which left her in a permanent state of bliss and peace. Before her transformation, Katie, a mother of three and a self-employed businesswoman, described herself as "completely depressed, suicidal, stuck in total pain and self-loathing." Sometimes she couldn't get up for days or weeks to bathe or brush her teeth. Eventually, her self-esteem became so low that she didn't feel she deserved to sleep in a bed and began sleeping on the floor.
One morning, she awoke on the floor when something crawled across her ankle. She opened her eyes and saw a cockroach—and in that exact same moment, something else awakened in her. She realized that all of her suffering came from her thoughts about her situation—my life is horrible, I don't deserve happiness—not from the situation itself. She began to laugh. It suddenly seemed so clear: when she believed her thoughts, she suffered, and when she didn't, she was happy. And she saw that it was the same for everybody. All suffering comes from believing our thoughts.
Although I don't come from Missouri, I have a "show me" kind of attitude about things. I wanted to know if Katie was really Happy for No Reason. Every time I saw her in a seminar, she certainly seemed happy enough. But that was in public. What was she like out of the spotlight? I decided to meet her and interview her for this book.
Katie is absolutely amazing, one of the most present, genuinely loving people I've ever known. She is the perfect example of what it means to be Happy for No Reason. Though she's been threatened at gunpoint, gone through a divorce, and faced losing her eyesight, she still remains anchored to a deep inner state of happiness. Listening to Katie and doing The Work has completely changed the way I "think about my thinking."
Since I've always had an inquisitive mind and a tendency to interrogate everyone I know, questioning my own mind came easily to me. Today I'm interested in, as they used to say on Dragnet, "just the facts, ma'am."
Once, for example, after a mild spat, I caught myself thinking, "Sergio should be less judgmental of me."
Wait a second, I wondered, is that true? Hmmm, not sure.
Can I absolutely know it's true? No, the evidence certainly wouldn't stand up in a court of law.
How do I feel when I think that thought? Contracted. Definitely an energy-killing thought.
Who would I be without that thought? Freer. More expanded. Happier.
Then I did the turnaround. I wrote down my initial thought: Sergio should be less judgmental of me. Then I played around with it, trying different versions to see if any rang true:
Sergio should not be less judgmental of me.
I should be less judgmental of Sergio.
I should be less judgmental of me.
As I went along, I found myself feeling compassion instead of judgment, and my initial thought—Sergio should be less judgmental of me—simply disappeared.
When you get into the habit of questioning your thoughts, you'll find that you don't have to try to control your mind or push out your painful thoughts. As time goes by, they simply lose the power to upset you. Your mind becomes peaceful, strong, and expanded, which automatically raises your happiness set-point. The following exercise is an opportunity to use The Work yourself.
Exercise
The Work Mini-Worksheet
Enter a belief or a judgment on the line below and then question it in writing using the following questions and turnaround:
Belief: _________________________________________
1. Is it true?
2. Can you absolutely know that it's true? (Can you really know what is best in the long run for your path or another person's path?)
3. How do you react when you believe that thought? What happens? (How do you treat yourself and others when you believe that thought?)
4. Who would you be without that thought? (How would you live life differently if you didn't believe that thought?)
Then turn the thought around.
______________________________________________
(Is that as true or truer?)
For each turnaround, find three genuine examples of how the turnaround is true in your life. This is not about blaming yourself or feeling guilty. It's about discovering alternatives that can bring you peace.
When we do The Work, we free ourselves from the effects of believing stressful thoughts such as "I'm not good enough," "He doesn't love me," "She doesn't understand me," "I'm too fat," "I need more money," and "Something terrible is going to happen." We can turn our stress, frustration, and anger into a freedom that we never dreamed possible.
For a more complete description of how to use The Work, go to www.thework.com.
Used by permission of Byron Katie.
Happiness Habit for the Mind #2
Go Beyond the Mind and Let Go
If you let go a little, you will have a little peace. If you let go a lot, you will have a lot of peace. If you let go completely, you will have complete peace.
—The Venerable Ajahn Chah, twentieth-century Buddhist monk
In Borneo, the natives have an ingenious technique for capturing the wild monkeys that raid their crops and stores of food. They take an empty coconut shell and make a small hole in it, just large enough for a monkey's hand. They put some rice into the coconut for bait and tie the coconut to the ground. The thieving monkey, smelling the food, comes to investigate. He sticks his hand inside the coconut to grab the rice but when he tries to pull his hand out, because it's clasped in a fist around the rice it won't fit through the hole anymore. To escape, the monkey must let go of the rice. Because they won't let go, the monkeys of Borneo remain trapped!
A lot of us are like those monkeys: trapped by our negative thoughts because we just won't let go of them. And the more we resist them, the more they stick around. It doesn't help to try pushing them away—they'll just keep coming back.
Another way to address our troubling thoughts is to go beyond our minds and connect with the feelings associated with the negative thoughts. It's the feeling that keeps the thought glued to the mind. When we welcome the feeling, accept it and then let it go, the thought will quite miraculously dissolve. An effective way to do this is through a simple, yet powerful technique called the Sedona Method.
### Lester's Miracle
The Sedona Method was discovered over fifty years ago by a man named Lester Levenson. In 1952, Lester, a physicist and successful entrepreneur, was forty-two years old. Although he was at the pinnacle of worldly success, he was very unhappy and unhealthy. He suffered from depression, an enlarged liver, kidney stones, spleen trouble, hyperacidity, and ulcers that had perforated his stomach and formed lesions. After having his second coronary, Lester's doctors sent him home to his Central Park South penthouse apartment in New York City to die.
But Lester was a man who loved challenges. So, instead of giving up, he decided to find some answers. He holed himself up in his apartment and did some serious soul searching. He found what he considered the ultimate tool for personal growth, a way of letting go of all inner limitations, that is the basis of the Sedona Method still taught today. He was so excited by his discovery that he used it intensively for a period of three months. By the end of that period, his body had become totally healthy again. What's more, he entered a state of profound peace and happiness that never left him. Instead of dying in just a few weeks as the doctors predicted, he lived to the age of eighty-four—another forty-two years.
### Dropping the Pen
I first learned about the Sedona Method from my dear friend Hale Dwoskin, who was one of Lester's students and today carries on Lester's work all over the world. Hale is amazing! He's like a laughing Buddha, and it's contagious. I can't be around him without wanting to giggle. He's definitely one of my Happy for No Reason mentors.
When I met Hale, letting go was not my strong suit. Sometimes I fought my negative thoughts and feelings, but mostly I was the queen of holding on to them, doggedly determined to figure them out, to understand where they came from and what they meant. Hale kept telling me, "Marci, just do the process." But I didn't think I could just let my thoughts and feelings go. It seemed too ridiculously simple.
What helped me finally get it was a little demonstration Hale showed me of how letting go works. Try it for yourself.
First, get a pen. Now hold the pen tightly in your hand. The pen represents your thoughts and feelings, and your hand is your awareness.
Notice that although gripping the pen is uncomfortable, after a little time it begins to feel familiar or "normal." Are you feeling that yet? In this same way, your awareness holds on tightly to your thoughts and feelings, and eventually you get used to holding on and don't even realize you're doing it.
Now open your hand and roll the pen around on your palm. Notice that your pen and your hand are not attached to each other. The same is true of your thoughts and feelings. Your thoughts and feelings are no more attached to you than the pen is attached to your hand. You are not your thoughts or feelings.
Now turn your hand over and let the pen go.
What happened? The pen dropped to the floor.
Was that hard? No—you simply stopped holding on!
That's what it means to let go.
Mariel Hemingway, the actress and granddaughter of writer Ernest Hemingway, has found the Sedona Method very helpful too. A week before our interview, I arranged to meet Mariel at a local bookstore where she was doing a signing for her most recent book, Healthy Living from the Inside Out. Mariel is lovely, funny, and authentic on all levels, a member in good standing of the Happy 100. In our interview Mariel told me the following story, which describes the freedom and happiness she's experienced as a result of going beyond her mind and letting go.
Mariel's Story
It Runs in the Family
I come from a family famous for its creative genius, beauty, love of the great outdoors—and problems. My grandfather, the writer Ernest Hemingway, struggled with depression for years and was also a notorious drinker. People are always telling me, "I once had a drink with your grandfather." This has happened so many times, in so many places, that I am convinced my grandfather had a drink with a significant percentage of the world's population.
In the end, his depression and drinking drove him over the edge. Four months before I was born, he killed himself, the fourth member of his immediate family to commit suicide. Since then, alcoholism, drug abuse, mental illness, and depression have continued to plague my family.
Although I escaped the "Hemingway curse" in most ways, my biggest struggle in life has been overcoming my self-hatred. For years, a nasty voice in my head was always saying, You're just not good enough. If a friend had talked to me in that horrible voice, I'd have certainly stopped being her friend.
I was especially hard on myself about how I looked. I was terribly self-conscious about my wide face, my flat chest, and my long, skinny legs. I thought I was ugly and detested my high-pitched voice. I was obsessed with my body.
This obsession also seems to run in the family. It's a little known fact that my grandfather Ernest used to weigh himself every day and write his weight on the side of the toilet in the bathroom of every house he lived in: March 8, 1945–185 pounds, and so on. My sister Margaux's body issues, including her bulimia, were better known, and most likely contributed to her depression and eventual suicide. It isn't hard to see the disastrous behavior pattern: either go crazy or kill yourself. Sometimes both. It was a grim prospect for a young girl growing up.
For a long time, I worried that I'd wake up one morning and have lost my mind like so many people in my family. So I became a control freak—especially around food. For years all I thought about was what I was going to eat next. It was so embarrassing. What a waste of time, I'd tell myself. But it was where my head would go because I couldn't face the deeper pain of being afraid to die or go crazy. My body and the food I ate were easier issues to deal with; at least they were controllable.
Of course, I knew better than to drink or do drugs—I'm a Hemingway, for heaven's sake!—but I was incredibly strict about exercise and what I put in my mouth. One positive side effect of my intense need for control was that I was afraid of throwing up, which kept me from being bulimic. What I mostly felt was an overriding negativity about myself that I never, ever, ever thought would go away. Like a black cloud obscuring my happiness, it was just a part of my life.
For many years I existed on a steady diet of espressos—the air and caffeine diet—and it damaged my health. I eventually cleaned up my act and started eating healthier and being gentler with myself about exercise, but I still felt the fear inside. If anything went wrong in my life, my need for control would rear its ugly head again.
I thought if I could control my body and my world, then that voice inside that was so filled with judgment would go away. And to top it off, I would judge myself for being so judgmental. Trying to get rid of my negative thoughts never worked; it only made me feel worse.
Then I learned a method for letting go of my thoughts and feelings called the Sedona Method. By asking myself some very simple questions, I could simply let go of even the stickiest feeling and thought. I suddenly understood that a major part of the problem was my conviction that my thoughts and feelings were attached to me. They aren't! My thoughts stick around because I resist and fight them, and hold on to the feelings that come along with them. What if I stopped holding on and simply let go?
Could it really be that easy?
At first I was resistant. Every flippin' second of my life, I was thinking I should be better. But right away, I found letting go helped lighten the intensity of my negative thoughts and feelings. The simple process of asking myself the Sedona Method questions—"Could I let go of this feeling?" "Would I let it go?" "When?"—seemed to pop the bubble of tension I felt around whatever problem I was facing. As I continued to practice, something loosened up inside. It was a relief to stop fighting my need for control and begin to accept myself the way I was. Though I'm not sure why it works, I know it does, and that's all that matters.
My favorite time to practice letting go is while I am out hiking. Like my father and grandfather, I've always loved being out in nature. When I find myself feeling unhappy about anything, I usually go outside.
I remember one time when something had set me off and I felt terrible about myself; I was convinced that I was unattractive—a huge, oversized blob. It was a gorgeous day, so I took off into the mountains. As I walked along the path that started in our backyard in Sun Valley, Idaho, I went through the letting-go process out loud. There I was, striding along the path, talking to myself. Thank God, no one was in earshot; they'd have thought I was a crazy woman.
When I got home, I looked at myself in the mirror and couldn't believe I was the same person that had set off on the walk. I was wearing the exact same shirt and shorts. I hadn't lost a pound, but I thought, You look fine. Just fine. Nothing was different physically; the only thing that had changed was that I'd let go of those feelings associated with self-judgment. I saw so clearly the power of my thoughts and feelings and how they had controlled me.
Although I still beat myself up at times, I forget how it once ruled my life. I used to beat myself up every six minutes; now it's only every six months. I've gone from being painfully obsessed with my body image to not even thinking about it. Though I still take good care of myself, it's because I want to, not out of a need to control my life.
Today, at age forty-five, I no longer have that nasty voice in my head. I'm kind to myself and talk to myself as a friend. When I finally saw that my thoughts, my feelings, and that horrible voice weren't nailed into me, I let them go—and found my true essence, the happiness I was born with. Now, most of the time, I feel the way I felt as a small child: as if I am floating through my life, not being judged, but just being.
### Dropping the Excess Baggage
Most of us can identify to some degree with Mariel's struggle with her negative self-worth. For some people this struggle plays out in body image, for others, relationships or career. No matter the area, the underlying feeling of not being enough destroys our happiness. We free ourselves when we stop fighting our thoughts and learn to drop them.
Like Mariel, I used to think it was my duty to wrestle to the ground every negative thought I had. It was on the trip to the Himalayas I described in the beginning of the book that I finally saw what I was doing to myself. As I watched that small Indian woman carrying my ninety-one-pound suitcase up the mountain on top of her head, I realized what a potent metaphor she'd given me. It was time to let go of all that unnecessary junk I was carrying around in my head! It was a burden for me and for the people around me. Learning to go beyond my mind and let go of my thoughts and feelings has been an easy and effective way for me to get rid of that dead weight.
### Putting the Sedona Method into Practice
The Sedona Method is based on two main premises:
1. Thoughts and feelings aren't facts and they're not you.
2. You can let them go.
At your core you already have the happiness you're seeking and all you need to do is uncover this natural happiness by letting go of the unhappiness or limitation that appears to be covering or obstructing it.
The tendency to hold on and attach ourselves to our unhappy thoughts and feelings is so strong it appears even in the way we use language. When we feel sad, we usually say, "I'm sad." When we feel unhappy, we usually say, "I'm unhappy." We are constantly reinforcing the belief of our being attached to our thoughts and feelings. The Sedona Method helps break that attachment.
The following exercise is an introduction to the Sedona Method and will show you how to use the questions that Mariel mentioned in her story.
Exercise
The Letting Go Process
Make yourself comfortable and focus inwardly. Your eyes may be open or closed.
Step 1: Focus on an issue that you would like to feel better about, and then allow yourself to feel whatever you are feeling in this moment. This doesn't have to be a strong feeling. In fact, if you are feeling numb, flat, cut off, or empty inside, those are feelings that can be let go of just as easily as the more recognizable ones. Just welcome the feeling and allow it to be as fully as you can.
This instruction may seem simplistic, but it needs to be. Most of us live in our thoughts, pictures, and stories about the past and the future, rather than being aware of how we actually feel in this moment. The only time that we can actually do anything about the way we feel (and, for that matter, about our businesses or our lives) is NOW.
Step 2: Ask yourself: Could I let this feeling go?
This question is merely asking you if it is possible to take this action. "Yes" and "No" are both acceptable answers. You will often let go even if you say "No." All the questions used in this process are deliberately simple. They are not important in and of themselves but are designed to point you to the experience of letting go.
Step 3: Ask yourself this simple question: Would I let this go? In other words: Am I willing to let go?
If the answer is "No," or if you are not sure, ask yourself: "Would I rather have this feeling, or would I rather be free?" Even if the answer is still "No," go on to Step 4.
Step 4: Ask yourself this simpler question: When?
This is an invitation to just let it go NOW. You may find yourself easily letting go. Remember that letting go is a decision you can make any time you choose.
Step 5: Repeat the preceding four steps as often as needed until you feel free of that particular feeling.
NOTE: You will probably find yourself letting go a little more at each step of the process. The results at first may be quite subtle, but if you are persistent, very quickly the results will get more and more noticeable. You may find that you have layers of feelings about a particular topic, so be patient. However, what you let go of is gone for good and you will feel lighter and more peaceful.
Portions of the preceding exercise were excerpted from The Sedona Method: Your Key to Lasting Happiness, Success, Peace and Emotional Well-being by Hale Dwoskin. Used by permission of The Sedona Method®, www.sedona.com.
Happiness Habit for the Mind #3
Incline Your Mind Toward Joy
What a wonderful life I've had! I only wish I'd realized it sooner.
—Colette, twentieth-century French novelist
One evening a Cherokee elder told his grandson about the battle that goes on inside people. He said, "My son, the battle is between the two 'wolves' that live inside us all. One is Unhappiness. It is fear, worry, anger, jealousy, sorrow, self-pity, resentment, and inferiority. The other is Happiness. It is joy, love, hope, serenity, kindness, generosity, truth, and compassion."
The grandson thought about it for a minute and then asked his grandfather, "Which wolf wins?"
The old Cherokee simply replied, "The one you feed."
Because of our inborn tendency to register our negative thoughts, feelings, and experiences more deeply than our positive ones, we often feed the wrong wolf. When your brain is Teflon for positivity, happiness goes "slip-sliding away." To be happier, you need to even up the score a bit. You can do this by Inclining Your Mind toward Joy.
I first heard this expression from James Baraz, when I took his fabulous course, "Awakening Joy," a ten-month experiential program for developing one's natural capacity for well-being and happiness. In the first class, James began by telling us that the course was about inclining our minds toward joy. He suggested we gently shift our focus toward giving more attention and energy to thoughts that serve us.
Inclining your mind toward joy works by roughening up the Teflon, so your positive experiences will stick more than the negative ones. When I started consciously inclining my mind toward joy, I began to feel happier right away by simply registering the large and small hits of happiness that I hadn't stopped to notice before. As I continued doing this over time, my happiness increased exponentially, because as we know from the Law of Attraction, when you notice and appreciate the happiness you already have in your life, you draw more happiness to you.
I've noticed that the one place it's often hard to incline our mind toward joy is in the area of our self-worth. Surveys show that two out of three adults in America have low self-esteem. My friend Lenora Boyle is a transformational life coach and teacher of the Option Method, a powerful process that helps people eliminate their negative beliefs. Having worked with thousands of men and women, she says that the most commonly held limiting belief is "I'm not good enough." From my years of giving self-esteem seminars, I know that having such a tough inner critic limits our experience of happiness.
Lisa Nichols, a dynamic motivational speaker and fellow contributor to The Secret, struggled with feelings of not being good enough for many years. Although today she is one of the Happy 100, she had to work to overcome her barriers to being happy. Her story shows how inclining her mind toward joy helped her overcome the many ANTs that filled her mind.
Lisa's Story
The Woman in the Mirror
For many years I was unhappy because I felt that the cards I had been dealt by life just weren't a winning hand. My mocha complexion, full lips, and round hips were not the ideals of beauty I saw on television or in the movies. I qualified for the free lunch program, and on top of that, I lived in South Central Los Angeles, a low-income neighborhood with a high crime rate.
I woke up every day feeling sad. I wanted things that I didn't have—things that were absolutely impossible. I wanted to look like Farrah Fawcett. Impossible. I wanted to get a car at sixteen, as some of my school friends did. Impossible. I wanted to live in Beverly Hills. Again, impossible. To compensate, I became a superachiever: I was captain of the track team, head cheerleader, senior editor on my yearbook, and on the student council. But it wasn't enough. The things that were wrong with me far outweighed the good—at least in my mind.
For decades, because all my thoughts were focused on what I didn't like about myself, I made a lot of very poor decisions. I sometimes became intimate with the wrong partner, and after these encounters, I'd often feel emptier than ever.
It wasn't until I was in my early thirties that I stopped looking to others for love and acceptance. One day a friend of mine pointed out that intimacy can mean "into me I see." Wow, that really woke me up! Could I find love by looking into myself?
I started standing in front of the mirror, looking into my own eyes, and asking, "Who is Lisa?" The answers were simple and honest. A woman who has full lips, round hips, mocha skin, and an Afro. A woman who has battled her weight for years. A woman who's left an abusive relationship. A woman who's in search of her spirituality and now has a relationship with God.
There were so many things to love about that woman. I looked in my eyes and said things like, "I'm proud that you . . ." and talked about the great things that I did. I listed things that I put on my résumé and things that I wouldn't dare put on my résumé: I'm proud that you moved away from your family to start your own business. I'm proud that you got out of an abusive relationship. I'm proud that you recognized that you have to work with your weight. I'm proud that you have never said a negative word—ever—about your son's father. I'm proud of the work you do to help so many teens. Focusing on these things felt wonderful!
I stood in front of the mirror and began to validate, love, and celebrate myself. I began to celebrate the big things and the little, itty-bitty things. "I'm proud that you did ten sit-ups this morning." Whatever the hugeness, whatever the smallness, I celebrated. My face would be wet with tears, but I kept saying "I'm proud that you . . ." until I ran out of things to say.
Before I did this exercise, I'd looked in the mirror, of course. I'd study my face, sighing at the little bumps on my forehead and hating how big my lips were and how frizzy my hair was. I looked at the external things that I felt people were judging—and that I had been judging too.
It was the same reflection, but now I saw different eyes staring back at mine. Eyes that looked for the beauty and goodness in myself—and found them. Over the years, taking the time to savor the good things I recognize about myself has made me stronger and able to love others more.
I still do the mirror exercise every night and I am amazed at how much love and compassion I feel for the woman in the mirror. Sometimes when I'm traveling, I'll even slip open my compact mirror in the back of a cab or wherever I am, and I'll do the exercise in my compact. There are probably rumors around the country that I'm a little strange, but that's okay. It's worth it to be a happy woman.
I had an experience recently that showed me how important this skill really is. A few months ago, I bought a new car. The next day, excited with my new toy, I was driving down the freeway. But as I drove onto the exit ramp, my car began to lose power and the engine started making a puddah-puddah, puddah-puddah sound.
"Oh no," I thought, "what's the matter with my brand new car?" Puddah-puddah, puddah-puddah, the car coughed and then died.
I was livid! I got out, and in my funky high-heeled sandals and diva shades, walked down the long exit ramp to the gas station at the bottom. I marched inside and told the attendant, "My car is broken. Would you please come see what's wrong with it?"
The attendant walked up the ramp with me, put the key in the ignition, and turned it. Then, giving me a disgusted look, he said, "Ma'am, your car's not broken. You're out of gas."
I was shocked. I immediately pulled out my cell phone and called the dealer. I said, "I just bought a car from you yesterday. I put a ton of money down on it. Now I just ran out of gas."
Without a second's hesitation the salesman said to me, "Ma'am, it is not my job to fill your tank." His answer changed my life. He was right. It was one of those epiphany moments: I saw that it wasn't anyone else's responsibility to fill my tank—in my car or in my life!
Today, keeping my attention on what I appreciate about myself helps me to fill up my own "love tank." The years of sending love to myself in the mirror have paid off handsomely. Not only do I love Lisa, I really, really like her too. I know my job is not to beat Lisa up. It's not to be unhappy about the way things are, but to love myself through the journey.
### The Mirror Exercise
Does the idea of standing in front of a mirror and appreciating your positive qualities feel uncomfortable and stupid? It did to me—which was a sign that I really needed to try it.
My cowriter Carol and I first learned the mirror exercise in 1990, when we took a week-long course from Jack Canfield on self-esteem. Jack assigned the exercise as homework every night, saying, "Make sure you do this behind a closed door so nobody walks by and thinks you're crazy." Carol and I were roommates at the course, so each night we took turns going into the bathroom, shutting the door, and whispering sweet nothings to our reflections: "I love you"; "You're beautiful"; "You have a loving heart."
The first night, I felt like a California New Age woo-woo nutcase, but soon I experienced a rush of sadness. I was such an expert at judging myself—why was it so hard to say nice things?
With practice, it gradually became easier to list reasons to love myself: "You're smart"; "You go out of your way to help others"; and so on. But the real power of this exercise came when I learned to express appreciation for myself for no reason—to look myself in the eye and simply love who I was, unconditionally.
If you're like most people, consciously recognizing the positive about yourself may feel conceited. After all, we're raised not to toot our own horns. So we end up not giving ourselves credit or acknowledgment, or, even worse, we beat ourselves up, which shuts down our hearts, contracts our energy, and—you guessed it—lowers our happiness set-points. Not surprisingly, people who are Happy for No Reason have a compassionate, encouraging, and validating attitude toward themselves. This isn't arrogance or self-centeredness; it's an appreciation and acceptance of who they are. Feeling this way about yourself is an important step in raising your happiness set-point.
### Register the Positive
To incline your mind toward joy, start registering your happy experiences more deeply. The first step is to make a conscious decision to look for them. You can make it a game you play with yourself. Have the intention to notice everything good that happens to you: any positive thought you have, anything you see, feel, taste, hear, or smell that brings you pleasure, a win you experience, a breakthrough in your understanding about something, an expression of your creativity—the list goes on and on. This intention activates the reticular activating system (RAS), a group of cells at the base of your brain stem responsible for sorting through the massive amounts of incoming information and bringing anything important to your attention. Have you ever bought a car and then suddenly starting noticing the same make of car everywhere? It's the RAS at work. Now you can use it to be happier. When you decide to look for the positive, your RAS makes sure that's what you see.
Adelle, one of the Happy 100, told me about a unique method she has for registering the positive. As she goes about her day, she gives away awards in her mind: the best-behaved dog award, the most colorful landscape design at a fast-food drive-through award, the most courteous driver award. This keeps her alert to the beauty and positivity that is all around her. Charmed by this idea, I tried it myself. I liked it so much, I was inspired to include it as the exercise at the end of this section.
Once you notice something positive, take a moment to savor it consciously. Take the good experience in deeply and feel it; make it more than just a mental observation. If possible, spend around thirty seconds soaking up the happiness you feel. If you want to accelerate your progress, take time every day to write down a few of your wins, breakthroughs, and things you appreciate about others—and about yourself. This will shift the balance of power in your mind, tilting the Velcro/Teflon ratio in your brain toward happiness.
In my interview with Happy 100 member Paul Scheele, a human development expert and the author of Natural Brilliance, he explained that registering the positive more deeply is easier when you engage the unconscious mind, which is responsible for 90 percent of our thoughts. Every day, we're subject to an onslaught of negative images and messages from the world that go into our minds on an unconscious level. An effective way to counteract these insidious negative thoughts is through the use of Paraliminal audio programs, which work on the level of our unconscious awareness. Paraliminal recordings use multiple voices, in combination with music and scientifically proven brain technology. These programs, which you listen to through stereo headphones, are carefully designed to work simultaneously with both sides of your brain and shift your conscious beliefs while you simply relax and listen. I've found using Paraliminals to be a remarkably easy way to incline my mind toward joy.
### Lean into the Thought That Makes You Feel Happier
The concept of inclining our mind toward joy is very similar to what Dr. Martin Seligman, who's considered the father of positive psychology, calls "learned optimism." In his book of the same name, Dr. Seligman, the director of the University of Pennsylvania Positive Psychology Center, draws on a vast body of research to support his thesis that it's possible to learn through practice to become more optimistic.
One of the most effective ways I've found to practice inclining your mind toward joy is leaning into the thought that makes you feel happier. The next time you're faced with a challenging situation that gives rise to negative thoughts and bad feelings, find an equally true thought about the situation that makes you feel better—and lean into it. The classic measure of optimism, seeing the glass as half full rather than half empty, is the perfect example of leaning into the equally true but happier thought.
Here's another example using a situation from my own life that I'm sure you've experienced as well. One day, as I bent over my computer, struggling to corral all my ideas into coherence, I had this negative, self-defeating thought: I can't get this project done in time.
I noticed how this thought made me feel: stressed out, panicked, and crummy, which didn't help matters.
Then I searched my mind for thoughts that were equally true and made me feel better, like I always manage to get things done. I can ask for help. The more I relax, the more the ideas flow through me. I inclined my mind toward joy by leaning into these thoughts. Rather than feeling contracted in my body, they made me feel relaxed and expanded.
Please notice that I'm not talking about forcing a positive thought or repeating an affirmation by rote in the mind—I can get this done in time. I can get this done in time. I can get this done in time—while you're still feeling that you can't. These still involve some degree of fighting with your negative thought. It's like having a shouting match in your mind: Who's going to win, the can't or the can? Obviously the winner will be the one that screams louder.
In contrast, when you lean into a happier thought, you're not trying to convince yourself of anything. You're simply shifting your focus from a part of your situation that makes you feel bad to a truthful part of your situation that creates a better feeling.
As the Cherokee grandfather pointed out, the wolf that wins is the one you feed.
Make a habit of feeding happiness.
Exercise
The Daily Happiness Awards
1. Throughout the day, look around you with an eye to giving out awards.
2. Be creative. For example, as you look at flowers, notice the one that could get the "Most Unusual Color" award or find one that's had the hardest struggle to survive, but made it, and give it the "Best Blossom of the Day" award. Look for extraordinary smiles, efficient service, or ingenious solutions to everyday challenges. There's no limit to the type or number of awards you can give in a day.
3. Invite other family members or friends to play this award game, and at the end of the day, tell each other the awards you've given out.
Summary and Happiness Action Steps
Thank God we don't have to believe everything we think! You strengthen the pillar of the mind by thinking in ways that support happiness: questioning your thoughts, going beyond the mind and letting go, and inclining your mind toward joy. Use the following action steps to practice the Happiness Habits for the Mind:
1. When you catch yourself thinking a negative thought, check to see whether it's telling you the truth by doing The Work.
2. To release sticky negative thoughts, practice the Letting Go Exercise.
3. To register the positive about yourself, try the Mirror Exercise once a day for at least a week, preferably twenty-one days.
4. Give Happiness Awards as you move through your day to keep your mind inclined toward joy.
5. Practice leaning into the thought that makes you feel happier.
## 5
## The Pillar of the Heart—Let Love Lead
I would rather have eyes that cannot see, ears that cannot hear, lips that cannot speak, than a heart that cannot love.
—Robert Tizon, author
Try a little experiment with me. Point to yourself. Now, notice where you're pointing. If you are like the thousands of people I've asked to do this, you're pointing to your heart. Nobody ever points to their head or belly button or kneecap. Why? Because we instinctively feel the heart is the essence of who we are.
Imagine looking deeply into the eyes of someone special and uttering the words, "I love you with all my head." It just doesn't have the same ring to it, does it?
Throughout history, people of all cultures have considered the heart, the seat of our emotions, to be central to human happiness and wisdom. In many traditions, the heart is referred to as the diamond, the jewel, or the lotus—all symbolizing the deepest and most valuable essence of our being.
In the center of our own body, there is a small shrine in the form of a lotus flower,
And within it can be found a small space.
The heavens and the earth are there; the sun, the moon, and the stars, fire and lightning and winds—the whole universe dwells within our heart.
—adapted from the Upanishads
Our hearts are where the "juice" in our life comes from, and when that juice is flowing, we feel great. When it's not, we feel unhappy.
I know this all too well from my own experience. For many years, I lived on and off with a severe, stabbing, and often debilitating pain in my heart. It began over twenty years ago, right after I ended a relationship with a man I loved. Although I didn't plan to marry him, I fell apart when he started dating a good friend of mine. I figured my heartache would mend with time, or at least when I met someone new. But it persisted. At times, I felt I couldn't take another breath because the pain was so intense; at other times, I was afraid I was having a heart attack. I visited many doctors, but no one could find a physical reason for the pain. My heart was heavy, both physically and emotionally. It was only when I began to practice the Happiness Habits for the Heart you'll learn in this chapter that my pain finally diminished.
My interviews with the Happy 100 revealed an important truth: Happy people let love lead in their lives. Although they have the same kind of fears, pains, and disappointments as the rest of us, they simply have different habits that allow them to keep their hearts open in their daily lives. When you practice these habits, you'll experience a more loving and open heart—one of the pillars of your Home for Happiness.
### The Heart Has an Energy Field
Think back to a happy time from your childhood: a special family outing, having fun with your best friend, or maybe getting your very first cat or dog. How do you feel recalling these memories? Chances are you feel a warmth and a sweet flow in your heart. This is your heart's energy.
When I was a child, my grandfather Poppa, whom I dearly loved, used to stay at our house every weekend. He'd arrive each Friday, bag of Hostess cupcakes in hand, and as soon as I saw him start walking up the steps to the front door, I'd practically vibrate with excitement and joy. I could feel the energy in his heart jumping out and meeting the energy in my heart. I couldn't wait to snuggle into his lap and hear more of his marvelous stories about the 1906 earthquake, his career in semi-pro baseball, and his many other adventures. Even as an eight-year-old, it was obvious to me that the heart has an energy that extends beyond the body.
Now science confirms my childhood experience: the heart does have a powerful energy field. The Institute of HeartMath, a highly esteemed research group whose work has been verified in studies conducted by Stanford University and Miami Heart Research Institute, among others, has found that our hearts generate an electromagnetic field around us that is several feet in diameter and five thousand times greater than the field generated by the brain.
One measure of your heart's activity that reflects your emotional state is called heart rate variability (HRV), which shows variations in the intervals between heart beats. In studies using EKG machines to measure HRV, Dr. Rollin McCraty and other HeartMath researchers have found that people's heart patterns look different when they're happy than when they're angry, frustrated, or sad. Look at the difference in the heart's patterns in these graphs.
Negative emotions cause erratic patterns called heart rhythm incoherence, which has a damaging effect on your body. When you feel angry, frustrated, or sad, stress hormones and cholesterol are released into your body, your heart pumps faster, and your blood pressure rises. In contrast, when you're feeling appreciative, loving, and emotionally balanced, you create heart rhythm coherence—smooth, even patterns in the heart rhythms.
According to research published by the Institute of HeartMath, heart rhythm coherence increases production of good hormones, such as the anti-aging hormone DHEA, normalizes blood pressure, improves cognitive function, and strengthens the immune system. In a famous study, researchers at the University of Kentucky analyzed journals that 180 nuns had kept from the time they were in their twenties and found that the sisters who expressed more positive emotions lived, on average, seven years longer than the sisters who expressed more negative emotions. Positive emotions not only feel good, but are good for you!
### Love and Fear
All of our emotions can be divided into two basic categories: Love and Fear. All the variations of love, such as gratitude, forgiveness, compassion, and appreciation, expand the heart and create heart wave coherence, while all the variations of fear, like anger, sadness, hurt, and guilt, contract the heart and create heart wave incoherence. At any given moment, either love or fear is running your life—and happiness or unhappiness will follow.
The HeartMath researchers discovered that when people focus on emotions such as appreciation, love, and gratitude, they're able to create more coherence in their heart rhythms at will. That means you can jumpstart the expansion of your heart any time you want. I'll show you how to do this later in the chapter using one of the techniques I learned at the Institute of HeartMath.
When I interviewed the Happy 100, I found that their ability to let love lead no matter what's going on in their lives is based on three powerful habits for opening the heart.
Happiness Habits for the Heart
1. Focus on Gratitude
2. Practice Forgiveness
3. Spread Lovingkindness
Happiness Habit for the Heart #1
Focus on Gratitude
If the only prayer you said in your whole life was "Thank you," that would suffice.
—Meister Eckhart, thirteenth-century German theologian
Have you ever felt your heart swell in gratitude? Have you ever wanted to spread your arms wide and shout, "Thank you, thank you, thank you!" Then you know what I mean when I say gratitude is a natural heart expander.
Yet it's easy to take things for granted. How much time during the day do you actually focus on gratitude, compared to the time you spend thinking about the problems in your life? We act as if gratitude and appreciation are our good china and our fancy tablecloth and bring them out only on really special occasions.
People who are Happy for No Reason don't necessarily have more in their lives to be grateful for; they simply focus more often on gratitude throughout their day. The difference is where they choose to put their attention.
The following story by Happy 100 member Rico Provasoli, a writer and chiropractor who has the energy and enthusiasm of ten people, illustrates the power of focusing on gratitude.
Rico's Story
Thank You for Everything
Imagine sitting in a dark room lit only by a candle that is almost burned down. As you watch, the candle flame flickers, splutters, seems to fade and then leaps up again—a drop of fragile light in the inky blackness around you. You see that it is only a matter of minutes before it is completely gone.
A number of years ago, my life was like that flame. A little longer, or the slightest puff of breath, and it would have been over. It was that close.
As a young man, I simply didn't know what I wanted to do with my life. At college, although my classmates were future elite corporate tycoons, high-profile politicians, and even a Supreme Court judge, I had no such ambitions. During my junior year abroad in France, I found my way to a Trappist monastery, where I spent nearly a year living as a monk. Although I loved my experience there, I soon realized I wasn't cut out for the monastic life. I left and began traveling around the world, sailing the high seas and visiting over fifty countries.
After a few years of adventuring, I ended up back in the States, where I eventually got married, had children, became a chiropractor, and built a clinic on the coast of Maine. I had a very good life, yet I wasn't happy. An inner restlessness dominated my mind and heart. For as long as I could remember, I had felt that way: aware most often of what I was missing in life, rather than grateful for what was working. After a period of deep soul-searching, my wife and I decided to divorce. I made sure that my ex-wife and children were taken care of, sold my practice, and began to travel abroad once more.
Then, in the course of a few months in 1991, my health began to unravel. I started to feel terrible all the time. I'd been sick before, but nothing like this. The list of symptoms kept lengthening, and no one could discover the cause. A friend of mine even hired an exorcist to try to cure me—candles, praying, the whole bit—to no avail. I was really in rough shape. This went on for almost ten years. Finally, in 2001, I hit rock bottom.
On Christmas Eve, I got a call from an eminent endocrinologist I'd recently seen: they had discovered a brain tumor in my anterior pituitary gland, and I would need immediate surgery. "Oh, and have a nice Christmas," he said as he hung up.
I was devastated. Hearing that you need to be opened up to have a tumor removed is never good news—on Christmas Eve or any eve. After spending a miserable Christmas Day, I went to the doctor's office to discuss the surgery. Right away, he told me that the pituitary tumor had been a misdiagnosis. Then he looked me in the eye and said, "You know, there's actually nothing wrong with you. It's all in your head. You're a hypochondriac."
It was the last straw. On top of all of my suffering, he was telling me that I was creating it for myself? The doctor prescribed antidepressants, but I resisted taking them. Finally, six months later, I was so desperate to feel like myself again that I agreed to give the antidepressants a try. After three days on the medication, I felt worse. In this state of total despair and hopelessness, I began to plan my death.
My sister-in-law, who is a psychotherapist, later explained that the particular antidepressant I took can sometimes prompt suicidal tendencies in the first fourteen days of use. At the time, all I knew was that I didn't want to live this way—couldn't live this way—any more. I didn't want to burden my family with the shame and the stigma of a suicide, so I figured I would take the ferry into the city and then simply walk in front of a bus. It would be a clean way out, just one of those terrible things that happens sometimes.
I sat at my kitchen table, contemplating my plan. I'd come to terms with the whole idea, accepting that it was the most sensible solution. It wasn't a passionate, charged decision; it felt very well reasoned. I thought, You know, this is it. I've had a full life. I don't need any more of this. I'm done. I felt a certain peace inside.
Then the mail arrived.
It was the day before Father's Day and there in the stack of mail was an envelope from my ex-wife, the mother of my children. I opened the envelope to find a card with beautiful artwork on the front that she had created herself. Inside the card, she had written a message that turned my world upside-down. She told me that she still felt deep love for me and was sincerely happy that I was the father of her children. She said that despite the challenges we'd been through, she had grown profoundly in my company and that she wished me nothing but wonderful health, good fortune, and happiness.
I was shocked. I thought, Wow! Wait a second. I'm going to kill myself? It was like a cosmic time-out. I saw that I'd been stuck again focusing on what was wrong in my life, rather than being grateful for what was working. But now my ex-wife's tremendous outpouring of love and gratitude had stopped me in my tracks. Her warm and genuine appreciation brought me back to my heart and helped open my eyes to all the good I had in my life. I abandoned my suicide plans and experienced the first taste of happiness I'd had in years.
These new feelings of gratitude carried me along like a wave. My life started turning around. Within twenty-four hours I met someone who recommended a tropical disease specialist. The specialist took one look at me and knew instantly what was wrong. Tests showed that I was infected with seven types of parasites, amoebas, and liver flukes—unwelcome guests I'd picked up on my travels abroad. I went through a course of powerful antiparasite medication, and to my great relief, I was cured. Finally, I felt like myself again. Actually, I felt better than my old self! The difference was my newfound appreciation and gratitude for my life. I was determined to keep this alive and, soon after, found two powerful tools that have helped me do this.
The first is a daily laughing practice: I simply spend ten minutes a day laughing. Although taking time out of my day to just sit and laugh sounds a little odd, my laughing sessions have been transformational. Along with the health benefits from the added endorphins, I find it cleanses me emotionally and keeps my happiness at a high level. I never miss a day.
Then, not long after I began the laughing practice, I received an anonymous fax that contained a mantra from a Zen master: "Thank you for everything. I have no complaints whatsoever." I decided to incorporate this expression of universal gratitude into my everyday routine as a daily renewal of gratitude and joy.
This phrase has become my anchor. Now, no matter what happens, I feel that I have no argument with life.
A few years ago, my sense of gratitude was put to the test when I got a phone call telling me that the bulk of my entire life savings had been embezzled. I had lost everything and would have to radically change my lifestyle. I set down the phone after hearing the news and stood for a moment, digesting the meaning of these events. My first response was "Oh, well." My years of ill health and my brush with suicide put it in perspective: in the larger scheme of things, it just wasn't that important.
I had learned from my days of open-water sailing that when trouble strikes, you keep moving—emergencies require action—so I immediately began to make calls: dissolving the lease on my home, arranging a move to a friend's house, and borrowing money to live on until I could figure out a Plan B.
An hour later, after I had set these things in motion, I was finally free to reflect on my situation. I smiled inwardly as the familiar words bubbled up inside of me: "Thank you for everything. I have no complaints whatsoever." Amazingly, it was true. Of course, I would have preferred things to be different, but I was at peace with what was. Then I sat down and began my laughing session.
Now, years later, through many ups and downs, I'm still laughing! A grateful heart has made all the difference.
### Why Gratitude Works
How can something as simple as gratitude be such a powerful tool for creating more happiness in our lives? The answer lies in the Law of Attraction. Remember the third principle of the Guiding Three: What you appreciate, appreciates. If you want more good in your life, rather than focusing your energy on the problems and obstacles, focus your attention on what's already good, what's working. This automatically draws more good to you.
I'm not suggesting you use gratitude as a way to deny, ignore, suppress, or sugar-coat painful feelings. Rather, gratitude is a way to incline your heart toward joy. Everyone has both challenges and blessings, but focusing your heart's energy on your blessings will make you far happier.
### The Good Vibes of Gratitude
Gratitude is not only fundamental to being Happy for No Reason, it's good for your health too. A recent study conducted by Dr. Michael McCullough at the University of Miami showed that people who describe themselves as feeling grateful tend to have more vitality and optimism, suffer less stress, and experience fewer episodes of clinical depression than the population as a whole.
In an experiment by Dr. Robert Emmons at the University of California–Davis, people who kept a "gratitude journal," a weekly record of things they felt grateful for, enjoyed better physical health, were more optimistic, exercised more regularly, and described themselves as happier than a control group who didn't keep journals.
These studies confirm that gratitude creates a specific energy that affects our bodies in a positive way. A clue to why this happens can be found in the thought-provoking work of Dr. Masuru Emoto.
I first heard of Dr. Emoto and his amazing photographs in the movie What the Bleep Do We Know? Using high-speed photography, Dr. Emoto showed that crystals in frozen water had dramatically different forms, depending on the kind of energy that was directed toward the water. In some of his experiments, Dr. Emoto had people gather in a circle around a container of water and send different feelings toward it. "Love and Thanks" is the beautiful crystalline structure that formed when feelings of love and gratitude were directed at the water.
Love and Thanks
In contrast, "You make me sick, I hate you" is the pattern that emerged after feelings of hate and negativity were directed at the water.
You make me sick, I hate you
Our bodies are composed of 70 to 80 percent water. Which pattern would you rather create in your body?
### Grateful for No Reason
If there's a VIP section in the Happy 100, Brother David Steindl-Rast would be in the front row! Brother David, a remarkably youthful Benedictine monk in his eighties, has written several books about the importance of gratefulness and is the founder of the Network for Grateful Living, a nonprofit organization that's committed to helping people awaken to gratefulness as the "core inspiration for personal change." I'd heard about Brother David for a number of years and had recently joined a monthly gratefulness group based on his work. When he'd agreed to come to our meeting, our host asked if I wanted to interview Brother David for my book. I jumped at the opportunity.
As I sat on the couch sipping tea with this delightful and wise man, I marveled at the love and joy he radiated. I started by asking him my standard questions, and within moments, I had tears in my eyes as he told me about the hardships he'd experienced during World War II and how they'd turned his heart toward gratefulness.
As a teenager in Austria during the Nazi occupation, he'd never expected to reach the age of twenty. Food was scarce—his family often lived on little more than soup made from weeds—and he was sure he'd be drafted and killed in combat. But he told me he'd been happy despite all the dangers and difficulties, because against the backdrop of impending death, he had seen life as the gift it was. That deep sense of appreciation has never left him.
Brother David offers a unique perspective on gratitude. I was particularly impressed when he said that gratefulness is not just about being grateful for the laundry list of things we have in our lives. "No matter how much or how little you have in life," he told me, "you can still be grateful." To Brother David, gratefulness means experiencing "great fullness," feeling full in every moment, appreciating exactly what is. What I call being Grateful for No Reason. He says, "Happiness is not what makes us grateful; it is gratefulness that makes us happy."
Brother David shared with me a wonderful exercise for increasing feelings of gratefulness in life. Each day, he picks a "theme for the day" to focus on. If he picks water, for example, every time he washes his hands, does the dishes, waters the plants, or brushes his teeth, he notices and appreciates the water and uses it as a reminder to be present in the moment in pure gratefulness. The next day, he might choose to focus on the sounds of traffic, and the next day something else. When I tried this powerful exercise myself, I loved it, and added it to the repertoire of gratitude techniques I practice regularly to keep my happiness level high.
### Gratitude in Action
Marc Bekoff is another one of the Happy 100 who wakes up each morning happy. He jumps out of bed, goes to the window, and says good morning to the mountains, the sunshine, and the birds and trees outside his window. Once, when a houseguest heard him do this, he asked, "Who are you talking to?"
"The day, of course!" Marc told him.
It may seem corny, but Marc, like all people who have high happiness set-points, truly delights in the good things in his life. Like all of us, he has challenges, yet he still greets every morning like this, setting the tone for the rest of his day.
Marc's happiness is always there. As I explained in Chapter 1, the state of Happy for No Reason is like a backdrop, remaining even in the midst of difficult emotional situations. For example, when Marc's mother was dying, he flew to her Florida home to spend time with her. She'd had a series of strokes and was paralyzed from the neck down. She couldn't speak, and it was hard to know how much she understood.
To stay centered, Marc often took long walks or runs. One day, he came back from his run with a beautiful bunch of fragrant pink flowers for his mother. Her middle name was Rose and she'd always had a deep fondness for flowers. When he walked into his parents' condominium, smiling broadly, a neighbor who was visiting gave him a surprised look and said to him, "How can you be happy when your mother is in the next room dying?"
Marc answered, "Because it's such a beautiful day and I know if she were more aware, my mother would love this day."
He went to his mother's bedside and put the flowers on her lap. She looked at them, and though she couldn't respond, Marc likes to think that she enjoyed their beautiful scent. Even when his mother was dying, Marc was still able to be grateful for the gift of her presence and to see the goodness in the world.
During our interview, I asked Marc how he remained happy while he was going through deep sadness. He told me, "Happiness is just a part of who I am. At my core, I'm always happy—and there are times when I'm also profoundly sad. They aren't mutually exclusive. The sadness doesn't take away the happiness."
### Saying Your Thank-Yous!
A year ago, at a get-together of a dozen girlfriends from college, I saw my old friend Therese Gibson. Therese had been one of the fun girls at school; she'd had an easy laugh and had always been up for an adventure. When she heard I was writing this book, she told me about the daily gratitude ritual she and her ninety-five-year-old father, Charlie, practice—they call it "saying their thank-yous"—that keeps them smiling and feeling good. Therese moved in with Charlie, who's still sharp as a tack, at a bad time in both their lives. Charlie's wife, Therese's mother, had just died, and Therese was at the tail end of a painful divorce. Money was tight and Therese says they were as glum as any two people could be. But both of them had heard that gratitude was a great way to feel better, so they decided to sit together for a few minutes each morning before Therese headed off to work and tell each other the three things they were grateful for in their lives.
"It was slow going in the beginning," Therese told me. "The first time we did it, I was feeling so low I had a hard time thinking of even one thing I was grateful for." Finally, she looked around the room and saw a vase she liked. She told Charlie, "I'm grateful for how pretty that vase is." It sounded silly, but it was the best she could do. Charlie wasn't any better at it, often waiting for Therese to give him a clue about what to say. But she and Charlie both noticed that even a thank-you for something superficial had a good effect.
Soon, their decision to focus on what was right in their lives began to pay off. Both Therese and Charlie started to feel happier and notice that more and more things were going their way. Even their money situation improved. Three thank-yous became five, became ten, and soon they had to stop listing the good things in their lives long before they ran out of things to say, or Therese would be late for work.
One day, they were feeling so light and happy after finishing their lists that Charlie, who'd always liked the musical Oklahoma!, started singing "Oh, What a Beautiful Morning." Therese joined in. It was the perfect expression of how being grateful made them feel. They added this song to their ritual and now, saying their thank-yous and singing together has become one of the highlights of their day.
What you're grateful for, you get more of. When you appreciate the happiness and love you already experience, more happiness and love come to you.
I've experienced myself how powerful gratitude is. After the heartbreak I described at the beginning of this chapter, a friend told me to write down five things I was grateful for each night before I went to bed for three weeks straight. I knew that psychologists say it takes twenty-one days to change a habit, so I agreed. At first I struggled, but my results kept me going. In fact, this simple little exercise worked so well that I continued doing it every night for the next three years, and over time, the pain in my heart eased.
I suggest you try the gratitude exercise yourself. Every night before you go to sleep, list five things that you're grateful for that day, and notice how you feel when you wake up the next morning.
### Tools from the HeartMath Experts
Appreciation and gratitude are a must if you choose to become the architect of increased happiness and your own fulfillment.
—Doc Childre, founder of HeartMath
On several occasions, I've made trips to the Institute of HeartMath, two hours from my home, in the heart of the redwood forest. There I met with the brilliant visionary and founder Doc Childre and some of his top researchers. The Institute of HeartMath is on the leading edge of research and technology in the area of the heart's influence on health and happiness. They've even developed a computer program and a hand-held device called the emWave that give you real-time feedback of your heart's rhythms to help you habituate heart rhythm coherence. I use these technologies regularly and find that they help my heart feel more open, radiant, and bathed in warmth.
You can use the following Quick Coherence Technique they developed any time you want to focus on the love and gratitude in your heart.
Exercise
The Quick Coherence Technique
Quick Coherence is a powerful emotion-refocusing technique that connects you with your heart power to help you release stress, balance your emotions, and feel better fast. Once you've learned the technique, it takes only a minute to do.
Step 1: Heart Focus
Gently focus your attention in the area of your heart. If you like, you can put your hand over your heart to help. If your mind wanders out of habit, just keep shifting your attention back to the area of your heart.
Step 2: Heart Breathing
As you focus on the area of your heart, pretend your breath is flowing in and out through that area. This helps your mind and energy to stay focused and your respiration and heart rhythms to synchronize. Breathe slowly and gently, until your breathing feels smooth and balanced, not forced. Continue to breathe with ease until you find a natural inner rhythm that feels good to you.
Step 3: Heart Feeling
As you continue to breathe, recall a positive feeling, a time when you felt good inside. Now try to reexperience the feeling. This could be a feeling of appreciation or care toward a special person, a pet, a place you enjoy, or an activity that was fun. Allow yourself to feel this good feeling of appreciation or care. If you can't feel anything, it's okay, just try to find a sincere attitude of appreciation or care. Once you've found a positive feeling or attitude, you can sustain it by continuing your heart focus, heart breathing, and heart feeling.
Used with permission courtesy of the Institute of HeartMath.
Happiness Habit for the Heart #2
Practice Forgiveness
To forgive is the highest, most beautiful form of love. In return, you will receive untold peace and happiness.
—Robert Muller, former Assistant Secretary-General to the United Nations
Sometimes it's hard to let love lead, especially when we've been hurt by someone. But whether the wound is big or small, you can't be truly happy until you forgive. No one can deny that sometimes people do terrible things or act in ways that appear cruel and wrong. Yet, even in instances like these, forgiveness is possible.
Many people think that feeling hatred, anger, and resentment toward the person who wronged them is a way to punish them—but IT'S EXACTLY THE OPPOSITE! Holding onto those emotions is like taking poison and expecting it to hurt the other person. It's you who's hurt. When you forgive, you heal your own anger and hurt and are able to let love lead again. It's like spring cleaning for your heart.
During my search for the Happy 100, one of my Chicken Soup for the Soul colleagues told me about Mary Lodge, whom she'd met through her work with prison inmates and their families. When I interviewed Mary, I was inspired by her strength and courage, and found that her story helped me reexamine the boundaries I place on my own forgiveness.
Mary's Story
Set Free
For years, my life was not what you'd call easy. Forced to stand up for myself on many occasions, including going through a divorce, I'd learned to be quite a scrapper. The truth is I found myself upset at people and situations—a lot. Unfortunately, this created a great deal of resentment and desire for revenge.
Then, one night in 1996, something happened that made all my previous upsets put together seem trivial in comparison. I woke up to the sound of the phone ringing at 3:00 in the morning. Filled with dread, I picked it up. It was my oldest son, Jay, telling me that my youngest son, eighteen-year-old Robbie, had been shot. "Mom, he's dead."
In that moment, I thought my life was over. The pain of losing Robbie was overwhelming. I wanted to crawl into a hole and never come out. But I knew I had to hold myself together for my other children, and to deal with the police, so I put my emotions on hold.
Shawn, the young man who had killed my son, was arrested and charged with murder. Shawn had known Robbie and shot him while the two had been arguing. He pleaded guilty, so there would be no trial, just a hearing at which the plea bargain would be arranged and his sentence handed down. I'd have to wait three long months for that hearing to arrive. I wasn't allowed to see or speak to Shawn throughout that time, which was probably wise—with my despair and fury at a boiling point, if I could've gotten my hands around his neck, I would have strangled him. This was my baby he'd shot!
The day of the hearing finally arrived, and I got my first glimpse of Shawn. As they led him into the dimly lit courtroom, he kept his eyes on the floor. Shadows masked his face, distorting his features and giving him a dark, sullen appearance. I felt a wave of white-hot anger shoot through me. Why had he done it? Shaking with emotion, I decided not to take the stand, but I made it clear to the judge that I wanted to speak to Shawn after the hearing was over.
Since Shawn had pleaded guilty, the verdict was no surprise and neither was the sentence: twenty to forty years in a state penitentiary. As the judge had promised, he summoned me to his chambers to meet with Shawn. I followed the bailiff down the hall, my heart beating faster with each step as I prepared to meet the young man who had taken my son's life. I'd waited a long time for the opportunity to let Shawn know how I felt about what he had done. Now, filled with rage and hatred, I had no idea what I was going to say, but I knew I wanted to let him have it.
I was frisked and led into a small, paneled office. Shawn stood trembling in the corner with his hands and feet shackled, wearing a baggy orange prison jumpsuit. His head was down, and although he was twenty years old, he was crying like a baby, sobbing his heart out. As I watched this boy, so forlorn—no parents, no friends, and no support—all I saw was another mother's son.
I asked the bailiff if I could approach Shawn. At this, Shawn looked up, revealing a childlike face stained with tears. Suddenly I found myself asking, "Can I give you a hug, Shawn?" He nodded his consent. The bailiff motioned me toward the prisoner, and I walked over to Shawn and put my arms around him. He just melted into my shoulder. It was the first compassion he'd had from anybody for a long, long time. As I stood there holding him, I felt my anger and hatred fall away.
Still, what came out of my mouth next surprised everyone, including me: "Shawn, I forgive you for this horrible thing you've done." Our eyes connected for a few moments. "I would rather my Robbie be where he is than be going to prison. I will pray for you every day." I asked Shawn to keep in touch with me, and then the bailiff escorted me from the room.
Soon after that, Shawn left for prison to begin serving his time. I felt little satisfaction at this. Robbie was gone and no sentence could bring him back, yet here was another boy whose life was destroyed.
Both his parents said they wanted nothing to do with him, so Shawn and I began corresponding. And for the first five years of his sentence, I was his only visitor. Five years ago, Shawn was transferred to a different prison, and the warden there does not allow victims' families to visit, but we still write often.
Some people don't understand how I can do it, but I've learned that forgiving doesn't mean condoning. I believe the compassion I felt in the judge's chambers that day was a gift from God. I know I could not have healed the deep, dark places of hatred and revenge, embedded within my heart and soul, had I not forgiven my son's murderer. Forgiveness set me free. It gave me the peace that I needed to get on with my life and eventually come to terms with Robbie's death.
Since then, I have become a Stephen minister, a type of lay minister, through my church. I help someone through a crisis or loss in his or her life simply by listening and just being there. It requires very little except commitment and compassion, two qualities I have found in great abundance in my heart since the day I forgave my son's killer.
I've learned that you only hurt yourself when you hold on to anger and resentment. Now, despite what happens, I feel truly peaceful and happy.
Hatred and revenge won't bring back my beloved son Robbie, but Shawn is someone's son too. The hatred has to stop somewhere. What better place to begin than with me?
### Why Forgive?
Few of us are faced with experiences as heartbreaking as Mary's, yet I chose to include her story for precisely that reason. If Mary could forgive her son's killer, then maybe we can forgive the people who've wronged us.
Why is it so hard to forgive? Here are the five main reasons. Do any sound familiar to you?
1. We think forgiveness means condoning the wrong behavior.
2. We think forgiveness means we have to let that person back into our lives.
3. We think feeling hatred for that person somehow gives us control, power, or strength.
4. We feel that if we forgive, we might get hurt again.
5. We want to punish the offender.
As it turns out, they're all off the mark. Forgiveness isn't about the person or people being forgiven—it's a gift you give yourself that allows your heart to stop being contracted. When you forgive, you release the toxic resentment and anger you're holding in your heart, finally freeing yourself to get on with your life. There is a well-known Tibetan Buddhist story that illustrates this point perfectly:
Two Tibetan monks meet each other a few years after being released from prison, where they had been tortured by their jailers.
"Have you forgiven them?" asks the first.
"I will never forgive them! Never!" replies the second.
"Well," says the first monk, "I guess they still have you in prison, don't they?"
It also helps to remember that forgiveness is neither an erasing of what happened nor a free ride for the perpetrator. Eva Kor, an Auschwitz survivor, publicly forgave the Nazis who had killed her family and used Eva and her twin sister as human guinea pigs for medical experiments. Her act of forgiveness was not meant to absolve the Nazis for their acts; it simply lifted the burden of pain and hatred Eva had been carrying for so long. She says this about forgiveness:
I believe with every fiber of my being that every human being has the right to live without the pain of the past. For most people there is a big obstacle to forgiveness because society expects revenge. . . . We need to honor our victims, but I always wonder if my dead loved ones would want me to live with pain and anger until the end of my life.
. . . I do it for myself. Forgiveness is really nothing more than an act of self-healing and self-empowerment. I call it a miracle medicine. It's free, it works, and has no side effects.
### Forgiveness as Medicine
Three years ago I sat in an auditorium, spellbound, as the director and cofounder of the Stanford University Forgiveness Project, Dr. Fred Luskin, spoke about his work in the field of forgiveness. Dr. Luskin, who travels all over the world to do his research, had recently brought together mothers in Northern Ireland who'd lost their sons on opposing sides of the conflict. Exuding love and compassion, he shared the miracles that forgiveness could bring.
His stories, which moved the entire audience to tears, demonstrated that people who forgive others are happier, have stronger, more loving relationships, and report fewer health problems and symptoms of stress. The medical community is starting to recognize the major role that anger and resentment play in creating disease and addictions. Dr. Luskin's research suggests that failure to forgive—holding hatred in your heart—is actually one of the risk factors in heart disease. Interestingly, he's found that people who just go through the internal process of forgiving their offender have immediate improvement in their cardiovascular, muscular, and nervous systems. So you don't even have to tell the other person that you've forgiven him or her to reap the benefits for yourself.
In his book Forgive for Good, Dr. Luskin describes Dana, who feels that the offense against her is unforgivable. Dr. Luskin tells Dana to imagine someone holding a gun to her head. Her only chance of survival lies in letting go of the anger and resentment she feels toward the person who wronged her. Now will she forgive? Put this way, Dana immediately says her pain isn't worth dying for—and finally realizes that she's been killing herself slowly by refusing to forgive.
Although no one's holding a gun to our head to get us to release pain and anger, our lives—and our happiness—truly depend on learning to let go of our pain and being able to forgive.
### Compassion Is the Key
As Mary discovered, if you can set aside your pain long enough to really look at the situation, you'll see that the people who hurt you are inevitably hurt themselves.
I once had a neighbor who was always yelling about something. My garbage cans were too close to her driveway. People parked their cars in front of her house. The neighborhood dogs were digging in her garden. Understandably, she wasn't my favorite person. One afternoon, I heard an ambulance coming down the street. I looked out my window and saw it turn into her driveway. The paramedics loaded the woman into the vehicle and, lights flashing, took off for the hospital. That day I found out that my neighbor was a very sick woman; she had a serious liver disease and chronic back problems and lived every day of her life in constant and excruciating pain. My anger at her disappeared instantly. From the day she came back from the hospital until she died a year and a half later, I not only felt differently toward her, but I found myself doing anything I could to help her.
When you understand the suffering of others, like magic, it transforms your negative feelings into compassion, and sets the stage for forgiveness to occur.
If you thinks this sounds great but are wondering how to let go and forgive, you've already cleared a big hurdle. Just being willing to consider forgiveness is sometimes the hardest part.
Here is a powerful exercise that can guide you through the forgiveness process:
Exercise
Forgiveness Process
1. Sit someplace where you will not be disturbed.
2. Close your eyes and think of someone you are holding anger, hatred, or resentment toward in your heart.
3. Take a couple of deep breaths and let yourself feel your feelings without having to do anything about them. Just notice them.
4. Now, realize that the person's hurtful action can't be changed. It's in the past and there is absolutely nothing that can be done to affect it now. Feel the finality of that.
5. Also realize that this person may never change. They are the way they are. Take a few deep breaths as you accept the truth of that.
6. Now, see that the person is the way they are—and did whatever they did—because they have some pain, some lack, some woundedness. They may not even realize it themselves, but it's there. People hurt others only because they are hurt themselves. See them through the eyes of compassion for their own suffering. Imagine they are a child that is hurting, lashing out at others in their own pain. Can you feel compassion for them?
7. Sit quietly for a minute or two more, just feeling the expansion that compassion—in any amount—brings to the heart.
NOTE: It's okay if you still feel angry; the purpose of this exercise is to begin to release the pain in your heart, not to excuse others for their actions. Keep repeating this exercise until you feel a shift, however small, in your heart. Your forgiveness will grow as you feel more compassion.
Happiness Habit for the Heart #3
Spread Lovingkindness
Those who bring sunshine into the lives of others cannot keep it from themselves.
—J. M. Barrie, nineteenth-century Scottish novelist
When your heart flows in love, you naturally feel happier. But what about those times when you don't feel loving? It may not take much to feel your heart close. When your surly teenager, an annoying coworker, or a dense sales clerk upset you, how do you live in the flow of love then?
You can restart your heart's flow by sending lovingkindness to anyone and everyone you see. It doesn't have to be dramatic; sometimes simply wishing others well switches on the pump that generates love in the heart. This creates a strong current of love and happiness, because a heart overflowing in love is continually being filled with more love.
When I sent out the Happy for No Reason survey, I was profoundly moved by a response I received from CJ Scarlet. I interviewed her further and found her story shows the tremendous effect that sharing love, either silently or in words and actions, can have on others and on yourself.
CJ's Story
The Flow of Love
I was forty-one years old, but felt like an old woman. For twelve years, I'd been living with lupus and scleroderma, both incapacitating autoimmune diseases. The steroids I took caused my weight to balloon. I needed my walker or cane to make it to the mailbox, and some days I had to crawl on my hands and knees to climb the stairs. Frequently, I couldn't drive and had to depend on my family for everything.
My doctors said that heart failure could happen at any time. I was terrorized by every chest pain, but at times wished death would come quickly to end my suffering. Almost every night I had nightmares of being sucked into black tornados or going down in flaming airplanes.
Reading self-help books and going to therapy had helped a little, but I was still in very bad shape. Buddhist writer Thich Nhat Hanh's books had touched me deeply, so when I had the opportunity to meet a Buddhist lama (teacher) who lived in my city, I jumped at the chance.
Using a cane, I hobbled up the walk to the house where the lama lived. I told him my tale of woe, fully expecting him to turn his compassionate gaze on me and offer his sympathy. Instead, he told me, brusquely, though not unkindly, "Stop feeling sorry for yourself, and start focusing on the happiness of others."
But," I objected, "I'm too sick. I can hardly help myself." Disappointed, I thought, He just doesn't understand.
I was convinced that I was too exhausted and too ill to help anyone; nevertheless, I began to pray for the happiness of others. I brought to mind people I knew, family members and friends, and imagined them happy, healthy, and at peace. Then I began to focus on the happiness of strangers. When passing people on the highway, I would look at them and wish for their well-being and prosperity. Finally, I was even able to extend my good wishes and help to people I actively disliked.
One day I was in my scooter, waiting in the grocery checkout line, when a woman came up behind me who was clearly having a bad day. Harassed and harried, she seemed irritated at everybody. Her cart was full to the brim and it was clear she wanted to get through the line as quickly as possible.
Normally, I would have tried to stay away from her bad vibes and negative energy. In fact, my first thought was, Look at how nasty she's being. I don't want to have anything to do with her.
Then I remembered the lama's advice. Okay, I thought, she's really having a tough day today. I know how that feels. Let me think about her happiness. What would help make her happier?
I turned to her and said, "It looks like you're in a hurry."
Startled by my overture, she said tersely, "Yes, I am. I'm running late."
I said, "Why don't you go in front of me?"
Looking at the few items in my scooter's cart, she quickly shook her head. "No, no, that's okay."
But I said, "Really, I'm not in a hurry. Go ahead."
The transformation was amazing. She went from being this angry person who radiated negativity and was probably going to chew out the clerk at the cash register to someone who felt validated, cared for, and appreciated. She steered her cart in front of mine, thanking me profusely, thanking the clerk, and when her groceries were all bagged up and ready to go, she left the store smiling.
I felt fabulous. Looking around, I noticed that everyone around me was smiling and friendly and talking to each other. "That was a nice thing to do." "Hope you have a great day today." We had all been affected by the interaction.
I started looking for more ways to think of the happiness of others. I smiled and rushed to help anyone in need, from giving money to homeless people to helping a woman arrange to buy a car so she could get a job. I bought a tank of gas for a woman who had no money and volunteered at the Red Cross during Hurricane Katrina. Every moment seemed to present an opportunity to improve the life of someone else, and I rejoiced in my physical, mental, and financial ability to do so.
I had done volunteer work before this—I had been very active in nonprofits for victim and child advocacy—but there had always been a trace of ego involved as I perceived those I helped as victims and myself as their rescuer. It had made me feel good about myself for helping. But this was different.
Now it was 100 percent about them. My focus was on their happiness directly. And when I wished them happiness, I felt a wave of love for them, which sometimes led to action and at other times was just a prayer, a heartfelt desire for their happiness. Very often they weren't aware of my intention at all. But it created a flow of love inside of me that grew stronger and stronger.
When I stopped focusing on whether other people were making me happy or meeting my needs and thought about what would make them happy, I started to see everyone as beautiful. I could understand that they were fighting their own heroic battles and trying to make their own way through life, and my heart went out to them. The more I focused on contributing to the happiness of others, the better I felt. And the better I felt, the happier I became. My dreams transformed into joyous celebrations.
Within a year my health was transformed. The pain that had wracked my body disappeared. I could once again take a deep breath, and I had more than enough energy to resume my life. Today, two years later, I feel better than I did at twenty-nine before I became ill. To my doctors' amazement, my once debilitating medical conditions are reversing themselves. I continue to live pain-free. I even work out at the local gym three times a week and am losing the steroid weight.
I met with the lama again, about a year after our first meeting. When I told him how my life was transformed from focusing on the happiness of others, he clapped his hands and smiled broadly at me. "Very good! Very good!" he kept saying.
His wise words had stopped my descent down a terrible slope and put me in touch with the most powerful force for good that there is: the loving compassion that resides in our hearts. It was the flow of that love that healed my body and today has become a bubbling, clear spring of happiness in my life.
### Wishing You Well
What a simple idea, and yet so life-changing. I've started practicing CJ's technique with great results. I used to hate waiting in line or getting stuck in traffic, but now I spend the time looking around and silently wishing everyone around me happiness, ease, comfort, and peace. Instead of fuming, I find myself smiling and feeling good.
Sometimes I go beyond just wishing people well in my heart. Recently I was at an airport, buying a burrito from a server who was crabby and grumpy. I was tempted to get a little crabby back, but instead, I thought about this guy on his feet all day long, cranking out burrito after burrito for impatient flyers off to exotic destinations while he was stuck in one place, baking under the warming lights. A pretty thankless job. I caught his eye. "Boy, you must get really tired of doing this all day long."
My server was completely taken aback. Someone had reached out to connect with him. He just melted. He gave me the biggest, warmest smile—and extra chips!
I think we all underestimate the power we have to make someone else's day just a little better simply by beaming them a little light and love. I suggest that you try CJ's technique yourself. You can also set some time aside each day to do the following short lovingkindness exercise. It's adapted from a Buddhist practice for culturing metta, which is defined as the strong wish for the welfare and happiness of others. This isn't exclusively a Buddhist concept; in Christianity, the term for this kind of unconditional love is agape. In Judaism, rachamim is the love that motivates us to give to others and includes empathy and care. This same ideal of love is expressed in Islam with the word mahabba, which means spiritual love for others and the divine.
Choosing to feel love and wish the best for everyone—a natural habit of happy people—can help you cultivate more happiness in your life.
Exercise
Lovingkindness Practice
This exercise expands your capacity for compassion by guiding you through the process of wishing lovingkindness to yourself and others.
1. Find a quiet place and sit comfortably. Close your eyes.
2. Take slow, deep breaths, being aware of the breath as it enters and leaves your body. Let your thoughts come and go easily.
3. Repeat the following phrases silently:
May I be safe.
May I be happy.
May I be healthy.
May I live with ease.
Continue feeling these wishes toward yourself for a minute or two or until you experience a sense of peace inside.
4. Now, move on to your friends and family. Picture one of them as you send the following wishes silently:
May you be safe.
May you be happy.
May you be healthy.
May you live with ease.
Continue sending these wishes until you feel a flow of love in your heart.
5. Now send these wishes to all the living beings around the world. Continue until you feel a sense of expansion in your heart.
Summary and Happiness Action Steps
You let love lead in your life by focusing on gratitude, practicing forgiveness, and spreading lovingkindness. When you strengthen the pillar of the heart, you'll feel more peaceful and compassionate. Use the following action steps to practice the Happiness Habits for the Heart:
1. Play the Gratitude Game. Before you go to sleep, think of five things you're grateful for that day. Write them down in a journal if you'd like.
2. Try Brother David's "Great-Fullness" Exercise. Pick a theme for the day, for example, water, and when anything reminds you of water, use it as a cue to experience gratitude for no reason in the moment.
3. Like Rico, make a point to laugh for a few minutes each day.
4. Do the HeartMath Quick Coherence exercise to habituate heart wave coherence.
5. Try the Forgiveness Process. Find the compassion in your heart that will make it easier to forgive.
6. Do the Lovingkindness Practice every day, and spread lovingkindness through your thoughts or actions wherever you are.
## 6
## The Pillar of the Body—Make Your Cells Happy
A sound mind in a sound body is a short but full description of a happy state in this world.
—John Locke, seventeenth-century English philosopher
Being happy is not just a state of mind—it's also a state of the body. In fact, our bodies are actually designed to support our happiness. The renowned neurophysiologist Dr. Candace Pert documented this mind-body-happiness link in her best-selling book, Molecules of Emotion, explaining that when we're happy, we're alive and buzzing with "happiness juices," the chemicals of body and brain that underlie our positive experiences.
There are no drugs more powerful than those you already have in your own head! More than 100,000 chemical reactions go on in your brain every second. Your brain contains a veritable pharmacopoeia of natural happiness-enhancing drugs: endorphins (the brain's painkiller, three times stronger than morphine), serotonin (which naturally calms anxiety and relieves depression), oxytocin (the bonding hormone), and dopamine (which promotes alertness and a feeling of enjoyment), among others. They're just waiting to be released to every organ and cell in your body. Because your brain's pharmacy is open twenty-four hours a day, you can create your own supply of these happiness chemicals any time you want. And when your cells are happy, you're happy.
In the previous two chapters we've looked at how your thoughts and feelings influence your happiness set-point. In this chapter, we'll look at how the way you eat, move, breathe, and rest, and even your facial expression can shift the balance of the brain's feel-good chemicals in your favor to overcome unhappiness.
