text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
These small barrel earrings, made from recycled sterling silver, are engraved with the Chinese character 'Ping', symbolizing peace. The symbol depicts a 'bamboo stem' balanced on each side by a single tongue of 'fire' representing the principle of balance between opposing forces. These earrings measure approximately 3/4" in length and 1/8" in width on on sterling silver earwire.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 381
|
\section{Introduction}
In a now widely accepted evolutionary sequence based on the shape of the
spectral energy distribution (SED) in the infrared, there are four stages in the
pre-main-sequence life of a low-mass star:
Class 0 objects are deeply embedded protostars with large submillimeter to
bolometric luminosity ratios, which are very faint at near-infrared wavelengths;
Class I objects are surrounded by infalling envelopes, but are less embedded;
Class II objects have cleared their envelopes and are surrounded by accretion
disks; and Class III objects have dispersed almost all of their circumstellar material
\citep{lada84, lada87,andre93}.
Therefore, Class I protostars are thought to be in one of the earliest evolutionary
stages. After the gravitational collapse of a molecular cloud core starts, the
infalling material forms an envelope around the central object. Since cloud
cores are usually rotating slowly, the material falls preferentially onto a disk,
which serves as a mass reservoir, most of which is eventually accreted by
the protostar.
Class I objects are thus surrounded by accretion disks, in addition to envelopes
extending to several 1000 AU. The radius of the disk is roughly equal to the
centrifugal radius ($R_c$), which defines the centrifugal barrier encountered by matter
falling in on the equatorial plane, if the specific angular momentum is conserved;
inside this radius the density distribution flattens, and the matter follows non-radial
trajectories, falling onto a disk. Material tends to accumulate close to $R_c$, which
is close to the protostar at the beginning of the collapse, but increases rapidly over
time \citep{terebey84,adams86}.
Besides accreting matter, young stars generate powerful outflows which are
launched along magnetic field lines; mass accretion onto the star and mass
loss in the form of outflows seem to be correlated \citep[e.g.,][]{hartigan95}.
In addition to collimated outflows, protostars might also emit a wide-angle
wind, which is expected to open up cavities in the envelope \citep{arce02}.
The presence of both collimated outflows and wide-angle stellar winds
explains the fact that some objects with large cavities have well-collimated
outflows. A partially evacuated cavity also forms as a natural
by-product of the collapse of a sheet \citep{hartmann94, hartmann96};
since molecular cloud cores are rarely spherically symmetric, this scenario
might be more realistic. In some cases, both stellar winds and the
flattening of the envelope are responsible for cavities oriented along the
rotational axis of a protostar.
The spectral index $n$ ($\lambda F_{\lambda} \propto {\lambda}^n$)
from 2 to 25 $\mu$m is used for the SED classification of protostars and
pre-main-sequence stars \citep{lada87,adams87,andre94}; Class I
objects have $n > 0$, Class II objects have $ -2 < n < 0 $, and Class III
objects have $ -3 < n < -2 $ ($n=-3$ is the spectral index of a photosphere
in the Rayleigh-Jeans limit).
Therefore, Class I objects are characterized by a strong infrared excess;
the dust in the envelope absorbs radiation from the central protostar and disk
and reemits it at infrared to mm wavelengths. The peak of the SED of this
type of objects usually lies in the mid- to far-infrared, depending on the density
of the envelope (and thus mass and infall rate), as well as the source luminosity
\citep{kenyon93a}.
Flat spectrum sources are likely objects surrounded by envelopes and disks
and therefore protostars, but with wide cavities in their envelopes and/or
seen close to pole-on \citep[e.g., DG Tau, T Tau;][]{furlan06a, calvet94}.
Some of the flat-spectrum sources could be transition objects (Class I/II),
consisting of a T Tauri star and its disk surrounded by remnant envelope
material (e.g., IRAS 04154+2823). Their SEDs typically show a double peak,
one centered in the near-IR and one in the far-IR.
In addition to a rising SED in the infrared due to this excess emission, Class I
protostars display various absorption features in their near- and mid-infrared
spectra due to silicates and ices in the envelope. However, many parts of the
infrared spectrum are not accessible from ground-based telescopes, which in
addition often lack the sensitivity to observe faint objects. The {\it Spitzer Space
Telescope} \citep{werner04} opened a new window onto low-mass
star formation; with unprecedented sensitivity, the precursors of low-mass
stars can be studied in various evolutionary stages.
Here we present 28 spectra from 5 to 36 $\mu$m of Class I objects in the
Taurus star-forming region observed with the Infrared Spectrograph\footnote{The
IRS was a collaborative venture between Cornell University and Ball Aerospace
Corporation, funded by NASA through the Jet Propulsion Laboratory and the
Ames Research Center.} \citep[IRS;][]{houck04} on board of {\it Spitzer}.
The 5--20 $\mu$m spectra of 5 of the 28 Class I objects were already
introduced in \citet{watson04}; these objects are IRAS 04016+2610,
04108+2803B, 04181+2654 B, 04239+2436, and DG Tau B.
Our IRS observations are part of a larger Infrared Spectrograph guaranteed-time
observing program; we presented the Class II and III objects from our sample of
150 young stellar objects in Taurus in an earlier paper \citep{furlan06a}.
Since the Taurus-Auriga star-forming region is young (1--2 Myr) and relatively
nearby \citep[140 pc;][]{kenyon94b, bertout99}, many of its objects have
been fairly well-studied at wavelengths covering almost the entire electromagnetic
spectrum. Here we use SEDs constructed with data from the literature and our IRS
spectra to constrain envelope models and to determine which parameters
describe the circumstellar environment of the Class I objects in our sample.
This paper is structured as follows: in \S\ \ref{obs_data_reduction} we
lay out our {\it Spitzer} IRS observations and data reduction; in \S\
\ref{sample} we present the IRS spectra and SEDs of the 28 Class I
objects in our sample; in \S\ \ref{env_models} we describe the
envelope models used to fit 22 of our Class I objects and give
detailed descriptions of the individual model fits; in \S\ \ref{discussion}
we discuss the different appearances of Class I objects and give a summary
of our models; and in \S\ \ref{conclusions} we present our conclusions.
\section{Observations and Data Reduction}
\label{obs_data_reduction}
Our Taurus targets were observed with the IRS on {\it Spitzer} during IRS
observing campaigns 3 (2004 February 6 to 8), 4 (2004 February 27 to March 5),
12 (2004 August 30 to 31; only 3 targets), and 24 (2005 September 12; only
IRAS 04166+2706). All targets were observed over
the full IRS range from 5 to 40 $\mu$m by using either the two low-resolution
IRS modules (Short-Low [SL] and Long-Low [LL], 5.2--14 $\mu$m and 14--38
$\mu$m, respectively, $\lambda$/$\Delta\lambda$ $\sim$ 90) or the SL
module and the two high-resolution modules (Short-High [SH] and Long-High
[LH], 10--19 $\mu$m and 19--37 $\mu$m, respectively,
$\lambda$/$\Delta\lambda$ $\sim$ 600), depending on the object's
expected mid-infrared flux. Thus, roughly half of our Class I objects were observed
with SL and LL, and the other half with SL, SH, and LH.
Our observations were carried out either in IRS staring mode, where the target
is placed on two nod positions along the slit, at 1/3 and 2/3 of the slit length, or
in mapping mode, with $2\times3$ step maps on the target (3 steps in the
dispersion direction and 2 steps in the spatial direction). More than half of the
objects (19 out of 28) were observed in mapping mode.
Data reduction was carried out using the SMART software tool \citep{higdon04}.
Spectra were extracted after bad pixels, including so-called rogue pixels,
were fixed by interpolation and after the sky in the low-resolution spectra was
subtracted using the off-order observation (see \citealt{furlan06a} for more
details). A variable-width column extraction was performed for low-resolution
spectra, while high-resolution spectra were extracted using a full-slit extraction.
The spectra were calibrated by dividing them by a calibrator spectrum
($\alpha$ Lac (A1 V) for low-resolution and $\xi$ Dra (K2 III) for
high-resolution observations), and by multiplying the result by the calibrator's
template spectrum \citep{cohen03}. Final spectra were obtained by averaging
the two nod positions (for staring mode observations) or the two central map
positions (for mapping mode observations). We also truncated all SH spectra below
13 $\mu$m and rebinned all high-resolution spectra to a resolution of
$\lambda$/$\Delta\lambda$ = 200 to achieve a more uniform representation
of the spectra.
As mentioned in \citet{furlan06a}, we applied scalar corrections to each
LH spectrum to account for the absence of sky subtraction in the high-resolution
modules. Some minor adjustments ($\lesssim$ 10\%) were also necessary for
SL to match it to the flux level of SH. For IRAS 04154+2823, 04158+2805,
04325+2402, and 04368+2557, a multiplicative factor of 1.2 had to be applied
to SL, while DG Tau B is the only Class I object in our sample to require an even
larger scaling: a multiplication by 1.35 of SL2 (5.2--7.5 $\mu$m) and by 1.21
of SL1 (7.4--14 $\mu$m).
The flux mismatches between short- and long-wavelength modules can be
attributed to the extended nature of our targets; even though all objects in
our sample appeared as point sources when observed with the IRS, we know
that most of our targets are somewhat extended due to the presence of an
envelope \citep[e.g.,][]{park02}. There are two effects caused by the
extension of the envelope influencing our IRS observations.
First, the LL and LH slits, which are 10.6\arcsec\ and 11.1\arcsec\ wide,
respectively, cover a larger area and thus more of the envelope emission than
the SL and SH slits, which are 3.6\arcsec\ and 4.7\arcsec\ wide, respectively.
However, the cold, outer envelope regions do not contribute much flux at the
IRS wavelengths; at a distance of 140 pc, the SL slit encompasses all emission
within 250 AU from the central source, while LH includes emission out to 780 AU,
sufficient to sample almost all of the material contributing to the IRS spectrum.
Only a few objects probably suffer from slit losses beyond about 30 $\mu$m,
evidenced by a change in slope, and thus decrease in flux level, around 30 $\mu$m;
this effect is most notable in IRAS 04108+2803 B, 04158+2805, 04181+2654 A,
04239+2436, and 04295+2251.
Second, and more importantly, is the fact that protostellar envelopes are usually
not spherically symmetric and that each of the IRS modules sample different
regions of a protostellar system, since the slits of the various modules are not
aligned parallel to each other and are also of different sizes (see, e.g., Fig.\
\ref{DGTauB_slits}). For example, a protostar seen edge-on might show a
dark central lane representing the disk and symmetric reflection nebulae
representing inner envelope regions; if SL is aligned along the disk, LL will
be oriented perpendicular to the dark lane and will sample more of the envelope
emission.
The larger scalar correction factors applied to some of Class I objects in our
sample (especially in the case of DG Tau B) are likely due to this second effect.
In particular, the centering and orientation of the narrow SL slit could result in
some of the envelope emission being missed by the SL observation (see Fig.\
\ref{DGTauB_slits}).
However, in most cases only a small fraction of the warm dust contributing at
IRS wavelengths is not included in the IRS observations, and therefore the spectra
we obtained are believed to be a good representation of the mid-infrared emission
of the inner envelope regions.
Our absolute spectrophotometric accuracy is of the order of 5--10\%,
depending on the module used and the pointing uncertainty. The relative
spectral accuracy is module-dependent; spectral features above the noise level
are likely real in SL, LL2 (14--21 $\mu$m) and SH, while in LL1 (20--36 $\mu$m)
and especially in LH artifacts from unresolved calibration issues often do not allow
reliable identifications of spectral features. Also the wavelength region where
SL and LL meet (13.5--14.5 $\mu$m) is affected by this higher uncertainty due
to possible artifacts at the order edges.
Some of the Class I objects we observed with IRS are members of multiple
systems (see Table \ref{tab_properties}). Except for GV Tau A and B,
they either are at separations that place all members well within the IRS slits,
or their separations are
large enough to ensure that only one component is observed. Since the infrared
emission from protostars is dominated by envelope emission, the multiplicity
of the central source should have no impact on the mid-infrared spectrum,
except when the configuration of the system is such that it influences the
emission from the envelope, e.g., by truncating it or causing a
departure from spherical symmetry in the illuminating radiation field.
We encountered reduction problems in a few objects; they are listed below. \\
{\it 04108+2803B.} This object is part of a mispointed mapping-mode
observation; by using the spectra from the central map position for SL
and SH and scaled spectra from the third map position for LH, we
created a continuous IRS spectrum. This stitching adds a higher uncertainty
to the absolute flux level in our spectra, but the overall shape of the spectrum
should be only minimally affected. \\
{\it GV Tau (A,B).} This binary is only separated by 1{\farcs}3; it was observed
with SL, SH, and LH in mapping mode, and it was not resolved in any of the modules.
Since this object consists of two point sources with likely some extended emission
surrounding them, their SH and LH calibration presented some problems. The final
spectrum was composed by using those nod and map positions for each module
in which the flux was highest, and in addition SH was scaled to match up with SL.
Due to this procedure, the absolute flux level of this spectrum carries a higher uncertainty.
In fact, we multiplied the entire spectrum by 0.85 to match fluxes at 11.7 and 17.9
$\mu$m with our, yet unpublished, ground-based measurements at these wavelengths. \\
{\it HH 30.} This object was too faint in SL2; the spectrum could only be extracted
in SL1 and in LL. \\
{\it LkHa 358.} The peak-up arrays contained bright to saturated sources during the
SL observations, which resulted in increased noise in the SL part of the spectrum. The
LL observation included 3 sources, LkHa 358, HL Tau, and XZ Tau, but since the
observation was centered on LkHa 358, the other two sources entered only partially
in the slit. In addition, all three sources were separated sufficiently in the spatial direction
to allow extraction of the individual spectrum of LkHa 358.
\section{The Sample}
\label{sample}
\subsection{IRS Spectra}
\label{IRS_spectra}
Our Taurus sample is based on objects analyzed by \citet{kenyon95} (from here on KH95);
a large fraction of our Class I targets has identifiers from the {\it IRAS} Point Source
Catalogue, since these objects were first identified as possible embedded protostars
in the {\it IRAS} surveys and confirmed in follow-up observations \citep[e.g.,][]
{beichman86, myers87, kenyon90}. Some objects that are part of our Class I
sample are identified as Class II objects in KH95: these objects are IRAS 04154+2823,
04158+2805, 04278+2253, CoKu Tau/1, HL Tau, IC 2087 IR, and LkHa 358.
We base our identification of these sources as Class I objects on the existence of
extended, near-infrared nebulosity and possibly outflows around these objects, or the
presence of a silicate absorption feature and a positive spectral index over the IRS range,
or the detection of ice absorption features in the 5--8 $\mu$m region as well as the
CO$_2$ ice feature at 15.2 $\mu$m.
Therefore, this sample includes all Taurus objects we observed with the IRS
that are thought to be at the protostellar evolutionary stage and therefore
surrounded, to varying extents, by envelopes, even though in a few cases
the SED slope does not lie in the typical range for Class I objects.
Figure \ref{ClassI_IRS} shows the IRS spectra of the 28 Class I objects
in our sample. Most have SEDs that are rising over the IRS spectral range,
and almost all of them show numerous absorption features in their
mid-infrared spectra, typical of embedded protostars \citep[e.g.,][]
{alexander03, gibb04, watson04}: wide bands due to amorphous silicates
at 9.7 and 18 $\mu$m (the latter one is much less prominent), and
features at 6.0 $\mu$m due to H$_2$O ice, at 15.2 $\mu$m due to
CO$_2$ ice, and at 6.8 $\mu$m due to organic compounds, possibly CH$_3$OH,
NH$_4^+$, or strongly polar H$_2$O ice \citep{vanDishoeck04, alexander03}.
In addition, the silicate absorption feature at
9.7 $\mu$m often displays a wider absorption shoulder at the longer-wavelength
end due to the broad H$_2$O ice libration band at $\sim$ 11-13 $\mu$m.
The long-wavelength wing of the CO$_2$ ice feature also suggests that
water ice is responsible, likely in the form of a mix with CO$_2$ ice
\citep{ehrenfreund99}.
Some protostars show an obvious self-absorbed silicate feature at 10 $\mu$m
(e.g., IRAS 04154+2823, 04181+2654 B; see Fig.\ \ref{ClassI_IRS}d),
where the silicate emission from inner
disk regions is absorbed in cooler envelope regions, causing a feature that
is more or less flat, with some emission and some absorption
characteristics \citep{mitchell81}. The self-absorbed features probably
indicate that the optical depth of the envelope along the line of sight is
not as large as for the cases where a clear silicate absorption feature is seen.
IRAS 04248+2612, 04264+2433, and CoKu Tau/1 (Fig.\ \ref{ClassI_IRS}e)
have a silicate emission feature,
but are surrounded by nebulosities and have an overall rising SED in the
infrared. Therefore, they are likely more evolved Class I objects seen through
low-density envelopes, which have mostly dispersed and thus reveal the central star
and its accretion disk. Based on the geometry of their near-infrared reflection nebulae,
IRAS 04248+2612 and CoKu Tau/1 are likely seen close to edge-on \citep{padgett99}.
\subsection{Spectral Energy Distributions}
\label{SED_section}
Most protostars are found near or are embedded within dense molecular cloud
cores \citep{myers87, onishi98}. Due to high extinction mostly due to their envelopes,
protostars are usually faint or even undetectable in the optical. In the near-infrared
and in the optical, the infalling envelope often appears as a reflection
nebula, seen in light scattered by dust grains in the inner parts of envelope cavities
\citep[e.g.,][]{tamura91, padgett99,park02}. In addition, Class I objects are often
extended in the sub-mm and mm \citep{chandler98,motte01}; at these wavelengths,
the thermal emission from cold dust in the envelope is detected.
To show the emission from protostellar systems at all wavelengths, in
particular how the emitted power is distributed at different wavelengths, we
generated spectral energy distributions with data from the literature.
In Figure \ref{ClassI_SED}, we show the SEDs of the
Class I objects in our sample, constructed with optical to mid-IR photometry
mostly from \citet{kenyon95}, the 2MASS $J$, $H$, and $K_s$ fluxes, the {\it IRAS}
12, 25, 60, and 100 $\mu$m fluxes, IRAC 3.6, 4.5, 5.8, and 8.0 $\mu$m
fluxes \citep{hartmann05,luhman06b}, sub-mm and mm fluxes, where available,
and the IRS spectrum. The {\it IRAS} fluxes are mostly from the {\it IRAS}
Faint Source Catalog; for CoKu Tau/1, GV Tau, HL Tau, and L1551 IRS 5 we adopted
the {\it IRAS} fluxes from \citet{weaver92}, and for the 12 and 25 $\mu$m fluxes
of DG Tau B and HH 30 the measurements from coadded {\it IRAS} scans by
\citet{cohen87}.
The fluxes at 350, 450, 800, 850, 1100, and 1300 $\mu$m are mostly
from \citet{moriarty94}, \citet{andrews05}, \citet{young03}, and
\citet{motte01}, while the few measurements at 160 $\mu$m were taken
by \citet{ladd91} with the Kuiper Airborne Observatory (see Table
\ref{tab_longwave} for a list of the long-wavelength data used in the
SED plots). The fluxes were not corrected for reddening, since the high
extinction of these objects is very likely intrinsic.
The SED plots show that most of the power of a Class I object is emitted in
the mid- to far-infrared; therefore, for many objects the IRS spectrum lies
close to the peak of the emission. For most objects, the SED is either rising
or flat over the IRS spectral range, with a deep silicate absorption feature at
10 $\mu$m as the most notable spectral feature. Only a few objects have
SEDs that seem to peak at $\sim$ 5 $\mu$m and decrease over the IRS
spectral range; they might be more evolved protostars and will be discussed
in \S\ \ref{special_obj}.
\subsection{Notes on Individual Objects Excluded in the Modeling}
\label{notes_objects}
\subsubsection{Class I Objects}
{\it GV Tau.} This object, also known as Haro 6-10, consists of two components
separated by 1{\farcs}3.
Like T Tau, the GV Tau system consists of a T Tauri star (component A) and
an embedded infrared companion (component B). The binary is surrounded by
a circumstellar disk and an almost edge-on circumbinary envelope, with outflows
oriented perpendicular to the disk plane \citep{menard93}. GV Tau B is likely responsible
for the outflows measured from this source \citep{chandler98, hogerheijde98}.
In the near-IR, the brightness ratio of the components varies considerably over
the timescale of a few years \citep{leinert89,menard93,koresko97}. It is likely
that the B component, which is more deeply embedded, varies more than the A
component in brightness and generates the deep 10 $\mu$m silicate absorption
feature, as well as the various ice features, observed in its mid-IR spectrum
(see Figure \ref{ClassI_IRS}c). \\
{\it HH 30.} Optical and near-IR HST images \citep{burrows96, cotera01}
reveal that this object is seen edge-on; the central star illuminates a flared
circumstellar disk, which obscures the protostar and is seen in scattered light.
In addition, a narrow jet is emitted perpendicular to the disk plane. Its appearance is
reminiscent of that of DG Tau B, which is also seen edge-on and emits a jet.
\citet{wood98} modeled the emission of HH 30 and concluded that, in addition to
a disk, only a tenuous envelope is required. The presence of a diffuse infalling
envelope, as well as outflows, indicates that HH 30 is a protostar, though likely
more evolved. \\
The {\it IRAS} fluxes of HH 30, determined by \citet{cohen87} from coadded
{\it IRAS} scans, are highly uncertain, since they are contaminated by emission from two
nearby sources, HL Tau and XZ Tau, which are much brighter. Our IRS spectrum
shows that the {\it IRAS} fluxes were overestimated by 1.5 to 2 orders of
magnitude (see Figure \ref{ClassI_SED}).
HH 30 is among the faintest objects in our Taurus sample; its IRS spectrum
suggests the presence of a deep silicate absorption feature, but its SED is
only poorly determined.
\subsubsection{Class I/II Objects}
\label{special_obj}
As indicated in \S\ \ref{SED_section}, a few objects in our sample have
SEDs that are not typical of protostars. IRAS 04278+2253 and IC 2087 IR
have clearly decreasing SEDs from 5 to 40 $\mu$m (see Fig.\ \ref{ClassI_IRS}d)
and would therefore be identified as Class II objects based on the slope of their
infrared SED.
Also IRAS 04154+2823 and LkHa 358 have SEDs that seem to peak around
5 $\mu$m and do not display prominent 10 $\mu$m silicate absorption features.
However, IC 2087 IR (also known as Elias 18) has ice absorption features in
the 5--8 $\mu$m spectral region, a 10 $\mu$m silicate absorption feature,
as well as a prominent CO$_2$ ice absorption at 15.2 $\mu$m.
It is likely surrounded by a circumstellar disk, probably seen edge-on given
its high visual extinction A$_V$ $\sim$ 18 \citep{shuping01}.
\citet{bowey03} measured the 10 $\mu$m spectrum of IC 2087 IR; they noted
a strong silicate absorption feature on a decreasing photosphere-like continuum.
The absorption features we detect in the IRS spectrum of IC 2087 IR, and the
fact that it is surrounded by a reflection nebulosity (IC 2087), suggest that it is
actually a protostar, surrounded by an accretion disk and some envelope material.
Further evidence for its earlier evolutionary stage are a molecular outflow
\citep[e.g.,][]{heyer87} and the presence of two Herbig-Haro knots, which are
not within the outflow, but are oriented along the axis of symmetry of the reflection
nebula \citep{gomez97}. It has been suggested that, since IC 2087 IR has a
rather weak outflow, it could be in transition between the embedded protostellar
and the T Tauri phase \citep{tegler95}.
In contrast to IC 2087 IR, IRAS 04278+2253 shows no silicate absorption
feature, only a weak CO$_2$ ice feature, and likely gas absorption features
in the 5--8 $\mu$m wavelength range, most prominently an H$_2$O
band centered at 5.7 $\mu$m. This feature, similar to what is observed in
some FU Ori objects \citep{green06}, is thought to arise in the upper layers
of hot inner disk regions that are heated by accretion. IRAS 04278+2253
is a 6{\farcs}8 binary, whose primary component is of G8 spectral type
and has a relatively high mass accretion rate \citep[$6.6 \times 10^{-6}$
M$_{\odot}$ yr$^{-1}$;][]{white04}. The secondary is of K7/M0 spectral
type \citep{white04} and thus clearly fainter than the primary.
Given this object's relatively large mass accretion rate, it is likely still a protostar,
but since no extended emission has been detected around it, there is likely only
little envelope material left.
LkHa 358 forms a small cluster of stars together with HL Tau and XZ Tau
(a flat-spectrum source, see \citealt{furlan06a}): all three objects are found
within a radius of less than 1\arcmin. All of them are likely embedded to some
extent; while HL Tau has an obvious absorption feature and an SED typical of
a Class I object, LkHa 358 only has a weak absorption feature at 10 $\mu$m,
and XZ Tau has silicate emission features at 10 and 20 $\mu$m and an overall
flat SED in the mid-IR. Since HL Tau is the brightest source of the group, it is
likely the dominating contribution to the IRAS source 04287+1807; therefore,
the {\it IRAS} fluxes for LkHa 358 are not known.
The IRS spectrum shows that LkHa 358 has a silicate feature that could be
partly self-absorbed, as well as some weak ice absorption features. The late
spectral type of LkHa 358 (M5.5 according to \citealt{kenyon98} and
\citealt{luhman00}) suggests that this object will end up as a very low-mass
star, with a mass close to that of a brown dwarf.
IRAS 04154+2823 is very faint and could be a more evolved protostar;
in fact, its 10 $\mu$m silicate feature seems to be self-absorbed,
with a broad absorption centered at 9.7 $\mu$m and emission between
10 and 12 $\mu$m, which is an indication of a tenuous envelope. Its
SED can be fitted reasonably well with a low-density envelope model (see Figure
\ref{model_04154}).
\section{Envelope Models}
\label{env_models}
\subsection{Model Description}
\label{model_descr}
In this paper, we generated envelope models for 22 of the 28 Class I objects we
observed in the Taurus star-forming region using the IRS on {\it Spitzer}
to characterize some of the envelope parameters that result in the observed
SED shapes. These models include the contributions of the central protostar,
the disk, and the envelope to compute the resulting SED of a protostellar system.
We used a modeling code based on two types of models that differ in the initial
configuration: so-called TSC models \citep[after][]{terebey84}, which assume
an initially spherically symmetric molecular cloud core, and sheet-collapse
models \citep{hartmann94, hartmann96}, which simulate the collapse of
elongated cloud cores. Both models result in flattened density distributions,
in which the emergent SED is strongly dependent on the inclination angle of the system.
Below, we will briefly introduce these two types of models.
\subsubsection{Envelope Component}
In the TSC models, the radiative equilibrium temperature is determined first by using
the angle-averaged density distribution for the infall region (\begin{math} \rho
\propto r^{-3/2} \end{math} for $r \gg R_c$, \begin{math} \rho \propto
r^{-1/2} \end{math} for $r \ll R_c$; $R_c$, the centrifugal radius, is where
material falling in on the equatorial plane lands and thus represents the outer disk
radius). In this region, departures from spherical symmetry become apparent only
at small distances, when material falls onto a disk; at large distances, the TSC model
can be described by spherical free-fall.
After the temperature distribution has been calculated, the flattened, axially symmetric
density distribution from \citet{terebey84} is used to compute the flux emitted by
the system \citep[][from here on KCH93]{kenyon93a}.
The models also include isotropic scattering. The source luminosity is not set
according to the mass infall rate, which means that the disk
accretion rate can be different from the mass infall rate of the envelope.
Outflow cavities along the poles can also be included in the models by specifying
a cavity semi-opening angle \citep{kenyon93b, calvet94}. Inside the bipolar
cavities, material has been evacuated; outside, material will fall onto a
disk with an inner radius \begin{math} R_{min} = R_c \: \sin^2{\theta} \end{math},
where ${\theta}$ is the half opening angle of the cavity, i.e. the angle between the
polar axis and the streamline of an infalling particle at the edge of cavity at large distances,
and $R_c$ is the centrifugal radius \citep{calvet94}. Thus, the shape of the cavities is
determined by the streamlines of the infalling material.
The inclusion of a cavity in the density distribution of the envelope causes
departures from spherical symmetry in the temperature distribution.
Scattered light from these cavities adds flux at near-infrared wavelengths,
but the cavities also remove hot, inner dust. Therefore, for larger cavities, the
net result is a decrease in near- and mid-infrared flux when the inclination angle
is larger than the opening angle of the cavity. However, when the inclination of
the system allows the observer to see inside the cavity and thus the star and
inner disk directly, the near-infrared emission increases \citep{calvet94}.
Sheet-collapse models start from the collapse of sheets that are initially in
hydrostatic equilibrium \citep{hartmann94,hartmann96}. This modification
to the spherical infall scenario produces partially evacuated cavities,
which, when aligned along the line of sight, reveal the inner parts of the
envelope and disk, thus increasing the object's optical and near-infrared emission.
Since the polar regions of the sheet collapse before the equatorial regions, polar
cavities, which are regions where low-density material is falling in, will result
\citep{hartmann94}. With sheet collapse, the opening angle of the cavity is
connected to the infall and not to the outflows; eventually, outflows, but also wide-angle
stellar winds \citep{arce02}, might easily remove the low-density material and fully
clear out the cavities, so in the end both effects, sheet collapse and mass loss from
a protostar, could be responsible for the formation of cavities.
\subsubsection{Disk Component}
The disk is modeled as an optically thick, steady accretion disk irradiated as a flat disk
by the central star ($T(r) \propto r^{-3/4}$; $T^4=T_{accretion}^4+
T_{irradiation}^4$), oriented perpendicular to the rotation axis of the envelope.
As the outer disk regions, which emit mostly in
the sub-mm and mm, are optically thin, and only the inner disk regions are
optically thick, the outer disk radius was set at 10 AU to represent the optically
thick part of the disk. However, the disk is expected to extend out to $R_c$;
emission from the protostellar envelope will be obscured if it lies behind the disk
along the line of sight. Therefore, to take this obscuration effect into account,
a second set of models was calculated with the disk radius set equal to $R_c$.
The latter set was used for the envelope and stellar emission, while the set with
the smaller disk radius was used for the disk emission. In this way the disk
component in our models represents only the optically thick disk regions;
the optically thin outer regions would likely add small amounts of additional flux
at sub-mm and mm wavelengths.
This simple disk model also does not include an inner disk wall at the dust
truncation radius, which would add flux at near-IR wavelengths, in particular
for systems with higher inclination angles $i$, where the disk emission is
reduced due to its $\cos(i)$ dependence.
In addition, since the disk model does not include an optically thin disk
atmosphere that generates a 10 $\mu$m silicate emission feature
\citep[see, e.g.,][]{dalessio99}, the shape of the silicate feature is only
determined by the envelope. The material in the envelope will extinguish the
disk and thus generate a 10 $\mu$m absorption feature in the continuum
emission arising from the disk, while the silicates in the envelope will produce an
emission feature when the density is low and an absorption feature for higher densities.
Thus, the silicate feature might not be modeled realistically for Class I objects
with low envelope densities, but should be a good approximation for more
typical, higher-density envelopes.
The inner disk radius is set by the star's magnetosphere, which channels the accretion
flow onto the star, and is roughly at the corotation radius, at a few stellar radii from the
stellar surface \citep{shu94}. However, in some models the inner disk radius was set
equal to the stellar radius, implying a boundary layer.
The accretion luminosity is set as the sum of the disk accretion luminosity, generated as
matter accretes through the disk, and of the hot component luminosity, produced by
the dissipation of the accretion shock on the stellar surface. For the cases where the
inner disk radius is equal to the stellar radius, half of the accretion luminosity is emitted
by the disk, and the other half by the boundary layer. For larger inner disk radii, the
accretion shock accounts for a larger fraction of the accretion luminosity than matter
accreting through the disk.
\subsubsection{Dust and Ices}
The dust in the envelope is composed of small silicate and graphite grains
\citep{draine84}, troilite (FeS), and water ice \citep{pollack94},
as well CO$_2$ ice \citep{ehrenfreund96}. The particle size distribution follows a
power law with particle radius $a$: $n(a) da \propto a^{-3.5} da$, with maximum sizes
of 0.3 $\mu$m. This maximum grain size corresponds to the typical value found
in the interstellar medium \citep[e.g.,][]{mathis77,weingartner01} and is motivated
by the notion that envelopes likely contain pristine material. Our data confirm this idea,
since the shape of the silicate absorption feature in most of our Class I objects indicates
that the dust is dominated by small grains \citep[see also][]{osorio03}.
The abundance of the various components of the dust mixture is
similar to the one adopted by \citet{osorio03}, who adjusted the abundances
given in \citet{draine84} and \citet{pollack94} to fit a model to the spectrum
of L1551 IRS 5. We do not include organics in our mixture; we assume that the
carbon is present in the form of graphite only. The fractional abundances with
respect to the mass of the gas for silicates, graphite, troilite, and H$_2$O ice are
$\zeta_{sil}=0.004$, $\zeta_{gra}=0.0025$, $\zeta_{tro}=0.000768$,
and $\zeta_{ice}=0.0005$, respectively. To get a better fit to the observed SEDs,
these fractional mass abundances could be varied until a good match was found,
but in order to limit the number of free parameters in the fit, we kept the abundances
of these components constant. Only the CO$_2$ ice abundance was adjusted for each
object to reproduce the depth of the CO$_2$ ice feature.
The opacities used in the current model should be regarded as approximations:
the water ice opacities used in the model do not generate a feature at 6.0 $\mu$m,
which is detected in our IRS spectra of Class I objects, and other ice opacities,
e.g. for methanol ice, which might be responsible for the feature at 6.8 $\mu$m,
are not included. Also, the 15.2 $\mu$m ice feature is treated as pure CO$_2$ ice,
but in reality is likely mixed with water and other ices, making the feature broader
\citep{ehrenfreund99}. Therefore, our envelope models are not able to reproduce
the majority of the ice features, and the most prominent ice feature, the CO$_2$
feature, can only be approximated in its depth, not its width. On the other hand,
since the purpose of this study is a characterization of the overall SEDs of Class I objects,
and thus of their envelope structure, we will neglect the fact that the ice features
will not be reproduced well or at all.
\subsubsection{Model Parameters}
The input parameters for the models are the total system luminosity
($L_{star}+L_{acc}$), the fraction of the total luminosity that arises from
the star ($\eta_{star}$), the radius of the star (assumed to be 2 R$_{\odot}$),
the minimum disk radius, the centrifugal radius (which is essentially the outer disk radius),
the inner and outer envelope radii, the flattening of the envelope (for sheet-collapse
models), and the reference density $\rho_1$, which is the density the envelope
would have at a radius of 1 AU for the limit $R_c=0$.
The latter parameter, $\rho_1$, is related to the mass infall rate $\dot{M}_i$
and the central mass $M_{\ast}$ \citep{kenyon93a}:
\begin{equation}
\label{rho1_equ}
\rho_1 = 5.3 \times 10^{-14} \left( \frac{\dot{M}_i}
{10^{-5} \: M_{\odot} \: \mathrm{yr}^{-1}} \right)
\left( \frac{M_{\ast}}{1 \: M_{\odot}} \right)^{-1/2}
\mathrm{g \: cm^{-3}}
\end{equation}
For a fixed central mass, $\rho_1$ determines the mass infall rate.
The inner radius of the envelope is determined by the dust destruction,
which, for typical dust made of silicates, is at the dust sublimation
temperature of $\sim$ 1500 K. If a cavity is included, the inner envelope
radius will be farther out.
The outer envelope radius is constrained particularly by sub-mm and mm observations,
since this parameter affects the SED almost exclusively beyond about 100 $\mu$m;
typical values are several 1000 AU.
For sheet-collapse models, the flattening of the envelope is described by
$\eta=R_{out}/H$, where $R_{out}$ is the outer radius of the envelope
and H is the scale height of the original sheet, an isothermal, infinite, flat layer
in hydrostatic equilibrium, and is defined by
\begin{math} H = c_s^2/(\pi G \Sigma), \end{math}
with $\Sigma$ equal to the surface density \citep{hartmann94}.
Over the course of the collapse of a flattened molecular cloud core, it is thought that
$\eta$ increases, reflecting an increase in the size of the cavity.
The emergent SED is computed with the disk radius (set equal to 10 AU for
the disk component, and to $R_c$ for the stellar and envelope components),
the inclination angle to the line of sight, and the opening angle of the cavity
as parameters. Scattering is also included, which causes an increase in
flux at near-infrared wavelengths ($\lesssim$ 3 $\mu$m).
The parameters that affect a modeled SED the most are $\rho_1$, $R_c$,
and the inclination angle. The higher the density, the deeper the silicate
absorption feature and the longer the wavelength at which the SED peaks.
A high inclination angle will also increase the silicate feature depth, in addition
to decreasing the emission below 8 $\mu$m and increasing the
emission in the far-infrared. The centrifugal radius, besides representing the
outer disk radius, also defines the region where infalling material piles up;
thus, for smaller $R_c$ high-density material can be found closer to
the central protostar, resulting in more mid- to far-infrared emission and
a deeper silicate absorption feature.
A sketch of our model envelope, with important model parameters identified,
is shown in Figure \ref{env_sketch}.
From previous studies \citep[e.g.,][]{kenyon93b}, typical values for
young protostars in the Taurus star-forming region are an $R_c$ of
$\sim$ 100 AU, and mass infall rates a few times 10$^{-6}$
M$_{\odot}$ yr$^{-1}$. Observations exclude centrifugal radii larger than
about 1000 AU and mass infall rates larger than 10$^{-6}$ M$_{\odot}$
yr$^{-1}$. From equation \ref{rho1_equ}, it follows that typical values
for $\rho_1$ are of the order of 10$^{-14}$ to 10$^{-13}$ g~cm$^{-3}$.
\citet{hartmann96} find that characteristic values of $\eta$ are between
2.5 and 3.5 for the main infall phase of the envelope, resulting in large
opening angles for the cavities. Over time, $\eta$ should increase, up to
a maximum value of $\sim$ 4, and the cavities become larger.
Inclinations are usually larger than 60\degr, but this could be a selection
effect, since the SEDs of low-inclination protostars with large cavities would
resemble the SEDs of T Tauri stars and therefore not be classified as Class I
objects \citep{kenyon93b}.
In the following section, we will introduce our envelope models for 22 Class I
objects of our sample. In Figures \ref{model_04016} to \ref{model_L1551}
(excluding Figures \ref{04108_2803AB} and \ref{04325_slits}),
we show the SED of each object together with the best-fit model (determined by eye)
and its components, the star ({\it dotted gray line}), the disk ({\it dash-dotted gray line}),
both extinguished by the envelope, and the envelope ({\it long-dashed gray line});
the sum of all components is represented by the thick, solid, gray line. All model
protostars were placed at a distance of 140 pc.
\subsection{Models of Individual Class I Objects}
\subsubsection{04016+2610}
\citet{padgett99} imaged this object in the near-IR with HST; it is surrounded
by a nebulosity on one side, with the protostar at its base. This might indicate
that we are observing IRAS 04016+2610 through the opening of a large cavity in
the envelope. A bipolar molecular outflow is oriented along the axis of symmetry
of the nebula \citep[e.g.,][]{gomez97, hogerheijde98}; its inclination angle was
estimated to be 60\degr\ by \citet{hogerheijde98}.
In Figure \ref{model_04016} we show an envelope model for IRAS 04016+2610
whose parameters were adjusted to yield a good fit of the near-IR to mm SED.
A sheet-collapse model with $\eta=1$ had to be adopted; TSC models
did not yield enough flux on the short- and long-wavelength sides of the
10 $\mu$m silicate feature. The IRS spectrum is fitted reasonably well between
about 8.5 and 25 $\mu$m; the discrepancies at the shorter wavelength side
might be resolved by adjusting the abundances of the dust and ices in the model,
while the lower IRS fluxes beyond 25 $\mu$m could be an aperture effect, with the
IRS slits truncating the regions contributing to the 20--40 $\mu$m region.
The model shown in Figure \ref{model_04016}, with a density of $4.5 \times
10^{-14}$ g cm$^{-3}$, an $R_c$ value of 100 AU, and an inclination angle
of 40\degr, is one of several that yielded a comparable good fit. In particular,
values for $\rho_1$ down to 3.5 $\times 10^{-14}$ g cm$^{-3}$ and
for $R_c$ down to 70 AU, as well as inclination angles within 30\degr--50\degr,
would yield acceptable fits. The lower values for $\rho_1$ and $R_c$ would
match previous modeling results of KCH93, who derived 3.2 $\times 10^{-14}$
g cm$^{-3}$ and 70 AU, respectively, for these two parameters. They would also
be roughly consistent with results from \citet{whitney97}, who derived an even smaller
$R_c$ of 50 AU and an inclination angle of 57\degr. On the other hand, our best-fit
model agrees very well with the results of \citet{eisner05} and of \citet{gramajo07}.
A much smaller $R_c$ would result in an SED that is too
narrow and contain too much flux in the 20--50 $\mu$m region, while a large $R_c$
would require larger inclination angles to match the 10 $\mu$m feature and
longer-wavelength IRS flux, but it would also decrease the emission below 8 $\mu$m.
\citet{stark06} require a larger centrifugal radius of 300 AU, combined with a larger
value for $\rho_1$ and an inclination angle of 65\degr, to reproduce the near-IR
images of IRAS 04016+2610.
Our derived range for the inclination angle, 30\degr--50\degr, is somewhat smaller
than the value of 60\degr\ derived from observations \citep{hogerheijde98,padgett99}
and modeling (KCH93). In addition, our derived cavity semi-opening angle of 5\degr\ is
smaller than expected from the images; for example, \citet{stark06} derive a value
of 25\degr\ for $\theta$. If we adopted a larger cavity, the emission from
near-IR wavelengths to about 20 $\mu$m would decrease and therefore not fit the shape
of most of the IRS spectrum.
Finally, our adopted luminosity of 4.5 L$_{\odot}$ is larger than the bolometric luminosity
of 3.7 L$_{\odot}$ (KH95), but necessary to yield enough flux.
\subsubsection{04108+2803 B}
This protostar is part of a binary; it is separated by about 21\arcsec\ from
component A, which is less embedded, displays a silicate emission feature,
but is about a factor of 5 fainter in the mid-infrared (see Figure \ref{04108_2803AB}).
Therefore, component B clearly dominates in the mid-IR; while 04108+2803 A
probably is a reddened T Tauri star, 04108+2803 B is a Class I object. Both
sources have no optical counterparts \citep[e.g.,][]{tamura91}.
According to \citet{young03}, IRAS 04108+2803 B is a Class I object based
on its bolometric temperature of 179 K, but would be classified as a Class 0
object based on its submillimeter luminosity. This IRAS source is not resolved in the
450 and 850 $\mu$m maps of \citet{young03}; since it is also very faint
at 1.3 mm \citep[39 mJy;][]{motte01} and does not have an outflow, it
is likely an evolved protostar with a smaller envelope.
The envelope model for IRAS 04108+2803 B requires relatively low density
(${\rho_1}=1.5 \times 10^{-14}$ g cm$^{-3}$), a small centrifugal
radius (R$_c=40$ AU), a moderate-size cavity ($\theta=$ 10\degr), and
an inclination angle of 40\degr\ (Figure \ref{model_04108B}). The outer
envelope radius was set at 6000 AU, which is larger than the projected binary
separation of 3000 AU, but necessary to fit the sub-mm and mm emission.
This suggests that component A is likely obscured by this envelope; alternatively,
IRAS 04108+2803 B could be surrounded by a smaller envelope, but a fairly
large circumstellar disk, which would add to the long-wavelength emission.
The IRS spectrum is fitted rather well with this envelope model; the change in slope
of the spectrum beyond 25 $\mu$m could be an effect of slit truncation.
The fit of the 10 $\mu$m feature and of the slope of the IRS spectrum
between 10 and 25 $\mu$m suggests that the values for the density and the
centrifugal radius are relatively well-constrained. Previous models by KCH93 and
also by \citet{whitney97} obtained higher values for $\rho_1$ and $R_c$
(about double our values), but with comparable inclination angles. \citet{eisner05}
derived a large value for $\rho_1$, too, but an $R_c$ similar to our value; their
best-fit model included a massive disk surrounded by a 500 AU envelope.
In our models, we can exclude $R_c$ values larger than about 50 AU, since they
would result in a decrease in flux from 15 to 40 $\mu$m due to an overall shallower
SED shape. A reference density up to $2.0 \times 10^{-14}$ g cm$^{-3}$ would
still fit the observed SED, if combined with a smaller inclination angle ($\sim$ 25\degr)
to reduce the depth of the silicate absorption feature.
The cavity semi-opening angle of 10\degr\ is required to fit the IRS spectrum; a
smaller opening angle would result in an increase in flux in the 10--25 $\mu$m
wavelength range. The inner disk radius was set at 3 stellar radii to decrease the
flux in the 2--8 $\mu$m region; this parameter does not affect the remaining parts
of the SED.
\subsubsection{04154+2823}
IRAS 04154+2823 might be a transition object between the Class I and II stage,
since it has an SED that is somewhat decreasing in the mid-IR, and it also does
not display a prominent silicate absorption feature. However, its spectrum shows
a clear CO$_2$ ice absorption feature at 15.2 $\mu$m, and its overall infrared
SED is relatively flat, suggesting that some envelope material should be present.
It is also very faint in the optical \citep{strom94}, possibly indicating larger
extinction along the line of sight.
It is somewhat reminiscent of IRAS 04158+2805, which is also faint and has a
relatively flat SED over the IRS spectral range.
This protostar has a low luminosity (L$_{bol}$ = 0.3--0.4 L$_{\odot}$; see
Table \ref{tab_lum_incl}); its envelope model requires a very low reference density
of ${\rho_1}=7.0 \times 10^{-15}$ g cm$^{-3}$ and a very small centrifugal
radius of 10 AU to reproduce the mid-IR spectrum as well as the sub-mm measurements
(Figure \ref{model_04154}). However, this model overestimates the flux between 20
and 100 $\mu$m and slightly underestimates the mm flux. A luminosity value
higher than about 0.4 L$_{\odot}$ would result in an increase in the long-wavelength
flux, which would fit the mm data better, but yield even more flux between 20
and 60 $\mu$m. An increase in $\rho_1$ would have a similar effect; in addition,
values of $\rho_1$ larger than about $8.0 \times 10^{-15}$ g cm$^{-3}$ can
be excluded given the constraint of the weak silicate absorption feature.
The low inclination angle ($i=20$\degr) is also well-constrained by the 10 $\mu$m
feature and the shape of the SED.
\subsubsection{04158+2805}
According to \citet{menard01} who imaged this source in the optical, IRAS
04158+2805 is likely surrounded by an edge-on disk and a bipolar nebula; a
jet is oriented perpendicular to the disk plane, in a geometry similar to HH 30.
\citet{park02} observed that this object is not extended in the near-IR or
at millimeter continuum wavelengths, and therefore it could be a heavily reddened
T Tauri star and not a Class I object. Recently, \citet{andrews07} modeled
the SED of IRAS 04158+2805 with only a large disk component.
However, \citet{kenyon98} concluded from their optical spectra and optical and
near-IR photometry that IRAS 04158+2805 is a Class I source. Unpublished HST
images at optical and near-IR wavelengths (programs 9103 and 10603, PI
K. Stapelfeldt and D. Padgett, respectively) show an extended, one-sided
conical nebulosity, confirming that this source is surrounded by an envelope.
We note that this object is of a late spectral type (M5--M6 according to
\citealt{luhman06a}; M6 according to \citealt{white04}); it will probably
become a very low-mass star, close to the hydrogen-burning mass limit.
The SED of IRAS 04158+2805 can be fitted with an envelope model with relatively
low density (${\rho_1}=2.0 \times 10^{-14}$ g cm$^{-3}$), small
centrifugal radius (R$_c=60$ AU), small cavity ($\theta=$ 5\degr) and
an inclination angle of 30\degr\ (Figure \ref{model_04158}). The
inner disk radius was set at 3 stellar radii to decrease the amount of flux
emitted at near-IR and mid-IR wavelengths out to 8 $\mu$m. The outer
envelope radius was set at 5000 AU to somewhat decrease the emission in the
sub-mm and mm. Nevertheless, the model overestimates the flux at wavelengths
longer than 30 $\mu$m, but lies in between the two measurements around 1 mm.
The discrepancy between the two long-wavelength data points could be explained
by the large difference of the two beam sizes (see Table \ref{tab_longwave});
the 880 $\mu$m measurement likely misses some extended emission.
We adopted a luminosity of 0.3 L$_{\odot}$, which is 50\% larger than the
bolometric luminosity we measured by integrating the fluxes fluxes of the SED;
a model with $L=$ 0.2 L$_{\odot}$, a smaller density ($\sim 1.0 \times
10^{-14}$ g cm$^{-3}$), and a centrifugal radius around 10 AU would yield
a comparable fit, but still overestimate the emission starting at 20 $\mu$m.
Overall, the model parameters are mainly constrained by the IRS spectrum;
the depth of the silicate feature and the shape of the IRS spectrum from 12 to
30 $\mu$m constrain the reference density $\rho_1$ and the cavity
semi-opening angle, respectively, reasonably well.
With the current reference density, a much smaller centrifugal radius would
result in even higher fluxes in the 30--60 $\mu$m region, while a value larger
than about 90 AU would decrease emission from about 15 to 40 $\mu$m.
A $\rho_1$ value of about 1.5 $\times 10^{-14}$ g cm$^{-3}$,
combined with an $R_c$ value of 50 AU and a slightly larger inclination angle of
40\degr\ would also yield an acceptable fit.
We can exclude an edge-on orientation based on the shape of the IRS spectrum.
Only the availability of accurate, far-IR to mm fluxes will improve the
model fit of IRAS 04158+2805.
\subsubsection{04166+2706}
This IRAS source is very faint in the near-IR; no source is detected in 2MASS,
but both \citet{kenyon90} and \citet{park02} detected a faint near-IR source
close to the {\it IRAS} position, which we adopted for the pointing of our IRS
observations. The latter authors describe the appearance of IRAS 04166+2706
as low surface brightness nebulosity. \citet{motte01} and \citet{tafalla04}
detected a dense, round core at mm wavelengths coinciding with the IRAS source.
Thus, this Class I object is likely deeply embedded.
It also has bipolar outflows \citep{bontemps96}, which are highly collimated and
of extremely high velocity \citep{tafalla04}, consistent with this object's early
evolutionary stage, possibly placing it at the Class 0 stage. The high degree of
symmetry of the red- and blueshifted gas of the outflows \citep{tafalla04}
suggests that this object is likely seen close to edge-on.
For the SED plot, we included the H and K magnitudes measured by \citet{kenyon90}.
The SED shape of IRAS 04166+2706 further indicates that it is likely seen at a large
inclination angle, close to edge-on. In fact, an envelope model that fits it well
(see Figure \ref{model_04166}) has an inclination angle of 85\degr. Smaller
inclination angles (70\degr-80\degr) would result in models that match the
observed spectrum from 15 to 30 $\mu$m more closely, but they would
overestimate the emission from the near-IR to 8 $\mu$m. Increasing the cavity
semi-opening angle by just 1\degr\ from its current value of 6\degr\ would
lower the flux between 8 and 30 $\mu$m, resulting in a better fit of the slope
of the spectrum beyond 12 $\mu$m, but a poorer fits of the silicate feature.
An inner disk radius of 5 stellar radii was adopted to lower the flux below 8 $\mu$m.
A reference density of $4.5 \times 10^{-14}$ g cm$^{-3}$ and a large centrifugal
radius ($R_c$ = 300 AU) are required to fit the depth of the silicate feature and also
the width of the SED. Values of $\rho_1$ and $R_c$ down to $3.5 \times 10^{-14}$
g cm$^{-3}$ and 200 AU, respectively, would also yield good fits; only the silicate feature
would be somewhat narrower. The best-fitting model of KCH93, who had fewer constraints
available, resulted in $\rho_1 = 1.0 \times 10^{-13}$ g cm$^{-3}$, $R_c$=70 AU,
and $i=30\degr$; a high-inclination model with the same $R_c$, but $i=90\degr$ and
$\rho_1 = 3.2 \times 10^{-14}$ g cm$^{-3}$ was also suggested as a fit.
This underlines the importance of the IRS spectrum to discriminate between different
models.
We note that our model does not fit the 850 and 1300 $\mu$m measurements,
which were done with a larger beam size (see Table \ref{tab_longwave}), but it
fits the data taken at similar wavelengths, but with $\sim$ 20\arcsec\ apertures.
This could indicate that some extended, cold dust outside of the infall region contributes
to the long-wavelength emission \citep[see also][]{jayawardhana01}.
\subsubsection{04169+2702}
This object, which is not detected at optical wavelengths, is surrounded by a
cometary reflection nebula in the near-IR and associated with Herbig-Haro
emission knots \citep{tamura91, gomez97}. It also has bipolar outflows
\citep[e.g.,][]{moriarty92, bontemps96}, which are perpendicular to an
elongated envelope structure, about 2200 AU $\times$ 1100 AU in size
and inclined at 60\degr\ from the line of sight \citep{ohashi97b}.
Measurements of continuum emission at 2.7 mm did not resolve the source on
scales of 2{\arcsec}--3{\arcsec}, probably indicating that this emission arises
from a disk of radius $\lesssim$ 150 AU \citep{ohashi97b}.
The SED of IRAS 04169+2702 is fitted well by a sheet-collapse envelope model
($\eta=1$) with moderate density ($\rho_1 = 3.2 \times 10^{-14}$
g cm$^{-3}$), a centrifugal radius of 100 AU, basically no cavity and
a high inclination angle (Figure \ref{model_04169}). Compared to the results
obtained by KCH93, both our reference density and centrifugal radii are smaller by
about a factor 3, and our inclination angle is larger (75\degr\ versus 30\degr).
\citet{whitney97} modeled this object with an $R_c$ of just 10 AU, a reference
density half our value, a cavity semi-opening angle of 11\degr, and an inclination
angle of 45\degr.
However, our model is better constrained. Both $\eta$ and the
cavity semi-opening angle are determined by the shape of the IRS spectrum
beyond about 12 $\mu$m. In particular, a TSC model would yield less flux from
about 10 to 100 $\mu$m given the same envelope parameters.
The large $R_c$ value is required to fit the long-wavelength part of the IRS spectrum,
while the value for the reference density is constrained by the shape of the silicate
absorption feature and the peak of SED. The 10 $\mu$m feature and the SED
shape also constrain the high inclination angle, which has to be larger than 70\degr\
to yield a good fit.
The luminosity used for the model ($L=1.5$ L$_{\odot}$) is twice as high
as the bolometric luminosity of 0.8 L$_{\odot}$ measured by KH95, but close
to the value we calculated by integrating the fluxes of the SED, 1.2 L$_{\odot}$,
and to the value of 1.4 L$_{\odot}$ determined by \citet{kenyon90}.
We can exclude a lower luminosity from our model fit, since it would not
yield enough emission to match the observations.
As for IRAS 04166+2706, our model does not fit the long-wavelength emission
measured with large aperture sizes, but it matches the observations taken with
smaller apertures. Again, we suggest that some cold dust just outside the infalling
envelope is responsible for the extended mm emission.
\subsubsection{04181+2654 A}
This IRAS source is a member of a wide binary system; component B lies to
the northwest of component A, at a projected distance of 31\arcsec, which
corresponds to about 4300 AU at the distance of Taurus. In addition, the IRAS
source 04181+2655 lies somewhat less than 1\arcmin\ to the northwest of
04181+2654 B. A bipolar outflow is centered on IRAS 04181+2655, but no
outflow has been detected from the IRAS sources to the south \citep{bontemps96},
which could be the result of a low inclination angle along the line of sight of these
two objects.
Of the two components of the 04181+2654 binary, the B component seems
to be more deeply embedded, since it is fainter than A by a factor of 12 in the
$J$ band, but only by about a factor of 2 in the $K$ band. Over the mid-IR
spectral range, component B is fainter than component A by a factor of 2 to 3.
The silicate feature of IRAS 04181+2654 B is seen partly in self-absorption,
indicating that it is oriented along the line of sight such that part of the silicate
emission originating in the inner disk is absorbed in the envelope.
\citet{motte01} measured the 1.3 mm flux of IRAS 04181+2654 A, which
is dominated by emission from an extended envelope.
According to \citet{park02}, both A and B are extended at mm continuum
wavelengths, but not in the near-IR. They suggest that both sources are protostars;
therefore, they are likely surrounded by an envelope which could be truncated in
its outer parts. Images at 0.9 $\mu$m obtained by \citet{eisner05} reveal a faint
nebulosity around IRAS 04181+2654 A, possibly tracing scattered light from an
envelope cavity; the B component is not detected.
The SED of IRAS 04181+2654 A can be fitted by an envelope model with a reference
density of $2.0 \times 10^{-14}$ g cm$^{-3}$, a centrifugal radius of 50 AU,
a cavity semi-opening angle of 7\degr, and an inclination angle of 20\degr\ (Figure
\ref{model_04181A}).
In order to fit the SED shape from the near-IR to 8 $\mu$m, we set the inner disk
radius to 5 stellar radii; even with this value, the near-IR fluxes are somewhat
underestimated. We chose an outer envelope radius of 10000 AU; such a large
envelope would encompass component B, too, but is necessary to yield enough
flux in the mm wavelength range. The fact that the model still underpredicts the
flux at 1.3 mm suggests that the mm measurement probably includes some
contribution from an envelope around component B.
The density is constrained by the depth of the silicate absorption feature,
which is rather shallow; it could be slightly larger, but then require a somewhat
smaller cavity semi-opening angle to decrease emission in the 8-25 $\mu$m range.
The slope of the spectrum from 12 to 30 $\mu$m constrains both the opening angle
of the cavity and the value of $R_c$. A larger $R_c$ would result in a broader SED,
but at the same time decrease the flux levels in the long-wavelength part of the IRS
spectrum. A smaller $R_c$ would generate a steep rise of the flux levels beyond
12 $\mu$m. KCH93 modeled both IRAS 04181+2654 A and B with an
envelope with a higher reference density, $3.2 \times 10^{-14}$ g cm$^{-3}$,
a somewhat larger $R_c$ (70 AU versus our 50 AU), and a comparable inclination angle.
\citet{whitney97} used the same value for $R_c$ as we did in our model, but with
almost double the values for reference density and inclination angle.
Overall, the model for IRAS 04181+2654 A is poorly constrained, given the lack
of sub-mm data points and the uncertainty of the 1.3 mm measurement
due to the presence of component B. We did not model that component,
since there are no sub-mm or mm measurements available that would constrain
the long-wavelength part of the SED.
\subsubsection{04239+2436}
In the near-infrared, a symmetric reflection nebula surrounds this IRAS source,
which is a binary separated by 0{\farcs}3 \citep{reipurth00}.
The brighter component (at 2.2 $\mu$m) of this system drives a large,
asymmetric Herbig-Haro jet, HH 300 \citep{reipurth00}; the asymmetry
suggest that the inclination angle of this object is closer to face-on than
edge-on. Maps of this Class I object at 450, 850, 1100, and 1300 $\mu$m
show that it is a point source with some diffuse emission from circumstellar
material \citep{chini01, eisner05}. Therefore, IRAS 04239+2436 is extended
from infrared to millimeter wavelengths \citep[see also][]{park02}, typical
for Class I objects.
The SED of IRAS 04239+2436 requires a model with relatively low density
($\rho_1 = 1.3 \times 10^{-14}$ g cm$^{-3}$) and very small centrifugal
radius (R$_c=$ 10 AU) (Figure \ref{model_04239}). A higher density would
increase the depth of the silicate feature, while a larger centrifugal radius would
decrease the emission in the 10 to 40 $\mu$m region and result in a shallow
silicate absorption feature. Increasing $\rho_1$ and $R_c$ up to about
$2.0 \times 10^{-14}$ g cm$^{-3}$ and 20 AU, respectively, would still
result in good fits if the luminosity was increased by about 20\%.
Previous models by KCH93 yielded a larger density, 3.2 $\times 10^{-14}$
g cm$^{-3}$, and a centrifugal radius of 70 AU, while models by \citet{whitney97}
resulted in values for $\rho_1$ and $R_c$ very similar to ours. \citet{eisner05}
derived a reference density almost twice our value and a centrifugal radius of 30 AU;
more recent results by \citet{gramajo07} found the same centrifugal radius as
\citet{eisner05}, but a reference density similar to ours.
Our inclination angle of 15\degr\ is smaller than the values derived by previous
modeling efforts, which range from 30\degr\ to 63\degr; a value of up to
$i=30\degr$ would yield an acceptable fit, albeit with somewhat decreased emission
from the near-IR to about 25 $\mu$m. The low inclination angle is well-constrained
by the SED, and it is also consistent with the near-IR morphology and observed
asymmetry of the Herbig-Haro jet \citep[see, e.g.,][]{reipurth00}.
With the current values for $R_c$ and $\rho_1$, the model does not reproduce
the shape of the silicate absorption feature very well, especially at the long-wavelength
side; this could be an effect of dust and ice composition, since the overall width of the SED
and shape of the IRS spectrum beyond 15 $\mu$m suggest such a small
$R_c$ value.
\subsubsection{04248+2612}
This IRAS source is actually a subarcsecond binary whose components are
roughly similar in brightness. Its IRS spectrum shows a silicate emission feature
and some very weak ice absorption features.
In the near-IR, it is surrounded by a bipolar reflection nebula with a somewhat
complex morphology, but likely seen close to edge-on \citep{padgett99}.
This object is also somewhat extended at 450 and 850 $\mu$m; the extension
parallel to the bipolar nebulosity seen in the near-IR is likely tracing dust heated
in an outflow \citep{young03}. \citet{white04} note that IRAS 04248+2612
is the lowest-mass protostar known that drives a molecular outflow and an HH
object \citep{moriarty92,gomez97}; they suggest it could actually be a substellar
Class I object. However, a recent spectral type determination of M4--M5 for this
object \citep{luhman06a} suggests that it will probably evolve into a very
low-mass star.
Since this Class I object displays a silicate emission feature, and in the near-IR the
central binary is detected, its envelope likely has low densities. Our model suggests
a reference density of $4.0 \times 10^{-15}$ g cm$^{-3}$, which is the lowest
$\rho_1$ value among the protostars modeled here, indicating an advanced stage
of evolution for this protostar. Our model further includes a relatively small
centrifugal radius of 30 AU, a cavity semi-opening angle of 15\degr, and an
inclination angle of 70\degr\ (see Figure \ref{model_04248}).
IRAS 04248+2612 is similar to CoKu Tau/1, which is also seen close to edge-on
through a low-density envelope. The short-wavelength emission is fitted very well; it is
dominated by the stellar component, which agrees with the fact that the central binary
is detected at near-IR wavelengths. The emission from far-IR to mm wavelengths is
underestimated; however, we found no set of parameters that could reproduce
both the mid-IR and long-wavelength parts of the SED. Thus, it is likely that the
sub-mm and mm emission is generated by an additional component, probably by
dust in a very extended region of cold dust outside the infall zone
\citep{jayawardhana01}.
Compared to modeling results by KCH93, who obtained two models of IRAS
04248+2612, one with $\rho_1$=$1.0 \times 10^{-13}$ g cm$^{-3}$ and
$R_c$=300 AU, and one with $\rho_1$=$3.2 \times 10^{-14}$ g cm$^{-3}$ and
$R_c$=70 AU, our model has a clearly lower density (by a factor of 8) and a smaller
centrifugal radius than the lower values of KCH93. Also models by \citet{whitney97}
resulted in a larger centrifugal radius (100 AU) and reference density ($1.5 \times
10^{-14}$ g cm$^{-3}$) than our values; however, they derived a large inclination
angle, as we did with our model. Recent modeling efforts by \citet{stark06} yielded
a model very similar to ours.
Based on our fit to the IRS spectrum, especially the presence of a silicate emission
feature, we can exclude a much higher density, and the fit to the shape of the IRS spectrum
between 15 and 25 $\mu$m also constrains the centrifugal radius to lie around 30 AU.
Comparable model fits could be achieved with $\rho_1$ values in the (3.0-5.0)
$\times 10^{-15}$ g cm$^{-3}$ range and $R_c$ values between 20 and 40 AU,
with smaller reference densities accompanied by smaller centrifugal radii. Our inclination
angle of 70\degr\ not only matches two previously published models of this object
\citep{whitney97, stark06}, but is also in good agreement with the near-IR morphology
observed by \citet{padgett99}.
\subsubsection{04264+2433}
The IRS spectrum of this IRAS source shows a prominent silicate emission feature,
but also possibly some ice features in the 5--8 $\mu$m wavelength region and an
overall rising SED in the mid-IR. Therefore, it is probably a protostar surrounded by
envelope material. This notion is further supported by the fact that IRAS 04264+2433
is surrounded by a reflection nebulosity and likely emitting a small bipolar HH jet
\citep{devine99}.
Similar to IRAS 04248+2612 and CoKu Tau/1, the detection of a silicate emission
feature in this protostar indicates that the envelope has low densities and is thus
likely at an advanced evolutionary stage. We can reproduce the SED of IRAS
04264+2433 fairly well with a model whose reference density is $5.0 \times
10^{-15}$ g cm$^{-3}$; in addition, the shape of the IRS spectrum between
about 13 and 20 $\mu$m constrains the value of the centrifugal radius, found to
be 30 AU (see Figure \ref{model_04264}). The inclination angle of 87\degr\
and cavity semi-opening angle of 13\degr\ were fine-tuned to yield a good fit over
the 5-20 $\mu$m range. The model overestimates the emission in the near-IR
somewhat, which can be mostly be attributed to the envelope component.
A previous model of IRAS 04264+2433 by KCH93 indicated a density twice as high
as our value, a centrifugal radius of only 10 AU, and an inclination angle of
60\degr. Despite the differences, the earlier model already gauged the properties
of this object, namely a low-density, highly inclined envelope with a small disk.
On the other hand, a model by \citet{whitney97} yielded different model parameters:
a centrifugal radius of 50 AU, a reference density of $3.75 \times 10^{-14}$
g cm$^{-3}$, and an inclination angle of 32\degr; only their cavity semi-opening
angle is similar to our value.
Our model would yield a comparable fit with a centrifugal radius of 20-40 AU, but
not with smaller or larger values for $\rho_1$.
In order to get enough flux from the far-IR to the mm wavelength range,
we had to increase the luminosity by a factor of 2 compared to the bolometric
luminosity of 0.37 L$_{\odot}$ measured by KH95. On the other hand,
integrating the measured fluxes of the SED yields a bolometric luminosity of 0.5
L$_{\odot}$; given the high inclination angle of this object, the luminosity
determined in this way is likely an underestimate of the true value.
\subsubsection{04295+2251}
This object, also known as L1536 IRS, is relatively faint; based on its bolometric
temperature of 270 K, it is classified as a Class I object, but its large sub-mm
luminosity would suggest it is a Class 0 object \citep{young03}. Given that
IRAS 04295+2251 seemed not to be extended in the near-IR or mm,
\citet{park02} indicated that it could be an edge-on disk; however, data
obtained by \citet{eisner05} shows that IRAS 04295+2251 is somewhat
extended at 1.3 mm and also at 0.9 $\mu$m, where scattered light is detected.
In addition, \citet{moriarty92} detected an outflow from this source, implying
an early evolutionary stage for this object.
Our envelope model for IRAS 04295+2251 reproduces the general shape of
the SED (see Figure \ref{model_04295}), but it does not represent a good
fit to the 10 $\mu$m silicate feature and somewhat underestimates the
near-IR and mm emission. However, at the longer wavelengths, our model
reproduces those fluxes measured with smaller apertures, 10\arcsec-20\arcsec
in size (see Table \ref{tab_longwave}); this suggests that a region of cold dust
outside of the infall region could be present.
A luminosity of 0.8 L$_{\odot}$ was necessary to obtain more emission
especially at longer wavelengths, even though the source's bolometric luminosity
is 0.5 L$_{\odot}$ (determined from integrating the fluxes of the SED). In the
literature, L$_{bol}$ values vary from 0.44 L$_{\odot}$ (KH95) to 0.64
L$_{\odot}$ \citep{myers87}.
A small centrifugal radius of 20 AU was adopted to reproduce the steep rise of the SED
past 15 $\mu$m; a value larger than 30 AU would result in a flatter SED in this wavelength
region, while a smaller $R_c$ would decrease the near-IR emission and cause an even
larger overestimate of the far-IR flux. A reference density up to 30\% larger than our
adopted value of $8.0 \times 10^{-15}$ g cm$^{-3}$ would still yield an acceptable
fit, but require lower inclination angles (40\degr-50\degr) to generate only a shallow
silicate absorption at 10 $\mu$m; these models would overestimate the emission
between 3 and 10 $\mu$m and in the far-IR.
A previous model of IRAS 04295+2251 by KCH93 resulted in a larger centrifugal
radius, 70 AU, and a smaller inclination angle, $i=$30\degr, but in a comparable
reference density. We can exclude an inclination angle smaller than about 60\degr\
with the current model parameters, since it would increase the emission in the
2-8 $\mu$m range and at far-IR wavelengths, where the model already
overestimates the observed emission.
\citet{whitney97} modeled this object with very different model parameters,
including a larger centrifugal radius and reference density, and an inclination angle
of only 18\degr. \citet{eisner05} used the same value for $\rho_1$,
$3.75 \times 10^{-14}$ g cm$^{-3}$, and a similar inclination angle as
\citet{whitney97}, but an $R_c$ of 30 AU, which is comparable to our result.
The relatively low reference density, combined with the inclination of 70\degr\ to the
line of sight and a cavity semi-opening angle of 5\degr, results in a self-absorbed silicate
feature at 10 $\mu$m, with part absorption, part emission characteristics. However, our
model does not reproduce the detailed shape of the observed 10 $\mu$m feature.
This is likely due to the fact that the disk component used in the model does not include
the optically thin disk atmosphere that would generate a 10 $\mu$m silicate emission feature,
which would then be subject to absorption by dust in the envelope. In the current model,
the disk component only contributes a silicate absorption feature (i.e., continuum extinguished
by the envelope), while the dust in the envelope is responsible for the emission component
at 10 $\mu$m (see the dash-dotted and long-dashed lines, respectively, in Figure
\ref{model_04295}). Also the shape of the inner cavity could affect the depth of
the silicate feature, for example by resulting in a hotter inner region for a given $\rho_1$.
It is likely that only a more complex model will yield a better fit, considering that this
object has a relatively low-density envelope and compact morphology, but large
sub-mm luminosity.
\subsubsection{04302+2247}
In the near-IR, this object is surrounded by two bright, symmetric nebulae separated
by a dark lane \citep{padgett99}. Models of the very symmetric quadrupolar structure
suggest that IRAS 04302+2247 is seen almost exactly at an inclination angle of 90\degr\
\citep{lucas97,wolf03}. The central source is likely obscured by a circumstellar disk,
and the near-IR nebulosity, seen in scattered light, traces the walls of a bipolar cavity
in the envelope \citep{lucas97, wolf03}. Outflows oriented perpendicular to
the dark lane were detected from this object \citep{moriarty92,lucas98}.
The edge-on orientation of IRAS 04302+2247 may explain why it is
invisible in the optical and very faint in the mid-IR. The SED of this object peaks close
to 100 $\mu$m and shows a pronounced 10 $\mu$m absorption feature.
The SED of IRAS 04302+2247 can be fitted with an envelope model with an
inclination angle of 89\degr, moderate density ($\rho_1 = 3.0 \times
10^{-14}$ g cm$^{-3}$), and a centrifugal radius of 300 AU (Figure
\ref{model_04302}). The deep silicate absorption feature and the steep
slope of the IRS spectrum beyond 10 $\mu$m require a high inclination
angle, consistent with the near-IR images. A higher density would also
increase the depth of the silicate feature, but it would decrease flux levels
over the 2--50 $\mu$m region, too.
The semi-opening angle of the cavity is relatively wide (22\degr), also
roughly consistent with the results of \citet{padgett99}. The shape of the
IRS spectrum beyond 12 $\mu$m, and the width of the SED, constrain the
value of $R_c$, which has to be large. Previous models by KCH 93, which
had much fewer constraints, derived a centrifugal radius of only 70 AU, but
a density about a factor of 3 higher than in our model, and an inclination
angle of 60\degr. \citet{whitney97} found an even smaller value for $R_c$,
10 AU, and a reference density half as large as our value, but their results
for cavity and inclination angles agree with ours. Recent models by
\citet{stark06} resulted in very similar parameters as for our model,
except for a somewhat smaller value for $\rho_1$.
The model emission is somewhat above the {\it IRAS} data points at 60 and
100 $\mu$m; this is likely due to the fact that our adopted luminosity of
1 L$_{\odot}$ is higher than the observed bolometric luminosity of 0.34
L$_{\odot}$ (KH95), which was determined from fewer flux measurements
($\lambda \leq$ 100 $\mu$m). Our model indicates that our adopted luminosity
might be slightly higher than the actual source luminosity.
\subsubsection{04325+2402}
This protostar is surrounded by a bipolar reflection nebula that is seen in scattered
light in the near-IR and shows a complex morphology \citep{hartmann99}.
It is likely a multiple system, consisting of a central, subarcsecond binary and a
faint companion about 8\arcsec\ away; both the binary and the companion are
surrounded by an accretion disk and are accreting from their envelopes
\citep{hartmann99}. The subarcsecond binary, which instead could just be the
result of scattering off a complex structure around a single star, is the likely source
of the reflection nebula detected in the near-IR \citep{hartmann99}.
This object is also associated with a molecular outflow \citep{moriarty92,hogerheijde98}.
Our IRS observations were centered on the central source (see Figure \ref{04325_slits}).
Even though the SL slit was oriented such that it also included the companion about 8\arcsec\
away, this component did not enter our extraction window, and we also did not detect it
separately; this is expected, since it is faint and likely seen edge-on \citep{hartmann99},
and even the spectrum of the bright central source is only about 40 mJy between 5 and
10 $\mu$m, with a steeply rising continuum beyond. In LL, which is oriented close to
perpendicular to SL, only the central source was included.
An envelope model that fits this object well requires a sheet-collapse model
($\eta=1.0$), moderate density ($\rho_1 = 3.0 \times 10^{-14}$ g cm$^{-3}$),
a centrifugal radius of 100 AU, a cavity semi-opening angle of 15\degr, and an inclination
angle of 80\degr\ (Figure \ref{model_04325}). Choosing a sheet-collapse model with
$\eta=1.0$ was necessary to fit the IRS spectrum in the 15-30 $\mu$m range.
The inclination angle is larger than the value of $\sim$ 60\degr\ inferred from the
orientation of the outflow by \citet{hogerheijde98}, but the shape of the IRS spectrum
requires a high-inclination model.
A previous model by KCH93 also determined $i=60$\degr, but a centrifugal
radius and a reference density three times as large as the values derived here.
Models by \citet{whitney97} resulted in an $R_c$ value of only 50 AU, but in
a reference density only somewhat larger than our value, and very similar values
for cavity and inclination angles (11\degr\ and 81\degr, respectively).
Compared to previous models, the IRS spectrum provides better constraints;
a larger $R_c$ would result in a flatter mid-IR SED, while a larger density would
generate a deeper silicate absorption feature.
The silicate feature is not well fitted with this model; it seems to be in absorption on
the short-wavelength side, but with an emission component on the long-wavelength side.
The binary nature of IRAS 04325+2402 could account for the peculiar shape of the
silicate feature, with one component possibly responsible for some silicate emission.
\subsubsection{04361+2547}
This Class I object, also known as TMR 1, is deeply embedded in its dense
environment and surrounded by a large reflection nebula, about 30\arcsec\
in size \citep{tamura91}. It is a close binary with a separation of 0{\farcs}31
\citep{terebey98}; the ``companion'' detected at a projected separation of
10\arcsec\ from this protostar, at the end of a nebulosity extending from the
source to the southeast, is likely a background star \citep{terebey00}.
IRAS 04361+2547 is extended in the near-IR, mid-IR and mm continuum
\citep{motte01, park02,luhman06b}. An outflow has been detected from
this object, but its morphology is not clearly bipolar, suggesting an inclination
angle of about 60\degr\ \citep{hogerheijde98}.
The steeply rising SED of IRAS 04361+2547 indicates that this object is seen at a
high inclination angle (Figure \ref{model_04361}). A good envelope model fit requires
a sheet-collapse model ($\eta=1.5$), an inclination angle of 80\degr, a cavity
semi-opening angle of 15\degr, an $R_c$ value of 100 AU, and a fairly typical
density ($\rho_1 = 2.0 \times 10^{-14}$ g cm$^{-3}$).
Modeling by KCH93 resulted in a reference density of 3.2 $\times 10^{-14}$ g
cm$^{-3}$ and a centrifugal radius of only 10 AU. Also, while our model suggests
a close to edge-on orientation, KCH93 derived an inclination angle of only 30\degr.
Later models by \citet{whitney97} and \citet{gramajo07} confirm our result of
a large inclination angle (and also our cavity opening angle), but they derive centrifugal
radii about half as large as our value, and generally a somewhat larger reference
density.
Our adopted reference density is required to fit the SED beyond 50 $\mu$m, as
well as to produce a silicate feature at 10 $\mu$m that is mostly dominated by
an emission component generated in the envelope (long-dashed line in
Figure \ref{model_04361}). The $R_c$ value of 100 AU was chosen to reproduce
the SED shape from about 15 to 50 $\mu$m. A larger $R_c$ would result in a
flatter SED, while a smaller value would result in more mid-IR flux and a pronounced
silicate absorption feature. Choosing $\eta_{star}=0.9$ and $\theta=$ 15\degr,
combined with the high inclination angle, depresses the emission below 10 $\mu$m.
Our model does not reproduce the deep ice features at 6.0 and 6.8 $\mu$m, but it
traces the continuum level between 5 and 8 $\mu$m (where emission from the
disk dominates). We note that the IRS fluxes are lower than the IRAC data points;
this is likely due to the extended emission detected in IRAC \citep{luhman06b}
and thus an aperture effect.
Below 2 $\mu$m, the near-IR scattered light component from
the envelope is likely overestimated.
We note that even though the luminosity of our model, 4.0 L$_{\odot}$, is larger
than the measured bolometric luminosity of 2.5 L$_{\odot}$ (see Table
\ref{tab_lum_incl}), it is necessary to yield enough flux at all wavelengths.
\subsubsection{04365+2535}
This IRAS source is also known as TMC 1A; it is optically invisible and thus
deeply embedded. It is surrounded by a cometary nebulosity in the near-infrared
\citep{tamura91, tamura96}, and it is extended both in the near-IR and at mm
wavelengths \citep{park02}. Both \citet{chandler96} and \citet{tamura96}
measured a bipolar outflow which is somewhat conical in shape due to the
blueshifted lobe being more prominent than the redshifted one. The near-IR
nebulosity is likely light from the central source scattered by dust in the outflows
\citep{tamura96}. Also, dust continuum measured at 790 $\mu$m and 1.1
mm is aligned with the outflow direction \citep{chandler98}. \citet{chandler96}
concluded that the inclination angle of this object lies between 40\degr and
68\degr, with a cavity of semi-opening angle between 15\degr\ and 21\degr.
The SED of 04365+2535 is fitted by an envelope model with relatively high density
($\rho_1 = 4.5 \times 10^{-14}$ g cm$^{-3}$), a centrifugal radius of 50 AU,
a small cavity, and an inclination angle of 30\degr\ (Figure \ref{model_04365}).
Compared to models by KCH93, both our reference density and centrifugal
radius are smaller, by factors of 0.45 and 0.17, respectively, and our model
suggests a more face-on orientation as opposed to the more highly inclined
model ($i=60$\degr) of KCH93 and the inclination angle determined from the
outflows \citep{chandler96}. More recent models by \citet{whitney97}
and \citet{gramajo07} resulted in a large inclination angle (70\degr-80\degr),
but in values for $R_c$ and $\rho_1$ very comparable to ours.
The high density is required to fit the deep silicate absorption feature, as well as to
shift the peak of the SED to longer wavelengths. Our model does not fit the sub-mm
and mm fluxes measured with a large aperture (40\arcsec-60\arcsec), but matches
the measurements done with apertures $\lesssim$ 20\arcsec (see Table
\ref{tab_longwave}). This suggest the presence of some extended, cold dust
beyond the infalling envelope.
A centrifugal radius of 50 AU was chosen to fit the emission from the silicate absorption
feature out to about 50 $\mu$m; a model with larger values for $\rho_1$ and $R_c$
($\sim 6.0 \times 10^{-14}$ g cm$^{-3}$ and 70 AU, respectively) would also yield
a good fit, but underestimate the emission in the 20-30 $\mu$m range somewhat. An
even larger centrifugal radius would decrease the emission from 15 to 40 $\mu$m.
The inclination angle of our envelope model is smaller than the value suggested by
\citet{chandler96} and by previous models; however, we can exclude a larger
inclination angle, since it would result in a steeper SED from 15 to 60 $\mu$m,
as well as less flux in the 2 to 10 $\mu$m region. We can also exclude a cavity
semi-opening angle larger than about 5\degr, since it would decrease the flux in
the 5-15 $\mu$m wavelength range.
\subsubsection{04368+2557}
This object has a bolometric temperature of less than 70 K, thus placing it in the
Class 0 domain, the earlier, deeply embedded protostellar stage \citep{chen95,
motte01}. It is very faint and extended in the near-IR \citep{park02}; no point
source is detected in 2MASS. The $K$-band data point in our SED plot is from
\citet{whitney97}, who only detected some faint reflection nebulosities at the
position of the IRAS source.
Since the central source is not detected in the near-IR, \citet{tamura96}
conclude that this object might be seen edge-on, obscured by a disk.
Our IRS observations were centered on the position determined by interferometric
observations at 2.7 mm by \citet{ohashi97a}, which also coincides with the
position of the source in high-resolution 7 mm VLA maps by \citet{loinard02}.
The latter authors resolved this IRAS source into two components separated
by 0{\farcs}17 and suggested that it is a binary in which one component is
surrounded by an almost edge-on disk.
In addition, this object might have a faint companion located 20\arcsec\ to the
northwest of the main source, but it was only detected at 800 $\mu$m
\citep{fuller96} and not confirmed at 1.3 mm \citep{motte01} and 2.7 mm
\citep{ohashi97a}.
IRAS 04368+2557 drives molecular outflows, which are located symmetrically
around the central source and also perpendicular to the elongated source detected
by \citet{loinard02}. \citet{hogerheijde98} inferred an inclination angle $i>$ 65\degr\
from observations of the outflows. \citet{ohashi97a} observed this object at high
resolution in the millimeter wavelength range and concluded that it is surrounded by
an elongated envelope $\sim$ 2000 AU in radius with large bipolar cavities, seen
edge-on.
The SED of IRAS 04368+2557 (Figure \ref{model_04368}) is similar to that of
IRAS 04302+2247 (Figure \ref{model_04302}); both objects have a deep silicate
absorption feature and SEDs that peak between 60 and 70 $\mu$m. Both are seen
close to edge-on, but the reference density of the IRAS 04368+2557 model fit is higher
($\rho_1 = 4.0 \times 10^{-14}$ g cm$^{-3}$), and the centrifugal radius smaller
($R_c$ = 200 AU). On the other hand, about 20\% changes in $\rho_1$ and $R_c$
yield comparable model fits, as long as the cavity semi-opening angle lies between
20\degr\ and 30\degr, and the source is seen edge-on. We can exclude a high
density as derived by KCH93 (3.2 $\times 10^{-13}$ g cm$^{-3}$), but agree
reasonably well with their result of $R_c$ (300 AU) and inclination angle
(60\degr-90\degr).
In the near-IR, our model reproduces the $K$-band measurement, but seems to
overestimate the 3.6 and 4.5 $\mu$m emission, as measured by IRAC
\citep{hartmann05}. However, this object is extended in IRAC images, and the
photometry from the 6\arcsec\ aperture shown in Figure \ref{model_04368}
does not capture all of the strong extended emission \citep[see also][]{hartmann05}.
As with IRAS 04302+2247, the values for $\rho_1$ and $R_c$ are fairly well
constrained by the depth of the silicate absorption feature, the shape of the IRS
spectrum beyond 10 $\mu$m, and the width and peak of the SED (the 100,
160, and 1300 $\mu$m measurements are likely overestimates due to aperture
effects). The similarity between the two IRAS sources suggests that some
Class I objects, seen at high inclination angles, could resemble Class 0 objects.
\subsubsection{04381+2540}
This IRAS source is also known as TMC 1; it is invisible in the optical and thus
deeply embedded \citep[e.g.,][]{moriarty92}. It is surrounded by a
near-infrared nebulosity seen in scattered light \citep{tamura91,terebey06}.
The conical shape of this near-infrared emission indicates that it likely traces an
evacuated outflow cavity; the CO molecular outflow has a semi-opening angle
between 13\degr\ and 19\degr\ and an inclination angle between 40\degr\
and 70\degr\ \citep{chandler96}.
Dust emission oriented perpendicular to outflows was detected at 790 $\mu$m
\citep{chandler98}. \citet{young03} measured extended 450 and 850 $\mu$m
emission from this object; based on the source's properties and model calculations,
they confirmed that this object is a Class I source. From radial intensity profile fits
to the 1.3 mm map, \citet{motte01} derived an outer envelope radius of 3900 AU.
Recently, \citet{apai05} found a 0{\farcs}6 companion to IRAS 04381+2540;
they suggest that this companion might be a young brown dwarf.
A good model fit for IRAS 04381+2540 (see Figure \ref{model_04381})
requires a somewhat higher luminosity than its bolometric luminosity of 0.7 L$_{\odot}$
(see Table \ref{tab_lum_incl}) to be able to reproduce the long-wavelength
emission. By increasing the inner disk radius to 5 stellar radii, the emission in the
near-IR and mid-IR out to 8 $\mu$m is decreased, resulting in a better fit. The
centrifugal radius of 70 AU and the reference density of $3.0 \times 10^{-14}$
g cm$^{-3}$ are well constrained by the depth of the silicate absorption feature
and the shape of the IRS spectrum from 15 to 30 $\mu$m. Decreasing $R_c$
to 60 AU and the inclination angle to 35\degr\ yields a nearly identical fit, but
we can exclude larger changes in these parameters, as well for the reference
density.
Previous modeling by KCH93 yielded a value of $1.0 \times 10^{-13}$ g cm$^{-3}$
for $\rho_1$ and 300 AU for $R_c$, and an inclination angle of 30\degr. However,
better-constrained models by \citet{whitney97} resulted in values for $\rho_1$
and $R_c$ similar to ours. \citet{eisner05} derived a smaller centrifugal radius
(30 AU) and also a larger reference density ( $6.75 \times 10^{-14}$ g cm$^{-3}$).
Our results for cavity semi-opening angle ($\theta=$10\degr) and inclination
($i=$40\degr) are at the lower limits of the range suggested by
\citet{chandler96}. Our value for $\theta$ agrees well with the modeling result
of \citet{whitney97}, but our inclination angle is smaller; on the other hand, this latter
quantity is consistent with the result of \citet{terebey06}, but not with their cavity
semi-opening angle of 40\degr. Higher values for both parameters can be
excluded based on the constraints placed by our IRS spectrum on the envelope models.
Finally, we note that we adopted an outer envelope radius of 10000 AU, which is larger
than the value derived by \citet{motte01}, but necessary to fit the long-wavelength
part of the SED.
\subsubsection{04489+3042}
Based on the M6 spectral type of IRAS 04489+3042, \citet{white04} indicated that
it could be substellar, a Class I brown dwarf. However, recent spectral type
determinations by \citet{luhman06a} suggest that its spectral type is M3--M4,
thus placing it into the low-mass star regime.
IRAS 04489+3042 is not extended in the near-IR or mm \citep{park02, motte01},
and no outflow has been detected \citep{gomez97}. \citet{park02} suggest it
could be in transition between Class I and II stage, probably seen close to pole-on.
Our preliminary envelope model fit for this object (Figure \ref{model_04489})
suggests a low density ($\rho_1 = 1.0 \times 10^{-14}$ g cm$^{-3}$),
a small centrifugal radius ($R_c$ = 15 AU), a very small cavity, and a low inclination.
Models by KCH93 yielded the same reference density as our model and a similar low
inclination angle, but a centrifugal radius of 70 AU. On the other hand, modeling efforts
by \citet{whitney97} resulted in larger values for all these three parameters.
We can exclude a centrifugal radius larger than about 25 AU, since it would decrease
the flux over the 10-60 $\mu$m spectral range. The low density and inclination angle
of our model are mainly set by the very weak silicate absorption feature and roughly
flat SED from about 5 to 15 $\mu$m.
The model does not yield a good fit of the details of the IRS spectrum below
about 15 $\mu$m, but the overall shape of the spectrum is reproduced reasonably
well. To fit the data point at 1.3 mm with the set of model parameters described above,
we adopted an outer envelope radius of only 1000 AU; the small envelope contributes
only little emission in the mm, where the emission from the disk dominates (see the
dash-dotted line in Figure \ref{model_04489}).
Since there are no sub-mm measurements available, and the IRS spectrum is relatively
flat with no strong silicate feature, the current model parameters are only weakly constrained.
\subsubsection{CoKu Tau/1}
Coku Tau/1 was identified as a Class II object by KH95, but it is surrounded by
filamentary reflection nebulae detected in the near-IR and suggested to represent
the walls of outflow cavities \citep{padgett99}. It is seen close
to edge-on, and, like IRAS 04248+2612, is a subarcsecond binary \citep{padgett99}.
Similar to IRAS 04248+2612 and 04264+2433, CoKu Tau/1 has a silicate emission
feature and an infrared SED that is rising from 3 to about 20 $\mu$m (see Figure
\ref{ClassI_SED}).
The appearance of CoKu Tau/1 in the near-IR suggests that it is surrounded by
a low-density envelope and an edge-on disk. It is likely a more evolved protostar,
at a stage when the envelope has begun dissipating.
The binary and the inner disk are extinguished by the outer disk, but since in the
near-IR the optical depth of the outer regions of the circumbinary disk decreases,
the two stars at the center are seen at near-IR wavelengths. As in the optical,
there is likely also some stellar light scattered by the envelope contributing to the
near-IR emission.
Our model of CoKu Tau/1 (see Figure \ref{model_CoKu_Tau1}) requires a low
density ($\rho_1=5.0 \times 10^{-15}$ g cm$^{-3}$), a small centrifugal
radius ($R_c$ = 40 AU), a cavity semi-opening angle of 5\degr, and a high
inclination angle ($i=80$\degr). We adopted a luminosity of 1.1 L$_{\odot}$,
which is slightly higher than the measured bolometric luminosity of 1.0 L$_{\odot}$
(see Table \ref{tab_lum_incl}). The inner disk radius was set at 7 stellar radii to
decrease the emission from the near-IR to 8 $\mu$m, which is still higher than the
observed fluxes, most likely due to an overestimate of the scattered light emission
from the envelope.
The shape of the IRS spectrum between 12 and 20 $\mu$m constrains the
centrifugal radius, while the low density is required to generate a silicate emission
feature and cause the SED to peak around 30 $\mu$m. Decreasing $\rho_1$
by about 20\% and increasing or decreasing $R_c$ by roughly the same percentage
yields comparable fits, provided the inclination angle lies in the 70\degr-80\degr\
range. A previous model by \citet{stark06} yielded a reference density smaller
by about a factor of three, a comparable values for $R_c$, but a somewhat smaller
inclination angle and a larger cavity opening angle. We can exclude a larger cavity,
since it would decrease the flux of the 10 $\mu$m silicate emission feature.
As mentioned earlier, the silicate emission feature at 10 $\mu$m is generated
by the envelope. The disk just contributes a weak silicate absorption feature,
which results from our simple disk model that includes only continuum emission,
extinguished by the envelope. A more realistic edge-on disk would yield a somewhat
stronger absorption feature; however, the envelope emission clearly dominates over
that of the disk at 10 $\mu$m, a fact that also applies to IRAS 04264+2433 and,
to a lesser extent, IRAS 04248+2612 (long-dashed gray lines in Figures
\ref{model_04248}, \ref{model_04264}, \ref{model_CoKu_Tau1}).
Thus, the silicate emission feature is not greatly affected by the shape of the silicate
feature originating from the disk, and our current models of low-density envelopes
have some validity.
\subsubsection{DG Tau B}
\label{DGTauB_model}
This object displays some similarity to HH 30; its near-IR HST images show
an edge-on disk and a bipolar reflection nebulosity (see Fig.\ \ref{DGTauB_slits}),
which was interpreted as the walls of an outflow cavity \citep{padgett99}.
In the optical, the nebula is visible, too, but the star is not detected
\citep{stapelfeldt97}. DG Tau B is also the source of a jet, which is oriented
perpendicular to the disk and along the axis of symmetry of the reflection nebula
\citep{eisloeffel98}. A molecular outflow, oriented along the redshifted optical
jet, has also been observed \citep{mitchell97}.
Our preliminary model for DG Tau B (see Figure \ref{model_DGTauB}) has an
inclination angle of 55\degr\ and a cavity semi-opening angle of 10\degr;
both values are smaller than expected from the optical and near-IR images.
However, a more highly inclined envelope would result in a steeply rising SED
over the mid-IR range, which would not match the observed spectrum.
The centrifugal radius (and thus outer disk radius) of 60 AU is poorly
constrained; a larger value would better fit the slope of the IRS spectrum beyond
20 $\mu$m, but result in a shallower silicate feature at 10 $\mu$m.
The deep silicate absorption feature requires a relatively high density ($\rho_1=3.5
\times 10^{-14}$ g cm$^{-3}$), but its long-wavelength wing is not fitted
very well. In addition, the model is not well-constrained at longer wavelengths, since
DG Tau B lacks any sub-mm measurements, and the {\it IRAS} fluxes at 60 and
100 $\mu$m are uncertain due to the proximity of DG Tau: the far-IR fluxes contain
the emission of both DG Tau B and DG Tau, which lies 1\arcmin\ to the
northeast of DG Tau B and is about two to three times brighter in the mid-IR.
Recent models by \citet{stark06} yielded quite different model parameters, except
for the reference density; they were able to fit the near-IR images of DG Tau B
adopting a luminosity of 0.2 L$_{\odot}$, a centrifugal radius of 300 AU, a $\rho_1$
value of $3.75 \times 10^{-14}$ g cm$^{-3}$, and cavity semi-opening and
inclination angles of 30\degr\ and 73\degr, respectively.
This set of parameters would not fit the IRS spectrum. In particular, we adopted a
luminosity of 2.5 L$_{\odot}$; a smaller luminosity, closer to the value derived from
integrating under the SED (1.8 L$_{\odot}$), would result in less far-IR flux and
require a smaller inclination angle to still reproduce the mid-infrared spectrum, which
would be inconsistent with the HST images, which indicate a more edge-on orientation.
\subsubsection{HL Tau}
Even though HL Tau is considered by some a classical T Tauri star surrounded
by an accretion disk, it is also embedded in an envelope \citep[e.g.,][]{motte01,
white04}; this is supported by our IRS spectrum, which shows the silicate feature
in absorption, as well as ice absorption features in the 5--8 $\mu$m range and
at 15.2 $\mu$m. HL Tau is also very bright at sub-mm and mm wavelengths,
and a rotating, elongated, $\sim$ 2000 AU long structure has been mapped in
$^{13}$CO \citep{beckwith90, sargent91}. \citet{stapelfeldt95} detected a
compact, one-sided reflection nebulosity in the optical; the star is not detected.
HL Tau drives a powerful, extended molecular outflow which was mapped in CO
emission \citep{monin96}. \citet{close97} observed HL Tau in the near-IR
with adaptive optics and resolved the inner accretion disk and bipolar outflow cavities.
They concluded that HL Tau is surrounded by an accretion disk of about 150 AU
radius, inclined at 67\degr, and an infalling envelope of about 1200 AU radius.
However, the environment of HL Tau is quite complicated, partly due to its proximity
to XZ Tau, which is likely at the origin of an expanding shell that compresses the
nebula around HL Tau \citep{welch00}.
The SED of HL Tau can be reproduced with an envelope model with a luminosity of
8.0 L$_{\odot}$, which is somewhat larger than its measured bolometric luminosity
(see Table \ref{tab_lum_incl}), a relatively high density ($\rho_1 = 4.5 \times
10^{-14}$ g cm$^{-3}$), an $R_c$ value of 100 AU, and flattening parameter
$\eta$ of 1.0, i.e., a sheet-collapse model (Figure \ref{model_HLTau}).
The latter quantity was adjusted to yield more flux in the 12 to 30 $\mu$m region
compared to the TSC models. A lower density would yield a 10 $\mu$m silicate
feature that is too narrow, while a density up to $6.0 \times 10^{-14}$ g cm$^{-3}$
would still yield a good fit. An even higher density would require a larger centrifugal radius,
which would result in an SED that overestimates the flux from about 25 to 200 $\mu$m.
The $R_c$ value is constrained by the shape of the IRS spectrum; we can rule out $R_c$
values smaller than 100 AU and larger than about 200 AU.
These results are also roughly consistent with previous modeling results by
\citet{calvet94}, who adopted a reference density of 3.2 $\times 10^{-14}$
g cm$^{-3}$ and a centrifugal radius of 200 AU.
We note that, despite the finding by \citet{close97} that the inclination
angle of HL Tau is 67\degr\ and that large cavities are present, our model
indicates virtually no cavity and a low inclination. The model by \citet{calvet94}
also suggests an outflow hole with a semi-opening angle of 10\degr; including
such a cavity in our model would reduce the flux over the IRS wavelength range,
but also increase the flux somewhat in the near-IR, where our model underestimates
the emission. A larger inclination angle would decrease the short-wavelength flux
even more and generate a deeper silicate absorption feature. On the other hand, our
value for the inclination angle is closer to the value of 30\degr\ determined by
\citet{calvet94}, and our disk radius ($\sim$ $R_c$) is also roughly consistent
with the value of 150 AU found by \citet{close97}.
Our model underestimates the emission at sub-mm and mm wavelengths, despite an
outer envelope radius of 10,000 AU. Models of HL Tau by \citet{dalessio97} have
shown that the outer disk requires higher temperatures to reproduce the SED shape
at longer wavelengths, implying irradiation of the circumstellar disk by the infalling
envelope. Since our model does not include this additional heating, our disk emission
is likely underestimated; thus, the ``missing'' flux at the longer wavelengths could be
attributed in part to this additional disk component. In addition, dust outside the
infall region could contribute to the mm emission.
\subsubsection{L1551 IRS 5}
This object is the most luminous Class I object in Taurus (L$_{bol}$ = 22-28
L$_{\odot}$; see Table \ref{tab_lum_incl}) and is therefore a well-studied
protostellar system. It consists of a 0{\farcs}3 binary, whose components
are each surrounded by a circumstellar disk, and of a circumbinary disk and a
flattened envelope \citep{looney97, rodriguez98}.
It is surrounded by an extended reflection nebula at optical and near-IR wavelengths
\citep{campbell88,tamura91,white00}, and it powers a highly collimated bipolar
molecular outflow \citep{snell85,moriarty92}. Even though no optical outburst
has been observed, L1551 IRS 5 is considered an FU Ori object, implying episodic
events of high disk accretion and outflow rates \citep{hartmann96a}.
The mid- and far-IR spectrum (2--200 $\mu$m) of this bright Class I object
was obtained by \citet{white00} using ISO; they detected gas-phase emission
lines and ice absorption features. We also detect strong ice absorption features
in our IRS spectrum, in addition to a very deep and broad silicate absorption feature
(see Figure \ref{ClassI_IRS}c).
L1551 IRS 5 has been modeled in detail by \citet{osorio03}, taking into account
emission from the circumstellar and circumbinary disks, as well as the envelope.
The model presented below does not include all the components of the system as
in \citet{osorio03} and should therefore be considered as preliminary.
L1551 IRS 5 is the most luminous object in our sample (adopted luminosity of
25~L$_{\odot}$). Its deep silicate absorption feature requires a high density
($\rho_1 = 7.0 \times 10^{-14}$ g cm$^{-3}$), and the steep SED beyond
15 $\mu$m a centrifugal radius of 100 AU (see Figure \ref{model_L1551}). An
$R_c$ value larger than about 150 AU would result in a decrease in flux between
15 and 30 $\mu$m and thus degrade the fit to the IRS spectrum. Our model indicates
that the cavity is small ($\theta=$\,5\degr), and the object is seen at an intermediate
inclination angle. Increasing $\rho_1$ would require a smaller inclination angle, but
a larger inclination is more in accordance with the observed outflows ($i$=65\degr;
\citealt{hogerheijde98}).
Compared to the models by KCH93, which suggested a reference density of
$\sim$ $10^{-13}$ g cm$^{-3}$, a centrifugal radius between 70 and 300 AU,
and an inclination angle between 30\degr\ and 60\degr, our model has a smaller,
but still high, reference density, an $R_c$ value towards the lower end of the KCH93
range, and an inclination angle in the middle of the range determined by KCH93.
\citet{whitney97} and, more recently, \citet{gramajo07}, modeled this object
with a reference density of only $3.75 \times 10^{-14}$ g cm$^{-3}$ ,
$R_c \sim$ 40 AU, $\theta$=20\degr, and $i \gtrsim$ 70\degr.
Our model parameters are different from the ones found by \citet{osorio03}, who
adopted an inclination angle of 50\degr, $\rho_1 = 4.0 \times 10^{-13}$ g cm$^{-3}$,
$R_c= 300$ AU, and a flattening parameter $\eta=2.5$ for the envelope. We could
obtain a comparable fit of the SED out to 30 $\mu$m by using $L=25$ L$_{\odot}$,
$\rho_1 = 2.5 \times 10^{-13}$ g cm$^{-3}$, $R_c= 100$ AU, $\eta=2.0$,
$\eta_{star}$=0.1, $\theta$=0.1\degr, and $i=$45\degr, but the far-IR emission
would be overestimated. On the other hand, the sub-mm and mm emission would
be better matched with the sheet-collapse model. Thus, the SED of L1551 IRS 5
can be reproduced by a TSC model, but the long-wavelength emission suggests the
presence of an additional, cold, likely elongated, dust component that is outside of
the infall region.
\section{Discussion}
\label{discussion}
\subsection{Identification of Class I Objects}
The mid-infrared spectra of Class I objects have quite different appearances; while some
differences are caused by individual envelope parameters like the density and size of the
infall region, the inclination angle also plays an important role. A Class I object viewed
pole-on through a cavity in the envelope can look like a Class II object with some
additional long-wavelength excess emission (as some FU Ori objects appear; see
\citealt{green06}), while an edge-on Class I object can have
the appearance of a Class 0 object. However, the envelope around a true protostar
will generate excess emission at mid-IR to mm wavelengths, resulting in a larger infrared
excess than expected from an accretion disk alone, and less excess when compared to
the large, cool envelopes around Class 0 objects that peak at sub-mm to mm wavelengths.
In addition, if seen within a narrow range of inclination angles, an edge-on Class II object
can appear similar to an embedded protostar \citep[see][]{dalessio99},
but it would lack additional excess emission caused by the envelope.
Observations at other wavelengths help in identifying the evolutionary state
of an object: as noted by \citet{park02}, true Class I sources are usually
extended in the near-IR and at sub-mm or mm wavelengths, the
former due to light scattered by dust in inner envelope regions, the latter due to
thermal emission by dust in the outer parts of the envelope. Furthermore, Class I
objects usually have molecular outflows or jets detected in the optical/near-IR and
especially at mm wavelengths.
Objects in transition between the Class I and Class II stage are expected to be
surrounded by less envelope material, in accordance with larger cavities in their
envelopes.
Of the 28 Class I objects presented in this paper, 7 were previously identified as Class II
objects in KH95: IRAS 04154+2823, 04158+2805, 04278+2253, CoKu Tau/1,
HL Tau, IC 2087 IR, and LkHa 358. While the SED shape of these objects might
indicate that they are T Tauri stars whose envelope has already mostly dissipated, the
presence of ice features in their mid-infrared spectrum, as well as the detection
of extended reflection nebulosities and outflows around some of these objects,
suggests that there is still some remnant envelope material.
IRAS 04154+2823, 04278+2253, 04489+3042, IC 2087 IR, and LkHa 358,
as well as some of the Class II objects presented in \citet{furlan06a}
(IRAS 04187+1927, DG Tau, DP Tau, FS Tau, HN Tau, T Tau, and XZ Tau),
could be objects in transition between the Class I and II stage,
when a protostar has almost completely cleared its envelope, but is
still surrounded by some nebulosity. In some cases, the ``evolved'' protostar
is still generating outflows, but due to envelope clearing it appears more and
more like a classical T Tauri star. In some other cases, like T Tau, HN Tau,
and XZ Tau, a binary consists of a T Tauri star and a more deeply embedded
source, which is likely an effect of the particular viewing geometry of the system:
either one component is more aligned with the outflow cavity and thus less
embedded, or the more extinguished component is actually a T Tauri star
oriented edge-on. Therefore, as stated earlier, the inclination angle is an
important factor in determining the appearance of Class I and II objects.
\subsection{Ices in the Envelopes of Class I Objects}
The ice and silicate absorption features detected in the IRS spectra are also a useful
tool to characterize Class I objects. The peak optical depth of these features reveals
their strength and can therefore be used as an indication for their origin.
To determine the optical depth of the CO$_2$ ice feature at 15.2 $\mu$m and that
of the silicate absorption feature around 10 $\mu$m, we first used a spline fit to
determine the underlying continuum. Then we derived the optical depth, $\tau$,
assuming $F_{obs} = F_c e^{-\tau}$, where $F_{obs}$ and $F_c$ are the
observed and the underlying continuum flux, respectively. The measured peak optical
depths of the CO$_2$ and silicate absorption features are shown in Figure
\ref{CO2_sil_taus}, together with previous measurements by \citet{gibb04} and
\citet{alexander03}, who determined the optical depths of silicate and ice absorption
features in ISO spectra of embedded, massive YSOs and low- and intermediate-mass
YSOs, respectively. For the \citet{alexander03} data, we used the same conversion
as in \citet{watson04} to convert the CO$_2$ equivalent widths to peak optical depths.
As already pointed out by \citet{watson04}, the low-mass Class I objects in
Taurus generally have large peak CO$_2$ optical depths, especially considering
that their peak silicate optical depth lies below $\sim$ 2 (the object with the
largest silicate and CO$_2$ optical depths is IRAS 04368+2557, a Class 0 object).
Most of the objects observed by \citet{alexander03} have similarly strong
CO$_2$ ice features for larger silicate optical depths, suggesting additional extinction
along the line of sight towards these objects.
It is expected that the two optical depths track each other along lines of sight through
molecular clouds (with some dispersion due to different conditions in different clouds),
since ice mantles are thought to grow in a similar manner onto dust grains in all
comparably shielded regions of molecular clouds.
The gray dashed line in Figure \ref{CO2_sil_taus} is drawn by eye to roughly
separate the region occupied by the Taurus Class I objects, where extinction from
the envelope dominates the optical depth, from that occupied by objects seen along
molecular cloud lines of sight, where the extinction is mostly due to intervening molecular
cloud material. The boundary is not well-defined, since some of the ISO observations
did include objects in which local extinction by an envelope dominates, but it separates
all but one of the Taurus Class I objects from the majority of the ISO targets.
Thus, we confirm the conclusions of \citet{watson04} that dust grains in protostellar
envelopes have larger ice mantles than grains in the ambient molecular cloud, implying
ice mantle growth inside the envelopes. We defer a more detailed study of the ice
absorption features, including the detailed composition and thus derived temperature
of the ices, to a future paper (Zasowski et al. 2008, in preparation).
We note that the objects with a peak silicate optical depth of 0 are protostars in which
the silicate feature is either in emission or displays both an emission and an absorption
component. As discussed in \S\ \ref{IRS_spectra}, the latter type of objects is
likely seen along lines of sight that suffer from less extinction by the envelope, caused
by a more face-on orientation or a lower-density envelope.
Therefore, for these objects the optical depth of the silicate feature, taken as a measure
of dust absorption in the envelope, is probably underestimated due to contamination
by the silicate emission arising from the disk.
The one Class I object that lies in the molecular cloud region of the plot is DG Tau/B;
it has a deep silicate, but only a weak CO$_2$ ice absorption feature. This could
indicate that grain mantles in its envelope are less rich in CO$_2$ ice, but on the
other hand this feature is superposed on the broad and deep silicate absorption at
18 $\mu$m, which adds additional uncertainty to the optical depth measurement
of the CO$_2$ feature.
The three objects identified as ``Class I/II objects'' in Figure \ref{CO2_sil_taus}
are IRAS 04278+2253, LkHa 358, and IC 2087 IR. The former two objects have
the lowest CO$_2$ feature strengths of our Taurus sample and virtually no silicate
absorption feature, which confirms that they are more evolved Class I objects.
IC 2087 IR has both CO$_2$ and silicate peak optical depths comparable to those
of typical Class I objects; thus, it is still likely embedded to some extent.
This shows that the optical depths of the silicate and ice absorption features can
help to recognize more evolved Class I objects, but the effects of the environment
and viewing angle also have to be taken into account.