Remember, when I refer to unhappiness in this book, I mean the garden-variety type of unhappiness and not clinical depression, which is a medical condition requiring professional intervention. If you're clinically depressed, it's important to work with a health care professional to address the chemical imbalances that may be a contributing factor. Nonetheless, strengthening the pillar of the body by practicing the habits in this chapter can be extremely helpful as an adjunct to standard treatment for depression.
### The Body's Happiness Robbers: Stress and Toxins
If our bodies are designed to support happiness, then why aren't we all happier? Just take a look at our lives. Most of us zip around like overcaffeinated bees, multitasking madly and grabbing meals on the run. Our stressful lifestyles, which include unhealthy eating and a lack of exercise and proper rest, hinder our ability to create happy cells.
Stress is a huge robber of happiness and health. Scientific evidence indicates that over 90 percent of all diseases are stress-related. So many of us are stressed out and exhausted, yet we ignore our symptoms and power through the day, taking medication to dull the pain without dealing with the deeper cause.
Then there are the environmental toxins we're exposed to every day: chemicals in our processed food, pesticides in our produce, hormones in our meat and milk, and polluted air and water. Yikes! When our cells don't get the support they need, rather than singing a joyful chorus of vitality and well-being, they starve and die, resulting in low energy and unhappiness. Fortunately, there are ways to inspire the choir to warble songs of well-being once more.
### Inspiring the Choir
There are many ways we can make our cells happy: we can stop bulldozing over our fatigue and discomfort, cleanse our bodies of accumulated toxins, reduce the new ones we take in, and use our brain's pharmacy to experience more happiness in our lives.
Studies abound showing how everyday activities—singing, listening to relaxing music, stroking a pet, getting a massage, enjoying a long hug, gardening—increase our happiness chemicals. Even smiling raises happiness levels.
Mona Lisa had the right idea. When we use our facial muscles to express emotion, we trigger specific brain neurotransmitters. Scientists studying the effects of Botox discovered that when depressed patients were treated for frown lines, they were no longer as depressed. Research by French physiologist Dr. Israel Waynbaum shows that frowning triggers the secretion of the stress hormones cortisol, adrenalin, and noradrenalin, chemicals that increase your blood pressure, weaken your immune system, and increase your susceptibility to depression and anxiety. Smiling decreases these substances and produces happiness chemicals such as endorphins and immune-boosting killer T-cells, relaxing muscles, reducing pain, and accelerating healing.
If you want to multiply the benefits of smiling, enjoy a good laugh. Dozens of studies confirm that laughter really is the best medicine. It too quells stress hormones and sends happiness hormones into high gear.
### Happiness: Just What the Doctor Ordered
Being happy is good for your health. Over the past two decades extensive research has shown that contentment strengthens the immune system and prevents disease:
• Happy people are 35 percent less likely to get a cold and produce 50 percent more antibodies in response to flu vaccines than the average person.
• Individuals who score high on happiness and optimism scales have a reduced risk of cardiovascular disease, hypertension, and infections.
• People who maintain a sense of humor, an indication of inner happiness, outlive those who don't, and the survival edge is particularly large for people with cancer. One study showed that a sense of humor cut a cancer patient's chance of premature death by about 70 percent.
Happiness and health create a positive feedback loop: improving one will automatically improve the other.
Our physical habits either cause our energy to contract, making us feel tired and sick, or bring about a state of expansion, supporting our happiness and well-being:
Though you may be doing everything you can to support your health, your body may still have its challenges. It's possible to be Happy for No Reason anyway. A number of people in the Happy 100 that I interviewed had health problems—even terminal disease—and remarkably, their pain and joy seemed to coexist. Their happiness showed up automatically, a state of peace and well-being underlying all of their experiences. Practicing the following Happiness Habits to make your cells happy will help reinforce that same unshakable state in you.
Happiness Habits for the Body
1. Nourish Your Body
2. Energize Your Body
3. Tune In to Your Body's Wisdom
Happiness Habit for the Body #1
Nourish Your Body
Tell me what you eat, and I will tell you what you are.
—Anthelme Brillat-Savarin, eighteenth-century French writer and epicure
For years, my cells were unhappy because I nourished them poorly. When I was very young, my diet consisted mainly of crisp bacon, juicy hamburgers, Hostess cupcakes (Poppa's weekly gift), crackers filled with a mysterious substance called "processed American cheese food product," and snowy-white Wonder Bread ("builds strong bodies twelve ways!"). Fruits and veggies came in only three flavors: carrots, celery, and apples, of which I ate the barest minimum so I'd have lots of room for McDonald's fries and shakes. I was always exhausted, so I'd rev myself back up with sugar, raiding the freezer for ice cream at all times of the day and night for an energy boost. Naturally, I'd enjoy my sundaes with a can of Tab—who wanted the extra calories from a regular Coke?
I fed my growing body so well that grow it did—an extra twenty pounds, which is a lot when you're five feet tall and sixteen years old. It wasn't just being chubby that bothered me; it was that I was tired, listless, and unhappy all the time. My temporary sugar highs were always followed by crashes that left me feeling more exhausted than ever.
When I got to college, I decided to clean up my food act. I drank my last can of Tab when I was seventeen, phased out the sugar, and became a vegetarian. When I switched from refined carbohydrates to whole foods, I was amazed to find that there are actually dozens of delicious fruits and vegetables.
At first this burst of virtuousness was prompted by my fear of wearing shorts in public. But I quickly discovered that any weight-loss triumphs paled in comparison to the sky-high levels of energy I felt once I'd gone off the bad stuff. I felt so much lighter, happier, and clearer—I felt more me. Now that I understand the relationship between my energy, mood, and food, I don't ever want to go back to the "Wonder Bread Years" and nonstop exhaustion.
It isn't always just a bad diet that causes unhappy cells. Sometimes a lack of balanced brain chemicals or hormones can also be a contributing factor. That was the case for my dear friend, actress Catherine Oxenberg, who at first glance seems to have it all: beauty, brains, a great husband, wonderful kids, and a fulfilling career. That's why it's so surprising for people to learn how she's struggled to overcome fatigue, illness, and recurring episodes of feeling blue, which she describes in the following story.
Catherine's Story
No More Sick and Tired
Most little girls dream of being a princess, but I can tell you from experience, it hasn't always been that wonderful. Though I was born a princess, descended from a long line of European royalty, I wasn't a happy little girl. When I was six, my mother, HRH Princess Elizabeth of Yugoslavia, divorced my father, a New York City businessman, and took my sister and me to live with her in London. At one point, I remember her saying to me, "Catherine, could you please stop moping around?"
I wanted to, but I just couldn't help the way I felt. Now, I believe that my "moping around" may have been the result of low levels of serotonin—a hormone that affects happiness. Later, as a teenager, I learned to make myself feel better by eating. On the outside, everything looked wonderful: I excelled in school, was admitted to Harvard, became a successful model and then an actress, making films and starring in the hit television series Dynasty. But I spent those same years self-medicating my inner pain by binge-eating and then throwing up. The horrible cycle of my bulimia made me feel even worse, and at one point, a doctor prescribed antidepressants in an effort to help me.
The antidepressants were not the miracle cure I'd hoped for, though I did feel a little bit better. Then I discovered I was pregnant, and my doctor and I agreed that I should stop taking the medication. After my daughter was born, I didn't go back on antidepressants. I continued to struggle with bulimia, on and off, for the next few years. With a lot of work on myself, I eventually managed to free myself from bulimia's terrible grip, though I still had a tendency to slide into unhappiness easily and often.
Not long after that, I met actor Casper Van Dien, fell in love, and got married. But within a year, my body started hurting everywhere; it was like having a mild case of the flu all the time. I was diagnosed with fibromyalgia, a chronic condition characterized by widespread pain in the muscles, ligaments, and tendons. This made sense to me: instead of feeling miserable mentally, I'd transferred the unhappiness to a physical level. I believe that in my case, the fibromyalgia was my body's way of experiencing depression. And as it turned out, when I was given a low dose of antidepressants, my pain was reduced considerably.
Although Casper and I already had three children between us, we knew we wanted to have a family together. When we decided to get pregnant, I stopped taking the medication. Within a month I was pregnant, and the fibromyalgia went away.
But, two babies later, the physical reality of being a forty-three-year-old mother of five children, two of whom were under three, began to take its toll on my already fragile health and happiness. I was exhausted and sick all the time. I couldn't focus or concentrate. And my memory! I was sure I had early-onset Alzheimer's disease.
Mornings were particularly awful. Even after a full night's sleep, I still felt tired, so Casper was the one who got up and made breakfast for the kids and got them off to school. I remember one morning, India, my eldest daughter, came into my bedroom to wake me. She wanted me to come eat breakfast with everyone. "Darling," I told her, "I'm exhausted. I feel really sick." She looked at me with a sad expression and said, "Mom, you're always sick," before she turned and left the room. Devastated, I curled into a ball and cried. She was right.
I knew there was something very wrong with me, yet no doctor could find the problem. My blood work appeared completely normal, and the only thing they'd recommend was that I go on antidepressants again, but I didn't like the troubling side effects. There had to be a way to feel better naturally, but I had no idea how. I was at my wits' end—when help came to me in a way I could never have imagined.
It was 3 a.m. and I was sitting on an airplane en route to Florida to visit my in-laws for Christmas. The red-eye flight had been a nightmare. The two little ones had refused to go to sleep, fighting, whining, and generally being impossible. Finally, everyone had settled down. I took a deep breath and closed my eyes too. Exhaustion and despair descended on me like a three-hundred-pound weight.
I couldn't sleep so I reached for the in-flight magazine in the seat pocket in front of me. As I flipped through it, my eye was caught by a heading in large type across the top of a page: ARE YOU SICK AND TIRED OF BEING SICK AND TIRED? I instantly got chills and felt my hair stand on end. I continued scanning down the page. Are you tired of being told by doctors that there's nothing wrong with you? That your blood work is "perfect" despite your symptoms of fatigue, lethargy, and mental cloudiness? Are you too tired to enjoy your life?
I felt tears come to my eyes. I'd been so ashamed, convinced that my problems were all in my head. It was such a relief to realize I wasn't the only one who felt this way! I kept reading.
It was the description of a clinic that specialized in a natural approach to balancing the body and mind that used supplements, natural hormones, and proper nutrition. When I began reading the story of a forty-five-year-old woman who'd tried this approach, I was floored. The woman could have been me. I had every single one of the symptoms she listed, but by following the clinic's recommendations, she'd gone from being miserable to feeling better than she'd ever felt before. I ripped the page out and put it in my purse. Then I leaned back in my seat, sending up a silent prayer of thanks for what I considered divine intervention, before finally falling asleep myself.
When we got home from Florida, I made an appointment for a consultation with the doctor featured in the magazine. I discovered that what I was experiencing was extremely common for women of all ages, and for men as well. The bottom line was that my hormones were out of balance, which had started a disastrous domino effect, affecting my thyroid and adrenal glands and seriously compromising my health. I began taking nutritional supplements and bioidentical (nonsynthetic) hormones.
My diet was also contributing to the problem, so on Dr. Hotze's recommendation, I eliminated all sugar, dairy, alcohol, and grains for a month. I ate moderate amounts of healthy fats in the form of nut butters, avocado, olive oil, and coconut oil, lots of vegetables, and a variety of proteins, such as organic meats, eggs, and fish.
At first, I didn't see much change, but after just two weeks, I stopped having that 3:00 afternoon slump. (I've since discovered that that's the time that your body stops making serotonin in preparation for the night's rest.) When the thirty days were up, I was so encouraged by how I felt that I decided to keep on going with my new nutrition, supplement, and hormone program. After another month, I had to admit the results were pretty miraculous. I felt transformed in body, mind, and spirit.
Today, though I've added more things back into my diet, I continue to have only small amounts of sugar and processed carbs, because I can definitely feel the difference when I eat them. In general, I just feel so healthy. I don't have fatigue anymore. I feel energized and have much more mental clarity; my strength has returned, and even my excess weight has just melted away. Best of all, I'm able to be there for my kids. I remember one day about a month after I started the program, my youngest came into the bedroom first thing in the morning and asked me to get up and make breakfast. I immediately threw back the covers and raced her to the kitchen. I was reaching for the cup of chai, caffeinated spiced tea that I'd always needed to wake myself up, when I suddenly realized I was already awake! I felt clear and full of energy. Since then, I've become a regular at the morning breakfast party at our house, and I don't know who's more excited about it—me or the kids.
How ironic that the "airplane trip from hell" turned out to be the beginning of my journey out of purgatory. Now, with my brain chemistry and hormones in balance, I'm able to experience the natural happiness that I see in the faces of my children. It was finding the key to a healthy, balanced body that unlocked the door to a happy life.
### The Power of the Plate
As Catherine discovered, a fundamental way to feel better is to get the natural chemicals in your body to function in a balanced way. What you eat can play a crucial role. Happy for No Reason isn't intended to be a diet book. (For some excellent books on nutrition and natural supplements to support your happiness, please refer to the resource section.) Still, I'd like to offer a few general guidelines to get you eating in a way that not only will improve your health, but will make your cells happy. Knowing the dramatic and cumulative impact that diet has on happiness has been one of my greatest motivators for sticking with this Happiness Habit.
#### The Make-Your-Cells-Happy Nourishment Guide
1. Go for Happiness-Building Foods—Fresh and Whole: A balanced diet of fresh, whole foods gives our brains the necessary raw materials to ensure that we produce abundant happiness juices every day. Without those building blocks, our biochemistry gets out of whack, causing elevated blood sugar levels, adrenal exhaustion, and a shortage of vital hormones that buffer stress in everyday life.
Eating whole foods means giving your body food that's as close as possible to the way nature made it. Avoid what I call "fake food" that comes out of a box or a can, which is many times removed from its original source by processing, preservation, and synthetic packaging. Shop the outer aisles at your supermarket, where you'll find the fruits and vegetables, fresh meats, fish, and poultry, not the inner ones where the boxed, canned, and packaged foods live. Buy whole grains, organic produce, and hormone-free meat, dairy, and poultry products. Yes, the food generally costs more, but you'll more than make up for it in fewer doctor visits and a healthier life.
ZIGGY ©2005 ZIGGY AND FRIENDS, INC.
Reprinted with permission of UNIVERSAL PRESS SYNDICATE.
All rights reserved.
2. Keep the Water Flowing: Being dehydrated is a no-no for happy cells. We need water because our bodies are mostly water! To fully take in the nourishment in our food, our bodies require the chemical elements of hydrogen and oxygen found in H2O (a fact I learned in high school biology and promptly forgot). People often feel hungry when what their bodies actually need is water. The next time you feel like reaching for a snack, take a sip or two of water first. Even better, do as the experts recommend: make it a habit to drink half your body weight in ounces each day. (A 120-pound person would drink 60 ounces.)
3. Lose the Happiness-Robbing Foods:
• Shake the Sugar: Do you know how hard it is to get health experts to agree on anything? Yet every one I interviewed agreed that the biggest happiness destroyer in our diets is white sugar. It's powerfully addictive and wreaks havoc with your brain, causing depression, anxiety, and that sluggish, low-energy state you recognize when your forehead hits the desk at 3 p.m. The synthetic sugar substitutes are unfortunately no better; numerous reports point to potentially negative side effects. When you take your sugar in its natural form, as in fresh fruit, the body can handle it better.
• Curb the Carbs: In the carb department, "whole" is the operative word. When you eat processed grains, such as white bread, white rice, or pasta and pastries made with white flour, you set off a high-glycemic alarm that causes your blood sugar levels to spike and crash, constantly altering your mood and energy levels. The Standard American Diet (SAD—what an appropriate acronym!) relies on too many starchy, refined carbs. Switching to whole grains like brown rice, quinoa, and millet is a simple but effective trick for feeling healthier and happier fast.
• Kick the Caffeine: When we need a lift, a lot of us reach for the caffeine jolt of coffee or soft drinks. Caffeine blocks the transmission of the brain chemical adenosine, which results in more adrenaline pouring into the bloodstream. After drinking a cup of coffee or a soda, you feel alert, motivated, and stimulated, but that high reaches its peak in 30 to 60 minutes, and then down you go. Instead, try decaffeinated green tea (regular green tea has caffeine), which not only offers powerful antioxidant protection to help make cells happy, but gives you a more sustainable level of energy.
### Super-Nourishment to Promote Happiness in the Brain and Body
To make our cells happy, it's sometimes helpful to boost our body's nourishment in specific ways. Nutritional psychology expert Julia Ross, the author of The Mood Cure, believes that much of our unhappiness is the result of "critically unmet nutritional needs." She's developed a program based on research that links our moods to the balance of our body's four key neurotransmitters that produce happiness. These neurotransmitters are fueled by nutrients called amino acids, and if your amino acid levels are sufficient and these four key neurotransmitter levels are high, you'll generally feel positive in life. If you are low in any of them, you may develop symptoms specific to that neurotransmitter deficiency.
The Four Happiness Neurotransmitters and Your Brain
If you're high in serotonin, you're positive, confident, flexible, and easygoing.
If you're sinking in serotonin, you'll tend to become negative, obsessive, worried, irritable, and sleepless.
If you're high in catecholamines (norepinephrin, dopamine, and adrenalin are in this group), you're energized, upbeat, and alert.
If your catecholamines have crashed, you can sink into a flat, lethargic funk.
If you're high in GABA (gamma-aminobutyric acid), you're relaxed and stress-free.
If there's a gap in your GABA, you'll be wired, stressed, and overwhelmed.
If you're high in endorphins, you're full of cozy feelings of comfort, pleasure, and euphoria.
If you're near the end of your endorphins, you may be crying during commercials and overly sensitive to hurt.
From The Mood Cure, by Julia Ross
At the end of this section, you'll find Julia Ross's Mood Type Questionnaire that can help you determine more accurately if you have adequate amino acid levels.
### Cleaning House
Many people's bodies are overloaded with "toxic gunk" that blocks the happiness juices from flowing. There are times when the best thing you can do for your body is to clean house and remove the gunk. Cleansing is a fast track to making your cells happy, but you have to do it properly. Cleanses usually include a temporary change in eating (or sometimes fasting), detoxification baths, and elimination herbs. According to Chinese medicine, each of our organs relates to a specific emotion—the liver relates to anger, the kidneys to fear, the lungs to sadness, and the spleen to worry—and there are specific types of cleanses for the different organs. Some experts recommend doing regular cleanses at the change of each season. IMPORTANT NOTE: Always consult your doctor before going on a cleanse; certain health conditions may make it an unwise option.
### Hormones and Happiness
Another major component of Catherine's health and happiness program involved hormones. While this aspect affects more women than men, it is an important factor to consider when anyone is experiencing unhappiness in life. Hormones are nourishment for the cells, affecting the functioning of every organ, but especially the brain, so hormonal imbalances can have a profound effect on mood and happiness.
Not all hormone replacements are alike. Many people prefer to take bioidentical hormones, compounds that have exactly the same chemical and molecular structure as hormones that are produced in the human body. According to the women's health expert Dr. Christiane Northrup, "Because bioidentical hormones are just like the hormones that our bodies were designed to recognize and utilize, their effects are more . . . consistent with our normal biochemistry—with less chance for unpredictable side effects at low replacement doses than with synthetic, non-bioidentical hormones." If you're struggling with any hormone issues, make sure to find a health care practitioner who doesn't believe in the one-size-fits-all approach.
Exploring the many ways you can nourish your body is a worthwhile investment of time that will pay rich dividends in your happiness.
Exercise
The Four-Part Mood Type Questionnaire
Write down the number next to each symptom listed below that you identify with. Total your score in each section and compare it to the cutoff score. If your score is over the cutoff, or if you have only a few of the symptoms but they bother you on a regular basis, learn more about the amino acid indicated in Julia Ross's book The Mood Cure or at her website: www.moodcure.com. The exciting news is that you can easily get the amino acids your brain needs from nutritional supplements, and you can experience your moods improve in a short period of time as your neurotransmitters get the fuel they need.
Part 1. Are You Under a Dark Cloud? If So, Your Serotonin Levels May Be Low
3) Do you have a tendency to be negative, to see the glass as half empty rather than half full? Do you have dark, pessimistic thoughts?
3) Are you often worried and anxious?
3) Do you have feelings of low self-esteem and lack confidence? Do you easily get to feeling self-critical and guilty?
3) Does your behavior often get a bit, or a lot, obsessive? Is it hard for you to make transitions, to be flexible? Are you a perfectionist, a neatnik, or a control freak? A computer, TV, or work addict?
3) Do you really dislike the dark weather or have a clear-cut fall/winter depression?
2) Are you apt to be irritable, impatient, edgy, or angry?
3) Do you tend to be shy or fearful? Do you get nervous or panicky about heights, flying, enclosed spaces, public performance, spiders, snakes, bridges, crowds, leaving the house, or anything else?
2) Have you had anxiety attacks or panic attacks (your heart races, it's hard to breathe)?
2) Do you get PMS or menopausal moodiness (tears, anger, depression)?
3) Do you hate hot weather?
2) Are you a night owl, or do you often find it hard to get to sleep, even though you want to?
2) Do you wake up in the night, have restless or light sleep, or wake up too early in the morning?
3) Do you routinely like to have sweet or starchy snacks, or wine in the afternoons, evenings, or in the middle of the night (but not earlier in the day)?
2) Do you find relief from any of the above symptoms through exercise?
3) Have you had fibromyalgia (unexplained muscle pain) or TMJ (pain, tension, and grinding associated with your jaw)?
2) Have you had suicidal thoughts or plans?
Total Score ____________ If your score is more than 12 in Part 1, you may have a serotonin deficiency.
Part 2. Are You Suffering from the Blahs? If So, Your Catecholamine Levels May Be Low
3) Do you often feel depressed—the flat, bored, apathetic kind?
2) Are you low on physical or mental energy? Do you feel tired a lot, have to push yourself to exercise?
2) Is your drive, enthusiasm, and motivation quota on the low side?
2) Do you have difficulty focusing or concentrating?
3) Are you easily chilled? Do you have cold hands or feet?
2) Do you tend to put on weight too easily?
3) Do you feel the need to get more alert and motivated by consuming a lot of coffee or other "uppers" like sugar, diet soda, or ephedra?
Total Score ____________ If your score is more than 6 in Part 2, your level of catecholamines may be low.
Part 3. Is Stress Your Problem? If So, Your GABA Levels May Be Low
3) Do you often feel overworked, pressured, or deadlined?
1) Do you have trouble relaxing or loosening up?
1) Does your body tend to be stiff, uptight, tense?
2) Are you easily upset, frustrated, or snappy under stress?
3) Are you easily chilled? Do you have cold hands or feet?
2) Do you tend to put on weight too easily?
3) Do you often feel overwhelmed, or as though you just can't get it all done?
2) Do you feel weak or shaky at times?
3) Are you sensitive to bright light, noise, or chemical fumes? Do you need to wear dark glasses a lot?
3) Do you feel significantly worse if you skip meals or go too long without eating?
2) Do you use tobacco, alcohol, food, or drugs to relax and calm down?
Total Score ____________ If your score is more than 8 in Part 3, you could be low in GABA.
Part 4. Are You Too Sensitive to Life's Pain? If So, Your Endorphin Levels May Be Low
3) Do you consider yourself or do others consider you to be very sensitive? Does emotional pain, or perhaps physical pain, really get to you?
2) Do you tear up or cry easily, for instance, even during TV commercials?
2) Do you tend to avoid dealing with painful issues?
3) Do you find it hard to get over losses or get through grieving?
2) Have you been through a great deal of physical or emotional pain?
3) Do you crave pleasure, comfort, reward, enjoyment, or numbing from treats like chocolate, bread, wine, romance novels, tobacco, or lattes?
Total Score ____________ If your score is more than 6, you may be low in endorphins.
Used by permission of Julia Ross.
Happiness Habit for the Body #2
Energize Your Body
There is a vitality, a life-force, an energy, a quickening that is translated through you into action. . . . Keep the channel open.
—Martha Graham, twentieth-century dancer and choreographer
You wouldn't expect your car to run well if you never shut it off or if you didn't fuel it. Yet we often neglect our bodies in similar ways and expect them to keep going like the Energizer Bunny. To access our natural joy and maintain balance in our lives, we need to energize our bodies through proper rest, breathing, and exercise.
Eastern systems of health and well-being have long recognized the presence of a vital life force or energy in the body. In China, this life force is called chi or qi, and in India, it's called prana. When this vital life force is increased in the body, it energizes the entire system, mobilizing healing forces and eliminating emotional blocks that dampen happiness.
I'd heard of Qigong, the ancient Chinese practice of cultivating the life energy in the body, but had never tried it. So I decided to take a class with Qigong master Chunyi Lin. I was impressed right away with Master Lin's energy, joy, and vitality. Although he is close to fifty, he looks like a man in his thirties. I interviewed Master Lin and immediately added him to the list of my Happy 100. His story shows how moving your body releases your vital energy and makes you happier.
Master Lin's Story
Moving Energy, Feeling Joy
I was born in the mountains of China. Both my parents had good jobs, and although we weren't rich, my parents were kind and loving people and we were happy.
When I was eight years old, the Cultural Revolution started. Chairman Mao, the Communist Party leader, wanted to get rid of everyone who had a different political opinion. Good people went to jail, teachers and professors were forced to work in the countryside farming, and hundreds and thousands of honest and learned people were killed. The whole country was in a terrible mess.
One night, as our family sat eating dinner, armed men came into our home. They grabbed my father, tied his hands behind him, and pushed him out of the house. They arrested him for what they eventually found were false charges. We couldn't see him for over six months; he was sent to jail and later, to a work compound.
Then one night, my mother disappeared. We didn't learn until later that she had been forced to flee for her life. Our nanny, an elderly widow my parents had taken in to live with us, was left alone to care for my two brothers, my sister, and me. We were still reeling from our mother's disappearance when, a few days later, a group of our neighbors, people my parents had liked and helped, stormed into our house holding weapons and ordered us out into the street. They did this to show their loyalty to the local people in power and to avoid sharing my parents' fate.
The wind was blowing with cold evening rain as we watched the neighbors seal off our house. We huddled together, four little children and an old woman, walking in the rain down our narrow street, trying to find shelter.
We were on the street for three days with no food. Nobody dared to take us in or give us anything to eat. On the third night, as we crowded together on the side of the street trying to keep warm, a gun battle broke out between two rival factions. Bullets were flying all around us and there were explosions everywhere.
Terrified, we ran from one house to the other, desperately trying to find a safe place to hide, but no one would let us in. As we pressed together in a doorway, shivering from fear, a very elderly lady opened the door. We recognized her as a landlord, who we'd been told by the government were all devils. Frightened, I jumped behind our nanny to hide.
The elderly lady looked at us and said, "Oh, you poor little children. With all this fighting, it's too dangerous for you to be outside. Please, come inside. You'll be safe."
At great personal risk to herself, this wonderful old lady fed us and then hid us, along with many others, in a storage shed at the back of her house. For weeks, we never ventured out of the shed for fear of discovery. Then finally, our parents managed to locate us.
They sent a family friend to escort us, and after a long and dangerous trip we made it to my grandmother's house in the country. We lived with my grandmother for over a year, before my parents determined that it was safe for us to come and live with them again.
My experiences during the Cultural Revolution colored my life for many years. Everything had been turned upside down. So-called good friends turned out to be the ones who wanted to take your life. So-called evil people turned out to be your angels. I didn't even trust my parents. In my child's mind, they had simply abandoned us. Feeling betrayed by everyone and everything, I withdrew from my family and friends.
In high school, I began studying Qigong, an ancient healing system of balancing energy in the body through movement, which is widely taught throughout China. When I was little, I'd heard many stories of Qigong masters and miraculous healings. I'd always loved those stories and wished that one day I could become a healer like that. During the Cultural Revolution, teaching or studying Qigong was forbidden. Anyone caught even practicing Qigong was arrested. It wasn't sanctioned by the communist government, so my teacher, a revered Qigong master, taught me in secret. I learned some very basic Qigong movements and found that when I did them, I felt so peaceful.
After high school, many students like me were taken by soldiers and forced to live on farms in the country. For five years, we worked like slaves, often without enough to eat. The work was very hard and I suffered many injuries and diseases in the country. I would often sneak off at night by myself to practice my Qigong. It was the only time of the day that I felt okay, and it helped keep me going.
Finally, Chairman Mao died and the Gang of Four was purged from power. Things began to return to normal in China. I was able to attend college and my life started to improve, but I was still a very depressed and angry young man. I didn't want to talk to anybody. I had no friends. Although I didn't act like it outwardly, I actually hated people. In fact, I hated everything. Sometimes I just wanted to kill myself and leave this world. I began searching hard, trying to find out how to be happy in life.
Then one day as I was playing basketball, I had an accident that severely damaged the cartilage in both my knees. I was in constant agony and could hardly walk, and nothing, including painkillers, could stop the pain.
I heard that a very powerful, nationally renowned Qigong master was coming to town to give a workshop and that many people, just by attending his workshop, had their serious problems go away. I didn't believe this was true, but I thought it couldn't hurt to try. Besides, I needed a miracle.
Fifteen thousand people attended the workshop, which was held in a soccer field. It started at noon. This master taught a very potent form of Qigong and led us through many different movements and meditations. I had amazing experiences during those hours. The energy I felt was like a volcano. My body rocked and shook as the Qi began to flow through the energy channels inside of me. First I cried just like a baby, and then I began to laugh uncontrollably. I laughed and laughed until my stomach muscles ached. I knew this was a Qi reaction, and after half an hour, it gradually stopped, leaving me comfortably tired.
Then I felt a tingly sensation, starting with my toes and working its way up the length of my body. This feeling was not only in the skin; it was in the muscle and in the bone. It felt as though my body was winter ice melting in the spring sunshine—it was so peaceful, so beautiful, so nurturing. The feeling of my own life force flowing unimpeded through my body was like a breeze sweeping through my bones. I felt so relaxed and happy. It wasn't the kind of happiness that comes from getting a birthday gift or earning a promotion at work. This happiness was from the bottom of my heart. At 7:30 that night, the master finally guided us through the ending exercise. I stood up—and found a miracle had happened! The swelling in my knees had disappeared and the pain was 80 percent gone. I began running and jumping on the soccer field like a little kid.
I continued to practice the very simple movements that I'd learned during the workshop, and after two months, all the pain in my knees was gone. What's more, by getting the energy or Qi moving in my body, I began to be flooded by feelings of peace and beauty, and everything in my life started working.
This flow of vital energy allowed me to feel the blockage in my heart that had caused the depression, anger, and emotional pain I had felt for so many years. Whenever I felt that block in my heart, I focused on feeling forgiveness. Over the months and years that followed, the impressions I'd held in my body from my many difficult experiences were released, and I felt happiness in every cell in my body. I became such an easygoing person. I was able to forgive all the people in my life who had hurt me and my family. The daily practice of increasing and balancing my energy through movement healed me completely—both physically and spiritually.
Naturally, I wanted to share Qigong with other people, so that they could have the same joy in their lives that I was experiencing. I developed my own system of Qigong and eventually began to teach this beautiful practice in China and then in America. I've seen how, when people move their bodies with conscious awareness, they get happier.
It's been nearly twenty years since that day on the soccer field when I first fully experienced the power of the Qi, and today I am flourishing. I have a wonderful family and I do exciting and meaningful work in the world, helping thousands of people to heal themselves. Every day is a good day—whatever happens.
### Motion with Meaning
Phys. Ed. class should have been called Happiness 101! Research overwhelmingly shows that people who exercise are happier. Like Master Lin, you can transform your life by making some form of exercise or movement, such as walking, running, swimming, dancing, Qigong, or yoga, a regular habit. When you exercise, your brain gets more oxygen, but even more important, it causes your body to produce valuable chemicals and hormones that impact energy, mood, and health. In a recent study done on athletes, the Harvard psychiatrist John Ratey found that dopamine, serotonin, and norepinephrine—those wonderful happiness chemicals—were all elevated after exercise. In fact, dozens of studies indicate that exercise is often as effective for reducing depression as the typical medications prescribed.
Exercise can also help relieve and prevent anxiety, creating a calming effect that lasts for about four hours after exercising. Try getting in a fight right after a rigorous workout. It's not likely to happen! Exercise also increases endorphins, the "bliss" neurochemicals responsible for that wonderful exercise high, by about 500 percent.
Dr. Henry S. Lodge, assistant clinical professor of medicine at Columbia University and coauthor of Younger Next Year, explains how exercise works at the cellular level to keep you vital and healthy. He says that every day you replace about 1 percent of your cells, getting a whole new body every three months. When you exercise, your muscles release specific substances that tell your cells to grow. When you sit around like a couch potato, your muscles let out a steady trickle of chemicals that tell your cells to die. What powerful motivation! If you started exercising in January, you could have a whole new body of strong, happy cells by April.
To turbocharge the effects of exercise, bring awareness to your movement. Moving with conscious intention is one of the reasons Qigong is so effective. You can take this same approach to any exercise. For example, as you set out on your next walk, Master Lin recommends telling yourself, "On my walk, all my channels are going to become clearer, I will open my heart to nature, and by the time I'm finished, my energy will be much stronger." He says this intention will dramatically boost the impact of movement on your happiness.
You can even energize your body without leaving your chair. Brian Siddhartha Ingle, an osteopath specializing in a neuromuscular education program called Hanna Somatics, told me that he sees daily how people's posture affects their energy and level of happiness. Next time you're feeling stressed, anxious, or blue, notice if your shoulders are coming up toward your ears. Instead of trying to pull your shoulders down, Dr. Ingle suggests exaggerating this movement by bringing your shoulders even closer toward your ears—then, SLOWLY, letting them come down. Do this three to four times and see if your stress and anxiety levels go down.
### Breath of Life
You can live for weeks without eating and a few days without water, but you can last for only a few minutes without breathing. Our breath is the most important fuel we have for energizing our bodies. For millennia, many traditions have understood the importance of skilled, mindful breathing to maintain optimal health and well-being. In the past thirty years, hundreds of clinical studies involving thousands of participants have been conducted showing how breathing techniques can relieve anxiety, depression, and chronic fatigue and increase mental clarity. Today, even our own FDA approves breath training as a recognized treatment for hypertension.
Stop and notice how you're breathing at this moment by putting your hand on your stomach. See if your hand goes out on your inhale and back in on your exhale. If you're like most people, you breathe from your chest cavity alone and your hand doesn't move at all. In my interview with Dr. John Douillard, an Ayurvedic practitioner and the author of Body, Mind, and Sport, he told me, "While shallow breathing will keep you alive, the oxygen isn't getting to the deeper cellular levels where it's needed to cleanse cellular pathways and open the way for happiness-enhancing chemicals to flow. Breathing deeply from the stomach is one of the most powerful waste-removal techniques we have; it increases the prana or Qi in the body and helps to create a happy, feel-good physical state. We have 26,000 opportunities a day, each time we breathe, to support our happiness."
The way we breathe is intimately connected with our emotions. Each emotional state has its own pattern of breathing: when we're anxious, we breathe quickly and shallowly; when we're sad, we sigh deeply; and when we're angry, we breathe in short, forceful bursts. But this connection works the other way too: if you start breathing as though you're anxious, sad, or angry, you stimulate the area of the brain associated with that emotion and quickly begin to experience it. The next time you're feeling agitated or anxious, try taking five to ten deep "belly breaths" in a row and notice the immediate calming effect it has on your body.
### Sleep Your Way to Happiness
Sleep is nature's way of recharging our batteries. Though we all know we're happier when we get enough sleep, ask most people whether they're getting eight hours a night, and they'll usually respond with a hollow laugh. Who has the time, they say, when there's the Letterman or Leno monologue to watch, bills to pay, children to tend to, problems to worry about?
When I was working as a consultant to Fortune 500 companies, I was always amazed to hear people talking about sleep as though it were a contest. One executive would say, "I get by on only five hours of sleep a night." Her colleague would say, "Yeah? I never get more than four hours." They'd brag that if their to-do lists got too long, they'd just lop off a couple more hours of sack time so they could accomplish more to get ahead. The person who slept the fewest hours a night won.
Getting too little sleep is definitely nothing to brag about. According to the Centers for Disease Control and Prevention, there's a strong correlation between the amount of restful sleep a person gets and his or her level of happiness. I was blown away when I read a December 2004 study in the journal Science reporting that the quality of our sleep has a greater influence on our ability to enjoy our day than household income or marital status. Who knew that snoozing trumps salaries and spouses?
I subscribe to the Ayurvedic wisdom (Ayurveda is the ancient medical tradition of India) that "an hour of sleep before midnight is worth two hours of sleep after midnight." I once went to see an Ayurvedic doctor, a little old man no taller than I was, who must have been at least a hundred years old. He was as wrinkled as a raisin but still had all his hair, a wide smile, and the twinkliest eyes I'd ever seen. He told me that if I was ever feeling unhappy or out of sorts, I should try going to bed before 10 (ideally by 9) three nights in a row, and then see how I feel. Every time I do that, I find a whole new world out there by the third day. I once again become an energetic, joyful, and rested person! I call this "catching the 10 o'clock angel train," and it's definitely one of my favorite happiness habits. Over the years, I've recommended this to lots of people, and everyone who's done it has thanked me profusely.
As an experiment for the next week, make getting a good night's sleep your priority. Letterman and Leno will soldier on without you, and you'll be so much happier.
To energize your body, return to these basics: keep your body moving, breathe deeply, and get enough rest. Here's a simple Qigong exercise you can use to raise your happiness level through conscious movement, regardless of whether you're sitting on a mountaintop or taking a break at work.
Exercise
Spring Forest Qigong—Breathing of the Universe
Don't be fooled by the seeming simplicity of this exercise. As Master Lin (pictured in the accompanying photos) says, "The most powerful is also the simplest." This exercise is subtle, yet extremely effective for opening blockages in the whole body, especially the lungs.
1. Relax in the proper position: Stand with your feet a bit more than shoulder-width apart, knees slightly bent, and look forward. Smile and relax. Draw your chin back slightly to straighten the spine. Drop your shoulders and move your elbows out a little. Open your hands and spread your fingers.
2. Breathe deeply: Slowly take three deep, gentle breaths through your nose. Imagine using your whole body to breathe. Visualize energy coming in and collecting in your belly. When you exhale, visualize any pain or sickness changing into smoke and shooting out from every cell into the universe.
3. Silently say the phrases: Close your eyes and silently say the phrases, "I am in the universe. The universe is in my body. The universe and I combine together." Take a moment to feel the quietness, the stillness of the universe.
4. Move your hands: As you inhale, slowly move your hands wide apart and to the sides. (Use the pictures as a guide.) As you exhale, slowly bring your hands back together (don't let your hands touch each other). When you open your arms, feel the energy expanding in the space between your hands. When you bring your hands toward each other, feel the energy compressing in the space between them.
Do the exercise for five to six minutes. When finished, take three slow, gentle, deep breaths, and allow yourself to relax for another minute or two.
Used by permission of Spring Forest Qigong.
Happiness Habit for the Body #3
Tune In to Your Body's Wisdom
We will be in tune with our bodies only if we truly love and honor them. We can't be in good communication with the enemy.
—Harriet Lerner, PhD, clinical psychologist and author
My interviews with the Happy 100 revealed that they consistently tune in to their body's wisdom when making choices that affect their health. Knowing when it's time to rest, time to drink more water, to take a bath or exercise—the self-nourishing activities that keep us in balance—is a habit all of us can develop.
Though our body knows exactly what is best for us at any given moment, we often don't listen. When we start loving and trusting the wisdom of our body, it's "the beginning of a beautiful friendship," one that makes our cells infinitely happier.
During my interview with Happy 100 member, author, and consciousness pioneer Gay Hendricks, he told me the following story that illustrates the powerful effect of living in tune with what your body really wants.
Gay's Story
The Magic Key
I was fat practically from the moment I was born. During my entire boyhood and well into adulthood, I thought being fat was a curse. I blamed it on fate, on God, on my parents. Sometimes I blamed it on all the enticing food I gorged myself with. But most of the time I blamed it on myself: my insatiable appetite, my lack of willpower, my craving for rich food. On really bad days I believed there was a pit inside of me—a fundamental fault so deep it felt bottomless. I'm not sure if being fat was making me unhappy, or being unhappy was making me fat. It didn't really matter—the fact was: I was very fat and very unhappy.
Things went this way until I was in my mid-twenties. Then one day during a graduate school psychology class exercise, a fellow student named George looked me in the eye and asked, "What is your weight all about?"
I was so taken aback that all I could do was stutter, "W-what do you mean?"
He simplified it for me: "Why are you so fat? Why are you trying so hard to kill yourself at such a young age?"
I was horrified. He was asking me about the one thing that must never be mentioned. Although my family had taken me to see a dozen different medical specialists in futile attempts to handle my weight problem, they never talked about it in front of me. My classmate did the one thing no one had ever done: he looked me in the eye and asked me why I was fat.
I froze up, breaking eye contact with him, and mumbled something about having glandular problems and a family history of obesity. He looked at me with pity tinged with disgust. "That's it?" he asked. "That's all you're going to say?"
When I didn't respond, he sighed, and then, to my relief, we continued our class work. The problem was that I couldn't sleep that night. I must have replayed his question in my mind a hundred times, but could find no answer.
Over the next week, I couldn't get his question out of my mind. I became more and more frazzled, my work became sloppy, and I argued incessantly with my wife. Then, one wintry day, I went out for a walk down a deserted stretch of country road.
Picture a pear-shaped, three-hundred-pound man trudging down the road, his body stuffed into an orange parka. The day shimmered, and all was peaceful and quiet around me. The only sound was the crunch of my boots on the dry snow as I tramped along the road, lost in thought.
Suddenly, both my feet shot straight out from under me and I fell. I'd stepped on a patch of ice that was covered with a fine dusting of snow. As the back of my head smacked on the icy road, I saw an explosion of stars and a blast of pain wracked my body.
In the very next moment, the pain disappeared, and I found myself having the strangest experience. Suddenly I was in two places at once! I was up in the air, looking down at my body, and at the same time looking up through my body toward the sky.
From this unique two-way vantage point, I could see deeply into myself for the first time. I saw that my mind, body, heart, and soul weren't as separate as I'd always believed. In a slow-motion, altered state of consciousness, I saw all the emotional defenses I had mounted on a daily basis to protect myself. Suddenly I had the answer to George's question: eating was my body's way to avoid pain! I had created an armor of fat that insulated me from my feelings of fear, sorrow, loss, and shame. The problem with this strategy is that it left me with a painful emptiness inside that further fueled my craving for food. It was the perfect vicious cycle.
Now as I lay there, I could feel my misery draining away as I simply relaxed into the spacious fullness that I realized was my soul. It permeated every part of me—my mind, body, emotions—and filled me to the brim with a vibrant contentment. I felt a strange sensation: I was completely satiated, maybe for the first time in my life!
This was the feeling of completeness I had sought with snacks, sandwiches, and sodas. I felt more than full; I felt nourished. I saw that by walling myself up inside to avoid painful feelings, I had lost touch with my body's wisdom and starved myself of spirit. No wonder I had a hunger no food could satisfy!
At that point, my eyes flickered open and the everyday world returned. I got up off the icy road, dusted myself off, and stood looking around. I felt a singing-humming exhilaration of clarity in me as I made a choice in every cell of my being. "From now on," I said, "I'm choosing to listen to my body's wisdom and truly nourish myself from the inside." Then, smiling with a newfound joy, I headed home.
Over the course of the next twelve months, I lost over a hundred pounds. I won't kid you and say it was easy. However, I can tell you sincerely that it was simple. My fall on the ice had shown me a magic key, which was just one action: Before I did anything, I tapped into my body's inner knowing and asked myself, "Would this truly feed me?"
For years, I'd known everything there was to know about protein and carbohydrates, calories and dieting—but nothing about nourishing myself from within. The magic key changed everything, including my relationship with food. The morning after my fall, I considered my standard breakfast: cereal with milk and a few spoonfuls of sugar. My favorite part was when I finished the cereal and got to the layer of undissolved sugar in the milk at the bottom of the bowl. I loved that sweet milky taste, even though I knew it made me feel bloated and irritable an hour later.
Force of habit pulled me toward the cabinet that held the cereal. But this time I paused and reminded myself of my commitment to listen to what my body really wanted. I looked at the cereal box and asked, "Would this truly feed me?" My body immediately said "No," with a slight contraction in my throat and stomach. It was like stepping on the brakes of a car and hearing the metallic squeal.
I went to the refrigerator and looked in. My eyes were instantly drawn to a box of fresh blueberries. They were the kind of thing I would never have eaten in a million years. Yet that day, the box of berries stood out like a beacon. They were actually quite beautiful. I took out the box and looked more closely. I picked up a blueberry and asked the same question, "Would this truly feed me?" My body said "Yes" with a sensation of expanding space and lightness inside me. I popped the blueberry in my mouth and chewed it slowly. It was absolutely delicious. To this day I can remember the burst of fresh aliveness in my mouth. I thought, "Could everything taste this good if it truly fed me?"
I took another blueberry and started to eat it, but stopped abruptly when I realized I was about to eat it simply because the first one had tasted so good. I asked myself again, "Would this particular blueberry also truly feed me?" When my body said "Yes," I popped the berry into my mouth. In response to my fourth inquiry, my body said "No." So, on the first morning of my new life, I had three blueberries for breakfast—and I felt great! When I got hungry an hour later, I just repeated the process. It was like magic; my stomach and every other part of my being felt full.
The magic key made life incredibly straightforward. It showed me what food to eat, told me when I needed to take a walk and when to turn off the light and go to sleep. It showed me which friends had to go and which ones I could continue to spend time with. It showed me the changes I needed to make in order to have a successful career. I've used it for everything, and it's always worked.
In retrospect, I think being fat was the best thing that ever happened to me. Struggling with my weight led me to an awakening that showed me what I had always wanted to know: how to be alive and in harmony with all things, including the one thing I'd never been in harmony with—myself. Today, being in touch with my body's wisdom in every moment has become an automatic choice, one that keeps me feeling vital and full of joy.
The magic key completely transformed my body and my life, showing me how to be nourished from the inside. My marriage, my career, my books and courses are all based on that nourishment. It took a sledgehammer-like blow to the head to knock the sense back into me, but it has led me to the life of my dreams.
### Body Talk
You can tune in to your body in any situation by asking yourself: Is this what my body really wants or needs? Or by posing specific questions: What am I hungry for? Am I listening to my body's cries for relaxation and relief from stress? What do I need to recharge my batteries? If you listen closely, your body will tell you what's best for your health and happiness every time. Sometimes, as Gay experienced, just mentally asking the question will bring an instant answer in the form of a physical sensation.
This is the perfect time to use your inner GPS to see whether something you're planning to do will expand or contract you. When you consistently choose the things that bring about a state of expansion in you, it dramatically increases your experience of happiness and well-being.
### Learning a New Language
Want to reverse the aging process? According to the life coach and author Martha Beck, PhD, when people connect with their bodies, they start to "age backwards." This connection comes, she says, when you accept all that's going on in your body, even the things you don't like. A lot of people don't listen to their bodies because they have strong negative feelings about them. In our interview, Dr. Beck told me that when you experience deep compassion for everything you're presently feeling instead of rejecting it, you come into harmony with your body and you can hear what it wants. Imagine feeling the same tenderness and concern for your body that you'd feel for a baby or a beloved pet in your care. Having this kind of acceptance for your body activates the areas of the brain associated with happiness, falling in love, and feeling a sense of oneness with everything around you.
When we learn our body's language and begin to treat ourselves gently, we feel more and more comfortable living in our own skin, which supports our experience of being Happy for No Reason.
Exercise
Tune In to Your Body's Wisdom
1. Sit quietly with your eyes closed. Take a few deep breaths in and out through your nose, relaxing your body.
2. Notice any place in your body where you feel discomfort or tension. Don't try to do anything, just be with whatever's happening in that part of your body. Discomfort is a message to us from our body, but we often ignore it, push it out, or simply try to make it go away with painkillers. (If there's no discomfort in your body, continue the exercise focusing on the body's pleasant sensations.)
3. Ask the part of the body that's uncomfortable what it needs to feel better. (Or ask the whole body what it needs for optimal health.)
4. Now just notice what happens. You may hear an answer in your head. Or you may get a feeling or a picture of what your body wants; for example, you may see a picture of yourself taking a walk, lying on the grass, or getting a massage. You may suddenly feel thirsty or hungry for a particular food. You may even just feel like laughing or crying.
5. When the process feels complete, send loving energy to your whole body, especially the uncomfortable part you just listened to. Thank your body for communicating with you.
Summary and Happiness Action Steps
When you practice nourishing your body properly, energizing it, and tuning in to its own wisdom, you make your cells happy, which strengthens the pillar of the body and supports greater happiness in life. Use the following action steps to practice the Happiness Habits for the Body:
1. Reexamine your eating habits to see whether they're supporting happy cells. Favor the happiness-building foods.
2. For one week, kick the caffeine, shake the sugar, and curb the carbs, and notice the effect on your body.
3. Drink enough water—half your body's weight in ounces—every day to see if you feel more energized.
4. Consult your health care professional to determine whether your hormones are in balance and to see whether your body would benefit from a change in amino acids or a fast.
5. Experiment with different forms of exercise to find ones that energize you, and use Master Lin's suggestion to bring conscious awareness to your movement.
6. Practice deep belly breathing, particularly when you're feeling stressed.
7. Catch the "10 o'clock angel train" for three days in a row and see how you feel.
8. Listen to your body's wisdom by checking in regularly to see whether you're giving it what it needs.
## 7
## The Pillar of the Soul—Plug Yourself In to Spirit
There are only two ways to live your life. One is as though nothing is a miracle. The other is as though everything is a miracle.
—Albert Einstein
Think of a time when you felt that life was miraculous: watching early-morning light fill a mountain valley, holding your newborn baby in your arms for the first time, or maybe looking up into the vast night sky ablaze with stars. Every one of us has had at least one moment when our appreciation for life was so intense, it became awe. It's at those times, when you feel both humbled and exalted, that you know you're plugged in to Spirit.
It doesn't matter what you call it—Spirit, Higher Power, Universal Source, Creative Intelligence, the Unified Field, Nature, or God—we're talking about the same thing. Plugging in to Spirit is the experience of feeling connected to an energy bigger than yourself. The more deeply you experience that connection, the richer and more joyful your life feels.
Our souls are the individual expression of this larger Spirit. When you experience that the beauty and mystery surrounding you is inside you as well, your life takes on another dimension. You realize you aren't just getting by in life, just going through the motions. Everything—your drive to work, your evening meal, your conversation with a friend, even your reaction to a piece of bad news—becomes permeated with a sense of grace. Living from that place of wonder and joy strengthens your connection to your soul, raising the fourth and final pillar of your Home for Happiness.
Although each of the Happy 100 have inspired me, the ones who stand out are the ones who have the greatest sense of oneness and unity with Spirit. They come from all different traditions: Christian, Jewish, Buddhist, Hindu, and Muslim. Others don't have a formal religion, yet experience a feeling of oneness with all life. The common thread that runs through all their experiences is a feeling of reverence, wonder, and gratitude for the remarkable gift of being alive. Happy 100 member Brian Hilliard, whose happiness set-point is off the charts, wakes up every morning with a big smile on his face. (His wife, Arielle, says he even smiles in his sleep!) Like so many of the people I interviewed, Brian attributes his state of perpetual well-being to his habit of tapping into the Source and feeling grateful for his life.
People who are Happy for No Reason don't always need to figure everything out or be in control; they live in the flow of life, trusting the underlying benevolence and wisdom of that larger wholeness.
Feeling separate from the Divine is what many consider the number one cause of human suffering and unhappiness. Looking back, it's clear to me that one of the main reasons I was unhappy when I was young was that I felt disconnected from Spirit. I knew there had to be more to life than my body, thoughts, and feelings. I truly had a hunger for the soul, but didn't know how to satisfy it. My first breakthrough in the spiritual arena came when I was sixteen and learned to meditate. It immediately started to relieve some of the existential angst and low-grade depression I'd felt since I was a little girl. Every morning, I'd get up twenty minutes early, settle into an oversized chair perfect for sitting cross-legged, and meditate before going to school. In the beginning, I often wondered if I was meditating correctly, but I felt so much expansion, peace, and well-being, I knew I was doing something right. Many times I felt a sense of coming home.
I was hooked. Meditation became as important a part of my daily routine as brushing my teeth. I'm convinced that meditating is what kept me from escaping my unhappiness through addictions and the other unhealthy behaviors I saw a lot of the other kids doing. Though I still had a way to go to be Happy for No Reason, it set me in the right direction. My parents noticed such a change in me that they learned to meditate too. Today, meditation still helps me stay connected to Spirit. Of all the things I've done to become happier, this is the step that's helped me the most.
Studies show that people who have a spiritual dimension in their lives—defined not as an affiliation with an organized religion, but an internal sense of the spiritual meaning in life—are happier than those who don't: they have happier marriages, are more effective parents, and in general feel better able to cope with whatever happens to them in life. Young people who consider themselves spiritual have higher grades and are less inclined to drink or do drugs.
Spirituality is good for your body as well. It's been found to improve blood pressure, strengthen the immune system, and reduce the rates of stroke, cancer, and heart disease. This has become so well accepted that 25 percent of medical schools in America now offer courses on spirituality and health!
The bottom line? When you feel connected to a Higher Power, you're happier and healthier and can handle the problems that come up in your life more easily.
### The World Is Too Much with Us
What keeps us from being plugged in to Spirit? If our souls are always accessible, why do so many people feel empty and dissatisfied?
Part of the problem lies in the hectic pace of life today and the emphasis on material achievement and accomplishment. The poet William Wordsworth wrote, "The world is too much with us; late and soon / Getting and spending, we lay waste our powers." (He wrote those lines in 1888. Can you imagine what he'd think of the world today?)
From the moment you were first able to walk and talk, the notion has been drummed into your head: "Don't just stand there! Do something!" You might even feel, like so many others, that unless you're being constructive, rushing around and accomplishing things, you aren't a worthy person.
On top of that, we seem to have a strong aversion to being bored. If we're not working or checking off the items on our to-do lists, we're amusing ourselves: reading magazines, books, or newspapers, playing computer games, doing crossword or sudoku puzzles, listening to radios and iPods, surfing the Net, watching television or movies. The list, like our TV screens, keeps getting bigger and bigger. Heaven forbid we have an unfilled moment in our day! We're settling for what entertains us, rather than going for what will deeply fulfill us.
Add our new global obsession with being available 24/7 wherever we are—did you know that one in six people on the planet owns a cell phone?—and you can see how easy it is to get lost in the material world and not give time to your soul. Life has become a multimedia, multidimensional, multitasking extravaganza!
This insane level of activity disconnects us from the inner silence that gives our lives meaning and depth. So how do you plug yourself in?
By making it a priority to experience your connection to Spirit. This requires routine time-outs from your busy life and a willingness to be quiet. In that silence, you can both listen and speak to your Higher Power, which cultivates a greater sense of acceptance, surrender, and trust.
When we make conscious connection to the soul, we're infusing the essence of being Happy for No Reason.
There are many ways to feel connected to your soul. It's a very personal thing, but from my research and interviews with the Happy 100, I've discovered some general guidelines. Practicing the following Happiness Habits can help anyone plug in to Spirit more easily.
Happiness Habits for the Soul
1. Invite Connection to Your Higher Power
2. Listen to Your Inner Voice
3. Trust Life's Unfolding
Happiness Habit for the Soul #1
Invite Connection to Your Higher Power
God is the friend of silence. See how nature—trees, flowers, grass—grows in silence; see the stars, the moon and the sun, how they move in silence. . . . We need silence to be able to touch souls.
—Mother Teresa
We're swimming in a sea of beauty and mystery—yet we often don't take the time to stop doing and notice. Turning inward is one of the most powerful ways you can nurture a conscious connection to your soul. Even just 15 minutes a day can make a huge difference. I know that for a lot of people that may seem like a tall order, but really, it's the best investment of your time you can make. When you feel stronger, happier, and more centered, it benefits everyone in your life—especially you.
There are many ways to cultivate a connection to a Higher Power. You can learn a formal meditation practice; you can take a walk and be in nature; you can sit in silence; listen to inspiring music; or you can use prayer to open the lines of communication with a power greater than yourself. It doesn't matter which you pick, as long as it works for you and you do it regularly. The Happy 100 have all found ways to keep their connection to Spirit flourishing in their everyday lives.
The actress Goldie Hawn is one of those "plugged-in" people. I've loved Goldie since the days when I first saw her in the comedy-variety show Laugh-In. The following story from her glowing memoir, A Lotus Grows in the Mud, describes her first experience of meditation. It's a beautiful account of connecting to your soul by closing your eyes and going inside.
Goldie's Story
The Space between Thoughts
I rejoice in the spaces between thoughts.
A beautiful woman leads me into a quiet room. The warm California breeze drifts through the open window, gently billowing the curtains and lifting my hair.
There is a lone chair in the room. She offers me the seat and whispers a secret mantra in my ear. Just before she leaves the room, she says, "Repeat this in your mind, over and over again." She closes the door behind her, leaving just me and my secret mantra.
I have always been drawn to unseen powers, to the mystical and the magical in life. With her help, I am about to discover the power of my own mind.
Closing my eyes, I feel the breeze lightly brushing my skin, while in my mind I dutifully repeat my mantra. I can smell incense burning in the room and the rose petals scattered all about me. This is my first experience of attempting to quiet my mind.
I chuckle to myself at first. What a cliché I am, sitting here in this room, in the seventies, with flower power at its peak, the latest celebrity to join the Transcendental Meditation bandwagon.
Whoops! That's a thought. Shhh. I have to go back to my mantra. She said thoughts would come in and out of my mind. "Just witness them," she whispered. "Don't judge them or give them any credence. Let them drift away, and then go back to your mantra."
It is not so easy to do.
The more I repeat the mantra, over and over, the more I feel my body relax. My breathing falls away to an almost imperceptible rate. My heart beats more slowly, and the blood it pumps through my veins lessens its pressure.
Thoughts roll into my busy mind again—people I must call, places I must go—and I push them away, hoping for a longer period of calm before the next wave of thoughts.
Listening to my mind saying the words of my mantra, sensing their rhythm and primordial sounds in my head, an inexplicable feeling begins to wash over me.
Deep inside, I feel I am going down and reconnecting to something I know, like an old friend, that deep place that is ever constant, ever joyous, ever alive with creativity. It is the deeper part of me that knows something. It is such a great connection, and fills me with such joy, that I feel like giggling.
Pushing the temptation aside, I carry on, wanting to feel it again. The more I repeat my mantra over and over, the more I let go. As my thoughts flow in and out, I become quieter and quieter in my mind.
My consciousness feels like a teabag being dipped into a glass of hot water and lifted out again. I can feel it becoming slowly saturated with nothingness. When I say nothingness, it is sort of a space in time in which no thought takes place.
Each time I repeat the mantra, the phenomenon becomes stronger, and the teabag becomes heavier and heavier, sinking deeper and deeper, its rich essences seeping into the water.
After a while—I can't say how long—I lose my sense of place. I can visualize the clear glass full of the rich goodness that is my life. I feel like I am merging my spirit with something that is very familiar to me, very safe, and it tickles my joy center.
I am filled with a sense of purity, such clarity, like I have never experienced before. There is no ego, no self, no thought. I am just here. Nothing matters. I am coming back to the purest state of being. I feel unadulterated bliss.
### Doing Nothing Can Bring You Everything
It isn't surprising to learn that the happiness Goldie radiates is rooted in her deeply spiritual nature. As a teacher of meditation, I've seen over and over how happiness grows in people's lives when they meditate regularly.
Meditation practices based in Eastern spiritual traditions became popular in the West in the '70s, but meditation has actually been a part of the Judeo-Christian and Native American traditions for thousands of years. The process of turning inward through meditation is universally recognized as a way to plug yourself in to Spirit.
Meditation takes many forms: focusing on a mantra or the breath, contemplation, visualization, or using sound. It's any process that quiets the mind and helps connect you to your source and innermost essence, that state of pure Truth and Love.
I love this beautiful story about how meditation works:
A wise teacher, instructing his students to meditate, told them, "The process is like filling a sieve with water." All of the students were confused by this statement. How it was possible to fill a sieve with water? Some thought it meant meditation was very difficult, and others thought it meant they could only expect temporary gains from their practice. Discouraged, they stopped meditating. One student, however, approached the teacher and asked him to explain.
The teacher took the student to the edge of the ocean, gave him a sieve, and told him to try to fill it with water. The student scooped the water into the sieve, but it immediately ran out. The teacher took the sieve from the student and said, "I will show you how." The teacher threw the sieve into the water, where it sank almost immediately. He told the student, "The sieve is full of water now and will stay that way forever. Meditation works the same way. It's not about scooping small amounts of Spirit into your individual life, but about dropping yourself into the ocean of Spirit and merging with that Spirit more and more each day."
### Meditation and the Happy Brain
Whether you are aware of them or not, whether you recognize them as spiritual or not, you probably have had the experiences of silence, or transcendence, or the Divine—a few seconds, a few minutes that seem out of time; a moment when the ordinary looks beautiful, glowing; a deep sense of being at peace, feeling happy for no reason. When these experiences come . . . believe in them. They reflect your true nature.
—Sri Sri Ravi Shankar, spiritual teacher and humanitarian
Though meditating is relaxing and enjoyable, its greatest value comes from the influence it has on your life outside of meditation. Hundreds of studies have been done over the past forty years showing the powerful effects of meditation on our bodies, minds, and emotions. Some of the first research, done in the early 1970s by physiologist Dr. Robert Keith Wallace, studied the effects of meditation, specifically the Transcendental Meditation (TM) technique, and found that it offered many physical and psychological benefits, including normalization of blood pressure, decreased anxiety, and better immune functioning. Many more studies on all different types of meditation followed, and today meditating has become an accepted form of stress management the world over.
Meditation does much more than just help you cope with stress. Some of the most exciting research being done today shows that meditating puts you on the fast track to being happy by enhancing activity in areas of the brain associated with happiness and compassion.
Psychologist Dr. Paul Ekman at the University of California San Francisco Medical Center tested Buddhist monks—expert meditators—and found that their meditation practice seemed to calm the brain's amygdala (the adrenaline on-switch I talked about in Chapter 4). What's more, in and out of meditation, the monks were more serene. They were far less likely to panic or get upset, no matter what happened to them.
Dr. Richard Davidson, whose research on neuroplasticity I mentioned in Chapter 4, has also done groundbreaking work with Buddhist monks, whom the Dalai Lama persuaded to volunteer for experiments studying the relationship between meditation, neuroplasticity, and brain activity. Dr. Davidson asked novice meditators as well as the monks, who'd spent more than 10,000 hours over the past thirty years practicing meditation, to engage in five different kinds of meditation while they measured their brain activity. The one that had the greatest effect was a "lovingkindness" meditation designed to focus on compassion. Dr. Davidson discovered that during meditation, the more experienced monks had much higher levels of brain activity in their left prefrontal cortex, indicating happiness, empathy, and other positive emotions, than in their right prefrontal cortex, the area associated with anxiety and depression. This positive style of brain functioning was also found outside the period of meditation. In her book, Train Your Mind, Change Your Brain, the science journalist Sharon Begley explains that the enduring effects of meditation are due to the brain's neuroplasticity: "Brain wiring responsible for negative emotions withers and that [wiring] responsible for compassion and happiness becomes stronger."
If you're bummed thinking you can't be happy because you haven't been a Buddhist monk for the past thirty years, don't be. Dr. Davidson's research showed that people who'd been meditating for only three months, twenty to thirty minutes a day, experienced significant physiological changes, reflecting greater happiness and health. Happily, you don't have to meditate for decades to have results.
### Different Strokes for Different Folks
Sitting down for a twenty- to thirty-minute meditation session each day is a great habit for plugging in to Spirit, but it isn't the only way to meditate. In our interview, Happy 100 member and Tibetan Buddhist lama Anam Thubten Rinpoche told me about a meditation practice that anyone can do, anytime. I call it the Pause Practice: Seven times throughout the day, simply pause and just "be." Become aware of your breath, and for a minute or two let yourself experience the present moment—the only time you can really experience happiness. When I do the Pause Practice regularly, I notice a greater sense of peace, perspective, and renewed energy.
You can also find great stillness and serenity simply by being in nature. Karen, one of the Happy 100, told me that whenever she takes a long walk alone in the woods or on the beach, she finds herself falling naturally into the rhythm of her breath as she walks along. Listening to the sound of the wind, the birds, and the water quiets her mind and heart. When she can't get away, just looking out the window for a few minutes at the trees or the clouds will often melt away tension and touch something deep inside her.
At the end of this section, you'll find a meditation exercise that anyone can use to experience more inner calm and relaxation.
### The Power of Prayer
Every spiritual tradition in the world includes prayer. For thousands of years, people have opened their hearts to a Higher Power through praying, a kind of hotline to the Divine.
Like meditation, prayer can take many different forms. In times of trouble, we pray for comfort, guidance, and healing, either for ourselves or for people we love. At other times, when beauty, love, or gratitude overwhelm us, we're moved to offer up prayers of thanksgiving and praise. What inspires us to pray isn't important—it's the prayer itself that plugs us in to Spirit.
Many studies show that praying has a strong impact on happiness, with thousands upon thousands of people reporting that prayer increases their sense of well-being, life satisfaction, and general happiness. Prayers can also have an influence on others, even from a distance. According to research, remote intercessory prayer has a positive effect on hospital patients' rates of recovery and healing.
In his book, Healing Words, Larry Dossey, MD, one of the world's foremost experts on the link between spirituality and health, describes experiments conducted by the Spindrift Organization of Salem, Oregon, measuring the effect of praying on simple biological systems such as sprouting seeds and yeast cultures. Repeated studies showed that seeds that were sent prayers germinated more quickly than identical seeds that weren't prayed for, and that the amount of prayer was also an important factor in effectiveness. The most fascinating finding of the Spindrift research is that nondirected prayer, which simply asks for God's will to be done or for the best thing to happen, was more powerful than directed prayer, which asks for a specific result.
You can see the power of prayer in these two photos by Masuru Emoto, whose work I introduced in Chapter 5. On the left is a picture of crystals formed in regular tap water. On the right, the same tap water after people had sent it prayers from a distance.
Tap water before prayer
Tap water after prayer
A remarkable demonstration of the beautiful energy prayer creates!
When you cultivate your connection to your Higher Power, you begin to recognize the presence of that Power underlying everything around you more and more. Making time each day to find what the poet T. S. Eliot called "the still point of the turning world" is a crucial piece of the Happy for No Reason puzzle.
Exercise
Light Meditation
The following exercise will give you a taste of that quiet place within that's the bedrock of inner peace and well-being.
1. Sit comfortably in a quiet place and close your eyes.
2. Take five or six deep breaths, noticing the breath as you inhale and exhale.
3. Picture a column of bright white light coming down from above your head, entering through the top of your head and filling up your head with light.
4. Feel that white light flowing from your head down through your neck and into your chest, lighting up your heart. Feel the warmth and expansion in your heart area.
5. Continue to see the light flowing down your arms, your torso, your spine, and your hips, filling those areas with light. See the light flowing down through your legs and into your feet, lighting up the soles of your feet.
6. Feel your entire body flooded with warm, bright light. In this state, sit for ten minutes and just allow any thoughts and experiences to arise. Don't try to push away thoughts or control your mind—simply be with whatever comes. If you'd like a focus for your attention, notice your breath as you inhale and exhale.
7. When you feel ready to end your meditation, become aware of any peace and serenity you feel. Rest in this peaceful feeling for a moment. Take a few deep breaths and savor this feeling. This is the essence of being Happy for No Reason.
8. Slowly open your eyes. Have the intention to carry this experience with you as you return to your normal activities.
Happiness Habit for the Soul #2
Listen to Your Inner Voice
To make the right choices in life, you have to get in touch with your soul. To do this, you need to experience solitude . . . because in the silence you hear the truth and know the solutions.
—Deepak Chopra, MD, physician, author, and speaker
In the previous chapter, we discussed tuning in to your body's wisdom; here, we'll focus on listening to your soul's wisdom to guide you. Your inner voice is the part of you that knows what to do—at all times and in all situations. For some people, it's like finding the wise parent you always wished you'd had. For others, the experience of inner knowing is vaster and more cosmic. Whatever the flavor, the Happy 100 listen to this inner voice.
When I read Elizabeth Gilbert's book Eat Pray Love, I was immediately charmed and inspired by her description of her first "conversation" with God. After finishing the book, it was clear to me that Liz was one of the Happy 100, and I called her to set up an interview. I'm happy to report that Liz is as delightful in life as she is on the page; our interview added wonderful material to what she'd already shared in her book. The following story describes the transformation Liz experienced when she learned to hear "the still, small voice" inside her.
Liz's Story
Hello, God. I'm Liz.
I was in the upstairs bathroom of the big house in the suburbs of New York which I'd recently purchased with my husband. It was a cold November, around three o'clock in the morning. My husband was sleeping in our bed. I was hiding in the bathroom for something like the forty-seventh consecutive night, and—just as during all those nights before—I was sobbing. Sobbing so hard, in fact, that a great lake of tears was spreading before me on the bathroom tiles, a veritable Lake Inferior of all my shame and fear and confusion and grief.
I don't want to be married anymore.
I was trying so hard not to know this, but the truth kept insisting itself to me. In daylight hours, I refused that thought, but at night it would consume me. What a catastrophe. How could I be such a jerk so as to proceed eight years into a marriage, only to leave it? We'd only just bought this house a year ago. Hadn't I wanted this nice house? Hadn't I loved it? So why was I haunting its halls every night now, howling like Medea? Wasn't I proud of all we'd accumulated—the prestigious home in the Hudson Valley, the apartment in Manhattan, the eight phone lines, the friends and the picnics and the parties, the weekends spent roaming the aisles of some box-shaped superstore of our choice, buying ever more appliances on credit? I had actively participated in every moment of the creation of this life—so why did I feel like none of it resembled me?
I don't want to be married anymore.
My husband was sleeping in the other room, in our bed. I equal parts loved him and could not stand him. The many reasons I didn't want to be this man's wife anymore are too personal and sad to share here. Much of it had to do with my problems, but a good portion of our troubles were related to his issues, as well. And because I would not ask anyone to believe that I am capable of reporting an unbiased version of our story, the chronicle of our marriage's failure will remain untold here.
I will also not discuss here all the reasons why I did still want to be his wife, or all his wonderfulness, or why I loved him and why I had married him and why I was unable to imagine life without him. Let it be sufficient to say that, on this night, he was still my lighthouse and my albatross in equal measure. The only thing more unthinkable than leaving was staying; the only thing more impossible than staying was leaving. I didn't want to destroy anything or anybody. I just wanted to slip quietly out the back door, without causing any fuss or consequences, and then not stop running until I reached Greenland.
This part of my story is not a happy one, I know. But I share it here because something was about occur on that bathroom floor that would change forever the progression of my life—almost like one of those crazy astronomical super-events when a planet flips over in outer space for no reason whatsoever, and its molten core shifts, relocating its poles and altering its shape radically, such that the whole mass of the planet suddenly becomes oblong instead of spherical. Something like that.
What happened was that I started to pray.
You know—like, to God.
Now this was a first for me. Culturally, though not theologically, I'm a Christian. Traditionally, I have responded to the transcendent mystics of all religions. I have always responded with breathless excitement to anyone who has ever said that God does not live in any one scripture or in a distant throne in the sky, but instead abides very close to us indeed—much closer than we can imagine, breathing right through our own hearts. I respond with gratitude to anyone who has ever voyaged to the center of that heart, and then has returned to the world with a report to the rest of us that God is an experience of supreme love.
In the middle of that dark November crisis, though, I was not interested in formulating my views on theology. I was interested only in saving my life. I had finally noticed that I seemed to have reached a state of hopeless and life-threatening despair, and it occurred to me that sometimes people in this state will approach God for help.
What I said to God through my gasping sobs was something like this: "Hello, God. How are you? I'm Liz. It's nice to meet you."
That's right—I was speaking to the creator of the universe as though we'd just been introduced at a cocktail party. But we work with what we know in this life, and these are the words I always use at the beginning of a relationship. In fact, it was all I could do to stop myself from saying, "I've always been a big fan of your work . . ."
"I'm sorry to bother you so late at night," I continued. "But I'm in serious trouble. And I'm sorry I haven't ever spoken directly to you before, but I do hope that I have always expressed ample gratitude for all the blessings that you've given me in my life."
This thought caused me to sob even harder. God waited me out. I pulled myself together enough to go on: "I'm not an expert at praying, as you know. But can you please help me? I'm in desperate need of help. I don't know what to do. I need an answer. Please tell me what to do. Please tell me what to do. Please tell me what to do . . ."
And so the prayer narrowed itself down to that simple entreaty—Please tell me what to do—repeated again and again. I don't know how many times I begged. I only know that I begged like someone who was pleading for her life. And the crying went on forever.
Until—quite abruptly—it stopped.
Quite abruptly, I found that I was not crying anymore. I'd stopped crying, in fact, in mid-sob. My misery had been completely vacuumed out of me. I lifted my forehead off the floor and sat up in surprise, wondering if I would now see some Great Being who had taken my weeping away. But nobody was there. I was just alone. But not really alone, either. I was surrounded by something I can only describe as a little pocket of silence—a silence so rare that I didn't want to exhale, for fear of scaring it off. I was seamlessly still. I don't know when I'd ever felt such stillness.