\subsection{Summary of the Models}
Envelope models aid in the identification and characterization of Class I objects;
when models fit the SED from the near-IR to the mm, they reveal both the
large- and small-scale structure of a protostar, its accretion disk and its envelope.
The IRS spectrum, which covers the SED from 5 to 40 $\mu$m, is a particularly
useful constraint, since the depth of the silicate feature at 10 $\mu$m is connected
to the density and inclination angle of the envelope, while the slope between 15 and
40 $\mu$m helps to constrain the envelope shape, centrifugal radius and in part the
size of the cavity.
Not all parameters can be uniquely defined by a model fit
to the IRS spectrum, but together with shorter- and longer-wavelength data,
the parameter space can be narrowed down considerably.
We point out that, given the large number of parameters for each model, we
did not determine the goodness of a model fit by ${\chi}^2$-minimization,
but rather by judging by eye, based on our experience on how the different
parameters affect the resulting model SED.
However, we justify the choice of specific parameters
and explain how the parameter space was narrowed. In addition, the more
constraints a model has, in particular the availability of sub-mm and mm flux
measurements and the source's luminosity, the more limited the choice of
parameters becomes. The fitting by eye is much more efficient than
${\chi}^2$-minimization, and in most cases we derive good fits that are
likely minima in ${\chi}^2$ space.
Typical uncertainties for our model parameters are listed in the footnote
of Table \ref{tab_model_fits}.
We also note that several assumptions made in the modeling code could be
refined to better reproduce the observations: e.g., the assumption that the
cavity follows the shape of the streamlines of infalling particles, the assumption
of a spherically symmetric density distribution when computing the equilibrium
temperature in the envelope, and the adoption of only the inner, optically thick
regions of a flat accretion disk as the disk component.
Some of our 22 models of Class I objects should be treated as preliminary, given
the assumptions made in the modeling code and/or some weaker constraints
due to the lack of data, but our models already reveal a large range in parameters
necessary to obtain fits (see Table \ref{tab_model_fits} for a summary of
the model parameters).
We computed models for the majority of Class I objects in our sample; however,
we did not attempt to model objects with poor observational constraints for their
SEDs, or objects which are likely more evolved Class I objects and for which these
envelope models might not apply. Thus, we did not attempt to model HH 30,
whose spectrum is very noisy and whose overall SED is poorly constrained by
the scarcity of measurements at other wavelengths. Also IRAS 04181+2654 B,
a member of a binary system, was not modeled, since it lacks long-wavelength
data (we modeled component A, which was observed at 1.3 mm). The three
sources with decreasing SEDs over the IRS spectral range, IRAS 04278+2253,
IC 2087 IR, and LkHa 358, were also not modeled.
Finally, no model was produced for GV Tau, since its IRS spectrum is uncertain,
and the fact that the two components are separated by 1{\farcs}3 (which
corresponds to about 180 AU at the distance of the Taurus star-forming region)
likely introduces asymmetries in the envelope of the more embedded component.
Our envelope models indicate centrifugal radii, and thus disk sizes, that cover
a range from 10 AU up to 300 AU, with a median value of 60 AU. Since the
centrifugal radius is thought to increase rapidly with time ($\propto t^3$),
we could conclude that the objects in our sample span somewhat different
ages (about a factor of 3). However, the main reason for the differences in
centrifugal radii is likely different initial conditions of the collapse, since the
initial centrifugal radius strongly depends on the mass of the central condensation
and the initial angular velocity of the cloud core
\citep[$R_c \propto M^3 {\Omega}^2$;][]{terebey84}.
In addition, some of the more deeply embedded protostars that have been identified
as Class 0/I objects, like IRAS 04166+2706 and 04368+2557, have large $R_c$
values, which would not be expected if the spread in centrifugal radii was only due
to time evolution.
Since a disk size of a few 100 AU is comparable to the radii of accretion disks found
around classical T Tauri stars \citep[e.g.,][]{dutrey96}, our median value for
$R_c$ (and thus outer disk radii) of 60 AU suggests that the accretion disks
found around protostars will expand, as is expected due to angular momentum
transfer in the accretion process.
We note that the disk component dominates the flux in the 3-8 $\mu$m
range for most of the models; for about 2/3 of these objects the fraction of
the luminosity contributed by the star, $\eta_{star}$, is either 0.1 or 0.2, i.e.,
80-90\% of the luminosity is generated by accretion. About half of the objects that
have no dominant disk contribution in the 3-8 $\mu$m range have luminosities
dominated by the star ($\eta_{star}$=0.8), and the other half have smaller
values for $\eta_{star}$, but generally large inclination angles. Thus, the
Class I objects in our sample whose luminosity is dominated by accretion (and
which likely have large mass accretion rates through their disks) generate SEDs
that are dominated by the disk in the 3-8 $\mu$m range, except if the disk
emission is diminished due to a high inclination angle.
The reference density $\rho_1$ varies from $4.0 \times 10^{-15}$ g
cm$^{-3}$ to $7.0 \times 10^{-14}$ g cm$^{-3}$, with a median value
of $3.0 \times 10^{-14}$ g cm$^{-3}$, very similar to the value found by
KCH93.
$\rho_1$ can be used to estimate the mass infall rate from the envelope onto
the disk by applying Equation
\ref{rho1_equ}; for a 0.5 M$_{\odot}$ object, the mass infall rates for the
objects in our Class I sample lie between $5.3 \times 10^{-7}$ and
$9.3 \times 10^{-6}$ M$_{\odot}$ yr$^{-1}$, about two orders
of magnitude larger than the accretion rates of classical T Tauri stars \citep[e.g.,][]
{gullbring98}. This would confirm previous results that Class I objects are at
an earlier evolutionary stage than T Tauri stars and still accreting a substantial
fraction of the final stellar mass \citep{muzerolle98}.
Even though the cavity semi-opening angle and, in some cases, the inclination
angle are usually less well-constrained than the reference density and centrifugal
radius, we observe that our models indicate a large range of inclination
angles, covering values from 15\degr\ to 89\degr\ (with a median
value of 55\degr), and a smaller range of cavity semi-opening angles
(0.1\degr\ to 27\degr, with a median of 6\degr).
In Table \ref{tab_lum_incl} we list the luminosities of the Class I objects
in our sample and their inclination angles. Both the values of bolometric luminosity
from the literature and those we derived by integrating under the SED are
actually apparent luminosities, since envelopes are not spherically
symmetric. Thus, we would expect the apparent bolometric luminosity to be larger
than the system luminosity (which enters as a model parameter) at low inclination
angles, while at high inclination angles the opposite should be the case.
The data in Table \ref{tab_lum_incl} shows that this trend is only weak for our
sample of Class I objects; in general, our model luminosities are higher than the
apparent bolometric luminosities.
This could suggest that we overestimated the luminosity of some of our
lower-inclination sources.
We note that most of our models do not require an initially flattened density
distribution ($\eta$ $\gtrsim$ 1); all but 5 of the 22 objects we modeled
can be fitted by TSC models, which assume an initially spherically symmetric
density distribution. However, we cannot exclude that sheet-collapse model
fits could also be found for the remaining 17 Class I objects for which only
TSC models have been considered so far. Thus, as a starting point for models,
it seems that star-forming cores in Taurus can be described by the approximation
of collapsing, singular isothermal spheres.
In a few cases we found that an additional component of cold dust seems to be
present, since certain models are able to reproduce the
sub-mm and mm fluxes measured in smaller apertures (10\arcsec-20\arcsec),
but not the measurements at similar wavelengths derived from large apertures
(40\arcsec-60\arcsec). In particular the 1.3 mm data points from \citet{motte01},
which correspond to fluxes integrated over a beam 60\arcsec\ in diameter,
are often higher than the model prediction.
As suggested by \citet{jayawardhana01}, cold dust of roughly constant density
outside of the infall region could contribute to the long-wavelength emission.
In these cases envelope models could aid in the distinction between emission from
infalling envelope material and emission from the surrounding dense cloud.
When comparing our modeling results with previous efforts, we note that our
results often differ considerably. However, in this work we have the additional
constraints through our IRS spectra, which, when combined with near-IR and
far-IR to mm data, allows us to determine the 9 parameters entering in the
envelope models. In particular the 10 $\mu$m silicate absorption feature
allows us to narrow down the parameter space. A next step in modeling
would involve reproducing the images of the Class I sources at the same
time as the SEDs to better take into account the envelope geometry and
asymmetries.
\section{Conclusions}
\label{conclusions}
After analyzing the IRS spectra and SEDs of 28 Class I objects in the Taurus
star-forming region, and generating envelope models for 22 of these objects,
we conclude the following:
${\bullet}$ Almost all protostars we observed display ice absorption features
in their mid-IR spectra, which originate in the envelopes around these young
stars. In particular, the CO$_2$ ice feature at 15.2 $\mu$m is ubiquitous
and strong among the Class I objects in our sample, signifying an origin in the
cold outer envelope regions.
The envelope is also generally responsible for a deep silicate absorption feature
at 10 $\mu$m.
The three objects with silicate emission features in their spectra, IRAS 04248+2612,
04264+2433, and CoKu Tau/1 are likely Class I objects seen through low-density
envelopes, which are in an advanced stage of dispersal and thus considered to be
more evolved.
${\bullet}$ The SEDs of Class I objects peak in the mid- to far-IR, where
the emission is dominated by the envelope. Objects with flat SEDs in the
infrared spectral range could be in transition between the Class I and II stage.
The inclination angle of protostellar systems strongly affects the appearance
of their mid-IR SED; Class I objects seen pole-on could appear as Class II
objects with additional long-wavelength excess emission, while edge-on
Class I objects have SED shapes similar to Class 0 objects, but with less
flux in the sub-mm and mm wavelength region.
${\bullet}$ The IRS spectra are instrumental in constraining envelope model
parameters, in particular the reference density $\rho_1$ (which is linked to the
mass infall rate) and centrifugal radius $R_c$. The depth of the silicate feature
and the shape of the spectrum between 12 and 30 $\mu$m provide constraints
which can be further tightened by model fits to near-IR, far-IR, sub-mm, and
mm observations. The median values for $\rho_1$ and $R_c$ for the Class I
objects we modeled are $3 \times 10^{-14}$ g cm$^{-3}$ and 60 AU,
respectively, with a relatively large range for both parameters.
The disk contribution dominates the emission in the 3-8 $\mu$m range for
most of the modeled Class I objects, confirming the importance of disk accretion
luminosity at this early evolutionary stage.
Most SEDs can be reproduced by TSC models, thus not requiring an initially
flattened density distribution of the collapsing cloud core; only 5 of the 22
objects we modeled needed sheet-collapse models for an adequate fit of
their SEDs.
The IRS spectra of Class I objects in Taurus display a variety in appearances
and add valuable constraints to envelope models. Fits to the
observed SEDs yield the parameters describing their physical structure,
and thus aid in confirming and refining our understanding of protostellar
cloud collapse and evolution.
\acknowledgments
Part of this work was carried out while the first author was supported by a
NASA Postdoctoral Program Fellowship at the NASA Astrobiology Institute at
the University of California, Los Angeles, administered by Oak Ridge Associated
Universities through a contract with NASA.
This work is based on observations made with the {\it Spitzer Space Telescope},
which is operated by the Jet Propulsion Laboratory, California Institute of Technology,
under NASA contract 1407. Support for this work was provided by NASA through
contract number 1257184 issued by JPL/Caltech. N.C. and L.H. acknowledge support
from NASA Origins grants NAG5-13210 and NAG5-9670, and STScI grant AR-09524.01-A.
P.D. acknowledges grants from PAPIIT, UNAM, and CONACyT, Mexico.
This publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the Infrared Processing
and Analysis Center/California Institute of Technology, funded by the National
Aeronautics and Space Administration and the National Science Foundation. It has also
made use of the SIMBAD and VizieR databases, operated at CDS (Strasbourg, France),
NASA's Astrophysics Data System Abstract Service, and of the NASA/ IPAC Infrared
Science Archive operated by JPL, California Institute of Technology (Caltech), under
contract with NASA.
Facilities: \facility{Spitzer(IRS)}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,963
|
Home More Filmy World Katy Perry posts Goddess Kali pic on Instagram, trollers can't keep calm!
Popular international singer Katy Perry has hogged the limelight but this time it's not for some fun news rather a controversial one. Katy, who has time and again shown her liking towards Indian culture got into trouble for posting Indian goddess Kali's picture on Instagram. Katy Perry has been slammed for sharing a picture of Hindu goddess Kali on Instagram, with many calling it disrespectful.
Music sensation Katy took to photo-sharing site Instagram and posted a Goddess Kali picture with a caption: "current mood". However, what followed was a huge debate among users.
"Don't disrespect Indian Goddess… this can't be called as mood.. have some values. Just delete this picture. (No hate for her)," one person wrote.
Another said it was not a good idea for the singer to use a religious image to represent her mood.
"This is a little offensive to the Hindu community because the meaning behind this photo and god symbolises something different. Not sure what you intended it for, but its supposed to symbolise strength, and courage," an Instagram user posted.
People trolled her for posting an Indian Goddess' picture, shaming her for doing so. Interestingly, Katy got married to stand-up comedian Russell Brand in Rajasthan in 2010. However, the couple split soon after that.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,282
|
\section{Excited baryon states in the history of the Universe}
\label{intro}
For this talk the organizers asked me to address what we learn about strong QCD (sQCD), i.e. QCD in the domain
where perturbative methods fail in describing nucleon resonances transitions.
Talking about a similar topic from the theory perspective,
Nathan Isgur said this in the concluding talk at $N*2000$: "{\it I am convinced that completing this chapter in
the history of science will be one of the most interesting an fruitful areas of physics for at least the next thirty
years.}" We are now 19 years into this 30 years prediction, and the physics of N*'s continues to go strong,
while many related issues remain to be explored.
As we are trying to make progress in the complex world of physical sciences, we should not lose sight of
what physics is all about: understanding the origin and the history of our universe, and the laws underlying
the observations. In this meeting we also address how excited states of the nucleon fit in to our understanding of the
forces and the dynamics of matter in the history of the universe. On the internet we
find beautiful representations of the phases through which the universe evolved from the
Big Bang (BB) to our times as shown in Fig.~\ref{universe}.
\begin{figure*}[h]
\centerline {
\includegraphics[width=4.5cm,height=7.0cm]{cp_universe_chronology_large.pdf}
\hspace{0.5cm}\includegraphics[width=6.5cm,height=7.0cm]{QCD_phase_transition.png}}
\caption{Left panel: The evolution of the Universe.
The line denoted as {\it Quark-hadron transition}, is where protons and neutrons are formed.
Existing electron accelerators as CEBAF, ELSA, and MAMI,
and colliders as BES III have sufficient energy reach to access this region and study
processes in isolation that occurred during this transition in the microsecond old universe and resulted in the freeze
out of baryons. Right panel: A generic phase diagram for the transition from the de-confined quark-gluon
state to the confined hadron state.}
\label{universe}
\end{figure*}
There are some marked events that have been of particular significance during the early phases of the
its history, such as the quark-gluon plasma of non-interacting colored quarks and gluons, and the
forming of protons and neutrons. During this transition dramatic events occur -
chiral symmetry is broken, quarks acquire mass dynamically, baryon
resonances occur abundantly, and colored quarks and gluons are confined. This crossover
process is governed by the excited hadrons, this is schematically
shown in the generic QCD phase diagram in Fig.~\ref{universe}. In this process
strong QCD (sQCD) is born as the process describing the interaction of colored quarks and gluons. These are the
phenomena that we are exploring with electron and hadron accelerators - the full
discovery of the baryon (and meson) spectrum, the role of chiral symmetry breaking and the
generation of dynamical quark mass in confinement. While we can not recreate the exact condition in
the laboratory, with existing accelerators we can explore these processes in isolation. With electron machines
and high energy photon beams in the few GeV energy range we search for undiscovered excitations of
nucleons and other baryons.
As the Universe expands and cools down the coupling of quarks to the gluon field becomes stronger and
quarks become more massive and form excited states in abundance. This eventually gives
way to the forming stable nucleons. In the heavy-quark sector It has been
demonstrated~\cite{Bazavov:2014xya,Bazavov:2014yba} that the entire complement of
excited states as predicted in the quark model~\cite{Capstick:1986bm} is needed to be included
in these calculation to
explain what is observed in "hot QCD" lattice calculations. Including only resonances from
the Review of Particle Physics (RPP) is insufficient to explain the computations within hot QCD.
Similar projections have been made in the light-quark sector that we will discuss
in the second part of this report. The close relationship of the baryon resonance spectrum and the
evolution of the early universe makes the experimental search for the "missing resonances" an even
more compelling experimental program.
\begin{figure*}[t]
\centering
\resizebox{2.0\columnwidth}{!}{\includegraphics{KLambda-crs_back.pdf}}
\caption{Invariant mass dependence of the $\gamma p \to K^+\Lambda$ differential cross section in the
backward polar angle range. There are 3 structures visible that may indicate resonance excitations,
at 1.7, 1.9, and 2.2 GeV. The blue full circles are based on the topology $K^+p\pi^-$,
the red open triangles
are based on topology $K^+p$ or $K^+\pi^-$, which extended coverage towards lower W at backward angles
and allows better access to the resonant structure near threshold. }
\label{KLambda-crs}
\end{figure*}
\section{Search for missing baryon states}
Vigorous spectroscopy programs are currently underway at various particle accelerators
in the quest for undiscovered excited mesons and baryons.
Experiments at electron machines such as CEBAF at JLAB in the US, ELSA at Bonn University in Germany,
and MAMI at the Johannes Gutenberg University at Mainz in Germany,
focus on the s-channel excitation of protons and neutrons to $\rm N^*$ and $\rm \Delta^*$ states.
The excited states of the nucleon have been studied experimentally since the 1950's~\cite{Anderson:1952nw}.
They contributed to the discovery of the quark model in 1964 by Gell-Mann and Zweig~\cite{GellMann1964,Zweig1964},
and were critical for the discovery of "color" degrees of freedom as first introduced by Greenberg~\cite{Greenberg:1964pe}.
The 3-quark quark structure of baryons resulted in the prediction of a wealth of excited states
with underlying spin-flavor and orbital symmetry of $SU(6) \otimes O(3)$. The predictions led to
a broad experimental effort to search for these states. Of the many
states predicted in the quark model, only a fraction have been observed, even today.
Searches for the "missing" states and detailed studies of the resonance structure
are now mostly carried out using electromagnetic probes and have been a major focus of
hadron physics for the past two decades \cite{Burkert:2004sk}. A broad
experimental effort has been underway for the past two decades, with measurements
of exclusive meson photoproduction and electroproduction reactions, including
many polarization observables. Precision data and the development of multi-channel
partial wave analysis procedures have resulted in the discovery of a series of excited states of the
nucleon in the mass range of 1.9 to 2.2 GeV, others have been upgraded in their status of likelihood
of existence as entered in the bi-annual Reviews of Particle Physics~\cite{Beringer:1900zz,Agashe:2014kda,Patrignani:2016xqp,Tanabashi:2018oca}.
We will discuss some of these states in the following sections.
\begin{figure*}[t]
\centering
\resizebox{2.0\columnwidth}{!}{\includegraphics{KSigma-crs_back.pdf}}
\caption{Invariant mass dependence of the $\gamma p \to K^+\Sigma^\circ$ differential cross section in the backward
polar angle range. }
\label{KSigma-crs}
\end{figure*}
Accounting for the complete excitation spectrum of the nucleon (protons and neutrons)
and understanding the effective degrees of freedom is perhaps the most important and
certainly the most challenging task of hadron physics. The experimental N* program currently
focusses on the search for new excited states in the mass range above 2 GeV
using energy-tagged photon beams in the few GeV range, and the study of the internal
structure of prominent resonances in meson electroproduction.
A quantitative description of baryon spectroscopy and the structure of excited
nucleons must eventually involve solving QCD for a complex strongly interacting
multi-particle system.
Recent advances in Lattice QCD led to predictions of the nucleon spectrum in QCD with
dynamical quarks~\cite{Dudek:2012ag}, albeit with still large pion
masses of 396 MeV. Lattice prediction can therefore only be taken as indicative of the
quantum numbers of excited states and not of the masses of specific states. In parallel,
the development of dynamical coupled channel models is being pursued with new vigor.
The EBAC group at JLab as well as others have shown~\cite{Suzuki:2009nj} that dynamical effects can result
in significant mass shifts of the excited states. As a particularly striking result, a very large
shift was found for the Roper resonance pole mass to 1365 MeV downward from its bare
mass of 1736 MeV. This result has clarified the longstanding puzzle of the incorrect
mass ordering of $N(1440){1\over 2}^+$ and $N(1535){1\over 2}^-$ resonances in the constituent
quark model. Developments on the phenomenological side go hand in hand with a
world-wide experimental effort to produce high precision data in many different channel
as a basis for a determination of the light-quark baryon
resonance spectrum. On the example of experimental results from CLAS, the strong impact
of precise meson photoproduction data is discussed.
Several reviews have recently been published on this and related
subjects~\cite{Klempt:2009pi,Tiator:2011pw,Aznauryan:2011qj,Aznauryan:2012ba,Crede:2013sze} where
many details can be found.
\section{Establishing the N* Spectrum}
\label{sec:1}
The complex structure of the light-quark (u \& d quarks) baryon excitation spectrum complicates the experimental
search for individual states. As a result of the strong interaction, resonances are wide, often 200 MeV to 300 MeV,
and are difficult to uniquely identify when only differential cross sections are measured. Most of the excited
nucleon states listed in the Review
of Particle Physics (RPP) prior to 2012 have been observed in elastic pion scattering
$\pi N \to \pi N$. However there are important limitations in the sensitivity to the
higher mass nucleon states that may have very small $\Gamma_{\pi N}$ decay widths, and
the extraction of resonance contributions then becomes exceedingly difficult in this channel.
Estimates for alternative decay channels have
been made in quark model calculations\cite{Capstick:1993kb} for various channels. This has
led to a major experimental effort at Jefferson Lab, ELSA, GRAAL, and MAMI
to chart differential cross sections and polarization observables for a variety of meson
photoproduction channels. At JLab with CLAS, many different final states have
been measured with high precision on the proton~\cite{Dugger:2005my,Dugger:2009pn,Mattione:2017fxc,Senderovich:2015lek,Williams:2009yj,Collins:2017sgu,Williams:2009aa,Williams:2009ab,Bradford:2006ba,Bradford:2005pt,Ho:2017kca,McCracken:2009ra,Paterson:2016vmc,Strauch:2015zob,Dey:2010hh,McNabb:2003nf,Golovatch:2018hjk}, many of them are now employed
in single- and in multi-channel analyses~\cite{Anisovich:2017pox}. Recently, the first measurements of open strangeness processes on neutron targets have been published~\cite{Compton:2017xkt,Ho:2018riy}.
\begin{figure}[t]
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{New-states.png}}
\caption{Evidence for 11 N and $\Delta$ states in RPP 2010 compared with RPP 2018~\cite{Tanabashi:2018oca}.}
\label{pdg2014}
\end{figure}
\begin{figure}[ht]
\resizebox{1.0\columnwidth}{!}{\includegraphics{N-D-2010.png}}
\resizebox{1.0\columnwidth}{!}{\includegraphics{N-D-2018.png}}
\caption{Top panel: Nucleon and $\Delta$ resonance spectrum below 2.2 GeV in RPP 2010~\cite{Beringer:1900zz}.
Bottom panel: Nucleon and $\Delta$ resonance spectrum below 2.2 GeV in RPP 2018~\cite{Tanabashi:2018oca}.
The green frames highlight the new states and states with improved start ratings compared to 2010. The light brown color
indicate 3* states, the dark color indicates 4* states.
The dashed frames indicate apparent mass degeneracy of states with masses near 1.7 GeV and 1.9 GeV and
different spin and parity. The Bonn-Gatchina analysis includes all the $K^+\Lambda$ and $K^+\Sigma^\circ$ cross section and
polarization data, as well as pion photoproduction data.}
\label{pdg}
\end{figure}
\begin{figure}[ht]
\resizebox{1.0\columnwidth}{!}{\includegraphics{LQCD-states.png}}
\caption{Nucleon resonance spectrum below 2.2 GeV from LQCD ~\cite{Edwards:2011jj}.
The new discovered Nucleon states from RPP 2018~\cite{Tanabashi:2018oca} are indicated by the red and blue ellipses with
their spin-parity assignments. The observed masses deviate from the predicted ones as the lattice calculation used 396MeV pion mass,
and no multi channel coupling is included.
The green ellipses are well-known states that come out in the LQCD work with higher masses.}
\label{lqcd}
\end{figure}
\subsection{New states from open strangeness photoproduction}
\label{KLambda}
Here one focus has been on measurements of $\gamma p \to K^+\Lambda$ and $\gamma p \to K^+\Sigma$. Using
a linearly polarized photon beam several polarization observables can be measured by analyzing the
parity violating decay of the recoil $\Lambda \to p \pi^-$. It is well known that the energy-dependence of
a partial-wave amplitude for one particular channel is influenced by other reaction
channels due to unitarity constraints. To fully describe the energy-dependence
of a production amplitude other reaction channels must be included in a coupled-channel approach.
Such analyses have been developed by the Bonn-Gatchina group\cite{Anisovich:2011fc},
at JLab\cite{JuliaDiaz:2007kz}, at J\"ulich\cite{Ronchen:2014cna} and other groups. More recent
measurements of single and double polarization data in the same channel~\cite{Paterson:2016vmc} are
shown in Fig.~\ref{Lambda-asymmetries}.
The data sets with the highest impact on resonance amplitudes in the mass range above 1.7~GeV have been
kaon-hyperon production using a spin-polarized photon beam and the polarization of the $\Lambda$ or $\Sigma^\circ$
is also measured by analyzing the parity violating decay $\Lambda \to p \pi^-$. The high precision cross section and
polarization data~\cite{Bradford:2006ba,Bradford:2005pt,McCracken:2009ra,Dey:2010hh,McNabb:2003nf} provide
nearly full polar angle coverage and span the $K^+\Lambda$ invariant mass range
from threshold to 2.9 GeV, hence covering the full nucleon resonance domain where new states could be discovered.
The backward angle $K^+\Lambda$ data in Fig.\ref{KLambda-crs} show clear resonance-like structures
at 1.7 GeV and 1.9 GeV that are particularly prominent and well-separated from other structures at backward angles, while
at more forward angles (not shown) t-channel processes become prominent and dominate the cross section.
The broad enhancement at 2.2~GeV may also indicate resonant behavior although it is less visible at more
central angles with larger background contributions.
The $K^+\Sigma$ channel also indicates significant resonant behavior as seen in Fig.~\ref{KSigma-crs}.
The peak structure at 1.9 GeV is
present at all angles with a maximum strength near 90 degrees, consistent with the behavior of a $J^P= {3\over 2}^+$
p-wave. Other structures near 2.2 to 2.3~GeV are also visible.
Still, only a full partial wave analysis can determine the underlying resonances, their masses and spin-parity.
The task is somewhat easier for the $K\Lambda$ channel, as the iso-scalar nature of the $\Lambda$ selects
isospin-${1\over 2}$ states to contribute to the $K\Lambda$
final state, while both isospin-${1\over 2}$ and isospin-${3\over 2}$ states can contribute to the $K\Sigma$ final state.
These cross section data together with the $\Lambda$ and $\Sigma$ recoil polarization and polarization transfer data
to the $\Lambda$ and $\Sigma$ had strong impact on the discovery of several new nucleon
states~\cite{Anisovich:2017bsk}. They also provided new evidence for several candidate states that had been observed previously but lacked
confirmation as shown in Fig.~\ref{pdg2014}. It is interesting to observe that four of the observed nucleon states have nearly
degenerate masses near 1.9~GeV, as seen in Fig.~\ref{pdg}.
Similarly, the new $\Delta$ state appears to complete a mass degenerate multiplet near 1.9~GeV as well. There is no obvious
mechanism for this apparent degeneracy. Nonetheless, all new states may be accommodated within the symmetric
constituent quark model based on $SU(6)\otimes O(3)$ symmetry group as far as quantum numbers are concerned. As discussed
in section~\ref{intro} for the case of the Roper resonance $N(1440){1\over 2}^+$, the masses of all pure quark model states
need to be corrected for dynamical coupled channel effects to compare them with observed resonances. The same applies to
the Lattice QCD predictions~\cite{Edwards:2011jj} for the nucleon and Delta spectrum.
\begin{figure}[t!]
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{Baryon-chemical-pot.png}}
\caption{The ratio of baryon chemical potential of strangeness versus all baryons for the RPP 2014 and RPP 2016. The hashed
grey area show the LQCD calculation in "hot QCD". The straight lines are calculations within a hadron gas model. The 2016 line,
which includes more N/$\Delta$ baryon states, moves closer to the LQCD area. Note that only 3* and 4* states are included.
If the newly discovered states in RPP 2018 (seven states that are now at 3* or 4* status) were included this line would be moving even closer to the LQCD area. }
\label{chemical-potential}
\end{figure}
Coming back to the evolution of the early universe, we may check what is the expected effect of these newly discovered
states on the evolution of the universe near the cross over temperature
$T_c \approx 155$~MeV? Figure~\ref{chemical-potential} shows the ratio of the baryon chemical potential of strangeness
carrying baryons over all baryons versus temperature near the crossover temperature between de-confined and confined
and conditions. The graph clearly shows the significant impact of the recently included new baryons by the PDG in the 2016
edition of the RPP in comparison to the 2012 edition. Furthermore, adding new states found since 2016 and now included in
the 2018 edition of RPP will bring the HRG model closer to the Lattice QCD calculation~\cite{Chatterjee:2017yhp}, demonstrating
the strong impact
excited nucleon states have in the transition from the phase of free colored quarks and gluons to quarks and gluons confined in
protons and neutrons.~\footnote{Increasing the number of excited baryons will lower ${\rm \mu_B}$ at a given temperature, and hence increase the ratio of $\rm{\mu_S \over \mu_B}$}
\begin{figure*}[ht]
\resizebox{2.0\columnwidth}{!}{\includegraphics{Lambda_asymmetries.png}}
\caption{Beam and Target polarization asymmetries (left panel), and beam-recoil polarization asymmetries (right panel) measured
by the CLAS collaboration in the $\gamma p \to K^+\Lambda$ channel. The projections from earlier analyses. shown in the
red and green bands, show large discrepancies with the data in the higher W range. A refit by the BnGa group (blue band) shows that
the discrepancies do not require new excited states but could be accommodated by just adjusting some parameters. }
\label{Lambda-asymmetries}
\end{figure*}
\begin{figure}[ht]
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{omega_fit.pdf}}
\caption{Phase motion of the partial wave fit to the $\gamma p \to p \omega$ differential cross section and spin density matrix elements. 3 resonant
states, the subthreshold resonance $N(1680){5\over 2}^+$, $N(2190){7\over 2}^-$, and the missing $N(2000){5\over 2}^+$ are needed to fit the data (solid line). Fits without $N(2000){5\over 2}^+$ (dashed-dotted line), or without $N(1680){5\over 2}^+$ (dashed line) cannot reproduce the data.}
\label{omega}
\end{figure}
\begin{figure}[t!]
\resizebox{1.0\columnwidth}{!}{\includegraphics{phi-p.pdf}}
\caption{Differential cross sections in a nearly full angular range for $\gamma p \to p \phi$ production. }
\label{phi}
\end{figure}
\begin{figure}[t!]
\vspace{-0.8cm}\hspace{-1.2cm}\resizebox{1.3\columnwidth}{!}{\includegraphics{phi_forward.pdf}}
\caption{Differential cross sections of $\gamma p \to p \phi$ production for the most forward angle bin. The two curves
refer to fits without (dashed) and with (dotted) a known resonance at 2.08 GeV included. }
\label{phi_forward}
\end{figure}
\subsection{Vectormeson photoproduction}
\label{Vectormeson}
Double-polarization measurements of single pion photoproduction data~\cite{Strauch:2015zob} have contributed significantly to the discovery of the high-mass $\Delta(2200)7/2^-$ state~\cite{Anisovich:2015gia}, although the state couples just at the 3.5\% level to
N$\pi$
Nevertheless in the mass range above 2.0~GeV resonances tend to decouple from simple final states like $N\pi$, $N\eta$, and
$K\Lambda$, the search for undiscovered high-mass states requires to consider more complex final states
with multi-mesons $N\pi\pi$ or vector mesons, such
as $N\omega$, $N\phi$, and $K^*\Sigma$. The study of such final states adds significant complexity as many more amplitudes can
contribute to these photoproduction processes compared to single pseudo-scalar meson production. As is the case for $N\eta$ production,
the $N\omega$ channel is selective to isospin $1\over 2$ nucleon states only.
The CLAS collaboration has collected a tremendous amount of data in the $p\omega$ channel~\cite{Williams:2009aa,Williams:2009ab}, including
single and double polarization measurements~\cite{Collins:2017vev,Akbar:2017uuk,Roy:2017qwv,Roy:2018tjs,Anisovich:2017rpe}. The CLAS collaboration performed a single channel
event-based analysis, whose results are shown in Fig.~\ref{omega}, and provided further evidence for the $N(2000){5\over 2}^+$. Also
a large amount of $p\phi$~\cite{Seraydaryan:2013ija,Dey:2014tfa} final states on differential cross sections and spin-density matrix elements have been published, although they have not systematically included in the more complex multi-channel analyses.
Photoproduction of $\phi$ mesons is also considered a potential source of new excited nucleon states in the mass range above 2 GeV.
Differential cross sections and spin-density matrix elements
have been measured for $\gamma p \to p \phi$ in a mass range up to nearly 3 GeV. In Fig.~\ref{phi} structures are seen near 2.2~GeV in the forward most angle bins and at very backward angles for both decay channels $\phi \to K^+K^-$ and
$\phi \to K_l^0K_s^0$, and with the exception of the smallest forward angle bin the structures are more prominent at backward angles.
Only a multi-channel partial wave analysis will be able to pull out any significant resonance strength.
Fig.~\ref{phi_forward} shows the differential cross section
$d\sigma /dt$
of the most forward angle bin. A broad structure at 2.2 GeV is present, but does not show the typical Breit-Wigner behavior of a single
resonance. It also does
not fit the data in a larger angle range, which indicates that contributions other than genuine resonances may be significant.
The forward and backward angle structures
may also hint at the presence of dynamical effects possibly due to molecular contributions such as diquark-anti-triquark
contributions~\cite{Lebed:2015dca}, the strangeness equivalent to the recently observed hidden charm $P_c^+$ states.
\begin{figure}[t!]
\centerline{\resizebox{1.0\columnwidth}{!}{\includegraphics{SU6.png}}}
\caption{Schematics of $SU(6)\otimes O(3)$ supermultiplets with some of the excited states that have been explored in
$ep\to e^\prime \pi^+ n$ ,$ep\to e^\prime p \pi^\circ$, $ep\to e^\prime p \eta$, and $ep \to e^\prime p \pi^+ \pi^-$
electroproduction experiments. The inset shows the helicity amplitudes and electromagnetic multipoles often used to
describe the data. Only the states highlighted in red are discussed here.}
\label{SU6}
\end{figure}
Another process that has promise in the search for new excited baryon states, including those with
isospin-${3\over 2}$ is $\gamma p \to K^*\Sigma$~\cite{sarantsev2015}.
In distinction to the vector mesons discussed above, diffractive processes do not play a role in this channel, which then
may allow better direct access to s-channel resonance production.
\begin{figure}[t!]
\centerline{\resizebox{1.0\columnwidth}{!}{\includegraphics{Delta_gm_ash.pdf}}}
\caption{(Color online) The magnetic N$\Delta$ transition form factor normalized to the dipole form.factor compared with the LF RQM with running
quark mass, and with DSE/QCD. Both are close to the data at high $Q^2$. At $Q^2 < 3$GeV$^2$ meson-baryon contributions are significant. }
\label{Delta-Gmn}
\vspace{0.5cm}
\centerline{\resizebox{1.0\columnwidth}{!}{\includegraphics{Delta-ratios.png}}}
\caption{(Color online) The ratios $R_{EM}$ and $R_{SM}$ for the $N\Delta(1232)$ transition.
The solid red curves are the LF RQM predictions, the green curves are from the DSE approach. References to data as in Fig.~\ref{Delta-Gmn}.}
\label{Delta-ratios}
\end{figure}
\section{Structure of excited nucleons}
\label{structure}
Meson photoproduction has become an essential tool in the search for new excited baryons. The exploration of the internal structure
of excited states and the effective degrees of freedom contributing to s-channel resonance excitation requires the use of electron beams,
where the virtuality ($Q^2$) of the exchanged photon can be varied to probe the spatial structure.
Electroproduction of final states with pseudoscalar mesons
(e.g. $N\pi$, $p\eta$, $K\Lambda$) have been employed with CLAS, leading to new insights into the scale dependence of
effective degrees of freedom, e.g. meson-baryon, constituent quark, and dressed quark contributions. Several excited states, shown
in Fig.~\ref{SU6} assigned to their primary $SU(6) \otimes O(3)$ supermultiplets have been studied.
The $N\Delta(1232){3\over 2}^+$
transition is now well measured in a large range of $Q^2$~\cite{Joo:2001tw,Ungaro:2006df,Frolov:1998pw}. Results on the
magnetic transition form factor $\rm G_{Mn}$ and on the quadrupole transition ratios are shown in Fig.~\ref{Delta-Gmn} and
Fig.~\ref{Delta-ratios}. Two of the prominent higher mass states, the Roper resonance
$N(1440){1\over 2}^+$ and $N(1535){1\over 2}^-$ are shown in Fig.~\ref{p11} and in Fig.~\ref{s11}, respectively, as representative
examples~\cite{Aznauryan:2008pe,Aznauryan:2009mx,Burkert:2019bhp} of a wide program at
JLab~\cite{Mokeev:2012vsa,Mokeev:2015lda,Fedotov:2018oan,Isupov:2017lnd,Denizli:2007tq,Armstrong:1998wg,Egiyan:2006ks,Park:2007tn,Park:2014yea}.
For these three states advanced relativistic quark model calculations~\cite{Aznauryan:2015zta} and QCD calculations from DSE~\cite{Segovia:2015hra} and
Light Cone sum rule~\cite{Anikin:2015ita} are available, for the first time employing QCD-based modeling of the excitation
of the quark core.
There is agreement with the data at $Q^2 > 1.5$~GeV$^2$ for the latter two states, while the meson-baryon contributions for
the $\Delta(1232)$ are more extended, and agreement with the quark based calculations is reached at $Q^2 > 4$~GeV$^2$. The calculations deviate significantly from the data
at lower $Q^2$, which indicates significant non quark core effects. For the Roper resonance such contributions have been described
successfully in dynamical meson-baryon models~\cite{Obukhovsky:2011sc} and in effective field theory~\cite{Bauer:2014cqa}. Calculations on the Lattice for the N-Roper transition amplitudes have been carried out with dynamical quarks~\cite{Lin:2011da}. The results agree
with the data in the range $Q^2 < 1.0$~GeV$^2$, where data and calculations overlap~Fig.~\ref{Roper-LQCD}.
\begin{figure}[t!]
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{Roper_A12.png}}
\caption{The transverse helicity amplitudes $A_{1/2}$ for the Roper resonance $N(1440){1\over 2}^+$. Data are from CLAS compared
to LF RQM with running quark masses (solid line), and with projections from the DSE/QCD approach (dotted line). The
shaded band indicates non 3-quark contributions inferred from the difference of the LF RQM curve and the CLAS data.
The red dashed line is the EFT calculation that describes the data at small $Q^2$.}
\label{p11}
\vspace{0.5cm}
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{Roper-LQCD.png}}
\caption{The $pN^+(1440)1/2^+$ transition amplitude $F_2(Q^2)$ from LQCD~\cite{Lin:2011da} compared to CLAS results. }
\label{Roper-LQCD}
\end{figure}
\begin{figure}[thb!]
\centering
\resizebox{1.0\columnwidth}{!}{\includegraphics{s11_a.pdf}}
\caption{The transverse amplitude $A_{1/2}$ for the $N(1535){1\over 2}^-$ resonance compared to LF RQM calculations (solid line)
and QCD computation within the LC Sum Rule approach.}
\label{s11}
\end{figure}
\begin{figure}[t!]
\centerline{\resizebox{0.9\columnwidth}{!}{\includegraphics{Charge-densities.png}}}
\caption{Left panels: $N(1440)$, top: projection of charge densities on $b_y$, bottom: transition charge densities when the proton
is spin polarized along $b_x$. Right panels: same for $N(1535)$. Note that the densities are scaled with
$b^2$ to emphasize the outer wings. Color code:negative charge is blue, positive charge is red. Note that all scales
are the same. }
\label{charge_densities}
\end{figure}
Knowledge of the helicity amplitudes in a large $Q^2$ allows for the determination of the transition charge densities on the light
cone in transverse impact parameter space ($b_x, b_y$)~\cite{Tiator:2008kd}. Figure~\ref{charge_densities} shows the comparison
of $N(1440){1\over 2}^+$ and $N(1535){1\over 2}^-$. There are clear differences in the charge transition densities between
the two states. The Roper state has a softer positive core and a wider negative outer cloud than $N(1535)$
and develops a larger shift in $b_y$ when the proton is polarized along the $b_x$ axis.
\begin{figure}[t!]
\centering
\resizebox{0.48\columnwidth}{!}{\includegraphics{d15_a12.pdf}}
\resizebox{0.48\columnwidth}{!}{\includegraphics{d15_a32.pdf}}
\caption{Helicity amplitude $A_{1/2}$ (left) and $A_{3/2}$ (right) for $N^+(1675){5/2}^-$ off proton target. }
\label{N1675}
\end{figure}
New electroproduction data on the Roper and several higher mass states have been obtained in the 2-pion channel,
specifically in $ep\to e'p\pi^+\pi^-$~\cite{Mokeev:2015lda}.
\subsection{The $N(1675){5/2}^-$ state - revealing the meson-baryon contributions}
In previous discussions we have assumed that meson-baryon degrees of freedom provide significant strength to the
resonance excitation in the low $Q^2$ domain where quark based approaches LF RQM, DSE/QCD, and LCSR calculations fail to
reproduce the transition amplitudes quantitatively. Our conclusion rests, in part, with this assumption.
But, how can we be certain of the validity of this assumption?
The $N(1675){5/2}^-$ resonance allows testing this assumption, quantitatively. Figure~\ref{N1675} shows
our current knowledge of the transverse helicity amplitudes $A_{1/2}(Q^2)$ and $A_{1/2}(Q^2)$, for proton target compared to
RQM~\cite{Aznauryan:2017nkz} and
hypercentral CQM~\cite{Santopinto:2012nq} calculations. The specific quark transition for a $J^P = 5/2^-$ state belonging to the
$SU(6)\otimes O(3)] = [70, 1^-]$ supermultiplet configuration, in non-relativistic approximation prohibits the transition from the
proton in a single quark transition.
This suppression, known as the Moorhouse selection rule~\cite{Moorhouse:1966jn}, is valid for the transverse transition
amplitudes $A_{1/2}$ and $A_{3/2}$ at all $Q^2$. It should be noted that this selection rule does apply to the
transition from protons but not from neutrons. Modern quark models, that go beyond single quark transitions,
confirm quantitatively the suppression resulting in very
small transition amplitudes from protons but large ones from neutrons. The measured helicity amplitudes off the protons are almost
exclusively due to meson-baryon contributions as the dynamical coupled channel (DCC) calculation indicates (dashed line).
The close correlation of the
DCC calculation and the measured data for the case when quark contributions are nearly absent,
supports the phenomenological description of the helicity amplitudes in terms of a 3-quark core that dominate at high $Q^2$ and
meson-baryon contributions that can make important contributions at lower $Q^2$.
\section{Conclusions and Outlook}
Over the past five years eight baryon states in the mass range from 1.85 to 2.15 GeV have been either discovered or
evidence for the existence of states has been significantly strengthened. To a large degree this is the result of adding
very precise photoproduction data in open strangeness channels to the data base that is included in multi-channel partial wave
analyses, especially the Bonn-Gatchina PWA. The possibility to measure polarization observables in these processes has been
critical. In the mass range above 2 GeV more complex processes such as vector mesons or $\Delta\pi$
may have sensitivity
to states with higher masses but require more complex analyses techniques to be brought to bear. Precision data in such
channels have been available for a few years but remain to be fully incorporated in multi-channel partial wave analyses processes.
The light-quark baryon spectrum is likely also populated with hybrid excitations~\cite{Dudek:2012ag} where the gluonic
admixtures to the wave function are dominating the excitation. These states appear with the same quantum numbers as
ordinary quark excitations, and can only be isolated from ordinary states due to the $Q^2$ dependence of their helicity
amplitudes~\cite{Li:1991yba}, which is expected to be quite different from ordinary quark excitations. This requires new
electroproduction data especially at low $Q^2$~\cite{LOI_Hybrids} with different final states and at masses above 2 GeV.
Despite the very significant progress made in recent years to further establish the light-quark baryon spectrum and explore
the internal structure of excited states, much remains to be done. A vast amount of precision data already collected
needs to be included in the multi-channel analysis frameworks, and polarization data are still to be analyzed.
There are approved proposals to study resonance excitations at much higher $Q^2$ and with higher precision at Jefferson Lab
with CLAS12~\cite{elou19,Burkert:2018nvj}, which may reveal the transition to the bare quark core contributions at short distances.
A new avenue of experimental research has recently been opened up with the data-based extraction of the first mechanical property of the proton - its internal pressure distribution~\cite{Burkert:2018bqq}. Mechanical properties of resonance transitions
have recently been explored for the $N(1535)3/2^- \to N(938)$ gravitational transition form factors calculations~\cite{Ozdem:2019pkg}. In order to access these new gravitational form factors experimentally, the nucleon to resonance transition GPDs
have to be studied. The framework for studying the $N \to N(1535)$ transition GPDs, which would enable experimental access to these mechanical properties, still remains to be developed.
\vspace{0.3cm}
\noindent I like to thank Inna Aznauryan and Viktor Mokeev for numerous discussions on the subjects discussed in this presentation.
\section{Acknowledgment}This work was supported by the US Department of Energy under contract No. DE-AC05-06OR23177.
This work was supported by
the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under Contract
No. DE-AC05-06OR23177.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,077
|
Q: OCI SDK For DotNet Hangs When Doing Requests I followed the example in described by Oracle here: https://docs.oracle.com/en-us/iaas/tools/dot-net-examples/51.3.0/vault/ListSecrets.cs.html and when I try any of the commands it sends a request and receives a response back with a response code of 200, but it does nothing after receiving a response. It just hangs. And I also only have 1 Vault and 1 Secret in that Vault. The logs stop with the log:
Setting Property Value from Header
Has any experienced this issue before?
I tried checking if it was the authentication, tried googling and tried other services like the KMSVaultService. I am expecting to get a value back when I call the ListSecrets method and for it not to hang.
A: The solution was to update the dependency packages. After doing that it started to work again
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,733
|
\section{Introduction}
Oblivious transfer (OT) \cite{qbc9,1-2OT} is known to be an essential
building block for two-party and multi-party protocols \cite{qi139}.
However, unconditionally secure OT was shown to be impossible even in
quantum cryptography, because the adversary can always cheat with the
so-called honest-but-curious attack \cite{qi499,qi677,qi725,qi797,qbc14}. To
evade the problem, the concept \textquotedblleft weak OT\textquotedblright\
was proposed recently \cite{qbc88}, in which the security goals of OT are
slightly modified, so that the honest-but-curious attack is no longer
considered a successful cheating. Even so, it was found that a security
bound exists for weak OT \cite{qbc88}, thus it cannot be unconditionally
secure either.
Nevertheless, we will point out below that the cheating strategy to weak OT
has its own limitation too. By making use of this limitation, we can build
quantum weak OT protocols which will violate the existing security bound
when the cheater is restricted to individual measurements as well as some
limited collective attacks. Therefore, while in principle the security bound
still applies to our protocols, in practice the attack will be very
difficult to be implemented.
Note that previously there was already a quantum OT protocol \cite{qi51}
which was considered secure against individual measurements \cite{qi138}.
However, the protocol calls for quantum bit commitment as a building block.
Thus its security is unreliable, as it is widely believed \cite{qi24,qi23}
that unconditionally secure quantum bit commitment does not exist.
In the next section, we will review the definitions of OT and weak OT, and
the existing security bound of the latter. Our protocol will be proposed in
Sect. III. Section IV is dedicated to the security analysis. We will show
why the security bound still applies in principle. On the other hand, we
will elaborate how to reach the maximal violation of the bound when only
individual measurements are allowed. It will also be shown that the bound
can still be violated for limited collective attacks. Some ideas on further
improvement of our protocol will be discussed in Sect. V. Finally, in Sect.
VI we summarize the result, and explain why it is important in practice to
find a secure protocol against individual measurements.
\section{Definitions and the security bound}
There are many variations of OT. The most well-known ones are all-or-nothing
OT \cite{qbc9} and one-out-of-two OT \cite{1-2OT}. Here we are interested in
the latter, which is defined as follows \cite{qi140}.
\bigskip
\textit{One-out-of-two Oblivious Transfer}
(i) Alice knows two bits $x_{0}$ and $x_{1}$.
(ii) Bob gets bit $x_{b}$ and not $x_{\bar{b}}$ with $Pr(b=0)=Pr(b=1)=1/2$.
(Here $\bar{b}$\ denotes the bit-compliment of $b$.)
(iii) Bob knows which of $x_{0}$ or $x_{1}$ he got.
(iv) Alice does not know which $x_{b}$ Bob got.
\bigskip
More rigorously, this definition indicates that a secure protocol should
guarantee that at the end of the process, Bob should get $x_{b}$ with
reliability $100\%$,\ i.e., the value he decoded matches Alice's actual
input with certainty. Meanwhile, the amount of information he gains on $x_
\bar{b}}$ should be arbitrarily close to zero, so that he has to guess $x_
\bar{b}}$ by himself, which results in a reliability $50\%$ for $x_{\bar{b}}$
since his guess stands a probability $1/2$ to be correct. However, as
pointed out in \cite{HeJPA}, in the literature there is the lack of a
self-consistent definition of OT specifically made for the quantum case.
This is because with quantum methods, it is possible that Bob may accept a
lower reliability of learning $x_{b}$, so that the reliability for $x_{\bar{
}}$ can be significantly raised. This is exactly what the honest-but-curious
attack \cite{qi499,qi677,qi725,qi797} achieves. In the above definition it
is vague whether such a result is considered as successful cheating, making
it hard to discuss the security of OT protocols in a precise way.
To mend the problem, weak OT is proposed \cite{qbc88}, with an improved
definition on Bob's cheating. Define the symbols
$P_{Alice}^{\ast }$: The maximum probability with which cheating-Alice can
get honest-Bob's choice bit $b$ and honest-Bob does not abort.
$P_{Bob}^{\ast }$: The maximum over $b\in \{0,1\}$ of the probability with
which cheating-Bob can get $x_{\bar{b}}$\ given that he gets $x_{b}$\ with
certainty and honest-Alice does not abort.
For every protocol there will always be $P_{Alice}^{\ast },P_{Bob}^{\ast
}\geq 1/2$, as a cheating party can do no worse than a random guess. Then
weak OT is defined as a kind of one-out-of-two OT which requires security
only against cheating-Bob who gets one of honest-Alice's bits with certainty.
Note that the names \textquotedblleft Alice\textquotedblright\ and
\textquotedblleft Bob\textquotedblright\ were used reversely in \cite{qbc88
, comparing with the literature on QOT \cit
{qbc9,qi139,qi499,qi677,qi725,qi797,qbc14,qi51,qi138,qi140,HeJPA,HeQOT,qi135
. Here we follow the literature and use the names in the above way.
It was proven \cite{qbc88} that the optimal security bound for any quantum
weak OT protocol i
\begin{equation}
2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2,
\end{equation
from which it follows that one of the two parties must be able to cheat with
probability at least $2/3$.
In brief, Ref. \cite{qbc88} obtained this bound with the following method.
First, consider Bob's cheating. Let $\rho _{b,x_{0},x_{1}}$ denote the
reduced state of his portion of the system. Since a weak OT protocol should
allow honest-Bob to learn $x_{b}$\ with certainty, there must be a
non-destructive measurement that enables him to do so without disturbing the
system. After Bob learned $x_{b}$\ with this measurement, his system will
still remain in the state $\rho _{b,x_{0},x_{1}}$. To gain some information
on the other bit $x_{\bar{b}}$, he performs the Helstrom measurement to
optimally distinguish the two states corresponding to $x_{\bar{b}}=0$\ and
x_{\bar{b}}=1$, respectively. Thus his cheating can be successful with
probabilit
\begin{equation}
P_{Bob}^{\ast }\geq \frac{1}{2}+\frac{1}{8}\Delta , \label{qbc88-pb}
\end{equation
wher
\begin{equation}
\Delta \equiv \frac{1}{2}(\sum\limits_{x_{0}\in \{0,1\}}\left\Vert \rho
_{0,x_{0},0}-\rho _{0,x_{0},1}\right\Vert _{Tr}+\sum\limits_{x_{1}\in
\{0,1\}}\left\Vert \rho _{1,0,x_{1}}-\rho _{1,1,x_{1}}\right\Vert _{Tr}).
\end{equation
Secondly, consider Alice's cheating. She implements a uniform superposition
over $x_{0}$, $x_{1}$\ of honest strategies by introducing two additional
private qubits for storing the values of $x_{0}$ and $x_{1}$. Then she
applies the controlled-unitary operation
\begin{equation}
controlled-U_{0}:\left\vert \psi _{1,1,x_{1}}\right\rangle \left\vert
1\right\rangle \left\vert x_{1}\right\rangle \rightarrow (I_{A}\otimes
U_{0,x_{1}})\left\vert \psi _{1,1,x_{1}}\right\rangle \left\vert
1\right\rangle \left\vert x_{1}\right\rangle
\end{equation
an
\begin{equation}
controlled-U_{1}:\left\vert \psi _{0,x_{0},1}\right\rangle \left\vert
x_{0}\right\rangle \left\vert 1\right\rangle \rightarrow (I_{A}\otimes
U_{1,x_{0}})\left\vert \psi _{0,x_{0},1}\right\rangle \left\vert
x_{0}\right\rangle \left\vert 1\right\rangle
\end{equation
for $x_{1}\in \{0,1\}$\ and $x_{0}\in \{0,1\}$, respectively, where
U_{0,x_{1}}$ and $U_{1,x_{0}}$\ satisf
\begin{eqnarray}
F(\rho _{1,0,x_{1}},\rho _{1,1,x_{1}}) &=&\left\langle \psi
_{1,0,x_{1}}\right\vert (I_{A}\otimes U_{0,x_{1}})\left\vert \psi
_{1,1,x_{1}}\right\rangle , \nonumber \\
F(\rho _{0,x_{0},0},\rho _{0,x_{0},1}) &=&\left\langle \psi
_{0,x_{0},0}\right\vert (I_{A}\otimes U_{1,x_{0}})\left\vert \psi
_{0,x_{0},1}\right\rangle .
\end{eqnarray
Here $F(\rho ,\xi )\equiv \left\Vert \sqrt{\rho }\sqrt{\xi }\right\Vert
_{Tr} $\ is the fidelity between $\rho $\ and $\xi $. Then the successful
probability for her cheating was shown \cite{qbc88} to b
\begin{equation}
P_{Alice}^{\ast }\geq \frac{1}{2}+\frac{1}{16}F, \label{qbc88-pa}
\end{equation
wher
\begin{equation}
F\equiv \sum\limits_{x_{0}\in \{0,1\}}F(\rho _{0,x_{0},0},\rho
_{0,x_{0},1})+\sum\limits_{x_{1}\in \{0,1\}}F(\rho _{1,0,x_{1}},\rho
_{1,1,x_{1}}).
\end{equation
Combining the Fuchs-van de Graaf inequalitie
\begin{equation}
1-\frac{1}{2}\left\Vert \rho -\xi \right\Vert _{Tr}\leq F(\rho ,\xi )\leq
\sqrt{1-\frac{1}{4}\left\Vert \rho -\xi \right\Vert _{Tr}^{2}}
\end{equation
with equations (\ref{qbc88-pb}) and (\ref{qbc88-pa}), the security bound
2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2$\ is finally obtained.
This bound was also shown to be optimal, as Ref. \cite{qbc88} exhibited a
family of protocols whose cheating probabilities can be made arbitrarily
close to any point on the $P_{Alice}^{\ast }$ versus $P_{Bob}^{\ast }$\
tradeoff curve.
\section{The protocol}
\subsection{Limitation of the cheating strategy in existing protocols}
Intriguingly, while the security bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast
}\geq 2$\ indicates that in a protocol where Bob cannot cheat (i.e.,
P_{Bob}^{\ast }=1/2$), Alice can guess Bob's choice $b$ at least with the
probability $P_{Alice}^{\ast }=3/4$, we must note that her cheating strategy
has a serious drawback. That is, once Alice applies the cheating, she will
not be able to determine the values of $x_{0}$, $x_{1}$. For example,
consider the Chailloux-Kerenidis-Sikora (CKS)\ protocol proposed in Sect. 4
of \cite{qbc14} (also presented in Sect. 3.2 of \cite{qbc88} with reverse
usage of the names \textquotedblleft Alice\textquotedblright\ and
\textquotedblleft Bob\textquotedblright ), as described below.
\bigskip
\textit{The CKS protocol}
1. Bob randomly chooses $b\in \{0,1\}$ and prepares the two-qutrit state
\left\vert \phi _{b}\right\rangle =(\left\vert bb\right\rangle +\left\vert
22\right\rangle )/\sqrt{2}$. He sends one of the qutrits to Alice.
2. Alice chooses $x_{0},x_{1}\in \{0,1\}$ and applies the unitary
transformation $\left\vert 0\right\rangle \rightarrow (-1)^{x_{0}}\left\vert
0\right\rangle $, $\left\vert 1\right\rangle \rightarrow
(-1)^{x_{1}}\left\vert 1\right\rangle $, $\left\vert 2\right\rangle
\rightarrow \left\vert 2\right\rangle $ on Bob's qutrit.
3. Alice returns the qutrit to Bob who now has the state $\left\vert \psi
_{b}\right\rangle =[(-1)^{x_{b}}\left\vert bb\right\rangle +\left\vert
22\right\rangle ]/\sqrt{2}$.
4. Bob performs the measurement $\{\Pi _{0}=\left\vert \phi
_{b}\right\rangle \left\langle \phi _{b}\right\vert ,\Pi _{1}=\left\vert
\phi _{b}^{\prime }\right\rangle \left\langle \phi _{b}^{\prime }\right\vert
,I-\Pi _{0}-\Pi _{1}\}$\ on the state $\left\vert \psi _{b}\right\rangle $,
where $\left\vert \phi _{b}^{\prime }\right\rangle =(\left\vert
bb\right\rangle -\left\vert 22\right\rangle )/\sqrt{2}$.
5. If the outcome is $\Pi _{0}$ then Bob learns with certainty that $x_{b}=0
, if it is $\Pi _{1}$ then $x_{b}=1$, otherwise he aborts.
\bigskip
This protocol can reach $P_{Bob}^{\ast }=1/2$, but it is insecure against
Alice's individual attacks. As shown in Sect. 4 of \cite{qbc14}, Alice's
optimal cheating strategy is simply to measure the qutrit she received in
step 2 using the computational basis. If she gets outcome $\left\vert
0\right\rangle $ ($\left\vert 1\right\rangle $) then she knows with
certainty that $b=0$ ($b=1$). If she gets outcome $\left\vert 2\right\rangle
$ then she guesses the value of $b$. Therefore on average, Alice can learn
Bob's $b$ correctly with the probability $P_{Alice}^{\ast }=3/4$. After the
measurement she returns the measured qutrit to Bob. Then Bob's state will be
either $\left\vert bb\right\rangle $ or $\left\vert 22\right\rangle $. With
any of these two states, the outcome of Bob's measurement in step 4 will
always be either $\Pi _{0}$ or $\Pi _{1}$. Hence he will never abort, so
that Alice's cheating cannot be detected at all.
However, we can see that Alice cannot control, nor she can learn what will
be the actual outcome of Bob's measurement in step 4, because $\left\vert
bb\right\rangle $ and $\left\vert 22\right\rangle $ can both be projected as
$\Pi _{0}$ or $\Pi _{1}$. As a result, at the end of the protocol Bob gets a
bit $x_{b}$, but its value is unknown to Alice. That is, once
dishonest-Alice gains the information on $b$, she loses the information on
x_{b}$. Now we prove that this result is general for any Alice's cheating
strategy.
\bigskip
\textbf{Theorem 1}: In the CKS protocol, Alice cannot learn $x_{b}$ with
reliability $1$ and gain a non-trivial amount of information on $b$
simultaneously, while escaping Bob's detection with probability $1$.
\textbf{Proof:} Let $\alpha $ denote the ancillary system that
dishonest-Alice introduced for her cheating, system $\beta $ denote the
qutrit that she received and then returns to Bob, and system $\beta ^{\prime
}$ denote the qutrit that Bob always keeps at his side. To ensure that Bob
will never abort in step 5 so that Alice can pass Bob's detection with
probability $1$, the state of $\beta \otimes \beta ^{\prime }$ at this stage
must be completely contained in the Hilbert space supported by $\left\vert
\phi _{b}\right\rangle _{\beta \beta ^{\prime }}$ and $\left\vert \phi
_{b}^{\prime }\right\rangle _{\beta \beta ^{\prime }}$. Therefore, for any
Alice's cheating strategy, at the end of step 3 the general form of the
quantum system shared by Alice and Bob can always be written a
\begin{equation}
T(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert \phi
_{b}\right\rangle _{\beta \beta ^{\prime }})=\lambda _{b}^{(0)}\left\vert
e_{b}^{(0)}\right\rangle _{\alpha }\left\vert \phi _{b}\right\rangle _{\beta
\beta ^{\prime }}+\lambda _{b}^{(1)}\left\vert e_{b}^{(1)}\right\rangle
_{\alpha }\left\vert \phi _{b}^{\prime }\right\rangle _{\beta \beta ^{\prime
}}, \label{resultant2}
\end{equation
for $b=0,1$, with $\left\vert e_{ini}\right\rangle _{\alpha }$\ ($\left\vert
e_{b}^{(0)}\right\rangle _{\alpha }$ and $\left\vert
e_{b}^{(1)}\right\rangle _{\alpha }$) being the normalized initial (final)
state(s) of $\alpha $, $\left\vert \lambda _{b}^{(0)}\right\vert
^{2}+\left\vert \lambda _{b}^{(1)}\right\vert ^{2}=1$, and $T$\ is the
operator that Alice applies for her cheating. Also, in step (4) of the
protocol, Bob learns that $x_{b}=0$ ($x_{b}=1$) if he gets $\left\vert \phi
_{b}\right\rangle _{\beta \beta ^{\prime }}$\ ($\left\vert \phi _{b}^{\prime
}\right\rangle _{\beta \beta ^{\prime }}$). Therefore, if Alice wants to be
able to learn $x_{b}$ with reliability $1$, she has to choose an operation
T $ which can ensure that $\left\vert e_{b}^{(0)}\right\rangle _{\alpha }$
and $\left\vert e_{b}^{(1)}\right\rangle _{\alpha }$\ are orthogonal.
By substituting $\left\vert \phi _{b}\right\rangle _{\beta \beta ^{\prime
}}=(\left\vert bb\right\rangle _{\beta \beta ^{\prime }}+\left\vert
22\right\rangle _{\beta \beta ^{\prime }})/\sqrt{2}$ and $\left\vert \phi
_{b}^{\prime }\right\rangle _{\beta \beta ^{\prime }}=(\left\vert
bb\right\rangle _{\beta \beta ^{\prime }}-\left\vert 22\right\rangle _{\beta
\beta ^{\prime }})/\sqrt{2}$ into equation (\ref{resultant2}), we can
rewrite it a
\begin{equation}
T(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert \phi
_{b}\right\rangle _{\beta \beta ^{\prime }})=(\left\vert f_{b}\right\rangle
_{\alpha }\left\vert bb\right\rangle _{\beta \beta ^{\prime }}+\left\vert
f_{b}^{\prime }\right\rangle _{\alpha }\left\vert 22\right\rangle _{\beta
\beta ^{\prime }})/\sqrt{2}, \label{resultant}
\end{equation
wher
\begin{eqnarray}
\left\vert f_{b}\right\rangle _{\alpha } &\equiv &\lambda
_{b}^{(0)}\left\vert e_{b}^{(0)}\right\rangle _{\alpha }+\lambda
_{b}^{(1)}\left\vert e_{b}^{(1)}\right\rangle _{\alpha }, \nonumber \\
\left\vert f_{b}^{\prime }\right\rangle _{\alpha } &\equiv &\lambda
_{b}^{(0)}\left\vert e_{b}^{(0)}\right\rangle _{\alpha }-\lambda
_{b}^{(1)}\left\vert e_{b}^{(1)}\right\rangle _{\alpha }, \label{fbx}
\end{eqnarray
To gain a non-trivial amount of information on $b$, equation (\ref{resultant
) indicates that Alice needs to distinguish the reduced density matrice
\begin{equation}
\rho _{b=0}\equiv (\left\vert f_{0}\right\rangle _{\alpha }\left\langle
f_{0}\right\vert +\left\vert f_{0}^{\prime }\right\rangle _{\alpha
}\left\langle f_{0}^{\prime }\right\vert )/2 \label{rho1}
\end{equation
an
\begin{equation}
\rho _{b=1}\equiv (\left\vert f_{1}\right\rangle _{\alpha }\left\langle
f_{1}\right\vert +\left\vert f_{1}^{\prime }\right\rangle _{\alpha
}\left\langle f_{1}^{\prime }\right\vert )/2. \label{rho2}
\end{equation
While there could exist an operation $T$, which can ensure $\rho _{b=0}\neq
\rho _{b=1}$ before Alice obtains $x_{b}$, we will show below that after
Alice performed any operation $M$ that can make her learn $x_{b}$ with
reliability $1$, $\rho _{b=0}$ and $\rho _{b=1}$ will become equal to each
other, so that they cannot be distinguished any more.
An important fact is that qutrit $\beta ^{\prime }$ is always kept at Bob's
side, so that it remains unchanged under Alice operation $T$. Thus we can
write $T=U_{\alpha \beta }\otimes I_{\beta ^{\prime }}$, where $U_{\alpha
\beta }$ applies on systems $\alpha $ and $\beta $ only, while $I_{\beta
^{\prime }}$\ is the identity operator on $\beta ^{\prime }$. Denotin
\begin{eqnarray}
U_{\alpha \beta }(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert
0\right\rangle _{\beta }) &=&\left\vert \Psi _{0}\right\rangle _{\alpha
\beta }, \nonumber \\
U_{\alpha \beta }(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert
1\right\rangle _{\beta }) &=&\left\vert \Psi _{1}\right\rangle _{\alpha
\beta }, \nonumber \\
U_{\alpha \beta }(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert
2\right\rangle _{\beta }) &=&\left\vert \Psi _{2}\right\rangle _{\alpha
\beta }, \label{U}
\end{eqnarray
then we have
\begin{eqnarray}
T(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert \phi
_{b}\right\rangle _{\beta \beta ^{\prime }}) &=&(U_{\alpha \beta }\otimes
I_{\beta ^{\prime }})(\left\vert e_{ini}\right\rangle _{\alpha }(\left\vert
bb\right\rangle _{\beta \beta ^{\prime }}+\left\vert 22\right\rangle _{\beta
\beta ^{\prime }})/\sqrt{2}) \nonumber \\
&=&(\left\vert \Psi _{b}\right\rangle _{\alpha \beta }\left\vert
b\right\rangle _{\beta ^{\prime }}+\left\vert \Psi _{2}\right\rangle
_{\alpha \beta }\left\vert 2\right\rangle _{\beta ^{\prime }})/\sqrt{2}).
\end{eqnarray
Comparing with equation (\ref{resultant}), we yiel
\begin{eqnarray}
\left\vert \Psi _{b}\right\rangle _{\alpha \beta } &=&\left\vert
f_{b}\right\rangle _{\alpha }\left\vert b\right\rangle _{\beta }, \nonumber
\\
\left\vert \Psi _{2}\right\rangle _{\alpha \beta } &=&\left\vert
f_{b}^{\prime }\right\rangle _{\alpha }\left\vert 2\right\rangle _{\beta }.
\end{eqnarray
The latter indicates that $\left\vert f_{b}^{\prime }\right\rangle _{\alpha
} $ does not depend on $b$, i.e.
\begin{equation}
\left\vert f_{0}^{\prime }\right\rangle _{\alpha }=\left\vert f_{1}^{\prime
}\right\rangle _{\alpha }. \label{jnt}
\end{equation
The
\begin{equation}
\left\langle f_{0}^{\prime }\right. \left\vert f_{1}^{\prime }\right\rangle
_{\alpha }=1. \label{f'f'1}
\end{equation
With equation (\ref{fbx}) we kno
\begin{eqnarray}
\left\langle f_{0}^{\prime }\right. \left\vert f_{1}^{\prime }\right\rangle
_{\alpha } &=&(\lambda _{0}^{(0)\ast }\left\langle e_{0}^{(0)}\right\vert
_{\alpha }-\lambda _{0}^{(1)\ast }\left\langle e_{0}^{(1)}\right\vert
_{\alpha })(\lambda _{1}^{(0)}\left\vert e_{1}^{(0)}\right\rangle _{\alpha
}-\lambda _{1}^{(1)}\left\vert e_{1}^{(1)}\right\rangle _{\alpha })
\nonumber \\
&=&\lambda _{0}^{(0)\ast }\lambda _{1}^{(0)}\left\langle e_{0}^{(0)}\right.
\left\vert e_{1}^{(0)}\right\rangle _{\alpha }-\lambda _{0}^{(0)\ast
}\lambda _{1}^{(1)}\left\langle e_{0}^{(0)}\right. \left\vert
e_{1}^{(1)}\right\rangle _{\alpha } \nonumber \\
&&-\lambda _{0}^{(1)\ast }\lambda _{1}^{(0)}\left\langle e_{0}^{(1)}\right.
\left\vert e_{1}^{(0)}\right\rangle _{\alpha }+\lambda _{0}^{(1)\ast
}\lambda _{1}^{(1)}\left\langle e_{0}^{(1)}\right. \left\vert
e_{1}^{(1)}\right\rangle _{\alpha } \label{f'f'}
\end{eqnarray
an
\begin{eqnarray}
\left\langle f_{0}\right. \left\vert f_{1}\right\rangle _{\alpha }
&=&(\lambda _{0}^{(0)\ast }\left\langle e_{0}^{(0)}\right\vert _{\alpha
}+\lambda _{0}^{(1)\ast }\left\langle e_{0}^{(1)}\right\vert _{\alpha
})(\lambda _{1}^{(0)}\left\vert e_{1}^{(0)}\right\rangle _{\alpha }+\lambda
_{1}^{(1)}\left\vert e_{1}^{(1)}\right\rangle _{\alpha }) \nonumber \\
&=&\lambda _{0}^{(0)\ast }\lambda _{1}^{(0)}\left\langle e_{0}^{(0)}\right.
\left\vert e_{1}^{(0)}\right\rangle _{\alpha }+\lambda _{0}^{(0)\ast
}\lambda _{1}^{(1)}\left\langle e_{0}^{(0)}\right. \left\vert
e_{1}^{(1)}\right\rangle _{\alpha } \nonumber \\
&&+\lambda _{0}^{(1)\ast }\lambda _{1}^{(0)}\left\langle e_{0}^{(1)}\right.
\left\vert e_{1}^{(0)}\right\rangle _{\alpha }+\lambda _{0}^{(1)\ast
}\lambda _{1}^{(1)}\left\langle e_{0}^{(1)}\right. \left\vert
e_{1}^{(1)}\right\rangle _{\alpha }. \label{ff}
\end{eqnarray
Now if Alice performs any operation $M$ on the state $T(\left\vert
e_{ini}\right\rangle _{\alpha }\left\vert \phi _{b}\right\rangle _{\beta
\beta ^{\prime }})$\ so that $x_{b}$ ($b=0,1$) is obtained with reliability
1$, the final state can still be written as equation (\ref{resultant2}),
except that the coefficients $\lambda _{0}^{(0)}$, $\lambda _{0}^{(1)}$,
\lambda _{1}^{(0)}$\ and $\lambda _{1}^{(1)}$\ cannot stay non-vanishing
simultaneously. Instead, one of $\lambda _{0}^{(0)}$ and $\lambda _{0}^{(1)}$
must become zero, and one of $\lambda _{1}^{(0)}$\ and $\lambda _{1}^{(1)}$\
must be zero too. In this case, we can see that in the right hand side of
either equation (\ref{f'f'}) or (\ref{ff}), only one of the coefficients
before the four terms $\left\langle e_{0}^{(0)}\right. \left\vert
e_{1}^{(0)}\right\rangle _{\alpha }$, $\left\langle e_{0}^{(0)}\right.
\left\vert e_{1}^{(1)}\right\rangle _{\alpha }$, $\left\langle
e_{0}^{(1)}\right. \left\vert e_{1}^{(0)}\right\rangle _{\alpha }$ and
\left\langle e_{0}^{(1)}\right. \left\vert e_{1}^{(1)}\right\rangle _{\alpha
}$ can remain non-vanishing. No matter which single term remains, there will
always be eithe
\begin{equation}
\left\langle f_{0}\right. \left\vert f_{1}\right\rangle _{\alpha
}=\left\langle f_{0}^{\prime }\right. \left\vert f_{1}^{\prime
}\right\rangle _{\alpha }.
\end{equation
o
\begin{equation}
\left\langle f_{0}\right. \left\vert f_{1}\right\rangle _{\alpha
}=-\left\langle f_{0}^{\prime }\right. \left\vert f_{1}^{\prime
}\right\rangle _{\alpha }.
\end{equation
Combining with equation (\ref{f'f'1}), they both giv
\begin{equation}
\left\vert f_{0}\right\rangle _{\alpha }\left\langle f_{0}\right\vert
=\left\vert f_{1}\right\rangle _{\alpha }\left\langle f_{1}\right\vert .
\end{equation
Substituting it and equation (\ref{jnt}) into equations (\ref{rho1}) and
\ref{rho2}), we finally obtai
\begin{equation}
\rho _{b=0}=\rho _{b=1}.
\end{equation
Thus they provide absolutely zero knowledge on $b$. Therefore, once Alice
performs the operation to learn $x_{b}$ with reliability $1$, she can no
longer gain any information on $b$. This ends the proof of Theorem 1.
\bigskip
The above proof does not exclude the existence of other cheating strategies,
in which Alice can learn $x_{b}$ with a reliability less than $1$, and/or
Bob may abort in step 5 with a non-vanishing probability. But this will do
no harm to our purpose, as it will be shown later in Sects. IV.B and IV.C.
In fact, besides the CKS protocol, some other QOT protocols \cit
{HeQOT,qi135} also display the same feature described in Theorem 1. Let
(3/4,1/2)$\textit{-protocol} denote any QOT protocol of this kind, i.e.,
both $P_{Alice}^{\ast }=3/4$ and $P_{Bob}^{\ast }=1/2$\ are satisfied
exactly, and Alice cannot determine $x_{b}$ with reliability $1$ once she
gain a non-trivial amount of information on $b$. It will be shown below that
though a $(3/4,1/2)$-protocol merely saturates the security bound
2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2$, it can be utilized to construct a
compound protocol which can eventually violate this bound when Alice is
limited to individual attacks.
\subsection{Our protocol}
Our method is to use such a $(3/4,1/2)$-protocol as a building block, with
which Alice transfers a series of bits $x_{0}^{(i)}$, $x_{1}^{(i)}$ (not the
final $x_{0}$, $x_{1}$ that she wants to transfer) to Bob. The values of
x_{0}^{(i)}$, $x_{1}^{(i)}$ are \textit{not} completely random. Instead,
they must be chosen according to a certain rule. Then Bob uses many of
x_{0}^{(i)}$, $x_{1}^{(i)}$ to check whether Alice can determine their
values correctly. Finally he uses one of the remaining pairs of $x_{0}^{(i)}
, $x_{1}^{(i)}$ and asks Alice to encode her $x_{0}$, $x_{1}$. The general
form of the protocol is as follows.
\bigskip
\textit{Protocol A: weak OT for transferring }$(x_{0},x_{1})$
A1. Alice and Bob discuss and agree on a set $S$ of classical $n$-bit
strings.
A2. Alice randomly chooses two strings\newline
$X_{0}=x_{0}^{(1)}x_{0}^{(2)}...x_{0}^{(i)}...x_{0}^{(n)}$ and
X_{1}=x_{1}^{(1)}x_{1}^{(2)}...x_{1}^{(i)}...x_{1}^{(n)}$ from $S$. Note
that at this stage, none of these $x_{0}^{(i)}$, $x_{1}^{(i)}$ have any
specific relationship with the two final bits $x_{0}$, $x_{1}$ (we call them
as \textit{target bits} thereafter) that Alice wants to transfer to Bob as
the goal of the weak OT.
A3. For each $i$ ($i=1,...,n$), Alice transfers $x_{0}^{(i)}$, $x_{1}^{(i)}$
to Bob using a $(3/4,1/2)$-protocol. Bob randomly chooses $b_{i}\in \{0,1\}$
and decodes $x_{b_{i}}$.
A4. Security check: among all these $n$ runs of the
$(3/4,1/2)$-protocol, Bob picks $m$ ($m<n$) runs randomly. For each of these
runs, he asks Alice to announce $x_{0}^{(i)}$ and $x_{1}^{(i)}$, and checks
whether they are consistent with the value of $x_{b_{i}}$ that he obtained
in the $(3/4,1/2)$-protocol. He also checks that there is at least two
strings $X_{0}^{\prime } $ and $X_{1}^{\prime }$ in set $S$, such that all
the $m$ bits $x_{0}^{(i)} $ ($x_{1}^{(i)}$) that Alice announced are
contained in $X_{0}^{\prime }$ ($X_{1}^{\prime }$).
A5. If Alice's announced values pass the above check, Bob picks one of the
remaining $n-m$ unchecked runs (which is denoted as the $\hat{i}$-th run) of
the $(3/4,1/2)$-protocol. This run should satisfy the requirement that
x_{0}^{(\hat{i})}=0$, $x_{0}^{(\hat{i})}=1$, $x_{1}^{(\hat{i})}=0$, and
x_{1}^{(\hat{i})}=1$ are all allowed by set $S$. That is, in set $S$ there
is at least one string which contains all the $m$ bits $x_{0}^{(i)}$ (
x_{1}^{(i)}$) that Alice announced in step A4 and $x_{0}^{(\hat{i})}=0$ (
x_{1}^{(\hat{i})}=0$), and an equal number of strings, each of which also
contains all these $m$ bits $x_{0}^{(i)}$ ($x_{1}^{(i)}$) but with $x_{0}^{
\hat{i})}=1$ ($x_{1}^{(\hat{i})}=1$) instead. This guarantees that the $m$
bits $x_{0}^{(i)}$ ($x_{1}^{(i)}$) announced in step A4 are insufficient for
Bob to deduce the value of $x_{0}^{(\hat{i})}$ ($x_{1}^{(\hat{i})}$) from
set $S$. Alice checks that this requirement is met after Bob told her the
value of $\hat{i}$.
A6. Alice completes the weak OT by using the $\hat{i}$-th run of the
(3/4,1/2)$-protocol\ to encode the target bits $x_{0}$, $x_{1}$. That is,
she announces $x_{0}\oplus x_{0}^{(\hat{i})}$ and $x_{1}\oplus x_{1}^{(\hat{
})}$ to Bob. Thus Bob can obtain either $x_{0}$ or $x_{1}$ depending on
whether he has obtained $x_{0}^{(\hat{i})}$ or $x_{1}^{(\hat{i})}$ in the
\hat{i}$-th run of the $(3/4,1/2)$-protocol.
\bigskip
\subsection{A concrete example}
To make our protocol easier for understanding and analyzing, here we provide
a concrete example of our above protocol where the CKS protocol is used as
the $(3/4,1/2)$-protocol and the explicit form of set $S$ is given.
\bigskip
\textit{Protocol B: a concrete example}
B1. Alice and Bob run the CKS protocol for $n=3k$\ times. Every $3$ runs of
the CKS protocol are grouped together and we call it as a \textit{triple run
. Let $x_{0}^{(i)}$, $x_{1}^{(i)}$ ($i=1,2,3$) denote the bits that Alice
transfers to Bob in a triple run. The values of the strings
X_{0}=x_{0}^{(1)}x_{0}^{(2)}x_{0}^{(3)}$ and
X_{1}=x_{1}^{(1)}x_{1}^{(2)}x_{1}^{(3)}$ cannot be completely random.
Instead, they are required to be chosen within the set $S=\{000,001,010,10
\} $.
B2. Security check: for every triple run, Bob randomly picks two runs of the
CKS protocol, denotes them as the $i_{1}$-th and $i_{2}$-th runs. The
remaining run that is not picked is denoted as the $i_{3}$-th run. Bob asks
Alice to reveal $x_{0}^{(i_{1})}$, $x_{1}^{(i_{1})}$ and $x_{0}^{(i_{2})}$,
x_{1}^{(i_{2})}$. If
x_{0}^{(i_{1})}=x_{1}^{(i_{1})}=x_{0}^{(i_{2})}=x_{1}^{(i_{2})}=0$\ then Bob
marks the corresponding triple run as a \textit{useful run}, as both
x_{0}^{(i_{3})}$ and $x_{1}^{(i_{3})}$\ can either be $0$ or $1$\ according
to the definition of set $S$, so that they may potentially be used for
encoding the target bits $x_{0}$, $x_{1}$ later. Else if any of
x_{0}^{(i_{1})}$, $x_{1}^{(i_{1})}$, $x_{0}^{(i_{2})}$, $x_{1}^{(i_{2})}$\
is $1$, Bob asks Alice to reveal $x_{0}^{(i_{3})}$, $x_{1}^{(i_{3})}$ too,
and checks whether both $X_{0}=x_{0}^{(1)}x_{0}^{(2)}x_{0}^{(3)}$ and
X_{1}=x_{1}^{(1)}x_{1}^{(2)}x_{1}^{(3)}$ belong to set $S$. He also checks
that none of Alice's announced values conflicts with what he decoded from
the CKS protocol.
B3. If Alice's data passes the above check, Bob picks one of the useful runs
and asks Alice to complete the weak OT using this run. Then Alice announces
x_{0}\oplus x_{0}^{(i_{3})}$ and $x_{1}\oplus x_{1}^{(i_{3})}$ to Bob, so
that he can obtain either the target bit $x_{0}$ or $x_{1}$ depending on
whether he has obtained $x_{0}^{(i_{3})}$ or $x_{1}^{(i_{3})}$ in the
corresponding run of the CKS protocol.
B4. For better security, Bob can further ask Alice to reveal
x_{0}^{(i_{3})} $ and $x_{1}^{(i_{3})}$\ of all the rest useful runs which
are not picked in step B3. Then he checks whether they conflict with what he
decoded from the CKS protocol.
\bigskip
\section{Security}
\subsection{The collective attack}
The above protocols A and B are, unfortunately, still restricted by the
security bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2$\ if Alice has
unlimited computational power to apply collective attacks. Taking Protocol B
as an example, her cheating strategy is as follows.
In step B1, for each triple run Alice introduces a $6$-qubit system
C=c_{0}^{(1)}c_{0}^{(2)}c_{0}^{(3)}c_{1}^{(1)}c_{1}^{(2)}c_{1}^{(3)}$ to
keep her choice of the strings $X_{0}=x_{0}^{(1)}x_{0}^{(2)}x_{0}^{(3)}$ and
$X_{1}=x_{1}^{(1)}x_{1}^{(2)}x_{1}^{(3)}$ at the quantum level. The state of
system $C$ is initialized a
\begin{eqnarray}
&&\left\vert
c_{0}^{(1)}c_{0}^{(2)}c_{0}^{(3)}c_{1}^{(1)}c_{1}^{(2)}c_{1}^{(3)}\righ
\rangle \nonumber \\
&=&\frac{1}{2}(\left\vert 000\right\rangle +\left\vert 001\right\rangle
+\left\vert 010\right\rangle +\left\vert 100\right\rangle ) \nonumber \\
&&\otimes \frac{1}{2}(\left\vert 000\right\rangle +\left\vert
001\right\rangle +\left\vert 010\right\rangle +\left\vert 100\right\rangle ),
\label{C}
\end{eqnarray
where the first (last) three qubits are corresponding to the string $X_{0}$
$X_{1}$). That is, it is a superposition of all the states allowed by set $S
.
In the $i$-th run ($i=1,2,3$) of the CKS protocol during a triple run, let
\beta ^{(i)}$\ denote the qutrit that Bob sent to Alice, taken from his
two-qutrit state $\left\vert \phi _{b}^{(i)}\right\rangle =(\left\vert
bb\right\rangle +\left\vert 22\right\rangle )/\sqrt{2}$. Alice uses
c_{0}^{(i)}$, $c_{1}^{(i)}$ as control qubits to determine her
transformation on $\beta ^{(i)}$. That is, on $c_{0}^{(i)}\otimes
c_{1}^{(i)}\otimes \beta ^{(i)}$ she applies the unitary transformatio
\begin{eqnarray}
T_{c_{0}^{(i)}c_{1}^{(i)}\beta ^{(i)}}
&=&\sum\limits_{x_{0}^{(i)},x_{1}^{(i)}=0}^{1}(\left\vert
x_{0}^{(i)}\right\rangle _{c_{0}^{(i)}}\left\langle x_{0}^{(i)}\right\vert
\otimes \left\vert x_{1}^{(i)}\right\rangle _{c_{1}^{(i)}}\left\langle
x_{1}^{(i)}\right\vert \nonumber \\
&&\otimes \left[
\begin{array}{ccc}
(-1)^{x_{0}^{(i)}} & 0 & 0 \\
0 & (-1)^{x_{1}^{(i)}} & 0 \\
0 & 0 &
\end{array
\right] _{\beta ^{(i)}}). \label{T}
\end{eqnarray
By doing so, Alice manages to finish the transformation $\left\vert
0\right\rangle \rightarrow (-1)^{x_{0}}\left\vert 0\right\rangle $,
\left\vert 1\right\rangle \rightarrow (-1)^{x_{1}}\left\vert 1\right\rangle
, $\left\vert 2\right\rangle \rightarrow \left\vert 2\right\rangle $ on
Bob's qutrit $\beta ^{(i)}$, just as it is required in the CKS protocol when
Alice is honest. The only difference is that in the current case,
x_{0}^{(i)}$\ and $x_{1}^{(i)}$\ do not have deterministic classical values.
Instead, they are kept at the quantum level.
In step B2 whenever Bob picks one run of the CKS protocol and asks Alice to
reveal the corresponding $x_{0}^{(i)}$\ and $x_{1}^{(i)}$, Alice measures
the qubits $c_{0}^{(i)}$ and $c_{1}^{(i)}$ in the computational basis
\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}$. If the result is
$\left\vert 0\right\rangle $ ($\left\vert 1\right\rangle $) then she
announces the corresponding $x_{0}^{(i)}$\ or $x_{1}^{(i)}$ as $0$ ($1$).
From equation (\ref{T}) it can be seen that Alice's announcement will never
conflict with the values Bob decodes from the CKS protocol. Now recall that
a useful run is defined as the triple run where
x_{0}^{(i_{1})}=x_{1}^{(i_{1})}=x_{0}^{(i_{2})}=x_{1}^{(i_{2})}=0$.
Therefore by combining equations (\ref{C}) and (\ref{T}), we know that the
state of $c_{0}^{(i_{3})}\otimes c_{1}^{(i_{3})}\otimes \phi _{b}^{(i_{3})}$
of any useful run at the end of step B2 become
\begin{eqnarray}
&&\left\vert c_{0}^{(i_{3})}\otimes c_{1}^{(i_{3})}\otimes \phi
_{b}^{(i_{3})}\right\rangle \nonumber \\
&=&\frac{1}{2}\sum\limits_{x_{0}^{(i_{3})},x_{1}^{(i_{3})}=0}^{1}\{\lef
\vert x_{0}^{(i_{3})}\right\rangle _{c_{0}^{(i_{3})}}\otimes \left\vert
x_{1}^{(i_{3})}\right\rangle _{c_{1}^{(i_{3})}} \nonumber \\
&&\otimes \frac{1}{\sqrt{2}}[(-1)^{x_{b}^{(i_{3})}}\left\vert
bb\right\rangle +\left\vert 22\right\rangle ]\}. \label{B2}
\end{eqnarray}
If a useful run is picked for the security check in step B4, Alice can
simply measure the qubits $c_{0}^{(i_{3})}$ and $c_{1}^{(i_{3})}$ in the
basis $\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}$ and reveal
$x_{0}^{(i_{3})}$ and $x_{1}^{(i_{3})}$\ correctly. On the other hand, if a
useful run is picked in step B3 to encode Alice's target bits $x_{0}$,
x_{1} $, then she will have the freedom to choose whether to measure
c_{0}^{(i_{3})}$ and $c_{1}^{(i_{3})}$ in the basis $\{\left\vert
0\right\rangle ,\left\vert 1\right\rangle \}$ and learn the values of $x_{0}
, $x_{1}$\ as an honest Alice does, or to learn the value of Bob's $b$ with
a certain probability instead. In the latter case, she measures
c_{0}^{(i_{3})}$ and $c_{1}^{(i_{3})}$ in the basis $\{\left\vert
+\right\rangle ,\left\vert -\right\rangle \}$, where $\left\vert \pm
\right\rangle =(\left\vert 0\right\rangle \pm \left\vert 1\right\rangle )
\sqrt{2}$. This is because equation (\ref{B2}) can be rewritten a
\begin{eqnarray}
&&\left\vert c_{\bar{b}}^{(i_{3})}\otimes c_{b}^{(i_{3})}\otimes \phi
_{b}^{(i_{3})}\right\rangle \nonumber \\
&=&\frac{1}{2}\sum\limits_{x_{b}^{(i_{3})}=0}^{1}\{(\sum\limits_{x_{\bar{b
}^{(i_{3})}=0}^{1}\left\vert x_{\bar{b}}^{(i_{3})}\right\rangle _{c_{\bar{b
}^{(i_{3})}})\otimes \left\vert x_{b}^{(i_{3})}\right\rangle
_{c_{b}^{(i_{3})}} \nonumber \\
&&\otimes \frac{1}{\sqrt{2}}[(-1)^{x_{b}^{(i_{3})}}\left\vert
bb\right\rangle +\left\vert 22\right\rangle ]\} \nonumber \\
&=&\frac{1}{\sqrt{2}}\left\vert +\right\rangle _{c_{\bar{b
}^{(i_{3})}}\otimes \sum\limits_{x_{b}^{(i_{3})}=0}^{1}\{\left\vert
x_{b}^{(i_{3})}\right\rangle _{c_{b}^{(i_{3})}} \nonumber \\
&&\otimes \frac{1}{\sqrt{2}}[(-1)^{x_{b}^{(i_{3})}}\left\vert
bb\right\rangle +\left\vert 22\right\rangle ]\} \nonumber \\
&=&\frac{1}{\sqrt{2}}\left\vert +\right\rangle _{c_{\bar{b
}^{(i_{3})}}\otimes \lbrack \left\vert -\right\rangle
_{c_{b}^{(i_{3})}}\otimes \left\vert bb\right\rangle +\left\vert
+\right\rangle _{c_{b}^{(i_{3})}}\otimes \left\vert 22\right\rangle ].
\end{eqnarray
We can see that if Alice finds the outcome of her measurement on
c_{0}^{(i_{3})}$ ($c_{1}^{(i_{3})}$) is $\left\vert -\right\rangle $, then
she knows with certainty that Bob's choice is $b=0$\ ($b=1$). This will
occur with the probability $1/2$. On the other $1/2$ case, the outcomes of
Alice's measurements on both $c_{0}^{(i_{3})}$ and $c_{1}^{(i_{3})}$ are
\left\vert +\right\rangle $, thus she has to guess the value of $b$ by
herself. Therefore, the average probability that Alice can learn Bob's $b$
correctly is still $P_{Alice}^{\ast }=3/4$, which is the same as that of the
original CKS protocol. As there is always $P_{Bob}^{\ast }\geq 1/2$ for any
protocol, we can see that the security bound $2P_{Alice}^{\ast
}+P_{Bob}^{\ast }\geq 2$ still holds in the current case.
\subsection{Security against individual attacks}
However, the above cheating requires the computational power to perform
collective operations on many qubits/qutrits. More rigorously, equations
\ref{C}) and (\ref{T}) indicate that at the end of step B1, in every triple
run Alice needs to make $6$ qubits and $3$ qutrits entangled together, even
if Bob's half of his two-qutrit states is not counted. Here we will show
that if Alice is limited to individual measurements, then the protocol can
be secure.
In this scenario, during each run of the CKS protocol, Alice is not allowed
to perform collective operations to entangle the qutrit $\beta $ she
received from Bob with her quantum ancillary system anymore. What she can do
is to handle $\beta $ individually. In general, such operations can be
modeled as a channel $C_{\alpha \beta }$ which takes $\beta $ as an input,
then outputs a single qutrit state and a classical register $\alpha $
containing her measurement outcome. The effect of the channel $C_{\alpha
\beta }$ can be written a
\begin{equation}
C_{\alpha \beta }(\left\vert e_{ini}\right\rangle _{\alpha }\left\vert
j\right\rangle _{\beta })=\sum\limits_{j^{\prime }=0}^{2}\lambda
_{jj^{\prime }}\left\vert e_{jj^{\prime }}\right\rangle _{\alpha }\left\vert
j^{\prime }\right\rangle _{\beta },
\end{equation
where $j=0,1,2$, and $\left\vert e_{ini}\right\rangle _{\alpha }$\ (
\left\vert e_{jj^{\prime }}\right\rangle _{\alpha }$) is the initial (final)
state of $\alpha $. Comparing with equations (\ref{resultant2}) and (\ref{U
), we can see that $C_{\alpha \beta }$ is actually a special case of the
general cheating operation $T$ studied in the proof of Theorem 1. The
specialty in the current case is that $\alpha $ is classical, while in
equation (\ref{resultant2}) it can be either classical or quantum.
Therefore, by formulating the resultant state of $C_{\alpha \beta }$ as
equation (\ref{resultant2}) and repeating the same proof in Sect. III.A, we
find that the result of Theorem 1 still applies here. That is, if Alice can
guess $b$ with nonzero bias by applying channel $C_{\alpha \beta }$ to Bob's
qutrit $\beta $ and Bob never aborts, then she cannot learn $x_{b}$ with
reliability $1$.
Moreover, as $\alpha $ is a classical register, there will be no alternative
bases for measuring it. That is, once Alice decides on what kind of channel
to apply, then the measurement basis for $\alpha $ is also fixed. No matter
when Alice will measure $\alpha $ and extract the information stored in it,
this information is already a deterministic classical object after the
channel is applied, and there is only one choice of the basis for extracting
it. This is different from an unlimited quantum attack, where Alice can
apply the cheating operation and delay the measurement, then at a later
time, if she wants to learn $x_{b}$, she measures $\alpha $ in a certain
basis, while if she wants to learn $b$, she measures $\alpha $ in another
basis. In the current case, even if the measurement could be delayed, there
is only one basis for Alice (otherwise it will become a collective attack).
Therefore, Alice needs to determine beforehand which basis to use, and picks
the corresponding channel to apply. Dishonest-Alice will surely choose a
basis which enables her to learn Bob's $b$ with a nonzero bias, because this
is the goal of her cheating. But then Theorem 1 guarantees that she cannot
know\ with certainty the value of $x_{b}$ that Bob actually obtained.
Consequently, if this run of the CKS protocol is picked for the security
check in Protocol B, Alice will stand a non-vanishing probability
\varepsilon $ to either announce a wrong value of $x_{b}$ or cause Bob to
abort (in case his measurement outcome is neither $\Pi _{0}$ nor $\Pi _{1}$
in step 5 of the CKS protocol).
Now suppose that Alice chooses to cheat in $pn$ ($1/n\leq p\leq 1$) runs of
the CKS protocol. While Alice can apply different strategies in these runs
so that the value of $\varepsilon $ can vary, we can define $\varepsilon
_{m} $\ as the minimum of $\varepsilon $ in any run that Alice cheats. Thus
1-\varepsilon _{m}$\ is the maximal probability for Alice to pass Bob's
check in a single run. Since at the end of Protocol B, $n-1$ runs of the CKS
protocol will be checked, there can be two possibilities.
(I) The only one run that is not checked is picked among the $pn$ runs that
Alice cheats. Since this run is used for encoding the target bits in step
B3, Alice can gain a non-trivial amount of information on $b$. As the CKS
protocol ensures that Alice can learn $b$ correctly with the probability
3/4 $ at the most, and the other $pn-1$ runs that Alice cheats are all
checked, the maximal probability for Alice to learn $b$ correctly and pass
the checks successfully in this case is\
\begin{equation}
P_{Alice}^{I}\leq \frac{3}{4}(1-\varepsilon _{m})^{pn-1}.
\end{equation}
(II) The only one run that is not checked is not picked among the $pn$ runs
that Alice cheats. As Alice acts honestly in this run, she can only get $b$
by guess, which can be correct with the probability $1/2$. Meanwhile, all
the $pn$ runs that Alice cheats are checked. Thus the maximal probability
for Alice to learn $b$ correctly and pass the checks successfully in this
case is\
\begin{equation}
P_{Alice}^{II}\leq \frac{1}{2}(1-\varepsilon _{m})^{pn}.
\end{equation}
Note that cases (I) and (II) occur with the probabilities $p$ and $1-p$,
respectively. Thus\ the total probability for Alice to pass the check while
learning $b$ correctly is\
\begin{eqnarray}
P_{Alice}^{\ast } &=&pP_{Alice}^{I}+(1-p)P_{Alice}^{II} \nonumber \\
&\leq &\frac{3}{4}p(1-\varepsilon _{m})^{pn-1}+\frac{1}{2
(1-p)(1-\varepsilon _{m})^{pn}.
\end{eqnarray
Sinc
\begin{eqnarray}
&&\frac{\partial }{\partial p}(\frac{3}{4}p(1-\varepsilon _{m})^{pn-1}+\frac
1}{2}(1-p)(1-\varepsilon _{m})^{pn}) \nonumber \\
&=&((\frac{3}{4}p+\frac{1}{2}(1-p)(1-\varepsilon _{m}))n\ln (1-\varepsilon
_{m}) \nonumber \\
&&+\frac{3}{4}-\frac{1}{2}(1-\varepsilon _{m}))(1-\varepsilon _{m})^{pn-1}
\nonumber \\
&<&0,
\end{eqnarray
higher $P_{Alice}^{\ast }$ can be obtained by lowering $p$. The lowest
nonzero $p$ is $p=1/n$, i.e., Alice cheats in $pn=1$ run only and hopes that
she is so lucky that this run is finally picked for encoding the target
bits. In this case\
\begin{eqnarray}
P_{Alice}^{\ast } &\leq &\frac{1}{2}+\frac{1}{4n}-\frac{\varepsilon _{m}}{2
(1-\frac{1}{n}) \nonumber \\
&\leq &\frac{1}{2}+\frac{1}{4n}.
\end{eqnarray
As a result, for any arbitrarily small positive constant $\zeta $, Bob can
choose $n>1/(4\zeta )$ and ask Alice to perform the corresponding Protocol
B, which can achieve $P_{Alice}^{\ast }<1/2+\zeta $.
On the other hand, Bob's cheating probability remains the same as that of
the CKS protocol. This is because in any useful run, the values of
x_{0}^{(i_{1})}$, $x_{1}^{(i_{1})}$, $x_{0}^{(i_{2})}$, $x_{1}^{(i_{2})}$
that Alice revealed are always $0$. As set $S$ is defined as
S=\{000,001,010,100\}$, any value of $x_{0}^{(i_{3})}$ and $x_{1}^{(i_{3})}
\ remains possible to Bob unless Alice reveals them. Thus the values of
x_{0}^{(i_{1})}$, $x_{1}^{(i_{1})}$, $x_{0}^{(i_{2})}$, $x_{1}^{(i_{2})}$ in
a useful run do not provide any information for Bob to deduce
x_{0}^{(i_{3})}$ and $x_{1}^{(i_{3})}$. Also, the values of $x_{0}^{(i)}$,
x_{1}^{(i)}$ in different triple runs are chosen independently, so that the
specific $x_{0}^{(i_{3})}$, $x_{1}^{(i_{3})}$\ finally chosen for encoding
the target bits $x_{0}$, $x_{1}$ are not affected by any $x_{0}^{(i)}$,
x_{1}^{(i)}$ from all the other runs. Consequently, Bob still has to decode
the target bits via the corresponding run of the CKS protocol, without any
help from other runs. Therefore, his cheating probability in our Protocol B
is still $P_{Bob}^{\ast }=1/2$, as what can be obtained in a single run of
the original CKS protocol \cite{qbc14}.
Putting things together, we can see that when Alice is limited to individual
measurements, in our Protocol B $2P_{Alice}^{\ast }+P_{Bob}^{\ast }$ can be
made arbitrarily close to $3/2$, which is the maximal violation of the
security bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2$ since the minimums
for $P_{Alice}^{\ast }$ and $P_{Bob}^{\ast }$\ are both $1/2$.
\subsection{Security against limited collective attacks}
If Alice is allowed to perform collective operations but it is restricted to
a limited number of quantum systems only, then the security bound
2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 2$ can also be violated to a certain
degree.
Here we consider the case where Alice's collective operations are limited to
the quantum systems in the same run of the CKS protocol only, i.e., the
qutrit $\beta ^{(i)}$ that Bob sends to her and the ancillary system she
introduces (e.g., it can contain the two control qubits $c_{0}^{(i)}$,
c_{1}^{(i)}$ for keeping $x_{0}^{(i)}$, $x_{1}^{(i)}$ at the quantum level).
She can still apply the transformation defined in equation (\ref{T}) or
other operations on these systems for cheating. Our discussion below will
remain valid as long as this ancillary system cannot be entangled with the
ancillary system for any other run (e.g., equation (\ref{C}) is not allowed).
In this scenario, since the potential cheating strategies could be
innumerous and much more complicated than the individual attacks, it is hard
to prove the exact security bound of our protocol. But at least here we can
obtain the loose upper and lower bounds of the security, which is $5/3\leq
2P_{Alice}^{\ast }+P_{Bob}^{\ast }<2$. It means that the probability for
successful cheatings is higher than that of the individual attacks, but it
still violates the security bound for the unlimited collective attack.
Let us prove the upper bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }<2$\ first.
After the end of step B1, from Alice's point of view, the general form of
the state of Alice's and Bob's combined system for each single run of the
CKS protocol i
\begin{eqnarray}
\left\vert \alpha \otimes \beta \otimes \beta ^{\prime }\right\rangle
&=&\lambda _{b}^{(0)}\left\vert e_{b}^{(0)}\right\rangle _{\alpha
}\left\vert \phi _{b}\right\rangle _{\beta \beta ^{\prime }}+\lambda
_{b}^{(1)}\left\vert e_{b}^{(1)}\right\rangle _{\alpha }\left\vert \phi
_{b}^{\prime }\right\rangle _{\beta \beta ^{\prime }} \nonumber \\
&&+\lambda _{b}^{(2)}\left\vert e_{b}^{(2)}\otimes \phi _{b}^{\prime \prime
}\right\rangle _{\alpha \beta \beta ^{\prime }},
\end{eqnarray
where the notations are the same as those in the proof of Theorem 1, with
the additional $\left\vert e_{b}^{(2)}\otimes \phi _{b}^{\prime \prime
}\right\rangle _{\alpha \beta \beta ^{\prime }}$, which represents the state
orthogonal to both $\left\vert e_{b}^{(0)}\right\rangle _{\alpha }\left\vert
\phi _{b}\right\rangle _{\beta \beta ^{\prime }}$ and $\left\vert
e_{b}^{(1)}\right\rangle _{\alpha }\left\vert \phi _{b}^{\prime
}\right\rangle _{\beta \beta ^{\prime }}$. Note that the actual system may
already collapse to one of the terms at the right hand side of the equation
due to Bob's measurement on $\beta \otimes \beta ^{\prime }$. But Alice can
still treat the whole state as the entangled form in this equation if she
has not measured $\alpha $. This is because Alice's and Bob's local
operations are commutable for the entangled system $\alpha \otimes \beta
\otimes \beta ^{\prime }$, so that it does not matter mathematically who
performs the measurement first.
Since Bob learns that $x_{b}^{(i)}=0$ ($x_{b}^{(i)}=1$) if he gets
\left\vert \phi _{b}\right\rangle _{\beta \beta ^{\prime }}$\ ($\left\vert
\phi _{b}^{\prime }\right\rangle _{\beta \beta ^{\prime }}$), otherwise he
aborts, the above equation can be understood a
\begin{eqnarray}
\left\vert \alpha \otimes \beta \otimes \beta ^{\prime }\right\rangle
&=&\lambda _{b}^{(0)}\left\vert e_{b}^{(0)}\right\rangle _{\alpha
}\left\vert x_{b}^{(i)}=0\right\rangle _{\beta \beta ^{\prime }}+\lambda
_{b}^{(1)}\left\vert e_{b}^{(1)}\right\rangle _{\alpha }\left\vert
x_{b}^{(i)}=1\right\rangle _{\beta \beta ^{\prime }} \nonumber \\
&&+\lambda _{b}^{(2)}\left\vert e_{b}^{(2)}\otimes abort \right\rangle
_{\alpha \beta \beta ^{\prime }},
\end{eqnarray
Comparing with equation (\ref{resultant2}), it is even more general since it
also includes the case where Bob may abort. Now if none of the coefficients
\lambda _{b}^{(0)}$\ and $\lambda _{b}^{(1)}$\ equals exactly to $1$, then
the value of $x_{b}^{(i)}$ is kept at the quantum level. That is, it will be
determined by the uncertainty in quantum measurement, so that Alice cannot
control with certainty which value can be\ obtained by Bob. Else if one of
\lambda _{b}^{(0)}$\ and $\lambda _{b}^{(1)}$\ equals exactly to $1$, then
the other one and $\lambda _{b}^{(2)}$\ obviously have to be zero, and the
value of $x_{b}^{(i)}$ becomes classically deterministic.
After the end of step B1, suppose that the values of $x_{0}^{(i)}$,
x_{1}^{(i)}$ in $pn$ ($n=3k$, $0\leq p\leq 1$) runs of the CKS protocol in
Protocol B are kept at the quantum level (the values of $\lambda _{b}^{(0)}
\ and $\lambda _{b}^{(1)}$\ depend on Alice's specific strategy, which can
be different in each run). In the rest $(1-p)n$ runs, $x_{0}^{(i)}$,
x_{1}^{(i)}$ are no longer kept at the quantum level after step B1, but take
deterministic classical values instead, so that Alice can ensure that the
values of both $X_{0}$ and $X_{1}$ are presented in set $S$. Then these
(1-p)n$ runs are in fact executed honestly, as Theorem 1 ensures that Alice
cannot use them to decode Bob's $b$. She can get $b$ only if one of the
other $pn$ runs of the CKS protocol dishonestly executed is picked in step
B3 for encoding the target bits $x_{0}$ and $x_{1}$ to complete the weak OT.
This will occur with the probability $p$. Even if Alice uses the optimal
cheating strategy so that she can still learn Bob's $b$ with the probability
$3/4$ (which is the maximum that can be obtained in the CKS protocol) for
such a single dishonest run, the probability for (this run to be chosen)
\textit{and} ($b$ is learned correctly) will drop down to $(3/4)p$. If any
other non-optimal cheating strategy was used in this run, the probability is
limited by this value too. In the rest $(1-p)$ occasions where one of the
(1-p)n$ honestly executed runs is chosen for encoding the target bits, Alice
can only get Bob's $b$ by guessing which has probability $1/2$ to be
correct. Therefore, the total probability for Alice to cheat successfully in
our Protocol B is bounded b
\begin{eqnarray}
P_{Alice}^{\ast } &=&[\frac{3}{4}p+\frac{1}{2}(1-p)]p_{c} \nonumber \\
&=&(\frac{1}{2}+\frac{1}{4}p)p_{c}, \label{pa}
\end{eqnarray
where $p_{c}$\ is the probability that Alice can pass the security checks.
Finding the tight bound for $P_{Alice}^{\ast }$\ requires a rigorous
evaluation of $p_{c}$, which will depend on the specific cheating strategy
Alice applies on the $pn$ runs. But we can show that there will always be
P_{Alice}^{\ast }<3/4$. This is because if $p<1$, for a loose evaluation we
can simply take the maximum $p_{c}=1$ which surely covers any strategy. Then
$P_{Alice}^{\ast }=1/2+p/4<3/4$. On the other hand, consider the case $p=1$.