Then I heard a voice. Please don't be alarmed—it was not an Old Testament Hollywood Charlton Heston voice, nor was it a voice telling me I must build a baseball field in my backyard. It was merely my own voice, speaking from within my own self. But this was my voice as I had never heard it before. This was my voice, but perfectly wise, calm and compassionate. This is what my voice would sound like if I'd only ever experienced love and certainty in my life. How can I describe the warmth of affection in that voice, as it gave me the answer that would forever seal my faith in the divine?
The voice said: Go back to bed, Liz.
I exhaled.
It was so immediately clear that this was the only thing to do. I would not have accepted any other answer. I would not have trusted a great booming voice that said either You Must Divorce Your Husband! or You Must Not Divorce Your Husband! Because that's not true wisdom. True wisdom gives the only possible answer at any given moment, and that night, going back to bed was the only possible answer. Go back to bed, said this omniscient interior voice, because you don't need to know the final answer right now, at three o'clock in the morning on a Thursday in November. Go back to bed, because I love you. Go back to bed, because the only thing you need to do for now is get some rest and take care of yourself until you do know the answer. Go back to bed, so that when the tempest comes, you'll be strong enough to deal with it. And the tempest is coming, dear one. Very soon. But not tonight. Therefore:
Go back to bed, Liz.
I call what happened that night, not a religious conversion, but the beginning of a religious conversation. The first words of an open and exploratory dialogue that would, ultimately, bring me very close to God, indeed.
In the years since then, I've found that voice again and again in times of code-orange distress. I've learned that the best way for me to reach it is through written conversation. I take out my most private notebook, which I keep next to my bed in case I'm ever in emergency trouble, and begin to write. And even during the worst of suffering, that calm, compassionate, affectionate and infinitely wise voice (who is maybe me, or maybe not exactly me) is always available for a conversation on paper at any time of the day or night.
At the beginning of my spiritual experiment, I didn't always have such faith in this internal voice of wisdom. I remember once reaching for my private notebook in a bitter fury of rage and sorrow, and scrawling a message to my inner voice—to my divine interior comfort—that took up an entire page of capital letters:
"I DO NOT BELIEVE IN YOU!!!!!!!!"
After a moment, still breathing heavily, I felt a clear pinpoint of light ignite within me, and then I found myself writing this amused and ever-calm reply:
"Who are you talking to, then?"
I have never doubted its existence again.
Today my connection to God, the voice inside my heart, is the foremost relationship in my life. And the way that I honor that relationship is to keep my life as calm as possible so that I can hear that voice. I consider this my primary devotional practice.
Still, it's not as if I'm laminated over with bliss. When crises come up, they shake me and shock me, just like everybody else. I definitely live in the real world, reacting to all the unexpected and unexplainable events that occur. The difference now is that I try not to react in a way that fights what is happening; instead I surrender to what's in front of me.
This doesn't mean that it's always easy or that when my life appears to be destroyed, I can run around being happy and excited about it. What it does mean, is that my job—and this is where prayer comes in—is to remain connected and conscious enough to ask the universe (or God or whatever I choose to call this power at that moment) the question, "What is it exactly that you're asking me to do here that I'm not getting yet? Open my eyes to see how I'm supposed to be using this."
Instead of praying as a lament, I pray as an information-gathering inquiry: "Can you please show me what I'm supposed to do right now?" I always assume there is something that I'm supposed to do or understand, even if I can't see it right at the moment.
What generally happens is that I get clearer about my reaction. It's like doing a CAT scan on myself that shows me my pockets of resistance—where I'm saying, "No, I don't accept that." Where I'm holding on and holding out. Where I'm saying, "I'm totally in accordance with the universe, and I really do trust the Divine force, but not this one thing—that's the deal breaker." And that's not surrender.
When I stop resisting what's happening and surrender, then I'm happy again. But that surrender only seems to come to me through that question and answer sesson, that prayerful inquiry.
Since that night in the bathroom, when I first I introduced myself to God, my life has turned around completely. Where there was neurosis and misery, there's now peace and fulfillment.
The loving wise voice is such a part of me now. When I find myself getting anxious or upset, that voice always asks me, "Are you really going to fall for this after all we've been through? Don't you know better? Haven't you learned?" I do know better and I have learned.
Which is why praying has become a moment-by-moment practice—a commitment I've made about how I'm going to live. It is this repeated experience of talking to God—asking and listening and then hearing the answer—that keeps me on the path toward greater and greater happiness. The other night I said, "How can I ever thank you, God?"
The calm, amused voice inside made me smile: "Keep in touch."
### Opening the Lines of Communication
Many of us have a tendency when making decisions to run around asking everybody and their mother what we should do, forgetting that whenever we check in and ask ourselves, we get a reliable answer. As Liz discovered, we each have an inner wisdom that's connected to Spirit, and it's available to us at all times.
Listening to your inner voice plugs you in to something bigger than yourself. You can ask questions of this inner voice about your life purpose, relationships, career—anything you want to know. You'll know you've made contact by the peace and utter fearlessness you experience in connection with the answers you receive.
There are many techniques for inner listening. Here are a few ways to pose a question and see what answers emerge:
Write it Down: Like Liz, some people find that inner listening works best on paper. When you have a question, sit quietly, inquire within, and then write whatever comes to you without censoring. No one is going to see this but you. Let the deepest part of your being flow through you.
Go to a Book: Another way to ask for guidance is to go to a book you are drawn to, open up randomly to a page, and see what message you discover there. You may pooh-pooh this technique, but I've found that it's often remarkably apt and helpful. Some people find that it allows them to break free of their own fixed ideas about a situation and see a new angle or perspective. I find it amazing how many times the exact answer I need is right there.
Look for Signs: Here's another one that may have your eyes rolling, but when I was writing my Chicken Soup books, I was astounded at how many people sent in stories about asking for signs. The most common sign people received came in the form of a bluebird or a cardinal. It got to the point where my coauthors and I had a separate file drawer for what we called the Bluebird/Cardinal Stories. The sheer number of these stories was impressive and made me look at asking for signs—and bluebirds and cardinals—in a whole new light.
Sergio and I were once stumped about whether or not we should rent a certain house. There were a lot of great things about this particular house, but we were genuinely torn. Thinking, Why not? I asked for a sign. As we walked toward the house to look at it one more time, I saw a dead bird, right in the middle of the path. It wasn't a bluebird or a cardinal, but it gave me enough pause to make me decide to wait until the next day before calling the landlord and taking the house.
First thing the next morning, I noticed a new listing in the paper for a house that sounded promising. Sergio and I ran right over to look. As we walked toward the front door, a beautiful deer crossed the path. Deer are my favorite animals, so I took it as a sign. The house was perfect. We immediately signed a lease and lived there happily for two years before buying our present house. Was it really a sign? Who can say? In this case, being open to a larger source of wisdom helped me get in touch with how I was really feeling about the house and guided me toward the better decision.
When you know that you can always go inside to find direction and wisdom, an unshakable peace and confidence dawns. The following exercise guides you through an inner listening process.
Exercise
Inner Listening
1. Find a quiet, comfortable place, and sit with a pen and paper ready.
2. At the top of the paper write down a question or issue you'd like guidance about. Phrase the question or issue as clearly as possible.
3. Close your eyes and take several deep breaths.
4. Ask your inner voice the question you wrote on the paper. It may take a few moments until you feel ready, but when you do, open your eyes and start writing whatever comes to you. It doesn't matter whether it makes sense or not. Keep writing until your hand feels like it won't move any longer, not reading what you have written as you go.
5. Now read over what you've written. You may be quite surprised at what wisdom has come out. Even a single word or phrase may be the key to your answer.
Happiness Habit for the Soul #3
Trust Life's Unfolding
The winds of grace are always blowing, but you have to raise the sail.
—Sri Ramakrishna, nineteenth-century Indian saint
I discovered in my interviews that the vast majority of the Happy 100 experience a sense of surrender in their lives and feel profoundly taken care of by a universal power. They trust that the universe is out to support them (Guiding Principle #2).
We try so hard to control everything in our lives that we forget the profound power of putting our faith in the universe and letting go. When you get into the habit of doing all that you can, then surrendering to a Higher Power and trusting that things will work out for the best, you'll experience a deeper sense of peace and well-being, hallmarks of being Happy for No Reason.
Reverend Michael Beckwith, founder of the Agape Spiritual Center in Los Angeles, is one of the Happy 100 whose life is lived in surrender. Anyone who has met Michael, been to one of his services, or seen him on TV can feel his deep connection to Spirit. Michael, who appears with me in The Secret, is also a colleague of mine on the Transformational Leadership Council, a group of 100 top transformational leaders who meet regularly to enhance our effectiveness and contribution to the world. Every time I'm with Michael, I feel so uplifted, I just want to shout Amen! Interviewing Michael was a treat, and his story, which follows, has stayed with me, inspiring me to let go and trust in new ways.
Michael's Story
Love-Beauty
As a child, I can remember feeling naturally attuned to the divinity all around me. Children, so fresh from their Source, often feel this way. There were times when I just knew I was standing face-to-face with my Higher Self, my God-essence—though I didn't call it that at the time.
An example of that connection occurred on my eleventh birthday, when my mother and I were visiting my grandmother. I was sent to the corner store to buy some groceries. As I stood in the bread aisle pondering which loaf to choose, all at once time suspended. Space dissolved, and my sight was no longer limited to Roman Meal versus Wonder Bread. I "saw" that a jar of baby food was about to fall in the next aisle. I tore around the corner just in time to catch it, much to the amazement of the man who had knocked it off the shelf. "How . . . how did you do that?" he stuttered.
"I don't know," I innocently replied, "I just saw it."
Walking home I felt an expansive oneness with all that was before me: the trees were singing, the grass had its own language—everything was alive, shimmering with a high-frequency life force. As I approached the house, my mother, grandmother, and uncle were standing on the stoop, affectionately smiling down at me. A mystical pain entered my heart as I realized that the boy they saw was not the Michael I knew myself to be. At that moment, I consciously shut down my cosmic connection and began conforming myself to labels that boxed me into being someone that everyone would be comfortable around. The price was the loss of my contact with Divinity.
I lived inside that box for another ten years. Whenever an expansion in awareness popped up, I managed to shut it back down. One of these incidents took place in my first year of junior high, when I was nominated to be school treasurer. I was terrified of public speaking and had to face a campaign trail that took me from class to class, and ultimately before the entire student body to present my four-point plan for "Banking with Beckwith." When the big day arrived, I sat in my seat, dreading the moment I would have to speak. When my name was called I walked up to the podium, forced open my mouth, and was shocked to hear myself talking about excellence and the greatness within each of us waiting to emerge—none of which was even one of my four points! My audience was as astonished as I was, evidenced by the hundreds of wide-eyed stares that followed me to my seat. I had been flung out of my box. To cover my embarrassment, for the next week I uncharacteristically exuded a bad attitude and got into fights—anything to conform to the unwritten preadolescent rule to be a regular guy. I wasn't yet ready to break my agreement with mediocrity.
The religious affiliation of my youth shed no light on my mystical experiences. In high school, disillusioned by ministers who didn't practice what they preached and dogma being taught that just didn't add up, I announced my atheism. By my freshman year of college at Morehouse, my atheism had evolved into agnosticism. I hadn't yet come into contact with a spiritual path that matched my inner experience.
From Morehouse I transferred to the University of Southern California as a psychology major. I'd grown up in an era when smoking marijuana was somewhat the norm, so, like many students, I smoked some pot. To supplement my income for this extra expense, I began selling a little pot myself. What started off as a simple cottage industry became a network that stretched from Los Angeles to Atlanta, Nashville, and New York, bringing in thousands of dollars a week.
While I was attending classes, writing term papers, and running my business, I was also entering a spiritual awakening. During this period I had visions and heard voices. But when I did a psychology internship working with mentally ill prisoners who also heard voices and saw visions, I began to consider my experiences pathological. My first antidote was to cut back on smoking weed, but this only intensified my experiences! Then something happened that made it impossible to keep turning my back on God.
For about a year I'd been having a recurring dream of being chased by three men. I'd always wake up before they grabbed me, but with each recurring dream they got a little closer. Finally, one night, they caught me.
As I struggled with my dream captors, I saw a tent with hundreds of people I knew standing in line to enter. I shouted to them for help, but they all turned their backs on me. While two of the men held me down, the third plunged a knife into my heart. The pain was excruciating. I screamed out, and then I died.
When I awoke from this dream, reality as I knew it had profoundly shifted. I clearly sensed a luminous presence connecting everything and everyone. It penetrated my soul with total and unconditional love; its beauty, as it moved through the animate and inanimate objects all around me, was beyond my power to describe. Still agnostic, I didn't give this presence a name—I simply called it Love-Beauty. The person who had spent so many years burying his connection to God had died; I could never fit back into that box again.
I began a voracious study of both Eastern and Western spirituality and mysticism. I discovered that when the culture and history of the world's religions are stripped away, what remains are universally applicable spiritual principles. These explorations in consciousness were the final motivation I needed to stop selling dope. There was one last shipment—the only one ever stored at my house—that required handling. But before I could get rid of it, I got busted.
Because of the size of my operation, the charges were serious, and I was facing significant jail time. Well-meaning friends offered their advice, from "Plea bargain" to "Take your money and leave the country." But I could no longer relate to myself as a drug dealer. My spiritual transformation had made me a new person and I knew, even beyond faith, that the new me wasn't going to jail.
Throughout the days of the trial I sat in the courtroom calmly reading books on spiritual subjects. On a day that seemed no different from any other, my lawyer suddenly shot out of his chair and made a point about a technicality. The judge called for a three-day recess. The day court resumed, I don't think I will ever forget the sound of the judge's gavel striking his podium as he pronounced: "Case dismissed!" However, he wasn't quite finished with me and asked that I approach his bench. "That was a lucky break, young man. I hope I never see you in my court again," he admonished.
I looked him straight in the eye and said, "You never will."
I went home. The wind was blowing fiercely, which caused me to look up at my neighbor's weathervane as I walked up the path to my door. I couldn't resist an urge to demand one last validation of my transformation, so, concentrating on the weathervane, with inner authority I said: "I believe. Heal any remainder of my unbelief. If the Presence is as real as I know it is, make the weathervane go against the wind, point in my direction and . . ."
Before I could complete my sentence the weathervane was pointing right at me, even though the wind current had not changed. I started crying. In that moment I completely surrendered my life to the Presence, knowing that this Presence was going to somehow make use of my life in its service.
By the age of twenty-seven, my hair was on fire to know truth. Whenever I heard about a teacher who lived the Presence authentically, I was there. From silent retreats, to long hours of meditation, to spiritual discourses, nothing broke the blissful tempo of my days on this profoundly private path. Until the day when, in a gentle yet firm tone, the voice within said, "It's time to go public."
Perhaps the greatest proof of my transformation was my resounding "Yes!" I left the anonymity of my solitary spiritual journey and enrolled in a New Thought–Ancient Wisdom school of ministry, clear that my purpose was not to become a minister, but rather to find a context in which to honor my "Yes" to the Presence. Besides, I wanted my weekends free! Eventually, I became a licensed spiritual counselor and for many years had a practice doing one-on-one prayer sessions and spiritual therapy.
Still, my inner voice relentlessly urged me to go more public. Little by little my resistance dissipated. As I spoke about universal wisdom to increasingly larger audiences, I experienced that I could stand in a place that was inwardly comfortable—the presence of God—and that when I got out of the way that presence could flow through me.
In 1985 an inner vision revealed a transdenominational, transreligious spiritual community where God's only religion—Love—was taught and celebrated. It was a foretelling of the Agape International Spiritual Center, which I founded in 1986 and which has grown to include thousands of members.
Today, the "Yes" is always there. Though I don't always know what serving the Presence will look like, I always have faith and trust in it. When I surrender to the Presence, it always works out.
The Presence is not going to let me down, because I'm here representing it. Like each one of us, I've been hired by the Universe to be myself.
### Having a Yes Mind
Sometimes, when I feel frazzled and tired of my self-appointed position as general manager of the Universe, I think of Michael and remind myself that I, too, am being taken care of by Love-Beauty. Though he's one of the lucky ones who was born feeling his oneness with Spirit, we all have the same connection. Our experience of Happy for No Reason grows exponentially when we trust that connection enough to let it direct our lives.
Throughout history, the wisest men and women have known that their highest good came from surrender—not as in losing a battle, but as in letting go of their personal limitations and aligning totally with an intelligence greater than themselves. Remember the Spindrift experiments? According to their research, the prayers that have the greatest effect are the ones that simply invite the presence and influence of a Higher Power rather than asking for a specific outcome. This demonstrates the power of that classic expression of surrender, "Thy will be done."
Trusting and letting go in this way brings enormous peace and freedom and creates what I call a "yes mind." Instead of saying no and resisting what's happening, you say yes. I had my first experience of the yes mind when I took improvisational acting classes many years ago.
One of the rules of improv is to respond instantly and enthusiastically—to say yes—to whatever another actor throws at you. During my first class, when my improv partner told me that I was an elderly rock star, I immediately began to play the part: strumming air guitar, clutching my lower back, and belting out lyrics as I gyrated like an eighty-year-old. In the next moment, the scene switched. Suddenly, I was supposed to be an alien coming to earth to rob a bank—and I had to go with it. I just said yes. The exercise broke my boundaries and freed my energy. I found it was a great practice for how to live.
When you say yes, it's like being a trapeze artist; you're able to let go of the trapeze you're holding on to and sail through the air toward the next one. Without letting go and then trusting that the next trapeze is there, that forward motion is impossible. Surrender and trust create a feeling of flow in every moment.
### Amazing Grace
One of my greatest spiritual mentors is Bill Bauman, a former priest and the most centered and grounded person I've ever met. Though Bill has a deceptively unassuming demeanor—he'd fit right in as a greeter at Wal-Mart—I like to call him the enlightened Mister Rogers, for beneath that comforting cardigan beats the heart of a spiritual hero. He's been the guardian angel of this book, and both Carol and I deeply appreciate his help and counsel.
Bill was one of the first of the Happy 100 I interviewed, one of the handful of people obviously Happy for No Reason I knew personally when I began the book. When I think of Bill, I think of a life filled with grace.
Grace, Bill explains, "is just a fancy word for the infinite, unconditional, all-responsive love of God flowing in our lives." He uses the analogy of the unconditional love a mother feels for a newborn child. "The mother sees no flaws in the child, only utter perfection and utter deservingness; in her love, she wants nothing more than to give and give and give. Now move that example up to the infinite level. Our Higher Power sees the perfection, the worthiness, the absolute wonder of who we are—and in that love, wants nothing more than just to give gifts, to give blessings, to give solutions."
In our interview, Bill told me about an experience that illustrates how he allows grace to flow in his life:
A number of years ago, my wife, Donna, and I decided to "start over" in a whole new area of the country. We left our old lives behind—including our old level of income. About a year after our relocation, we found ourselves low on funds; we were doing our best, but there just wasn't enough money coming in yet.
We decided to take a day off and spend some time at a nearby Trappist monastery, hanging out in their church and opening ourselves to grace.
As Donna and I sat together in the chapel, we didn't ask for anything specific; we didn't say "Please give us money"; we just invited grace. We sat quietly, with an attitude of openness and receptivity, and put ourselves in the presence of Divine Grace, in the presence of Abundance, in the presence of the maternal, all-loving expression of Universal Power. I remember thinking, "I don't have the ideal solution. Your response is always so much bigger than what I'd come up with anyway. I put myself in total connection to You. Whatever You send as the perfect response to this situation, I'm open to."
That was Phase One. After an hour or so, we left the monastery and started Phase Two, which was to just let it all go. We went about our lives, trusting that the perfect response would happen.
The very next day, we received a call from our accountant telling us that he'd just come across a mistake he'd made and was sending us a check for $6,000. That same afternoon, Donna got a call from the local college, offering her a summer teaching job and a full-time position for the next year.
Then, two days later, we got a $3,500 check in the mail from an insurance company we'd never heard of, paying on a policy we never knew we had. An hour later, another phone call came in, offering me a consulting job.
I was surprised and I wasn't. Though I had no doubt that something would happen, the generosity and speed of the response was astounding. I guess that's why they call it amazing grace.
When I asked Bill how we can all learn to live in grace, he said, "By practicing surrender. Look for opportunities to be open to blessings, without defining too clearly what they should be, and then trust and let go."
For those of us locked in the "I'm in charge" and "I have to take control" approach to life, surrendering starts by simply inviting ourselves to believe that our Higher Power will respond to our needs and being open to the flow of grace when it comes. This gentle invitation is a great tool for letting go and trusting life to unfold—gracefully.
People living in that perpetual state of surrender and trust often report increased experiences of synchronicity in their lives: amazing coincidences, uncanny "luck," unexpected aid, and perfect timing—simply being in the right place at the right time. When you experience a lot of synchronicities, it's a sign that you are plugged in to Spirit.
Exercise
Inviting Grace
This simple exercise reinforces the No Tension step of the Secret Formula you learned in Chapter 2: Intention, Attention, No Tension.
1. Sit quietly and write a letter to your Higher Power about a situation in your life that you would like assistance with (for example, relationship, health, job). Write from your heart.
2. Ask that the perfect people, places, and circumstances be brought to you for the deepest fulfillment of that situation. You may want to ask for right understanding or forgiveness, if appropriate.
3. Put the letter somewhere meaningful to you where it won't be disturbed. Some people like to put their letter in a special book, in a sacred place in their home, or under a rock in their garden. You won't be looking at it again for a while.
4. Now, let go. Turn the situation over to a Higher Power. Relax, feel grateful, and invite yourself to believe that the universe is always supporting you.
5. Notice what happens over the next days or weeks. You may want to look at your letter in a month or two and see how the universe responded.
Summary and Happiness Action Steps
When you plug in to Spirit by taking time for silence, listening to your inner voice, and trusting life's unfolding, your experience of Happy for No Reason inevitably grows. Plugging in to Spirit strengthens the final pillar of your Home for Happiness. Use the following action steps to practice the Happiness Habits for the Soul:
1. Make silence, meditation, or prayer part of your daily routine.
2. Do the Pause Practice seven times a day for a week and see how you feel.
3. Look for guidance by posing questions in a journal, opening to a page in a book, or asking for a sign.
4. Practice surrender: invite yourself to believe that the universe is out to support you, send out an intention for your higher good, and then let go.
## 8
## The Roof—Live a Life Inspired by Purpose
There are two great days in a person's life—the day we are born and the day we discover why.
—William Barclay, twentieth-century Scottish theologian
I'm sure you've heard at some time in your life, "We're all here for a purpose." Yet if you polled the first 100 people you met on the street about their purpose, most of them would probably answer with a sigh, "I don't have a clue." Although there are people who sail through life graced with a sense of direction and clarity, many others feel they've missed the boat.
Happy people feel they are on the planet for some purpose. In my interviews with the Happy 100, I heard the same thing again and again: they live inspired, moment to moment, by a sense of purpose and meaning.
So, what is your purpose? Contrary to popular belief, your purpose isn't your job or profession—it's much bigger than that. It's an overarching intention to do what's meaningful to you in your life. For example, my purpose is sharing my love and energy by inspiring others to lead their highest life possible. Right now I happen to be a transformational speaker and writer, but I could fulfill that purpose in any number of other ways. I could be a teacher, a musician, a secretary, a doctor, a gardener. Living inspired by purpose can take many forms—the real key is to first discover the sense of meaning and purpose within you.
The research on happiness clearly shows that people who are deeply committed to whatever gives their life meaning are much happier than those who don't have this greater sense of purpose. According to the psychologist Edward Diener, who researches subjective well-being in the Department of Psychology at the University of Illinois at Urbana–Champaign, one of the vital ingredients for happiness is "having a meaning in life . . . and having goals embedded in your long-term values that you're working for, but also that you find enjoyable." Health and longevity studies show that when people live with a sense of purpose, no matter how big or small, they live longer and healthier lives.
### Purpose, Meaning, and Work
Your purpose is the roof of your Home for Happiness. When you don't have that sense of purpose, it's like having a leaky roof; it can rain unhappiness over every other aspect of your life. Unfortunately there seem to be a lot of leaky roofs in the world today. According to a 2005 study released by the research company Harris Interactive, only 20 percent of working Americans are passionate about their work. That means that four out of five people aren't inspired by what they do. This statistic makes it clear why the largest number of heart attacks occur on Monday mornings. People would literally rather die than go to work.
Many people work at jobs just to pay the bills, feeling they don't have the freedom, skills, or opportunity to find employment they really like. They resign themselves to a life without a sense of deeper purpose, making the best of a bad situation and living for the weekends. Others spend years searching for the perfect job, like Goldilocks, trying to find something that's just right. Of those, a lucky few do find careers that are fulfilling, but unless they're inspired by a sense of inner purpose, that fulfillment is fragile. Their happiness depends on their job, so if they lose their job or retire, they feel adrift and quickly lose that sense of direction and satisfaction in life.
The statistics on retirement bear this out. It's been reported that the happiest retirees are those who still feel a sense of purpose even after their career is over. These people often transfer their skill set to another setting. A retired banker might volunteer at a job training center, or an employee who was always handy might become the neighborhood Mr. Fix-it. The important thing is to let your purpose lead the way.
### Job, Career, or Calling
Do you consider your daily activity a job, a career, or a calling? The following story illustrates the difference:
One day an old woman walked up to a dusty building site where three strong, young men were working hard laying bricks. She walked up to the first man and asked him what he was doing. He replied rather rudely, "Can't you see? I'm laying bricks. This is what I do all day—I just lay bricks." She then asked the second man what he was doing. He replied, "I'm a bricklayer and I'm doing my work. I take pride in my craft, and I'm happy that what I do here feeds my family." As she walked up to the third man, she could see that his eyes were full of joy and his face was as bright as the day. When she posed the same question to him, he replied with great enthusiasm, "Oh, I'm building the most beautiful cathedral in the whole world."
It isn't the activity in your life that defines your feeling of purpose, it's your perspective. Organizational psychologist Amy Wrzesniewski of New York University conducted research on people's work orientations based on these three categories: job, career, or calling. She found that, regardless of the job itself, those who feel they're following a calling experience greater satisfaction from their work and more happiness in their lives.
### If You're Not in the Job You Love, Love the Job You're In
People who are Happy for No Reason, whether or not they're in their ideal career or calling, bring a sense of purpose with them wherever they go and to whatever they're doing—even the most mundane of tasks. Whether they're changing the oil in the car or preparing a family meal, they're still inspired by purpose. Their sense of purpose isn't "out there." In fact, if you took away whatever a happy person is currently inspired to do, he or she would simply find something else to be inspired by.
I once heard a story about the great conductor Arturo Toscanini. On his eightieth birthday, someone asked Toscanini's son, Walter, what his father ranked as his most important achievement. Walter answered, "For my father there can be no such thing. Whatever he happens to be doing at the moment is the biggest thing in his life—whether it's conducting a symphony or peeling an orange."
As we were writing this chapter, Carol told me how a sense of purpose once saved her from misery in a job:
When I graduated from college with a degree in literature, there wasn't a huge demand for literature grads in the marketplace. Although I wasn't sure what I wanted to do with my life, I needed to pay my rent, so I took a position as a receptionist for a busy stock brokerage firm. While the job had a lot of perks, there was one big problem: I hated being a receptionist. Answering a phone all day was alternately stressful and boring. Within a month, I loathed getting up in the morning, and my unhappiness at work was coloring my whole life. I knew I had two choices: I could find another job or find a way to like the one I had. I decided to do both. While I looked for other employment, I searched for a way to be happier where I was.
I challenged myself to become "the best receptionist in the world." Because I'd always had a very strong sense of wanting to serve others and make a difference, I wrote the word SERVICE in big letters across the blotter on my desk. I answered the phone with a smile in my voice, learned to recognize frequent callers' voices and address them by name. I'd heard the quote from cosmetic tycoon Mary Kay Ash, "Pretend that every single person you meet has a sign around his or her neck that reads, Make Me Feel Important," and I put that concept into practice. I joked around with the brokers and other staff and generally made my work day a party. Not only did my happiness level skyrocket, but within a month, I was promoted to a more interesting job within the brokerage. Though I eventually found a way to make my living that suited me much better, I've never forgotten how I managed to turn that job—and my happiness level—around.
Being connected to purpose expands you and helps you feel inspired in each moment, which naturally leads to more success. As Albert Schweitzer said, "Success is not the key to happiness. Happiness is the key to success. If you love what you are doing, you will be successful."
Look at the symptoms of contraction and expansion to see if you are "on course" in the purpose department.
Living an inspired life supports your happiness and the happiness of all those who are touched by you.
Happiness Habits for a Life of Purpose
1. Find Your Passion
2. Follow the Inspiration of the Moment
3. Contribute to Something Greater Than Yourself
Happiness Habit for a Life of Purpose #1
Find Your Passion
When we learn to say a deep, passionate yes to the things that really matter, then peace begins to settle onto our lives like golden sunlight sifting to a forest floor.
—Thomas Kinkaide, artist
All right, you're saying, I'm sold! I want to live inspired by purpose. But how do I start? The clues that lead you to your unique and individual purpose are yours for the asking. All you need to do is stop.
That's right—just stop.
Take the time to step out of your busy life and investigate your inner landscape. Being totally honest and fearless, ask yourself: What am I passionate about? What do I love to do? What truly matters to me? The answers—your passions—are like a bread-crumb trail leading you to your purpose.
A great way to get in touch with your passions is to take The Passion Test, developed by Happy 100 member Janet Attwood. Janet has been one of my dearest friends for twenty-five years, and I've seen how using The Passion Test has transformed her life and the lives of so many others. She's a shining example of the joy that comes from this inner connection to purpose. Her story describes the path that led her to discover The Passion Test, now a #1 best-selling book, coauthored with her ex-husband, best friend, and business partner, Chris Attwood.
Janet's Story
Singing and Dancing under the Streetlight
There was once a little girl who loved everyone and everything—especially her mommy. The little girl and her mother used to read stories, watch movies, laugh and sing together for hours. For both of them, every moment they were with each other was filled with the most delicious joy.
In those happy days, the little girl would sometimes go outside in the evening and sing and dance under the streetlight on the sidewalk in front of her house. She imagined herself performing someday in front of thousands of people, touching their hearts.
When the little girl was about seven, her mother started drinking. Gone was the gentle sweet voice that had so lovingly sung her to sleep and tucked her into bed every single night of her life. Instead, the little girl heard her mother and father screaming at each other at the top of their lungs. No matter how many promises the little girl's mother gave, she continued to drink. Things just got worse and worse, until one day the little girl's mother was committed to a mental institution.
When she was finally released, she just started drinking again. After that, the little girl's mother moved away, going in and out of halfway houses. The little girl lost track of her mother and used to read the obituaries in the paper, terrified she'd find her mother's name there. Unable to understand the anger she felt, the little girl took it out on everyone around her—and on herself.
As the years passed, the little girl became more and more unhappy.
At seventeen she was physically abused.
At eighteen she was strung out on drugs.
At nineteen she was living with heroin addicts and riding with the Hell's Angels. Life became one long rhythm of drugs, sex, and depression. The little girl knew if she didn't change her life, she would end up just like her mother . . .
That little girl was me.
At twenty, determined to avoid my mother's fate, I cleaned up my act. With the help of my brother, I found a place to live, got a job, and learned to meditate. I began reading transformational books and listening to personal growth tapes. My life took a turn for the better, but I still felt something was missing. The jobs I took were all right, some better than others, but they barely paid the bills. At one point, I found myself working in Silicon Valley, recruiting disk drive engineers. This was not the job for me. I felt even more discouraged, and I wondered where I was going with my life.
One day I saw a poster for a seminar in San Francisco called "Yes to Success." Maybe a little shot of positive attitude will turn my job around, I thought.
The day of the seminar, I watched, intrigued, as Debra Poneman, an elegantly dressed woman in her early thirties, stepped to the front of the room and began to speak. Within a matter of minutes she had us all completely captivated and excited.
Debra told us about a survey that had been done of 100 of the most successful people in the United States, showing that all of these enduringly successful people had one thing in common.
"Can anyone guess what that one thing was?" she asked, looking around at the audience.
When no one raised their hand, she continued. "Each and every one of them had achieved the five most important elements in their lives they felt were necessary for their own success and fulfillment."
In that moment, the earth stood still.
"Would you please repeat that?" I asked, hardly able to contain myself.
"The one thing they all had in common was that each and every one of them had achieved the five most important elements in their lives they felt were necessary for their own success and fulfillment."
Inside my head a voice screamed, "That's it! That's all I have to know. I just have to figure out what I feel are the five most important elements to living an ideal life, and then I can become just like those hundred successful people."
As I sat watching Debra speak, I thought about the elements I felt were necessary to create my ideal life—and realized that sharing transformational knowledge, in just the way Debra was doing, was what I wanted to do more than anything else in the whole world. The realization was so intense, it was almost painful. I felt as if I were birthing a whole new life! Then, in the very next moment, I became aware of an incredible inner calm, followed by the most peaceful feeling of happiness I had ever experienced. It came from the core of my being as I connected with what really mattered to me.
Driving home with the radio blaring, singing at the top of my lungs, I knew something miraculous was about to happen. I just didn't know what.
Within three weeks I'd quit my job as a recruiter and convinced Debra to hire me to give "Yes to Success" seminars. I never looked back. I spent the following years getting clearer and clearer about the five elements that would bring me my ideal life. I experimented, sometimes following what I thought I should do (not so good), and other times following my heart's desire (always good). When I was doing what I loved from the inside, the outside flowed in ever-widening streams of success and happiness.
This experience of clarifying the five things that mattered most to me was the beginning of The Passion Test, the simple process I developed for finding your purpose in life by identifying what you truly love and care about most. I firmly believe that what you love and God's will for you are one and the same.
A few years later, after taking hundreds of people through The Passion Test and seeing the dramatic results so many had, my best friend, business partner, and ex-husband, Chris Attwood, and I wrote a book called The Passion Test that became a #1 national best-seller. We also cofounded Healthy Wealthy n Wise Magazine, which has become the largest online transformational magazine in the world.
Today, I facilitate The Passion Test all over the world and work with thousands of people from all walks of life. I am living my passions, and life has truly become a magical adventure. The impact of it all really hit me some months ago when I was speaking to 200 homeless women in transition in Miami. These women had quit believing, quit dreaming, and given up. They mostly thought their life was over, except for the painful process of trying to eke out a mundane existence.
I shared with them some of the stories of my early life and then I took them through The Passion Test. I asked them to start making choices every day in favor of what they most cared about, what most lit their fire. I promised them that if they did this, little by little, they would find the energy to move toward a happier life. They would find the greatness that I knew was inside each of them.
As they gave me a standing ovation, excited about what they had discovered about themselves, tears welled up in my eyes. I thought of that little girl singing and dancing under the streetlight so many years ago, who had just wanted to touch people's hearts. I was living that little girl's dreams. The journey had brought me here to give the gift of love to these beautiful women, so much like my own mother.
### Do What You Love, Love What You Do
The important thing is not to think much but to love much; and so do that which best stirs you to love.
—St. Teresa of Avila, sixteenth-century Spanish nun and mystic
What is it that makes your heart expand? What makes your soul sing? Most people are so busy going through the motions of their day that they don't pay attention. Sometimes it's subtle, but the clues to your passion always involve being interested, drawn, and intrigued.
The psychologist and author Mihaly Csikszentmihalyi, who is considered one of the world's leading researchers on positive psychology, calls that feeling of pure enjoyment you experience when you're fully involved doing something, "flow." Time may stop, or hours go by in what seem like minutes. You're naturally focused and not easily distracted. According to the University of Pennsylvania's Positive Psychology Center, one of the hubs of happiness research in this country, engaging in an activity that produces flow is so gratifying that people are willing to do it for its own sake, rather than for external incentives such as money. Looking for the places you experience flow in your life will help you steer toward your passions.
Janet says, "Once you've identified what you truly care about, choose in favor of your passions every day." Life is made up of choices; when you choose over and over to do the things that matter to you, it leads you to express your unique purpose in all areas and helps you draw more of what you want into your life—there's that Law of Attraction again!
I learned about doing what you love and loving what you do from my father. Dad absolutely loved being a dentist. He retired at age seventy-two, reluctantly. He wanted to find a new outlet for his talents, so he analyzed what he loved about dentistry. He realized it wasn't about putting fillings in people's mouths—it was that he loved working in intricate ways with his hands, in ways he felt were artistic.
So, at age seventy-two, Dad took up needlepoint—and loved it. He became a master needlepoint artist, winning awards throughout California. I remember going home one day for a visit when he was about eighty-five. He'd just begun the biggest and most intricate needlepoint project I've ever seen, a detailed depiction of the Tree of Life.
I asked him, "Dad, how long is this going to take you to finish?"
And he said, "Honey, I figure at the pace I'm going it's going to take me about four years."
Imagine, an eighty-five-year old man beginning a four-year project—but his passion for expressing his artistry gave him a strong sense of purpose. And did he complete that project? You bet he did! It was his greatest work of all. Today, it hangs proudly on the wall of my mother's living room in the same house that my parents shared for fifty-three years.
My dad taught me that feeling a sense of purpose allows you to bring joy to whatever it is you're doing.
Exercise
Identifying Your Passions
This exercise by Janet Attwood and Chris Attwood is the first step in getting clear about what really matters to you. When you go through the complete Passion Test, you will discover how to identify your top five passions and how to align your life with them. You can find the complete test in the book The Passion Test, or go to www.PassionTestOnline.com
1. On a piece of paper list at least ten things that will make your life and your work ideal. Complete the sentence, "When my life is ideal, I am____________." For example, "When my life is ideal, I am inspiring others with my love of writing" or "I am feeling healthy, fit, and energetic" or "I am enjoying healthy relationships with my friends and family."
If you're stuck, think of things you absolutely don't want to have in your life, then turn them around. For example, if you feel, "When my life is ideal, I will never be around people who lie, cheat, or steal," turn it around to "When my life is ideal, I will always be surrounded by people who are honest, have the highest integrity, and love to give."
2. Now, think of four people you know who are not passionate about what they're doing in their lives. What do they talk about? Where is their attention focused? How do they treat the people they spend time with? List at least five behaviors you notice in these people.
Do you see any of these behaviors in yourself? Can you see how any of these behaviors might sabotage your ability to live a purposeful life?
3. List five things you can do in the coming week to start changing these behaviors and begin aligning your life with the things you wrote down in Step 1 so you can live the passionate, purposeful life you deserve.
Used by permission of Janet Attwood and Chris Attwood.
Happiness Habit for a Life of Purpose #2
Follow the Inspiration of the Moment
When you follow your bliss . . . doors will open where you would not have thought there would be doors; and where there wouldn't be a door for anyone else.
—Joseph Campbell, twentieth-century scholar and professor
When you're clear about your passions, you're lit by a fire inside that shows you what to do in each moment. You are led to inspired action. You know what you want to do in life, but you may not know how it will happen. Inspiration will lead you to the how.
Following your inspiration doesn't mean you do only the things that are easy. Inspiration gives you the courage and persistence to do whatever has to be done to fulfill your purpose—even if it's challenging or scary. When you're inspired, you act from a sense of inner purpose, not out of obligation or to get others' approval.
Rhonda Byrne is a shining example of someone who radiates happiness and is absolutely led by the inspiration of her soul. This guided her to create the film and book The Secret, which the New York Times calls the biggest phenomenon in the history of self-help. Rhonda was also named by Time magazine as one of the 100 most influential people in the world in 2006. Rhonda told us her story in an interview, and Carol and I then wrote the following account of the inspiration behind the remarkable unfolding of her film's release.
Rhonda's Story
Spreading The Secret
I'd always been a happy person. I had a wonderful family, dear friends, and a successful career as a television producer that I deeply enjoyed. But in 2004, after making six movie-length television specials in the course of twelve months, I was exhausted. Plus, my father had just died, and on top of dealing with my own grief, I was sick with worry about my mother, who was taking Dad's death especially hard. One night I got off the phone with her and was in such despair, I couldn't stop crying.
My twenty-three-year-old daughter, seeing how upset I was, handed me a book saying, "Mom, I think this will really help." It was The Science of Getting Rich by Wallace Wattles. I was mystified how a book about getting rich could help me deal with my sadness about my mother, but I opened it and began to read.
With every sentence my astonishment grew. The book was about getting rich all right, but money was only one part of it. The Science of Getting Rich tells you how to be overflowing with happiness and abundance on every level of your life. Though I'd never read anything like it, I knew right away it was absolutely true.
By the time I finished reading, I was a totally different person. It was like looking up and noticing the sun for the first time. I saw everything in a new light.
For the next few weeks, all I did was study—reading book after book as I traced back through history the ideas I'd been introduced to by Wattles. By the end of those weeks, I knew I'd stumbled onto the most valuable secret in the world: What you think, feel, say, and do, you attract to yourself. In truth, we all create our own reality!
When I put this principle to practice in my own life, my entire existence was transformed. Before long, I knew I wanted to share this secret with as many people as possible. With my television and film background, I thought the best way to do this would be to create a film.
I spent the better part of the next year working with my production company to make The Secret, which is what I decided to call the film. The whole experience was a fabulous adventure. We used the principles of the Law of Attraction that we were presenting in the film at every stage of its creation and development—for scripting, props, interviews, even the distribution.
From the very beginning, I had it in my mind that we would distribute the film through the conventional paths: either through cinemas or television networks. But before we'd even finished filming, the television avenue completely closed and wasn't a possibility. Then, once we'd completed the project, the cinema avenue closed as well. We'd talked to television networks and movie studios around the world, but we'd come to a dead end. There we were, sitting with this film we wanted to get out into the world with no idea how to do it.
I felt stuck until I realized I was getting caught up in the how, trying to figure out the way my intention should unfold. I knew from the Law of Attraction that my job was to stay focused on the what I wanted to create: to feel the gratitude and joy that would come from my successful outcome, and to trust and have faith that the way would be shown. So I let go of all my worries and plans about the distribution. This left me totally in the dark, unable to see the path ahead of me. Still, I kept holding to my purpose and feeling the joy in my heart.
The moment I let go of the how, an extraordinary sequence of events brought a company called Vividas into our lives. At the time, Vividas was the pioneer in the field of online streaming video, a breakthrough technology that allowed viewers to watch videos on their computer without downloading the material first. Up to that point, it had been used only for short videos like the movie trailers Vividas produced for motion picture studios. No one had ever done a whole film in this way before, but Vividas was game. Remarkably, we'd looked all over the world for distribution, but ended up working with a company located only two streets away from our offices in Australia! We collaborated with their experts, and The Secret became the first movie in history to be streamed online. What was even more amazing was that this new technology allowed us to release the film worldwide within a twenty-four-hour period, something I'd dreamed about and known would happen, but had been told was impossible.
The Secret carved a whole new path for movies to be released. Films usually go into cinemas or retail stores first, but The Secret became a phenomenon through online streaming video and Internet sales of the DVDs and then went into retail stores and other conventional outlets. After our success, movie studios and distribution companies everywhere came to us, asking for our template for releasing a movie in this way. People thought we'd known exactly the path we were taking, but we hadn't. It all came from trusting and feeling our way through the joy within us.
When I discovered the Law of Attraction, my purpose became clear to me: to be in joy and through every action and word to share that joy with billions. I've learned to pay special attention to my inner purpose—to be in that joy.
Sometimes I'll take a step in a particular direction and I'll get blocked very quickly. It's as if the Universe is saying to me, "Whoa, you just hold on here, Bessie. Where do you think you're going?" Then I just come back to the trust and joy and gratitude and say, "Okay, you lead me." I wait and from that place of joy, I soon see a whole different path open up.
That happened to me this year. I was so eager to move into the next film, excited about everything I wanted to include in it. I stepped forward in all of my passion and excitement and was instantly blocked. I knew straight away the Universe was saying, "Rhonda, you need to look after this baby—before you go off giving birth to another baby. Let The Secret grow and mature, and give everything that you can to this baby first. You are jumping the gun." I absolutely knew it with every fiber of my being. So I shifted my attention back.
The "new baby" is still there and it's burning away inside me, but I have a calm and a peace about it. I've let go of any schedule for it to happen. I'm certain that I'll feel the timing in my heart, and will know exactly—in the very second—when it's right to move on that.
When I was writing the book The Secret, I waited to feel the push inside me that would let me know the time was right. I didn't write a single word until I felt that the whole project had already been done. I kept feeling the outcome and feeling the outcome—which was more joy, more clarity, and more love. When I felt the real push to begin, I never sat down at the computer without first having tears of gratitude, love, and joy streaming down my face. Every day my heart was in a total meltdown and when that happened, my mind got out of the way and the creative power just flowed through me. Through joy, that joy spoke, so it could share itself with the world.
### Your Job Is the What, Not the How
When you're inspired by purpose like Rhonda, you just need to keep following what you're led to do. You can absolutely trust that inspiration will carry you from one step to another. In the film The Secret, Jack Canfield talks about the experience of traveling in a car at night, the road lit only 200 feet in front of you by the car's headlights. He says that even though you can't see your destination, the lit portion ahead is all you need to stay on the road and get where you're headed. In life, the fire of your inspiration acts as those headlights, allowing you to see what's next. Your job is to follow that light.
### Precession: Small Steps, Big Results
When you do what you're inspired to do in the moment, you're often unaware of the effects your actions will have or where they'll lead you. A honeybee buzzing from flower to flower isn't aware that it's cross-pollinating the plants and making life on earth possible. As far as it's concerned, it's just drawn to one flower after another to collect nectar, which will be converted to honey.
Rosa Parks, the African American woman who refused to take a back seat on a public bus, had no idea that her one courageous act would be the start of the civil rights movement in the South. She simply took the step that felt right to her at that moment, making what she thought was a small statement about freedom.
Buckminster Fuller, the famous architect and visionary, used the term "precession" to describe this phenomenon of a series of small steps leading to an astonishing, unforeseen conclusion.
I've experienced precession in my own life many times. If you had told me twenty years ago that I'd be a best-selling author, I would have thought you were crazy. I only wanted to be a speaker. I hated writing!
In fact, in my job at the Austrian crystal company, everyone in the company knew my well-established method for writing important business letters and memos. I would delay writing a letter for as long as I possibly could. (The office staff would know this by how clean my desk was: the cleaner my desk, the more I'd been procrastinating.) Finally, after putting the letter off as long as I could and struggling unsuccessfully to get a few sentences down, I'd go to my buddy Jay's office and cajole him into writing the letter for me. I was convinced that I was the worst writer in the world.
When I left that job to become a corporate trainer, the only position I could find was teaching business writing. On the surface, I thought, No way! What kind of cosmic joke is this? But on a deep level, it felt like the right thing to do. I took the job, learned all about writing and editing, and found I actually did have some skill and talent. Six years later, when I had the idea for Chicken Soup for the Woman's Soul, I was prepared: I had the exact skills I needed to co-create that book. I certainly didn't plan for it, but that's where life took me.
Some folks hear the fanfare of trumpets when they have their Big Idea. Many don't. You may feel you have a special and unique gift but haven't quite tumbled on how to express it. Rest assured: if you keep your sights on the lighted road, the opportunity is bound to sneak up on you when you least expect it.
Exercise
Acting from Inspiration
I like to think of inspiration as being in spirit, being led by my soul throughout the day. A wonderful way to act from inspiration is to start your day by asking yourself the following questions, which are adapted from the book A Course in Miracles. Here's what you do:
1. Close your eyes, and take a few deep, slow breaths.
2. Ask yourself these three questions:
What would Spirit have me do?
Where would Spirit have me go?
What would Spirit have me say, and to whom?
3. As you move through your day, use the answers you receive to "light the 200 feet in front of you" on your way.
Happiness Habit for a Life of Purpose #3
Contribute to Something Greater Than Yourself
I don't know what your destiny will be, but one thing I do know: the only ones among you who will be really happy are those who have sought and found how to serve.
—Albert Schweitzer, physician and humanitarian
The happiest people contribute to something greater than themselves in life. When Stewart Emery interviewed people with enduring success and happiness for his book Success Built to Last, he found that their goals weren't fame, wealth, or power. People with those goals inevitably ended up feeling empty and unhappy. My interviews with the Happy 100 showed me that people who are Happy for No Reason may be famous, wealthy, and powerful, but those are outcomes of living fully and passionately, engaged in meaningful service to a larger cause. Oprah Winfrey once said, "I never went for the money, I just said, God use me. Show me how to take who I am, who I want to be, and what I can do, and use it for a purpose greater than myself."
Happy 100 member Lynne Twist is also someone who passionately devotes her life to a greater purpose. She's sometimes referred to as a modern-day Mother Teresa. I've been fortunate to spend time, personally and professionally, with Lynne, and whenever I'm around her, I'm so moved by the way her beauty and goodness shine from her in everything she does. She's often so full of gratitude that her eyes fill with tears. In our interview, Lynne shared the story of how she found her call to serve.
Lynne's Story
The Calling
As a teenager, I led a double life. To most people I appeared a typical 1950s high school girl. I earned straight A's, was a cheerleader and homecoming queen, and even dated the captain of the football team. But there was also another Lynne. This Lynne was deeply religious: she woke every morning before dawn to go to morning Mass, idolized Mother Teresa, and entertained dreams of becoming a nun. Which was the real me? They both were.
This double life began with the death of my father, whom I'd adored. A big-band leader like Glenn Miller, he'd turned our home into a joyful, fun place, filled with musicians, dancing, and singing. Then, two days before my fourteenth birthday, my father died of a heart attack, peacefully, while he slept. He was only fifty years old.
His death was incomprehensible to me. My shock and grief made me look for a deeper meaning to my life, and I turned to God and the church. It was then that my aspirations for a life of service—a life that made a difference—started.
I hid my spirituality from all the kids I knew because it wasn't cool to be religious, but I tried to bridge the gap between my inner and outer life by becoming involved in community service projects and roping in all my friends to help. We tackled all kinds of jobs, from collecting clothes for a local fund-raising sale to tutoring disadvantaged kids who had dropped out of school. It was so satisfying to help other people, and we all had fun doing it.
After high school, I went to Stanford, where studying the mystical poetry of Rilke and Rumi, among others, took the place of my daily churchgoing. I was still searching for the mission I was put on earth to accomplish when I fell in love with Bill Twist. We married during my senior year, and after graduation, in very short order, our daughter and two sons were born.
Although those were happy years, my search for greater meaning was always simmering below the surface. It spurred me to take a personal growth seminar called est, which I found profoundly transformational, and also to study with the renowned inventor, designer, and futurist Buckminster Fuller, whose books I'd read in college and admired greatly. As a young man of thirty-two, Bucky had contemplated suicide. Deciding against it at the last moment, he'd said to himself, "I may be a throw-away person, but perhaps I can take this throw-away life and devote it to making a difference." (It was actually Bucky who coined the phrase, "Make a difference with your life.") He'd launched an experiment to see if one ordinary individual could change the world and benefit all humanity.
In 1977, I helped to introduce Bucky to the founder of the est program, Werner Erhard, and from that relationship, the Hunger Project—the commitment to end world hunger by the year 2000—was born.
When I heard about the project a few days later, it created an inner earthquake in me. I knew without any doubt that this was what I was meant to do. I stepped out of "the movie of my life starring me" and became a supporting player in a much larger film. Suddenly, my personality and my agenda took a back seat to this greater calling. This commitment woke me up in the morning, showed me what to wear and where to go. It gave me an eloquent voice and the words to speak.
I took on a leadership role at the Hunger Project and soon found myself stretched to the max. Once again I was leading a double life—now a suburban mom and a crusader—although this time it wasn't a secret. I had a wonderful husband and three kids, six, eight, and ten, and was committed to being there for them; I was also committed to doing whatever it took to end world hunger by the year 2000. My two commitments lived side-by-side—sometimes literally. People from places like Bangladesh, Sweden, Japan, and Ethiopia often stayed in our home with us during their Hunger Project training in the States, and because I traveled constantly, I brought the kids and Bill with me whenever possible. Other families went to Disneyland and Aspen for vacations; we went to Zimbabwe and Indonesia.
We had enough money so that I could hire a terrific housekeeper, which helped enormously, but I still made a concerted effort to always be home on weekends. This sometimes meant flying to India on a Monday and flying back home on Friday! I felt the strain of always being pulled in two directions.
One Saturday, distraught that I had missed my daughter's choral concert and my son's championship soccer game, I called a family meeting. We sat on the floor in a circle and I told Bill and the kids, "I'm feeling so guilty that I didn't make the Halloween costumes this year, and that I missed the concert and the game. I need your permission to go on. I'm so dedicated to my work with the Hunger Project, but I feel torn that I can't do everything." By the time I finished, I was sobbing.
Summer, my eight-year-old daughter, came over and put her arms around me and said, "Mom, if you can help end world hunger, we don't want you driving us to the orthodontist. Someone else can do that." She went on, "We have the coolest life and the most amazing people living with us. We're so lucky and so proud of you."
My two sons and my husband joined in, putting their arms around Summer and me, as my husband said, "Go for it. We're just thrilled with your commitment; it lights up our lives." We hugged and cried and laughed, and in that moment, the schism between the two things I held so dear closed up and any lingering traces of a double life vanished.
Ending world hunger became a family commitment after that. The kids volunteered at the office, often lying on the floor beneath my desk to do their homework, and we all became world citizens. I realize now that my kids didn't have to have two separate lives, one "normal" and the other "spiritual." We melded the two—and have been all the better for it.
People often think that living a life of meaning means sacrifice, but for me, it's been exactly the opposite. I've had opportunities to do things and meet people I could never have imagined. When I finally met my childhood hero, Mother Teresa, our connection was immediate and natural. She called herself "God's pencil" and felt that God wrote his story to the world through her and people like her. I could relate to that—I also felt I was being used by God to write the end of hunger and the end of the suffering that goes along with it.
Living a committed life gave me the opportunity not only to work alongside Mother Teresa, but also to sit with the Dalai Lama and to have an ongoing relationship with Archbishop Desmond Tutu and Nelson Mandela. These were people that I never in a million years thought I'd know, let alone have an opportunity to work with. It's also provided me the equally valuable privilege of being in the presence of wise and courageous people all over the world. Following the famine of 1984–1985, I sat around a dry well for five days and nights with a group of Ethiopian mothers whose children had died of starvation. These women and others like them inspired me to keep going for the goal we'd set.
When we started the Hunger Project in 1977, 44,000 people a day were dying of hunger and starvation, most of them children under five years old—and those numbers were rising every year. Today, though the world's population has grown by over 50 percent, the number of deaths from hunger and starvation is 19,000 people a day, less than half of what it was in 1977. It's still too many, but it's a staggering accomplishment.
I always thought that I'd be working for the Hunger Project for the rest of my life: I'd either end hunger or die trying. But in 1994, to my great surprise, I heard something new calling me. At first, I ignored it, thinking it was just a distraction from the work in front of me, but its voice was so persistent and so persuasive, in the end it made me look inside and take a new direction in my life. In 1996, Bill and I founded the Pachamama Alliance to work with the indigenous peoples of South America to preserve the rain forests and to create a new global vision of sustainability for all life. Scientists predict that without the rain forests, South America will become barren, creating a hunger crisis that will affect millions of people and endanger the health of the planet as a whole. This goal is not as far from my original mission as it may seem: rather than working to end world hunger, I'm working to ensure that it doesn't happen in the first place.
Over the past thirty years, giving myself to the larger purpose that calls me, in whatever form it appears, has led me to a life happier than any I could ever have imagined.
### Making a Difference with Your Life
It's not necessary to be a Mother Teresa or a Lynne Twist to contribute to something greater than yourself. When you discover what matters to you, your everyday actions can serve others and the world in ways large and small.
It really doesn't matter what you choose to serve. All of us are drawn to different causes. For some people, it's wildlife; for others it's social justice, ending poverty, or making sure all children have access to the arts. The details aren't important. When you serve a purpose larger than yourself it brings more meaning and joy to your life. I've experienced this myself being a part of the local Big Brothers/Big Sisters program. Spending time with my fantastic little sister, Leah, brings me great delight.
Your desire to make a difference can influence even routine decisions, such as what you eat, where you shop, and what kind of car you drive. I have a friend who drives a hybrid and eats only organic produce that she buys at the local farmer's market. She calls this voting with her pocketbook, spending her money in the way that supports what she believes in.
And you don't have to be wealthy to make donations that yield generous results. Even giving $10 or $20 to charitable organizations can feed a family for a week, buy a cow or goat, help fund a small business, or buy seeds to grow crops, making a huge difference in others' lives. Contributing to something greater than yourself doesn't have to involve money; it can be an exchange of your time, your interest, or your caring.
Research has consistently shown that giving of yourself to others—altruism—is associated with greater well-being, health, and happiness, as long as you don't "overgive." Giving of yourself doesn't mean being codependent, trying to fill an inner emptiness, or serving others at our own expense. The service I'm talking about comes from joy, inspiration, and purpose and supports more peace and well-being in your life.
Exercise
Call to Serve Visualization
You can do this visualization by having someone guide you through it or by listening to your own recorded voice leading you through the steps. As an alternative, you can read all the instructions first and then lead yourself through the exercise silently.
1. Sit or lie down in a quiet place where you'll be undisturbed. Close your eyes and take slow, deep belly breaths, relaxing your whole body.
2. Allow yourself to feel lighter and lighter and more expanded—until you experience being so light that you feel and see yourself begin to float above your body.
3. Imagine now that you are high above your body, hovering over the earth. You look down at the planet beneath you—you see a beautiful, shimmering blue sphere. You can see the oceans and continents and vast cloud formations below. You look more closely and see the mountains, forests, valleys, and cities.
4. You see the billions of people and animals that all live interconnectedly on earth. You feel connected to all the life teeming beneath you. You feel you are a part of a bigger design. You put your attention on the question: How am I being called to serve?
5. You find yourself being drawn toward a place on the planet that holds a particular significance or fascination for you. You see a situation that moves you to serve. You may be shown someone, somewhere, or something that is familiar to you, or it may be completely new. (You may be drawn to things such as helping animals, working to cure a particular disease, preserving the oceans, or serving children in your area or in a developing nation.) Be curious and adventurous. See where you land, the kind of surroundings you find yourself in, and what's there for you to do. You may receive one or multiple glimpses of the opportunities that await you. Be open to all possibilities.
6. When you feel complete with your experience, express gratitude and gradually feel yourself returning to the room where you are sitting or lying down. Reflect on how you can use the images you received to be led to where you can be of service. See what unfolds in your life.
Summary and Happiness Action Steps
When you live inspired by purpose, you choose in favor of your passions, let the fire of inspiration light your path, and contribute to something greater than yourself in ways large and small. This puts the roof on your Home for Happiness. Use the following action steps to practice the Happiness Habits for a Life of Purpose:
1. Look for ways to turn your job into a career and your career into a calling. What can you change in your present circumstances to feel more inspired by your purpose?
2. Take the complete Passion Test every six months to stay aligned with what truly matters to you.
3. Begin each day by asking yourself, "What would be meaningful to do today?" Then be led by inspiration throughout the day.
4. Ask yourself, "How can I be of best service to others?" Take a step to volunteer locally: call your local nursing home, food bank, animal shelter, literacy program, Sierra Club chapter—the list is endless. Find out what you can do to help. Even an hour once or twice a month can make a difference in your life and the lives of others.
## 9
## The Garden—Cultivate Nourishing Relationships
Whoever is happy will make others happy, too.
—Mark Twain, writer and humorist
I love sitting in my garden. My favorite spot is on a bench in a sunny corner where I have the perfect view of the flowers and trees in my yard. It's a wonderful way to unwind after a crazy day or just to spend a few minutes appreciating the beauty around me.
The people in your life are the garden surrounding your Home for Happiness. When you look around that garden, do you see beautiful roses and dahlias, or a neglected patch of weeds? Our relationships affect us in a similar way—either uplifting us, or dragging us down. Happy people cultivate relationships in their lives that nourish and support their happiness.
### People Power
Scores of studies in the field of positive psychology demonstrate that having good social relationships is one of the strongest predictors of happiness. Large-scale surveys conducted by the University of Chicago's National Opinion Research Center found that people with five or more close friends (not including family members) are 50 percent more likely to describe themselves as "very happy" than those with fewer strong relationships. Another survey of 800 people showed that those who valued wealth, success, and status over close friends and a loving relationship with a significant other were twice as likely to be "fairly" or "very" unhappy.
Edward Diener, the father of happiness research, and Martin Seligman, the father of positive psychology, conducted a study in 2002 focusing on two groups of people: those with the highest scores on a standard measure of happiness and those with the lowest. They discovered that the one trait the happy group had in common, which the unhappy group didn't share, was having close trusting relationships.
In my interviews, I found the same trait. Although the Happy 100 vary greatly in the number of relationships they have, each one of those relationships is a healthy one that supports their happiness. What sets the Happy 100 apart is that they don't depend on others to make them happy. When you're Happy for No Reason, you enjoy being with your friends and family, but you also enjoy spending time alone, in your own company. You bring your happiness to your relationships rather than trying to extract happiness from them.
### Emotional Contagion
One of the most exciting discoveries that neuroscientists have made in the past decade is that our brains are actually wired to relate to others. Everyone with whom we interact, or even just nod to as we pass on the street, stimulates a "neural bridge" that connects us. Our brains contain "mirror neurons" that fire in or out of synch with the people around us.
Have you ever caught yourself unintentionally mimicking the facial expression, posture, body language, or speech rhythm of a person you're having a conversation with? Or yawning in response to someone else's yawn—even though you're not tired? When you observe someone else performing an action, these neurons "mirror" that action inside your brain as though you were performing it yourself!
Mirror neurons have also been linked to our ability to empathize with the emotions of others, explaining why, when a person who's angry or upset walks into a room, everyone in the room can feel it. Or why just seeing someone else overcome with emotion is enough to make our own eyes fill with tears. Some researchers now believe that autism, which causes an impaired ability to relate to others, may involve damaged mirror neurons.
Our emotions are contagious. According to the internationally known psychologist Daniel Goleman, the author of Emotional Intelligence and Social Intelligence, emotions spread from one person to another much like a cold. And while it can be good to "catch" an uplifting feeling, it can be damaging to take on others' feelings of anger, jealousy, anxiety, or hate.
Dr. Goleman says that the more connected we are with someone emotionally, the greater the influence they have on us—especially over time. I've caught some great emotions from those closest to me, particularly Sergio. I didn't consciously have happiness on the checklist of qualities for my ideal mate, but I certainly feel blessed that Sergio is a deeply happy guy. He's earned his way into the Happy 100 fair and square, and in fact, is often bubbling over in song. At first, I thought he was just trying to impress me when, in the shower, he'd belt out beautiful Italian love songs I could hear all the way across the house. But even today, when he no longer needs to play the impressing game, singing joyfully and loudly in the shower is still part of his daily routine. And it doesn't stop there. Sergio sings when he's making breakfast, when he's doing the laundry (yes, he does the laundry), when he's returning emails—pretty much all day long. Sometimes I even have to come out of my office when I'm on a business call to ask him to turn down the volume. But I wouldn't trade being around his joyous influence for anything in the world. It has such a fabulous impact on my own happiness.
Think about the people in your life. Do you want to "catch their emotions"?
### Hey, Girlfriend!
Our relationships have a biochemical effect on our bodies. When we make healthy connections with people, our brains flood our cells with happiness chemicals, and when we have unhealthy social interactions, harmful chemicals are released.
According to the latest research, women's biochemical wiring makes them more likely than men to seek out relationships with others. Though men and women both release adrenaline and cortisol when under stress, scientists found in a landmark UCLA study that in order to buffer those stress chemicals, women's brains release oxytocin, the bonding hormone I talked about in Chapter 6. This is why women who are going through a rough time often want to gather with other women or have a good, long yak-fest with a close girlfriend. Or why they may feel drawn to caring for their children or pets. Researchers call this behavior "tending and befriending." It is stimulated by oxytocin and in turn creates more oxytocin. The more women tend and befriend, the more oxytocin they release, producing a calming influence and further lowering their stress.
There's no question that friends help women to live happier and healthier lives. The famed Nurses' Health Study from Harvard Medical School found that the more friends a woman has, the less likely she is to develop physical problems as she ages and the better chance she has to lead a joyful life. In fact, the results were so significant that the researchers concluded that not having close friends or confidants was as detrimental to a woman's health as smoking cigarettes or being overweight.
My colleague John Gray, the author of Men Are from Mars, Women Are from Venus, whom I interviewed for this book, has reviewed extensive research about the relationship between stress, hormones, and gender. He told me that though women release oxytocin when faced with stress, men don't have this same biochemical response. According to the studies, when men are stressed and release cortisol, it lowers their dopamine and testosterone levels, causing frustration and depression. Dr. Gray says that men are biologically hardwired to seek out ways to stimulate the production of those neurochemicals by problem-solving, taking action, and overcoming risk and danger, rather than talking or taking care of others. The lower amount of oxytocin in their system makes them less interested in bonding with friends.
### Relationships and Your Energy
Until you're firmly established in the state of Happy for No Reason, your happiness is affected by the energy of the people around you. When you surround yourself with relationships that support you, your energy expands. When you have a lot of toxic people in your life, your energy contracts.
Practicing the following relationship habits will help raise your happiness set-point.
Happiness Habits for Relationships
1. Tend to Your Relationships
2. Surround Yourself with Support
3. See the World as Your Family
Happiness Habit for Relationships #1
Tend to Your Relationships
You become the average of the five people you associate with most.
—Jim Rohn, author and motivational speaker
We tend to our gardens by weeding, watering, and planting to make sure they flourish. We can tend to our relationships by spending more time with people who support our happiness and less time with those who erode it.
Of course, the happier you are inside, the less you're adversely influenced by your external environment. In my interviews with the Happy 100, I saw that they handle toxic people when they have to, but they also limit their interactions with them when they can.
Happy 100 member, author, and life coach Martha Beck, whom you met in Chapter 6, went through a profound personal transformation that brought her to a state of deep happiness. In the following story, she describes her experience learning to recognize the relationships in her life that support her.
Martha's Story
My Great Big Unrelated Family Reunion
I'd just moved to Phoenix, and, feeling isolated, I attended a book signing, hoping to meet other bookish people. The signing was about as interesting as mud, so I decided to leave. This turned out to be strangely difficult. As I walked away, I literally felt something trying to turn me around and push me back into the building. All the way home, I fought an intense impulse to turn the car around, go back, and meet someone.
A week later I walked into a coffee shop and saw Annette. I recognized her immediately, but not because she'd been at that book signing. She had been, but I hadn't seen her. I recognized Annette because I just knew her. It felt as though long ago, our souls had made an agreement to meet in Phoenix, Arizona, at that appointed time. This feeling was so strong it would have scared me—except that I'd felt it several times before, with different people, in different places.
I was about fifteen when I first got the feeling I'd met only a small percentage of the people I loved most. Unlike many teenagers, I found sex, drugs, and rock 'n' roll much less interesting than English literature and biology. I had lots of equally nerdy friends, a group of peers who pressured me only to study for the SATs. Still, I felt oddly misplaced. I'd walk the halls of my large public high school like a zebra separated from its herd, scanning for other striped creatures.
"There are more of us," I kept thinking. "Where are the others?"
Every now and then I'd meet someone—in class, at a debate meet, at the mall—who drew my attention like a magnet. These people fairly glowed in the dark; I couldn't take my eyes off them. They were of all ages and both genders. This wasn't about romantic attraction—I just . . . recognized them.
As I matured, these incidents grew even more noticeable. On my first day of college, feeling desperately scared and lonely, I walked into an art studio and instantly recognized the professor as one of my life's most important mentors. Later that day, I was sitting at a bus stop, drawing in a sketchbook, when a nicely dressed stranger glanced at my work.
"Listen," she said, "there's an art class you should be taking."
"I know," I said. "I'm already in it."
The stranger looked me in the eyes. There was no question we were talking about the same class. She nodded. Her bus arrived, and she got on. I never saw her again—though, of course, I knew and loved her. Still do. We'd kept our appointment.
If this is starting to sound weird to you, think how it felt to me. The longer I lived, the more intensely I "recognized" people I'd never met. What's more, I began to realize that the connection was often mutual: the strangers I knew, knew me. All of them had questions in their eyes, as though they were continually searching for lost loved ones in a crowd.
In my late twenties, I felt more and more at odds with my family and community of origin. Both were steeped in a particularly rigid religious orthodoxy that made my relationships with them feel increasingly unhealthy to me. I got therapy—good therapy—and the more diligently I looked within myself and learned to distinguish between what felt right to me and what felt wrong, the more I understood that spending less time with that family and community wasn't a failing on my part, but a necessary, healing, healthy choice.
And I realized why I'd felt a warm, living connection with people I barely knew or had only seen.
It was because they were my family.
Now, I'm so comfortable with the notion of family as a spiritually linked network of souls that it never surprises me to meet a new brother, sister, mother, father, daughter, or son. My nearest and dearest tend to enter my life in extravagantly miraculous ways.
Annette, for example, became my first writer friend, and soon thereafter, we invited two other writers, Dawn and Thora, to form a writing group. The night before our first meeting, I dreamed that a Navajo medicine woman handed me a blue stone butterfly and said, "Dineh," which is Navajo for "the people." I didn't think about this much until I met Thora. Then I got gooseflesh; she looked exactly like the medicine woman in my dream. Then Dawn happened to mention her twin sister, Denae—which she pronounced exactly like the Navajo word dineh. At this point, I told the group about my odd dream. When I mentioned the blue stone butterfly, Annette burst out laughing. She opened her purse and pulled out a blue stone butterfly.
Without that writing group, I would never have finished a book at all. With their support, I finished a manuscript that eventually found an agent and a publisher. When I went to New York to meet my new editor, Betsy, I realized within about thirty seconds that she was my favorite sister.
After our first Manhattan lunch, I sent her a small ceramic turtle as a gift, along with a note explaining that turtles have always symbolized how we in the writing world need to go through life: sure, slow, steady, knowing when to pull our heads in. "I got the turtle," Betsy told me later, "and I thought, she knows." Of course I knew. I'd been doing this long enough to recognize when I was keeping an appointment with one of my nearest and dearest.
My life seems to have turned into a long family reunion. These days, it's filled with so many loved ones, such a huge and varied tribe, that I sometimes find myself weeping helplessly with joy, wonder, and gratitude. When I meet someone I love, we rarely even bother to pretend we don't recognize one another.
At a conference, I meet another presenter who doesn't introduce himself. We simply fall into each other's arms, thrilled that we've both shown up for our appointment. "Hey," he says, beaming at me, "I have a book you need." I take the book, knowing that it contains exactly the information and inspiration for which I've been praying. I may never see him again, but we can feel one another's presence in the world.
On a book tour in Germany, a man I've never met grabs my hand and says, "Du." I smile and say, "You." We both burst out laughing, thrilled to finally meet one another. "We don't speak the same language in our minds," he says in German, "but we do in our hearts." I don't speak German. I understand him perfectly.
In Africa, I walk into a one-room school, meet the teacher and several members of the small Shangaan village. I recognize literally everyone.
"I'm adopting this school," I say to the teacher. "I want to help you get anything you need."
She nods, matter-of-factly, and just says, "Yes."
Neither of us needs to add, "I'm so glad we kept this appointment."
Having reached a certain age, and lived through thousands of moments like these, I've learned to simply enjoy them without frantically wondering what it all means. Having a soul-family is more than enough reward in itself. But I still have a great many questions. Does everyone have a huge family of unrelated individuals, who may have nothing in common physically, but recognize one another by the texture of their souls? Is there some task each spiritual clan is meant to complete?
I don't know for sure, but I have a hunch the answer to each of these questions is "Yes." If I'm wrong, if my life history of recognizing family members proves to be nothing more than delusion, I won't mind. As delusions go, this one is wondrous, delightful, sweet, and harmless. But just in case I'm right—just in case this all sounds strangely familiar to you—the time may come when we see one another at a coffee shop, at a bookstore, in a tiny one-room school in some remote village, and feel that instantaneous thrill of recognition.
If this should come to pass, and you see me first, I have just one request: Don't be shy. I've been waiting to meet you for a very long time.
### Building Up Emotional Immunity
Spending time with those you love—family, friends, or pets—can reset the balance of your biology toward greater happiness, so it's important to make wise choices about the company you keep.
Not everyone in our life will always nourish us. I'm sure you've had the experience of feeling good, until you're around a friend, relative, or coworker who drags you down. It's emotional contagion again: you're picking up their neuronal vibes. Staying out of range of emotional bullies and happiness vampires who suck the life out of you is the easiest way to avoid negative emotional contagion.
Usually it's obvious which people are toxic to be around: they're the complainers, the discouragers, the people whose criticism aims to wound. Some are harder to spot: they're self-absorbed, fearful, judgmental, and manipulative. They may even mean well; still, you come away from contact with them feeling drained and frustrated.
To populate your world with uplifting people and minimize your connection with people who are toxic, use your inner GPS. Close your eyes, take a deep breath, and imagine each person in your life. Who expands you and who contracts you?
Even when you've identified the toxic people in your life, the problem is that it's not always possible to stop spending time with them; you may work with happiness vampires or may even be related to them. What do you do then?
Learning to create appropriate boundaries is a must. As Dr. Phil says, "We teach people how to treat us"—by what we accept or don't accept from them. When you have to be around toxic people, here are some ways to bolster your immunity:
1. Break the chain reaction: Now that you know about mirror neurons, put them to work to your advantage. If you have to talk to an angry or negative person, consciously soften your gaze, keep your expression neutral, use body language that's the opposite of his or hers. Don't reflect the other person's tension, or your body will hijack you by mirroring that negativity.
2. Put up an invisible barrier: When you can't leave and you're being bombarded by toxic emotions, the author and UCLA psychiatrist Judith Orloff recommends imagining an invisible wall or a shield around yourself. This will give you a feeling of emotional protection and may help to buffer your desire to respond in kind.
3. Stay on your side of the road: Don't try to change the other person. It may be tempting to think that you can help others by trying to save them or pointing out the "error of their ways," but it rarely ever works. The most effective way to influence others is to model the behavior you'd like to see in them.
### Shower the People You Love with Love
The best way to keep relationships happy, healthy, and supportive can be summed up in one word: appreciation. (What you appreciate, appreciates.) When we demonstrate our appreciation for the support we receive from others, it reinforces that behavior and deepens our connection to them.
Appreciation is a basic human need, both at home and at work. In fact, according to the U.S. Bureau of Labor, 40 percent of employees leave their job not because of salary or workload, but because they don't feel appreciated.
Too often, we take our closest personal relationships for granted as well, giving little energy and paying little attention to the people who mean the most to us. Psychologist John Gottman did a famous study of marital happiness using what he calls "the magic ratio" to predict whether 700 newly married couples would remain married or get divorced. Dr. Gottman said that couples who had a ratio of five positive interactions for every one negative interaction would stay together. In a follow-up done ten years later, a staggering 94 percent of the couples he had said would divorce were, in fact, no longer married.
Judith W. Umlas, author of The Power of Acknowledgment, says, "One of the most important things a person can do to raise his or her happiness level is to acknowledge those around them. According to a recent Gallup Management Journal article, when someone is acknowledged, dopamine is released—a neurochemical that's directly linked to being happy!"
In 2004, Dr. Donald O. Clifton and his grandson, Tom Rath, wrote How Full Is Your Bucket? based on over fifty years of comprehensive research done by social scientists and the Gallup Organization. Its message is easy to understand and apply: The most effective way to motivate, connect to, and inspire others, what the authors call "filling up a person's bucket," is through specific and sincere positive recognition. And when you fill someone else's bucket, it raises your own happiness level as well.
When I want to amp up the appreciation, I love using the following technique, called The Appreciation Practice. It's a wonderful way to enrich your relationship with anyone—spouse, child, friend, or coworker. Sergio and I practice this almost every night before going to sleep, and it always leaves me smiling.
Exercise
The Appreciation Practice
1. Begin by acknowledging something you appreciate about the other person (for example, "You make me laugh," "I feel supported by you," "You're kind"). When you're finished, switch roles. Go back and forth at least five times or as long as you like.
2. Now repeat the exercise, only this time acknowledge one thing you appreciate about yourself. Then let the other person have a turn. Go back and forth at least five times or as long as you like.
Happiness Habit for Relationships #2
Surround Yourself with Support
You have to do it by yourself,
And you can't do it alone.
—Martin Rutte, corporate consultant and speaker
We sometimes need more support than our friends and family can provide. When we're having a tough time or deciding to go for our dreams, the people closest to us may commiserate, take our side, or tell us we're fine the way we are, rather than offer us the strength and honesty we need to move forward.
Often, the best way to surround yourself with support is to join or form a group that meets regularly, whose sole purpose is to provide guidance and straight feedback—and keep you from sliding into your old patterns of victimhood.
During our research, Carol met the woman who told us the following story and was instantly impressed by her friendly, loving, and open manner. When Carol told her about the book we were writing on happiness, the woman said she had learned how to be truly happy many years ago and kindly agreed to be interviewed. Although Molly Baker is not her real name, her inner peace and well-being are the genuine article. Molly's story is a beautiful example of the role a support group can play in moving us toward greater experiences of happiness.
Molly's Story
That's What Friends Are For
John (a pseudonym) and I met in October and he proposed in December. I said yes on the spot. I didn't need to think about it. I was very attracted to him. Besides being an Ivy League graduate and having a good job, John was funny, popular, and a born leader—everything I felt I wasn't. Plus, I was twenty-one and ready to get married. This was the early 1950s, and I had no career goals except to be a wife and mother. Still, when John and I married ten months later, we really didn't know each other.
I soon discovered that my new husband drank—often and to excess.
That first year of marriage was a very unhappy time for me. John, a high-functioning alcoholic, was busy making a name for himself at his company. He worked late most nights, and then went out for business meetings, arriving home at all hours, almost always "under the influence." Yet, no matter how much he drank, he never seemed to have a hangover. He invariably got to the office on time and never missed a day of work. His boss had no idea of his problem.
I, on the other hand, was painfully aware of John's drinking. When John drank, his entire personality changed: he became wobbly, uproarious, sloppy, and stumbling. I was very uncomfortable being around him. There were car accidents—luckily only the cars were seriously damaged—and trips to the emergency room to stitch up his injuries. I was ashamed to tell my parents and sisters what was really going on in my life.
When I complained to John that he was drinking too much, he always denied it, claiming that I was being stupid or ridiculous. The sad thing was that I believed him. What's more, I saw that getting mad at John about his drinking just made him defensive and uncomfortable, which served only to make him drink more. I knew things were terribly wrong, but I didn't know what to do. What made it so hard was that at his core, John was such a good man. When he wasn't drunk, I loved being with him and respected him. My unhappiness and unexpressed anger grew steadily—as did my loneliness.
When our son, and then our daughter, arrived, things became more bearable. The years flew by as I hid on the hamster wheel of activity, keeping myself too busy to dwell on my situation. From the outside, we looked like a normal family, but in private, my relationship with John was steadily deteriorating.
When I complained to my two closest girlfriends about John, they always commiserated. It felt good to vent, but wallowing around with them in the same muck year after year never caused me to grow or change.
By the time we had been married twenty-three years, John's drinking was worse than I had ever imagined it could get, yet he still completely denied his illness. I felt so fundamentally empty, I didn't know how much longer I could take it.
One night I woke up and found that I was sobbing with fury and pounding on John's chest. All the anger I had been denying was spilling out in my sleep. John, snoring off the evening's drinking, never even woke. Afterward, I lay quiet for a long time, facing the fact that I was one sick cookie. I was drowning in resentment and despair, but the fear of being alone, of having to support myself, and of upsetting my parents and my children, even though they were already grown, paralyzed me.
Then one day, a few weeks later, a friend of mine told me about a support group for people who were involved with alcoholics. "We're both married to drunks. Let's go!" she said. I agreed eagerly.
We arrived at the church where the Al-Anon meeting was being held and walked into a room with folding chairs set up in a circle in the middle. Once everyone was settled, the meeting began.
I was immediately struck by everyone's openness. Each person seemed to be accepted as they were, and I felt an unconditional love flowing around the circle, enveloping the people I assumed were the regulars, as well as the newcomers. It was like coming in out of the cold; my whole body relaxed in the warmth of this gathering. Aside from introducing myself, I didn't have to say another word—unless I wanted to. I sat listening intently as people shared experiences that I knew all too well.
I began going to the meetings two or three times a week. I couldn't get enough of the love and acceptance that was given so freely. There was no criticism or judgment from anyone, and, equally important, they didn't allow me to stay stuck in self-pity. Ever so gently, the group helped me turn my focus away from John and his actions to finding the strength and confidence to turn my life around—whatever that meant for me. There was no advice given. I just heard story after story from people who had been where I was and had moved forward in their lives.
After each meeting, people always came up to me and hugged me, whispering words of encouragement into my ear. I was like a sponge, absorbing it all. Little by little, drop by drop, I began to feel whole.
It still wasn't easy at home. When John was drunk, I still felt victimized by his behavior and beaten down by his denial of any problem, but after each of our encounters, instead of stewing in my own helplessness, I shared my experience with my group and came away focused on my own worthiness and one step closer to finding serenity within myself.
Late one night, about a year and half after I first discovered my "circle of support," John came home at 2 o'clock in the morning, very drunk. He was fumbling around our bedroom, talking loudly, and being totally obnoxious.
I looked at this man who had been my husband for over twenty years and all I felt was compassion and a calm conviction that I didn't want to be in the room with him while he was in this state. I marveled at the complete peace I felt inside. I still well up with emotion remembering that moment: there was no more fear.
Clearly and evenly, I said, "John, I'm going to sleep in the other room tonight." And I got up and walked down the hall to the spare bedroom.
He followed me into the room and began arguing. "Oh, come on, what are you doing? Come back to the bedroom." His usual denial routine. I looked directly into his eyes. "No, John. I'm going to sleep here. I'll see you in the morning." There was not a trace of agitation or anger in my voice—which surprised me as much as it did him. He left the room and I went to sleep, feeling more strength and well-being than I'd ever felt before.
The next morning, I sat down with John and told him, "John, I can no longer accept your behavior. I want a separation." At this, all the color drained from his face, leaving him as white as the snowy starched shirt he was wearing.
I continued, "I am moving out today. I need to take some time to be quiet, and to figure out who I am and what I want."
I think John was in shock, because all he did was stand up and say, "Okay," before heading out the door for work.
That day I moved into a friend's house, which was empty since she was away on vacation. For three weeks, I reveled in my alone-ness and the newfound pleasure of my own company. As always, I continued to attend my Al-Anon meetings. Even outside the meetings, the people in the group were my "life-support system." If I got shaky, I'd call one or the other of them and they would remind me of how far I'd come and share their own experiences, strength, and hope.
At the end of the three weeks, John called me. Surprisingly accommodating, he offered to move to a friend's apartment and let me move back into our house. I accepted happily, and we went about our separate ways for another few weeks. I had finally decided that it was time to find a job and an apartment when John called and said he needed to talk to me.
When we met, John asked me, "Molly, what is it that you want?"
Having talked about this on many occasions in the presence of my group, I was completely clear about my answer. I smiled at him and said, "John, I know I love you. I know I respect you. I don't want a divorce, but I am not willing to accept your behavior when you are drunk."
He was silent for a moment, and then said, "That's all I wanted to know." Then he stood up and said good-bye.
Three days later, he called. I will never forget his words, "Molly, I am an alcoholic, I'm in AA, and I want to come home."
That was over thirty years ago.
Since that day, John hasn't had a drink and I haven't missed a meeting. Now, at more than five decades of marriage and counting, John and I are enjoying our life together—our separate interests and pursuits, as well as our deeply satisfying shared time. I love and respect my husband and feel his love and respect for me. I am grateful every day for the people in my life who were there for me. When they helped me take my eyes off John and others, and to concentrate on me—my attitudes and actions—miracles happened. I was able to find self-worth and serenity and become the empowered person I am today.
Except when I am ill or out of town, I still attend a local Al-Anon gathering once a week. The unconditional love and effective support I get there is as compelling now as it has ever been. As an elder in the Al-Anon community, I try to spend time with the newcomers, especially the shy and tentative people. I am well aware of how much courage it takes to come through that door for the first time—and I do all I can to put them at ease. I know deeply the magic of these rooms and the power of a support group to help people find their way through life.
### The Wind beneath Your Wings
Though I've never attended a 12-step program myself, I know that, as Molly found, they are extraordinarily helpful for millions of people. Any group that reminds you of your soul's deepest truth when you forget is worth its weight in gold. I know this because I was a member of a women's support group for many years that was a godsend, keeping me on track on my search for true happiness.
In 1987 I took a life-changing seminar called Self-Empowerment Training, led by a wonderful therapist named Ali Najafi. At the end of the course, Ali recommended we form groups to keep the momentum of empowerment going in our lives. Five women—Holly, Jennifer, Sandy, Janice, and my dear friend and cowriter Carol—joined me in forming an extraordinary support group and quickly became my family. As the years passed, we welcomed new members Lane and Kami. Every week, we met at a different house—the host of the week provided treats—and we went around the circle twice, first talking about what we felt our "wins" were for the past week and then sharing our goals for the upcoming week. We asked for support for the changes we wanted to make and encouraged each other as much as we could. Everything said in our meetings was kept strictly confidential, and we were vigilant about giving each person an equal amount of time—though we'd bend the rules if someone had a crisis that needed more discussion. We went through many things together: marriages (including Holly's to my ex-boyfriend, the one who'd been the catalyst for my broken heart), births, and divorces. I saw how truly bonded we'd become when we had to face a shared tragedy: the sudden death of one of our members.
One cold snowy January afternoon, Sandy was killed in a car accident. Because she was single and had no other close family, it fell to us, her support group, to arrange her funeral. It turned out to be a celebration not only of a woman we all loved dearly, but also of the gratitude and love we felt for each other. When I moved away ten years later, I immediately found another women's group to join. This type of support has been so important in my life that I seek it out whenever I can.
The importance and power of support is the same all over the world. Zainab Salbi, whom you met in Chapter 3, told me a beautiful story that illustrates this. As part of her work with war survivors in Bosnia and Herzegovina, Zainab once met with a group of women in a very small village. One woman who was experiencing a lot of sadness confessed that she was being beaten regularly by her husband. She didn't want to leave her husband; she just wanted him to stop hitting her. The worst part was that she blamed herself for his abusive behavior. The other women in the group hugged her, cried with her, and admitted they were all going through the same thing. The group decided to take action. They told the woman to have her son alert them the next time her husband started to hit her.
The very next day, her husband started beating her. Her son went to the window, as instructed, and yelled out to all the other women in the group, "Help! My mother needs help!" Each woman dropped what she was doing. Dressed in their headscarves and sneakers, they all marched over, surrounded the house, and began shouting to the husband, "If you are going to hit her, you have to hit all of us! Are you going to beat us up too?" Astounded and ashamed, the man stopped. The other men, peeking from their windows, saw how the women banded together, and after that day, the incidence of domestic violence in that village dropped dramatically.
### Creating Your Happiness Dream Team
Surrounding yourself with support can take many forms. Nancy Fursetzer, a member of the Happy 100, told me about her unique support group, which includes Albert Einstein, Helen Keller, Mother Teresa, Gandhi, Goethe, Abraham Lincoln, Lao-tzu, and many more great men and women of the past and present. Nancy has collected their quotes and posted them around her home and office, some displayed in frames, others written on notes stuck on mirrors, next to her computer, her phone, and the kitchen sink. Everywhere she turns, Nancy has inspiring reminders 24/7 from her personal Dream Team, proving that getting support doesn't have to be limited by time and space!
The value of a support group is that you create a TEAM, where Together Everyone Achieves More. This is based on the ancient principle that when two or more people gather for a common purpose or goal, they magnify their efforts and can more quickly and easily achieve the outcomes they desire. You can use the following exercise to hold your own support group meetings.
Exercise
Support Group Meeting Format
1. Form your group by finding five to seven people you trust and respect. Determine your meeting times (once or twice a month is recommended). Alternate who will lead each meeting, and decide who will keep track of the time during the meeting.
2. Begin the meeting with the leader sharing an invocation or an inspiring quote.
3. Each person takes three to four minutes to share with the group his or her "wins" and accomplishments since the last meeting.
4. Each person takes ten to fifteen minutes to share with the group his or her goals or intentions or to ask for support from the rest of the group.
5. Each person states an action he or she will take before the next meeting to move toward his or her intentions (for example, you will exercise three times a week for thirty minutes).
6. Let the leader close the meeting with a statement of gratitude and positive intention.
It's important that everyone maintain the following guidelines: listen without interruption, avoid victim behaviors (blame/shame/complain), offer suggestions only when solicited, and keep everything that's shared completely confidential.
Happiness Habit for Relationships #3
See the World as Your Family
Human beings are more alike than we are un-alike. . . . Try to spread your giving of yourself . . . to people who may not even look like you. You belong to everybody, and everybody belongs to you.
—Maya Angelou, author and poet
What I noticed in my interviews with the Happy 100 is that they see the whole world as their family. Their love, empathy, compassion, and caring, which are natural outgrowths of being Happy for No Reason, aren't limited to their relatives and friends, but extend to all of humanity. Nationality, race, religion—none of these creates a boundary. People who are Happy for No Reason see that people everywhere are just like them, and that we all want the same things: love and happiness. Because they always feel they're part of a larger family, happy people make it a habit to give whatever they can, wherever they are.
The Happy 100 believe that their happiness is one of the greatest gifts they can give to the world. In my interview with Liz Gilbert, whose story is in Chapter 7, she told me that after her book Eat Pray Love was published, she was often asked by reporters, "Don't you think it was a little selfish to travel around the world looking for yourself?" She always answered, "You know what? I actually think it would have been a little selfish to spend the rest of my life in narcissistic, depressive, anxious misery. That person adds nothing to society, adds nothing to any room that she enters, adds nothing to the people whom she touches. The best community service I can possibly offer the world is to stay healthy and sane."
During my search for the Happy 100, a business colleague told me about his friend, Happy Oasis. When I heard her name I was sure it would be an interesting interview. And it was. Happy truly lives up to her name. In the extraordinary story that follows, Happy describes an experience she had over twenty years ago while traveling in Asia that taught her what it truly means to see the world as your family and help others by sharing your own happiness.
Happy's Story
The Smiling Man
I've always been a free spirit. When I finished high school in 1983, instead of attending the Ivy League university my parents expected me to go to, I took off for Australia. I spent the next few years traveling and earning money, as I literally worked my way across Australia and Southeast Asia, pursuing my own course of study in anthropology.
When I left the United States, I was a naïve, sheltered eighteen-year-old. I considered myself a happy person, but looking back, it was more a case of ignorance being bliss. Until my travels in the Third World, I simply hadn't been aware of the extreme suffering of humanity.
I went to Bangladesh hoping to spend time with a group of tribal people who lived there. When I arrived in the capital city, Dhaka, I discovered that besides being monsoon season, there was a widespread famine that was causing a lot of illness and death all over the country.
One morning, I boarded a bus in Dhaka, heading for the more remote areas of the country. Looking around at the other passengers, I saw that I was the only Westerner on the bus. In addition, I was a blond, blue-eyed and very young-looking woman in a Muslim country where I had scarcely seen any women at all. I'd tried to dress in the Muslim style, wrapping a black sarong around my head and covering my arms and legs as much as I could, but I knew I was very conspicuous, and it made me a little uncomfortable.
We left the city and soon were driving through farmland and small villages. It had been raining heavily for days, and at one point, the bus had to pull off the rapidly flooding road onto a field to avoid disaster. As I watched, the road washed away and the small village nearby began to flood. The land the bus was parked on was quickly surrounded by water, and I saw crowds of people making their way toward our football-field-size island of higher ground. Soon there were hundreds of extremely skinny people, most of them children, barefoot and dressed in rags, lying on the ground outside the bus. I realized to my horror that they were dying; the dysentery and lack of food were taking their toll.
Soon I was the only one who had not disembarked to see what was going on or perhaps to help. I sat on the bus by myself, wondering what I could do to help. My first naïve impulse was to cash the $2,000 in traveler's checks I had in my money belt—meant to finance the next year or two of my traveling—and buy food for these people. But I quickly realized that there was no way to get to a bank.
Then I thought, I can use the $150 in Bangladeshi currency I have to buy everyone a meal from the grocery store. But looking at the flooded shanty village and rice paddies surrounding our little island, the reality dawned: there was no food to be bought anywhere.
My next desperate thought was, Surely, the Red Cross will show up soon. They have to. But the rain kept pouring down, and within an hour I had to accept that the Red Cross wasn't coming. Judging from what I had seen of the country so far, I doubted they ever came.
I started to cry, whimpering to myself. I wasn't concerned about my own survival; I was carrying a small knapsack with water, a little food, and a change of clothing. After my year traveling, I was used to living a very rugged life. But I felt so helpless, believing there was nothing I could do.
I heard something and looked up to see a man wearing only a loincloth stepping on to the bus. He was frail, skinny, and although he was probably only thirty-five, he looked very old. He hobbled toward me, gripping the tops of the wooden benches of the bus for support. When he reached me, he stood gazing at me, then stretched out his hand and touched what was showing of my hair under my head covering. Normally, I didn't let strange men touch my hair, but I was distracted by his eyes. They looked like the eyes of a ghost or an angel—of someone who had already died. Then, as he pulled his hand away, I noticed his fingers, or what was left of them. They were scaly and stubby, half the length of ordinary fingers. I froze: The man was a leper. Before I could react, he turned and hobbled off the bus, disappearing into the crowd outside. Shaken by the encounter, I sat in silence, feeling even more helpless and out of my element than before.
I was still upset when, a few minutes later, another man came up beside the bus and stood outside my window, gazing in at me. He looked like everybody else I'd met in Bangladesh—scantily clad, very skinny, barefoot—except he was smiling broadly.
It suddenly seemed horribly inappropriate to me that this man was smiling under these circumstances. Through my tears, I snapped at him, "How can you smile in such a situation?"
To my surprise, he answered me in perfect Queen's English, "A smile is all I have to give, madam."
I was shocked at the power of these nine simple words. They turned my world—and all my ideas about helping others—upside down. But before I could respond, he motioned to me, saying, "Come. Come with me."
I got off the bus and went with him into the rain. We spent the next ten hours walking through the field, singing to one person after the other as they lay dying. Kneeling beside each person, the Smiling Man, as I called him, sang beautiful, soul-stirring Muslim chants while I sang Christian songs I had learned at summer camp.
Our singing—well, mostly the Smiling Man's, as he was so much better at this than I was—calmed people and seemed to give them peace. Sometimes he would stroke their foreheads; at other times he would touch their shoulders and encourage me to do the same. I was shy about this when it was a man in front of us, since in Muslim countries a woman isn't supposed to touch men she is not related to, but under the circumstances it seemed okay. We would do this until there was a sign the person felt better, such as the slight curve of a smile, or some response that showed relaxation. Then we'd leave them to journey into the other realm called death by themselves, or stay beside them until they were gone. As the hours ticked by, we went around and around that field, singing to dozens and dozens of people.
At one point, as we picked our careful way through the maze of bodies, I recognized the man from the bus—the leper who had touched my hair—lying motionless on the ground. I stopped and took a closer look. With his eyes closed, he seemed to blend in with the earth beneath him. Then, with a jolt, I realized he was dead. I felt a pang of sorrow; I wished we had found him sooner. Sending up a silent prayer, I turned away and hurried to catch up with the Smiling Man, who was already kneeling next to a child and beginning to sing.
The Smiling Man and I talked on and off throughout the day. There were times when I felt overwhelmed by the scene around me and began to cry. Mostly he ignored my tears, but at one point he said to me, "We have a reason to cry and we aren't crying. You have nothing to cry about. Why are you crying?" It was said kindly, but with a touch of fatherly sternness. It was his way of asking me to take on the role of leadership, as if saying, "Gather yourself. Let's do what we can."
The rain finally subsided. The driver called out and people began to board the bus. I said good-bye to the Smiling Man and went back to my seat. As I sat there, waiting to leave, I knew I'd probably never see him again. Still, he'd become my hero. I admired his wisdom and what I can only describe as gumption. Without a penny, without a dollar—without any material item—he'd eased the suffering of hundreds of people by offering his love and joy. I made a silent vow to be like that smiling man.
In the years since then, I have made it a priority in my life to be as happy as I can, to share that happiness with as many people as possible, and to treat everyone I meet as family. Even here in America, at the grocery store, at the bank, wherever you go, you often don't know what's going on with the people you see. Somebody could be terribly depressed, and by just smiling, opening up, and reaching out—giving of myself the way the Smiling Man did—I've found I really can offer some relief and light. That's the reason I changed my name to Happy Oasis: to be an oasis of happiness for everyone. And the wonderful side effect is that I get to carry that oasis inside myself wherever I go. It's the basis of the high level of happiness I enjoy today.
The Smiling Man showed me that giving love to others isn't complicated or difficult. I know firsthand that when a smile is all you have to give—it can be enough.
### The Power of Connection
Smiles are universally recognized signs of friendliness and loving intention. Even a brief smile can have an enormous impact. I heard Caroline Myss tell a story from her book Invisible Acts of Power that illustrates this perfectly. The story is about a young man who'd become so despondent that he'd decided to return to his apartment to commit suicide. As he stood on a street corner waiting for a car to pass, the woman behind the wheel looked directly at him and flashed him a huge smile. The smile was so warm and caring that it convinced him there was still goodness in the world, and he turned away from ending his life. No matter who or where you are, a sincere smile bridges even the largest age and cultural differences and creates a sense of connection.
Robert Biswas-Diener, often referred to as the Indiana Jones of positive psychology because he's traveled to remote regions around the world studying happiness, has found that connection is a powerful influence on happiness, among even the poorest citizens of the world. Biswas-Diener, in collaboration with his father, Edward Diener, examined the life satisfaction of homeless people and slum dwellers in Calcutta. The results were fascinating. Good social relationships and healthy bonds with family made residents of the Calcutta slums more resilient and better able to withstand the negative effects of dire poverty.
It was Robert who introduced me to Happy 100 member Roko Belic, a young documentary filmmaker whose first documentary, Genghis Blues, made with his brother, Adrian, was nominated for an Oscar in 2000 and won many awards. Roko's current project is a film on happiness called Happiness Revolution that's taken him to many countries including Brazil, India, Namibia, and Japan, to document how people around the world experience—or don't experience—happiness. Based on what he's found on his travels, Roko also believes that a sense of belongingness is vital for sustaining well-being and happiness.
Roko told me in our interview that he'd gone to Japan because he'd heard from many sources that, though a materially wealthy country, Japan wasn't doing so well emotionally. Riding the subway in Tokyo, he was shocked to see that 80 percent of the commuters were actually asleep or trying to sleep. Because work is the highest priority in most people's lives there, they tend to work outrageously long hours, sometimes as many as twenty hours a day. This dedication to productivity has taken its toll, not only in sleep deprivation, but in the amount of connection people feel with each other.
But Roko had also heard that some of the world's oldest people live in Japan, specifically Okinawa. Familiar with the research showing that happy people live longer, he and his crew left Tokyo and went to a small village on the island of Okinawa to see if the Okinawans' longevity had anything to do with being happy.
There, Roko found a small pocket of happiness. Many people he met were in their nineties, and although they spent their days farming in the hot sun and living what we Westerners would consider a rudimentary lifestyle, the joy in their lives was palpable. What made this even more striking is that a large number of the residents were older women who'd lost their husbands and children when Okinawa was razed during World War II. But far from being bitter or sad about their losses, these women radiated good humor and happiness.
The key was the strong sense of connectedness that seemed to include all the generations. Every Friday night, the villagers got together and had a dance. While a band played, everyone, from the young children to the old ladies, danced together to traditional music. Roko said that everyone had a good time, even the teenagers whose American counterparts would be too "cool" for gatherings like these. The Okinawans' high level of joy shows how powerful a sense of community is for supporting individual happiness.
What if that sense of community included the whole world? Imagine feeling as comfortable with anyone, anywhere, as you do hanging out with your closest friends or family. That's really what it means to see the world as your family. Palestinian American Naomi Shihab Nye, celebrated poet and essayist, shares an experience of this she had in the Albuquerque airport terminal:
After learning my flight had been detained four hours, I heard the announcement: "If anyone in the vicinity of Gate 4-A understands any Arabic, please come to the gate immediately."
Well, one pauses these days. But Gate 4-A was my own gate, so I went there. An older woman in full traditional Palestinian dress, just like my grandmother wore, was crumpled on the floor, wailing loudly.
"Please help," said the flight service person. "Talk to her. What is her problem? We told her the flight was going to be four hours late, and she did this!"
I stooped to put my arm around the woman and spoke to her haltingly: "Shu dow-a, shu-biduck habibti, stani stani schway, min fadlick, sho bit se-wee?"
The minute she heard my words, however poorly used, she stopped crying. It turned out she thought our flight had been canceled entirely, and she needed to be in El Paso for a major medical treatment the following day. I said, "No, no, we're fine, you'll get there, just late. Who is picking you up? Let's call him."
We called her son and I spoke with him in English. I told him I would stay with his mother till we got on the plane and, once aboard, would ride next to her. Then, while we were waiting to board, we called her other sons. Then we called my dad, and he and she spoke for a while in Arabic. They found out, of course, they had ten shared friends.
Then I thought—just for the heck of it—why not call some Palestinian poets I know and let them chat with her? This all took up about two hours. She was laughing a lot by then, telling about her life, patting my knee, answering questions. She had pulled a sack of homemade mamool cookies—little powdered sugar crumbly mounds stuffed with dates and nuts—out of her bag and was offering them to all the women at the gate.
To my amazement, not a single woman declined one. It was like a sacrament. The traveler from Argentina, the mom from California, the lovely woman from Laredo—we were all covered with the same powdered sugar. And smiling. There is no better cookie.
And then the airline broke out free beverages from huge coolers, and two little girls from our flight, one African American, one Mexican American, ran around serving us all apple juice and lemonade. They were covered with powdered sugar too.
I noticed that my new best friend—by now we were holding hands—had a potted plant poking out of her bag, some medicinal thing with green furry leaves. Such an old country traveling tradition. Always carry a plant. Always stay rooted to somewhere.
And I looked around that gate of late and weary ones, and thought, This is the world I want to live in. The shared world. Not a single person in this gate—once the crying of confusion stopped—seemed apprehensive about any other person. They took the cookies. I wanted to hug all those other women too.
This can still happen anywhere. Not everything is lost.
The more we experience even total strangers as our family, the happier we'll all be. The following exercise will help you get in the habit of seeing the whole world as your family.
Exercise
The World Is My Family
Spend a day relating to each person you meet as if he or she were your own mother, or your child, or your dearest relative. Do this at work, when you're shopping, participating in a group, doing an errand. Make people feel important, loved, valued, respected, and appreciated, and do it as an active choice, with intention to put something wonderful into the world. At the end of the day, notice how you feel. It may surprise you that after a day of caring for everyone you meet, you reap the benefit of feeling as happy and peaceful as you intended for others to feel.
Summary and Happiness Action Steps
When you cultivate nourishing relationships, you use emotional contagion to your advantage, surround yourself with support, and see the world as your family. This creates a beautiful garden around your Home for Happiness. Use the following action steps to practice the Happiness Habits for Relationships:
1. Use your inner GPS to identify your nourishing and toxic relationships, the "roses and weeds" in your garden.
2. Use the emotional immunity boosters when you need to interact with toxic people.
3. Do the Appreciation Practice daily for one week.
4. If appropriate, check out a 12-step program that can help you with your situation.
5. Create a support group and set up a schedule of regular meetings.
6. See the world as your family by focusing on the ways we are more alike than different and radiating lovingkindness to everyone you meet.
# PART III
# Happy for No Reason Ever After
Our business is to be happy.
—His Holiness the Dalai Lama
## 10
## The Happy for No Reason Plan for Life
It is only possible to live happily ever after on a day-to-day basis.
—Margaret Bonnano, writer
By now you know that "happily ever after" isn't just for fairy tales or for only the lucky few. From reading the compelling research and the experiences of truly happy people—perhaps even trying some of the exercises—you know that when you practice the 21 Happiness Habits, you too can join the Happy 100 and experience inner peace and well-being as the backdrop for everything else in your life.
Here's a summary of the seven steps you've just taken to build your Home for Happiness, along with the habits that go along with them.
Happiness Habits
The Foundation—Take Ownership of Your Happiness
1. Focus on Solutions: To become more empowered, build on what's already working to improve any situation in your life.
2. Look for the Lesson and the Gift: Instead of blaming others or making excuses, find the lesson and the blessing in each circumstance.
3. Make Peace with Yourself: Lighten your load by accepting the past and moving forward.
The Pillar of the Mind—Don't Believe Everything You Think
4. Question Your Thoughts: Examine your beliefs to determine if your mind is telling you the truth.
5. Go Beyond the Mind and Let Go: Free yourself from your negative thoughts and feelings.
6. Incline Your Mind toward Joy: Lean into the thoughts that support your happiness.
The Pillar of the Heart—Let Love Lead
7. Focus on Gratitude: Put your attention on what you appreciate to expand the energy of your heart.
8. Practice Forgiveness: Release your resentment and anger toward others to free your heart.
9. Spread Lovingkindness: Practice "beaming" love and good wishes to everyone you encounter.
The Pillar of the Body—Make Your Cells Happy
10. Nourish Your Body: Balance your brain and body chemistry through optimal nutrition and supplementation.
11. Energize Your Body: Use movement, breath, and proper rest to increase the vital life force in your body.
12. Tune In to Your Body's Wisdom: Love and honor your body and listen to its needs.
The Pillar of the Soul—Plug Yourself In to Spirit
13. Invite Connection with Your Higher Power: Find silence through prayer, meditation, or time in nature to experience your relationship to your Higher Power.
14. Listen to Your Inner Voice: Make connection with your soul's wisdom to guide you in life.
15. Trust Life's Unfolding: Open to grace and put yourself in the flow of life.
The Roof—Live a Life Inspired by Purpose
16. Find Your Passion: Discover what truly matters to you and align yourself with what makes your heart sing.
17. Follow the Inspiration of the Moment: Focus on what you want and allow yourself to be led to how it will unfold.
18. Contribute to Something Greater Than Yourself: Respond to your inner calling to serve others in ways great and small.
The Garden—Cultivate Nourishing Relationships
19. Tend to Your Relationships: Appreciate the people in your life and use emotional contagion to enhance your happiness.
20. Surround Yourself with Support: Create a support system that helps you stay focused on living your best life.
21. See the World as Your Family: Feel a sense of love and connection to all of humanity.
To download a free Happiness Habits miniposter, go to www.HappyforNoReason.com/bookgifts.
Making the Happiness Habits second nature requires practice. It takes time and repetition for your brain to create the neural pathways that will support a higher level of happiness. Here are some tips to help you on the way:
1. Remember the Guiding Three.
2. Take baby steps.
3. Set up a support system.
### Remember the Guiding Three
You can employ the first principle of the Guiding Three—Whatever expands you makes you happier—by using your inner GPS to inform your choices in life every day.
Engage the second principle—The universe is out to support you—by asking yourself whenever necessary, "If there were a higher purpose to what's happening, what would it be?"
The best way to use the third principle—Whatever you appreciate, appreciates—is my Secret Formula that you learned in Chapter 2: intention, attention, and no tension.
Intention: Keep your intention to be Happy for No Reason clear and lively in your awareness. Put your written Happy for No Reason intention from Chapter 2 in a place where you'll see it on a regular basis.
Attention: Keep the momentum going by regularly putting your attention on developing the Happiness Habits and doing the action steps at the end of each chapter designed to help you practice them.
No Tension: The state of peace and well-being you want is already there inside you. Relax, let go, and trust that it's all unfolding.
The process is like planting a flower: you plant it (your intention), you water and fertilize it (your attention), and then you relax (no tension), knowing that over time you'll begin seeing the blooms. You can let go—the process is happening.
### Take Baby Steps: Overcoming Your Resistance to Change
To make the quickest progress, you don't have to take huge leaps. You just have to take baby steps—and keep on taking them. In Japan, they call this approach kaizen, which translates literally as "continual improvement." Using kaizen, great and lasting success is achieved through small, consistent steps. It turns out that slow and steady is the best way to overcome your resistance to change.
Most of us resist change—even when it's good for us! It's the reason that so many people have exercise bikes stuck in their garages, rarely used gym memberships, and that case of low-carb, low-calorie, low-fat canned diet shake gathering dust in the pantry. This resistance is based in our physiology. Our brain often views change with suspicion, if not downright fear. To overcome that resistance, keep the changes you make small and gradual enough to fly under your brain's "fear radar." When you set goals that are easily achievable, your brain's fear response isn't activated.
What's more, as you continue to take small steps, the brain begins creating new neural pathways that support each new Happiness Habit. Soon your new habits become hardwired and you find yourself easily and automatically practicing your desired behaviors.
### Set Up a Support System: Happiness Loves Company
To make this process even more effortless and enjoyable, you can invite other people to join you. Enlist the support of a coach, mentor, friend, or group of friends. Practicing the Happiness Habits with others helps us own them on a deeper level.
The psychiatrist Dr. William Glasser, the author of many books, including Every Student Can Succeed, has spent years studying how people learn. Expanding on the theories of the twentieth-century educator Edgar Dale, he says:
We Learn . . .
10% of what we read
20% of what we hear
30% of what we see
50% of what we see and hear
70% of what we discuss with others
80% of what we experience personally
95% of what we teach to someone else
This means you can increase the impact of reading Happy for No Reason sevenfold by simply discussing the information you've learned in this book with other people. Here are some great ways to get support from others:
1. Find a Happy for No Reason Buddy: Like an exercise partner, another person who's practicing the Happiness Habits can help keep you on track. When someone else is cheering you on and depending on you for help to stay focused, it makes your goal of being Happy for No Reason a priority in your life. Besides, having a buddy always makes everything more fun.
2. Form a Happy for No Reason Support Group: Multiply the buddy effect by starting a Happiness Habit support group, where you gather regularly—face to face, online, or by phone—to advise, listen, and encourage each other as you all raise your happiness set-points. You can spend each meeting focusing on a different habit. Your group will add their attention to your Happy for No Reason intention, increasing its power exponentially. My prediction is that you'll end each meeting feeling uplifted and blessed to find fellow travelers who share your goal of increasing the level of joy in the world. (Visit www.happyfornoreason.com to locate a support group near you.)
3. Consult with a Happy for No Reason Life Coach: Many people are using life coaches and finding them very helpful in supporting the achievement of their goals. Contact a Happy for No Reason life coach who is specially trained to guide you in raising your happiness set-point. (Visit www.happyfornoreason.com to find out more about Happy for No Reason life coaches.)
4. Seek Out Happy for No Reason Mentors: I consider the Happy 100 my Happy for No Reason mentors and know how incredibly valuable their influence is in my life. Being around people who already experience a deep inner state of happiness will help you build your Home for Happiness more quickly and easily. In some traditions, "being in the company of the wise" is considered one of the most powerful steps you can take to enhance happiness in life.
### The World Is as You Are
How the world appears to you is determined by the color of the glasses you look through. When you're happy, you see happiness all around you. When you're unhappy, you find unhappiness everywhere. I love the following fable which illustrates this beautifully.
Long ago in a faraway village, there was a place known as The House of a Thousand Mirrors. A happy little dog learned of this place and decided to visit. When he arrived, he bounced happily up the stairs to the doorway of the house. He looked through the doorway with his ears lifted high and his tail wagging as fast as it could. To his great surprise, he found himself staring at a thousand other happy little dogs with their tails wagging just as fast as his. He smiled a great smile, and was answered with a thousand great smiles just as warm and friendly. As he left the house, he thought to himself, "This is a wonderful place. I will come back and visit it often."
In this same village, another little dog, who was not happy like the first one, decided to visit the house. He slowly climbed the stairs and hung his head low as he looked into the door. When he saw the thousand unfriendly-looking dogs staring back at him, he growled at them and was horrified to see a thousand little dogs growling back at him. As he left, he thought to himself, "This is a horrible place, and I will never come back here again."
Everyone in the world is your mirror. When you're Happy for No Reason, the world reflects that happiness back to you.
### Be Part of the Happiness Revolution
As you build your Home for Happiness, it won't be hard to find others to join you—there's a happiness revolution going on. Happiness is showing up everywhere, in magazines, newspapers, books, and on TV. There are even ad campaigns and huge billboards touting phrases like "Dare to Be Happy." More and more people want to discover how to be truly happy in their lives now. I read recently that happiness today is what self-esteem was for the 1990s and self-actualization was for the 1970s. This is good news because the more people who have their attention on happiness, the more momentum is created for everyone.
The entire world doesn't have to be happy for there to be a fundamental shift in the happiness level on the planet. There have been numerous studies conducted that indicate that when even just 1 percent of a population experiences greater peace, well-being, and coherence by meditating together regularly, it has an effect on the entire community, lowering the crime rate, number of accidents, acts of violence, and incidents of disease.
Learning to be Happy for No Reason puts you on the leading edge of the happiness revolution and is a powerful way to contribute to the world. You become like a beacon, lighting up your own life and the lives of everyone around you. This idea is beautifully expressed by an ancient Chinese proverb that still holds true today:
When there is light in the soul, there is beauty in the person.
When there is beauty in the person, there is harmony in the house.
When there is harmony in the house, there is order in the nation.
When there is order in the nation, there is peace in the world.
### Happy for No Reason—Our Personal Fulfillment
In the introduction, I mentioned how blessed both Carol and I feel to have been able to write this book. Spending time interviewing the Happy 100 and focusing all of our attention on the subject of happiness has made us both happier, healthier, and kinder people (at least according to our spouses!). The perspective of Happy for No Reason permeates our interactions with people, our perception of the world around us—everything! I'm amazed at the power of those four little words, Happy for No Reason, to uplift and remind people of the happiness they already have inside.
This was confirmed for me recently during a conversation I had when I was ordering a lamp over the phone. While the woman processed my information, we started talking about happiness. When I mentioned the title of my book, the woman became very animated, saying, "Happy for No Reason! I love that! Sometimes I feel happy and I don't know why, but I've never known what to call it. Happy for No Reason is the perfect description." The next day I received a surprise email from her that included this paragraph:
At the end of the day as I drove home after speaking with you, I kept hearing "Happy for No Reason" in my head. I couldn't help but smile—and found myself smiling for miles.
The ripple effect of Happy for No Reason continued: reading her email made me smile too.
For Carol, this idea of being happy from the inside had been percolating for many years. When the title Happy for No Reason came to me and I told Carol about it, she instantly agreed that it was the perfect way to describe that state beyond ordinary happiness.
The next day she called me, very excited, and said that she had been going through some old journals of hers and had come across the lyrics to a song she'd written in 1984. She had completely forgotten about the song, but when she read it again that day, it had given her goose bumps. When she told me the lyrics, I too was blown away by the synchronicity:
Driving my car, yellow lines flashing by,
New York winter and a New York sky
Then for no reason special that I can see,
Suddenly I'm happy as a person can be.
Hot soapy water, doing the dishes,
Staring at the clouds and making wishes.
Then for no reason special that I can see,
Suddenly I'm happy as a person can be.
Joy inside me blossoms like a rose,
Something sweet and warm flows.
Then for no reason special that I can see,
Suddenly I'm happy as a person can be.
Cue the Twilight Zone theme song: Doo-doo-DOO-doo. Doo-doo-DOO-doo. We laughed, but both of us knew it was no accident we were doing this book together!
Working on Happy for No Reason has totally transformed Carol's life. She says that although she considered herself pretty happy before, the knowledge of Happy for No Reason has given her a level of everyday peace and joy she'd had only glimpses of earlier. She feels a new confidence knowing that no matter what happens to her, she has a great toolbox full of Happiness Habits that can help her find her way back to the happiness she has inside.
For me, writing Happy for No Reason has been the fulfillment of my childhood longing: to find deep happiness and to share it with others. Building my Home for Happiness has erased the angst and emptiness that weighed me down for so many years. I feel extraordinary gratitude for the steadily growing experience of Happy for No Reason in my life.
In his last days, my dad, my original Happy for No Reason role model and the inspiration for this book, gave me one final gift.
Celebrating his ninety-first birthday at a dinner at home with the entire family, my father ate what turned out to be his last meal. It was also the last time he was up and about before he passed away, peacefully, a week later. Though we didn't realize what he was doing at the time, that evening he made a very specific and deliberate point to take each one of his children—my sister, my brother, and me—one by one over to the Tree of Life needlepoint, his final masterpiece, which was hanging on the living room wall.
As he smiled lovingly and pointed at the framed needlepoint, I knew he was trying to convey something important. He could barely talk, but now I believe that with this final gesture he was trying to communicate to me, "You're the next generation in the Tree of Life, and I want you to pass on the message of my life. I have lived a deeply happy life. Please live that happiness yourself and help others to live it too."
This book is my way of passing on my dad's message.
My deepest desire is that each one of us becomes full of light, love, and happiness in our own life and through that we create a world of peace.
May we all be Happy for No Reason!
## RECOMMENDED RESOURCES
### Programs to Help You Be Happy for No Reason . . . Now!
Now that you know it's possible to be Happy for No Reason, you've set your sights on experiencing it more and more in your life—and the sooner the better. To help speed your progress, I created the following programs that provide greater support than you can get from a book alone.
#### Happy for No Reason Seminar
In the Happy for No Reason Seminar I will personally guide you in applying the principles, habits, and tools in this book to immediately transform your life and lay the foundation for lasting happiness. You will put into practice the seven steps of building your inner Home for Happiness and raise your happiness set-point so that peace and well-being are established as your everyday reality.
#### Happy for No Reason Coaching
However grounded we are in the principles we've learned, life has a way of throwing curveballs that continually challenge us. My coaching program is designed to assist you in integrating the Happiness Habits into your everyday life to make your happiness truly unshakable. Coaching will help you develop new strategies for thinking, feeling, and acting that support your happiness. It is the fastest and surest way to overcome old habits and begin living the life of your dreams.
#### Happy for No Reason Paraliminal CD
The Happy for No Reason Paraliminal CD is like an automatic, perpetual happiness generator. Based on the breakthrough technologies of neurolinguistic programming and whole-brain learning, Paraliminal CDs help you acquire new behaviors and more positive feelings in minutes. I cocreated the CD with my friend Paul R. Scheele, the world leader in this technology and truly the master of the mental makeover. Just put on the stereo headphones and sit back and listen to the words and music, and you'll be guided from the inside out to experience more happiness in your life.
#### Happy for No Reason Interview Series
The knowledge in this book won't change your life until you make it an integral part of your daily reality. My in-depth interviews with the Happy 100 and other happiness experts will show you these concepts, habits, and tools in action and bring them vividly to life. Seeing them applied in the context of people's varied real-life experiences will help you integrate them and inspire you with a vision of possibilities.
#### Happy for No Reason CD Course
I've created a CD course—a personal learning program—that you can use in your home or office. As you listen, you will go step-by-step through life-changing processes that will shift your awareness, change your habits, and lead you to a richly happy life.
#### Happy for No Reason Keynote Presentations
In my twenty years of delivering keynote addresses to countless corporations, associations, and professional and nonprofit organizations, I have learned that a keynote presentation can transform and galvanize individuals and organizations with measurable and often startlingly powerful results. It is my joy to offer my services in the inspiration and transformation of your audience or organization. Where individuals are happy and productive, any level of success is possible.
For more information on these and other programs, please visit
www.HappyForNoReason.com/Programs.
### General Happiness Books
The Art of Happiness, by His Holiness the Dalai Lama (New English Library, 1999).
Authentic Happiness: Using the New Positive Psychology to Realize Your Potential for Lasting Fulfillment, by Martin Seligman (Free Press, 2004).
Feel Happy Now!, by Michael Neill (Hay House, 2008).
Happier: Learn the Secrets to Daily Joy and Lasting Fulfillment, by Tal Ben Shahar (McGraw-Hill, 2007).
Happiness: Developing Life's Most Important Skill, by Matthieu Ricard (Little, Brown, 2006).
Happiness Makeover: How to Teach Yourself to Be Happy and Enjoy Every Day, by M. J. Ryan (Broadway, 2005).
How We Choose to Be Happy: The 9 Choices of Extremely Happy People—Their Secrets, Their Stories, by Rick Foster and Greg Hicks (Perigee Trade, 2004).
The Joy Diet: 10 Daily Practices for a Happier Life, by Martha Beck (Crown, 2003).
Positivity Psychology Coaching: Putting the Science of Happiness to Work for Your Clients, by Robert Biswas-Diener and Ben Dean (Wiley, 2007).
Stumbling on Happiness, by Dan Gilbert (Vintage, 2007).
What Happy People Know: How the New Science of Happiness Can Change Your Life for the Better, by Dan Baker and Cameron Stauth (St. Martin's Griffin, 2004).
### Building Your Home for Happiness Books
#### Chapter 3 (Personal Power)
The Game of Life (Hay House Classics), by Florence Scovel Shinn (Hay House, 2006).
How Did I Get Here? Finding Your Way to Renewed Hope and Happiness When Life and Love Take Unexpected Turns, by Barbara DeAngelis (St. Martin's Griffin, 2006).
The Power of Now: A Guide to Spiritual Enlightenment, by Eckhart Tolle (New World Library, 2004).
The Secret, by Rhonda Byrne (Atria Books/Beyond Words, 2006).
The Solutions Focus: Making Coaching and Change Simple, by Paul Z. Jackson and Mark McKergow (Nicholas Brealey Publishing, 2007).
#### Chapter 4 (Mind)
Biology of Belief: Unleashing the Power of Consciousness, Matter, and Miracles, by Bruce H. Lipton (Mountain of Love, 2005).
Loving What Is: Four Questions That Can Change Your Life, by Byron Katie and Stephen Mitchell (Three Rivers Press, 2003).
The Sedona Method: Your Key to Lasting Happiness, Success, Peace and Emotional Well-Being, by Hale Dwoskin (Sedona Press, 2003).
#### Chapter 5 (Heart)
Count Your Blessings: The Healing Power of Gratitude and Love, by John Demartini (Hay House, 2006).
Forgive for Good, by Frederic Luskin (HarperSanFrancisco, 2003).
The HeartMath Solution: The Institute of HeartMath's Revolutionary Program for Engaging the Power of the Heart's Intelligence, by Doc Lew Childre and Howard Martin (HarperSanFrancisco, 2000).
The Hidden Messages in Water, by Masaru Emoto and David A. Thayne (Atria, 2005).
#### Chapter 6 (Body)
The Diet Cure, by Julia Ross (Penguin, 2000).
Essential Cleansing for Perfect Health, by Brenda Watson (Renew Life Press, 2006).
Everything You Need to Know to Feel Go(o)d, by Candace Pert and Nancy Marriott (Hay House, 2006).
The Four-Day Win: End Your Diet War and Achieve Thinner Peace, by Martha Beck, PhD (Rodale, 2007).
Good Night: The Sleep Doctor's 4-Week Program to Better Sleep and Better Health, by Michael Breus, PhD (Dutton Adult, 2006).
Hormones, Health, and Happiness: A Natural Medical Formula for Rediscovering Youth with Bioidentical Hormones, by Steven F. Hotze, MD (Wellness Central, 2007).
Mars and Venus Diet and Exercise Solution: Create the Brain Chemistry of Health, Happiness, and Lasting Romance, by John Gray (St. Martin's Press, 2003).
The Mood Cure: The 4-Step Program to Take Charge of Your Emotions—Today, by Julia Ross (Penguin, 2003).
Natural Highs: Supplements, Nutrition, and Mind-Body Techniques to Help You Feel Good All the Time, by Hyla Cass, MD, and Patrick Holford (Avery, 2003).
Women's Bodies, Women's Wisdom: Creating Physical and Emotional Health and Healing, by Christiane Northrup, MD (Bantam, 2006).
#### Chapter 7 (Soul)
Awakening into Oneness: The Power of Blessing in the Evolution of Consciousness by Arjuna Ardagh (Sounds True, 2007).
How to Know God: The Soul's Journey into the Mystery of Mysteries, by Deepak Chopra (Three Rivers Press, 2001).
Inspiration: Your Ultimate Calling, by Wayne Dyer (Hay House, 2007).
No Self, No Problem, by Anan Thubten (Dharmata Press, 2006).
A Return to Love: Reflections on the Principles of "A Course in Miracles," by Marianne Williamson (Harper Paperbacks, 1996).
#### Chapter 8 (Purpose)
Beneath a Vedic Sun: Discover Your Life Purpose with Vedic Astrology, by William Levacy (Hay House, 2006)
The Better World Shopping Guide, by Ellis Jones (New Society Publishers, Canada, 2006).
Finding Your Own North Star: Claiming the Life You Were Meant to Live, by Martha Beck (Three Rivers Press, 2002).
The Passion Test: The Effortless Path to Discovering Your Destiny, by Janet Bray Attwood and Chris Attwood (Hudson Street Press, 2007).
The Power of Full Engagement: Managing Energy, Not Time, Is the Key to High Performance and Personal Renewal, by Jim Loehr and Tony Schwartz (Free Press, 2004).
Success Built to Last: Creating a Life That Matters, by Jerry Porras, Stewart Emery, and Mark Thompson (Plume, 2007).
The Success Principles: How to Get from Where You Are to Where You Want to Be, by Jack Canfield and Janet Switzer (Collins, 2006).
#### Chapter 9 (Relationships)
Getting the Love You Want: A Guide for Couples, by Harville Hendrix (Pocket Books, 2005).
Lasting Love: The 5 Secrets of Growing a Vital, Conscious Relationship, by Gay Hendricks and Kathlyn Hendricks (Rodale, 2004).
Men Are from Mars, Women Are from Venus: The Classic Guide to Understanding the Opposite Sex, by John Gray (Harper Paperbacks, 2004).
#### Chapter 10
One Small Step Can Change Your Life: The Kaizen Way, by Robert Maurer, PhD (Workman, 2004).
### Tools and Techniques
Agape International Spiritual Center—Rev. Dr. Michael Beckwith
www.agapelive.com
The Agape International Spiritual Center is a transdenominational spiritual community whose doors are open to all seekers in search of authentic spirituality, personal transformation, and selfless service to humankind.
AIM Program of Energetic Balancing (AIM)
www.aimprogram.com
Developed by Stephen Lewis, AIM uses sophisticated computers to assist people in removing energy blockages and in raising their happiness thermostat. The AIM Program is free to any person with autism or Down syndrome.
The Art of Living Foundation
www.artofliving.org
This international nonprofit educational and humanitarian organization offers workshops teaching meditation and breathing techniques that calm the mind, release stress, clear the body of toxins, and energize the whole system in minutes.
B.E.S.T.—Morter HealthSystem
www.morter.com
Morter HealthSystem developed and teaches the Bio Energetic Synchronization Technique—B.E.S.T.—a physical energy-balancing procedure to reestablish the full healing potential of the body using its natural healing abilities. B.E.S.T. helps balance the autonomic nervous system and supports true, vibrant health.
The Biocybernaut Institute—Dr. James Hardt
www.biocybernaut.com
The Biocybernaut Institute offers Advanced Brian Wave Neurofeedback training. People are gently guided to their own self-discovery through programs that train them how to open their hearts and do deep forgiveness work that will enhance their life experiences and alpha brain waves.
The Canfield Training Group—Jack Canfield
www.jackcanfield.com
The Canfield Training Group offers life-changing programs that focus on living The Success Principles, raising self-esteem, and optimizing peak performance. Jack Canfield, America's Success Coach, helps you get from where you are to where you want to be.
Center for Soulful Living—Bill Bauman
aboutcsl.com
The Center for Soulful Living is a worldwide community of people dedicated to living in soul-centered and grace-filled ways. Their primary intention is to help people connect with their own inner wisdom, power, and brilliance.
Cortical Field Re-Education
www.corticalfieldreeducation.com
Cortical Field Re-Education (CFR), developed by Harriet Goslins, is a revolutionary approach to improving health and well-being by enhancing communication between the brain and the body. It is an integrative program that encompasses not only physical healing, but the emotional, spiritual, and energetic levels that accompany healing.
The Demartini Method
www.drdemartini.com
Derived from quantum physics, The Demartini Method is a predetermined set of questions and actions that neutralizes emotional charges and brings balance to your mind and body. The process allows you to break through to new levels of inspiration, creativity, and performance.
Emotional Freedom Technique (EFT)
www.emofree.com
EFT uses gentle tapping on key meridian points on the body to release just about every emotional, health, and performance issue you can name. At www.emofree.com you can download a free manual that will teach you how to use this technique as well as watch a video demonstration of EFT.
Hanna Somatics
www.livingsomatics.com
Hanna Somatics is a program consisting of slow, gentle movements that improve the function of the nervous system. The benefits include a significant reduction of pain and an increase of ease and happiness in the body.
Healthy Wealthy nWise
www.healthywealthynwise.com
Healthy Wealthy nWise is one of the largest online transformational magazines in the world. Their Passion interview live teleconferences are free. Past interviews have included Stephen R. Covey, Wayne Dyer, Marci Shimoff, David Lynch, John Gray, Byron Katie, Paula Abdul, Neale Donald Walsch, Rhonda Byrne, and Willie Nelson
The Hendricks Institute—Gay and Kathlyn Hendricks
www.hendricks.com
The Hendricks Institute teaches core skills for conscious living. Its focus is on assisting people in opening to more creativity, love, and vitality through the power of conscious relationship and whole-person learning.
Holosync
www.centerpointe.com
Holosync is a sophisticated form of neuro-audio technology that easily and effortlessly produces the electrical brain wave patterns of deep meditation every time. It is a scientifically proven brain technology that gives you all the benefits of meditation.
The Institute of HeartMath
www.heartmath.com
The mission of HeartMath is to establish heart-based living and global coherence by inspiring people to connect with the intelligence and guidence of their own hearts. The HeartMath system is composed of research, programs, products, and technologies to improve health, well-being, and personal fulfillment.
Intentional Living Program—Dr. Sue Morter
www.drsuemorter.com
Dr. Morter's Intentional Living Program is a unique mind/body approach and method that empowers people to break through limiting blocks to live the life of their dreams. This experiential and educational program works on the level of repatterning the brain and body to achieve optimal functioning and success.
John Douillard's Life Spa
www.lifespa.com
Life Spa is a Boulder, Colorado, center for Ayurveda and Panchakarma, designed to rejuvenate, detoxify, and balance the deeper tissues of the body. Their programs transform the body at the cellular level.
Lefkoe Institute: Making Change Easier—Morty Lefkoe
www.lefkoeinstitute.com
The Lefkoe Institute helps individuals who want to make lasting changes in their behavior and/or emotions in a gentle yet effective way. Using the Lefkoe Method unwanted beliefs are literally unwired for good.
Lindwall Foundation—Freedom through Releasing
www.lindwallreleasing.org
The Lindwall Releasing Process is a powerful process for letting go of negative memories or limiting thoughts. The process involves allowing a past disturbance to surface, confronting the situation as needed, then releasing the negative impact and emotional content by speaking affirmations.
The Mood Cure Website—Julia Ross
www.moodcure.com
The Mood Cure is based on the pioneering work done at Julia's California clinic, Recovery Systems, since 1988. This comprehensive, natural approach uses targeted brain-fueling amino acids, combined with a high-protein, healthy fat, veggie-rich diet, to produce improved mood in days.
Nancy Fursetzer
www.Positician.com
Certified Positician® coach. Positicians look for the best in everything and teach others to do the same. Positician certification available.
Network for Grateful Living—Brother David Steindl-Rast
www.gratefulness.org
A Network for Grateful Living (ANG*L) is a nonprofit organization dedicated to gratefulness as the core inspiration for personal change, international cooperation, and sustainable activism in areas of universal concern.
Option Method—Lenora Boyle
www.changelimitingbeliefs.com
Lenora Boyle has been helping people to be happier since 1991 through her interactive seminars, private coaching practice, and teleclasses. She teaches seminars using the Option Method that specializes in going to the root of the problem: the limiting belief or past conditioning.
Paraliminals—Learning Strategies
www.learningstrategies.com/Paraliminal/Home.asp
Paraliminals are a neurolinguistic programming and whole-brain learning technology that increase your personal power by activating your "whole mind" with a precise blend of music and words. Each session is carefully scripted to give you life-changing results.
The Passion Test
www.thepassiontest.com
Janet and Chris Attwood's Passion Test is an incredibly simple yet profound tool that has been given to thousands of people all over the world. The Passion Test is the perfect way to align yourself with those things that you are most passionate about so that you can share your special gifts.
Patanjali Kundalini Yoga Care
www.kundalinicare.com
Patanjali Kundalini Yoga Care is a spiritual guidence service for sincere seekers based on traditional kundalini science. Provides assessments, individual recommendations for spiritual practice, and follow-up support and guidence.
Positive Psychology Services—Robert Biswas-Diener
www.intentionalhappiness.com
Positive Psychology Services offers a wide range of coaching, training, and speaking services based on the new science of positive psychology.
Psych-K
www.psych-k.com
Psych-K helps you rewrite the software of your mind by changing self-limiting beliefs into beliefs that support you. Psych-K develops whole-brain communication, which quickly and easily changes subconscious beliefs.
Releasing the Inner Magician (RIM)—Dr. Deborah Sandella
www.innermagician.com
The RIM method is an advanced mind-body method for releasing the past and rewiring sabotaging thoughts and feelings. RIM helps you tap your Inner Magician, the divine intuitive power within, that dissolves pain to reveal the natural strength and passion of your spirit.
The Sedona Method—Hale Dwoskin
www.sedona.com
The Sedona Method is a unique program for making lasting, positive changes in your life. The technique supports you in quickly shifting your state of consciousness from one of contraction, stress, and resistance to one of expansion, relaxation, and acceptance.
Solutions Focus—Paul Z. Jackson and Mark McKergow
www.thesolutionsfocus.com
Solutions Focus is the new wave of positive, minimal change for people and organizations—the proven way to find what works as simply and gently as possible. They offer trainings, books, articles, speeches, consultations, and much more.
The Soulmate Kit—Arielle Ford
www.soulmatekit.com
The Soulmate Kit is a step-by-step guide comprising a DVD, three CDs, and a 104-page workbook designed for both men and women. It guides users through a deep and detailed program for clearing out the emotional baggage of the past while creating and magnetizing Mr. or Ms. Right into your life.
Soul Medicine Institute—Dawson Church
www.soulmedicineinstitute.org
Soul Medicine Institute is dedicated to encouraging the understanding that a vibrant spiritual connection is essential to wellness. Soul Medicine Institute facilitates training, education, and research into the role of intention, consciousness, and energy in healing.
Soulwave Institute—Katie Darling
www.soulwave.org
Katie Darling's work explores the scientific and spiritual idea that everything is a wave, including you and me, existing within an oceanic field. Soulwave trainings offer a direct experience of this that clears the way for powerful healing, joy, aliveness, and a new kind of wisdom called dynamic intelligence.
Spring Forest Qigong—Chunyi Lin
www.springforestqigong.com
Spring Forest Qigong is a revolutionary technique based on a healing practice that is thousands of years old, revised and enhanced for a twenty-first-century world. Qigong Master Chunyi Lin created Spring Forest Qigong with one goal in mind: putting the power in your hands to fully realize that you were born a healer.
Switched-On Seminars: Brain Performance Optimization—Jerry Teplitz
www.teplitz.com
Switched-On Seminars focus on rewiring the circuitry of your brain to create new levels of success. Seminars in Selling, Network Marketing, Golf, and Management integrate the left and right hemispheres of your brain to help stop your brain from being triggered by unpleasant past experiences and allow you to easily adapt to new opportunities and changes.
Tapas Acupressure Technique (TAT)
www.tatlife.com
TAT is a leading-edge energy meridian healing technique to end stress, create vibrant good health, and live a happy life. On the www.tatlife.com website you can download a free booklet on how to do TAT for yourself.
Vedic Behavior and Trend Analysis Systems—William R. Levacy
www.vedicsky.com
Vedic Behavior and Trend Analysis Systems helps analyze behavior and forecast events so you can anticipate good outcomes for your actions. It provides a road map for effective living, in tune with the laws of nature.
The Work—Byron Katie
www.thework.com
The Work is a simple yet powerful process of inquiry that teaches you to identify and question the stressful thoughts that cause all the suffering in the world. People who do The Work faithfully report life-changing results.
Yes To Success Seminars—Debra Halperin Poneman
www.yestosuccess.com
This is the seminar that launched the careers of many top transformational leaders including Deepak Chopra, Janet Attwood, and Marci Shimoff! Founded in 1981, Yes to Success, Inc., provides a system for uncovering your true purpose and then gives you the tools you need to realize that purpose. Learn how to live each and every day joyfully and profoundly.
### Additional Happiness Resources
#### Happiness Websites, Newsletters, E-zines, Blogs, and Films
www.goodmorningworld.org
Receive Good Morning World's Idea Dream daily email messages for a better world. The daily email includes one idea dream from over 64,000 Ideas and Dreams for a Better World collected since October 2004.
www.happinessclub.com
The Happiness Club's mission is to promote the benefits of being happy through meetings, newsletters, and an informative website to the people in your community and around the world.
www.thehappinessshow.com
The Happiness Show is a website offering free happiness information, resources, and over 130 twenty-eight-minute episodes in streaming audio and video for all Internet connections.
www.happinessproject.typepad.com
The Happiness Project website is the blog of author Gretchen Rubin, who spent a year test-driving every principle, tip, theory, and scientific study she could find on happiness. On this daily blog, she recounts some of her adventures and insights as she grapples with the challenge of being happier.
www.happinessfilm.com
The Happiness Revolution. Academy Award–winning filmmaker Roko Belic presents a full-length documentary on happiness filmed in more than fourteen countries, in such locations as the Louisiana bayou, the desert of Namibia, the beaches of Brazil, and the gardens of Findhorn. This film showcases amazing findings about happiness from a variety of sources, from a scientist performing brain scans in his lab to a rickshaw puller in the streets of Calcutta.
www.reflectivehappiness.com
The Reflective Happiness Program gives you direct access to the minds that founded positive psychology. The program consists of Happiness Building Exercises, Happiness Tests and Questionnaires, access to the Positive Psychology Community Forum, Newsletter, and Book Club, as well as Questions and Answers with Dr. Martin Seligman, the father of positive psychology.
www.wisebrain.org
The Wise Brain website developed by Rick Hanson, PhD, and Rick Mendius, MD, provides information about the neuroscience of psychological well-being and spiritual growth. You can subscribe to the free Wise Brain Bulletin, a twice-monthly e-zine that features happiness tools and methods and contemplative wisdom.
### Happiness Classes and Conferences
Awakening Joy
www.awakeningjoy.info
Awakening Joy is a ten-month experiential course led by James Baraz, founding teacher of Spirit Rock Meditation Center, designed to develop your natural capacity for well-being and happiness.
University of Pennsylvania Positive Psychology Center
www.authentichappiness.sas.upenn.edu
The Positive Psychology Center offers questionnaires, resources, articles, and a master's degree in applied positive psychology.
Annual Positive Psychology Summit
www.gallupippi.com
The Gallup Institute for Global Well-Being offers an annual global well-being forum featuring the International Positive Psychology Summit and New Insights from the Gallup World Poll.
## ACKNOWLEDGMENTS
This book was a collaborative effort in so many ways. I feel deeply grateful for the extraordinary support I received and for the invaluable contributions of friends, family, colleagues, the Happy 100, and experts in a variety of fields. Carol and I would like to thank many wonderful and generous people who have shared in this journey with us.
From Marci:
Carol, for giving your heart, mind, and soul to this project with total devotion and working tirelessly to create the best book possible. Thank you for being there for me as a gifted writer, a brilliant collaborator, and a loving friend. I couldn't have done this book without you, nor would I have wanted to.
Sergio, for being my rock, my hero, and my love. Thank you for your unwavering support, your profound insights, and for continually showing me in many ways, big and small, that love is always most important. I am so blessed to share my life with a man who is truly Happy for No Reason!
My extraordinary family: Mom and Dad, the two best parents in the world, for their unconditional love and for always believing in me. My sister, Lynda, for being the other "spiritual one." My brother, Paul, who taught me everything I know (at least that's what he tells me). My wonderful Aunt Marian, fabulous sister-in-law, Susan, and great nephews and niece, Aaron, Vickie, Jared, and Tony. Maija Snepste, who's a loving addition to our family. Leah Basch for being such an awesome "little sister." And to Francesca, Max, and Silvia Baroni, for sustaining me with your love and those superb Sunday night Italian dinners at your home.
Bonnie Solow, my literary agent and dear friend, who has seen us through this entire book-writing process with love, grace, encouragement, incredible patience, and consummate skill. Thank you for holding my hand and nourishing my heart. You are the best literary agent in the world and a most extraordinary friend. And to Lulu for spreading joy everywhere.
Jack Canfield, my phenomenal mentor and Chicken Soup "soul" brother, for guiding me, inspiring me, and so generously sharing with me all your extraordinary wisdom and knowledge. I wouldn't be where I am today without you. I love you and deeply value our friendship and the tremendous impact you've had on my life. Inga Canfield, my "fun fairy" and wise sage all in one package: thank you for being a great friend. Mark Victor Hansen, Patty Aubery, Russ Kamalski, Patty Hansen, and the entire Chicken Soup for the Soul family for always being there for me with wisdom, love, and support. Jennifer Hawthorne, for starting on this writing path with me. May bluebirds always surround you with happiness.
Pete Bissonette, Paul Scheele, and Learning Strategies for being my wonderful partners in bringing a new level of happiness to the world. I love working with you and feel fortunate to be on the same team with you. My colleagues on the Transformational Leadership Council, I am constantly awed by your individual and collective brilliance, sheer goodness, and commitment to make a difference in the world. I so deeply appreciate your support. Deborah Rozman, Howard Martin, Doc Childre, and my other great friends at the Institute of HeartMath for your contribution in the sphere of human development and for generously assisting me in my work.
Maharishi Mahesh Yogi, my first spiritual teacher, for giving me the gift of meditation, life-changing knowledge, and profound, ageless wisdom.
Janet (Jani) Attwood for being my best cheerleader, dear friend, and favorite person to laugh with. Your loving heart is as big as the universe. Chris Attwood, thank you for your great wisdom and your open heart. You nailed it every time I called for help. Jani and Chris, it has been so much fun playing in the book world with you at the same time—you are my family.
Bill Bauman, for being my personal guardian angel and spiritual mentor. Your expanded vision, deep love, and unique sense of humor are just a few of the reasons you are a constant blessing in my life. Bill Levacy, for always inspiring me in the right direction and at the right time. You've guided me impeccably for the past fifteen years—you're a star! Nandu, for helping me stay on course and keeping the energy high.
Catherine Oxenberg, my dear soul sister, your spiritual perspective and loving friendship bring so much joy and peace in my life. I love you. Debra Poneman for being my first mentor, my great friend, and for helping me say Yes to Success.
Marianne Williamson, for being an exquisite role model. I love how you contribute to the world, and your magnificence as a leader, a visionary, and a friend. Hale Dwoskin and Amy Edwards, for encouraging me to remember the truth and always being available with your love and wisdom. Arielle Ford and Brian Hilliard, for sharing your love and skill in living. Gayatri Schiffer and Brian Siddhartha Ingle, for keeping joy, order, and spirit in our home. Jill Lublin and Steve Lillo, for being so willing to jump in to help. Your fantastic skills, insights, and care saved the day many times. Stephen M. R. Covey, for our conversation that encouraged me more than you know.
Swami Chandrasekharanand Saraswati and Joan Shivarpita Harrigan for your brilliant teachings and practices that have made such a difference in my life.
My phenomenal friends who have given me love and support throughout this journey: Lenora Boyle, AlexSandra Leslie, Suzanne Lawlor, Mary Weiss, Stewart and Joan Emery, Peggy O'Neill, Barbara Stanny, Elinor Hall, Ron Hall, Jeddah Mali, Robert Kenyon, Renee Skop, Lynn Robertson, Cindy Buck, John and Bonnie Gray, Yakov Smirnoff, Donna Bauman, Diane Alabaster, Suelena Pamplin, Bruce Allen, Master George Yau, Michael Laughrin, Janet Switzer. My women's groups: Holly Moore, Lane Cole, Janice Peterson, Sandy Kopff (smiling down), Kami Mailloux, Terri Tate, Elyn Kopeck, Katherine Revoir, and my Monday morning visions group. And to all my friends who saw articles on happiness and sent them to me (you know who you are).
From Carol:
Marci, for inviting me to join you on this life-changing project. You've added to my life in so many ways and I will always be deeply grateful. My wonderful husband, Larry Kline, who cheered me on every step of the way, patiently accepting month after month of nonstop work. Your love and support mean more to me than you can ever know. My stepchildren, Lorin and McKenna—your sense of self and spirit of adventure continually inspire me. My mother, Selma, who is always so loving and encouraging. My sister, Bobbie, the other author in the family, who listened understandingly and was always willing to help me with grammar questions. My "sensible" brother, Burt, for giving me wise counsel whenever I called. My other sibs and their spouses, Jim and Di, Wilbur, Pam, Holly and Charlie, who took up the slack. I owe you—and love you all. Cindy Buck, for being my loving friend, cheerleader, and confidante. I couldn't have done it without you, Cindala. My women's group: Karen Joost, my serene heart-sister, and Toni D'Orr, the embodiment of kindness, for so freely sharing your insights and affection. I love you more with each passing year. My adored friends: Debbie Pogel—the other Shaina Rivkah, Josie Batorski, Peggy O'Neill, Ceci Balmer, Nina Falk, Lane Cole, Holly Moore, Betsy Dockhorn, Georgia Nemkov, Christian Wolfbrandt, Stephanie Hewitt, the Woolfs, Marcy Luikart, Kathy Bennett, Laurie Edgcomb, and Wilma Melville; your friendship and support keep me going and make life so sweet.
From Both of Us:
Our amazing editors: Betsy Rapoport, who's not only superbly talented at what she does, but is funny, kind, and oh so wise. You were a joy to work with! Cindy Buck, whose editorial brilliance and valuable input improved this book tremendously, and whose friendship we prize deeply. We love you, Cin. Nancy Marriott, your hard work, unflagging enthusiasm, and positivity were a blessing to us. And Kristin Loberg, your skill, good humor, and steadiness were matched only by your vast experience and competence.
Katina Griffin, Marci's executive assistant, for kindheartedly going above and beyond the call of duty to support us and this book. Sue Penberthy, Marci's loyal assistant and bookkeeper, who for eleven years has nourished her with "mother" energy and always kept the money flowing. Suzanna Gratz, for your dedication and persistence juggling a multitude of balls in such a loving fashion. Laurie Kalter and Megan Woolever, for your great support, thorough research, and crackerjack project management. And D'ette Corona, who kept our permissions process in order and whose remarkably even keel continues to amaze us. You are a gem.