As the collective attacks are limited to the quantum systems in each single
run of the CKS protocol, the values of $x_{0}^{(i)}$, $x_{1}^{(i)}$ in
different runs will not correlate with each other. Then in the current case,
since $x_{0}^{(i)}$, $x_{1}^{(i)}$ in all the $n$ runs are kept at the
quantum level, any one of them can turn out to be either $0$ or $1$ during
the measurement in the security check. The outcome is determined
independently in each single run by quantum uncertainty, thus Bob cannot
ensure with probability $100\%$\ that $X_{0}$ and $X_{1}$ will always take
the legitimate values in set $S$ in every single run in the security check.
Therefore, we have $p_{c}<1$ when $p=1$. Then equation (\ref{pa}) gives
P_{Alice}^{\ast }=(1/2+1/4)p_{c}<3/4$. Namely, no matter $p<1$ or $p=1$,
P_{Alice}^{\ast }$ cannot equal exactly to $3/4$, i.e., it is always lower
than that of the original CKS protocol.
Meanwhile, Bob's cheating probability still equals to that of the CKS
protocol, i.e., $P_{Bob}^{\ast }=1/2$, since the specific $x_{0}^{(i_{3})}$,
$x_{1}^{(i_{3})}$\ finally chosen for encoding the target bits $x_{0}$,
x_{1}$ are not affected by any $x_{0}^{(i)}$, $x_{1}^{(i)}$ from all the
other runs, as it is elaborated in the previous subsection. Combining this
P_{Bob}^{\ast }$\ with $P_{Alice}^{\ast }<3/4$, we can see that under the
limited collective attack, our Protocol B can obtai
\begin{equation}
2P_{Alice}^{\ast }+P_{Bob}^{\ast }<2\times \frac{3}{4}+\frac{1}{2}.
\label{protocol B}
\end{equation
Thus the upper bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }<2$\ is proven.
Now we prove the lower bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq 5/3$.
This is because there exists the following cheating strategy for Alice. In
every triple run, she only keeps one pair of $x_{0}^{(i)}$, $x_{1}^{(i)}$ (
i=1,2,3$) at the quantum level by using the collective operation described
by equation (\ref{T}). The other two pairs of $x_{0}^{(i)}$, $x_{1}^{(i)}$
are all taken as $0$ beforehand, and the corresponding two runs of the CKS
protocol are executed honestly. Then all triple runs can pass the security
check with certainty. Meanwhile, when a useful run is finally picked for
encoding the target bits, the pair $x_{0}^{(i)}$, $x_{1}^{(i)}$ kept at the
quantum level stands a probability $1/3$ to be chosen. Thus we have $p=1/3$.
Substitute it into equation (\ref{pa}) and we yiel
\begin{equation}
2P_{Alice}^{\ast }+P_{Bob}^{\ast }=\frac{5}{3}, \label{lower}
\end{equation
so that this lower bound can be reached even when Alice is restricted to the
limited collective attack. But we do not know whether this bound is tight at
the present moment, as it is unclear whether there may exist an even better
cheating strategy.
\section{Potential improvements}
By observing the above cheating strategy that led to the lower bound
equation (\ref{lower}), we can see that the probability $p=1/3$ comes from
the specific set $S$ used in Protocol B, which is made of $3$-bit strings
only. In the more general form, i.e., our Protocol A, we can expect that
choosing a more complicated set $S$ may further reduced the value of $p$.
For example, set $S$ can be chosen as a classical error-correcting code,
e.g., the binary linear $(n,k,d)$-code \cite{qi43}. That is, $S$ is taken as
a set of classical $n$-bit strings. Each string is called a codeword. This
set of strings has two features. (a) Among all the $2^{n}$ possible choices
of $n$-bit strings, only a particular set of the size $\sim 2^{k}$\ ($k<n$)
is selected to construct this set. (b) The distance (i.e., the number of
different bits) between any two codewords in this set is not less than $d$\
$d<n$). With these features, it can be expected that by increasing $n$ while
fixing $k/n$ and $d/n$, a dishonest Alice will have to introduce a much
larger number of entangled control qubits for keeping more pairs of
x_{0}^{(i)}$, $x_{1}^{(i)}$ at the quantum level, so that
X_{0}=x_{0}^{(1)}x_{0}^{(2)}...x_{0}^{(i)}...x_{0}^{(n)}$ and
X_{1}=x_{1}^{(1)}x_{1}^{(2)}...x_{1}^{(i)}...x_{1}^{(n)}$ will appear as
legitimate strings in set $S$ no matter which bits are picked for the
security check. Therefore with a properly chosen $S$, Protocol A may further
lower the successful probability of limited collective attacks, and also
raises the difficulty of implementing these attacks. However, the rigorous
security bound will depend heavily on the structure of the specific set $S$
used in the protocol. This analysis is left for future research.
\section{Summary and remarks}
Thus we show that the security bound $2P_{Alice}^{\ast }+P_{Bob}^{\ast }\geq
2$ for weak OT can be violated for an Alice with limited computational
power. As a rigorously checkable example, we proposed Protocol B which
reaches the maximal violation $2P_{Alice}^{\ast }+P_{Bob}^{\ast }\rightarrow
3/2$ when only individual measurements are allowed. For attacks using
collective operations on a limited number of quantum systems, there can
still be $P_{Alice}^{\ast }<3/4$ while $P_{Bob}^{\ast }=1/2$. An even lower
value of $P_{Alice}^{\ast }$ could be expected from Protocol A.
Note that Ref. \cite{qbc88} obtained the security bound without limiting
Alice to individual measurements. Thus our protocols does not really break
the bound in principle. But it still has great practical significance. This
is because in practice, any quantum storage devices can keep the quantum
states faithfully for a limited time only. Suppose that $\tau $ is the
maximal storage time available with state-of-the-art technology. Then during
step B1 of Protocol B, Bob can require that every run of the CKS protocol is
separated from each other by a time interval larger than $\tau $, so that
any ancillary system that a dishonest Alice may introduce to entangle with
Bob's qutrit will suffer from errors, making Alice unable to pass the
security check. In this case, Alice has to finish the measurement on Bob's
qutrit (if she does not want to perform the honest unitary transformation)
in each single run of the CKS protocol before the next run begins. Thus her
cheating is actually reduced to individual measurements. So we can see that
as long as our protocol is proven secure against individual measurements,
then it naturally implies that we can use it as a secure protocol in
practice.
This work was supported in part by
the NSF of Guangdong province.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,864
|
professional bodies, trade unions, farmers and people living with HIV/AIDS.
HIV/AIDS, a representative from the Coalition Against Privatization of Healthcare S!
end, there was a closing session with a summary of conclusions.
This report summarizes the key views and conclusions of the Workshop.
such as in services, investment, government procurement and intellectual property.
future Malaysia-US FTA on the basis of existing US FTAs.
provisions related to investment, government procurement and competition policy.
participation are important, given the possible far-reaching effects on society.
by 2010 under an FTA.
policy instrument to protect against artificially cheap products coming in from the US.
difficulties for its trading partners.
fostering a mature industrialized economy.
by committing at the WTO or in an FTA.
instruments, are automatically opened to US competition.
sectors as non-tariff barriers and areas where it wants to see market access.
foreign investors such as transfer of technology.
prevent or reduce their policy space to determine their own investment policies.
provisions, due to banning of dangerous chemicals.
crisis, due to its requirements to allow free transfers of capital.
companies to bid on equal terms with local companies for government contracts.
Intellectual Property Rights (TRIPS) and is afforded flexibilities in its implementation.
obliged under the TRIPS Agreement) to 70 or 95 years after the death of the author.
and allow five-year monopolies even when there is no patent, via data exclusivity.
spending to more than double in the first year ! of their US FTA and get worse after that.
meet patentability criteria, but allows the prohibition of patenting of plants and animals.
amended if Malaysia ratifies UPOV 1991.
farmers giving their views on the proposed Malaysia-US FTA.
macroeconomic stability, national development and social equity.
Most of the manufacturers of pharmaceuticals in Malaysia are generics manufacturers.
as generic medicines are cheaper.
shown in the case of Mexico under NAFTA.
many more will die if medicine prices are increased.
equitable distribution of healthcare could be prevented.
environmental laws. In addition, there are many other environmental concerns that arise.
could be abused in the name of protecting the environment.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,254
|
Altair acquires S-FRAME analysis and design software
A model of a multistory hybrid mass timber building in S-FRAME's S-TIMBER software object view (left) and the resultant column force view (right)
Altair, a global leader converging simulation, HPC, and AI, has announced its acquisition of S-FRAME Software, a structural analysis software platform used by engineers to evaluate a structure's ability to withstand external loads (like wind, water, and snow) and meet design code requirements around the world.
For more 30 years, high-end architectural clients have utilised Altair's sophisticated technology – which is also used to design high-tech modern aircraft and automobiles – to develop some of the world's most innovative and iconic structures. Utilising S-FRAME Software's finite element structural analysis and code support with Altair's high-fidelity structural optimisation solutions will allow architects and civil engineers to be more innovative and bring their visions to life while adhering to local code requirements.
According to CIMdata, the leader in product lifecycle (PLM) education and research, "the AEC segment will be one of the fastest-growing segments of the PLM market with a 14.7 percent CAGR to $11 billion in 2025".
"We are excited to expand our presence in the rapidly growing AEC market to lead the transformation from rule-based design to predictive, high-performance design, fueled by simulation," said James R. Scapa, founder and chief executive officer, Altair. "By using highly sophisticated performance-based, predictive simulation, architects and designers will be able to be more creative, fueling innovation while increasing safety."
As an AEC innovator for many years, S-FRAME Software's integrated solutions make it possible to analyse, design, and detail structures regardless of geometric complexity, loading conditions, nonlinear effects, design code requirements, or material type including steel, concrete, composites, or mass timber – one of the hottest in sustainable building materials. S-FRAME Software has been a leading advocate for the use of timber in building design as the material offers sustainability through greenhouse gas reduction and other environmental benefits like insulation as well as architectural aesthetics. S-FRAME Software also allows users to produce comprehensive and detailed design reports that include clause references and intermediate results for multiple design code support.
"Joining the Altair team is a strategic way to accelerate our growth and become part of a full solutions suite with their advanced structural design offerings," said Marinos Stylianou, chief executive officer, S-FRAME Software. "We are confident our combined offerings will be able to empower engineers, designers, and builders to reach new heights."
Founded in 1981, S-FRAME Software is based in British Columbia, Canada, and serves a global client base.
S-FRAME Software will be available via Altair Units, Altair's patented, subscription-based licensing model, which allows organizations to pay only for what their employees need, when they need it.
Previous articleCortec presents MilCorr® VpCI® Shrink Film for extreme outdoor climates
Next articleFurniture China 2021 postponed to December
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,362
|
## CONTENTS
1. Introduction
2. UNIT 01: CREATING WITH KIDS AND CLAY
1. How This Book Is Organized
2. Basic Supplies
3. Varieties of Clay
4. Other Supplies
5. Tips for Working with Kids and Clay
6. Cleanup and Caring for Your Supplies
3. UNIT 02: AIR-DRY CLAYS AND LEARNING THE BASICS
1. Lab 1: Shoe-Stamped Pottery
2. Lab 2: Coffee and a Donut
3. Lab 3: Textured Landscape Plaque
4. Lab 4: Cupcake Containers
5. Lab 5: Stack of Pancakes
6. Lab 6: Coil Press Pot
7. Lab 7: Monogram Wall Hanging
8. Lab 8: Dropped Cone Sculpture
9. Lab 9: Wax Resist Leaf Dish
10. Lab 10: Line Relief Pencil Cup
11. Lab 11: Hamburger Holder
4. UNIT 03: CLAY SCULPTURE
1. Lab 12: Clay Coat of Arms
2. Lab 13: Pie Safe
3. Lab 14: Egyptian Sarcophagus
4. Lab 15: Magical Forest Friend Door
5. Lab 16: Miniature Cactus Garden
6. Lab 17: Garden Gnome Home
7. Lab 18: Bobblehead Pets
8. Lab 19: Castle on the Hill
9. Lab 20: Crayon and Pencil Sculpture
10. Lab 21: Woodland Creature Portraits
11. Lab 22: Viking Ships
12. Lab 23: Marionette Puppets
13. Lab 24: Animals Masquerade Masks
5. UNIT 04: POLYMER CLAY AND LEARNING THE BASICS
1. Lab 25: Crystallized Pinch Pots
2. Lab 26: Color Mixing Clock
3. Lab 27: Fun Fortune Cookies
4. Lab 28: Light-Up Lightning Bugs
5. Lab 29: All-About-Me Mobile
6. UNIT 05: POLYMER CLAY SCULPTURE
1. Lab 30: Tie-Dye Turtles
2. Lab 31: Miniature Accordion Books
3. Lab 32: Sushi for Supper
4. Lab 33: Glitter Bugs
5. Lab 34: Monster Magnets
6. Lab 35: Day of the Dead Heads
7. Lab 36: Backpack Charms
8. Lab 37: Game Pieces and Dice for Your Own Game
9. Lab 38: Let Your Light Shine Nightlight
10. Lab 39: Funky Faux-saic
11. Lab 40: Desktop Dinos
7. UNIT 06: MAKING YOUR OWN CLAY
1. Lab 41: Simple No-Cook Clay
2. Lab 42: Stamped Clay Ornaments
3. Lab 43: Scented Clay Acorns
4. Lab 44: Make Your Own Play Dough
5. Lab 45: Clay Ice Pops
6. Lab 46: Jurassic Fossils with Sand Clay
7. Lab 47: Glitter Clay Beads
8. Lab 48: Make Your Own Silly Putty
9. Lab 49: Salt Dough Recipe and Rose Sculpture
10. Lab 50: Salt Dough Bird Sculpture
11. Lab 51: Salt Dough Texture Fish
12. Lab 52: Candy Clay
8. Resources
9. Acknowledgments
10. About the Author
11. Index
## Guide
1. Cover
2. Start of Content
3. Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
## INTRODUCTION
Nearly twenty years ago, I was in college, filling my days with studio art classes and my nights with hours of painting. While it was an absolute dream come true for me, my parents, who were paying for my education, weren't so certain the dream would become a reality. When my parents suggested I also take some art education classes, I was seriously doubtful. What did I know about teaching art to children? Fast-forward to the present: I've been teaching all this time. I still have much to learn, but I know one thing for certain: Kids love to create as much as I do.
If you were to ask the hundreds of children I've taught over the years what their favorite art activity is, they'd respond with an enthusiastic "clay!" There's something about working with your hands, creating something three-dimensional and functional, not to mention getting messy, that really captures every child's heart and imagination. When my students come back to visit, I always hear that they still have that coil mug, clay fish, or snowman sculpture on display in their homes.
In this book, we'll explore the many different clay mediums available that do not require kiln fire. From air-dry and paper clays to polymer and clays you can make with household ingredients, this book introduces 52 clay labs to experience and inspire. Have fun and happy creating!
##
CREATING WITH KIDS AND CLAY
Children are born artists.
They instinctually know how to dig their fingers into clay or to put crayon to paper (or the walls in your home, if you're not careful!) and create something magical. You might not understand the images you see, but when children proudly explain it to you, their imagination is evident.
This book is for children who love to get a little messy, work with their hands, and create memories. All that's to say: This book is for the artist who lives in us all!
### HOW THIS BOOK IS ORGANIZED
This book is divided by units of non-fired clays: air-dry, polymer, and homemade. When creating with these various types of clay bodies, young artists will notice the differences between them and may take a liking to one over the other. Some clay is water-based and sticky, while others are not. Despite the differences in ingredients and texture, almost all of the clay used in these labs is interchangeable. Go ahead and explore, using your preferred clay in any lab!
These labs allow kids to explore each medium, learn new techniques, and be inspired. Each creative idea leads to another, picking up momentum like a ball rolling downhill. However, the labs do not have to be completed in the order presented. Allow your young artists to create what piques their interests. One child might want to create a project exactly as it's made in this book. Another might choose to take the idea in a completely new direction. All directions are good ones.
Within many of the labs, there are adaptive suggestions for the developing artist. When younger siblings work alongside an older brother or sister, it can be frustrating for the young ones to match what they perceive as a certain standard. Coax them gently. Teach them that developing strong hand muscles and skill takes time. Introduce them to the adaptive methods suggested. Most of all, relax, create, and have fun together!
### BASIC SUPPLIES
Let's talk about the tools and materials you'll need to start your own clay lab. Here is a basic list of what you'll need for your clay adventures:
#### A COVERED AND CLUTTER-FREE WORKSPACE
I like to work on my dining room table. However, I don't want the clay to stain or damage my table, so I always place a large sheet of clean white paper down before I work. A roll of paper is my preferred table covering because it allows me to simply tear off a sheet, work, and recycle it when finished. Newspaper, a plastic placemat, or a large sheet of cardboard would work as well. Canvas is a good choice, too, because it can be washed and reused.
#### DROP CLOTH
You'll need this if you're working in a room with carpeting. Dropped pieces of clay ground into the fibers of a carpet can be difficult to remove.
#### SMOCK OR APRON
The clays used in this book require only soap-and-water cleanup. However, young artists often have the habit of wiping their hands on their clothing. A smock or an apron will keep them a little cleaner.
#### BABY WIPES
For children who might not like the texture of some of the clays or the sensation of dried clay on their hands, baby wipes are a handy solution.
#### ROLLING PIN OR LARGE JAR
For these projects, you'll often need to flatten a slab of clay. A rolling pin will do the job, but a large jar works just as well. However, pounding clay flat is my students' favorite option. I tell them to aim for "cookie thickness." Once flat, flip the clay over to the smooth, non-pounded side. Allow your artists to pick their favorite flattening method.
#### TEXTURED MATERIALS
One of the fun things about clay is that it can capture texture. I use the soles of my shoes, burlap, old sweaters, and even lace to create texture. Save pieces of textured plastic, cardboard, knits, and more for playing with clay.
#### SKEWERS
Skewers are pointed wooden sticks for making kabobs. They're inexpensive and available at many grocery stores. We use them in our labs for cutting clay and poking holes. We also use them for the slip and score method of joining two pieces of clay.
Slip-and-Score
Slip is another word for watery clay and score means to scratch. To join two pieces, score the surfaces of both (a skewer works well), moisten the surfaces with water, and press the two pieces together. In this book, the slip-and-score method is done with toothbrushes and skewers.
#### OLD TOOTHBRUSH
When working with air-dry clays, toothbrushes work very well for slipping, scoring, and attaching clay pieces. To attach pieces of air-dry clay to one another, gently scrub both pieces of clay with a slightly dampened toothbrush. A toothbrush is also a handy tool for creating texture on clay.
#### WATER IN A SPILLPROOF CUP
With the exception of polymer clay, all of the clays in this book will need water as you work with them. Water acts as glue for many projects and is also useful for smoothing surface cracks. Keep just a small amount of water available in a shallow bowl to prevent spills. Dog bowls are great alternatives to water cups because they are created to be spillproof. You can find them inexpensively at your local dollar store.
#### TOOTHPICKS, PAPERCLIPS, PENCILS, AND STICKS
The scratched, poked, and embossed details of many projects in this book were created by using everyday household objects. Keep a creative eye open and put together a box of items that will add to your clay explorations.
#### GARLIC PRESS
An inexpensive garlic press is a fun tool to keep on hand. When used with air-dry clay, it creates a spaghetti-like texture that can be used for hair, beards, grass, or foliage.
#### DENTAL FLOSS
Some air-dry clay comes in thick blocks. An easy and effective way to cut clay from a large block is with dental floss. Simply wrap the floss around your index fingers and pull it through the clay to slice off a nice even piece.
#### BABY OIL
Polymer clay can dry out and become crumbly and hard to work with. If this happens, simply put a few drops of baby oil on your hands and massage it into the clay. As you work, the clay will return to its soft, pliable state.
#### SOAP, WATER, AND A GENTLE BRUSH
Some of the darker shades of polymer clay can temporarily stain hands. Use warm soapy water and a gentle brush to easily remove the coloring.
#### DULL BUTTER KNIFE
This is the perfect tool for cutting clay cleanly, with no danger to small hands.
### VARIETIES OF CLAY
The wide variety of clays to choose from can be overwhelming. The key is to allow your child to try several types. Each has its own texture and feel, qualities, and curing method. Exploring many options allows your artist to discover his or her favorites.
#### POLYMER CLAY
Polymer clays are widely available in a tempting variety of colors. But you don't have to purchase every polymer color to make your creations. In our Polymer Unit, we'll show you how to create your own colors as well as exploring the medium to its fullest!
Polymer clay is hardened by baking it in an ordinary kitchen oven or toaster oven. Directions vary by manufacturer, and the baking must be done with adult supervision. Overbaking can cause fumes. Be sure to allow polymer projects to cool thoroughly before handling.
Polymer clay is nontoxic, but all the same, take certain precautions with it. Kids should wash their hands after using it, and they should not try to taste it. Tools that are used for preparing food should not be used with the clay, and polymer crafts should not be used for serving food.
Store leftover polymer clay in a glass container. Alternatively, you can sort the clay by color in a paper egg carton. Don't use plastic containers or plastic wrap because they can cause a reaction with polymer clay.
#### AIR-DRY CLAY
Craft stores sell many varieties of air-dry clay. Each has a unique set of properties, but you don't need to purchase them all! Buy one at a time and explore each to its fullest to decide which you like best. Many of the labs lend themselves to several types of clay, not just the one suggested.
The down side to air-dry clay is that it takes a long time to dry, which can be difficult for impatient young artists. To expedite the drying process, set the clay in the sun on a warm day. While drying, be certain to turn the clay over and allow the back side to dry as well. If the clay is not rotated, mold can grow on the damp underside. When you're finished working with air-dry clay, wrap it tightly with plastic wrap and store it in a sealed bag. This will prevent the clay from drying out.
#### POLYFORM MODEL AIR/AIR-DRY MODELING CLAY
This clay comes in both white and terra cotta. When it dries, it looks much like kiln-fired clay, but projects made from this clay may take up to a week to dry. It must be kept in an airtight zipper-lock bag when not in use.
#### PLUS MODELING CLAY
This clay looks and feels like kiln-fired clay. It's sticky and messy, just like the real thing! Like the Polyform clay, it's water-based and should be kept in an airtight bag. Again, allow for several days of drying time and don't forget to rotate to avoid mold.
#### CREATIVE PAPERCLAY MODELING MATERIAL
This clay is very lightweight and fun to work with. It's prone to drying out as you work, so keep water on hand to smooth out cracks. Be sure to store it in an airtight zipper-lock bag.
#### HOMEMADE CLAY
What do you do when you don't have store-bought clay available to work with? You make your own, of course! In fact, a lot of the fun of homemade clay is making it. You will find that each recipe has its special crafting properties. You might enjoy some more than others, but one thing is for certain, you'll have a good time with it! Almost all of the homemade clay recipes are no-cook and no-bake. When baking is called for, be certain an adult supervises.
Each type of homemade clay has its own recipe. Read over each recipe carefully to be sure you have all of the necessary items. You'll also need a mixing bowl, measuring cups and spoons, an apron, and a clean work surface.
### OTHER SUPPLIES
#### WATERCOLOR, ACRYLIC, OR GOUACHE PAINTS
Just like clay, different paints have different qualities. For the beginner artist, watercolor is a wonderful choice because it is easy to use, mix into new colors, and clean up. Watercolor is translucent, which does not lend itself to complete coverage when painting. For that, acrylic paint works well. Small bottles of acrylic paint in a variety of colors are available at craft stores. However, acrylic doesn't always wash off textiles and other surfaces, so keep that in mind. Gouache is a versatile paint that offers the best of both worlds: it can be translucent or opaque depending on how it is applied. It is also washable.
Once you finish creating with air-dry or homemade clays, you might want to add color. Watercolors work nicely if you want the color of the clay to show through. Gouache is more opaque and allows you to blend colors as you would with watercolor. Acrylic is great for complete color coverage.
#### MOD PODGE OR CLEAR SEALER
To seal and protect finished pieces, brush on a matte or gloss sealer. When you're working with children, use a nontoxic, acrylic-based sealant such as Mod Podge. Acrylic-based sealants are fume free, they dry quickly, and they wash up with soap and water. Best of all, they are safe for children to use, too. But acrylic-based sealants are not water- and weatherproof, so make sure your young artists don't leave their finished projects outdoors.
### TIPS FOR WORKING WITH KIDS AND CLAY
There is always a learning curve when trying something new. This can cause frustration for young artists and lead to a chorus of "I can'ts." To prevent that from happening, chat with the children before each session. Let them know that they will be learning something new, which takes time and patience. Have a "no-negative-talk" rule during creativity time. A small change in vocabulary from "I can't" to "I can if I keep trying" makes a world of a difference.
When working with a group of children, teach them the buddy system. If they see a buddy who is struggling, allow them to reteach the instructions. I have found that children are often better at explaining directions to each other than I am because they speak the same language.
Don't do the work for the children. If you decide to create alongside of children, never pick up their pieces and work on them. This indicates that you are the artist and they aren't. Simply get a new piece of clay and go over the steps slowly with them on your own piece. They will be happier with a finished work of art that they created themselves.
Try to save your child's works of art, even if just temporarily. Not everything your child creates will be a masterpiece. However, even the lesser pieces are important in showing growth. Jot down the date the work was created. Later, gather up the projects and arrange them chronologically. Your little artist will be motivated to continue creating.
Air-dry and homemade clay can become fragile once dry if it is rolled or pinched too thin. I like to use the measurement "as thick as a cookie" for rolling or pounding clay flat. This rule does not apply to polymer clay.
A little goes a long way when working with clay. Instead of allowing your child to work with an entire package of air-dry or polymer clay, cut off a small portion for use and add more as needed.
### CLEANUP AND CARING FOR YOUR SUPPLIES
Working with clay and managing the mess is the art of controlled chaos. Cover the work surface, the floor if necessary, and your young artist. Take heart in knowing that these clays clean up with soap and water. Most of the fun is in the mess-making!
I'm a firm believer in having a dedicated space to create in. Even if it's a small desk, a corner of the dining table, or a TV tray, having a space available with the basic supplies, a variety of clays, and this book neatly stored will inspire kids to create.
Teaching good cleanup techniques will ensure that supplies have a long art-making life. Skewers, rolling pins, brushes, and palettes should be washed thoroughly. If time is an issue, have a plastic tub with warm soapy water available to allow tools to soak.
In my art room, my students work on a surface covered in paper with a plastic placemat on top. They use a damp soapy sponge to wipe down their placemats. After several days, the paper is recycled and replaced. This keeps tables clean and cleanup a snap.
Have a small broom and dust pan available for your child to sweep up any clay that may have fallen to the floor. A small handheld vacuum would work well for carpeted rooms.
##
AIR-DRY CLAYS AND LEARNING THE BASICS
A wide variety of air-dry clays
are available. It's fun to explore their different properties with the labs in this chapter. Some of your young artists will prefer one variety over others, so keep in mind that you can use any air-dry clay in any of the following labs.
Air-dry clay is water-based, so keep a spillproof cup of water on the work table in case the clay becomes dry and difficult to manipulate. Drying time varies for different brands of air-dry clay and for different thicknesses. Rotate finished projects as they dry to ensure even and thorough drying.
| SHOE-STAMPED POTTERY
---|---
One of the coolest things about clay is its ability to capture textures. In this lab, you'll experiment with creating textures in clay by using the indentations on the soles of shoes. Use as many shoes as you can find to create a variety of textures to decorate the rim of this unique piece of pottery. But beware: The dirt stuck in the bottoms of shoes will end up stuck in your clay, so use shoes with clean soles.
Tools & Materials
air-dry clay (I used paper clay for this lab.)
shoes
terra cotta pot
water
paintbrushes
acrylic paint
metallic acrylic paint
1 Begin by rolling several balls of clay in your hands. To roll a ball of clay, place it between your hands. Gently add pressure as you rotate your hands in circles. (Fig. 1)
2 Find shoes with clean soles that have a variety of patterns and textures. Firmly press each ball of clay into the indentations in the soles. Flatten each piece slightly and then peel it off. You'll see the texture in the clay. (Fig. 2 & 3)
3 Set out the terra cotta pot. Moisten the back of one of the pieces of clay with a little water and press it gently onto the rim of the pot. Continue until the rim is covered with overlapping stamped pieces. (Fig. 4)
4 Allow the pot to dry for several days. Paint the stamped pieces along the rim with acrylic paint. Once it's dry, gently paint over the color with a metallic acrylic paint to give it a bit of shine and show off the shoe-stamped texture. (Fig. 5)
| COFFEE AND A DONUT
---|---
In this lab, you'll learn many of the basics for sculpting with clay: how to create a pinch pot, a slab or a flattened piece of clay, and a coil or rolled piece of clay. You'll put all of these new skills together to create a coffee cup, saucer, and donut!
Tools & Materials
white water-based air-dry clay (I used Amaco Marblex.)
water
toothbrush
crayons
watercolor paint
paintbrushes
puffy paint
Mod Podge
1 Begin by placing a ball of clay the size of a small orange in the palm of your hand. With your other hand, place your thumb on top and your fingers around the back. Push your thumb firmly into the clay until it gets close to the bottom but does not go all the way through. (Fig. 1)
2 Remove your thumb from the clay. The opening you've created is the start of a pinch pot. To make the pot bigger, put your thumb back into the opening and use your fingers to pinch the sides. Turn the pot as you pinch until the sides and the bottom are the thickness of a cookie.
3 If the pinch pot loses its shape, make a 1" (2.5 cm) tear in the clay. Overlap the two torn edges and smooth the seam with your fingers. Repeat on the opposite side of the pot. (Fig. 2)
4 To make the handle for the cup, pinch off a piece of clay the size of a gumball. Roll it up and down in your hands to make a snake of clay. The snake should be as thick as your finger and even all the way across.
5 Roll one end of the snake to make a coil. Bend the snake into a handle shape and hold it up to your pinch pot. Trim the snake if necessary and attach it: Scratch the pinch pot with a wet toothbrush where you want the handle to go. Push the handle onto the scratched spot and smooth it until it sticks. Set the cup aside. (Fig. 3)
6 To create the plate for the donut, start with a piece of clay the size of a small orange. Roll it into a ball. Place it on your work surface and pound it flat. Stop pounding the clay when it's as thick as a cookie and level all the way across. Set it aside. (Fig. 4)
7 For the donut, start with a golf ball–sized piece of clay. Roll it up and down in your hands. Stop when it is 6" (15 cm) long. Bend it to make the letter _O_. To connect the ends for the _O_ , rub both with a wet toothbrush, overlap, and smooth together. Set aside all three clay pieces to dry for about a week. (Fig. 5)
8 Use crayons to draw designs and watercolors to add designs to the cup and plate and paint the donut. To make the donut look more realistic, add sprinkles or icing with puffy paint. Allow the paint to dry.
9 To seal and protect your treasures, add a coat of Mod Podge. (Fig. 6)
| TEXTURED LANDSCAPE PLAQUE
---|---
In this lab, you'll learn the skills to create a landscape of your choice, such as an underwater scene, a sunset on a beach, or a mountainous view—use your imagination! Hang your creation on the wall or show it off on a small easel.
Tools & Materials
air-dry clay (I used Amaco Marblex.)
skewer
textured fabric
rolling pin or jar
toothbrush
water
garlic press
watercolor paint or gouache
paintbrushes
Mod Podge
1 Start with a piece of clay the size of an orange. Roll or pound the clay into a flat slab that is even and as thick as a cookie. If you like, use the skewer to trim the edges into a shape, such as a rectangle or circle, or leave it rough. (Fig. 1)
2 Tear off a fresh piece of clay the size of a golf ball. Pound it flat onto the fabric. This piece can be paper thin. (Fig. 2)
3 Tear the textured clay in half lengthwise. This will be the "land" for your landscape. Gently place it on the first slab to see how it looks and then remove. Create hills and landscape features with different textures. (Fig. 3)
4 Attach the landscape pieces to the slab by scratching the surfaces with a wet toothbrush and pressing them gently into place. (Fig. 4)
5 Think about adding a sun, moon, stars, or clouds to the sky. Pound the clay flat and use a skewer to cut out these shapes. Use the toothbrush to attach them to the slab.
6 Place a small ball of clay in the back of the garlic press and squeeze to create grass. Attach it to the slab with a moistened toothbrush. (Fig. 5)
7 Consider adding a house, a family pet, or trees to the landscape. Again, cut those out of flattened clay, scrub the backs gently with a wet toothbrush, and attach. (Fig. 6)
8 If the plaque is to be hung, poke two holes at the top with the skewer. Allow it to dry for up to a week and then feed a string through the holes.
9 Paint the plaque with watercolor or gouache. Allow the paint to dry.
10 Seal and protect your plaque with Mod Podge. (Fig. 7)
| CUPCAKE CONTAINERS
---|---
Who says your clay projects can't look good enough to eat? These fun cupcakes aren't just for decoration. They're also containers that can hold your tiniest treasures. In this lab, you'll use the pinch pot and coil-making skills you have learned. Any air-dry clay would work well here.
Tools & Materials
air-dry clay (I used ACTIVA Plus Clay Natural Self-Hardening White.)
skewer
water
toothbrush
acrylic paint in several colors
paintbrushes
Mod Podge
1 Pinch off a piece of clay about the size of a golf ball. Roll it into a ball.
2 Place the ball in the palm of one hand. Press the thumb of your other hand firmly into the center of the ball. Be certain not to go all the way through the clay with your thumb. (Fig. 1)
3 You now have the start of a pinch pot. Begin to pinch the outer rim of the pot all the way around until it is as thick as a cookie. This will be the top of the cupcake. (Fig. 2)
4 Set the cupcake top on your work surface like a dome and get ready to decorate it. To create icing, flatten a piece of clay the size of a gumball between your fingers. Place it on the table. Use a skewer to carve icing shapes using a wavy line. Scrub the top of the pinch pot with a wet toothbrush and add the icing. (Fig. 3 & 4)
5 Create sprinkles by rolling small pieces of clay. Use the wet toothbrush to attach the sprinkles in place. Roll a small ball of clay to add to the top for a cherry. (Fig. 5)
6 For swirled icing, roll a long snake of clay. Roll it into a coil by bending it around itself. Attach the start of the coil to the top of the pinch pot. Then wrap the rest of the snake around the pinch pot. (Fig. 6)
7 To create the bottom of the cupcake, create a smaller pinch pot, so that the top of the cupcake fits over it. (If the bottom is too wide, follow step 3 in Lab 2, to make it narrower.) Press this pinch pot firmly onto the surface of a table so that it stands without toppling over. Using a skewer, draw lines to show the texture of a cupcake liner. (Fig. 7)
8 Allow the top and bottom of the cupcake to dry separately for several days.
9 Use acrylic paint to decorate the cupcake and all of the decorations. Allow the paint to dry. Seal the cupcake with Mod Podge.
| STACK OF PANCAKES
---|---
A tall stack of pancakes is a delicious treat. With air-dry clay, you can serve up a stack as tall as you like with as much syrup, butter, and fruit as you desire. The best part is, this flap-jack stack also serves as a fun container to hold your treasures! This pot is made by stacking coils around the edge of a circular base.
Tools & Materials
air-dry clay (I used DAS air-dry modeling clay.)
toothbrush
water
skewer
paint in golden brown and other colors
paintbrushes
Mod Podge
1 Begin by pinching off a piece of clay about the size of a baseball. Flatten the clay by first squeezing it between your hands and then pounding it flat. It should be the size, shape, and thickness of an actual pancake. (Fig. 1)
2 This will be the base of the pancake pot. To make the stack, pinch off a fresh piece of clay the size of a gumball and roll it into a ball. Roll it between your hands to make it longer. Place it on your work surface and roll it from the middle out to the ends to make a snake of clay. It should be about as thick as a finger. (Fig. 2)
3 Start adding the sides to your pancake pot. Use the toothbrush and water to gently scrub around the outer edge of the base and along one side of the snake. Coil the snake around the edge of the base, pressing the wet, scrubbed surfaces together. (Fig. 3 & 4)
4 If the clay snake cracks, gently rub water into the cracks. If it breaks apart, overlap the two ends and roll them back together.
5 Continue rolling out more snakes and adding them to the pancake pot until it is at the height you want. (Fig. 5)
6 To make the lid, repeat step 1. It should be slightly wider than the pot. (Fig. 6 & 7)
7 Use the skewer to cut out a small square piece of clay. Attach it to the lid. Use your extra clay to create fruit by molding the shapes in your hand. Attach them to the side of the stack of pancakes. (Fig. 8)
8 Dry the lid on a flat surface, not the top of the stack of pancake. Allow the pot to dry for up to a week before painting.
9 Paint the pancakes golden brown and allow the paint to dry. Seal the pancakes with Mod Podge.
| COIL PRESS POT
---|---
This lab teaches you how to make a pot by pressing it into a mold. Many objects around your home can be used as molds. Unless they are terra cotta, they will need to be lined in plastic wrap so the clay does not stick. Press the clay on the inside of the mold, not the outside, because clay shrinks as it dries. The type of coils you created in Lab 5 will be used again here. This time, you'll learn to bend and shape the coils to create interesting designs!
Tools & Materials
small, wide mouthed bowl
plastic wrap
air-dry clay (I used ACTIVA Plus Clay Natural Self-Hardening White.)
water
acrylic paint
paintbrushes
Mod Podge
1 Cover the interior of the bowl in plastic wrap to prevent the clay from sticking.
2 Pinch off a Ping-Pong ball–size piece of clay and roll it into a ball. Roll the clay up and down between your hands until it resembles a hot dog, Place it on your work surface and roll it with your fingers to stretch and lengthen it like a snake. Start in the middle and work your hands out to the ends. The snake should be as thick as a thumb. (Fig. 1)
3 Stop and check the snake every now and then. It should be the same thickness all the way across. If it breaks, don't worry about it! Just overlap the two ends and resume rolling.
4 Begin coiling the snake into a spiral, using your work surface for support. Start with the center of the coil, bending the snake into a tight _C_ shape, and then round and round, with the spiral layers pressed together. (Fig. 2)
5 Press the coil into the bottom of the bowl. (Fig. 3)
6 Follow steps 2 through 4 to create four more coils. Add them, one at a time, to the inside of the bowl, beginning where the previous coil ended and continuing up the sides of the bowl. (Fig. 4)
7 Press the coils in place inside the bowl. Fill in any gaps with small pieces of clay. (Fig. 5)
8 With a wet finger, smooth the clay inside the bowl. (Fig. 6)
9 Set it aside to allow the clay to dry for several days. Gently remove the bowl from the mold.
10 Paint the bowl with a variety of colors and allow them to dry. Seal the bowl with Mod Podge.
| MONOGRAM WALL HANGING
---|---
In Lab 1, you pressed clay into the soles of your shoes for a fun texture. Let's find out what happens when you press clay into textured fabric. This will provide a background for a wall hanging featuring your initials. I used burlap, lace, and doilies to create mine.
Tools & Materials
textured fabric
air-dry clay (I used Crayola Air-Dry Clay.)
rolling pin or jar
large cookie cutters
toothbrush
skewer
water
paint (I used fluorescent tempera cakes.)
paintbrushes
Mod Podge
string or ribbon
1 Place the textured fabric on your work surface. Pinch off a piece of clay the size of a small orange. Stand up to press the clay firmly into the fabric. Roll or pound the clay to flatten it. Stop when the clay is as thick and level as a cookie. (Fig. 1)
2 Gently peel the clay off of the fabric and place it textured-side up on your work surface. Select a cookie cutter and press it through the clay. (Fig. 2)
3 Gather the extra clay and roll it into a ball between your hands. Then roll it on your work surface until it forms a skinny snake.
4 Shape the snake into the first letter of your name. Place it on the piece of clay cut with the cookie cutter. If it doesn't fit, roll it into a ball and try again. If it's difficult making the letter, draw it on a piece of paper. Then follow the shape of the letter with the clay. (Fig. 3)
5 To make the letter stick, use a wet toothbrush to gently scratch the surface of the cookie shape. Firmly press the letter onto the scratched surface. (Fig. 4)
6 Use the skewer to poke two holes at the top of the cookie shape. (Fig. 5)
7 Allow the wall hanging to dry for about a week. When dry, paint the letter. For a fun finish, water down the paint. Dip a large brush into the watery paint and gently tap the back of the brush over the wall hanging for a splattered paint look. (Fig. 6)
8 Once the paint is dry, seal and coat the wall hanging with Mod Podge.
9 Thread the string or ribbon through the holes of your wall hanging. Hang it up and enjoy it!
| DROPPED CONE SCULPTURE
---|---
What's the worst thing that can happen when you get a cone of your favorite ice cream? You drop it, of course! This fun sculpture is a take on that idea, but it's also a functional container that can hold and hide small items. It's truly a sweet treat to sculpt.
Tools & Materials
air-dry clay
burlap or textured fabric
rolling pin or jar
water
toothbrush
skewer
acrylic paint
paintbrushes
Mod Podge
1 Pinch off a piece of clay the size of a small orange and roll it into a ball.
2 Lay out a piece of burlap or another fabric with texture. Place the ball of clay onto the fabric and smash it flat.
3 Flatten it further by rolling the rolling pin or jar in one direction. Turn the piece of fabric and flatten the clay again, rolling in the opposite direction. Stop when the clay is as thick as a cookie. (Fig. 1)
4 Carefully remove the clay from the fabric.
5 Bend the clay into a cone shape, pointed at one end and wide at the other. Using water and a toothbrush, slip and score the seam. (Fig. 2)
6 Turn the cone point-side up to make sure it can stand on the work surface. Set the cone aside.
7 To make the ice cream, follow steps 1 through 3 in Lab 2 for making a pinch pot. (Fig. 3)
8 Turn the pinch pot upside down. Scrub the top with a wet toothbrush. Attach the cone to the open side of the pinch pot. (Fig. 4)
9 To create the splat of ice cream and toppings, use a piece of clay the size of a Ping-Pong ball. Flatten it with your hands and then use the rolling pin to flatten it further.
10 Place the upside down ice cream cone on top. Using a skewer, cut a wavy line through the flattened piece of clay so that it resembles a splat. (Fig. 5)
11 Using scraps of leftover clay, add details such as sprinkles and a cherry to the splat.
12 Once complete, allow it to dry for several days. Dry the splat and the ice cream cone separately so they do not stick together.
13 When the pieces are dry, paint them with a variety of colors. Allow the paint to dry.
14 Seal and protect your dropped cone sculpture with Mod Podge. Hide treasures underneath the upturned ice cream cone!
| WAX RESIST LEAF DISH
---|---
In this lab, you'll create a shallow dish out of air-dry clay leaves. You'll use real leaves as guides for cutting out the shapes. You'll also press them into the clay to capture the texture of the leaves' veins. The special clay technique you'll experiment with is called _wax resist_. You can use any air-dry clay for this project, though terracotta clay adds a nice fall color to the leaves.
Tools & Materials
air-dry clay
leaves, real or artificial
rolling pin or jar
skewer
shallow dish
plastic wrap
water
crayons
watercolor paint
paintbrushes
Mod Podge
1 Pinch off a Ping-Pong ball–size piece of clay. Roll the clay into a snake, about 3" (7.5 cm) long. Flatten it between your hands.
2 Place a leaf on the table with the vein-side up. Lay the flattened clay on top of the leaf and press it firmly. Roll the clay flat with the rolling pin or jar. The rolled clay should be as thick as a cookie so that it does not become fragile when it dries. (Fig. 1)
3 Turn the clay over—leaf-side up. Using the skewer, cut around the edge of the leaf, all the way through the clay. Holding the stem, gently peel the leaf off of the clay to reveal the texture of the veins. Repeat steps 1 through 3 to create three leaves. (Fig. 2)
4 Cover the shallow dish with plastic wrap. (This will be the mold for your project. The plastic wrap will prevent the clay leaves from sticking to the dish.) Begin laying the clay leaves onto the dish using the slip and score technique. (Fig. 3)
5 Firmly press the leaves onto the plastic-covered dish.
6 Allow the leaves to dry completely for a couple of days. When the clay is no longer cool to the touch and is lighter in color, it is dry.
7 Remove the clay from the dish.
8 Color the leaves with crayons, and then paint them with watercolors. The wax of the crayons will resist the watercolor paint, which is where _wax resist_ gets its name. (Fig. 4 & 5)
9 To seal and protect the piece, coat it in Mod Podge when the paint is dry.
Hint: For the best cutting results, keep the skewer vertical and make sure it cuts all of the way through to the work surface.
| LINE RELIEF PENCIL CUP
---|---
You've been using many textures in your clay labs. For this pencil holder, you'll create your own raised design. A raised design in a work of art is called a "relief." You'll need a Styrofoam tray for this project. If you use one that once held meat or other food, wash it in hot, soapy water first.
Tools & Materials
ink pen
Styrofoam tray
butter knife
air-dry clay
rolling pin or jar
cornstarch (if needed)
ruler
skewer
water
toothbrush (optional)
watercolor paint or metallic acrylic paints
paintbrushes
clear varnish
varnish brush
1 My favorite part of this project is coming up with the design. Here, I drew several different lines. These lines created shapes, filled with patterns.
2 Use the pen to draw a design on the Styrofoam tray. Go over the lines at least twice. Press down on the pen to make sure the lines are indented, but not cut through, the Styrofoam. (Fig. 1)
3 Use the butter knife to cut off a 2" (5 cm) slab of clay. Press the clay firmly onto the plate design.
4 Roll the clay to cookie thickness with the rolling pin. After rolling the length of the plate, turn the plate and roll the clay in the other direction to cover the entire plate with clay. (Fig. 2)
5 Slowly peel the clay from the plate. The clay may stick a little because it's moist. If this happens, remove the clay, wash the plate, and try again. This time, sprinkle cornstarch on the surface of the plate to prevent sticking.
6 Using the ruler and a skewer, cut the clay into a rectangle. Dampen your fingers and smooth the rough edges. (Fig. 3)
7 Slip and score one edge of the rectangle with a toothbrush or skewer. (See here for the slip-and-score technique.) Gently roll the clay into a cylinder shape and join the two short ends of the rectangle. Smooth the edge of the clay. (Fig. 4)
8 To create the base for the pencil holder, use a piece of clay the size of a golf ball. Flatten the clay until it is the same size as the bottom of the cylinder. Slip and score the edge of the base and connect it to the cylinder. (Fig. 5)
9 Allow the cup to dry for several days.
10 Use watercolor paint or metallic acrylic paint to decorate it. Allow the paint to dry. Seal and protect the piece with clear varnish. (Fig. 6)
| HAMBURGER HOLDER
---|---
This hamburger is delicious to look at, but hidden inside are two secret compartments. Keep tiny treasures under the hamburger buns. In this lab, you'll use your knowledge of pinch pot, slab, and coil construction. Just be careful that no one accidently tries to eat your sculpted treat!
Tools & Materials
air-dry clay (I used Crayola Air Dry Clay.)
skewer
toothbrush
water
gouache or watercolor paint
paintbrushes
Mod Podge
1 Pinch off a piece of clay the size of a small orange. Roll it into a ball and shape it into a pinch pot. This will be the bottom bun of the hamburger. (Fig. 1)
2 To make the pot as shallow as a bun, turn it over and gently tap it on the surface of the table.
3 Pinch off Ping-Pong ball–size pieces of clay for the burger, cheese, tomato slice, and lettuce. Roll these pieces of clay into balls and flatten them onto the work surface. (Fig. 2)
4 For the hamburger, keep the clay thick. For the lettuce, pound it thin and pinch the edges to give it a wrinkled look. Use the skewer to cut the piece of cheese into a square. (Fig. 3)
5 Rub the top surface of the lettuce and the bottom surface of the burger with the toothbrush and a little water. Press them together. Do the same with the top surface of the burger and the bottom surface of the tomato. Repeat with the top surface of the tomato and the bottom surface of the cheese.
6 Create pickles with small rounds of clay and attach them onto the cheese.
7 To create the top bun, repeat steps 1 and 2. Attach tiny bits of clay to the top of the bun for the sesame seeds. (Fig. 4)
8 Roll a skinny snake of clay for a squirt of ketchup. Attach it to the top of the cheese, as in step 6. (Fig. 5)
9 Allow the two buns and the burger stack to dry separately for up to a week.
10 Paint the burger and buns with gouache or water-color paints. Allow the paint to dry. Seal and protect the pieces with Mod Podge. (Fig. 6)
##
CLAY SCULPTURE
You have now learned to create
a ball of clay, a rolled and coiled snake of clay, and a slab or flattened piece of clay. You also know to attach clay pieces with the slip-and-score method, which is simply brushing the clay with a wet toothbrush or skewer. In the following labs, you'll see that you can create anything you can dream up with a ball, coil, and slab. Take your knowledge a bit further by exploring the variety of textures you can add to your clay creations.
| CLAY COAT OF ARMS
---|---
A coat of arms is a personal logo, made up of symbols that represent the person. During the Middle Ages, knights decorated their shields with their coats of arms. That way, even if a knight was covered up with armor on a battlefield, he could be identified by his coat of arms. Think about what special things you would choose to symbolize your unique personality. Then make the symbols out of clay for your very own coat of arms!
Tools & Materials
dental floss
air-dry clay (I used Amaco Marblex.)
skewer
toothbrush
water
crumpled newspaper
metallic acrylic paints
paintbrushes
Mod Podge
1 Cut a 12" (30.5 cm)-long piece of dental floss and pull it tight between your hands. Use it to cut a toast-thick slice from the brick of clay. (Fig. 1)
2 Pound the clay until it is as thick and level as a cookie. Turn the clay over to the smooth side.
3 Cut a shield shape from the clay with the skewer. Begin by cutting a wide V-shape for the bottom of the shield. The top could be a straight line or two curved lines that meet together in the middle. (Fig. 2 & 3)
4 Pinch two small pieces from the brick of clay. Roll each into a skinny snake. Use these strips to divide the shield into four sections. Use the toothbrush to attach the clay strips in place. (Fig. 4)
5 Now it's time to add special symbols. To create shapes such as stars and hearts, pound a piece of clay flat, turn it smooth-side up, and cut out the shapes with the skewer. Attach the shapes to the shield. (Fig. 5)
6 To create animals, visit Lab 21. Make letters as shown in Lab 7.
7 For a scroll, pound a long, thin rectangle of clay. Roll the ends of the rectangle inward, like a scroll. Attach the scroll to the shield. Write words on the scroll with the skewer. (Fig. 6)
8 Crumple a piece of newspaper and drape the shield over it. This will give it a three-dimensional shape as it dries.
9 If you are going to hang the shield, use the skewer to poke two holes near the top. Allow the shield to dry for a week or more.
10 To give the shield the look of metal, paint it with metallic acrylics. Paint the background first and then paint the details. Allow the paint to dry. Seal and protect it with a coat of Mod Podge. (Fig. 7)
| PIE SAFE
---|---
_Trompe l'oeil_ is French for "fool the eye." In art, that means an artist has created something so realistic that you are fooled into thinking it's real. Try your hand at this by creating a fun pie safe: a wedge-shaped lidded box that looks just like a slice of pizza, pie, or cake!
Tools & Materials
air-dry clay (I used Crayola Air Dry Clay.)
rolling pin or jar
skewer
toothbrush
water
garlic press
gouache or acrylic paint
paintbrushes
Mod Podge
1 Pinch off a piece of clay the size of a small orange. Squeeze it flat between your hands. Flatten it further with the rolling pin or jar until it is cookie thickness.
2 With the skewer, lightly draw two diagonal lines, like a capital _A_ , in the clay. The shape should resemble a triangle with a curved bottom. Use the skewer to cut out the shape. (Fig. 1)
3 Repeat step 1. Place the triangle from step 2 on top of the slab. Trace and cut out the shape with the skewer. These pieces will be the top and bottom of the pie safe.
4 Roll long snakes of clay as thick as a finger. Use the snakes to build up the sides of the container on the bottom triangle. Use a wet toothbrush to scrub the edges of the triangle and attach each layer. (Fig. 2)
5 Continue adding layers, overlapping the ends as you go, until the sides are the desired height and are level.
6 Use the other triangle to make the lid. For a pizza or pie slice, add a rim of clay along the short edge of the triangle for the crust. (Fig. 3)
7 For a pizza slice, squeeze clay through the garlic press to create cheese. Scrub the surface of the triangle with the toothbrush and gently press the clay "cheese" onto it. (Fig. 4 & 5)
8 Use extra clay and the skewer to cut out and add toppings, such as pepperoni, mushrooms, and peppers. Scrub the back of each topping with the wet toothbrush and gently attach it to the top of the pizza.
9 For a fruit pie filling, make rolled balls of clay. For cake icing, see here. (Fig. 6)
10 Allow the top and bottom to dry for a week or more.
11 Paint the pie safe with an opaque paint, such as gouache or acrylic. When dry, coat and seal the pie safe with Mod Podge. (Fig. 7)
| EGYPTIAN SARCOPHAGUS
---|---
Have you ever stopped to look at the mummies when you visited a museum? Ancient Egyptian pharaohs and queens were buried in beautiful burial boxes called _sarcophagi_. The boxes were often painted to look like the pharaoh or queen inside. For our lab, we'll use a paper-based clay called CelluClay to create our own sarcophagus and mummy!
Tools & Materials
CelluClay
large bowl
measuring cup
1 cup (235 ml) warm water
2 plastic banana split containers
gold acrylic paint
black acrylic paint
paintbrushes
1 Pull off about a quarter of the CelluClay paper pulp from the block and place it in the large bowl. Gently break it up with your fingers, until it looks like shredded paper.
2 Pour about 1 cup (235 ml) of warm water into the bowl. Knead the wet pulp until it has a clay texture. There should be no dried bits of pulp remaining. If it's too watery, knead in a little more clay. (Fig. 1)
3 Press clumps of the clay over the outside of the containers. Overlap bits of clay to cover both containers entirely. One will be the top of the sarcophagus and the other will be the bottom. (Fig. 2)
4 To make the head for the top of the sarcophagus, roll a piece of clay the size of a gumball. Place it a little below the top of the sarcophagus. Press your thumbs into the head shape to create eye sockets. (Fig. 3)
5 To make the headdress, roll a snake of clay. Drape it over the head.
6 To make the arms, roll two more pieces, a little thicker than the first. Place these under the head and fold them across the chest. (Fig. 4)
7 Add a lump of clay at the bottom of the sarcophagus for feet.
8 To create a mummy to place inside the sarcophagus, roll a piece of clay about as thick as a hot dog. Bend the bottom for feet. To make the arms, follow step 6. (Fig. 5)
9 Set the two sarcophagus pieces and the mummy aside to dry for several days.
10 Paint the sarcophagus gold and allow it to dry. Add details, hieroglyphs, and designs in black with a small paint brush. (Fig. 6 & 7)
| MAGICAL FOREST FRIEND DOOR
---|---
Do you believe in magic? Some people say magic only enters our lives if we invite it in. What better way to invite magic in than by creating a magical forest-friend door? Place your finished door in your bedroom, next to the cookie jar, or on the stairs to the attic—you never know who might open the little door and take a step outside. In this lab, you'll use all of the skills you've learned so far to make magic happen!
Tools & Materials
air-dry clay
textured fabric, such as burlap, lace, or doilies
rolling pin or jar
water
skewer
toothbrush
watercolor or gouache paints
paintbrushes
Mod Podge
1 Make the basic shape for the door. Break off a baseball-sized piece of air-dry clay. Flatten the clay with your hands and place it on a piece of textured fabric.
2 Roll the clay with the rolling pin or jar to make a rectangle about as thick as a cookie. Carefully peel the textured fabric from the clay. (Fig. 1)
3 Roll the bottom edge of the clay upward to create a front step for the door. If the clay dries or begins to crack, rub a little water into it with your fingers to smooth the dryness.
4 Use the skewer to lightly draw an upside down _U_ for the door. (Fig. 2)
5 Roll small balls of clay and gently flatten them between your fingers. Use a wet toothbrush to attach them around the door. (Fig. 3)
6 Think of details that you can add to the door. Turn a little slab into a window, a ball into a doorknob, or a skinny snake into climbing ivy. (Fig. 4)
7 Set the finished door aside to dry for several days. You can place it on crumpled tissue or foam so air will reach underneath and help it to dry completely.
8 Paint it with watercolors or gouache paints. Try painting light colors on top of dark colors or dark colors on top of light colors for a different effect. Allow the paint to dry. Seal and protect the forest-friend door with Mod Podge. (Fig. 5)
| MINIATURE CACTUS GARDEN
---|---
Here's a garden that doesn't need sun or water, just your creative touch! For this lab, you'll create your own collection of potted cacti. Each of your plants will have its own little pot made from a slab of clay. Have fun exploring different textures, shapes, and sizes for your cactus garden!
Tools & Materials
air-dry clay (I used terra cotta colored clay.)
textured fabric, such as lace or burlap
rolling pin or jar
water
skewer
acrylic paints in several shades of green
paintbrushes
Mod Podge
1 Pinch off a Ping-Pong ball–size piece of air-dry clay. Flatten it, place it on the textured fabric, and roll it with the rolling pin. The slab should be as thick as a cookie. (Fig. 1)
2 Roll the bottom edge of the clay to create a rim. (Fig. 2)
3 Turn the slab into a pot by bending it into a cylinder and overlapping the ends. (Fig. 3)
4 Pinch the bottom of the cylinder to close it. Flatten the bottom by gently tapping it on the table. (Fig. 4)
5 Place a ball of clay inside the pot to act as soil for your plants.
6 Look at images of cacti online. Some are shaped like flattened circles while others are like connected triangles. Flatten a piece of clay to cookie thickness and use the skewer to cut out the shapes you want.
7 Decorate the plants by poking dents in the surface with the skewer. Add them to the pots and allow them to dry for a couple of days. (Fig. 5)
8 When the cactus garden is dry, paint it different shades of green. Allow the paint to dry. Seal your cactus garden with Mod Podge. (Fig. 6)
| GARDEN GNOME HOME
---|---
Gnomes are small but mighty creatures, fabled to guard Earth's treasures. This lab allows you to stretch your imagination while creating a unique and textured miniature abode.
Tools & Materials
butter knife
air-dry clay (I used terra cotta colored clay.)
textured fabric, such as burlap, lace, doilies, or old sweaters
rolling pin or jar
skewer
toothbrush
water
watercolor, gouache, or acrylic paint (I used watercolor paint.)
paintbrushes
Mod Podge
1 Use the butter knife to cut two 1" (2.5 cm) slabs of clay.
2 Lay out the textured fabric. Place the clay on top of the fabric and press it down firmly. Use the rolling pin to roll the clay flat to cookie thickness. (Fig. 1)
3 Peel the clay from the fabric and set it aside. Repeat steps 1 and 2 for the second piece.
4 Make the roof for the house. Take one of the pieces from step 2 and fold it into a cone, narrow at the top and wide at the bottom, with the texture on the outside. Set it aside. (Fig. 2)
5 Now take the other piece from step 2 and bend it into a cylinder. The top and bottom should have the same size opening, with the texture on the outside. This will be the bottom part of the house. (Fig. 3)
6 Gently place the cone roof on the house. If it doesn't fit, simply shape the cylinder again to make it narrower. Use a toothbrush and water to attach the roof to the house. Use your fingers to smooth and attach the two pieces together. (Fig. 4)
7 Now use your imagination to decorate the gnome home! What kind of door will you make? How about a window with a flower box? Use the skewer to scratch the outlines of a door or window into the clay. Add a chimney with another piece of clay. Allow the gnome home to dry for several days in a safe place.
8 Paint the gnome home with watercolor, gouache, or acrylic paint. I like to layer different colors so the textures are more visible. Allow the paint to dry. Seal the gnome home with Mod Podge. (Fig. 5)
| BOBBLEHEAD PETS
---|---
To create a bobblehead pet, you'll use the skills you learned in earlier labs for pinch pots and textures. You can create any bobblehead creature you can imagine, from a couple of sweet owls, like I've made here, to monsters or dragons.
Tools & Materials
dull knife
air-dry clay
textured fabric, such as burlap
rolling pin or jar
ruler
skewer
toothbrush
water
watercolor, gouache, or acrylic paint
paintbrushes
Mod Podge
### MAKE THE BASE OF THE BOBBLEHEAD
The base is made as a cylinder with a pointed top for the head to bobble on.
1 Cut about 2" (5 cm) of clay from the short side of the clay block. Lay it on a piece of textured fabric and press it down to flatten it. (Fig. 1)
2 Use the rolling pin to flatten the clay some more. Roll it up and down and side to side until it is cookie thickness.
3 Peel the clay off of the fabric. Lay it texture-side up on your work surface. (Fig. 2)
4 Using a ruler and skewer, measure and cut the clay into a rectangle about 4" × 7" (10 × 18 cm). (Fig. 3)
5 Lift the clay and bend it into a cylinder shape, overlapping the short ends. Use the skewer to slip and score the clay to hold the seam in place. (Fig. 4)
6 Pinch around the top edge of the cylinder to start shaping it into a point. Continue pinching until the point is 1/2" (1.25 cm) long and about as thick as an uncapped marker. If it is too thin, it may break when dry. (Fig. 5 & 6)
### MAKE THE BOBBLEHEAD
1 Roll a ball of clay the size of a golf ball.
2 Create a pinch pot as shown in Lab 2, shown here and here. Place the ball in the palm of your hand and cup your fingers around it. Press the thumb of your other hand deeply into the ball. (Fig. 7)
3 Set the ball indented-side up on your work surface. Use your fingers to create a hollow head, pinching the clay to cookie thickness all the way around. (Fig. 8)
4 Turn the head over and set it on top of the base to make sure it fits. Remove the head and place it on your work surface.
5 Now make the bobblehead's face. Using additional clay, roll balls for the eyes, cut a triangle shape for the nose, and add texture with the skewer. Attach the pieces by the slip-and-score method (shown here) with the wet toothbrush or skewer. (Fig. 9)
6 Add details to the body of the bobblehead. Use additional clay to create wings, feet, and a belly. The possibilities are endless! (Fig. 10 & 11)
7 Allow the bobblehead pet to dry for a couple of days. When it is no longer cool to the touch, it's dry.
8 Paint your creation with watercolor, gouache, or acrylic paint. Allow the paint to dry. Seal and protect your creation with Mod Podge. (Fig. 12 & 13)
| CASTLE ON THE HILL
---|---
Long ago, castles or even entire towns were built on hilltops with moats around them. A moat is a body of water that surrounds an area. Moats were built to provide protection for the townspeople living inside the walls of the castle. You can create your very own castle on a hill complete with a surrounding moat in this lab.
Tools & Materials
dental floss
air-dry clay (I used Amaco Marblex.)
textured fabric
rolling pin or jar
newspaper
garlic press
toothbrush
water
ruler
skewer
watercolor or gouache paint
paintbrushes
Mod Podge
1 Cut a 12" (30.5 cm)–long piece of dental floss and pull it tightly between your hands. Use the dental floss to cut a toast-thick slice from the brick of clay.
2 Lay a piece of textured fabric on your work surface. Place the slab of clay on top. Roll or pound the clay to a level cookie thickness. Peel the slab from the fabric and place it texture-side up on your work surface. (Fig. 1)
3 Crumble a piece of newspaper into a ball the size of a small orange. Flatten one side by pressing it onto your work surface. Drape the slab of clay, texture-side up, over the newspaper so that it forms a hill. (Fig. 2)
4 Gently roll the edges of the clay toward the hill, creating a moat. (Fig. 3)
5 Squeeze several small pieces of clay through a garlic press to create grass for the moat's shore. Roll small balls of clay for rocks. Scrub the moat's shore with a wet toothbrush and attach the grass and rocks. (Fig. 4)
6 To create the castle, flatten another piece of clay to cookie thickness. Using the ruler and skewer, measure and cut the clay into a rectangle. Mine measures 31/2" × 7" (9 × 18 cm).
7 Bend the short sides of the rectangle inward to make a cylinder. To attach, scrub the edges with a wet toothbrush. Overlap the two pieces of clay and smooth with your finger and water. (Fig. 5)
8 Create castle turrets at the top of the cylinder by using the skewer to cut out small rectangles. Reuse the small triangles by attaching them to the sides of the cylinder to look like building stones. (Fig. 6 & 7)
9 To create a door, use the skewer to cut an upside down _L_ at the bottom of the cylinder. Gently pull the L outward to open the door. (Fig. 8)
10 Attach the castle to the top of the hill with a little water. To secure it, roll a snake of clay to go around the base. Scratch the surface of the castle and hill with a wet toothbrush and attach the snake of clay. (Fig. 9 & 10)
11 Further embellish the castle with bits of clay for ivy and more stones. Allow it to dry for up to a week.
12 Use watercolor or gouache to paint the castle, hill, and moat. Allow the paint to dry. Coat your creation with Mod Podge. (Fig. 11)
| CRAYON AND PENCIL SCULPTURE
---|---
Imagine that you had a giant crayon or pencil. How fun would that be? Many artists draw inspiration from familiar objects when creating their art. This form of art is called _pop art_. The word _pop_ comes from _popular culture_ , or items that surround us in everyday. Pencils and crayons are a part of everyday life, so why not use them as subjects for a big pop art sculpture?
Tools & Materials
4" × 7" (10 × 18 cm) piece of thin cardboard from a cereal box
masking tape
scissors
paper towel roll, empty
CelluClay
warm water
large bowl
wax paper (optional)
pencil
acrylic paint
paintbrushes
Mod Podge
1 Create the tip of the crayon or pencil by bending the cardboard into a cone. Pull the upper-right corner toward the center of the cardboard. The widest part should be about the width of the paper towel roll. Tape the cardboard to hold the shape. (Fig. 1)
2 The bottom of the cone will be uneven. Use the scissors to trim it so that the edge is flat. (Fig. 2)
3 Make cuts about every 1/2" (1.3 cm) around the bottom of the cone so that it fits over the opening of the towel roll. (Fig. 3)
4 Tape the cone to one end of the roll. Also place a large piece of tape across the opposite end of the paper towel roll to close it off. (Fig. 4)
5 Following the manufacturer's directions, mix up a batch of CelluClay with the water in the bowl. You'll need a ball of clay about as big as a baseball. CelluClay can be very dusty, so it's best to mix it in the kitchen or outdoors.
6 Flatten small balls of clay between your hands. Lay them onto the sculpture. Overlap additional pieces slightly when adding more. Continue until the entire sculpture is covered. (Fig. 5 & 6)
7 If it's a sunny day, place the sculpture outside on a sheet of wax paper. Otherwise, allow it to dry indoors for up to a week.
8 If you're making a pencil, draw a line around the tip for the pencil lead. Draw a line around where the point meets the paper towel roll to indicate the wood portion of the pencil. Draw additional lines to indicate the metal band and eraser at the end of the pencil. Paint the different sections with the appropriate colors and allow the paint to dry. (Fig. 7–9)
9 If you're making a crayon, draw a line where the cone at the top meets the paper towel roll. This will be where the crayon's wrapper starts. Draw another line around the opposite end of the crayon to show where the paper wrapper ends. Paint the crayon the color of your choice and allow the paint to dry. (Fig. 10 & 11)
10 Brush on a coat of Mod Podge to add a little shine. (Fig. 12)
| WOODLAND CREATURE PORTRAITS
---|---
I live near a forest and share my backyard with many woodland creatures. In the morning, bunnies bounce by in search of the perfect blades of grass. During the day, squirrels and chipmunks visit our bird feeder. At night, deer like to rest under a tree while raccoons knock over our trashcan. It's fun to have all of these furry friends nearby. They are the inspiration behind these woodland creature portraits, which you'll make with modeling clay.
Tools & Materials
air-dry clay (I used Plus Modeling Clay.)
water
skewer
cardboard
scissors
acrylic paint in gold and brown
paintbrushes
watercolors
Mod Podge
glue
1 Pinch off a Ping-Pong ball–size piece of clay and roll it into a ball. Use your fingers to pinch and shape the clay to create a muzzle—that's a nose—for your animal. A fox or deer has a long muzzle. A bunny or squirrel has a shorter one. (Fig. 1)
2 Press your thumbs into the clay to create the eye sockets. (Fig. 2)
3 Stretch and pull the clay to create ears for your animal. Depending on the animal, the ears might be long or short, and they might start at the top of the head or on the sides. If your clay starts to dry out and crack, work a little water into the clay. (Fig. 3)
4 Hold the skewer horizontally and use the point to make an opening for the animal's mouth.
5 Now use the point of the skewer to scratch fur on the animal's head. Poke holes for nostrils and whiskers. Then gently draw the eyes and other details. (Fig. 4)
6 To give the creature a little character, add a bowtie or decorate its head with flowers and leaves. If you're making a deer, poke a couple of holes in the top of the head so you can add twigs for antlers. Set the animal portrait aside to dry for several days. (Fig. 5)
7 While the clay is drying, cut out shapes from cardboard to use as mounts. Paint the mounts with gold and brown acrylic paint to make the faux wood background. Allow the paint to dry.
8 Paint the animals with layers of watercolors to create a furry textured look. Allow the paint to dry. (Fig. 6)
9 Paint the animals with Mod Podge. Once dry, glue the animal portrait to the faux wood base and display!
| VIKING SHIPS
---|---
The Vikings sailed in wooden boats called _longships_ , which often had the head of a dragon at the front and the tail at the back. Vikings arranged their shields along the outside of the ship both for protection and to conserve space. Often, the sails featured symbolic designs.
Tools & Materials
air-dry clay (I used DAS air-dry modeling clay.)
toothbrush
water
skewer
gouache or acrylic paint
paintbrushes
paper
scissors
pencils or crayons
glue
Mod Podge
1 Pinch off a piece of clay the size of a golf ball. Roll it into a ball and then roll it into a hotdog shape, about 4" (10 cm) long.
2 Flatten the clay until it's as thick as a cookie. This is the base of the ship. (Fig. 1)
3 Build the ship taller by the coil method. Begin with a piece of clay the size of a gumball. Roll it between your hands to make a snake of clay as thick as a finger. (Fig. 2)
4 Attach the snake of clay to the edge of the ship base using the wet toothbrush method. Continue adding layers until the ship is the desired height. (Fig. 3 & 4)
5 Make the ship narrower by gently pressing in the sides. Make it a bit longer by gently pulling at the ends. (Fig. 5)
6 To make shields, roll and flatten small balls of clay. Use the skewer to decorate the shields. Use the wet toothbrush to attach the shields to the ship. (Fig. 6)
7 Create the ship's dragon head with a thick piece of clay snake. Use the skewer to create nostrils and eyes. Pinch the clay to create spikes. Use the dull end of the skewer to create scales. Create a tail by the same method. (Fig. 7)
8 Attach the dragon head and tail to the ship with the wet toothbrush. (Fig. 8)
9 Poke a hole in the center of the boat. This is where the mast will be glued. Allow the ship to dry for up to a week. Paint it when it's dry.
10 Cut out a paper sail. Draw or paint a design on it. (Fig. 9)
11 Glue the top of the skewer upright into the hole in the ship. Allow it to dry. Poke a tiny hole at the top and bottom of the sail and slide it onto the skewer.
12 Seal and protect the Viking ship with Mod Podge.
| MARIONETTE PUPPETS
---|---
A marionette is a puppet that seems to come to life when you move its strings. Create marionettes in the forms of animals and people and perform your very own show!
Tools & Materials
air-dry clay (I used paper clay.)
water
skewer
acrylic paint
paintbrushes
toilet paper tube, one per marionette
hole punch
yarn or twine
two twigs 10" or 12" (25.5 or 30.5 cm) long
glue
1 Create the marionette's head by rolling a piece of clay the size of a Ping-Pong ball. If you're creating an animal, gently pinch the clay forward to create a muzzle, as in Lab 21. Create ears either by pinching and pulling the clay or by sculpting the ears separately and attaching them with water. Press your thumbs into the clay to make eye sockets. (Fig. 1 & 2)
2 If you're creating a person, create the eyes and nose separately and attach them with a little water.
3 Use the skewer to poke a hole through the head, going from the top to the bottom. Twist the skewer back and forth while gently pushing it through. Set the head aside. (Fig. 3)
4 If you're making an animal, make the feet by rolling four small pieces of clay into balls. Use the skewer to poke a hole through each of the balls. Allow the feet to dry. (Fig. 4)
5 If you're making a person, roll out four little tube shapes for the arms and legs. Use the skewer to poke a hole through each limb from side to side. Allow the limbs to dry. (Fig. 5)
6 When the pieces have dried, you're ready to paint. You'll need the paper tube, too—it will form the marionette's body. Gather all the pieces, decide on the colors, and paint the head, feet/limbs, and body with acrylic paint. Allow the paint to dry. (Fig. 6 & 7)
7 To assemble an animal, punch two holes at the front of the tube and two holes in the same positions at the back. Thread yarn or twine through the punched holes on the front of the animal's body. Now thread the yarn or twine through one of the feet and tie a double knot at the bottom to secure. Do the same for the other foot. Repeat this process for the back of the animal's body. (Fig. 8 & 9)
8 To attach the animal's head, punch a hole in the tube above one set of feet. Cut two lengths of yarn or twine about 18" (45.5 cm) long. Double knot one end of the yarn. Thread the yarn through the hole inside the tube so that the knot is on the inside. Slide the animal head down the string from the top. (Fig. 10)
9 Punch a hole above the feet at the opposite end of the tube. Double knot the end of the other piece of string. Thread it through the hole as before, knotting it where it emerges from the hole. This string and the one attached to the head will control the puppet.
10 If you're making a person, follow steps 8 and 9, adding the arms outside the tube and the legs inside. To attach the head, punch two holes directly above the holes created for the arms. Using a 26" (66 cm) piece of yarn, thread it through both holes. Bring the strings together and tie a double knot. Slide the head over the two strings.
11 Make the marionette controls by tying two twigs together in a cross. Glue the knot to secure it and let dry. Tie and knot a string around the intersection of the two sticks to secure it. (Fig. 11)
12 If you're making an animal, tie the ends of the yarn pieces to two ends of the cross stick with a double knot. Add a dot of glue to the knots and allow it to dry before operating. (Fig. 12)
13 If you're making a person, attach the doubled string to the center of the cross sticks. Tie a double knot, anchor it with glue, and let dry.
| ANIMAL MASQUERADE MASKS
---|---
During a holiday called _Carnival_ , celebrated around the world, there are days of parties and parades. People enjoy going to these events wearing fun masks to hide their identities. Make your own carnival mask and see whether anyone recognizes you!
Tools & Materials
CelluClay
water
large bowl
plastic mask
poster board
scissors
masking tape
pencil
acrylic paint
paintbrushes
1 Following the steps in Lab 14, mix the CelluClay with water in the bowl.
2 Set out the masks. Using poster board and scissors, cut out shapes for animal ears. Triangles would be perfect for a fox or a cat. Round ears work well for a panda. Tape the ears to the masks. (Fig. 1)
3 Press the clay mixture over the entire mask evenly. Smooth it with your fingers. (Fig. 2)
4 Smooth more clay over the ears. Set the mask aside and allow it to dry for several days. (Fig. 3)
5 Draw an animal face design on the mask and paint it. When the paint has dried, celebrate by hosting a masquerade ball! (Fig. 4 & 5)
##
POLYMER CLAY AND LEARNING THE BASICS
Polymer clay is a wonderful
medium to explore. Because it is not water based, it is less messy than other clays. And no matter what you want to make, there's a polymer clay color to match. However, you don't need to purchase them all. In this unit, you'll learn that a lot of the fun of working with polymer clay is in marbling or blending colors. Like the other types of clay previously explored, polymer clay is great for grabbing texture.
| CRYSTALLIZED PINCH POTS
---|---
The magic of this lab is that you'll be a scientist as well as an artist when you try it! You're going to grow crystals inside your clay pinch pot. You'll be amazed at the beautiful results of this science experiment.
Tools & Materials
three or four colors of polymer clay
white glue
paintbrushes
alum powder (You can buy it in the spice section of a grocery store.)
1 cup (235 ml) water
microwave-safe bowl
food coloring
spoon
newspaper
1 Pinch off a gumball-size piece of each color of clay. Roll them into snakes of equal length.
2 Twist the snakes together, and then form them into a ball. (Fig. 1 & 2)
3 Roll the ball of clay into a snake. Twist it again and form it into a ball. You can repeat this as often as you'd like. The more times the clay is rolled, twisted, and formed into a ball, the more the colors will swirl. (Fig. 3)
4 Place the ball of clay in your palm. Press your thumb deep inside it and then pinch the edges to create a small pot. Bake the pot in the oven according to the manufacturer's directions. Allow it to cool. (Fig. 4)
5 When the pot has cooled, paint a layer of glue inside it. Sprinkle alum powder onto the wet glue until it is covered. Allow the glue to dry for several hours. (Fig. 5)
6 Pour the water into the bowl and bring it to a boil in the microwave. Add 1/2 cup (110 grams) of alum powder and again, bring it to a boil. Stir a few drops of food coloring into the mixture. Stir the mixture and allow it to cool. (Fig. 6 & 7)
7 Place your pinch pot in the solution and allow it to sit overnight. In the morning, gently remove the pot from the solution and place it on newspaper to dry. Enjoy the sparkles!
8 You can save and reuse the alum powder solution. Simply reheat it in the microwave until the alum powder has dissolved. Results of this experiment may vary from very large to very small crystals.
| COLOR MIXING CLOCK
---|---
You don't have to own every rainbow color of polymer clay. You can make them yourself from the three primary colors—red, yellow, and blue. In this lab, you'll use twelve different colors to make the numbers on a clock. You'll need a battery-powered clock kit, which comes with numbers and hands. These can be found at craft and hardware stores and are very easy to assemble.
Tools & Materials
dull knife
polymer clay in red, yellow, and blue
discarded CD
battery-powered clock-making kit
1 Cut two small pieces of clay from each of the three colors. Roll each piece into a ball. Place them on a table to form a triangle. These are the three primary colors. (Fig. 1)
2 To create the secondary colors, combine the primaries: red and yellow to make orange; yellow and blue for green; and red and blue for purple. (See box.)
3 Cut gumball-size pieces of red and yellow from the polymer blocks. Individually, roll each color into a snake. Starting at the top, twist two colors together. Squeeze the twist into a ball. Roll it into a snake again, twist it, and squeeze it into a ball. Watch the colors blend to make orange. Repeat until the clay is the color you want. Use the same process to make green and purple. (Fig. 2 & 3)
4 To make the tertiary colors, blend the primary and secondary colors. See the box below right.
5 Make a small ball of each color about the size of a gumball. Place the CD shiny-side up on your work surface. Arrange the balls, equally spaced, around the edge of the CD. Press them into place. Gently press the numbers of the clock into the clay. (Fig. 4–6)
6 Bake the clock at a slightly lower temperature than recommended in the manufacturer's directions. This will bake the clay—without melting the numbers or the CD. Let cool when done.
7 Assemble the clock's mechanism according to the directions that come with the kit.
Start with Three Colors, End with Twelve!
Primary Colors
1. red
2. yellow
3. blue
Secondary Colors
4. orange (red + yellow)
5. green (blue + yellow)
6. purple (blue + red)
Tertiary Colors
7. red-orange (red + orange)
8. yellow-orange (yellow + orange)
9. yellow-green (yellow + green)
10. blue-green (blue + green)
11. blue-purple (blue + purple)
12. red-purple (red + purple)
| FUN FORTUNE COOKIES
---|---
When you eat at a Chinese restaurant, you are served fortune cookies with tea at the end of the meal. Our clay version of a fortune cookie will make a sweet gift for a family member or friend. Be sure to write a happy fortune for someone special to discover.
Tools & Materials
polymer clay, in brown and white
small strips of paper
markers or pens
jewelry hoop, optional
1 Start with the fortunes. Write some fun fortunes or kind words on small strips of paper. (Fig. 1)
2 Now make the cookies. For a light brown cookie color, roll out snakes of brown and white polymer clay. Twist the two snakes together and then form them into a ball. Continue to do this until the colors are well blended. End with the clay in a ball.
3 Flatten the ball between your fingers. Press the clay onto your work surface and flatten it into a circle. (Fig. 2)
4 With your fingers tucked inside, bend the clay up like a taco. Keep your fingers inside to create a pocket in the clay and give the cookie a three-dimensional shape.
5 Gently pinch around the top and one side of the taco shape. Leave one side open. That is where the fortune will slide in. (Fig. 3)
6 Starting from the pinched edge, gently bend the two ends of the cookie toward each other until the inside is a _V_ shape. (Fig. 4)
7 Bake the clay cookie according to the manufacturer's directions. When cool, slide the fortune inside and deliver it to a friend!
VARIATIONS
1 Follow steps 1 through 3. Before folding the clay into a taco shape, place a thinly flattened piece of white clay on top of it, just to one side of the center. Allow the white clay to hang out of one end of the cookie just a little. Then follow steps 4 through 7. The white clay will have the appearance of paper. (Fig. 5)
2 Add a small jewelry hoop to the cookie before it's baked and use it as a charm for a necklace. (Fig. 6)
| LIGHT-UP LIGHTNING BUGS
---|---
Some people call them lightning bugs; others call them fireflies. No matter what you call them, during the summer months, these small insects light up the night. They are beautiful to watch and even more fun to create.
Tools & Materials
polymer clay, in a variety of colors
skewer
paper
scissors
quick dry tacky glue
paintbrush
LED, battery-operated lights (Christmas lights that are not LED lights will be too hot for this project.)
glass jar (optional)
1 Roll a small piece of clay between your hands to make a ball. Shape the ball into an oval about 1/2" (1.3 cm) long.
2 Create the wings of the insect by using the skewer to cut the oval in half lengthwise. Gently separate these two pieces and flatten to form wings. (Fig. 1)
3 Roll a small ball of clay for the bug's head. Attach the wings to the head. (Fig. 2)
4 Roll two tiny white balls of clay and press them into the bug's head for eyes. Add even smaller balls of black clay to the eyes for the pupils.
5 Create as many lightning bugs as there are lights or as many as desired. (Fig. 3)
6 Follow the manufacturer's instructions to bake the clay bugs. Once they've cooled, cut pieces of paper about the same size as the bugs and set aside.
7 Brush a dollop of glue to the back of each bug. Position each bug on the strand of lights so it has a bulb near its backside. (Fig. 4)
8 Place a piece of the cut paper over the back of the bug so that the light is sandwiched in between. Press gently so as not to break the bug and count to 30. This will allow time for the glue to set. (Fig. 5 & 6)
9 Once all of the bugs have been added, light up the lightning bugs! For fun, keep these on display in a glass jar and use as a light on a nightstand.
| ALL-ABOUT-ME MOBILE
---|---
A mobile is a sculpture that moves and changes its form when it's hung in the air. Gather treasures from outside to attach to your mobile. Watch your work of art shift and transform while it moves.
Tools & Materials
sticks, feathers, leaves
acrylic paint
paintbrushes
polymer clay, in three or four colors
skewer
cookie cutters (optional)
small metal clay stamps (optional)
yarn or string
12" (30.5 cm)-long stick
glue
1 Collect items from outside, such as sticks, feathers, and leaves. Paint lines, designs, and patterns on these treasures.
2 Follow the steps in Lab 25 to create blended clay with your favorite colors. Roll them into small balls and flatten some of them. (Fig. 1)
3 Use the skewer or cookie cutters to cut shapes into the clay. Add designs with a metal stamp, if desired. (Fig. 2 & 3)
4 Use the skewer to poke a hole at the top of each shape. Poke holes through the remaining small balls to make beads. (Fig. 4)
5 Bake the clay, including the balls still on the skewer, according to the manufacturer's directions. Allow the clay to cool.
6 Lay out the mobile design. To string the beads, cut a 10" (25.5 cm) length of yarn. Make a double knot at one end. Slide the beads down the string until the knot stops them. Tie the string to the hanging stick with a double knot. (Fig. 5)
7 To add the other designs, cut a piece of yarn double the length of the bead yarn. Fold it in half. Place the folded yarn inside the hole of each piece. This will create a loop of yarn. Thread the ends of the yarn through each loop. Secure it to the hanging stick with a double knot. (Fig. 6)
8 To add leaves and feathers, attach yarn with a double knot. Secure them to the hanging stick with a drop of glue on all of the knots. Let dry. (Fig. 7)
9 To hang, cut a long piece of string and tie it to the ends of the hanging stick with a double knot.
##
POLYMER CLAY SCULPTURE
Now that you know how to
explore color and texture in polymer clay, let's dive into sculpting. Because polymer clay is more expensive than other clays, sculptures created from polymer clays are smaller. However, many of the labs previously covered in this book could also be created in polymer clay. In this unit, you'll learn how to work on a smaller scale and also how to create the structures underneath a sculpture.
| TIE-DYE TURTLES
---|---
In this lab, you'll explore a fun technique that creates the colors and patterns of a tie-dye T-shirt. The funky design could be added to many of your sculptures. Today you'll be adding it to a rock and turning it into a turtle!
Tools & Materials
round rocks
polymer clay, in several colors
skewer
1 Walk outside and collect a few rocks that are round in shape. The rocks do not have to be smooth. Wash and dry them. (Fig. 1)
2 Work an entire block of clay between your hands until it's soft and easy to squeeze. The color you use does not matter because it will not be seen.
3 Roll the softened clay into a ball and flatten it into a thin circle. Cover the upper part of the rock with the clay. Only the bottom of the rock should be exposed. Patch on pieces of flattened clay, if needed, to cover the areas, so that there is a smooth surface for the turtle's back. (Fig. 2)
4 To create the tie-dye, pick a polymer color and roll it into a ball the size of a large gumball. Flatten it into a thin circle with your fingers. (Fig. 3)
5 Repeat, making a slightly smaller circle of clay with another color. Center it on the larger circle of clay. With your finger, smear the colors of the inner circle to the outer. (Fig. 4)
6 Pick another color, roll it into a smaller ball, and flatten it. Center it on top of the other colors and smear the inner circle to the previous circle. Do not smear all the way to the first circle.
7 Continue adding smaller circles and smearing with shorter strokes. Once complete, drape your tie-dye over the rock and tuck it underneath. (Fig. 5)
8 Roll out four small pieces of clay the size of a peanut for the turtle's feet. Attach two on each side. (Fig. 6)
9 To create a head, roll a thick coil of clay and bend into a cane shape. Attach it to the rock. Use the skewer to press into the clay to create the mouth. Roll tiny balls of clay for the eyes and pupils and attach. (Fig. 7)
10 Follow the manufacturer's directions for baking the clay turtle, leaving the rock inside. Allow the turtle to cool thoroughly before touching it.
| MINIATURE ACCORDION BOOKS
---|---
An accordion book gets its name because it can expand just like the musical instrument. The book's paper is folded back and forth, like a zigzag that can open to be quite long. The accordion book was invented in Asia for folding scrolls that had become too long to be rolled. You can create a unique accordion book to hold all your interesting ideas!
Tools & Materials
polymer clay, in several colors
rolling pin
two 2" × 3" (5 × 7.5 cm) pieces of paper
skewer
cookie cutters
mini cookie cutters (optional)
one 18" × 3" (45.5 × 7.5 cm) piece of paper
glue
two 8" (20.5 cm) pieces of ribbon
1 Follow the steps in Lab 26 to blend two or three pieces of clay together for a swirled effect. Flatten the clay until it is slightly bigger than one of the 2" × 3" (5 × 7.5 cm) pieces of paper.
2 Place the paper on top of the flattened clay and trace it with a skewer, cutting the clay into a rectangle. Repeat steps 1 and 2. One piece of clay will serve as the book's cover and the other as the back. (Fig. 1)
3 Decorate the front and back covers any way you desire. Use cookie cutters to cut out flattened pieces of another color of clay. Press them onto the covers. (Fig. 2 & 3)
4 Follow the manufacturer's instructions for baking the clay. Allow the clay to cool.
5 Begin the pages of the book. Set the 18" (45.5 cm) piece of paper in front of you. Make the first fold 2" (5 cm) wide. Then continue folding the sheet back and forth until complete.
6 Drizzle glue around the inside edges of the book's front cover. Place the accordion-folded paper on top. On the inside front cover, draw a 1" (2.5 cm) line of glue that extends from the center to the outer edge. Place one end of a piece of ribbon on top of the line of glue. Then place a 2" × 3" (5 × 7.5 cm) piece of paper on top. The paper will sandwich the ribbon. Repeat step 6 on the back cover. Allow the book to dry while open. (Fig. 4–7)
7 Use the book for writing and sketching. Use the ribbons to tie the book closed.
| SUSHI FOR SUPPER
---|---
In display cases on the outsides of restaurants in Japan, you'll find realistic looking fake food. These food sculptures are called _sampuru_ , which means "sample." Let's create our own Japanese food sculptures with clay sushi!
Tools & Materials
polymer clay, in several colors, including black and white
rolling pin or jar
garlic press
skewer
ruler or strip of stiff cardboard
Mod Podge
fine green glitter
1 Sushi is often wrapped in nori, or dried seaweed, which is black. Pinch off a piece of black polymer clay the size of a gumball. Roll it into a short snake and flatten it between your fingers. Use the rolling pin to flatten and stretch it. (Fig. 1)
2 To create rice, place white polymer clay in the garlic press and squeeze. Place strings of the white clay on top of the nori. (Fig. 2)
3 Some sushi rolls include carrots, cucumber, and avocado. Place orange clay inside the garlic press and squeeze. Add the orange strings to the middle of the sushi. Add green for vegetables and pink for fish. (Fig. 3)
4 Gently roll the sushi. If any clay is sticking out of the roll, trim it off with a ruler edge. (Fig. 4)
5 To make uramaki, create more rice. Attach the rice on the outside of the sushi. (Fig. 5)
6 Nigiri is one type of fresh fish served on rice. We will make two types of nigiri: salmon and shrimp. Create rice and compress it into an oval. Blend orange and white clay to create salmon. Roll the blended colors into a short snake and squeeze it flat. Place it on top of the mound of rice. (Fig. 6)
7 Create another mound of rice as in step 6. To make the shrimp, use a light pink colored clay. Roll it into a snake and fold it in half. While holding one end, begin to twist it. Flatten the two ends for the tail. Add it to the mound of rice. (Fig. 7)
8 Create chopsticks, wasabi, ginger, and a soy sauce bowl using the methods you've learned.
9 Bake all of the clay sushi pieces according to the manufacturer's directions. Allow the clay to dry.
10 Mix a little Mod Podge with a sprinkle of fine green glitter. Paint it on the nori to give it a little shimmer. (Fig. 8)
| GLITTER BUGS
---|---
When creating with polymer clay, you'll often end up with a lot of leftover pieces. You might be tempted to throw them away, but don't. You can use them for fun sculptures like these glitter bugs. You can also create these insects without the little gemstones.
Tools & Materials
polymer clay
skewer
small metal-backed gemstones
1 Decide what size bug to make. Roll a piece of clay into a ball that would make a good bug body. Shape it as you like and set it aside. (Fig. 1)
2 Pick about four pieces of scrap clay and roll them into small snakes, all about the same length. (Fig. 2)
3 Twist the snakes, creating a rope of clay. Once twisted, form it into a ball. (Fig. 3)
4 Do this several times until the mix of colors is pleasing. Flatten the clay. Cut it with the skewer into a circle, larger than the bug body. Cut the circle in half. Attach the two pieces to the body as the bug's wings. (Fig. 5)
5 Add details, such as small dots of clay, stripes of color, and parts of the bug's body. Try making other insects like dragonflies and bees. (Fig. 6 & 7)
6 Shape a piece of green clay into a ball. Flatten it and cut it into a leaf shape. Place the bug on the leaf.
7 Press gemstones into the polymer clay.
8 Follow the manufacturer's directions for baking the clay bugs.
| MONSTER MAGNETS
---|---
Doesn't every refrigerator need a three-eyed monster? How about a mustached Cyclops or a big-eyed, buck-toothed creature? Explore creating a variety of silly monster features that you can change, rearrange, and stick to any metal surface!
Tools & Materials
polymer clay, in a variety of colors
skewer
self-adhesive magnets
scissors
metallic pen or marker
glue
1 Start making an eye by cutting and rolling a piece of clay into a ball the size of a gumball. Flatten the ball. (Fig. 1)
2 Make a smaller ball of clay with another color. Flatten it and place it on top of the first piece to create the iris—the colorful part of the eye.
3 For more color, add tiny pieces of flattened clay on top of the iris. (Fig. 2)
4 Add a small dot of clay for the pupil in the center of the eye and a white highlight. (Fig. 3)
5 If desired, make another eye, a little different than the first. Or make some variations: Connect two circles to make attached eyes. Add a flattened circle cut in half to create an eyelid. Make some eyelashes! (Fig. 4)
6 Make a mustache by flattening a piece of clay into a shape you like. Add texture with the skewer. You can create a fancy mustache by overlapping two pieces of clay shaped like wings. For a mouth, flatten a large ball of clay. Use the skewer to cut the outer edge of the flattened clay into the shape of a mouth. Flatten, cut out, and add different colors of clay for teeth and a tongue. (Fig. 5)
7 Bake the clay according to the directions on the package. Allow the clay to cool.
8 Trace the creature features onto a piece of magnet and cut out the shape. Attach the features to the magnet by peeling and attaching the sticky back. If you are not using self-adhesive magnets, simply attach the features to the magnet with glue. (Fig. 6–8)
| DAY OF THE DEAD HEADS
---|---
The Day of the Dead or _el Dia de los Muertos_ is a Mexican holiday that is celebrated by many people around the world. During this multi-day holiday, people remember friends and family members who have passed away. But this is not a sad day! There are parades with giant _calaveras_ , or handmade skulls, and music in the streets. Create your own fun and funky calavera in this lab.
Tools & Materials
8" (20.5 cm) square of aluminum foil
one block of white polymer clay per head
polymer clay, in a variety of colors, including black
small metal backed gems (optional)
gold acrylic paint (optional)
paintbrushes
1 Crumple the foil into an oval-shaped ball about the size of a golf ball. (Fig. 1)
2 Warm a block of white polymer clay between your hands and squeeze it to soften it. Flatten the clay on your work surface. (Fig. 2)
3 Fold the clay over the foil ball, flattening any lumps. Use small pieces of clay to fill in any exposed spots. (Fig. 3)
4 Gently press eye sockets into the clay head with your fingers. Roll small pieces of black clay and place them into the sockets. (Fig. 4)
5 To make flowers, roll a small piece of clay into a snake. Flatten it between your fingers. Starting at one end, roll the clay. Pinch it on one end so that it resembles a rose.
6 Use a skinny snake of black clay for the mouth. Add smaller pieces of clay along the mouth for the teeth. Shape a clay heart and place it upside down on the skull for the nose. (Fig. 5–7)
7 Get creative! Add a mustache, a floral crown, glasses, or a hat to your dead heads. (Fig. 8)
8 To create a colorful background, roll three colors of clay snakes and twist them together. Form them into a ball, and then roll them into a snake again. Repeat so that the colors in the clay swirl. (Fig. 9)
9 Flatten the piece of clay and add it to the back of the skull's head.
10 Follow the manufacturer's baking instructions. Allow the clay to cool and decorate it with gems and/or paint.
| BACKPACK CHARMS
---|---
Charms give you a fun way to showcase your interests—your favorite food, sport, hobby, or anything else. In this lab, you'll use the sculpting skills you've learned in previous creations to make unique charms to add to your backpack or a necklace or bracelet. You'll need a variety of colors of polymer clay to create these mini-masterpieces. Everyone's choices will be different for this lab, so use your imagination to create your charm.
Tools & Materials
pencil
paper
variety pack of polymer clay
scissors
skewer
jump rings
1 Jot down your favorite foods, hobbies, and other things. Sketch a few designs before creating. My favorite thing is art, so I decided to sculpt a paint palette.
2 Blend together two colors to create the wood for the palette. When the colors have the appearance of wood grain, flatten it between your fingers. (Fig. 1)
3 On a piece of paper, draw a small palette shape about the size of a quarter and cut it out. Place the paper template on the flattened clay and trace around it with the skewer. (Fig. 2)
4 To make the paintbrush, roll a tiny snake of clay about 1" (2.5 cm) long. Add a little bit of black to the top for the brush and a bit of silver between the brush and the handle. Gently press the brush onto the palette. (Fig. 3)
5 Add little dollops of clay in different colors around the edge of the palette as paint.
6 Before baking, press a jump ring into the clay. This will be where the charm clips are added.
7 Follow the manufacturer's directions to bake your charms. Allow the clay to cool before handling or wearing them.
TIP: Go back through the labs in this book. Look for the directions for creating donuts, fortune cookies, pizza, cupcakes, rainbows, and other fun items. Sculpt these on a miniature scale and add the jump ring before baking. (Fig. 4–6)
| GAME PIECES AND DICE FOR YOUR OWN GAME
---|---
A board game is fun to play. But it's even more fun when you create your very own! Begin by thinking of a theme for your game: aliens, monsters, robots, snack foods, zoo animals—you name it. Then think of a clever environment for your characters: a different planet, a refrigerator, or a wild safari. You'll be putting these fun ideas of yours together to create a game for you to play with family and friends!
Tools & Materials
variety pack of polymer clay
skewer
9" × 12" (23 × 30.5 cm) poster board
watercolor paint
water cup
paintbrushes
permanent markers
10 pieces of 1"× 2" (2.5 × 5 cm) paper
1 Decide on a theme and create three to six game pieces. Start with pieces of clay no bigger than a gumball. Roll the clay into a ball. Press it firmly onto a table so that it has a flat base. (Fig. 1)
2 Add features to the pieces. If they are monsters, add eyes, mouths, and silly details. For animals, add the characteristics that make them unique. (Fig. 2)
3 To create the dice, use a piece of clay the size of a gumball. Roll it into a ball and then shape it into a cube between your fingers. (Fig. 3)
4 With small pieces of clay in a contrasting color, add dots to the dice. Each side of the dice should have a different number of dots from one to six.
5 Follow the manufacturer's directions to bake the game pieces and the dice. Allow the clay to cool.
6 Design the game board any way you want. For a splatter paint board, paint the entire surface with water. Then, using watercolors, dip the brush into the paint and gently tap the back of the brush to splatter the paint. Allow it to dry completely. (Fig. 4)
7 Using a permanent marker, draw a "start" spot for the game. Make this spot big enough for all the pieces to fit into. (Fig. 5)
8 Draw a line for the path of the game. Draw another line about 1" (2.5 cm) from the first one to create parallel lines. Divide the path into spaces with short lines. It will resemble a sidewalk.
9 Write silly directions in these spaces such as "go back to start" or "take three steps forward." (Fig. 6)
10 Create a small deck of draw cards with more directions on them. Don't forget to add a couple of spaces that state "draw a card." Give the game a name! Play the game!
| LET YOUR LIGHT SHINE NIGHTLIGHT
---|---
Create your own nightlight using some of the polymer clay left over from other creations. Never throw away your scraps of polymer clay because you can always use it for fun projects like this one! Customizing your own nightlight will add a little bit of bright to your night.
Tools & Materials
polymer clay, in various colors
skewer
inexpensive nightlight
small metal polymer stamps
glue
1 Pinch off three gumball-sized pieces of different colors clay. Roll each into a snake.
2 Hold the snakes at one end and twist them until they look like a candy cane. (Fig. 1)
3 Squash the twisted cane and form it into a ball. Roll it into a long snake and twist it again so that the colors swirl. (Fig. 2)
4 Flatten the clay on your work surface. Place the nightlight on top. Trace around the edge of the nightlight with the skewer to cut out the shape. Set the nightlight aside. (Fig. 3 & 4)
5 Create a design in the clay with the metal clay stamps. (Fig. 5)
6 Bake the clay according to manufacturer's instructions. Allow it to cool.
7 Use glue to adhere the design to the night light. Plug it in!
| FUNKY FAUX-SAIC
---|---
Mosaics are works of art created from small pieces of glass, stone, or clay. Usually, the bits and bobs are attached to a surface with glue, and grout is added to fill in the in between spaces. Our mosaic—on a door sign or terra cotta pot—is created differently. That's why ours is a _faux_ or fake mosaic!
Tools & Materials
wood or terra cotta item, such as wooden door sign or terra cotta pot
polymer clay
skewer
scissors
metal polymer stamps (optional)
large baking sheet
glue
paintbrushes
Mod Podge (optional)
1 Choose something to cover with faux-saic. Wood surfaces and terra cotta pottery work well.
2 Prepare the fake mosaic tiles by flattening the polymer clay and cutting it into strips with the skewer. (Fig. 1)
3 Use scissors or polymer stamps to cut the clay into small shapes.
4 Place the tiles on a large baking sheet and spread them out. If some are touching, it's fine. Follow the manufacturer's instructions to bake the clay tiles. Allow the clay to cool. Gently break apart any tiles that have stuck together.
5 Plan a design for the surface you want to cover. Brush glue on small portions of the surface. (Fig. 2 & 3)
6 Arrange the tiles in place on the glued sections, leaving just a little bit of space in between. Repeat, brushing on glue and arranging tiles on top. (Fig. 4 & 5)
7 Once all the tiles are in place, allow the glue to dry overnight.
8 To ensure that the tiles will stay in place, add a layer of Mod Podge over the entire surface.
| DESKTOP DINOS
---|---
Many sculptures have something called an _armature_ underneath the outer shape. An armature functions like the skeleton that supports the shape. For your Desktop Dino, you will create an armature from aluminum foil.
Tools & Materials
1 piece of aluminum foil about 10" × 10" (25.5 × 25.5 cm)
polymer clay
skewer
acrylic paint
paintbrushes
Mod Podge
fine glitter (optional)
1 Gently crumple the foil until it resembles a snake. Bend it into a _U_ shape. Pinch one end for the dinosaur's tail. (Fig. 1)
2 The other end of the foil will be the dinosaur's head. To create an armature that will have an open mouth, gently tear the foil about 1" (2.5 cm). Crumple the top and bottom of the torn piece until it resembles the open mouth of a dinosaur.
3 Choose any color clay for the dinosaur. To blend two colors, roll them both into snakes and twist one around the other. Form the twist into a ball. Roll the ball into a snake. Continue twisting and rolling until the clay has created a swirled pattern or has blended completely. (Fig. 2)
4 Pinch off pieces of the clay and squish them into thin pieces for the dinosaur's skin. Drape the pieces over the armature. Gently press them into place. (Fig. 3)
5 Continue flattening the clay and adding it to the aluminum until the entire armature is covered. (Fig. 4)
6 To create the short front legs of a Tyrannosaurus Rex, roll two small pieces of clay and add them on either side of the dinosaur's body. (Fig. 5)
7 To create the dinosaur's back legs, roll longer, thicker pieces and add them to the sides. Bend the feet so that the dinosaur can stand. (Fig. 6)
8 Create a scaly texture by rolling small balls of clay and pressing them onto the dinosaur's skin. Add eyes and teeth with small pieces of clay.
9 Before baking, be certain that the dinosaur can stand. If it falls over, adjust the head and legs until it balances.
10 Follow the manufacturer's directions to bake the clay, with the aluminum inside. Allow the clay to cool.
11 Paint some details onto your dinosaur and allow to dry. Add gloss with Mod Podge or mix fine glitter with Mod Podge before brushing it on.
##
MAKING YOUR OWN CLAY
In the following labs, you get
to be a chef, a scientist, and an artist as you mix and create with your own clay. Most of the clay recipes call for a handful of basic household ingredients. None of them require cooking, and only a few will need some time in an oven on low heat. When using food coloring to color your homemade clay, be certain to cover your work space and wear gloves so you don't stain your hands.
Most of all, have fun mixing up your batches of clay!
| SIMPLE NO-COOK CLAY
---|---
The best thing about homemade clay is making it! Because we'll be using cooking supplies, it's best to create this clay in the kitchen for easy cleanup. This clay recipe is simple and requires very few ingredients. You can also reuse this clay by keeping it in an airtight zipper-lock bag. If the clay becomes dry while you're working, simply add more water. If you create something you'd like to keep, leave it out to air-dry.
Tools & Materials
large mixing bowl
measuring cups
2 cups (250 g) all-purpose flour
3/4 cup (216 g) salt
3/4 cup (175 ml) warm water
zipper-lock bag
1 In the mixing bowl, measure and add all three ingredients. (Fig. 1 & 2)
2 Use your hands to knead or squeeze the ingredients together. Do this for two to three minutes or until all of the ingredients combine like a ball of dough. (If it's too hard to knead it all at once, split the ball of dough in half and continue to knead.) (Fig. 3 & 4)
3 Once the ball of clay is smooth, it is time to create! To combine pieces of clay, simply apply water as if it were glue. To save your creation, set it out to dry overnight. You can store unused clay in an airtight zipper-lock bag to be used again in the future.
| STAMPED CLAY ORNAMENTS
---|---
Making ornaments is a fun and easy way to create small gifts and explore the textures of rubber stamps. Once these ornaments are dry, they can be painted and strung up with colorful ribbon or twine.
Tools & Materials
Simple No-Cook Clay (Lab 41)
rubber stamps
skewer
watercolor paint
paintbrushes
ribbon or twine
1 Pinch off a piece of clay about the size of a gumball. Roll it into a smooth ball. Flatten the ball between the palms of your hands. (Fig. 1 & 2)
2 Press a stamp into the flattened clay. Stamps that have a deep design work best. Stamps with small fine details do not always work well with this clay, but it's fun to experiment.
3 If you are not pleased with the stamped design, roll it up and try it again. If the clay begins to crack, rub in a little bit of water. (Fig. 3)
4 Poke a hole in the top of the ornament with the skewer. (Fig. 4)
5 Allow the ornament to dry overnight. Turn it over after several hours to ensure that both sides dry.
6 Paint the ornaments with watercolor. Allow the paint to dry. (Fig. 5 & 6)
7 String a ribbon or twine through the hole for hanging.
| SCENTED CLAY ACORNS
---|---
Here is a perfect fall project. You won't believe how wonderful these acorns smell!
Tools & Materials
Simple No-Cook Clay (Lab 41)
spices, such as ginger, cinnamon, cloves, pumpkin, and apple pie spice
measuring spoons
small bowl
acorn tops
string or twine
glue
1 Divide the clay into Ping-Pong ball–size pieces.
2 Sprinkle about 1/4 teaspoon of spice into the bowl. Roll each of the clay balls around in the bowl. (Fig. 1 & 2)
3 Work the clay by smashing it between your hands and rolling it into a ball again. (Fig. 3)
4 For a stronger scent, sprinkle more spice into the bowl and repeat. If too much spice is added, it can cause the clay to dry out. This can be remedied by adding just a touch of water to the clay. (Fig. 4 & 5)
5 Set out the acorn tops. Press a scented ball of clay inside each acorn top. Allow them to dry overnight. (Fig. 6)
6 To hang the scented acorns, glue a length of string to the tops and allow it to dry.
| MAKE YOUR OWN PLAY DOUGH
---|---
This recipe for play dough smells delicious because of the Kool Aid. However, because it is made with salt, you won't want to taste it! If you keep this clay in resealable containers, you can play with it again and again. However, in Lab 45, you'll see how you can make a colorful keepsake with your Kool Aid play dough.
Tools & Materials
21/2 cups (313 g) all-purpose flour
1/2 cup (144 g) salt
1 tablespoon (19 g) cream of tartar
large bowl
small bowls (the same number as you have Kool-Aid packets)
variety of unsweetened Kool-Aid packets
1 cup (235 ml) water
spoon
wax paper
3 resealable containers
1 Mix the flour, salt, and cream of tartar together in a large bowl. (Fig. 1)
2 Divide the flour mixture evenly among the smaller bowls. No measurement is needed, simply eyeball the amount to see whether it looks even.
3 Pour a different color Kool-Aid packet into each of the flour mixtures. (Fig. 2)
4 Slowly pour water into the bowl. (Fig. 3)
5 Mix with a spoon. The mixture should be clumpy and slightly sticky. It should not crumble. If it does, add a small amount of water. (Fig. 4 & 5)
6 Dump the mixture from one bowl onto the wax paper. Use your hands to knead the clay until the color is evenly blended and there are no lumps in the clay. (Fig. 6)
7 Follow these steps for the different colors of Kool Aid.
8 Store Kool-Aid play dough in separate resealable containers for future use. (Fig. 7)
| CLAY ICE POPS
---|---
When you grow tired of using your play dough, you can create these fun ice pop sculptures! They look so real that you might even be able to fool your friends into thinking they're edible. Adding clear glitter really gives the ice pops a frosty appearance.
Tools & Materials
Play Dough, in several colors (Lab 44)
ice pop sticks
wax paper
Mod Podge
paintbrushes
clear glitter
1 Follow the recipe for making Play Dough in Lab 44. Pick two different colors of the Play Dough to swirl together for one ice pop. Each of the two pieces should be about the size of a Ping-Pong ball.
2 Roll each piece of play dough into a long snake. They should be the same length. (Fig. 1)
3 Put the two snakes next to each other. With one hand, hold them together at the top. With the other hand, begin twisting the two snakes together. The twist should look a little like a candy cane. (Fig. 2)
4 Shape the twist into a ball, and then roll it into a snake again. The two doughs should be starting to mix and swirl together. To blend more, pinch the snake at the top and twist with the other hand as in step 3. (Fig. 3)
5 Form the clay into a ball again and then roll it into a short snake. Gently pressing it onto your work surface, shape the clay to resemble an ice pop shape. (Fig. 4)
6 Slide an ice pop stick into the bottom of the dough. (Fig. 5)
7 Allow the clay to dry on waxed paper overnight. Turn the ice pop over every so often so that it dries evenly.
8 To add some icy sparkle, paint a coat of Mod Podge onto one side of the ice pop. Sprinkle with clear glitter. (Fig. 6)
9 Allow the Mod Podge and glitter to dry for a few hours before repeating step 8 on the back and sides.
| JURASSIC FOSSILS WITH SAND CLAY
---|---
The Jurassic period was a time when dinosaurs roamed the Earth. In this lab, you will use sand to create your own clay, and you'll also be able to create dinosaur fossils! You and your friends can have fun burying and excavating these fossils just like paleontologists.
Tools & Materials
large bowl
measuring cups
2 cups (708 g) sand
11/2 cups (188 g) all-purpose flour, plus more for dusting
11/4 cups (360 g) salt
spoon
1 cup (235 ml) warm water
glitter (optional)
wax paper
plate or cookie sheet
toy cars and dinosaurs
watercolor paint, in black or brown
paintbrushes
1 In a large bowl, stir the sand, flour, and salt together with the spoon. (Fig. 1)
2 While stirring, slowly add the water. When the clay can no longer be mixed with the spoon, use your hands. (Fig. 2)
3 Split the clay into small pieces about the size of a golf ball.
4 For a bit of sparkle, press the clay into a container of glitter and work it into the clay. (Fig. 3)
5 Place wax paper on a plate and cover it with a sprinkle of flour. Flatten the clay onto the paper.
6 Use plastic toys such as dinosaurs and cars. Press the dinosaur feet firmly into the clay. Try driving a toy car through the clay. (Fig. 4–6)
7 If it's a sunny day, place the plate in the sun to dry. Flip the clay pieces over every couple of hours. The clay can also be dried indoors for a couple of days.
8 Once the clay is dry, use brown and black watercolor to paint it. Paint the indentations one color and the rest of the clay a different color. This will make the fossils more visible. (Fig. 7)
9 Take the fossils outside and bury them. Have fun excavating your dinosaur fossils!
| GLITTER CLAY BEADS
---|---
This lab uses the recipe from Lab 41, but this time, you'll add color and sparkle to your clay. You'll also learn how to mix the three primary colors—red, yellow, and blue—to create the secondary colors—orange, green, and purple. This recipe makes a lot of clay. Store the extra in airtight zipper-lock bags.
Tools & Materials
Simple No-Cook Clay (Lab 41)
zipper-lock bags
red, yellow, and blue food coloring
fine glitter, in an assortment of colors
disposable gloves (optional)
skewer
scissors
string, yarn, or twine
1 Make the Simple No-Cook Clay recipe from Lab 41. Knead the clay with your hands until it's smooth.
2 Divide the clay into three equal parts. Place the clay into individual zipper-lock bags. Add three or four drops of food coloring to each to make one red, one yellow, and one blue. (Fig. 1)
3 Squeeze a bag until the food coloring and the clay are evenly mixed. If the color is too light, add more food coloring and continue squeezing bag. (Fig. 2)
4 Sprinkle in glitter the same color as the clay. Squeeze the bag until eveything is well mixed. Remove it from the bags.
5 The food coloring used on the clay may stain your hands. Wear disposable gloves before handling the clay. Roll each color of clay into a ball. (Fig. 3)
6 To make the beads for a rainbow necklace, pinch off a piece of red clay the size of a gumball. Roll into a ball. Slowly slide it onto the pointed end of a skewer. (Fig. 4)
7 To create orange, pinch off a piece of yellow clay and a smaller piece of red. Roll the two colors between your hands until orange is created. Slide it onto the skewer.
8 To create green, use yellow and blue. Use a little more yellow than blue to create green.
9 To create purple, use equal parts red and blue.
10 Slide all of the beads onto the skewer in rainbow order: red, orange, yellow, green, blue, and purple. Allow the beads to dry overnight. (Fig. 5)
11 Cut a piece of string, yarn, or twine so that it is about 24" (61 cm) long.
12 Slide the beads onto the string in rainbow order. Use a double knot to tie the ends of the string. Enjoy wearing your necklace. (Fig. 6)
| MAKE YOUR OWN SILLY PUTTY
---|---
Silly Putty is not clay that you can sculpt. Instead, it is great for working up those hand muscles by squeezing and playing with it. But the best part about this Silly Putty is making it. It is messy, sticky fun!
Tools & Materials
4 oz (120 ml) bottle of Elmer's Glue All
mixing bowl
measuring cup
food coloring
spoon
1/2 cup (120 ml) Sta Flo concentrated liquid starch
wax paper
zipper-lock bag
1 Pour the entire contents of the bottle of glue into the bowl. Add one or two drops of food coloring and stir. (Fig. 1 & 2)
2 Pour the liquid starch into the bowl. Let the glue, starch, and food coloring sit in the bowl for five minutes. (Fig. 3)
3 Cover your work surface with wax paper. Dump the mixture onto the wax paper.
4 Pull and squeeze the Silly Putty for five to ten minutes. At first it will be very sticky! Keep pulling, squeezing, and rolling it into a ball until it is no longer so tacky. (Fig. 4–6)
5 Once the Silly Putty is mixed thoroughly, it can bounce, and it can even pull prints from the newspaper.
6 Store your Silly Putty in an airtight zipper-lock bag to play with later.
| SALT DOUGH RECIPE AND ROSE SCULPTURE
---|---
This recipe is similar to the one in Lab 41, but the artwork you make from it will be baked. This will harden the clay and allow you to paint it right away. To make less clay, divide all of the ingredients in half.
Tools & Materials
large bowl
measuring cups
2 cups (250 g) flour
1/2 cup (144 g) salt
spoon
3/4 cup (175 ml) water, plus more to sculpt
wax paper
skewer
cookie sheet
nonstick cooking spray
acrylic paint
paintbrushes
Mod Podge
zipper-lock bag
1 In a large bowl, mix together the flour and salt with the spoon.
2 Slowly add the water and mix it with the spoon. When you can no longer mix it with the spoon, knead the dough with your hands. The dough will have a lumpy texture. (Fig. 1)
3 Cover your work surface with wax paper. Pinch off a piece of dough the size of a large gumball. Roll into a ball. Flatten it between your fingers.
4 To create the center of the rose, start by rolling the edge of the clay inward. Continue rolling until you reach the other side. (Fig. 2)
5 To add petals to the rose, pinch off another piece of clay, roll it into a ball, and flatten it. With your finger, add a couple of drops of water to the flattened clay and add it to the outside of the rose center. The water will act like glue. (Fig. 3)
6 Continue rolling, flattening, and adding clay circles with a drop of water to the outside of the rose until it is the size that you want. Press it gently onto the waxed paper. This will create a flat bottom so the rose can stand. (Fig. 4)
7 Pinch a piece of clay and shape it into a leaf. Use the point of the skewer to add veins to the leaf. Add a drop of water to the leaf and stick it to the outside of the rose. (Fig. 5)
8 Preheat the oven to 250°F (120°C, or gas mark 1/2). Spray the cookie sheet with nonstick cooking spray. Place the rose on the cookie sheet and bake for 25 to 30 minutes. Allow it to cool.
9 To create a rose that looks more three dimensional, paint the rose one color. Then lightly brush white over the top edges of the rose petals. This will create a tint or a lighter version of the rose color. Allow the paint to dry. (Fig. 6)
10 Seal the rose's surface with Mod Podge.
11 Store any extra clay in the zipper-lock bag for later use. It will keep for up to three days if refrigerated.
| SALT DOUGH BIRD SCULPTURE
---|---
These fun birds make great wall hangings. With their silly expressions and vibrant colors, they are sure to brighten up your day. The same dough used in Lab 49 is used in this lab. You can also create bird sculptures with air-dry clay from a craft store.
Tools & Materials
Salt Dough Recipe (Lab 49)
textured fabric, such as burlap or lace
wax paper
skewer
small dish of water
cookie sheet
nonstick cooking spray
acrylic paint
paintbrushes
Mod Podge
string, yarn, or twine
glue
1 Make the Salt Dough Recipe from Lab 49.
2 Use your hands to roll a piece the size of a small orange. Press the clay onto the textured fabric. (Fig. 1)
3 Peel the clay off the fabric slowly. Place it on a sheet of wax paper.
4 Cut the circle of dough in half with the skewer. One half will be the bird's body. The other half is extra clay. (Fig. 2)
5 From the extra clay, pinch off two small pieces for the bird's eyes. Put a dab of water on the back of the eyes to act as glue. Press smaller pieces of clay on top of the circles for the pupils.
6 From the extra clay, pinch off a peanut-sized piece. Roll it into a small snake. Bend the snake in half and place it below the eyes for the beak. (Fig. 3)
7 To create a wing with a small ball of clay, repeat steps 2, 3, and 4. Use half a circle for the wing. (Fig. 4)
8 Poke two holes at the bottom of the bird. Be certain the holes are not too close to each other, nor too close to the bottom edge. Poke two holes at the top of the bird, one at each end. (Fig. 5)
9 For feet, roll two gumball-size pieces of clay. Poke a hole through the balls with the skewer. (Fig. 6)
10 Preheat the oven to 250°F (120°C, or gas mark 1/2). Spray a cookie sheet with nonstick cooking spray. Bake the birds and beads for 25 to 30 minutes and allow to cool.
11 Paint the bird and beads to decorate. Allow the paint to dry. Add a coat of Mod Podge. (Fig. 7)
12 To assemble the bird, cut two lengths of string about 8" (20.5 cm) long. For the feet, thread the string through the two holes at the bottom of the bird. Thread the two beads for feet. Tie a double knot to secure the beads. (Fig. 8)
13 Thread the string through the front top holes for a hanger—knot the ends. Secure all knots with a drop of glue.
| SALT DOUGH TEXTURE FISH
---|---
Using the recipe from Lab 49, create these expressive clay fish. They're simple and fun to make. They would also make decorations and gifts for someone who loves to fish!
Tools & Materials
Salt Dough Recipe (Lab 49)
textured fabric, such as burlap or lace
wax paper
small dish of water
skewer
cookie sheet
nonstick cooking spray
acrylic paint
paintbrushes
Mod Podge
1 Make the Salt Dough Recipe from Lab 49.
2 When you've mixed the dough, pinch off and roll a piece the size of a small orange. Press it onto the textured fabric.
3 Peel the clay off of the fabric slowly. Place it on a piece of wax paper. (Fig. 1)
4 Create eyes for the fish by rolling two balls of clay the size of a gumball. Flatten them. Add a little water to the back of the flattened clay with your fingers. Press the eyes next to each other on one side of the clay circle.
5 For the pupils, roll smaller pieces of clay. Add a little water to the back. Place these on the eyes. (Fig. 2)
6 To make the mouth, roll a gumball-size piece of clay into a snake. Shape the snake into a circle. Place it under the eyes. (Fig. 3)
7 To make the tail and fin, roll a Ping-Pong ball–size piece of clay into a ball. Flatten it into a circle on a piece of fabric. Use the skewer to cut it in half. (Fig. 4)
8 Add one half of the circle to the back of the fish for the tail. Add the other half in between the fish face and the tail. This will be the fin. (Fig. 5 & 6)
9 Cut a small piece of clay into a _U_ shape. Press it into place at the top of the fish's head.
10 Preheat the oven to 250°F (120°C, or gas mark 1/2). Spray the cookie sheet with nonstick cooking spray. Place the fish on the cookie sheet and bake for 25 to 30 minutes. Once it's out of the oven, allow it to cool before painting.
11 Use acrylic paint to decorate your fish. Allow the paint to dry. Seal and protect your fish with Mod Podge. (Fig. 7)
| CANDY CLAY
---|---
This is the only clay in this book that you can actually eat! It looks and feels like many of the other clays in this book. Look at the other labs for ideas on what to sculpt. Or use cookie cutters to create decorations to add to sweet treats such as cupcakes and cookies. You'll have a blast creating an edible sculpture.
Tools & Materials
package of Candy Melts
microwave-safe bowl
spoon
measuring cup
1/2 cup (176 g) light corn syrup
disposable gloves (optional)
food coloring
wax paper
sprinkle of flour
cookie cutters (optional)
cellphane gift bags (optional)
ribbon (optional)
1 Empty the entire package of Candy Melts into the bowl. (Fig. 1)
2 Microwave on the defrost setting for two minutes. Use the spoon to stir. Continue to microwave until the candy has completely melted.
3 Mix in the corn syrup and stir until blended. (Fig. 2)
4 Pour the candy clay onto a strip of waxed paper. Allow it to cool and harden for a few hours or place it in the refrigerator for 30 minutes. (Fig. 3)
5 To add color to the clay, break off a piece and work it between your hands. It may be hard at first. Working with it for a couple of minutes will soften the clay.
6 Add one drop of food coloring in the center of the clay. Fold the clay inward toward the drop and work the color into the clay by kneading it. If you want to prevent your hands from being stained, wear disposable gloves. (Fig. 4 & 5)
7 Place a sheet of wax paper on the table and sprinkle it with flour. Press the clay onto the wax paper. Use cookie cutters to cut out shapes or letters. (Fig. 6 & 7)
8 Use the candy clay to decorate sweet treats. Or place them in a cellophane bag with a bow and give them away as edible gifts!
## RESOURCES
### BOOKS FOR AIR-DRY CLAY CREATIONS
The following books were written for young artists to explore kiln-fired clay. However, the projects in these books can just as easily be created with air-dry clays.
_Ceramics for Kids_ by Mary Ellis
_The Great Clay Adventure_ by Ellen Kong
_The Kids 'N' Clay Ceramics Book_ by Kevin Nierman and Elaine Arima
_Exploring Clay with Children_ by Chris Utley and Mal Magson
_Clay in the Primary School_ by Peter Clough
_Super Simple Clay Projects_ by Karen Latchana Kenney
### BOOKS FOR POLYMER CLAY CREATIONS
_Kawaii Polymer Clay Creations_ by Emily Chen
_The Polymer Clay Techniques Book_ by Sue Heaser
_Modeling Clay Animals_ by Bernadette Cuxart
_Modeling Clay Fantasy Characters_ by Bernadette Cuxart
_Crazy Clay Creatures_ by Maureen Carlson
### ONLINE RESOURCES
Here are some great online resources for the budding sculptor.
Clay Kids is a wonderful site with great Claymation videos as well as plenty of tips and tricks.
www.claykids.com
Crayola has a wide variety of kid friendly projects available on their website. Simply search clay to find an assortment of ideas to choose from.
www.crayola.com
## ACKNOWLEDGMENTS
To the folks at Quarry books who helped hold this clueless girl's hand in the writing and photography of this book. I wouldn't have been able to do it without y'all.
To my husband, Mitch, for understanding my disappearing to the "clay room" each evening and weekend only to reappear with clay under my nails and stuck in my hair. Thank you for giving me the time and space to make big messes.
To my mom for her excellent cheerleading skills. You are always there to encourage me, and I love you for it.
To my sweet art teacherin' friends, who are always there to offer advice, encouragement, and hugs. The world is truly a better place with good friends.
And a big special thank you to my sweet students, who have taught me just as much about creativity, imagination, and mess making over the years as I've attempted to teach them. Teaching art is truly the best job there is, and I'm thrilled that I have the opportunity to do it!
## ABOUT THE AUTHOR
Cassie Stephens has been teaching art to kindergarten through fourth grade students in the Nashville area for close to twenty years. Prior to art teaching, she was a terrible waitress, a miserable corn detasseler, and a lazy egg factory employee. A fateful voyage from the Midwest to the South, with her grandmother in tow, landed her an art educator position in Tennessee. She's been creating big messes with small people ever since.
If you ask Cassie's students what they love about art, you'll usually get one of two responses: "It's fun!" and "Because we play with clay!" Knowing just how much her students enjoy creating with clay, Cassie has introduced her students to a wide variety of projects, many of which you can find in the pages of this book. What she has discovered is that showing students ideas in working with clay is fun, but the real magic is in seeing just where her students will take these ideas and what they will create.
When not elbows deep in mess making merriment, you can find Cassie blogging all about her adventures in art education at cassiestephens.blogspot.com.
## INDEX
acorns, as Scented Clay Acorns,
acrylic paints,
air-dry clay
Bobblehead Pets,
Castle on the Hill,
Clay Coat of Arms,
Coffee and a Donut,
Coil Press Pot,
Cupcake Containers,
Dropped Cone Sculpture,
fragility of,
Garden Gnome Home,
Hamburger Holder,
introduction to,
Line Relief Pencil Cup,
Magical Forest Friend Door,
Marionette Puppets,
Miniature Cactus Garden,
Monogram Wall Hanging,
Pie Safe,
Shoe-Stamped Pottery,
Stack of Pancakes,
Textured Landscape Plaque,
Viking Ships,
Wax Resist Leaf Dish,
Woodland Creature Portraits,
animals
Animal Masquerade Masks,
Bobblehead Pets,
Desktop Dinos,
Jurassic Fossils with Sand Clay,
Marionette Puppets,
Salt Dough Bird Sculpture,
Salt Dough Texture Fish,
Tye-Dye Turtles,
Woodland Creature Portraits,
aprons,
baby oil,
baby wipes,
beads
All-About-Me Mobile,
Glitter Clay Beads,
Salt Dough Bird Sculpture,
body
Day of the Dead Heads,
Egyptian Sarcophagus,
Monster Magnets,
books, as Miniature Accordion Books,
buddy system,
buildings
Castle on the Hill,
Garden Gnome Home,
Magical Forest Friend Door,
Textured Landscape Plaque,
butter knives,
candy, as Candy Clay,
CelluClay
Animal Masquerade Masks,
Crayon and Pencil Sculpture,
Egyptian Sarcophagus,
charms, as Backpack Charms,
cleanup, ,
clock, as Color Mixing Clock,
coat of arms, as Clay Coat of Arms,
Creative Paperclay,
dental floss,
dinosaurs
Desktop Dinos,
Jurassic Fossils with Sand Clay,
drop cloths,
fish, as Salt Dough Texture Fish,
floss,
food
Backpack Charms,
Candy Clay,
Clay Ice Pops,
Coffee and a Donut,
Cupcake Containers,
Dropped Cone Sculpture,
Fun Fortune Cookies,
Hamburger Holder,
Pie Safe,
Stack of Pancakes,
Sushi for Supper,
game, as Game Pieces and Dice for Your Own Game,
garlic press,
gouache paints,
homemade clay
Candy Clay,
Clay Ice Pops,
fragility of,
Glitter Clay Beads,
introduction to,
Jurassic Fossils with Sand Clay,
Make Your Own Play Dough,
Make Your Own Silly Putty,
Rose Sculpture,
Salt Dough Bird Sculpture,
Salt Dough Recipe,
Salt Dough Texture Fish,
Scented Clay Acorns,
Simple No-Cook Clay,
Stamped Clay Ornaments,
insects
Glitter Bugs,
Light-up Lightning Bugs,
jars,
knives,
magnets, as Monster Magnets,
masks, as Animal Masquerade Masks,
mobiles, as All-About-Me Mobile,
Mod Podge,
mosaic, as Funky Faux-saic,
negativity,
nightlight, as Let Your Light Shine Nightlight,
ornaments, as Stamped Clay Ornaments,
paints,
paperclips,
pencils,
pencil cup, as Line Relief Pencil Cup,
plants
All-About-Me Mobile,
Castle on the Hill,
Garden Gnome Home,
Magical Forest Friend Door,
Miniature Cactus Garden,
Rose Sculpture,
Scented Clay Acorns,
Textured Landscape Plaque,
Wax Resist Leaf Dish,
plaques
Clay Coat of Arms,
Magical Forest Friend Door,
Monogram Wall Hanging,
Salt Dough Bird Sculpture,
Textured Landscape Plaque,
Woodland Creature Portraits,
play dough
Clay Ice Pops,
Make Your Own Play Dough,
plus modeling clay
Coil Press Pot,
Cupcake Containers,
introduction to,
Woodland Creature Portraits,
polymer clay
All-About-Me Mobile,
Backpack Charms,
Color Mixing Clock,
Crystallized Pinch Pots,
Day of the Dead Heads,
Desktop Dinos,
Fun Fortune Cookies,
Funky Faux-saic,
Game Pieces and Dice for Your Own Game,
Glitter Bugs,
introduction to,
Let Your Light Shine Nightlight,
Light-up Lightning Bugs,
Miniature Accordion Books,
Monster Magnets,
Sushi for Supper,
Tye-Dye Turtles,
pop art, as Crayon and Pencil Sculpture,
pottery
Coffee and a Donut,
Coil Press Pot,
Crystallized Pinch Pots,
Cupcake Containers,
Egyptian Sarcophagus,
Hamburger Holder,
Line Relief Pencil Cup,
Pie Safe,
Shoe-Stamped Pottery,
Stack of Pancakes,
Viking Ships,
Wax Resist Leaf Dish,
puppets, as Marionette Puppets,
rolling pins,
salt dough
Rose Sculpture,
Salt Dough Bird Sculpture,
Salt Dough Recipe,
Salt Dough Texture Fish,
sand clay, in Jurassic Fossils with Sand Clay,
sealants,
ships, as Viking Ships,
Silly Putty, as Make Your Own Silly Putty,
skewers,
skulls, as Day of the Dead Heads,
slip-and-score method,
smocks,
soap,
spillproof cups,
sticks,
text
All-About-Me Mobile,
Candy Clay,
Clay Coat of Arms,
Fun Fortune Cookies,
Miniature Accordion Books,
Monogram Wall Hanging,
textured materials,
toothbrushes,
toothpicks,
water,
watercolor paints,
workspace,
DEDICATION
This book is dedicated to all of the bright and budding young artists I've had the privilege to share the joy of mess making with over the past twenty years. Together we have learned, laughed, and grown. Here's to many more messes and masterpieces!
© 2017 Quarto Publishing Group USA Inc.
Original artwork and project designs © 2017 Cassie Stephens
First published in the United States of America in 2017 by
Quarry Books, an imprint of
Quarto Publishing Group USA Inc.
100 Cummings Center, Suite 265-D
Beverly, Massachusetts 01915-6101
Telephone: (978) 282-9590
Fax: (978) 283-2742
QuartoKnows.com
Visit our blogs at QuartoKnows.com
All rights reserved. No part of this book may be reproduced in any form without written permission of the copyright owners. All images in this book have been reproduced with the knowledge and prior consent of the artists concerned, and no responsibility is accepted by producer, publisher, or printer for any infringement of copyright or otherwise, arising from the contents of this publication. Every effort has been made to trace the copyright holders and ensure that credits accurately comply with information supplied. We apologize for any inaccuracies that may have occurred and will resolve inaccurate or missing information in a subsequent reprinting of the book.
Digital edition published in 2017.
Digital edition: 978-1-63159-403-8
Softcover edition: 978-1-63159-270-6
Library of Congress Cataloging-in-Publication Data available.
Design and Page Layout: Laura H. Couallier, Laura Herrmann Design
Cover Image: Patrick F. Smith Photography
Photography: Patrick F. Smith Photography: pages 6, , , , , , , and . All other photography by Cassie Stephens.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 5,715
|
Cesar je departement v Kolumbii. Leží na severovýchodě země při hranicích s Venezuelou. Departement byl vytvořen v roce 1967 a sousedení s regiony La Guajira y Magdalena, Bolívar, Santander a Norte de Santander. Ze severu sem zasahuje pohoří Sierra Nevada de Santa Marta, ze západu Serranía del Perijá, nejvýznamnějšími řekami jsou Magdalena a její pravobřežní přítok Cesar.
Před osídlením Španěly zde sídlily indiánské kmeny. Prvním dobyvatelem byl Pedro de Badillo v roce 1529 a v roce 1631 Němec Ambrosio Alfinger. Klima je zde teplé. Hlavním městem departementu je Valledupar.
Externí odkazy
Cesar
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,148
|
A Mission To Help People!
Have You Heard About The Simplest Way To Sell a House?
More and more people are discovering the value and convenience of a direct property sale.
We are "real" home buyers that have the ability and funds to purchase your house, and have CASH in your hands in just a few days (or as soon as the title and closing company can close)!!
We are NOT "wholesalers" that get a contract on a house, then try to find a real buyer to sell the contract to.
Don't take our word for it, check out our great reviews and A+ Rating with the BBB!
This is to let you know how much we appreciate all that you have done for us. When this all began we thought for sure that we would end up in foreclosure, but due to all your hard work for us and working so closely with our mortgage company, you were able to save us from a devastating end. If in the future I know of someone with similar problems I would certainly recommend your service.
Give our Team a Call! A no-hassle, no-obligation offer might be your solution!
Listing Your House Is Not The Only Way To Achieve a Successful Sale!
There are more important things in life than worry about that property. We can eliminate the stress by buying directly from you. Contact us now for a fair offer.
You will have to consistently clean the house for showings.
If you do find a buyer, you will likely have to wait on the buyer's lender to come through with the financing. This can take weeks and even fall through in some cases!
At the closing table, you will have to pay numerous fees, including closing costs and your agent's commissions.
You will have no guarantees on how long the process will take or how much you'll make!
Working with The Sierra Group LLC is much different. We want to work with you directly, without agents, banks, or lenders. We have the funds to pay you now, so there is no waiting on your money, and you don't ever have to worry about the sale falling through.
* No Seller or Buyer closing costs to pay. We pay all closing costs.
* Flexible possession and closing dates.
We have happily helped house sellers in all sorts of situations. Some fell behind on payments and were going to lose their house to foreclosure. Others received an inheritance they weren't prepared for. And others were going through a divorce and wanted to sell quickly. No matter what you are dealing with, we are able to help you without any additional stress or judgment.
Our primary goal is to help local homeowners out of a situation they no longer want to be in. Get your fair and honest offer now! There is no hassle or obligation!
The Sierra Group LLC is currently paying cash for houses in your area! If you have thought about selling, find out just how lucrative a direct sale can be for you!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,288
|
Q: How to create a user and assign a password in mysql What is going on with the below? Why will mysql not let me in? I am using 5.6
root@ip-10-93-145-98:/home/ubuntu# echo "grant all privileges on *.* to 'really'@'%' identified by 'stupid';" | mysql -u root -ptest101
Warning: Using a password on the command line interface can be insecure.
root@ip-10-93-145-98:/home/ubuntu# mysql -u really -pstupid
Warning: Using a password on the command line interface can be insecure.
ERROR 1045 (28000): Access denied for user 'really'@'localhost' (using password: YES)
A: Run a FLUSH PRIVILEGES after the GRANT ...
Try to login again after that.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,381
|
"I used them and they are fantastic. An electricians dream."
"Hard to find these generally in wholesalers and tools shops. Can be ordered but only if you know there available in the first place. Until last week I'd not seen this saw. One of my workmates has the 20 and 25mm saws he'd picked up from Edmundsons at one of the London airports whilst working there. I used them and they are fantastic. An electricians dream. So much so that I've ordered the 'Gang of Three' - 20 25 and the 32mm. Around £43 with postage. Asked said workmate how long he'd had and been using his and at 3 years and counting, that's got to be great value. All I can add is 'Get along Postie'!"
"They'll cut a neater cleaner hole, and quicker too - Worth the money."
"Far and away better than general service hole saws us trade people usually buy. These are 21st century gear. Used with care they'll cut a neater cleaner hole, and quicker too! Recommend cutting oil as this will extend the life by miles. Worth the money."
"Best holesaw for cutting tray,trunking and panels just don't cut galvanised boxes."
"Good if not better than the red ones."
"nice product, good if not better than the red ones. ideal for drilling sheet steel, trunking etc as they dont fly through the hole once drilled due the the shank on the body."
"Arrived as stated without hassle. We wanted the saw primarily to cut drain holes in the stern of a boat so it had to tackle fairly thick fibreglass and chipboard. Managed everything and, unlike some cheaper holesaws, it still has plenty of life left. Not the cheapest but worth the money."
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,377
|
# **FINANCIAL STATEMENTS DEMYSTIFIED**
**A SELF-TEACHING GUIDE**
**BONITA K. KRAMER
CHRISTIE W. JOHNSON**
Copyright © 2009 Bonita K. Kramer and Christie W. Johnson. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.
ISBN: 978-0-07-154388-0
MHID: 0-07-154388-0
The material in this eBook also appears in the print version of this title: ISBN: 978-0-07-154387-3, MHID: 0-07-154387-2.
eBook conversion by codeMantra
Version 2.0
All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps.
McGraw-Hill Education eBooks are available at special quantity discounts to use as premiums and sales promotions or for use in corporate training programs. To contact a representative, please visit the Contact Us page at www.mhprofessional.com.
**TERMS OF USE**
This is a copyrighted work and McGraw-Hill Education and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill Education's prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms.
THE WORK IS PROVIDED "AS IS." MCGRAW-HILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill Education and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill Education nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill Education has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill Education and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.
From Bonita K. Kramer:
For Mark, David, and Kyle—all of whom have demystified the meaning
of life in my mind.
From Christie W. Johnson:
For Pat and Swithin—you believe in me, always.
From both of us:
And for Gil W. Crain, Ph.D.,
who unexpectedly passed away in July of 2008.
A great friend, colleague, mentor, and professor
who encouraged and supported us in all our endeavors.
## **CONTENTS**
**ACKNOWLEDGMENTS**
**INTRODUCTION**
**CHAPTER 1 Four Basic Financial Statements**
**CHAPTER 2 Basic Concepts**
**CHAPTER 3 The Income Statement**
**CHAPTER 4 The Balance Sheet: Classifications and Concepts**
**CHAPTER 5 The Balance Sheet: A Closer Examination**
**CHAPTER 6 Statement of Cash Flows**
**[CHAPTER 7 Reading the Financial Statements:
The Auditors' Reports and Financial
Statement Footnotes](chapter7.html#ch7)**
**CHAPTER 8 Analysis of Financial Statements**
**CHAPTER 9 Additional Issues**
**CHAPTER 10 Fraudulently Misstated Financial Statements**
**Final Exam**
**Answers to Quiz and Final Exam Questions**
**Index**
## **Acknowledgments**
_From Bonita K. Kramer_ :
I doubt I would have undertaken this project without the support of my coauthor, Christie Johnson. Christie is an extraordinary writer who has a wealth of information about financial accounting. She has been a faculty member at Montana State University since I was an undergraduate accounting student here many years ago. Then, as now, she had a stellar reputation as an educator, and to coauthor this book with her as a colleague truly has been one of the highlights of my career.
Additionally, however, I could not have written this book without the patience of my husband, Mark, who is my best friend, and our boys, David and Kyle. While my husband never doubted my ability to write this book, he was concerned about the time I would invest in it.
Finally, I want to thank my parents, Rod and Betty Peterson, for their encouragement throughout my life to continue my education and to pursue various professional challenges over the years. They never doubted my abilities and always quietly cheered me on. Yet they also instilled in me a sense of what is truly important in life, and I am grateful.
_From Christie W. Johnson_ :
Bonita flatters me with her remarks describing why our collaboration on this project took place, but I should set the record straight. The opportunity actually arose from Bonita's reputation as an expert in the area of fraud education and contributions to this specialized field and her outstanding reputation as a teacher and scholar. She has garnered numerous teaching and research-related honors during her career as an accounting educator. Bonita is a teaching role model and an incredibly patient and effective mentor when it comes to publishing endeavors. She has a talent for teaching an old dog (like me) some new tricks, and I was honored by her invitation to collaborate. I appreciate and am very grateful for this opportunity, one I probably would not have enjoyed had it not been for Bonita's faith and interest in working with me.
Bonita and I both very much enjoy getting to know our students in person and teaching to a live audience. We are motivated to come up with many different ways to illustrate or explain a concept, knowing that individual student learning styles vary considerably. If we see a few foreheads and eyebrows scrunching, we have a visual cue that our students are struggling to understand a difficult or important concept. In a classroom setting, we can react to these cues, ask and answer questions, and meet with students one on one.
In writing _Financial Statements Demystified,_ it was our joint goal to connect with individuals we probably won't meet in person, a prospect both exciting and a bit daunting! We organized this book by presenting the topics and concepts we both felt to be most relevant to your understanding of financial accounting and reporting while avoiding the accounting system details found in most traditional textbooks. We hope that you find our explanations and illustrations provide useful information and that the benefits you gain by reading this book exceed the cost of its purchase!
I too would like to acknowledge my family's support in this endeavor. My husband, Pat, was remarkably patient, especially during marathon writing episodes when I would become so focused and paused just long enough to share the meals he prepared. My daughter, Swithin, plays an important supporting role in my writing and publishing endeavors. In an interesting reversal of roles, I now seek Swithin's input and advice. I think she must take great delight when she has an opportunity to wield a bold-line red pen, much like the one I drew out of my bag and used to critique her written work when she was in junior high and high school.
Bonita's parents and my own instilled many virtues we both appreciate and honor, and they believed deeply in our talents and abilities. While we each grieved over the loss of a parent last year, we knew they would be proud of our collaborative efforts on this book.
## **INTRODUCTION**
This book is intended for people who want to understand the information provided in financial statements. We don't describe all the accounting details of preparing financial statements, which is an approach typically employed in many accounting courses. Since many accounting students, upon graduation, will hold positions requiring that they prepare their employers' financial statements or audit their clients' financial statements, the preparer approach employed in college makes sense for accounting majors. However, that is not our goal here. Instead, we focus on a user approach to understanding financial statements.
Because of the user approach we take in this book, you don't need any accounting courses or background to learn this material. Instead, this book is meant to help people base decisions on the information provided in financial statements.
What kinds of decisions might you be thinking of making in which financial statements can be useful? Well, perhaps you are an investor thinking about your future retirement and are wondering whether you might like to purchase some stock in a particular company. Or perhaps you already own stock in a company and are trying to decide if this is a good time to sell that stock. Or perhaps you are a creditor wondering whether you should lend some money to an organization. Perhaps you are a union official involved in negotiations with a business and want to have a thorough understanding of the business's financial health. Perhaps you own a business and, regardless of its size, want to understand the financial statements compiled by your accountants better to help you make future business decisions. Perhaps you are a manager in an organization that has its own accounting department, which routinely prepares financial statements for your use, and you want to comprehend better the statements they are sending to your office. Or perhaps you are about to enter a graduate business program without an accounting undergraduate degree and want a reference guide to the content and terminology of financial statements.
We hope this book will answer some of your basic questions about financial statements, explain common terminology and concepts, and demonstrate the interrelationships among the four basic financial statements so that you will be better able to understand and make decisions on the basis of the information provided in financial statements.
To accomplish these goals, we first provide a general overview of the four financial statements. Next, we describe some basic concepts and principles that are built into all financial statements. After that, we explain each of the four financial statements and their typical contents in detail, with examples. Armed with this understanding, we then explain how to read financial statements, including the auditor's report, and discuss some common ratios you can compute on the basis of the financial information. Our last section discusses the techniques typically employed by people who are intent on fraudulently misstating their financial statements and gives you guidance on some red flags that might alert you to the possibility that the financial statements should not be replied on.
We encourage you to grab the annual report of any company you are interested in and pore over it as you read this book. We think you'll be surprised and pleased to discover how understandable the financial statements really are!
## **CHAPTER 1
Four Basic Financial Statements**
### **Why Are Financial Statements Necessary?**
Financial statements are the output of the accounting process, a formal way of communicating financial information that can be used by a variety of parties in making decisions about a business. For example, understanding financial statements can help you decide whether you want to invest in, lend money to, or grant credit to a company. The owners of an existing business use information derived from financial statements in planning and evaluating business activities. Union officials may use financial statements in their negotiations with management on behalf of the employees. The results reported in financial statements may help a company determine the size of bonuses paid to top management. In short, financial statements are a critical source of information for most business decisions.
In this book, we assume that you know nothing about the accounting process or the different types of financial statements that result from that process. However, even if you already know something about accounting or financial statements, you may find this book useful in providing a more complete view of financial statements, their interrelationships, and the accounting concepts that underlie their preparation.
What will you learn if you read this book? Our purpose is to assist you in understanding financial statements by acquainting you with accounting and financial statement terminology, the underlying concepts on which financial statements are based, and the form and content of those statements and their interrelationships. You will develop skills in reading financial statements, including the accompanying footnotes, and understanding the related auditor's report. Once we have covered those topics, you will be able to develop skills in identifying fraudulently misstated financial statements, a topic covered in the final chapter.
### **Objectives of Financial Reporting**
Before we introduce you to the four basic financial statements, you should understand the overall purpose of summarizing accounting information in formal financial statements. The accounting profession has identified the objectives of financial reporting as follows:
• To provide information that is useful in making investment and credit decisions
• To provide information that is useful in assessing cash flow prospects
• To provide information about the resources of a business, claims to those resources, and changes in them
Note that "to provide information" is the focus of these objectives. In other words, the information provided by financial statements can be useful in answering the following types of questions:
• Is the business profitable?
• Do the operating activities of the business generate sufficient cash flows?
• Has the business grown? Was the growth achieved through its own profits, additional investments by its owners, or borrowing?
• Does the business generate sufficient cash flows to repay amounts borrowed? To return profits to its owner or owners?
• Does the business generate sufficient cash flows to pay its current obligations?
• Is the business likely to continue to be able to pay its obligations as they become due over the long run?
• What are the amounts and types of the resources of the business? What are the amounts and types of claims against those resources?
Many of the answers to these questions can be found directly in the financial statements and indirectly through the relationships between certain items in the financial statements and other information disclosed in the notes to those statements. The answers to these questions can help you decide whether to purchase a business or invest in another business by becoming a partner or stockholder or perhaps help you decide whether to continue to operate or sell your existing business. You may decide whether to lend money or extend credit to a business or determine why it will be difficult or easy for you to get a business loan or establish credit. Financial statement information can help you determine the likelihood of being repaid if you lend money or extend credit or gauge the likelihood of receiving an adequate return on your investment in your own business or investments in others. As you progress through this book, it should become clear that the more you learn about financial statements and the concepts on which they are based, the more understandable and useful they become.
### **The Four Basic Financial Statements**
At this point, we want to introduce you very briefly to the four basic financial statements: (1) the balance sheet, (2) the income statement, (3) the statement of changes in owner's equity, and (4) the statement of cash flows. Later chapters will discuss each type of statement and the underlying concepts in more detail.
#### **THE BALANCE SHEET**
The balance sheet presents the assets, liabilities, and residual equity of the owner or owners of a business. It is a snapshot of the business, showing its financial position at a specific point in time. In fact, the balance sheet sometimes is referred to as the statement of financial position.
Assets are the economic resources of the business. They may be tangible—something you can see and touch—or they may represent rights the business possesses. Examples of assets include cash, accounts receivable, inventories of goods to be sold to customers, supplies, prepaid expenses, land, buildings, equipment, investments, and natural resources.
Liabilities are obligations of the business, or creditor claims against the assets of the business. Examples of liabilities include accounts payable to suppliers, customer deposits for goods or services to be provided, salaries or wages payable, taxes payable, rent payable, mortgage notes payable, and interest payable.
Equity is the excess of the assets over the liabilities, or the owners' claims against the assets in the business. Sometimes the terms _net assets_ and _net worth_ are used to describe the owner's residual interest. Depending on the form in which the business is organized, this residual interest may be called owner's equity (for a sole proprietorship: a one-owner business), owners' equity (for a partnership in which there is more than one owner), or stockholders' equity (for a corporation) (see Figure 1.1).
**Figure 1.1** Terms for Equity under Different Business Forms
The balance sheet is based on a basic formula that is called the balance sheet equation or sometimes the accounting equation (Figure 1.2):
**Figure 1.2** Balance Sheet Equation
This formula always must hold true; in other words, assets always equal liabilities plus owner's equity. It is probably apparent now how the balance sheet gets its name: It always must be in balance!
A simple balance sheet for a corporation appears in Figure 1.3. Each financial statement includes a heading indicating the name of the business, the type of statement, and the date or period of time pertaining to the statement. Assume the business was formed on January 1, 20x1, with assets, liabilities, and owner's equity, as shown in Figure 1.3.
The assets of this business consist of cash and supplies, which are economic resources of the business. The liabilities consist of a single note payable, which is an
**Figure 1.3** Example of a Balance Sheet, Beginning of Year
obligation (debt) of the business. At this point, owner's equity consists of one account called SDK, Capital.
Because this is a new business, it appears the owner contributed $1,200 that makes up part of the cash balance and possibly was used to purchase the supplies and that the remainder of the cash was obtained through borrowing. It is also possible that all the cash contributed by the owner went into the company's cash account and the money borrowed was used to purchase supplies, with the remainder going into the cash account. The balance sheet does not tell you exactly which assets each party contributed, but that's not necessary to understand the company's financial position. Instead, you can tell that there are two claims against the $2,000 of assets of the business: the claim of the party who lent the business $800 and the residual claim of the owner of $1,200.
Note that total assets of $2,000 are equal to the sum of the claims against the assets, which total $2,000; Assets = Liabilities + Owner's Equity ($2,000 = $800 + $1,200). Stated another way, total assets minus the creditors' claims against the assets equals owner's equity ($2,000 – $800 = $1,200).
#### **THE INCOME STATEMENT**
The second basic financial statement we are introducing you to—the income statement—presents the revenues and expenses of the business over a period of time. Thus, it is not a snapshot as of one date, as the balance sheet is, but more of a motion picture. The income statement presents the results of the primary operating activities of the business, which provides services or the production and/or sale of goods. The income statement also computes the net income or profit for the period (if revenues > expenses) or the net loss (if expenses > revenues). It is probably apparent now how the income statement gets its name: It computes income for the period (assuming the company was profitable). If a company has suffered a loss for the period, the statement still is referred to as the income statement. Sometimes, however, this statement is called the profit and loss statement.
Revenues are the resources that flow into the business primarily from the provision of services or the production and/or selling of goods. Examples of revenues include consulting fees earned, commissions earned, rental fees earned, and sales revenue earned. Expenses are the costs associated with generating revenues. Examples of expenses are salaries and wages, rent, utilities, supplies used, cost of goods sold, insurance, interest, and taxes.
Be careful not to equate revenues with cash inflows. The accounting profession requires that revenues presented on the income statement represent amounts that have been "earned" regardless of whether those amounts actually were received in cash during that period. The same concept holds true for expenses in that expenses are not necessarily the same thing as cash outflows. The expenses reported on the income statement represent costs that have been "incurred" regardless of whether those costs actually were paid in cash during that period. We'll cover this concept in more detail in Chapter 2 and again in Chapter 6. For now, we want to show you an example of what the income statement might look like (Figure 1.4).
**Figure 1.4** Example of an Income Statement
This income statement is for the same company whose balance sheet we presented earlier. It shows the revenues earned and the expenses incurred during the company's first year of activities. The revenues consist of fees earned from providing consulting services to customers during the year regardless of whether the customers have paid for those services in cash yet. The expenses consist of wages, equipment rental, supplies, and interest. These expenses were incurred while the company performed consulting services even though the expenses may not have been paid in cash during that time period. As a result of providing consulting services, the business earned a profit of $22,880.
#### **THE STATEMENT OF OWNER'S EQUITY**
The third financial statement we will introduce you to is called the statement of owner's equity. Since stockholders are the owners of a corporation, it stands to reason that this statement is referred to as the statement of stockholders' equity for a corporation.
The purpose of this statement is to report the changes in each separate component of owner's equity and in total owner's equity for a period of time. Sometimes this statement is called the statement of changes in owner's equity. Equity is increased by additional investments of cash or other assets by the owner or owners. Net income also increases equity. Because additional owner investments or net income increases equity, one can deduce what decreases equity: withdrawals by or a distribution of assets to the owner or a net loss (Figure 1.5).
**Figure 1.5** Items That Affect Owner's Equity
The statement of owner's equity helps link the owner's equity shown in the balance sheet at the beginning of the period with that shown at the end of the period.
The statement of changes in owner's equity for the business appears next. Notice that it presents the amounts and causes of the change in owner's equity from the beginning to the end of the first year (Figure 1.6).
**Figure 1.6** Example of a Statement of Owner's Equity
At the beginning of the year, the owner's residual equity in the business was $1,200. The owner's equity of the business increased during the year because of the net income earned by the business of $22,880 and the contribution of equipment by the owner. The owner's equity decreased by $18,000 during the year as a result of the owner withdrawing $18,000 of business assets (probably cash) for personal use.
#### **THE STATEMENT OF CASH FLOWS**
The fourth and last financial statement that we are going to introduce you to is the statement of cash flows. This statement presents a summary of the cash flows of a business during a specific period. It shows the amounts and causes of the change in the cash balance during that time and links last year's balance sheet to this year's balance sheet by reconciling the cash balance at the beginning of the year with the cash balance at the end of the year. We'll show that in more detail in Chapter 6.
How is the statement of cash flows different from the income statement? Remember that the income statement calculates the net income (or loss) by subtracting expenses from revenues. These revenues and expenses are not necessarily equal to the cash inflows and outflows, because revenues are recorded when earned and expenses are recorded when incurred. The statement of cash flows focuses on cash. Thus, for example, some revenue earned by selling goods to a customer who hasn't paid yet would appear on the income statement but not on the statement of cash flows.
Why is this distinction important? A company can be profitable yet still be cash-poor if most of its revenues are earned but not yet received in cash. However, a company needs cash to pay its bills, and so the statement of cash flows gives financial statement readers a good idea of the likelihood that the company will be able to pay its debts as they become due and remain solvent in the near term.
The cash flows are presented in three categories: operating, investing, and financing. Examples of what are included in each category are shown in Figure 1.7.
**Figure 1.7** The Three Categories of Cash Flow Activities and Examples of Each One
The statement of cash flows for the business that we've used in this chapter, SDK Consulting, is shown in Figure 1.8.
Note that the statement of cash flows shows the cash flows from operating, investing, and financing activities; the change in the cash balance from the beginning to the end of the year, which may be an increase or a decrease; and the noncash investing and financing activities. Why is the last element included? These items may affect cash flows in future years, and so this information helps readers of the financial statement to predict whether cash flows will differ significantly in the future as a result of these items.
How do we interpret SDK Consulting's statement of cash flows for this year? It shows that operating activities—in other words, what the company is in the business of doing—increased cash by a net amount of $20,580, which consisted of cash collected from customers and cash paid for the various expenses during the period. You may have noticed that the amounts of cash received and paid for the various operating activities differ from the amounts of revenues and expenses shown for those items in the income statement. Again, that is the case because the income statement reports revenues earned, regardless of whether the cash has been received, and expenses when incurred, regardless of whether the cash has been paid.
The statement of cash flows also shows that investing activities, in this case the purchase of equipment, used $5,000 of the cash available. Financing activities used a net amount of $14,800 of the cash available, consisting of cash received from
**Figure 1.8** Example of a Statement of Cash Flows
additional borrowing, repayment of borrowed money, and cash withdrawn by the owner. In addition, other equipment was acquired through a contribution made to the business by the owner, which did not use any of the cash available to the business. Overall, the cash balance increased by $780, from $1,800 at the beginning of the year to $2,580 at the end of the year.
### **Disclosure of Other Information**
Financial statements are more useful to people outside the business when explanations of other relevant information are provided. Examples of the type of additional information commonly disclosed include the methods of accounting used (particularly where there are alternative ways of accounting for certain events), guarantees of the debt of other parties, and explanations of unusual or significant events reported in the statements. This additional information usually is presented in the notes to financial statements, which often are referred to simply as the footnotes. The footnotes may require several pages for a complex business, but they contain a great deal of relevant information.
### **Relationships Between the Four Basic Statements**
As you reviewed each of the financial statements, you may have noticed that each statement presents different information but that the statements are interrelated. We began with the balance sheet presenting the assets, liabilities, and owner's equity of a new business at the time it was formed. During the course of the year, the business engaged in many activities. The operating activities of the business were summarized and presented in the income statement, which showed the revenues earned, the expenses incurred, and the resulting net income. Cash receipts and payments arising from operating activities were summarized and presented in the statement of cash flows.
The business also engaged in other activities, including borrowing and repaying loans, purchasing equipment, the owner's contribution of additional equipment to be used by the business, and a withdrawal of cash by the owner. The cash receipts and payments from these investing and financing activities were presented in the statement of cash flows, along with disclosure of the investing and financing activities that did not affect cash. The relationships among the financial statements are shown in Figure 1.9.
**Figure 1.9** How the Four Financial Statements Are Related
Why is there a dotted line from the income statement's net income to the statement of cash flows? As you'll learn in Chapter 6, there are two different methods of preparing the statement of cash flows. One method begins with the net income reported on the income statement, which is not the method we briefly introduced in this chapter. We'll introduce you to both methods in Chapter 6.
When you review the SDK Consulting balance sheet presented earlier in this chapter, you will notice that the cash balance shown there ($1,800) doesn't equal the ending cash balance shown on the statement of cash flows ($2,580). This apparent discrepancy is not an error because the balance sheet we have presented earlier in Figure 1.3 was for January 1, 20x1, when the business was formed. The statement of cash flows shows the cash balance one year later, on December 31, 20x1. Using the information we have presented for SDK Consulting, the ending balance sheet would look like the one shown in Figure 1.10.
**Figure 1.10** Example of a Balance Sheet, End of Year
If you are feeling adventurous and are wondering where the balances for all the amounts on the December 31, 20x1 balance sheet came from, continue reading this paragraph. Otherwise, be content to note the interrelationships among the four financial statements that we just illustrated and come back to this paragraph later. Here is how we derived the balances:
• Cash of $2,580: See the ending cash balance on the statement of cash flows.
• Accounts receivable of $3,000: This is the amount you should receive in the near future from your customers because they owe you for services you already have provided or goods you already have sold to them. If you look at the income statement, you'll notice that we generated revenues from consulting of $40,000, but the statement of cash flows indicates that only $37,000 was collected in cash from customers. Thus, our customers must owe us the remaining $3,000.
• Supplies on hand of $0: We began the business with $200 of supplies, but the income statement shows that we expensed $4,000 of supplies. The statement of cash flows indicates that we spent cash of $3,800 on supplies for the year. Thus, we used up (expensed) all the supplies we started with ($200) plus all the supplies we purchased during the year ($3,800).
• Equipment of $6,000: Per our beginning balance sheet, we owned no equipment (which is probably why we show some equipment rental expense on the income statement). The statement of cash flows reports that we purchased equipment costing $5,000 and also that the owner made an additional $1,000 investment of equipment in the business (which is also shown on the statement of changes in owner's equity).
• Wages payable of $500: The beginning balance sheet shows that we owed no wages (which makes sense because we just started business that day). However, the income statement reports that we expensed $10,000 in wages, and the statement of cash flows shows that we paid $9,500 cash for wages. This means we still owe our employees $500 for wages they have earned.
• Notes payable of $4,000: The beginning balance sheet reports that we started with a note payable of $800. The statement of cash flows indicates that we borrowed more money with another note, receiving $4,000. Also, the statement of cash flows indicates that we paid $800 on a note payable that we owed (probably the original note), meaning we still owe $4,000 in total as of the end of the year.
• SDK, Capital, of $7,080: See the ending balance on the statement of owner's equity.
Note, of course, that assets = liabilities + owner's equity on the December 31, 20x1, balance sheet (again, that equality always must hold because the balance sheet must balance).
Information overload? Actually, we think you will be surprised by how much you have learned already. You learned that the business was profitable, its operating activities provided a positive cash flow, it was able to pay its current obligations, it was able to return profits to the owner, and it has grown. By reviewing each of the four financial statements, you have learned important information about the past activities of the business and its financial position at the end of the year. This information would be useful to the owner in planning and controlling the activities and finances of the business and to other parties making investment and credit decisions about the business. Later chapters will present expanded discussions of each of the financial statements and the concepts underlying their preparation and use.
### **Quiz for Chapter 1**
1. The balance sheet equation is:
a. Cash – liabilities = owner's equity
b. Assets = liabilities + stockholders' equity
c. Cash = liabilities + stockholders' equity
d. Assets = liabilities – owners' equity
2. An example of an asset is:
a. Salaries payable
b. Common stock
c. Supplies on hand
d. Rental fees earned
3. The balance sheet shows the assets, liabilities, and owners' equity:
a. For a point in time
b. For a period of time
c. For the future
d. Since the company was established
4. The income statement:
a. Computes a company's net worth
b. Presents a company's financial position at a point in time
c. Shows a company's cash inflows and outflows for a period of time
d. Presents a company's revenues and expenses for a period of time
5. If a business bills its clients $100,000 for services performed during the year but collects only $80,000 in cash, revenues reported on the income statement will be:
a. $20,000
b. $80,000
c. $100,000
d. $180,000
6. If a business incurs $40,000 of expenses associated with operating the business during the year, of which only $35,000 was paid in cash, expenses on the income statement will be:
a. $5,000
b. $35,000
c. $40,000
d. $75,000
7. Decreases in owner's equity are caused by:
a. Net income
b. An additional investment of cash during the year
c. Net income minus net loss
d. Withdrawals by the owner
8. Which of the following is not one of the categories shown on the statement of cash flows?
a. Cash flows from business activities
b. Cash flows from operating activities
c. Cash flows from investing activities
d. Cash flows from financing activities
9. If a business reports revenues of $200,000 and expenses of $180,000 and had $170,000 cash received from customers and $145,000 cash payments for expenses, its cash flow from operating activities is:
a. $10,000
b. $20,000
c. $25,000
d. $55,000
10. Accounts receivable represents:
a. The amounts of cash received from customers for services or goods provided
b. The amount owed by customers for goods or services received but not yet paid for
c. The amount of revenue earned from providing goods or services to customers
d. An expense of the business as a result of borrowing
## **CHAPTER 2
Basic Concepts**
We introduced you to the form and basic content of the four financial statements in Chapter 1, where we also showed you the interrelationships among the financial statements. Before we go into more detail about each financial statement, it's important to understand some of the basic concepts that underlie the preparation of the statements.
Why are any basic concepts necessary? Financial statements may be prepared for use by many different parties, including business owners and managers, investors, creditors, and regulatory bodies such as the Securities and Exchange Commission. Preparing general-purpose financial statements, which is our focus in this book, helps ensure that there is some degree of consistency in the information presented to users outside the business. In other words, anyone comparing the financial statements of ABC Company and DEF Company doesn't have to wonder whether ABC Company prepared its financial statements by using a set of concepts completely different from those used by DEF Company.
In Chapter 1 we introduced you to the objectives of financial reporting (why we prepare financial statements). Understanding how we prepare those statements helps a reader interpret and understand them. To begin, we will introduce you to the qualitative characteristics of accounting information. These characteristics are the bridge between the why and the how (see Figure 2.1).
**Figure 2.1** How Qualitative Characteristics of Accounting Connect the Why with the How of Financial Statement Preparation
### **Qualitative Characteristics of Financial Statements**
Useful financial statements should contain information that is relevant, reliable, comparable, and consistent. What is meant by these terms?
Relevant information helps users make predictions of future company performance on the basis of past information and provides feedback about past predictions. Relevant information is important, useful information to the readers of a financial statement.
Reliable information can be verified, is neutral rather than biased in favor of the business, and represents the substance (true nature) of the underlying events. Reliable information is free from errors.
Information that is presented or reported in a similar manner for different companies is comparable. Comparable information is useful because it allows one to make comparisons among different companies and identify real similarities or differences among those companies.
Consistent financial information is presented in the same manner from one period to the next for a particular company. Consistency helps ensure that the financial statements of one company are comparable from one period to the next. A reader of financial statements should be able to assume that any differences among a company's financial statements between years are due to real economic events, not to a simple change in the accounting treatment.
### **Basic Concepts, Assumptions, and Principles
Used in Financial Statements**
General-purpose financial statements are based on several basic concepts, assumptions, and principles. We will introduce you to these topics next, along with the related accounting terminology. In later chapters, you will apply these topics and terms to specific examples. The basic concepts, assumptions, and principles we will introduce you to in this chapter are as follows:
• Business entity assumption
• Monetary unit assumption
• Objectivity
• Going concern assumption
• Historical cost principle
• Time period assumption
• Revenue recognition principle
• Matching principle
• Accrual basis of accounting versus cash basis of accounting
• Full disclosure principle
• Materiality
• Conservatism
• Generally accepted accounting principles (GAAP)
#### **BUSINESS ENTITY ASSUMPTION**
The business, whether it is formed as a sole proprietorship, a partnership, or a corporation, is considered a separate and distinct entity from the owner or owners. Because the business is considered a separate entity, the financial statements reflect only the transactions of the business. The owner's personal transactions are not included in the business records. For example, if the owner purchases a vehicle for his or her family's use, the vehicle is not reported in the business's records or financial statements.
#### **MONETARY UNIT ASSUMPTION**
Only transactions or events that can be measured in terms of money are reported on financial statements. Because of this assumption, some important events may not be recorded by a business. For example, if a company hires the best employee workforce, it is not possible to assign a dollar value to this event and it therefore will not appear directly on the financial statements. Indirectly, however, the effect of this superior workforce may be reflected on the financial statements through increased profitability over time, but the reader of those statements may not be able to see this cause-and-effect relationship.
In the United States, financial statements are prepared using the dollar as the unit of measure, and it is assumed that the value of the dollar remains relatively stable. Although inflation affects the value of the dollar over time, the effects of inflation are not reflected in financial statements. Some people consider this a limitation of financial statements because the length of the ruler used to measure events continually changes with changes in the purchasing power of the dollar.
#### **OBJECTIVITY**
All the transactions entered in the accounting records should be based on objective, verifiable evidence rather than on something subjective, such as one's opinion. For example, assume that a company buys a piece of land for a price far below what the company believes the land is worth. Regardless, the land is recorded in the company's accounting records at the actual price paid (which can be verified or supported by various documents). The land is not recorded at how much the company believes it is worth (an opinion that undoubtedly will vary from individual to individual).
#### **GOING CONCERN ASSUMPTION**
Accounting methods are based on the assumption that the business will remain in existence forever unless there is evidence that it will not continue indefinitely. For example, this assumption may be questioned when a business is suffering financially and the liquidation (sale) of its assets is imminent. Among other things, the going concern assumption affects the values of assets on the balance sheet, which we discuss next.
#### **HISTORICAL COST PRINCIPLE**
Goods and services purchased are recorded at the cost when they were acquired, which is referred to as the historical cost or original cost. Consistent with the objectivity principle, cost is an objective, verifiable measure. Market values are not used often as a basis for recording. Why? Because of the going concern assumption, we assume that the company's assets will continue to be used in the business and not be sold, making the current market values less relevant.
Using the example above of a company purchasing land for a price far below what the company believes the land is worth, the land will remain on the company's books at the price the company actually paid for it. Even if there is indisputable evidence that the land is worth much more than that, the land will remain on the company's books at its historical cost.
#### **TIME PERIOD ASSUMPTION**
Financial statements are more useful if they present information for short periods rather than at the conclusion of a company's existence. The time period assumption (sometimes called the periodicity assumption) implies that a company can divide its business activities and transactions into shorter periods.
The time period covered by the statements is usually one year, although quarterly or monthly financial statements often are prepared. Many businesses use the calendar year, January 1 through December 31, for financial reporting purposes. Other companies may use a fiscal year, which covers the natural cycle of the business and ideally ends on a date when business activity is at a low point. The fiscal year is also 12 months in length. For example, a ski resort's fiscal year might end shortly after the end of the ski season, whereas a retailer's fiscal year may end in January after the holiday shopping season.
If financial statements were not prepared until the conclusion of the business's life, it would be fairly easy to measure the firm's total profit or loss. However, in the meantime it would be difficult to make business decisions without timely financial information. When the business uses short periods to report its activities, the information is more timely, but it is necessary to make judgments about the period in which certain business activities should be reported and when estimates of amounts should be made. The shorter the period is, the more difficult the estimates and judgments can be. For example, if you provide services to a customer in year 1 but do not receive cash from the customer until year 2, you must determine in which period the revenue was earned and will be reported. A similar judgment must be made about expenses incurred in one period but paid in another. The accounting profession has established the revenue recognition principle and the matching principle to provide guidelines.
#### **REVENUE RECOGNITION PRINCIPLE**
Revenue generally is considered earned and reported in the income statement in the period in which the business provided the service or sold the goods even though the customer may not pay for the services or goods until a later time. This principle applies even if the customer paid for the services or goods in advance. In this way, the income statement is more useful in assessing the accomplishments of the business during a specific period. In other words, it is easier to determine how successful the company was at generating a profit when the focus is on the earning process, not on the timing of the cash receipts. Figure 2.2 shows a few examples.
**Figure 2.2** Revenue Recognition Examples
#### **MATCHING PRINCIPLE**
Costs incurred by the business generally are recorded as expenses on the income statement in the period in which the related revenue is produced. In other words, revenues are matched with the costs and efforts associated with producing those revenues, resulting in the net income or loss during the specific period. Figure 2.3 shows examples of the matching principle.
**Figure 2.3** Matching Principle Examples
These expenses are all being matched with the revenues they helped produce. For example, the employees who worked for the company in December helped generate revenues in some capacity, such as by performing services for customers or selling goods to customers.
#### **ACCRUAL BASIS OF ACCOUNTING VERSUS
CASH BASIS OF ACCOUNTING**
When the income statement presents the revenues earned and the expenses incurred during a time period (without regard to when the cash actually was received or paid), the company is using the accrual basis of accounting. The accounting profession views the accrual basis as the most appropriate way to present income statement information. If the income statement merely reports the cash receipts and cash payments during a period, the company is using the cash basis of accounting. Why does the accounting profession prefer the accrual basis? It believes that the accrual basis of accounting better presents the underlying efforts and accomplishments of a business, which are not necessarily reflected in cash flows. Let's use a simplified example to demonstrate this.
Assume that a new company started business operations this year.
• It performed services worth $100,000 for customers, who have paid $40,000 so far.
• The company paid its employees $60,000 for wages and owes its employees another $5,000 for work they have performed, which will be paid on the next payday.
What is the net income generated by the business in its first year (see Figure 2.4)?
**Figure 2.4** Net Income Calculation under the Accrual Basis of Accounting versus the Cash Basis of Accounting
Notice that under the accrual basis of accounting the company has generated a profit of $35,000 in its first year of operations, whereas under the cash basis of accounting the company appears to be having problems since it suffered a loss of $20,000. The difference is due to the timing of the recognition of the revenues and related expenses. By following the revenue recognition and the matching principles introduced earlier in this chapter, the company is using the accrual basis of accounting. The accounting profession believes that this more fairly reflects the economic reality of the company's operating results. The company earned more revenues than it incurred in expenses, and assuming that this trend continues in the future, the company will continue to be profitable over time.
Cash flow is important, of course. A company needs cash to pay its bills as they come due. The operating activity section of the statement of cash flows can be used to assess cash flows, and this is much like an income statement prepared using the cash basis. By presenting both an income statement using the accrual basis of accounting and a statement of cash flows, the company allows a financial statement reader to assess both profitability and cash flows from operating activities.
#### **FULL DISCLOSURE PRINCIPLE**
Financial statements are more useful when significant information is disclosed adequately to the readers. The concept of full disclosure affects whether certain events are shown individually or combined with other items in the financial statements, how and where they are shown, the types of descriptions appearing in the financial statements, and the degree of explanatory material provided in the footnotes to the financial statements. Materiality, which is discussed next, also affects the way information is presented or disclosed.
#### **MATERIALITY**
Financial statement information and disclosures are presented for material (significant) items. Information is considered material if it would influence the decision of a reasonable financial statement user. Immaterial or insignificant items might not be disclosed separately. Whether an issue is material is a matter of judgment.
#### **CONSERVATISM**
When significant uncertainty surrounds the proper accounting treatment of an item or event, accountants apply the concept of conservatism by selecting the treatment that is least likely to overstate net income or assets. For example, if a business is uncertain about its ability to collect a portion of its accounts receivable (amounts customers owe the company), it may record an estimated loss from uncollectible accounts and report the accounts receivable on the balance sheet at the net amount it expects to collect.
Conservatism does not mean that preparers of financial statements intentionally understate assets or net income. Instead, it is a guideline that is used when there is doubt. If there is no doubt, there is no reason to apply this concept. The logic behind the concept is that once the uncertainty is resolved, if it is determined that net income or net assets are actually a higher amount, the financial statement readers will be pleasantly surprised. In contrast, if the opposite situation occurs and it is determined that net income or net assets are actually a lower amount than initially was reported, the financial statement users are likely to be upset. Good news is better than bad news.
#### **GENERALLY ACCEPTED ACCOUNTING PRINCIPLES**
The phrase "generally accepted accounting principles"—often simply stated as GAAP (pronounced "gap")—are the guidelines and standards developed by the accounting profession that must be followed in preparing financial statements. There is no single authoritative source of GAAP, although primarily GAAP is developed by an organization called the Financial Accounting Standards Board (FASB, pronounced "fazzbee"). GAAP is not carved in stone, and as circumstances change or new business transactions evolve, GAAP will change. We will be explaining GAAP to you throughout this book, for example, in Chapter 3 when we discuss how to compute earnings per share on the income statement. In that instance, you will notice that GAAP can be fairly specific; you also will find that GAAP can provide broad guidelines, such as the revenue recognition principle and the matching principle.
### **Quiz for Chapter 2**
1. Which of the following is one of the qualitative characteristics of financial reporting?
a. Material
b. Relevant
c. Objective
d. Conservative
2. Reliable information:
a. Can be verified
b. Is neutral
c. Is free from errors
d. All of the above
3. The business entity assumption requires that the owner's personal assets and liabilities must be included on the business's financial statements.
a. True
b. False
4. The monetary unit assumption:
a. Recognizes that the value of a dollar changes over time
b. Requires that financial statements be restated to show the effects of inflation
c. Means that only transactions and events that can be measured in terms of money are reported on financial statements
d. Requires judgment in assigning dollar values to nonmonetary transactions such as hiring a superior workforce
5. The time period assumption:
a. Recognizes that a company can divide its business activities and transactions into shorter periods, such as a year or less, and still provide useful financial statements
b. Requires that financial statements cover a period of at least one year to be useful
c. Requires that companies use a calendar year (January 1–December 31) for financial statement reporting purposes
d. Requires that reporting periods must be of sufficient length that no judgments or estimates are built into the financial statements
6. The objectivity concept:
a. Allows opinions to be used in developing financial statement data
b. Requires that transactions entered into the accounting records be based on verifiable evidence
c. Would allow a company to record assets at the values the company believes the assets are worth
d. Allows opinions to be used in developing financial statement data as long as the company's accountants agree with the opinions
7. The conservatism concept in accounting:
a. Encourages the overstatement of assets and revenues
b. Encourages the understatement of liabilities and expenses
c. Supports the selection of an accounting method that will understate assets or net income when there is doubt about the proper accounting treatment of an item or event
d. Supports the selection of an accounting method that will overstate liabilities or expenses when there is doubt about the proper accounting treatment of an item or event
8. The assumption that the business will remain in existence forever unless there is evidence to the contrary is called the:
a. Going concern assumption
b. Business entity assumption
c. Monetary unit assumption
d. Time period assumption
9. The historical cost principle requires that goods and services purchased be recorded at the:
a. Original cost
b. Original cost with adjustments to market value at the end of each reporting period
c. Original cost with adjustments to market value at the end of each reporting period only if there is indisputable proof that the market value is reliable
d. Original cost adjusted for inflation
10. The matching principle requires that expenses:
a. Be recorded when the cash revenue is received
b. Be recorded when the cash is paid
c. Be recorded as soon as the company knows it will incur those expenses
d. That are incurred by a business be recorded in the same period in which the related revenue is recognized
## **CHAPTER 3
The Income Statement**
The material in Chapters 1 and introduced you to the basic form and content of financial statements and the underlying concepts of financial reporting. Each of the types of financial statements will be discussed in more detail in Chapters 3 through , along with examples and references to the basic concepts. This chapter covers the income statement in more detail, using financial statements from Example Company to provide illustrations.
### **Basic Format**
The income statement shows the revenues earned, the expenses incurred, and the resulting net income or net loss during a period of time. The basic form and content of the income statement for Example Company are shown in Figure 3.1.
**Figure 3.1** Basic Income Statement Format
An income statement prepared using the accrual basis of accounting reports revenues earned and expenses incurred during a period, based on the revenue recognition and matching principles. In other words, revenues do not necessarily equal the cash receipts, nor do the expenses necessarily equal the cash payments made by the business during that period. The income statement covers a specific period, generally one year, but also may be prepared for shorter periods.
#### **REVENUES**
Revenues are resources flowing into the business, primarily from providing services or selling purchased goods to customers. A company also can manufacture its own goods to sell to customers. Revenues generally provide cash or a future claim to cash, for example, with accounts receivable or a note receivable.
Assume you own your own house-cleaning business. You earn revenue every time one of your employees cleans a customer's house. There are three ways a customer might pay you: (1) in advance, (2) as soon as your employee finishes performing the service, or (3) shortly after you bill the customer for the service (see Figure 3.2).
**Figure 3.2** Examples of When to Record Revenue
In situation 1, if the customer pays you in advance, your company will record the cash when you receive it, but it will be recorded as a liability or a debt (unearned revenue), which appears on the balance sheet. Why is this considered unearned revenue? If your business doesn't fulfill its obligation to clean the house, your company will have to refund the customer's money. However, once your employee has performed the cleaning service, you will remove the liability from your books and record it as earned revenue.
In situation 2, if the customer pays immediately upon the completion of the house cleaning, the cash you receive will also be recorded as revenue.
However, in the third situation, if the customer does not pay you immediately but asks that you send a bill, your company still has earned the revenue. Your business performed the agreed-upon service, and the customer owes your company money. Thus, you will record the revenue as being earned, but you can't record any cash as coming in. Instead, you will record this as an account receivable in your books. An account receivable is an asset, and like all assets, it will appear on the balance sheet. It represents money owed to your business by your customers for goods you already have sold to them or services your business already has performed. In other words, accounts receivable represents a claim to cash that your company has against someone, and you should receive the cash sometime in the future.
Note that in all three situations, revenue is considered to be earned once your company has performed the service, regardless of when the cash is received. This approach is in accordance with the revenue recognition principle introduced in Chapter 2, which is also part of the accrual basis of accounting.
#### **EXPENSES**
Expenses are costs associated with providing services or selling goods. Examples include:
• Salaries or wages paid to employees
• Rent paid for office space
• Advertising
• Insurance
• Utilities (e.g., electricity, telephone)
• Office supplies used
• The company's cost of inventory sold to customers
• Interest
• Taxes
According to the matching principle, costs are reported as expenses of the same period in which the revenue was earned or in the period in which the expenses were incurred, regardless of when those costs or expenses were paid in cash. In other words, we try to match the expenses, or costs of doing business, with the revenues that the expenses helped generate. This matching process helps develop a more accurate picture of the company's operating success (or failure) during a period of time.
Let's go back to the example in which you own your own house-cleaning business. Some of your expenses may be paid in advance, whereas others may be paid after you incur them (see Figure 3.3).
**Figure 3.3** Examples of When to Record Expense
If your company pays a cost of doing business in advance, such as the insurance in situation 1, you must record the cash payment when you made it or your cash account will not reflect the actual amount of cash you have on hand. However, you have not incurred the expense yet because the time period for which you have paid has not passed. For example, if you decided to change insurance carriers, your old insurance company would have to refund you a prorated amount of the premium you had paid but for which you had not yet received coverage. Therefore, an expense that your company has paid for in advance is called a prepaid expense, which is an asset (and will appear on the balance sheet). However, as time passes, your prepaid expense no longer is prepaid but is actually an incurred expense. The insurance company provided you with insurance coverage during this period, and once the 12 months have passed, you must remove the balance from the prepaid insurance expense account and add it to the insurance expense account (which will appear on the income statement).
However, in the second situation your company did not pay for a cost of doing business in advance but instead paid shortly after receiving the bill for the past month of service. Even though you have not paid your electric bill, you have incurred the expense. Your business used the electricity provided by the electric company, and even though you have not paid the bill yet, your company owes the money to the electric company. To reflect this transaction, you will record the expense, but there will be no cash going out until you actually make the payment. Instead, you will record this as an account payable in your books. An account payable is a liability or a debt (and will appear on the balance sheet). It represents money your business owes for goods or services another party already has provided to you. It is an account you must pay in the future.
Note that in both situations, an expense is recorded as incurred once the cost of doing business has happened, regardless of when the cash is paid. This approach is in accordance with the accrual basis of accounting, which was introduced in Chapter 2.
According to the concepts of materiality and full disclosure, each significant revenue and expense should be shown separately in the income statement, but insignificant or immaterial items may be combined. The footnotes may provide additional information about items reported on the income statement.
#### **NET INCOME (OR LOSS)**
Net income or net loss is the difference between revenues and expenses and reflects the results of all the income-producing activities of the business (see Figure 3.4).
**Figure 3.4** Net Income (Loss) Formula
There are other terms for net income, all of which cover the same concept. These other terms include _earnings, profit_ , and simply _income_.
### **Income Statement Format for
A Business That Sells Goods**
The income statements presented in Chapter 1 and in the earlier part of this chapter were for simple service businesses. If a company purchases goods (products or inventory) for resale to customers or manufactures the goods that it sells to customers, additional items must be reported in the income statement.
An example of an income statement for Example Company appears in Figure 3.5.
**Figure 3.5** Income Statement Example
Example Company purchases goods from other sources (or manufactures its goods) for resale to its customers. This basic income statement form may be used for both sole proprietorships and partnerships. The income statement for a corporation is similar and is discussed later in this chapter.
#### **NET SALES**
As we have discussed, sales revenue represents revenue earned from the sale of goods during the time period, whether the customers paid cash at the time of sale, paid in advance, or will pay in the future. Recall that the cash advances collected from customers, such as customer deposits and gift certificates that were not earned during the period, will be shown on the balance sheet as a liability: unearned revenues. Revenues that were earned during the period but have not been collected in cash (such as sales on credit) will be shown on the balance sheet as an asset: accounts receivable.
More detailed reporting of sales revenue may be necessary when significant or material amounts of merchandise are returned by customers or when cash discounts are taken by customers for early payment on account. For example, the sales revenue for Example Company may appear as it does in Figure 3.6.
**Figure 3.6** Net Sales Revenue Detail
In this example, gross sales revenue represents the total sales of the company for the year. However, some of those sales were returned by customers, and so it would be misleading to report only the gross sales revenue on the income statement. A separate account is created to record the sales returns: sales returns and allowances. Allowances also are recorded in this account, and they represent reductions in the amount owed by the customer. The company might agree to grant an allowance because the customer was not happy with the purchase for some reason (e.g., slightly inferior goods, wrong color). Since the original sale was recorded as part of the gross, or total, sales revenue, it would be misleading not to reduce the gross sales revenue by any allowances granted.
Sales discounts are reductions in the amount owed by the customer in return for prompt payment. For example, a company may offer terms of 2/10, n/30, which is read as "two ten, net thirty." These terms mean that the seller will give the buyer 2 percent off the total sale if the buyer pays within 10 days of the invoice date, typically the date of the bill. Otherwise, the entire amount owed is due within 30 days of the invoice date, with no discount.
Why would a seller offer a sales discount? A company needs cash to pay its own bills. It can't pay its bills with a sale it made on account (an account receivable) even though that is an asset and represents cash the company should be receiving in the near future. Thus, the seller prefers to receive cash as soon as possible.
Why would a buyer take advantage of a sales discount? Discounts are a great deal for the buyer. By paying the bill only 20 days early (since the entire amount is owed within 30 days with no discount), the buyer gets 2 percent off. Interest rates typically are stated on an annual basis, such as the interest rate your bank pays you on your savings account. However, this discount interest rate pertains to a 20-day period only. To see how much interest the buyer would be paying to keep his or her cash for only an extra 20 days, convert the discount interest rate to an annual basis. For ease in computation, assume there are 360 days in a year, which translates to eighteen 20-day periods. Without considering compounding interest, this equals an annual interest rate of approximately 36 percent (18 periods times 2 percent), an astronomically high rate by most standards.
Why are separate accounts kept for sales discounts and sales returns and allowances? It would be possible simply to reduce sales revenue for those items, but if that is done, this information is lost. Keeping separate accounts for these items allows management (and the readers of the financial statements) to determine, for example, if sales returns seem to be excessively high, which may indicate that the company is experiencing a problem with its products.
#### **COST OF GOODS SOLD**
Cost of goods sold sometimes is referred to as cost of sales, and the name of the account aptly describes what it represents. This account is an expense and includes the cost of the merchandise sold to customers during the year, not necessarily the amount of cash paid by the company for the merchandise. Some of this merchandise may have been purchased in a prior period, or the company still may owe money to its suppliers for some of the merchandise. Calculating cost of goods sold and reporting it on the income statement is another illustration of the matching principle that was discussed in Chapter 2 and is shown in Figure 3.7.
**Figure 3.7** Examples of When to Report Cost of Goods on the Income Statement
In situation 1, the accrual basis of accounting will result in the same reporting as the cash basis of accounting since the purchase, sale, and cash flows all occurred during the same period (year 1).
In situation 2, however, the company sold some merchandise that it had not paid for yet and for which it had not collected all the cash from its customers yet. Under the matching principle followed with the accrual basis of accounting, the revenue is to be recorded when it is earned (i.e., sold to the customers) and the expense is to be recorded in the same period as its related revenue (the sales of the merchandise).
Following the matching principle, situation 3 requires that the revenue be recorded when sold to the customers in years 1 and 2 (i.e., when earned) and the related expense be matched against that revenue (i.e., the goods sold in year 1 are expenses of year 1, and the goods sold in year 2 are expenses of year 2).
What happens to goods that are purchased but not yet sold? These goods are not shown on the income statement as part of cost of goods sold but rather are an asset called inventory that is shown on the balance sheet.
More detailed reporting of the cost of goods sold section of the income statement for Example Company appears in Figure 3.8.
**Figure 3.8** Cost of Goods Sold Detail
The formula shown in the figure makes logical sense. Essentially, what the formula for cost of goods sold states is that the inventory you have on hand at the beginning of the year plus the additional inventory you purchase during the year equals the inventory you have available to sell. Subtracting the inventory you still have on hand at the end of the year (it wasn't sold if it is still on hand) from what was available to sell must equal the amount of inventory sold. Since all the components of the formula are stated in dollars, the end result is not the number of units sold but the cost of the inventory sold. Specifically, it is the cost to your company: what your business paid for the inventory, not what your company is selling it for to your customers.
#### **GROSS PROFIT**
Gross profit often is referred to as gross margin and represents the markup on the goods sold. This markup is the difference between what your company paid to purchase the inventory you will sell to your customers and the price you will charge your customers for the inventory they purchase from you. Your company should earn a gross profit. If not, it means you are selling your inventory for less than or exactly what it cost, and your company obviously won't be in business for very long in this case.
#### **OPERATING EXPENSES**
The only expense we have placed on the income statement up to this point is the cost of goods sold. Before any income or loss is computed, the other operating expenses must be deducted from the gross profit.
Operating expenses represent expenses arising from the primary business activity, which in this case is the sale of inventory to customers. These expenses are deducted from gross profit to determine operating income (or operating loss). Typical operating expenses may include the ones shown in Figure 3.9.
**Figure 3.9** Examples of Operating Expenses
Where do the owner's personal expenses belong? According to the business entity assumption, personal expenses of the owners that were paid by the business are not included in the income statement but instead are shown in the statement of owner's equity as withdrawals. Similarly, salaries withdrawn by the owner or owners of a sole proprietorship or partnership usually are treated as withdrawals rather than expenses of the business because the owner is not considered an employee. Instead, the salary withdrawn by the owner is considered a distribution of the profits of the business to the owner.
Operating expenses often are categorized further as selling expenses and general and administrative expenses. Immaterial expenses may be combined rather than shown separately, particularly in financial statements used by parties other than management.
As with cost of goods sold, the matching principle applies to operating expenses. The salaries and wages expense consists of the salaries and wages earned by employees during the period even though the business may not pay the employees until a later time. For example, assume the income statement was for the calendar year, which ended on Tuesday, December 31, 20x1, and the weekly pay period ends on Friday, January 3, 20x2, which is in the next accounting period. Further, assume the business is closed on weekends. This year's salaries and wages expense includes two days of wages expense incurred (Monday and Tuesday) even though the wages have not been paid (see Figure 3.10).
**Figure 3.10** Example of Accrued Wages Expense (Matching Principle)
What happens with the liability for the two days of wages owed by the company to its employees as of December 31, 20x1? That debt will appear on the balance sheet as salaries and wages payable. When it is paid on Friday, January 3, 20x2, cash will be decreased and the liability will be decreased (since nothing more is owed), but there will be no impact on expense. This is an example of an accrued expense: one that is incurred but not yet paid.
Another example of the matching principle is the treatment of rent expense. Rent expense consists of the cost of renting space for the business that was incurred during the accounting period even though the rent may have been paid in advance. For example, assume the lease requires one year of rent to be paid on December 1, 20x1 and the accounting time period ends on December 31. How much rent expense will appear on the income statement for the year ending December 31, 20x1?
The income statement will include the cost of 1 month (or 1/12) of the lease as rent expense, and the remaining 11 months (11/12) will be shown on the balance sheet as an asset: prepaid rent (see Figure 3.11). This is an example of a prepaid expense: one that is paid but not yet incurred.
**Figure 3.11** Example of Prepaid Rent (Matching Principle)
Most other expenses are treated similarly; they are reported as expenses of the period in which they were incurred. As a practical matter, most operating expenses are both incurred and eventually paid in the same accounting period, except for those that fall close to the beginning or end of the accounting period. Consequently, accountants realize the importance of looking for and analyzing prepaid and accrued expenses at the end of the accounting period to make sure they are reported properly.
Although almost all operating expenses require cash payments at or near the time they are incurred, certain types of operating expenses do not require cash payments. Depreciation expense is a common example of an operating expense that does not use any of the business's cash that is based on a different application of the matching principle. When a business purchases an asset that will be used in the company for a long time, such as office equipment, part of the cost of the asset is treated as an operating expense in each of the years in which it is used rather than treating its entire cost as an expense in the year in which it is purchased. The accounting profession considers this the proper treatment of these types of assets because the assets are used in the business for many years, helping the business generate revenues. In other words, it is another application of the matching principle. The annual expense is referred to as depreciation and will be explained in more detail in Chapters 4 and , along with other similar expenses, when the related balance sheet accounts are discussed.
The operating expense section of the income statement for Example Company is shown in Figure 3.12.
**Figure 3.12** Operating Expense Detail
#### **OPERATING INCOME (OR LOSS)**
Operating income represents the profit earned by the business as a result of its primary business activity: revenues and expenses arising from the sale of goods. An operating loss is the opposite, representing the loss suffered by the business from its primary business activity. Other items affecting the overall profit of the business for the time period are shown in a separate category: other revenues and expenses.
#### **OTHER REVENUES AND EXPENSES**
This category of revenues and expenses sometimes is referred to as other income and expenses. It represents the activities affecting the profit or loss of a particular period which are not the result of the primary business activity (the sale of goods in the Example Company case). If material, the results of these activities should be separated from the normal operating activities, with each material item presented separately. Examples of what might be shown in this section of the income statement are listed in Figure 3.13.
**Figure 3.13** Examples of Other Revenues and Other Expenses
The gains and losses on sales of business assets which are included in the above list of examples are based on the difference between the selling price of an asset and its cost. If an asset sells for more than it cost, a gain results. Conversely, if an asset sells for less than it cost, the company has suffered a loss.
The other income and expense section of the income statement for Example Company is shown in Figure 3.14.
**Figure 3.14** Other Income and Expense Detail
Net income (or net loss) is the last item on the income statement. It represents the net result of all revenues and other income earned during the period minus all costs and expenses incurred during the period, whether from the primary business activity or from other activities. Pulling together all the components of the income statement that we have discussed, Figure 3.15 shows a more detailed version of the income statement of Example Company.
**Figure 3.15** Example of Detailed Income Statement
By using the Example Company income statement, you can answer several questions, as shown in Figure 3.16.
**Figure 3.16** Examples of Questions That Can Be Answered from the Income Statement
However, with the income statement alone there are some questions you cannot answer, such as the ones shown in Figure 3.17.
**Figure 3.17** Examples of Questions That Cannot Be Answered from the Income Statement
### **Income Statement Format for A Business
That Manufactures Its Products**
When a business manufactures its goods for resale to its customers, additional information about manufacturing costs incurred may be presented in a separate schedule that is referred to as the manufacturing statement. The manufacturing statement presents a summary of the principal manufacturing costs, as shown in Figure 3.18.
**Figure 3.18** Example of a Manufacturing Statement
What do each of these elements represent?
• Goods in process at the beginning of the period represents the cost of raw materials, direct labor, and overhead incurred on goods in prior periods that have not been completed as of the beginning of the year. Additional materials, labor, and overhead costs are incurred during the year to finish manufacturing those goods as well as other goods started during the period. Goods in process at the end of the year represents the cost of materials, labor, and overhead on incomplete goods.
• Raw materials used represents the cost of materials that become part of the finished product during the year. For example, in an automobile manufacturing plant, raw materials would include steel, glass, tires, and so on.
• Direct labor is the cost of salaries and wages of employees directly involved in manufacturing goods. For example, the wages of employees who work on the assembly line in an automobile manufacturing plant installing windshields in the vehicles would be classified as direct labor. The salary of the company's chief executive officer cannot be traced directly to the manufacture of vehicles, and so that amount would not be classified as direct labor.
• Overhead represents all the indirect costs of production. Indirect costs cannot be traced directly to manufactured goods and include the salaries of supervisors, office supplies, depreciation of equipment or buildings, utilities, maintenance, marketing, advertising, salaries of company accountants and attorneys, and so forth. Each significant type of overhead cost may be disclosed separately.
• Cost of goods manufactured represents the cost of goods completed during the period that are ready for sale, similar to the cost of goods purchased by a business that purchases goods for resale.
The costs associated with manufactured goods would be shown as an expense of the accounting period in which the goods were sold because of the matching principle. The cost of unused raw materials on hand, the costs of partially completed goods, and the costs of unsold finished goods would be shown on the balance sheet as inventories. A manufacturer typically has three types of inventories: (1) raw materials, (2) goods in process or work in process, and (3) finished goods. The cost of these inventories will become part of cost of goods sold in future periods as the goods are completed and sold.
### **Income Statement for a Corporation**
We will discuss corporations again in Chapter 4 when we go over the balance sheet and in more detail in Chapter 5; however, it is useful to see how the income statement for a corporation differs from the income statement for a sole proprietorship (Example Company) discussed earlier in this chapter (see Figure 3.19).
**Figure 3.19** Example of a Corporation's Income Statement
The basic content of the corporation's income statement in Figure 3.19 is similar to what was presented for Example Company in Figure 3.5, with two basic differences: (1) income taxes and (2) earnings per share.
#### **INCOME TAXES**
A corporation is a separate legal entity, and as a separate entity it must pay income taxes on its profits. For a sole proprietorship and a partnership, each owner reports his or her share of the company's profits on his or her personal tax return, but the business itself does not pay taxes.
Note in Figure 3.19 that income before income taxes is $100,000: the same amount that was reported as net income in our earlier example for a sole proprietorship (Figure 3.5). Since this business entity is a corporation, income tax expense is shown as a cost of doing business on the corporation's income statement. Income tax expense represents income taxes incurred by the business entity on its accounting income regardless of whether the taxes have been paid in cash or will be paid in the future. The taxes incurred are calculated in accordance with the Internal Revenue Service's tax code, which is beyond the scope of this book. (Maybe there will be a book called _Taxes Demystified_ someday.)
#### **EARNINGS PER SHARE**
A corporation is required to disclose the earnings associated with a share of common stock (the unit of ownership of a corporation). In a corporation, the stockholders are the owners of the company. In a sole proprietorship or a partnership, common stock does not exist, and so any kind of earnings per share calculation would be meaningless.
Earnings per share (EPS) is the corporation's net income divided by the number of shares of common stock outstanding, and so it represents a measure of the earnings associated with a single share of stock (the basic unit of ownership). In the above example, it was assumed there were 10,000 shares of common stock outstanding (i.e., out in the hands of the stockholders; in other words, owned by stockholders), so when the corporation earned $77,500 its EPS was $7.75.
EPS is a favorite measure of profitability among investors and Wall Street analysts, and companies pay close attention to their EPS for a specific period and from period to period because reported EPS can affect a corporation's stock price. There are two different types of EPS that can be reported on the income statement, which we will introduce you to next.
##### **Basic or Simple Earnings Per Share**
All corporations will report a basic (or simple) EPS. The formula is shown in Figure 3.20.
**Figure 3.20** Basic or Simple Earnings per Share Formula
Dividends are a distribution of the profits of a corporation to its stockholders. Preferred stock is a type of ownership interest that is different from that represented by common stock, and this concept will be explained more fully in the discussion of the balance sheet in Chapter 5. The numerator in the formula in Figure 3.20 represents the year's net income that is available to the common stockholders. Some corporations may have no preferred stock outstanding, in which case the numerator is simply net income.
The denominator represents the number of shares of common stock outstanding during the year, weighted by the period in which the shares were outstanding. What do we mean by "outstanding"? We mean that the shares are owned by the common stockholders: The shares have been issued to the stockholders, and they hold the shares. (There is some other terminology associated with shares of stock that we will introduce in Chapter 5 when we cover the stockholders' equity in the balance sheet in more detail.)
If a corporation has 10,000 shares of common stock outstanding all year long, the computation of the weighted average number of shares is quite simple: 10,000. But what happens if a corporation issues additional shares of stock during the year or possibly buys back some shares (called treasury stock and introduced in Chapter 5)? In these cases, the number of shares outstanding must be weighted by the fraction of the year during which they are outstanding. This sounds more complicated than it really is. Here is an example:
• Assume that Your Corporation, Inc., has 100,000 shares of common stock outstanding as of January 1, 20x1 (and the corporation has a fiscal year end of December 31).
• On April 1, 20x1, another 30,000 shares were issued for cash.
_Question:_ What is the weighted average number of shares outstanding?
_Answer:_ 122,500, which is calculated as (100,000 × 3/12) + (130,000 × 9/12) How do we arrive at that number? The corporation began the year with 100,000 shares of common stock outstanding, and those shares were outstanding for three months. On April 1, another 30,000 shares were issued for cash, and so now a total of 130,000 shares was outstanding for the rest of the year (nine more months).
Note that in this example, the total number of shares outstanding at the end of the year (130,000) is not the denominator in the EPS calculation. That is the case because the additional assets (specifically, cash) received in April when the additional 30,000 shares were sold were not available to help the corporation become more profitable for the entire 12 months. Thus, EPS is computed more accurately by using the weighted average number of shares outstanding rather than the ending number of shares outstanding.
##### **Diluted Earnings Per Share**
Many but not all corporations also report diluted earnings per share. Diluted EPS will always be less than basic EPS. Diluted EPS represents the worst-case scenario for what could happen to EPS if additional shares of common stock were issued when a corporation also has issued certain other securities that can be exchanged for shares of common stock on a future date based upon the wishes of the holders of those securities. The actual computation of diluted EPS is more complicated than we describe here, and so, for our purposes, it is important that you understand that diluted EPS is a hypothetical worst-case scenario showing the lower amount of EPS that could result if additional common stock was issued. However, these convertible securities have not actually been exchanged for shares of common stock at this point.
The accounting profession requires that diluted EPS be presented for reasons of full disclosure, a concept that was explained in Chapter 2. If these other securities were actually converted into shares of common stock during the following year, the financial statement reader should not be surprised to see a decrease in EPS because additional shares of common stock are now outstanding.
Figure 3.21 shows the basic EPS and diluted EPS amounts reported by two well-known publicly traded corporations: Target and Kellogg's.
**Figure 3.21** Comparison of Basic Earnings per Share and Diluted Earnings per Share for Two Corporations
However, not all companies will report diluted EPS if they have what is called a simple capital structure; in other words, these companies do not have any potentially dilutive securities that if converted into shares of common stock could reduce (dilute) EPS. For example, Tootsie Roll, Inc. reports only basic EPS and explains in its financial statement footnotes that because the company has a simple capital structure with no potentially dilutive securities issued, it does not report a diluted EPS.
#### **WHAT ABOUT DIVIDENDS?**
You may have noticed throughout our discussion of the income statement that we did not show dividends declared and/or paid to stockholders as an expense. The reason for this is that dividends are not an expense of the business. Expenses are a cost of doing business and are incurred in providing services or selling goods or performing any other activity associated with the company's primary business activity. Dividends are a distribution of some of a corporation's profits to its owners—the stockholders—usually in the form of cash. As such, dividends do not belong on the income statement; instead, they appear on the statement of changes in stockholders' equity. For a sole proprietorship or a partnership, withdrawals represent a distribution of some of the business profits to the owner or owners, and, similar to dividends, owner withdrawals appear on the statement of changes in owner's equity and not on the income statement.
### **Variations in Income Statement Format**
Income statements may be presented in different formats. Regardless of the format presented, you'll notice that the basic content of the income statement has not changed: it contains the revenues, expenses, and any gains or losses experienced by the company for the period of time being reported.
### **Multiple-Step Versus Single-Step Income Statement**
The most common variations in income statement format are multiple-step versus single-step income statements. Our earlier examples were generally multiple-step income statements with several categories before reaching the bottom-line net income figure. The multiple-step format emphasizes the relationship of sales and cost of goods sold (through the computation of gross profit) and the operating versus nonoperating results.
The income statement presented next is in the single-step format, using the same amounts as before. The single-step format calculates the bottom-line net income figure in one single step by grouping all the revenues and other income together, and grouping the expenses and losses. The single-step format does not emphasize the distinction between the primary operating and nonoperating status of each item (see Figure 3.22).
**Figure 3.22** Example of a Single-Step Income Statement for a Corporation
The full disclosure and materiality concepts affect the amount of detail in the income statement and the number and depth of explanations or detailed schedules needed in the footnotes to the financial statements.
### **Quiz for Chapter 3**
1. Which of the following items belong on the income statement?
a. Assets and liabilities
b. Revenues and expenses
c. Assets and equity
d. Unearned revenues and prepaid expenses
2. Net income is computed as:
a. Gross margin minus cost of goods sold
b. Cost of goods sold minus operating expenses
c. Revenues minus expenses
d. Assets minus liabilities
3. Gross profit:
a. Is the same as net income
b. Represents the excess of net sales revenue over cost of goods sold
c. Is the excess of the cash received on sales over the cash paid for goods sold
d. Equals revenues minus operating expenses
4. When a company provides a service to a customer, the revenue is recorded when:
a. The cash is received
b. The company performs the service
c. The company bills the customer
d. It is certain the customer is not going to complain about the quality of the service
5. Which of the following is not an example of an expense?
a. Dividends
b. Utilities
c. Rent
d. Depreciation
6. Which of the following forms of business will report income tax expense on its income statement?
a. Sole proprietorship
b. Partnership
c. Corporation
d. All of the above
7. Cost of goods sold equals the cost of merchandise:
a. Purchased this year, whether with cash or by credit
b. Purchased with cash this year
c. Sold to the customers this year for which cash payment has been received
d. Sold to the customers this year
8. An accrued expense:
a. Is one that has been incurred but not yet paid
b. Is not recorded unless it has been paid in cash
c. Is paid in advance but not yet incurred
d. Occurs only with the cash basis of accounting
9. Which of the following is not one of the inventory classifications that a manufacturing business will have on its manufacturing statement?
a. Overhead
b. Raw materials
c. Work in process
d. Finished goods
10. Earnings per share:
a. Is calculated for all forms of business (sole proprietorships, partnerships, and corporations)
b. Is computed for both common stock and preferred stock
c. Is always presented in two forms: basic and diluted
d. Is a profitability measure that shows the earnings associated with a single share of common stock
## **CHAPTER 4
The Balance Sheet:**
Classifications and Concepts
Chapters 1 and introduced you to the basic form and content of financial statements and the underlying concepts of financial reporting; that was followed by a more detailed discussion of the income statement and related concepts in Chapter 3. We will take a closer look at the balance sheet and the statement of owners' equity in this chapter and again in Chapter 5.
We begin this chapter with a brief overview of the basic concepts, form, and content of the balance sheet. The next section focuses on the concepts and guidelines used to classify the various assets and liabilities into meaningful and useful categories and provides brief descriptions of the items you are likely to find in each category. The final section of the chapter discusses how owners' equity is presented in the balance sheet and in the statement of owners' equity.
By the end of this chapter, you will have a better understanding of important balance sheet concepts and classifications, but there is still much more to learn in your quest to understand financial statements. We will revisit each of the balance sheet classifications in Chapter 5 and will discuss the unique characteristics and financial reporting impact of many items in much more detail. The examples and concepts in both chapters will broaden your understanding of not only the balance sheet but also the income statement and financial statement disclosures.
As was stated above, this chapter begins with a brief overview of the basic concepts, form, and content of the balance sheet.
### **Basic Format**
The balance sheet presents the assets, liabilities, and equity of the business owner or owners, representing the financial position of a business at a moment in time. The balance sheet also is referred to as a statement of financial position.
**Balance Sheet = Statement of Financial Position**
Recall that the income statement presents the results of business transactions that took place during a specific period; this is somewhat like reporting the events as you watch a video unfold. Reading the balance sheet, by contrast, is somewhat like seeing a still frame shot as you hit the pause button; it is a snapshot of the assets, liabilities, and equity at that moment in time. It is important to note that the balance sheet may provide a much different view if you hit pause at a different moment in time.
**Income Statement = Events Taking Place During a Period of Time
Balance Sheet = Snapshot Taken at a Moment in Time**
A simple balance sheet for Example Company appears in Figure 4.1, followed by a brief overview of some basic terms and concepts.
**Figure 4.1** Example Company Balance Sheet December 31, 20x1
• Assets represent the economic resources of a business.
• Parties with an interest in or a claim to the business assets are referred to as either creditors or owners.
• Liabilities are obligations of the business or creditor claims to the assets.
• Equity represents the residual interest or the owners' claims to the assets of the business.
As was discussed in Chapter 1, the balance sheet is based on a simple equation, presenting the assets, liabilities, and equity of a business at a specific moment in time:
Sometimes the balance sheet is presented with assets on the left and liabilities and equity on the right, further illustrating the equality of the balance sheet equation, as is shown in Figure 4.2.
**Figure 4.2** Example Company Balance Sheet December 31, 20x1
When the balance sheet equation is rearranged, it accentuates the residual nature of the owners' interest in the business and why owners' equity often is referred to as net assets or net worth.
The term _net assets_ is used more commonly in business conversations, though the term _net worth_ is used more commonly in referring to an individual's wealth. Bill Gates is a noteworthy front-runner.
### **Assets**
Assets represent economic resources of the business, which may be tangible (such as cash, supplies, and equipment) or intangible (such as patents and goodwill). Assets also may represent amounts due from other parties, such as accounts and notes receivable, or prepaid expenses. Assets initially are reported on the balance sheet at their cost, which is an objective, verifiable measure of their value at the time they are acquired.
### **Liabilities**
Liabilities represent obligations of the business such as accounts and notes payable or accrued expenses such as salaries, taxes, and interest payable. Liabilities also may include obligations to provide services or goods in the future for which a customer may have paid in advance. Liabilities generally are recorded at cost; this is determined by the cost of the goods or services received for which payment is due. Liabilities may be viewed as creditor claims to the assets of the business.
### **Equity**
As was discussed above, equity represents the owners' interest in or claims to the assets of the business. However, as was discussed earlier, the claims of creditors (suppliers, lenders, and others) come first, before the claims of the owners; thus, owners' equity is a residual interest in the assets owned by a business.
**Assets – Liabilities (Creditor Claims) = Equity**
A business can be structured using one of many different forms of ownership, each with its own unique advantages and disadvantages. It may be a sole proprietor-ship owned and operated by a single owner or be structured as a partnership or a corporation with multiple owners (this is discussed further in Chapter 5). In a sole proprietorship or partnership, the equity of each owner is referred to as _capital_ , and the term _stockholders' equity_ applies to a corporation.
Although you may tend to think of owners as individuals like yourself, other companies, pension and mutual funds, and a variety of other parties may hold ownership positions in a company. Regardless of the business structure or the nature of the parties with ownership, owners' equity in the balance sheet includes amounts the owners originally invested in the business and subsequent profits which were reinvested (retained) by the business rather than returned to the owners.
**Owners' Equity = Investments + Profits Retained (Reinvested)**
### **Classified Balance Sheet**
The information presented in a balance sheet is much more meaningful and useful if significant items are shown separately and related items are grouped in some sort of classification scheme, just as an income statement is more useful when the results of routine, ongoing operating activities (sales, cost of goods sold, and operating expenses) are presented first and clearly distinguishable from nonrecurring or somewhat incidental events. In accordance with the basic concepts of materiality and full disclosure that were discussed in Chapter 2, information is significant and thus material if it would influence the decision of a reasonable financial statement user. Just as each material revenue and expense is reported separately (immaterial items may be combined), each material asset and liability should be shown separately in the balance sheet, and the information is more useful when related items appear together or are reported under descriptive categories.
A condensed version of a classified balance sheet appears in Figure 4.3. As you will notice, the balance sheet has several categories listed under the assets, liability, and equity headings. We will discuss the reasons for this classification scheme below.
**Figure 4.3** Example Company Balance Sheet December 31, 20x1
The discussion which follows first focuses on the current asset and current and long-term liability classifications. Other asset and owners' equity classifications are discussed in the final portion of the chapter.
### **Current Assets**
Current assets include cash, along with other assets that generally will be converted to cash or sold or used within a year and are listed on the balance sheet in order of liquidity, a relative term that indicates how soon or how easily these assets will produce cash the business can use to purchase more assets and/or pay off its short-term obligations when they come due.
The current asset section of the balance sheet typically includes the items in Figure 4.4, listed in order of liquidity. Only brief explanations of each item are provided below; we will focus on the unique characteristics of current assets and other categories of assets in much more detail later in this chapter.
**Figure 4.4** Examples of Current Assets
• **Cash** : currency and coins on hand, amounts on deposit, undeposited cash receipts, and cash register and petty cash funds
• **Cash equivalents** : a special category of investments that can be sold quickly and easily; often combined with cash on the balance sheet
• **Short-term investments** : temporary investments of excess cash to earn interest or dividends
• **Accounts receivable** : amounts owed by customers who purchased goods or services on credit, perhaps due in 30 days; may be referred to as trade receivables
• **Notes receivable** : formal promissory notes with specific maturity dates, interest rates, and repayment terms
• **Accrued receivables** (or **accrued revenues** ): amounts a company has earned but not yet collected, such as interest on loans or investments, or rent
• **Related party receivables** : amounts owed to the company by affiliated companies, such as subsidiaries; reported separately from customer receivables
• **Inventories** : merchandise held for resale or, for a manufacturing business, raw materials on hand, work in process, and unsold finished goods
• **Prepaid expenses** : supplies and postage on hand or rent or insurance premiums paid in advance
### **Current Assets and Liquidity**
As was mentioned earlier, current assets are presented in order of liquidity. Cash is listed first as it is immediately available for any purpose. Short-term investments can be sold immediately if the business runs short on cash. If a company extends credit to its customers, it can expect to collect most of its accounts receivable within the credit period (often 30 days) or perhaps earlier if customers take advantage of the cash discount terms discussed in Chapter 3 and to collect the principal and interest when due on short-term notes. In addition, some businesses sell or borrow against their accounts and notes receivable to generate cash more quickly. Cash, short-term investments, and receivables are referred to as _quick current assets_ as they are the most liquid and generate cash in a very short time frame.
However, inventory is another matter. For a business that sells merchandise or manufactures and sells goods, it may take a considerable amount of time before the goods actually are sold (if at all) and the customer accounts are collected. Inventory is not as liquid as most other current assets and thus appears near the bottom of the list of current assets.
#### **CURRENT ASSETS: MORE DETAILS**
The guideline used to distinguish current assets is actually more specific than was mentioned earlier. Current assets consist of cash or other assets that will be converted to cash, sold, or consumed within a year or within the operating cycle of the business, whichever is longer. Most customers will pay for the goods or services they received at some point, the inventory will be sold, and the company eventually will receive cash from its customers; other receivables, such as interest earned on loans and investments, will be received in cash; and prepaid expenses such as supplies, rent, and insurance will be used or consumed.
#### **CURRENT ASSETS AND THE OPERATING CYCLE**
The term _operating cycle_ refers to the length of time it takes for an investment in inventory to be converted to cash; this means the length of time it takes for inventory to be sold and for customer accounts to be collected.
**Operating Cycle: Inventory → Sale (Accounts Receivable) → Cash**
For some businesses, such as a produce store or a meat shop, the operating cycle may be only a few days, whereas the operating cycle for a retail clothing store may be a few months if it sells seasonal apparel. At the other end of the spectrum, the operating cycle for a construction company specializing in large-scale projects that take years to complete or a company producing aged liquor products may span several years. Therefore, within the current asset classification, there can be considerable diversity depending on the nature of the business involved. Along with key financial ratios you will learn about in Chapter 8, the length of time it takes to sell the product and collect the amounts owed by customers is an important consideration in evaluating a company's liquidity.
#### **CURRENT VERSUS LONG-TERM LIABILITIES**
From the discussion above, it makes sense that current liabilities are generally obligations that must be fulfilled or debts that must be repaid within a year and that long-term liabilities are not due for at least a year or perhaps will not be settled in full for many years. The distinction between current and long-term assets and liabilities is significant. Such classifications provide valuable information about liquidity, which refers not only to how quickly current assets will generate cash but also to a company's ability to meet its short-term obligations as they become due.
The current liability section of a balance sheet typically includes the items shown in Figure 4.5. Only brief explanations of each item are provided in this chapter; we will focus on the unique characteristics of both current and long-term liabilities in much more detail in Chapter 5.
**Figure 4.5** Examples of Current Liabilities
• **Accounts payable** : amounts the company owes to suppliers and others for goods and services purchased on credit
• **Notes payable** : formal promissory notes with specific maturity dates, interest rates, and repayment terms
• **Estimated warranty obligations** : estimated costs of repairs and replacements for products sold under warranty
• **Unearned revenues** : amounts a company received from customers in advance, including sales of gift certificates, subscriptions, future rent, and customer deposits for special orders
• **Accrued liabilities** (or **accrued expenses** ): expenses for which the company has not yet paid, such as salaries and wages, interest on notes payable, and utilities
• **Payroll tax liabilities** : payroll taxes withheld from employees and employer payroll taxes which have not yet been paid to the taxing authorities
• **Income taxes payable** : income taxes the company owes on the basis of its estimated or completed income tax return for the year
• **Dividends payable** : cash dividends declared by a corporation's board of directors to distribute a portion of corporate earnings to the stockholders
• **Current portion of long-term debt** : the amount due and payable in the coming year on a long-term loan, such as the next year's monthly payments on a 30-year mortgage note payable
#### **CURRENT LIABILITIES AND LIQUIDITY**
If current assets consist of cash and other assets that will generate cash soon and current liabilities are obligations that must be settled or debts that must be repaid soon, the relationship between current assets and current liabilities seems quite important. In fact, much attention is focused on this in evaluating a company's liquidity: its ability to meet its currently maturing obligations.
For example, have you ever worried about making your monthly mortgage payment on time because your next payday falls a couple days after the due date? Or maybe you just changed jobs, and although you will be paid a higher salary, your first paycheck will be delayed. Your house is worth $300,000 and your loan balance is $160,000, and so you could pay off the loan in full easily if you sold your house. Your net worth is $140,000 and you are solvent, meaning that you have more assets than liabilities. But you plan to live in your house, not sell it; you will have your paycheck soon, but you need a few more days to make this month's mortgage payment.
But what if you have been laid off and can't find a new job? At first, it might seem like a temporary cash crunch, but when you still haven't landed a new job after several months, it is a much bigger problem. You aren't very liquid, and if this goes on for too long, you no longer will be solvent.
Similarly, a business may face a temporary cash crunch, but over the long run, if it continues to have problems coming up with enough cash from its normal operations to pay its bills on time (or in your case, a regular paycheck), it (you) cannot survive indefinitely. Suppliers may discontinue extending credit and demand to be paid in cash at the time of purchase or service. Banks may become unwilling to make even a short-term loan when the borrower's ability to repay it on time, if at all, is in question. Even if the bank approves a loan, it may be at a high interest rate to compensate for this additional risk.
#### **WORKING CAPITAL AND THE CURRENT RATIO**
_Working capital_ is a term used commonly to describe the relationship between current assets and current liabilities; it provides an indication of a company's short-term liquidity: its ability to pay its current obligations when due using cash it generates from its routine operations. Working capital is the excess of current assets (which generate cash) over current liabilities (which use cash). For example, a company with $300,000 of current assets and $200,000 of current liabilities has $100,000 of working capital.
It would seem that this company is in a better position to be able to repay its short-term obligations on time and has a fairly good cushion in case something unexpected arises than is a company that also has $300,000 of current assets but has $295,000 of current liabilities and only $5,000 of working capital.
The relationship between current assets and current liabilities also can be expressed as a ratio that indicates the amount of current assets for every dollar of current liabilities, also indicating whether there is an acceptable cushion. In this case, one would say the company has 1.5 times as many current assets as current liabilities, or $1.50 of current assets for every $1.00 of current liabilities.