Sarah Clarehart, for your wholehearted dedication, saintly patience, and graphic design genius. Randall Heath and Liz Howard, for your skillful artistry. Joe Burull and Jerry Downs, for your keen photographic eye and your hard work making Marci look her best! Aylene Rhiger, for your website wizardry. Jerry Teplitz, for your muscle-testing mastery. Paul Hoffman, for making our message sing! Scott de Moulin and Dallyce Brisbin, for your generous and brilliant coaching.
Our publishing "dream team," the fabulous and talented people at Free Press: Dominick Anfuso, for being the most easygoing, charming, reassuring, and positive editor on the planet. We feel so lucky to have worked with you on this project. Martha Levin, for your confidence in this book. Maria Auperin, for your kindness, patience, and competence helping us every step of the way. Leah Miller, for your joyful care in coordinating so many details for this book. Eric Fuentecilla, for your excellent work on the cover. Eleni Caminis, for your printing heroism. And the wonderful marketing and publicity team, including Suzanne Donahue, Carisa Hays, and Heidi Metcalfe, for helping get this book in the hands of so many readers.
The Happy 100, for giving so freely of your time and experience to help others find the same happiness you radiate so beautifully. The many people who responded to the Happy for No Reason survey. We read each and every one, and they provided us such valuable information.
The subject matter experts in the various fields who patiently answered our many questions. In addition to the many experts already quoted in the book, we're grateful to the following people: Arjuna Ardagh, Jim Bunch, Katie Darling, Dr. John Demartini, Amy Edwards, Pralaya Gordon, Joan Shivarpita Harrigan, Dr. Steven Hotze, the late Isa Lindwall, Dr. Elena Loboda, Sue Morter, Christian Opitz, George Ortega, Swami Chandrasekharanand Saraswati, Susan Seifert, Mark and Bonita Thompson, and Jerry White. Thank you for explaining complex scientific processes in a way that we nonscientists could easily understand and for sharing brilliant insights in the field of happiness. Your contributions provide the objective framework for the concept of Happy for No Reason and the steps of building your Home for Happiness.
Robert Biswas-Diener, one of the kindest men ever, who shared his time and expertise with us whenever we called on him.
Our guinea pigs, those people who took the time to read and give us feedback on our manuscript: Chris Attwood, Janet Attwood, Sergio Baroni, Lane Cole, Oriana Green, Lisa Hotchkiss, Katrina Hunt, Brian Siddhartha Ingle, Suzanne Lawlor, Peggy O'Neill, Gayatri Schiffer, Barbara Stanny, and Kevin Twohy. Your input was immensely useful.
Because of the size and scope of this project, we may have left out the names of some people who contributed along the way. If so, please forgive us, and know that we really do appreciate you very much.
We are so grateful and love you all!
## GIVING BACK
In the spirit of giving back, we are delighted to donate a portion of the author proceeds from Happy for No Reason to the following two worthy causes:
Operation Smile. Founded in 1982, Operation Smile is a worldwide children's medical charity whose network of global volunteers are dedicated to helping improve the health and lives of children and young adults. Since its founding, Operation Smile has treated more than 100,000 children born with cleft lips, cleft palates, and other facial deformities. In addition to contributing free medical treatment, Operation Smile trains local medical professionals in its twenty-five partner countries and leaves behind crucial equipment to lay the groundwork for long-term self-sufficiency.
Operation Smile, Inc.
6435 Tidewater Drive, Norfolk, VA 23509
Phone: (888) OPSMILE, Fax: (757) 321-7660
Website: www.operationsmile.org
The Pachamama Alliance. The Pachamama Alliance is dedicated to empowering indigenous people to preserve their territories and way of life, and thereby protect the natural world for the entire human family. Their main strategy includes supplying rainforest peoples with the tools and resources necessary to support the continued strength and vitality of their communities and culture. The indigenous people offer a new way of seeing and living in the world that is inherently interconnected and sustainable. Together, this reciprocal partnership is working to bring forth an environmentally sustainable, spiritually fulfilling, and socially just human presence on this planet.
The Pachamama Alliance
1009 General Kennedy Avenue, PO Box 29191, San Francisco, CA 94129
Phone: (415) 561-4522
Web site: www.pachamama.org
## ABOUT MARCI SHIMOFF
Marci Shimoff is a celebrated transformational leader and happiness expert who has inspired millions of people around the world with her message of the infinite possibilities that life holds. One of the nation's foremost motivational experts and a top-rated professional speaker, she has delivered programs for a large variety of audiences and organizations, including numerous Fortune 500 companies. For over twenty years, she has received wide acclaim for sharing her breakthrough methods for personal fulfillment and professional success.
Marci is also one of the best-selling nonfiction authors of all time and the woman's face of the biggest self-help book phenomenon in history, Chicken Soup for the Soul, which has reached more than 150 million people. She is the coauthor of six of the top-selling titles in the series, including Chicken Soup for the Woman's Soul and Chicken Soup for the Mother's Soul. Her books have sold more than 13 million copies worldwide, in thirty-three languages, and have been on the New York Times best-seller list for a total of 108 weeks, with four titles reaching #1 for a total of twelve weeks. Her books have also topped the USA Today and Publishers Weekly best-seller lists.
In addition, Marci is a featured teacher in the international movie and book phenomenon The Secret, offering her insights on the key principles to creating lasting success and fulfillment. A popular and engaging media personality, Marci has appeared on more than 500 national and regional television and radio shows and has been interviewed for more than 100 newspaper articles throughout North America. Her work has been published in national women's magazines, including Ladies' Home Journal and Woman's World.
President and cofounder of The Esteem Group, Marci delivers keynote addresses and seminars on self-empowerment and peak performance to corporations, women's associations, and professional and nonprofit organizations. She received her MBA in organizational behavior from UCLA and also completed a one-year advanced certification program to become a stress management consultant.
Marci is a founding member and serves on the Executive Committee of the Transformational Leadership Council, a group of 100 top leaders serving over 10 million people in the self-development market. Marci is dedicated to fulfilling her vision and life's purpose of helping people to live more empowered and joy-filled lives.
To find out more about Marci's keynote presentations, books, or seminar programs, you can contact her at:
The Esteem Group
369-B Third Street #314
San Rafael, CA 94901
Phone: 415-789-1300
Fax: 415-789-1309
www.marcishimoff.com
www.happyfornoreason.com
## ABOUT CAROL KLINE
Carol Kline is the coauthor of five books with over 5 million sold, in the best-selling Chicken Soup for the Soul series, including Chicken Soup for the Dog Lover's Soul and Chicken Soup for the Cat Lover's Soul, and the #1 New York Times best-selling Chicken Soup for the Mother's Soul 2. In 2006, she cowrote You've Got to Read This Book: 55 People Tell the Story of the Book That Changed Their Life, with Jack Canfield and Gay Hendricks.
A freelance writer and editor since 1980, Carol, who has a BA in literature, specializes in narrative nonfiction and self-help. She has written for newspapers, newsletters, and magazines, and, in addition to her own Chicken Soup books, has also contributed stories and her editing talents to many other books in the Chicken Soup for the Soul series.
Carol is also a speaker, self-esteem facilitator, and animal welfare advocate. In addition, she has taught stress-management systems to the general public since 1975. At present, she is at work on several writing projects on a variety of topics.
To write to Carol or to inquire about her writing or speaking services, please use the following contact information:
Carol Kline
Carol Kline, Inc.
P.O. Box 521
Ojai, CA 93024
E-mail: ckline@happyfornoreason.com
For more information, visit
www.HappyforNoReason.com
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## BIOGRAPHIES OF THE HAPPY 100 WHOSE STORIES APPEAR IN HAPPY FOR NO REASON
Janet Attwood is the coauthor of the #1 national best-selling book, The Passion Test, coauthor of From Sad to Glad, and the cofounder of the #1 online transformational magazine, Healthy Wealthy n Wise. An expert on passion, she presents her Passion Test programs all over the world. Janet's "Empowered Women CD Series" is now being broadcast in women's homeless shelters across the United States. www.janetattwood.com
Martha Beck is a New York Times best-selling author and a monthly columnist at O: The Oprah Magazine. Called by NPR "the best-known life coach in America," her books include The Four-Day Win: End Your Diet War and Achieve Thinner Peace and Expecting Adam. www.liveyournorthstar.com
Michael Bernard Beckwith is the founder of the Agape International Spiritual Center in Los Angeles. Dr. Beckwith conducts retreats and is a teacher of meditation, a seminar leader, and the originator of the Life Visioning Process, which he teaches throughout the country. His books include Inspirations of the Heart, Forty Day Mind Fast Soul Feast, A Manifesto of Peace, and Living from the Overflow. www.agapelive.com
Rhonda Byrne is the producer and creator of the worldwide book and film phenomenon The Secret and was listed among Time magazine's 100 Most Influential People in the World in 2007. Originally from Australia, Rhonda is currently in the United States, where she is continuing her work on projects that will uplift, inspire, and bring joy to the world. www.thesecret.tv
Chellie Campbell is the author of The Wealthy Spirit and Zero to Zillionaire, and created the popular Financial Stress Reduction® Workshop to help people become successful at producing more income, managing their money, and having more time off for fun! A professional speaker, seminar leader, and poker champion, Chellie teaches her workshops in the Los Angeles area and gives programs throughout the country. www.chellie.com
Aerial Gilbert is the outreach manager at Guide Dogs for the Blind and is also an avid athlete. She rows regularly on San Francisco Bay and has competed successfully in numerous rowing events. Aerial shares her personal experiences in public presentations to fraternal organizations, businesses, schools, and other groups. www.guidedogs.com
Elizabeth Gilbert is the author of numerous books, including the best-selling Eat Pray Love, which has recently been optioned by Paramount Pictures for a film starring Julia Roberts. She has also written for the New York Times Magazine, Real Simple, and O: The Oprah Magazine. Liz currently lives in New Jersey and is at work on a new book. www.elizabethgilbert.com
Mariel Hemingway is an actress who has been pursuing her passion for yoga and health for more than twenty-two years and is now seen as a voice of holistic and balanced living. Her latest book, Mariel Hemingway's Healthy Living from the Inside Out, is a how-to guide to finding one's balance and health through self-empowering lifestyle techniques. www.marielhemingway.com
Gay Hendricks, PhD, has served for more than thirty years as one of the major contributors to the fields of relationship transformation and body-mind therapies. Along with his wife, Dr. Kathlyn Hendricks, Gay is the author of many best-sellers, including Conscious Loving, Spirit-Centered Relationships, and The Corporate Mystic, as well as the founder of the Hendricks Institute, which offers seminars in North America, Asia, and Europe. www.hendricks.com
Chunyi Lin is a certified international Qigong Master, founder of Spring Forest Qigong, and the coauthor of #1 Amazon.com best-seller, Born a Healer. In addition, Master Lin has created a series of home learning materials for students, including videos, guided audio meditations, and reference manuals. His vision is "a healer in every family and a world without pain." www.springforestqigong.com
Mary G. Lodge is a mother to five children, eleven grandchildren, and five great-grandchildren. Her hobbies include writing, painting, and gardening and she enjoys speaking on the subject of forgiveness. Contact her at LodgeDoor@aol.com
Lisa Nichols, featured teacher in The Secret, is a dynamic international motivational speaker and powerful advocate of personal empowerment. Coauthor of the Chicken Soup for the African American Soul series, Lisa is the founder and CEO of Motivating the Teen Spirit, LLC, which is recognized by many as the most comprehensive empowerment skills program available today for teen self-development. www.lisa-nichols.com
Happy Oasis is the founder and CVO, Chief Visionary Officer, of Raw Spirit Festival, Earth's largest raw health, ecosustainability, and world peace festival in Sedona, Arizona. Traveling abroad for decades as an adventure anthropologist and adopted by tribes, Happy authored Uncivilized Ecstasies and Bliss Conscious Communication. www.RawSpiritFest.com
Catherine Oxenberg is a European princess and an award-winning actress who has appeared in numerous films and television programs and is best known for her role in the popular television series Dynasty. In 2006, she and her husband, actor Casper Van Dien, starred in the TV series Watch Over Me. She is also the proud mother of five beautiful children. www.catherineoxenberg.com
Rico Provasoli is a retired chiropractor, now full-time writer. His recent titles include Golf between the Ears and Please Don't Tell My Guru. He lives in northern California. He continues to sail, laugh, and give living thanks for everything. He travels as a public speaker leading seminars on Relief from the Inner Critic. www.ricoprovasoli.com
Zainab Salbi is the founder and CEO of Women for Women International, the 2006 recipient of the Conrad N. Hilton Humanitarian Prize. Zainab is also the author of Between Two Worlds: Escape from Tyranny: Growing Up in the Shadow of Saddam and The Other Side of War: Women's Stories of Survival and Hope. www.womenforwomen.org
CJ Scarlet, having triumphed over a life-threatening illness, is now an author, motivational speaker, and personal coach dedicated to helping others achieve the happiness and prosperity they deserve. CJ's first inspirational book, Neptune's Gift, is now available online at www.cjscarlet.com
Lynne Twist, author of the best-selling book The Soul of Money, is a global activist, fund-raiser, speaker, and consultant who has dedicated her life to global initiatives that serve the best instincts in all of us, including ending world hunger, protecting the world's rainforests, and creating a sustainable future for all life. www.soulofmoney.org
## PERMISSIONS
We are grateful to the various people who shared with us their compelling and uplifting stories for inclusion in this book. In addition, we thank the following individuals and organizations who have given us permission to reprint their material:
Grumpy Gene cartoon. Printed by permission of Hagen Cartoons.
Money cartoon. Printed by permission of Patrick Hardin.
Dog cartoon. Printed by permission of Mike Twohy; © 1992 The New Yorker Magazine.
Pumpkin cartoon. Printed by permission of Mark Anderson.
The Solutions Focus Technique. Printed by permission of Paul Z. Jackson and Mark McKergow.
Blaming cartoon. Printed by permission of Martha Campbell.
M-Power March. Printed by permission of Morter HealthSystems.
Bad programming cartoon. Copyright 2001 by Randy Glasbergen. [www.glasbergen.com](ch04.html#page_86)
The Work Mini-Worksheet. Printed by permission of Byron Katie.
The Letting Go Exercise. Printed by permission of The Sedona Method®.
Dear Diary cartoon. Printed by permission of Mike Twohy; © 1996 The New Yorker Magazine.
Heart Rhythm graphs. Printed by permission of the Institute of HeartMath.
Good beat cartoon. Printed by permission of Jonny Hawkins.
Dr. Emoto's water photos. Printed by permission of I.H.M. Co., LTD.
The Quick Coherence® Technique. Printed by permission of the Institute of HeartMath. Quick Coherence is a registered trademark of the Institute of HeartMath.
Smile transplant cartoon. Copyright 2001 by Randy Glasbergen. [www.glasbergen.com](ch06.html#page_150)
Ziggy ©2005 Ziggy and Friends, Inc. Reprinted with permission of Universal Press Syndicate. All rights reserved.
"Your Brain's True and False Emotional Chemistry" from The Mood Cure by Julia Ross, copyright © 2002 by Julia Ross. Used by permission of Viking Penguin, a division of Penguin Group (USA) Inc.
"The Four-Part Mood Type Questionnaire" from The Mood Cure by Julia Ross, copyright © 2002 by Julia Ross. Used by permission of Viking Penguin, a division of Penguin Group (USA) Inc.
Breathing the Universe Exercise. Printed by permission of Spring Forest Qigong.
Cell phone cartoon. Copyright 2001 by Randy Glasbergen. [www.glasbergen.com](ch07.html#page_187)
Postcard. From A Lotus Grows in the Mud by Goldie Hawn, with Wendy Holden, © 2005 by Illume, LLC. Used by permission of G. P. Putnam's Sons, a division of Penguin Group (USA) Inc.
Dr. Emoto's water photos. Printed by permission of I.H.M. Co., LTD.
Hello, God. I'm Liz. Adapted from Eat, Pray, Love by Elizabeth Gilbert, ©2006 by Elizabeth Gilbert. Used by permission of Viking Penguin, a division of Penguin Group (USA) Inc. Additional material is from an interview with the author.
The Identifying Your Passions Exercise used by permission of Janet Attwood and Chris Attwood.
Infectious smile cartoon. Copyright 2001 by Randy Glasbergen. [www.glasbergen.com](ch09.html#page_245)
Skippy cartoon. Printed by permission of Jonny Hawkins.
Gate 4-A Albuquerque Airport story. Printed by permission of Naomi Shihab Nye.
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Claim Your FREE HAPPY FOR NO REASON TOOLS at www.HappyforNoReason.com/bookgifts
Congratulations! You're on the Happy for No Reason journey, and you'll continue to grow happier every day if you put the knowledge you've gained in this book into practice. To help speed you on your way, I've created the following gifts for you:
Free Audio Recording: Highlights of the Happy 100 Interviews
Listen to some of the most inspiring and enlightening excerpts from my interviews with the Happy 100. There's a special power in hearing these people share their remarkable stories and insights about creating more happiness in life.
Free Happy for No Reason Workbook
This downloadable 26-page workbook includes the Happy for No Reason Questionnaire, all 21 of the Happiness Habit Exercises, and special bonus material.
Free Happy for No Reason eZine
Every few weeks I will send you a fun and inspiring ezine that includes practical tips and the latest breakthroughs and ideas on unconditional happiness.
BONUS: Free Happiness Habits Miniposter
This beautiful one-page summary of the 21 Happiness Habits is perfect to hang on your vision board or display in a special place where you will see it daily as a reminder.
For these and other free gifts, visit: www.HappyforNoReason.com/bookgifts
To access free gifts, users must register by providing their first name and email address. Offer subject to availability.
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All rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information, address Atria Books Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.
First Atria Paperback edition July 2013
ATRIA PAPERBACK and colophon are trademarks of Simon & Schuster, Inc.
The Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event, contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.
Designed by Nancy Singer Olaguera/Sarah Clarehart
Cover design by Sarah Clarehart
Cover photograph by Joe Burull
The Library of Congress has cataloged the Free Press hardcover edition as follows:
Shimoff, Marci.
Happy for no reason : 7 steps to being happy from the inside out / Marci Shimoff with Carol Kline.
p. cm.
Includes bibliographical references.
1. Happiness. I. Kline, Carol, date. II. Title.
BJ1481.S556 2008
158—dc22
2007029803
ISBN 978-1-4165-4772-3
ISBN 978-1-4165-4773-0 (pbk)
ISBN 978-1-4165-5398-4 (ebook)
|
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"redpajama_set_name": "RedPajamaBook"
}
| 6,073
|
<?php
if(!defined('sugarEntry') || !sugarEntry) die('Not A Valid Entry Point');
/*********************************************************************************
* Description: Defines the Spanish language pack for the base application.
* Portions created by REDK Ingeniería del Software S.L..
* All Rights Reserved.
* Contributor(s): REDK Software Engineering (www.redk.net)
********************************************************************************/
$mod_strings = array (
'LBL_YOURS' => 'Suyo',
'LBL_IN_DATABASE' => 'En Base de datos',
'LBL_CONFLICT_EXISTS' => 'Hay un Conflicto Para - ',
'LBL_ACCEPT_DATABASE' => 'Aceptar Base de datos',
'LBL_ACCEPT_YOURS' => 'Aceptar Suyo',
'LBL_RECORDS_MATCH' => 'Los Registros Coinciden',
'LBL_NO_LOCKED_OBJECTS' => 'No hay Objetos Bloqueados',
);
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 513
|
Q: React state based on cookie not updating state of material ui switch on page refresh I am using react cookie and able to successfully retrieve cookie:
const [cookies, setCookie] = useCookies(['darkTheme']);
console.log("initial cookie is ",cookies.darkTheme) /// logs "true" as string
const [darkState, setDarkState] = useState(Boolean(cookies.darkTheme));
console.log("initial darkstate is ", darkState) /// logs true as boolean as expected
Then in material ui switch, I try to use darkState
{console.log("about to assign switch state as ", darkState)/// logs true}
<Switch checked={darkState} onChange={handleThemeChange} name="darkSwitch" />
Here is the handlethemechange function although the problem is with the initial load even before the function is called.
const handleThemeChange = () => {
console.log("in theme change and darkState will be ", !darkState)
darkState?setDarkState(false):setDarkState(true);
setCookie('darkTheme', darkState, { path: '/' });
console.log("in theme change cookie is ", cookies.darkTheme)
};
The above does not work and checked={darkState} always seems to be false although the console log just above it says its boolean value true.
If I just change checked={darkState} to checked={true}, the switch is checked and works but obviously, I do not want to just hard code true.
I am doing this in a child component so I cannot use getInitialProps option thats available in next js. I am trying to avoid using componentwillMount as I read on SO there are issues with it.
My guess is the cookie value although logs correctly in console, is not reaching the switch in time?
A: The problem is with the way next js pre renders everything so while things look good in the console logs, the form components do not get default value in time.
The solution is to use useEffect which can be used like componentDidMount.
In my case, I am putting the cookie value into the state of the switch so that it gets a state based on last set cookie value.
useEffect(() => {
setDarkState(cookies.darkTheme == "true");
}, [darkState]);
Technically, this alone should solve the issue but I had to use a ref at the top of the component as well to make it work.
const switched = useRef(cookies.darkTheme == "true"? true: false);
const [darkState, setDarkState] = useState(switched);
I do not have an explanation for why I had to first assign the value to a ref and then put it in useState. May be someone can suggest why.
A: It seems to work fine for me: https://stackblitz.com/edit/react-sfjkeq?file=src/App.js
If you try to line 10 to const [darkState, setDarkState] = useState(Boolean(!cookies.darkTheme)); it's changing the state of the Switch component.
Could you past more of your code in you post?
|
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"redpajama_set_name": "RedPajamaStackExchange"
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| 4,279
|
The Toy Chest Celebrates 35 Years
Patti Hess, Joyce Powell, Cathy Martin, David Martin
photo by Cindy Steele
The Toy Chest
Celebrates 35
by Susan W. Showalter
Cathy and David Martin's backgrounds as preschool and elementary schoolteachers, antique dealers and especially their love of community made them ideal candidates to become the new owners of The Toy Chest in 1989 when Helen and Ed Hollis decided to sell their business. This year they celebrate The Toy Chest's 35th anniversary.
"The Hollis family opened The Toy Chest in 1970 on Main Street," explained Cathy. "Ed spent several years creating a model circus and visitors could view his masterpiece. The building and circus were destroyed in a fire in the 1980's then it reopened in its former location."
The Toy Chest has been in its present location at the Artists Colony Shops on Van Buren Street since 1988. The large variety of merchandise makes the shop feel more like a very big store. Attention to detail is obvious. The store sells several original toys.
"We worked with the company B. Dazzle to create a puzzle which features a picture of Bean Blossom's covered bridge," said Cathy. "We also offer a Farkel game that has a log cabin and The Toy Chest name on it. We are happy to offer both Brown County and Indiana related toys and books by Brown County authors Portia Harvey Sperry (Abigail), Gwenyth Swain (I Wonder, as I Wander and Chig and the Second Spread) and Patricia Frechtman (The Browne Twins Series). A music CD, Heart Shaped Rocks (Parent's Choice Award), composed and produced by staff member by Pamela Jay, is also available at the store."
The Martins, who are members of the Brown County Chamber of Commerce and the American Specialty Toy Retailing Association, have experienced changes in the toy industry over the years. Now there are the imports from China and somewhat lower pricing. Nostalgic toys, retro toys and family games have become more popular and more customers seek toys not offered by the mass market. Cathy and David personally restock the store themselves each night but they have several employees who handle sales.
Joyce Powell, store manager, has enjoyed working at The Toy Chest since 1987. When asked what her favorite aspects of the store are she quickly responded, "the people. Also, I love sharing the historic traditions of Steiff creations. I like offering Steiff products because they're such quality toys." Joyce looks forward to family news updates from the numerous repeat customers.
"This is a nice place to work…a fun place to work," said Patti Hess who has worked at the store for 15 years. "We have such neat traditional toys that people don't find everywhere. We have Thomas the Train and beautiful Madame Alexander Dolls. We have puzzles, games and marionettes. We carry other unique tin wind-up toys and steam engines."
"To be successful in retail is like anything else," said David. "You have to work hard, you have to have some kind of a flair for it and you have to have some luck. It's a combination of all of that."
David was Chairman of the Brown County Library Board of Directors for several years and was instrumental in building the new building. He received the John Rudd Community Service Award in 2004.
"Some of my favorite things about the store are the nested hand painted dolls from Russia and our books," said Cathy who, along with David, reads each book before it is placed in the store. "I also am appreciative of enthusiastic customers and being able to help people. Three grandmothers from the Chicago area, claiming to be toy connoisseurs, told us this is the best toy store they had ever been in!"
According to David, the Russian nesting dolls were made with foot powered wood lathes then hand painted by specialized artists when he and Cathy bought the store. Since then, the dolls are made in small cottage type industries with several people working together.
"We appreciate the importance of play experiences in children's development. Our children and we collected antique toys," said Cathy, who helped establish the Brown County Parks and Recreation Board. She served on the Brown County Foundation scholarship award committee, is an Indiana Heritage Arts Board Member and is active in Psi Iota Xi philanthropic sorority.
Cathy met David when each was working in the library in their hometown of Ft. Wayne. They moved here after taking teaching jobs in Columbus. Their two children, Seth and Becky, are graduates of Brown County High School. Seth and his wife have two-month old twins. (Just imagine…parents and grandparents who own a toy store!)
"Shops come and go," said Cathy. "I am glad that there are still primarily mom and pop stores. People come in everyday saying that they love Brown County and they look forward to coming to our store. The original customers are actually bringing in their kids. These are some of the things I love about having the store," Cathy said as her smile and eyes sparkled.
The Toy Chest opens daily at 10:00 a.m. and closes at 5:00 p.m. on Sunday through Thursday but is open until 7:00 p.m. on Friday and Saturday. For more information please contact The Toy Chest at 812 988-2817.
|
{
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| 4,115
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Waring Garden Apartment is located at the address 5320 Adobe Falls Rd in San Diego, California 92120. They can be contacted via phone at (619) 286-6154 for pricing, hours and directions.
For maps and directions to Waring Garden Apartment view the map to the right. For reviews of Waring Garden Apartment see below.
|
{
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| 9,574
|
«Наковальня или молот» (, ) — политический двухсерийный художественный фильм с элементами детектива совместного производства Болгарии-ГДР-СССР, снятый в 1972 году.
Премьера фильма состоялась: 8 сентября 1972 (София), 9 сентября 1972 (Берлин) и 29 октября 1973 (Москва).
Сюжет
Фильм о драматических событиях 1933 года во время знаменитого лейпцигского судебного процесса над болгарским коммунистом Георгием Димитровым, обвиненного со своими товарищами в поджоге Рейхстага, после прихода к власти нацистов в Германии.
В центре сюжета — схватка в суде между Димитровым и Герингом, представителями коммунистической и нацистской идеологий.
В ролях
Стефан Гецов — Георгий Димитров (дублировал Алексей Алексеев)
Аня Пенчева — Магдалена (дублировала Вера Бурлакова)
Ханс Петер Райнеке — ван дер Люббе
Ханс Хардт-Хардтлофф — Клюге — Геббельс (дублировал Александр Белявский)
Всеволод Сафонов — советский полпред Евгений Жариков — Игорь Герд Михель Хеннебер — Константин фон Нейрат Жанна Прохоренко — Люба Вильям Полонный — Герман Геринг Ханньо Хассе — Артур Небе'' (дублировал Юрий Леонидов)
Награды
1972 — «Золотая роза» МКФ в Варне — приз за лучшую мужскую роль (Стефан Гецов)
1975 — «Золотой Лачено» МКФ в Авеллино
См.также
«Урок истории» (1956)
Ссылки
Фильм «Наковальня или молот» на сайте imdb
Фильмы-драмы Болгарии
Фильмы студии «Мосфильм»
Фильмы-драмы, основанные на реальных событиях
Исторические фильмы Болгарии
Фильмы-драмы Германии
Фильмы-драмы СССР
Исторические фильмы Германии
Исторические фильмы СССР
Георгий Димитров
Фильмы ГДР
Фильмы Христо Христова
Фильмы на болгарском языке
Фильмы Болгарии 1972 года
|
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| 7,642
|
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\put(-20,13){\tiny University of Maryland Elementary Particle
Physics University of Maryland Elementary Particle Physics University of
Maryland Elementary Particle Physics}
\put(-20,-241.5){\tiny University of Maryland Elementary
Particle Physics University of Maryland Elementary Particle Physics
University of Maryland Elementary Particle Physics}
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\begin{document}
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\begin{center}
\vglue .04in
{\large\bf Type -B/ -O Bosonic String Sigma-Models}
\\[.6in]
S. James Gates, Jr.\footnote{gates@umdhep.umd.edu} \\[.1in]
{\it Department of Physics\\
University of Maryland at College Park\\
College Park, MD 20742-4111, USA\\
${~~}$\\
{\rm {and}}\\
${~~}$\\
{\rm {V.G.J. Rodgers}}\footnote{vincent@hepaxp.physics.uiowa.edu}\\
Department of Physics and Astronomy\\
University of Iowa\\
Iowa City, Iowa~~52242--1479}
\\ [1.8in]
{\bf ABSTRACT}\\[.002in]
\end{center}
\begin{quotation}
{We provide world sheet non-supersymmetrical actions that describe the
coupling of a bosonic string to the tachyon and massless states of
both the type-B and type-O theories. The type-B theory is derived as a
truncation and chiral doubling of the Ramond-Ramond sector in our previous
model that connected the (1,0) heterotic string to a 10D, type-IIB
supergravity background. The type-O theory then follows from a
``fermionization'' of the type-B theory.}
\endtitle
\sect{Introduction}
~~~~Approximately two years ago, we pointed out a curious feature \cite{Idea}
of (1,0) supergravity NSR non-linear $\sigma$-models and type-IIB supergravity.
Although totally unexpected at the time, we formulated a coupling of a
heterotic string \cite{HET} to a background that consisted
of the massless bosonic fields of the type-IIB supergravity theory. The
reason for this to be unexpected was that it had always been thought that
the type-IIB supergravity theory (whose superspace description is well
known \cite{Typ2BSG}) was only associated with the type-IIB superstring
of Green and Schwarz \cite{GS2}. From this point of view it was very
``unnatural'' to find a heterotic model associated with the massless
bosonic fields of type-IIB supergravity.
On the other hand, the construction of a (1,0) NSR non-linear $\sigma$-model
coupled to a type-IIB supergravity background cleared up an earlier puzzle.
Several years prior to the work in \cite{Idea}, it was noted that the (1,0)
heterotic string nonlinear $\sigma$-model possessed exactly the right properties
to allow a coupling to a 4D, N = 8 supergravity background also \cite{GN}.
This 4D result can now be viewed as the simple dimensional reduction of
the type-IIB heterotic string nonlinear $\sigma$-model.
In order to accomplish the result in \cite{Idea}, we introduced
a feature which had not appeared in the context of stringy non-linear
$\sigma$-models prior to that time. The idea was that when certain combinations
of p-forms occur in the Ramond-Ramond sector of a superstring theory, they
correspond to the introduction of WZNW models on the world sheet of the
superstring where the currents of the WZNW model do not represent internal
symmetries but instead are associated with the Clifford algebra of the the
target spacetime. This concept has reappeared recently in discussions of
Dirichlet p-branes and type-IIB supergravity \cite{Nil}.
Since the model of \cite{Idea} is a (1,0) theory, when analyzed in terms of
its left and right handed degrees of freedom, one finds the expected
result that the left-handed sector is supersymmetric while the
right-handed sector is non-supersymmetric. This observation
is the key to the construction of the non-supersymmetric nonlinear
$\sigma$-model for the type-B string. In the language of 2D field theory
we well know how to take an ordinary boson and separate it into
its left-handed and right-handed segments by use of chiral bosons
\cite{Sieg}. Once this is done, the original left-handed supersymmetric
sector may be disgarded. However, in order to have a consistent
closed bosonic string, it is required to have some non-supersymmetric
left-handed sector. A new left-handed sector may be introduced
by constructing the ``mirror'' of the right-handed sector of the
original (1,0) type-IIB heterotic string non-linear $\sigma$-model.
There are two steps required for constructing the mirror. First,
all the 2D fields that depended only the $\sigma^{\mm}$-coordinate
on-shell must be replaced by fields that depend on $\sigma^{\pp}$-coordinate
on-shell\footnote{In the language of conformal field theory this
corresponds to $X(z) \to {\Tilde X} ({\bar z})$.}. Also the target
spacetime Clifford algebra of the original Ramond-Ramond sector of
the (1,0) model must be replaced by its target spacetime chiral
reflection\footnote{All the undotted 10D spinor indices must be
replaced by dotted 10D spinor indices.}. This mirror is the `glued'
back to the original right handed sector and the result is the type-B
nonlinear $\sigma$-model with precisely the couplings to the massless
spectrum of the type-B string.
\sect{Review of NSR Supergravity-Heterotic Sigma-Models in (1,0) Superspace}
~~~~Although the original formulation of the heterotic string \cite{HET}
was {\it {not}} as a (1.0) superfield theory, with the work of \cite{BGM},
it became possible to show that the action given by
\begin{equation}
S_{HET-1} ~=~ { 1 \over 4 \pi {\a}' } {\int d^2\sigma d\zeta^-} E^{-1} \Big\{ ~ i
\eta_{\un m \un n} ~ (\nabla_+ { X}^{\un m}) \, (\nabla_{\mm}
{X}^{\un n}) \, - \, (\, \eta_-{}^{\hat I } \nabla_+ \eta_-{}^{\hat
I} \,) ~ \Big\} ~~~,
\end{equation}
upon gauge-fixing to eliminate the (1,0) supergravity fields yields
the action of Gross et.~al. in the fermionic formulation (i.e. ${\hat I}
= 1, ..., 32$). A second formulation (also given by Gross et.~al.)
utilizing chiral bosons \cite{Sieg} whose (1,0) superspace form is
\begin{equation}
\eqalign{
S_{HET-2} ~=~ {\int d^2\sigma d\zeta^-} &E^{-1} \Big\{ \, { 1 \over 4 \pi {\a}' } [ \, i
\eta_{\un m \un n} ~ (\nabla_+ { X}^{\un m}) \, (\nabla_{\mm} {\bf
X}^{\un n}) \, ] ~+~ {~~~~~~~~~~~~~~~~~~} \cr
&{~~~~~~~~} i {1 \over 2 }[\, (\nabla_+ {\Phi_{R}}^{ \hat a } )\,(
\nabla_{\mm}{\Phi_{R}}^{ \hat a }) ~+~ \Lambda_{\mm}{}^{\pp} ( \nabla_+
{\Phi_{R}}^{\hat a} ) \, ( \nabla_{\pp}{\Phi_{R}}^{\hat a })~ ]
\, \Big\} ~~~, }
\end{equation}
where ${\hat a} = 1, ...16$ was also described. Neither of these two
formulations allows for a manifest realization of the $E_8 \otimes E_8$
symmetry of heterotic string theory. To accomplish this, it is necessary
to carry out a non-abelian bosonization \cite{GaSg} of the fermionic
superfields ($\eta_- {}^{\hat I}$) in the first action above\footnote{The
action in (2.2) may be regarded as an abelian bozonization of (2.1).}.
\begin{equation}
\eqalign{
S_{HET-3} ~=~ { 1 \over 4 \pi {\a}' } &{\int d^2\sigma d\zeta^-} E^{-1} [ \, i \eta_{\un m
\un n} ~ (\nabla_+ { X}^{\un m}) \, (\nabla_{\mm} {X}^{\un n})
\, ] {~~~~~~~~~~~~~~~~~~}\cr
-{1 \over 2 \pi } &{\int d^2\sigma d\zeta^-} E^{-1} i {\scriptstyle{1 \over 2}} Tr \{ ~ R_+ R_{=} ~+ i
\L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} R_+ \nabla_+ R_+ \cr
&{~~~~~~~~~~~~~~~~~~~~~~~~~} ~+ {\scriptstyle{2\over3}} \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }}
\{ \, R_+ \, , \, R_+ \, \} ~ R_+ \cr
~+ &\int_{0}^1 d \, y ~[ \, ( {d \widetilde U \over dy} \widetilde
U^{-1} \, ) [ \, \nabla_{=}((\nabla_+\widetilde U ) \widetilde U^{-1})
~-~ \nabla_{+}((\nabla_{=}\widetilde U ) \widetilde U^{-1} ) \,
]~ \} ~~~. ~~~~~~}
\end{equation}
where the following definitions are used,
\begin{equation}
R_a ~\equiv~ U^{-1} \nabla_a U ~~~,~~~ U ~ \equiv ~ exp\Big[ \, i
{\Phi_{R}}^{\hat a } t_{\hat a } \Big] ~~~.
\end{equation}
In this last equation, the quantities ${\Phi_{R}}^{\hat a }$ constitute
496 righton (1,0) superfields containing the same number of component
rightons. From this last equation it is clear that in order to give
a complete specification of the action in (2.3), it is necessary to
introduce a set of matrices (above denoted by $t_{\hat a }$) that form
a Lie algebra. In fact, the action of (2.3) permits us to associate a
{\it {distinct}} world sheet action with {\it {each}} consistent 10D
heterotic string. It has long been known \cite{het10s} that the
groups that lead to tachyon-free 10D heterotic strings are; $E_8
\otimes E_8$, $SO(32)$ and $SO(16) \otimes SO(16)$. Thus the matrices
$t_{\hat a }$ may be chosen to provide representations of these algebras
and there is no need to introduce winding modes to construct the
$E_8 \otimes E_8$ representation.
No background fields at all appear in the actions above. The way to introduce
background spacetime fields is of course well known \cite{cups}. The NS-NS
bosonic fields ($g_{ \un m \un n}$, $b_{ \un m \un n} $ and $\Phi$) are made
to appear by replacing the first line of (2.3) by
\begin{equation}
\eqalign{
S_{NS} &=~ { 1 \over 2 \pi {\a}' } {\int d^2\sigma d\zeta^-} E^{-1} [ i {\scriptstyle{1 \over 2}} ( g_{ \un m
\un n}(X) ~ + ~ b_{ \un m \un n}(X) ) ~ (\nabla_+ { X}^{\un m})
(\nabla_{=} { X}^{\un n}) ~ ] \cr
&{~~~~~~~}+~ {\int d^2\sigma d\zeta^-} E^{-1} \Phi(X) {\S}^+ \cr
&\equiv~ \int d^2 \sigma d\zeta^- E^{-1}
[i \frac12 (\eta_{\un a \un b} + b_{\un a \un b}(X))
\P_{+} {}^{\un a} \P_{=} {}^{\un b}
~+~ \Phi(X) \S^+ ~ ] ~~, }
\end{equation}
where $\P_{+} {}^{\un a} \equiv ( 1/ \sqrt{ 2 \pi {\a}'})( \nabla_+ {
X}^{\un m}) e_{\un m} {}^{\un a} $ and
$\P_{=} {}^{\un a} \equiv ( 1/ \sqrt{ 2 \pi {\a}'})( \nabla_= {
X}^{\un m}) e_{\un m} {}^{\un a} $.
The introduction of the R-R bosonic fields (i.e. the spacetime gauge
fields $A_{ \un a} {}_{ \hat I \hat J }$) for the internal symmetry
groups can be carried out for the $SO(32)$ and $SO(16) \otimes SO(16)$
theories quite easily \cite{SEN}. The last term in (2.1) is replaced by,
\begin{equation}
S_{R} ~=~ - {\scriptstyle{1 \over 2}} \, {\int d^2\sigma d\zeta^-} E^{-1}[~ \eta_-{}^{\hat I } \nabla_+
\eta_-{}^{\hat I} ~+~ \P_{+} {}^{\un a} \, \eta_-{}^{\hat I } A_{\un a}
{}_{ \hat I \hat J }(X) \eta_-{}^{\hat J } ~ ] ~~~.
\end{equation}
However, the introduction of the R-R bosonic fields in the $E_8
\otimes E_8$ theory can {\it {only}} be done by modifying the
last three terms in (2.3). These must be replaced by \cite{BDG},
\begin{equation}
\eqalign{
{S'}_R ~=~ - {1 \over 2 \pi }
&{\int d^2\sigma d\zeta^-} E^{-1} i {\scriptstyle{1 \over 2}} Tr \{ ~ ( R_+ ~+~ 2 \Gamma_+) R_{=} \cr
& ~~+ i \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} (R_+ ~+~ \Gamma_+ ) \nabla_+ (R_+ ~+~ \Gamma_+) \cr
&~~+ {2\over3} \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} \{ \, (R_+ ~+~ \Gamma_+) \, , ~
(R_+ ~+~ \Gamma_+) \, \} ~ (R_+ ~-~ \frac12 \Gamma_+) \cr
&~~+ \int_{0}^1 d \, y ~[ \, ( {d \widetilde U \over dy} \widetilde
U^{-1} \, ) [ \, \nabla_{=}((\nabla_+\widetilde U ) \widetilde U^{-1})
~-~ \nabla_{+}((\nabla_{=}\widetilde U ) \widetilde U^{-1} ) \,
]~ \} ~~~. }
\end{equation}
where $\Gamma_+ ~ \equiv ~ \P_{+} {}^{\un a} A_{\un a} {}^{\hat a}
(X) t_{\hat a}$. Since the non-abelian bosonized theory offers the
most complete description, we will only utilize it in the subsequent
discussion.
In addition to these three 10D tachyon-free heterotic string theories,
there are also a number of other non-supersymmetric theories that contain
a tachyon \cite{het10s}. These include $SO(32)$, $E_8 \otimes SO(16)$, $SO(24)
\otimes SO(8)$, $(E_7 \otimes SU(2))^2$, $SU(16) \otimes U(1)$,
$E_8$. Although it appears not generally known, the (1,0) supergeometry
also permits the introduction of a coupling of the tachyon to
the worldsheet of the heterotic string. This is accomplished
by a slight generalization of a result presented some time
ago \cite{BGM}. The construction requires the introduction of
a world sheet minus spinor superfield $\Psi_-$ that appears
in the action
\begin{equation}
S_{tachyon} ~=~ {\int d^2\sigma d\zeta^-} E^{-1} [~ - {\scriptstyle{1 \over 2}} (\, \Psi_-
\nabla_+ \Psi_- \,) ~+~ i T(X) \Psi_- ~] ~~~.
\end{equation}
In the limit where $T(X) = 1$, this action introduces a cosmological
constant on the worldsheet. Note that each of the non-supersymmetric
models corresponds to a distinct choice of the matrix generators
$t_{\hat a}$ for the groups listed above.
Up until our work of \cite{Idea}, as far as was known, the models
above described all 10D (1,0) supergravity NSR heterotic $\sigma$-models.
With this work we proposed the addition of one more such model.
The introduction of this final model, begins with the observation that
the $E_8 \otimes E_8$ algebra in (2.4) may be replaced by another
(non-compact) Lie algebra so that the new group elements take the
form
\begin{equation}
U \, \to\, U_{R-1} ~\equiv~ exp\Big[\, \Phi_R \d_{\a} {}^{\b}
~+~ \frac 12 \Phi_{\un a \un b\, R} (\sigma^{\un a \un b})_{\a}^{~\b}
~+~ \frac 1{24} \Phi_{\un a \un b \un c \un d\, R} (\sigma^{\un a
\un b \un c \un d})_{\a}^{~\b} \, \Big]
\end{equation}
where the 10D Pauli matrices above correspond to the right-handed
spacetime chiral projection of the 10D gamma matrices! This structure
introduces 256 (1,0) righton superfields ($\Phi_R$, $\Phi_{\un a \un b
\, R}$ and $\Phi_{\un a \un b \un c \un d\, R}$) instead of the
familiar 496 (1,0) righton superfields of the $E_8 \otimes E_8$
model. The $\sigma$-model associated with this alternative construction
is obtained by using the group elements in (2.9) to replace those in
(2.4). As well the quantity $\Gamma_+$ below (2.7) must be replaced by
\begin{equation}
\Gamma_+ ~\to ~ \Pi_+ {}^{\un a} \Big[ \,
(\nabla_{\un a} {\rm A} )~
\d_{\a} {}^{\b} ~+~ \frac 12 {\rm F}_{\un a \un b \un c}~
(\sigma^{\un b \un c})_{\a}^{~\b} ~+~ \frac 1{24} {{\rm F}^{(+)}}_{\un a
\un b \un c \un d \un e}~ (\sigma^{\un b \un c \un d \un
e})_{\a}^{~\b} \, \Big]
\end{equation}
The R-R bosonic fields here are precisely what is needed so that their
addition to the NS-NS fields describes the bosonic spectrum of 10D,
type-IIB supergravity. The $\sigma$-model we have described can easily
be studied upon performing dimensional compactification. These
actions provide an intrinsic description of the models that were
recently studied by Maharana \cite{2B}.
Thus, we argued that among the family of 10D heterotic string theories,
by different choices of the matrix generators we could describe all
previous known models in a uniform manner. As we found
there seemed to exist one more member of this family that corresponds
to replacing the internal compact group generators by the non-compact
chiral 10D Pauli matrices and gives rise to a heterotic string model
with a N = 2B supersymmetry.
Finally there is an intriguing interpretation that we can give to
(2.3) as modified in (2.9) and (2.10). Within the context of superfield
theories, it is well known that it is possible to use low N superfields
to realize a theory with a higher than N supersymmetry. Some examples
of this are the use of 4D, N = 1 superfield perturbation theory to
describe 4D, N = 4 supersymmetric Yang-Mills theory \cite{GRS} or to
describe 4D, N = 2 supersymmetric Yang-Mills theory \cite{G}. In the
same way, our work suggests that it is possible to realize the N = 2B
superstring as a theory of N = 1 superstrings coupled to a certain
``matter superstring'' containing the R-R sector p-forms.
\sect{10D Green-Schwarz Heterotic Sigma-Models}
~~~~In the last section, we used (1,0) superfields to review the
situation of (1,0) supergravity heterotic $\sigma$-models. Since it is
our eventual goal to show that the last model above is very closely
related to the type-B and type-O bosonic string theories, as a first
step we begin to eliminate the (1,0) world sheet supersymmetry by
going to a Green-Schwarz type formulation. The relevant starting points
are (2.5) and (2.7). The action in (2.5) is replaced by ${\cal S}_{GS}$
$$
S_{GS} ~=~\int d^2 \sigma \,V^{-1}\Bigl [ - {\scriptstyle {\frac 12 }} \P_{\pp}^{~
\un a} \P_{\mm~\un a}~+~ \int_{0}^{1}dy \, \hat {\P}_{y}~^{\un C}\hat{\P}_{
\pp}^{~\un B}\hat{ \P}_{\mm}~^{\un A}\hat{G}_{\un A\un B\un C} ~+~
\Phi(Z) {\cal R}(V) ~\Bigr ] ~~~, ~~~ $$
$$\P _{\pp}~^{\un A}~=~V_{\pp}^{ ~m}( \partial_{m}Z^{\un M}) E_{\un M}~^{\un
A}~~,~~
\P _{\mm}~^{\un A}~=~V_{\mm}~^{m}(\partial_{m}Z^{\un M}) E_{\un M}~^{
\un A} ~~,$$
\begin{equation}
\hat{Z}^{\un M}~=~Z^{\un M}(\sigma,\t,y)~~,~~\hat{\P}_{y}{}^{\un
A}~=~(\partial_{y}\hat{Z}^{\un M})E_{\un M} {}^{\un A} ({\hat Z})~~,~~
\hat{G}_{\un A\un B\un C} = G_{\un A\un B\un C}(\hat{Z})~.
\end{equation}
here $Z^{\un M}(\t, \sigma)$ is the superstring coordinate
($Z^{\un M}(\t, \sigma) \equiv (\Theta^{\mu}(\t, \sigma), X^{\un m}(\t,\sigma))$,
$G_{\un A \un B\un C}$ is the field strength supertensor for a super
2-form $b_{\un A \un B}(Z)$, $\Phi(Z)$ is the spacetime dilaton and
${\cal R}(V)$ is the world sheet curvature tensor. Since only in the
cases of the $E_8 \otimes E_8$ or $SO(32)$ theories is there spacetime
supersymmetry\footnote{The only reason to use a Green-Schwarz formulation
is precisely the presence of spacetime \newline ${~~~~~}$ supersymmetry.},
we do not at this stage have to worry about the introduction of a
term to accommodate the tachyon that occurs for the other heterotic
strings.
The replacement action for (2.7) can be obtained in the following
manner. The action in (2.7) is a (1,0) superfield action, which
using standard techniques, we have many times previously analyzed in
terms of its component fields. The component field formulation contains
fermions (none of which are dynamical), so to pass to an action
that is compatible with the Green-Schwarz type action above in (3.1),
we simply set all fermions to zero. This leaves the action
\begin{equation}
\eqalign{
S_R = - { 1 \over 4 \pi }
\int d^2 \sigma \, &V^{-1} ~Tr \{~ ({\cal D}_{\pp} U^{-1} ) ({\cal D}_{\mm} U )
~+~ \l_{\mm} {}^{\pp} ( U^{-1} D_{\pp} U )^2 \cr
+~ & \int_{0}^1 dy ( {\tilde U}^{-1} { d~~ \over dy } {\tilde U})
[~ ({\cal D}_{\pp} {\tilde U}^{-1} ) ({\cal D}_{\mm} {\tilde U} ) \,-\, ({\cal D}_{\mm}
{\tilde U}^{-1} ) ({\cal D}_{\pp} {\tilde U} ) ~ ] ~ \cr
- 2 & \, \P_{\pp}~^{\un B} \, \Gamma_{\un B}^{~~\hat a}
t_{\hat a} ( U^{-1} {\cal D}_{\mm} U ) ~ \} ~~ ,~~ }
\end{equation}
with $D_{\pp} U \equiv {\cal D}_{\pp} U - i \P_{\pp}~^{\un B} {\Gamma}_{\un B}^{~~\hat a}
U t_{\hat a}$. The quantity $U \equiv exp [i {\phi}^{\hat a }_R (\t ,
\sigma) t_{\hat a }]$ is an element of an {\it {a}} {\it {priori}} arbitrary
group. The matrices $t_{\hat a}$ generate a compact Lie algebra for the
right-gauge group ${\cal G}_R$ where $\hat a = 1 , \dots ,d_G$, $ [ t_{
\hat a },~t_{\hat b } ] = i f_{\hat a \hat b}{}^{\hat c} t_{\hat c }$,
$f_{\hat a \hat b \hat c} f^{\hat a \hat b}{}_{\hat d} = c_2 {\d}_{\hat
c \hat d }$, and $ Tr (t_{\hat a} t_{\hat b } ) = 2 k \d_{\hat a \hat
b}$. Above, we have used the notation (${\cal D}_{\pp}, \, {\cal D}_{\mm}$)
to denote the world-sheet two-dimensional gravitationally covariant
derivative. To describe the two tachyon-free 10D theories, we pick
${\cal G}_R = E_8 \otimes E_8$ or $SO(32)$, respectively.
At this stage, we once again replace the compact group generators and
their corresponding 2D righton fields as described in (2.9) but here
all the superfields (the $\Phi$'s) are replaced by a 2D
Duffin-Kemmer-Petiau field $\phi_{\a}{}^{\b}$
\begin{equation}
\phi_{\a}{}^{\b} ~=~ \phi_R \d_{\a} {}^{\b}
~+~ \frac 12 \phi_{\un a \un b\, R} (\sigma^{\un a \un b})_{\a}^{~\b}
~+~ \frac 1{24} \phi_{\un a \un b \un c \un d\, R} (\sigma^{\un a
\un b \un c \un d})_{\a}^{~\b} ~~~.
\end{equation}
One final step is that the spacetime gauge superconnection and
generators on the last line of (3.2) must undergo the replacement
\begin{equation}
\Gamma_{\un A}^{~~\hat a} t_{\hat a} ~\to ~ \Big[ \,
(\nabla_{\un A} {\rm A} )~ \d_{\a} {}^{\b} ~+~ \frac 12 {\rm F}_{\un A
\un b \un c}~ (\sigma^{\un b \un c})_{\a}^{~\b} ~+~ \frac 1{24} {{\rm
F}^{(+)}}_{\un A \un b \un c \un d \un e}~ (\sigma^{\un b \un c \un d
\un e})_{\a}^{~\b} \, \Big] ~~~.
\end{equation}
Thus, we recover (within the context of a Green-Schwarz formulation)
the exact same result as seen from the NSR formulation. Namely, a
heterotic 10D, N = 1 Green-Schwarz $\sigma$-model with manifest $E_8
\otimes E_8$ symmetry is replaced by a model where the spectrum of
field strength superfields in the R-R sector is exactly what one needs
to describe the type-IIB theory.
\sect{On to Bosonic Type-B and Type-O Theories}
~~~~In this final section we complete the journey from the heterotic
type-IIB $\sigma$-model to the purely non-supersymmetric type-B and
type-O models. This is done in several stages. First since these
final models are bosonic, we can begin with (3.1) and simply set
the Grassmann superstring coordinates ($\Theta$) identically to zero.
Since we want to allow coupling to the tachyon, we also add one additional
term to find the NS-NS sector of the type-B theory takes the form
\begin{equation}
\eqalign{
S_{NS}^B &=~ { 1 \over 2 \pi {\a}' } \int d^2 \sigma ~ V^{-1} [ {\scriptstyle{1 \over 2}}
( g_{ \un m \un n}(X) ~ + ~ b_{ \un m \un n}(X) ) ~ (\nabla_{\pp}
{X}^{\un m}) (\nabla_{\mm} { X}^{\un n}) ~ ] \cr
&{~~~~~~~}+~ \int d^2 \sigma ~ V^{-1} \, [~ \Phi(X) {\cal R} ~+~ T(X) ~ ] \cr
&\equiv~ \int d^2 \sigma V^{-1}
[ \frac12 (\eta_{\un a \un b} + b_{\un a \un b}(X))
\P_{\pp} {}^{\un a} \P_{\mm} {}^{\un b}
~+~ \Phi(X) {\cal R} ~+~ T(X) ~ ] ~~~. }
\end{equation}
This looks almost exactly like (2.5). There is one important
difference of course. All of the quantities in (2.5) are (1,0)
superfields whereas all quantities appearing here are ordinary 2D
fields.
For the righton R-R sector of the type-B theory, we take the action
of (3.2) and simply set to zero the Grassmann superstring coordinates
($\Theta$) to zero.
\begin{equation}
\eqalign{
S_{R-1}^B = - { 1 \over 4 \pi }
\int d^2 \sigma \, &V^{-1} ~Tr \{~ ({\cal D}_{\pp} U_R^{-1} ) ({\cal D}_{\mm} U_R )
~+~ \l_{\mm} {}^{\pp} ( U_R^{-1} D_{\pp} U_R )^2 \cr
+~ \int_{0}^1 dy &( {\tilde U_R}^{-1} { d~~ \over dy } {\tilde U_R})
[~ ({\cal D}_{\pp} {\tilde U_R}^{-1} ) ({\cal D}_{\mm} {\tilde U_R} ) \,-\, ({\cal D}_{\mm}
{\tilde U_R}^{-1} ) ({\cal D}_{\pp} {\tilde U_R} ) ~ ] ~ \cr
- 2 \, \P_{\pp}~^{\un a}
&\Big[ \, (\nabla_{\un a} {\rm A}^1 ) ~+~ \frac 12 {\rm F}_{\un a \un b
\un c}^1 ~ (\sigma^{\un b \un c}) ~+~ \frac 1{24} {{\rm F}^{(+)}}_{\un a
\un b \un c \un d \un e}~ (\sigma^{\un b \un c \un d \un e}) \, \Big] \,
( U_{R}^{-1} {\cal D}_{\mm} U_R ) ~ \} ~~ ,~~ }
\end{equation}
The actions in (4.1) and (4.2) cannot comprise the entirety of the
type-B theory. We obtained the latter of these directly from the
GS action of the previous section. To complete the type-B theory
we must construct the ``mirror'' to (4.2) above. In the mirror action
all indices of the $\pp$ type are exchanged for the $\mm$ type and
vice-versa. However, this is not sufficient. In addition we must
replace the non-supersymmetric limit of $U_{R-1}$ by its mirror also.
This is done as follows.
In 10D, the notion of chiral spinors exists. In fact, the matrices that
appear in (2.9) are in such a basis. There also exist 10D matrices
where we exchange the handedness of the spacetime spinor indices
(i.e. $\a \to {\dot \a}$ etc.). Thus we may introduce $U_{L}$
according to the definition,
\begin{equation}
U_{L} ~\equiv~ exp\Big[\, \phi_L \d_{\dot \a} {}^{\dot \b}
~+~ \frac 12 \phi_{\un a \un b\, L} (\sigma^{\un a \un b})_{\dot \a}^{~
\dot \b} ~+~ \frac 1{24} \phi_{\un a \un b \un c \un d\, L} (\sigma^{\un a
\un b \un c \un d})_{\dot \a}^{~\dot \b} \, \Big] ~~~
\end{equation}
which introduces 256 lefton fields ($\phi_L$, $\phi_{\un a \un b
\, L}$ and $\phi_{\un a \un b \un c \un d\, L}$). Like its right
handed mirror counterpart, the argument of the exponential function
above can be shown to form a Lie algebra. In fact, its composition
law is exactly the same as that given for the right handed one in
ref. \cite{Idea}, with the exception that the term involving the
Levi-Civita tensor has the opposite sign. The complete ``mirror''
action to (4.2) is given by
\begin{equation}
\eqalign{
S_{R-2}^B = - { 1 \over 4 \pi }
\int d^2 \sigma \, &V^{-1} ~Tr \{~ ({\cal D}_{\pp} U_L^{-1} ) ({\cal D}_{\mm} U_L )
~+~ \l_{\pp} {}^{\mm} ( U_L^{-1} D_{\mm} U_L )^2 \cr
-~ \int_{0}^1 dy &( {\tilde U_L}^{-1} { d~~ \over dy } {\tilde U_L})
[~ ({\cal D}_{\pp} {\tilde U_L}^{-1} ) ({\cal D}_{\mm} {\tilde U_L} ) \,-\, ({\cal D}_{\mm}
{\tilde U_L}^{-1} ) ({\cal D}_{\pp} {\tilde U_L} ) ~ ] ~ \cr
- 2 \, \P_{\pp}~^{\un a}
&\Big[ \, (\nabla_{\un a} {\rm A}^2 ) ~+~ \frac 12 {{\rm F}}_{\un a
\un b \un c}^2 ~ ({\Bar \sigma}^{\un b \un c}) ~+~ \frac 1{24} {{\rm
F}^{(-)}}_{\un a \un b \un c \un d \un e}~ ({\Bar \sigma}^{\un b \un c
\un d \un e}) \, \Big] \, ( U_{L}^{-1} {\cal D}_{\mm} U_L ) ~ \} ~~ .~~ }
\end{equation}
On the final line above, we have placed bars above the 10D Pauli matrices
to indicate that these are the ones with the dotted spacetime spinor
indices. So the complete non-linear $\sigma$-model for the type-B
theory is just
\begin{equation}
S_{Tot}^B ~=~ S_{NS}^B~+~ S_{R-1}^B~+~S_{R-2}^B ~~~.
\end{equation}
It can be seen that the complete spectrum of spacetime fields includes
the graviton ($g_{ \un m \un n}$), axion ($b_{ \un m \un n} $), dilaton
($\Phi$), tachyon ($T$), two scalars (${\rm A}^1$ and ${\rm A}^2$
)\footnote{An interesting feature to note is that the scalars $A^1$ and $A^2$
possess ``moduli'' (i.e. we may \newline ${~~~~~}$ shift them by constants
without affecting the action) unlike the scalars that usually appear
\newline ${~~~~~}$ in Lefton-Righton Thirring Models \cite{BDG}.}, two
Kalb-Ramond fields (${\rm A}_{ \un m \un n}^1$ and ${\rm A}_{\un m \un
n}^2$), 4-form ${\rm A}_{ \un a \un b \un c \un d}^+$ with a self-dual
field strength ${{\rm F}^{(+)}}_{\un a \un b \un c \un d \un e}$ and a
second 4-form ${\rm A}_{\un a \un b \un c \un d}^-$ with an anti-self-dual
field strength ${{\rm F}^{(-)}}_{\un a \un b \un c \un d \un e}$.
The key to our successful construction of the model described
immediately above is the suggestion that currents associated with
internal symmetries can sometimes be traded for currents associated with
the Clifford algebra of the spacetime spinors. We first did this in
the 10D supersymmetric theory. If this conjecture is accepted, then
the reverse is likely to also be true. With this as a working assumption,
we will now show that the type-O string \cite{Sag} propagating in
the presence of its tachyon and massless modes can be constructed
from the corresponding type-B theory.
So we once more will trade some of the currents. In particular
we note that by keeping only the middle terms in the exponential
of (3.3) and (4.3) we also maintain the structure of a Lie group.
So the simple idea is to trade the currents associated with the
0-forms and 4-forms and replace them by currents associated with
$SO(32)$. This is all easily done for the $\sigma$-models and leads
to the following type-O action. The final answer is rather
tedious to write only because it possesses a number of different R-R
sectors but takes the form
\begin{equation}
S_{Tot.}^{O} ~\equiv ~ S_{NS}^{O} ~+~ S_{R-1}^O ~+~ S_{R-2}^O
~+~ S_{R-3}^O ~+~ S_{R-4}^O ~~~.
\end{equation}
Below we give each sub-action.
\begin{equation}
\eqalign{
S_{NS}^{O} &=~ \int d^2 \sigma V^{-1} [ \frac12 \,\eta_{\un a \un b}
\P_{\pp} {}^{\un a} \P_{\mm} {}^{\un b} ~+~ \Phi(X) {\cal R}
~+~ T(X) ~ ] ~~~. }
\end{equation}
\begin{equation}
\eqalign{
S_{R-1}^O = - { 1 \over 4 \pi }
\int d^2 \sigma \, &V^{-1} ~Tr \{~ ({\cal D}_{\pp} U_{R}^{-1} ) ({\cal D}_{\mm} U_{R} )
~+~ \l_{\mm} {}^{\pp} ( U_{R}^{-1} D_{\pp} U_{R} )^2 \cr
+~ \int_{0}^1 dy &( {\tilde U_{R}}^{-1} { d~~ \over dy } {\tilde
U_{R}})
[~ ({\cal D}_{\pp} {\tilde U_{R}}^{-1} ) ({\cal D}_{\mm} {\tilde U_{R}} ) \,-\, ({\cal D}_{\mm}
{\tilde U_{R}}^{-1} ) ({\cal D}_{\pp} {\tilde U_{R}} ) ~ ] ~ \cr
- \, \P_{\pp}~^{\un a} &\Big[ \, {\rm F}_{\un a \un b \un c}^1 ~
(\sigma^{\un b \un c}) \, \Big] \, ( U_{R}^{-1} {\cal D}_{\mm} U_{R} ) ~
\} ~~ ,~~ }
\end{equation}
\begin{equation}
\eqalign{
S_{R-2}^O = - { 1 \over 4 \pi }
\int d^2 \sigma \, &V^{-1} ~Tr \{~ ({\cal D}_{\pp} U_{L}^{-1} ) ({\cal D}_{\mm} U_{L} )
~+~ \l_{\mm} {}^{\pp} ( U_{L}^{-1} D_{\pp} U_{L} )^2 \cr
-~ \int_{0}^1 dy &( {\tilde U_{L}}^{-1} { d~~ \over dy } {\tilde
U_{L}})
[~ ({\cal D}_{\pp} {\tilde U_{L}}^{-1} ) ({\cal D}_{\mm} {\tilde U_{L}} ) \,-\, ({\cal D}_{\mm}
{\tilde U_{L}}^{-1} ) ({\cal D}_{\pp} {\tilde U_{L}} ) ~ ] ~ \cr
- \, \P_{\mm}~^{\un a} &\Big[ \,{\rm F}_{\un a \un b \un c}^2 ~
({\bar \sigma}^{\un b \un c}) \, \Big] \, ( U_{L}^{-1} {\cal D}_{\pp} U_{L} ) ~
\} ~~ ,~~ }
\end{equation}
\begin{equation}
\eqalign{
S_{R-3}^O ~\equiv ~ \int d^2 \sigma \, &V^{-1} \Big[ \, i \, \psi_- (
\, {\cal D}_{\pp} ~+~ i \P_{\pp}~^{\un a} \, A_{\un a}^{\hat a} ~t_{\hat a}
\, ) \psi_- ~ \Big] ~~~,~~{~~~~~~~~~~~~~~~~~~~~} }
\end{equation}
\begin{equation}
\eqalign{
S_{R-4}^O ~\equiv ~ \int d^2 \sigma \, &V^{-1} \Big[ \, i \, \psi_+ (
\, {\cal D}_{\mm} ~+~ i \P_{\mm}~^{\un a} \, {\tilde A}_{\un a}^{\hat a}
~t_{\hat a} \, ) \psi_+ ~ \Big]
~~~.~~ {~~~~~~~~~~~~~~~~~~~~} }
\end{equation}
In the latter two equations above, we have introduced 64 right-moving
and 64 left-moving Majorana fermions ($\psi_-$ and $\psi_+$, respectively)
on the worldsheet. These are the degrees of freedom that were traded by
removing the lefton and righton 0-forms and 4-forms in $U_R$ and $U_L$ in
the type-B theory.
The complete spectrum of spacetime fields includes the graviton
($g_{ \un m \un n}$), dilaton ($\Phi$), tachyon ($T$), two sets
of $SO(32)$ gauge fields ($A_{\un a}^{\hat a}$ and $ {\tilde A}_{\un
a}^{\hat a}$) and two Kalb-Ramond fields (${\rm A}_{\un m \un n}^1$
and ${\rm A}_{\un m \un n}^2$). This is the spectrum of the type-O
bosonic string at low orders.
\noindent
{\bf {Acknowledgment; }} \newline \noindent
${~~~~}$One of the authors (SJG) wishes to acknowledge M. T. Grisaru,
M. Gaberdiel, Nikita Nekrasov and B. Zweibach and for useful discussions.
Additional thanks go to R. Whittlesay and N. Buckner who acted as
the catalysts for these discussions.
\newpage
\noindent{{\bf {APPENDIX A: Kalb-Ramond Matter Field and
4D String $\sigma$-models }}}
One interesting consequence of the construction of the model in
ref. \cite{Idea} is that a compactification of the model reveals
how 4D Kalb-Ramond matter fields can be coupled to the world
sheet in NSR or GS string $\sigma$-models. The idea is that the
previous work utilizing Lefton-Righton Thirring Models \cite{BDG,LRTM}
can appropriately be modified to include Kalb-Ramond fields as
long as the Kalb-Ramond fields do not carry any internal
charges. In this brief appendix we will demonstrate how this
construction is carried out within the confines of the 4D, N = 8
(1,0) supergravity NSR $\sigma$-model. We only pick this choice
because of its simplicity. Any model with smaller values of N
(or even larger values of D) can be treated by the same techniques
as those we use below.
The 4D, N = 8 (1,0) supergravity NSR $\sigma$-model is described by
using the NSR sector in (2.5) but the R-R sector is replaced by
$$
\eqalign{
S_{R-5} = \int d^2 \sigma d\zeta^- E^{-1} i\frac12 [~
( &L_{=}^{\hat \a} + \Gamma_{=}^{\hat \a} ) ( L_+^{\hat\a}
- \L_+ {}^{=} (L_{=}^{\hat\a} + \Gamma_{=}^{\hat\a}))
+ L_+^{\hat\a} \Gamma_{=}^{\hat\a} \cr
+ & ( R_+^{\hat I} + 2 \Gamma_+^{\hat I}) R_{=}^{\hat I}
- i \L_{=}{}^{\pp} (R_+^{\hat I} + \Gamma_+^{\hat I} )
\nabla_+ (R_+^{\hat I}+\Gamma_+^{\hat I}) \cr
+ & 4 S^{\a \hat I} \scriptstyle R_{=}^{\hat I} \scriptstyle L_+ ^{\hat \a}
- ~ 4 \L_+{}^{=} (M^{-1})^{\hat I\hat K}\Phi^{\hat \a \hat I}
\Phi^{\hat \a \hat J} \S_{=}^{\hat J} \S_{=}^{\hat K} \cr
- & 4 i \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} \Phi^{\hat \a \hat I} \scriptstyle L_+^{\hat \a} \nabla_+
( \Phi^{\hat \b \hat I} \scriptstyle L _+^{\hat \b} )~] ~~~, \cr}
\eqno(A.1)
$$
$$
\eqalign{
\scriptstyle L _+ ^{\hat \a} = &\ L _+ ^{\hat \a} - \L _+{}^{=} ( L_{=}^{\hat \a}
+\Gamma_{=}^{\hat\a}) ~~, ~~ L_{A}^{\hat \a} \equiv \nabla_{A}
\varphi_L {}^{\hat \a} ~~, ~~ R_{A}^{\hat I} \equiv
\nabla_{A} \varphi_R {}^{\hat I} ~~, \cr
\scriptstyle R _{=} ^{\hat I} = &\ R _{=} ^{\hat I} - i [\L _{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }}
\nabla _+ (R_+ ^{\hat I} + \Gamma_+^{\hat I}) + \frac12 (\nabla _+ \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }})
(R_+ ^{\hat I} + \Gamma _+ ^{\hat I}) ] ~~, \cr
\S_{=}^{\hat I} = &\ \scriptstyle R_{=}^{\hat I} -2 i [ \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }}
\nabla_+ (\Phi^{\hat \b \hat I} \scriptstyle L_+^{\hat \b} )+\frac12 (\nabla_+
\L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} ) \Phi^{\hat \b \hat I}
\scriptstyle L_+^{\hat \b} ] ~~, \cr
~~ ( M )^{\hat I \hat J} = & \ \d^{\hat I \hat J} -4 i ( \nabla_+
\L _+{}^{=}) \L_{=}{}^{{{ =\hskip-3.75pt{\lin}}\hskip3.75pt }} \Phi^{\hat
\a \hat I} \Phi^{\hat \a \hat J} ~~ ,} \eqno(A.2)
$$
$$
\eqalign{
\Gamma_{=}{}^{\hat \a} & \equiv \P_{=}{}^{\un a} A_{\un a}{}^{\hat \a}(X)
~~, ~~ A_{\un a}{}^{\hat \a} = ({\widetilde A}_{\un a}^{[ij]})~~, \cr
\Gamma_+{}^{\hat I} & \equiv \P_{+} {}^{\un a} A_{\un a}{}^{\hat I}
(X)~~, ~~
A_{\un a}{}^{\hat I} = (A_{ \un a}, A_{\un a}^{[i^{\prime}j^{\prime}]},
A_{ \un a}^{[i^{\prime}j^{\prime}] [ k^{\prime } l^{\prime} ]} ) ~~.
\cr } \eqno(A.3)
$$
$$
\eqalign{
\Phi_{\hat \a \hat I} & \equiv ~ ( ~ \Phi_{[ij]} ,~
\Phi_{[ij] [i^{\prime}j^{\prime}]} ,~ \Phi_{[p^{ \prime}
q^{\prime}] [ i^{\prime} j^{\prime}] [ k^{\prime} l^{\prime}]}
,~ \d_{ i^{\prime} [i } \d_{j ] j^{\prime} }
{\widetilde \Phi}_{[ k^{\prime} l^{\prime}]} -
\d_{ k^{\prime} [i } \d_{j ] l^{\prime} }
{\widetilde \Phi}_{[ i^{\prime} j^{\prime}]} ~) ~~~. }
\eqno(A.4)
$$
We see in addition to the graviton, axion and dilaton which
appear in the NSR sector, there also appear 28 spin-1 fields
$({\widetilde A}_{\un a}^{[ij]}, \, A_{ \un a}, A_{\un a}^{[i^{\prime}
j^{\prime}]}, \, A_{ \un a}^{[i^{\prime}j^{\prime}] [ k^{\prime }
l^{\prime} ]} )$ and 68 scalar fields $(\Phi_{[ij]} , \,
\Phi_{[ij] [i^{\prime}j^{\prime}]} ,~ \Phi_{[p^{ \prime}
q^{\prime}] [ i^{\prime} j^{\prime}] [ k^{\prime} l^{\prime}]}
, \, {\widetilde \Phi}_{[ k^{\prime} l^{\prime}]} )$ which
complete the bosonic spectrum to that of 4D, N = 8 supergravity.
We now wish to carry out a duality transformation on the world
sheet of the (1,0) string whereby some of the scalar fields
are replaced by Kalb-Ramond fields. We will use the scalar
field ${\widetilde \Phi}_{[ k^{\prime} l^{\prime}]}$ for the
purpose of illustration.
We first set to zero both ${\widetilde \Phi}_{[ k^{\prime} l^{\prime}]}$
and $\phi_R^{[ k^{\prime} l^{\prime}]}$. The former operation
eliminates the spacetime scalar ${\widetilde \Phi}_{[ k^{\prime}
l^{\prime}]}$ from among the background fields and the latter
operation eliminates the modes of the string which describe this
state. At this intermediate point, the theory will be inconsistent.