We will discuss several key benchmarks commonly used to analyze a business, including working capital and other measures used to assess a company's liquidity, in Chapter 8.
#### **CURRENT LIABILITIES: MORE DETAILS**
As with current assets, the guideline used to distinguish current liabilities is more specific than was mentioned earlier. Current liabilities are obligations that are due within a year or within the operating cycle of the business, whichever is longer, and are settled by using current assets or through the creation of other current liabilities.
We discussed the concept of the operating cycle earlier and defined it as the length of time required to convert an investment in inventory back into cash. You already understand that a produce store or a meat shop may have an operating cycle of a few days, whereas a clothing retailer may have four operating cycles within a year if it stocks seasonal apparel and a construction contractor or liquor producer may have an operating cycle that spans several years.
But along with the operating cycle, there are other factors that must be considered in deciding under which category some liabilities should be reported. Sometimes certain liabilities are reported in both the current and long-term liability classifications of the balance sheet. For example, if a business has a 30-year mortgage note payable which it used to finance the purchase of land and its office building, one most likely would classify the mortgage note payable as a long-term liability. However, if the business is required to make monthly installment payments on the loan, the monthly principal payments for the coming year should be classified as a current liability, whereas the remaining principal balance of the loan should be classified as long-term.
Thus, you might find the _current portion of long-term debt_ included among the current liabilities, which represents the portion of the loan principal balance that must be repaid within the coming year. The remaining loan balance will be reported as a long-term liability as this amount is not payable for at least one year in the future. The current versus long-term classification depends on when obligations are due and what resources will be used to settle those obligations.
To complicate matters, the second part of the definition of a current liability also requires a bit more explanation. Current liabilities are obligations which will be settled by using current assets or through the creation of other current liabilities. Let's assume your business sells gift certificates or gift cards that the recipient can present to receive products you sell in your store. Is this a current liability? Yes: The obligation will be settled by using a current asset: the merchandise inventory chosen when the gift certificate is redeemed.
Perhaps your business has a 90-day note payable to the bank, but rather than requiring you to pay it off and negotiate another new loan, the bank permits you to replace it with another 90-day loan. Is this a current liability? Yes: The obligation will be settled through the creation of another current liability: another 90-day note payable.
Another example pertains to dividends, a way corporations return part of the business's profits to their owners (stockholders). A corporation can declare a cash dividend to be paid to the stockholders at a specified future date; this sounds like, and is, a current liability. But if it is a _stock dividend_ , meaning that the corporation plans to give the owners more shares of stock in the corporation, it is not a current liability because it will not be settled by using cash but rather with equity: more shares of stock.
Let's turn to another example that requires a bit more thought. Corporations often borrow substantial amounts of money for long periods by issuing bonds payable (discussed in more detail in Chapter 5). The corporation generally is required to pay the interest owed each year (usually every six months) but does not have to pay back the amount borrowed for many years, maybe not for 100 years, as in the case of the Walt Disney Company bonds issued a few years ago. For the first 99 years, the bonds payable would appear as a long-term liability on Disney's balance sheet. But where is this liability reported at the end of year 99?
If Disney must repay the bonds within the coming year, it seems that it should be classified as a current liability. In most cases, bonds payable maturing within the coming year would be a current liability, but what if Disney created a fund, setting money aside each year to pay off the bonds when due, and the fund could not be used for any other purpose? Should the bonds payable within the coming year be reported as a current liability in this case? Probably not; Disney has a fund set aside for this purpose and will not use its current assets to pay the obligation.
The system of classifying liabilities into current and long-term categories provides valuable information even though one has to think carefully about the sources of money that are available to settle some types of obligations.
#### **LONG-TERM LIABILITIES**
Although it might have been more logical to discuss all the asset classifications first and then discuss liability and equity categories, we hope you can appreciate the importance of the current versus noncurrent asset and liability distinctions as they are covered in this sequence. As was mentioned earlier, we will provide a much more detailed discussion of liabilities and equity in Chapter 5, but it is important to give you a preliminary idea of what you might see under the long-term liability classification of a balance sheet. That section of a balance sheet typically includes the items in Figure 4.6.
**Figure 4.6** Examples of Long-Term Liabilities
• **Notes payable** : formal promissory notes with specific maturity dates, interest rates, and repayment terms; generally not due for at least one year
• **Mortgage notes payable** : the principal balance of a note payable used to finance the purchase of plant assets, with the assets pledged as collateral for the loan
• **Capital lease obligations** : the obligation to make future lease payments under the terms of a long-term capital lease agreement which is, in substance, similar to financing the acquisition of an asset with long-term debt
• **Pension and postretirement obligations** : obligations to provide pension and other benefits that employees have earned and are entitled to receive in the future
• **Bonds payable** : a common form of long-term borrowing used by corporations to raise substantial amounts of money for long periods; interest payments generally are paid semiannually, but the principal amount borrowed may not be due for many years in the future
• **Deferred income tax liabilities** : income taxes that may be due in future years, arising from differences in accounting and taxable income
### **A Quick Review**
Let's take a moment to review the key concepts discussed so far in this chapter. The balance sheet presents assets, liabilities, and equity, providing a snapshot of a company's financial position at a moment in time, and sometimes is referred to as a statement of financial position. The way individual items are classified and the order in which they are listed provide useful information.
Current assets include cash, short-term investments, receivables, inventory, and prepaid expenses. Along with cash, the other current assets are expected to generate cash in the near term, generally within a year, and are listed in order of liquidity, a relative term indicating how soon or how easily these assets will produce cash. Liabilities are obligations that must be settled or debts that must be repaid and are classified as either current or long-term.
Current liabilities are generally due within a year and will be settled by using current assets. The relationship between current assets and current liabilities is useful in evaluating a company's liquidity. Working capital—the excess of current assets over current liabilities—and the current ratio and other measures provide information about a company's ability to meet its current obligations.
The next section will take a closer look at other asset categories, and then the chapter will conclude by discussing the owners' equity portion of the balance sheet.
### **Noncurrent Assets**
A balance sheet presents several other types of asset categories that appear below the current asset heading, typically including those being used in business operations or investments held to achieve a particular long-term business objective. They may be tangible physical assets the business owns or has the right to use, intangible assets which convey specific legal rights to the business and have value even though one cannot touch or feel them, funds set aside to repay long-term debt, or perhaps investments in other companies to maintain a beneficial business relationship.
These assets generally are not held for sale and thus are not a ready source of cash, in contrast to short-term investments, inventory, and other items presented in the current asset classification. The terms _noncurrent_ and _long-term, long-lived,_ or _fixed_ assets are all broad references commonly used to describe this group of assets.
The asset section of the balance sheet for Example Company introduced earlier in this chapter appears in Figure 4.7, followed by a discussion of each major category.
**Figure 4.7** Example Company Balance Sheet (partial) December 31, 20x1
#### **LONG-TERM INVESTMENTS**
Investments appearing in this category often are made to achieve a long-term business objective or as a result of a contractual requirement. The investments are expected to be held for more than a year, perhaps for many years, unlike investments classified as current assets, which are viewed as a ready source of cash to cover short-term needs.
Long-term investments in equity and debt securities (such as stocks and bonds) of other companies may be held to generate dividend and interest income and gains from market value appreciation. Investments also may be made to establish and maintain a beneficial long-term business relationship or may represent a substantial ownership position in another company or in a joint venture arrangement. Land or other assets held for investment or future expansion purposes and funds set aside and invested to provide for the eventual payment of long-term debt or pension obligations also are included in long-term investments. Figure 4.8 shows the types of investments that may be classified as long-term.
**Figure 4.8** Examples of Long-Term Investments
• **Available for sale securities** : investments in equity or debt securities (stocks and bonds) of other companies which are expected to generate income from cash dividends and interest and/or market price appreciation. Such investments also may be referred to as _marketable securities_ as they can be purchased and sold readily in the financial markets and can be classified as either current or long-term, based on the company's intent.
• **Held to maturity debt securities** : investments in debt securities, such as corporate or government bonds, which the investor intends to hold until they mature, perhaps many years from now.
• **Bond sinking fund** : a fund established to repay bonds payable or other long-term debt at maturity. The fund may consist of investments in stocks and bonds, but unlike other long-term investments, the earnings and proceeds from the sale of these investments can be used only to repay a specific obligation at maturity; thus, it is reported separately from other long-term investments.
• **Pension fund assets** : funds set aside and invested for the purpose of meeting future obligations for employee pension plans and similar obligations.
• **Investments in affiliated companies** : stock investments made for the purpose of establishing and maintaining a beneficial long-term business relationship, perhaps with an important supplier or distributor. These investments represent substantial ownership positions in other companies, typically between 20 and 50 percent.
• **Joint venture investments** : companies may join forces, with each party contributing unique expertise and/or resources, to take on a large-scale or risky endeavor which none could finance or conduct alone. The venture parties may establish and invest in a separate entity or invest resources to be used in joint venture activities which are governed by a legal agreement.
• **Land held for investment purposes** : land purchased for future expansion or for speculative purposes with the expectation that it will appreciate in value. Land currently being used in business operations would be reported under property, plant, and equipment.
#### **PROPERTY, PLANT, AND EQUIPMENT**
Property, plant, and equipment, sometimes referred to as _plant assets_ or _fixed assets_ , includes tangible physical assets currently being used in business operations. Although these assets can be sold or exchanged, they generally are not held for sale in the normal course of business. Along with land, buildings, equipment, furniture, and fixtures, a building under construction for a company's own use may be reported in this classification, and even assets the company is leasing may be included if the lease terms are in substance similar to ownership. The property, plant, and equipment section of a balance sheet typically includes the items in Figure 4.9.
**Figure 4.9** Examples of Property, Plant, and Equipment
As was discussed earlier, these assets initially are valued at their cost when acquired and a portion of the cost is reported in the income statement as _depreciation expense_ during each year of use, in accordance with the matching principle. Most plant assets depreciate; they wear out or become less efficient or obsolete and eventually must be replaced, with the exception of land, which is permanent in nature and is not depreciated. A depreciable asset such as equipment is reported in the balance sheet as shown in Figure 4.10.
**Figure 4.10** Balance Sheet Reporting for a Depreciable Asset
The original (historical) cost of the equipment was $100,000. Accumulated depreciation of $28,000 represents the total amount of depreciation expense to this point. Equipment (net) is the undepreciated cost, which often is referred to as _book value_. Equipment also may be presented in the manner shown in Figure 4.11.
**Figure 4.11** Balance Sheet Reporting for a Depreciable Asset, Net
In Chapter 5, we will discuss factors used to determine the cost of a plant asset and the concepts and methods used to calculate depreciation expense.
#### **NATURAL RESOURCES**
Natural resources on the balance sheet include ownership or legal rights to minerals, petroleum, timber, and other creations of nature that can be extracted or harvested from the land and then sold in an unprocessed state or processed and sold. A lumber or paper products company may own its forestland or purchase the rights to harvest trees from the owner of the land or the government. An oil producer may lease the land from the property owner and make royalty payments in exchange for the drilling and production rights.
A company engaging in removing natural resources is quite similar to a manufacturing business except that the raw materials are acquired from nature rather than purchased from suppliers. As was discussed in Chapter 3, the inventory reported for a manufacturing company consists of the raw materials, direct labor, and overhead costs incurred to manufacture its product. These costs are not reported as operating expenses during manufacturing; instead, they are expenses when the inventory is sold and will be reflected in the amount reported as cost of goods sold in the income statement.
However, the natural resource producer does not purchase its raw materials; it acquires them from nature. The accounting concept of _depletion_ is similar to the concept of depreciation that was discussed earlier. A part of the cost of the natural resource is allocated to each period of production, and the depletion cost in each period is treated in the same manner as the raw material cost for a manufacturer. A mining company may report its natural resources on the balance sheet as is shown in Figure 4.12, identical to the manner used to report equipment in the earlier example.
**Figure 4.12** Balance Sheet Reporting for a Natural Resource
The cost of the natural resource was $200,000,000. Accumulated depletion of $50,000,000 represents the total amount of depletion expense to this point. However, as will be discussed further in Chapter 5, there are several unique costs associated with acquiring and developing a natural resource, with some costs capitalized and others expensed. We also will discuss how the amount of depletion may be determined.
#### **INTANGIBLE ASSETS**
Intangible assets represent the cost of assets acquired by a business that lack physical substance, often consisting of legal or contractual rights such as copyrights, patents, franchises, licenses, and trademarks or trade names. If a company negotiates the purchase of an existing business, the acquired company's customer mailing list may be treated as an intangible asset even though it may be difficult to pinpoint its precise value.
Goodwill is also an intangible asset, something unidentifiable associated with the ability of the business to generate favorable earnings from the quality of management, the reputation of the business, the loyalty of customers, or other factors. As a result, one might be willing to acquire the business at a price higher than the value of the specific assets it owns, which is the accounting concept of goodwill. Brief explanations of some common intangible assets appear in Figure 4.13.
**Figure 4.13** Examples of Intangible Assets
Although intangible assets will be discussed further in Chapter 5, there are a couple of important things that should be mentioned now. First, the intangible assets presented on the balance sheet are only those which have been purchased or acquired. For example, a company may have developed its own valuable customer mailing list, but that would not appear on its balance sheet as an intangible asset because it was internally developed. However, if it was sold to another company, the purchaser would include the cost of the mailing list as an intangible asset. The same is true with goodwill; it may exist but is not recorded as an intangible asset unless it is purchased.
Another example is a company which spent $5 million on research and development (R&D) activities and ultimately acquired a new patent for which it paid $10,000 in legal and registration fees. Only the $10,000 legal and registration fees would appear as the cost of the patent on its balance sheet; the $5 million R&D costs were expensed. Maybe the patent has a value of $20 million because of the exclusive right and competitive advantage it provides, but it appears on the balance sheet at its direct cost of $10,000. However, if it is sold to another party for its value, the purchaser will show the patent on its balance sheet at $20 million, which is the purchaser's acquisition cost.
These examples illustrate a key concept pertaining to intangible assets. Internal development costs, including R&D costs, are expensed; only the direct costs of acquiring intangible assets are included on the balance sheet.
Just as with other long-lived assets, for most intangible assets, a portion of the cost is treated as an expense of each period of use. The term used for intangible assets is _amortization expense_ , but the concept is similar to that of depreciation of plant assets and depletion of natural resources discussed earlier.
### **Owners' Equity**
This chapter concludes with a discussion of the owners' equity portion of the balance sheet and the related statement of owners' equity. Recall that the balance sheet is based on a simple equation, presenting the assets, liabilities, and equity of a business at a specific moment in time, as shown below.
Or, rearranged:
In the rearranged form, this equation emphasizes that creditor claims (liabilities) to the assets come before those of the owners and the residual nature of the owners' interest in the business, often referred to as _net assets_ or _net worth_ for this reason.
Most businesses finance their formation and growth with some combination of funds provided by owners and borrowed funds. It is unlikely that banks would make loans or suppliers would extend credit to a company unless the owners also had invested their own resources in the business. It is expected that the company will repay creditors when due and repay the principal and interest on loans when they mature, before the owners withdraw cash or other assets from the business for their own use, again emphasizing the residual nature of their interest in the business.
A business can be structured as a sole proprietorship, owned and operated by a single owner, or have two or more owners and be structured as a partnership, a corporation, or variations of these two forms. Individuals, other companies, pension funds, mutual funds, and many other parties can hold an ownership interest in a company.
Regardless of the business structure or the nature of the parties with ownership, owners' equity in the balance sheet has two components: (1) the amounts owners invested in the business and (2) the profits earned by the business which were reinvested (retained) rather than being distributed to or withdrawn by the owners. We will provide two examples based on the information shown in Figure 4.14, which shows total owners' equity of $400,000 at the end of the first year, 20x1.
In sole proprietorships and partnerships, owners' equity is referred to as _capital_ , which is presented as a single amount for each owner and includes both investments made by the owners and profits retained by the business.
Figure 4.15 shows how owners' equity will appear in the balance sheet if the business is a partnership. Swithin's capital balance of $100,000 represents her initial investment of $20,000 and an $80,000 share of the company's profit during its
**Figure 4.14** Information Used to Illustrate Reporting Owners' Equity December 31, 20x1
first year of business. Patrick's capital balance of $300,000 represents his investment of $60,000 and his $240,000 share of the profit. We assume that profits are allocated in the proportion of capital invested, although partners can allocate profits in any manner to which they agree.
**Figure 4.15** Balance Sheet Reporting for a Partnership December 31, 20x1
In the case of a corporation, owners' equity is referred to as _stockholders' equity_. Owners receive shares of stock as evidence of their ownership of and investment in the corporation, typically common stock, although there are other types. Figure 4.16 shows how owners' equity is presented if the business is structured as a corporation.
**Figure 4.16** Balance Sheet Reporting for a Corporation December 31, 20x1
In the examples in Figures 4.15 and 4.16, note that total owners' equity is $400,000 in both cases; however, there are two primary differences. First, the amounts invested by the stockholders and the profits retained by the corporation are shown separately rather than as a single amount. Second, the total equity of an individual stockholder is not shown; this makes sense when you realize that ownership in a public corporation can change frequently as shares of stock are traded on a daily basis in the financial markets.
Building on this example, Figure 4.17 illustrates how owners' equity events in the second year of operations would be presented in the statement of changes in owners' equity. During 20x2, the second year, Swithin and Patrick each invested additional capital, earned a share of the profits, and withdrew a portion of the profits for personal use.
**Figure 4.17** Statement of Changes in Partnership Equity For the Year Ended December 31, 20x2
If the company is structured as a corporation, Figure 4.18 shows the same events, but as changes in common stock and retained earnings rather than as changes in the individual capital accounts of each owner.
**Figure 4.18** Statement of Changes in Stockholders' Equity For the Year Ended December 31, 20x2
Of the $80,000 net income earned during the year, $40,000 was distributed to the stockholders in the form of cash dividends and the remainder was retained by the corporation. By the end of 20x2, the second year, retained earnings of $360,000 represents the cumulative amount of profits earned to date that was retained or reinvested in the business.
### **Summary**
This chapter discussed the concepts and guidelines used to classify the various assets and liabilities into meaningful and useful categories and provided brief descriptions of each of the items you probably would find in each category. We also introduced you to some basic owners' equity concepts and illustrated how owner transactions are reported in the financial statements, using simple examples for both partnerships and corporations.
Building on this foundation, Chapter 5 will focus on specific and unique characteristics of many of the balance sheet items discussed here. We feel these additional concepts are important so that you understand not only the balance sheet but also how they pertain to certain revenues or expenses reported in the income statement. We hope you find Chapter 5 to be a useful reference when you run across a particular financial statement item and need a refresher.
### **Quiz for Chapter 4**
1. A company has $300,000 of liabilities and total assets of $700,000. What is the dollar amount of its owner equity?
a. $ 300,000
b. $ 400,000
c. $ 700,000
d. $1,000,000
2. Current assets include cash and other assets, such as:
a. Receivables that will be collected in cash
b. Merchandise inventory that will be sold
c. Supplies on hand that will be used
d. All of the above if each occurs within a year or within the operating cycle of the business, whichever is longer
e. None of the above
3. Current liabilities include:
a. Unearned revenues
b. Payroll tax liabilities
c. Accounts payable to suppliers
d. Interest payable
e. All of the above
4. Which of the following is true?
a. Current assets are presented in the balance sheet in order of liquidity.
b. Working capital is the dollar amount of excess of current assets over current liabilities.
c. The ratio of current assets to current liabilities can be used to evaluate a company's liquidity.
d. A company is liquid if it is able to meet its current obligations as they become due.
e. All of the above
5. Investments in marketable securities such as stocks and bonds can be classified as either current assets or long-term investments, depending on the intent and expected holding period.
a. True
b. False
6. Equipment purchased at an original cost of $7,000 has a book value of $3,200. A total of $3,800 depreciation expense has been taken on the equipment in the past.
a. True
b. False
7. Which of the following pertaining to intangible assets is true?
a. Patents, copyrights, and trademarks represent legal rights conveyed to the holder for a specific period of time.
b. Contractual arrangements such as franchise and licensing agreements are classified as intangible assets.
c. Goodwill is the amount paid to buy another company in excess of the fair value of the assets that can be specifically identified.
d. All of the above are true.
e. None of the above are true.
8. Capital is the term used to describe the equity of an individual owner if the company is a sole proprietorship or partnership.
a. True
b. False
9. The amount of profit a company distributes to its owner(s) is referred to as:
a. Dividends if the company is a corporation
b. Withdrawals if the company is a sole proprietorship or partnership
c. Both of the above are true.
d. Neither of the above is true.
10. At the beginning of the year, a corporation had $300,000 of common stock and $700,000 of retained earnings. During the year, the corporation issued $200,000 of additional common stock, earned net income of $500,000, and paid cash dividends of $100,000. Total stockholders' equity at year end is:
a. $1,200,000
b. $1,500,000
c. $1,700,000
d. $1,600,000
e. Some other amount
## **CHAPTER 5
The Balance Sheet:**
A Closer Examination
In Chapter 4, we discussed the form and content of the balance sheet, along with the guidelines that are used to classify the different assets, liabilities, and types of equity. These categories are useful in gaining a better understanding of the financial position of a company at a specific point in time. Much of the discussion focused on the current asset and current liability distinctions. Those are important factors in evaluating a company's liquidity: its ability to meet its current obligations when due, using cash generated from its ongoing operating activities. Chapter 4 also provided examples and brief discussions of the other major asset, liability, and owners' equity categories typically presented in the balance sheet.
Building on this foundation, we will now focus on specific and unique characteristics of many balance sheet items that were discussed only briefly in Chapter 4. These important concepts affect not only the information reported in the balance sheet but also the income statement, and so we will discuss and illustrate how these relate to and affect certain revenues and expenses. You can return to this chapter as a useful reference source when you run across a particular financial statement item and need a quick refresher. The classified balance sheet in Figure 5.1 shows the balance sheet format and the major categories of assets, liabilities, and stockholders' equity that were discussed in Chapter 4. The business presented in this example is legally structured as a corporation and its owners are referred to as stockholders.
**Figure 5.1** Balance Sheet Format and Categories for a Corporation Example Company Balance Sheet December 31, 20x1
### **Current Assets**
Current assets generally consist of cash or other assets which will be converted to cash or sold or used within a year or during the operating cycle of the business, whichever is longer. For a company that sells merchandise, the operating cycle is the length of time it takes for inventory to be sold and for customer accounts to be collected:
**Operating Cycle: Inventory → Sale (Accounts Receivable) → Cash**
Most businesses complete one or more operating cycles within a year and use the one-year benchmark when classifying assets as current, although the cycle is longer for some businesses, as was discussed in Chapter 4.
Current assets are presented in order of _liquidity,_ a relative term indicating how soon or how easily those assets will produce cash the business can use to purchase more assets and/or pay its obligations when due. Working capital—the excess of current assets over current liabilities—and the ratio of current assets to current liabilities are measures that are used to evaluate a company's liquidity.
Figure 5.2 shows the items typically found in the current asset classification, presented in order of liquidity. We described each item briefly in Chapter 4 and will discuss some of them in more detail later in this chapter.
**Figure 5.2** Balance Sheet Reporting: Current Assets
#### **VALUATION OF CURRENT ASSETS**
The initial balance sheet value of a current asset generally is based on the historical cost concept; the cost when acquired for items such as short-term investments, inventory, and prepaid expenses; or the amount of cash the company is entitled to collect on accounts, notes, and other receivables. However, the accounting concept of conservatism applies to all assets, including current assets. If a receivable is unlikely to be collected or if inventory is obsolete or can be sold only at a substantially reduced price, balance sheet values should be reduced. Current assets appear on the balance sheet at a conservative value—the lower of the initial amount or the current value—with one exception. Investments in marketable securities are based on current market values, whether below or above the initial cost, as these investments can be sold and converted to cash immediately at the current market price.
#### **CASH**
_Cash_ includes amounts on deposit with banks or other financial institutions, currency and coins on hand, cash and checks which have not been deposited, and cash register and petty cash funds (cash on hand that can be used to pay for small or unexpected expenditures without writing a business check). Postdated checks and IOUs from employees are receivables, not cash. Postage stamps are supplies or prepaid expenses, not cash. A bank overdraft is a current liability and represents amounts owed for checks written which exceed the amount on deposit.
Cash balances that appear in the current asset classification are presumed to be immediately available for any purpose. Separate account balances are combined into a single amount; however, an overdraft in one account should be reported as a current liability unless the overdraft can be legally offset against another account balance. Sometimes there are restrictions on the use or availability of amounts on deposit. Depending on the nature and significance of those restrictions, the restrictions may be described in a financial statement footnote, the balance on deposit may be classified elsewhere on the balance sheet and excluded from current assets, or both.
#### **CASH EQUIVALENTS**
_Cash equivalents_ are actually a special category of short-term investments that are almost like cash and typically are combined with actual cash amounts on the balance sheet. For an investment to be considered a cash equivalent, it must be highly liquid and readily marketable (it can be converted to cash quickly and easily via financial market mechanisms), and if the investment had to be sold, the amount of cash received would not be substantially different from the amount invested (meaning that the value of the investment is not likely to change substantially during this short period). In light of these parameters, cash equivalents are viewed as a relatively risk-free investment of excess cash; they are investments that can be sold quickly and easily with little risk of changes in value if an urgent cash need arises.
#### **SHORT-TERM INVESTMENTS**
_Short-term investments_ and _marketable securities_ are generic terms used to describe a variety of investments a company may hold. An investment generally is considered marketable if it can be purchased or sold in the public financial markets, but a company may hold other types of investments too. This general category may include investments in stocks and bonds of other companies, U.S. Treasury bills and bonds, mutual funds, money market funds, certificates of deposit, and commercial paper. Many of these types of investments can be classified as either short-term or long-term, some may be classified as cash equivalents and combined with cash on the balance sheet, and others may be underlying investments held in a fund to repay long-term debt or pension obligations. _Equity securities_ and _debt securities_ are other terms used to describe some investments.
##### **Equity Securities**
An _equity security_ generally represents some form of ownership in another company, such as owning the common stock of a corporation:
**Equity Security = Ownership = Common Stock**
You are not guaranteed to make money on a common stock investment. In theory, the market price of that investment should rise if the company does well, although there are many other factors which influence stock prices, not all of which are based on a company's performance. At its discretion, the board of directors may declare a cash dividend to be paid to the stockholders, distributing a portion of the corporation's profits to its owners, with the remaining earnings retained or reinvested in the company. A more complete discussion of corporations, stock, and dividends appears in the material on stockholders' equity later in this chapter.
##### **Debt Securities**
A _debt security_ represents a borrower-lender relationship:
**Debt Security = Borrowing = Bonds Payable**
If you purchase a debt security such as a corporate bond, you are directly or indirectly making a loan to the company for which it has promised to pay a stated amount of interest, usually semiannually, and to repay the amount loaned on a specific maturity date, often many years in the future.
In the meantime, the market value of the bond investment may change over time with changes in interest rates caused by general economic conditions and changes in investor perceptions of risk for a particular company. A more complete discussion of corporate bonds appears in the material on long-term liabilities later in this chapter.
Equity, debt, and other forms of securities are publicly traded on a daily basis in the financial markets, with regulated mechanisms bringing together parties with money to invest and companies that need to raise capital via stockholder investments or borrowings. When you (or a company) purchase a marketable debt or equity security, you can hang on to it for years, sell it later the same day, or sell it at any other time of your choosing, based on its current market value.
##### **Balance Sheet Classification: Investments in Equity and Debt Securities**
It is important to note that most of these investments could be classified as either current assets or long-term investments. In part, the distinction between the two classifications is based on the company's judgment regarding the nature of these investments. If the investments are included among the current assets, you generally would conclude that these are investments of idle cash, with the expectation that the investments will generate dividends or interest income and/or increase in value and be sold at a profit when the idle cash later is needed to finance ongoing business activities.
However, if these investments are classified as long-term, the company may be taking a longer-term view and hoping to maintain an ongoing business relationship with another company (perhaps a supplier) or to support a joint venture with another company when neither company can provide adequate financing or assume the full risk of a particular business venture, whether a domestic or a foreign endeavor.
Although specific types of investments may be classified as either short-term or long-term, the purpose and intent of such investments is a key factor in deciding under which classification the investments will be reported. The purpose and nature are revealed further by specific financial statement terms; investments in marketable securities may be presented in the balance sheet as _trading, available for sale_ , or _held to maturity_ securities.
##### **Trading Securities**
The term _trading securities_ typically is applied to companies which invest in and actively manage a portfolio of equity and debt security investments, with frequent purchase and sales transactions (trading), taking advantage of short-term changes in market price to produce earnings, along with the earnings from dividends and interest. These investments are classified as current assets and appear immediately below cash and cash equivalents because they can be sold readily to generate cash if needed. The dividends and interest earned, along with the gains or losses from the sale of these investments, are reported as other income or expense in the income statement.
Trading securities initially are valued at their cost but later are presented on the balance sheet on the basis of the current market price of the portfolio of investments, even if that price is higher than cost. This treatment is a rare exception to the concept of conservatism; most assets are reported at cost, or at less than cost if their value has declined, but are not adjusted above cost if their value has increased. However, this practice may be justified because the investments can be sold readily.
Let's assume that the investments a company owns at year end were purchased for $100,000 but the current market value is $130,000. If the investments were sold at this current market price, the company would report a _gain_ on the sale of its investments of $30,000. But it still owns the investments, and the balance sheet value reported is $130,000. Where is the $30,000 increase in market value reported? In the income statement, almost as though the investments had been sold.
Assume that the company also earned $10,000 in dividends and interest on its trading securities portfolio during the year, sold some investments at a $2,000 loss, and adjusted its year-end investment portfolio to market value. The financial statements would present the information shown in Figure 5.3.
**Figure 5.3** Trading Securities: Financial Statement Reporting
The increase in market value is considered an _unrealized gain_ (or unrealized holding gain) because the investments were not sold. Similarly, if the market price of the investments had declined, an unrealized holding loss would be reported in the income statement. Even though the gain or loss is unrealized, it was largely a matter of discretion whether to sell or hold the investments at this time; they could have been sold readily if the company desired.
##### **Available for Sale Securities**
Although some companies hold trading security portfolio investments, a more common purpose for investments in marketable securities is to generate dividend or interest income or to gain from market price appreciation rather than carrying excessive cash balances during low points in the business cycle. If necessary, such investments can be sold, but they also can be held for months or even years and are referred to logically as _available for sale securities_. The current versus the long-term classification of these investments provides an indication of the company's purpose and intent.
Just as with trading securities, available for sale securities are valued and presented on the balance sheet on the basis of current market prices, and the dividends and interest earned and gains and losses from sales are reported in the income statement. However, in the case of available for sale securities, an unrealized (holding) gain resulting from the increase in the market price of the investments is not included in the income statement; instead, it is reported as _accumulated other comprehensive income_ , which is a unique and separate component of stockholders' equity. When gains and losses are reported in the income statement and included in net income for the period, they ultimately are reflected in the year-end retained earnings included in total stockholders' equity. However, the unrealized gain in our example was excluded from net income and from year-end retained earnings; as a result, the unrealized gain must be reported as a separate component of year-end stockholders' equity. We will discuss these concepts again in the last section of this chapter on stockholders' equity.
If the company classified the investments in the example above as available for sale securities, the financial statements would appear as shown in Figure 5.4.
**Figure 5.4** Available for Sale Securities: Financial Statement Reporting
##### **Held to Maturity Securities**
A bond investor (bondholder) is directly or indirectly making a loan to the corporation in return for a stated amount of interest that typically is paid semiannually and a stated amount that will be paid back on a specified future date. As is discussed in more detail in the section on long-term liabilities in this chapter, the bondholder is guaranteed to receive a stated amount of interest, but if the stated interest is too low, an investor pays less for the bond; likewise, if the stated interest rate is higher than current conditions indicate, the bond must be purchased at a higher price.
Bond prices fluctuate over time, influenced by current market conditions, the state of the economy, and the perceived risk of a particular corporation. However, at the time of the initial investment, the price paid by the bondholder locks in a long-term rate of return if the bond is held to maturity. The investor can either hold the bond until it matures to earn that rate or sell the bond to another investor at its current market price, again based on the market conditions existing at the time.
When bond investments are classified as trading or available for sale securities, the stated interest payments the investor receives will be reported as income in the income statement for the year, the bond investment on the balance sheet will be based on its current market value, and the change in the value of the bond investment will be reported either in the income statement or in stockholders' equity, as was discussed above.
However, if the investor intends to hold the bond investment until it matures, it generally will be classified as a long-term investment, specifically as a _held to maturity_ security. Unlike trading and available for sale securities, a held to maturity investment is not adjusted to market value at year end because the company does not expect to sell the investment in the near term; the current market price of the bond is less relevant than it is for other investments the company can or intends to sell in the near term.
The purpose and form of some other types of investments will be discussed later in this chapter in the material on long-term investments.
#### **ACCOUNTS RECEIVABLE**
Accounts receivable represents the amounts due from customers for services rendered or goods sold in the past. Receivables from customers also are referred to as _trade receivables_ ; they are the result of routine business transactions in which goods and services are traded for the right to collect cash immediately or not long afterward. A company may increase its sales volume by allowing customers to buy on credit, but in that case there is a risk that some customers will not pay their balances when due. Accounts receivable initially are recorded on the basis of the amount of sales or services revenue to be received, but when the accounting concept of conservatism is applied, the balance sheet value should reflect the amount the company expects to realize or actually collect in cash.
##### **Estimated Bad Debts Expense and the Allowance for Doubtful Accounts**
Estimated _bad debts expense_ (or uncollectible accounts expense) should be included in the operating expenses in the income statement, matching sales earned in the current period with the associated expense of uncollectible accounts. Estimates generally are based on past experience, current credit policies, and economic conditions. Companies may estimate uncollectible accounts by using a percentage of sales revenue or examine the age of year-end receivables based on the due date, paying close attention to past due accounts, knowing that the likelihood of collection diminishes the longer an account has been past due.
For example, in 20x1, a company's first year of operations, it made credit sales of $500,000, collected $450,000 on account, and has $50,000 in year-end receivables. It estimates that $5,000 of its year-end receivables will be uncollectible. The balance sheet at year end reports the amounts legally due from customers less an allowance for estimated uncollectible accounts, as shown in Figure 5.5.
**Figure 5.5** Balance Sheet Reporting: Accounts Receivable
Accounts receivable (net) of $45,000 represents the amount of cash the company expects to collect; this often is referred to as _net realizable value_. Alternatively, the company may report only the net amount of receivables, with the amount of the allowance shown parenthetically, as in Figure 5.6.
**Figure 5.6** Balance Sheet Reporting: Accounts Receivable—Net
Although the company expects that it will collect only $45,000 of its current accounts receivable, let's assume it still continues to pursue collection in the next year, 20x2, but to no avail, and finally writes the $5,000 of accounts receivable off its books in 20x2. Should the $5,000 of bad debts expense appear in the 20x1 or the 20x2 income statement? According to the matching principle, it is an expense of 20x1, matched with the sales revenue earned in that period, not an expense of some future year when the decision is made to write the specific accounts off the books. The financial statement results for 20x1 and 20x2 would be as shown in Figure 5.7. The example highlights the proper matching of the bad debts expense associated with sales revenue earned in 20x1, and ignores any additional sales revenue the company earned in 20x2.
**Figure 5.7** Matching Bad Debts Expense with Sales Revenue
Contrary to the accounting concept of matching, some companies may use the _direct write-off method_ and report bad debts expense in the income statement in the year in which the specific accounts are written off. Although this method is used for income tax purposes, it is contrary to the matching principle and should be used in financial statements only if the amounts are immaterial (insignificant).
If the company's estimate was accurate, the net amount of receivables both before and after the write-off is $45,000, which is the amount the company expected to collect from its year-end receivables, as shown in Figure 5.8.
**Figure 5.8** Balance Sheet Effects: Writing Off a Specific Customer Receivable
Is it easy to estimate uncollectible accounts accurately? Even companies which have been in business for years and have used consistent credit policies may be affected by unexpected events. For example, Hurricane Katrina probably had an impact on the likelihood of collecting receivables from customers in New Orleans. Easy credit, declining real estate values, and rising homeowner defaults on mortgage loans in recent years have had far-reaching effects on lenders, investors and the economy. At the time of this writing, many financial institutions have failed, credit has dried up, and the United States is in the midst of an unprecedented financial and economic crisis.
##### **Allowances for Cash Discounts, Sales Returns, and Collection Expenses**
As was discussed in Chapter 3, many companies offer cash discounts to credit customers as an incentive for early payment. For example, if the credit terms are 2/10 net/30 on a $10,000 account balance, the customer is offered a 2 percent cash discount of $200 if the account balance is paid within 10 days (2/10), but the customer must pay the full amount due of $10,000 in 30 days (net/30). Although the company would receive less than it billed, if receivables are collected soon, it may be able to take advantage of cash discounts on amounts it owes to its own suppliers.
Customers often are permitted to return items purchased on credit or may be granted a price allowance on damaged goods they otherwise might return. A company may engage the services of another company to process its receivables or pursue past-due accounts. As with cash discounts, sales returns and allowances and collection expenses reduce the amount of cash the company expects to generate from its receivables. Depending on the nature and significance of these amounts, the company may review its year-end receivables and establish additional allowances, somewhat like the allowance for uncollectible accounts. In that case, it will report a more conservative value for its receivables, reflecting the amount of cash it expects to generate, and the estimated discounts, returns, and collection expenses will be matched with the revenue earned in the year of the sales, not in a later time period when they are taken.
#### **NOTES RECEIVABLE**
A company may offer relatively informal credit terms to its customers, but if credit is extended for a longer time or if a customer's account is past due, the company may require the customer to sign a formal promissory note specifying the maturity date, the repayment terms, and any assets pledged as collateral for the loan. Although a business may make a cash loan to an individual or another company, this would not be a common practice unless lending activities are a significant part of ongoing and routine business operations.
Formal promissory notes offer two advantages over the somewhat informal credit terms associated with accounts receivable. First, the note is a legal document which offers the company better protection in the event of default; second, the note typically specifies that it be repaid with interest.
Let's assume a company sold merchandise to a customer and accepted a $2,000 promissory note on December 1, 20x1. The stated annual interest rate on the note is 12 percent, and the $2,000 principal and interest owed on the note are due when the note matures in six months, on June 1, 20x2. The total amount of interest that will be received when the note matures is calculated as follows:
The 12 percent interest rate on the note is an annual interest rate, but the note term is only six months, and so only $120 in interest will be received. How should these events be reported in the December 31, 20x1, financial statements? The company has earned one month of interest even though it will not collect the note or the interest until 20x2.
According to the revenue recognition principle, a portion of the total interest that will be received is to be reported in the income statement each year: one month in 20x1 and the other five months in 20x2. On the December 31, 20x1, balance sheet, notes receivable of $2,000 is the principal amount of the note, and the $20 of interest earned but not yet collected is an accrued receivable, as shown in Figure 5.9.
**Figure 5.9** Income Statement and Balance Sheet Reporting: Notes Receivable and Interest
In some industries, companies borrow against or sell their notes and accounts receivable to generate cash more quickly. In secured borrowing arrangements, the company pledges its receivables as collateral for the loan and then describes the arrangements in the footnotes to the financial statements.
In other cases, it may sell its receivables either _with recourse_ or on a _without recourse_ basis. If a customer defaults and does not pay the balance when due, if the receivable was sold without recourse, the party purchasing the receivable bears the risk of loss and must estimate the bad debts expense, as was discussed above.
However, if a customer defaults on a receivable sold with recourse, the party selling the receivables must reimburse the purchaser for the loss. In this case, the seller must estimate the amount of uncollectible receivables it sold with recourse, and the _recourse obligation_ , which is the estimated amount the seller may owe the buyer, will be reported as a liability on the seller's balance sheet. Additional information about these arrangements must be provided in the footnote disclosures.
#### **ACCRUED RECEIVABLES (ACCRUED REVENUES)**
According to the matching principle, the income statement should include revenues earned during the time period even though the cash may not be received until a future date. A _ccrued receivables_ (accrued revenues) represent amounts yet to be collected for revenues a company has earned, such as interest earned on investments and notes receivable, or revenues unrelated to routine business operations such as amounts billed for rent or other incidental services provided. An illustration of accrued interest was presented in the earlier section on notes receivable, and other examples and concepts pertaining to accrued receivables were discussed in Chapters 2 and .
#### **RELATED PARTY RECEIVABLES**
Related party receivables may include amounts owed to the company by affiliated companies, such as advances or loans to subsidiaries. It also may include receivables from company executives or employees. As these receivables are very different from the trade receivables from company customers, they should be disclosed separately on the balance sheet, and additional information may be included in the financial statement's footnotes. Recent business scandals have heightened the public's awareness of such matters as unscrupulous executives have gained personally by taking advantage of their companies' resources.
#### **INVENTORIES**
_Merchandise inventory_ is the term applied to companies which acquire products for resale. The cost of any unsold goods on hand is included in the current asset classification of the balance sheet as it will generate cash when it is sold and customer accounts are collected. As was discussed in Chapter 3, according to the matching principle, the cost of inventory appears as an expense in the income statement in the period in which it is sold, referred to as _cost of goods sold_.
##### **Inventory Cost**
Inventory generally is valued at its historical cost on the balance sheet, which includes the invoice price and transportation charges on goods purchased less any returns, price allowances and cash discounts taken for early payment. Transportation costs paid for the delivery of purchased inventory are referred to as _transportation-in_ or _freight-in;_ these costs are included in the total cost of the inventory rather than as operating expenses. A detailed schedule showing how cost of goods sold is determined, which also might be presented in the income statement, appears in Figure 5.10.
**Figure 5.10** Detailed Schedule of Cost of Goods Sold
##### **Manufacturing Inventories**
A manufacturer presents three categories of inventory as current assets on its balance sheet: raw materials and supplies inventory, goods in process inventory, and finished goods inventory. _Raw materials and supplies inventory_ are items that will be used to manufacture products. _Goods in process inventory_ , as the name implies, are products which are not complete, and the inventory amount consists of the manufacturing costs incurred to date. _Finished goods inventory_ is similar to merchandise inventory, the term used to describe inventory purchased for resale and are completed goods on hand that are ready for sale.
The cost to manufacture a product consists of direct (raw) materials, direct labor, and manufacturing overhead costs. _Direct materials_ are the raw materials which become part of the finished product. _Direct labor_ is the cost of the salaries and wages of those directly involved in manufacturing or assembling the product. _Manufacturing (or factory) overhead_ consists of all the indirect costs associated with the manufacturing process, such as supervisory wages, building and equipment depreciation, insurance, utilities, and supplies. The three components of manufacturing costs are shown in Figure 5.11.
**Figure 5.11** Manufacturing Costs
Unlike selling and administrative expenses, which are reported as operating expenses in the time period incurred, these manufacturing costs are referred to as _product costs_ and are reflected in the cost of inventory on the balance sheet until the products are sold, as shown in Figure 5.12.
**Figure 5.12** Manufacturing Costs: Inventories and Cost of Goods Sold
##### **Gross Profit**
The difference between the selling price and the cost of the inventory (cost to purchase or manufacture) is referred to as _gross profit_ or _gross margin_. For example, if an item of inventory was purchased for $100 and sold for $125, the gross profit is $25:
For the company to be profitable, whether the inventory consists of goods purchased for resale or manufactured goods, individual selling prices and the total sales volume must generate a sufficient amount of gross profit to cover operating and other ongoing expenses. The length of time it takes to manufacture and sell the inventory and collect receivables—in other words, the operating cycle—is an important aspect of a company's liquidity.
##### **Inventory Systems and Controls**
Inventory quantities may be determined by using a _periodic inventory system_ where a physical count is conducted at least once a year, or by using a _perpetual inventory system_ which keeps a running balance of inventory purchased, sold, and on hand. If a perpetual inventory system is used, the recorded inventory quantity also should be verified periodically by a physical count. In preparing monthly or quarterly financial statements, it may be necessary to estimate inventory balances unless a perpetual inventory system is used. Although more complex and costly, a perpetual inventory system provides much better inventory control and makes it easier to identify errors and losses from theft. Technological advances such as bar code scanners and point of sale registers make the use of a perpetual system much more feasible than it was in the past.
##### **Inventory Cost-Flow Assumptions**
Let's consider the case of a campus bookstore which sells clothing and other items imprinted with the university name or logo. It makes frequent purchases and sales of its most popular logo items, but the cost varies a bit each time an item is reordered, perhaps because of quantity price breaks, transportation costs, and/or cash discounts for early payment; all these factors are considered in determining inventory cost. If the items it sells always could be purchased at exactly the same price, it would be simple to determine the cost of ending inventory and the cost of the goods sold. However, if the cost to purchase (or manufacture) a particular item varies, what is the right dollar amount for ending inventory and cost of goods sold? It depends.
As an example, let's take a closer look at a particular style of sweatshirt the bookstore always keeps in stock. As shown in Figure 5.13, the bookstore started the year with 300 of these sweatshirts, which were purchased last year at a cost of $10 each. It placed two reorders during the year, first purchasing 800 sweatshirts for $12 each and later purchasing another 500 at $15 each. A total of 1,600 sweatshirts were available for sale during the year, which were purchased at a total cost of $20,100.
At year end, a physical inventory count was taken; 600 sweatshirts were still on hand, and the other 1,000 were sold during the year. How much of the $20,100 total
**Figure 5.13** Cost of Inventory Available for Sale
cost should be assigned to the 600 sweatshirts in year-end inventory? How much to the 1,000 sweatshirts that were sold? It depends.
Figure 5.14 is a visual representation of the inventory purchased at different prices and the quantities sold and on hand at year end, along with the questions we will answer. As we will explain in the discussion below, the bookstore might use a specific identification system and keep track of each individual sweatshirt, or it might take an easier approach and use a cost-flow assumption to determine the cost of the inventory sold during the year and the cost of the inventory on hand at year end.
**Figure 5.14** Cost of Inventory Available for Sale
##### **Specific Identification**
It is logical to assume that a car dealer knows specifically which cars were sold, which are still on the lot, and the specific cost of each one. Dealers of rare and collectible art and jewelers who sell expensive one-of-a-kind items are also likely to keep very specific records for each item purchased and sold. Each would use a _specific identification system_ to keep track of individual inventory items and could readily determine the cost of the goods sold and the cost of inventory on hand.
However, our campus bookstore and many other companies make frequent purchases and sales of identical low-cost items, and manufacturers produce large quantities of those items. Keeping track of each individual item would be tedious and time-consuming, and the cost could exceed the potential benefits of such an elaborate system (the cost-benefit concept that was introduced earlier in Chapter 2). Instead, the bookstore might base its cost of goods sold and inventory values on an assumption about which items were sold, which remain in inventory, and their cost. In fact, most companies employ a _cost-flow assumption_ for items which are similar or low-cost and are not nearly as easily or readily identified as are vehicles, original art, or one-of-a-kind jewelry items. It might adopt a FIFO, LIFO, or weighted average cost-flow assumption.
It is very important to understand these concepts because the assumption a particular company employs may have a significant impact on the amounts reported in its financial statements. Companies that might be otherwise identical could present very different financial statement results as a result of choosing from one of several acceptable accounting methods.
##### **First-in, First-out (FIFO)**
The _first-in, first-out (FIFO)_ assumption is based on the notion that the first goods purchased (first-in) are the first that will be sold (first-out) and that the last goods purchased will remain in inventory. To visualize this concept, think about buying milk or other perishable items from a grocery store. The grocery store attempts to sell milk and other perishable items on a FIFO basis by placing the "old" milk at the front of the shelf and the "new" milk at the back. This plan works if the customer buys a carton of old milk from the front of the shelf and the new milk is still on hand. You also might think about a jelly bean dispenser, a jar that is filled from the top but whose contents are dispensed from a spout at the bottom. The old jelly beans are dispensed first, and the new ones are still in the jar.
From a physical-flow standpoint, most companies sell their inventory on a FIFO basis, trying to sell the old stock of a specific inventory item first, with the new stock of that item still on hand. If the bookstore uses a FIFO assumption for this style of sweatshirt that it always keeps in stock, the dollar amount for cost of goods sold and for ending inventory would be determined as shown in Figure 5.15.
**Figure 5.15** FIFO: Cost of Goods Sold and Ending Inventory Calculations
As noted in Figure 5.15, the first items purchased were the first items sold and the last items purchased remain in inventory at year end. The bookstore sold all 300 of the sweatshirts purchased for $10 each and 700 of those purchased for $12. Ending inventory is assumed to consist of the sweatshirts most recently purchased, which includes the remaining 100 sweatshirts purchased for $12 and the 500 that were purchased for $15. A bit earlier, we mentioned that a company might determine the cost of inventory on hand and sold using a _cost-flow_ assumption (such as the FIFO example above), which is not necessarily the same as the _physical flow_ (how the actual physical inventory changes as in the jelly bean jar example). Cost-flow assumptions and financial statement results will be discussed again as we conclude our inventory section.
##### **Last-in, First-out (LIFO)**
The _last-in, first-out (LIFO)_ concept is based on the assumption that the last goods purchased (last-in) will be the first sold (first-out) and that the first goods purchased will remain in inventory. To visualize this concept, you might think about the time you went to the hardware store and purchased nails from a large wooden keg or metal bin. You most likely grabbed a handful of nails from the top, as did each customer who came after you. When the keg ran low, the hardware store dumped the new shipment of nails into the keg, but the old nails remained at the bottom. Lumber, firewood, crushed rock, and landscaping bark also may be sold in this manner. In the case of the jelly bean jar, instead of using the dispenser at the bottom of the jar, you might decide to grab a big handful from the top of the jar and eat the new jelly beans, with the old ones still left in the jar. Or you might reach to the back of the milk shelf and buy the freshest milk, not the milk that has been on the shelf for a few days that the grocery store wants you to buy first.
If the bookstore uses a LIFO assumption for this particular style of sweatshirt that it always keeps in stock, the dollar amount for cost of goods sold and for ending inventory will be determined as shown in Figure 5.16.
As noted in Figure 5.16, the last items purchased were assumed to be the first items sold and the earliest items purchased remain in inventory at year end. The bookstore sold all 500 of the sweatshirts purchased for $15 each and 500 of those purchased for $12. Ending inventory consists of the 300 earliest purchased sweatshirts that were on hand at the start of the year and were purchased for $10 apiece and 300 of the ones purchased for $12.
By now, you might be wondering if the bookstore actually would sell this particular sweatshirt in this way, selling the new shipment first and keeping the ones it purchased months or maybe even years ago on hand. Probably not, but it still might use a LIFO cost-flow assumption for reasons that will be discussed later.
**Figure 5.16** LIFO: Cost of Goods Sold and Ending Inventory Calculation
##### **Weighted Average**
This cost-flow assumption is based on the use of a _weighted average_ unit cost for all inventory items available for sale during the year rather than on an examination of the individual prices associated with items purchased at a particular time. When you pull into a gas station to fill up your car, you pump gas from a large reservoir that usually is stored in an underground tank. You might visualize the weighted average assumption by thinking about how the old gas becomes mixed with the new gas when the distributor arrives and replenishes the storage tank. Or you could go back to the example of the jelly bean jar; if you shook up the jar and then grabbed a handful of jelly beans, the old and new jelly beans would be mixed; you would be eating jelly beans from this mixture, and the ones remaining in the jar also would be a mixture of old and new jelly beans.
If the bookstore uses a weighted average cost-flow assumption for this particular style of sweatshirt that it always keeps in stock, the dollar amounts for cost of goods sold and ending inventory would be determined as shown in Figure 5.17.
Note that the unit cost of $12.56 was determined by using the total cost and the total quantity of all items available for sale—$20,100 total cost ÷ 1,600 units = $12.56—rather than a simple average of the unit costs of $12.33 [($10 + $12 + $15) ÷ 3]. The weighted average calculation considers that half of the units were purchased for $12 and fewer were purchased at the $10 and $15 prices.
**Figure 5.17** WEIGHTED AVERAGE: Cost of Goods Sold and Ending Inventory Calculations
##### **Physical Flow versus Cost-Flow Assumptions**
Most companies attempt to sell each particular item on a FIFO basis by selling the old stock and keeping the new stock of that item on hand. However, a company may adopt any one of the three cost-flow assumptions illustrated and described above regardless of the physical flow of its inventory purchases and sales. In theory, the cost-flow assumption a company uses should achieve a better matching of the sales revenue earned and the associated cost. However, companies that are otherwise identical might present substantially different results for cost of goods sold and ending inventory in their respective financial statements, depending on the cost-flow assumption adopted.
##### **A Comparison of Inventory Cost-Flow Assumptions**
From our discussion and examples illustrating the different cost-flow assumptions above, you learned that each method produces different values for inventory and for cost of goods sold. A comparison of the three methods discussed above is presented in Figure 5.18, assuming the business is legally structured as a corporation and is subject to income taxation.
As you review the financial statement results, you should note that inventory prices were rising in our example; each new purchase was made at a higher cost than the one before it. If prices are in an upward trend, FIFO will produce the highest balance sheet value for inventory because the most recent, highest-priced goods are assumed to be on hand; it also will produce the highest profit because cost of goods sold is based on the assumption that the earliest, lower-cost goods are the first ones sold. LIFO, by contrast, reports the lowest inventory value based on the
**Figure 5.18** Inventory Cost-Flow Assumptions—Comparison of Financial Statement Effects
assumption that the earliest goods purchased at the lowest prices remain in inventory, and the lower profit is based on the assumption that the most recently purchased goods, those purchased at higher prices, were the first ones sold. As you would expect, the weighted average amounts fall somewhere in between.
Because there could be such different financial statement results between otherwise identical companies, it is important to find out which assumption a particular company is using, but how would one know? The full-disclosure concept introduced earlier in Chapter 2 requires a company to explain the policies and assumptions it is using in the footnotes to the financial statements.
##### **Why LIFO?**
In the discussion above, you learned that when prices are rising, the FIFO assumption produces the highest value for inventory on the balance sheet (based on the most recent purchases at higher prices) and cost of goods sold is lower (based on previous purchases at lower prices), and so a FIFO company is likely to report a higher profit than would a similar company using LIFO. Also recall that for most businesses, the FIFO assumption more closely resembles the actual physical flow of merchandise; the old stock is sold first, and the new stock is on hand. Therefore, you might wonder why a company would use a LIFO assumption in such circumstances.
One might conclude that companies prefer to use the FIFO assumption over LIFO in periods when prices are rising. In that case, why do over one-third of public companies use the LIFO assumption? LIFO is an acceptable method for tax return purposes; if prices are rising, LIFO results in lower taxable income and a lower tax bill. As a result, a company identical to others in all other respects might receive a hefty tax break by using a LIFO assumption.
Although there are many circumstances in which a company can use different methods to determine accounting income versus taxable income, LIFO is an exception. Tax law requires a company using LIFO for tax purposes to employ this method in its financial statements as well; many refer to this as the _LIFO conformity rule_. Companies using LIFO must feel that the potential tax savings outweigh reporting lower profit and inventory values in their financial statements. Figure 5.19 presents a comparison of the income tax effects for each method and highlights the LIFO method income tax savings, during a period of rising inventory prices and an assumed income tax rate of 40 percent.
**Figure 5.19** Income Tax Savings: LIFO Inventory Cost-Flow Assumptions
##### **Lower of Cost or Market**
There is one final inventory valuation concept that should be mentioned here: the _lower of cost or market (LCM)_ concept. Assume that the bookstore's year-end inventory consists of sweatshirts it purchased for $15 each. Maybe some of the sweatshirts in the window display were faded by the sun and can be sold for only $13 in that condition. Or maybe the price of cotton has declined and to compete with off-campus shops which purchased the same sweatshirts more recently at a cost of $14 and are selling them for $24, the bookstore may have to lower its selling price from $25 to $24 regardless of its cost.
When the value of inventory on hand has declined because the inventory is shopworn or obsolete or when future selling prices must be reduced to pass on cost savings or to stay competitive, according to the concept of conservatism, the inventory should be written down to the _lower of cost or market_. Inventory valued on a FIFO basis is more likely to be affected by such price declines than is a LIFO inventory value based on previous, lower prices. Footnote disclosures indicate whether the company is valuing inventory at the lower of cost or market. If a substantial writedown in value occurs, the footnotes also may indicate the amount and in what income statement classification the loss is included.
#### **PREPAID EXPENSES**
We conclude our discussion of current assets with examples of prepaid expenses, the final category under this heading. Prepaid expenses typically consist of the cost of supplies and postage on hand and rent or insurance premiums paid in advance. As the supplies and postage are used or the rental or insurance period lapses, these costs will be shown as operating expenses in the income statement in accordance with the matching principle discussed and illustrated in Chapters 2 and .
### **Long-Term Investments**
Investments appearing in this category often are made to achieve a long-term business objective or as a result of a contractual requirement; they are expected to be held at least a year but can be held for years or decades. Long-term investments may include equity and debt securities (such as stocks and bonds) of other companies and land or other assets held for investment rather than operating purposes and funds set aside and invested to provide for the eventual repayment of certain kinds of long-term debt or pension obligations. Substantial investments in affiliated or other companies may be made to build and maintain a beneficial long-term business relationship or may represent a company's share of a separate joint venture with other companies or parties. The long-term investment section of a classified balance sheet may include the items shown in Figure 5.20.
In the discussion of short-term investments near the beginning of this chapter, we discussed the concepts and characteristics of stock and bond investments in detail as well as the specific terminology used in the financial statements. You learned that many of these investments could be classified as either current or long-term, and rather than repeating that discussion, we will review the basic concepts and then focus on the unique characteristics of some other forms of long-term investments.
**Figure 5.20** Examples of Long-term Investments
#### **EQUITY AND DEBT SECURITIES**
The term _marketable securities_ is used to describe investments in equity and debt securities of other companies that can be sold readily in the financial markets. An investment in common stock consists of the purchase of ownership in another corporation and is referred to as an _equity security_. The investor may receive cash dividends on the investment if the corporation distributes a portion of its profits to the stockholders or benefit from an increase in the market price of the stock if the corporation reinvests its profits wisely and prospers but can suffer a loss if the stock price declines.
An investment in a corporate bond is a _debt security_ as bonds are actually a form of long-term borrowing that corporations and others use to raise substantial amounts of capital. The investor is entitled to receive a stated amount of interest each year and the principal amount owed when the bonds mature, which may be many years in the future. However, the investor also can choose to sell the bond to another party rather than hold it to maturity, and the value of the bond may change over time as a result of changing market conditions and perceptions of risk.
#### **CLASSIFICATION**
Investments in marketable securities are classified on the balance sheet as _trading, available for sale_ , or _held to maturity_ securities. Actively managed trading security portfolios are considered current assets. However, stock and bond investments classified as available for sale may appear under either the current or the long-term classification. Held to maturity securities, as the term implies, are investments in bonds or other debt securities that the investor intends to hold until they mature on a specified future date (equity securities—stock—do not have a maturity date).
#### **COST, MARKET VALUE, AND EARNINGS**
Long-term investments initially are recorded at their cost; dividends and interest received and gains and losses from sales are classified as other income. Recall that trading and available for sale securities are presented on the balance sheet at current market values. _Unrealized holding gains_ occur if the market value is greater than the cost of the investment; _unrealized holding losses_ occur when the market value falls below cost. Unrealized holding gains and losses are included in net income if the investments are classified as trading securities, but if the investments are classified as available for sale, the unrealized gains or losses are excluded from net income and instead reported as _accumulated other comprehensive income_ in stockholders' equity. Accumulated other comprehensive income is explained later in this chapter.
Held to maturity securities are not adjusted to market value because they are not expected to be sold but instead held until the maturity date; thus, the current market value is less relevant than it is for other categories of investments. We will discuss corporate bonds in more detail in the section on long-term liabilities later in this chapter.
#### **BOND SINKING FUND**
The terms and conditions for long-term borrowing arrangements can be very complex, and the borrowing agreement may specify that a _sinking fund_ be established to repay the debt at maturity. The funds set aside for this purpose are invested, perhaps in stocks and bonds, as was discussed earlier. The earnings from the investments in the fund are reported as other income; the earnings are reinvested, and eventually the investments are sold and the funds are used to retire the debt. However, because the investments and proceeds from this fund can be used only for a designated purpose, the sinking fund is reported separately from other long-term investments.
#### **PENSION FUND ASSETS**
A company may elect or be required to establish a fund to cover its future pension obligations. Unfortunately, companies with unfunded or underfunded pension plans are often in the news these days, as is the Social Security system. The concepts pertaining to a pension fund are similar to those for the bond sinking fund discussed above.
#### **INVESTMENTS IN AFFILIATED COMPANIES**
A company also may purchase stock investments to establish and maintain a long-term business relationship with another company. For example, if a company wishes to ensure the availability and quality of a key component of its manufactured product, it may make a substantial investment in one of its suppliers to gain influence over its operations or even wish to gain control of the company. The financial reporting for these types of investments differs from those discussed earlier, primarily based on the level of ownership acquired.
#### **20 TO 50 PERCENT OWNERSHIP: EQUITY METHOD**
The _equity method_ is used when the investor owns 20 to 50 percent of another company and has significant influence over its operating and/or financial policies and decisions. In this case, the investor's income statement includes its share of the net income of the company it owns, not the cash dividends it receives. For example, assume that Manufacturing Company purchases 40 percent of Supplier Corporation's common stock on January 1, 20x8. Supplier Corporation's statement of stockholders' equity for 20x8 appears in Figure 5.21.
**Figure 5.21** Supplier Corporation Statement of Stockholders' Equity For the Year Ended December 31, 20x8
On January 1, 20x8, Supplier Corporation's stockholders' equity (net assets) was $10,000,000, and Manufacturing Company purchased a 40 percent ownership stake, or 40 percent of the net assets, for $4,000,000. During 20x8, Supplier Corporation earned a $1,000,000 profit, distributed $200,000 of that profit to the stockholders as cash dividends, and retained the remaining $800,000 of its net income for the year. How much did Manufacturing Company earn from this investment? How much of this actually was received in cash?
Using the equity method for this investment, Manufacturing Company would include the information presented in Figure 5.22 in its financial statements.
According to the equity method, Manufacturing Company earned $400,000—its 40 percent share of Supplier Corporation's $1,000,000 net income—and received $80,000 in cash—its 40 percent share of the $200,000 income distributed as cash dividends. Supplier Corporation retained $800,000 of its 20x8 net income, of which Manufacturing Company's share is 40 percent, or $320,000, which is added to the amount reported for its investment in the year-end balance sheet. By year
**Figure 5.22** Equity Method Investment: Manufacturing Company Financial Statements (Partial)
For the Year Ended December 31, 20x8
end, Supplier Corporation's net assets are $10,800,000, and Manufacturing Company has a 40 percent interest, or $4,320,000, which is the amount Manufacturing Company reports as its investment in its year-end balance sheet when using the equity method.
When an investor owns 20 to 50 percent of another company and has significant influence over that company's operating and/or financial decisions, under the equity method, the investor recognizes income as it is earned rather than when the income is distributed later (if at all) as cash dividends. Its share of the reinvested earnings is reflected as an increase in the balance sheet value of the investment.
#### **JOINT VENTURES**
Joint ventures are formed to combine resources and unique expertise for a particular undertaking. A company may establish or invest in a joint venture to complete a large-scale project, conduct research, or finance and/or share the risk for activities it might be unable or unwilling to accomplish on its own. A separate entity may be established in which each party invests, participates, and/or helps finance, and many joint ventures are structured as partnerships (discussed later in this chapter). Other joint ventures may be based on legal agreements without the formation of a separate entity. The venture parties commonly use the equity method described above to report their investments in joint ventures.
#### **GREATER THAN 50 PERCENT OWNERSHIP AND CONTROL: CONSOLIDATION**
When a company owns more than 50 percent of another corporation's common stock, it essentially controls that corporation as its majority owner, although effective control can be achieved by other means. The term _parent_ is used to describe a company which owns a controlling interest in another company, and the term _subsidiary_ is used for the company it controls. In these circumstances, the investor generally must prepare _consolidated financial statements_ that present the combined results as though it were a single entity with branches or divisions. Transactions between the separate companies—for example, when a manufacturer purchases raw materials from its subsidiary supplier—are eliminated, as are receivables and payables between the two companies. If the parent company has 100 percent ownership, the subsidiary is _wholly owned_. However, a parent company can control another company if it owns the majority of its outstanding voting stock, and the subsidiary in this case would be referred to as _majority-owned_. Even with less than 100 percent ownership, the parent company's consolidated financial statements will present the combined results as a single economic entity, but the income and net assets of the subsidiary are allocated between the _controlling interest_ (parent) and the _noncontrolling interest_ , and these amounts will be identified separately in the consolidated income statement and in consolidated stockholders' equity. For example, the consolidated financial statements might appear as shown in Figure 5.23.
**Figure 5.23** Financial Statement Reporting: Consolidated Financial Statements (Partial)
We are now ready to take a closer look at the major categories of assets a company acquires and uses in its business operations. We will discuss the characteristics and concepts pertaining to plant assets, natural resources, and intangible assets in the next three sections of this chapter.
### **Property, Plant, and Equipment**
The property, plant, and equipment asset classification includes tangible physical assets currently being used in business operations. The terms _plant assets, fixed assets_ , and _long-lived assets_ also are used to describe assets in this category. Along with land, buildings, equipment, vehicles, and furniture and fixtures, a building currently under construction for a company's own use may be reported in this classification. Even assets the company is leasing may be included if the lease terms are in substance similar to ownership which is discussed later in this chapter.
Although these assets could be sold or exchanged, they are held for use and not generally held for sale in the normal course of business. New or used equipment, if held for resale by a dealer, would be presented as inventory under current assets. Land purchased for speculative purposes or as a possible site for future expansion would be classified as a long-term investment. A real estate developer would classify the land and its office building as a plant asset, while a land or real estate development held for sale would be classified elsewhere.
The property, plant, and equipment classification of the balance sheet probably would include the items shown in Figure 5.24.
**Figure 5.24** Balance Sheet Reporting: Property, Plant, and Equipment
#### **COST OF PROPERTY, PLANT, AND EQUIPMENT**
According to the historical cost concept, the balance sheet value for plant assets initially is based on the cost when acquired, which is presumed to be the current market value at that time. Acquisition cost includes reasonable costs necessary to acquire the asset and prepare it for use. For example, the cost of equipment might include not only the invoice price but also the costs of shipping, installation, and trial runs to make sure the equipment is working properly. The cost of a building might include the construction permit, architectural fees, and excavation of the building foundation, along with the more obvious costs associated with the materials, labor, and overhead costs required for constructing the new building. The cost of land purchased as a building site might include the cost to remove an unwanted structure, the cost to clear or drain the site, and any other necessary costs to prepare the land for construction of the new building.
#### **CAPITAL VERSUS REVENUE EXPENDITURES**
The amounts spent to acquire a plant asset and prepare it for use, as well as the cost of later additions or significant improvements which extend its useful life, are referred to as _capital expenditures_. One might say that a particular amount should be capitalized; that means that those expenditures are included in the cost of a particular plant asset and eventually depreciated.
By contrast, the costs of routine repairs and maintenance of plant assets are operating expenses, while abnormal repair costs, perhaps arising from vandalism or damages caused by fire or natural disasters, are more likely to be presented in the income statement as losses rather than as operating expenses. The term _revenue expenditures_ is used to describe these types of costs as they are matched with the current period's revenues in the income statement and are not included in the asset cost.
#### **CAPITALIZED INTEREST**
When a company finances the purchase of a plant asset with borrowed funds, the interest on the loan is reported as interest expense in the income statement. However, if specific conditions are met, for example, when a company constructs a building or another specific plant asset for its own use, the interest incurred during the construction period may be capitalized and included in the cost of the building rather than expensed immediately. As a result, once the asset under construction has been completed and placed in use, the interest that was capitalized during the construction period is reflected in the asset cost and will be included in the depreciation expense during the period of use.
#### **CAPITAL LEASES**
In specific cases, assets the company does not own but leases under the terms of a long-term contract are included in plant assets on the balance sheet. If the substance of the lease agreement more closely resembles that of financing the purchase of a long-lived asset with a long-term borrowing arrangement, the leased asset will be capitalized and included on the balance sheet, much like the purchase of a plant asset that was financed by issuing long-term debt. The "cost" of the leased asset will be based on the present value of the future lease payments, and the lease obligation will be reported as a long-term liability in a manner similar to that used for a long-term mortgage note payable. Lease obligations and some related concepts are discussed further in the section on long-term liabilities later in this chapter.
#### **MATERIALITY**
Two final accounting concepts, materiality and cost-benefits, often are applied when one is attempting to differentiate between capital and revenue expenditures. For example, a metal wastebasket purchased for $20 may last for decades (or centuries) and theoretically should be included in plant assets, but it may be expensed immediately as an insignificant amount; the benefit of keeping detailed records for waste-baskets and other immaterial items probably is not worth the additional cost.
#### **DEPRECIATION**
Just as supplies are treated as an expense when they are used and not when they are purchased, the cost of plant assets are considered to be expenses during the periods of use in accordance with the matching principle. Although plant assets are not used up the way supplies are, the amount invested in plant assets represents a significant cost of doing business. When assets are used in revenue-producing activities, the matching principle requires that a portion of the cost of plant assets be recorded as an expense during each of the periods in which they are used rather than as an expense in the period in which they were acquired. This concept of cost allocation is referred to as _depreciation_.
Your personal experience with depreciation probably differs from the accounting concept. For example, after you purchase a car and drive it for a couple of years, you probably would say it has depreciated, and you might then look online to find its book value in case you want to trade it in or sell it. However, for financial statement purposes, depreciation is considered to be cost allocation, treating a portion of the original cost as an expense during each year in which the asset is used. Accounting depreciation does not necessarily measure the actual decline in value, nor does the accounting book value of an asset necessarily indicate its current value if it were sold or traded in. In fact, some assets actually appreciate or increase in value.
While land is a permanent asset, most other plant assets wear out, lose their value, become obsolete, and/or eventually must be replaced. As was discussed in Chapter 4, a depreciable asset is presented on the balance sheet as shown in Figure 5.25.
In this example, the total acquisition cost of the equipment was $100,000. It is customary to show both the original cost and the _accumulated depreciation_ amount, which is the total amount of past depreciation expense: $28,000 in this example. The $72,000 difference is reported as equipment (net), which is referred to as the _book value_ of the equipment. Alternatively, this information may be presented in the balance sheet in the manner shown in Figure 5.26.
**Figure 5.25** Balance Sheet Reporting: Depreciable Asset
**Figure 5.26** Balance Sheet Reporting: Depreciable Asset Net
#### **BOOK VALUE**
The term _book value_ , as it is used in accounting, is simply the cost of a plant asset that has not been depreciated yet, not an indication of its current value, which could be substantially different for reasons that will be discussed below. Since the company is using the equipment and does not intend to sell it, its current market value is less relevant than would be the market value of an investment the company can or does plan to sell.
Even though the book value is not a measure of an asset's current value, knowing the original cost and accumulated depreciation can be useful in judging how soon an asset may need to be replaced and providing at least a notion of the magnitude of the replacement cost. For example, assume Company A and Company B each report equipment with a net book value of $72,000 on their respective balance sheets as shown in Figure 5.27.
**Figure 5.27** Book Value: A Comparison
Although the equipment book value for both companies is $72,000, you should notice that Company B paid $300,000 to purchase its equipment and has taken $228,000 in depreciation. You might then conclude that Company B may need to replace its equipment sooner and perhaps at a much higher cost than Company A. You might wonder if Company B has the cash or the ability to borrow the money it will soon need to replace the equipment.
#### **DEPRECIATION METHODS**
The accounting concept of depreciation allocates a portion of the cost of an asset to expense during each period of use in accordance with the matching principle. Different methods and assumptions can be used in determining annual depreciation expense, just as there are different methods for determining inventory and cost of goods sold. Although the total depreciation expense taken on a particular asset would be the same regardless of the depreciation method used, the amount of the expense taken in individual years may be significantly different.
Depreciation expense concepts and some common alternative methods in the context of plant assets will be discussed in detail below, and you will find that similar concepts also apply to the depletion of natural resources and the amortization of intangible assets, which are discussed in later sections of this chapter.
#### **DEPRECIATION BASE**
All depreciation methods are based on the cost of the asset, an estimate of its salvage or residual value, and its estimated life or output. The _salvage or residual value_ is an estimate of the sale, trade-in, or scrap value of the asset at the end of its useful life. The depreciation base, or the total amount of depreciation expense to be taken, is based on the original cost less its estimated salvage value. In all the examples below, it is assumed that the amount capitalized as the cost of equipment is $100,000, the estimated salvage value is $30,000, and the depreciation base—the total amount of depreciation expense—is $70,000 for all methods.
**Cost $100,000 – Salvage Value $30,000 = Depreciation Base $70,000**
#### **USEFUL LIFE OR PRODUCTIVE OUTPUT**
A company estimates the useful life or productive output of a particular asset on the basis of expected wear and tear, obsolescence, and the company's normal replacement policies. If it is company policy to replace its delivery vans every five years, even though a van may have a serviceable life of 20 years, it will be depreciated using a five-year life or the expected mileage for the five years it will be used by the business.
#### **STRAIGHT-LINE DEPRECIATION**
_Straight-line (SL)_ depreciation is a simple and widely used method. An equal amount of depreciation expense is taken during each year of use, as shown in Figure 5.28. If one plotted the amount the annual depreciation expense on a line graph, there would be a straight line, hence the name of this method. The $14,000 of annual depreciation generally will be reported as an operating expense in the income statement, but if the equipment is used to manufacture or assemble products, the depreciation will be included in the manufacturing overhead costs discussed earlier. On the balance sheet, the book value of the equipment will decrease steadily by $14,000 each year, and the book value will be $30,000 at the end of its useful life which is the original estimate of its salvage value, as shown in Figure 5.28.
**Figure 5.28** Straight-Line Depreciation
The book value of this equipment is simply the cost that has not been depreciated yet and is not meant to be an indication of its value if it was sold; also, accounting depreciation is a matter of allocating a portion of the equipment cost to expense during each year of use and is not meant to measure the decline in the value of the equipment that year.
Recall that the $30,000 salvage value was simply an estimate made when the equipment was purchased. If the equipment in this example was sold at the end of year 5, the company would not necessarily receive $30,000. If the equipment was sold for $25,000, a _loss on sale of equipment_ of $5,000 would be reported in the income statement. If it was sold for $32,000, a _gain on sale of equipment_ of $2,000 would be reported. The loss or gain would be reported below normal operating results as other income or expense.
#### **UNITS OF PRODUCTION DEPRECIATION**
The _units of production (UP)_ or similar output- or activity-based depreciation methods use some measure of productivity or actual use. For example, a vehicle may be depreciated on the basis of mileage, heavy construction equipment on the basis of engine hours, or a photocopier on the basis of the number of copies made. Let's assume that the equipment in the example above is used in mining operations and is depreciated on the basis of tons of mineral ore extracted and that an estimated 350,000 tons will be mined using this equipment. The calculation of the depreciation rate per ton of ore and the amounts of depreciation expense for each year are shown in Figure 5.29.
**Figure 5.29** Units of Production Depreciation
When the units of production method is used, depreciation expense varies each year; however, this method matches the cost of the equipment with its actual productivity or use, unlike the straight-line method, which simply uses an equal amount of depreciation expense each year. The units of production method makes good sense for certain types of assets, but in most cases it would be hard to justify using this method in depreciating a building.
#### **DECLINING BALANCE DEPRECIATION**
Compared with straight-line depreciation, the _declining balance (DB)_ method results in higher amounts of depreciation expense in the early years and lower amounts in the later years of use and is referred to as an _accelerated method_ of depreciation for that reason. If an asset is more productive and efficient when it is relatively new and requires more repairs and maintenance expenditures and is less efficient as it ages, this concept of matching the cost of an asset to the periods of benefit makes sense for certain types of assets. It is also important to have a basic understanding of the declining balance method because income tax depreciation for many assets is based on some form of this method.
With declining balance methods, a fixed percentage of book value is used as the depreciation expense each year; as the book value declines each year, so does the amount of depreciation expense; this explains the name of these methods. If a company used the straight-line method for equipment with a five-year life, it would depreciate one-fifth of the cost, or 20 percent, each year. A _double declining balance method_ involves using twice the straight-line rate: 40 percent in this case. The calculations of the rate and the annual depreciation expense for the double declining balance method are shown in Figure 5.30.
**Figure 5.30** Double Declining Balance Depreciation Method
Just as in the previous examples, a total of $70,000 depreciation expense is taken ($100,000 cost – $30,000 salvage). However, salvage value is omitted from the initial calculation of depreciation expense even though the company cannot depreciate the equipment below its estimated salvage value. As a result, only $6,000 of depreciation expense can be taken in year 3 and none in years 4 and 5 because the asset is fully depreciated.
#### **A COMPARISON OF DEPRECIATION METHODS**
In the discussion and examples illustrating the different depreciation methods above, you learned that total depreciation expense is the same with each financial statement method; however, the depreciation expense and resulting book value differ each year. A comparison of the three methods discussed is presented in Figure 5.31.
**Figure 5.31** Comparison of Depreciation Methods
A total of $70,000 of depreciation expense was taken for all three methods; however, the straight-line version was based on an even amount each year, the units of production version was based on the actual use of the asset, and the declining balance version accelerated the amount of depreciation to be taken, with higher expense in the early years. The book value of the equipment at the end of its five-year life was $30,000 in all three methods; however, the balance sheet value for each year differed considerably. Just as with inventory cost-flow assumptions, the choice of a depreciation method can have a significant impact on the financial statement results and should be taken into consideration in evaluating a company's performance. The methods used and other information must be disclosed in the footnotes to the financial statements.
#### **OTHER DEPRECIATION METHODS**
Companies may use several other methods to depreciate plant assets, depending on the nature of the assets involved, materiality, the pattern of use, and other factors. The _sum-of-the-years-digits_ method is another type of accelerated depreciation, although it is not illustrated here as it is not as widely used as it was in the past. Rather than depreciating individual assets, some companies base annual depreciation expense on a group of similar assets, for example, a leasing company's fleet of vehicles or a utility company's power poles.
#### **INCOME TAX DEPRECIATION METHODS**
Companies must use the methods specified in the tax law to prepare their income tax returns. Financial statement and income tax depreciation methods are similar in some respects but differ in others. Currently, tax law is based on the Modified Accelerated Cost Recovery System (MACRS), and many assets are depreciated using a declining balance concept, although optional straight-line methods are permitted in particular cases.
Tax law provides incentives which encourage companies to invest in productive assets. Salvage value is ignored in many cases, and that produces higher depreciation expense deductions; also, the useful lives that are specified in the tax law are often shorter than those which would be used for financial statement depreciation. When a company prepares its income tax return, it uses a specified tax life for the asset and the percentage of cost it can deduct each year can be found in a table that is based on a variation of the declining balance method discussed above. Figure 5.32 shows the annual depreciation deduction that can be taken in the tax return for the asset in the examples cited above.
Higher and earlier tax depreciation deductions reduce tax obligations that otherwise are payable in the early years after the purchase of a depreciable asset. As a result, there may be differences between income as reported in a corporation's income statement and the taxable income based on the corporation's tax return, leading to deferred income taxes, which is briefly discussed later under long-term liabilities. Figure 5.33 illustrates the difference between depreciation expense in the corporation's income statement and the depreciation expense deducted each year on its income tax return.
**Figure 5.32** Income Tax Depreciation—MACRS
**Figure 5.33** Comparison of Financial Statement and Income Tax Depreciation
#### **IMPAIRMENT LOSSES**
You probably noticed that the concept of conservatism came up often in our discussion of the various asset categories. If the market value of investments declined, they were reported on the balance sheet at the lower market value. When it seemed doubtful that customer receivable balances would be collected in full, an allowance for estimated uncollectible accounts was established, and receivables were reported at the lower net realizable value. If inventory on hand was damaged, became obsolete, or could be sold only at significantly reduced prices, if at all, the inventory was written down to the lower of cost or market.
In the face of uncertainty, by taking a conservative approach, a company is less likely to overstate its income or its assets. The concept of conservatism also applies to long-lived assets—not just plant assets but also natural resources and intangible assets, as will be discussed later. Long-lived assets are reported at a conservative value on the balance sheet. They initially are valued at their cost, but only certain costs can be capitalized. Plant assets other than land are depreciated, and the balance sheet values decline over time, yet any appreciation or increase in the actual economic value of a plant asset in use is ignored completely in the financial statements unless that asset is sold to another party for a higher value. Book value is simply the cost that has not been depreciated yet, not the current market value, and the book value of a particular asset can vary with the depreciation method used. For all these reasons, the balance sheet values are already pretty conservative, but what if even this conservative value is too high?
##### **Impairment Indicators**
Assume a company constructs an office building using high-quality materials and craftsmanship. All the offices have huge windows with views of the nearby hills and forests and the magnificent mountain ranges in the distance, and there are beautifully landscaped outdoor areas surrounding the building with running paths and a trout pond. Despite the high rent, the owner has leased all the offices and has a long waiting list of prospective tenants in case a vacancy arises.
Business was great for the first five years, until the county opened a 24-hour gravel pit operation on the property adjacent to the office building. Gravel pits, dirt and dust, noisy machinery, heavy truck traffic, and obstructed views caused many tenants to move out as soon as their lease terms expired. Those who remained agreed to do so at a fraction of the rent they had been paying. No one on the waiting list seemed interested anymore.
Regardless of its book value, the office building is certainly worth far less than it was before the gravel pit operations began. The future rent the owner expects to earn may be enough to cover the operating costs but not enough to recover all the costs originally invested in the building. Even though the owner plans to keep leasing the offices in the building, it is certainly worth less than before if the owner decides to sell out.
These events may require the company to report an _impairment loss_ in its income statement. An impairment loss occurs when the value of a long-lived asset has declined below its book value, if future earnings are not sufficient to recover the cost, or the asset could be sold only for less than its book value. Although you might not hear the term _impairment loss_ in a news broadcast, huge write-downs, sometimes as much as a 50 percent reduction in book value of a company's long-lived assets, are definitely newsworthy items.
### **Natural Resoruces**
Natural resources include ownership of or rights to minerals, petroleum, timber, and other creations of nature that can be extracted or harvested from the land and then sold in an unprocessed state or processed further and sold. For example, a logging company may harvest trees and sell them to a lumber mill, but it also may have its own lumber mill which produces heartwood for furniture and construction-grade lumber such as two-by-fours but then sell other timber by-products to companies which produce plywood, paper, or landscape products made from tree bark.
#### **COST OF NATURAL RESOURCES**
There are unique costs associated with acquiring and developing a natural resource; some are capitalized as part of the cost of the natural resource, others may be capitalized as buildings or equipment, and some will be expensed. Costs to purchase land containing a natural resource or to acquire the legal rights to natural resources generally are capitalized as the cost of those resources.
_Exploration costs_ to determine if a natural resource exists or if it exists in a sufficient quantity to be economically feasible to develop and extract generally are expensed in most industries, with one major exception. Oil and gas exploration companies are permitted to capitalize their exploration costs as part of the natural resource, using either the full-cost or the successful-efforts approach.
In the _full-cost approach_ , all exploration costs are capitalized, even costs associated with dry holes, whereas with the _successful-efforts approach_ , only the costs associated with producing wells are capitalized and the other exploration costs are expensed. The approach used has an impact not only on the balance sheet value of the natural resource but also on the amount of depletion that will be taken each year, and so it is important to determine which method a company is using by reading the footnotes to the financial statements.
The costs to develop a natural resource can be _intangible_ in nature, such as drilling, blasting, and tunneling or building roads, and these costs generally are capitalized as part of the cost of the natural resource. _Tangible costs_ such as trucks to haul mineral ore and an office trailer generally are capitalized as plant assets and depreciated if those items can be used in other ways or at other locations. By contrast, the costs of tangible equipment that cannot be moved or has no other use, such as a permanent drilling rig or underground pumps, probably will be capitalized as part of the natural resource.
Many companies are legally obligated to restore the property at the conclusion of extractive operations. Future _restoration costs_ can be substantial and are a component of the overall cost of operations even though some restoration costs may not be incurred until operations cease many years later. The company must estimate the future costs it will incur to satisfy any legal requirements and report that restoration (or retirement) obligation in the balance sheet as a long-term liability; the costs will be reflected as an additional component of the depletion cost recorded as the resource is extracted.
#### **DEPLETION OF NATURAL RESOURCES**
_Depletion_ is similar to the concept of depreciation used for plant assets, matching the cost of a long-lived asset to the periods of use or benefit. As the natural resource is removed or depleted, a portion of the capitalized natural resource cost will be allocated to each period, using a units of production approach. The depletion rate is calculated on the basis of the cost of the resource less any residual value and the estimated quantity of the resource that will be extracted. The depletion taken each year is based on the actual quantity produced.
As was discussed above, there are costs unique to natural resource assets, some of which become part of the depletion base, while others are capitalized as plant assets, and other costs are expensed. If the company owns the land containing the resource, depletion will be based on the cost of the resource less the estimated residual value of the land. It is often difficult to make a reliable estimate of the quantity or output of some natural resources, particularly mineral deposits and offshore petroleum reserves, and the company may seek the opinion of outside experts. Estimating the board-feet of lumber that could be produced from a tract of standing timber would be less difficult.
**Natural Resource Cost – Residual Value = Depletion Base
Depletion Base ÷ Estimated Units = Depletion Rate per Unit**
Natural resource producers are similar to manufacturers, except that the raw materials for their products are derived from nature rather than purchased from suppliers. The depletion cost each period is treated in the same manner as the direct material cost for a manufacturing company. Thus, the depletion cost for a particular year is not an operating expense; it is included in the cost of inventory on the balance sheet and later in the cost of goods sold in the income statement, as shown in Figure 5.34.
**Figure 5.34** Depletion Cost: Inventory and Cost of Goods Sold
Thus, depletion becomes part of the total cost of the natural resource product produced, along with the labor and overhead costs when determining the cost of the inventories on hand and the cost of the products sold. Depletion rules for income tax purposes may be significantly different from those described above.
#### **NATURAL RESOURCES ON THE BALANCE SHEET**
Natural resources are shown in the balance sheet at cost less the amount of depletion taken in previous periods (see Figure 5.35), similar to plant assets, and often appear in the property, plant, and equipment classification. Additional information about natural resources and restoration obligations is provided in the footnotes to the financial statements, and oil and gas producers have other specific disclosure requirements.
**Figure 5.35** Balance Sheet Reporting: Natural Resources
### **Intangible Assets**
_Intangible assets_ are quite different from the other asset categories discussed in this chapter, and so we begin with a scenario to get you to think about some of their unique characteristics. Your grandparents married young and moved to a new town to start a family and a business. You are now running a prosperous business that has been in your family for three generations, and out of the blue, you are approached by someone who wants to buy your family business. What is it worth?
As a starting point, you could have the land and building appraised, find trade-in values for your vehicles and equipment, and look up how much it would cost to buy the inventory in your store at today's prices and the market price of your short-term investments.
However, are there some things you can't quite pinpoint, in addition to those you can touch or identify, that make your business more valuable? We think so. It is an established and very profitable business with loyal customers and a sterling reputation spanning three generations. Your employees are knowledgeable and have worked in this business for years, as did their parents and grandparents. People in surrounding states recognize your business name; everywhere you go, you see people wearing T-shirts and hats printed with the unique logo you designed for your store, including people who are not regular customers.
We think your business has some _intangible assets_ , and we're sure you think so, too. If you decide to sell the company, we hope the buyer thinks like the rest of us and offers you a higher price as a result! These intangible characteristics have become increasingly significant; in some cases 80 to 90 percent of the price paid to acquire an existing company is for such intangible things, in stark contrast to the past, when the value of manufacturing facilities or other tangible assets acquired was a primary component of the price offered.
Even though financial accounting guidelines for intangible assets have evolved in recent years, you should have at least a basic understanding of these guidelines and their limitations and inconsistencies, which are even more important in light of the importance of intangible assets in the current business era.
#### **CATEGORIES OF INTANGIBLE ASSETS**
The term _intangible assets_ can refer to a diverse and broad range of items. Some can be identified readily and separately, particularly those which represent legal or contractual rights, whereas others arise from research or good business practices and are associated with the business as a whole. Intangible assets may be developed internally or purchased from others.
##### **Separately Identifiable Intangible Assets**
You are probably familiar with some common intangible assets, such as patents, copyrights, and franchises, but accounting guidelines include examples of many others, which are described briefly below:
• **Marketing-related** : trademarks and trade names, certification marks, trade dress (such as a unique product color or shape), a distinctive newspaper masthead, internet domain names, and noncompetition agreements
• **Customer-related** : customer lists, confirmed customer orders, and contracts
• **Artistic-related** : literary works, operas and plays, musical works, photographs and art, movies and television programs
• **Contract-based** : franchise, licensing, royalty, and lease agreements; broadcast rights; advertising, construction, management, and service contracts; and use rights for drilling, water, air, timber, and routes
• **Technology-based** : patented technology, computer software, databases, and trade secrets (such as a secret recipe)
##### **Goodwill**
In thinking about what your business might be worth in the example above, there seem to be some very intangible qualities, such as its long-standing reputation, loyal employees and customers, and sustained profitability. These characteristics make your business more valuable than the sum of its parts but cannot be separated from the business as a whole. You might say your business has _goodwill_ , another category of intangible assets. The prospective buyer may offer you a higher price to purchase the business because of the goodwill it has developed over time.
Is this goodwill reported as an intangible asset on your balance sheet? No, but it would appear on the balance sheet if the company was purchased by the prospective buyer because there would be an objective measure of its value: the excess amount paid over the tangible and intangible assets that could be identified specifically. For example, if you sold the business for $800,000 and the specific assets were worth $600,000, goodwill of $200,000 would appear on the buyer's balance sheet. Tremendous amounts of goodwill often are recorded by large public corporations which have purchased other large corporations recently.
#### **COST OF INTANGIBLE ASSETS**
As implied by the example involving goodwill, there seems to be a difference in whether an intangible asset will appear on the balance sheet, depending on if it was internally developed or it was purchased. The same is true with other types of intangible assets.
Research and development costs may produce intangible benefits to the business but generally are expensed rather than recorded as assets because uncertainty exists about the future benefits provided by those expenditures. As was discussed in Chapter 4, a company that spent $5 million on research and development activities, ultimately leading to a patent registered at a cost of $10,000, would capitalize only the $10,000 direct patent cost, while the $5 million of R&D would be expensed. This conservative approach is used because it is difficult to determine how much of the R&D costs produced a direct benefit.
However, if another party paid $20 million to purchase the patent for the competitive edge it would gain, the patent would appear on the purchaser's balance sheet at $20 million, an objective measure of its cost. In summary, only the direct costs to acquire or maintain an intangible asset are capitalized; internal development costs are expensed.
#### **AMORTIZATION OF INTANGIBLE ASSETS**
The matching principle requires that the cost of an intangible asset be expensed over its useful life to the business. The term _amortization_ is used to describe the cost allocation for an intangible asset, just like the concepts of depreciation for plant assets and depletion for natural resources.
Many intangible assets have a definite legal life or a limited economic life, such as patents, copyrights, and franchises. An intangible asset with a definite, finite life should be amortized over the shorter of its legal life or its estimated economic or useful life to the business.
However, there are a few intangible assets that are not amortized. Certain types of intangible assets with a short legal life can be renewed, such as a trademark which can be renewed every 10 years and is considered to have an indefinite life. If the company intends to and is able to renew this type of intangible asset indefinitely, it is not amortized. Goodwill is another example of an intangible asset that is not amortized. However, if an intangible asset is not amortized, the company must evaluate the balance sheet value every year. If its value has declined, an _impairment loss_ will be reported in the income statement and the intangible asset will be reported in the balance sheet at this more conservative value as we discussed earlier in the plant asset section.
As with plant assets and natural resources, amortization of intangible assets is based on the cost, residual value (if any), and estimated useful life, but with a couple of differences. For most intangible assets, it is assumed that the residual or salvage value is zero unless there is some type of commitment to sell the intangible at the end of its useful life. The useful life should be based on the shorter of the legal life or the economic life. For example, a patent may provide legal protection for 20 years, but if the product patented probably will become obsolete after 5 years, the shorter period should be used.
The method used to amortize an intangible asset should reflect its pattern of use or benefit. For example, a product patent might be amortized using the units of production method rather than the straight-line method. The calculation of amortization expense would be much like the depreciation methods illustrated earlier in this chapter.
Intangible assets are presented on the balance sheet in the same manner as plant assets and natural resources, showing the cost and accumulated amortization separately, or the net amount. Additional information about the types of intangible assets, their manner of acquisition, and accounting policies can be found in the footnotes to the financial statements.
### **Current Liabilities**
Current liabilities are obligations arising from past events which must be paid or settled within a year or within the operating cycle of the business, whichever is longer. These obligations are settled by using current assets (such as cash or merchandise inventory) or by the creation of other current liabilities (such as renewing a 90-day note).
As was discussed in Chapter 4, the distinction between current assets and current liabilities is very important in evaluating a company's ability to meet its short-term obligations as they come due, as is the proper classification of liabilities as either current or long-term. Current assets are expected to generate the cash resources needed to sustain ongoing business activities, cover routine expenses, and pay suppliers and lenders on time to maintain a healthy credit standing. Liquidity depends not only on how quickly a company's current assets generate cash but also on whether the amounts will be sufficient to cover its short-term obligations and provide an adequate cushion if something unexpected arises.
#### **DETERMINING AND VALUING CURRENT LIABILITIES**
The amounts reported for current liabilities generally are based on the amounts payable in cash or in goods and services. If an obligation arises from a bank loan, purchasing goods from a supplier on credit, monthly utility bills, or employee salaries and payroll tax liabilities, it is easy to determine that the obligation exists and identify the party to be paid and the amount owed.
However, not all potential obligations are that straightforward. Sometimes it is difficult to determine if a liability actually exists, and if it does exist, it may be difficult to determine the precise amount, the specified party involved, or both. As one would expect, the accounting concept of conservatism also applies to liabilities. As a result, some liabilities are based on estimated amounts, whereas other potential obligations may be omitted from the balance sheet and described only in the footnotes to the financial statements.
#### **CONTINGENT LIABILITIES**
A _contingent liability_ is a potential obligation arising from a past transaction, but whether an actual liability exists depends on or is contingent upon a future outcome. For example, assume your company sells goods to a customer and accepts a one-year note receivable. Rather than waiting a year to collect the note and interest, you sell the note to the bank with recourse. If the customer defaults and does not pay the bank when the note matures, your company is obligated to pay the bank. Does a liability exist? At this moment, the existence of this liability is contingent on a future outcome: whether the customer pays the bank at maturity or defaults on the loan.
Accounting guidelines for contingent liabilities are based on the likelihood that a liability exists and the ability to estimate the amount. If the liability is probable and one can make a reasonable estimate of the amount, it will be included as a liability on the balance sheet and often described in the financial statement footnotes. However, if it is only reasonably possible that there will be an unfavorable outcome, it will be described in the footnotes and not appear on the balance sheet. It is common to see a caption for contingent liabilities on the balance sheet, but with no dollar amount. This form of disclosure is meant to call the reader's attention to information discussed in the footnotes.
Returning to the example of selling a note receivable with recourse, if you think it is probable that the customer will default and that you will have to pay the bank, you will report the amount you expect to pay as a liability in your balance sheet and the loss in your income statement. If it is reasonably possible but not probable that the customer will default, footnote disclosures will describe the sale with recourse and state that it is reasonably possible the customer will default and also will provide an estimate of the amount of the recourse liability.
As one would imagine, contingent liabilities are challenging and controversial issues, and considerable judgment is involved in determining how they should be treated in financial statements, particularly those arising from pending litigation and environmental obligations.
#### **ESTIMATED LIABILITIES**
An estimated liability may involve uncertainty about the amount owed or the specific party to whom there is an obligation. To illustrate an estimated liability, consider the case of a manufacturer that provides warranty coverage on its products and that approximately 5 percent of the products it sold in past years were returned for replacement under the warranty. Does a liability exist? Probably.
The obligation under the warranty may be viewed as a contingent liability, but on the basis of past experience it is probable that the company will incur warranty replacement costs on some of the products sold this year, too. Even though it is not known precisely how many products will be returned for replacement or the specific customers involved, drawing upon past experience it is possible to make a reasonable estimate of the overall warranty costs that will be incurred. The estimated warranty costs for products sold should be treated as expenses and thus matched with the sales revenue earned during the current year, and the _estimated warranty obligation_ for products that are still under warranty coverage but have not yet been returned for replacement should be reported as liabilities in the year-end balance sheet. The accounting treatment for warranties will be discussed again later in this section of the chapter.
Chapter 4 discussed common types of current liabilities, and we will discuss a few of them in more detail below or refer you to other sections of this chapter where a similar example can be found. Current liabilities typically include the items in Figure 5.36.
**Figure 5.36** Balance Sheet Reporting: Current Liabilities
#### **ACCOUNTS PAYABLE**
_Accounts payable_ also are referred to as _trade payables_ ; they arise from normal business trade transactions and typically represent amounts owed to suppliers for goods and services purchased on credit. The normal credit period may be 30 days, but it can be longer in certain industries. If the credit terms are n/10 EOM, the amount billed is due within 10 days after the end of the month. As was discussed earlier in the material on accounts receivable, cash discount terms may be offered as an incentive for early payment. For example, if the credit terms are 1/10 net/30 on a $10,000 account balance, if the balance is paid within 10 days of the invoice, a 1 percent cash discount of $100 can be taken, otherwise one must pay the full $10,000 within 30 days (net/30).
If a company's current assets are not generating cash quickly enough for the company to take advantage of the discounts offered by its suppliers, it might consider a short-term borrowing arrangement in order to do so; the 1 percent discount, if annualized, is an 18 percent effective annual interest rate. For example, assume the company will have the money to pay the full invoice price of $10,000 when it is due in 30 days, but would need to borrow money to be able to pay the invoice 20 days earlier and receive the cash discount of $100. If it borrowed $10,000 at 18% interest for 20 days, it would pay $100 of interest ($10,000 × 18% × 20/360 days = $100). It would pay $100 interest to receive a $100 cash discount and break even. But if it could borrow the money for less than 18%, it would come out ahead.
When cash discounts are offered, a company might record its purchase of merchandise at the full invoice price of $10,000 and deduct the cash discount taken to arrive at an inventory cost of $9,900. Another company might record the purchase at $9,900 if it plans to take the discount and instead monitor the amount of the discounts it missed as this can be costly.
#### **NOTES PAYABLE**
Notes payable are formal promissory notes with specific maturity dates, interest rates, and repayment terms. Rather than repeating examples here, we refer you to the discussions of notes receivable in an earlier section and the notes payable and bonds payable sections in later sections of this chapter.
#### **ESTIMATED WARRANTY OBLIGATIONS**
Concepts pertaining to contingent and estimated liabilities were discussed in the introduction to current liabilities in this chapter. Obligations and/or unearned revenues arising from warranties and similar arrangements pose some interesting and challenging accounting issues due to the uncertainties involved.
When a company provides warranty coverage for the products it sells, the warranty coverage provided can be viewed from at least two different perspectives. In the case of the manufacturing company that was discussed earlier in this section, the estimated warranty costs were considered to be an additional expense associated with the product sales, even though the precise number of products that would be returned under the warranty and the specific customers involved were unknown at that time.
In that case, drawing upon its past experience, the manufacturer estimated the warranty costs it expected it would incur and treated those costs as expenses of the current year which were matched with the sales revenue earned that year. The obligation for estimated warranty costs on products that were still under warranty coverage but had not yet been returned for replacement was reported as a liability in the year-end balance sheet. This treatment is referred to as an _expensed warranty approach_. You should note that given the uncertainty involved, considerable judgment is used when estimating the costs and the remaining unsettled warranty obligation, which can provide unethical managers an opportunity to paint a more favorable picture of the company than may actually exist.
Another perspective to warranty coverage can be taken which is referred to as a _sales warranty approach_. In this approach, the warranty coverage is viewed as a separate sales component, much like a service contract, which is discussed further in the unearned revenue section below.
#### **UNEARNED REVENUES**
_Unearned revenues_ represent obligations that must be fulfilled in the future for which the company has been paid in advance. Common examples of unearned revenues include customer deposits for special orders, amounts received from sales of gift cards and certificates, online and print subscriptions, health club memberships, and future rental payments. You may find potentially returnable deposits, such as a lease security deposit, included in unearned revenues.
The revenues associated with these advance collections will be reported in the income statement in the period in which the goods or services are provided, or the customer's rights to a returnable deposit have expired or the customer has not met the conditions specified in the arrangement. Gift cards are increasingly popular, and many merchants have found that they provide substantial amounts of income and no cost when gift recipients fail to use the cards or have a few unused dollars remaining.
Referring back to the warranty example discussed earlier, a company may treat a portion of the selling price of its product as a separate warranty component, much like a service contract, or separately sell extended warranty coverage. The amount assigned to the warranty is unearned revenue that later will be earned over the period covered by the warranty. Rebates, coupons, premiums, frequent flyer miles and other award programs pose unique issues in determining the amount of revenue to be recognized at the time of sale versus at some future date.
#### **ACCRUED LIABILITIES (ACCRUED EXPENSES)**
_Accrued liabilities_ (or accrued expenses) represent expenses which have been incurred but are unpaid, such as salaries and wages, interest on notes payable (even if the note is long-term), and utilities. If employees earn vacation, sick pay, or personal leave days, according to the matching principle, the amounts earned each period should be included in the operating expenses in the income statement, and the estimated liability for days earned but not yet taken will be a current liability on the balance sheet.
#### **PAYROLL TAX LIABILITIES**
_Payroll tax liabilities_ include the income tax and FICA (Social Security and medicare) taxes the employer withheld from employee paychecks and additional payroll taxes imposed on the employer that have not been deposited or paid to the taxing authorities. The amount reported as salaries and wages expense in the income statement represents employees' gross pay, and salaries and wages payable on the year-end balance sheet represents the net amount owed to employees after withholdings. However, the employer must remit not only the payroll taxes withheld from employees, but also the additional employer payroll taxes it owes. The amount reported as payroll tax expense in the income statement includes the employer's matching amount of FICA taxes and state and federal unemployment taxes.
#### **INCOME TAXES PAYABLE**
Corporations are taxable entities and so any unpaid federal and state income taxes based on quarterly estimates or the completed income tax return are reported as current liabilities. However, income taxes on profits earned by sole proprietorships and partnerships are personal liabilities of the owner or owners, not liabilities of the business entity. Differences between accounting income and taxable income may result in deferred income taxes, which are discussed in the section on long-term liabilities later in this chapter.
#### **DIVIDENDS PAYABLE**
A corporation may distribute a portion of its earnings to the stockholders by declaring a cash dividend. There is no guarantee that common stockholders will receive cash dividends even if the corporation has paid regular cash dividends in the past; dividends are declared at the discretion of the board of directors. There are usually several days between the date a dividend is declared and the time it actually is paid. If a cash dividend has been declared but not been paid yet, current liabilities will include the amount of the dividend payable. As will be discussed in a later section, corporations may also issue preferred stock, which does carry a stated dividend rate, and preferred stock dividends must be paid before the common stockholders may receive cash dividend distributions. However, even the stated amount of the preferred stock dividend is not a liability until the board of directors formally declares that the dividend will be paid. The board of directors instead may declare a stock dividend that will be paid in additional shares of stock rather than in cash. This is not a current liability as it will not be settled with current assets or current liabilities; the dividend declared will be reported in stockholders' equity.
#### **CURRENT PORTION OF LONG-TERM DEBT**
As will be discussed below in the section on long-term liabilities, some borrowing arrangements require that a portion of the debt be repaid each year, such as an installment loan or serial bonds payable. If this is the case, the principal amount due and payable in the coming year is included in current liabilities and the remaining principal balance is reported as a long-term liability and presented on the balance sheet as shown in Figure 5.37, along with additional footnote disclosures.
**Figure 5.37** Balance Sheet Reporting: Long-Term Installment Loans
### **Long-Term Liabilities**
While current liabilities are obligations which are generally due within a year and are settled by using current assets such as cash or merchandise inventory, long-term liabilities are borrowings or other obligations which are not due or payable for at least one year and perhaps will not be settled in full until many years in the future.
In Chapter 4, we briefly discussed examples of obligations one might find in the long-term liability classification. We now will discuss examples of several of the long-term liabilities shown in Figure 5.38.
**Figure 5.38** Balance Sheet Reporting: Long-Term Liabilities
#### **NOTES PAYABLE**
A company may negotiate a loan from a financial institution or another lender to cover its short-term or long-term cash needs and sign a promissory note which specifies the interest rate, maturity date, and repayment terms. It may be an unsecured loan; if the borrower defaults and does not repay the note when due, no specific assets are pledged as collateral for the loan. The lender may require monthly payments which include principal and interest (see Mortgage Note Payable example below), require that only period interest payments be made, or allow the borrower to wait until the loan matures to repay both the principal and the interest that has accumulated.
As an example, assume a company signed a $10,000, 12 percent, five-year promissory note on January 1, 20x1. It was required to make semiannual interest payments only on July 1 and January 1 of each year, but wait to repay the principal amount due when the loan matures. The semiannual interest payment would be calculated as follows:
**Principal** | × | **Interest rate** | × | **Time** | **=** | **Interest**
---|---|---|---|---|---|---
$10,000 | × | 12% | × | 6/12 months | = | $600
Assume the company made the first interest payment on July 1, 20x1, but will not make the second interest payment until January 1, 20x2, the interest payment due date. When the December 31, 20x1, balance sheet is prepared, interest payable of $600 will be classified as a current liability, and the $10,000 note payable will be classified as a long-term liability, as shown in Figure 5.39.
**Figure 5.39** Balance Sheet Reporting: Notes and Interest Payable December 31, 20x1
In the final year before the loan matures, in most cases the note payable will be classified as a current liability as it is due within a year and will be repaid (settled by using current assets). But what if the company renegotiates the loan, and rather than repaying when it matures, it signs a new five-year note? In this case, the note may continue to be reported as a long-term liability as it will not be settled in cash but instead replaced with another long-term note.
#### **MORTGAGE NOTES PAYABLE**
A _mortgage note payable_ is another form of long-term borrowing that typically is used to finance the purchase of land and/or buildings. A mortgage is a _secured_ loan; the property financed by the loan is pledged as collateral and can be sold to cover the amount owed to the lender if the borrower defaults. Mortgage loans typically require monthly installment payments of principal and interest. Assume the principal balance of a 15-year mortgage loan is $100,000, and $3,000 of this balance is due within the coming year. The mortgage loan will appear in both the current liability and long-term liability classifications, as shown in Figure 5.40.
**Figure 5.40** Balance Sheet Reporting: Mortgage Loan Repaid in Installments December 31, 20x1
#### **DISCOUNTED NOTES PAYABLE**
Before we discuss what we mean by a _discounted note payable_ , let's start with a more straightforward example. Assume that Company A purchases an acre of land in a business park for $10,000 and signs a promissory note payable to the bank. The principal amount of the note is $10,000, it has a 10 percent stated interest rate, and it matures in two years. However, the 10 percent interest will be compounded and added to the amount owed and will be paid when the note matures in two years rather than being paid each year. Figure 5.41 shows how the interest is calculated
**Figure 5.41** Calculation of Compound Interest and Maturity Value of a Note Payable
and shows that a total of $12,100 will be repaid in two years: the $10,000 principal balance of the loan and the $2,100 compound interest. Note that no interest was paid in 20x1 and so the 20x2 interest is calculated on the total amount owed; in other words, the interest was compounded.
Assume Company B purchases an adjacent acre of land in the same business park and signs a note payable, promising to pay $12,100 in two years with zero interest. Does it seem realistic that Company B could borrow for two years and pay no interest? Probably not. There is an implied component of interest in this transaction even though none was stated specifically. If the market value of the land is $10,000, the additional $2,100 paid when the note matures is the implied interest or imputed interest that has accumulated during the two years.
The substance of Company B's loan is identical to that of Company A, although the forms of the two notes differ. In Company B's case, we would say that we are _discounting_ the note to reflect the implied interest and that the _present value_ of the note is $10,000. Both companies would record the land at its cost of $10,000, and both would pay $12,100 when the notes mature, which includes $2,100 of compound interest. However, on the date the note was signed, Company B would report the note on its balance sheet as shown in Figure 5.42.
**Figure 5.42** Balance Sheet Reporting: Discounted Note Payable—Company B January 1, 20x1
Company B's 20x1 income statement would include $1,000 of interest expense ($10,000 × 10 percent), and the December 31, 20x1, balance sheet would present the new ending balance of the loan as shown in Figure 5.43.
**Figure 5.43** Balance Sheet Reporting: Discounted Note Payable—Company B December 31, 20x1
Company B's 20x2 income statement would include $1,100 of interest expense ($11,000 × 10 percent), and the December 31, 20x2, balance sheet would present the new ending balance of the loan, the total amount that will be repaid, as shown in Figure 5.44.
**Figure 5.44** Balance Sheet Reporting: Discounted Note Payable—Company B December 31, 20x2
Financial statements include many other applications of the concepts similar to those described in the examples above. Although a detailed discussion of these concepts is beyond the scope of this book, similar concepts are applied to capital lease obligations and bonds payable, which are discussed below.
#### **CAPITAL LEASE OBLIGATIONS**
A company may enter into rental or lease agreements for equipment, buildings, or other assets as an alternative to financing an actual purchase of those assets on its own. A rental or lease arrangement may be for a short duration, such as renting specialized equipment for a few days, but for buildings and other assets the lease term may span many years. There are many considerations involved in a lease versus buy decision, along with many complex financial reporting issues that are beyond the scope of this book, but we will discuss a couple of examples briefly below.
Let's first examine an example of what might be called an _operating lease_. A company decides to rent office space and signs a one-year lease agreement in December 20x1, pays the first two months in advance, and makes monthly rent payments for the remainder of the lease term. As was discussed in Chapter 3, when financial statements are prepared on December 31, 20x1, one month of rent expense will be reported in the income statement, and the other month paid in advance will be reported as a current asset, prepaid rent, and not an expense until the following year in accordance with the matching principle.
Now let's consider a company that needs its own office building and can either finance the purchase of the building with a 25-year mortgage note payable and make monthly loan payments or enter into a 25-year agreement to lease another office building and make monthly lease payments. If it decides to borrow money and purchase a building, the building will be included in plant assets, and the mortgage note payable will be included in liabilities on the balance sheet.
But what if the company instead decides to lease the other building and is legally obligated to make 25 years of lease payments, along with other terms and conditions? That's a pretty big commitment. If this arrangement is considered to be an _operating lease_ , the monthly lease payments would be reported as rent expense in the income statement, but you would not find the obligation for the future lease payments included among the liabilities; in other words, this would be an _off–balance sheet_ arrangement (now almost a household term thanks to Enron and its clever arrangements to conceal its activities).
In contrast to an operating lease, a _capital lease_ is the term used to describe lease arrangements that more closely resemble financing the purchase of an asset with long-term debt. Regardless of the form of the written agreement, the economic substance should be considered carefully because it may indicate that the arrangement is like a purchase in disguise in an effort to keep debt off a company's balance sheet.
If the 25-year lease arrangement above was determined to be a capital lease, it would be treated much like the actual purchase of the office building with a mortgage note payable. The capital lease obligation would represent the present value of the future lease payments (using concepts similar to the discounted note payable discussed earlier) and would be treated very much like a mortgage note payable, with a portion of each lease payment applied to pay down the capital lease obligation (much like principal payments on a loan) and the remainder of each payment is much like the interest portion of a mortgage loan payment. The building would be capitalized as though it had been purchased, included in plant assets, and depreciated.
#### **BONDS PAYABLE**
Bonds are a common form of financing that are used not only by large corporations but also by federal and state governments and agencies, school districts, and many other parties which need to borrow substantial amounts of money for a long period of time. When you purchase a U.S. savings bond, you are lending your money to the government. If you purchase a school district bond, you are making a loan to help pay for the construction of new schools or other improvements. If you purchase a corporate bond, you are lending money the corporation will use to finance expansion and growth either internally or to acquire other companies to achieve external growth.
This book is focused on the financial statements of business entities, and so we will emphasize concepts pertaining to corporate bonds, although many of the concepts are similar for bonds issued by other parties.
_Debentures_ are a form of unsecured bonds (no specific collateral), while other types of bonds may pledge assets or specific revenue sources as collateral. Some bond agreements require the company to make periodic payments into a _sinking fund_ which will be used to repay the bonds on the maturity date.
One advantage of this form of long-term borrowing is the company's ability to raise substantial amounts of money that it may not have to repay for years or even decades without relying upon a single lender, such as a bank, to do so. The denomination of a typical corporate bond is $1,000; in theory, if the corporation needs to borrow $10,000,000, it could borrow $1,000 each from 10,000 different parties. Another advantage is that a bondholder (the investor) can sell the bond investment to other parties and not have to wait to be repaid when the bonds mature, which might be many years in the future. Just like stock investments, bonds are purchased and sold daily in the financial markets.
Although the principal (or maturity value) of the bond may not be repaid for many years in the future, a stated amount of interest is owed and generally is paid semiannually. For example, if the corporation issues $10,000,000 of bonds with a stated annual interest rate of 4 percent and pays the interest semiannually, every six months the cash interest payment would be calculated as shown below:
**Principal** | × | **Interest Rate** | × | **Time** | **=** | **Interest**
---|---|---|---|---|---|---
$10,000,000 | × | 4% | × | 6/12 months | = | $200,000
Amounts received by the corporation at the time it issues bonds may differ from the amount it is required to repay when the bonds mature to adjust for differences in the interest rate specified for the bonds and market rates of interest at the time of borrowing. Thus, bonds payable are shown in the balance sheet at their present value, based on implied or imputed interest, similar to the example of the discounted note payable that was discussed earlier in the chapter.
For example, if 4 percent bonds were issued when market rates of interest were higher than 4 percent, the bondholders might pay only $9,200,000 to purchase the bonds but if the bondholders hold their investment to maturity they still will be paid the full $10,000,000 when the bonds mature which compensates them for the additional interest they expected to earn. We would say the bonds are issued at a _discount_ , based on concepts similar to the discounted note payable example that was discussed earlier. (The amount paid for the bonds, $9,200,000 in this case, is not a random number. It is based on a mathematical calculation using the timing and amounts of cash payments and an applicable interest rate, a topic beyond the scope of this book.) The bonds would be presented on the balance sheet as shown in Figure 5.45.
**Figure 5.45** Balance Sheet Reporting: Bonds Payable Issued at a Discount
By contrast, if the market rate of interest at the time the bonds are issued is less than 4 percent, the corporation will issue the bonds at a higher price; for example, the bonds might sell for $10,800,000. (Again, this is not a random number; it is computed using calculations that provide the bondholders with a lower interest rate that the market is willing to accept at the time.) This lower effective interest rate of return is achieved because the company will repay only $10,000,000 at maturity, reducing its overall effective interest cost. We would say the bonds are issued at a _premium_ , and the bonds would be presented in the balance sheet as shown in Figure 5.46.
**Figure 5.46** Balance Sheet Reporting: Bonds Payable Issued at a Premium
#### **COVENANTS, COLLATERAL, AND SINKING FUNDS**
The terms and conditions for long-term borrowing arrangements can be very detailed and complex, and the agreement may specify conditions that must be met throughout the term of the loan or may impose restrictions. These conditions and restrictions, which are referred to as loan _covenants_ , may limit the amount of cash dividend payments to the stockholders, require the company to meet certain financial benchmarks (including certain ratios that will be discussed in Chapter 8), or require it to pledge specific assets as collateral for the loan in the event of default. Some agreements require that money be set aside and invested in a fund that will be used to repay the loan at maturity, such as the _sinking fund_ that was discussed in the section on long-term investments. The financial statement footnotes for long-term debt will provide considerable additional information, including key covenants, pledged assets, stated and effective interest rates, maturity dates, and scheduled payments or sinking fund requirements for each of the next five years.
#### **DEFERRED INCOME TAXES**
Corporations are considered legal entities separate from their owners and a corporate entity must file tax returns and pay federal and state income taxes on its profits. A corporation makes quarterly estimates of its taxable income for the current year and makes quarterly estimated tax payments; any unpaid taxes owed for the current year are reported under current liabilities as _income taxes payable_.
By contrast, amounts reported as _deferred income tax liabilities_ represent income taxes which may become payable in later years and are related to differences in accounting and taxable income. According to the matching principle, the amount reported as _income tax expense_ in the current year's income statement represents the tax due on the basis of the revenues and expenses including in accounting income for that year.
However, according to tax law, not all revenues are taxable income in the same year they are considered to have been earned under financial accounting concepts. Likewise, not all expenses are tax-deductible in the same year they were incurred and reported in the income statement.
Accounting for income tax liabilities is a very complex issue that is beyond the scope of this book, but a brief example may help explain the concept. Deferred income taxes commonly arise when the depreciation methods used in the financial statements (discussed earlier in this chapter) differ from the depreciation methods and tax deductions specified in the tax law, although there are many other examples.
Assume a corporation recently purchased equipment and capitalized a cost of $100,000. It expects to use this equipment for 10 years, after which the trade-in value is presumed to be zero, and even if it could be sold for its scrap value, that would be a negligible amount. It will depreciate $10,000 per year for a total of $100,000, using the straight-line method of depreciation explained earlier in this chapter.
According to the matching principle, income tax expense in the income statement is based on the accounting income earned in that period that will be subject to income taxation but not necessarily reported on the tax return for that particular year. Assume income before depreciation is $500,000 and the income tax rate is 40 percent. As shown in Figure 5.47, the company will report income tax expense of $196,000 in its 20x1 income statement, based on accounting depreciation of $10,000.
**Figure 5.47** Income Statement: Calculation of Income Tax Expense
However, for income tax purposes, assume that this asset qualifies for a shorter five-year life and that depreciation is based on the MACRS method discussed earlier in this chapter (Figure 5.32). A total of $100,000 of depreciation will be taken, with a depreciation deduction in the income tax return of $20,000 in the first year of use. Taxable income and income taxes currently payable would be determined as shown in Figure 5.48.
**Figure 5.48** Tax Return: Calculation of Income Tax Currently Payable
Assuming the company has not paid its 20x1 income taxes yet, its year-end balance sheet would include the information shown in Figure 5.49.
**Figure 5.49** Balance Sheet: Income Tax Liabilities
The amount reported as _income taxes payable_ of $192,000 represents the income tax the corporation is obligated to pay for the 20x1 income tax year. Most corporations pay estimated income taxes owed on a quarterly basis and so the amount reported as a current liability may be only the unpaid fourth quarter installment (or $48,000 in this example).
The _deferred income tax liability_ of $4,000 at December 31, 20x1, represents the additional income taxes the corporation expects to pay in a future year (or years) when taxable income is greater than accounting income. In the early years of the asset's useful life, the tax deduction for depreciation is greater than depreciation expense in the income statement, and the deferred income tax liability will increase as the corporation postpones paying income taxes it eventually will owe. However, in later years, when the income tax deduction is lower (or zero), the income taxes that were postponed or deferred become payable.
The total amount of depreciation for both income tax and accounting purposes is $100,000 in this example. If all other factors are held constant, the total amount of income tax paid would equal the total amount of income tax expense that was reported in the income statement for the ten years the asset is used and depreciated. However, in individual years the timing and the amounts differ, which is aptly referred to as a _timing difference_. You can think of the deferred income tax liability arising in 20x1 as the timing difference between tax depreciation ($20,000) and income statement depreciation ($10,000) times the 40 percent income tax rate, which is $4,000; in other words, the income taxes that were deferred or postponed for the time being but must be paid in the future.
You also may find _deferred income tax assets_ on the balance sheet. In this case, certain revenues may be taxable income before the revenue is earned according to the revenue recognition principle and reported in the income statement, and certain expenses reported in the income statement may not be tax-deductible until a later year. In this case, it is almost as though the corporation had prepaid its income tax, but this prepayment is referred to as a deferred income tax asset on the balance sheet because it is quite different from other prepaid expenses such as rent or insurance. The financial accounting treatment of deferred income taxes is a very complex area.
### **Owners' Equity and Changes in Owners' Equity**
Owners' equity represents the residual claims of the owners to the assets of the business and sometimes is referred to as _net assets_ or _net worth_. The balance sheet shows the total amount of owners' equity at a point in time and includes the amounts invested by the owners and the profits retained or reinvested in the business. The statement of owners' equity shows any additional investments made during the year, the profit or loss, and the amounts withdrawn by or distributed to the owners.
We will begin by discussing the basic characteristics and concepts of owners' equity for three types of business entities: sole proprietorships, partnerships, and corporations. In the final portion of this chapter, we will discuss corporations in more detail.
#### **SOLE PROPRIETORSHIPS**
A _sole proprietorship_ is a business with a single owner, often the person who manages the business. Legally, the business is not considered separate from the owner; the owner is personally responsible for business actions and can be personally liable for business debts. However, according to the business entity assumption as discussed in Chapter 2, business activities are reported separately from the personal activities of the owner. Personal expenses of the owner paid by the business and the owner's salary are treated as withdrawals and not as business expenses. The results of business activities are included on the owner's personal income tax return, and the owner personally pays income taxes on business profits; the business entity is not taxed.
In the case of a sole proprietorship, the owner's equity is referred to as _capital_ , which includes the owner's investments and profits retained by the business presented as a single amount on the balance sheet, as shown in Figure 5.50.
**Figure 5.50** Sole Proprietorship: Balance Sheet (Partial) December 31, 20x1
An additional financial statement, the statement of owner's equity, shows the cause of any changes in owner's equity that occurred during the year. Figure 5.51 shows the statement of owner's equity for 20x2, the second year of business operations.
**Figure 5.51** Sole Proprietorship Statement of Owner's Equity For the Year Ended December 31, 20x2
This statement shows that the owner invested $20,000 of additional capital in the business during the year, the increase in owner's equity when the business earned a profit of $80,000, and that the owner withdrew $40,000 of the profits for personal use, but the remainder of the earnings were reinvested or retained by the business. The year-end capital balance of $460,000 includes the amounts invested by the owner as well as the profits which were retained or reinvested in the business.
#### **PARTNERSHIPS**
A general partnership is a voluntary association of two or more owners which can be based on a simple verbal agreement but ideally should be based on a written contract. Partners may consist of individuals, other business entities, estates or trusts, or other parties. In fact, many business joint venture activities are structured as partnerships.
Similar to a sole proprietorship, a general partnership is not considered to be a legal entity separate from its owners, although the business is considered a separate accounting entity. The partners are personally responsible for business actions and can be held personally liable for business debts. Personal expenses paid by the partnership and the partners' salaries are treated as withdrawals and not as business expenses.
The partnership files an informational income tax return and reports each partner's share of the income to the taxing authorities. However, the partnership entity is not taxed on its income; each partner's share of the income is reported and taxed on each partner's separate income tax return.
If the business in the earlier example had been structured as a partnership with two owners, the 20x2 financial statements would include the information presented in Figure 5.52. From Chapter 4, recall that the partnership had two owners: Swithin invested and owned a one-fourth interest and Patrick invested and owned the other three-fourths of the partnership. Partnership income can be divided among the partners in any manner to which they agree, but in the absence of an agreement, each is considered an equal partner and is entitled to an equal share of the profits. However, in this example, we assumed that the partners agreed to divide the profits on the basis of the relative capital invested, with one-fourth (¼) going to Swithin and three-fourths (¾) to Patrick. The withdrawals might be the share of the partnership profit each partner is allowed to withdraw in lieu of a salary.
**Figure 5.52** Statement of Partnership Equity For the Year Ended December 31, 20x2
Balance Sheet (Partial) December 31, 20x2
##### **Partnerships and Related Business Forms**
The discussion above was based on what might be called a _general partnership_ , but there are many partnership-like variations for entities with multiple owners. Some are more like partnerships and others are more like corporations, with a range of variations in between. Each form has its unique advantages and disadvantages, often pertaining to which owners have the authority to manage the business and make decisions, the risk of personal liability of the owners, or differences in the way business income and distributions are taxed. The characteristics, advantages, and disadvantages of each of these business forms should be investigated and understood before one selects a particular form of business entity, based on expert accounting and legal advice.
Although a detailed discussion of various forms of entities is beyond the scope of this book, we provide short explanations for the more common ones below:
• **General Partnership** : In a general partnership, each partner has the right to conduct business on behalf of the partnership, is entitled to a share of the profits, and has a residual interest in the net assets, but can be held personally liable for actions against the partnership and for partnership debts.
• **Limited Partnership** : There must be at least one general partner with decision-making authority and personal risk in a limited partnership entity. The limited partners are more like passive investors; they provide capital to the business and usually are guaranteed a return on their investments. They do not participate in the management of the business and are not personally liable for partnership debts.
• **Limited Liability Partnership (LLP)** : A limited liability partnership is similar to a general partnership in terms of the rights to participate in management, a share of the profits, and a residual interest in net assets. However, the individual partners generally are protected from personal liability unless it is caused by a partner's own personal negligence or malpractice.
• **Limited Liability Company (LLC)** : A limited liability company is a bit more like a corporation as the owners are not liable for business debts but similar to a general partnership in the distribution and taxation of profits. The owners of an LLC are referred to as members.
#### **CORPORATIONS**
A _corporation_ is both a separate legal entity and a separate accounting entity and is formed subject to the laws of the state in which it is incorporated. Ownership in a corporation is evidenced by shares of stock; the equity of an individual stockholder is based on the proportion of the total stock that is owned. If the corporation is _publicly held_ , its shares of stock may be purchased and sold readily by investors, and the individual owners of the corporation may change frequently. A _nonpublic_ or _closely held_ corporation may establish limitations pertaining to the type of parties permitted to own stock or the conditions for selling stock to other parties. A corporation may have a single stockholder, a few owners or hundreds, and the owners may be individuals, partnerships, estates or trusts, corporations, or other parties.
Because a corporation is a legal entity that is separate from its owners, the stockholders do not participate in its management or operations unless they also happen to be officers or employees. Stockholders are not personally liable for business activities or business debts. A corporation files a tax return and also pays income taxes on the income earned by the corporation. In addition, the stockholders also may be individually subject to income taxation on earnings distributed as cash dividends, other distributions, and gains and losses from the sale of their stock investments.
In the case of a corporation, owners' equity is referred to as _stockholders' equity_. Owners receive shares of stock as evidence of their ownership and investment in the corporation, typically _common stock_ , although there are other types. A public corporation may have hundreds of stockholders, and ownership can change daily as shares of stock are traded in the financial markets. As a result, a corporation reports the amounts originally invested (contributed) by the stockholders and the profits retained by the corporation rather than the equity of individual owners. Figure 5.53 shows how stockholders' equity is presented if the business is structured as a corporation.
**Figure 5.53** Corporation Balance Sheet (Partial) December 31, 20x1
Although the equity of an individual stockholder is not presented in the financial statements, this can be determined by calculating the _book value_ of a share of stock. In this case by dividing total stockholders' equity of $400,000 by 40,000 shares of stock, the book value is $10 for one share of stock. The equity of an individual stockholder who owns 10,000 shares, or one-fourth of the stock, is $100,000 (10,000 shares × $10 book value per share), which is equivalent to the amount of Swithin's capital balance in the previous partnership example.
Figure 5.54 shows the statement of stockholders' equity that would be presented in the corporation's 20x2 financial statements. Stockholders invested an additional $20,000 by purchasing additional common stock. The corporation earned a profit of $80,000, of which $40,000 was distributed to its stockholders in the form of cash dividends and the remaining profit earned in 20x2 was retained or reinvested. At December 31, 20x2 the corporation has $100,000 of common stock which represents the amounts stockholders invested, and retained earnings of $360,000 which represents the cumulative amount of profits earned to date that were retained by the corporation.
**Figure 5.54** Statement of Stockholders' Equity For the Year Ended December 31, 20x2
Balance Sheet (Partial) December 31, 20x2
Building on the basic stockholders' equity concepts above, the final portion of this chapter will focus on additional concepts and terminology likely to be found in the financial statements of a corporation.
#### **COMMON STOCK**
The corporate charter indicates the types and amount of stock a corporation is authorized to issue. Most corporations have _common stock_ or _capital stock_. Common stockholders are the true owners of the business and are entitled to one vote for each share of stock owned. Stockholders do not participate directly in the management of the business unless they also happen to be directors, officers, or employees.
Stockholder participation is based on exercising voting rights to elect the board of directors and certain other matters. One may control a corporation by owning more than 50 percent of the stock, although control may be achieved with lesser ownership if other stockholders do not unite and vote as a unit or by other means. Common stockholders have the right to receive a proportionate share of cash dividends if and when they are declared by the board of directors, but common stock does _not_ carry a stated or guaranteed cash dividend. Common stockholders also have the first right to purchase a proportionate share of any additional stock issued, and to receive a proportionate share of corporate assets upon liquidation, but only after creditor claims and other claims have first been paid.
#### **PREFERRED STOCK**
Corporations may also be authorized to issue other types of stock, such as _preferred stock_ , which may appeal to a different set of investors than those who invest in common stock. Despite its name, preferred stock is not necessarily a better or a preferred type of investment; rather, the term preferred is used for a category of stock which conveys specific or preferential rights that common stockholders do not receive. Preferred stockholders do not have voting rights; however, there is a stated cash dividend that will be paid to preferred stockholders and must be paid before distributions to the common stockholders can be made. In most cases, preferred stock dividend rights are _cumulative_ ; if a corporation is unable to pay the required preferred stock dividend in a particular year, the amounts accumulate and must also be paid before any distributions to the common stockholders. If a corporation has past-due or _dividends in arrears_ , it will disclose this information in the financial statement footnotes.
Preferred stockholders have other special rights; they will be paid a stated amount for their stock if the corporation retires or redeems it, and if the corporation liquidates and creditors have been paid, the preferred stockholders will be paid for their equity before any distributions can be made to the common stockholders.
#### **PAR OR STATED VALUE**
The corporate charter may specify a _par value_ or _stated value_ associated with a share of stock. Par value is a legal concept that generally specifies the minimum amount stockholders must invest in the corporation at the time the stock is first issued and a minimum amount of capital the corporation must maintain. In theory this provides some protection to corporate creditors whose claims can be paid only from corporate assets (and not from the personal assets of the owners as might happen in the case of a sole proprietorship or partnership). For common stock, the par or stated value is usually a nominal amount and should not be viewed as an indication of the current market price or value of one share; in fact, the par value of some companies common stock is one penny. The amount stockholders invest typically is greater than the par value of the common stock, and the excess is reported separately as _contributed capital in excess of par value_.
For preferred stock the par value is the amount on which the stated dividend will be based. For example, a share of preferred stock with par value of $100 and a 10 percent dividend rate will be paid a $10 cash dividend each year.
#### **RETAINED EARNINGS, DEFICITS, AND RESTRICTIONS**
Let's use a simple example to illustrate what we mean by retained earnings or reinvested earnings and explain why even a profitable corporation may not distribute its profits to its stockholders. Assume a corporation's net income for its first year of operations was $100,000, and it has $100,000 in cash in the bank at year-end. Rather than paying cash dividends to its stockholders now, the company can grow bigger and faster if it reinvests those earnings in a building it needs, and so it purchases a building with the $100,000 in cash. At year end it has $100,000 of retained earnings but no cash because its earnings were reinvested in the building it purchased.
Next year its net income is $200,000, but it still has no cash; it paid for all of its operating expenses but has not collected the $200,000 its customers still owe from this year's sales even though it expects it will collect this amount early next year. At this point, the corporation has earned a total of $300,000 of net income and has $300,000 of retained earnings, but still it has not paid the stockholders any cash dividends. However, if the corporation reinvests its earnings and continues to perform well, the market price of its stock may increase over time and the stockholders will be better off in the long run than they would be if they received cash dividends now and purchased some other type of investment.
A company may have profits but no cash. It may have retained earnings but no cash. It may have profits and cash but can create greater value in the long run by reinvesting its earnings and cash instead of paying cash dividends.
Instead of retained earnings, a corporation may have a _deficit_. Most people associate this term with news broadcasts about the economy and the federal government. Deficit spending means that the government is spending more than it brings in from income tax revenues and other sources. A trade deficit means that the value of goods the United States imports exceeds the value of the goods it exports to other countries. In the case of a corporation, a deficit means that since the time of incorporation its cumulative losses have exceeded its cumulative profits and this is definitely not a good signal.
Retained earnings that are earmarked for particular purposes or are legally unavailable for distribution to the owners are disclosed in the financial statements or the footnotes. Some borrowing agreements may restrict the amount of earnings available for dividends until the loans are repaid, but a corporation may restrict its earnings voluntarily as a signal of its plans for future expansion or for other reasons.
#### **DIVIDENDS**
Dividends are a distribution of past or present earnings to the stockholders, but they are not expenses of the corporation and do not appear on the income statement. Dividends are similar to owner withdrawals in a sole proprietorship or a partnership; however, it is the board of directors, not the owners, which determines the amount and timing of dividend payments. Preferred stockholders are entitled to a specified dividend stated on the stock certificate: a fixed dollar amount or a percentage of the par value.
Common stock ordinarily does not have a specified dividend rate. The amount of dividends declared is at the discretion of the board of directors even if the corporation has paid regular cash dividends in the past.
Rather than distributing cash, the corporation may declare a _stock dividend_ , and each stockholder receives a proportionate amount of the stock distributed. When a company's stock price continues to rise, it may declare a _stock split_ to keep the price of a share in a more reasonable zone. For example, if the market price of one share of stock has risen to $200, the corporation may declare a two-for-one stock split. Each owner will have twice as many shares as before the split, but each share will be worth half as much because the market price would be expected to decline to $100 after the stock split.
#### **TREASURY STOCK**
Just like any other investor, a corporation can purchase shares of stock in other companies or even its own stock. _Treasury stock_ represents a corporation's own shares of stock which were issued to its stockholders but have been reacquired by the corporation. As was discussed in other sections of this chapter, investments in the stock of other companies may be found under several different balance sheet classifications, depending on the purpose and intended holding period. However, when a corporation purchases (reacquires) its own stock from existing stockholders, it is not considered an asset; instead, it is shown as a deduction from total stockholders' equity as less stock owned by outside investors.
A corporation may purchase its own stock for many reasons. If the market price is falling, it can purchase the shares at a low price and perhaps sell (reissue) them later when the price goes up. It may purchase shares to avoid a hostile takeover of the company or to distribute to key executives later as additional compensation or to sell to employees as part of an employee stock ownership plan. Shares held in the treasury have no voting or dividend rights; treasury stock represents shares that were issued but are no longer _outstanding_ for voting and dividend purposes.
Treasury stock is shown as a deduction from total stockholders' equity in the balance sheet, along with disclosure of the number of treasury shares held and any restrictions placed on retained earnings. If the shares later are reissued at a price higher or lower than their price when purchased, those gains or losses are reflected as increases or decreases in stockholders' equity, not in the income statement. A corporation cannot report profits or losses from dealing in its own stock.
#### **ACCUMULATED OTHER COMPREHENSIVE INCOME**
From our earlier discussions of short-term and long-term investments in marketable securities, you learned that _trading_ and _available for sale securities_ are adjusted to market value on the year-end balance sheet. The _unrealized holding gain or loss_ on trading securities is reported in the income statement; however, if the investments are classified as available for sale, the unrealized gain or loss is reported in stockholders' equity, not included in net income or in the year-end retained earnings.
Accounting guidelines firmly support the notion that all revenues and/or gains and expenses and/or losses be reported in the income statement, with proper classifications and disclosures. However, there are a handful of exceptions, including the unrealized holding gain or loss on available for sale securities and some others which are beyond the scope of this book. Since these gains and losses bypass the income statement, thus are not reflected in retained earnings at year-end, stockholders' equity includes these unique types of gains and losses in a separate caption, referred to as _accumulated other comprehensive income_.
On the basis of the additional concepts discussed above, Figure 5.55 provides a more detailed illustration of the stockholders' equity portion of the balance sheet. Note that the amounts invested by the preferred and common stockholders are shown first under the heading of _contributed capital_ and the income reinvested in the corporation is shown separately as _retained earnings_.
**Figure 5.55** Balance Sheet: Stockholders' Equity December 31, 20x2
As you review Figure 5.55, note that each type of stock is shown separately, along with some additional information. For example, the corporate charter authorized the corporation to issue 1,000 shares of preferred stock, but to date it has issued only 100 of the 1,000 shares authorized. It can issue additional preferred stock at a future date without having to amend its charter. The preferred stock has a par value of $100 and a 10 percent stated dividend rate; the annual dividend requirement is $10 per share, and preferred stock dividends must be paid before the common stockholders receive any distributions.
The corporate charter authorized the issue of 50,000 shares of common stock, but only 5,000 of those shares have been issued to date; additional stock can be issued at a later date if the corporation needs to raise more capital to finance growth and expansion. The par value of the common stock, typically a nominal amount, is only $5 per share; the amount reported as common stock represents the par value of the 5,000 shares issued or $25,000. However, when the common stock was issued, the market price must have been much higher than $5 per share. The common stockholders contributed (invested) $65,000 more than the par value, for a total investment of $90,000; thus, the 5,000 shares of common stock were issued at an average market price of $18 a share.
Sometime after the common stock originally was issued, the corporation reacquired 100 shares of its own stock, which appears as a deduction from stockholders' equity, not as an investment in the asset section of the balance sheet. The $3,000 amount reported as _treasury stock_ is based on the cost at the time the stock was reacquired. Thus, the market price when the 100 shares were reacquired must have been at an average price of $30 per share.
Finally, note that the corporation reports $30,000 as _accumulated other comprehensive income_. Perhaps this is the unrealized holding gain on available for sale securities that was discussed in an earlier section of this chapter; this signifies that the company has investments in available for sale securities for which the current market value is $30,000 greater than their cost.
### **Summary**
This chapter has covered a lot of ground. Building on the foundation introduced in Chapter 4, we focused on the specific and unique characteristics of many balance sheet items and the relationship of those items to income statement reporting. You also learned that financial statements include assumptions and estimates, the statements do not always include precise information, and alternative acceptable accounting methods for items such as depreciation and inventory cost produce different financial statement results.
We are now ready to move on to the last of the four financial statements: the statement of cash flows. Once we have completed the study of the financial statements, the remainder of the book will acquaint you with how to read and use financial statements and discuss ways in which a company may misstate its results fraudulently.
### **Quiz for Chapter 5**
1. According to the matching principle, bad debts expense should appear in the income statement in the time period in which the:
a. Sales revenue was earned
b. Specific customer account balance was written off as uncollectible
2. Customers owe a company $100,000. If the company reports accounts receivable (net) of $94,000 on its balance sheet:
a. The estimated amount the company expects to collect from its receivables is $94,000.
b. The company has established an allowance for doubtful accounts of $6,000
c. The company included an estimate of uncollectible accounts in the operating expenses reported in its income statement.
d. All of the above are true.
e. None of the above are true.
3. On November 1, year 1, a company accepts a $2,000, 12 percent, six-month note receivable from a customer. When preparing its financial statements at December 31, year 1, which of the following is true?
a. $40 of interest income will be reported in the income statement.
b. $40 will be included in the accrued receivables in the balance sheet.
c. When the note matures, the company is entitled to collect $2,000 and $120 interest.
d. All of the above.
e. None of the above.
_Use the following information to answer questions #4 & #5:_
During the year, a company purchased three units of inventory for $2,000, $2,400, and $2,500, respectively. It sold two units and has one left in ending inventory. Determine the amounts that would be reported as cost of goods sold and ending inventory by using the cost-flow assumptions in the next two questions.
4. Using FIFO, the amounts would be:
a. Cost of goods sold $4,400 and ending inventory $2,500
b. Cost of goods sold $4,900 and ending inventory $2,000
c. Cost of goods sold $4,600 and ending inventory $2,300
5. Using LIFO, the amounts would be:
a. Cost of goods sold $4,400 and ending inventory $2,500
b. Cost of goods sold $4,900 and ending inventory $2,000
c. Cost of goods sold $4,600 and ending inventory $2,300
6. A cost that is capitalized rather than being expensed immediately is:
a. Included in the cost of a plant asset and will be reflected in depreciation expense later
b. The cost of an addition made to an existing asset, such as a new wing to an office building
c. Significant improvements or substantial repairs/replacements made to an existing asset which are expected to extend its useful life
d. All of the above
e. None of the above
7. Equipment was purchased at an original cost of $200,000. Its useful life is estimated to be 5 years or 600,000 units, and its salvage value is estimated to be $50,000. Using the straight-line method, year 2 depreciation expense will be:
a. $30,000
b. $40,000
c. $60,000
d. $80,000
8. A company incurred $2 million of research and development costs which ultimately led to a patent on a new cancer treatment medication, and the patent was registered at a cost of $100,000. The balance sheet value of the patent will be:
a. $2,100,000
b. $2,000,000
c. $100,000
9. All but which of the following are true pertaining to corporate bonds?
a. The issuing corporation must rely on a single lender to raise the money it needs.
b. The principal or maturity value of the bonds will be repaid on a specified future date.
c. The issuing corporation promises to pay a stated amount of interest on the bonds, usually twice a year.
d. The amount the corporation receives at the time of borrowing can be greater or less than the amount it must repay at maturity to adjust for differences in the stated interest rate and the market rates of interest.
e. All of the above are true.
10. Which of the following is true regarding a partnership?
a. The partnership entity pays income tax on its profits.
b. Partners can be held personally liable for business debts.
c. Ownership is limited to individual persons only; a partnership cannot be owned by other companies or parties.
d. All of the above are true.
e. None of the above are true.
## **CHAPTER 6
Statement of Cash Flows**
In this chapter we conclude our detailed discussions of each separate financial statement by taking a closer look at the statement of cash flows. We will review the basic concepts and format of the statement of cash flows and then discuss each of the cash-flow classifications in greater detail. The chapter concludes with a comprehensive example that illustrates all four of the financial statements Case Company prepares at year end to comply with generally accepted accounting principles (GAAP). Despite the relative simplicity of this example, it will reinforce several of the basic financial reporting concepts introduced and discussed in various ways throughout the first five chapters. The Case Company example also will illuminate how information provided within each individual financial statement is related to that provided in one or more of the other financial statements.
### **Basic Format**
The statement of cash flows presents a summary of the cash flows of the business during a period of time and the change in the cash balance. This financial statement helps explain the change in the cash balance as well as changes in other assets and liabilities on the balance sheet and shows why net income differs from the cash generated or used in routine business activities. Cash flows are classified as arising from operating, investing or financing activities, and certain noncash activities also are disclosed.
Figure 6.1 presents a condensed version showing the basic form and content of the statement of cash flows for Case Company. Cash flows from each significant activity are disclosed separately under the classifications of operating, investing, and financing activities. The net increase or decrease in cash during the year is also shown, along with the resulting year-end cash balance which is also the amount reported as cash in the year-end balance sheet.
**Figure 6.1** Case Company Statement of Cash Flows
For the Year Ended December 31, 20x1
#### **OPERATING ACTIVITIES**
_Operating activities_ are those arising from primary business activities. Revenues and expenses reported in the income statement usually generate or use cash, though the period in which revenue is earned or an expense is incurred may differ from the timing of the actual receipt or payment of cash, and some revenues and expenses do not involve cash. Receipts from cash sales, collections on receivables, and dividends and interest received on loans and investments as well as cash payments to suppliers and employees and for operating expenses, interest, and income taxes are included in net operating cash flows.
#### **INVESTING ACTIVITIES**
_Investing activities_ pertain to assets acquired and used by the company, such as plant assets, natural resources, and intangibles, as well as investments. Cash flows from investing activities include cash receipts and payments arising from the purchase or sale of these types of assets.
#### **FINANCING ACTIVITIES**
A company finances its formation and growth with owners' investments and borrowed funds. Cash flows from _financing activities_ include cash receipts and cash payments arising from borrowing and repaying loans, cash invested in the business by the owners, and cash withdrawals or dividends paid to the owners.
#### **NONCASH INVESTING AND FINANCING ACTIVITIES**
Not all investing or financing activities involve a cash receipt or payment; for example, the purchase of equipment financed with a note payable is both an investing activity (equipment purchase) and a financing activity (borrowing). These activities are disclosed but do not appear in the actual statement of cash flows.
### **Case Company: A Comprehensive Example**
Case Company will be used as a comprehensive example in this chapter. Although this business is a sole proprietorship, the simplest form of business organization, the applicable accounting concepts are identical to those involving partnerships and corporations even though the specific terminology or underlying transactions may differ slightly in those cases.
### **Cash Flows From Operating Activities**
Although the income statement is viewed as the primary statement for reporting the results of operating activities, cash flows arising from these activities are also of great importance as they affect the liquidity and financial flexibility of the business. The statement of cash flows presents cash receipts and cash payments arising from the related revenues and expenses reported in the income statement, although the actual cash receipt or payment may occur in a different period.
Cash receipts from operating activities include cash sales and collections from customers on account, dividends received from investments in stock, and interest received on loans and investments. Cash payments for operating activities include cash paid to suppliers for raw materials or goods acquired for resale, payments to employees, and payments for other operating expenses, taxes, and interest on loans.
A company may need to find temporary cash resources to sustain its routine business operations in the short run. In a given year, if it is unable to generate a sufficient amount of cash sales and customer collections to cover its routine operating expenses, pay its suppliers on time and have enough left to accumulate the funds needed to pay the interest and maturity value of long-term obligations or return profits to its owners, it will face difficulties in borrowing money and/or attracting new owner investments. Operating cash flows are vital to business sustenance and success; the statement of cash flows provides valuable information about cash generated and used in ongoing operations.
#### **CASH FLOWS FROM OPERATING ACTIVITIES: DIRECT METHOD VERSUS INDIRECT METHOD**
A company can present its net cash flow from operating activities by using either a direct method or an indirect method approach in the statement of cash flows. The _direct method_ presents the specific amounts of cash received and cash paid for each significant item and the resulting net cash flow arising from operating activities. By contrast, the _indirect method_ shows only the net effect of items which caused net income and net operating cash flows to differ. We will explain and provide examples to illustrate these concepts below.
We use the condensed income statement for Case Company in Figure 6.2 to discuss basic concepts and illustrate how operating cash flows are reported using both the direct and indirect approaches.
**Figure 6.2** Case Company Income Statement (Condensed)
For the Year Ended December 31, 20x1
##### **Direct Method**
If Case Company uses the _direct method_ , net operating cash flows will be based on the specific amounts of cash received and paid for all significant (material) income statement items, as shown in Figure 6.3.
The detailed cash-flow information presented in the direct method approach is useful in assessing differences between revenues and related cash receipts and between expenses and related cash payments. The information provided can be understood readily by persons with little background in accounting and is useful to
**Figure 6.3** Direct Method: Cash Flows from Operating Activities
For the Year Ended December 31, 20x1
managers in evaluating budgeted and actual cash flows. For example, you can see that Case Company earned $800,000 sales revenue, but only $782,000 was received in cash this year. It had cost of goods sold and operating expenses of $696,000 but its cash payments for these items was $733,000 during the year. We will discuss the reasons for these differences in later examples.
##### **Indirect Method**
If Case Company uses the _indirect method_ , net operating cash flows will be determined by using the differences between net income and cash flows for particular items. For example, the difference between revenues and actual cash collections is related to the increase or decrease in customer receivables. Similarly, the difference between expenses and cash payments is related to prepayments and expenses payable. The format in Figure 6.4 is not exactly what one would see in an actual financial statement, but it illustrates the concept of the indirect method, which will be discussed and illustrated in more detail later in this chapter.
**Figure 6.4** Indirect Method: Cash Flows from Operating Activities
For the Year Ended December 31, 20x1
As you can see, both methods arrived at net cash flows from operating activities of $45,000. The direct method shows the specific amounts of actual cash receipts and payments, while the indirect method shows the differences between revenues earned and cash received and the differences between expenses incurred and actual cash payments for those expenses. Although most revenues earned and expenses incurred will also be received or paid in the same financial statement period, some cash flows may occur before the revenues are earned or the expenses incurred and some cash flows may occur in a later period.
#### **CASH FLOWS FROM OPERATING ACTIVITIES: COMPREHENSIVE EXAMPLES**
We expand our discussion of operating cash flows in the next few examples, using more specific information and the more detailed version of Case Company's income statement presented in Figure 6.5.
**Figure 6.5** Case Company Income Statement
For the Year Ended December 31, 20x1
##### **Example 1: Sales Revenue versus Cash Receipts from Customers**
Case Company started the year with $13,000 of accounts receivable from the prior year's sales, all of which were collected in cash this year. It earned $800,000 of sales revenue this year, all from credit sales. Of that amount, $769,000 was collected in cash and the remaining $31,000 is in year-end accounts receivable. A visual representation of the information is presented in Figure 6.6.
**Figure 6.6** Sales Revenue versus Cash Receipts
Figure 6.7 shows how this information might appear in the financial statements. The income statement reports the $800,000 of sales revenue earned during the year in accordance with the revenue recognition principle. In the statement of cash flows, the direct method shows the amount actually collected in cash of $782,000, which includes the beginning accounts receivable of $13,000 and $769,000 of the current year's credit sales.
**Figure 6.7** Financial Statements (Partial): Sales Revenue versus Cash Receipts
By contrast, the indirect method shows only the $18,000 difference between sales revenue ($800,000) and actual cash receipts ($782,000) as a deduction from net income due to the increase in accounts receivable. An increase in accounts receivable means that not all of this year's revenue earned from credit sales were collected in cash; thus, operating cash flows were $18,000 less than net income. If receivables had decreased, rather than increased, this amount would be added to net income signifying that cash collections were greater than the amount of sales revenue earned.
##### **Example 2: Cost of Goods Purchased versus Cash Paid to Merchandise Suppliers**
Case Company started the year with $36,000 of accounts payable from the prior year's purchases of merchandise, all of which were paid in cash this year. It purchased $525,000 this year, all on credit. Of this amount, $509,000 was paid in cash and the remainder of $16,000 is in year-end accounts payable. A visual representation of the information is presented in Figure 6.8.
**Figure 6.8** Purchases versus Cash Payments
Figure 6.9 shows how this information might appear in the financial statements. Recall that the cost of inventory purchased is not an expense in the income statement until the inventory has been sold. As will be discussed in the example to follow, the amount reported as cost of goods sold is based on not only the $525,000 of inventory purchased, but also the amounts of inventory on hand at the beginning and end of the year.
The direct method shows the cash payments to suppliers of $545,000 as an operating cash outflow; this amount includes the payment of the $36,000 accounts payable owed at the beginning of the year and $509,000 cash payments for some, but
**Figure 6.9** Financial Statements (Partial): Purchases versus Cash Payments
not all, of the inventory purchased on credit during the year. The indirect method shows only the $20,000 difference between the amount of inventory purchased on credit ($525,000) and the cash payments to suppliers ($545,000) as a deduction from net income. A decrease in accounts payable means that the company paid more cash to suppliers than the amount of inventory it purchased during the year. If accounts payable instead had increased, it would signify that cash payments to suppliers were less than the amount of credit purchases, and this difference would be added to net income.
##### **Example 3: Cost of Goods Purchased versus Cost of Goods Sold**
The previous example focused on the difference between cash payments to suppliers and inventory purchased on credit during the year, but any difference between the amount of inventory purchased versus the amount sold also must be considered. In previous chapters you learned that inventory costs are expenses when the inventory is sold and matched with the sales revenue reported in the income statement for that period, and not as expenses in the period when purchased. Continuing on with the previous example, the difference between the amount of inventory purchased versus the amount sold is also considered and illustrated below.
Case Company started the year with $40,000 of merchandise inventory which it purchased during the previous year and had $525,000 of additional credit purchases of inventory this year; thus, during the year a total of $565,000 of inventory was on hand and available for sale. Of this amount, goods which cost $500,000 were sold and the remaining $65,000 were still on hand and included in year-end inventory. A visual representation of the information is presented in Figure 6.10.
**Figure 6.10** Purchases versus Cost of Goods Sold
Figure 6.11 shows the financial statement effects for cost of goods sold and cash payments to suppliers, taking into consideration the differences between credit purchases versus cash payments to suppliers, as well as the additional effects of the difference between the amounts purchased versus the amount of inventory sold that was discussed above in Example 3 (the continued example).
The $45,000 difference between cost of goods sold of $500,000 (treated as an expense in the income statement) and the $545,000 cash paid to suppliers (treated as an operating cash outflow in the statement of cash flows) arose for two reasons. First, Case Company's cash payments to suppliers on account were $20,000 more the cost of the inventory it purchased on credit, which resulted in a $20,000 decrease in accounts payable by year-end (see Figure 6.8). Second, Case Company purchased $25,000 more inventory than it sold during the year, which resulted in a $25,000 increase in merchandise inventory by year-end (see Figure 6.10).
When reported operating cash flows are based on the direct method, the specific amount of cash paid to suppliers of $545,000 is shown. By contrast, if Case Company reports operating cash flows based on the indirect method, the adjustments to net income reveal that: 1) cash payments to suppliers exceeded the amount purchased on credit – accounts payable decreased and 2) inventory purchased on credit exceeded the amount sold – inventory increased. Figure 6.11 shows the income statement and the statement of cash flows effects for both the direct and the indirect methods.
**Figure 6.11** Financial Statements (Partial):
Cost of Goods Sold versus Cash Payments
##### **Example 4: Depreciation Expense**
Depreciation expense is a bit different from most other types of operating expenses reported in the income statement and is also handled in a different manner in the statement of cash flows. Recall that depreciation, depletion, and amortization allocate a portion of the cost of an asset as an expense during each period of use according to the matching principle and that these do not represent operating expenses that will require the payment of cash. Any cash paid at the time of purchase is considered an investing and not an operating cash flow, and if the purchase was financed with borrowed funds, the principal amounts repaid are considered financing, not operating activities. Depreciation expense is used in determining net income, but as a noncash expense it is either omitted from operating activities in the direct method approach or added to net income as a noncash item in the indirect method approach, as shown in Figure 6.12.
Concepts similar to those discussed in the examples above also apply to most other types of revenues and expenses. Visual representations and financial statement results for Case Company's other revenues and expenses are presented and briefly discussed in the examples which follow.
**Figure 6.12** Financial Statements (Partial): Depreciation Expense
##### **Example 5: Salaries and Wages Expense versus Cash Payments to Employees**
Case Company incurred salary and wage expenses of $130,000, of which $125,000 was paid in cash and the remaining $5,000 is owed to employees and included in year-end salaries and wages payable. It also paid the $4,000 of salaries it owed to employees at the end of the prior year. The $1,000 increase in salaries and wages payable represents the difference between the current year's salary and wages expense ($130,000) and the cash payments made to employees ($129,000). A visual representation of this information appears in Figure 6.13, and the financial statement results are presented in Figure 6.14.
**Figure 6.13** Salaries Expense versus Cash Payments
**Figure 6.14** Financial Statements (Partial): Salaries Expense versus Cash Payments
##### **Example 6: Supplies and Postage Expense versus Cash Payments for Supplies Purchased**
Case Company used $7,000 of supplies during the year but purchased and paid for only $6,000 of supplies; thus, its supplies inventory decreased by $1,000. It used the $2,000 of supplies on hand at the beginning of the year and $5,000 of the $6,000 of the supplies purchased and has $1,000 of supplies on hand at year-end. A visual representation of this information appears in Figure 6.15, and the financial statement results are presented in Figure 6.16.
**Figure 6.15** Supplies Expense versus Cash Payments
**Figure 6.16** Financial Statements (Partial):
Supplies Expense versus Cash Payments
##### **Example 7: Insurance Expense versus Cash Payments for Insurance**
Case Company purchased $11,000 of insurance during the year, but only $9,000 was an expense of this year; thus, its year-end prepaid insurance increased by $2,000. The prepaid insurance at the beginning of the year was used, but only $6,000 of the additional insurance purchased during the year was used, and so year-end prepaid insurance is $5,000. A visual representation of this information appears in Figure 6.17, and the financial statement results are presented in Figure 6.18.
**Figure 6.17** Insurance Expense versus Cash Payments
**Figure 6.18** Financial Statements (Partial):
Insurance Expense versus Cash Payments
##### **Example 8: Property Tax Expense versus Cash Payments for Property Taxes**
Case Company incurred property tax expense of $5,000, paid $4,000 of this amount in cash and the remaining $1,000 is in year-end property taxes payable. Property tax payments totaled $7,000, which included the payment of $4,000 of the current year's property tax expense and the $3,000 it owed from the prior year. The $2,000 decrease in property tax payable signifies that the cash payments ($7,000) were greater than the current year's property tax expense ($5,000). A visual representation of this information appears in Figure 6.19, and the financial statement results are presented in Figure 6.20.
**Figure 6.19** Property Tax Expense versus Cash Payments
**Figure 6.20** Financial Statements (Partial):
Property Tax Expense versus Cash Payments
##### **Example 9: Revenues Earned and Received and Expenses Incurred and Paid in the Same Period**
Other revenues and expenses in Case Company's income statement were received or paid in cash during the same year, such as the $1,000 of interest income, the $5,000 of interest expense, and the $20,000 for advertising and $15,000 for telephone and utility expenses. The specific amounts of cash received or paid for these revenues and expenses are presented as operating cash flows when using the direct method approach. If operating cash flows are reported using the indirect method approach, there is no need to show adjustments to net income because the amounts reported as revenues and expenses under the accrual basis of accounting are identical to the amounts of the cash received or paid during the same time period.
##### **Example 10: Nonoperating Gains and Losses**
In addition to the net income and net operating cash flow differences discussed and illustrated earlier, there is one more concept you may encounter when reviewing a company's statement of cash flows: the treatment of _nonoperating gains and losses_. Although Case Company did report interest income and interest expense separately from the revenues and expenses arising from its routine and ongoing operations, it did not have any _gains_ or _losses_ to report this year, and so a different and unrelated example will be used to explain how nonoperating gains and losses are treated in the statement of cash flows.
As was discussed in earlier chapters, when a company sells a plant asset or an investment it owns, the difference between the selling price and the book value of the asset or investment is reported as a gain or as a loss in its income statement. As you learned earlier in this chapter on the statement of cash flows, purchases and sales of assets used by the business or investments it owns are considered to be investing activities, and not operating activities; thus, the cash received from the sale of such an asset or investment is classified as an investing activity cash flow and does not appear in the operating cash flows when the direct method approach is used. However, when the indirect method approach is used to report operating cash flows, any nonoperating gains that are included in the amount of net income reported in the income statement should be deducted, and any nonoperating losses should be added to net income. The cash received from the sale of an asset or investment, whether sold at a gain or at a loss, is classified as an investing cash inflow.
For example, assume both accounting income and operating cash flows were $100,000, excluding any effects from the sale of equipment. If equipment with a book value of $20,000 was sold for $25,000, a $5,000 gain from the sale would be included in the amount of net income reported in the income statement. In the statement of cash flows, the sale of equipment is considered an investing activity, and the $25,000 cash proceeds from the sale should be reported as an investing cash inflow. If the indirect method is used to report operating cash flows, the $5,000 nonoperating gain must be deducted from net income as it did not produce an operating cash flow. The financial statements would include the information shown in Figure 6.21. Notice that the net _operating_ cash flow of $100,000 was not affected by the equipment sale; the $25,000 cash received from the equipment sale was an _investing_ activity cash flow, not an operating cash flow.
**Figure 6.21** Financial Statements (Partial): Sale of Equipment and Gain
If instead the equipment had been sold for $18,000, a $2,000 loss from the sale would have been deducted in arriving at the amount of net income reported in the income statement. Similar to the previous example, the $18,000 cash proceeds from the sale should be reported in the statement of cash flows as an investing activity. If the indirect method is used to report operating cash flows, the $2,000 nonoperating loss which reduced net income but did not result in a cash outflow must be added back to net income to arrive at the proper amount of operating cash flows. The financial statements would include the information shown in Figure 6.22. As in the previous example, the net _operating_ cash flows of $100,000 has not changed; the equipment sale generated an $18,000 _investing_ activity cash flow and did not affect operating cash flows.
**Figure 6.22** Financial Statements (Partial): Sale of Equipment and Loss
### **Case Company Operating Cash Flows**
The examples above illustrated and explained why the amount of a specific revenue or expense that was reported in the income statement may differ from the amount of the related cash receipt or cash payment during the same time period. Let's take a look at the operating activity section of Case Company's statement of cash flows which incorporates the specific revenue and expense examples that were discussed separately above. Figure 6.23 shows net cash flows from operating activities using the _direct method_ approach. Figure 6.24 also presents net operating cash flows, but as they would be reported using the _indirect method_ approach.
**Figure 6.23** Direct Method: Operating Cash Flows
**Figure 6.24** Indirect Method - Operating Cash Flows
### **Which Method to Use: Direct or Indirect?**
The accounting profession encourages the use of the direct method, with additional disclosure using the indirect method in a supplemental schedule that is referred to as the _schedule reconciling net income to net cash flows from operating activities_. However, it is permissible to use only the indirect method in the statement of cash flows. In fact, and perhaps unfortunately, the indirect method is used more widely in published financial statements. Although the indirect method does make sense and provides valuable information, it is not as straightforward or easily understood as the direct method.
### **Accrual Basis Versus Cash Basis**
In summary, when you compare the income statement results with the operating activities section of the statement of cash flows, there are some similarities and some differences. The income statement and the operating activities section of the statement of cash flows are similar in that both present information about the primary business activities: the income earned and cash flows from providing services or selling goods.
However, the income statement and the cash flow statement differ in that the income statement shows revenues earned, expenses incurred, and the resulting net income or loss without regard to cash flows, using the accrual basis concept. Preparing the operating activities section of the statement of cash flows is much like preparing the income statement, but using the cash basis of accounting. Although cash basis information is important and valuable, an income statement prepared on the cash basis generally is not considered the best measure of a company's operating performance.
In the next section, we will discuss investing and financing activities and the way they are reported in the statement of cash flows, as well as the concept of cash equivalents.
### **Investing Activities**
_Investing activities_ may be viewed as transactions involving the assets of the business; they generally consist of the purchase and sale of investments and assets used by the business. Investing activities also may include cash loans and collections of the principal amounts of those loans, although the dividends and interest collected on loans and investments are classified as operating activities. The investing activities section of a company's statement of cash flows may include the items shown in Figure 6.25.
**Figure 6.25** Statement of Cash Flows Reporting: Investing Activities
Note that net cash flows from investing activities is a negative amount; in the example above, the company used more cash for investing activities than it took in. It might have financed the purchase of additional investments and other assets internally, using excess cash generated from its routine operating activities. However, it might have been necessary to find and use other resources to finance the acquisition of new plant assets and other investments.
### **Financing Activities**
_Financing activities_ may be viewed as transactions involving the parties which provide resources to the business: the creditors, and the owners. Transactions with creditors include borrowing and repaying the principal amount of loans, although the interest on loans is treated as an operating activity. Transactions with owners include investments by the owners, distributions to the owners such as withdrawals and cash dividends, the purchase of treasury stock, and the retirement of outstanding stock. The financing activities section of a company's statement of cash flows may include the items shown in Figure 6.26.
It was mentioned in the earlier discussion of investing activities that a company can finance its growth internally or through external means. In the short run, a company may finance expansion and growth or even overcome liquidity problems using capital provided by owners (sole proprietor, partners, or stockholders) or using borrowed funds (bank loans, mortgage loans, bonds payable). However, if a company is to be viable in the long run it ultimately must be able to repay the amounts
**Figure 6.26** Statement of Cash Flows Reporting: Financing Activities
borrowed, distribute dividends, and/or increase the market price of its stock to reward its investors. If the company cannot sustain itself through profits and cash flows it should be able to generate from its ongoing and routine business activities, it will be difficult or perhaps impossible, to attract new capital investments or to find additional lending sources.
On the basis of the examples of investing and financing activities illustrated in Figures 6.25 and 6.26, in the short run, it appears that this company has financed a significant portion of its plant asset expansion by issuing bonds payable and by issuing (selling) additional common stock. Will it be able to repay the interest and principal amounts it has borrowed? Will it be able to distribute some of its profits to stockholders via cash dividends and/or generate a higher market price for its stock and reward its investors in this way? A review of a company's statement of cash flows can provide some important hints or clues, as would a review of any of the other financial statements. However, key questions are more likely to be answered by analyzing the financial statements and footnote disclosures in their entirety, by taking a look at how things are developing for a company over time, and how a given company compares to a competitor. The additional topics covered in Chapters 7 to 10 will provide you with some additional tools and further insight that might help you in finding answers to some of your important questions about a company's health.
### **Gross Versus Net Cash Flows**
Cash flows from investing and financing activities generally are based on the amounts of cash received or paid for significant items; this is more relevant and useful information than a simple report of the net amount of cash received or paid for each item. For example, if $10,000 was paid to purchase equipment and other equipment was sold for $2,000, under investing activity cash flows, both the cash payment of $10,000 and the cash receipt of $2,000 would be presented, rather than just the net cash paid of $8,000. Similarly, if $60,000 was borrowed and $5,000 was repaid, both the borrowing and the repayment would be shown rather than the net cash receipt of $55,000.
There is an exception which permits net reporting for certain types of borrowing activities if the turnover is quick, the amounts are large, and the maturities are three months or less. For example, assume your banker permits you to borrow to meet seasonal cash requirements with a series of one-month loans. During the course of the year, you take several such loans for a total of $100,000 and repay a total of $95,000. In this case, it is permissible to report only the net amount of cash borrowings of $5,000 because the turnover of the loans was quick and the amounts of the loans were relatively large.
### **Noncash Investing and Financing Activities**
The statement of cash flows reports only those activities which result in receipts or payments of cash. However, a business may engage in investing or financing activities which do not directly result in cash receipts or cash payments; information about such activities is relevant and useful because it can help the financial statement user to predict future cash flows pertaining to these items. Consequently, the accounting profession requires disclosure of significant noncash investing and financing activities. Noncash investing and financing activities include the purchase of assets financed with a note or mortgage payable, acquiring the long-term rights to the use of an asset via a capital lease agreement, repayment of debt with noncash assets, and settling a debt by issuing stock instead of a cash repayment. For example, a company's financial statements or footnotes disclosures for noncash investing and financing activities may include the items shown in Figure 6.27.
Let's take a look at the investing and financing activities Case Company presents in its statement of cash flows and its supplemental disclosures, as shown in Figure 6.28.
As you can see, Case Company's only investing activity was the purchase of equipment, for which it used $1,000 in cash; the remainder was financed through
**Figure 6.27** Schedule of Noncash Investing and Financing Activities
**Figure 6.28** Investing and Financing Activities
borrowing, specifically, a $4,000 note payable. The financing activities included $10,000 of additional cash invested by the owner, $50,000 of cash withdrawn by the owner, and $5,000 of cash paid on the principal balance of the long-term mortgage note payable (the interest paid was an operating activity). The cash flow information combined with the noncash disclosures can assist you in determining the cause of changes in asset, liability, and equity accounts which may be explained only partially by the other financial statements.
### **Cash and Cash Equivalents**
The final section of the statement of cash flows shows the change in cash and cash equivalents during the year, and the amount of cash and cash equivalents at year-end. Cash consists of cash on hand or on deposit with banks or other financial institutions. A business may invest its idle cash in a variety of ways to generate earnings until the cash is needed for business activities. Cash equivalents, as discussed in Chapter 5, are a special category of temporary investments of cash in highly liquid, readily marketable securities that may be converted into known amounts of cash. Cash equivalents generally consist of investments in U.S. Treasury bills, money market funds, and commercial paper, if such investments are purchased within three months of their original maturity date.
In the balance sheet, cash and cash equivalents are combined and shown as a single amount, the change in the total amount of cash and cash equivalents is shown, and the policies used to determine cash equivalents should be disclosed. The change in cash and cash equivalents and accounting policy disclosure for Case Company appear in Figure 6.29.
**Figure 6.29** Change in Cash and Cash Equivalents
Even though cash equivalents are comprised of specific types of investments, purchases and sales of cash equivalents are not reported because they are considered to be much like actual cash on hand. For example, assume a business has $4,000 in cash. If it invests $1,000 in a money market fund and has $3,000 in cash, total cash and cash equivalents has not changed; it is still $4,000. If it sells $200 of the investment in the money market fund, it still has $4,000 in cash and cash equivalents; $3,200 cash and $800 of cash equivalents. However, the interest and dividends received from investments in cash equivalents are reported in the statement of cash flows under operating activities. The $1,000 of interest income Case Company reported was from its investments in cash equivalents; it had no other investments during the year.
### **Pulling It All Together**
As we near the conclusion of this chapter, we provide two additional figures which will help you understand how the information presented in each individual financial statement relates to one or more of the other statements. Figure 6.30 focuses on the differences between the revenues and expenses as reported in the income statement and the cash receipts and cash payments reported as operating activities in the statement of cash flows. Figure 6.31 focuses on the comparative balance sheet and shows the amount of the change for each asset, liability and total owner's equity
**Figure 6.30**
**Figure 6.31**
In Figure 6.30, the revenue (expense) column shows the amounts reported in the income statement, resulting in accrual basis net income of $100,000. The cash receipts (payments) column shows the amounts reported as operating cash flows using the direct method, resulting in a net operating cash inflow of $45,000, which can be viewed as cash basis net income. Notice that net operating cash flows were $55,000 less than accrual basis net income. The differences between individual revenue/expense and cash receipt/payment amount s represent the adjustments to net income that are shown when reporting net operating cash flows using the indirect method approach.
Figure 6.31 focuses on the comparative balance sheet and shows the amount of the change; it also provides a brief explanation for each item. As you will notice, the changes in each item were reported in or explained by one or more of the other financial statements. The balance sheet shows the financial position at a point in time. The transactions and events that took place during the year were reported in the income statement, the statement of owner's equity, and/or the statement of cash flows and produced the amounts reported in the year-end balance sheet.
We conclude this chapter by presenting the complete set of financial statements Case Company might include in its application for a bank loan or in a year-end annual report to stockholders if Case Company was a corporation instead of a sole proprietorship. In this example, we assume that Case Company has elected to use the direct method to report its operating cash flows and to present the indirect method and its noncash investing and financing activities in separate schedules or footnote disclosures. Of course, there would be many additional disclosures for other financial statement items included in the accompanying financial statement footnotes, along with the information provided in this example. This complete set of financial statements for Case Company is shown in Figure 6.32.
### **Summary**
We introduced you to basic financial accounting and reporting concepts in Chapters 1 and and then discussed and illustrated each financial statement in much more detail and made frequent reference to how the basic financial accounting and reporting concepts applied in each case. The income statement was discussed in Chapter 3, the balance sheet and the statement of owners' equity were discussed in Chapters 4 and , and we concluded with a discussion of the statement of cash flows in this chapter.
Building on this foundation, in the chapters which follow we will discuss some additional financial reporting issues and the information found in a company's annual report, including the report of the independent auditor, and will introduce you to financial statement analysis tools and techniques. We will conclude the book with a discussion of fraudulently misstated financial statements.
**Figure 6.32** Case Company - Financial Statements
### **Quiz for Chapter 6**
1. Cash flows from investing activities include all of the following except:
a. Purchases of plant assets
b. Dividends and interest earned on investments
c. Sales of investments
d. Collection of the principal amount of a note receivable
e. All of the above are investing activities.
2. Cash flows from financing activities include all of the following except:
a. Issuing common stock
b. Cash dividends paid to stockholders
c. Repayment of bonds payable at maturity
d. Payment of interest on bonds and notes payable
e. All of the above are financing activities.
3. Which of the following is true regarding operating cash flows?
a. Cash sales and customer collections on accounts receivable are included in the amount reported as operating cash flows.
b. Specific amounts of cash receipts and cash payments will be reported as operating activities using the direct method.
c. If cash received from customers differs from the amount of sales revenue reported in the income statement, the difference will be reported as an adjustment to net income using the indirect method.
d. All of the above.
e. None of the above.
4. Sales revenue for the year was $500,000. Beginning accounts receivable was $10,000, and ending accounts receivable was $15,000. Cash receipts from customers were:
a. $500,000
b. $495,000
c. $505,000
5. Salaries expense for the year was $100,000. Beginning salaries payable was $10,000, and ending salaries payable was $15,000. Cash payments for salaries were:
a. $100,000
b. $ 95,000
c. $105,000
6. Rent expense for the year was $40,000. Beginning prepaid rent was $10,000, and ending prepaid rent was $15,000. Cash payments for rent were:
a. $40,000
b. $35,000
c. $45,000
7. Which of the following is true regarding the reporting of depreciation, depletion, and amortization costs?
a. These amounts would be ignored in preparing the operating cash flow section using the direct method, as they are noncash expenses.
b. These amounts would be added to net income in preparing the operating cash flow section using the indirect method, as these expenses reduced net income but did not have an associated cash outflow.
c. Both of the above are true.
d. Neither of the above is true.
8. Which of the following is an example of a noncash investing and/or financing activity?
a. Issuing bonds payable to finance the purchase of land and buildings
b. Retiring long-term debt by issuing common stock
c. Trading in an old delivery vehicle for a new model
d. All of the above
e. None of the above
9. Which of the following is true pertaining to cash equivalents?
a. Cash equivalents are a particular category of highly liquid, readily marketable investments.
b. Cash and cash equivalents often are combined and presented as a single amount on the balance sheet.
c. The statement of cash flows should include the change in the total amount of cash and cash equivalents from the beginning to the end of the year.
d. All of the above are true.
e. None of the above are true.
10. The term _accrual basis of accounting_ means that the income statement covering a particular period will include the revenues earned and expenses incurred during that period regardless of the period in which the company received cash from its customers or paid cash for operating expenses.
a. True
b. False
## **CHAPTER 7
Reading the Financial Statements:**
The Auditors' Reports and Financial Statement Footnotes
In the previous chapters, we discussed the four basic financial statements in detail, explaining common terminology and concepts and exploring their interrelationships. In this chapter, we discuss ways to read the set of financial statements with a focus on understanding the auditors' opinions and the financial statement footnotes.
### **Annual Reports and Form 10-Ks**
All publicly traded companies in the United States are required under federal securities laws to publish financial reports every year and submit them to the Securities and Exchange Commission (SEC). Privately held companies are not subject to this requirement, since the public cannot purchase shares of their stock. Consequently, the general public has no need to know the financial details of these companies.
The annual report required by the SEC is referred to as a Form 10-K (a Form 10-Q is submitted for quarterly financial information), and it contains a large amount of detailed information about a corporation. It is not unusual for a corporation's Form 10-K to exceed 100 pages. Anyone can access these filings on the SEC's Web site (www.sec.gov) or the individual corporation's Web site (usually under a tab titled "Investors," "Investor Relations," or something similar).
The 10-K includes the financial statements of the public company, along with additional required information. However, companies do not have the flexibility to choose what type of information they will submit to the SEC or where in the Form 10-K they will provide information. The SEC wants to enhance the comparability of financial information from company to company and dictates what must be included in Form 10-K. Thus, the type of information provided by a company will be similar in content and location from one Form 10-K to another, even among a variety of industries.
For example, let's look at the Form 10-Ks of the following four companies, which are engaged in different types of business operations:
• Target Corporation, which operates large-format general merchandise and food discount stores, for the fiscal year ended February 3, 2007
• Cabela's Incorporated, a large direct marketer and leading specialty retailer of fishing, hunting, camping, and related outdoor merchandise, for the fiscal year ended December 30, 2006
• Coca-Cola Enterprises, Inc., the world's largest Coca-Cola bottler, which markets, sells, manufactures, and distributes a variety of nonalcoholic beverages, for the fiscal year ended December 31, 2006
• Kimberly-Clark Corporation, which principally manufactures and markets a wide range of health and hygiene products, for the fiscal year ended December 31, 2006
You'll find that their Form 10-K tables of contents are identical, as shown in Figure 7.1.
Although a Form 10-K is the annual report a public company is required to file with the SEC, it is not the same as the annual report sent to the shareholders. Shareholders' annual reports often give the appearance of a magazine, with glossy paper, brightly colored photographs and illustrations, and a lot of marketing-type information about the company. Financial statements are included in these annual reports but often appear toward the back
**Figure 7.1** Form 10-K Table of Contents for Target Corporation, Cabela's Incorporated, Coca-Cola Enterprises, Inc., and Kimberly-Clark Corporation
of the reports on nonglossy paper, and there typically isn't as much financial data as there is in a Form 10-K. However, some companies reproduce the Form 10-K in its entirety as part of the shareholders' annual report.
### **Audited versus Nonaudited Financial Statements**
The annual financial statements submitted by public companies to the SEC are required to be audited. Private companies are not required by legislation to have their financial statements audited; however, if a private company is applying for a loan, it is likely that the bank will require audited financial statements before approving a loan. Otherwise, the bank will have to take the risk of accepting the financial statements as management has prepared them.
A company will include the report of its independent auditors on the financial statements in both the Form 10-K and the shareholders' annual report. The audit report often is referred to as the _auditors' opinion_ , because the auditors are providing an opinion about whether the financial statements are presented fairly in all material respects, based on the evidence gathered during the audit process.
Why is the audit report important? Essentially, management is responsible for designing an accounting information system that will capture all of its business transactions and compile the financial statements. However, anyone outside the company, such as a current or potential investor or creditor, may be concerned about whether management has put the financial statements together without bias. Could management have intentionally or even unknowingly presented some events more optimistically than is warranted? Absolutely. Worse, could management have intentionally included some fictitious revenues, omitted some actual expenses, or performed some other accounting trick so that the company's financial results would appear better than in fact they are? Yes, if management is so inclined. The independent auditors' report is designed to alleviate that concern and give the public assurance that the financial statements can be relied on. _Assurance_ is a key word: In fact, it is common in the accounting profession to refer to auditing as an assurance service or attest service because the auditors are providing assurance or attesting (vouching for) to the fairness of the presentation of the information on the financial statements.
It is important to understand exactly how much assurance the auditors are providing in regard to the financial statements and to understand what their opinion means. For quite some time there has been a disconnect between what the accounting profession requires of auditors and what many in the general public believe auditors do. This misunderstanding has been referred to as the expectations gap. We'll attempt to narrow this gap in the next few paragraphs by discussing a few key auditing concepts.
### **Independent Auditors**
The purpose of the audit report is to give the public confidence that it can rely on the financial statements prepared by management. However, what assurance does the public have that it can rely on what the auditors have to say? To enhance the public's confidence in the audit report, auditors are required to be _independent_ with respect to the client (the company whose financial statements are being audited). Independence is a critical aspect of auditing. It is so important that if for some reason a public accounting firm discovers during an audit that the firm is not independent with respect to its client, it is required to stop auditing and issue a very short (one sentence) report stating that the public accounting firm is not independent and therefore it cannot and is not issuing any kind of opinion on the financial statements.
To enhance the public's confidence in its work, the audit report is signed with the name of the public accounting firm that performed it, not with the names of the individual auditors on the engagement team. In other words, the entire firm is standing behind their opinion.
To underscore its independence, the public accounting firm is required to include the word _independent_ in the title of its audit report. The rules for determining whether a firm is or is not independent are fairly complex but are in place essentially to help ensure that auditors are able to conduct their audits in an objective, unbiased, and impartial manner. In fact, the auditing profession defines independence as the ability to conduct an audit in an unbiased and objective manner. This definition does not imply that auditors automatically assume that management is dishonest, but they also should not assume that management is honest; professional skepticism is to be exercised during the conduct of an audit.
To help ensure independence, there are rules forbidding an auditor's direct financial interest of any amount in a client (even owning one share of stock in a client is prohibited), as well as rules pertaining to the type of employment that an auditor's dependents and other relatives may have with a client. The accounting profession traditionally has been concerned with auditors being not only independent in fact—truly independent with respect to their clients—but also independent in appearance because perception is as important as reality. Thus, some of the independence rules may appear harsh, but the auditing profession believes the public's confidence in their work is worth it.
One factor that complicates the issue of independence is the fact that the audit fee is paid by the client to the auditors. Certainly, if the client becomes unhappy with the auditors for any reason, it can terminate the relationship and hire another public accounting firm to conduct the audit. Critics wonder how independent auditors truly can be when the client writes the check. This issue became even more controversial in the late 1990s, when many public accounting firms provided consulting services to their audit clients for large additional fees. For example, the public accounting firm that audited Enron earned millions more a year in consulting revenue than in audit fees. Many people were concerned that this type of relationship could cause auditors to agree with any questionable financial reporting on the part of their audit clients for fear of losing not only the audit fee but also the substantial consulting revenue. Primarily as a result of the numerous widely publicized corporate scandals of the late 1990s and early 2000s, Congress enacted the Sarbanes-Oxley Act of 2002, which greatly limited the types of nonaudit services public accounting firms can offer to their audit clients.
### **Reasonable Assurance versus Absolute Assurance**
An audit report provides _reasonable assurance_ that the financial statements are free of material misstatements, not absolute assurance. In fact the term _reasonable assurance_ must be used in the audit report. However, reasonable assurance is never defined precisely in the professional auditing standards. Auditors understand that the term refers to the fact that audits cannot provide 100 percent guarantees that there are no material misstatements in the financial statements, although this is a concept that the public frequently misunderstands. When auditors have obtained reasonable assurance is a matter left to the auditors' judgment.
Why do auditors provide reasonable assurance as opposed to absolute assurance? It is impossible for auditors to audit everything to provide such a guarantee. Even if they audited every single transaction throughout the year, which would be impossible in terms of time and cost, auditors could not know definitely whether there were unrecorded transactions that remained undetected. Thus, it is economically not feasible and impractical to expect auditors to look at everything: they cannot provide absolute assurance about financial statements.
### **Material Misstatements in Financial Statements**
For all practical purposes, auditors cannot be held responsible for finding every misstatement of any amount in the financial statements. Would it really matter to most financial statement readers if the company's inventory was overstated by $1? $10? $100? At some point, it will matter because the misstatement could cause reasonable financial statement users to make decisions different from what they would have decided if the misstatement had not existed. It is at that point that misstatements are considered to be material.
Materiality varies from company to company. For example, overstating inventory by $10,000 may be material for a small business owner. However, when Wal-Mart presents its financial statements in millions of dollars, the $33,685 inventory amount appearing on its January 31, 2007, balance sheet is actually $33,685,000,000 (or nearly $34 billion), a $10,000 overstatement of inventory surely is immaterial.
Because materiality will vary from company to company and even from year to year for a specific company, it is not defined in terms of dollars or percentages throughout the professional auditing standards. Once again, like reasonable assurance, materiality is a decision left to the auditors' judgment. Also, like reasonable assurance, the word _material_ is required by professional auditing standards to be in the auditors' report.
### **The Standard Audit Report**
With these basic auditing concepts in mind, let's look at a standard audit report. For our example, we'll use Wal-Mart again (see Figure 7.2).
**Figure 7.2** Ernst & Young LLP's Audit Report for Wal-Mart Stores, Inc., Financial Statements as of January 31, 2007
The audit report contains quite a bit of information. The professional auditing standards require fairly uniform wording, however, so that financial statement readers can assess the type of opinion being given without getting bogged down in variations in terminology and phrasing. Note in Figure 7.2 that the terms we discussed above— _independent, reasonable assurance_ , and _material_ —are all present. Note also that the audit opinion is signed with the public accounting firm's name, not the names of the individual auditors who worked on the engagement, because the entire firm is standing behind the opinion.
The opinion shown in Figure 7.2 is called an _unqualified audit opinion_ , and it is the best opinion the auditors can give. Often, this is referred to as a _clean opinion_. What the auditors are saying is that the financial statements, in their opinion, follow generally accepted accounting principles (GAAP) in all material respects. The auditors are giving this opinion without any qualifications. That is why the opinion is given the name _unqualified_. A qualification is a discovery by the auditors of some kind of problem; we'll explain that shortly.
To determine whether a company received this best opinion, look for the paragraph that begins with "In our opinion..." and determine whether the auditors are stating that without any exceptions the financial statements "present fairly, in all material respects" (see Figure 7.3).
**Figure 7.3** Different Types of Audit Opinions
The vast majority of audits of public companies result in this best opinion. If the auditors have found a significant misstatement in the financial statements, they will discuss it with management and the company will have an opportunity to adjust the books to the auditors' satisfaction. In almost all cases, management will opt to do this so that it can acquire the best opinion. Why does management strongly prefer to receive an unqualified opinion? Because anything less can affect the reputation of the company adversely and consequently decrease the company's stock price.
The Wal-Mart audit opinion contains the opinion paragraph as the third paragraph. Sometimes public accounting firms begin the report with the opinion because of its importance. However, the key phrases you are looking for do not vary regardless of the placement of the opinion paragraph.
### **Modifications to the Standard Audit Report**
In a few circumstances, auditors may give an unqualified opinion but add an extra paragraph to the report. For example, auditors are required to add an extra paragraph after the opinion paragraph if the company did not apply the same generally accepted accounting principles consistently from last year to this year. Also, auditors must add an extra paragraph after the opinion paragraph if they have serious doubt about the company's ability to continue as a going concern for a period of up to one year past the balance sheet date. Auditors also are allowed to add an extra paragraph to emphasize any matter they wish, but that choice is always at the discretion of the auditor.
### **Consistency**
Auditing standards require the addition of a _consistency_ paragraph to indicate to the readers that if the financial statements are different this year (for example, either a higher or lower net income than last year), it could be because the use of accounting principles differed between the two years. In other words, the financial statements may not be directly comparable. However, this does not imply that the company is engaged in anything improper. Companies are allowed to switch from one generally accepted accounting principle (GAAP) to another as long as the change, its effects, and management's justification for the change are disclosed. Further, when the accounting profession issues a new standard that changes the way certain types of transactions previously were accounted for and requires that all companies follow the new standard, consistency comes into play and the auditors must include the additional paragraph again.
For example, take another look at the consistency paragraph in Wal-Mart's audit opinion:
As discussed in Note 13 to the consolidated financial statements, effective January 31, 2007, the Company adopted Statement of Financial Accounting Standards No. 158, _Employers' Accounting for Defined Benefit Pension and Other Postretirement Plans._
The auditors added this paragraph to the standard unqualified audit opinion because they are required to do so under auditing standards. Wal-Mart's management properly adopted this new GAAP and properly disclosed it (in Note 13 to the financial statements). If it had not done that and the effects were material, the auditors would have had to give a qualified or possibly an adverse opinion on the financial statements.
Because the accounting profession frequently issues new accounting pronouncements, auditors often include consistency paragraphs in their reports.
### **Going Concern**
Recall that in Chapter 2 we discussed the going concern assumption and explained that accounting methods are based on the assumption that the business will remain in existence indefinitely unless there is evidence to the contrary. Auditors cannot predict the future any better than the average person can, but they are required under auditing standards to add a _going concern paragraph_ to an audit opinion if they uncovered evidence during the audit that gives them "substantial doubt" about the company's ability to continue as a going concern for a period of up to one year beyond the balance sheet date. The content of the paragraph is up to the auditors' judgment, but the two terms _substantial doubt_ and _going concern_ must be included.
The opinion on the historical financial statements still can be unqualified (the best opinion) because the financial statements relate to the past and the opinion states whether they are presented in conformity with GAAP. The auditor will refer the financial statement readers to the financial statement footnote (prepared by management) that discusses the circumstances that have given rise to the going concern issue as well as its plans to mitigate those circumstances. However, the absence of a going concern paragraph is never a guarantee from the auditors that the company will be in existence for a year beyond the balance sheet date.
### **Emphasis of a Matter**
Auditors have the option to add an extra paragraph after the opinion to stress any matter they believe should be emphasized. The professional auditing standards provide only a handful of examples of cases in which auditors may choose to exercise this option, for example, when the company has experienced an unusually important event subsequent to the date of the financial statements. (This subsequent event would have to have been accounted for adequately and/or disclosed by management; otherwise, GAAP has not been followed and the auditors' opinion probably will be affected.)
Thus, it is important to read the entire auditors' opinion, not just the key phrases in the opinion sentence.
### **Qualified Opinion**
When the auditors give a _qualified opinion_ , the words _except for_ will be included in the opinion sentence. The auditor is stating that except for the effects of either not following GAAP or not being able to perform a key auditing procedure, the financial statements are presented fairly in all material respects. The auditors will include an extra paragraph—termed an _explanatory paragraph_ —before the opinion paragraph to explain to the readers why they believe the financial statements have not been presented completely fairly.
A qualified opinion results from either a material (significant) departure from GAAP or a material limitation in the auditors' ability to conduct the audit in accordance with professional auditing standards. For example, auditors generally are required to observe the client's physical inventory count. If a company conducts its inventory count at the end of its fiscal year, for example, on December 31, and for some reason doesn't hire the auditors until later, say, mid-January, it is not possible for the auditors to observe the count and comply with the auditing standards. If the auditors cannot satisfy themselves about the ending inventory balance (assuming it is material) through any alternative auditing procedures, there is a limitation on the scope of their audit and they must qualify their opinion.
Is a qualified opinion useful? Yes, but it's not as good as an unqualified opinion, and so management generally prefers not to receive a qualified opinion. The auditors are stating that except for the item they have explained, the financial statements have been presented fairly in all material respects. It's up to the readers to decide whether the qualification is important to them and whether they can use the financial statements for their decision-making purposes.
### **Adverse Opinion**
An _adverse opinion_ , as the name implies, is the worst audit opinion that can be given. The auditors are stating that the financial statements are not presented fairly in all material respects, not even a portion of them. This is a strong statement coming from auditors, and it essentially means that the auditors are telling the readers that they cannot rely on the financial statements for their decision-making purposes. Adverse opinions rarely are given because management does not want them for obvious reasons and will work closely with the auditors to make any adjustments the auditors consider necessary to avoid an adverse opinion.
As with a qualified opinion, if the auditors are giving an adverse opinion, there will be at least one explanatory paragraph before the opinion paragraph explaining to the readers why the auditors have come to this conclusion.
### **Disclaimer of Opinion**
A _disclaimer of opinion_ is no opinion. Disclaimers are given, as explained earlier in this chapter, when the auditors discover that they are not independent with respect to the client. This discovery should happen rarely because before the acceptance of the audit engagement, the auditors should have investigated and determined the status of their independence diligently.
A disclaimer also is given when there is a highly material (very significant) limitation on the scope of the audit, in other words, when the auditors are unable for some reason to perform some essential audit procedure or procedures and are unable to satisfy themselves about the account balance or balances through any alternative auditing procedures. A disclaimer does not result from the auditors' knowledge that the financial statements are materially misstated; instead, the auditors are unable to form an opinion because of a lack of knowledge that results from their inability to perform one or more essential audit procedures.
### **Management's Responsibilities versus
Auditors' Responsibilities**
The audit report makes it clear that the financial statements, including the footnotes, are management's responsibility. The auditors are not responsible for preparing the financial statements or for any aspect of capturing the accounting data throughout the year that eventually generate the account balances reported on the financial statements. Instead, the auditors' responsibility is to express an opinion that is based on the audit work.
### **Auditors' Report on Internal Control Over
Financial Reporting**
Before the effective date of the Sarbanes-Oxley Act of 2002 (SOX), audit reports were somewhat shorter than they are today. In response to the numerous corporate scandals that occurred shortly before the act was passed (e.g., Enron), Section 404 of SOX requires that auditors also provide an opinion on the effectiveness of public companies' internal control over financial reporting. The logic behind this requirement is essentially that if a public company has effective internal controls over financial reporting, there is a reduced risk of fraudulently misstated financial statements being published and relied on by the investing public.
The internal control report can be separate or can be combined with the audit opinion on the financial statements. The Wal-Mart example provided in Figure 7.2 is a separate report; note that the auditors refer to their separate report on internal controls over financial reporting. The actual internal control report is on the page that follows the audit opinion in Wal-Mart's annual report.
Internal controls are processes and procedures that management puts in place to help ensure the accuracy of financial reporting as well as to safeguard assets and records and to help ensure compliance with all applicable laws and regulations. Controls are designed to help prevent, detect, and correct any errors (or fraud) that might occur. An example of an internal control is that somebody who has no cash handling or recording responsibilities should reconcile the monthly bank statement with the records to help ensure that there are no mistakes in recording cash. If this responsibility was assigned to someone who also handled cash and recorded cash transactions, it could be a simple matter to steal some cash and then cover up the theft by juggling the books and fudging the bank reconciliation.
Auditors of public companies now are required to perform enough additional audit work to allow them to express an opinion on the effectiveness of the company's internal controls over financial reporting as of the fiscal year end. Management is required to assess the effectiveness of those controls and report their assessment. For the first several years after passage of the SOX Act, auditors also were required to express an opinion on management's assessment.
Thus, if the auditors issued a combined report, there were three opinions stated throughout the report:
• An opinion on whether the financial statements are presented fairly in all material respects, in accordance with GAAP
• An opinion on whether management's assessment of the effectiveness of the company's internal controls over financial reporting as of the fiscal year end is fairly stated in all material respects, based on criteria related to internal controls that were developed by the accounting profession
• An opinion on whether the company maintained in all material respects effective internal control over financial reporting as of the fiscal year end, based on the same criteria used for management's assessment
As of late 2007, however, the Public Company Accounting Oversight Board eliminated the requirement for auditors to give the second opinion listed above (the opinion on whether management's internal control assessment was fairly stated). Management still must give its assessment, and the auditors still give their own opinion on the effectiveness of the internal controls (as well as on the financial statements themselves), but the additional opinion by the auditors on management's internal control assessment no longer is considered necessary.
However, the phrases we have discussed: _present fairly, except for,_ or _do not present fairly_ (see Figure 7.3) still will be present in regard to the opinion on the financial statements.
### **Footnotes to the Financial Statements**
The footnotes (or notes) to the financial statements can be very extensive, running several pages in an annual report. Generally accepted accounting principles require that certain types of disclosures be made in the footnotes, and so this information should be considered an integral part of the financial statements. When the auditors express an opinion on the financial statements, it includes the footnote disclosures as well. An unqualified opinion by the auditors means that not only are the company's financial statements presented fairly in all material respects in conformity with GAAP, so are the footnotes.
Footnotes are required because of the full disclosure concept discussed in Chapter 2. The accounting profession believes that financial statements are more useful when significant information is disclosed adequately to the readers. If this additional important information were included on the face of the financial statements, it would be difficult to read the statements because of the volume of information. Consequently, this information is included in the notes to the financial statements.
The first footnote required by GAAP is a summary of significant accounting policies in which management explains which GAAP it chose to use when alternatives exist. For example, this first note will answer questions such as the following:
• What method or methods of depreciation are used for property, plant, and equipment?
• Over how long a period are the fixed assets depreciated?
• Which inventory valuation method is used?
• How are advertising costs recognized for financial reporting purposes?
• When are research and development costs recognized?
• What policies does the company follow in recognizing revenue?
• What principally makes up the company's accounts receivable?
The remaining footnotes can appear in any order. For example, the headings of the footnotes for two public companies, Wal-Mart and Kellogg's Company, as presented in their recent annual reports are shown in Figures 7.4 and 7.5.
Figures 7.4 and 7.5 present only the footnote headings and subheadings; the actual footnotes are 17 and 24 pages long, respectively, in the annual reports.
**Figure 7.4** Wal-Mart, Inc., 2007 Annual Report Footnote Headings
**Figure 7.5** Kellogg's Company 2006 Annual Report Footnote Headings
### **Quiz for Chapter 7**
1. Which of the following are required to file annual reports on a Form 10-K to the Securities and Exchange Commission?
a. Not-for-profit and governmental entities
b. Private companies
c. Public companies
d. All of the above
2. Annual reports to shareholders and annual reports filed with the SEC are one and the same.
a. True
b. False
3. Form 10-Ks are confidential financial information and are not available to the public.
a. True
b. False
4. Which of the following is the best opinion auditors can give a company on its financial statements?
a. An unqualified opinion
b. A qualified opinion
c. An adverse opinion
d. A disclaimer of opinion
5. Auditors are allowed to give an opinion on the client's financial statements even if the auditors are not independent with respect to the client as long as that fact is fully disclosed.
a. True
b. False
6. Auditor independence means the auditors:
a. Are not paid by the client but instead are paid by the federal government
b. Must maintain an objective, impartial, and unbiased attitude toward the client both in fact and in appearance
c. Automatically assume management is dishonest
d. Can own only immaterial (small) amounts of stock in an audit client
7. A qualified audit opinion:
a. Is the worst opinion and means that the financial statements are not presented fairly in all material respects, not even a portion of them
b. Is no opinion because the auditors were unable to complete the audit
c. Means that except for the effects of not following GAAP or not being able to perform a key auditing procedure, the financial statements are presented fairly in all material respects
d. Means that under no circumstances should anyone rely on these financial statements.
8. The Sarbanes-Oxley Act of 2002 applies only to public companies, not to private companies.
a. True
b. False
9. Which of the following statements is true regarding materiality?
a. Auditors provide absolute assurance that all material errors and fraud are detected.
b. Materiality is defined as being at least 10 percent of assets.
c. The determination of what is material is up to the auditors.
d. Auditors provide reasonable assurance that all immaterial errors and fraud are detected.
10. Footnotes to the financial statements:
a. Are required under GAAP
b. Are usually brief and insignificant
c. Are not audited by the auditors
d. Are required because of the materiality concept
## **CHAPTER 8
Analysis of Financial Statements**
In this chapter, we discuss various techniques readers can use to analyze a set of financial statements.
### **Horizontal Analysis versus Vertical Analysis**
Sometimes it can be difficult to interpret in a meaningful way all the dollar amounts presented in a set of financial statements. For example, if one company has liabilities of $10,000 and another company has liabilities of $10,000,000, is the first company less risky? Maybe, maybe not. It depends in part on the size of the company (how much in assets does each company have?) and the company's industry (what is normal?).
A useful way to analyze financial statements is to perform either a horizontal analysis or a vertical analysis of the statements. These types of analysis help a financial statement reader compare companies of different sizes, which can be difficult to do when the dollar amounts vary significantly, and evaluate the performance of a company over time.
The horizontal and vertical analysis approaches are similar in that the dollar amounts reported are converted to percentages. The approaches differ in the base used to compute the percentages.
### **Horizontal Analysis**
_Horizontal analysis_ focuses on trends and changes in financial statement items over time. Along with the dollar amounts presented in the financial statements, horizontal analysis can help a financial statement user to see relative changes over time and identify positive or perhaps troubling trends. We will use the income statement information for H. A. Example Company in Figure 8.1 to explain how one might prepare a three year horizontal analysis.
**Figure 8.1** Income Statements for Horizontal Analysis
In one horizontal analysis approach, a base year is selected and the dollar amount of each financial statement item in subsequent years is converted to a percentage of the base year dollar amount. Assuming 20x1 is the base year, 20x2 and 20x3 revenues were 108% and 120% of the base year amount, as shown in the following calculations:
Similar computations would be made for H. A. Example Company's remaining income statement items as shown in Figure 8.2.
**Figure 8.2** Horizontal Analysis Example
Some interesting trends can be noted from this analysis. The dollar amounts and percentages for each financial statement item increased each year, but the trends for each item differed. For example, in 20x3 when revenues were 120% of the base year amounts, cost of goods sold was less—only 115% of the base year amount. Perhaps H. A. Example Company raised its selling prices and/or its inventory cost declined? Notice that 20x3 net income was 187% of the base year amount; even though operating expenses had increased to 123% of the base year amount, this was more than offset by the favorable trends in revenues and cost of goods sold.
In addition to base year comparisons, dollar and percentage changes from one year to the next could also be analyzed. For example, 20x2 revenues increased by $8,000 or 8% over the previous year, and 20x3 revenues increased by $12,000 or 11.1% over the previous year, as shown in the following calculations:
How is the base year selected? That's up to the individual performing the analysis. Fraud examiners who are investigating a case of fraudulent financial reporting, for example, probably will select the last year in which they believe no fraud occurred as the base year in order to estimate the extent of the fraud. In other situations, the choice will depend to some degree on the purpose for which the reader is using the financial statements. Are you trying to decide whether to buy (or sell) stock now that a company has experienced a significant change such as new management or the introduction of a new product line? Then perhaps the base year will be the last year before the change. Essentially, the choice of the base year is up to the individual financial statement user.
Are these proportional increases that we calculated for H. A. Example Company good? Perhaps the competitors in the same industry are increasing even more. To interpret the proportional changes, the reader will need additional information, such as the industry averages and/or the changes for a particular company that the financial statement reader also is considering for investment purposes.
### **Vertical Analysis**
_Vertical analysis_ sometimes is referred to as common-size analysis because all of the amounts for a given year are converted into percentages of a key financial statement component. For example, on the income statement, total revenue is 100% and each item is calculated as a percentage of total revenue. On the balance sheet, total assets are 100% and each asset category is calculated as a percentage of total assets. In the balance sheet equation, total assets are equal to total liabilities plus equity; thus, each liability and/or equity account is also calculated to be a percentage of this total (i.e., total liabilities and equity are 100%). Vertical or common-size analysis allows one to see the composition of each of the financial statements and determine if significant changes have occurred.
We will use the balance sheet information for V. A. Example Company in Figure 8.3 to explain how one might prepare a three year vertical analysis.
**Figure 8.3** Balance Sheets for Vertical Analysis
After each year's total asset amount is set as 100 percent (or total liabilities plus stockholders' equity, since the amounts must balance), the various accounts' dollar amounts are converted to a percentage of the total assets. When the calculation is complete, the sum of the percentages for the individual asset accounts must equal 100 percent. The sum of the percentages for the various liability and equity accounts also will equal 100 percent (see Figure 8.4).
**Figure 8.4** Vertical Analysis Example
Vertical analysis of a balance sheet will answer questions relating to asset, liability, and equity accounts, such as the following:
• What percentage of total assets is classified as current assets? Current liabilities comprise what percentage of total liabilities and stockholders' equity?
• Inventory makes up what percentage of total assets? Is this changing significantly over time? If so, is it increasing or decreasing? (These answers could lead to additional questions such as the following: If it is increasing, could this indicate that the company is having trouble selling its inventory? If so, is this because of increased competition in the industry or perhaps obsolescence of this company's inventory?)
• Accounts receivable makes up what percentage of total assets? Is this changing significantly over time? If so, is it increasing or decreasing? (These answers might lead to additional questions such as the following: If it is increasing, could this indicate that the company is having trouble collecting its receivables? If it is decreasing, could this indicate that the company has tightened its credit policy? If so, is it possible that the company is losing sales that it might have made with a less strict credit policy?)
• What is the composition of the capital structure? In other words, total liabilities make up what percentage of total assets and total stockholders' equity makes up what percentage of total assets?
Vertical analysis of an income statement helps answer questions such as the following:
• What percentage of revenues is cost of goods sold?
• What is the gross profit percentage?
• What is the mix of expenses (in terms of percentages) that the company has incurred in this period?
Since total revenues usually are set at 100 percent, vertical analysis of the income statement essentially shows how many cents of each sales dollar are absorbed by the various expenses. For example, if total revenues were $200,000 and total wage expense was $50,000, total wage expense would equal 25 percent of total revenues. In other words, for every $1 in sales earned, 25 cents goes to employee wages.
In the case of V. A. Example Company, the organization appears to be fairly stable over the three years of data we have. Again, these percentages won't provide you with a lot of insight in and of themselves. The analysis is more meaningful when the percentages are compared with competitors' or industry averages or for a long period of time for one company. However, if some unreasonable fluctuations are noted for one company over time and/or the percentages are significantly different from industry averages, the possibility of fraudulent financial reporting should be considered, a concept we'll cover in Chapter 10.
### **Financial Statement Ratios**
Another method to analyze financial statement information involves the use of ratios. Ratios are useful in showing the relationships among financial statement data and also can be useful in comparing one company with another or analyzing the same company over time.
Ratios often are classified into different categories. Within each category, the ratios measure a common element, such the company's profitability or ability to pay its current debts. Sometimes the terminology of these different categories varies, but even with differing terms, the concepts are the same (see Figure 8.5).
**Figure 8.5** Categories of Ratios
We'll discuss the more common financial ratios next and calculate them using the Ratio Analysis Example Company financial statements in Figures 8.6 and 8.7.
**Figure 8.6** Income Statement for Ratio Analysis
**Figure 8.7** Balance Sheet for Ratio Analysis
### **Liquidity Ratios**
Debts cannot be paid with assets such as accounts receivable and buildings but must be paid with cash. Thus, ratios that help measure a company's ability to pay its current debts as they become due often are called liquidity ratios since the term _liquidity_ means nearness to cash. Cash is the most liquid asset a company can have; you can't get closer to cash than having cash itself. Accounts receivable are less liquid; they will be converted into cash once the customer pays the company. Inventory is even less liquid; it must be sold first, and if it is sold on account, time must pass before the resulting account receivable is converted into cash. If a company can pay its debts when they are due, the company is said to be solvent, and so liquidity ratios sometimes are referred to as solvency ratios.
A company's balance sheet may report that it has $1,000,000 in cash (or $10,000,000 or perhaps only $1,000). Is that enough to pay the bills that will be coming due in the near future, such as a year or less? It's impossible to answer that question with information on only the dollar amount of cash on hand. The answer depends on the amount of current liabilities that are also on the balance sheet. Furthermore, the company may have accounts receivable or other current assets such as short-term investments that can be converted easily into cash and used to pay bills as the need arises. Thus, answering the question requires a comparison of the relationship between various amounts on the balance sheet. Useful ratios for making this comparison include the current ratio and the quick ratio.
#### **CURRENT RATIO**
The current ratio is calculated as follows:
**Current Ratio = Current Assets ÷ Current Liabilities**
Recall from Chapter 5 that current assets are assets that will be converted into cash, sold, or used typically within a year and include assets such as cash, short-term investments, accounts receivable, interest receivable, inventory, and prepaid expenses. Note that prepaid expenses never will be converted into cash but will be expensed or used up within a year. Current liabilities are the debts that will be paid or settled within a year by providing goods or services and include items such as accounts payable, the portion of any long-term debt (such as a mortgage) that is due typically within a year, interest payable, wages payable, taxes payable, accrued liabilities, and unearned revenues.
The current ratio for Ratio Analysis Example Company is computed as follows:
**Current Ratio** | **=** | **Current Assets** | **÷** | **Current Liabilities**
---|---|---|---|---
|
**=** | **$2,428,000** | **÷** | **$4,020,000**
|
**=** | **.60 to 1**
|
This computation means that for every $1 of debts that will become due within a year, the company currently has 60 cents of assets that will be converted into cash or used up within the year with which to pay those debts. There are no hard and fast rules about the range in which any ratio should fall; one wants to consider a variety of factors rather than any single ratio. However, a current ratio of at least 1.0 implies exactly enough current assets with which to pay the debts that will come due in a year, with no cushion. Even then, the composition of the current assets is important because if a significant portion consists of prepaid expenses, those expenses will not be converted into cash to pay the bills. It is not unusual for banks, when writing loan agreements for a business, to include requirements that must be met or the loan will be due back in full immediately. Often the requirements state that certain minimum ratios must be maintained, such as a current ratio of 2.0 or 3.0, to help ensure continued liquidity but also allow a cushion. Does this imply that the higher a current ratio, the better? To an extent; however, if the current ratio becomes too high, that also can be cause for concern. Perhaps a company is keeping too much of its assets in liquid form when possibly the organization could become more profitable if, for example, the assets instead were placed in long-term investments and/or used to purchase additional fixed assets by which the company could increase operating activities.
In the case of Ratio Analysis Example Company, one probably would want to read management's discussion and analysis of the company's financial condition and results of operations, which is contained in the Form 10-K as well as the annual report to shareholders. Perhaps the current ratio of .60 is not troublesome as the company is expecting a significant inflow of cash early in the following year, as is normal in its industry. In contrast, perhaps there is increased competition in the industry, and the company is having trouble.
#### **QUICK RATIO**
The quick ratio (sometimes called the acid test ratio) is a more conservative liquidity measure. It gets its name from the fact that the numerator contains the assets that can be turned into cash most quickly. The formula is
In other words, inventory and prepaid expenses are removed from the current assets total so that the most liquid current assets remain in the numerator. For this reason, creditors often consider this ratio more meaningful than the current ratio.
Why are inventory and prepaid expenses removed? Inventory is slower to convert to cash: It first must be sold, and if it is sold on credit, the receivable must be collected, all of which will take some time. Further, some of the inventory may be obsolete or otherwise difficult to sell. Also, prepaid expenses will not be converted into cash; these are expenses, such as advertising or rent, which the company has paid for in advance. As time passes and the benefits of the payments are received (e.g., the advertising services), the company will follow the matching principle discussed in Chapter 2 and gradually write off the prepaid expenses from the balance sheet to an expense on the income statement.
For our example company, the quick ratio is computed as follows:
**Quick Ratio** | **=** | **($411,000 + $945,000)** | **÷** | **$4,020,000**
---|---|---|---|---
|
**=** | **.60 to 1**
|
Quick ratio that is much smaller than a current ratio indicates that the company has quite a bit of resources tied up in inventory and/or prepaid expenses. Conversely, when the two ratios are close to equal, it indicates that the company does not have much inventory and/or prepaid expenses. In Ratio Analysis Example Company's case, the quick ratio indicates that there is only 34 cents available for every $1 of debts that are soon to be due. Before pushing the panic button, one would want to look at recent previous years of the company and also read management's discussion and analysis, which also is contained in the annual report to shareholders. Has something unusual happened recently?
### **Profitability Ratios**
Profitability ratios measure a company's success (or failure) in earning a profit. Recall that profit exists when revenues exceed expenses. Profitability ratios are not the same as liquidity ratios because a company can be profitable but have liquidity problems. How can this be the case? Perhaps a company has made many credit sales rather than cash sales, and so the revenue is recorded and profits have increased. But perhaps many customers have not paid what they owe the company, and so the cash hasn't flowed in yet. Bills cannot be paid with accounts receivable but must be paid with cash. Thus, it is possible for a company to be profitable and insolvent at the same time.
Liquidity ratios are important for a company's short-term success (how long can a company stay in business if it cannot pay its bills when they are due?). Profitability ratios are important for a company's long-term success (how long can a company stay in business if its expenses exceed its revenues on a continuous basis?).
Profitability ratios compute the relationship between profit and some other financial statement item, such as assets, sales, equity, or shares of common stock outstanding. The more common profitability ratios include the following:
• Gross margin on sales
• Profit margin on sales (return on sales)
• Return on assets
• Return on equity
• Earnings per share
• Dividend payout ratio
• Price-earnings ratio
#### **GROSS MARGIN ON SALES**
Recall from Chapter 3 and our discussion of the income statement that gross margin (also called gross profit) is the excess of sales revenue over the cost of goods sold. Thus, gross margin represents the amount of sales revenue left, after covering the cost of the product sold, to pay all remaining business expenses and have some profit left over.
Gross margin ratios vary widely from industry to industry. In the Ratio Analysis Example Company case, the gross margin on sales is computed as follows:
**Gross Margin on Sales** | **=** | **Gross Margin** | **÷** | **Net Sales**
---|---|---|---|---
|
**=** | **$4,825,000** | **÷** | **$10,907,000**
|
**=** | **44%**
|
This computation indicates that for every dollar in net sales that the company generates, the product sold cost 56 cents and there is 44 cents left to cover all remaining expenses and, the company hopes, constitute a profit.
#### **PROFIT MARGIN ON SALES**
Profit margin on sales takes the bottom line (net income or net profit) after all expenses are subtracted from revenues and compares it with net sales revenue. Again, this can vary widely from industry to industry. This ratio sometimes is called return on sales and is computed as follows:
**Profit Margin on Sales = Net Income ÷ Net Sales**
For our company, the return on sales is computed as follows:
**Profit Margin on Sales = $979,000 ÷ $10,907,000 = 9%**
Thus, for every dollar in sales revenue, 9 cents remains for bottom-line profit and 91 cents is spent on goods sold and all other expenses.
#### **RETURN ON ASSETS**
The return on assets (ROA) ratio measures how effectively management is using its assets to generate a profit. It is computed as follows
**Return on Assets = Net Income ÷ Average Total Assets**
For the Ratio Analysis Example Company (using year-end assets for simplicity), the return on assets is
**Return on Assets = $979,000 ÷ $10,715,000 = 9.1%**
This ratio is interpreted to mean that for every dollar invested in assets, the company earned 9.1 cents. Is 9.1 percent an adequate return on the investment the company has made in its assets? Comparison to the return on assets over time for this company, to competitors, and to the industry average would be useful. It should at least exceed the return that could be earned by investing in low-risk investments; otherwise it might make more sense for the company to cease doing business and instead invest in these alternatives (and earn a higher return).
#### **RETURN ON EQUITY**
The return on equity (ROE) ratio measures how well management is maximizing the return on the stockholders' (owners') investment in the business. For this reason, the ratio sometimes is referred to as return on investment. It is computed as follows:
**Return on Equity = Net Income ÷ Average Stockholders' Equity**
For the Ratio Analysis Example Company (using year-end stockholders equity for simplicity), the return on equity is
**Return on Equity = $979,000 ÷ $2,071,000 = 47.3%**
This computation indicates that for every dollar the stockholders have invested in the company, 47.3 cents was earned this year; that is a fairly impressive return. However, as with all ratios and other types of analysis, one will want to consider other benchmarks. Furthermore, risk is an important consideration. "Higher risk, higher reward" is not a cliché: If an investment in a particular company presents a greater risk, the higher degree of uncertainty should be rewarded with a greater potential return.
#### **EARNINGS PER SHARE**
Earnings per share (EPS) was discussed briefly in Chapter 3 in the analysis of the income statement. It represents a measure of the earnings associated with a single share of stock.
This ratio is unique in that under generally accepted accounting principles require that it be presented by public companies on their income statements; no other ratios are required to be reported. Many investors and creditors track EPS closely; consequently, management also pays it close attention because it is a highly visible profit-ability number and can affect a company's stock price.
The formula for basic (or simple) EPS is
Preferred stock is a different type of ownership interest from that represented by common stock and nearly always involves a preference in regard to dividends (a fixed amount of dividends must be paid to the preferred stockholders each year; further, the preferred stockholders are first in line to receive dividends, and so common stockholders receive nothing until the preferred stockholders are paid). Dividends are not an expense or a cost of doing business but a distribution of the profits of the business to its owners. Since they are not an expense, they are not reported on the income statement. Therefore, subtracting dividends from net income does not double-count the dividends. Instead, with this deduction, the numerator in the formula above represents the year's net income that is available to the common stockholders. Some corporations may have no preferred stock outstanding, in which case the numerator is simply net income.
The denominator represents the number of shares of common stock outstanding during the year, weighted by the period of time in which the shares were outstanding. _Outstanding_ means the shares are owned by the common stockholders; the shares have been issued to the stockholders, and they hold the shares. These shares must be weighted by the period in which they were outstanding; otherwise, the resulting EPS number will be distorted. For example, if a company reported net income of $1,000,000 for the year (but had no preferred stock outstanding) and had 1,000,000 shares of common stock outstanding as of year end, it appears the EPS would be $1.00. But what if 750,000 shares were issued on the last day of the fiscal year? In that case, only 250,000 shares were outstanding during the year, and so the EPS would be a more impressive $4.00 ($1,000,000 250,000). The capital (money) that was generated by selling the remaining 750,000 shares was not available throughout the year for the company to use to generate profits, and so those shares should be weighted accordingly (in this case, with less than a day left in the year, not weighted at all).
For the Ratio Analysis Example Company, assume that the par value of the common stock is 25 cents per share. With $105,000 reported on the balance sheet as the amount of common stock issued, this translates to $105,000 $0.25, or 420,000 shares. Assume that these shares were outstanding since the beginning of the year, therefore, no weighting by period of time outstanding is required. Thus, basic EPS is computed as follows:
**$979,000 ÷ 420,000 = $2.33**
Since the balance sheet does not report any preferred stock outstanding, there cannot be any preferred dividends to subtract from the net income in the numerator.
If the example company had any potentially dilutive securities present, the company also must report a second type of EPS: diluted EPS. Examples of potentially dilutive securities include bonds or preferred stock shares issued that give the holder the choice of converting the bonds or shares into a preestablished number of shares of common stock. Stock options are also potentially dilutive and often are given to employees as a form of additional compensation. Options give the employee the choice to purchase a set number of shares of stock at a preestablished price and can result in considerable compensation to the employee when the market price is above the preestablished purchase price. Diluted EPS takes into consideration how these additional securities can reduce the reported EPS number, primarily through the increased number of common shares outstanding in the denominator, even though the securities have not been exchanged for shares of common stock. The computation is more complex than that for basic EPS and typically is covered in more advanced courses in financial accounting. If a company has dilutive securities, that company is required under generally accepted accounting principles (GAAP) to present dilutive EPS along with basic EPS on the income statement. Since the example company does not have any convertible bonds, convertible preferred stock, or stock options issued, no dilutive EPS computation is required.
#### **DIVIDEND PAYOUT RATIO**
The dividend payout ratio is often of interest to investors because it indicates how much of a company's wealth is distributed to the common stockholders. Preferred stockholders typically have a fixed, stated amount that will be received as dividends each year, and so this ratio is calculated for the common stockholders as follows:
The denominator represents the net income available to common stockholders.
In the Ratio Analysis Example Company, recall that there is no preferred stock outstanding per the balance sheet. The dividends paid during the year to the common stockholders would appear on the statement of stockholders' equity, and if the dividends were all paid in cash during the year (rather than declared but not yet paid), they also will appear as a cash outlay on the statement of cash flows. We did not present either of those two statements for the example company, and so if we assume that $450,000 was paid during the year to the common stockholders, the dividend payout ratio becomes
**$450,000 ÷ $979,000 = 46%**
Once a company begins to distribute dividends, investors often expect to receive them consistently. Further, some investors will expect a relatively high payout ratio to receive what they consider an acceptable return on their investment.
#### **PRICE-EARNINGS RATIO**
The price-earnings ratio (PE ratio) is computed as its name implies:
**PE Ratio = Market Price per Share of Common Stock ÷ Earnings per Share**
This ratio gives readers an idea how much the earnings of the company cost an investor who wishes to purchase shares of the company's stock. The market price tells the investor the cost of a share but by itself does not give the investor an idea of the quality of that purchase. Is the price reasonable considering the profitability of the company?