In order to restore the consistency, new modes on the world
sheet must be introduced. We can do this by introducing a new
righton WZNW model precisely of the form of the R-R terms in
(2.3). However, the group element that corresponds to (2.4)
is here described by
$$
U_{R-2} ~\equiv~ exp\Big[\, \frac 12 \Phi_{\un a \un b\, R}^{[ k^{\prime}
l^{\prime}]} ~(\sigma^{\un a \un b}) \otimes {\cal T}_{[ k^{\prime}
l^{\prime}]} \, \Big]
\eqno(A.5)
$$
where now $\sigma^{\un a \un b}$ denotes the right-handed Pauli matrix
Lorentz generator for 4D spacetime. Also in the above expression
${\cal T}_{[ k^{\prime} l^{\prime}]}$ denotes a matrix representation
of a $U(1)^6$ group. Note that the fact that this is an abelian
group is absolutely critical. The matrices $\sigma^{\un a \un b} \otimes
{\cal T}_{[ k^{\prime} l^{\prime}]}$ clearly form an algebra.
This would not necessarily be the case if ${\cal T}_{[ k^{\prime}
l^{\prime}]}$ represented some non-abelian group. So the duality on
the worldsheet correspond to
$$
{\widetilde \Phi}_{[ k^{\prime} l^{\prime}]} ~ \to ~
\Phi_{\un a \un b\, R}^{[ k^{\prime} l^{\prime}]} ~~~.
\eqno(A.6)
$$
Finally to complete the introduction of the 4D spacetime Kalb-Ramond
field, we replace the $\Gamma_+$ in (2.7) by
$$
\Gamma_+ ~\to ~ \Pi_+ {}^{\un a} \, F_{\un a \, \un b \, \un c}^{[ k^{\prime}
l^{\prime}]} ~(\sigma^{\un b \un c}) \otimes {\cal T}_{[ k^{\prime}
l^{\prime}]} ~~~.
\eqno(A.7)
$$
We note that interestingly enough, the reduction of the type-IIA
theory is expected to possess precisely six matter Kalb-Ramond
fields. Thus, the final model that we have discussed (or possibly
its mirror with respect to the duality transformation described above)
might present an intrinsic approach to investigating heterotic
type-IIA duality totally within the confines of heterotic
string theory.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,538
|
Raumbild-Verlag ("Raumbild") was a German publishing outfit which focused exclusively on stereoscopic imagery, usually accompanied by expository text. Founded in Dießen in the 1930s by Otto Wilhelm Schönstein (1891–1958), Raumbild produced 6x13cm photographic stereo pairs designed to be used with the proprietary fold-out "Photoplastikon" viewer produced for the company, some of which featured Zeiss Jena lenses. Quickly aligning itself with the propaganda arm of the NSDAP, Raumbild moved around several times before settling in Munich in 1939. After the defeat of Germany in the Second World War, Raumbild changed directions in terms of content, while still focusing on subjects relevant to German identity. In 1996, the German Historical Museum obtained the remaining archives of Raumbild, which are now available to researchers.
Founder
Otto Wilhelm Schönstein was born in Nürnberg in 1891, and in his early life became a textile merchant. He was fascinated by stereoscopic photography, and in his early 40s he decided to transition from hobbyist into professional purveyor of stereoscopic images. After the Second World War, Schönstein was tried and released for his complicity in Nazi propaganda; he lived in relative obscurity, publishing new 6x13cm stereoview sets under pseudonyms until his death in 1958.
Establishment, early efforts, and association with Nazism
Schönstein incorporated Raumbild on 14 January 1935. Although his first publishing efforts were failures due to negative reviews from Reichsstelle zur Forderung des deutschen Schrifttums, his later work with Raumbild throughout the mid-1930s and through the mid 1940s was very popular in Germany. The country was swept up in nationalistic fervor; Schönstein would capitalize on this in his works after changing his focus from travel photography to Germanic subjects. He quickly came to the attention of the Nazi Party, and his company became a de facto photographic arm of the Propaganda Ministry of the Party. This association led to unprecedented access for Schönstein and his employees and collaborators, beginning with the 1936 Summer Olympics in which he collaborated with Leni Riefenstahl. The resulting book & stereograph collection, Die Olympischen Spiele, was a commercial success, at which point Heinrich Hoffmann, Adolf Hitler's personal photographer, became a silent partner with a 50% stake in Raumbild for a 5,000 RM initial investment and a 10,000 RM loan. Hoffmann became the official photographer, while Schönstein – regarded as an inferior lensman – curated and produced the various sets released by the company. This partnership cemented a longstanding relationship between Raumbild and the Third Reich which lasted until the end of the Second World War, although production of new materials ceased in 1942 as the war efforts made continuing photography difficult.
Postwar activities
After the war, Raumbild stopped producing materials related to the NSDAP. During denazification, which continued until 1953, Schönstein continued to create photographs and photographic sets; after a 6-month probationary period in 1946 related to his collusion with the Third Reich, Raumbild resumed production. Schönstein produced sets with the intention of selling to American occupying forces; this provided the bulk of his income for the remainder of his life. In 1951, most of Schönstein's assets were seized as a judgement against his actions during the War; nevertheless, he would produce one of his final major works through Raumbild, covering the 1952 Olympics. During this period, Raumbild was primarily publishing the works of other photographers, primarily in color, and the subject matter ranged from landscapes to architecture to nudes. In early 1958, Raumbild was declared defunct, and Schönstein would have to sell; the company passed to Siegfried Brandmüller, a former employee of Schönstein's who would run the company for the next 27 years. The 23-year-old Brandmüller kept Schönstein on as an employee, which greatly embarrassed the former Raumbild giant; he would die just six months later. After a few initial successes by Brandmüller, Raumbild went dormant for a long time, although a number of attempts were made to re-launch the company; all of these failed. Raumbild folded for good in the mid-1980s. The Raumbild archives (outside of those seized by the American government and now in the public domain) changed hands a number of times and are now owned by the German Historical Museum.
Publications
It is difficult to produce a definitive list of Raumbild's output, as some works were produced in extremely small editions, and numerous smaller manufacturers attached the Raumbild name to unauthorized sets not actually affiliated with the company. Below is a partial list of known works produced by Raumbild-Verlag. Because Raumbild views are designed for a specific type of viewer, they cannot be directly transposed into anaglyphs; some cropping is required.
Early works
1935, Venedig, ein Raumerlebnis ("Venice, a space experience"), a travel series on Venice, which was rejected by the NDSAP Propaganda Ministry Photographers: Schönstein, Kurt Lothar Tank
1935, Unnamed series on Rome and the Vatican, never completed
Nazi propaganda
1936, Die Olympischen Spiele ("The Olympic Games"), a 100-card set & book featuring the 1936 Summer Olympics in Berlin, as well as the Winter Olympics and Chess Olympiad. Photographers: Schönstein, Heinrich Hoffmann
1936, Münchner Karneval ("Munich Carnival"), a 10-card set. Photographer: Schönstein
1936, Reichsparteitag der Ehre ("Reich Party of Honor"), a 100-card set & book focused on the 1936 Nuremberg Rally. Photographer: Hoffmann
1937, Tag der Deutschen Kunst ("Day of German Art"), a 100-card set & book showcasing German works of art. Photographer: Hoffmann
1937, Die Weltausstellung Paris 1937 ("The World's Fair Paris 1937"), a 100-card set & book featuring the 1937 World's Fair. Photographer: Hoffmann
1937, Reichsparteitag der Arbeit ("Reich Party of Labor"), a 100-card set & book featuring the 1937 Nuremberg Rally. Photographer: Hoffmann
1938, Deutsche Gaue ("German Authority"), a 100-card set & book focused on German identity, primarily with patriotic German scenic and architectural images. Photographer: Hoffmann
1938, HITLER / MUSSOLINI: der Staatsbesuch des Führers in Italien ("The State Visit of the Leader in Italy"), a 100-card set & book about Adolf Hitler's official visit to Mussolini's Italy. Photographer: Hoffmann
1939, Großdeutschlands Wiedergeburt – Weltgeschichtliche Stunden an der Donau ("Greater Germany Rebirth – World Historical Hours on the Danube"), a 120-card set & book commemorating the reunification of Germany and Austria. Photographer: Hoffmann
1938, Parteitag Großdeutschland ("Party Congress Greater Germany"), a 100-card set & book centered around the 1938 Nuremberg Rally. Photographer: Hoffmann
1939, Der Traditionsgau München-Oberbayern ("The Traditional Munich & Upper Bavaria"), a very limited 100-card set & book focusing on the seat of power for the Third Reich, given to elite Party members. Photographer: Hoffmann
1939, Der Erste Großdeutsche Reichskriegertag ("The First Greater-German State War Veteran's Day"), a 100-card set & book featuring a military celebration in Kassel. Photographer: Hoffmann
1939, Die Soldaten des Führers im Felde – Der Feldzug in Polen 1939 ("The Soldiers of the Führer in the Field – The Campaign in Poland 1939"), a 100-card set & book focused on the Wehrmacht invasion of Poland. Photographers: Hoffmann and Hugo Jäger
1940, Der Kampf im Westen ("The Struggle in the West"), a 100-card set & book featuring battle photography after the capture of Poland. Photographers: unknown (various) There is a rare, late-1944 version of this publication in existence, which removed a photograph of Erwin von Witzleben after the failed 20 July plot.
1942, Die Kriegsmarine ("The Wartime Navy"), a 100-card set & book that covers German naval prowess before and during the War. Photographers: unknown (various)
1942, Fliegen und Siegen ("Flying and Victory"), a 100-card set & book that covers the German Luftwaffe before and during the War. Photographers: unknown (various)
Postwar productions
1949, Martin's Kunstmappen ("Martin's Art folders"), Series I: Outdoor Nudes (20-card set), series II: Indoor Nudes (20-card set). Photographer: Hans S. Martin
1952, Deutsche Fußball-Olympiamannschaft im Wettstreit der Nationen ("German Olympic football team in the competition of nations"), a 50-card set featuring the German football team at the 1952 Olympic Games. Photographer: Schönstein.
1952, Stadt und Landschaft der US-Zone. ("U.S. Occupied Zone of Germany"), 120-card set & book featuring Germany during denazification. Photographer: unknown
1959, Grock lacht über Gracht. 30 card boxed set featuring famed Swiss clown Grock. Photographer: Kurt Gelsner.
References
Publishing companies of Germany
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,333
|
{"url":"http:\/\/clay6.com\/qa\/55886\/a-point-d-is-on-the-side-bc-of-an-equilateral-triangle-abc-such-that-dc-lar?show=55887","text":"Comment\nShare\nQ)\n\n# A point D is on the side BC of an equilateral triangle ABC such that $DC=\\large\\frac{1}{4}$$BC$.Then $AD^2$=____$CD^2$\n\n$\\begin{array}{1 1}16\\\\13\\\\12\\\\10\\end{array}$\n\n$AD^2=13 CD^2$","date":"2019-05-20 00:38:25","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\": 0.8639537692070007, \"perplexity\": 1554.6445773672563}, \"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\/1558232255251.1\/warc\/CC-MAIN-20190520001706-20190520023706-00486.warc.gz\"}"}
| null | null |
{"url":"http:\/\/mathhelpforum.com\/discrete-math\/171185-what-does-relation-mean-print.html","text":"# What does this relation mean?\n\n\u2022 Feb 13th 2011, 05:10 PM\nMSUMathStdnt\nWhat does this relation mean?\nWhat does it mean to multiply an element of a group by a subgroup?\n\nFrom my Discrete class notes:\nLet $G$ be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let $H$ be a subgroup of $G$.\n\n$\\forall a,b \\in G$, define relation $R$ as $aRb$ if $aH=bH$.\nRelation $R$ is called a 'coset'.\n\nWhat does that relation, coset, mean? I interpret the definition as \" $a$ relation $b$ if $a$ left times a subgroup equals $b$ left times the same subgroup\". But that is kind of meaningless. What is an element times a subgroup?\n\nJeff\n\u2022 Feb 13th 2011, 05:17 PM\npickslides\nYes this is a coset, its purpose or application is probably not that important at this stage.\n\nI would make sure you understand how to find aH and Ha and the differences. (aH = Ha only when H is a normal subgroup of G)\n\nDo you have an example?\n\u2022 Feb 13th 2011, 05:48 PM\nMSUMathStdnt\nNo example. I typed everything in my notes (and I'm listening to my recording of the lecture now, there is nothing more in what the Prof. said).\n\nI guess my question was more along the lines of, what is a coset? What is $aH$? Is it just multiplying $a$ times $H$? If yes, what does it mean to multiply an element by a set? Does that simply mean multiplying $a$ times every element of the set? If yes, doesn't that imply that the coset is not a subset of the subgroup.\n\nThe textbook does not have any information on cosets and I'm loathe to look at Wikipedia (it often confuses me worse).\n\nThanks,\nP.S. What happens if I agree with you? (re: your signature) :D\n\u2022 Feb 13th 2011, 05:53 PM\nPlato\nQuote:\n\nOriginally Posted by MSUMathStdnt\nFrom my Discrete class notes:\nLet $G$ be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let $H$ be a subgroup of $G$.\n$\\forall a,b \\in G$, define relation $R$ as $aRb$ if $aH=bH$.\nRelation $R$ is called a 'left coset'.\nWhat does that relation, coset, mean?\n\nFirst of all understand what $aH$ means.\n$aH=\\{ah:h\\in H\\}$ in other words $aH$ is simply a set obtained by operating $a$ on each element of $H$.\nThat is called a left-coset of $H$ generated by $a$.\n\nNow we say that $aRb$ if and only if $aH=bH$.\ni.e. the generate the same left-coset.\n\u2022 Feb 13th 2011, 05:58 PM\nMSUMathStdnt\nQuote:\n\nOriginally Posted by Plato\nFirst of all understand what $aH$ means.\n$aH=\\{ah:h\\in H\\]$ in other words $aH$ is simply a set obtained by operating $a$ on each element of $H$.\nThat is called a left-coset of $H$ generated by $a$.\n\nNow we say that $aRb$ if and only if $aH=bH$.\ni.e. the generate the same left-coset.\n\nOK. I was starting to suspect that. Thanks for confirmation and explanation.\n\nP.S. Was this a typo? Is there an extra 0 in there?\n$aH=\\{ah:h\\in H\\]$\n\nShould read: $aH=\\{ah:h\\in H\\}$ right?\n\u2022 Feb 13th 2011, 06:01 PM\nPlato\nYes, see my edit.","date":"2017-01-24 14:17:28","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\": 44, \"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\": 0.8479273915290833, \"perplexity\": 466.18074725059114}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560284411.66\/warc\/CC-MAIN-20170116095124-00320-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{Introduction}
Publishers are increasingly moving away from ad-based models, because of the well-known failures~\cite{2018adcost} of ad-based internet funding models. The most common adopted alternative is for sites to deploy ``paywalls''.
``Paywalls'' here are a broad term for monetization systems where visitors are charged subscription fees to access site content, sometimes after being able to sample a small amount of content for free.
The upsides of paywall systems are well understood (i.e., they promise to enable the continued creation of high-quality content). Less understood are the risks and larger implications of an increasingly ``walled'' web. Possible risks include reducing societal access to news and information and the privacy harms of the increased user tracking needed to enforce paywalls.
This work aims to improve the understanding of the popularity, risks and benefits of paywalls online. To introduce the topic, we first (i) describe why the web is increasingly moving away from ``open'' models to ``paywalled'' models, (ii) outline
why this transition is an important topic of study for the research community,
and then (iii) present the structure of the rest of the paper.
\subsection{The Move from Ads to Paywalls}
Digital advertising is the current dominant monetization method for web
publishers, and funds much of the web. Publishers sell advertisements along
page content; middle parties buy these ad slots and fill them with images and
content provided by clients and ad-agencies. This process is usually
programmatic, based of user's personal (i.e., behavioral) data, and completed via
real-time programmatic auctions~\cite{rtbPrices17,pachilakis2019no}.
Web sites are increasingly unsatisfied from this ad-based funding system, for
many reasons. First, the system is dominated by two parties, Google and
Facebook, who jointly harvest more than 70\% of global ad
revenues~\cite{adOligopoly,adDuopoly}, reducing the publisher's ``take'' for ad placements through market power. Second, ad-based funding systems suffer from significant and increasing rates of
fraud~\cite{adfraud,adfraud2,Liu:2014:DDC:2616448.2616455,Zarras:2014:DAM:2663716.2663719,malvertising},
depriving web sites of further funding. Third, behavioral advertising
systems are increasingly incompatible with individual and legal privacy
demands~\cite{trackersWWW2016,vallina2016tracking,Leung:2016:YUA:2987443.2987456,DBLP:journals/corr/abs-1805-10505,razaghpanah2018apps}.
Last, users increasingly use ad blocking tools, for a variety of privacy, performance, and aesthetic reasons~\cite{adlergic,nithyanand2016adblocking},
further depriving publishers of revenue. As a result, ad revenues have decreased in recent years. Both big and small publishers are
coming up short on advertising revenue, even if they are long on visitors
traffic. Accounts of publisher-loss under ad-based funding models contain figures as high as~95\%~\cite{techdirtAdLoss}.
The difficulties of ad-based funding systems have pushed publishers to
alternative funding models, including
donations~\cite{wikipediaFunds1,wikipediaFunds2} or in-browser
crypto-mining~\cite{truthMiners2018}. The most common alternative though is ``paywalls'',
where users pay publishers directly to
access the content they create~\cite{drum2019}
Figure~\ref{fig:screenshot} shows a representative example of a paywall system.
Paywalls so far have a mixed record as funding systems for publishers.
Publishers with large, loyal audiences and high-quality content tend to
be successful with this subscription strategy, with The New York
Times~\cite{nyTimesPaywall}, Wired~\cite{wiredPaywall}, The Financial Times~\cite{ftPaywall} and The Wall Street Journal~\cite{wsjPaywall} as
prominent successful examples. The success of paywalls for smaller and more targeted sites (e.g., local news), or sites with
less affluent audiences, is less clear.
It is important to note that the rapid growth of paywalls has drawn the attention of big tech companies like Google, Facebook and Apple, who have started building platforms to provide or support paywall
services~\cite{paywallsApple,paywallsGoogle,paywallGoogle2,paywallsFacebook}, in an effort to claim their share of the market.
\subsection{Understanding the State of Paywalls}
Creating a sustainable system to fund news and related content is an important goal, and paywalls seem to be a promising (partial) solution to the problem. However, this move from ``open'' to ``walled'' business strategies brings significant, understudied risks. For example, paywalls (implicitly or otherwise) may impose a ``class system'' on the web~\cite{paywallsDemocracy1,paywallsDemocracy2}, potentially driving information-seeking visitors who cannot afford to pay for subscriptions to
badly-sourced, less-vetted, or even intentionally false (but free) new sources.
Despite the importance of the rise of paywalls to the web, it is surprising how little the topic has been studied by the research community. Important open
questions include how popular paywall systems are, what policies paywalls
impose, how users are tracked for paywall enforcement, and whether paywalls are effective at protecting premium content.
\subsection{Contributions}
In this work, we aim to improve the understanding of paywall systems through the first systematic study of paywalls on widely-used web sites. This work makes the following contributions to the understanding of paywall systems on the web:
\begin{enumerate}[labelindent=1cm, leftmargin=0.5cm]
\item A \textbf{novel system for programmatically determining if a site is using a paywall}, though the combination of multiple crowd-sourced data sets and tools.
\item A \textbf{case study of how a popular paywall library operates},
from how a publisher deploys it, how the paywall identifies
users, to how the configured content access policy is enforced.
\item A \textbf{large-scale measurement of paywall popularity}, including
what kinds and what countries account for most paywall use, and how paywall use has changed over time.
Example results include finding that paywall use has increased dramatically over time ($2\times$ more paywalls every 6 months) and that paywall adoption differs by country (e.g., 18.75\%\xspace in US, 12.69\%\xspace in Australia) and industry.
\item An \textbf{in-depth, large scale analysis of deployed paywall policies}, including subscription costs, how paywall adoption impacts the hosting website, how robust paywalls are to evasion, the mechanisms paywalls use to prevent users from viewing protected content, and the privacy implications of paywalls.
\item A \textbf{classifier for deterring whether a site is using a paywall}
for use on sites not considered by crowd-sourced resources, for future long
term, web scale measurements of paywall adoption and behavior.
\end{enumerate}
\begin{figure}[t]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{./figs/Screenshot1.png}
\caption{Truncated article in Wall Street Journal.}
\label{fig:screenshot1}
\end{subfigure}%
\hfill
\begin{subfigure}{.5\textwidth}
\centering
\vspace{0.3cm}
\includegraphics[width=0.8\linewidth]{./figs/Screenshot3.png}
\caption{Obscured article in Miami Herald.}
\label{fig:screenshot2}
\end{subfigure}\vspace{-0.2cm}
\caption{Examples of raised paywalls in major news sites. Paywalls may be enforced in different ways to deny access to articles to non-subscribed users.}
\label{fig:screenshot}\vspace{-0.4cm}
\end{figure}
\section{Background}
\label{Background}
Paywalls are an increasingly popular monetization strategy for web sites, as publishers attempt to become less dependent on advertising. Figure~\ref{fig:screenshot} shows a typical paywall, where a publisher is blocking access to content until the user pays a fee. To enforce access control, paywalls track the engagement of the user with the publisher content: i.e., how much time they spend on a web site, how many articles they have read, how many times a user has visited the website.
\subsection{Types of Paywalls}
We group paywalls into two categories, based on how restrictive they are: (i) \emph{hard paywalls}, where users cannot gain access the site without first purchasing a subscription (e.g., monthly or annual subscriptions) and (ii) \emph{soft paywalls} that allow limited, free-of-charge viewing for a specific amount of time or number of visit (e.g., ~5 free articles per month per user).
\point{Hard Paywalls}
Hard paywalls require subscriptions before visitors can access content (e.g., Financial Times requires a subscription before the user can read any article). Such a strategy runs the risk of deterring users and thereby diminishing the publisher's influence over all. As reported in the press~\cite{declinedTraffic}, The Times experienced a~90\% drop in traffic after introducing a hard paywall.
\point{Soft or Metered Paywalls}
Soft (or metered) paywalls limit the number of articles a viewer can read before requiring a paid subscription. Soft paywalls use the free articles as a strategy to entice users to subscribe. Soft paywalls require some method (often a JavaScript snippet on the user-side) for measuring either the number of articles a user has accessed, or the time a user spends in browsing the website's articles.
\begin{figure}[tb]
\centering
\vspace{-.9cm}
\includegraphics[width=1.18\linewidth]{./figs/tinypass-diagram.pdf}\vspace{-.5cm}
\caption{High level overview of the core functionality of a paywalled website powered by Tinypass.}\vspace{-.55cm}
\label{fig:tinypassOverview}
\end{figure}
As with hard paywalls, a publisher's web traffic can also be affected by the installation of soft paywalls (e.g., traffic to the New York Times declined by~5\% to~15\% one month after the installation of its soft paywall~\cite{NYTdecreasedTraffic1,NYTdecreasedTraffic2}). Overall, though, fewer users are discouraged by soft paywalls. Prior studies~\cite{hardVsSoft} have found, on average, retention rates for publishers with soft paywalls reaching~58.5\%, compared to only~15--20\% for publishers with hard paywall.
\section{Paywall Case Study}
\label{sec:casestudy}
This section provides a detailed case study of a popular third-party paywall system. We provide this case study (i) to introduce the reader to how paywalls work, and (ii) to document the kinds of privacy-affecting behaviors paywalls often rely on to impose their policies.
We select Piano's Tinypass paywall-as-a-service product~\cite{tinypass} for our case study for several reasons. First, it is one of the most popular third-party paywall providers (Tinypass owns~38.2\% of the market, as measured in Figure~\ref{fig:thirdPartyLibs_popularity}), so understanding how this system works provides a good understanding of the kinds of paywall code users are likely to experience. And second, Tinypass can be deployed as a configurable, paywall-as-a-service, allowing publishers (blogs, news sites, magazines, etc.) to impose a variety of paywall policies, both hard and soft.
\subsection{Tinypass: The protocol}
At some point prior to the user's visit, a site owner creates an account at Tinypass, where they describe the subscription policies they wish to enforce. Tinypass generates the keys and identifiers used to enforce the paywall and track visitors.
Once a site owner installs Tinypass on their site, the paywall works in the following six stages, with numbers corresponding to Figure~\ref{fig:tinypassOverview}:
\point{Step one}
The user's browser makes a request to a website where the site owner has installed Tinypass.
\point{Step two}
The website responds with the HTML of their page, including a reference to the Tinypass JavaScript\xspace library, hosted on Tinypass's servers. The content provider's response may also include optional, customized parameters that allow Tinypass to integrate with other services, like Facebook and Google Analytics. At the time of this writing, Tinypass's code is hosted at \url{https://code.tinypass.com/tinypass.js}.
\point{Step three}
The referenced JavaScript causes the browser to request code from Tinypass's server, which responds with a bootstrapping system, providing basic routines for fetching the main implementation code, helper libraries, and utilities for rate limiting and fingerprinting. Depending on the particular deployment, minified versions of this code also includes common utilities like CommonJS-style dependency tools or cryptography libraries.
\point{Step four}
The browser executes the complete Tinypass library, and the full (post-bootstrap) Tinypass library performs a number of privacy-relevant checks. First, Tinypass attempts to determine if a site visitor is actually an automated browser (e.g., Puppeteer, WebDriver client). Tinypass attempts to determine if the user has an ad-blocker installed. Interestingly, Tinypass not only detects if the user currently has an ad-blocker installed, but also if the visitor has changed their ad-blocker usage (e.g., the user had an ad-blocker installed on a previous visit but no longer does, or vice versa).
\begin{figure}[t]
{\footnotesize
\begin{lstlisting}[caption=Excerpt of Tinypass's fingerprinting JavaScript.,label={lst:fingerprint}]
var _getFingerprint = function () {
if (fingerprint) {
return fingerprint;
}
var fingerprint_raw = _getLocality();
fingerprint_raw += _getBrowserPlugin();
fingerprint_raw += _getInstalledFonts();
fingerprint_raw += _getScreen();
fingerprint_raw += _getUserAgent();
fingerprint_raw += _getBrowserObjects();
fingerprint = murmurhash3.x64hash128(fingerprint_raw);
util.debug("Current browser fingerprint is: " + fingerprint);
return fingerprint;
};
\end{lstlisting}\vspace{-0.7cm}
}
\end{figure}
Tinypass then generates a user fingerprint, implemented with the code hosted at \url{https://cdn.tinypass.com/api/libs/fingerprint.js}. The Tinypass fingerprinting library (shown in part in Listing~\ref{lst:fingerprint}) hashes together a number of commonly known semi-unique identifiers (installed plugins, preferred language, installed fonts, screen position, user agent, etc.) to build a unique identifier, hashed together using the MurmurHash3 hash algorithm~\cite{murmur3}). The result is an identifier that is consistent across cookie-clears, and so can re-identify users attempting some evasion techniques. Tinypass also reads, if available, a first-party cookie the library also uses to identify users. When available, this cookie is used in place of the above fingerprint, to track how much content the user has visited.
\begin{figure}[tb]
{\footnotesize
\begin{lstlisting}[caption=Excerpt of returned Tinypass end point data (meter is Tinypass's terminology for a counter describing how much more non-paywalled content a user can view).,label={lst:metrics}
]
...
"trackingId": "{jcx}H4sIAAAAAAAAAI2QW2vCQBCF_8s...",
"splitTests": [],
"currentMeterName": "DefaultMeter",
"activeMeters": [
{
"meterName": "DefaultMeter",
"views": 0,
"viewsLeft": 4,
"maxViews": 4,
"totalViews": 0
}
],
...
\end{lstlisting}\vspace{-0.5cm}
}
\end{figure}
\point{Step five}
Next, the Tinypass library gathers the above information, combines it with information about the page, derived fingerprinting values, the date, and other similar data, and \texttt{POST}s them to a Tinypass endpoint~\footnote{\url{https://experience.tinypass.com/xbuilder/experience/execute?aid=*}}, which records information about the page view. The server then returns a JSON string describing a variety of information about the page view, and excerpt of which is presented in Listing~\ref{lst:metrics}. This JSON string includes a wide variety of both user-facing and program-effecting values, including how many more pages the user is able to visit before the paywall is triggered, possibly new identifiers to rotate on the browsing session, whether the user has logged in and is known to Tinypass (e.g. the user logged in on a different domain owned by the same publisher).
\point{Step six}
Finally, the Tinypass code running on the browser enforces the described paywall policy. The code, client-side, uses the response data to decide how to respond to the page view, possibly by obscuring page content or presenting a subscription offer dialog (by default, Tinypass offers pre-made-but-configurable modal and ``inline'' dialogues the website can check from). In the pages we observed, Tinypass only enforced subscription requirements (i.e., preventing users from viewing content) after the above check was completed. A side effect of this implementation decisions is that Tinypass's restrictions can be circumvented by simply blocking the Tinypass library (see Section~\ref{sec:circumvention}).
\section{Current Paywall Deployments}
\label{sec:results}
In this section, we present a large-scale measurement of paywall deployments on the web. The measurements presented give a broad assessment of how often paywalls are used (by country and by industry). We then present a variety of measurements of how deployed paywalls operate, including the access policies they enforce, their enforcement mechanisms, and how robust these paywalls are to circumvention. The section begins with a description of how we gathered there relevant datasets for measurements, and then proceeds in the above described order.
\begin{figure}[t]
\centering
\vspace{-0.4cm}
\includegraphics[width=.7\linewidth]{./measurements/popularity_growth_half}\vspace{-.2cm}
\caption{Growth of paywall deployments per 6 months. Note that the y-axis
depicts the growth-rate, and not absolute numbers.}\vspace{-.2cm}
\label{fig:popularity_growth}
\end{figure}
\begin{figure}[b]
{\small
\centering
\begin{tabular}{lr}
\toprule
\textbf{Data} & \textbf{Volume} \\
\midrule
Paywalled websites from bypassing extensions & 147\xspace \\
Third-party paywall libraries & 43\xspace \\
Unique paywalled sites & 1,710\xspace \\
Countries the paywalled sites originate from & 61\xspace \\
\bottomrule
\end{tabular}
}
\vspace{-0.3cm}
\caption{Summary of our crowdsourced dataset labeling which websites use
paywalls.}
\vspace{-0.3cm}
\label{tbl:dataCollected}
\end{figure}
\subsection{Dataset}
\label{sec:dataset}
To conduct the measurements described in this section, we built
an oracle to determine whether a web site uses a paywall. While seemingly
a simple question, the diversity of paywall libraries, enforcement
mechanisms, access policies and varying verbiage makes this a difficult
question to answer without significant human intervention.
To solve this problem, we draw on two existing crowd-sourced datasets:
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering\vspace{-.5cm}
\includegraphics[width=1.1\linewidth]{./measurements/alexaTopic}\vspace{-.3cm}
\caption{Type of industry or type of content, paywalled websites deliver.}
\label{fig:alexaTopic}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering
\vspace{.2cm}
\includegraphics[width=1\linewidth]{./measurements/prevalenceTop6_newsTop.eps}\vspace{-.3cm}
\caption{Portion of news sites using paywalls per country. Paywall adoption reaches 18.75\%\xspace in US and 12.69\%\xspace in Australia.}
\label{fig:popularityPerCountry}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.3cm}
\includegraphics[width=1\linewidth]{./measurements/alexaRank_cdf}\vspace{-0.2cm}
\caption{Distribution of the popularity of each paywalled website in our
dataset based on the Alexa ranking.}
\label{fig:alexaRank}
\end{minipage}
\vspace{-.4cm}
\end{figure*}
\point{A) Extensions}
First, we extract rules from several popular browser extensions~\cite{bypassExt1,bypassExt2,bypassExt3,bypassExt4} designed to help users circumvent paywalls. By examining the source code of these extensions, we are able to (directly or indirectly) identify 147\xspace{} websites that the tools' authors and maintainers label as using paywalls.
\point{B) Filter lists}
Second, we use a popular, crowd-maintained filter list that identifies
third-party paywall libraries~\cite{fanboyPaywalls} so that they can
be blocked with common filter-list consuming tools (e.g., AdBlock Plus, uBlock Origin). This list includes filter rules for blocking resources related to a variety of internet ``annoyances''; we extract the subset of the list specifically targeting paywalls. This gives us a list of~43\xspace{} third-party paywall libraries. We query for each entry of this paywall libraries list in two existing, current web crawl archives (i.e., HTTPArchive~\cite{wayback} and PublicWWW~\cite{publicWWW}). We found 1,563\xspace{} sites using one of these paywall libraries and we labeled them as ``paywalled''.
We combine the above two approaches (i.e., paywalled domains labeled by browser
extensions and paywalled sites including third-party paywall libraries)
to identify 1,710\xspace{} unique paywall-using domains, from
61\xspace{} countries. This dataset, summarized in Figure~\ref{tbl:dataCollected}, comprises our complete dataset of paywalled sites used as an oracle in this section, and we provide open-sourced\footnote{\url{https://gist.github.com/panpap/68af1c99b49366dfce4044a354f6e1b8}}.
\subsection{Paywall Popularity}
We start by measuring how popular paywalls are, across several dimensions.
We use the domains identified in Section~\ref{sec:results} as the set of paywalled sites, and all other sites on the web as not paywalled.
\subsubsection{Increase in Paywall Use}
\label{sec:growth}
We first measure whether paywall use has increased over time. We find sites
in our dataset started using paywalls in~2015, with overall paywall use strictly
increasing since. Paywall use has increased at a rate between~120\% and~230\%
every six months since~2015 until recently. In the first six months of~2019,
paywall use quadrupled, and has grown by a further~\textit{180\% } during the
first two months in the second half of~2019. These measures are summarized
in Figure~\ref{fig:popularity_growth}.
We measure paywall growth over time by applying our paywall oracle
(described in Section~\ref{sec:dataset}) to archived versions of the same
sites in the Wayback Machine web archive~\cite{wayback}. We use these
archived versions of each site to approximate date each site adopted a paywall.
The precise methodology is as follows:
\begin{enumerate}[leftmargin=0.5cm,topsep=0pt]
\item We build the set of paywall library related URLs and domains using the technique described in Section~\ref{sec:dataset}.
\item We fetch the most recent archive of each paywalled website in our dataset from the Wayback machine and check whether that historical version is using a paywall.
\item If we observe the site using a paywall, we fetch the next-most-recent version of the site from the Wayback machine (e.g., we move back one recording in time) and re-check.
\item We continue this process until we encounter a version of the web site that no longer is using a paywall. Once we encounter a non-paywalled version of the site, we note the date that version of the site and record it as when the site began using a paywall. \vspace{-0.1cm}
\end{enumerate}
\point{Limitations}
We note two limitations of the above approach, and why we do not believe they significantly impact our findings. It is possible that earlier versions of sites used different types and providers of paywalls than current sites, and so our paywall detection oracle may be missing historical paywall use. While possible, we do not think this limitation significantly impacts the results for two reasons: (i) prior research~\cite{vastel2018filters} has found that filter lists (like the ones we use for paywall library detection) rarely delete rules, and so that paywall-targeting filter lists would identify both current and historical paywalls.
What is more, (ii) we manually evaluated a random sample of commits from the git history of the paywall-targeting portion of the filter list and we found no rule deletions. This gives us further confidence, though not certainty, that filter rules that would identify paywalls on previous versions of paywalled sites have not been removed.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/enforcingStrategie}
\caption{Popularity of the different paywall enforcing policies. Most of the publishers prefer to obfuscate (48.2\%\xspace) or truncate (44.5\%\xspace) the article the user has not yet access to.}
\label{fig:enforcingStrategies}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{0.4cm}
\includegraphics[width=1.05\linewidth]{./measurements/freeArticles_both_cdf}
\vspace{-0.4cm}
\caption{Distribution of the number of free articles allowed per user. The median paywalled website allows 3.5\xspace articles, the median soft-paywalled website allows 4\xspace articles and all hard-paywalled do not allow \emph{any} free article.}
\label{fig:freeArticles}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.6cm}
\includegraphics[width=1.05\linewidth]{./measurements/cost_subscr_cdf}
\vspace{-0.5cm}
\caption{Cumulative distribution of the subscription cost per website for a 12-month content access.}
\label{fig:cost_subscr_cdf}
\end{minipage}
\end{figure*}
The second possible limitation is that our approach might miss sites that
used to have paywalls, but no longer do. We believe such cases to be rare.
We observed no instances of sites using paywalls, removing the paywalls,
and then re-establishing it. This suggests (though does not prove)
that sites do not commonly abandon paywall strategies once they have adopted them.
\subsubsection{Paywall Use by Site Type}
We measure what types of content paywalled sites provide.
We find that most~(80.3\%\xspace{}) paywalled sites provide some form
of news content, whether targeted at the local, regional, or world-level.
Figure~\ref{fig:alexaTopic} provides summary of this measurement.
For this measurement, we use the sites identified as using paywalls from
Section~\ref{sec:dataset} with information available from the Alexa Top Sites
service. The Alexa Top Sites classifies domains into one of~17 different
classes (i.e., news, sports, business, arts, society). Three categories
describe news content, though at different levels of focus (e.g., ``World'',
``Regional'' or, generically, ``News''). We group these together for our
measurements, since they are thematically very similar. The remaining 14
categories account for just 19.7\%\xspace{} of paywalled sites.
\subsubsection{Paywall Use by Country}
Next, we measure which countries have the highest rates of paywall use. Because news sites account for most paywall use, we focus this measurement on news sites. We find that US news sites have been the quickest to move to paywalls, followed by Australia, France, Canada and Germany. Figure~\ref{fig:popularityPerCountry} summarizes our findings. Since our oracle does not identify all websites with paywalls, Figure~\ref{fig:popularityPerCountry} presents only the lower bound of the existing paywalled sites.
We measure rates of paywall use by country by first retrieving the Alexa the Top~10,000 websites per country. We filter the list and remove all non-news sites. Then, we calculate the percentage of paywall-using news sites, as a fraction of all news sites, per country. We find that 18.75\%\xspace of US news sites use paywalls, 12.69\%\xspace of Australian new sites, and less than ~7\% in all other countries.
\subsubsection{Paywall Use by Popularity}
Next, we measure whether there is any clear relationship between paywall deployment and the popularity of a website. We did not observe any such relationship. We
anticipate that most paywalled sites would be popular (as measured by Alexa Top Sites), as a successful paywall would require a significant number of
subscribers, which in turn would require a significant amount of baseline
visitors. Instead, we find that only 8.54\%\xspace{} of paywall-using sites
are among the~10,000 most popular sites on the web. The median paywall-using
site is ranked 365,316\xspace{}. The full distribution
of the popularity of paywall using sites is presented in Figure~\ref{fig:alexaRank}.
\begin{figure}[t]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.9\linewidth]{./measurements/thirdPartyLibs_popularity}
\vspace{-0.7cm}
\caption{Popularity of third-party paywall libraries in our dataset. A
small number of paywall implementations account for the majority of
third-party paywall deployments.}
\label{fig:thirdPartyLibs_popularity}\vspace{-0.2cm}
\end{figure}
\subsection{Paywall Libraries}
\label{sec:3plibs}
A significant number of sites rely on third-parties for their paywall
implementations. These third-parties sell ``paywall-as-a-service'' products,
where publishers pay fee to have the third-party manage and
enforce the paywall on the publisher's site. We observe
that a small number of paywall providers account for the vast majority
of paywall deployments, with Piano and Tecnavia being the most popular
paywall providers (23.5\%\xspace{} and 21.0\%\xspace{} market
share, respectively). The full distribution of third-party paywall
market share is depicted in Figure~\ref{fig:thirdPartyLibs_popularity}.
This consolidation of paywall implementation and enforcement is significant,
for a variety of reasons. First, market consolidation may effect the amount of
income content-makers can receive for their content (popular third-party
paywall providers receive 10-15\% of each sold subscription). Second, provider
consolidation may make large scale circumvention easier, as circumventors
need to target a smaller number of systems (see Section~\ref{sec:circumvention}). Third, a small number of paywall providers
tracking users across a large number of websites has clear privacy implications
(see Section~\ref{sec:privacy}).
We measure the popularity and consolidation of third-party paywall libraries
by crawling each paywalled site in our dataset and observing which resources
from known paywall providers were fetched. We find that at least
25\%\xspace of paywalled websites outsource their paywall
functionality to third-parties. The distribution of third party paywall
use follows a rough power-law distribution.
\subsection{Paywall Polices}
Next, We measure the distribution of policies enforced by paywalls. We find that paywalls vary widely by type, enforcement mechanism, and how much, if any, content visitors can view before needing to pay. For these measurements, we randomly sample ~115\xspace paywall-using websites from our dataset for manual evaluation.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/cost_per_country}
\caption{Min, 15th percentile, median, 85th percentile, and max annual subscription costs for paywalls, by country.}
\label{fig:cost_per_country}
\end{minipage}
\hfill
\begin{minipage}{0.33\linewidth}
\centering\vspace{0.2cm}
\includegraphics[width=1.05\linewidth]{./measurements/alexa_bounce_cdf}\vspace{-0.1cm}
\caption{Distribution of the bounce rate per website. The median paywalled site has slightly higher bounce rate (68.4\%\xspace) contrary to the non-paywalled (67.5\%\xspace).
\label{fig:alexa_bounce_cdf}\vspace{-0.2cm}
\end{minipage}
\hfill
\begin{minipage}{0.33\linewidth}
\centering\vspace{0.4cm}
\includegraphics[width=1.05\linewidth]{./measurements/pageviews_cdf}
\caption{Distribution of the daily page views per visitor per news site. In median values, an average visitor browses on a daily basis 13.42\%\xspace less pages on a paywalled news site.}\vspace{-0.3cm}
\label{fig:pageviews}
\end{minipage}
\end{figure*}
\subsubsection{The different types of Paywalls}
First, we observe that 66.7\%\xspace of paywalls are ``soft'' (i.e., allow some free content access), 15.7\%\xspace are ``hard'' (i.e., allow no free access), with the remaining~16.6\%\xspace paywalled sites using a ``hybrid'' strategy (i.e., some content is free, some requires payment, based on the author/time of publication/topic, etc.). Some ``hybrid'' publishers use machine learning or other dynamic approaches to determine per-user whether an article should be locked or not~\cite{mlpaywall,vidoraML,algorithmicPaywall}.
\subsubsection{Enforcement Mechanism}
We also measure the distribution of paywall enforcement techniques. Despite the heterogeneity of the paywall implementations, we see only three approaches used to enforce paywalls: (i) truncating article text, (ii) obfuscating the article with popups, or (iii) redirecting users to a subscription page. We measure the popularity of each of the above approaches in our manually evaluated set; Figure~\ref{fig:enforcingStrategies} presents the results. The largest percentage (48.2\%\xspace) of the websites obfuscate or truncate~(44.5\%\xspace) the article the user has not yet access to. Only a few~(7.3\%\xspace) redirect the user to a login/subscribe page.
\subsubsection{Allowed Free Content}
We also measure the distribution of how much content users can view before triggering a (hard or soft) paywall. For the ~15.7\% of sites that use a hard-paywall strategy, visitors cannot view \emph{any} articles for free. For soft paywalls, this number varies by publisher. Figure~\ref{fig:freeArticles} plots the distribution of the free articles users could consume before hitting a paywall in the websites we tested. Overall, the median paywalled website allows 3.5\xspace articles. All hard paywalled websites do not allow \emph{any} access to articles, when the median soft-paywalled website allows 4\xspace articles to be read for free. A significant number of soft paywalls~(30\%) that allow~2 or fewer articles to be read before triggering enforcement.
\subsubsection{Paywalls Cost}
\label{sec:cost}
Next, we measure the distribution of paywall subscription costs. We find that most paywall subscriptions are monthly, that the median annual cost for paywall access is 108 USD, and that subscription costs seem to be highest in Germany. All of these measurements were conducted through a manual evaluation of 105\xspace paywalled sites. We sampled 20 sites from each of the top~6 paywall using countries. For 12 sites, we were not able to access the site or determine the subscription costs
We first measure the distribution of subscription options for users. ~82.86\%\xspace of paywall sites provide a monthly subscription option and ~35.23\%\xspace of sites provide an annual one. Hence,~64.76\%\xspace of the paywalled sites provide \emph{only} a monthly subscription option and ~17.14\%\xspace \emph{only} an annual one.
Next, we measure the distribution of purchasing an annual subscription to a site's content. The median observed annual subscription cost is 108 USD. ~22\% of sites charge less than 60 USD, ~21\% of sites charge more than~180 USD. Figure~\ref{fig:cost_subscr_cdf} presents the full distribution of annual subscription costs. We note that the subscription rates we observe are lower than those estimated by previous work (around~189 USD on average)~\cite{paywallNiemanlab}, possibly reflecting a general decrease in costs. We measure the distribution of annual costs by manually noting the annual subscription cost in the local currency. For sites that do not offer an annual subscription, we sum the cost of twelve monthly subscriptions. We then convert all costs to USD for comparison purposes.
Finally, we measure how subscription costs differ by country. As depicted in Figure~\ref{fig:cost_per_country}, we plot the min, the~15th percentile the median, the~85th percentile and the max of the annual subscription cost across the different countries. The median prices of subscriptions in Australia and Germany are highest (193 and 190 USD, respectively). Subscription costs vary widely by site, too. In Germany and the United States, for example, the most expensive paywalls cost ~$2.63\times$ and~$3.51\times$ more than the median rate, respectively.
\subsection{How Paywalls Affect Site Use}
\label{sec:alexa}
Paywalls affect how users interact with the site. Recent studies~\cite{kim2019newspapers}, by monitoring the pageviews of 36 news sites before and after paywall deployment, report that they lose nearly~30\% of their daily traffic and a loss of pageviews, ranging from a~10\% to~55\%.
In this section, we measure differences between how sites interact with paywalled and non-paywalled sites. We find that users view less pages on paywalled sites, stay for shorter periods of time and link to pages less. Interestingly, we did not see a significantly difference to the bounce rate between paywalled and non-paywalled sites~\footnote{We do not address the issue of causation; its possible, for example, that the types of site likely to use paywalls have lower \emph{dwell times} already, so that the use of a paywall is a more a result of lower dwell time than the cause. We leave disentangling cause and effect for future work.}.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/dailyTimeOnSite_normalized_cdf}
\caption{Distribution of the average time a visitor spends daily per news site. In median values visitors tend to spend daily $2.46\times$\xspace more time per site rank on non-paywalled websites.}
\label{fig:dailyTimeOnSite}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.3cm}
\includegraphics[width=1.05\linewidth]{./measurements/siteLinks_normalized_cdf}
\caption{Distribution of the incoming site links per news site. Paywalled sites get significantly (18.9$\times$\xspace) less site links per rank in median values.}
\label{fig:siteLinks}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering
\vspace{-.3cm}
\includegraphics[width=1.05\linewidth]{./measurements/paywallsBypassApproaches.pdf}
\vspace{-.6cm}
\caption{Success rate of the different paywall bypassing approaches. Clearing the cookie jar alone can bypass~75\% of the paywalls.}
\label{fig:paywallsBypassFeatures}
\end{minipage}
\end{figure*}
\subsubsection{Bounce Rate}
We find that paywalled new sites have slightly higher bounce rates~\footnote{The percentage of visitors who visit a site and then leave, rather than continuing to view other pages within the same site.} than non-paywalled news sites. The distributions of bounce rates is depicted in Figure~\ref{fig:alexa_bounce_cdf}.
The median paywalled news site has slightly higher bounce rate (68.4\%\xspace) contrary to the median non-paywalled (67.5\%\xspace). However, we see that for ~30\% of the websites in the two categories the difference is~2-7\% higher. To compare bounce rates, we used the Alexa Top Sites data, and compared the bounce rates for the paywalled news sites in our data set with the Alexa top ~1K news sites.
\subsubsection{Daily Page Views}
Next, we measure the number of pages the average visitor performs daily on the websites and we compare how this changes for the paywalled and non-paywalled news sites. In Figure~\ref{fig:pageviews}, we plot the cumulative distribution of these page views per website in our dataset. Users visit on average 13.42\%\xspace less pages on paywalled news sites than non-paywalled new sites.
\subsubsection{Average Time spent on Site}
Figure~\ref{fig:dailyTimeOnSite} compares the distribution of the median time users spend on paywalled and non-paywalled websites, normalized by popularity (based on its Alexa rank). We find that visitors spend daily $2.46\times$\xspace more time per on non-paywalled news sites.
\subsubsection{Content Popularity and Link Rate}
Finally, we measure the impact of paywalls on how often sites link to the paywalled sites. Since site linking may be affected by the popularity of the website, in Figure~\ref{fig:siteLinks}, we plot the cumulative distribution of the number of site links (or backlinks) per news site normalized by its Alexa rank. We observed paywalled sites being linked to significantly less (18.9$\times$\xspace) often than non-paywalled sites.
\subsection{Paywalls and Privacy}
\label{sec:privacy}
Most behavioral advertising systems require users to pay for content with their privacy; users are tracked in behavioral advertising systems, and can view ``free'' content.
Paywalls have the possibility of changing this system. Since users are directly paying for content, one might hope users would no longer face the privacy harms associated with behavioral advertising systems. Unfortunately, we see that this is not the case. People \textit{do not} generally receive a tracker free version of site content when paying for subscriptions. Instead, paywall systems seem to serve as an \textit{additional} monetization mechanism on top of existing, privacy harming, ad systems.
\begin{figure}[b]
\centering
\vspace{-0.2cm}
{\small
\begin{tabular}{l|rr|rr}
\toprule
& \multicolumn{2}{c}{\bf Vanilla User} & \multicolumn{2}{c}{\bf Premium User} \\
{\bf News site} & {\bf Ads} & {\bf Tracking} & {\bf Ads} & {\bf Tracking}\\ \midrule
heraldsun.com.au & 171 & 13 & 169 & 9\\
miamiherald.com & 123 & 12 & 112 & 11\\
wsj.com & 63 & 4 & 61 & 4\\
kansascity.com & 61 & 9 & 56 & 6 \\
ft.com & 20 & 0 & 11 & 0\\
salon.com & 138 & 5 & 0 & 1 \\
japantimes.co.jp & 109 & 12 & 98 & 8 \\
leparisien.fr & 125 & 10 & 81 & 4 \\
independent.co.uk & 11 & 6 & 10 & 6 \\
spectator.co.uk & 18 & 2 & 14 & 2 \\
\bottomrule
\end{tabular}\vspace{-0.2cm}
\caption{Requests captured for vanilla and premium user. User continues receiving the same amount of trackers and ads in the content she receives even if she has paid for it.}
\label{tbl:withAndWithout}
}\vspace{-0.2cm}
\end{figure}
We measure whether paying for paywall access improves user privacy (i.e., removes the need for sites to try and monetize through tracking) by purchasing subscriptions to 10\xspace randomly selected paywalled news sites. Our goal is to examine the types of network requests issued before and after paying for the subscription.
We create two scenarios, the vanilla (non-subscribed) user, and the premium (subscribed) user. For each selected site, we create an account and purchased a subscription before the starting the measurement. We also select 5 child pages on each site for evaluation.
Then, we enable the popular Disconnect plugin~\cite{disconnect} in monitoring and no-blocking mode, and browse each selected child page on each site under each of the two personas, in the same order, and observe the issued network requests. Figure~\ref{tbl:withAndWithout} presents the average number of ad- and tracking- related requests encountered in each scenario. In the vast majority of cases, there is no significant difference in terms of ad- or tracking- related web requests.
\section{Paywall Circumvention}
\label{sec:circumvention}
Paywalls must be robust to circumvention if they are going to be a successful monetization scheme for websites. If paywalls can be easily avoided, then content producers will wind up in the same situation they are in with ads and ad-blockers; declining revenues as circumvention tools become more popular. We find that \textit{all observed paywalls are trivial to circumvent}.
We evaluate how robust paywalls are to circumvention in two steps: (i) we categorize the approaches of several popular paywall circumvention strategies, and (ii) we test each strategy on 32\xspace paywalled news sites, we randomly select from our dataset. This subset comprises~28 soft and~4 hard paywalls on popular websites like Wired, Bloomberg, Spectator, Washington Post, Irish Times, Medium, Build, Japan Times, Statesman, and Le Parisien.
\subsection{Evasion Evaluated}
We test the robustness of each paywall system by using Chrome version~71. For each evaluated site, (a) we browse different pages till we trigger the paywall, and then (b) we test a variety of bypassing approaches to circumvent the paywall and get access to the ``protected'' article.
Figure~\ref{fig:paywallsBypassFeatures} lists the evaluated paywall-circumvention strategies, which includes pre-packaged tools, fingerprint evasion techniques, and third-party services. Specifically, we consider:
\begin{enumerate}\itemsep=-1pt
\item changing the screen size dimensions
\item hiding the user's IP address
\item changing the user agent string
\item using an ad blocker extension
\item enabling ``Reader Mode''
\item using the Pocket web service\footnote{or similar ``reader'' services like ``JustRead''~\cite{justRead} and ``Outline''~\cite{outline}}
\item enabling Incognito/Private Mode
\item emptying the cookie jar
\item blocking HTTP requests for popular paywall libraries
\end{enumerate}
Overall, we are able to bypass \emph{all of the soft paywalls and none of the hard paywalls}. Hard paywalls perform their enforcement server-side, when the soft paywalls perform their policy enforcement client-side, and thus their access control is circumventable.
\begin{figure}[tb]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.65\linewidth]{figs/crawlingFlowchart.pdf}
\vspace{-0.2cm}
\caption{Data collection steps of our paywall detector's crawling component.
\vspace{-0.5cm}
\label{fig:visitedSites}
\end{figure}
\subsection{Evasion Approaches Analyzed}
Many of the evaluated evasion approaches are rarely successful. For example, changing the screen size or the IP address of the user rarely circumvents a soft paywall~(4\% effectiveness). A moderate number of soft paywalls~(12\%) is flummoxed by modifying the browser's user agent string.
The majority~(75\%) of soft paywalls is bypassable by resetting the cookie jar (in some cases erasing the first-party cookie only is insufficient, since it is automatically re-spawned by fingerprinting JavaScript code, as seen in Section~\ref{sec:casestudy}). As a result, switching into browsers' ``private browsing'' modes is also sufficient to bypass most paywalls. Some paywalled sites refuse to render content in ``reader modes'' or ``private browsing'' modes, either first party (e.g., reader modes shipped with Safari and Firefox) or third-party (e.g., services like Pocket~\cite{pocket}). Such detection schemes are uncommon though; switching into reader-mode, for example, circumvents paywall enforcement in~60\% of the cases.
Ad-blocking extensions, in their default configurations, have little-to-no effect on paywalls. However, by using the list of known paywall libraries from Section~\ref{sec:results} and by blocking requests to these domains we are able to bypass~48\% of the paywalls without breaking the website's main functionality.
Third-parties like Google Search, Twitter, Reddit and Facebook, can also be used to gain access to some paywalled articles. Some paywalls give visitors from these large third-party systems unfettered access to their content, in pay-for-promotion initiatives. By spoofing the referrer field of the HTTP GET requests, some paywalls are vulnerable to exploiting a controversial policy~\cite{firstclick} where publishers (for promotion purposes) allow access to articles when the visitor comes from one of these platforms (by clicking on a tweet, a post, a Google search result etc.)~\cite{referrerBypass}. These mechanisms can provide access to hard paywalled articles. As a result, some publishers (e.g., Wall Street Journal) have stopped allowing such special access through their paywalls~\cite{wsJGoogleRef}.
\section{Paywall Detection}
\label{sec:detection}
\begin{figure}[t]
\centering
{\small
\vspace{-0.2cm}
\begin{tabular}{lr}
\toprule
{\bf Metric} & {\bf Value} \\ \midrule
Precision & 77\%\xspace{} \\
Recall & 77\%\xspace{} \\
F-Measure & 75\%\xspace \\
AUROC & 0.74\xspace \\
\bottomrule
\end{tabular} \vspace{-0.2cm}
}
\caption{Weighted average of the performance of our RF classifier, after k=5 cross-fold validation.}
\label{table:model-performance}
\vspace{-0.5cm}
\end{figure}
This section presents the design and evaluation of a ML-based detection system
whose goal is to determine whether a site uses a paywall. Our paywall detector
consists of two components: (i) a crawling component that visits a subset of
pages on a site, records information about each page's execution, and extracts
some ML features; and (ii) a classifier, that uses the extracted features to
predict if the site uses a paywall.
We present this classifier as a partial solution for the problem of measuring
changes in the adoption and behavior of paywalls over time. We propose this ML
approach as a \emph{complement} to the crowd-sourced approach described in
Section~\ref{sec:dataset}. The classifier can be used to automatically gauge
paywall prevalence. This automated approach can help identify and quantify
paywalls that have not been identified by crowd-sourced
lists, such as paywalls deployed by unpopular or region-specific sites.
Before describing the classifier in detail, we note two things. First,
the classifier is designed to help detect broad, web-scale trends in
paywall use and behavior, not to detect at real time paywall use on any single specific site.
Second, an important finding of this classifier is that there is far greater
diversity in paywall behavior and implementation logic than we expected at the
start of the effort. We expect this to be a useful starting point for future
studies.
\subsection{Crawling Methodology}
\label{sec:methodology-crawling}
The data collection step of our paywall detector, depicted in
Figure~\ref{fig:visitedSites}, begins with three crawls of the target website:
(i) the \emph{initial crawl} that collects a list of child pages on the website,
(ii) the \emph{cookie jar crawl}, where each child page is crawled
sequentially in the same browsing session and (iii) the \emph{clean
crawl}, where each child page is crawled with a fresh browsing session (i.e., a
``clean'' cookie jar).
This strategy replicates viewing patterns that might cause a paywall to be
triggered, and then attempts to detect the paywall's presence by looking for
page content that was visible on previous visits, but is no longer visible. For
each page crawled, the crawler records the final state of the DOM, which DOM
elements are visible, which page elements are positioned to obscure others (e.g.,
modal dialogs), and other page execution data only available at runtime.
\subsection{Feature Extraction}
We selected features that target both immediately triggering paywalls and
paywalls that trigger after viewing multiple pages. These features aim to
capture an intuition about how paywalls behave, and can fall into three rough
categories: textual features, structural features, and visual features.
\point{Text features} These features consider the text of the page, targeting
text and idioms associated with paywalls. The crawler looks for the phrases
``subscribe'', ``sign up'' and ``remaining'' (translated into 87 languages) in
(i) the ``readermode''\xspace subset of the page, (ii) any overlay or popup elements (e.g.,
elements that have, or are children of elements that have, z-index values
greater than zero), and (iii) elsewhere in the page. These three checks
are performed both in ``cookie jar''\xspace and the ``clean crawl''\xspace recordings of each page.
Several text features use a ``readermode''\xspace version of page, the subsection of the document
identified as the page's ``main content'', or the content stripped of page
``boilerplate'' elements (e.g., advertisements, navigation elements, decorative
images). While there are many different ``readermode''\xspace identification
strategies~\cite{ghasemisharif2018speedreader}, in this work we use Mozilla's
\emph{Readability.js}~\cite{readability} implementation, because of its
popularity and ease of use. We expect using other ``readermode''\xspace strategies would work
roughly as well.
\point{Structural features}
These features target page structure (i.e., HTML), independent
of specific page text or presentation. Structural features include whether the
website has a RSS or ATOM feed, changes in the number of text nodes present in
the page between its ``cookie jar''\xspace and ``clean crawl''\xspace versions, how many
measured pages contain a ``readermode''\xspace subset, and the average and maximum
difference in the amount of text in the document in ``readermode''\xspace, between ``cookie jar''\xspace and ``clean crawl''\xspace
measurements.
\point{Visual features}
These features focus on visual aspects of measured pages, and how those visual
aspects change between the ``cookie jar''\xspace and ``clean crawl''\xspace measurements for each child page. The
detector measures how many text nodes are obscured and the average and maximum
change in obscured text nodes between the two measurements for each page.
Additional display features are the number, and change in, text nodes in the
browser viewport, and number of text nodes (regardless of text content)
appearing in overlay (i.e., $z$-index great than zero) page elements. These
features identify paywalls that prevent users from reading page content
through popups or similar methods.
\subsection{Classifier Accuracy}
\label{sec:classification}
Our paywall detector uses a \emph{random forest} classifier, specifically the
\textit{RandomForestClassifier} implementation provided by the popular
SciKit-Learn~\cite{sklearn} python library. Classification parameters were
selected through~5-fold evaluation using the entirety of the aforementioned
extracted features. As a ground truth, we use a subset of the paywalled sites the oracle identified (Section~\ref{sec:dataset}). To assess the accuracy of the classifier we use a different subset of the oracle's data and a set of non-paywalled websites we manually generate. The paywall detector achieves an
average precision of~77\%\xspace, recall of~77\%\xspace and an area
under the receiver operating characteristics~(AUROC) of~0.74\xspace. These
results are encouraging and suggest that our approach can be used to gauge
paywall prevalence on the web. They also indicate that paywalls
vary in behavior more than we anticipated, and that more complex features
may be needed to further improve accuracy.
\section{Related Work}
\label{RelatedWork}
In~\cite{cornia2017pay}, authors perform an empirical study of the pay models (freemium and paywall models) in European news. In particular, they manually analyzed a small dataset of~171 of the most important news organizations in France, Poland, Germany, Italy, Finland, and UK. Their results show that~66\% percent of the newspapers operate a pay model and that the average price for a monthly subscription is~13.64 Euros when prices in general range from~2.10 to~54.27 Euros/month. In our measurements, 3 years after, the average monthly subscription cost~10.93 Euros, when specifically in Germany it is 20.48 (was 19.75) Euros and in France it is 12.54 (was 13.97) Euros.
In~\cite{myllylahti2017content}, authors explore the content that news publishers consider worthy of placing behind a paywall. They analyze~614 articles from the leading Australasian financial newspapers (i.e., the Australian Financial Review (AFR) and the National Business Review (NBR)). Results show that publishers consider hard (or fast-paced) news and opinion pieces as the most valuable news commodity. In addition, as presented, AFR locked~86\% of its content compared to NBR's~41\%.
In~\cite{carson2015behind}, authors analyze selected paywalled news sites in US, UK and Australia to compare the type, pricing and audience uptake. Results show that paywalls are part of newspapers' toolkit for bringing in new revenue but there is no evidence to suggest they can be a standalone solution. However, in this political economic environment for mastheads, digital advertising revenues alone are also insufficient to meet the cost of providing quality journalism.
\section{Conclusion}
\label{sec:conclusion}
Despite the seemingly important implications, paywalls impose on the free web, as an internet phenomena, they have been understudied. This paper aims to address this blind spot by conducting the first large scale study of paywalls on the web. Our results show that paywall use increases over time ($2\times$ more paywalls every 6 months), its adoption differs by country (e.g., 18.75\%\xspace in US, 12.69\%\xspace in Australia), and besides the privacy implications, paywalls fail to reliably protect publishers content. Finally, we present the design of a novel, automated system for detecting whether a site uses a paywall. We hope this work can be a significant first step in understanding the phenomena of paywalls.
\section{Conclusion}
\label{sec:conclusion}
Despite the seemingly important implications, paywalls impose on the free web, as an internet phenomena, they have been understudied. This paper aims to address this blind spot by conducting the first large scale study of paywalls on the web. Our results show that paywall use increases over time ($2\times$ more paywalls every 6 months), its adoption differs by country (e.g., 18.75\%\xspace in US, 12.69\%\xspace in Australia), and besides the privacy implications, paywalls fail to reliably protect publishers content. Finally, we present the design of a novel, automated system for detecting whether a site uses a paywall. We hope this work can be a significant first step in understanding the phenomena of paywalls.
\subsection{Paywalls and Privacy}
\label{sec:privacy}
Most behavioral advertising systems require users to pay for content with their privacy; users are tracked in behavioral advertising systems, and can view ``free'' content.
Paywalls have the possibility of changing this system. Since users are directly paying for content, one might hope users would no longer face the privacy harms associated with behavioral advertising systems. Unfortunately, we see that this is not the case. People \textit{do not} generally receive a tracker free version of site content when paying for subscriptions. Instead, paywall systems seem to serve as an \textit{additional} monetization mechanism on top of existing, privacy harming, ad systems.
\begin{figure}[b]
\centering
\vspace{-0.2cm}
{\small
\begin{tabular}{l|rr|rr}
\toprule
& \multicolumn{2}{c}{\bf Vanilla User} & \multicolumn{2}{c}{\bf Premium User} \\
{\bf News site} & {\bf Ads} & {\bf Tracking} & {\bf Ads} & {\bf Tracking}\\ \midrule
heraldsun.com.au & 171 & 13 & 169 & 9\\
miamiherald.com & 123 & 12 & 112 & 11\\
wsj.com & 63 & 4 & 61 & 4\\
kansascity.com & 61 & 9 & 56 & 6 \\
ft.com & 20 & 0 & 11 & 0\\
salon.com & 138 & 5 & 0 & 1 \\
japantimes.co.jp & 109 & 12 & 98 & 8 \\
leparisien.fr & 125 & 10 & 81 & 4 \\
independent.co.uk & 11 & 6 & 10 & 6 \\
spectator.co.uk & 18 & 2 & 14 & 2 \\
\bottomrule
\end{tabular}\vspace{-0.2cm}
\caption{Requests captured for vanilla and premium user. User continues receiving the same amount of trackers and ads in the content she receives even if she has paid for it.}
\label{tbl:withAndWithout}
}\vspace{-0.2cm}
\end{figure}
We measure whether paying for paywall access improves user privacy (i.e., removes the need for sites to try and monetize through tracking) by purchasing subscriptions to 10\xspace randomly selected paywalled news sites. Our goal is to examine the types of network requests issued before and after paying for the subscription.
We create two scenarios, the vanilla (non-subscribed) user, and the premium (subscribed) user. For each selected site, we create an account and purchased a subscription before the starting the measurement. We also select 5 child pages on each site for evaluation.
Then, we enable the popular Disconnect plugin~\cite{disconnect} in monitoring and no-blocking mode, and browse each selected child page on each site under each of the two personas, in the same order, and observe the issued network requests. Figure~\ref{tbl:withAndWithout} presents the average number of ad- and tracking- related requests encountered in each scenario. In the vast majority of cases, there is no significant difference in terms of ad- or tracking- related web requests.
\section{Background}
\label{Background}
Paywalls are an increasingly popular monetization strategy for web sites, as publishers attempt to become less dependent on advertising. Figure~\ref{fig:screenshot} shows a typical paywall, where a publisher is blocking access to content until the user pays a fee. To enforce access control, paywalls track the engagement of the user with the publisher content: i.e., how much time they spend on a web site, how many articles they have read, how many times a user has visited the website.
\subsection{Types of Paywalls}
We group paywalls into two categories, based on how restrictive they are: (i) \emph{hard paywalls}, where users cannot gain access the site without first purchasing a subscription (e.g., monthly or annual subscriptions) and (ii) \emph{soft paywalls} that allow limited, free-of-charge viewing for a specific amount of time or number of visit (e.g., ~5 free articles per month per user).
\point{Hard Paywalls}
Hard paywalls require subscriptions before visitors can access content (e.g., Financial Times requires a subscription before the user can read any article). Such a strategy runs the risk of deterring users and thereby diminishing the publisher's influence over all. As reported in the press~\cite{declinedTraffic}, The Times experienced a~90\% drop in traffic after introducing a hard paywall.
\point{Soft or Metered Paywalls}
Soft (or metered) paywalls limit the number of articles a viewer can read before requiring a paid subscription. Soft paywalls use the free articles as a strategy to entice users to subscribe. Soft paywalls require some method (often a JavaScript snippet on the user-side) for measuring either the number of articles a user has accessed, or the time a user spends in browsing the website's articles.
\begin{figure}[tb]
\centering
\vspace{-.9cm}
\includegraphics[width=1.18\linewidth]{./figs/tinypass-diagram.pdf}\vspace{-.5cm}
\caption{High level overview of the core functionality of a paywalled website powered by Tinypass.}\vspace{-.55cm}
\label{fig:tinypassOverview}
\end{figure}
As with hard paywalls, a publisher's web traffic can also be affected by the installation of soft paywalls (e.g., traffic to the New York Times declined by~5\% to~15\% one month after the installation of its soft paywall~\cite{NYTdecreasedTraffic1,NYTdecreasedTraffic2}). Overall, though, fewer users are discouraged by soft paywalls. Prior studies~\cite{hardVsSoft} have found, on average, retention rates for publishers with soft paywalls reaching~58.5\%, compared to only~15--20\% for publishers with hard paywall.
\subsection{Paywall Polices}
Next, We measure the distribution of policies enforced by paywalls. We find that paywalls vary widely by type, enforcement mechanism, and how much, if any, content visitors can view before needing to pay. For these measurements, we randomly sample ~115\xspace paywall-using websites from our dataset for manual evaluation.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/cost_per_country}
\caption{Min, 15th percentile, median, 85th percentile, and max annual subscription costs for paywalls, by country.}
\label{fig:cost_per_country}
\end{minipage}
\hfill
\begin{minipage}{0.33\linewidth}
\centering\vspace{0.2cm}
\includegraphics[width=1.05\linewidth]{./measurements/alexa_bounce_cdf}\vspace{-0.1cm}
\caption{Distribution of the bounce rate per website. The median paywalled site has slightly higher bounce rate (68.4\%\xspace) contrary to the non-paywalled (67.5\%\xspace).
\label{fig:alexa_bounce_cdf}\vspace{-0.2cm}
\end{minipage}
\hfill
\begin{minipage}{0.33\linewidth}
\centering\vspace{0.4cm}
\includegraphics[width=1.05\linewidth]{./measurements/pageviews_cdf}
\caption{Distribution of the daily page views per visitor per news site. In median values, an average visitor browses on a daily basis 13.42\%\xspace less pages on a paywalled news site.}\vspace{-0.3cm}
\label{fig:pageviews}
\end{minipage}
\end{figure*}
\subsubsection{The different types of Paywalls}
First, we observe that 66.7\%\xspace of paywalls are ``soft'' (i.e., allow some free content access), 15.7\%\xspace are ``hard'' (i.e., allow no free access), with the remaining~16.6\%\xspace paywalled sites using a ``hybrid'' strategy (i.e., some content is free, some requires payment, based on the author/time of publication/topic, etc.). Some ``hybrid'' publishers use machine learning or other dynamic approaches to determine per-user whether an article should be locked or not~\cite{mlpaywall,vidoraML,algorithmicPaywall}.
\subsubsection{Enforcement Mechanism}
We also measure the distribution of paywall enforcement techniques. Despite the heterogeneity of the paywall implementations, we see only three approaches used to enforce paywalls: (i) truncating article text, (ii) obfuscating the article with popups, or (iii) redirecting users to a subscription page. We measure the popularity of each of the above approaches in our manually evaluated set; Figure~\ref{fig:enforcingStrategies} presents the results. The largest percentage (48.2\%\xspace) of the websites obfuscate or truncate~(44.5\%\xspace) the article the user has not yet access to. Only a few~(7.3\%\xspace) redirect the user to a login/subscribe page.
\subsubsection{Allowed Free Content}
We also measure the distribution of how much content users can view before triggering a (hard or soft) paywall. For the ~15.7\% of sites that use a hard-paywall strategy, visitors cannot view \emph{any} articles for free. For soft paywalls, this number varies by publisher. Figure~\ref{fig:freeArticles} plots the distribution of the free articles users could consume before hitting a paywall in the websites we tested. Overall, the median paywalled website allows 3.5\xspace articles. All hard paywalled websites do not allow \emph{any} access to articles, when the median soft-paywalled website allows 4\xspace articles to be read for free. A significant number of soft paywalls~(30\%) that allow~2 or fewer articles to be read before triggering enforcement.
\subsubsection{Paywalls Cost}
\label{sec:cost}
Next, we measure the distribution of paywall subscription costs. We find that most paywall subscriptions are monthly, that the median annual cost for paywall access is 108 USD, and that subscription costs seem to be highest in Germany. All of these measurements were conducted through a manual evaluation of 105\xspace paywalled sites. We sampled 20 sites from each of the top~6 paywall using countries. For 12 sites, we were not able to access the site or determine the subscription costs
We first measure the distribution of subscription options for users. ~82.86\%\xspace of paywall sites provide a monthly subscription option and ~35.23\%\xspace of sites provide an annual one. Hence,~64.76\%\xspace of the paywalled sites provide \emph{only} a monthly subscription option and ~17.14\%\xspace \emph{only} an annual one.
Next, we measure the distribution of purchasing an annual subscription to a site's content. The median observed annual subscription cost is 108 USD. ~22\% of sites charge less than 60 USD, ~21\% of sites charge more than~180 USD. Figure~\ref{fig:cost_subscr_cdf} presents the full distribution of annual subscription costs. We note that the subscription rates we observe are lower than those estimated by previous work (around~189 USD on average)~\cite{paywallNiemanlab}, possibly reflecting a general decrease in costs. We measure the distribution of annual costs by manually noting the annual subscription cost in the local currency. For sites that do not offer an annual subscription, we sum the cost of twelve monthly subscriptions. We then convert all costs to USD for comparison purposes.
Finally, we measure how subscription costs differ by country. As depicted in Figure~\ref{fig:cost_per_country}, we plot the min, the~15th percentile the median, the~85th percentile and the max of the annual subscription cost across the different countries. The median prices of subscriptions in Australia and Germany are highest (193 and 190 USD, respectively). Subscription costs vary widely by site, too. In Germany and the United States, for example, the most expensive paywalls cost ~$2.63\times$ and~$3.51\times$ more than the median rate, respectively.
\section{Introduction}
\label{Introduction}
Publishers are increasingly moving away from ad-based models, because of the well-known failures~\cite{2018adcost} of ad-based internet funding models. The most common adopted alternative is for sites to deploy ``paywalls''.
``Paywalls'' here are a broad term for monetization systems where visitors are charged subscription fees to access site content, sometimes after being able to sample a small amount of content for free.
The upsides of paywall systems are well understood (i.e., they promise to enable the continued creation of high-quality content). Less understood are the risks and larger implications of an increasingly ``walled'' web. Possible risks include reducing societal access to news and information and the privacy harms of the increased user tracking needed to enforce paywalls.
This work aims to improve the understanding of the popularity, risks and benefits of paywalls online. To introduce the topic, we first (i) describe why the web is increasingly moving away from ``open'' models to ``paywalled'' models, (ii) outline
why this transition is an important topic of study for the research community,
and then (iii) present the structure of the rest of the paper.
\subsection{The Move from Ads to Paywalls}
Digital advertising is the current dominant monetization method for web
publishers, and funds much of the web. Publishers sell advertisements along
page content; middle parties buy these ad slots and fill them with images and
content provided by clients and ad-agencies. This process is usually
programmatic, based of user's personal (i.e., behavioral) data, and completed via
real-time programmatic auctions~\cite{rtbPrices17,pachilakis2019no}.
Web sites are increasingly unsatisfied from this ad-based funding system, for
many reasons. First, the system is dominated by two parties, Google and
Facebook, who jointly harvest more than 70\% of global ad
revenues~\cite{adOligopoly,adDuopoly}, reducing the publisher's ``take'' for ad placements through market power. Second, ad-based funding systems suffer from significant and increasing rates of
fraud~\cite{adfraud,adfraud2,Liu:2014:DDC:2616448.2616455,Zarras:2014:DAM:2663716.2663719,malvertising},
depriving web sites of further funding. Third, behavioral advertising
systems are increasingly incompatible with individual and legal privacy
demands~\cite{trackersWWW2016,vallina2016tracking,Leung:2016:YUA:2987443.2987456,DBLP:journals/corr/abs-1805-10505,razaghpanah2018apps}.
Last, users increasingly use ad blocking tools, for a variety of privacy, performance, and aesthetic reasons~\cite{adlergic,nithyanand2016adblocking},
further depriving publishers of revenue. As a result, ad revenues have decreased in recent years. Both big and small publishers are
coming up short on advertising revenue, even if they are long on visitors
traffic. Accounts of publisher-loss under ad-based funding models contain figures as high as~95\%~\cite{techdirtAdLoss}.
The difficulties of ad-based funding systems have pushed publishers to
alternative funding models, including
donations~\cite{wikipediaFunds1,wikipediaFunds2} or in-browser
crypto-mining~\cite{truthMiners2018}. The most common alternative though is ``paywalls'',
where users pay publishers directly to
access the content they create~\cite{drum2019}
Figure~\ref{fig:screenshot} shows a representative example of a paywall system.
Paywalls so far have a mixed record as funding systems for publishers.
Publishers with large, loyal audiences and high-quality content tend to
be successful with this subscription strategy, with The New York
Times~\cite{nyTimesPaywall}, Wired~\cite{wiredPaywall}, The Financial Times~\cite{ftPaywall} and The Wall Street Journal~\cite{wsjPaywall} as
prominent successful examples. The success of paywalls for smaller and more targeted sites (e.g., local news), or sites with
less affluent audiences, is less clear.
It is important to note that the rapid growth of paywalls has drawn the attention of big tech companies like Google, Facebook and Apple, who have started building platforms to provide or support paywall
services~\cite{paywallsApple,paywallsGoogle,paywallGoogle2,paywallsFacebook}, in an effort to claim their share of the market.
\subsection{Understanding the State of Paywalls}
Creating a sustainable system to fund news and related content is an important goal, and paywalls seem to be a promising (partial) solution to the problem. However, this move from ``open'' to ``walled'' business strategies brings significant, understudied risks. For example, paywalls (implicitly or otherwise) may impose a ``class system'' on the web~\cite{paywallsDemocracy1,paywallsDemocracy2}, potentially driving information-seeking visitors who cannot afford to pay for subscriptions to
badly-sourced, less-vetted, or even intentionally false (but free) new sources.
Despite the importance of the rise of paywalls to the web, it is surprising how little the topic has been studied by the research community. Important open
questions include how popular paywall systems are, what policies paywalls
impose, how users are tracked for paywall enforcement, and whether paywalls are effective at protecting premium content.
\subsection{Contributions}
In this work, we aim to improve the understanding of paywall systems through the first systematic study of paywalls on widely-used web sites. This work makes the following contributions to the understanding of paywall systems on the web:
\begin{enumerate}[labelindent=1cm, leftmargin=0.5cm]
\item A \textbf{novel system for programmatically determining if a site is using a paywall}, though the combination of multiple crowd-sourced data sets and tools.
\item A \textbf{case study of how a popular paywall library operates},
from how a publisher deploys it, how the paywall identifies
users, to how the configured content access policy is enforced.
\item A \textbf{large-scale measurement of paywall popularity}, including
what kinds and what countries account for most paywall use, and how paywall use has changed over time.
Example results include finding that paywall use has increased dramatically over time ($2\times$ more paywalls every 6 months) and that paywall adoption differs by country (e.g., 18.75\%\xspace in US, 12.69\%\xspace in Australia) and industry.
\item An \textbf{in-depth, large scale analysis of deployed paywall policies}, including subscription costs, how paywall adoption impacts the hosting website, how robust paywalls are to evasion, the mechanisms paywalls use to prevent users from viewing protected content, and the privacy implications of paywalls.
\item A \textbf{classifier for deterring whether a site is using a paywall}
for use on sites not considered by crowd-sourced resources, for future long
term, web scale measurements of paywall adoption and behavior.
\end{enumerate}
\begin{figure}[t]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{./figs/Screenshot1.png}
\caption{Truncated article in Wall Street Journal.}
\label{fig:screenshot1}
\end{subfigure}%
\hfill
\begin{subfigure}{.5\textwidth}
\centering
\vspace{0.3cm}
\includegraphics[width=0.8\linewidth]{./figs/Screenshot3.png}
\caption{Obscured article in Miami Herald.}
\label{fig:screenshot2}
\end{subfigure}\vspace{-0.2cm}
\caption{Examples of raised paywalls in major news sites. Paywalls may be enforced in different ways to deny access to articles to non-subscribed users.}
\label{fig:screenshot}\vspace{-0.4cm}
\end{figure}
\section{Paywall Circumvention}
\label{sec:circumvention}
Paywalls must be robust to circumvention if they are going to be a successful monetization scheme for websites. If paywalls can be easily avoided, then content producers will wind up in the same situation they are in with ads and ad-blockers; declining revenues as circumvention tools become more popular. We find that \textit{all observed paywalls are trivial to circumvent}.
We evaluate how robust paywalls are to circumvention in two steps: (i) we categorize the approaches of several popular paywall circumvention strategies, and (ii) we test each strategy on 32\xspace paywalled news sites, we randomly select from our dataset. This subset comprises~28 soft and~4 hard paywalls on popular websites like Wired, Bloomberg, Spectator, Washington Post, Irish Times, Medium, Build, Japan Times, Statesman, and Le Parisien.
\subsection{Evasion Evaluated}
We test the robustness of each paywall system by using Chrome version~71. For each evaluated site, (a) we browse different pages till we trigger the paywall, and then (b) we test a variety of bypassing approaches to circumvent the paywall and get access to the ``protected'' article.
Figure~\ref{fig:paywallsBypassFeatures} lists the evaluated paywall-circumvention strategies, which includes pre-packaged tools, fingerprint evasion techniques, and third-party services. Specifically, we consider:
\begin{enumerate}\itemsep=-1pt
\item changing the screen size dimensions
\item hiding the user's IP address
\item changing the user agent string
\item using an ad blocker extension
\item enabling ``Reader Mode''
\item using the Pocket web service\footnote{or similar ``reader'' services like ``JustRead''~\cite{justRead} and ``Outline''~\cite{outline}}
\item enabling Incognito/Private Mode
\item emptying the cookie jar
\item blocking HTTP requests for popular paywall libraries
\end{enumerate}
Overall, we are able to bypass \emph{all of the soft paywalls and none of the hard paywalls}. Hard paywalls perform their enforcement server-side, when the soft paywalls perform their policy enforcement client-side, and thus their access control is circumventable.
\begin{figure}[tb]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.65\linewidth]{figs/crawlingFlowchart.pdf}
\vspace{-0.2cm}
\caption{Data collection steps of our paywall detector's crawling component.
\vspace{-0.5cm}
\label{fig:visitedSites}
\end{figure}
\subsection{Evasion Approaches Analyzed}
Many of the evaluated evasion approaches are rarely successful. For example, changing the screen size or the IP address of the user rarely circumvents a soft paywall~(4\% effectiveness). A moderate number of soft paywalls~(12\%) is flummoxed by modifying the browser's user agent string.
The majority~(75\%) of soft paywalls is bypassable by resetting the cookie jar (in some cases erasing the first-party cookie only is insufficient, since it is automatically re-spawned by fingerprinting JavaScript code, as seen in Section~\ref{sec:casestudy}). As a result, switching into browsers' ``private browsing'' modes is also sufficient to bypass most paywalls. Some paywalled sites refuse to render content in ``reader modes'' or ``private browsing'' modes, either first party (e.g., reader modes shipped with Safari and Firefox) or third-party (e.g., services like Pocket~\cite{pocket}). Such detection schemes are uncommon though; switching into reader-mode, for example, circumvents paywall enforcement in~60\% of the cases.
Ad-blocking extensions, in their default configurations, have little-to-no effect on paywalls. However, by using the list of known paywall libraries from Section~\ref{sec:results} and by blocking requests to these domains we are able to bypass~48\% of the paywalls without breaking the website's main functionality.
Third-parties like Google Search, Twitter, Reddit and Facebook, can also be used to gain access to some paywalled articles. Some paywalls give visitors from these large third-party systems unfettered access to their content, in pay-for-promotion initiatives. By spoofing the referrer field of the HTTP GET requests, some paywalls are vulnerable to exploiting a controversial policy~\cite{firstclick} where publishers (for promotion purposes) allow access to articles when the visitor comes from one of these platforms (by clicking on a tweet, a post, a Google search result etc.)~\cite{referrerBypass}. These mechanisms can provide access to hard paywalled articles. As a result, some publishers (e.g., Wall Street Journal) have stopped allowing such special access through their paywalls~\cite{wsJGoogleRef}.
\subsection{Data collection}
\label{sec:dataset}
In this study we collect the needed data in two ways: First, we collect 5 different paywall bypassing browser extensions and we extracted the paywalled websites they advertise they can bypass. This gives us~147\xspace unique paywalled websites.
Second, we manually collect a list of domains that provide third party paywall libraries. We enhance this list with first and third party paywall libraries appearing in the latest version of EasyList\cite{fanboyPaywalls}. This gives us a list of~43\xspace unique paywall libraries. We use the libraries appearing in this list to query web archives like httparchive via BigQuery~\cite{httparchive} and PublicWWW~\cite{publicWWW} and find websites that use such paywall libraries.
These two methods give us a dataset of~1,710\xspace paywalled websites as summarized in Table~\ref{tbl:dataCollected}.
\begin{figure}[tb]
\centering
\begin{tabular}{lr}
\toprule
\textbf{Data} & \textbf{Volume} \\
\midrule
Paywalled websites from bypassing extensions & 147\xspace \\
Third-party paywall libraries & 43\xspace \\
Unique paywalled sites & 1,710\xspace \\
Countries the paywalled sites originate from & 61\xspace \\
\bottomrule
\end{tabular}
\vspace{-0.3cm}
\caption{Summary of our dataset.}\vspace{-0.3cm}
\label{tbl:dataCollected}
\end{figure}
\section{Paywall Detection}
\label{sec:detection}
\begin{figure}[t]
\centering
{\small
\vspace{-0.2cm}
\begin{tabular}{lr}
\toprule
{\bf Metric} & {\bf Value} \\ \midrule
Precision & 77\%\xspace{} \\
Recall & 77\%\xspace{} \\
F-Measure & 75\%\xspace \\
AUROC & 0.74\xspace \\
\bottomrule
\end{tabular} \vspace{-0.2cm}
}
\caption{Weighted average of the performance of our RF classifier, after k=5 cross-fold validation.}
\label{table:model-performance}
\vspace{-0.5cm}
\end{figure}
This section presents the design and evaluation of a ML-based detection system
whose goal is to determine whether a site uses a paywall. Our paywall detector
consists of two components: (i) a crawling component that visits a subset of
pages on a site, records information about each page's execution, and extracts
some ML features; and (ii) a classifier, that uses the extracted features to
predict if the site uses a paywall.
We present this classifier as a partial solution for the problem of measuring
changes in the adoption and behavior of paywalls over time. We propose this ML
approach as a \emph{complement} to the crowd-sourced approach described in
Section~\ref{sec:dataset}. The classifier can be used to automatically gauge
paywall prevalence. This automated approach can help identify and quantify
paywalls that have not been identified by crowd-sourced
lists, such as paywalls deployed by unpopular or region-specific sites.
Before describing the classifier in detail, we note two things. First,
the classifier is designed to help detect broad, web-scale trends in
paywall use and behavior, not to detect at real time paywall use on any single specific site.
Second, an important finding of this classifier is that there is far greater
diversity in paywall behavior and implementation logic than we expected at the
start of the effort. We expect this to be a useful starting point for future
studies.
\subsection{Crawling Methodology}
\label{sec:methodology-crawling}
The data collection step of our paywall detector, depicted in
Figure~\ref{fig:visitedSites}, begins with three crawls of the target website:
(i) the \emph{initial crawl} that collects a list of child pages on the website,
(ii) the \emph{cookie jar crawl}, where each child page is crawled
sequentially in the same browsing session and (iii) the \emph{clean
crawl}, where each child page is crawled with a fresh browsing session (i.e., a
``clean'' cookie jar).
This strategy replicates viewing patterns that might cause a paywall to be
triggered, and then attempts to detect the paywall's presence by looking for
page content that was visible on previous visits, but is no longer visible. For
each page crawled, the crawler records the final state of the DOM, which DOM
elements are visible, which page elements are positioned to obscure others (e.g.,
modal dialogs), and other page execution data only available at runtime.
\subsection{Feature Extraction}
We selected features that target both immediately triggering paywalls and
paywalls that trigger after viewing multiple pages. These features aim to
capture an intuition about how paywalls behave, and can fall into three rough
categories: textual features, structural features, and visual features.
\point{Text features} These features consider the text of the page, targeting
text and idioms associated with paywalls. The crawler looks for the phrases
``subscribe'', ``sign up'' and ``remaining'' (translated into 87 languages) in
(i) the ``readermode''\xspace subset of the page, (ii) any overlay or popup elements (e.g.,
elements that have, or are children of elements that have, z-index values
greater than zero), and (iii) elsewhere in the page. These three checks
are performed both in ``cookie jar''\xspace and the ``clean crawl''\xspace recordings of each page.
Several text features use a ``readermode''\xspace version of page, the subsection of the document
identified as the page's ``main content'', or the content stripped of page
``boilerplate'' elements (e.g., advertisements, navigation elements, decorative
images). While there are many different ``readermode''\xspace identification
strategies~\cite{ghasemisharif2018speedreader}, in this work we use Mozilla's
\emph{Readability.js}~\cite{readability} implementation, because of its
popularity and ease of use. We expect using other ``readermode''\xspace strategies would work
roughly as well.
\point{Structural features}
These features target page structure (i.e., HTML), independent
of specific page text or presentation. Structural features include whether the
website has a RSS or ATOM feed, changes in the number of text nodes present in
the page between its ``cookie jar''\xspace and ``clean crawl''\xspace versions, how many
measured pages contain a ``readermode''\xspace subset, and the average and maximum
difference in the amount of text in the document in ``readermode''\xspace, between ``cookie jar''\xspace and ``clean crawl''\xspace
measurements.
\point{Visual features}
These features focus on visual aspects of measured pages, and how those visual
aspects change between the ``cookie jar''\xspace and ``clean crawl''\xspace measurements for each child page. The
detector measures how many text nodes are obscured and the average and maximum
change in obscured text nodes between the two measurements for each page.
Additional display features are the number, and change in, text nodes in the
browser viewport, and number of text nodes (regardless of text content)
appearing in overlay (i.e., $z$-index great than zero) page elements. These
features identify paywalls that prevent users from reading page content
through popups or similar methods.
\subsection{Classifier Accuracy}
\label{sec:classification}
Our paywall detector uses a \emph{random forest} classifier, specifically the
\textit{RandomForestClassifier} implementation provided by the popular
SciKit-Learn~\cite{sklearn} python library. Classification parameters were
selected through~5-fold evaluation using the entirety of the aforementioned
extracted features. As a ground truth, we use a subset of the paywalled sites the oracle identified (Section~\ref{sec:dataset}). To assess the accuracy of the classifier we use a different subset of the oracle's data and a set of non-paywalled websites we manually generate. The paywall detector achieves an
average precision of~77\%\xspace, recall of~77\%\xspace and an area
under the receiver operating characteristics~(AUROC) of~0.74\xspace. These
results are encouraging and suggest that our approach can be used to gauge
paywall prevalence on the web. They also indicate that paywalls
vary in behavior more than we anticipated, and that more complex features
may be needed to further improve accuracy.
\section{Paywall Case Study}
\label{sec:casestudy}
This section provides a detailed case study of a popular third-party paywall system. We provide this case study (i) to introduce the reader to how paywalls work, and (ii) to document the kinds of privacy-affecting behaviors paywalls often rely on to impose their policies.
We select Piano's Tinypass paywall-as-a-service product~\cite{tinypass} for our case study for several reasons. First, it is one of the most popular third-party paywall providers (Tinypass owns~38.2\% of the market, as measured in Figure~\ref{fig:thirdPartyLibs_popularity}), so understanding how this system works provides a good understanding of the kinds of paywall code users are likely to experience. And second, Tinypass can be deployed as a configurable, paywall-as-a-service, allowing publishers (blogs, news sites, magazines, etc.) to impose a variety of paywall policies, both hard and soft.
\subsection{Tinypass: The protocol}
At some point prior to the user's visit, a site owner creates an account at Tinypass, where they describe the subscription policies they wish to enforce. Tinypass generates the keys and identifiers used to enforce the paywall and track visitors.
Once a site owner installs Tinypass on their site, the paywall works in the following six stages, with numbers corresponding to Figure~\ref{fig:tinypassOverview}:
\point{Step one}
The user's browser makes a request to a website where the site owner has installed Tinypass.
\point{Step two}
The website responds with the HTML of their page, including a reference to the Tinypass JavaScript\xspace library, hosted on Tinypass's servers. The content provider's response may also include optional, customized parameters that allow Tinypass to integrate with other services, like Facebook and Google Analytics. At the time of this writing, Tinypass's code is hosted at \url{https://code.tinypass.com/tinypass.js}.
\point{Step three}
The referenced JavaScript causes the browser to request code from Tinypass's server, which responds with a bootstrapping system, providing basic routines for fetching the main implementation code, helper libraries, and utilities for rate limiting and fingerprinting. Depending on the particular deployment, minified versions of this code also includes common utilities like CommonJS-style dependency tools or cryptography libraries.
\point{Step four}
The browser executes the complete Tinypass library, and the full (post-bootstrap) Tinypass library performs a number of privacy-relevant checks. First, Tinypass attempts to determine if a site visitor is actually an automated browser (e.g., Puppeteer, WebDriver client). Tinypass attempts to determine if the user has an ad-blocker installed. Interestingly, Tinypass not only detects if the user currently has an ad-blocker installed, but also if the visitor has changed their ad-blocker usage (e.g., the user had an ad-blocker installed on a previous visit but no longer does, or vice versa).
\begin{figure}[t]
{\footnotesize
\begin{lstlisting}[caption=Excerpt of Tinypass's fingerprinting JavaScript.,label={lst:fingerprint}]
var _getFingerprint = function () {
if (fingerprint) {
return fingerprint;
}
var fingerprint_raw = _getLocality();
fingerprint_raw += _getBrowserPlugin();
fingerprint_raw += _getInstalledFonts();
fingerprint_raw += _getScreen();
fingerprint_raw += _getUserAgent();
fingerprint_raw += _getBrowserObjects();
fingerprint = murmurhash3.x64hash128(fingerprint_raw);
util.debug("Current browser fingerprint is: " + fingerprint);
return fingerprint;
};
\end{lstlisting}\vspace{-0.7cm}
}
\end{figure}
Tinypass then generates a user fingerprint, implemented with the code hosted at \url{https://cdn.tinypass.com/api/libs/fingerprint.js}. The Tinypass fingerprinting library (shown in part in Listing~\ref{lst:fingerprint}) hashes together a number of commonly known semi-unique identifiers (installed plugins, preferred language, installed fonts, screen position, user agent, etc.) to build a unique identifier, hashed together using the MurmurHash3 hash algorithm~\cite{murmur3}). The result is an identifier that is consistent across cookie-clears, and so can re-identify users attempting some evasion techniques. Tinypass also reads, if available, a first-party cookie the library also uses to identify users. When available, this cookie is used in place of the above fingerprint, to track how much content the user has visited.
\begin{figure}[tb]
{\footnotesize
\begin{lstlisting}[caption=Excerpt of returned Tinypass end point data (meter is Tinypass's terminology for a counter describing how much more non-paywalled content a user can view).,label={lst:metrics}
]
...
"trackingId": "{jcx}H4sIAAAAAAAAAI2QW2vCQBCF_8s...",
"splitTests": [],
"currentMeterName": "DefaultMeter",
"activeMeters": [
{
"meterName": "DefaultMeter",
"views": 0,
"viewsLeft": 4,
"maxViews": 4,
"totalViews": 0
}
],
...
\end{lstlisting}\vspace{-0.5cm}
}
\end{figure}
\point{Step five}
Next, the Tinypass library gathers the above information, combines it with information about the page, derived fingerprinting values, the date, and other similar data, and \texttt{POST}s them to a Tinypass endpoint~\footnote{\url{https://experience.tinypass.com/xbuilder/experience/execute?aid=*}}, which records information about the page view. The server then returns a JSON string describing a variety of information about the page view, and excerpt of which is presented in Listing~\ref{lst:metrics}. This JSON string includes a wide variety of both user-facing and program-effecting values, including how many more pages the user is able to visit before the paywall is triggered, possibly new identifiers to rotate on the browsing session, whether the user has logged in and is known to Tinypass (e.g. the user logged in on a different domain owned by the same publisher).
\point{Step six}
Finally, the Tinypass code running on the browser enforces the described paywall policy. The code, client-side, uses the response data to decide how to respond to the page view, possibly by obscuring page content or presenting a subscription offer dialog (by default, Tinypass offers pre-made-but-configurable modal and ``inline'' dialogues the website can check from). In the pages we observed, Tinypass only enforced subscription requirements (i.e., preventing users from viewing content) after the above check was completed. A side effect of this implementation decisions is that Tinypass's restrictions can be circumvented by simply blocking the Tinypass library (see Section~\ref{sec:circumvention}).
\section{Related Work}
\label{RelatedWork}
In~\cite{cornia2017pay}, authors perform an empirical study of the pay models (freemium and paywall models) in European news. In particular, they manually analyzed a small dataset of~171 of the most important news organizations in France, Poland, Germany, Italy, Finland, and UK. Their results show that~66\% percent of the newspapers operate a pay model and that the average price for a monthly subscription is~13.64 Euros when prices in general range from~2.10 to~54.27 Euros/month. In our measurements, 3 years after, the average monthly subscription cost~10.93 Euros, when specifically in Germany it is 20.48 (was 19.75) Euros and in France it is 12.54 (was 13.97) Euros.
In~\cite{myllylahti2017content}, authors explore the content that news publishers consider worthy of placing behind a paywall. They analyze~614 articles from the leading Australasian financial newspapers (i.e., the Australian Financial Review (AFR) and the National Business Review (NBR)). Results show that publishers consider hard (or fast-paced) news and opinion pieces as the most valuable news commodity. In addition, as presented, AFR locked~86\% of its content compared to NBR's~41\%.
In~\cite{carson2015behind}, authors analyze selected paywalled news sites in US, UK and Australia to compare the type, pricing and audience uptake. Results show that paywalls are part of newspapers' toolkit for bringing in new revenue but there is no evidence to suggest they can be a standalone solution. However, in this political economic environment for mastheads, digital advertising revenues alone are also insufficient to meet the cost of providing quality journalism.
\section{Current Paywall Deployments}
\label{sec:results}
In this section, we present a large-scale measurement of paywall deployments on the web. The measurements presented give a broad assessment of how often paywalls are used (by country and by industry). We then present a variety of measurements of how deployed paywalls operate, including the access policies they enforce, their enforcement mechanisms, and how robust these paywalls are to circumvention. The section begins with a description of how we gathered there relevant datasets for measurements, and then proceeds in the above described order.
\begin{figure}[t]
\centering
\vspace{-0.4cm}
\includegraphics[width=.7\linewidth]{./measurements/popularity_growth_half}\vspace{-.2cm}
\caption{Growth of paywall deployments per 6 months. Note that the y-axis
depicts the growth-rate, and not absolute numbers.}\vspace{-.2cm}
\label{fig:popularity_growth}
\end{figure}
\begin{figure}[b]
{\small
\centering
\begin{tabular}{lr}
\toprule
\textbf{Data} & \textbf{Volume} \\
\midrule
Paywalled websites from bypassing extensions & 147\xspace \\
Third-party paywall libraries & 43\xspace \\
Unique paywalled sites & 1,710\xspace \\
Countries the paywalled sites originate from & 61\xspace \\
\bottomrule
\end{tabular}
}
\vspace{-0.3cm}
\caption{Summary of our crowdsourced dataset labeling which websites use
paywalls.}
\vspace{-0.3cm}
\label{tbl:dataCollected}
\end{figure}
\subsection{Dataset}
\label{sec:dataset}
To conduct the measurements described in this section, we built
an oracle to determine whether a web site uses a paywall. While seemingly
a simple question, the diversity of paywall libraries, enforcement
mechanisms, access policies and varying verbiage makes this a difficult
question to answer without significant human intervention.
To solve this problem, we draw on two existing crowd-sourced datasets:
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering\vspace{-.5cm}
\includegraphics[width=1.1\linewidth]{./measurements/alexaTopic}\vspace{-.3cm}
\caption{Type of industry or type of content, paywalled websites deliver.}
\label{fig:alexaTopic}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering
\vspace{.2cm}
\includegraphics[width=1\linewidth]{./measurements/prevalenceTop6_newsTop.eps}\vspace{-.3cm}
\caption{Portion of news sites using paywalls per country. Paywall adoption reaches 18.75\%\xspace in US and 12.69\%\xspace in Australia.}
\label{fig:popularityPerCountry}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.3cm}
\includegraphics[width=1\linewidth]{./measurements/alexaRank_cdf}\vspace{-0.2cm}
\caption{Distribution of the popularity of each paywalled website in our
dataset based on the Alexa ranking.}
\label{fig:alexaRank}
\end{minipage}
\vspace{-.4cm}
\end{figure*}
\point{A) Extensions}
First, we extract rules from several popular browser extensions~\cite{bypassExt1,bypassExt2,bypassExt3,bypassExt4} designed to help users circumvent paywalls. By examining the source code of these extensions, we are able to (directly or indirectly) identify 147\xspace{} websites that the tools' authors and maintainers label as using paywalls.
\point{B) Filter lists}
Second, we use a popular, crowd-maintained filter list that identifies
third-party paywall libraries~\cite{fanboyPaywalls} so that they can
be blocked with common filter-list consuming tools (e.g., AdBlock Plus, uBlock Origin). This list includes filter rules for blocking resources related to a variety of internet ``annoyances''; we extract the subset of the list specifically targeting paywalls. This gives us a list of~43\xspace{} third-party paywall libraries. We query for each entry of this paywall libraries list in two existing, current web crawl archives (i.e., HTTPArchive~\cite{wayback} and PublicWWW~\cite{publicWWW}). We found 1,563\xspace{} sites using one of these paywall libraries and we labeled them as ``paywalled''.
We combine the above two approaches (i.e., paywalled domains labeled by browser
extensions and paywalled sites including third-party paywall libraries)
to identify 1,710\xspace{} unique paywall-using domains, from
61\xspace{} countries. This dataset, summarized in Figure~\ref{tbl:dataCollected}, comprises our complete dataset of paywalled sites used as an oracle in this section, and we provide open-sourced\footnote{\url{https://gist.github.com/panpap/68af1c99b49366dfce4044a354f6e1b8}}.
\subsection{Paywall Popularity}
We start by measuring how popular paywalls are, across several dimensions.
We use the domains identified in Section~\ref{sec:results} as the set of paywalled sites, and all other sites on the web as not paywalled.
\subsubsection{Increase in Paywall Use}
\label{sec:growth}
We first measure whether paywall use has increased over time. We find sites
in our dataset started using paywalls in~2015, with overall paywall use strictly
increasing since. Paywall use has increased at a rate between~120\% and~230\%
every six months since~2015 until recently. In the first six months of~2019,
paywall use quadrupled, and has grown by a further~\textit{180\% } during the
first two months in the second half of~2019. These measures are summarized
in Figure~\ref{fig:popularity_growth}.
We measure paywall growth over time by applying our paywall oracle
(described in Section~\ref{sec:dataset}) to archived versions of the same
sites in the Wayback Machine web archive~\cite{wayback}. We use these
archived versions of each site to approximate date each site adopted a paywall.
The precise methodology is as follows:
\begin{enumerate}[leftmargin=0.5cm,topsep=0pt]
\item We build the set of paywall library related URLs and domains using the technique described in Section~\ref{sec:dataset}.
\item We fetch the most recent archive of each paywalled website in our dataset from the Wayback machine and check whether that historical version is using a paywall.
\item If we observe the site using a paywall, we fetch the next-most-recent version of the site from the Wayback machine (e.g., we move back one recording in time) and re-check.
\item We continue this process until we encounter a version of the web site that no longer is using a paywall. Once we encounter a non-paywalled version of the site, we note the date that version of the site and record it as when the site began using a paywall. \vspace{-0.1cm}
\end{enumerate}
\point{Limitations}
We note two limitations of the above approach, and why we do not believe they significantly impact our findings. It is possible that earlier versions of sites used different types and providers of paywalls than current sites, and so our paywall detection oracle may be missing historical paywall use. While possible, we do not think this limitation significantly impacts the results for two reasons: (i) prior research~\cite{vastel2018filters} has found that filter lists (like the ones we use for paywall library detection) rarely delete rules, and so that paywall-targeting filter lists would identify both current and historical paywalls.
What is more, (ii) we manually evaluated a random sample of commits from the git history of the paywall-targeting portion of the filter list and we found no rule deletions. This gives us further confidence, though not certainty, that filter rules that would identify paywalls on previous versions of paywalled sites have not been removed.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/enforcingStrategie}
\caption{Popularity of the different paywall enforcing policies. Most of the publishers prefer to obfuscate (48.2\%\xspace) or truncate (44.5\%\xspace) the article the user has not yet access to.}
\label{fig:enforcingStrategies}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{0.4cm}
\includegraphics[width=1.05\linewidth]{./measurements/freeArticles_both_cdf}
\vspace{-0.4cm}
\caption{Distribution of the number of free articles allowed per user. The median paywalled website allows 3.5\xspace articles, the median soft-paywalled website allows 4\xspace articles and all hard-paywalled do not allow \emph{any} free article.}
\label{fig:freeArticles}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.6cm}
\includegraphics[width=1.05\linewidth]{./measurements/cost_subscr_cdf}
\vspace{-0.5cm}
\caption{Cumulative distribution of the subscription cost per website for a 12-month content access.}
\label{fig:cost_subscr_cdf}
\end{minipage}
\end{figure*}
The second possible limitation is that our approach might miss sites that
used to have paywalls, but no longer do. We believe such cases to be rare.
We observed no instances of sites using paywalls, removing the paywalls,
and then re-establishing it. This suggests (though does not prove)
that sites do not commonly abandon paywall strategies once they have adopted them.
\subsubsection{Paywall Use by Site Type}
We measure what types of content paywalled sites provide.
We find that most~(80.3\%\xspace{}) paywalled sites provide some form
of news content, whether targeted at the local, regional, or world-level.
Figure~\ref{fig:alexaTopic} provides summary of this measurement.
For this measurement, we use the sites identified as using paywalls from
Section~\ref{sec:dataset} with information available from the Alexa Top Sites
service. The Alexa Top Sites classifies domains into one of~17 different
classes (i.e., news, sports, business, arts, society). Three categories
describe news content, though at different levels of focus (e.g., ``World'',
``Regional'' or, generically, ``News''). We group these together for our
measurements, since they are thematically very similar. The remaining 14
categories account for just 19.7\%\xspace{} of paywalled sites.
\subsubsection{Paywall Use by Country}
Next, we measure which countries have the highest rates of paywall use. Because news sites account for most paywall use, we focus this measurement on news sites. We find that US news sites have been the quickest to move to paywalls, followed by Australia, France, Canada and Germany. Figure~\ref{fig:popularityPerCountry} summarizes our findings. Since our oracle does not identify all websites with paywalls, Figure~\ref{fig:popularityPerCountry} presents only the lower bound of the existing paywalled sites.
We measure rates of paywall use by country by first retrieving the Alexa the Top~10,000 websites per country. We filter the list and remove all non-news sites. Then, we calculate the percentage of paywall-using news sites, as a fraction of all news sites, per country. We find that 18.75\%\xspace of US news sites use paywalls, 12.69\%\xspace of Australian new sites, and less than ~7\% in all other countries.
\subsubsection{Paywall Use by Popularity}
Next, we measure whether there is any clear relationship between paywall deployment and the popularity of a website. We did not observe any such relationship. We
anticipate that most paywalled sites would be popular (as measured by Alexa Top Sites), as a successful paywall would require a significant number of
subscribers, which in turn would require a significant amount of baseline
visitors. Instead, we find that only 8.54\%\xspace{} of paywall-using sites
are among the~10,000 most popular sites on the web. The median paywall-using
site is ranked 365,316\xspace{}. The full distribution
of the popularity of paywall using sites is presented in Figure~\ref{fig:alexaRank}.
\begin{figure}[t]
\centering
\vspace{-0.3cm}
\includegraphics[width=0.9\linewidth]{./measurements/thirdPartyLibs_popularity}
\vspace{-0.7cm}
\caption{Popularity of third-party paywall libraries in our dataset. A
small number of paywall implementations account for the majority of
third-party paywall deployments.}
\label{fig:thirdPartyLibs_popularity}\vspace{-0.2cm}
\end{figure}
\subsection{Paywall Libraries}
\label{sec:3plibs}
A significant number of sites rely on third-parties for their paywall
implementations. These third-parties sell ``paywall-as-a-service'' products,
where publishers pay fee to have the third-party manage and
enforce the paywall on the publisher's site. We observe
that a small number of paywall providers account for the vast majority
of paywall deployments, with Piano and Tecnavia being the most popular
paywall providers (23.5\%\xspace{} and 21.0\%\xspace{} market
share, respectively). The full distribution of third-party paywall
market share is depicted in Figure~\ref{fig:thirdPartyLibs_popularity}.
This consolidation of paywall implementation and enforcement is significant,
for a variety of reasons. First, market consolidation may effect the amount of
income content-makers can receive for their content (popular third-party
paywall providers receive 10-15\% of each sold subscription). Second, provider
consolidation may make large scale circumvention easier, as circumventors
need to target a smaller number of systems (see Section~\ref{sec:circumvention}). Third, a small number of paywall providers
tracking users across a large number of websites has clear privacy implications
(see Section~\ref{sec:privacy}).
We measure the popularity and consolidation of third-party paywall libraries
by crawling each paywalled site in our dataset and observing which resources
from known paywall providers were fetched. We find that at least
25\%\xspace of paywalled websites outsource their paywall
functionality to third-parties. The distribution of third party paywall
use follows a rough power-law distribution.
\subsection{How Paywalls Affect Site Use}
\label{sec:alexa}
Paywalls affect how users interact with the site. Recent studies~\cite{kim2019newspapers}, by monitoring the pageviews of 36 news sites before and after paywall deployment, report that they lose nearly~30\% of their daily traffic and a loss of pageviews, ranging from a~10\% to~55\%.
In this section, we measure differences between how sites interact with paywalled and non-paywalled sites. We find that users view less pages on paywalled sites, stay for shorter periods of time and link to pages less. Interestingly, we did not see a significantly difference to the bounce rate between paywalled and non-paywalled sites~\footnote{We do not address the issue of causation; its possible, for example, that the types of site likely to use paywalls have lower \emph{dwell times} already, so that the use of a paywall is a more a result of lower dwell time than the cause. We leave disentangling cause and effect for future work.}.
\begin{figure*}[t]
\centering
\begin{minipage}{0.32\linewidth}
\centering
\includegraphics[width=1.05\linewidth]{./measurements/dailyTimeOnSite_normalized_cdf}
\caption{Distribution of the average time a visitor spends daily per news site. In median values visitors tend to spend daily $2.46\times$\xspace more time per site rank on non-paywalled websites.}
\label{fig:dailyTimeOnSite}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering\vspace{-0.3cm}
\includegraphics[width=1.05\linewidth]{./measurements/siteLinks_normalized_cdf}
\caption{Distribution of the incoming site links per news site. Paywalled sites get significantly (18.9$\times$\xspace) less site links per rank in median values.}
\label{fig:siteLinks}
\end{minipage}
\hfill
\begin{minipage}{0.32\linewidth}
\centering
\vspace{-.3cm}
\includegraphics[width=1.05\linewidth]{./measurements/paywallsBypassApproaches.pdf}
\vspace{-.6cm}
\caption{Success rate of the different paywall bypassing approaches. Clearing the cookie jar alone can bypass~75\% of the paywalls.}
\label{fig:paywallsBypassFeatures}
\end{minipage}
\end{figure*}
\subsubsection{Bounce Rate}
We find that paywalled new sites have slightly higher bounce rates~\footnote{The percentage of visitors who visit a site and then leave, rather than continuing to view other pages within the same site.} than non-paywalled news sites. The distributions of bounce rates is depicted in Figure~\ref{fig:alexa_bounce_cdf}.
The median paywalled news site has slightly higher bounce rate (68.4\%\xspace) contrary to the median non-paywalled (67.5\%\xspace). However, we see that for ~30\% of the websites in the two categories the difference is~2-7\% higher. To compare bounce rates, we used the Alexa Top Sites data, and compared the bounce rates for the paywalled news sites in our data set with the Alexa top ~1K news sites.
\subsubsection{Daily Page Views}
Next, we measure the number of pages the average visitor performs daily on the websites and we compare how this changes for the paywalled and non-paywalled news sites. In Figure~\ref{fig:pageviews}, we plot the cumulative distribution of these page views per website in our dataset. Users visit on average 13.42\%\xspace less pages on paywalled news sites than non-paywalled new sites.
\subsubsection{Average Time spent on Site}
Figure~\ref{fig:dailyTimeOnSite} compares the distribution of the median time users spend on paywalled and non-paywalled websites, normalized by popularity (based on its Alexa rank). We find that visitors spend daily $2.46\times$\xspace more time per on non-paywalled news sites.
\subsubsection{Content Popularity and Link Rate}
Finally, we measure the impact of paywalls on how often sites link to the paywalled sites. Since site linking may be affected by the popularity of the website, in Figure~\ref{fig:siteLinks}, we plot the cumulative distribution of the number of site links (or backlinks) per news site normalized by its Alexa rank. We observed paywalled sites being linked to significantly less (18.9$\times$\xspace) often than non-paywalled sites.
\subsection{Data collection}
\label{sec:dataset}
In this study we collect the needed data in two ways: First, we collect 5 different paywall bypassing browser extensions and we extracted the paywalled websites they advertise they can bypass. This gives us~147\xspace unique paywalled websites.
Second, we manually collect a list of domains that provide third party paywall libraries. We enhance this list with first and third party paywall libraries appearing in the latest version of EasyList\cite{fanboyPaywalls}. This gives us a list of~43\xspace unique paywall libraries. We use the libraries appearing in this list to query web archives like httparchive via BigQuery~\cite{httparchive} and PublicWWW~\cite{publicWWW} and find websites that use such paywall libraries.
These two methods give us a dataset of~1,710\xspace paywalled websites as summarized in Table~\ref{tbl:dataCollected}.
\begin{figure}[tb]
\centering
\begin{tabular}{lr}
\toprule
\textbf{Data} & \textbf{Volume} \\
\midrule
Paywalled websites from bypassing extensions & 147\xspace \\
Third-party paywall libraries & 43\xspace \\
Unique paywalled sites & 1,710\xspace \\
Countries the paywalled sites originate from & 61\xspace \\
\bottomrule
\end{tabular}
\vspace{-0.3cm}
\caption{Summary of our dataset.}\vspace{-0.3cm}
\label{tbl:dataCollected}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,319
|
{"url":"https:\/\/docs.octeract.com\/so1000-how_to_set_options","text":"# Option and Option Files on Octeract Engine\n\nOcteract Engine allows users to customise their solving by changing solver options according to their preference. Let\u2019s take a look at what exactly is meant by solver options and how to change these options using an options file.\n\nOcteract Engine accepts custom options through a plain text file. The user can create an empty text file in the format octeract.opt. Within this file the options can be specified, as explained in the section below.\n\n## Setting Custom Options in an Option File\n\nIt\u2019s important to note that all options are case-sensitive. Each option requires a new line and they are set using the following syntax:\n\nOPTION_NAME = option_value\n\nTo show this better, here\u2019s an example.\n\nLet\u2019s assume that the user wants to:\n\n\u2022 turn off the optimisation based bounds tightening technique (OBBT, default behaviour is on)\n\u2022 set infinity to $$10^6$$ (default is $$10^7$$)\n\u2022 adjust the convergence tolerance to 0.1 (default is $$10^{-3}$$ ) and\n\u2022 allow Octeract Engine to run for 60 seconds (default is $$2\\times10^9$$ seconds)\nThe contents of the options file should look like the text below:\n\n \n#my octeract custom option file example\nUSE_OBBT = false\nINFINITY = 1.e6\nCONVERGENCE_TOLERANCE = 0.1\nMAX_SOLVER_TIME = 60\n\n\n\nFor further information on how to change specific options, have a look at the Knowledge Base on the left sidebar.\n\n## Using Custom Option Files in Third Party Software\n\nWhen a *third party software is used, in conjunction with Octeract Engine, the user should follow the steps below:\n\n\u2022 Create a text file with the name octeract.opt and save it in the location where the third party software will be run. The name should be exact and in lower case.\n\u2022 Define custom options within octeract.opt.\nA quick note on using a set of options defined, in the options file octeract.opt, when there is a change in location of the file being run.\n\nIf the user would like to use octeract.opt for a file in a different location (to where the options file is currently located), the options file octeract.opt will need to be added to this location.\n\nFor example, if a user chooses to run e.g. a Pyomo file by launching Pyomo at location\/some\/path\/in\/the\/users\/computer\/ and this location includes an Octeract Engine options file octeract.opt. Octeract Engine will read octeract.opt automatically.\n\nIf the user wants to launch Pyomo for, the same or a different file at locationa\/different\/path\/in\/the\/users\/computer\/ using the same set of options as before, the octeract.opt file needs to be copied from location \/some\/path\/in\/the\/users\/computer\/ to a\/different\/path\/in\/the\/users\/computer\/.\n\nTherefore, the directory from which the third party software is launched is the correct location of the octeract.opt file.\n\n*Third party software that Octeract Engine supports: Pyomo, Python, AMPL, GAMS, JuMP and C++.\n\n## Using Custom Option Files from Terminal\n\nFor Octeract Engine usage, via a Terminal environment on Ubuntu\/Linux\/Unix or Command Prompt (CMD) on Windows, the user can choose how to use custom option files from the two methods mentioned below:\n\nMethod 1: the user can specify the custom options within a standard octeract.opt file in the directory where Octeract Engine will be run from.\n\nMethod 2: the user can append the following to the usual Octeract Engine call:\n\n-o \/path\/to\/options\/file\/octeract.opt","date":"2022-06-27 11:20:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 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\": 0.2983902394771576, \"perplexity\": 3460.7600706767157}, \"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-2022-27\/segments\/1656103331729.20\/warc\/CC-MAIN-20220627103810-20220627133810-00458.warc.gz\"}"}
| null | null |
\section{Introduction}
\label{sec:Introduction}
Resource-constrained devices (e.g., low-end sensors, smart-cards, RFID-tags) play an important role in emerging ubiquitous systems like Internet of Things and Systems (IoTS) and Wireless Sensor Networks (WSNs). Using service oriented architecture (SOA) further broadens the horizons of IoTS, opening up many applications where resource-constrained devices can participate as service consumers and/or providers.~\cite{TSC_Main,TSC_1,TSC_2} It is vital for such systems to operate securely and efficiently. Hence, it is necessary to provide authentication and integrity for resource-constrained devices. For instance, guaranteeing the integrity and authentication of financial transactions in a smart-card or RFID-tag is critical for any commercial application. However, this is a challenging task due to the memory, processor, bandwidth and battery limitations of these devices.
It is also important to be able to publicly verify the authentication tags produced by resource-constrained devices. This enables any resourceful device (e.g., a laptop or a base station) to publicly audit transactions and system status.
Symmetric cryptography primitives such as Message Authentication Codes (MACs) are computationally efficient and therefore are preferred for resource-constrained devices in small-scale systems. However, such primitives are not scalable for large-distributed systems and are not publicly verifiable~\cite{PKC:WNSInvestigation:Lopez:2006,LogFASFC2012}. They also cannot achieve the non-repudiation property, which is necessary for various applications (e.g., transportation payment systems, medical implants, logical/physical access with tiny devices) that may need public auditing.
Digital signatures rely on public key infrastructures for the signature verification~\cite{JonathanKatzModernCrytoBook,BroadcastSecureBookPerrig}. They are publicly verifiable, scalable for large systems and achieve non-repudiation. Therefore, they are highly useful authentication tools for security-critical applications such as medical devices, payment systems, secure auditing in embedded devices and security systems (e.g., building access control). However, existing digital signatures also have some limitations that might prevent them to be fully practical for highly resource-constrained devices.
In the following, we first briefly discuss some prominent digital signature alternatives and their limitations when employed on resource-limited devices. We then present our contributions by summarizing the desirable properties of our scheme, followed by its limitations.
\subsection{Limitations of Signer Efficient Signatures} \label{subsec:RelatedWork}
The existing digital signature alternatives do not offer small private key size, small signature size, and high efficiency at the signer, at the same time. We elaborate on some of these alternatives below.
\noindent \textbf{Traditional Signatures:} RSA~\cite{RSA}, one of the most well-known signature schemes, is computationally efficient at the verifier's side. However, it requires an {\em expensive operation}\footnote{\small{We refer to operations such as modular exponentiation~\cite{JonathanKatzModernCrytoBook}, elliptic curve scalar multiplication~\cite{ECCGuide} or pairing~\cite{PairingCrypto} as expensive operations.}} at the signer's side and have large key/signature sizes. Hence, it may not be suitable for resource-constrained embedded devices (e.g., medical implants).
Elliptic Curve (EC) based schemes are highly popular on such devices due to their small key and signature sizes, along with better efficiency compared to RSA~\cite{ECDSA, BLS_asiacrypt, Schnorr91}. Various different curves, and signature schemes on these curves have been proposed, which offer improved computational efficiency and security~\cite{Curve25519Base, FourQ, Kummer, Ed25519, SchnorrQ}. Some of these curves are also implemented in embedded devices~\cite{Ed255198bit, Kummer8bit, FourQ8bit} such as 8-bit AVR microprocessors. However, these signature schemes still require an expensive operation (i.e., EC scalar multiplication) at the signer's side. This requirement may hinder an efficient adoption of these schemes to low-end microprocessors with critical battery limitations (e.g., medical implants).
\noindent \textbf{Online/Offline Signatures:} Many techniques have been proposed to improve the efficiency of traditional signatures. These include online/offline signatures that eliminate expensive operations in signature generation via pre-computed tokens generated offline~\cite{ImprovedDSAEurocrypt94, OnlineOfflineEvenBase1989}. Although these schemes are computationally efficient, they incur large storage overhead to the signer. Later, Shamir et al. in~\cite{OfflineOnline_ImprovedShamir_2001} proposed an improved online/offline signature that is more space efficient. However, by nature, these schemes require linear storage with respect to the number of signatures that can be generated, which is impractical for storage-limited signers.
\noindent \textbf{One-time Signatures (OTSs):} These schemes rely on one-way functions without trapdoors and offer very efficient computations~\cite{OTS_Lamport_79,HORS_BetterthanBiBa02,BroadAuth:Law:2013:Comparison:AsiaCCS}. Specifically, Lamport~\cite{OTS_Lamport_79} proposed the first one-time signature scheme, where for each bit of the hash of the message, two hash outputs were stored as the public key that resulted in a very large size. Then, in HORS signature scheme~\cite{HORS_BetterthanBiBa02}, special message encoding techniques have been considered to significantly reduce the public while preserving the computational efficiency. Hash-based schemes also offer post-quantum security that is lacked in most of the traditional signatures. On the other hand, they have a large signature and very large public key sizes. Some EC-based OTSs also exist~\cite{Zaverucha} that offer small signature size, but with a trade-off between private key size and signature generation efficiency. Moreover, in OTSs, a private/public key pair can be used only once. This may require costly private key generations at the resource-limited device for each message to be signed.
\noindent \textbf{Multiple-time Signatures:} For these schemes, after $K$ signatures, the key pair must be re-generated. Therefore, {\em we refer to these schemes as $K$-time signatures}. Inherently, OTSs (e.g., Lamport~\cite{OTS_Lamport_79}, HORS~\cite{HORS_BetterthanBiBa02}) can be used as $K$-time signatures if $K$ key pairs are generated at the key generation phase. Some hash-based multiple-time signatures were proposed~\cite{OTS_r_time_PieprzykWX03, OTS_r_time_HORSED_2006, XMSS, TVHORS} based on HORS signature~\cite{HORS_BetterthanBiBa02}. However, these schemes either suffer from large key/signature sizes~\cite{OTS_r_time_PieprzykWX03, OTS_r_time_HORSED_2006, XMSS}, or provide security only for a short-limited amount of time~\cite{TVHORS}. Some stateless hash-based schemes were also proposed~\cite{SPHINCS}, extending these multiple-time signatures to full-time (traditional) signature schemes. Although it is shown that they can be implemented on resource-constrained devices~\cite{ArmedSPHINCS:Hulsing:PKC:2016}, it is highly computationally costly at the signer's side, and it also requires the transmission of large signatures (e.g., up to 41 KB).
Our proposed scheme also falls into this category and inherits some of the limitations of $K$-time signatures (e.g., after $K$ signatures, the key must be re-generated). However, due to the unique construction of our scheme that leverages Fiat-Shamir transform with compact and efficient elliptic curves, it offers the highest signer efficiency among all the aforementioned $K$-time signatures (including the use of OTSs as $K$-time signatures). For instance, as shown in Table~\ref{tab:AVR}, our scheme outperforms HORS~\cite{HORS_BetterthanBiBa02} (most efficient counterpart) $6\times$ at signature generation and $12 \times$ at signature size on an 8-bit microprocessor.
\noindent \textbf{Lattice-based and Code-based Signatures:} The main advantage of these schemes is {\em post-quantum security}. Although some of these schemes offer computational efficiency, they are still relatively computationally expensive (e.g., require heavy operations such as Gaussian Sampling~\cite{BLISS}) and key/signature sizes might be prohibitive for resource-constrained devices~\cite{BLISS, dilithium, pqsigRM, CodeBasedCourtois, Tachyon}. Since these schemes provide long-term security (security against quantum attackers), they might be ideal for resource-constrained devices in the future, if the sizes are reduced.
\vspace{1mm}
\noindent\fbox{%
\parbox{.48\textwidth}{%
{\em There is a need for a signer efficient digital signature that prioritizes the signer efficiency by achieving optimal private key and signature sizes without requiring any expensive operations for the signature generation.}
}%
}
\subsection{Our Contribution}
In this paper, we create a new multiple-time signature scheme, which we refer to as {\em \underline{S}igner \underline{E}fficient \underline{M}ultiple-time \underline{E}lliptic \underline{C}urve \underline{S}ignature (\ensuremath {\texttt{SEMECS}{\xspace}})}. We summarize some desirable properties of \ensuremath {\texttt{SEMECS}{\xspace}}~below. A detailed performance analysis is given in Section~\ref{sec:PerformanceAnalysis}
$\bullet$~{\em \underline{High Computation \& Energy Efficiency at the Signer}}: \ensuremath {\texttt{SEMECS}{\xspace}}~only requires {\em two hash function calls, a single modular multiplication, and a modular subtraction} to generate a signature. Therefore, its cost is even comparable to symmetric hash-based MACs that are not suitable for large and distributed systems. \ensuremath {\texttt{SEMECS}{\xspace}}~offers very fast signature generation, (1.23 microseconds on an i7 Skylake processor). On a resource-constrained processor, this translates into high energy efficiency. Our experiments confirmed that \ensuremath {\texttt{SEMECS}{\xspace}}~signature generation~has 6$\times$ lower energy consumption compared to its closest counterpart and 118$\times$ lower than Ed25519~\cite{Ed25519, Ed255198bit} on 8-bit AVR microprocessor (see Table~\ref{tab:AVR}).
$\bullet$~{\em \underline{Compact Private Key \& Signature}}: \ensuremath {\texttt{SEMECS}{\xspace}}~only requires storing a 32-byte private key (that can be derived from a 16-byte seed with a PRF) and incurs an additional 32 Bytes to the message as the signature, for $\kappa = 128$-bit security level. \ensuremath {\texttt{SEMECS}{\xspace}}~has two signature components of 32 Bytes, where one of them is used to recover the first 32 Bytes of the message. Therefore, the transmission overhead is just 32 Bytes, that is optimal for EC-based digital signatures (as in BLS~\cite{BLS_asiacrypt}). Thus, \ensuremath {\texttt{SEMECS}{\xspace}}~is also lightweight in terms of signer storage and transmission. Due to its small signature size, \ensuremath {\texttt{SEMECS}{\xspace}}~is also very energy efficient in terms of communication, in addition to its high energy efficiency in signer computation. Moreover, since \ensuremath {\texttt{SEMECS}{\xspace}}~does not require any EC operation at the signer's side, the signer does not need to store any curve parameters and codes. This is specifically important for resource-constrained devices that have limited space for the code (e.g., AVR ATmega 2560 has 256 KB).
\begin{table*}[t!]
\centering
\caption{Signer-side performance of \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts on 8-bit AVR microprocessor} \label{tab:AVR}
\vspace{-2mm}
\begin{threeparttable}
\begin{tabular}{| c || c | c | c | c | c | c | }
\hline
\textbf{Scheme} & $K$ & \specialcell[]{\textbf{Signature Generation} \\ \textbf{ Time (}CPU cycle\textbf{)}} & \specialcell[]{\textbf{Private Key} \\ \textbf{(Byte)}} & \specialcell[]{\textbf{Signature } \\ \textbf{Size (Byte)}} & \specialcell[]{\textbf{Computation energy} \\ \textbf{(mJ)}} & \specialcell[]{\textbf{Communication energy} \\ \textbf{($\mu$J)}}\\ \hline \hline
\multicolumn{7}{|c|}{\em \textbf{Full-time signatures}} \\ \hline
SPHINCS~\cite{SPHINCS} & $2^{\kappa}$ & 2 681 600 389 & 1088 & 41000 & 16760.00 & 6115.56 \\ \hline
ECDSA~\cite{ECDSA} & $2^{\kappa}$ & 48 188 992 & 32 & 64 & 301.18 & 9.55 \\ \hline
Ed25519~\cite{Ed25519, Ed255198bit} & $2^{\kappa}$ & 23 211 611 & 32 & 64 & 145.07 & 9.55 \\ \hline
$\mu$Kummer~\cite{Kummer, Kummer8bit} & $2^{\kappa}$ & 10 404 033 & 32 & 64 & 65.03 & 9.55 \\ \hline
SchnorrQ~\cite{SchnorrQ, FourQ8bit} & $2^{\kappa}$ & 3 740 000 & 32 & 64 & 23.38 & 9.55 \\ \hline \hline
\multicolumn{7}{|c|}{\em \textbf{$K$-time signatures}} \\ \hline
\multirow{2}{*}{HORS~\cite{HORS_BetterthanBiBa02}} & 1 & \multirow{2}{*}{1 180 618} & \multirow{2}{*}{16} & \multirow{2}{*}{384} & \multirow{2}{*}{7.38} & \multirow{2}{*}{57.28} \\
& $2^{17}$& & & & & \\ \hline
\multirow{2}{*}{HORSE~\cite{OTS_r_time_HORSED_2006}} & 1 & 1 180 618 & 16384 & \multirow{2}{*}{384} & 7.38 & \multirow{2}{*}{57.28} \\
& $2^{17}$& 19 644 106 & 278528 & & 122.78 & \\ \hline
\multirow{2}{*}{XMSS~\cite{XMSS}} & 1 & 10 233 600 & \multirow{2}{*}{16} & 2080 & 63.96 & 310.25 \\
& $2^{17}$& 101 509 850 & & 2592 & 634.44 & 386.62 \\ \hline
\multirow{2}{*}{Zaverucha et al.~\cite{Zaverucha}} & 1 & \multirow{2}{*}{6 250 660} & \multirow{2}{*}{16} & \multirow{2}{*}{48} & \multirow{2}{*}{39.07} & \multirow{2}{*}{7.16} \\
& $2^{17}$& & & & & \\ \hline \hline
\multirow{2}{*}{\ensuremath {\texttt{SEMECS}{\xspace}}} & 1 & \multirow{2}{*}{ \textbf {195 776} } & \multirow{2}{*}{\textbf {32} } & \multirow{2}{*}{\textbf {32}} & \multirow{2}{*}{ \textbf{1.22}} & \multirow{2}{*}{ \textbf{4.77}} \\
& $2^{17}$& & & & & \\ \hline
\end{tabular}
\begin{tablenotes}[flushleft] \scriptsize{
\item The cost of hash-based schemes are estimated based on the cost of a single hash operation.
}
\end{tablenotes}
\end{threeparttable}
\vspace{-5mm}
\end{table*}
$\bullet$~{\em \underline{Open-source Implementation and Comprehensive Analysis}}: We fully implemented \ensuremath {\texttt{SEMECS}{\xspace}}~on a laptop and the signature generation of \ensuremath {\texttt{SEMECS}{\xspace}}~on an 8-bit AVR microprocessor. We open-sourced all of our implementations for broad testing, benchmarking and adoption purposes. We also analyzed and compared the efficiency of \ensuremath {\texttt{SEMECS}{\xspace}}~with a wide variety of efficient signature schemes (see Section~\ref{sec:PerformanceAnalysis}) on both platforms.
$\bullet$~{\em \underline{Provable Security with a Tight Reduction}}: We prove that \ensuremath {\texttt{SEMECS}{\xspace}}~is existentially unforgeable against chosen-message attacks in Random Oracle Model (ROM)~\cite{RandomOracleModel93}. In Section~\ref{sec:SecurityAnalysis}, we show that \ensuremath {\texttt{SEMECS}{\xspace}}~has a tight reduction to the Discrete Logarithm Problem (DLP), without the need for the forking lemma~\cite{ForkingLemma}, as Fiat-Shamir type signatures do. In our security analysis, we exploit the fact that \ensuremath {\texttt{SEMECS}{\xspace}}~is a multiple-time signature, and therefore it has higher security for a limited number of queries (as the nature of multiple-time signatures).
All the above properties show that \ensuremath {\texttt{SEMECS}{\xspace}}~is potentially an ideal alternative to provide authentication and integrity for resource-constrained devices.
\noindent \textbf{Differences of this work with its preliminary version appeared in WiSec 13'~\cite{ETA}}: (i) In this work, we developed a new signature scheme that we refer to as \ensuremath {\texttt{SEMECS}{\xspace}}~that reduces the signature/private key size of our preliminary scheme \ensuremath {\texttt{ETA}}{\xspace}~\cite{ETA}, and thereby achieves optimal signature and key sizes for an EC-based signature scheme. Moreover, \ensuremath {\texttt{SEMECS}{\xspace}}~generates private key components deterministically, and therefore offers improved security against weak pseudo-random number generators. (ii) In this work, we provided a full-fledged open-source implementation of \ensuremath {\texttt{SEMECS}{\xspace}}~on 8-bit AVR ATmega 2560 microprocessor with a comprehensive energy consumption analysis and comparison. We also gave comprehensive performance comparison of \ensuremath {\texttt{SEMECS}{\xspace}}~with some of the most recent and efficient digital signatures (including but not limited to Ed25519~\cite{Ed25519}, SchnorrQ~\cite{FourQ, SchnorrQ}, SPHINCS~\cite{SPHINCS}, XMSS~\cite{XMSS}, HORS~\cite{HORS_BetterthanBiBa02}). (iii) In this work, we provided an improved security proof that achieves a significantly tighter security reduction compared to that of \ensuremath {\texttt{ETA}}{\xspace}.
\noindent \textbf{Limitations}: Despite all its merits, \ensuremath {\texttt{SEMECS}{\xspace}}~also has its limitations that are inherent to multiple-time signatures: (i) It can sign up to a pre-determined $K$ messages, but then needs to be bootstrapped. (ii) It is a stateful signature scheme. (iii) In \ensuremath {\texttt{SEMECS}{\xspace}}, the public key size is linear with respect to $K$. This requires verifiers to be storage resourceful.
\noindent \textbf{Potential Use-cases}: Remark that for our envisioned applications, the signer computational/storage/communication efficiency is much more important than the verifier storage efficiency alone. Furthermore, these applications permit verifiers to be storage resourceful (e.g., a cloud server for medical systems, base stations in WSNs, control centers in cyber-physical systems). Similarly, it is feasible to perform the key generation phase offline in these applications. In the following, we discuss some of the potential applications for \ensuremath {\texttt{SEMECS}{\xspace}}.
Medical implants are equipped with resource-constrained microprocessors (e.g., 8-bit AVR~\cite{IMDProcessor}, as we used in our experiments) that need to report sensitive data to doctors, hospital servers, etc. Symmetric key authentication (MACs) is usually preferred for these systems. However, these mechanisms lack non-repudiation, and public verification that are highly desirable for some medical systems, because of digital forensics and legal issues~\cite{Medical:PersonalHealthDevice:Analysis:Standard:Rubio:2016, MedicalDevice:Survey:2015:Camara2015272}. Thus, there is a need for low-cost public key primitives (e.g., authentication) for these systems~\cite{Ozmen_IOT_SP}. \ensuremath {\texttt{SEMECS}{\xspace}}~can be considered as an ideal alternative for medical implants due to its signer efficiency. Our experiments on 8-bit AVR showed that \ensuremath {\texttt{SEMECS}{\xspace}}~consumes less energy compared to its counterparts. In practice, this translates into a longer battery life that is critical for medical implants.
Additionally, \ensuremath {\texttt{SEMECS}{\xspace}}~is highly suitable to provide authentication for SOA based IoT systems. SOA based IoT infrastructures are comprised of networked, resource-constrained devices~\cite{TSC_Main,TSC_1} that require efficient authentication mechanisms. Some essential applications of SOA based IoT includes but not limited to e-health, smart product management and smart events for emergency management~\cite{TSC_1,TSC_Healthcare_IoT_Services1,TSC_Healthcare_IoT_Services2}. Similarly with medical implants, non-repudiation and public verification are critical for these applications. Moreover, a server (or a broker - i.e., coordinators that operate between the server and the devices) is usually utilized the connected resource-constrained devices to provide these services~\cite{TSC_2}. Servers and brokers are usually equipped with higher-end processors, compared to the IoT devices, that has expandable memories. Therefore, in such SOA based IoT applications, we believe that the server or broker can tolerate the storage of a larger public key in exchange of a significantly higher signer efficiency that translates into longer battery life for resource-constrained devices.
Many secure WSN protocols such as clone detection~\cite{NodeReplication_Journal_ContiPMM11}, secure code dissemination~\cite{SelugeHyun:2008} and secure logging~\cite{SUHaSAFSS11} include a low-end signer that reports to resourceful servers, and base stations. \ensuremath {\texttt{SEMECS}{\xspace}}~can substantially increase the lifespan of WSNs by serving as the authentication mechanism for such protocols. Moreover, \ensuremath {\texttt{SEMECS}{\xspace}}~can be deployed in some token-based logical/physical access control systems.
\section{Preliminaries}
\label{sec:Prelim}
In this section, we first give our notation and definitions. We then describe our system and security models.
\subsection{Definitions and Algorithms} \label{subsec:DefSecModel}
\noindent\textbf{Notation.} $||$ denotes the concatenation operation.
$|r|$ denotes the bit length of variable $r$. $r \stackrel{\$}{\leftarrow}\mathcal{S}$ denotes that variable $r$ is
randomly and uniformly selected from set $\mathcal{S}$. We denote by
$\{0,1\}^{*}$ the set of binary strings of any finite length. $\mathcal{A}^{\mathcal{O}_0,\ldots,\mathcal{O}_{i}}(\cdot)$
denotes algorithm $\mathcal{A}$ is provided with oracles
$\mathcal{O}_0,\ldots,\mathcal{O}_{i}$. For example,
$\mathcal{A}^{\mathit{SGN.Sig}_{\ensuremath {\mathit{sk}}{\xspace}}}(\cdot)$ denotes that algorithm
$\mathcal{A}$ is provided with a {\em signing oracle} of signature
scheme $\mathit{SGN}$ under a private key \ensuremath {\mathit{sk}}{\xspace}. $H_i: \{0,1\}^{*} \rightarrow \mathbb{Z}_{q}^{*}, i\in\{0,1\}$ are distinct Full Domain Hash Functions~\cite{FullDomainHashExact2000}, where $q$ is a large prime.
\begin{definition}
\label{Def:CEMGenericDefiniton}
A $K$-time signature scheme \ensuremath {\texttt{SGN}}{\xspace}~is comprised of a tuple of three algorithms
$(\mathit{Kg},$ $\mathit{Sig},\mathit{Ver})$ defined as follows:
\begin{enumerate}[-]
\item $(\ensuremath {\mathit{sk}}{\xspace},\ensuremath {\mathit{PK}}{\xspace})\leftarrow
\ensuremath {\texttt{SGN.Kg}}{\xspace}(1^{\kappa},K)$,: The key generation
algorithm takes the security parameter $1^{\kappa}$ and the maximum
number of messages to be signed $K$ as the input. It returns a private/public
key pair $(\ensuremath {\mathit{sk}}{\xspace}_0,\ensuremath {\mathit{PK}}{\xspace})$ as the output.
\item $\sigma_{j}\leftarrow \ensuremath {\texttt{SGN.Sig}}{\xspace}(\ensuremath {\mathit{sk}}{\xspace}_j,M_j)$: The signature generation algorithm takes the private key $\ensuremath {\mathit{sk}}{\xspace}_j,~0\leq j \leq K-1$ and a message $M_j$ to be signed as the input. It returns a signature
$\sigma_{j}$ on $M_j$ as the output, and then updates $\ensuremath {\mathit{sk}}{\xspace}_j$ to $\ensuremath {\mathit{sk}}{\xspace}_{j+1}$.
\item $b\leftarrow \ensuremath {\texttt{SGN.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},M_j,\sigma_{j})$: The signature verification algorithm takes $\ensuremath {\mathit{PK}}{\xspace}$, message $M_j$ and its corresponding signature $\sigma_{j},~0\leq j \leq K-1$ as the
input. It returns a bit $b$, with $b=1$ meaning {\em valid}, and
$b=0$ otherwise.
\end{enumerate}
\end{definition}
\ensuremath {\texttt{SEMECS}{\xspace}}, and its preliminary version \ensuremath {\texttt{ETA}}{\xspace}~in~\cite{ETA}, are inspired from the Schnorr signature scheme~\cite{Schnorr91}, which is described in the algorithm below.
\vspace{-2mm}
\begin{algorithm}[h!]
\caption{Schnorr Signature Scheme}\label{alg:schnorr}
\hspace{5pt}
\begin{algorithmic}[1]
\Statex $\underline{(y,Y) \leftarrow \ensuremath {\texttt{Schnorr.Kg}}{\xspace}(1^{\kappa})}$:
\vspace{3pt}
\State Generate large primes $q$ and $p>q$ such that $q | (p-1)$.
\State Select a generator $\alpha$ of the subgroup $G$ of order $q$ in $\mathbb{Z}_{p}^{*}$.
\State \Return a private/public key pair $(y\stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*},Y\leftarrow
\alpha^{y} \bmod p)$ and system-wide parameter $\ensuremath {\mathit{I}}{\xspace}\ensuremath {\leftarrow}{\xspace}(q,p,\alpha)$ as the output.
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{(s,e)\leftarrow \ensuremath {\texttt{Schnorr.Sig}}{\xspace}(y,M)}$: Given $y$, compute signature $\sigma$ on a message $M$ as follows:
\vspace{3pt}
\State $R\ensuremath {\leftarrow}{\xspace}\alpha^{r} \bmod p$.
\State $e\ensuremath {\leftarrow}{\xspace} H_0(M||R)$.
\State $s\ensuremath {\leftarrow}{\xspace}(r-e\cdot y) \bmod q$, where $r\stackrel{\$}\ensuremath {\leftarrow}{\xspace} \mathbb{Z}_{q}^{*}$.
\State \Return $\sigma = (s,e)$.
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{b\leftarrow \ensuremath {\texttt{Schnorr.Ver}}{\xspace}(Y,M,\langle s,e \rangle)}$:
\vspace{3pt}
\State $R'\ensuremath {\leftarrow}{\xspace} Y^{e}\alpha^{s} \bmod p$.
\If{$e=H_0(M||R')$} \Return $b=1$
\Else~\Return $b=0$
\EndIf
\end{algorithmic}
\end{algorithm}
\subsection{Models} \label{subsec:SecModel}
We give our system and security models as below.
\noindent \textbf{System Model}: There are two types of entities in the system.
\begin{enumerate}
\item {\em Resource-constrained Signers}: Signers are storage, computational, bandwidth and power limited devices (e.g., medical implants, wireless sensors, RFID-tags). The objective of \ensuremath {\texttt{SEMECS}{\xspace}}~is to minimize the cryptographic overhead of signers.
\item {\em Resourceful Verifiers}: Storage resourceful verifiers (e.g., a laptop, base station) that can be any (untrusted) entity.
\end{enumerate}
We assume that the key generation/distribution is performed {\em offline} before deployment. For instance, a key generation center can generate private/public keys and distribute them to each entity in the system. Otherwise, the signer can also perform the key generation, before deployment, when it does not have any battery limitations.
\noindent \textbf{Security Model}: A standard security notion for a signature scheme is Existential Unforgeability under Chosen Message Attack (\ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace})~\cite{ModernCryptoBellareRogaway}. We define $K$-time {\em EU-CMA} experiment (in random oracle model~\cite{RandomOracleModel93}) for \ensuremath {\texttt{SGN}}{\xspace}~as below. In this experiment, Adversary $\mathcal{A}$ is provided with two oracles: (i) A {\em random oracle} \ensuremath {\mathit{RO}(.)}{\xspace}~from which $\mathcal{A}$~ can request the hash of any message $M$ of their choice up to (polynomially unbounded) $K'$ messages. (ii) A signing oracle $\ensuremath {\texttt{SGN.Sig}}{\xspace}_{sk}(.)$~from which $\mathcal{A}$~ can request a \ensuremath {\texttt{SGN}}{\xspace}~signature on any message $M$ of their choice up to (pre-determined constant) $K$ messages.
\begin{definition}
\label{Def:CEMSecurityModel}
{\em EU-CMA experiment} for \ensuremath {\texttt{SGN}}{\xspace}~is as follows:
\noindent Experiment $\mathit{Expt}_{\ensuremath {\texttt{SGN}}{\xspace}}^{\ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}}(\mathcal{A})$
\begin{enumerate}[]
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item $(sk_0,PK)\leftarrow
\ensuremath {\texttt{SGN.Kg}}{\xspace}(1^{\kappa},K)$
\item $(M^{*},\sigma^{*})\leftarrow \mathcal{A}^{\ensuremath {\mathit{RO}(.)}{\xspace},\ensuremath {\texttt{SGN.Sig}}{\xspace}_{sk}(.)}(PK)$,
\item If $\ensuremath {\texttt{SGN.Ver}}{\xspace}(PK,M^{*},\sigma^{*})=1$ and $M^{*}$ was not queried to \ensuremath {\texttt{SGN.Sig}}{\xspace}, return $1$, else, return $0$.
\end{enumerate}
\noindent
The EU-CMA-advantage of $\mathcal{A}$~ is defined as
\begin{eqnarray*}
\ensuremath {\mathit{Adv}_{\sgn}^{\EUCMA}(\mathcal{A})}{\xspace}=Pr[\mathit{Expt}_{\ensuremath {\texttt{SGN}}{\xspace}}^{\ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}}(\mathcal{A})=1].