In the Ratio Analysis Example Company case, assume that the current market price per share of common stock is $42. The PE ratio is computed as follows:
**$42 $2.33 18 times**
This result means that investors are paying 18 times the current earnings to purchase a share of stock. Is this reasonable? As with all ratios, one will want to compare this to other benchmarks, such as the PE ratio of competitors and the industry average. There is no single benchmark for what a PE ratio "should" be, and the rules of thumb have changed over time.
### **Asset Management Ratios**
Asset management ratios (also called turnover, efficiency, and sometimes activity ratios) indicate the company's effectiveness at using its various assets. The asset management ratios we will discuss here are the following:
• Days sales outstanding
• Accounts receivable turnover
• Inventory turnover
• Days inventory supply
• Asset turnover
#### **DAYS SALES OUTSTANDING**
This ratio calculates the number of days it takes the company to collect its accounts receivables on average. The lower this number is, the better, because a company cannot pay its bills with receivables, only with cash.
The formula is as follows:
For the Ratio Analysis Example Company, the days sales outstanding (sometimes called days of accounts receivable, receivable days, or average collection period) is
**($945,000 × 365) ÷ $10,907,000 = 31.6 days**
This calculation gives you an idea of how liquid the receivables are. Recall that we are including accounts receivable in the current ratio and quick ratio computations, and so if we consider the accounts receivable to be liquid, it is useful to have an idea of exactly how liquid they are.
#### **ACCOUNTS RECEIVABLE TURNOVER**
The accounts receivable turnover ratio is closely related to the days sales outstanding. Instead of computing the average number of days that accounts receivable are outstanding, this ratio calculates the number of times accounts receivable are collected during the year. In the example above, we calculated the days sales to be 31.6; since there are 365 days in a year, this translates to the receivables turning over about 11.5 times a year (365 ÷ 31.6). In other words, it measures the number of times on average the company collects its accounts receivable during the year. The formula for accounts receivable turnover is
**Accounts Receivable Turnover = Net Sales ÷ Average Accounts Receivable**
When computing receivables turnover, there are some important things that should be considered. First, if a significant amount of the net sales consists of cash sales, those sales should be excluded since they never result in accounts receivable (assuming the reader can gather this information from the financial statements). Often, the reader will not be able to obtain this information. However, if the proportion of cash and credit sales remains fairly constant, the ratio will not be affected by using total net sales.
Second, since the balance sheet shows account balances at a point in time, simply using the end of year (balance sheet) amount for accounts receivable in the denominator could lead to a distorted result if the ending balance is not representative of the typical balance. A more representative result can be obtained by using an average accounts receivable balance, which is calculated as follows:
**(Beginning Receivables + Ending Receivables) ÷ 2 = Average Accounts Receivable.**
Thus, as with all ratio analysis, the reader will not want to rely on any single measure or even a handful. The more pieces of the puzzle one looks at, the better the overall resulting picture will be.
#### **INVENTORY TURNOVER**
This ratio indicates how quickly on average inventory is sold (i.e., turns over). How long does inventory sit on shelves before it is sold? The formula is
**Inventory Turnover = Cost of Goods Sold ÷ Average Inventory**
Note that the denominator consists of average inventory, and not year-end inventory for reasons similar to those we discussed for receivables turnover above.
For the Ratio Analysis Example Company (assuming that beginning inventory as determined from the previous year's balance sheet is $717,000), the inventory turnover is
**$6,082,000 ÷ [(717,000 + 824,000)/2] = 7.9 times**
Thus, inventory was sold and replaced 7.9 times that year. Inventory turnover and its variation—days inventory supply, which is discussed next—are an indication of the liquidity of inventory.
### **Days Inventory Supply**
A variation of inventory turnover is days inventory supply, which is a measure of the average number of days sales for which the company has inventory available. It is calculated as follows:
**Days Inventory Supply = (Average Inventory × 365) ÷ Cost of Goods Sold**
For the Ratio Analysis Example Company, days inventory supply is calculated as follows:
**([($717,000 + $824,000)/2] × 365) ÷ $6,082,000 = 46.2 days**
If we've already calculated the inventory turnover, the days inventory supply for the year is more easily calculated this way:
**365 ÷ Inventory Turnover = 365 ÷ 7.9 times = 46.2 days**
This calculation indicates that if we stop purchasing inventory, we have enough still on hand to maintain sales for about 46 days. Is this sufficient? As was indicated earlier, it depends on the industry. If the company sells jewelry or other expensive luxury items, the days inventory supply of 46 suggests that business is good and the product is moving quickly. However, if the company sells fresh fruit or other perishable items, the days inventory supply of 46 suggests that business is sluggish. Thus, inventory turnover and its variation days supply of inventory are useful in determining whether a company is carrying too much inventory, which can be caused by poor inventory management or sluggish sales.
If a company is carrying too much inventory, it may mean that the inventory is becoming obsolete, and the company will not be able to sell the inventory without reducing prices. Too much inventory also strains a company's resources, as it represents a use of cash and can lead to additional storage and display costs. However, if a company is carrying too little inventory, that can indicate a problem in production or purchasing. Further, the company may not be able to meet customer demand and as a result could lose sales now and in the future.
#### **ASSET TURNOVER**
The asset turnover ratio is computed as follows:
**Asset Turnover = Net Sales ÷ Average Total Assets**
Average total assets is computed as follows:
**Average Total Assets = (Beginning Total Assets + Ending Total Assets) ÷ 2**
The beginning total assets will be the total assets shown on the prior year's balance sheet, since last year's ending balance equals this year's beginning balance.
Since all assets are included in the computation, this is a more general indication of how efficiently a company uses its assets to generate sales. Specifically, it indicates how many dollars of sales are produced for every dollar invested in assets.
For the Ratio Analysis Example Company, the asset turnover (assuming beginning total assets equals $10,575,000) is
**$10,907,000 ÷ [($10,575,000 + $10,715,000)/2] = 1.02 times**
This calculation indicates that the company generated $1.02 in sales in the year for every $1.00 it invested in assets. Whether this is an acceptable amount again depends on the industry as this ratio can vary widely from one industry to another. Generally, a company with a low asset turnover needs more capital (whether from stockholders or from creditors) to generate additional sales. In contrast, a company with a higher asset turnover can generate additional sales with a smaller influx of capital.
### **Leverage Ratios**
Leverage ratios (sometimes called coverage or capital structure ratios) measure a company's long-term ability to pay its debts as they become due. Leveraging means using other people's money to generate profit for a company. The leverage ratios we will introduce are
• Debt-to-equity ratio
• Debt ratio
• Times interest earned
#### **DEBT-TO-EQUITY RATIO**
The debt-to-equity ratio, as the name implies, is computed as follows:
**Debt-to-Equity Ratio = Total Liabilities ÷ Total Stockholders' Equity**
This ratio calculates how much debt a company has in relation to its owners' equity. If the ratio is 1.0, the company has exactly the same amount of debt as it has owners' equity. With a ratio greater than 1, the company has more debt than owners' equity; in other words, the company owes more than it owns. The higher the ratio becomes, the more heavily burdened with debt the company is. Creditors often prefer that this ratio be low because it suggests that the company is less likely to default as it can pay back its loan with the owners' investments if necessary. Usually if the ratio is low, it indicates that there is greater long-term financial safety and more flexibility to borrow in the future, if necessary. (In fact, these leverage ratios sometimes are referred to as safety ratios.)
For the Ratio Analysis Example Company, the debt-to-equity ratio is
**$8,644,000 ÷ $2,071,000 = 4.2 times**
This computation indicates that the company has about 4.2 times as much debt as equity. Thus, the company owes much more to others than the stockholders own. You probably can anticipate our next statement: This computation is more meaningful when compared with a benchmark such as the industry average, a particular competitor, and/or this company over time. Further, no single ratio will paint a complete picture.
#### **DEBT RATIO**
The debt ratio measures the amount of debt in relation to the total assets of the company. Recall from our discussion in Chapter 1 that the assets came from somewhere: either from outsiders (what you owe) or from owners (what you own). The debt ratio is computed as follows:
**Debt Ratio = Total Liabilities ÷ Total Assets**
For the Ratio Analysis Example Company, the debt ratio is
**$8,644,000 ÷ $10,715,000 = 0.80 or 80%**
This computation indicates that about 80 percent of the company's assets are financed by creditors. The higher this ratio is, the greater is the risk that the company may have trouble meeting its debt obligations.
#### **TIMES INTEREST EARNED**
Interest must be paid on debt and it must be paid on time or the company risks having its creditors call in the loan. To get an indication of a company's ability to pay the annual interest it owes, times interest earned can be calculated. The formula is as follows:
**Times Interest Earned = Income before Interest Expense and Taxes ÷ Interest Expense**
Note that the bottom-line net income figure is not used in the numerator, since that already has interest expense deducted. Thus, using the net income figure would distort the computation of the company's ability to pay the interest it owes. Since income taxes are based on income after all expenses, using income before taxes as well helps one compute a more meaningful ratio.
For the Ratio Analysis Example Company, times interest earned is computed as follows:
**Times Interest Earned = $1,753,000 ÷ $307,000 = 5.7 times**
The numerator, income before interest and taxes, was computed as follows:
Thus, for this period the company earned 5.7 times what it had to pay in interest. Although there is no single benchmark above which this ratio should be, one hopes the ratio will be greater than 1. Otherwise, the company is earning exactly what it must pay for interest, and that probably will cause concern among the lenders.
### **Summary**
As you undoubtedly appreciate by now, numerous financial ratios can be computed from the financial statements (even more than we have presented here). Figure 8.8 summarizes the ratios we've introduced throughout this chapter. The key is to understand the information provided by the ratios and focus on the ones that are most meaningful for your particular decision, along with any other relevant information.
**Figure 8.8** Summary of Ratios
### **Quiz for Chapter 8**
1. Horizontal analysis:
a. Helps one see the relative changes in accounts over time
b. Begins by setting the dollar amount of total assets to equal 100 percent, and then everything else on the balance sheet is computed as a percentage of total assets
c. Also is called common-size analysis
d. Requires that one compare subsequent years with the immediately preceding year
2. When conducting a horizontal or vertical analysis, one will want to compare the results with:
a. Competitors
b. Industry averages
c. The company over a period of time
d. Any or all of the above
3. Liquidity ratios measure a company's:
a. Ability in the near future to pay its debts as they become due
b. Success or failure at earning a profit
c. Effectiveness at using various assets
d. Long-term ability to pay debt as it becomes due
4. Leverage ratios are also known as:
a. Turnover ratios
b. Activity ratios
c. Efficiency ratios
d. Capital structure ratios
5. Current assets divided by current liabilities is called the:
a. Quick ratio
b. Acid test ratio
c. Current ratio
d. Return on assets
6. Profitability ratios and liquidity ratios are the same.
a. True
b. False
_Use the following information to answer questions 7 through 10._
Annie Cooper Corporation reported the following information on its most recent financial statements:
Net Sales: $1,000,000
Cost of Goods Sold: $750,000
Operating Income: $120,000
Net Income: $100,000
Total Assets: $500,000
Total Stockholders' Equity: $400,000
7. What is Annie Cooper Corporation's gross margin on sales?
a. 10%
b. 12%
c. 25%
d. 33%
8. What is Annie Cooper Corporation's return on sales?
a. 10%
b. 12%
c. 25%
d. 33%
9. What is Annie Cooper Corporation's return on assets?
a. 10%
b. 20%
c. 24%
d. 25%
10. What is Annie Cooper Corporation's return on equity?
a. 15%
b. 25%
c. 30%
d. 62.5%
## **CHAPTER 9
Additional Issues**
Now that we have covered the four financial statements in detail, we will introduce a few additional issues that you occasionally may run into on financial statements. These items do not occur on a regular basis, however, and in fact sometimes are referred to as irregular items. These issues may affect any of the financial statements; they include the following:
• Discontinued operations
• Extraordinary gains and losses
• Changes in accounting principles
• Changes in estimates
• Correction of errors
### **Discontinued Operations**
Sometimes you may find an item on the income statement called a gain or loss from _discontinued operations_. This type of item occurs when a company stops operating a particular component of its business, and upon its disposal, the company has no significant continuing involvement with that component. Examples are shown in Figure 9.1.
**Figure 9.1** Examples of Discontinued Operations
Generally accepted accounting principles (GAAP) require that this be shown separately on the income statement after income from continuing operations is computed. Why can it not be combined with other revenues and expenses? Because it is logical for the reader of a financial statement to assume that unless there is evidence to the contrary, next year's net income will be reasonably close to this year's net income. However, by definition, discontinued operations will not affect net income in the future in either a positive or a negative direction. Thus, the impact on net income this year should be disclosed separately. When this is done, income from continuing operations also will be identified separately, and this information should be useful to anyone trying to approximate what next year's net income will be.
Taxes must be paid on all income, including income from discontinued operations. The tax effect of discontinued operations also must be separated, and the result is that an after-tax gain or loss from discontinued operations will be shown on the income statement (often described as _net of tax_ on the income statement). If taxes have to be paid on income, why would there be any tax effect of a loss? If there is a loss from discontinued operations, there will be a tax benefit—or a reduction in tax expense—in total for the company.
When a company has discontinued operations, two items are shown: the gain or loss from the component's operations during the period before disposal and the gain or loss from the actual disposal of the component. These amounts can be combined on the face of the income statement, with disclosure of the two items in the footnotes. For example, in Figure 9.1, only one of the four companies displayed both items on the face of its income statement. However, the other three companies provided the required information in the footnotes.
### **Extraordinary Gains and Losses**
Extraordinary gains and losses do not occur nearly as frequently as discontinued operations; in fact, by definition they are quite rare. To be classified as an extraordinary item, the event must be material (large in dollar amount) and must be unusual and occur infrequently. In fact, it may be easier to rule out what cannot be treated as an extraordinary gain or loss than it is to specifically identify what is. Examples that do _not_ qualify as extraordinary items include the following:
• The writing off of accounts receivable as uncollectible
• Gains or losses from discontinued operations (these are classified as discussed above)
• The effects of a strike (whether against the company or against its competitors or major suppliers)
• Gains or losses from selling or abandoning business property or equipment
None of these items qualify as extraordinary because they are considered to be usual business activities.
However, if the loss from abandoning business property occurred because of a major natural disaster that normally does not occur in the area (e.g., a hurricane, an earthquake, a flood), it might qualify as an extraordinary item. Whether the catastrophe normally occurs in the location is a key criterion to consider (see Figure 9.2).
**Figure 9.2** Examples of Possible Extraordinary Items
The accounting profession requires that extraordinary items be classified separately toward the bottom of the income statement rather than included in the regular ongoing income statement accounts so that readers will recognize that the financial impact is an anomaly this year that shouldn't be expected to occur in the foreseeable future. Consequently, it should be easier for readers to predict what future net income and cash flows will be, and that may be useful information for their decision-making purposes.
It has become increasingly rare to find an item classified as an extraordinary gain or loss. Several recent analyses of public companies' financial statements have found that perhaps only 2 percent or less report an extraordinary gain or loss.
### **Changes in Accounting Principles**
A change in an _accounting principle_ occurs when a company switches from one generally accepted accounting principle to another (i.e., from one GAAP to a different GAAP). When management selects a particular GAAP to follow, such as the last-in, first-out (LIFO) inventory costing method (discussed in Chapter 5), over other acceptable methods, management is not obligated to use that principle for the rest of time. However, it generally is expected that once a company selects a particular accounting principle to follow, it will not change principles year after year arbitrarily and capriciously. Many financial statement readers would frown on this practice because it suggests that management may be trying to play games with the company's bottom-line net income figure since the selection of accounting principles affects how and when revenues and expenses are recorded, thus affecting net income. Further, it makes comparability next to impossible.
When management chooses to change its selection of an accounting principle, how-ever, it is required to provide details in the financial statement footnotes that state the nature of the change and explain why the new method is preferable. The details also will include the financial impact on affected accounts, income from continuing operations, and net income. This information must be provided to aid financial statement readers in making predictions about future earnings levels and cash flows, especially when it may otherwise be difficult to determine if changes in a company's performance are due to actual economic events versus those caused by a change in the method used to account for an event. Without this information, it also is difficult to compare the company's past financial statements with those of the present. To aid in comparability, the accounting profession now requires that when a change occurs, the company adjust all prior year financial statements that are presented as though the newly adopted principle had been used in those years as well.
Recall from Chapter 7 that auditors are required to add a paragraph to their opinion to bring attention to situations in which a company has not applied accounting principles consistently this year compared to the prior year. Paradoxically, this paragraph often is referred to as a _consistency paragraph_ even though it is included only when the company has _not_ been consistent in its use of accounting principles from the prior year. The auditors' paragraph will refer the reader to the financial statement footnote in which management has provided the details of the change.
When the accounting profession issues a new accounting standard requiring a change in the proper accounting for a particular type of transaction, organizations have to comply with that standard. Otherwise, assuming that the effect of noncompliance is material, the auditors must give the company something less than the most desirable audit opinion (the unqualified opinion). Because the Financial Accounting Standards Board issues new accounting standards often, it is not unusual to see changes in accounting principles (see Figure 9.3).
**Figure 9.3** Examples of Changes in Accounting Principle
Note that in some cases it is clear from the auditors' consistency paragraph that the change is driven by the issuance of a new professional standard, whereas in other cases it is not as clear but can be determined by reading the financial statement footnotes.
What kinds of changes would not be considered by the accounting profession to be changes in accounting principle? Examples include the following:
• Changing from non-GAAP to GAAP (such as from using the cash basis to using the accrual basis of accounting); this is considered a correction of an error.
• Changing the salvage value or service life used in depreciation calculations for property, plant, and equipment; this is a change in estimate.
• Correctly applying an accounting principle this year when it had been applied incorrectly in the past; this is a correction of an error.
• Increasing the write-down of inventory as a result of obsolescence because of new information obtained; this is a change in estimate (discussed next).
### **Changes in Estimates**
Although on the surface it may appear that accounting is quite black and white in nature, in reality many estimates are inherent in the process of preparing financial statements. Examples of how estimates are built into financial statement numbers include the following:
• Salvage value and useful life used for calculating depreciation
• Percentage of net credit sales or accounts receivable that will be uncollectible
• Percentage of products sold that will be returned under warranty and the dollar amount of the warranty repairs and replacements
• Amount to write down inventory because of obsolescence
Because estimates are essentially educated guesses, they are bound to change as time passes and the company gains experience or as new information arrives. How are changes in estimate handled? For example, with the sudden economic recession that developed in late 2008, undoubtedly many companies will increase their estimate of the bad debts that will arise from the year's sales. Imagine the confusion and the resulting lack of confidence in reported financial information if companies were required to revise and reissue past financial statements every time an estimate was changed. For that reason the accounting profession decided that any changes should be incorporated into the financial statements currently and prospectively, in other words, in the period in which the change occurred and any future periods affected. No prior period statements are restated.
The use of estimates is disclosed in the first footnote (summary of significant accounting policies). For example, the 2006 annual report of Johnson & Johnson includes as part of the first financial statement footnote under the heading "Use of Estimates" the following:
The preparation of consolidated financial statements in conformity with accounting principles generally accepted in the U.S. requires management to make estimates and assumptions that affect the amounts reported. Estimates are used when accounting for sales discounts, rebates, allowances and incentives, product liabilities, income taxes, depreciation, amortization, employee benefits, contingencies and intangible asset and liability valuations. For instance, in determining annual pension and post-employment benefit costs, the Company estimates the rate of return on plan assets, and the cost of future health care benefits. Actual results may or may not differ from those estimates.
For an example of a current and prospective application, assume the following facts:
• A company purchases a piece of equipment for $65,000 and estimates its useful life at six years, with a salvage value at the end of that time of $5,000. ($60,000 total depreciation; $10,000 per year)
• The company depreciates the equipment for four years under the straight-line depreciation method. ($40,000 total depreciation taken to date)
• In the fifth year, the company estimates the useful life to be a total of eight years (instead of six) but still believes the salvage value will be $5,000 at the end of that time. ($65,000 − $40,000 taken − $5,000 salvage = $20,000 remaining depreciation for next four years)
How is this change in estimate reflected in the financial statements (see Figure 9.4)?
**Figure 9.4** Example of Prospective Implementation of a Change in Accounting Estimate
In the fifth year, when the estimate changes, the amount shown as depreciation expense on the income statement for this equipment will be $5,000 rather than $10,000 as it had been in each of the previous four years. Assuming no further changes in estimates, by the end of the eighth year, the equipment will be fully depreciated ($60,000). Since it cost $65,000, it will have a book value of $5,000 (the estimated salvage value at the end of the eighth year).
Depending on how the estimate changes, the affected calculation can be increased or decreased, and the amount of change can be material. If the effect is material, companies will disclose the change in estimates in the footnotes. Similarly, if the change affects multiple periods, companies are required to disclose the effect of the change. In light of the numerous estimates built into financial statement data and the fact that management does not have a crystal ball with which to see accurately into the future, it is not surprising that changes in estimates are very common.
### **Correction of Errors**
Just as changes in estimates are very common, so are errors. Regardless of the checks and balances a company may have built into its accounting system, no system (or group of accountants) is perfect. Examples of errors include the following:
• Recording expenses in the wrong period
• Incorrect application of an accounting principle
• Recognizing revenue in the wrong period
• Math errors such as incorrect computation (and recording) of sales discounts
• Misclassifying an item such as recording an accounts receivable as a note receivable
• Changing from non-GAAP to GAAP, such as switching from the cash basis to the accrual basis of accounting
Errors are unintentional honest mistakes. Intentional errors are considered to be financial statement fraud, as is discussed in Chapter 10. As soon as a company discovers it has made an error, it must correct the error. When the error affects prior financial statements, they must be restated if they are presented again. If this occurs, the auditors' report will include a short paragraph bringing this to the reader's attention since the correction represents a lack of consistency. For example, in the 2006 financial report of Sara Lee Corporation, the auditors included the following paragraph in their opinion:
As discussed in Note 1 to the consolidated financial statements, the Corporation has restated its fiscal 2005 and 2004 consolidated financial statements.
Management provides the details about the error or errors and the correction in the financial statement footnotes.
### **Quiz for Chapter 9**
1. Discontinued operations are:
a. Shown on the balance sheet as a liability
b. Shown on the balance sheet as an asset
c. Shown on the income statement as a separate item
d. Shown on the income statement but combined with other revenues and expenses
2. Income and expenses from discontinued operations have no impact on income taxes.
a. True
b. False
3. Extraordinary gains or losses:
a. Must be unusual and occur infrequently
b. Occur infrequently but can be a normal business activity as long as they are highly material
c. Often are listed on financial statements
d. Are shown as an asset on the balance sheet
4. Which of the following is an example of an extraordinary item?
a. Gains or losses from discontinued operations
b. Losses suffered from the effects of a strike against the company
c. Losses suffered from abandoning business property and equipment because of damage from a major volcanic eruption that previously had not occurred in the area for over a century
d. Writing off an accounts receivable as uncollectible
5. Changes in accounting principle:
a. Rarely occur because once management selects an accounting principle to follow, it is not allowed to change it except in extremely unusual circumstances
b. Are reflected in the current year and future years only; past financial statements are not restated
c. Require no additions or changes to the standard audit report
d. Must be disclosed by management in the financial statement footnotes, stating the nature of the change, explaining why the new method is preferable, and including the financial impact on affected accounts, income from continuing operations, and net income
6. Which of the following is an example of a change in accounting principle?
a. Changing the service life for equipment in depreciation calculations
b. Switching from the LIFO method to the FIFO method of inventory valuation
c. Implementing the accrual basis of accounting after using the cash basis of accounting in prior years
d. Increasing the amount of bad debt expense recorded because of a downturn in the economy
7. A change in estimate requires that prior year financial statements be restated.
a. True
b. False
8. A change in estimate rarely occurs, as very few estimates are built into financial statements.
a. True
b. False
9. If a company discovers that its prior year financial statements contain an error, they must be restated before they are presented again.
a. True
b. False
10. Which of the following is an example of an error?
a. Increasing the salvage value of equipment for depreciation calculations as a result of better information
b. Inadvertently recording revenue this year when it should have been recorded in the following year
c. Intentionally understating expenses in order to be able to report a higher net income
d. Switching from one generally accepted accounting principle to another because management now believes that the new principle is preferable and will result in an improvement in financial reporting
## **CHAPTER 10
Fraudulently Misstated Financial Statements**
Now that you are familiar with the information contained in financial statements, we conclude this book by discussing the topic of fraudulently misstated financial statements.
Not all cases of fraud are discovered and not all discovered frauds are reported, and so there are no solid statistics on the extent to which fraud occurs. Thus, all the statistics are estimates, primarily based on various national and international surveys of different organizations and surveys of people certified in fraud investigation. However, all the surveys are consistent in finding that fraud is pervasive and knows no geographical boundaries. Further, fraud can occur in all types of organizations—private companies, public companies, not-for-profit organizations, universities and other educational institutions, and governmental agencies—and in organizations of all sizes.
The Association of Certified Fraud Examiners (ACFE), headquartered in Austin, Texas, periodically surveys a random sample of Certified Fraud Examiners (CFEs) discussing recent fraud cases they have investigated. These survey responses form the basis for the publication titled _Report to the Nation on Occupational Fraud and Abuse_ , first published in 1996 and with subsequent surveys done in 2002, 2004, 2006 and 2008. (These reports can be downloaded free at http://www.acfe.com/fraud/report.asp.)
In its surveys, the ACFE classifies fraud in three broad categories:
1. Asset misappropriation, or the theft of an organization's assets, such as embezzling cash or stealing inventory
2. Fraudulent financial reporting, or the intentional falsification of financial statement information, such as overstating revenues
3. Corruption, or using one's position in an organization wrongfully to obtain a personal benefit, such as accepting kickbacks (cash from a vendor in exchange for having the organization make purchases from that vendor)
The ACFE results suggest that although it is not common, fraudulent financial reporting is occurring more frequently (see Figure 10.1).
**Figure 10.1** Rate of Occurrence for Different Categories of Fraud According to the Association of Certified Fraud Examiners Fraud Survey
* Some of the fraud cases studied involved multiple methods of defrauding the organization; thus totals may be greater than 100 percent.
Not only has the occurrence of financial statement fraud increased, as might be expected, the median dollar misstatement caused by fraudulent financial reporting is significantly greater than the median dollar loss in either of the other two fraud categories (see Figure 10.2).
**Figure 10.2** Median Fraud Loss/Misstatement for Different Categories of Fraud According to the Association of Certified Fraud Examiners Fraud Survey
* These data were not published in the 1996 report.
This disparity in median fraud amounts is caused by the fact that the financial statements are reporting for the entire company, and if management is engaged in fraudulent financial reporting, the amounts will be material; in other words, the fraud is intended to deceive readers into making decisions they would not have made if they had had complete and honest information. Asset misappropriations and corruption typically are perpetrated by an individual who usually does not steal an amount that would be material to the financial statements.
Thus, although fraudulent financial reporting appears not to happen often, when it does occur, it can be devastating.
The ACFE survey results are consistent with those of KPMG, one of the "Big Four" international public accounting firms. KPMG conducted national fraud surveys in 1994, 1998, and 2003. Although comparable data from the 1994 survey were not gathered, KPMG reported that fraudulent financial reporting was the category of fraud that occurred the least but that it increased from 3 percent to 7 percent between 1998 and 2003. Further, the firm reported that the average annual cost of fraudulent financial reporting is by far the highest in all fraud categories, estimated at nearly $258 billion in the 2003 survey.
### **The Fraud Triangle**
In Chapter 9 we discussed how the accounting profession requires that errors in the financial statements be corrected. In this chapter, we are not concerned with errors, which are unintentional and honest mistakes, but with fraud. Fraud involves intent, specifically, the intent to deceive. There is no such thing as unintentional or accidental fraud.
Fraud researchers have discovered that three elements are necessary for fraud to occur: pressure, perceived opportunity, and rationalization. These three elements combined make up what is referred to as the fraud triangle, and all three must be present simultaneously for a fraud to occur (see Figure 10.3). The fraud triangle applies to all three fraud categories: asset misappropriation, fraudulent financial reporting, and corruption.
**Figure 10.3** The Fraud Triangle
Pressure is the motive to commit fraud, and it may be financial or nonfinancial in nature. Examples of financial pressures include the following:
• Desire to live beyond one's means
• A gambling habit
• Alcohol or drug addiction
• Investment losses
• Excessive personal debts
Examples of nonfinancial pressures include the following:
• A disgruntled employee who wants to get even with the employer
• Perceived peer or family pressure to be successful
• Low self-worth and a belief that one will be well liked if one has more money
• A desire to beat the system
Upper management perpetrates fraudulent financial reporting, not an individual employee, as is often the case with asset misappropriation. Management is in a position to override any internal controls that are in place to help ensure accurate financial reporting, something that would be nearly impossible for an individual employee to do. In fact, fraudulent financial reporting usually is referred to as management fraud, whereas asset misappropriation is termed employee fraud.
Often, management's motive for engaging in this type of fraud is to report better financial results than have occurred to maintain or increase the company's stock price. The stock price may be an important factor in determining the compensation that will be paid to upper management because those individuals may hold a large number of stock options or because their compensation may be tied in other ways to the stock price (or other performance measures that are reported in the financial statements, such as net income and total revenues).
Other reasons to perpetrate this type of fraud include obtaining financing (or obtaining the financing on better terms, such as a lower rate of interest), appearing to meet loan requirements (such as maintaining a minimum current ratio), and encouraging investment in the company through stock purchase (such as by making the company appear to be less risky because of low debt levels). Those in upper management may engage in fraudulent financial reporting to appear more successful than they actually are because of perceived family pressure to do well or the desire to impress stockholders, especially if upper management is relatively new to the company. It is possible, however, that management will report results worse than the real results to pay less in income taxes or to be able to show gradual improvement in results over time.
The second element in the fraud triangle—perceived opportunity—refers to the idea in the perpetrator's mind that he or she will be able to commit fraud and get away with it. Most rational people will not commit fraud if they believe they will be caught. With this element, _perceived_ is the key word: There may be internal controls in place that will detect the fraud shortly after it starts, but if the perpetrator is not aware of the controls, he or she will begin to defraud the organization. Of course, the flip side is also true: If a potential perpetrator believes there are controls in place that in reality do not exist that will detect the fraud he or she is considering committing, the person will not commit the fraud.
Management is responsible for establishing the company's internal control system, and so with this thorough knowledge of the system, its members are in a perfect position to override the controls that are in place. Further, because members of management supervise other employees, they often are able to persuade or coerce subordinates (such as accounting personnel) to go along with a fraudulent financial reporting scheme.
The third element of the triangle—rationalization—is unique to fraud perpetrators. Rationalization means that fraudsters must be able to explain in their own minds why their actions are not criminal. In other words, they must be able to make their actions consistent with their personal code of conduct. Common rationalizations include the following:
• I'm only borrowing the money; I'll pay it back.
• Everybody does it.
• I'm not hurting anyone.
• People would understand if they knew how much I need this.
• It's not that serious.
• They owe it to me. I deserve to get paid more.
Members of management may rationalize their fraudulent financial reporting as being only temporary, until the company overcomes a short-term setback. Or they may believe that accounting rules are flexible (we can bend them, but we can't break them). They also may rationalize that no one will be hurt by the inflated stock price because they are too clever to be caught. Regardless of the rationalization, ethical standards are lacking when fraud occurs.
All three elements—pressure, perceived opportunity, and rationalization—must be present simultaneously for fraud to occur. If any of the elements are missing, fraud will not occur. For example, if no pressure or motive to commit fraud exists, why would it occur? Why would a person commit fraud if he believed he would get caught, in other words, if the perceived opportunity was lacking? And how could a person commit fraud if her moral values were strong and she could not rationalize the act? Thus, if any one of the elements is removed, the risk of fraud occurring is minimized greatly.
The fraud triangle concept that all three elements must be present for fraud to occur is similar to that for fire. Heat, fuel, and oxygen all must be present simultaneously for fire to exist. Take away any one of those elements, and the fire is out.
### **The Auditors' Responsibility for Detecting Fraud**
Fooling the auditors is certainly the trick to master when one is engaging in fraudulent financial reporting. On the surface, this may seem difficult, but numerous fraud surveys indicate that auditors seldom detect fraud (see Figures 10.4 and 10.5).
**Figure 10.4** Sources of Fraud Detection per KPMG Fraud Surveys
* The totals exceed 100 percent because many respondents experienced more than one fraud. Tips came from employees, customers, vendors, regulatory or law enforcement agencies, and anonymously.
**Figure 10.5** Sources of Fraud Detection per ACFE Fraud Surveys
* Totals exceed 100 percent because for many frauds, there was more than one method of detection.
Again, since not all cases of fraud are discovered and not all discovered frauds are reported, any fraud statistic is an estimate. However, these surveys are consistent in finding that tips are the number one method of discovering a fraud. Further, although these results of detection methods pertain to all categories of fraud, not just fraudulent financial reporting, it is interesting to note how seldom auditors (internal or external) detect fraud.
What exactly is the auditors' responsibility for detecting fraud? External auditors—the auditors from a public accounting firm who are hired by a company to audit its financial statements and render an opinion on them—are required under professional auditing standards to plan and conduct the audit to provide reasonable assurance about detecting material misstatements regardless of whether the material misstatement is due to errors or to fraud. However, recall that the dollar amount of some frauds may not have a material effect in the financial statements for the company as a whole. Note the following points:
• The auditor has no responsibility to detect immaterial fraud (where materiality is defined in relation to the financial statements as a whole); recall that an item is considered material if its misstatement or omission would cause a reasonable person to make a decision different from the decision that would be made if the item had been stated correctly.
• The auditors' responsibility to detect material frauds is no different from their responsibility to detect material errors (unintentional misstatements).
• The auditors do not provide absolute assurance or any kind of guarantee that all material fraud will be detected.
Internal auditors are employees of the company and as such are not considered independent in the required sense for rendering an opinion on financial statements. Therefore, they cannot audit a company's financial statements. However, they provide useful services to management, such as assisting in the design and implementation of internal controls and monitoring the effectiveness of those controls over time.
The professional standards for internal auditors require that those auditors have sufficient knowledge to identify fraud indicators; however, they are not expected to have the same level of fraud expertise as an individual whose primary responsibility is fraud investigation. Internal auditors can assist in fraud deterrence by evaluating the effectiveness of the internal control system, identifying areas of the organization that may be vulnerable to fraud, and implementing controls to help minimize that risk.
In light of the different functions of external and internal auditors, it is not surprising that few fraud cases are detected by external auditors or that more instances are caught by internal auditors, who work year-round at the same company on a closer and more detailed level. The moral of the story is that auditors, both internal and external, cannot be relied on to detect fraud. However, audited financial statements are still preferable to unaudited financial statements (some assurance is better than no assurance).
Further, recall from our discussion in Chapter 7 on reading the auditors' report that the Sarbanes-Oxley Act of 2002 (SOX) was passed in response to the numerous widely publicized corporate accounting scandals that had occurred from the 1990s until that time (e.g., Enron, WorldCom, Xerox, Tyco, Waste Management, Global Crossing). Section 404 of the Act requires auditors to provide an opinion on the effectiveness of public companies' internal control over financial reporting. The logic behind this requirement is essentially that if a public company has effective internal controls over financial reporting, there is a reduced risk of fraudulently misstated financial statements being published and relied on by the investing public.
### **Methods of Fraudulently Misstating
Financial Statements**
The majority of cases of fraudulent financial reporting involve making a company's financial health appear better than it really is. There are a number of methods by which this can be accomplished:
• Overstating revenues
• Understating expenses
• Understating liabilities
• Overstating assets
• Improper disclosures
Often, a combination of methods is used. For example, a company may "hold the books open" and record revenue this year when the revenue should be recorded in the following year (overstating revenues). At the same time, the company may delay the recognition of some expenses and record them in the following year when the proper recognition of the expenses should have occurred this year (understating expenses).
#### **OVERSTATING REVENUES**
Because net income equals revenue minus expenses, any time revenues are overstated, net income also will be overstated (see Figure 10.6).
**Figure 10.6** Effect on Net Income of Fraudulently Overstating Revenues
Revenue can be overstated by doing the following:
• Recording fictitious revenue
• Recognizing revenue prematurely
• Understating sales returns
Fictitious revenue can be created by recording sales to customers who do not exist (ghost customers) or by recording inflated sales to actual customers. If the latter method is chosen, management will need to take care not to overbill the actual customers, and so although fake documentation will be required, careful records must be kept. The lengths to which some managers go to perpetrate this kind of fraud can be amazing (see Figure 10.7).
**Figure 10.7** Example of Fraudulent Financial Reporting with Fictitious Revenues: ZZZZ Best
Recognizing revenue early—a timing difference—means recording revenue before GAAP allows it to be recorded. Recall from Chapter 2 that the revenue recognition principle states that revenue generally is considered earned and reported in the income statement in the period in which the business provided the service or sold the goods even though the customer may not pay for the services or goods until a later time. For example, it would not affect a company's financial statements to record revenue on July 13 that was earned on July 25, when the fiscal year end is December 31. Thus, this type of fraud occurs around the end of the fiscal year. Companies may record their revenue prematurely at that time if it appears that they may not reach their desired earnings target. Of course, this is like robbing Peter to pay Paul because the revenue cannot be recorded in the subsequent year when it truly is earned since in that case it would be double-counted.
Understating sales returns is another technique that can be used to overstate revenues. Recall from Chapter 3 on the income statement that sales returns are deducted from gross sales to arrive at net sales revenue. Thus, if the amount of sales returns is understated, net sales revenue and net income will be overstated (see Figure 10.8).
**Figure 10.8** Example of Fraudulent Financial Reporting with Understated Sales Returns: Regina Carpet Cleaning Company
#### **UNDERSTATING EXPENSES**
Understating expenses is a fraudulent technique that has the same effect on net income as overstating revenues (see Figure 10.9).
**Figure 10.9** Effect on Net Income of Fraudulently Understating Expenses
Because net income equals revenue minus expenses, any time expenses are understated, net income will be overstated. Expenses can be understated by
• Postponing expense recognition
• Capitalizing expenses (i.e., recording an expense as an asset)
Recognizing expenses in a subsequent accounting period violates the matching principle. Recall from Chapter 2 that the matching principle requires that costs incurred by the business generally be recorded as expenses on the income statement in the period in which the related revenue is produced. In other words, revenues are matched with the costs and efforts associated with producing those revenues, resulting in the net income or loss during the specified period. If the expenses are recorded in an accounting period later than the one in which the related revenues were generated, net income will be overstated.
Capitalizing expenses—improperly recording a cost as an asset that will appear on the balance sheet rather than as an expense of the period that will appear on the income statement—is a variation of recognizing expenses later.
Costs that provide benefits to a company over a number of accounting periods, such as the purchase of a company vehicle or manufacturing equipment, should not be expensed immediately in the period of purchase. That action would violate the matching principle since those assets will help the company generate revenues over future periods. Instead, these costs are capitalized—recorded as an asset—and gradually expensed over time as the assets are used. This expense is called depreciation expense and represents nothing more than the allocation of the asset's cost over the periods benefited by the use of the asset (depreciation in accounting terminology has nothing to do with writing an asset down to market value). Figure 10.10 illustrates this concept.
**Figure 10.10** Depreciation Expense Example
If a company is engaged in fraudulent financial reporting, it may decide to capitalize expenses improperly to overstate income. In the earlier example in this chapter, the company had purchased a piece of equipment and properly capitalized it and depreciated it over its seven-year useful life. But what if the $70,000 outlay had been for employees' wages and the company decided to capitalize that expense? Sound improbable? Capitalizing operating expenses was a technique used by WorldCom to misstate its financial statements by a reported $3.8 billion.
The proper accounting technique would be to expense the $70,000 for wages in that one period since there is no future benefit; the employees worked that year and helped generate revenues that year. However, if the company instead decided to record the $70,000 payment as the purchase of computer equipment, the effect on the financial statements would be to report much less expense (only through gradual depreciation) than would be the case if the entire $70,000 had been expensed and affected net income that year. Thus, the effect of improperly capitalizing expenses is to do the following:
• Overstate assets (which makes the balance sheet look better)
• Understate expenses (which makes net income and the income statement look better)
• Overstate stockholders' equity (net income increases year-end equity and so assets and equity are both overstated, and, the more equity, the better)
#### **UNDERSTATING LIABILITIES**
The fewer liabilities—or the less debt—a company has, the healthier and less risky it appears. Because the balance sheet must remain in balance, it will appear that more of the company is owned than owed (see Figure 10.11).
**Figure 10.11** How Fraudulently Understating Liabilities Affects the Balance Sheet
This effect occurs because if a liability is understated, there is usually an expense that is not recorded (giving rise to the debt). As was explained in the previous section, if expenses are understated, net income is overstated, which in turn overstates equity. Note that these fraudulent financial reporting schemes are not necessarily mutually exclusive.
Liabilities can be understated by
• Completely omitting some of them from the financial statements or
• Recording them at an amount lower than what is proper
Any liability can be understated if the company decides to do so. For example, a company may decide to omit recording contingent liabilities: liabilities that become debts contingent on a future event occurring. GAAP states that contingent liabilities must be booked if they are "probable" and the amount can be "reasonably estimated." For example, a company must estimate and record a liability for future warranty obligations if it sells products with a warranty. It is probable that some products will fail or be defective in some manner during the warranty period and will be returned. Further, the amount can be estimated reasonably on the basis of past experience.
Another type of liability that a company may omit is unearned revenues. Recall that when a company receives cash from a customer in advance of performing a service or providing goods, the revenue has not been earned. Instead, the cash is recorded as received (an asset has increased) and a liability is also recorded (unearned revenue). The company must either provide the customer goods or services or return the money back to the customer if for some reason it does not provide the service or the goods. Warranty service contracts, sales of gift cards, and frequent flier and other awards programs fall into this category of unearned revenues.
Depending on how bold the company is in its fraudulent financial reporting scheme, it may decide to stash unpaid bills in desk drawers or filing cabinets until it can pay them and fail to include these amounts as liabilities. It is more difficult for auditors to detect an unrecorded item that should be included than to detect an error of an item that has been recorded but using the wrong amount. If an item has been recorded, it can end up being selected to appear in the auditors' sample of items to test. In contrast, if an item has not been recorded, it is not possible for the auditors to select that item for testing from the population of recorded items.
#### **OVERSTATING ASSETS**
Overstating assets will achieve the same objectives as understating liabilities. The more assets a company has, the healthier it appears. Because the balance sheet must remain in balance, it will appear that more of the company is owned than owed (see Figure 10.12).
**Figure 10.12** Example of How Fraudulently Overstating Assets Affects the Balance Sheet
This effect occurs because if an asset is overstated, the company is not going to record a corresponding nonexistent or overstated debt for the nonexistent or overstated asset. Rather, there is usually a revenue that also was recorded. As was explained earlier in this chapter, if revenues are overstated, net income is overstated, which in turn overstates equity. Again notice how these fraudulent financial reporting schemes are not necessarily mutually exclusive.
Any asset can be overstated, but commonly the following assets are targeted:
• Inventory
• Accounts receivable
• Fixed assets (property, plant, and equipment)
For a manufacturing or retail business, inventory usually constitutes a large proportion of total assets. Auditors are required under professional auditing standards to observe a client's inventory count physically at the end of the fiscal year if inventory is material to the financial statements. This requirement is from one of the first auditing standards issued, which resulted from a major inventory fraud that occurred in 1938 at McKesson & Robbins. Despite this requirement, the auditors still can be fooled if management is intent on doing that (see Figures 10.13 and 10.14).
**Figure 10.13** Example of Fraudulent Financial Reporting—Overstated Assets: Phar-Mor, Inc.
**Figure 10.14** Example of Fraudulent Financial Reporting—Overstated Assets: Crazy Eddie
Accounts receivable also may be overstated. Recall that when a receivable is recorded, so is the related revenue. Thus, not only are assets overstated, so are revenues (and net income and therefore equity).
Auditors nearly always will confirm a sample of receivables, however. When auditors confirm information, they are requesting that an independent third party, such as a customer, directly verify to the auditors, usually in writing, that some specific information is correct (e.g., the amount the customer owes the client as of the end of the fiscal year). However, the auditors can be fooled if management is so inclined. In the ZZZZ Best case (Figure 10.7), management persuasively talked the auditors out of confirming its fictitious receivables by pressuring them not to "harass" its customers. Instead, management was able to persuade the auditors to accept internal documentation relating to its receivables, which of course had been manufactured by management.
Fixed assets also may be overstated, but, in a method that is unique to them, management may intentionally understate the depreciation expense each year, such as by using an unrealistic service life. The smaller expense translates into a larger net income, which increases equity. The accumulated depreciation contra asset account that appears on the balance sheet also is understated so that the net plant and equipment added on the asset side of the balance sheet is overstated. Thus, the balance sheet still balances.
#### **IMPROPER DISCLOSURES**
Recall that GAAP requires that detailed and extensive footnotes accompany financial statements. Thus, there are probably limitless ways in which management can mislead financial statement readers via the footnotes. Damaging information known to management may not be mentioned, or facts can be misstated. Those actions certainly do not conform to the full disclosure principle discussed in Chapter 2, but any method of misstating financial statements fraudulently does not conform to any concept of financial reporting.
The Enron case is a classic example of improper disclosure. Enron developed _special purpose entities (SPEs)_ to finance growth and report profits without reporting the related debt on its balance sheet. According to the accounting rules in place at the time, as long as at least 3 percent of an SPE's financing was provided by outside investors, the SPE's financial statements did not have to be combined (consolidated) with Enron's financial statements. Thus, Enron could use the capital raised by the SPE without having to report the related debt on its balance sheet. This debt avoidance was significant since the SPEs were structured to meet the minimum 3 percent rule, meaning that the remaining 97 percent of the capital typically was contributed by loans from banks. Later, investigators discovered that even the minimal 3 percent rule wasn't always followed by Enron. For example, the outside investors providing the 3 percent capital sometimes were individuals within Enron (e.g., Andrew Fastow, Enron's chief financial officer, who was involved in the creation and operation of several of these entities and named some of them after his children). Fastow reportedly received $30 million in profits on his investments in the Enron SPE investments that he oversaw.
Enron used the these entities for other purposes as well. It sold some of its assets at grossly overstated amounts to its SPEs, enabling it to report substantial paper gains on its income statement. Further, the SPE loans often were collateralized with Enron common stock. Thus, those SPEs essentially were a convoluted business relationship by which Enron could look financially better off than it was in reality, and Enron's management made sure that the related financial statement footnotes did not portray the true nature of Enron's relationship with the these entities. Wall Street analysts who had years of experience reading financial statements and footnote disclosures found the Enron footnotes pertaining to the SPEs confusing. If those experts could not understand the information provided in the footnotes, how could the average financial statement reader understand it?
### **Red Flags**
How can an investor or a potential investor detect possible fraudulent financial reporting in a company's financial statements? First, keep in mind that fraudulent financial reporting is the exception, not the rule. In the five surveys conducted by the Association of Certified Fraud Examiners, fraudulent financial reporting represented only 5.0 to 10.6 percent of the fraud cases reported by the respondents (see Figure 10.1). This statistic does not mean that fraudulent financial reporting happens up to 10 percent of the time, since these results were not derived from all financial statements produced in any one year, only from fraud cases investigated by the respondents for the most recent year.
Unfortunately, there is no single foolproof method of determining whether financial statements are fraudulently misstated; if there were, all auditors would be aware of that method and use it regularly. However, sometimes a careful look at ratios and their trend over time can be useful, especially when the ratios are compared with industry averages.
### **Nonsensical Ratios**
Keeping in mind the various techniques used to report financial statements fraudulently (overstate revenues, understate expenses, understate liabilities, overstate assets, and/or improperly disclose information), an analysis of the ratios that would be affected by these techniques can help identify potentially misstated financial statements. Any changes in these ratios from one year to the next and over a period of time should make sense. Further, it's reasonable to expect the ratios to be similar to industry averages unless there is a logical explanation for why they are not. Thus, if a company's financial ratios are showing unusual fluctuations that cannot be explained rationally, especially if the ratios are out of line with industry averages, it may be a good idea to reconsider relying on those financial statements.
Industry averages can be obtained fairly easily online, although some Web sites require that an individual pay a subscription fee for their services. On a search engine such as Google, type in "industry averages" and hundreds of thousands of hits will emerge; there are plenty of resources available. One free Web site that provides some industry averages is www.finance.yahoo.com.
For example, in the ZZZZ Best fraud (see Figure 10.7), the financial statements were reporting gross profits of approximately 45 to nearly 60 percent (see Figure 10.15).
**Figure 10.15** Examples of Nonsensical Trends and Ratios from ZZZZ Best Co., Inc., Financial Statements
However, the industry average was around 10 percent. Recall that nearly all of the sales revenue and related accounts receivable were fictitious in this case. Consequently, the accounts receivable grew from $0 to nearly $700,000 the next year and approximately $2,500,000 only three months after that. Sales increased dramatically in one year from slightly over $1.24 million to just under $4.85 million, a 291 percent increase. In a company doing such booming business, cash increased only from approximately $30,000 to $87,000 during that period. However, almost half of the revenues that year were from only one insurance restoration job (which, of course, was a fake). How could a financial statement reader know this? When an internationally renowned public accounting firm—ZZZZ Best's auditors—couldn't detect the fraud, it's hard to criticize any financial statement reader. But perhaps the idea that this relatively new company was founded and run by a 20-year-old history major with minimal business education and experience should have raised some suspicions. The identity and basic background of a company's CEO should not be a mystery to financial statement readers.
Other red flags were also present. For example, during the year when nearly $4.85 million in net sales was reported, general expenses of approximately $1.13 million also were reported on the income statement. Yet three months later the quarterly financial statements showed nearly $5.4 million in net sales (for three months) while general expenses decreased to approximately $622,000. Total assets on the balance sheet also were skyrocketing. In one year the reported total assets climbed from $178,000 to almost $5.1 million, an increase of over 2,700 percent. Further, the next quarter's financial statements listed total assets of slightly over $8.2 million, which is an increase of over 4,500 percent from the base year only 15 months earlier. This is phenomenal, unheard-of growth.
Ratio and trend analysis also reveal some red flags on the Regina Company, Inc.'s, financial statements (see Figure 10.8). Recall that in this case the vacuum cleaners with the revised product design featuring internal plastic parts were being returned at a fast rate because the plastic parts were melting when the vacuum was operated. Management chose not to book those sales returns properly, which would have reduced the income statement's reported net sales (Figure 10.16).
**Figure 10.16** Examples of Nonsensical Trends and Ratios from the Regina Company Financial Statements
Consequently, the income statements for three consecutive years showed net sales increasing from approximately $76 million to $128 million to $181 million (increases of 68 percent and 138 percent over the first year). At the same time, cost of goods sold was reported at $46 million, $71 million, and $95 million (increases of 53 percent and 105 percent over the first year). As a result, the gross profit margin changed from approximately 39.3 percent to 44.8 percent to 47.6 percent. Were these increasing gross profit margins due to a more efficient manufacturing process for the revised vacuums, or did they occur because the cost to manufacture vacuum cleaners with plastic parts was lower than the manufacturing costs for the earlier models with steel parts?
Either explanation is rational, but when one also looks at days inventory supply for Regina, a concern arises. Specifically, for the same years, the days inventory supply changed from 77 to 76 to 113. Does this suggest sluggish sales? Inventory increased from $9.7 million to $19.6 million to $39.1 million during those three years, when reported total assets changed from $43.2 million to $65.2 to $118.1 million.
Perhaps many sales were on account, and if that was the case, how quickly were the accounts receivable being collected? In Regina's case, the accounts receivable increased from approximately $14 million to $28 million to $51 million. The resulting days sales outstanding increased from 69 days to 79.1 days to 102.9 days. Was it taking longer to collect the receivables because customers were disputing what they owed for those returned faulty products?
### **The Exception, Not the Rule**
It bears repeating that fraudulent financial reporting is the exception, not the rule. Although numerous corporate accounting reporting scandals have been publicized widely in the last several years, those companies account for only a fraction of 1 percent of all published financial statements.
### **Quiz for Chapter 10**
1. It is impossible to know exactly how much fraud occurs, and as a result, all fraud statistics are estimates.
a. True
b. False
2. The Association of Certified Fraud Examiners classifies fraud into three broad categories. Which of the following is not one of those categories?
a. Embezzlement
b. Asset misappropriation
c. Fraudulent financial reporting
d. Corruption
3. According to the Association of Certified Fraud Examiners, fraudulent financial reporting is the most frequently occurring type of fraud.
a. True
b. False
4. Which of the following is not one of the three elements of the fraud triangle?
a. Pressure
b. Greed
c. Perceived opportunity
d. Rationalization
5. According to recent fraud surveys conducted by both the Association of Certified Fraud Examiners and the Big Four public accounting firm KPMG, tips are the most common manner in which frauds are detected.
a. True
b. False
6. Which of the following statements is true? Auditors provide:
a. Reasonable assurance that they have detected immaterial fraud
b. Absolute assurance that they have detected material fraud
c. Reasonable assurance that they have detected material fraud
d. More assurance that they have detected material fraud than that they have detected material errors
7. Fraudulent financial reporting usually is done to improve a company's financial position. Which of the following approaches is not a common method by which to accomplish this objective?
a. Overstate revenues
b. Overstate liabilities
c. Understate expenses
d. Improper disclosures
8. One method of overstating net income is to overstate sales returns.
a. True
b. False
9. Expenses can be understated by capitalizing expenses.
a. True
b. False
10. One useful method to detect fraudulently misstated financial statements is to compare reported results with industry averages.
a. True
b. False
## **Final Exam**
1. Financial statements can be used for a variety of business purposes, including:
a. Deciding whether to invest in a company
b. Deciding whether to lend money to a company
c. Determining the amount of bonuses to be paid to a company's top executives
d. All of the above
2. Which of the following is not one of the four basic financial statements?
a. Statement of changes in working capital
b. Income statement
c. Balance sheet
d. Statement of cash flows
3. Which of the following financial statements reports the amount of cash as of the end of the fiscal year?
a. Income statement and balance sheet
b. Balance sheet and statement of cash flows
c. Income statement and statement of cash flows
d. Statement of cash flows and statement of changes in owners' equity
4. Which of the following financial statements reports an organization's financial position at a specific point in time?
a. Income statement
b. Balance sheet
c. Statement of cash flows
d. Statement of changes in owners' equity
5. Revenues are the:
a. Excess of assets over liabilities
b. Economic resources of the business
c. Resources flowing into the business primarily from providing services or producing and/or selling goods
d. Cash flowing into the business
6. If a company reports $500,000 of revenues (of which $350,000 was received in cash) and $100,000 of expenses (of which $25,000 was paid in cash), the amount of net income reported for the period will be:
a. $250,000
b. $325,000
c. $400,000
d. $425,000
7. The ending balance of owners' equity as computed on the statement of owners' equity also appears on the:
a. Statement of cash flows
b. Balance sheet
c. Income statement
d. All three of these financial statements
8. The accounting profession requires that accounting information be comparable. This concept means that the information:
a. Helps financial statement users make predictions of future company performance on the basis of past information
b. Can be verified
c. Is presented or reported in a similar manner for different companies
d. Is presented in the same manner from one period to the next for a particular company
9. Financial statements are based on the assumption that they reflect only the business's transactions and do not include any of the owner's personal transactions.
a. True
b. False
10. Financial statements are prepared in accordance with GAAP, which stands for:
a. Generally antiquated accounting practices
b. Government approved accounting policies
c. Generally accepted accounting principles
d. Generally acceptable auditing procedures
11. Under the revenue recognition principle, revenue is reported on the income statement when:
a. Cash is received for the goods sold or the services provided
b. A contract is signed to provide the services or sell the goods
c. The goods are sold or the services are provided
d. Cash is paid for the goods or services
12. Companies are required to prepare their financial statements on the cash basis of accounting.
a. True
b. False
13. Many guidelines and standards have been developed by the accounting profession that must be followed in preparing financial statements. Primarily, these standards are developed by an organization called the:
a. Financial Accounting Standards Board
b. Society of Professional Accountants
c. Financial Statements Standards Committee
d. Governmental Accounting Oversight Board
14. Accounts receivable is:
a. An asset
b. A liability
c. A revenue
d. An expense
15. For one year, a company has received $130,000 in cash from customers and spent $125,000 in cash for expenses. For the same period, it reports revenues of $275,000 and expenses of $150,000. What is the reported net income on the income statement?
a. $5,000
b. $125,000
c. $130,000
d. $275,000
16. Unearned revenues:
a. Are reported on the income statement
b. Represent cash received from a customer in advance of performing a service or selling goods
c. Equal the cash received during the year
d. Are an asset reported on the balance sheet
17. Expenses are reported on the:
a. Income statement
b. Statement of cash flows
c. Balance sheet
d. Statement of changes in owners' equity
18. Which of the following is not an expense?
a. Employee salaries
b. Rent for office space
c. Office supplies used
d. Prepaid advertising
19. The matching principle requires that:
a. Expenses paid in cash be matched against cash revenues, with the resulting difference reported on the income statement
b. Costs of doing business be reported in the same period as the revenues they helped generate
c. Every item listed on the balance sheet be matched against its counterpart on the income statement
d. A matching footnote be provided for every item reported on the financial statements
20. _Earnings, profit, income_ , and _net income_ are different terms for the same concept.
a. True
b. False
21. If a company purchased inventory during the year that cost $250,000, had inventory on hand at the beginning of the year that cost $100,000, had inventory on hand at the end of the year that cost $30,000, and paid $225,000 cash for the inventory purchases, the cost of goods sold for the year would equal:
a. $295,000
b. $320,000
c. $325,000
d. $380,000
22. Gross margin represents:
a. The markup on the goods sold
b. The difference between gross sales revenue and net sales revenue
c. The excess of profit margin over operating expenses
d. The difference between cost of goods available for sale and ending inventory
23. For a company that manufactures the goods it sells, overhead represents:
a. The cost of salaries and wages of employees directly involved in manufacturing the goods
b. The cost of materials that become part of the finished product during the year
c. The costs incurred on goods manufactured in prior periods that have not been completed as of the beginning of the year
d. All the indirect costs of production
24. To determine the amount of dividends paid during the year, one would look at:
a. The income statement
b. The statement of cash flows
c. The balance sheet
d. The income statement and the statement of cash flows.
25. Earnings per share is a measure of profitability for common stock, not preferred stock.
a. True
b. False
26. Current assets are:
a. Presented on the balance sheet in order of liquidity
b. Assets that will be converted to cash or sold or used within a year or the operating cycle of the business, whichever is longer
c. Generally valued at their cost at the time acquired
d. Reported at a value below the original cost if the value has declined
e. All of the above
27. Investments classified as long-term would include all of the following except:
a. A corporate bond which is expected to be held until it is repaid at maturity
b. Seasonally idle cash which was invested in the common stock of another corporation
c. Land purchased for speculative purposes
d. Investments in stocks and bonds set aside to repay bonds payable at maturity
e. All of the above are long-term investments
28. Investments in marketable securities were purchased for $80,000, and the year-end market value is $90,000.
a. The investments will be reported on the balance sheet at the market value of $90,000 if they are classified as trading or available for sale securities.
b. The $10,000 increase in market value is referred to as an unrealized holding gain.
c. If the investments are classified as available for sale, the unrealized gain will be reported in stockholders' equity, not in the income statement.
d. All of the above are true.
e. None of the above is true.
29. For most companies, dividends and interest earned and gains or losses from the sale of investments are included in the routine revenues when reported in the income statement.
a. True
b. False
30. Which of the following is true regarding an investment in a corporate bond?
a. The bondholder will receive a stated amount of interest each year.
b. The market price of the bond can fluctuate over time.
c. The bondholder will be repaid a stated amount if the investment is held to maturity.
d. The bondholder can sell its bond investment to another party.
e. All of the above are true.
31. A company extends 2/10, n/30 credit terms to its customers. Which of the following is true regarding a customer invoice in the amount of $3,000?
a. The normal credit terms require the customer to pay the $3,000 balance within 30 days of the invoice date.
b. If the customer pays the invoice within 10 days, a cash discount of $60 can be taken.
c. Both of the above are true.
d. Neither of the above is true.
32. Which of the following would a manufacturing company include in the cost of goods in process inventory on its balance sheet?
a. The cost of materials used to manufacture the product
b. Wages paid to the assembly line workers
c. Depreciation on manufacturing equipment
d. All of the above
e. None of the above
33. For which of the following companies would the physical flow of merchandise inventory most closely resemble the FIFO assumption?
a. Car dealer
b. Meat shop
c. Gasoline station
d. All of the above
e. None of the above
34. A company using the weighted average cost-flow assumption purchased 200 units at $10 each and 800 units at $11 each. The weighted average unit cost is:
a. $10.80 per unit
b. $10.50 per unit
35. During the year, a company purchased three units of inventory and paid $4,000, $4,800, and $5,000, respectively. It sold two units and has one left in ending inventory. Determine the amounts that would be reported using the FIFO cost-flow assumption.
a. Cost of goods sold $8,800 and ending inventory $5,000
b. Cost of goods sold $9,800 and ending inventory $4,000
36. In a period of rising inventory prices, which of the following inventory cost-flow assumptions would produce the highest net income?
a. FIFO
b. LIFO
c. Weighted average
d. Cannot be determined without specific price information
37. In a period of rising inventory prices, which of the following inventory cost-flow assumptions would produce the lowest income tax liability?
a. FIFO
b. LIFO
c. Weighted average
d. Cannot be determined without specific price information
38. Inventory may be valued at the lower of cost or market when goods on hand are:
a. Damaged or shopworn
b. Obsolete
c. Replacement cost has declined and future selling prices must be reduced regardless of the cost paid at the time of purchase
d. All of the above
e. None of the above
39. Investments in marketable securities for which the earnings and ultimate proceeds from the sale of such investments will be used to pay the maturity value of bonds payable should be classified as a long-term investment.
a. True
b. False
40. A company owns 30 percent of the common stock of Corporation X and has significant influence. Corporation X had net income of $100,000 and paid cash dividends of $10,000. Using the equity method, the investor would report $3,000 of dividend income in its income statement for the year.
a. True
b. False
41. When a company owns 80 percent of another corporation's common stock and has control:
a. The investor is referred to as a parent company
b. The company in which the investor owns stock is referred to as a subsidiary company
c. The investor's financial statements will reflect the combined results of the two companies as though it were a single entity
d. The consolidated financial statements will show the amount of net income and stockholders' equity attributable to the 20 percent noncontrolling interest
e. All of the above
42. Which of the following might be included in the property, plant, and equipment classification in the balance sheet?
a. Land purchased for speculative purposes
b. Farm equipment held for resale by a farm implement dealer
c. Building under construction for the company's own use
d. All of the above
e. None of the above
43. Which of the following would _not_ be included in the amount reported as the cost of equipment on the balance sheet?
a. The cost to repair damages caused by vandalism
b. Delivery charges on equipment purchased
c. Costs to install and adjust equipment before its use in manufacturing operations
d. The invoice price less the amount of the cash discount taken for early payment
e. All of the above would be included
44. The cost of a building that a company is constructing for its own use probably would include:
a. Architectural design fees
b. Materials and labor costs
c. Depreciation on equipment used in construction activities
d. Interest on funds borrowed to finance the construction of the building
e. All of the above
45. In regard to the accounting concept of depreciation:
a. The amount reported as depreciation expense each year measures the decline in the current market value of a plant asset
b. The book value of the asset on the year-end balance sheet is an estimate of its current market value if it was sold on this date
c. Both of the above
d. Neither of the above
46. The units of production depreciation method might be used for which of the following assets?
a. Delivery vehicles
b. Machinery used to crush mineral ore produced in mining operations
c. Automated packaging equipment used in manufacturing operations
d. All of the above
e. None of the above
_Use the following information to answer Questions #47 & #48:_
Equipment was purchased at an original cost of $500,000. Its useful life is estimated to be five years or 100,000 units, and its salvage value is estimated to be $20,000.
47. The total amount of depreciation expense that can be taken during the five years of use is:
a. $500,000
b. $480,000
48. Using the units of production method, the depreciation rate per unit is:
a. $5.00 per unit
b. $4.80 per unit
49. If a company reports an impairment loss in its income statement, this signifies that the value of a long-lived asset declined below its book value and that future earnings from the use or sale of that asset are not sufficient to recover its cost.
a. True
b. False
50. Which of the following could be capitalized as the cost of a natural resource?
a. The cost to purchase rights to harvest timber on state land
b. Exploration costs incurred by oil and gas companies
c. Blasting and tunneling costs incurred to construct mine shafts
d. All of the above
e. None of the above
51. The depletion cost taken in a particular year:
a. Is treated in a manner similar to that used for the raw material costs incurred by a manufacturing company
b. Is a component of the cost of the inventory of the natural resource product on the balance sheet until it is sold
c. Will be a component of the amount reported as cost of goods sold in the income statement when the natural resource product eventually is sold
d. All of the above
e. None of the above
52. Which of the following is true regarding goodwill?
a. Goodwill is associated with the value of a company as a whole and cannot be identified separately.
b. Goodwill can be included among the intangible assets on the balance sheet only if it arose from the acquisition of another company.
c. Goodwill represents the excess paid to acquire a company over the fair value of the identifiable net assets acquired.
d. All of the above are true.
e. None of the above is true.
53. The term _depreciation expense_ applies to plant assets. What term is used to describe a similar concept pertaining to intangible assets?
a. Depletion
b. Amortization
54. Current liabilities:
a. Are obligations which must be paid or settled within a year or the operating cycle of the business
b. Will be settled using cash or other current assets or by the creation of other current liabilities
c. May include obligations for which the amount or specific party owed may not be known at the time the balance sheet is prepared.
d. All of the above
e. None of the above
55. Which of the following is true regarding contingent liabilities?
a. Probable contingent liabilities the amount of which can be estimated must be reported as liabilities in the balance sheet.
b. Contingent liabilities are disclosed in the footnotes only and should not be reported as liabilities in the balance sheet.
c. Contingent liabilities are ignored in preparing the financial statements and footnotes because of the uncertainty of whether an actual liability exists at this time.
d. All of the above are true.
e. None of the above is true.
56. Which of the following would be reported as accrued liabilities in the balance sheet?
a. Vacation pay employees have earned but not taken
b. Rental fees owed for the temporary use of snow removal equipment
c. Interest owed on a promissory note payable
d. The unpaid monthly telephone bill
e. All of the above
57. A one-year, $6,000, 9 percent note payable was signed on November 1, year 1. Which of the following is true?
a. $540 of interest will be paid when the note matures in year 2.
b. $90 of interest will be included in accrued liabilities on the December 31, year 1 balance sheet.
c. $450 of interest expense will be reported in the year 2 income statement.
d. All of the above are true
e. None of the above is true.
58. Which of the following is true regarding a capital lease?
a. The substance of the lease agreement resembles that of financing the purchase of an asset with long-term debt.
b. The asset under lease will be included in plant assets and depreciated.
c. The obligation for future lease payments will be included in long-term liabilities.
d. All of the above are true.
e. None of the above is true.
59. How should the balance of a long-term mortgage note payable which is repaid in monthly installments be reported in the balance sheet?
a. As a long-term liability
b. As a current liability
c. The principal payments due in the coming year should be reported as a current liability, and the remaining principal balance should be reported as a long-term liability.
60. If at the time a corporation issues bonds payable the current market rate of interest is higher than the specified interest rate the corporation has promised to pay to its bondholders:
a. The corporation will receive less cash than it has promised to repay on the maturity date
b. The bonds will be issued at a discount
c. The bonds will be reported on the balance at the amount the corporation has promised to repay at maturity less the applicable discount
d. All of the above
e. None of the above
61. When a corporation issues bonds at a discount, this signifies that the market rate of interest applicable at the time the bonds are issued is:
a. Higher than the stated interest rate the corporation has promised to pay
b. Lower than the stated interest rate the corporation has promised to pay
62. When a corporation issues bonds at a premium, this signifies that the market rate applicable at the time the bonds are issued is:
a. Higher than the stated interest rate the corporation has promised to pay
b. Lower than the stated interest rate the corporation has promised to pay
63. When a corporation reports deferred income tax liabilities on its balance sheet, this may signify that it paid less income tax this year because of a higher tax depreciation deduction but that additional taxes are expected to become payable in a future year.
a. True
b. False
64. Which of the following is not true of a business formed and operated as a sole proprietorship?
a. The owner's salary is treated as a business expense.
b. The owner will pay personal income taxes on the business profits.
c. The capital balance includes the owner's investments and the profits retained by the business.
d. All of the above are true.
e. None of the above is true.
65. Which of the following is true regarding a corporation?
a. The amount of each individual stockholder's equity is shown in the financial statements.
b. Stockholders have a right to participate in managing and operating the corporation.
c. The corporation files an income tax return and pays income taxes on its profits.
d. All of the above are true.
e. None of the above is true.
66. The terms _par value_ and _market value_ are synonymous and are used interchangeably.
a. True
b. False
67. All of the following are true regarding treasury stock except:
a. Treasury stock represents shares of the corporation's own common stock that the corporation has reacquired
b. Treasury stock is reported as an investment on the balance sheet
c. Shares held in the treasury do not have voting or dividend rights
d. The balance sheet value for treasury stock is based on the cost at the time it is reacquired
e. All of the above
68. If a corporation issues common stock and receives an amount greater than the par value, its stockholders' equity will include amounts for both common stock and contributed capital in excess of par value.
a. True
b. False
69. A stated amount of cash dividends must be paid on which of the following?
a. Common stock
b. Preferred stock
c. Treasury stock
d. All of the above
e. None of the above
70. At the beginning of the year, a corporation had $600,000 of common stock and $1,400,000 of retained earnings. During the year, the corporation issued $400,000 of additional common stock, earned net income of $1,000,000, and paid cash dividends of $200,000. Total stockholders' equity at year end is:
a. $2,400,000
b. $3,000,000
c. $3,400,000
d. $3,200,000
e. Some other amount
71. The indirect method of reporting operating cash flows shows the differences between the amounts of revenues and expenses included in the amount of reported net income and the related cash receipts and payments associated with those revenues and expenses.
a. True
b. False
72. Cash dividends paid by a corporation to its stockholders will be reported in the statement of cash flows as a(n):
a. Operating activity
b. Investing activity
c. Financing activity
73. Information that would explain the change in the amount reported as equipment (net) from the beginning to the end of the year can be found by reviewing:
a. The depreciation expense reported in the income statement
b. Purchases and sales of equipment reported as investing activities in the statement of cash flows
c. The supplemental schedule of noncash investing and financing activities presented along with the statement of cash flows
d. All of the above
e. None of the above
74. An increase in accounts receivable indicates that the amount of cash collected from customers is:
a. Greater than the sales revenue earned during the year
b. Less than the sales revenue earned during the year
75. An increase in accounts payable indicates that the amount of cash paid to suppliers is:
a. Greater than the amount of inventory purchased during the year
b. Less than the amount of inventory purchased during the year
76. A Form 10-K:
a. Is a tax form that must be filed by all businesses
b. Is an annual report required by the Securities and Exchange Commission to be filed by all public companies
c. Allows companies great flexibility in terms of what information is to be reported
d. Is limited to containing a company's financial statements
77. Financial statements filed with the Securities and Exchange Commission must be audited.
a. True
b. False
78. In an auditors' report, the auditors provide an opinion about whether the financial statements:
a. Are presented fairly in all material respects in accordance with GAAP
b. Are presented accurately in accordance with GAAP
c. Are presented fairly regardless of materiality
d. Were audited properly in accordance with auditing standards
79. Auditors provide a guarantee that audited financial statements are correct.
a. True
b. False
80. Auditors are required, without exception, to be independent with respect to the company whose financial statements are being audited.
a. True
b. False
81. Independence when auditing financial statements is important because it helps ensure that the audit is conducted with an unbiased, objective viewpoint, giving the public greater confidence in the auditors' report.
a. True
b. False
82. An audit can be conducted in accordance with professional standards and an unqualified opinion can be given, yet the financial statements still can contain large errors or fraud that went undetected by the auditors.
a. True
b. False
83. A qualified audit opinion:
a. Means that the auditors are qualified to render this opinion
b. Is the best audit opinion that can be given
c. Is the same as a clean opinion
d. Means the auditors have discovered some kind of problem with the financial statements
84. A disclaimer of opinion by the auditor means:
a. The financial statements should not be relied on by anyone because the auditors believe they contain large misstatements
b. The financial statements can be relied on
c. The auditors have discovered some problem with the financial statements, but other than that, the statements can be relied on
d. For some reason, such as the inability to gather enough evidence pertaining to one or more matters, the auditors are not rendering any kind of opinion
85. Since the passage of the Sarbanes-Oxley Act of 2002, auditors of public companies are required to issue an opinion on:
a. The effectiveness of the company's internal control over financial reporting as of the end of the fiscal year
b. The business decisions made by management during the year
c. Management's qualifications to run the company
d. The efficiency of the company's operations
86. The auditors' opinion on the financial statements includes the footnote disclosures.
a. True
b. False
87. Vertical analysis:
a. Involves computing changes in an account from one year to the next for several subsequent years
b. Involves converting all the amounts on a financial statement for one year to a percentage of some other amount, such as total revenues
c. Allows one to see the relative change in any account over time
d. Requires that several ratios be computed
88. Horizontal analysis can help answer questions such as:
a. What percentage of total assets is classified as current assets?
b. What is the gross profit percentage?
c. Accounts receivable this year are what percentage of the base year's accounts receivable?
d. Inventory makes up what percentage of total assets?
89. Profitability ratios include:
a. Return on assets and gross margin on sales
b. The current ratio and the quick ratio
c. Times interest earned and the debt-to-equity ratio
d. Inventory turnover and days' sales outstanding
90. If a company reports net sales of $500,000, cost of goods sold of $300,000, total operating expenses of $50,000, cash of $400,000, and total assets of $1,000,000, the profit margin on sales is:
a. 30%
b. 50%
c. 60%
d. 80%
91. On which financial statement might one find a gain or loss from discontinued operations?
a. Income statement
b. Statement of cash flows
c. Balance sheet
d. Statement of changes in owners' equity
92. Extraordinary gains or losses:
a. Are shown separately on the income statement, after the effect of taxes
b. Appear on financial statements much more frequently than does a change in accounting principles
c. Include natural disasters (e.g., hurricanes, floods) regardless of the geographical location
d. Require no footnote disclosures
93. When a company changes an estimate of some information built into the financial statements (e.g., the useful life of a plant asset that is being depreciated):
a. It must immediately restate the past five years of financial statements to reflect the new estimate
b. It is allowed to ignore the new estimate and continue using the old estimate
c. It should use the new estimate in the year it occurs and subsequent years
d. It reports the change in estimate expense on the income statement
94. According to the Association of Certified Fraud Examiners, of the three general categories of fraud, fraudulent financial reporting occurs the least but is the most costly.
a. True
b. False
95. Fraudulent financial reporting requires intent on the part of management to deceive the readers of the financial statements.
a. True
b. False
**Which of the following statements is false?**
96. Management may feel pressure to perpetrate fraudulent financial reporting to:
a. Maintain or increase the company's stock price
b. Obtain financing at better rates of interest
c. Temporarily survive some short-term setbacks
d. Appear to meet loan requirements, such as maintaining minimum benchmarks of certain ratios
97. Most frauds are detected by auditors.
a. True
b. False
98. Auditors have the same level of responsibility to detect material frauds as the responsibility to detect material errors (honest mistakes).
a. True
b. False
99. Which of the following is not one of the common methods by which to overstate revenues fraudulently?
a. Record fictitious revenue
b. Understate sales returns
c. Hold the books open and recognize revenue early
d. Overstate sales discounts
100. Researchers have coined the term _fraud rectangle_ to describe the four elements necessary for fraud to occur.
a. True
b. False
## **Answers to Quiz and Final Exam Questions**
### **CHAPTER 1**
1. b
2. c
3. a
4. d
5. c
6. c
7. d
8. a
9. c
10. b
### **CHAPTER 2**
1. b
2. d
3. b
4. c
5. a
6. b
7. c
8. a
9. a
10. d
### **CHAPTER 3**
1. b
2. c
3. b
4. b
5. a
6. c
7. d
8. a
9. d
10. d
### **CHAPTER 4**
1. b
2. d
3. e
4. e
5. a
6. a
7. d
8. a
9. c
10. d
### **CHAPTER 5**
1. a
2. d
3. d
4. a
5. b
6. d
7. a
8. c
9. a
10. b
### **CHAPTER 6**
1. b
2. d
3. d
4. b
5. b
6. c
7. c
8. d
9. d
10. a
### **CHAPTER 7**
1. c
2. b
3. b
4. a
5. b
6. b
7. c
8. a
9. c
10. a
### **CHAPTER 8**
1. a
2. d
3. a
4. d
5. c
6. b
7. c
8. a
9. b
10. b
### **CHAPTER 9**
1. c
2. b
3. a
4. c
5. d
6. b
7. b
8. b
9. a
10. b
### **CHAPTER 10**
1. a
2. a
3. b
4. b
5. a
6. c
7. b
8. b
9. a
10. a
### **FINAL EXAM**
1. d
2. a
3. b
4. b
5. c
6. c
7. b
8. c
9. a
10. c
11. c
12. b
13. a
14. a
15. b
16. b
17. a
18. d
19. b
20. a
21. b
22. a
23. d
24. b
25. a
26. e
27. b
28. d
29. b
30. e
31. c
32. d
33. b
34. a
35. a
36. a
37. b
38. d
39. a
40. b
41. e
42. c
43. a
44. e
45. d
46. d
47. b
48. b
49. a
50. d
51. d
52. d
53. b
54. d
55. a
56. e
57. d
58. d
59. c
60. d
61. a
62. b
63. a
64. a
65. c
66. b
67. b
68. a
69. b
70. d
71. a
72. c
73. d
74. b
75. b
76. b
77. a
78. a
79. b
80. a
81. a
82. a
83. d
84. d
85. a
86. a
87. b
88. c
89. a
90. a
91. a
92. a
93. c
94. a
95. a
96. c (this is a rationalization)
97. b
98. a
99. d
100. b
## **Index**
_Please note that index links point to page beginnings from the print edition. Locations are approximate in e-readers, and you may need to page down one or more times after clicking or tapping a link to get to the indexed material._
### **A**
ABC Company,
Accelerated depreciation,
Accounting concept of conservatism, , ,
Accounting concept of depletion,
Accounting information, , ,
Accounting majors,
Accounting methods, , , ,
Accounting process, –4
Accounting profession, , , –27, , , , –197, –202, –206, –241,
Accounting treatment, , , ,
Accounts payable, , , , –168,
Accounts receivable, turnover ratio,
Accounts receivable, , , , –90, , –166, –219, , , , ,
Accrual basis of accounting, , –26, –33, , , , ,
Accrued liabilities, ,
Accrued receivables, –94
Accurate financial reporting,
Acquisition cost, , ,
Adverse opinions,
Allowances, , , ,
Amortization, , , –129, ,
Amortization expense, ,
Analyzing prepaid and accrued expenses,
Assets = liabilities + owner's equity, –7
Asset management ratios,
Asset turnover ratio,
Assets, –7, –10, , , –58, –70, , , , –206, , –216, –222, , –230, , –265
accounts receivable, , , –61, , –93, , , , ,
buildings, , , , , , , ,
cash, , –15, , –69, , , , , –221, –228, –240, , , ,
equipment, , –13, , –72, , –119, –126, , , –176, , –242, , ,
investments, –5, , , –70, , –89, , –107, , , , –142, –162, , –180, –220, ,
land, , –23, , –71, , –113, –124, , –138
Association of Certified Fraud Examiners (ACFE),
Assumptions, , , , –104, , , ,
Assurance, , , , –252
Audit report, –201,
Audited financial statements, ,
Auditor's report, , , –208
Auditors' opinion, , ,
### **B**
Bad debts expense, –91,
Balance sheet, –77, –155, , –202, –215, –220, ,
assets, –70, –86, , –216, –222, –262, –265
liabilities, –58, –83, –216, –220, –258,
residual equity, ,
Bank overdraft,
Basic concepts, , –27, , ,
Basic financial accounting and reporting concepts,
Bill Gates,
Bond sinking fund,
Bonds payable, , , , , , –180
Book value, , –114, , –119, –123, , –175,
Borrowed money,
Business assets, , , ,
Business decisions, , ,
Business loans,
Business owners, ,
Business records,
Business transactions, , , ,
Business's financial health,
### **C**
Cabela's Incorporated, –195
Capital expenditures,
Capital lease, –140,
Case Company, –174, , , ,
Cash balance, , –12, , , ,
Cash basis of accounting, , –26, ,
Cash equivalents, , , , , –183
Cash flow activities,
financing, –11, , , , , –140, –161, , –182, , ,
investing, –11, , , –161, , –176, –181, , , ,
operating, –11, –40, –52, , , , , , , –164, –183, , , ,
Cash flows, –5, –15, –26, , , , –166, –175, –183, , , , –239
Cash inflows, ,
Cash outflows,
Cash receipts, , , , , , –162 –166, , ,
Cause-and-effect relationship,
Certified Fraud Examiners (CFEs),
Claims of creditors,
Coca-Cola Enterprises, Inc., –195,
Common stock investment,
Common stock outstanding, –49, ,
Common stock, –51, –76, , , –109, –151, , , , , –224, ,
Common stockholders, , , –151, –155, –225
Company's books, –23
Comparable information,
Compounding interest,
Concept of conservatism, , , , , , –122,
Consolidated financial statements, , , ,
Consulting services, ,
Company liquidity,
Contingent liability, –131
Contractual rights, ,
Contribution of equipment,
Controlling, , –110
Corporation, , , , , , , , –76, , , –109, , , –155, , –195,
Corporations, –50, , , , , , , , , –145, –148, –151, ,
nonpublic/closely held,
publicly held,
Cost of goods sold, , –40, , , , –72, , –103, , , , –168, ,
Cost of sales,
Covenants,
Credit decisions, ,
Credit purchases, –168
Credit sales, , –166, ,
Creditors' claims,
Current assets, –68, –86, –95, –106, , , , –136, , –220
Current assets and liquidity, ,
Current liabilities, –65, , , , ,
Current liabilities and liquidity,
Current portion of long-term debt,
Current ratio, , –220, ,
### **D**
Days inventory supply, ,
Debt, , , , , , , –69, –91, , –107, , , , , –146, –149, , , –219, , –231, –249, –259,
Debt securities, , –85, –106
Declining balance (DB),
DEF Company,
Deferred income tax liabilities/assets, ,
Depletion of natural resources, ,
Depreciation expense, , –71, –113, , –120, , , , ,
Depreciation methods, –116, –120, ,
Diluted EPS, –51,
Direct labor, ,
Direct method, of accounting, , –166, –169, –175, –178,
Discontinued operations, –237
Distribution of assets,
Dividend payout ratio,
Dividends, , , , , –87, –109, , –153, , –161, –180, , –225
Dividends payable, –63
### **E**
Earnings, , , –50, , , –86, , , , , , , , –154, , , –226, , ,
Earnings per share (EPS), ,
Economic resources, –6, –57
Enron, , , , , –262
Equity method, –109
Equity securities, –85,
Estimated liability, ,
Estimated warranty obligations,
Eventual payment,
Example Company, –32, –36, , –45, –48, , –57, , , , –226, –231, , , ,
Expenses, , –11, , –26, –44, , , , , , –92, , –113, , –133, –146, –164, , –170, , , , –222, , , –258, ,
insurance, , –34, , , , , ,
interest, , , , , –110, , , ,
rent, , , , , ,
salaries and wages, , –41, , , , –134,
taxes, , , , , , –145, –161, , , , ,
Extraordinary gains and losses,
### **F**
Fictitious revenue, ,
Financial Accounting Standards Board (FASB), ,
Financial information, –3, , , ,
Financial institutions, ,
business decisions of, , ,
invest in, ,
lend money to,
Financial position, , , , , , ,
Financial reporting, , , , , –56, , , , , , –206, , , –252, –255, –262,
assessing cash flow,
investment and credit decision, ,
resources of a business, ,
Financial statement disclosures,
Financial statement footnotes, , –206, –240,
Financial statements, –15, –206, –231, –265
Financing activities, , , –161, –182,
borrowing of,
repaying of,
First-in, first-out (FIFO),
Fiscal year, , , , , , , , ,
Form -K, –196,
Fraud triangle, –250
asset misappropriation, –248
corruption, –248
fraudulent financial reporting, , –252
Fraudulently misstated financial statements, –265
Full disclosure principle, ,
Future retirement,
### **G**
Generally accepted accounting principles (GAAP), , , , ,
Gross margin, , , –222
Gross profit, , , , , , –265
Gross sales revenue,
### **H**
Historical cost concept, ,
Historical cost, –23, , , ,
Holiday shopping season,
Horizontal analysis, –213
House-cleaning business, ,
### **I**
Impairment loss, –123,
Immaterial expenses,
Income, accumulated other, Income statement, –52, ( _see also_ Profit and Loss Statement)
Income tax depreciation method,
Income tax expense, , –144
Income taxes, , , , –145, , , , ,
Income-producing activities,
Indirect method, of accounting, –164, –169, –175, ,
Information
overload,
relevant, ,
reliable,
Insurance carriers,
Intangible assets, , –73, , , , –129
copyrights, –73, –128
goodwill, , –73, –128
patents, , –73, –128
Internal Revenue Service's tax code,
Inventories of goods,
Inventory cost, , , , , , ,
Inventory cost-flow assumption, , ,
Inventory systems,
Investing activities, , –161, , –179
Investment, , , , –70, –89, , –109, , , –142, , –155, –162, –175, –180, –220, , , –249,
Investments in affiliated companies, –70,
Investments in stocks and bonds,
certificates of deposit,
commercial paper, ,
money market funds, ,
mutual funds, ,
### **J**
Johnson & Johnson,
Joint venture investments, –70
Joint ventures,
### **K**
Kellogg Company,
Key benchmarks,
Key financial ratios,
Kimberly-Clark Corporation, –195,
KPMG,
### **L**
Last-in, last-out (LIFO),
Lease, , –71, –113, , , –140,
Legal and registration fees,
Leverage ratios, –230
Liabilities, –7, –58, –68, , –83, , –135, , –216, –220, –258,
Liquidation, ,
Liquidity ratios, , , ( _see also_ Solvency ratios)
Long-term liability, , –66, , , –137
Lower of cost or market (LCM) concept,
### **M**
Manufacturing costs, , –96,
Manufacturing inventories,
finished goods, ,
goods in process, ,
raw materials and supplies,
Market values, , ,
Marketable securities, –84, –87, , ,
Matching principle, , –27, –33, , –42, –91, , , , , , , ,
Materiality, , , , , , , , ,
McKesson & Robbins,
Merchandise inventory, , –95, , , –168
Misstatement in the financial statements,
Modified Accelerated Cost Recovery System (MACRS),
Money market fund, , –183
Mortgage notes payable,
Multiple-step income statements,
Mutual funds, , ,
### **N**
Natural resources, , –73, , , –125, –129,
legal rights to, , ,
other creations of, ,
Net assets, , , , , –110,
Net income, –10, –27, , , –49, –108, –152, –163, –169, , , , –225, , , , –259,
Net loss, –9, , ,
Net sales, , , –227, , –255,
Net worth, , , , ,
Nonaudited financial statements,
Noncash investing, , , ,
Nonoperating gains and losses,
Nonrecurring,
Notes payable, , , –133
Notes receivable, , , –94,
### **O**
One-owner business, ( _see also_ Sole Proprietorship)
Operating activities, , , , , , , –162, , –176, –179, ,
Operating cash outflow, ,
Operating cycle, –62, , , ,
Operating expenses, –40, , , , , , , , , , ,
administrative, ,
general,
selling, ,
Operating income, ,
Operating loss, ,
Overhead cost, , , , ,
Overstating revenues, , ,
Ownership, –49, , –71, –75, , , , –111, , –150,
controlling interest, , –110
parent company, –110
subsidiary, –110
### **P**
Partnership, , , , , , , , , , –149, –152,
Payroll tax liabilities, ,
Pension and mutual funds,
Pension and postretirement obligations,
Pension fund assets, –70
Pension funds,
Periodicity assumption,
Personal transactions,
Planning, ,
Plant assets, , , , –113, , , –129, –140, ,
Postdated checks and IOUs,
Preferred stock, , , –225
Preferred stockholders, –152, ,
Premium, , , ,
Prepaid expenses, , , , , –84, , , –221
Prepaid rent, ,
Price-earnings (PE) ratio,
Primary business activity, , –44,
Principal manufacturing cost,
Principles, , , –27, , , , , , , –236, ,
Professional auditing standards, , , –203, ,
Profit and loss statement,
Profit margin on sales,
Profit, ,
Profitability ratios,
Property tax expense,
Property, plant, and equipment, , ,
Public accounting firm, –197, –201, , ,
Public Company Accounting Oversight Board,
Public financial markets,
Purchasing equipment,
Purchasing power,
### **Q**
Qualified opinion, –204
Quick current assets,
Quick ratio, –221,
### **R**
Raw materials, , –72, , , ,
Real economic events,
Reasonable assurance, , ,
Regina Company, Inc.,
Relevant information, ,
Reliable information,
Repayment, , , , , , , , ,
of loans, ,
Residual interest, , , ,
Residual nature of the owners' interest, ,
Retained earnings, , , –154
Return on assets,
Return on equity (ROE),
Revenue expenditures, –113
Revenue recognition principle, , , , , , ,
Revenues, , –32, –44, , , , –94, , , , , , –222, , ,
commissions,
consulting fees,
rental fees,
sales revenue, , –37, –90
Royalty payments,
### **S**
Salaries and wages payable, , ,
Sale securities, –89, , –155
Sales discount, , –242
"two ten, net thirty,"
Sales returns, , , –255,
Salvage value, –116, , , ,
Sara Lee Corporation,
Sarbanes-Oxley Act of 2002 (SOX), ,
Savings account,
SDK Consulting, –8, –12,
SDK, Capital, ,
Securities and Exchange Commission (SEC), ,
Securities, held to maturity, , , –107
Selling price, , , –105, ,
Separately identifiable intangible assets,
Shareholders' annual reports,
Short-term liquidity,
Single note payable,
Single-step income statements,
Sole proprietorship, , –48, , , , , –146, , ,
Solvency ratios,
Statement of cash flows, , , –190,
Statement of financial position, ,
Statement of owners' equity, , , ,
Stock dividend, , , , ,
Stockholder, , –49, , , , , –155, , –225, –230,
Stockholders' equity, , , , , , , –150, –155,
Straight-line (SL) depreciation,
Sum-of-the-years-digits method,
### **T**
Tangible physical assets, , ,
Target Corporation, –195,
Tax law, , ,
Trading securities, –87, ,
Transportation costs, ,
Treasury stock, , , ,
Treatment of rent expense,
### **U**
U.S. Treasury bills, ,
Underlying concepts of financial reporting, ,
Unearned revenues, , –133, ,
Units of production (UP),
Unrealized gain/loss, –88, ,
### **V**
Vertical analysis, –212, –215
### **W**
Wal-Mart, –199, –202, –206
Walt Disney Company,
Weighted average, –50, , ,
Withdrawals, , , , –147, , ,
Workforce, –22
Working capital, , ,
WorldCom, ,
### **Z**
ZZZZ Best, ,
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 2,447
|
Book Tickets...
Your Movie ALADDINANDRE RIEU 2019 MAASTRICHT CONCERT- SHALL WE DANCEANNAANNABELLE COMES HOMEAPOCALYPSE NOW - FINAL CUTCHILD'S PLAY (2019)DEAD DON'T DIEDESPICABLE ME 2EPICFAST & FURIOUS PRESENTS: HOBBS & SHAWHOW TO TRAIN YOUR DRAGONMATILDAMEN IN BLACK: INTERNATIONAL MIDSOMMARPADDINGTONPULP FICTION RESERVOIR DOGS SCREAM (1996)SHREKSHREK 2SPIDER-MAN: FAR FROM HOMESTUBERTHE ANGRY BIRDS MOVIE 2THE HUSTLETHE LION KING THE LION KING 3DTHE QUEEN'S CORGITHE SECRET LIFE OF PETS 2TOY STORY 4X-MEN: DARK PHOENIXYESTERDAY
RISE: The Story Of Augustines
Director: Todd Howe
Cast: William McCarthy (Himself), Robert Allen (Himself), Eric Sanderson (Himself)
Feature Runtime: 1 hr 20 mins
Pre-Show: Approx. 20 mins
In August of 2009 singer William McCarthy's younger brother James, in the midst of a lengthy incarceration on charges of attempted murder, was found dead of an apparent suicide inside Napa State Hospital. Just one week later, McCarthy's band Pela imploded, leaving both him and band mate Eric Sanderson devastated, in financial ruin, and with a half-finished album that may never see the light of day.
RISE: The Story of Augustines chronicles the journey of two men who faced their demons, refused to fall and established a lifelong brotherhood as they struggled to finish their seminal debut record, all while trying to maintain their own sanity and pay tribute to James. McCarthy's songwriting during the toughest year of his life would come to be known as his defining work. The album Rise Ye Sunken Ships stands as a testament both to the pain of loss and the celebration of life. The two resurfaced with a new name, Augustines and, together with drummer Robert Allen, who shepherded William and Eric through their darkest of times, rose up to become one of the most celebrated independent bands in the world.
This is a truly compelling story of struggle, perseverance, brotherhood and hope.
Press To Watch Trailer
Home/ Movies This Week/RISE: The Story Of Augustines
Book Tickets for RISE: The Story Of Augustines
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,771
|
Well-loved (and well-used) trombones, trumpets, violins and even an accordion can be found buzzing, squawking and howling during the hour-long "petting zoo." Fun to do, and fun to watch.
The music petting zoo will set up shop from 2 to 3 p.m. Saturday, April 16, at the Cambria Library, 1043 Main St.
All ages are welcome. For details, call 927-4336.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,438
|
{"url":"https:\/\/learn.careers360.com\/jobs\/question-gaurav-spends-30-of-his-monthly-income-on-food-articles-40-of-the-remaining-on-conveyance-and-clothes-and-saves-50-of-the-remaining-if-his-monthly-salary-is-rs18400-how-much-money-does-he-save-every-month-7494\/","text":"## Filters\n\nQ&A - Ask Doubts and Get Answers\nQ\n\n# Gaurav spends 30% of his monthly income on food articles, 40% of the remaining on conveyance and clothes and saves 50% of the remaining. If his monthly sa\n\nGaurav spends 30% of his monthly income on food articles, 40% of the remaining on conveyance and clothes and saves 50% of the remaining. If his monthly salary is Rs.18,400, how much money does he save every month?\n\n\u2022 Option 1)\n\n3864\n\n\u2022 Option 2)\n\n4903\n\n\u2022 Option 3)\n\n5091\n\n\u2022 Option 4)\n\n6789\n\n\u2022 Option 5)\n\n5849\n\nSaving = \u00a0 $\\inline 18400\\times \\left ( \\frac{70}{100} \\right )\\times \\left ( \\frac{60}{100} \\right )\\times \\left ( \\frac{50}{100} \\right )$\n$\\inline \\Rightarrow$\u00a0 3864 .","date":"2019-07-19 20:43:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7552573680877686, \"perplexity\": 12189.687498535259}, \"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-2019-30\/segments\/1563195526359.16\/warc\/CC-MAIN-20190719202605-20190719224605-00098.warc.gz\"}"}
| null | null |
Max Steiner (født Maximilian Raoul Walter Steiner 10. maj 1888, i Wien, Østrig-Ungarn, død 28. december 1971, i Hollywood, Los Angeles, USA) var en Oscar-belønnet østrigsk-amerikansk komponist af musik til teateropsætninger og film. Han er nok mest kendt for filmmusikken til klassikerne Casablanca, King Kong (1933) og Borte med blæsten.
Max Steiner modtog 26 Oscar-nomineringer for sit bidrag til filmmusikken gennem sin karriere og vandt tre af dem, for The Informer (1935), Now, Voyager (1942) og Since You Went Away (1944).
Han har en stjerne på Hollywood Walk of Fame, på 1551 Vine Street.
Film, han blev Oscar-nomineret for
The Gay Divorcee (1934)
The Lost Patrol (1934)
The Garden of Allah (1936)
Jezebel (1938)
Borte med blæsten (1939)
The Letter (1940)
Sergeant York (1941)
Casablanca (1942)
The Adventures of Mark Twain (1944)
Rhapsody in Blue (delt med Ray Heindorf, 1945)
Night and Day (delt med Ray Heindorf, 1946)
Life with Father (1947)
My Wild Irish Rose (delt med Ray Heindorf, 1947)
Johnny Belinda (1948)
Beyond the Forest (1949)
The Flame and the Arrow (1950)
The Miracle of Our Lady of Fatima (1952)
The Jazz Singer (delt med Ray Heindorf, 1953)
The Caine Mutiny (1954)
Battle Cry (1955)
Andre film (udvalg)
King Kong (film fra 1933)
Eksterne henvisninger
Max Steiner hos AmericanComposers.com
Filmkomponister fra USA
Oscar for bedste musik
Hollywood Walk of Fame
Personer fra Wien
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,983
|
Lutka Pink (wł. Ludwika Pinkusiewicz) (ur. 1906 w Warszawie, zm. 1998 w Nowym Jorku) – polska malarka, od 1937 roku tworząca we Francji.
Życiorys
Młodość w Polsce
Urodziła się w ortodoksyjnej rodzinie żydowskiej jako Sznajdla Łaja Pinkusewicz. W późniejszych latach utrzymywała, że urodziła się w 1916 r. i ten rok podawała we wszystkich dokumentach. Ukończyła pięcioletnie żydowskie gimnazjum żeńskie "Chawaceles" (hebr. Lilia), a od 1928 r. studiowała w warszawskiej Szkole Sztuk Pięknych, mimo iż dziewczęta z ortodoksyjnych żydowskich rodzin bardzo rzadko kontynuowały naukę po gimnazjum. Jej wykładowcami byli Mieczysław Kotarbiński, Władysław Skoczylas, Felicjan Szczęsny Kowarski oraz Michał Walicki; uczęszczała również na wykłady Karola Stryjeńskiego, a także uczestniczyła w organizowanych przez niego obozach sportowo-malarskich. Podczas nauki jej prace były prezentowane na wystawach zbiorowych w Instytucie Propagandy Sztuki.
Dyplom ukończenia studiów otrzymała w 1934 r., a następnie poświęciła się sztuce. W tym okresie powstawały pejzaże, martwe natury i portrety, tworzyła lekkimi pociągnięciami pędzla, przywiązując wagę do światła i materii. Większość obrazów z tego okresu zaginęła podczas II wojny światowej.
Pobyt we Francji
Podczas studiów poznała Kazimierza Zielenkiewicza, z którym się związała, a w 1938 r. zaręczyła się. W tym samym roku Zielenkiewicz otrzymał stypendium artystyczne i oboje wyjechali do Paryża, gdzie Tadeusz Natanson zapoznał ich z Édouardem Vuillardem, którego twórczość oboje cenili. Odbyli również krótką podróż do Włoch, a po powrocie do Francji Lutka Pink uległa fascynacji pracami Pierre'a Bonnarda, które cechowała jedność koloru i światła.
W 1940 r. poślubiła Kazimierza Zielenkiewicza, a ponieważ była Żydówką wobec wkroczenia do Francji hitlerowców zmuszona była się ukrywać. Oboje skorzystali z zaproszenia Jana Wacława Zawadowskiego i przebywali w jego posiadłości w Orcel koło Aix-en-Provence, tworząc pod silnym wpływem Paula Cezanne'a. W tym okresie ich prace powstawały w pozornie podobny sposób: Lutka tworząc pejzaże i martwe natury stosowała mocne, barwne plamy tworzone grubymi pociągnięciami pędzla, ale bryły były rozbite, a barwy rozproszone. Do Paryża powrócili w 1946 r. Wiedząc, że jej cała rodzina zginęła podczas Holokaustu, postanowiła pozostać we Francji na stałe. W 1952 r. związek z Kazimierzem Zielenkiewiczem uległ zakończeniu, a para zdecydowała się na separację zakończoną w 1956 r. rozwodem. W tym czasie Lutka prowadziła intensywne życie artystyczne, a sławę w środowisku twórców przyniósł jej ekstrawagancki tryb życia, witalność i rzadko spotykany temperament. Porzuciła sztukę przedstawiającą, skupiając się na taszyzmie i abstrakcji, tworzyła dużo i oryginalnie, wzbudzając ciekawość krytyków. Jej prace uczestniczyły w wielu odbywających się we Francji wystawach, m.in. Salonach Majowych, Biennale na Lazurowym Wybrzeżu oraz Salonach Młodych Malarzy. Została doceniona przez wziętego krytyka sztuki Pierre'a Restany'ego, który w jednym ze swoich felietonów określił Lutkę jako drugą najlepszą artystkę w Paryżu. Ona sama pytana o motto swojej twórczości mawiała: radość życia, nic tylko radość życia. Potrafiła uzyskać efekt wibrowania kolorów na płaszczyźnie, a wiele jej prac oddaje harmonię i rytmiczność, radość tworzenia sprawia, że forma ulega pozornemu rozsypaniu na płaszczyźnie, przy jednoczesnej zwartej konstrukcji.
U schyłku życia artystka coraz częściej mówiła o przeprowadzce do Izraela, który wielokrotnie odwiedzała. Zmarła podczas pobytu w Nowym Jorku, ale zgodnie z jej życzeniem spoczęła 30 marca 1998 w Jerozolimie na cmentarzu na Górze Oliwnej.
Przypisy
Bibliografia
Zofia Dubowska "Lutka Pink - pożegnanie", Gazeta Wyborcza 13 maja 1998
sylwetka w serwisie Art-Price
Atelier Lutka Pink Richelieu-Drouot, salle 4, vendredi 24
Linki zewnętrzne
Portret Lutki Pink
Absolwenci Szkoły Sztuk Pięknych w Warszawie
Polacy we Francji
Polscy malarze pochodzenia żydowskiego
Urodzeni w 1906
Zmarli w 1998
Ludzie urodzeni w Warszawie
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,012
|
Dobrivoje Trivić (szerb cirill betűkkel: Добривоје Тривић; Ševarice, 1943. október 26. – Újvidék, 2013. február 26.) Európa-bajnoki ezüstérmes szerb labdarúgó.
Pályafutása
Klubcsapatban
A válogatottban
1966 és 1969 között 13 alkalommal szerepelt a jugoszláv válogatottban. Részt vett az 1968-as Európa-bajnokságon.
Sikerei, díjai
FK Vojvodina
Jugoszláv bajnok (1): 1965–66
Olympique Lyon
Francia kupa (1): 1972–73
Francia szuperkupa (1): 1973
Jugoszlávia
Európa-bajnoki döntős (1): 1968
Források
1943-ban született személyek
2013-ban elhunyt személyek
Jugoszláv labdarúgók
Szerb labdarúgók
Labdarúgó-középpályások
Az FK Vojvodina labdarúgói
Az Olympique Lyonnais labdarúgói
A Toulouse FC labdarúgói
Az 1968-as Európa-bajnokság labdarúgói
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,410
|
Q: POSTGRES ERROR: could not load library I am not able to perform any operation on my database. The error I'm getting is :
"error: could not load library "C:/Program Files/PostGreSQL/9.1/lib/postgis-2.0.dll": Invalid access to memory location."
The error popped up only today morning and it was fine until yesterday.
Thanks in advance for your help. :D
A: A system restart fixed the thing. :D
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,811
|
Manuscripts Online (http://www.manuscriptsonline.org/) enables you to search a diverse body of online primary resources relating to written and early printed culture in Britain during the period 1000 to 1500. The resources include literary manuscripts, historical documents and early printed books which are located on websites owned by libraries, archives, universities and publishers. The Manuscripts Online API enables users to connect programmatically to the search engine, using GET parameters, and retrieve search results in an XML format.
The Manuscripts Online API is available at https://www.hrionline.ac.uk/Manuscripts/search.jsp.
At least one keyword, person or place name must be specified. Any number of parameters may be combined but, unless otherwise stated, each parameter can only have single value - see URL-encoded Strings for details on how to search for multiple values or phrases. Input parameters should be appended as parameters to the URL (e.g. https://www.hrionline.ac.uk/Manuscripts/search.jsp?rs=katherine&sr=bc,ci,te. Further examples of API queries can be found in Example API Queries).
kw Keyword Any URL-encoded String. Details about the format of valid URL-encoded search strings can be found beneath the table.
kwv Keyword Name Variant One of the following, if this parameter is not specified, variants are not searched.
pc Place Name Any valid URL-encoded String.
pcv Place Name Variant One of the following, if this parameter is not specified, variants are not searched.
rs Person Name Any valid URL-encoded String. The search will provide matches that contain at least one person name that matches all the criteria - i.e. a search for George* Clarence*, will search for any matches will contain a person name which contains both George and Clarence (in that order) and other characters in place of the wildcard. This field can also be used to search for people referred to by specific titles, e.g. Bishop of London, as well as given and family names.
rsv Person Name Variant One of the following, if this parameter is not specified, variants are not searched.
mi Reference Any URL-encoded String. Details about the format of valid URL-encoded search strings can be found beneath the table.
dtf Cited Date From Any date in the form yyyy or yyyy-mm-dd. Zero may be used (for example 1425-00-00 is equivalent to 1425 or 1425-03-00 may be used to search for March, 1425). The date search includes dates mentioned in the document as well as dates associated with the document. The date from should be equal to or less than the date to (if a date to is specified).
dtt Cited Date To Any date in the form yyyy or yyyy-mm-dd. Zero may be used (for example 1425-00-00 is equivalent to 1425 or 1425-03-00 may be used to search for March, 1425). The date search includes dates mentioned in the document as well as dates associated with the document. The date to should be equal to or more than the date from (if a date from is specified).
sdf Source Date From Any date in the form yyyy or yyyy-mm-dd. Zero may be used (for example 1425-00-00 is equivalent to 1425 or 1425-03-00 may be used to search for March, 1425). The date search includes dates mentioned in the document as well as dates associated with the document. The date from should be equal to or less than the date to (if a date to is specified).
sdt Source Date To Any date in the form yyyy or yyyy-mm-dd. Zero may be used (for example 1425-00-00 is equivalent to 1425 or 1425-03-00 may be used to search for March, 1425). The date search includes dates mentioned in the document as well as dates associated with the document. The date to should be equal to or more than the date from (if a date from is specified).
The order in which the resources are specified does not matter and does not affect the returned results.
ct Category A letter code. More than one category may be specified; each category code should be separated by a comma (URL-encoded as %2c). Some resources and / or results may match more than one category - for example, British History Online falls into both the "Literary Manuscripts" category and the "Official Documents (government, civic, legal, religious)" category.
ft Format A single letter code. More than one format may be specified; each format code should be separated by a comma (URL-encoded as %2c).
Search strings may contain several words, separated by spaces (URL-encoded as +) - which will result in an OR search, matching texts that contain any of the words. An exact match may be carried out by enclosing a multi-word phrase in double quotes, or the user may specify that one or more of the words must be present by using a plus (URL-encoded as %2b) in front of the required word(s). Results containing certain words can be excluded by placing a minus sign(-) in front of the words. + and - can not be used to mandate or exclude words within exact phrases (i.e. within double quoted phrases) but can be used before phrases to indicate a required or excluded phrase.
* can be used as a wildcard in the middle or end of a word or phrase (e.g. ta*lor returns taylor and tailor), but not at the start; the more letters you specify the better.
The results are returned as a UTF-8 encoded XML document with a root element, CHSP. The XML document consists of two main subsections - a Q element, which details the query submitted and a RES element which provides the results, and also facets which can be used to drill down through the results. Results are generally returned in channels, each channel representing a single collection / data source.
The REFINE section contains details of search facets, each detailed with a PARAM element, that can be used to drill down into the results. Each result set can be drilled down via by document category ("ct"), document format ('ft'), date ("dt" ),and availability ("ac" - whether the source material is available to all users or only to those with a subscription). Each PARAM element has three attributes - name(ct, ft, dt or ac); value (a String description of the limits of the facet), and match (the number of documents that match this facet). As documents may match more than one facet description, the total number of matches may exceed the total number of results.
Fewer than 20 results are returned in total - a single, mixed channel is returned.
Results are only returned from one source collection, either because the query specified only one collection or only one collection contained matches.
Unless fewer than 20 results are returned, each channel relates to one specific source collection. The CHANNEL element has attributes sn (start index for results), en (end index for results), ip(items per page), m (total number of matches for this channel), type, sid (electronic resource id, equivalent to the sr codes for queries, or "nyi" if a mixed channel - further details on the sr codes can be found in the Input Parameters table). The channels are ordered by Lucene scores - the one with the highest scoring first result is returned first. This means that the channels have no fixed order.