\end{eqnarray*}
The EU-CMA-advantage of \ensuremath {\texttt{SGN}}{\xspace}~is defined as
\begin{eqnarray*}
\ensuremath {\mathit{Adv}_{\sgn}^{\EUCMA}(t,K',K)}{\xspace}=\max_{\mathcal{A}}\{Adv_{\ensuremath {\texttt{SGN}}{\xspace}}^{\ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}}(\mathcal{A})\},
\end{eqnarray*}
where the maximum is over all $\mathcal{A}$ having time complexity
$t$, making at most $K'$ queries to \ensuremath {\mathit{RO}(.)}{\xspace}~and at most $K$ queries to \ensuremath {\texttt{SGN}}{\xspace}.
\end{definition}
The security of \ensuremath {\texttt{SEMECS}{\xspace}}~relies on the intractability of {\em Discrete Logarithm
Problem (DLP)}~\cite{ModernCryptoBellareRogaway}, which is defined below.
\begin{definition} \label{Def:DL}
Given a cyclic group $G$ of order prime $q$ and a generator $\alpha$ of
$G$, let $\mathcal{A}$ be an algorithm that returns an element of
$\mathbb{Z}_{q}^{*}$. Consider the following experiment:
\vspace{1mm}
\noindent Experiment $\mathit{Expt}_{G}^{\ensuremath {\mathit{DL}}{\xspace}}(\mathcal{A})$
\begin{enumerate}[]
\setlength{\parsep}{0pt}
\item $y \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}$,
\item $Y\leftarrow \alpha^{y} \bmod p$,
\item $y'\leftarrow \mathcal{A}(Y)$,
\item If $\alpha^{y'} \bmod p = Y$, return $1$, else, return $0$.
\end{enumerate}
The {\em DL-advantage of $\mathcal{A}$} in this experiment is defined as
\begin{eqnarray*}
\ensuremath {\mathit{Adv}_{G}^{\dl}(\mathcal{A})}{\xspace}=Pr[\mathit{Expt}_{G}^{\ensuremath {\mathit{DL}}{\xspace}}(\mathcal{A})=1].
\end{eqnarray*}
The {\em DL-advantage of $(G,\alpha)$} in this experiment is defined as
\begin{eqnarray*}
\ensuremath {\mathit{Adv}_{G}^{\dl}(t)}{\xspace}=\max_{\mathcal{A}}\{\ensuremath {\mathit{Adv}_{G}^{\dl}(\mathcal{A})}{\xspace}\},
\end{eqnarray*} where the maximum is over all $\mathcal{A}$ having
time complexity $t$.
\end{definition}
\section{The Proposed Scheme}
\label{sec:ProposedSchemes}
Some DLP-based signatures (e.g., ECDSA~\cite{ImprovedDSAEurocrypt94}, Schnorr~\cite{Schnorr91}) can eliminate expensive operations from the signature generation by pre-computing the component $R=\alpha^{r} \bmod p$ for a random $r\ensuremath \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}{\xspace}$ during the key generation. The signer stores $(r,R)$ and then use them to compute signatures during the online phase, without any expensive operation. However, this approach incurs linear storage to the signer's side (i.e., one token per message).
It is highly desirable to construct a multiple-time signature scheme, which has constant signer storage and yet avoids expensive operations. However, this is a challenging task due to the nature of the aforementioned schemes. That is, in these schemes, the token $R$ is directly used during the signature computation and therefore its storage cannot be trivially off-loaded to the verifier's side. This forces the signer either to store or to compute a token for each message.
\subsection{Preliminary Scheme: Efficient and Tiny Authentication}
In our preliminary work {\em Efficient and Tiny Authentication (ETA)}~\cite{ETA}, we designed a signature scheme that can shift the storage of ephemeral public keys to the verifier's side without disrupting the security and verifiability of signatures. We outline our preliminary scheme \ensuremath {\texttt{ETA}}{\xspace}~in Algorithm~\ref{alg:eta} for the sake of completeness.
In the following, we focus on our newly proposed \ensuremath {\texttt{SEMECS}{\xspace}}~digital signature scheme and also highlight the differences between \ensuremath {\texttt{ETA}}{\xspace}~and our improved scheme \ensuremath {\texttt{SEMECS}{\xspace}}.
\subsection{Signer Efficient Multiple-time Elliptic Curve Signature (\ensuremath {\texttt{SEMECS}{\xspace}})} \label{subsecOMECS}
We first discuss the challenges of eliminating ephemeral keys from the signature generation in Schnorr-like signatures, which is an important step to achieve signer optimal elliptic curve signatures. We then explain our strategies in \ensuremath {\texttt{SEMECS}{\xspace}}~towards addressing these challenges.
\begin{algorithm}[t!]
\caption{Efficient and Tiny Authentication (\ensuremath {\texttt{ETA}}{\xspace}) Scheme}\label{alg:eta}
\hspace{5pt}
\begin{algorithmic}[1]
\Statex $\underline{(\ensuremath {\mathit{sk}}{\xspace}_0,\ensuremath {\mathit{PK}}{\xspace},\ensuremath {\mathit{I}}{\xspace})\ensuremath {\leftarrow}{\xspace}\ensuremath {\texttt{ETA.Kg}}{\xspace}(1^{\kappa},K)}$:
\vspace{3pt}
\State $(y,Y,\langle q,p,\alpha \rangle) \ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{Schnorr.Kg}}{\xspace}(1^{\kappa})$.
\State $r_0\ensuremath \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}{\xspace}$
\For{$j=0,\ldots,K-1$}
\State $R_j\ensuremath {\leftarrow}{\xspace} \alpha^{r_j} \bmod p$.
\State $r_{j+1}\ensuremath {\leftarrow}{\xspace} H(r_{j})$.
\State Generate verification tokens as $v_j\ensuremath {\leftarrow}{\xspace} H(R_j)$.
\EndFor
\State \Return The private and public key, as $\ensuremath {\mathit{sk}}{\xspace}_0\ensuremath {\leftarrow}{\xspace}(y,r_0)$ and $\ensuremath {\mathit{PK}}{\xspace}\ensuremath {\leftarrow}{\xspace}(Y,\ensuremath {\overrightarrow{v}}{\xspace}=v_0,\ldots,v_{K-1})$, respectively.
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{\sigma_j\ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{ETA.Sig}}{\xspace}(\ensuremath {\mathit{sk}}{\xspace}_{j},M_j)}$: Given $\ensuremath {\mathit{sk}}{\xspace}_j=(y,r_j)$, compute signature $\sigma_j$ on a message $M_j$ as follows:
\vspace{3pt}
\State $x_j\ensuremath \stackrel{\$}{\leftarrow}{\xspace}\{0,1\}^{\kappa}$.
\State $e_j\ensuremath {\leftarrow}{\xspace} H(M_j||j||x_j)$.
\State $s_j\ensuremath {\leftarrow}{\xspace} r_j-e_j\cdot y \bmod q$.
\State The signature $\sigma_j$ on $M_j$ is $\sigma_j\ensuremath {\leftarrow}{\xspace} (s_j,x_j,j)$.
\State Update $r_j$ as $r_{j+1}\ensuremath {\leftarrow}{\xspace} H(r_j)$, erase $r_j$ (to save memory).
\If{$j> K-1$} \Return $\perp$ (i.e., the limit on the number of signatures is exceed).
\Else~\Return $\sigma_j$
\EndIf
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{b \ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{ETA.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},M_j,\sigma_j)}$: If $j \ge K$ then return $b=0$ and {\em abort}. Otherwise, continue as following:
\vspace{3pt}
\State $R_j'\ensuremath {\leftarrow}{\xspace} Y^{H(M_j||j||x_j)}\cdot \alpha^{s_j}$.
\If{$v_j=H(R_j')$} \Return $b=1$
\Else~\Return $b=0$
\EndIf
\end{algorithmic}
\end{algorithm}
\subsubsection{Challenges of Removing Ephemeral Key from Signature Generation}
In Schnorr-like signatures~\cite{ECDSA, Ed25519, SchnorrQ}, an expensive operation is required to compute the ephemeral key $(R=\alpha^{r} \bmod p,r\ensuremath \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}{\xspace})$. This ephemeral key is an essential part of the signature generation and proof of security, and therefore it is a challenging task to remove it from signing without disrupting the security. For example, $R$ is committed to the signature as $s\ensuremath {\leftarrow}{\xspace} r-H(M||R)\cdot y \bmod q$ in Schnorr signatures~\cite{Schnorr91}. The ephemeral key enables programming of random oracle and also used in Forking Lemma~\cite{ForkingLemma} in the security proof of Schnorr-like signatures~\cite{Ed25519, SchnorrQ}.
\subsubsection{Eliminating Expensive Operations from Signature Generation} We first pre-compute $K$ ephemeral keys as $r_j\ensuremath {\leftarrow}{\xspace} H_0(y||j)$, $R_j\ensuremath {\leftarrow}{\xspace} \alpha^{r_j} \bmod p$ and store their hash commitments at the verifier as $\beta_j \ensuremath {\leftarrow}{\xspace} H_1(R_j)$ for $j=0,\ldots,K-1$ (Steps 3-7 in Algorithm~\ref{alg:somecs} $\ensuremath {\texttt{SEMECS.Kg}}{\xspace}$). This permits the derivation of $r_i$ to be used in signature $s_j$ deterministically without requiring any expensive operation, which will later to be verified by its corresponding $\beta_j$. Since $R_j$ is not required in the signature generation, we avoid expensive operations, but only rely on a few hash calls and a single modular addition/multiplication.
Our next step is to ensure that the correctness and provable security are still achieved in the absence of the ephemeral key in the signature generation. In \ensuremath {\texttt{ETA}}{\xspace}~\cite{ETA}, we mimicked the role of $R_j$ in $e_j$ by replacing it with a random number $x_j\ensuremath {\leftarrow}{\xspace}\{0,1\}^{\kappa}$ as $e_j\ensuremath {\leftarrow}{\xspace} H(M_j||j||x_j)$ (Steps 1-2 in Algorithm~\ref{alg:eta} $\ensuremath {\texttt{ETA.Sig}}{\xspace}$). However, this requires the explicit transmission of an extra $\kappa$-bit randomness and therefore is not optimal in terms of signature size. Moreover, this random number must be generated online, so requires a strong random number generator to be present in a low-end device.
In the following (Section \ref{subsubsec:Compactkey}), we first outline how \ensuremath {\texttt{SEMECS}{\xspace}}~improves the signature generation of \ensuremath {\texttt{ETA}}{\xspace}~by reducing the private key and signature sizes. We then elaborate on how \ensuremath {\texttt{SEMECS}{\xspace}}~achieves the correctness and a tight security reduction in Section \ref{subsubsec:VerificationReduction}.
\begin{algorithm}[t!]
\caption{Signer Efficient Multiple-time Elliptic Curve Signature (\ensuremath {\texttt{SEMECS}{\xspace}}) Scheme}\label{alg:somecs}
\hspace{5pt}
\begin{algorithmic}[1]
\Statex $\underline{(\ensuremath {\mathit{sk}}{\xspace}_0,\ensuremath {\mathit{PK}}{\xspace})\ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{SEMECS.Kg}}{\xspace}(1^{\kappa},K)}$:
\vspace{3pt}
\State $(y,Y,\langle q,p,\alpha \rangle) \ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{Schnorr.Kg}}{\xspace}(1^{\kappa})$.
\For{$j=0,\ldots,K-1$}
\State $r_j\ensuremath {\leftarrow}{\xspace} H_0(y||j)$.
\State $R_j\ensuremath {\leftarrow}{\xspace} \alpha^{r_j} \bmod p$.
\State $z_j\ensuremath {\leftarrow}{\xspace} H_1(y||j)$.
\State $\gamma_j\ensuremath {\leftarrow}{\xspace} z_j \oplus H_0(R_j)$.
\State $\beta_j \ensuremath {\leftarrow}{\xspace} H_1(R_j)$.
\EndFor
\State \Return $\ensuremath {\mathit{sk}}{\xspace}_0\ensuremath {\leftarrow}{\xspace} y$ and $\ensuremath {\mathit{PK}}{\xspace}\ensuremath {\leftarrow}{\xspace}(Y,\alpha,\ensuremath {\overrightarrow{v}}{\xspace}=( \langle \gamma_0,\beta_0 \rangle,\ldots, $ $\langle \gamma_{K-1},\beta_{K-1} \rangle,K)$.
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{\sigma\ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{SEMECS.Sig}}{\xspace}(\ensuremath {\mathit{sk}}{\xspace}_j, M_j)}$: Given $\ensuremath {\mathit{sk}}{\xspace}_j=(y,j)$ compute the signature as follows:
\vspace{3pt}
\If{$|M_j| < |q|$ } set $(\ensuremath{\overline{M}}_j = M_j, \ensuremath{\widetilde{M}}_j=0)$,
\Else~ split $M_j$ into two as $(\ensuremath{\overline{M}}_j||\ensuremath{\widetilde{M}}_j)$ such that $|\ensuremath{\overline{M}}_j|=|q|$.
\EndIf
\State $r_j\ensuremath {\leftarrow}{\xspace} H_0(y||j)$.
\State $z_j\ensuremath {\leftarrow}{\xspace} H_1(y||j)$.
\State $c_j\ensuremath {\leftarrow}{\xspace}\ensuremath{\overline{M}}_j \oplus z_j$.
\State $e_j \ensuremath {\leftarrow}{\xspace} H_0(c_j||\ensuremath{\widetilde{M}}_j)$.
\State $s_j\ensuremath {\leftarrow}{\xspace} r_j-e_j\cdot y \bmod q$.
\If{$j> K-1$} \Return $\perp$ (i.e., the limit on the number of signatures is exceeded).
\Else~\Return The signature $\sigma_j$ on $M_j$ is $\sigma_j\ensuremath {\leftarrow}{\xspace} (s_j,c_j)$, where the sender transmits $(\sigma_j,\ensuremath{\widetilde{M}}_j)$ to the receivers.
\EndIf
\end{algorithmic}
\algrule
\begin{algorithmic}[1]
\Statex $\underline{b \ensuremath {\leftarrow}{\xspace} \ensuremath {\texttt{SEMECS.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},\ensuremath{\widetilde{M}}_j,\sigma_j)}$: If $|c_j| > |q|$ or $j \ge K$ then \ensuremath {\texttt{SEMECS.Ver}}{\xspace}~return 0 and {\em aborts}. Otherwise, continue as following:
\vspace{3pt}
\State $ R_j'\ensuremath {\leftarrow}{\xspace} Y^{H_0(c_j||\ensuremath{\widetilde{M}}_j)}\cdot \alpha^{s_j} \bmod p$.
\If{$\beta_j \neq H_1(R_j')$} \Return $b=0$.
\Else~\Return $b=1$ and recover the message $M_j$ as follows:
\State $ \ensuremath{\overline{M}}_j \ensuremath {\leftarrow}{\xspace} \gamma_j \oplus H_0(R_j') \oplus c_j$.
\If{$\ensuremath{\widetilde{M}}_j=0$} set $M_j=\ensuremath{\overline{M}}_j$.
\Else~set $M_j=(\ensuremath{\overline{M}}_j||\ensuremath{\widetilde{M}}_j)$.
\EndIf
\EndIf
\end{algorithmic}
\end{algorithm}
\subsubsection{Achieving Compact Key and Signature Sizes} \label{subsubsec:Compactkey}
Our idea is to embed randomness into the message itself by creating a ``randomized message recovery" strategy, thereby avoiding an explicit transmission of randomness.
We first split message $M_j$ into two pieces as $(\ensuremath{\overline{M}}_j||\ensuremath{\widetilde{M}}_j)$ such that $|\ensuremath{\overline{M}}_j|=|q|$ and $\ensuremath{\widetilde{M}}_j$ is the rest of message. If $|M_j| < |q|$ then we simply set $\ensuremath{\overline{M}}_j = M_j$ and $\ensuremath{\widetilde{M}}_j=0$ (Steps 1-2 in Algorithm~\ref{alg:somecs} $\ensuremath {\texttt{SEMECS.Sig}}{\xspace}$). We then deterministically derive $z_j\ensuremath {\leftarrow}{\xspace} H_1(y||j)$, generate a randomness as $c_j\ensuremath {\leftarrow}{\xspace}\ensuremath{\overline{M}}_j \oplus z_j$ and compute the hash of the message as $e_j \ensuremath {\leftarrow}{\xspace} H_0(c_j||\ensuremath{\widetilde{M}}_j)$. Finally, we compute $s_j\ensuremath {\leftarrow}{\xspace} r_j-e_j\cdot y \bmod q$ (Step 7 in Algorithm~\ref{alg:somecs} $\ensuremath {\texttt{SEMECS.Sig}}{\xspace}$).
Our signature $\sigma_j$ on $M_j$ is $\sigma_j = (s_j,c_j)$, where the sender transmits $(\sigma_j,\ensuremath{\widetilde{M}}_j)$ to the receivers. Remark that, we only transmit $c_j$ that carries $|q|$-bit part of the message since $c_j\ensuremath {\leftarrow}{\xspace}\ensuremath{\overline{M}}_j \oplus z_j$. Therefore, the only component of the signature that introduces cryptographic transmission overhead is $s_j \in \mathbb{Z}_{q}^{*}$, which is optimal for an elliptic curve based signature scheme\footnote{$c_j$ does not offer confidentiality. After $\sigma_j=(s_j,c_j)$ is released, the message and $z_j$ are publicly recovered to permit signature verification.}. This is as small as some of the most compact signatures (e.g., BLS~\cite{BLS_asiacrypt}) but without requiring expensive operations at the signer's side. Morever, it is also smaller than SchnorrQ~\cite{FourQ, SchnorrQ} and \ensuremath {\texttt{ETA}}{\xspace}~\cite{ETA} that transmit $e_j$ and $x_j$, respectively, as an extra information on top of $s_j$.
\ensuremath {\texttt{SEMECS}{\xspace}}~achieves a small private key $y \in \mathbb{Z}_{q}^{*}$, which is identical to that of traditional Schnorr-like signatures~\cite{ECDSA, Ed25519, SchnorrQ} and only a half of the size that of \ensuremath {\texttt{ETA}}{\xspace}'s private key. The small and constant private key size is achieved by generating the random values with a deterministic function (e.g., a hash function) just using a seed value ($y$). Therefore, the signer doesn't need to store all the random values generated at key generation, but only stores the seed and deterministically derives all random values from it ($\ensuremath {\texttt{SEMECS.Sig}}{\xspace}$ Step 3-4). Moreover, unlike \ensuremath {\texttt{ETA}}{\xspace}, \ensuremath {\texttt{SEMECS}{\xspace}}~signature generation does not require any fresh randomness and therefore avoids potential hurdles of weak pseudo-random number generators on the signer device~\cite{ATTACKprng, PRNG-Nguyen}.
\begin{figure*}[!t]
\centering
\includegraphics[width=.88\linewidth]{SEMECS_Cropped.pdf}
\caption{High-level description of \ensuremath {\texttt{SEMECS}{\xspace}}~algorithms.}
\label{fig:somecs}
\vspace{-5mm}
\end{figure*}
\subsubsection{Signature Verification and Tight Security Reduction}\label{subsubsec:VerificationReduction}
The verifier first checks the range of randomness $c_j \in \mathbb{Z}_{q}^{*}$ and the limit on number of permitted signatures. The verifier then computes $R_j'\ensuremath {\leftarrow}{\xspace} Y^{H(c_j||\ensuremath{\widetilde{M}}_j)}\cdot \alpha^{s_j}$ and checks whether it matches with $\beta_j=H_1(R_j') \in \ensuremath {\mathit{PK}}{\xspace}$. If it does not hold, the verifier returns $b=0$. Otherwise, the verifier returns $b=1$ and uses auxiliary value $\gamma_j$ to recover the $q$-bit piece of message $\ensuremath{\overline{M}}_j$ from $c_j$ as $ \ensuremath{\overline{M}}_j \ensuremath {\leftarrow}{\xspace} \gamma_j \oplus H_0(R_j') \oplus c_j$ forming the original message as $M_j = (\ensuremath{\overline{M}}_j||\ensuremath{\widetilde{M}}_j)$.
Note that the verifier should either know which public key component ($\beta_j$) it should use at $\ensuremath {\texttt{SEMECS.Ver}}{\xspace}$ Step 2 or have a simple search operation among all $\beta$s to see if there is one that matches the calculated $H_1(R_j')$. Therefore, there is a trade-off between a verifier computation and transmission overhead. However, both of these costs are almost negligible. Since $j$ is a value up to $K$ ($K = 2^{17}$ in our experiments), the transmission of it only incurs 2-3 Bytes of extra overhead. If the verifier computation is preferred, this only adds an overhead of a binary search operation, that has a complexity of $log_2(K)$. In the binary search option, we basically assume that the verifier stores the public key sorted, and after the value $H_1(R_j')$ is calculated, binary search is made on sorted $\beta$s.
We now elaborate the design rationale behind the use of two separate verification tokens $(\beta_j,\gamma_j)$ in \ensuremath {\texttt{SEMECS}{\xspace}}, as opposed to only one token $v_j$ in \ensuremath {\texttt{ETA}}{\xspace}, for $j=0,\ldots,K-1$.
(i) In Schnorr-like schemes, the randomness incorporated into message hashing is released with $s_j$ but {\em not before}. This is useful to construct an indistinguishable simulation in the security proof of Schnorr-like signatures\footnote{ In our security proof for \ensuremath {\texttt{SEMECS}{\xspace}}~in Theorem \ref{the:Theorem1}, the simulator $\mathcal{F}$~ programs random oracle \ensuremath {\mathit{RO}(.)}{\xspace}~such that the probability that adversary $\mathcal{A}$~ querying \ensuremath {\mathit{RO}(.)}{\xspace}~on $c_j||M_j$ {\em before} querying it to the signature oracle $\ensuremath {\texttt{SEMECS.Sig}}{\xspace}_{sk}$ is as difficult as random guessing $c_j \in \mathbb{Z}_{q}^{*}$. Hence, the probability that simulator $\mathcal{F}$~ aborts during the query phase is negligible in terms of $\kappa$ (see success probability analysis in Theorem \ref{the:Theorem1}). }. In \ensuremath {\texttt{SEMECS}{\xspace}}, $c_j$ that randomizes the message hash as $H_0(c_j||\ensuremath{\widetilde{M}}_j)$, is computed from $z_j$ as $c_j\ensuremath {\leftarrow}{\xspace}\ensuremath{\overline{M}}_j \oplus z_j$. Our idea is to store $z_j$ at the verifier's side as $\gamma_j \ensuremath {\leftarrow}{\xspace} z_j \oplus H_0(R_j)$ so that it can be recovered {\em only after} $s_j$ is released. We avoid an online transmission of $z_j$ but yet randomize the message hash via $c_j$ including $q$-bit part of the message $\ensuremath{\overline{M}}_j$ (with no extra transmission overhead). After $\sigma_j = (c_j,s_j)$ is released, the verifier computes $z_j$ from $\gamma_j$ via $H_0(R_j)$. Note that $\beta_j=H_1(R_j)$ does not reveal $z_j$ but yet permits the verification of $ R_j'\ensuremath {\leftarrow}{\xspace} Y^{H_0(c_j||\ensuremath{\widetilde{M}}_j)}\cdot \alpha^{s_j} \bmod p$.
(ii) In \ensuremath {\texttt{SEMECS}{\xspace}}, we present an improved security proof with a reduction to DLP with a much tighter bound compared to that of \ensuremath {\texttt{ETA}}{\xspace}. The security of \ensuremath {\texttt{ETA}}{\xspace}~is reduced to Schnorr signatures, whose security proof relies on Forking Lemma~\cite{ForkingLemma}. Intuitively, if there is an adversary $\mathcal{A}$~ making at most $K'$ \ensuremath {\mathit{RO}(.)}{\xspace}~queries, and forging signatures with probability $\epsilon$, then the Forking Lemma states that one can compute discrete logarithms with constant probability by rewinding the forger $O(K'/\epsilon)$ times. Therefore, the security reduction loses a factor $O(K')$ that can be very large~\cite{Seurin2012}.
Our key observation is that, since \ensuremath {\texttt{SEMECS}{\xspace}}~is a $K$-time signature with pre-determined ephemeral public keys, we can avoid using Forking Lemma and obtain a reduction to DLP. That is, the hash of ephemeral keys are committed at the key generation phase as $\{\beta_j\}_{j=0}^{K-1} \in \ensuremath {\mathit{PK}}{\xspace}$. At the forgery phase, if $\mathcal{A}$~ outputs a forgery on \ensuremath {\mathit{PK}}{\xspace}~as $(M^{*},$ $\sigma^{*})$, where $\sigma^{*}=(s_{j}^{*},c_{j}^{*}),~0\leq j \leq K-1$, by validity condition, this forgery has to be on a $\beta_j \in \ensuremath {\mathit{PK}}{\xspace}$. Therefore, $\mathcal{F}$~ can extract private key $y$ {\em without} a need of rewinding $\mathcal{A}$~. This permits us to avoid a large factor of $O(K')$ but only need a small constant factor $O(K)$ in our security reduction. We stress that this is possible due to special $K$-time nature of \ensuremath {\texttt{SEMECS}{\xspace}}, but it does {\em not} apply to polynomially unbounded Schnorr signature variants as proven in~\cite{Seurin2012}.
The detailed description of \ensuremath {\texttt{SEMECS}{\xspace}}~is given at Algorithm~\ref{alg:somecs} and further outlined in Figure~\ref{fig:somecs}.
\section{Security Analysis}
\label{sec:SecurityAnalysis}
We prove that \ensuremath {\texttt{SEMECS}{\xspace}}~is a $K$-time {\em EU-CMA} signature scheme in Theorem ~\ref{the:Theorem1} (in the random oracle model~\cite{RandomOracleModel93}). We ignore terms that are negligible in terms of $\kappa$.
\begin{theorem} \label{the:Theorem1}
$\ensuremath {\mathit{Adv}_{\somecs}^{\EUCMA}(t,K',K)}{\xspace} \le \ensuremath {\mathit{Adv}_{G}^{\dl}(t')}{\xspace}$, where $t'=O(t)+(2K)\cdot O(\kappa^{3})+(6K+K')\cdot\ensuremath {\texttt{RNG}}{\xspace}$.
\end{theorem}
\noindent {\em Proof:} Let $\mathcal{A}$~ be a \ensuremath {\texttt{SEMECS}{\xspace}}~attacker. We construct a {\em DL-attacker} $\mathcal{F}$~ that uses $\mathcal{A}$ as a sub-routine. That is, we set $(y'\ensuremath \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}{\xspace}, Y'\ensuremath {\leftarrow}{\xspace} \alpha^{y'} \bmod p)$ as defined in $\mathit{DL\mhyphen experiment}$ (i.e., Definition~\ref{Def:DL}) and then run the simulator $\mathcal{F}$~ by Definition~\ref{Def:CEMSecurityModel} (i.e., \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment) as follows:
\vspace{2mm} \noindent \underline{{\em Algorithm $\mathcal{F}(Y')$}}
\begin{enumerate}[-]
\setlength{\itemsep}{2pt}
\setlength{\parskip}{1pt}
\setlength{\parsep}{1pt}
\item \underline{{\em Setup:}} $\mathcal{F}$~ keeps three lists \ensuremath {\overrightarrow{\mathcal{M}}}{\xspace},~\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},~and \ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}, all initially empty. \ensuremath {\overrightarrow{\mathcal{M}}}{\xspace}~is a message list that records each $M_j$ queried to \ensuremath {\texttt{ETA.Sig}}{\xspace}~oracle. $\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}[j]$ and $\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[j]$~record $(M_j,i)$ queried to \ensuremath {\mathit{RO}(.)}{\xspace}~oracle and its corresponding \ensuremath {\mathit{RO}(.)}{\xspace}~answer $(h_j,i)$, respectively, for cryptographic hash functions $H_i, i \in \{0,1\}$. $(h_j,i)\ensuremath {\leftarrow}{\xspace} \ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[M_j,i]$ denotes the retrieval of \ensuremath {\mathit{RO}(.)}{\xspace}~oracle answer of $(M_j,i)$ that has been queried before. If $(M_j,i)$ has not been queried before then $\perp \ensuremath {\leftarrow}{\xspace} \ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[M_j,i]$. $\mathcal{F}$~ sets counters $(l\as0,n\as0)$ and continues as follows:
\begin{enumerate}[$\bullet$]
\item {\em $h\ensuremath {\leftarrow}{\xspace}\ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}(M,l,\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace},i)$}: $\mathcal{F}$~ implements a function \ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}~to handle \ensuremath {\mathit{RO}(.)}{\xspace}~queries. That is, cryptographic functions $H_i, i \in \{0,1\}$ are modeled as random oracles via \ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}. If $\exists{j}:(M,i)=\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}[j]$ then \ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}~returns $\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[j]$. Otherwise, it~returns $h\ensuremath \stackrel{\$}{\leftarrow}\mathbb{Z}_{q}^{*}{\xspace}$ as the answer for given $H_i$, assigns $(\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}[l]\ensuremath {\leftarrow}{\xspace} (M,i),~\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[l]\ensuremath {\leftarrow}{\xspace} (h,i))$ and $l \ensuremath {\leftarrow}{\xspace} l+1$.
\item $\mathcal{F}$~ creates a simulated \ensuremath {\texttt{SEMECS}{\xspace}}~public key \ensuremath {\mathit{PK}}{\xspace}~as follows:
\begin{enumerate}[-]
\item $Y\ensuremath {\leftarrow}{\xspace} Y'$,
\item For $j=0,\ldots,K-1$,
\begin{enumerate} [a)]
\item $(s_j,e_j,z_j)\ensuremath \stackrel{\$}{\leftarrow}{\xspace} \mathbb{Z}_{q}^{*}$
\item $R_j\ensuremath {\leftarrow}{\xspace} Y^{e_j}\cdot \alpha^{s_j} \bmod p$
\item $\gamma_j\ensuremath {\leftarrow}{\xspace} z_j \oplus \ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}(R_j,l,\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace},0)$
\item $\beta_j \ensuremath {\leftarrow}{\xspace}\ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}(R_j,l,\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace},1) $
\end{enumerate}
\item $\ensuremath {\mathit{PK}}{\xspace}\ensuremath {\leftarrow}{\xspace}(Y,\alpha,\ensuremath {\overrightarrow{v}}{\xspace}=( \langle \gamma_0,\beta_0 \rangle,\ldots, $ $\langle \gamma_{K-1},\beta_{K-1} \rangle,K)$.
\end{enumerate}
\end{enumerate}
\item \underline{\em Execute $(M^{*},\sigma^{*})\ensuremath {\leftarrow}{\xspace}\mathcal{A}^{\ensuremath {\mathit{RO}(.)}{\xspace},\ensuremath {\texttt{SEMECS.Sig}}{\xspace}_{sk}(.)}(\ensuremath {\mathit{PK}}{\xspace})$}:
\begin{enumerate}[$\bullet$]
\item \underline{Queries:} $\mathcal{F}$~ handles $\mathcal{A}$~'s queries as follows:
{\em (i)} $\mathcal{A}$~ queries \ensuremath {\mathit{RO}(.)}{\xspace}~on a message $M$ for $H_i, i\in\{0,1\}$. $\mathcal{F}$~ returns $h\ensuremath {\leftarrow}{\xspace}\ensuremath {\mathit{H}\mhyphen\mathit{Sim}}{\xspace}(M,l,\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace},i)$.
{\em (ii)} $\mathcal{A}$~ queries \ensuremath {\texttt{SEMECS.Sig}}{\xspace}~oracle on a message $M_n$. If $n>K-1$ then $\mathcal{F}$~ rejects the query (i.e., the query limit is exceeded). Otherwise, $\mathcal{F}$~ continues as follows:
\begin{enumerate}[a)]
\setlength{\itemsep}{3pt}
\setlength{\parskip}{1pt}
\setlength{\parsep}{1pt}
\item If $|M_n| < |q|$ set $(\ensuremath{\overline{M}}_n = M_n, \ensuremath{\widetilde{M}}_n=0)$, else split $M_n$ into two as $(\ensuremath{\overline{M}}_n||\ensuremath{\widetilde{M}}_n)$ such that $|\ensuremath{\overline{M}}_n|=|q|$.
\item $\mathcal{F}$~ generates $c_n \ensuremath {\leftarrow}{\xspace} z_n \oplus \ensuremath{\overline{M}}_n$ and checks if $(c_n||M_n,0)\in \ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}$. If it holds then $\mathcal{F}$~ {\em aborts} (i.e., the simulation fails). Otherwise, $\mathcal{F}$~ continues as follows.
\item $\mathcal{F}$~ inserts $(\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}[l]\ensuremath {\leftarrow}{\xspace} (c_n||M_n,0),\ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[l]\ensuremath {\leftarrow}{\xspace} (e_n,0)$).
\item $\mathcal{F}$~ returns $\sigma_n \ensuremath {\leftarrow}{\xspace} (s_n,c_n)$ to $\mathcal{A}$~, sets $\ensuremath {\overrightarrow{\mathcal{M}}}{\xspace}[n]\ensuremath {\leftarrow}{\xspace} M_n$ and then increments $(n\ensuremath {\leftarrow}{\xspace} n+1,l\ensuremath {\leftarrow}{\xspace} l+1)$.
\end{enumerate}
\item \underline{Forgery of $\mathcal{A}$~:} Eventually, $\mathcal{A}$~ outputs a forgery on \ensuremath {\mathit{PK}}{\xspace}~as $(M^{*},$ $\sigma^{*})$, where $\sigma^{*}=(s_{j}^{*},c_{j}^{*}),~0\leq j \leq K-1$. By definition \ref{Def:CEMSecurityModel}, $\mathcal{A}$~ wins the $K$-time \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment for \ensuremath {\texttt{SEMECS}{\xspace}}~if $\ensuremath {\texttt{SEMECS.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},$ $M^{*},\sigma^{*})=1$ and $M^{*} \notin \ensuremath {\overrightarrow{\mathcal{M}}}{\xspace}$ hold. If these conditions hold, $\mathcal{A}$~ returns $1$, else, returns $0$.
\end{enumerate}
\item \underline{Forgery of $\mathcal{F}$~}: If $\mathcal{A}$~ loses in the $K$-time \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment for \ensuremath {\texttt{SEMECS}{\xspace}}, $\mathcal{F}$~ also loses in the \ensuremath {\mathit{DL}}{\xspace}~experiment, and therefore $\mathcal{F}$~ {\em aborts} and returns $0$. Otherwise, if $(c_{j}^{*}||M^{*}) \in \ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}$ then $\mathcal{F}$~ {\em aborts} and returns $0$ (i.e., $\mathcal{A}$~ wins the experiment without querying \ensuremath {\mathit{RO}(.)}{\xspace}~oracle). Otherwise, $\mathcal{F}$~ sets $s_{j}^{*} \ensuremath {\leftarrow}{\xspace} \ensuremath {\overrightarrow{\mathcal{L'}}}{\xspace}[c_{j}^{*}||M^{*}]$ and continues as follows:\\
Recall that $R_{j} \equiv Y^{e_{j}} \cdot \alpha^{s_j} \bmod p$ holds, where $0\leq j \leq K-1$. Moreover, since $\ensuremath {\texttt{SEMECS.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},M^{*},\sigma^{*})=1$ holds, $R_{j} \equiv Y^{{e_j}^{*}}\cdot \alpha^{s_{j}^{*}} \bmod p$ also holds. Therefore, we write the following equations:
\begin{eqnarray*}
R_{j} \equiv Y^{e_j}\cdot \alpha^{s_j} \bmod p, \\
R_{j} \equiv Y^{{e_j}^{*}}\cdot \alpha^{s_{j}^{*}} \bmod p,
\end{eqnarray*}
$\mathcal{F}$~ then extracts $y'=y$ by solving the below modular linear equations (note that only unknowns
are $y$ and $r_j$), where $Y=Y'$ as defined in simulation:
\begin{eqnarray*}
r_j & \equiv & y'\cdot e_j+ s_j \bmod q,\\
r_j & \equiv & y'\cdot e_{j}^{*} + s_{j}^{*} \bmod q,
\end{eqnarray*}
Note that $Y'\equiv \alpha^{y'} \bmod p$ holds, since $\mathcal{A}$~'s forgery is valid and non-trivial on $Y'=Y$. Therefore, by Definition \ref{Def:DL}, $\mathcal{F}$ wins the $\mathit{DL\mhyphen experiment}$.
\end{enumerate}
The execution time and probability analysis of the above experiment are as follows:
\noindent \underline{{\em Execution Time Analysis}}: In this experiment, the running time of $\mathcal{F}$~ is that of $\mathcal{A}$~ plus the time it takes to respond $q_H$ \ensuremath {\mathit{RO}(.)}{\xspace}~queries and $K$ \ensuremath {\texttt{ETA.Sig}}{\xspace}.
\begin{itemize}
\item {\em Setup phase:} $\mathcal{F}$~ draws $3K$ random numbers, performs $2K$ modular exponentiations, $K$ XOR operations, and then invokes \ensuremath {\mathit{RO}(.)}{\xspace}~$2K$ times by drawing additional $2K$ random numbers. Hence, the total cost of this phase is $(2K)\cdot O(\kappa^{3})+(5K)\cdot \ensuremath {\texttt{RNG}}{\xspace}$, where $O(\kappa^{3})$ denotes the cost of modular exponentiation and \ensuremath {\texttt{RNG}}{\xspace}~denotes the cost of drawing a random number. We omit the costs of XOR operations.
\item {\em Query phase:} $\mathcal{F}$~ draws $K$ random numbers to handle $\mathcal{A}$~'s \ensuremath {\texttt{ETA.Sig}}{\xspace}~queries, whose cost is $K\cdot \ensuremath {\texttt{RNG}}{\xspace}$. $\mathcal{F}$~ also draws $K'$ random numbers to handle $\mathcal{A}$~'s \ensuremath {\mathit{RO}(.)}{\xspace}~queries, whose cost is at most $K'\cdot \ensuremath {\texttt{RNG}}{\xspace}$.
\end{itemize}
Therefore, the approximate total running time of $\mathcal{F}$~ is $t'=O(t)+(2K)\cdot O(\kappa^{3})+(6K+K')\cdot\ensuremath {\texttt{RNG}}{\xspace}$.
\vspace{2mm}
\noindent \underline{{\em Success Probability Analysis}}:
$\mathcal{F}$~ succeeds if all below events occur.
\begin{enumerate}[-]
\setlength{\itemsep}{0pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{0pt}
\item \ensuremath {\overline{\mathit{E1}}}{\xspace}: $\mathcal{F}$~ does not abort during the query phase.
\item \ensuremath {\mathit{E2}}}{\xspace: $\mathcal{A}$~ wins the $K$-time \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment for \ensuremath {\texttt{SEMECS}{\xspace}}.
\item \ensuremath {\mathit{\overline{E3}}}{\xspace}: $\mathcal{F}$~ does not abort after $\mathcal{A}$~'s forgery.
\item \ensuremath {\mathit{Win}}{\xspace}: $\mathcal{F}$~ wins the $K$-time \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment for \ensuremath {\mathit{DL}}{\xspace}{\em-experiment}.
\item $Pr[\ensuremath {\mathit{Win}}{\xspace}] = Pr[\ensuremath {\overline{\mathit{E1}}}{\xspace}]\cdot Pr[\ensuremath {\mathit{E2}}}{\xspace|\ensuremath {\overline{\mathit{E1}}}{\xspace}]\cdot Pr[\ensuremath {\mathit{\overline{E3}}}{\xspace}|\ensuremath {\overline{\mathit{E1}}}{\xspace} \wedge \ensuremath {\mathit{E2}}}{\xspace]$
\end{enumerate}
$\bullet$ {\em The probability that event \ensuremath {\overline{\mathit{E1}}}{\xspace}~occurs}: During the query phase, $\mathcal{F}$~ aborts if $(M_j||x_j)$ $\in\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace},~0\leq j \leq K-1$ holds, {\em before} $\mathcal{F}$~ inserts $(c_j||M_j)$ into \ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}~(i.e., the simulation fails). This occurs if $\mathcal{A}$~ guesses the randomized output $c_j$ and then queries $(c_j||M_j)$ to \ensuremath {\mathit{RO}(.)}{\xspace}~{\em before} querying it to \ensuremath {\texttt{SEMECS.Sig}}{\xspace}. The probability that this occurs is $\frac{1}{2^{|q|}}$, which is negligible in terms of $\kappa$. Hence, $Pr[\ensuremath {\overline{\mathit{E1}}}{\xspace}]=(1-\frac{1}{2^{|q|}})\approx 1$.
$\bullet$ {\em The probability that event \ensuremath {\mathit{E2}}}{\xspace~occurs}: If $\mathcal{F}$~ does not abort, $\mathcal{A}$~ also does not abort since the simulated view of $\mathcal{A}$~ is {\em indistinguishable} from the real view of $\mathcal{A}$~ (see the indistinguishability analysis). Therefore, $Pr[\ensuremath {\mathit{E2}}}{\xspace|\ensuremath {\overline{\mathit{E1}}}{\xspace}]=\ensuremath {\mathit{Adv}_{\somecs}^{\EUCMA}(t,K',K)}{\xspace}$.
$\bullet$ {\em The probability that event \ensuremath {\mathit{\overline{E3}}}{\xspace}~occurs}: $\mathcal{F}$~ does not abort if the following conditions are satisfied:
\begin{enumerate} [i]
\item $\mathcal{A}$~ wins the \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}~experiment for \ensuremath {\texttt{SEMECS}{\xspace}}~on a message $M^{*}$ by querying it to \ensuremath {\mathit{RO}(.)}{\xspace}. The probability that $\mathcal{A}$~ wins without querying $M^{*}$ to \ensuremath {\mathit{RO}(.)}{\xspace}~is as difficult as a random guess.
\item After $\mathcal{F}$~ extracts $y'$ by solving modular linear equations, the probability that $Y' \not\equiv \alpha^{y'} \bmod p$ is negligible in terms $\kappa$, since $(Y=Y') \in \ensuremath {\mathit{PK}}{\xspace}$ and $\ensuremath {\texttt{SEMECS.Ver}}{\xspace}(\ensuremath {\mathit{PK}}{\xspace},M^{*},\sigma^{*})=1$. Hence, $Pr[\ensuremath {\mathit{\overline{E3}}}{\xspace}|\ensuremath {\overline{\mathit{E1}}}{\xspace} \wedge \ensuremath {\mathit{E2}}}{\xspace]=\ensuremath {\mathit{Adv}_{\somecs}^{\EUCMA}(t,K',K)}{\xspace}$.
\end{enumerate}
Omitting the terms that are negligible in terms of $\kappa$, the upper bound on {\em \ensuremath {\mathit{EU}\mhyphen\mathit{CMA}}{\xspace}-advantage of \ensuremath {\texttt{SEMECS}{\xspace}}}~is as follows:
\begin{eqnarray*}
\ensuremath {\mathit{Adv}_{\cem}^{\EUCMA}(t,K',K)}{\xspace} & \le & \ensuremath {\mathit{Adv}_{G}^{\dl}(t)}{\xspace},
\end{eqnarray*}
\vspace{2mm}
{\em \underline{Indistinguishability Argument}}: The real-view of $\mathcal{A}$~ is comprised of the public key $\ensuremath {\mathit{PK}}{\xspace}=(Y,\alpha,p,q,\ensuremath {\overrightarrow{v}}{\xspace}=( \langle \gamma_0,\beta_0 \rangle,\ldots, $ $\langle \gamma_{K-1},\beta_{K-1} \rangle,K)$, the answers of $\ensuremath {\texttt{SEMECS.Sig}}{\xspace}_{sk}(.)$ as $\ensuremath {\overrightarrow{\mathcal{\sigma}}}{\xspace}=(s_j,c_j)$ for $j=0,\ldots,K-1$, and the answer of \ensuremath {\mathit{RO}(.)}{\xspace}~as $\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}=(h_0,\ldots,h_{K'-1})$ on corresponding $H_i, i\in\{0,1\}$, respectively. That is, $\ensuremath {\overrightarrow{\mathcal{A}}}{\xspace}_{\mathit{real}}=\langle \ensuremath {\mathit{PK}}{\xspace},\ensuremath {\overrightarrow{\mathcal{\sigma}}}{\xspace},\ensuremath {\overrightarrow{\mathcal{L}}}{\xspace}\rangle$, where all values are generated/computed by \ensuremath {\texttt{SEMECS}{\xspace}}~algorithms as in the real system. All variables in \ensuremath {\overrightarrow{\mathcal{A}}}{\xspace}~are computed from the values $\{y,r_n,z_n,h_j,\alpha,p,q\}_{n=0,j=0}^{K-1,K'-1}$. Hence, the joint probability distribution of all other variables in \ensuremath {\overrightarrow{\mathcal{A}}}{\xspace}~are determined by the joint probability of these values. All these are random in $\mathbb{Z}_{q}^{*}$. Therefore, the joint probability distribution of \ensuremath {\overrightarrow{\mathcal{A}}}{\xspace}~is,
{\small
\begin{eqnarray*}
Pr[\ensuremath {\overrightarrow{A}_{\mathit{real}}}{\xspace}=\overrightarrow{a}] &= & Pr[\overline{y} = y | \overline{r}_0 = r_0 \wedge,\ldots,\overline{h}_{K'-1} = h_{K'-1}]\\
& = & \frac{1}{(q-1)^{1+2K+K}\cdot(p-1)^{2}}
\end{eqnarray*}
}
We denote the simulated view of $\mathcal{A}$~ is as \ensuremath {\overrightarrow{A}_{\mathit{sim}}}{\xspace}, and it is equivalent to \ensuremath {\overrightarrow{A}_{\mathit{real}}}{\xspace}~except that in the simulation, values $(s_j,e_j,z_j,c_j)$ for $j=0,\ldots,K-1$ are randomly selected from $\mathbb{Z}_{q}^{*}$. Note that the joint probability distribution of these variables are identical to original signature and hash outputs (since hash function is modeled as RO). Hence, we write $Pr[\ensuremath {\overrightarrow{A}_{\mathit{real}}}{\xspace}=\overrightarrow{a}]=Pr[\ensuremath {\overrightarrow{A}_{\mathit{sim}}}{\xspace}=\overrightarrow{a}]$.
\hfill$\square$
\begin{table*}[t!]
\vspace{-2mm}
\centering \caption{Private/public key sizes, signature size and signature generation/verification costs of \ensuremath {\texttt{SEMECS}{\xspace}}~and
its counterparts} \label{tab:analytic}
\vspace{-2mm}
\begin{tabular}{|@{}c@{}||@{}c@{}|@{}c@{}|@{}c@{}||@{}c@{}|@{}c@{}|}
\hline \multicolumn{1}{|c||}{\multirow{2}{*} {\textbf{Scheme}}} & \multicolumn{3}{c||}{\bf{Signer}} & \multicolumn{2}{c|}{\bf{Verifier}} \\
\cline{2-6} & \specialcell[]{\textbf{Private Key} \\ \textbf{Size}} & \specialcell[]{\textbf{Signature} \\ \textbf{Size}} & \specialcell[]{\textbf{ Signature} \\ \textbf{ Generation}} & \specialcell[]{\textbf{Public Key} \\ \textbf{Size}} & \specialcell[]{\textbf{ Signature} \\ \textbf{ Verification}} \\ \hline \hline
\multicolumn{6}{|c|}{\em \textbf{Full-time signatures}} \\ \hline
SPHINCS~\cite{SPHINCS} & $ n_S $ & \specialcell[]{ $ n_S(k_S({|t_S|} $ \\ $- x_S + 1) + 2^{x_S}) $ } & $ (2t_S - 1) \cdot H $ & $ n_S $ & \specialcell[]{$ (k_S((\log t_S) - x_S + 1)$ \\ $ + 2^{x_S} -1) \cdot H $} \\ \hline
ECDSA~\cite{ECDSA} & $|q|$ & $2|q|$ & $EMul$ &$|q|$ & $1.3\cdot EMul$ \\ \hline
Ed25519~\cite{Ed25519} & $|q|$ & $2|q|$ & $EMul$ & $|q|$ & $1.3\cdot EMul$ \\ \hline
Kummer~\cite{Kummer} & $|q|$ & $2|q|$ & $EMul$ & $|q|$ & $1.3\cdot EMul$ \\ \hline
SchnorrQ~\cite{SchnorrQ} & $|q|$ & $2|q|$ & $EMul$ & $|q|$ & $1.3\cdot EMul$\\ \hline \hline
\multicolumn{6}{|c|}{\em \textbf{$K$-time signatures}} \\ \hline
HORS~\cite{HORS_BetterthanBiBa02} & $\kappa$ & $\kappa \cdot u$ & $ (u + 1) \cdot H$ & $t\cdot|H| \cdot K$ & $(u+1)\cdot H$ \\ \hline
HORSE~\cite{OTS_r_time_HORSED_2006} & $(\kappa \cdot t \cdot log_{2}(K))$ & $\kappa \cdot u$ &~ $(u \cdot log_{2}(K)+1)\cdot H$ &$t\cdot|H|$ & $(u+1)\cdot H$ \\ \hline
XMSS~\cite{XMSS} & $\kappa$ & \specialcell[]{$(l + log_{2}(K)) $ \\ $ \cdot |H|$} & \specialcell[]{$(((log_{2}(K)+2)\cdot (log_{2}(K)$ \\ $ + l \cdot (w+2)))/2 + 4 \cdot log_{2}(K)) \cdot H$} & \specialcell[]{$(2(log_{2}(K) + log_2(l))$ \\ $+1) \cdot |H|$} & \specialcell[]{$(log_{2}(K)+ l $\\ $\cdot (w + 1)) \cdot H $} \\ \hline
Zaverucha et al.~\cite{Zaverucha} & $\kappa$ & $\kappa + |q|$& $(m/2) \cdot (Add_q + H)$ & $m \cdot |q| \cdot K$& $1.3\cdot EMul$ \\ \hline \hline
\ensuremath {\texttt{SEMECS}{\xspace}} & $|q|$ & $|q|$ & $2\cdot H + Mul_q + Sub_q$ & $(2K+1)\cdot |q|$& $1.3\cdot EMul$ \\ \hline
\end{tabular}
\flushleft{\scriptsize{
$K$ denotes the number of signatures that can be generated using a single key pair for $K$-time signature schemes.
$Emul$ and $Eadd$ denote the costs of EC scalar multiplication over modulus $p'$, and EC addition over modulus $p'$, respectively. ECDSA~\cite{ECDSA}, Ed25519~\cite{Ed25519}, Kummer~\cite{Kummer}, and SchnorrQ~\cite{SchnorrQ} only differ from each other in terms of the underlying curve. The operations that are required are the same for these schemes. $H$ and $Mul_{q}$ denote a cryptographic hash and a modular multiplication over modulus $q$, respectively. We omit the constant number of negligible operations if there is an expensive operation (e.g., integer additions are omitted if there is an $Emul$). We use double-point scalar multiplication for verifications of ECC based schemes ($1.3 \cdot Emul$ instead of $2 \cdot Emul$~\cite{ECCGuide}). $t_S$, $k_S$ and $x_S$ are SPHINCS~\cite{SPHINCS} parameters where $t_S$ is the number of secret key elements, $k_S$ is the number of revealed secret key elements and $x_S$ is a small integer. SPHINCS~\cite{SPHINCS} parameter $n_S$ denotes the bit length of hashes. Zaverucha et al.~\cite{Zaverucha} parameter $m$ should be selected such that $m\choose m/2$ $ \geq 2^{2\kappa}$. Integers $t$ and $u$ denote the parameters used in HORS~\cite{HORS_BetterthanBiBa02} and HORSE~\cite{OTS_r_time_HORSED_2006}. $w$ is the Winternitz parameter and $l$ is the tree parameter in XMSS~\cite{XMSS}.
\textbf{Remark:} For HORS~\cite{HORS_BetterthanBiBa02} and Zaverucha et al.~\cite{Zaverucha}, similarly to \ensuremath {\texttt{SEMECS}{\xspace}}, we deterministically generate the necessary private key components from a seed (i.e., using a keyed hash) to have a small constant private key that can be deployed to low-end devices.
}
}
\vspace{-5mm}
\end{table*}
\section{Performance Analysis and Comparison}
\label{sec:PerformanceAnalysis}
In this section, we first present the analytical analysis of \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts. Then, we present the results of our experiments on a commodity laptop and an 8-bit AVR embedded processor. Our evaluation metrics include key sizes, signature size, and computation costs. On the 8-bit microprocessor, we focus on the signer cost (i.e., private key, signature size, signature generation) since our system model includes resource-constrained devices as signers. For our counterparts, we consider state-of-the-art $K$-time signature schemes as well as some traditional (full-time) signatures.
\vspace{1mm}
\noindent \textbf{Remark:} Our envisioned applications require high signer efficiency to be practical on resource-constrained devices. Hence, {\em optimizing the online signer efficiency is the essential performance objective for \ensuremath {\texttt{SEMECS}{\xspace}}}. Recall that we assume verifiers are resourceful entities, which is a reasonable assumption for our envisioned applications (see Section \ref{sec:Introduction}). Also note that in \ensuremath {\texttt{SEMECS}{\xspace}}~system model, private/public keys are generated before the system deployment (see Section \ref{subsec:SecModel}). Hence, the key generation cost (i.e., {\em offline cost}) is not a critical performance metric for \ensuremath {\texttt{SEMECS}{\xspace}}.
\subsection{Analytical Performance Analysis}
Here, we describe the analytical costs of our scheme, where the online costs are summarized in Table~\ref{tab:analytic}.
\noindent \textbf{Key Generation: }Key generation of \ensuremath {\texttt{SEMECS}{\xspace}}~requires $K$ EC scalar multiplications that is higher than its full-time counterparts. For instance, for EC-based signature schemes (e.g., ECDSA~\cite{ECDSA}, Ed25519~\cite{Ed25519}, Kummer~\cite{Kummer}, and SchnorrQ~\cite{SchnorrQ}) keys are generated with only one EC scalar multiplication. However, it is comparable to its $K$-time counterparts as their key generation also depends on $K$. Note that in our system model, key distribution is performed before the deployment. Thus, we believe that this does not pose a limitation for our considered use-cases.
\noindent \textbf{Signer Overhead: }In \ensuremath {\texttt{SEMECS}{\xspace}},~signer stores a small private key that is the same size as its full-time elliptic curve counterparts. The private key of some $K$-time signatures can be deterministically derived from a $\kappa$-bit seed, which is 2$\times$ smaller than that of \ensuremath {\texttt{SEMECS}{\xspace}}. However, this makes a small difference in practice (i.e., 16 Bytes vs 32 Bytes). Signature generation of \ensuremath {\texttt{SEMECS}{\xspace}}~only requires 2 hash function calls, a single multiplication, and subtraction under$\mod q$. This introduces a significantly smaller overhead compared to its alternatives. The counterparts of \ensuremath {\texttt{SEMECS}{\xspace}}~either require expensive operations (i.e., EC scalar multiplication) or a very large number of hash function calls for signature generation. Only HORS~\cite{HORS_BetterthanBiBa02} and Zaverucha et al.~\cite{Zaverucha} have comparable signature generation speed. However, when we generate the private key components from a seed, these hash function calls dominate the signature generation cost for these schemes due to their large private key size.
\noindent \textbf{Signature Transmission: }Signature size of \ensuremath {\texttt{SEMECS}{\xspace}}~is the smallest compared to its counterparts. Note that since the signature component $c_j$ contains the information to recover the first $|q|$ Bytes of the message, we do not consider its transmission as an overhead. \ensuremath {\texttt{SEMECS}{\xspace}}~only requires additional $|q|$ Bytes to be transmitted. Since the transmission of signatures introduces an overhead to the energy consumption of signer (and verifier), we believe it is essential to minimize its size.
\noindent \textbf{Verifier Overhead: }In \ensuremath {\texttt{SEMECS}{\xspace}}~the public key is linear with the messages to be signed with a single key pair. Therefore, it increases as $K$ increases (similar to its $K$-time counterparts except HORSE~\cite{OTS_r_time_HORSED_2006}). Considering that the verifier device is a resourceful device (e.g., server, command center) in \ensuremath {\texttt{SEMECS}{\xspace}}~applications, we believe this is tolerable. The signature verification of \ensuremath {\texttt{SEMECS}{\xspace}}~requires an EC double scalar multiplication (can be accelerated with Shamir's trick~\cite{ECCGuide}).
\noindent \textbf{Parameters: } {\em We selected parameters to provide $\kappa = 128$-bit security for both \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts.} For elliptic-curve based schemes (including \ensuremath {\texttt{SEMECS}{\xspace}}), we selected $|q| = 256$-bit. For Zaverucha et al., we selected $m=260$, for HORS and HORSE, we selected $t = 1024$ and $u = 24$ to provide the desired security level. For XMSS and SPHINCS, we used the parameters suggested in the base papers. We refer the interested readers to the base papers of these schemes for the detailed explanation of their parameter choice.
\begin{table*}[t!]
\centering
\vspace{-2mm}
\caption{Experimental performance comparison of \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts on a commodity hardware} \label{tab:Laptop}
\vspace{-2mm}
\begin{threeparttable}
\begin{tabular}{| c || c | c | c | c | c | c | c | }
\hline
\textbf{Scheme} & $K$ & \specialcell[]{\textbf{Signature Generation}\\ \textbf{ Time (}CPU cycle\textbf{)}} & \specialcell[]{\textbf{Private Key}\textsuperscript{$ \mathparagraph$} \\ \textbf{(Byte)}} & \specialcell[]{\textbf{Signature }\\ \textbf{Size (Byte)}} & \specialcell[]{\textbf{Signature Verification}\\ \textbf{ Time (}CPU cycle\textbf{)}} & \textbf{Public Key}\textsuperscript{$\ddagger$} & \specialcell[]{\textbf{End-to-End} \\ \textbf{Delay (}CPU cycle\textbf{)}} \\ \hline \hline
\multicolumn{8}{|c|}{\em \textbf{Full-time signatures}} \\ \hline
SPHINCS~\cite{SPHINCS} & $2^{\kappa}$& 37 466 005 & 1088 & 41000 & 1 051 562 & 1056 & 38 517 567 \\ \hline
ECDSA~\cite{ECDSA} & $2^{\kappa}$ & 1 510 320 & 32 & 64 & 1 932 650 & 32 & 3 442 970 \\ \hline
Ed25519~\cite{Ed25519} & $2^{\kappa}$ & 146 620 & 32 & 64 & 286 750 & 32 & 433 370 \\ \hline
Kummer~\cite{Kummer} & $2^{\kappa}$& 58 450 & 32 & 64 & 98 560 & 32 & 157 010 \\ \hline
SchnorrQ~\cite{SchnorrQ} & $2^{\kappa}$ & 30 481 & 32 & 64 & 54 241 & 32 & 84 722 \\ \hline \hline
\multicolumn{8}{|c|}{\em \textbf{$K$-time signatures}} \\ \hline
\multirow{2}{*}{HORS~\cite{HORS_BetterthanBiBa02}} & 1 & \multirow{2}{*}{16 823} & \multirow{2}{*}{16} & \multirow{2}{*}{384} & \multirow{2}{*}{8 975} & 32 KB & \multirow{2}{*}{25 798} \\
& $2^{17}$& & & & & 4 GB & \\ \hline
\multirow{2}{*}{HORSE~\cite{OTS_r_time_HORSED_2006}} & 1 & 16 823 & 16384 & \multirow{2}{*}{384} & \multirow{2}{*}{8 975} & \multirow{2}{*}{32 KB} & 25 798 \\
& $2^{17}$& 280 247 & 278 528 & & & & 287 503 \\ \hline
\multirow{2}{*}{XMSS~\cite{XMSS}} & 1 & 137 856 & \multirow{2}{*}{16} & 2080 & 115 239 & 416 & 253 095 \\
& $2^{17}$& 1 367 431 & & 2592 & 120 983 & 1504 & 1 488 414 \\ \hline
\multirow{2}{*}{Zaverucha et al.~\cite{Zaverucha}} & 1 & \multirow{2}{*}{89 180} & \multirow{2}{*}{16} & \multirow{2}{*}{48} & \multirow{2}{*}{52 872} & 4160 & \multirow{2}{*}{142 052} \\
& $2^{17}$& & & & & 520 MB & \\ \hline \hline
\multirow{2}{*}{\ensuremath {\texttt{SEMECS}{\xspace}}} & 1 & \multirow{2}{*}{\textbf{2 425}} & \multirow{2}{*}{\textbf{32}} & \multirow{2}{*}{\textbf{32}} & \multirow{2}{*}{52 872} & 96 & \multirow{2}{*}{\textbf{55 297}} \\
& $2^{17}$& & & & & 8 MB & \\ \hline
\end{tabular}
\begin{tablenotes}[flushleft]\scriptsize{
\item $\ddagger$ The sizes are in terms of \textbf{Bytes}, if otherwise not specified.
\item $ \mathparagraph $ System wide parameters \ensuremath {\mathit{I}}{\xspace}~(e.g., p,q,$\alpha$) for each scheme are included in their corresponding codes, and private key size denote to specific private key size.
\item The cost of hash-based schemes are estimated based on the cost of a single hash operation.
}
\end{tablenotes}
\end{threeparttable}
\vspace{-3mm}
\end{table*}
\subsection{Performance Evaluation on Commodity Laptop}
We implemented \ensuremath {\texttt{SEMECS}{\xspace}}~on a laptop and compared its cost to its state-of-the-art counterparts.
\noindent \textbf{Hardware Configurations and Software Libraries: } As our commodity hardware, we used a laptop equipped with an Intel i7 Skylake 2.6 GHz CPU with 12 GB RAM.
We implemented \ensuremath {\texttt{SEMECS}{\xspace}}~on FourQ curve~\cite{FourQ} to offer fast verification. We used the open-source implementation of this curve which can be found at\footnote{\url{https://github.com/Microsoft/FourQlib}}. We used our hash function as blake due to its high efficiency and high security~\cite{blakeHash}. Specifically, we used blake2s due to its optimization on low-end devices. We open-source our implementations at
\begin{center}
\fbox{\url{www.github.com/ozgurozmen/SEMECS}}
\end{center}
We ran the open-sourced implementations of our counterparts on our hardware setting, if possible. For the hash-based constructions, we conservatively simulated their costs with blake2s hash function, to be fair with them.
\noindent \textbf{Experimental Results: } Table~\ref{tab:Laptop} shows the benchmarks and specific key/signature sizes for \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts. We observe that \ensuremath {\texttt{SEMECS}{\xspace}}~is 7$\times$ faster than its closest counterpart (HORS~\cite{HORS_BetterthanBiBa02}) in terms of signature generation. Specifically, {\em it takes only $1.23$ microseconds to generate a signature with \ensuremath {\texttt{SEMECS}{\xspace}}}. Moreover, it has a compact private key of 32 Bytes and the smallest signature (32 Bytes) among its counterparts. Signature verification of \ensuremath {\texttt{SEMECS}{\xspace}}~is also fast since we used the optimized FourQ~\cite{FourQ} curve to implement our scheme. Therefore, only HORS~\cite{HORS_BetterthanBiBa02} and HORSE~\cite{OTS_r_time_HORSED_2006} offer faster verification. The main limitation of \ensuremath {\texttt{SEMECS}{\xspace}}~is its public key size. Specifically, when $K = 2^{17}$, which { \em allows signing a message in every 20 minutes for 5 years without a key replacement}, the public key size is 8 MB. However, this is much smaller than some of the most efficient $K$-time counterparts such as HORS, and Zaverucha et al., that have 4 GB and 520 MB public key, respectively. We also implemented the key generation of \ensuremath {\texttt{SEMECS}{\xspace}}~on this experimental setting and observed that generating the key for $K= 2^{17}$ takes $1.75$ seconds.
\subsection{Performance/Energy Evaluation on 8-bit Microprocessor}
We fully implemented the signature generation of \ensuremath {\texttt{SEMECS}{\xspace}}~on an 8-bit microprocessor to assess its energy and time efficiency on low-end embedded devices.
\noindent \textbf{Hardware Configurations and Software Libraries: } We used 8-bit AVR ATmega 2560 microprocessor to measure the signer efficiency of our scheme compared to its counterparts. We selected this low-end device due to its low energy consumption and extensive use in practice, especially in IoT applications and medical devices~\cite{IMDProcessor, ATmega2560Medical, ATmega2560Medical2}. It is equipped with 256 KB flash memory, 8 KB SRAM, 4 KB EEPROM, and its maximum clock frequency is 16 MHz.
We implemented \ensuremath {\texttt{SEMECS}{\xspace}}~using Rhys Weatherley's crypto library~\cite{weatherley} that offers high-speed operations for low-end devices. Specifically, we used its blake2s implementation and modified its reduction algorithm to compute$\mod q$ (where $q$ is FourQ parameter) using Barrett reduction. We also open-source our 8-bit implementations at the link given above to facilitate the test and broad adoption of \ensuremath {\texttt{SEMECS}{\xspace}}.
We used the results of our counterparts that were given in 8-bit AVR microprocessors, if possible. For instance, we used the results provided in~\cite{Ed255198bit} for Ed25519,~\cite{Kummer8bit} for $\mu$Kummer and~\cite{FourQ8bit} for SchnorrQ. We ran the ECDSA implementation of microECC~\cite{microECC} on our hardware. Similar to the laptop implementation, we measured the cost of a single hash (blake2s) call on our microprocessor and conservatively estimated the hash-based schemes' cost.
\noindent \textbf{Experimental Results: }As summarized in Table~\ref{tab:AVR}, our analysis confirmed that \ensuremath {\texttt{SEMECS}{\xspace}}~is highly efficient at the signer's side. Signature generation of \ensuremath {\texttt{SEMECS}{\xspace}}~is performed with less than 200 thousand cycles, which is 6$\times$ faster than HORS~\cite{HORS_BetterthanBiBa02} and 19$\times$ faster than SchnorrQ (its fastest counterparts). In addition to this, \ensuremath {\texttt{SEMECS}{\xspace}}~requires a small private key and a signature size that is the smallest among its counterparts. This makes \ensuremath {\texttt{SEMECS}{\xspace}}~very desirable for applications that include resource-limited signers.
\begin{figure*}[t]
\centering
\begin{subfigure}[t]{.5\textwidth}
\includegraphics[width=\linewidth]{pulse.pdf}
\caption{Energy of Signature Generation vs Pulse Sensor}
\label{fig:areaPulse}
\vspace{-1mm}
\end{subfigure}%
\begin{subfigure}[t]{.5\textwidth}
\includegraphics[width=\linewidth]{pressure.pdf}
\caption{Energy of Signature Generation vs Pressure Sensor}
\label{fig:areaPressure}
\vspace{-1mm}
\end{subfigure}
\caption{Energy consumption of signature generation vs IoT sensors} \label{fig:areaBattery}
\vspace{-4mm}
\end{figure*}
\noindent \textbf{Energy Consumption Analysis: }We analyzed the energy consumption of \ensuremath {\texttt{SEMECS}{\xspace}}~and its counterparts on our experimental setting and compared with the energy consumption of two common IoT sensors (a pulse and a pressure sensor). We first derived a generic energy consumption estimation (as in~\cite{MICAZenergy} that offers an estimation for MICAz) for 8-bit AVR ATmega 2560 based on our \ensuremath {\texttt{SEMECS}{\xspace}}~implementation and used it to estimate the energy consumption of our counterparts (similarly with~\cite{FourQ8bit} that uses~\cite{MICAZenergy}). We also calculated how much energy is required to operate IoT sensors. We took into consideration (i) energy drawn by the sensor (ii) energy drawn by the microprocessor to read data from the sensor and (iii) energy drawn by the microprocessor during the idle time.
We powered our microprocessor with a 2200 mAh power pack. This allowed us to use an ammeter/power meter connected between the battery and the microprocessor. We measured 5V of voltage and 20mA of current on load, which is verified by the datasheet of the processor\footnote{\url{http://www.atmel.com/Images/Atmel-2549-8-bit-AVR-Microcontroller-ATmega640-1280-1281-2560-2561\_datasheet.pdf}}. Then, we used the formula $E = V \cdot I \cdot t$ to calculate the energy consumption in Joules (as in~\cite{Precomputation:LowCostSig:Ateniese:NDSS2013}). We also considered the potential deployment of nRF24L01 Single Chip 2.4 GHz Transceiver to 8-bit ATmega for signature transmission. Based on its datasheet, we also estimated the energy consumption of signature transmission with this low-power transceiver. Specifically, nRF24L01 operates at 3.3 V, 11.3 mA and support a transmission rate of 2Mbps. {\em Our results showed that 8-bit AVR ATmega 2560 consumes roughly $6.25nJ$ per cycle of computation and $18.65nJ$ per bit of transmission.}
We also calculated how much energy is necessary to operate IoT sensors. Specifically, we used a pulse sensor\footnote{\url{https://pulsesensor.com/}} (that could serve as an example of a medical sensor) and a BMP183 pressure sensor\footnote{\url{https://cdn-shop.adafruit.com/datasheets/1900_BMP183.pdf}} (that could be an example of a daily IoT application). In our energy calculations, we considered a sampling frequency of 10 seconds for the pulse sensor and 10 minutes for the pressure sensor, due to the difference/urgency in their usage. Figure~\ref{fig:areaBattery} shows how many percentage of the battery is spent on the IoT sensor, compared with that of the cryptographic operations (i.e., signing) of different schemes. One can observe that for pulse sensor (see Figure~\ref{fig:areaPulse}), HORS and SchnorrQ require the $5.16\%$ and $14.71\%$, whereas, with \ensuremath {\texttt{SEMECS}{\xspace}}, this is decreased to a negligible level ($0.89\%$). For the pressure sensor, the energy consumption of \ensuremath {\texttt{SEMECS}{\xspace}}~is only $3.14\%$ where the closest counterpart is $16.41\%$. This shows that preferring \ensuremath {\texttt{SEMECS}{\xspace}}~as the authentication mechanism in 8-bit AVR microprocessors significantly reduces the impact of cryptographic operations on battery life.
Based on this analysis, we noticed that \ensuremath {\texttt{SEMECS}{\xspace}}~outperforms its counterparts for both computation energy and communication energy at the signer's side. We believe that this is essential in practice to extend the battery lives of critical embedded devices such as implantable medical devices.
\section{Conclusion}
\label{sec:Conclusion}
In this paper, we proposed a new signature scheme called \ensuremath {\texttt{SEMECS}{\xspace}}, which achieves several desirable properties that are critical for resource-constrained devices. Specifically, \ensuremath {\texttt{SEMECS}{\xspace}}~only requires two hash, a modular multiplication, and a modular subtraction to compute a signature. Moreover, it has a constant-small private key and signature, that is optimal for an EC-based signature scheme. Our experiments on both laptop and 8-bit AVR confirmed the energy and computational efficiency of \ensuremath {\texttt{SEMECS}{\xspace}}. Therefore, we believe \ensuremath {\texttt{SEMECS}{\xspace}}~is an ideal alternative for providing authentication and integrity services for resource-constrained devices.
\section*{Acknowledgments}
\addcontentsline{toc}{section}{Acknowledgments}
This work is supported by the NSF CAREER Award CNS-1652389. We would like to thank Rouzbeh Behnia for his valuable comments.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,662
|
Q: Only can select one contact, how to get multi selection using ContactUI? DESIRED: I'm trying to be able to select multiple contacts in ContactsUI like this:
CURRENT: I cant find a way to get the multiple selection buttons in the table view cells
As you can see from the code belowI'm using the contactPickerView so I dont know what I'm doing wrong :/
import UIKit
import ContactsUI
import Contacts
class FindContactsVC: UIViewController, CNContactPickerDelegate {
//button to open contactUI
@IBAction func contactsButton(_ sender: Any) {
let cnPicker = CNContactPickerViewController()
cnPicker.delegate = self
self.present(cnPicker, animated: true, completion: nil)
}
//Works, but only allows me to select one at a time
func contactPicker(_ picker: CNContactPickerViewController, didSelect contact: CNContact) {
print(contact.phoneNumbers)
print("WYASDFY")
}
//This function never runs :/
func contactPicker(picker: CNContactPickerViewController, didSelectContacts contacts: [CNContact]){
print(contacts)
print("plz work!")
contacts.forEach { contact in
for number in contact.phoneNumbers {
let phoneNumber = number.value
print("number is = \(phoneNumber)")
}
}
}
override func viewDidLoad() {
super.viewDidLoad()
}
}
Thanks ^.^
A: I haven't checked other parts of your code, but at least, you need to implement the second delegate method properly:
func contactPicker(picker: CNContactPickerViewController, didSelectContacts contacts: [CNContact]){
↓
func contactPicker(_ picker: CNContactPickerViewController, didSelect contacts: [CNContact]) {
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,026
|
Dr. Howard Jacob is a leader in the genetics and genomics fields. He and his team were the first to save a patient's life through genomic sequencing. His passion for improving the lives of critically ill patients has been the catalyst for his determination to bring whole genome sequencing into the clinical setting to affect patient care. Dr. Jacob has published more than 250 peer-reviewed articles focused on genetic mapping of complex diseases and building genomic resources to understand the impact of genetic variation.
Hear him speak at the School of Pharmacy about the growing role of genome sequencing in medication management and the pivotal role that pharmacists play in pharmacogenomics and precision medicine.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,692
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