Each channel has a channel name (CN) which either identifies it as a mixed channel or provides a brief textual description of the source collection. Each channel then consists of a number of R elements, each of which contains a single (numbered) result or match. Within the R Element, the T element provides the match title, one or more U elements the provides the URL (the link to the document in the source collection) ) with attribute CHOICE denoting the url as either default or a giving its source name, and the S element - a text snippet which usually contains some highlighted terms (within b tags) that match the search query. As many of the source collections require the user to have a subscription, the URL may display an error rather than direct the user to the source material if the user is not currently logged in, via a personal or institutional subscription.
Invalid use of codes or invalid data will usually result in the return of an empty result set, such as this. Where another error occurs, an error document is returned. Please contact us if you encounter unexpected errors.
The HRI developer currently responsible for maintaining the Manuscripts Online API is Katherine Rogers. Technical support may also be also be requested from hri-support@sheffield.ac.uk. Further information on HRI Digital can be found at http://hridigital.shef.ac.uk/.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,686
|
\section{Introduction and summary of results}
Since the Universe expands exponentially fast during inflation, gradients are
exponentially erased and can be neglected. At the same time, the exponential
stretching of spatial lengths
{\bf classicalizes} the physics and allows a {\bf classical} treatment.
One can therefore consider a homogeneous and classical inflaton field which thus
determines self-consistently a homogenous and isotropic Friedman-Robertson Walker
metric sourced by this inflaton.
This treatment is valid for early times well after the Planck time
$ t = 10^{-44} $ sec., at which the quantum fluctuations are expected to be large and
thus a full quantum gravity treatment is required.
\medskip
In this paper we study the entire coupled evolution of the inflaton field $ \phi(t) $ and
the scale factor $ a(t) $ of the metric for generic initial conditions,
fixed by the values of $ \phi(t_0) $ and $ d\phi(t_0)/dt $ at a given
initial time $ t_0 $.
\medskip
We show that the {\bf generic} early universe evolution has three stages: a
decelerated fast-roll stage followed by an inflationary fast-roll stage and then
by a slow-roll inflationary regime which is an attractor always reached for
{\bf generic} initial conditions. This evolution is valid for all regular inflaton potentials.
In addition, we find a particular (extreme)
slow-roll solution starting from $ t = -\infty $ in which the fast-roll stages
are absent.
\medskip
The evolution backwards in time from
$ t_0 $ reachs generically a mathematical singularity at some time $ t = t_* $
where the scale factor $ a(t) $ vanishes, and the Hubble parameter becomes singular.
We find the general behaviour of the inflaton and the scale factor
near the singularity as given by eqs. (\ref{sing1})-(\ref{asegs}) and determine the
validity of the classical approximation, namely $ (H/M_{Pl})^2 \ll 1 $.
It must be stressed that such mathematical singularity is attained extrapolating
the classical treatment where it is {\bf no more valid}. The singularity
is never reached in the validity region of the classical treatment and therefore
such mathematical singularity {\bf is not a real physical phenomenon} here.
Quantum loops effects turns to be less than 1\% for $ t - t_* > 10^{-42} $ sec and therefore
the classical treatment of the inflaton and the space-time can be trusted
well before the begining of inflation.
\medskip
The fast-roll (both decelerated and inflationary) and slow-roll regimes are analyzed in
detail, with both the exact numerical evolution and an analytic approximation,
and the whole equation of state evolution in the three regimes.
We consider here the double well (broken symmetric) fourth order inflaton potential
since it gives the best description of the CMB+LSS data \cite{mcmc,biblia}
within the Ginsburg-Landau effective theory approach we follow.
\medskip
The characteristic time scale of the fast-roll era turns to be
$ t_1= (1/m) \; \sqrt{V(0)/[3 \; M^4] } \sim 10^4 \; t_{Planck} $ where
$ V(0) $ is the double well inflaton potential at zero inflaton field,
$ m $ is the inflaton mass and $ M $ the energy scale of inflation.
The time scale of the inflaton in the extreme slow roll solution goes as
the inverse of $ t_1 $, namely $ 1/[m^2 \; t_1] $.
\medskip
We study the {\bf whole} evolution of the curvature and tensor fluctuations along
the three succesive regimes: decelerated fast-roll followed by inflationary
fast-roll and then inflationary slow-roll, and compute the power spectrum
by the end of inflation.
The fluctuations feel a {\bf singular attractive} potential
near the $ t = t_* $ singularity
(as in the case of a particle in a central singular potential) with {\bf exactly} the
{\bf critical} strength ($ -1/4 $) for which the fall to the centre becomes possible.
Precisely, the logarithmic behaviour of the fluctuations
for $ t \to t_* $ eq.(\ref{sr0}) describes the fall to $ t = t_* $ for the
critical strength of the potential $ W_\mathcal{R} $ felt by the fluctuations.
\medskip
We generalize the Bunch-Davies initial conditions (BDic) to the present case in
which the potential felt by the fluctuations can never be neglected.
In general, the mode functions for large $ k $ behave as
free modes since the potential $ W_\mathcal{R} $
becomes negligible in this limit except at the singularity
$ t=t_* $. One can then impose
Bunch-Davies conditions for large $ k $ which corresponds to
assume an initial quantum vacuum Fock state, empty of curvature excitations
\begin{equation}\label{BDkIntro}
S_\mathcal{R}(k;\eta) \buildrel{k \to \infty }\over=
\frac{e^{-i \; k \; \eta}}{\sqrt{2 \; k}}
\end{equation}
and therefore
$$
\frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) \buildrel{k \to \infty }\over=
-i \; k \; S_\mathcal{R}(k;\eta_0)\; .
$$
Here $ \eta $ stands for the conformal time: $ d\eta = dt / a(t) $.
Eq.(\ref{BDkIntro}) fulfils the Wronskian normalization
(that ensures the canonical commutation relations)
\begin{equation}\label{wronskianIntro}
W[S_\mathcal{R},S^*_\mathcal{R}]= S_\mathcal{R} \;
\frac{dS^{*}_\mathcal{R}}{d \eta} -
\frac{dS_\mathcal{R}}{d \eta} \; S^*_\mathcal{R} = i \; .
\end{equation}
In asymptotically flat (or conformally flat) regions of the space-time
the potential felt by the fluctuations $ W_\mathcal{R}(\eta) $ vanishes and the fluctuations
exhibit a plane wave behaviour for {\bf all} $ k $ (not necesarily large).
This is not the case in strong gravity fields or near curvature singularities
as in the present cosmological space-time where $ W_\mathcal{R}(\eta) $ can
never be neglected at fixed $ k $. However, we can choose Bunch-Davies initial
conditions (BDic) at $ \eta = \eta_0 $ (or equivalently, $ t = t_0 $) by imposing
\begin{equation}\label{BD1Intro}
\frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) = -i \; k \;
S_\mathcal{R}(k;\eta_0) \quad {\rm for ~ all} \; k \; .
\end{equation}
That is, we consider the initial value problem for the mode functions
giving the values of $ S_\mathcal{R}(k;\eta) $ and
$ dS_\mathcal{R}/d \eta $ at $ \eta = \eta_0 $.
This condition combined with the Wronskian condition
eq.(\ref{wronskianIntro}) implies that
\begin{equation}\label{BD2Intro}
|S_\mathcal{R}(k;\eta_0)| = \frac1{\sqrt{2 \; k}} \quad , \quad
\left| \frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) \right| =
\sqrt{\frac{k}2} \; .
\end{equation}
which is equivalent to eq.(\ref{BDkIntro}) for large $ k $.
\medskip
The power spectrum at the end of slow-roll inflation $ P_\mathcal{R}(k) $
gets dynamically modified by the effect of the preceding fast-roll eras
through the transfer function of initial conditions $ D(k) $:
\begin{equation} \label{powRIntro}
P_\mathcal{R}(k)= P^{BD}_{\mathcal{R}}(k)\left[1+ D(k) \right] \; ,
\end{equation}
$ D(k) $ accounts for the effect of both the initial conditions
and the fluctuations evolution during fast-roll (before slow-roll).
$ D(k) $ depends on the time $ t_0 $ at which BDic are imposed.
\medskip
The power spectrum $ P^{BD}_{\mathcal{R}}(k) $ corresponds to start the evolution
with pure slow-roll from $ t_0 \to-\infty $ and
with BDic eq.(\ref{BD1Intro})-eq.(\ref{BD2Intro}) imposed
there at $ t_0 \to-\infty $, that is $ \eta_0 =-\infty $.
$ P^{BD}_{\mathcal{R}}(k) $ is given by its customary pure slow-roll expression,
\begin{equation}\label{pbdIntro}
\log P^{BD}_{\mathcal{R}}(k) = \log A_s(k_0) + (n_s-1) \; \log\frac{k}{k_0} +
\tfrac12 \; n_{run} \; \log^2\frac{k}{k_0} +{\cal O}\left(\frac1{N^3}\right) \; .
\end{equation}
where $ N $ is the number of inflation efolds since the pivot CMB scale $ k_0 $ exits
the horizon. We take here $ N=60 $.
Actually, BDic can be imposed at $ \eta = \eta_0 = -\infty $ if and only if
the inflaton evolution {\bf also} starts at $ \eta = \eta_0 = -\infty $.
This {\bf only} happens for a {\it particular} inflaton solution: the {\it extreme slow-roll}
solution that we explicitly present and analyze in sec. \ref{esr}.
In the extreme slow--roll case the fast-roll eras are absent,
BDic are imposed at $ t_0 \to-\infty $ (that is $ \eta_0 =-\infty $), then
$ D(k) = 0 $ and $ P_\mathcal{R}(k) = P^{BD}_{\mathcal{R}}(k) $. Only in this case
the fluctuation power spectrum at the end of inflation is the usual power spectrum
$ P^{BD}_{\mathcal{R}}(k) $ eq.(\ref{pbd}).
\medskip
When BDic are imposed at {\bf finite times} $ t_0 $, the spectrum {\bf is not} the
usual $ P^{BD}_{\mathcal{R}}(k) $ but it gets modified by
a non-zero transfer function $ D(k) $
eq.(\ref{powR}). The power spectrum $ P_{\mathcal{R}}(k) $ vanishes at $ k = 0 $
and exhibits oscillations which vanish at large $ k $ [see figs. \ref{fig:DBDC} and \ref{fig:BDs}]
\medskip
Generically, the power spectrum vanishes at $ k = 0 $ and we thus have
\begin{equation}\label{propgIntro}
1 + D(k) \buildrel{k \to 0 }\over= {\cal O}(k^{n_s+1}) \quad .
\end{equation}
as shown in sec. \ref{dcero}.
$ D(k) $ presents a first peak for $ k \sim 2/\eta_0 $ and then oscillates asymptotically
with decreasing amplitude such that
\begin{equation}\label{propgIntro2}
D(k) \buildrel{k \to \infty }\over= {\cal O}\left(\frac1{k^2}\right) \; .
\end{equation}
\medskip
We solved numerically the fluctuations equation
with the BDic eq.(\ref{BD}) covering both the fast-roll and slow-roll
regimes, namely for different initial times $ t_0 $ ranging from the singularity
$ \tau = \tau_* $ till the transition time $ \tau_{trans} $
from fast-roll to slow-roll. That is to say,
we solved the fluctuations evolution for BDic imposed at different times in the three eras
and we compare the resulting power spectra among them.
We computed the corresponding transfer function, $ D(k) $
for the BDic imposed at the different eras.
We depict $ 1 + D(k) $ vs. $ k $ for the different
values of the time $ t_0 $ where BDic are imposed in figs. \ref{fig:DBDC}.
\medskip
When the BDic are imposed during the fast--roll stage well {\bf before} it ends,
$ D(k) $ changes much more significantly than along the extreme slow roll solution.
This is due to two main effects: the
potential felt by the fluctuations is attractive during fast--roll, and $ \eta_0 $, (far from
being almost proportional to $ 1/a(\eta) $), tends to the constant value
$ \eta_\ast $ as $ \tau \to \tau_\ast^+ $ and $ a(\eta)\to 0 $. The numerical
transfer functions $ 1+ D(k) $ obtained from eqs.(\ref{curvapot}) and (\ref{powR})
are plotted in figs.~\ref{fig:DBDC}.
\medskip
We have also computed $ D(k) $ analytically with BDic at finite times $ \eta_0 $, and a simple
form is obtained in the scale invariant case,
which is the leading term in the slow-roll expansion:
\begin{equation}\label{dsr32Intro}
D(k) = \frac{\cos 2x}{x^2} - \frac{\sin 2x}{x^3} + \frac{\sin^2 x}{x^4}
\quad , \quad x \equiv k \; \eta_0 \; .
\end{equation}
\medskip
Different initial times $ t_0 $ lead essentially to a rescaling of $ k $
in $ D(k) $ by a factor $ \eta_0 $ since the
conformal time $ \eta $ is almost proportional to $ 1/a(\eta) $ during slow-roll
[see figs.~\ref{fig:DBDC} and below eq.(\ref{dsr})].
By virtue of the dynamical attractor character of slow--roll,
the power spectrum when the BDic are imposed at a finite time
$ t_0 $ cannot really distinguish between the extreme slow--roll solution or any other
solution which is attracted to slow--roll well before the time $ t_0 $.
\medskip
Using the transfer function $ D(k) $ we obtained, we computed the change on the
CMB multipoles $ \Delta C_{\ell}/ C_{\ell} $ for $ \ell =1, 2 $ and $3$ as
functions of the starting instant of the fluctuations $ t_0 $.
We plot $ \Delta C_{\ell}/ C_{\ell} $ for $ 1 \leq \ell \leq 5 $
vs. $ t_0 - t_\ast $ in fig. \ref{dcl}. We see that $ \Delta C_{\ell}/ C_{\ell} $ is {\bf positive} for
small $ t_0 - t_\ast $ and {\bf decreases} with $ t_0 $ becoming then
{\bf negative}. The CMB quadrupole observations indicate a large {\bf suppression}
thus indicating that $ t_0 - t_\ast \gtrsim 0.05/m \simeq 10100 \; t_{Planck}$.
\medskip
The fact that choosing BDic leads to a primordial power and its respective
CMB multipoles which correctly {\bf reproduce} the observed spectrum
justifies the use of BDic.
\medskip
Besides finding a CMB quadrupole suppression in agreement with observations
\cite{biblia}-\cite{quamc},
we provide here {\bf predictions} for the dipole and $ \ell \leq 5 $-multipole
suppressions. Forthcoming CMB observations can provide better data to confront our
CMB multipole suppression predictions.
It will be extremely interesting to measure the primordial dipole and compare
with our predicted value.
\section{The pre-inflationary and inflationary fast-roll eras}\label{uno}
The current WMAP data are validating the single field slow-roll scenario
\cite{WMAP5}. Single field slow-roll models provide an appealing,
simple and fairly generic description of inflation. This
inflationary scenario can be implemented using a scalar field,
the \emph{inflaton} with a Lagrangian density (see for example ref. \cite{biblia})
\begin{equation}
\mathcal{L} = a^3(t)\left[\frac{\dot{\varphi}^2}2 -
\frac{(\nabla\varphi)^2}{2 \, a^2(t)}-V(\varphi) \right] \; ,
\end{equation}
where $ V(\varphi) $ is the inflaton potential. Since the universe
expands exponentially fast during inflation, gradient terms
are exponentially suppressed and can be neglected.
At the same time, the exponential stretching of spatial lengths
classicalize the physics and permits a classical treatment.
One can therefore consider an homogeneous and classical inflaton field
$ \varphi(t) $ which obeys the evolution equation
\begin{equation}\label{eqno}
{\ddot \varphi} + 3 \, H(t) \; {\dot \varphi} + V'(\varphi) = 0
\end{equation}
in the isotropic and homogeneous FRW metric which is sourced by the inflaton
\begin{equation}\label{FRW}
ds^2= dt^2-a^2(t) \; d\vec{x}^2
\end{equation}
$ H(t) \equiv {\dot a}(t)/a(t) $ stands for the Hubble parameter.
The energy density and the pressure for a spatially homogeneous inflaton
are given by
\begin{equation}\label{enerpres1}
\rho = \frac{\dot{\varphi}^2}2+ V(\varphi)
\quad , \quad p =\frac{\dot{\varphi}^2}2-V(\varphi) \; .
\end{equation}
Threfore, the scale factor $ a(t) $ obeys the Friedmann equation,
\begin{equation}\label{frinf}
H^2(t) = \frac1{3 M^2_{Pl}} \left[\frac12 \; \dot \varphi^2 +
V(\varphi)\right] \; .
\end{equation}
In order to have a finite number of inflation efolds, the inflaton
potential $ V(\varphi) $ must vanish at its absolute minimum
\begin{equation}\label{minV}
V'(\varphi_{min})=V(\varphi_{min}) = 0
\end{equation}
Otherwise, inflation continues forever.
\medskip
We formulate inflation as an effective field theory within the
Ginsburg-Landau spirit \cite{1sN,gl,biblia}. The theory of the second order
phase transitions, the Ginsburg-Landau theory of superconductivity,
the current-current Fermi theory of weak interactions, the sigma model of
pions, nucleons (as skyrmions) and photons are all successful
effective field theories. Our work shows how powerful is
the effective theory of inflation {\bf to predict observable quantities}
that can be or will be soon contrasted with experiments.
\medskip
The effective theory of inflation should be the
low energy limit of a microscopic fundamental
theory not yet precisely known. The energy scale of inflation $ M $ should be
at the Grand Unified Theory (GUT) energy scale in order to
reproduce the amplitude of the CMB anisotropies \cite{biblia}.
Therefore, the microscopic theory of inflation
is expected to be a GUT in a cosmological space-time.
Such a theory of inflation would contain many fields of various spins.
However, in order to have a
homogeneous and isotropic universe the expectation value of the
energy-momentum tensor of the fields must be homogeneous and isotropic.
The inflaton field in the effective theory may be a coarse-grained average of
fundamental scalar fields, or a composite (bound state) of fundamental fields
of higher spin, just as in superconductivity. The inflaton does
not need to be a fundamental field, for example it may emerge as
a condensate of fermion-antifermion pairs $ < {\bar \Psi} \Psi> $
in a GUT in the cosmological background. In
order to describe the cosmological evolution is enough to consider
the effective dynamics of such condensates. The relation between
the effective field theory of inflation and the
microscopic fundamental GUT is akin to the relation between the
effective Ginzburg-Landau theory of superconductivity and the
microscopic BCS theory, or like the relation of the $O(4)$ sigma
model, an effective low energy theory of pions, photons and chiral
condensates with quantum chromodynamics (QCD) \cite{quir}.
Vector fields have been considered to describe inflation in
ref.\cite{gmv}. The results for the inflaton should not be very different from the
effective inflaton description since the energy-momentum tensor
of the vector field is to be taken homogeneous and isotropic.
Namely, we are always in the presence of a scalar condensate.
Since the mass of the inflaton is given by $ M^2/M_{Pl} \sim 10^{13} $GeV
\cite{biblia}, massless fields alone cannot describe inflation which
leads to the observed amplitude of the CMB anisotropies.
\medskip
The classical inflaton potential $ V(\varphi) $ gets modified by
quantum loop corrections.
We computed relevant quantum loop corrections to inflationary dynamics
in ref. \cite{biblia,effpot}. A
thorough study of the effect of quantum fluctuations reveals that
these loop corrections are suppressed by powers of $
\left(H/M_{Pl}\right)^2 \sim 10^{-9} $ where $ H $ is the Hubble parameter during
inflation \cite{biblia,effpot}. Therefore, quantum loop corrections
are very small, a conclusion that validates the reliability of the classical
approximation and of the effective field theory approach to
inflationary dynamics. In particular, the (small) one-loop
corrections to the potential in an inflationary universe are very
different from the Coleman-Weinberg form \cite{biblia,effpot}.
\medskip
We choose the inflaton field initially homogeneous which
ensures it is always homogeneous. The fluctuations around
are small and give small corrections to the homogeneity
of the Universe. The rapid expansion of the Universe,
in the inflationary regimes, takes care
of the classical fluctuations, quickly flattening an eventually
non-homogeneous condensate.
\subsection{The complete inflaton evolution through the different eras}
It is convenient to use the dimensionless variables to analyze
the inflaton evolution equations eqs.(\ref{eqno})-(\ref{frinf}),
\cite{biblia}:
\begin{equation} \label{tau}
\tau = m \; t \quad , \quad h \equiv \frac{H}{m}
\quad , \quad \phi = \frac{\varphi}{M_{Pl}} \; .
\end{equation}
The inflaton potential has then the universal form
\begin{equation}\label{v1}
V(\varphi) = M^4 \; v\left(\frac{\varphi}{M_{Pl}}\right) \; ,
\end{equation}
where $ M $ is the energy scale of inflation and $ v(\phi) $ is a dimensionless function.
Without loss of generality we can set $ v'(0)=0 $ \cite{biblia}. Moreover, provided
$ V''(0)\neq 0 $ we can set without loss of generality $ |v''(0)|=1/2 $.
Namely, we have for small fields,
\begin{equation}\label{vpol}
v(\phi) \buildrel{\phi \to 0 }\over= v(0) \mp \frac12 \; \phi^2 +
{\cal O}(\phi^3)
\end{equation}
where the minus sign in the quadratic term corresponds to new inflation
and the plus sign to chaotic inflation.
In these dimensionless variables, the energy density and the pressure
for a spatially homogeneous inflaton are given from eq.(\ref{enerpres1}) by
\begin{equation}\label{enerpres}
\frac{\rho}{M^4} = \frac12 \left(\frac{d\phi}{d \tau}\right)^2 + v(\phi)
\quad , \quad \frac{p}{M^4} = \frac12 \left(\frac{d\phi}{d \tau}\right)^2
-v(\phi) \; ,
\end{equation}
and the coupled inflaton evolution equation (\ref{eqno})
and the Friedmann equation (\ref{frinf}) take the form \cite{biblia},
\begin{eqnarray} \label{evol}
&& \frac{d^2 \phi}{d \tau^2} + 3 \; h \; \frac{d\phi}{d \tau} +
v'(\phi) = 0 \quad , \cr \cr
&& h^2(\tau) = \frac13\left[\frac12 \left(\frac{d\phi}{d \tau}\right)^2
+ v(\phi) \right] \quad .
\end{eqnarray}
These coupled nonlinear differential equations completely define the time
evolution of the inflaton field and the scale factor once the initial
conditions are given at the initial time $ \tau_0 $. Namely, the initial
conditions are fixed by giving two real
numbers, the values of $ \phi(\tau_0) $ and $ d\phi(\tau_0)/d\tau $.
\medskip
It follows from eqs.(\ref{evol}) that
\begin{equation}\label{aseg}
\frac{d^2 a}{d \tau^2} = \frac13\left[
v(\phi)-\left(\frac{d\phi}{d \tau}\right)^2\right] = -\frac12 \left(p
+\frac13 \; \rho \right) \; .
\end{equation}
When $ d^2 a/d \tau^2 > 0 $ the expansion of the universe accelerates
and it is then called inflationary.
\medskip
The derivative of the Hubble parameter is {\bf always} negative:
\begin{equation}\label{hpu}
\frac{dh}{d \tau} = -\frac12 \left(\frac{d\phi}{d \tau}\right)^2 \; .
\end{equation}
Therefore $ h(\tau) $ decreases monotonically with increasing $ \tau $.
Conversely, if we evolve the solution backwards in time from $ \tau_0, \;
h(\tau) $ will generically {\bf increase} without bounds. Namely, at some time
$ \tau = \tau_* , \; h(\tau) $ can exhibit a singularity
where simultaneously $ a(\tau_*) $ vanishes.
\medskip
In fact, the equations (\ref{evol}) admit the singular solution for
$ \tau \to \tau_* $,
\begin{equation}\label{sing1}
\phi(\tau)\buildrel{\tau \to \tau_* }\over= \sqrt{\frac23} \;
\log\frac{\tau-\tau_*}{b} \to -\infty \quad , \quad
h(\tau) \equiv \frac{d}{d \tau} \log a(\tau)
\buildrel{\tau \to \tau_* }\over= \frac1{3 \; (\tau-\tau_*)}
\to +\infty \; ,
\end{equation}
where $ b $ is an integration constant. The energy density $ \epsilon(\tau) $ and equation of state
take the limiting form,
\begin{equation}\label{pe2}
\rho(\tau)\buildrel{\tau \to \tau_* }\over=\frac1{3 \; (\tau-\tau_*)^2}
\to +\infty
\quad , \quad \frac{p(\tau)}{\rho(\tau)}\buildrel{\tau \to \tau_* }\over=1
\; .
\end{equation}
Namely, the limiting equation of state is $ p \buildrel{\tau \to \tau_* }\over= + \rho $.
We have in this regime
\begin{equation}\label{acero}
a(\tau) \buildrel{\tau \to \tau_* }\over= C \; (\tau-\tau_*)^\frac13
\to 0 \; ,
\end{equation}
where $ C $ is some constant.
That is, the geometry becomes singular for $ \tau \to \tau_* $.
The behaviour near $ \tau_* $ is non-inflationary, namely decelerated, since
\begin{equation}\label{asegs}
\frac{d^2 a}{d \tau^2}\buildrel{\tau \to \tau_* }\over= -\frac29 \;
(\tau-\tau_*)^{-\frac53} \to -\infty \; .
\end{equation}
For $ \tau \to \tau_* $, near the singularity,
the potential $ v(\phi) $ becomes negligible
in eqs.(\ref{evol}). Therefore, eqs.(\ref{sing1})-(\ref{asegs}) are valid for
all regular potentials $ v(\phi) $.
\medskip
The evolution starts thus by
this decelerated fast-roll regime followed by an inflationary fast-roll
regime and then by a slow-roll inflationary regime \cite{biblia}.
Recall that the slow-roll regime is an {\bf attractor} \cite{bgzk}, and therefore
the inflaton always reaches a slow-roll inflationary regime for
{\bf generic} initial conditions. We display in fig. \ref{phs} the inflaton flow
in phase space, namely $ d \phi/d \tau $ vs. $ \phi $ for different initial conditions.
The number of efolds of slow-roll inflation $ N_{sr} $ is determined by the
time when the inflaton trajectory reaches the red quasi-horizontal line of slow-roll regime
[see fig. \ref{phs}].
We see that $ d \phi/d \tau $ decreases steeply with $ \phi $. This implies
that $ N_{sr} $ is mainly determined by the initial value of $ \phi $
with a mild (logarithmic) dependence on the initial value of $ d \phi/d \tau $
The inflaton flow described by eq.(\ref{sing1}) results
\begin{equation}\label{fifis}
{\dot \phi}(\tau) \buildrel{\tau \to \tau_* }\over=\sqrt{\frac23} \;
\frac{e^{-\sqrt{\frac32} \; \phi(\tau)}}{b}
\end{equation}
which well reproduce the almost vertical blue and green lines in fig. \ref{phs}.
\medskip
\begin{figure}[h]
\includegraphics[width=16.cm]{flow.eps}
\caption{The complete inflaton flow in phase space.
$ d \phi/d \tau $ vs. $ \phi $ for different initial conditions.
We see that the inflaton always reaches a slow-roll regime for generic initial
conditions represented by a red quasi-horizontal line. Hence,
the slow-roll line is an attractor.
Ultimately the inflaton reaches asymptotically the absolute minima $ d\phi/d \tau = 0 ,
\; \phi = \phi_{min} = \sqrt{8 \, N/y} = 19.52\ldots$. The number of efolds of slow-roll
inflation $ N_{sr} $ increases for decreasing initial $ \phi> 0 $ when $ d\phi/d \tau > 0 $
initially. The $ \phi> 0 , \; d\phi/d \tau > 0 $ trajectories corresponding to $ N_{sr} > 63 $
are colored in green.}
\label{phs}
\end{figure}
The inflationary regimes are characterized by
the slow-roll parameters $ \epsilon_v $ and $ \eta_v $ \cite{biblia}
\begin{equation}\label{epsv}
\epsilon_v = \frac1{2 \; h^2} \;
\left(\frac{d \phi}{d \tau}\right)^2
\quad , \quad \eta_v = \frac{v''(\phi)}{v(\phi)} \; .
\end{equation}
The slow-roll behaviour is defined by the condition $ \epsilon_v < 1/N $.
Typically, $ \epsilon_v \lesssim 1/N $ during slow-roll. More generally
accelerated expansion (inflation) happens for $ \epsilon_v < 1 $ while
we have decelerated expansion for $ \epsilon_v > 1 $ as follows
from eqs.(\ref{enerpres})-(\ref{aseg}) and (\ref{epsv}).
The parameter $ \eta_v $ is also of the order $ 1/N $ during slow-roll and
it is generically of order $ 1/N $ during fast-roll except when the potential $ v(\phi) $
vanishes.
\medskip
Eq.(\ref{hpu}) implies a monotonic decreasing of the expansion rate of the
universe. There are {\bf four stages} in the universe evolution
described by eqs.(\ref{evol}):
\begin{itemize}
\item{The non-inflationary fast-roll stage starting at the singularity
$ \tau = \tau_* $ and ending when $ d^2 a/d \tau^2 $ becomes positive
[see eq.(\ref{aseg})].}
\item{The inflationary fast-roll stage starts when $ d^2 a/d \tau^2 $
becomes positive and ends at $ \tau = \tau_{trans} $ when $ \epsilon_v $
becomes smaller than $ 1/N $ [see eq.(\ref{epsv})].}
\item{The inflationary slow-roll stage follows, and it continues as long as
$ \epsilon_v < 1/N $ and $ d^2 a/d \tau^2 > 0 $. It ends when
$ d^2 a/d \tau^2 $ becomes negative at $ \tau = \tau_{end} $.}
\item{A matter-dominated stage follows the inflationary era.}
\end{itemize}
The four stages described above correspond to the evolution for generic
initial conditions or, equivalently, starting from the singular
behaviour eqs.(\ref{sing1}). In addition, there exists a special (extreme)
slow-roll solution starting at $ \tau = -\infty $ where the fast-roll stages are
absent. We derive this extreme slow-roll solution in sec. \ref{esr}.
\medskip
As shown in refs. \cite{mcmc,biblia} the double well (broken symmetric)
fourth order potential
\begin{equation}\label{bini}
V(\varphi) = \frac14 \; \lambda \;
\left(\varphi^2-\frac{m^2}{\lambda}\right)^2=
- \frac12 \, m^2 \; \varphi^2 + \frac14 \; \lambda \;\varphi^4 +
\frac{m^4}{4 \, \lambda}
\end{equation}
provides a very good fit for the CMB+LSS data, while at the same time being
particularly simple, natural and stable in the Ginsburg-Landau sense.
This is a new inflation model with the inflaton rolling from the vicinity
of the local maxima of $ V(\varphi) $ at $ \varphi = 0 $ towards the
absolute minimum $ \varphi = m/\sqrt{\lambda} $.
The inflaton mass $ m $ and coupling $ \lambda $
are naturally expressed in terms of the
{\bf two} relevant energy scales in this problem: the energy scale of inflation $ M $
and the Planck mass $ M_{Pl} = 2.43534 \; 10^{18}$ GeV,
\begin{equation}
m=\frac{M^2}{M_{Pl}} \;,\quad \lambda = \frac{y}{8 \, N} \;
\left(\frac{M}{M_{Pl}}\right)^4 \; .
\end{equation}
Here $ N \sim 60 $ is the
number of efolds since the cosmologically relevant modes exit the horizon
till the end of inflation and $ y \sim 1 $ is the quartic coupling.
The MCMC analysis of the CMB+LSS
data combined with the theoretical input above yields the value
$ y \simeq 1.26 $ for the coupling \cite{mcmc,biblia}.
$ y $ turns to be {\bf order one} consistent with the Ginsburg-Landau
formulation of the theory of inflation \cite{biblia}.
This model of new inflation yields as most probable
values: $ n_s \simeq 0.964 ,\; r\simeq 0.051 $ \cite{mcmc,biblia}.
This value for $ r $ is within reach of forthcoming CMB observations.
For $ y > 0.431946\ldots $
and in particular for the best fit value $ y \simeq 1.26 $,
the inflaton field exits the horizon in the negative
concavity region $ V''(\varphi) < 0 $ intrinsic to new inflation \cite{biblia}.
We find for the best fit \cite{mcmc,biblia},
\begin{equation}\label{masas}
M = 0.543 \times 10^{16} \quad {\rm GeV ~ for ~ the ~ scale ~ of ~
inflation ~ and} \quad m = 1.21 \times 10^{13} \quad {\rm GeV ~ for ~
the ~ inflaton ~ mass.}
\end{equation}
\medskip
We consider from now on the quartic broken symmetric potential eq. (\ref{bini})
which becomes using eq.(\ref{v1})]
\begin{equation}\label{vnue}
v(\phi) = \frac{g}4 \left(\phi^2 - \frac1{g}\right)^{\! 2}
= -\frac12 \; \phi^2 + \frac{g}4 \; \phi^4 + \frac1{4 \; g} \quad {\rm where} \quad
g = \frac{y}{8 \; N} \; .
\end{equation}
We have two arbitrary real coefficients characterizing the initial
conditions. We can choose them as $ b $ and $ \tau_* $ [see eq.(\ref{sing1})].
A total number of slow-roll inflation efolds $ N_{sr} \simeq 63 $ permits to
explain the CMB quadrupole suppression \cite{quadru,quamc,biblia}. Such
requirement fixes the value of $ b $ for a given coupling $ y $.
We integrated numerically eqs.(\ref{evol}) with eq.(\ref{sing1}) as
initial conditions.
We find that $ b = 4.745272\ldots \; 10^{-5} $ yields 63 efolds of inflation
during the slow-roll era for $ y = 1.26 $, the best fit to
the CMB and LSS data.
We find that $ b $ is a monotonically increasing function of the coupling
$ y $ for fixed number of slow-roll efolds. At fixed coupling, $ b $
increases with the number of slow-roll efolds.
We display in fig. \ref{ylnb} $ b $ as a function of $ y $ and the number of
slow--roll inflation efolds $ N_{sr} $.
\begin{figure}[h]
\includegraphics[width=16.cm]{logb.eps}
\caption{Left panel: the coefficient $ b $ characterizing the initial conditions
vs. the quartic coupling $ y $ for $ N_{sr} = 63 $ efolds of slow-roll
inflation. Right panel: b vs. $N_{sr}$ for $y=1.26$. The preferred values
$y=1.26$ and $N_{sr}$ are highlighted in both panels.}
\label{ylnb}
\end{figure}
For this value of $ y $ and 63 efolds of inflation
during the slow-roll, fast-roll ends by $ \tau = \tau_{trans} = 0.2487963\ldots $.
In figures \ref{evolu}, we depict
$ \log a(\tau), \; \log h(\tau), \;
\phi(\tau), \; \log|{\dot \phi}(\tau)|, \; \log[ N \; \epsilon_v(\tau)] $
and $ p(\tau)/\rho(\tau) $ vs. $ \tau $ till a short time after the end of inflation.
We define the time $ \tau_{end} $ when inflation
ends by the condition $ {\ddot a}(\tau_{end}) = 0 $ which
gives $ (\tau_{end}- \tau_*) = 18.2547816\ldots $.
\begin{figure}[h]
\includegraphics[width=16.cm]{fastplots.eps}
\caption{Time evolution during during the three eras: non-inflationary fast-roll,
inflationary fast-roll and slow-roll and beyond the end of inflation (MD era).
$ \log a(\tau) , \; \log h(\tau), \; \phi(\tau), \; \log|{\dot \phi}(\tau)|, \;
\log [N \; \epsilon_v(\tau)] $ and $ p(\tau)/\rho(\tau) $ vs. $ \tau $.
$ a(\tau) $ grows monotonically reaching 63 efolds by the end of inflation.
$ h(\tau) $ diverges for $ \tau \to \tau_* = -0.8499574\ldots $ according to
eq.(\ref{sing1}) and decreases fast during fast-roll
($ \tau \leq \tau_{trans} = 0.2487963\ldots $). Then, $ h(\tau) $
decreases slowly during slow-roll as discussed in sec. \ref{solusr}.
We depict $ h(\tau) $ for short times ($ 0 < \tau- \tau_* < 0.3 $) in fig. \ref{apefi}.
$ {\dot \phi}(\tau) $ diverges for $ \tau \to \tau_* $ according to
eq.(\ref{sing1}) and decreases fast during fast-roll becoming
very small during slow-roll. After the fast-roll stage where
the inflaton field grows according to eq.(\ref{sing1}), $ \phi(\tau) $
slowly rolls toward its absolute minimum at $ \phi_{end} = \sqrt{8 \, N/y} =
19.52\ldots $.
$ \log [N \; \epsilon_v(\tau)] $ vs. $ \tau- \tau_* $.
We have that $ \epsilon_v(\tau_*) = 3 $ according to
eqs.(\ref{sing1}) and (\ref{epsv}). $ \epsilon_v(\tau) $
decreases fast during fast-roll becoming of the order $ 1/N $. We {\bf define}
the end of fast-roll (and beginning of slow-roll) by the condition
$ N \; \epsilon_v(\tau) \equiv 1 $ which gives $ \tau_{trans} - \tau_* = 0.2487963\ldots $.
The equation of state $ p(\tau)/\rho(\tau) $ fastly decreases during fast-roll from
the value $ p/\rho = +1 $ for $ \tau \to \tau_* $ [see eq.(\ref{pe2})]
passing through $ p/\rho = -1/3 $ at the beginning of fast-roll
inflation [see eq.(\ref{aseg})],
$ \tau = \tau_s = \tau_* + 0.0573 $, and reaching $ p/\rho = -1 $ by the
beginning of slow-roll. $ p/\rho $ vanishes again near the end of slow-roll inflation
by $ \tau_{end} = \tau_* + 18.698\ldots $.}
\label{evolu}
\end{figure}
\bigskip
Furthermore, we study in this paper the curvature and tensor fluctuations
during the {\bf whole} inflaton evolution in its three succesive
regimes: non-inflationary fast-roll,
inflationary fast-roll and inflationary slow-roll.
\medskip
The equation for the scalar curvature fluctuations take in conformal time
$ \eta $ and dimensionless variables the form \cite{biblia}
\begin{equation}\label{fluces}
\left[\frac{d^2}{d\eta^2}+k^2-
W_\mathcal{R}(\eta)\right]S_\mathcal{R}(k;\eta) =0 \; .
\end{equation}
where $ d\eta = d\tau / a(\tau) $,
\begin{equation}\label{defwz}
W_\mathcal{R}(\eta) \equiv \frac1{z} \; \frac{d^2 z}{d \eta^2}
\quad {\rm and} \quad
z(\eta) \equiv \frac{a(\eta)}{h(\eta)} \; \frac{d\phi}{d \tau} \; .
\end{equation}
In cosmic time $ \tau $,
eq.(\ref{fluces}) takes the form
\begin{equation}\label{flutau}
\left[\frac{d^2}{d\tau^2} + h(\tau) \; \frac{d}{d\tau} +
\frac{k^2}{a^2(\tau)} - V_\mathcal{R}(\tau)\right]S_\mathcal{R}(k;\tau)
=0 \; .
\end{equation}
where
\begin{eqnarray}\label{wtau}
&& V_\mathcal{R}(\tau) \equiv
\frac{W_\mathcal{R}(\tau)}{a^2(\tau)} = h^2(\tau) \;
\left[ 2 - 7 \; \epsilon_v + 2 \; \epsilon_v^2 -
\sqrt{8 \; \epsilon_v} \; \frac{v'(\phi)}{h^2(\tau)} -
\eta_v \; (3 - \epsilon_v)\right] = \cr \cr
&& = h^2(\tau) \; \left[ 2 - 7 \; \epsilon_v + 2 \; \epsilon_v^2\right]
- 2 \; \frac{d \phi}{d \tau} \; \frac{v'(\phi)}{h(\tau)} - v''(\phi)
\quad ,
\end{eqnarray}
and $ \epsilon_v $ and $ \eta_v $ are given by eq.(\ref{epsv}).
\medskip
We display $ V_\mathcal{R}(\tau) $ vs. $ \tau $ in fig. \ref{potas}
for the best fit value of the coupling $ y = 1.26 $ and 63 efolds of slow-roll inflation.
\bigskip
The equation for the tensor fluctuations take in conformal time
$ \eta $ and dimensionless variables the form \cite{biblia}
\begin{equation}\label{Sten}
S^{''}_{T}(k;\eta)+\left[k^2-
\frac{a''(\eta)}{a(\eta)}\right]S_{T}(k;\eta) = 0 \; .
\end{equation}
\begin{figure}[h]
\includegraphics[width=16.cm]{fpot.eps}
\caption{The potential $ V_\mathcal{R}(\tau) $ felt by the fluctuations.
Upper plot: $ V_\mathcal{R}(\tau) $ vs. $ (\tau- \tau_*) $
in the stage where $ V_\mathcal{R}(\tau) $ is {\bf repulsive} ($ V_\mathcal{R}(\tau) > 0 $)
which happens for $ (\tau- \tau_*) > 0.114 $. Notice that $ V_\mathcal{R}(\tau) $
slowly decreases during the slow-roll stage as $ V_\mathcal{R}(\tau) \simeq 2 \;
h^2(\tau) + 1 + O(1/N) $ according to eq.(\ref{wtau}) and fig. \ref{evolu}.
Lower plots: Comparison of the exact (numerical) evolution and the analytic approximations
eq.(\ref{vafr}) during fast-roll and slow-roll. Left lower plot:
$ (\tau- \tau_*)^2 \; V_\mathcal{R}(\tau) $ vs. $ \tau- \tau_* $
in the stage where $ V_\mathcal{R}(\tau) $ is
{\bf atractive} ($ V_\mathcal{R}(\tau) < 0 $) from the exact (numerical) calculation
and from the analytic approximation eq.(\ref{vafr}). This happens for
$ 0 \leq (\tau- \tau_*) < 0.114 $. Notice that $ \displaystyle
\lim_{\tau\to \tau_*} (\tau- \tau_*)^2 \; V_\mathcal{R}(\tau) = -1/9 $ according
to eq.(\ref{wep0}). Lower right plot: $ V_\mathcal{R}(\tau) $ vs. $ \tau- \tau_* $ when
$ V_\mathcal{R}(\tau) > 0 $ from the exact (numerical) calculation
and from the analytic approximation eq.(\ref{vafr}).}
\label{potas}
\end{figure}
\begin{figure}[h]
\includegraphics[width=16.cm]{fastplots2.eps}
\caption{Comparison of the exact (numerical) evolution (blue continuous line)
and the analytic approximations (red dashed line)
eq.(\ref{hafr}) during fast-roll and slow-roll for
$ \log a(\tau) , \; \log h(\tau), \; \phi(\tau), \; \log|{\dot \phi}(\tau)|, \;
\epsilon_v(\tau) $ and $ p(\tau)/\rho(\tau) $ vs. $ \tau - \tau_* $. The
exact $ \ln a(\tau) $ and $ \ln h(\tau) $ are close to the approximation eq.(\ref{hafr}).
The scale factor is normalized to unit at $ \tau = 0 $, sixty efolds before the end of inflation.
The exact (numerical) equation of state $ p(\tau)/\rho(\tau) $ is quite close to
the analytic approximation eq.(\ref{epsfr}) both during fast-roll and slow-roll.
The same happens for the exact (numerical) inflaton field $ \phi(\tau), \; {\dot \phi}(\tau) $
and the analytic approximation eq.(\ref{solfr}).}
\label{apefi}
\end{figure}
\subsection{Inflaton and scale factor behaviour near the initial mathematical
singularity}\label{persi}
In order to find the behaviour of $ \phi(\tau) $ and $ a(\tau) $
near the initial singularity we write
\begin{equation}\label{sing2}
\phi(\tau) = \sqrt{\frac23} \; \log\frac{\tau-\tau_*}{b} + \phi_1(\tau)
\quad , \quad
h(\tau) = \frac1{3 \; (\tau-\tau_*)} + h_1(\tau) \; .
\end{equation}
Inserting now eqs.(\ref{sing2}) into eqs.(\ref{sing1}) yields
for $ \phi_1(\tau) $ and $ h_1(\tau) $ the non-autonomous
differential equations
\begin{eqnarray}\label{h1fi1}
&& {\ddot \phi}_1 +\left(\frac1{\tau-\tau_*} + 3 \; h_1 \right){\dot \phi}_1
+ \frac{\sqrt6}{\tau-\tau_*} \; h_1 - \phi_1 -
\sqrt{\frac23} \; \log\frac{\tau-\tau_*}{b} + g
\left(\sqrt{\frac23} \; \log\frac{\tau-\tau_*}{b} + \phi_1\right)^3 = 0 \\ \cr
&& h_1^2 + \frac2{3\; (\tau-\tau_*)} \; h_1 - \frac{{\dot \phi}_1}6 \;
\left(\sqrt{\frac23} \; \frac2{\tau-\tau_*} +{\dot \phi}_1 \right)+\frac16 \;
\left(\sqrt{\frac23} \; \log\frac{\tau-\tau_*}{b} + \phi_1\right)^2
-\frac{g}{12} \left(\sqrt{\frac23} \; \log\frac{\tau-\tau_*}{b} +
\phi_1\right)^4 - \frac1{12 \; g} = 0 \; , \nonumber
\end{eqnarray}
where $ \dot \phi $ stands for $ d\phi/d\tau $.
The asymptotic solution of eqs.(\ref{h1fi1}) for $ \tau \to \tau_* $
turns to have the dominant form
\begin{equation}\label{defP}
\phi_1(\tau) \buildrel{\tau \to \tau_* }\over=(\tau-\tau_*)^2 \;
P_4^{\phi}\left(\log\frac{\tau-\tau_*}{b}\right)
\quad , \quad
h_1(\tau) \buildrel{\tau \to \tau_* }\over= (\tau-\tau_*) \;
P_4^{h}\left(\log\frac{\tau-\tau_*}{b}\right)
\end{equation}
where $ P_4^{\phi}(z) $ and $ P_4^{h}(z) $ are fourth degree polynomials
in their arguments. The polynomials turn to be of fourth degree
because the inflaton potential is of fourth degree.
Their explicit expressions follow after calculation
\begin{eqnarray}\label{expli}
&&\phi_1(\tau) \buildrel{\tau \to \tau_* }\over=-\frac{(\tau-\tau_*)^2}{\sqrt6} \;
\left[\frac{g}{18} \left(\log^4\frac{\tau-\tau_*}{b} +
\frac23 \; \log^3\frac{\tau-\tau_*}{b}
-\frac{11}3 \; \log^2\frac{\tau-\tau_*}{b}+ \frac{49}9 \; \log\frac{\tau-\tau_*}{b}-
\frac{439}{54}\right) \right. \cr \cr
&& \left. - \frac16 \; \left(\log^2\frac{\tau-\tau_*}{b} + \frac13 \;
\log\frac{\tau-\tau_*}{b} - \frac78 \right)+ \frac1{8 \; g} \right] \; , \cr \cr
&& h_1(\tau) \buildrel{\tau \to \tau_* }\over=\frac{\tau-\tau_*}9
\left[\frac{g}{18} \left(6 \; \log^4\frac{\tau-\tau_*}{b} -
8 \; \log^3\frac{\tau-\tau_*}{b}
+ 8 \; \log^2\frac{\tau-\tau_*}{b} - \frac{11}3 \; \log\frac{\tau-\tau_*}{b} +
\frac{146}9 \right) \right. \cr \cr
&& \left. - \log^2\frac{\tau-\tau_*}{b}+ \frac23 \;
\log\frac{\tau-\tau_*}{b} - \frac19 + \frac3{4 \; g} \right]
\end{eqnarray}
As a consequence, the scale factor near the singularity takes the form
\begin{equation}
a(\tau) \buildrel{\tau \to \tau_* }\over= C \; (\tau-\tau_*)^\frac13 \left[1 +
(\tau-\tau_*)^2 \;
P_4^{a}\left(\log\frac{\tau-\tau_*}{b}\right) \right] \; .
\end{equation}
where the coefficients of the fourth order polynomial $ P_4^{a} $ can be
obtained from eqs.(\ref{sing1}) and (\ref{expli}).
\subsection{Quantum loop effects and the validity of the classical inflaton picture}
When $ \tau \to \tau_* $ quantum loop corrections are expected to become very large
spoiling the classical description. More precisely, quantum loop
corrections are of the order $ (H/M_{Pl})^2 $ \cite{biblia}. From eqs.(\ref{tau}) and
(\ref{sing1}) the quantum loop corrections are of the order
$$
\left(\frac{H}{M_{Pl}}\right)^2 \buildrel{(\tau-\tau_*) \ll 1}\over=
\left[\frac{m}{3 \; (\tau-\tau_*) \; M_{Pl}}\right]^2 =
\left(\frac{1.66 \; 10^{-6}}{\tau-\tau_*}\right)^2= \frac19 \;
\left(\frac{\tau_{Planck}}{\tau-\tau_*}\right)^2
$$
where we used $ m = 1.21 \; 10^{13} $ GeV \cite{biblia}.
The characteristic time is here the Planck time
$$
\tau_{Planck} = m \; t_{Planck} = \frac{m}{M_{Pl}} = 2.703 \; 10^{-43} \; {\rm sec} \times m
= 4.97 \; 10^{-6} \; .
$$
Namely, the quantum loop corrections are less than 1\% for times
\begin{equation}\label{corrQ}
(\tau-\tau_*) > \frac{10}3 \; \tau_{Planck} = 1.66 \; 10^{-5} \; .
\end{equation}
Therefore, for times $ (\tau-\tau_*) > 10^{-5} $ the classical treatment of the inflaton
and the space-time presented in sec. \ref{uno} and \ref{persi} can be trusted and
we see that the classical description has a wide domain of validity.
The use of a classical and homogeneous inflaton field is justified
in the out of equilibrium field theory context as the quantum formation
of a condensate during inflation. This condensate turns to obey the classical evolution
equations of an homogeneous inflaton \cite{eri}.
We see from eq.(\ref{sing1}) that the inflaton field becomes negative for
$ \tau \to \tau_* $. But since a condensate field should be always positive, the
classical and homogeneous inflaton picture requires
$$
\tau-\tau_* > b
$$
For the best fit coupling $ y = 1.26 $ and 63 efolds
of inflation we have $ b = 4.745272\ldots \; 10^{-5} = 9.55 \; \tau_{Planck} $
which is consistent with eq.(\ref{corrQ}).
By comparing this value of $ b $ with eq.(\ref{corrQ}) we see that the quantum loop
corrections are negligible in the stage where the condensate is already formed.
We can obtain a lower bound on $ b $ since $ b $ increases with the
number of inflation efolds $ N_{sr} $ at fixed inflaton potential
and since $ N_{sr} $ cannot be smaller than
the lower bound provided by flatness and entropy \cite{biblia}.
\medskip
Although all inflationary solutions obtained evolving backwards in time from
the slow-roll stage do reach a zero of the scale factor,
such mathematical singularity is attained extrapolating
the classical treatment where it is {\bf no more valid}.
In fact, one never reaches the singularity in the validity region of the
classical treatment.
In summary, the classical singularity at $ \tau=\tau_* $
{\bf is not a real physical phenomenon} here.
\medskip
The classical description with the homogeneous inflaton is very good for
$ \tau-\tau_* > 10 \;\tau_{Planck} $ well before the beginning of inflation.
\subsection{The fast-roll regime: analytic approach}
As we see from fig. \ref{evolu} the inflaton field $ \phi(\tau) $ is much
smaller than $ d\phi/d \tau $ during fast-roll. We can therefore
approximate the coupled inflaton evolution equation and Friedmann equation
eqs.(\ref{evol}) as
\begin{eqnarray} \label{evfr}
&& \frac{d^2 \phi}{d \tau^2} + 3 \; h \; \frac{d\phi}{d \tau}
= 0 \quad , \cr \cr
&& h^2(\tau) =
\frac13\left[\frac12 \left(\frac{d\phi}{d \tau}\right)^2 + \frac1{4 \; g}
\right] \quad .
\end{eqnarray}
Or, in a compact form,
\begin{equation}
\frac{d^2 \phi}{d \tau^2} + \sqrt{\frac32} \; \frac{d\phi}{d \tau} \;
\sqrt{\left(\frac{d\phi}{d \tau}\right)^2 + \frac1{2 \; g}}=0 \; ,
\end{equation}
which has the exact solution
\begin{equation} \label{solfr}
\frac{d\phi}{d \tau} = \sqrt{\frac23} \; \frac1{\tau_1 \;
\sinh\left(\displaystyle\frac{\tau-\tau_*}{\tau_1}\right)}
\quad , \quad \phi(\tau) = \sqrt{\frac23} \;
\log\left[\frac{2 \; \tau_1}{b} \;
\tanh\left(\frac{\tau-\tau_*}{2 \; \tau_1}\right)\right] \quad ,
\end{equation}
where $ \tau_1 $ turns out to be the characteristic time scale
\begin{equation}\label{tau1}
\tau_1 = 2 \; \sqrt{\frac{g}3} = \sqrt{\frac{y}{6 \;N}} \; .
\end{equation}
We find for the best fit to CMB and LSS data, $ y = 1.26 $ and $ N = 60 $,
\begin{equation}\label{vtau1}
\tau_1 = 0.0592 = 11910 \; \tau_{Planck} \; ,
\end{equation}
well after the Planck scale $ \tau_{Planck} = 4.97 \; 10^{-6} $.
The integration constant in eq.(\ref{solfr}) matches with the small
$ \tau -\tau_* $ behaviour eq.(\ref{sing1}). The Hubble parameter and
the scale factor are here
\begin{equation} \label{hafr}
h(\tau) = \frac1{3 \; \tau_1} \; \coth u
\quad , \quad
a(\tau) = C \; \left[ \tau_1 \; \sinh u \right]^{\! \frac13}
\quad , \quad u \equiv \frac{\tau-\tau_*}{\tau_1} \quad ,
\end{equation}
where the integration constant was chosen to fulfil eq.(\ref{acero}).
The scale factor eq.(\ref{hafr}) interpolates between the non-inflationary
power law behaviour eq.(\ref{acero}) for $ \tau -\tau_* \to 0 $ and the eternal
inflationary de Sitter behaviour for $ \tau -\tau_* \gg \tau_1 $.
Since we have set $ v(\phi) $ equal to
constant, slow-roll De Sitter inflation {\bf never stops} in this approximation.
Namely, neither matter-dominated nor radiation-dominated eras are reached
in this approximation.
\medskip
We can eliminate the variable $ u $ between $ \phi $ and $ d\phi/d \tau $ in eq.(\ref{solfr})
with the result
\begin{equation}\label{fifip}
\frac{d\phi}{d \tau} = \sqrt{\frac23} \; \left[ \frac{e^{-\sqrt{\frac32} \; \phi(\tau)}}{b}
- \frac{b}{4 \; \tau_1^2} \; e^{ \sqrt{\frac32} \; \phi(\tau)} \right] \; .
\end{equation}
This equation generalizes eq.(\ref{fifis}) which corresponds to
the first term here and describes the behaviour for $ \tau -\tau_* $.
Notice that
$$
-\infty < \phi(\tau) < \sqrt{\frac23} \;
\log\left[\frac{2 \; \tau_1}{b}\right] \quad , \quad 0 < \frac{d\phi}{d \tau} < +\infty
$$
and that $ b /[2 \;\tau_1] = 4.0105 \; 10^{-4} $.
\medskip
The evolution described by eqs.(\ref{solfr})-(\ref{hafr}) starts from the
mathematical singularity at $ \tau = \tau_* $ with monotonically decreasing
$ d\phi/d \tau $ and $ h(\tau) $ and a monotonically increasing
$ \phi(\tau) $ from its initial value $ \phi(\tau_*) = -\infty $.
\medskip
Slow-roll is reached asymptotically for large $ \tau $ since
$ d\phi/d \tau $ vanishes for $ \tau -\tau_* \to \infty $.
\medskip
We find for the parameter $ \epsilon_v $ [eq. (\ref{epsv})] and for the equation of
state,
\begin{equation}\label{epsfr}
\epsilon_v(\tau) = \frac3{1 + \sinh^2 u}
\quad , \quad \frac{p(\tau)}{\rho(\tau)} =
\frac2{\cosh^2 u} - 1 \quad .
\end{equation}
We see that $ \epsilon_v(\tau) $ monotonically decreases with $ \tau $
and vanishes for $ \tau-\tau_* \to \infty $.
The equation of state $ p/\rho $ smoothly interpolates between $ + 1 $ at $ \tau = \tau_* $
(extreme non-inflationary fast-roll)
and $ - 1 $ (slow-roll inflation) for $ \tau - \tau_* \to \infty $, passing by
$ p/\rho = -1/3 $ (the beginning of fast-roll inflation) at $ \tau - \tau_* = 0.0573 $.
\medskip
The potential $ V_\mathcal{R}(\tau) $ eq.(\ref{fluces}) felt by the fluctuations
takes here the form
\begin{equation}\label{vafr}
V_\mathcal{R}(\tau) = \frac1{6 \; g} \left[1 - \frac1{2 \; \sinh^2 u}
- \frac9{\cosh^2 u} \right] \quad , \quad u = \frac{\tau-\tau_*}{\tau_1} \; .
\end{equation}
\medskip
The limiting values of $ h(\tau), \; \phi(\tau) $ and
$ V_\mathcal{R}(\tau) $ for $ \tau \to \infty $ give a reasonable
approximation to the numerical results. We have
\begin{equation}\label{infi}
h(\infty) = \frac1{3 \; \tau_1} =
\sqrt{\frac{2 \; N}{3 \; y}} \quad , \quad
\phi(\infty) = \sqrt{\frac23} \;
\log\left[\frac{2 \; \tau_1}{b} \right] \quad , \quad
\frac{d\phi}{d\tau}(\infty) = 0 \quad , \quad
V_\mathcal{R}(\infty) = \frac{4\; N}{3 \; y} \; .
\end{equation}
The characteristic time scale $ \tau_1 $ is generically a small number since
according to eq.(\ref{tau1}) $ \tau_1 \sim 1/\sqrt{N} $. The value of $ \tau_1 $
for the best fit value for $ y $ is given in eq.(\ref{vtau1}).
\medskip
The end of fast-roll $ \tau_{trans} $ can be estimated in this approximation
by using eq.(\ref{epsfr}) for $ \epsilon_v(\tau) $
setting $ \epsilon_v(\tau_{trans}) = 1/N $. This gives,
$$
\epsilon_v(\tau) \simeq 12 \; e^{-\frac{2 \; \tau_{trans}}{\tau_1}}
= \frac1{N} \quad , \quad \tau_{trans} \simeq \frac12 \; \tau_1 \;
\ln(12 \; N) = 0.195 \; .
$$
This approximated value for $ \tau_{trans} $ should be compared with the
exact numerical result $ \tau_{trans} = 0.2487963\ldots $. $ h(\tau_{trans}) $ and
$ V_\mathcal{R}(\tau_{trans}) $ differ in less than $ 1\% $
from their values at $ \tau = \infty $ given by eq.(\ref{infi}).
\medskip
In figs. \ref{apefi} we plot
$ \ln a(\tau), \; \ln h(\tau), \; \phi(\tau), \; \ln |{\dot \phi}(\tau)|,
\epsilon_v(\tau) $ and $ \; p(\tau)/\rho(\tau) $ computed
{\bf numerically} and computed using the analytic expressions
eqs.(\ref{solfr})-(\ref{epsfr}). We compare in figs. \ref{potas}
the exact potential $ V_\mathcal{R}(\tau) $ with the analytic approximation eq.(\ref{vafr}).
We see that the simple analytic formulas
eqs.(\ref{solfr})-(\ref{vafr}) provide a very good approximation
during the fast-roll regime $ \tau \leq t_{trans} = 0.2487963\ldots $.
In particular, eq.(\ref{solfr}) provides an excellent approximation to
$ \phi(\tau) $ as shown in fig. \ref{apefi}.
In particular, the analytic formulas eqs.(\ref{solfr})-(\ref{vafr})
become {\bf exact} near the singularity at $ \tau = \tau_* $.
\subsection{The fast-roll regime: numerical solution}
To construct a singular solution we can integrate eqs.~\eqref{evol}
backwards in time starting from initial conditions of strong non-inflationary fast--roll
type, namely
$$
K \equiv \frac{\dot\phi^2}{2 \, v(\phi)} \gg 1 \; ,
$$
producing a given total number $ N_{sr} $ of slow--roll inflationary
efolds. For instance, we start from some $ \phi $ and $ \dot\phi $ such that
$ K = 10^4 $. The time extent backwards from this moment
has to be limited so that, integrating back and forth, the required relative accuracy of
$ 10^{-12} $ is preserved. We furthermore impose that $ N_{sr}=63 $.
\medskip
We adopt the convention that conformal time $ \eta $ vanishes from below
when inflation ends and that $ a(\tau=0)=1 $ when there are still $ N=60 $
efolds till the end of inflation. This choice of the scale factor normalization
seems the most natural.
Then, $ \eta $ has a finite non-zero limit $ \eta_\ast $ as
$ \tau $ approaches the time $ \tau_\ast $ of the singularity,
since $ a( \tau)\simeq C \; (\tau- \tau_\ast)^{1/3} $ as $ \tau\to \tau_\ast $
according to eq.(\ref{acero}). That is,
$$
\eta = \int_{\tau_{end}}^\tau \frac{d\tau'}{a(\tau')}=
\eta_\ast + \int_{\tau_\ast}^\tau \frac{d\tau'}{a(\tau')} \; .
$$
The numerics of a fast--roll solution of this type are in Table I
where a relative accuracy of $ 10^{-12} $ is preserved.
Using the asymptotic behaviour eq.(\ref{sing1}) as $ \tau\to \tau_\ast^+ $
we obtain from Table I:
$$
\tau_\ast = -0.8499574\ldots \quad , \quad b = 4.745272\ldots
10^{-5} \quad {\rm and} \quad \eta_\ast = -15.605614\ldots \; .
$$
Slow--roll begins at
$ \tau_{trans} = \tau_\ast+0.2487963\ldots = -0.6011611\ldots$.
\medskip
The initial value of the ratio
$$
\displaystyle \frac{d\varphi/dt}{\varphi} = m \; \frac{\dot\phi}{\phi}
$$
has the dimension of mass. The natural mass scale in the problem
is here the energy scale of inflation $ M $. Therefore, assuming this ratio of the
order $ M $ yields
$$
\frac{\dot\phi}{\phi} < \frac{M}{m} \sim 10^3 \; .
$$
Hence, it s natural to
start the fast--roll evolution with $ \dot\phi/\phi < 10^3 $.
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& $ K=5.3458 \ldots \; 10^7$ & $ K=10^4 $ & inflation start: $ \ddot a=0^-$
& fast-roll $\to$ slow-roll & $ a=1 $ & inflation end: $ \ddot
a=0^+$ \\
\hline \hline
$ \tau $ & $-0.8499493\ldots$ & $-0.8493593\ldots$ & $-0.7746494\ldots$
& $-0.6011611\ldots$ & 0 & $17.4048242\ldots$ \\
\hline
$ \phi $ & $ -1.4401237\ldots$ & $2.0690604\ldots$ & $5.9342489\ldots$
& $6.4783577\ldots$& $6.7484076\ldots$& $18.5586530\ldots$ \\
\hline
$ \dot\phi $ & $100391.035\ldots$ & $1365.05241\ldots$ & $8.8601670\ldots$
& $0.9182661\ldots$& $0.3974015\ldots$& $0.94150557\ldots$ \\
\hline
$ \log a $ & $-7.0325621\ldots$ & $-5.5999353\ldots$ & $-3.9142151\ldots$
& $-2.9999999\ldots$& $0$ & $60$ \\
\hline
$h$ & $40984.4689\ldots$& $557.30817\ldots$ & $6.2650841\ldots$
& $5.0295509\ldots$& $4.9653990\ldots$& $0.6657449\ldots$ \\
\hline
$ \eta $ & $-15.6050091\ldots$ & $-15.376218\ldots$& $-15.3549996\ldots$
& $-4.0169827\ldots$& $-0.2020609\ldots$ & $0$ \\
\hline
\end{tabular}
\caption{Fast-roll solution with $ N_{sr} = 63 $ efolds of slow-roll
inflation. Recall that $ \tau = 4.97 \; 10^{-6} \; (t/t_{Planck}) $.}
\end{table}
\section{The slow-roll inflationary era}
\subsection{The extreme slow-roll solution}\label{esr}
There always exist a special solution of eqs.(\ref{evol}) that starts
at $ \tau = -\infty $ with vanishing inflaton, vanishing scale factor
but nonzero Hubble parameter. More precisely, eqs.(\ref{evol}) can
be approximated for small $ \phi $ and $ \dot \phi $ as
\begin{eqnarray} \label{evola}
&& \frac{d^2 \phi}{d \tau^2} + 3 \; h \; \frac{d\phi}{d \tau} -\phi = 0
\quad , \cr \cr
&& h^2(\tau) = \frac13 \; v(0) \quad .
\end{eqnarray}
where we used eqs.(\ref{vpol}) and (\ref{evol}).
Eqs.(\ref{evola}) admit the asymptotic solution for $ \tau \to -\infty $
\begin{equation}\label{asiex}
\phi(\tau) \buildrel{\tau \to -\infty }\over= C_0 \;
e^{\alpha \; \tau} \to 0
\quad , \quad h(\tau)\buildrel{\tau \to -\infty }\over=
\sqrt{\frac{v(0)}3}
\quad , \quad a(\tau)\buildrel{\tau \to -\infty }\over=
e^{\sqrt{\frac{v(0)}3} \; \tau} \to 0 \; ,
\end{equation}
where $ C_0 $ is an integration constant, $ v(0) = 2 \, N/y $ for the double-well
potential eq.(\ref{vnue}) and
$$
\alpha \equiv \frac12 \left[ \sqrt{3 \; v(0)+4}-
\sqrt{3 \; v(0)}\right] > 0 \; .
$$
Notice that $ \alpha $ can be expressed in terms of the fast-roll characteristic time-scale
$ \tau_1 $ [eq.(\ref{tau1})],
$$
\alpha = \frac1{2 \; \tau_1}\left[ \sqrt{1 + 4 \; \tau_1^2} - 1 \right] \simeq \tau_1
$$
since $ \tau_1 \simeq 0.0592 \ll 1 $ [see eq.(\ref{vtau1})].
\medskip
It must be noticed that the characteristic time scale of the inflaton evolution
in the extreme slow-roll solution for early times [see eq.(\ref{asiex})]
$$
\frac1{\alpha} \simeq \frac1{\tau_1} \gg 1 \; ,
$$
turns to be the {\bf inverse} of the
characteristic time scale $ \tau_1 $ of the fast-roll solution and to be very large.
On the contrary, the characteristic time scale of the scale factor evolution in the same regime is
very short
$$
\sqrt{\frac3{v(0)}} = 3 \; \tau_1 \ll 1 \; .
$$
The fast-roll stages both non-inflationary and inflationary are absent in this solution.
The extreme slow-roll solution only possesses the slow-roll inflationary stage followed by the
matter dominated era.
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|}
\hline
& inflation start & $ a=1 $ & inflation end: $ \ddot a=0^+ $ \\
\hline \hline
$ \tau $ & $-344.9514017\ldots$& 0 & $17.40482446\ldots$ \\
\hline
$ \phi $ & $ 10^{-8} $ & $ 6.7484118\ldots$ & $ 18.5586530\ldots$ \\
\hline
$ \dot\phi $ & $\alpha\,10^{-8}=5.8937108453...10^{-10}$ &
$0.3973384\ldots$ & $0.94150557\ldots$\\
\hline
$ \log a $ &$-1938.4867948\ldots$& 0 & 60 \\
\hline
$ h $ & $ \sqrt{2 \, N/(3 \, y)} =
5.6361006\ldots$ & $4.9653973\ldots$ & $0.6657449\ldots$ \\
\hline
$ \eta $ & $-\infty~$(f.a.p.p) & $-0.2020610\ldots$ & 0 \\
\hline
\end{tabular}
\caption{Relevant quantities of the extreme slow-roll inflaton solution
for the coupling $ y = 1.2592226\ldots$.
We adopt the convention that $ a(\tau=0)=1 $ when there are still $ N=60 $
efolds till the end of inflation. Recall that $ \tau = 4.97 \; 10^{-6} \; (t/t_{Planck}) $.}
\end{table}
For the value of the coupling $ y = 1.2592226\ldots$, we get
for the extreme slow--roll solution
\begin{equation}\label{eq:phend}
\alpha = 0.058937108\ldots \;,\quad
\phi_{\rm end} = 18.5586530\ldots \; ,\quad
\dot\phi_{\rm end} = 0.9415055\ldots
\end{equation}
In table II we display the values of the relevant magnitudes for this
extreme slow-roll solution.
\medskip
Except for the extreme slow--roll solution,
all solutions are of fast-roll type and come from singular values of $ \phi $ and $ h $
according to eq.(\ref{sing1}) as $ \tau\to \tau_\ast^+ $
for some finite $ \tau_\ast $ characteristic of each particular
solution. The slow-roll stage
(which starts when $ \epsilon_v=1/N $ from above, and ends when
again $ \epsilon_v=1/N $ from below)
of all distinct solutions turns to be almost identical to that of the
extreme slow--roll case as one could expect for an attractor.
\subsection{The inflaton during slow-roll inflation: analytical solution}\label{solusr}
In the slow-roll regime higher time derivatives can be neglected in the
evolution eqs.(\ref{evol}) with the result
\begin{equation}\label{sr1}
3 \, h(\tau) \; {\dot \phi} + v'(\phi) = 0 \quad , \quad
h^2(\tau) = \frac{v(\phi)}3 \; .
\end{equation}
These first order equations can be solved in closed form as
\begin{equation} \label{Nefo}
N[\phi] = -\int_{\phi}^{\phi_{end}} \;
v(\phi') \; \frac{d \phi'}{dv} \; d\phi' \; \; .
\end{equation}
where $ N[\phi] $ is the number of e-folds since the field $ \phi $ exits the
horizon till the end of inflation (where it takes the value $ \phi_{end} $).
Eq.(\ref{Nefo}) indicates that $ N[\phi] $ scales as $ \phi^2 $ and
hence the field $ \phi $ is of the order $ \sqrt{N} \sim \sqrt{60} $.
Therefore, we proposed as universal form for the inflaton potential
\cite{1sN,biblia}
\begin{equation} \label{V}
v(\varphi) = N \; M^4 \; w(\chi) \; ,
\end{equation}
\noindent where $ \chi $ is the dimensionless, slowly varying field
\begin{equation}\label{chifla}
\chi = \frac{\varphi}{\sqrt{N} \; M_{Pl}} = \frac{\phi}{\sqrt{N}} \; .
\end{equation}
The equations of motion (\ref{evol}) in the field $ \chi $ become
\begin{eqnarray} \label{evol2}
&& {\cal H}^2({\hat \tau}) = \frac13\left[\frac1{2\;N}
\left(\frac{d\chi}{d {\hat \tau}}\right)^2 + w(\chi) \right] \quad
{\rm with} \quad {\cal H} = \frac{h}{\sqrt{N}} \; , \cr \cr
&& \frac1{N} \; \frac{d^2
\chi}{d {\hat \tau}^2} + 3 \; {\cal H} \; \frac{d\chi}{d {\hat \tau}} + w'(\chi) = 0
\quad .
\end{eqnarray}
and $ \hat \tau $ stands for the rescaled dimensionless time
$$
{\hat \tau} \equiv \frac{\tau}{\sqrt{N}} = \frac{m \; t}{\sqrt{N}} \; .
$$
To leading order in the slow-roll approximation (neglecting $ 1/N $
corrections), eqs.(\ref{evol2}) are solvable in terms of quadratures
\begin{equation}\label{tausr}
{\hat \tau} -{\hat \tau}_{trans} = - \int_{\chi({\hat \tau}_{trans})}^\chi d\chi' \;
\frac{\sqrt{3 \; w(\chi')}}{ w'(\chi')}\; ,
\end{equation}
where $ {\hat \tau}_{trans} $ stands for the beginning of slow-roll inflation
and we used that
\begin{equation}\label{hsr}
{\cal H}({\hat \tau}) = \sqrt{\frac{w(\chi)}3} +
{\cal O}\left(\frac1{N}\right) \; ,
\end{equation}
For the broken symmetric potential eq.(\ref{bini}), from eqs.(\ref{enerpres}),
(\ref{tausr}) and (\ref{hsr}), we find
\begin{eqnarray}\label{chislr}
&& \chi({\hat \tau}) = \chi({\hat \tau}_{trans}) \; e^{\sqrt{\frac{y}6} \;
({\hat \tau}-{\hat \tau}_{trans})}
+{\cal O}\left(\frac1{N} \right)
= \sqrt{\frac8{y}} \; e^{-\sqrt{\frac{y}6} \; ({\hat \tau}_{end}-{\hat \tau})}
+{\cal O}\left(\frac1{N} \right)
\; , \\ \cr
&& {\cal H}({\hat \tau}) =\sqrt{\frac2{3 \; y}}
\left[ 1 - e^{-\sqrt{\frac{2 \; y}3} \; ({\hat \tau}_{end}-{\hat \tau})}
\right] +{\cal O}\left(\frac1{N} \right) \; , \cr \cr
&& \frac{p}{\rho}({\hat \tau}) = -1 + \frac{y}{6 \; N} \;
\frac1{\sinh^2\left[\sqrt{\frac{y}6}({\hat \tau}_{end}-{\hat \tau})\right]}
+{\cal O}\left(\frac1{N^2} \right)\; , \label{ecesta} \\ \cr
&& {\rm for} \quad {\hat \tau}_{trans} \leq {\hat \tau} \leq {\hat \tau}_{end} = \sqrt{\frac3{2 \; y}} \;
\ln \left[\frac8{\chi^2({\hat \tau}_{trans}) \; y}\right] +
{\cal O}\left(\frac1{\sqrt{N}} \right) \label{taufin} \; .
\end{eqnarray}
Inflation ends when the equation of state becomes
$ p/\rho = -1/3 $ [see eq.(\ref{aseg})].
According to eq.(\ref{ecesta}), this happens when
$ {\hat \tau}_{end} -{\hat \tau} \sim {\cal O}\left(1/\sqrt{N} \right) $. Therefore,
expressions eqs.(\ref{chislr})-(\ref{ecesta}) are valid as long as
$$
{\hat \tau}_{trans} \leq {\hat \tau} \leq {\hat \tau}_{end} - {\cal O}\left(\frac1{\sqrt{N}} \right)
\quad {\rm where} \quad {\cal O}\left(\frac1{\sqrt{N}}\right) > 0 \; .
$$
That is, eqs.(\ref{chislr}) hold while the inflaton is not very near the
minimum of the potential $ \chi_{end} = \sqrt{8/y} $.
\medskip
By integrating the Hubble parameter $ {\cal H}({\hat \tau}) $
we obtain for the scale factor $ a({\hat \tau}) $
\begin{eqnarray} \label{asr}
&& \log \frac{a({\hat \tau})}{a({\hat \tau}_{trans})} = \sqrt{\frac2{3 \; y}} \; N \;
({\hat \tau} -{\hat \tau}_{trans})-
\frac{N}8 \; \chi^2({\hat \tau}_{trans}) \left[e^{\sqrt{\frac{2 \; y}3} \;
({\hat \tau}-{\hat \tau}_{trans})} - 1 \right] =\\ \cr
&&= \sqrt{\frac{2 \; N}{3 \; y}} \; m \; (t -t_{trans})-\frac18 \;
\left[\frac{\varphi(t_{trans})}{M_{Pl}}\right]^2 \;
\left[e^{\sqrt{\frac{2 \; y}{3 \; N}} \; m \; (t-t_{trans})} - 1 \right] \; ,
\nonumber
\end{eqnarray}
where we used eqs.(\ref{tau} ) and (\ref{chislr}). It must be noticed
that $ a({\hat \tau}) $ is not exactly a de Sitter scale factor, even in the large
$ N $ limit at fixed $ {\hat \tau} $.
At the end of inflation the number of efolds is $ \ln a \simeq
64 $, the inflaton is near its minimum
$$
\chi = \sqrt{\frac8{y}} \simeq 2.52 \quad ,
$$
$ \dot \chi $ starts to oscillate around zero and $ \cal{H}({\hat \tau}) $
begins a rapid decrease (see figs. \ref{evolu}). At this time the inflaton field
is no longer slowly coasting in the $ w''(\chi) < 0 $ region but rapidly
approaching its equilibrium minimum. When inflation ends, the inflaton is
at its minimum value up to corrections of order $ 1/\sqrt{N} $. Therefore, we
see from the Friedmann eq.(\ref{evol2}) and eqs.(\ref{chislr}) that
\begin{equation}\label{reduH}
\frac1{N} \left(\frac{d\chi}{d {\hat \tau}}\right)^{\! 2}({\hat \tau}_{end}) =
{\cal O}\left(\frac1{N}\right) \quad , \quad
w(\chi({\hat \tau}_{end})) = {\cal O}\left(\frac1{N}\right) \quad
{\rm and ~~ therefore,} \quad {\cal H}({\hat \tau}_{end}) =
{\cal O}\left(\frac1{\sqrt{N}} \right) \; ,
\end{equation}
while $ {\cal H}({\hat \tau}_{trans}) = {\cal O}(1) $. Namely, the Hubble parameter
decreases by a factor of the order $ \sqrt{N} \sim 8 $ during slow-roll
inflation. We see in fig. \ref{evolu} that the exact $ \cal{H}({\hat \tau}) $ decreases
by a factor six during slow-roll inflation,
confirming the slow-roll analytic estimate.
We can compute the total number of inflation efolds $ N_{tot} $ to leading
order in slow-roll inserting the analytic formula for $ {\hat \tau}_{end} $
eq.(\ref{taufin}) in eq.(\ref{asr}) with the result,
\begin{equation}\label{ntotsr}
N_{tot} = \frac{N}{y} \left\{\ln\left[ \frac8{\chi^2({\hat \tau}_{trans}) \; y}\right] - 1 +
\frac18 \; y \; \chi^2({\hat \tau}_{trans}) \right\} +
{\cal O}\left(\frac1{\sqrt{N}} \right) \; .
\end{equation}
We have verified the slow-roll analytical results
eqs.(\ref{chislr})-(\ref{ntotsr}) comparing them with the numerical
solution of eqs.(\ref{evol}). Both results are concordant up to the error
estimation in each case: $ {\cal O}\left(1/N \right) $ or
$ {\cal O}\left(1/\sqrt{N} \right) $.
\medskip
The field $ \phi $ as a function of the dimensionless time $ \tau $
eq.(\ref{chislr}) takes the form
$$
\phi(\tau) = \phi({\tau}_{trans}) \; e^{\sqrt{\frac{y}{6 \; N}} \;
(\tau-\tau_{trans})}
$$
and then
$$
{\dot \phi(\tau)} =\sqrt{\frac{y}{6 \; N}} \; \phi(\tau) \; .
$$
For $ y \simeq 1.26 $ and $ N = 60 $ we get $ \sqrt{y/[6 \; N]} = 0.0577 $
in agreement with the slope of the red quasi-horizontal slow-roll line in the phase
space flow fig. \ref{phs}.
\section{Complete Fluctuations evolution and fast-roll
effects on the power spectrum.}
\subsection{Scalar and tensor fluctuations near the initial singularity.}
In order to study the curvature and tensor fluctuations in this regime,
it is important to evaluate the parameter $ \epsilon_v $ and the potential
felt by the fluctuations $ V_\mathcal{R} $.
Inserting eqs.(\ref{sing2}) and (\ref{defP}) into eqs.(\ref{epsv}) and
(\ref{wtau}) yields near the initial singularity
\begin{eqnarray}\label{wep0}
&& V_\mathcal{R}(\tau)\buildrel{\tau \to \tau_* }\over=
- \frac1{9 \; (\tau-\tau_*)^2} \left[1 + (\tau-\tau_*)^2 \;
P_4^{V}\left(\log\frac{\tau-\tau_*}{b}\right) \right] \quad , \quad
\epsilon_v \buildrel{\tau \to \tau_* }\over= 3 \left[1 + (\tau-\tau_*)^2 \;
P_4^{\epsilon}\left(\log\frac{\tau-\tau_*}{b}\right) \right] \\ \cr
&&
W_\mathcal{R}(\eta)\buildrel{\eta \to 0 }\over=-\frac1{4 \; \eta^2}
\left[1 + \eta^3 \; P_4^{W}\left(\log \eta\right)\right] \; .
\end{eqnarray}
where
$$
\eta \buildrel{\tau \to \tau_* }\over= \frac32 \; (\tau-\tau_*)^\frac23
$$
is the conformal time for
$ \tau \to \tau_* $ and $ P_4^{V}(x), \; P_4^{\epsilon}(x) $ and $ P_4^{W}(x) $
are polynomials of degree four in $ x $ .
We see that the fluctuations feel a {\bf singular attractive} potential
near the $ \eta = 0 $ singularity. Actually, the behaviour of
$ W_\mathcal{R}(\eta) $ for $ \eta \to 0 $ is {\bf exactly} the
{\bf critical} strength ($ -1/4 $) for which the fall to the centre becomes possible in a
central and attractive singular potential \cite{ll}.
We find from eqs.(\ref{flutau}) and (\ref{wep0}) for the fluctuations near
the singularity
\begin{equation}\label{sr0}
S_\mathcal{R}(k;\eta) \buildrel{\eta , \eta_0 \to 0 }\over= \sqrt{\frac{\eta}{\eta_0}}
\left[ {\cal A}_\mathcal{R}(k) + {\cal B}_\mathcal{R}(k)
\; \log \frac{\eta}{\eta_0} \right] \; ,
\end{equation}
where $ \eta_0 $ is the time when the initial conditions will be imposed,
$ {\cal A}_\mathcal{R} $ and $ {\cal B}_\mathcal{R} $ are complex constants constrained by the
Wronskian condition (that ensures the canonical commutation relations)
\cite{biblia}
\begin{equation}\label{wronskian}
W[S_\mathcal{R},S^*_\mathcal{R}]= S_\mathcal{R} \;
\frac{dS^{*}_\mathcal{R}}{d \eta} -
\frac{dS_\mathcal{R}}{d \eta} \; S^*_\mathcal{R} = i \; .
\end{equation}
Namely,
\begin{equation}\label{conwr}
2 \; {\rm Im}[ {\cal A}_\mathcal{R} \; {\cal B}_\mathcal{R}^* ] = \eta_0 \; .
\end{equation}
Precisely, the logarithmic behaviour for $ \eta \to 0 $ of the wave
function eq.(\ref{sr0}) describes the fall to $ \tau - \tau_* = 0 $ for the
critical strength of the potential $ W_\mathcal{R}(\eta) $.
For larger attractive strengths the wave function eq.(\ref{sr0})
shows up an oscillatory behaviour \cite{ll}. Notice, however the physical
nature of the process:
here we have a time evolution near a classical singularity
at a given time while in the potential case one has particles falling
(or emerging) from a point in space where the potential is singular.
\medskip
In general, the mode functions for large $ k $ must behave as
free modes (plane waves) since the potential $ W_\mathcal{R}(\eta) $
in eq.(\ref{fluces}) becomes negligible in this limit except at the singularity
$ \tau=\tau_* $. One can then impose
Bunch-Davies conditions for large $ k $ which corresponds to
assume an initial quantum vacuum Fock state, empty of curvature excitations
\cite{biblia}
\begin{equation}\label{BDk}
S_\mathcal{R}(k;\tau) \buildrel{k \to \infty }\over=
\frac{e^{-i \; k \; \eta}}{\sqrt{2 \; k}}
\end{equation}
and therefore
$$
\frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) \buildrel{k \to \infty }\over=
-i \; k \; S_\mathcal{R}(k;\eta_0)\; .
$$
Eq.(\ref{BDk}) fulfils the Wronskian normalization eq.(\ref{wronskian}).
In asymptotically flat (or conformally flat) regions of the space-time
the potential felt by the fluctuations vanish and the fluctuations
exhibit a plane wave behaviour for {\bf all} $ k $ (not necesarily large).
This is not the case near strong gravity fields or curvature singularities
as in the present cosmological space-time where $ W_\mathcal{R}(\eta) $ can
never be neglected at fixed $ k $. However, we can choose Bunch-Davies initial
conditions (BDic) at $ \eta = \eta_0 $ by imposing
\begin{equation}\label{BD}
\frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) = -i \; k \;
S_\mathcal{R}(k;\eta_0) \quad {\rm for ~ all} \; k \; .
\end{equation}
That is, we consider the initial value problem for the mode functions
giving the values of $ S_\mathcal{R}(k;\eta) $ and
$ dS_\mathcal{R}/d \eta $ at $ \eta = \eta_0 $.
Notice that eq.(\ref{BD}) combined with the Wronskian condition
eq.(\ref{wronskian}) implies that
$$
|S_\mathcal{R}(k;\eta_0)| = \frac1{\sqrt{2 \; k}} \quad , \quad
\left| \frac{dS_\mathcal{R}}{d \eta}(k;\eta_0) \right| =
\sqrt{\frac{k}2} \; .
$$
which is equivalent to eq.(\ref{BDk}) for large $ k $.
\medskip
Since the mode functions $ S_\mathcal{R}(k;\eta) $ are defined up to
an arbitrary constant phase we can write eq.(\ref{sr0}) valid
near the metric singularity as
\begin{equation}\label{S}
S_\mathcal{R}(k;\eta) \buildrel{\eta , \eta_0 \to 0 }\over=
\sqrt{\frac{\eta}{2 \; k \; \eta_0}}\left[1 - \left(\frac12 + i \; k \;
\eta_0\right) \; \log \frac{\eta}{\eta_0} \right] \; .
\end{equation}
In eq.(\ref{sr0}) this corresponds to the coefficients,
$$
{\cal A}_\mathcal{R} = \frac1{\sqrt{2 \; k}} \quad , \quad
{\cal B}_\mathcal{R} = \frac1{\sqrt{2 \; k}}
\left(\frac12 + i \; k \; \eta_0\right) \quad .
$$
We have in cosmic time,
\begin{equation}\label{Stau}
S_\mathcal{R}(k;\tau) \buildrel{\tau , \tau_0 \to \tau_* }\over=
\frac1{\sqrt{2 \; k}} \;
\left(\frac{\tau-\tau_*}{\tau_0}\right)^\frac13
\left[1 - \left(\frac13 + i \; k \;
\tau_0^\frac23 \right) \; \log \frac{\tau-\tau_*}{\tau_0} \right] \; ,
\end{equation}
where $ \tau_0 - \tau_* = \left(2 \; \eta_0/3\right)^\frac32 $
for $ \tau \to \tau_* $.
Namely, imposing the BD initial condition (BDic) eq.(\ref{BD}) at small
$ \eta_0 $ where the small $ \eta $ behavior eq.(\ref{sr0}) applies,
yields specific values for the coefficients of the linearly
independent solutions $ \sqrt{\eta} $ and $ \sqrt{\eta} \, \log\eta $
that we can read from eqs.(\ref{S})-(\ref{Stau}).
\medskip
For general $ \tau_0 $ (i. e., $ \tau_0 $ not near $ \tau_* $), the mode functions for
$ \tau \to \tau_* $ take the form
\begin{equation}
S_\mathcal{R}(k;\tau) \buildrel{\tau \to \tau_* }\over=
\frac1{\sqrt{2 \; k}} \;
\left(\frac{\tau-\tau_*}{\tau_0}\right)^\frac13
\left[X(k,\tau_0) - \left(Y(k,\tau_0) + \frac{i \; k \;
\tau_0^\frac23}{X(k,\tau_0)}\right) \log \frac{\tau-\tau_*}{\tau_0}
\right] \; ,
\end{equation}
where we imposed eq.(\ref{conwr}) and we have from eq.(\ref{Stau}),
$$
X(k,\tau_*) = 1 \quad {\rm and} \quad Y(k,\tau_*) = \frac13 \quad .
$$
Notice that $ X(k,\tau_0) > 0 $ for $ \tau_0 \to \tau_* $ as we see from
eq.(\ref{Stau}). Our numerical calculations show that $ X(k,\tau_0) > 0 $
{\bf for all} $ \tau_0 $ and $ k $.
\subsection{The primordial power spectrum,
Scalar curvature fluctuations and the CMB+LSS data.}\label{priflu}
\begin{table}
\begin{tabular}{|c|c|c|}
\hline
Inflation (fast-roll) starts & $ {\ddot a}(\tau_s) = 0 $ & $ \tau_s = \tau_* + 0.0753090 $ \\
\hline $ V_\mathcal{R}(\tau) $ becomes positive & $
V_\mathcal{R}(\tau_+) = 0 $ & $ \tau_+ = \tau_* + 0.114 $ \\
\hline End of fast-roll & $ N \; \epsilon_v(\tau_{trans}) = 1 $ &
$ \tau_{trans} = \tau_* +0.2487963\ldots $ \\
\hline Maximum of $ V_\mathcal{R}(\tau) $ & $ V_\mathcal{R}'(\tau_M) = 0 $
& $ \tau_M = \tau_* +0.3503 \; , \; V_\mathcal{R}(\tau_M) = 51.196 $ \\
\hline End of Inflation & $ p(\tau_{end}) = 0 $ &
$ \tau_{end} = \tau_* + 18.2547816 $ \\ \hline
\end{tabular}
\caption{Selected time values for $ y = 1.26 $ and $ N_{sr} = 63 $ efolds
of slow-roll inflation. Notice that slow-roll starts exactly when fast-roll ends.
Recall that $ \tau = 4.97 \; 10^{-6} \; (t/t_{Planck}) $.}
\end{table}
The power spectrum of curvature perturbations $ \mathcal{R} $ is given by the
expectation value $ <\mathcal{R}^2> $ in the state with general initial conditions
\cite{biblia}
\begin{equation}\label{pot1}
< \mathcal{R}^2(\ensuremath{\vec{x}},\eta)> = \displaystyle \left(\frac{m}{M_{PL}}\right)^2
\int_0^{\infty} \frac{\displaystyle |S_\mathcal{R}(k;\eta)|^2}{\displaystyle z^2(\eta)} \;
\frac{k^2 \, dk}{2 \, \pi^2} \; .
\end{equation}
where $ z(\eta) $ is given by eq.(\ref{defwz}).
Notice in eq.(\ref{pot1}) the factor $ (m/M_{PL})^2 $ in the physical power
spectrum expressed in terms of the dimensionless quantities used here.
The power spectrum at time $ \eta $ is customary defined as
the power per unit logarithmic interval in $ k $
$$
<\mathcal{R}^2(\ensuremath{\vec{x}},\eta)> =\int_0^{\infty} \frac{dk}{k} \; P_\mathcal{R}(k,\eta) \; .
$$
Therefore, the scalar power for general initial conditions
is given by the fluctuations behavior by the end of
inflation \cite{biblia} ,
\begin{equation}\label{curvapot}
P_\mathcal{R}(k) = \left(\frac{m}{M_{PL}}\right)^2
\; \frac{k^3}{2 \; \pi^2} \; {\displaystyle \lim_{\eta \to 0^-}}
\left|\frac{S_\mathcal{R}(k;\eta)}{z(\eta)} \right|^2 \; .
\end{equation}
The mode functions $ S_\mathcal{R}(k;\eta) $ obey the fluctuations equation (\ref{fluces})
where the potential $ W_\mathcal{R}(\eta) $ [eq.(\ref{wtau})]
during slow-roll and to leading order in $ 1/N $ takes the simple form \cite{biblia} ,
\begin{equation} \label{eqnz}
W_\mathcal{R}(\eta)= \frac2{\eta^2} \left[1+ \frac32\; (3
\; \epsilon_v-\eta_v) \right] = \frac{\nu^2_\mathcal{R}-
\frac14}{\eta^2} \quad , \quad \nu_\mathcal{R} = \frac32+ 3 \,
\epsilon_v -\eta_v + {\cal O}\left(\frac1{N^2}\right) \; .
\end{equation}
In the slow-roll regime we can consider $ \epsilon_v $ and $ \eta_v $ [see eq.(\ref{epsv})]
constants in time in eq.(\ref{eqnz}). During slow-roll,
the general solution of eq.(\ref{fluces}) is then given by
\begin{equation} \label{bogo}
S_\mathcal{R}(k;\eta) = A_\mathcal{R}(k) \;
g_{\nu_\mathcal{R}}(k;\eta) + B_\mathcal{R}(k) \; g^*_{\nu_\mathcal{R}}(k;\eta) \; ,
\end{equation}
with
\begin{equation}\label{gnu}
g_{\nu }(k;\eta) = \frac12 \; i^{\nu +\frac12} \;
\sqrt{-\pi \eta} \; H^{(1)}_{\nu }(-k \; \eta) \; ,
\end{equation}
$ A_\mathcal{R}(k), \; B_\mathcal{R}(k) $ are constants determined by the
initial conditions and $ H^{(1)}_{\nu}(z) $ is a Hankel function.
The Wronskian of the solutions $ S_\mathcal{R}, \; S_\mathcal{R}^* $ is given by
eq.(\ref{wronskian}) and
$$
W[g_{\nu},g_{\nu}^*]=i
$$
This generically determines that
\begin{equation}\label{abc}
|A_\mathcal{R}(k)|^2 - | B_\mathcal{R}(k)|^2 = 1 \; .
\end{equation}
For wavevectors deep inside the Hubble radius $ | k \, \eta | \gg 1
$ the mode functions $ g_{\nu}(k;\eta) $ have the asymptotic behavior
\begin{equation}
g_{\nu}(k;\eta) \buildrel{\eta \to
-\infty}\over= \frac1{\sqrt{2 \, k}} \; e^{-i \; k \; \eta} \quad , \quad
g^*_{\nu}(k;\eta) \buildrel{\eta \to -\infty}\over=
\frac1{\sqrt{2k}} \; e^{ i \; k \; \eta} \; , \label{fnuasy}
\end{equation}
while for $ \eta \to 0^- $, they behave as:
\begin{equation}\label{geta0}
g_\nu(k;\eta)\buildrel{\eta \to 0^-}\over=
\frac{\Gamma(\nu)}{\sqrt{2 \, \pi \; k}} \; \left(\frac2{i \; k \;
\eta} \right)^{\nu - \frac12} \; .
\end{equation}
In particular, in the scale invariant case $ \nu=\frac32 $ which is
the leading order in the slow-roll expansion, the mode functions
eqs.(\ref{gnu}) simplify to
\begin{equation}\label{g32}
g_{\frac32}(k;\eta) =
\frac{e^{-i \; k \;\eta}}{\sqrt{2k}}\left[1-\frac{i}{k \;\eta}\right]\; .
\end{equation}
As we see from eq.(\ref{defwz}), $ z(\eta) $ obeys eq.(\ref{fluces}) for $ k = 0 $
and therefore $ z(\eta) $ in the slow-roll regime behaves as
\begin{equation}\label{zeta2}
z(\eta) = \frac{z_0}{ (-k_0 \; \eta)^{\nu_\mathcal{R}-\frac12}} \; ,
\end{equation}
where $ z_0 $ is the value of $ z(\eta) $ when the pivot scale $ k_0 $ exits the horizon,
that is at $ \eta = -1/k_0 $. Combining this
result with the small $ \eta $ limit eq.(\ref{geta0}) we find from
eqs.(\ref{curvapot}) and (\ref{zeta2}),
\begin{equation} \label{powR}
P_{\mathcal{R}}(k) = P^{BD}_{\mathcal{R}}(k)\left[1+ D(k) \right] \; ,
\end{equation}
where we introduced the transfer function for the initial
conditions of curvature perturbations:
\begin{equation} \label{DofkR}
D(k) = 2 \; | {B}_\mathcal{R}(k)|^2 -2 \;
\mathrm{Re}\left[A_\mathcal{R}(k) \; B^*_\mathcal{R}(k)\,i^{2\nu_\mathcal{R}-3}\right] \; .
\end{equation}
$ D(k) $ is obtained imposing BDic at $ \tau = \tau_0 $ according
to eq.(\ref{BD}).
\medskip
Notice as shown in sec. \ref{dcero} that the transfer function $ D(k) $ enjoys the properties
\begin{equation}\label{propg}
1 + D(k) \buildrel{k \to 0 }\over= {\cal O}(k^{n_s+1}) \quad , \quad
D(k) \buildrel{k \to \infty }\over= {\cal O}\left(\frac1{k^2}\right) \; .
\end{equation}
$ D(k) $ accounts for the effect in the power spectrum both of the initial conditions
and of the fluctuations evolution during fast-roll (before slow-roll).
$ D(k) $ depends on the time $ \tau_0 $ at which BDic are imposed.
\medskip
If one chooses the extreme slow-roll solution presented in sec. \ref{esr} and imposes BDic at
$ \tau_0 = -\infty $ (that is, $ \eta_0 = -\infty $) then $ D(k) = 0 $ and the fluctuation
power spectrum at the end of inflation is the usual power spectrum
$ P_\mathcal{R}(k)=P^{BD}_{\mathcal{R}}(k) $.
\medskip
$ P^{BD}_{\mathcal{R}}(k) $ is given by its customary slow-roll expression,
\begin{equation}\label{pbd}
\log P^{BD}_{\mathcal{R}}(k) = \log A_s(k_0) + (n_s-1) \; \log\frac{k}{k_0} +
\tfrac12 \; n_{run} \; \log^2\frac{k}{k_0} +{\cal O}\left(\frac1{N^3}\right) \; .
\end{equation}
We solved numerically the fluctuations equation (\ref{flutau}) in
cosmic time with the BDic eq.(\ref{BD}) covering both the fast-roll and slow-roll
regimes. We started at
initial times $ \tau_0 $ ranging from the vicinity of
$ \tau = \tau_* $ till the transition
time $ \tau_{trans} = 0.2487963\ldots $ from fast-roll to slow-roll.
We computed the transfer function $ D(k) $ from the mode functions
behaviour deep during slow-roll inflation from eqs.(\ref{curvapot}) and (\ref{powR})
\cite{biblia}.
In figs. \ref{fig:DBDC} we depict $ 1 + D(k) $ vs.
$ k $ for twelve values of the time $ \tau_0 $ where BDic are imposed.
\begin{figure}[h]
\includegraphics[width=16.cm]{DBDs4}
\caption{Numerical transfer function $ 1 + D(k) $.
Lower left panel: Numerical transfer function $ 1 + D(k) $ for
BDic at $ \tau = \tau_0 = \tau_\ast + \Delta \tau $, for different $ \Delta \tau $
values as given in the picture. We see here that the peak of $ 1 + D(k) $ grows and moves for
larger $ k $ as $ \tau_0 $ increases. Here $ N_{sr} = 63 $.
Lower right panel: The transfer function $ 1+D(k) $ when the BDic eq.(\ref{BD}) are imposed
during slow--roll at finite times $ \tau_0 $ and $ N_{sr} $ efolds of slow--roll
have still to occur.
Upper panels: Numerical transfer function $ 1 + D(k) $ for
BDic at $ \tau = \tau_0 = \tau_\ast + \Delta \tau $, for different values of
$ \Delta \tau $ as given in the picture. We get stronger oscillations in
$ 1 + D(k) $ for decreasing $ \tau_0 $ in
the range $ \Delta \tau < 0.04 $. Here $ N_{sr} = 63 $.}
\label{fig:DBDC}
\end{figure}
\medskip
Notice that when BDic are imposed at {\bf finite times} $ \tau_0 $, the spectrum
{\bf is not} the usual $ P^{BD}_{\mathcal{R}}(k) $ but it gets modified by
a non-zero transfer function $ D(k) $
eq.(\ref{powR}). The power spectrum $ P_{\mathcal{R}}(k) $ vanishes at $ k = 0 $
and exhibits oscillations which vanish at large $ k $ [see figs. \ref{fig:DBDC}
and \ref{fig:BDs}].
\begin{figure}[h]
\includegraphics[width=14.cm]{BDs}
\caption{Power spectrum with BDic eq.(\ref{BD}) imposed during slow--roll
when $ N_{sr} $ efolds of slow--roll inflation have still to occur. We see here the decrease of
the power spectrum $ P_\mathcal{R} $ as $ k^{n_s-1} $ multiplied by the oscillations
of $ 1 + D(k) $. See eqs.(\ref{powR}) and (\ref{pbd}) and figs. \ref{fig:DBDC}.
The non-oscillatory black curve corresponds to the usual power with BDic at $ \eta_0 =
-\infty $ eq.(\ref{pbd}) decreasing as $ k^{n_s-1} $. The later are imposed the BDic,
the smaller is the number of slow-roll efolds $ N_{sr} $ and the whole $k$-spectrum
shifts to larger $ k $.}
\label{fig:BDs}
\end{figure}
\medskip
During slow-roll different initial times $ \tau_0 $ lead essentially to a rescaling of $ k $
in $ D(k) $ by a factor $ \eta_0 $ since the
conformal time $ \eta $ is almost proportional to $ 1/a(\eta) $ during slow-roll
[see figs.~\ref{fig:BDs}- \ref{fig:DBDC} and below eq.(\ref{dsr})].
By virtue of the dynamical attractor character of slow--roll,
the power spectrum when the BDic are imposed at a finite time
$ \tau_0 $ cannot really distinguish between the extreme slow--roll solution
(for which slow--roll starts from the very beginning $ \eta_0 = -\infty $)
or any other solution which is attracted to slow--roll well before the time $ \tau_0 $.
\subsection{Accurate numerical computation of the power spectrum and the
transfer function $ D(k) $ of initial conditions.}
In order to accurately calculate $ n_s $ we proceed as follow.
We match the solution $ S_\mathcal{R}(k;\eta) $ with the slow--roll
solution $ g_{\nu_\mathcal{R}}(k;\eta) $ eq.(\ref{gnu}) at the time $ \tau_0 $ when
$ N_{sr} $ efolds
of slow--roll have still to occur. $ \eta $ and $ \nu_\mathcal{R} $ are computed at this time
$ \tau_0 $. In practice, this corresponds to setting
$ A_\mathcal{R}(k)=1, \; B_\mathcal{R}(k) = 0 $
(and therefore $ D_\mathcal{R}(k) = 0 $) in the Bogoliubov transformation eq.(\ref{bogo}).
Then, we integrate numerically the fluctuations
equations eq.(\ref{flutau}). By construction, this produces the standard spectra
$ P^{BD}_{\mathcal{R}}(k) $ eq.(\ref{pbd}) that quickly stabilize as $ N_{sr} $ is
increased a few efolds above $ N=60 $.
\medskip
It is convenient to introduce the quantity
\begin{equation}\label{defl}
L_s \equiv \log\left[\left(\frac{M_{PL}}{m}\right)^2 \; A_s(k_0=m)\right] \; ,
\end{equation}
with $ k_0=m $ when $ a(\eta)=1 $, that is $ N=60 $ efolds before inflation ends.
In table IV we provide $ L_s $ for several values of $ N_{sr} $.
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
~~$N_{sr}$ ~~&~~ $L_s$ ~~&~~ $n_s$ ~~&~~ $ n_{run} $ \\
\hline \hline
~~61~~&~~ $4.6585381\ldots$ ~~&~~ $0.9637013\ldots$ ~~&~~ $-0.0000701\ldots$ \\
\hline
~~63~~&~~ $4.6583004\ldots$ ~~&~~ $0.9641135\ldots$ ~~&~~ $-0.0001639\ldots$ \\
\hline
~~65~~&~~ $4.6584371\ldots$ ~~&~~ $0.9642483\ldots$ ~~&~~ $-0.0002165\ldots$ \\
\hline
~~67~~&~~ $4.6584463\ldots$ ~~&~~ $0.9642444\ldots$ ~~&~~ $-0.0002165\ldots$ \\
\hline
~~69~~&~~ $4.6584469\ldots$ ~~&~~ $0.9642448\ldots$ ~~&~~ $-0.0002167\ldots$\\
\hline
\end{tabular}
\caption{Exact values of $ L_s = \log[\left(M_{PL}/m\right)^2 \; A_s(k_0=m)] $
for several values of $ N_{sr} $ from the numerical calculation. The exact values of
$ n_s $ vary little with $ N_{sr} $ and are close to the slow-roll approximation value.
Also $ n_{run} $ is close to the value in the slow-roll approximation.}
\end{table}
\medskip
To transform this $ k_0 $ in a wavenumber today we need:
\begin{itemize}
\item {the total redshift from 60 efolds before inflation ends till today
[since we choose $ a( \tau = 0 ) = 1 $ when there are still $ N=60 $
efolds till the end of inflation].}
\item {the value of $ m $ as determined by the observed value of the amplitude $ A_s(k_0) $.}
\end{itemize}
Let $ k_0^{\rm CMC} $ be the
value of the pivot scale of CosmoMC [that is 50 (Gpc)$^{-1}$ today] 60 efolds
before the end of inflation. Then, we have from eqs.(\ref{pbd}) and (\ref{defl}),
\begin{equation}\label{Asm}
\log A_s(k_0=m) = L_s + 2 \, \log\frac{m}{M_{PL}} = L_s^{\rm CMC} +
(n_s^{\rm CMC}-1) \; \log\frac{m}{k_0^{\rm CMC}} + \tfrac12 \; n_{run} \;
\left[\log\frac{m}{k_0^{\rm CMC}}\right]^2 + {\cal O}\left(\frac1{N^3}\right) \; ,
\end{equation}
where $ L_s^{\rm CMC} \equiv \log A_s^{\rm CMC} (k_0^{\rm CMC}) $ and $ n_s^{\rm CMC} $
are best fit values in a given CosmoMC run. Since the running index $ n_{run} $ is
$ {\cal O}\left(1/N^2\right) $, we get for $ m $,
\begin{equation}
\left(\frac{m}{M_{PL}}\right)^2 = \left(\frac{m}{k_0^{\rm CMC}}\right)^{n_s^{\rm CMC}-1}
\exp\,(L_s^{\rm CMC} - L_s)\left[1+ {\cal O}\left(\frac1{N^2}\right)\right] \; .
\end{equation}
The wavevectors at $ a = 1 $ (60 efolds before inflation ends) and today are related by
\cite{biblia}
\begin{equation}\label{ka1}
k^{a=1} = \frac{e^{60}}{a_r} \; k^{today} \; ,
\end{equation}
where $ a_r $ is the scale factor by the end of inflation
\begin{equation}\label{var}
a_r = 2.5 \; 10^{-29} \; \sqrt{\frac{10^{-4}M_{PL}}{H_{60}}} \; ,
\end{equation}
and $ H_{60} $ is the Hubble parameter 60 efolds before inflation ends.
We thus have for the pivot wavenumber at $ a = 1 $
\begin{equation} \label{vk0}
k_0^{\rm CMC} \simeq 1.46\ldots \, \sqrt{\frac{H_{60}}{10^{-4}M_{PL}}} \times 10^{15}~{\rm GeV}
\end{equation}
and
\begin{equation*}
\left(\frac{m}{M_{PL}}\right)^{2-(n_s^{\rm CMC}-1)/2} =
\left(\frac{16.67\ldots}{\sqrt{h_{60}}}\right)^{n_s^{\rm CMC}-1} \,
\exp\,(L_s^{\rm CMC} - L_s) \quad , \quad
{\rm where} \quad h_{60} \equiv \frac{H_{60}}{m} \; .
\end{equation*}
Notice the small $ 1/N $ correction $ (n_s^{\rm CMC}-1)/2 $ in the exponent of
$ m/M_{PL} $. Eq.(\ref{Asm}) yields for the best fit CosmoMC run
$ L_s^{\rm CMC} = -19.9808\ldots $ and $ n_s = 0.9635\ldots $ \cite{biblia}:
\begin{equation*}
m \simeq 4.8114\ldots 10^{-6}\, M_{PL} =
1.1717 \ldots 10^{13}~{\rm GeV}
\end{equation*}
The exact values given above in Table IV
$$
A_s = \left(\frac{m}{M_{PL}}\right)^2 \; \exp(L_s) \; , \quad n_s \quad
{\rm and} \quad n_{run}
$$
are obtained taking into account
the fast-roll and slow--roll stages in the numerical calculation.
We can compare them to their slow--roll (leading $ 1/N $) analytic
counterparts for the double-well quadratic plus quartic potential,
\cite{biblia}
\begin{equation*}
A_s = \frac{N^2}{12\pi^2} \left(\frac{m}{M_{PL}}\right)^2 \,
\frac{(1-z)^4}{y^2 \, z} \;,\quad
n_s = 1 - \frac{y}{N} \, \frac{3 \, z + 1}{(1-z)^2} \;,\quad
n_{run} = \frac{y^2 \, z}{N^2 \, (1-z)^4}\left(24 \, z^2 - 35 \, z +3\right)
\end{equation*}
where $ N=60 , \; z = 0.117446 $ and $ y = z-1-\log z = 1.2592226\ldots $, that is
\begin{equation*}
A_s = \frac{N^2}{12\pi^2} \; \left(\frac{m}{M_{PL}}\right)^2 \;
\exp(4.59536898\ldots) \; , \quad n_s = 0.9635620\ldots \; , \quad
n_{run} = -0.0000664 \ldots
\end{equation*}
The figure in the exponent is to be compared with the $ L_s $ values in Table IV.
The agreement with Table IV is quite good, especially for $ n_s $.
\medskip
\begin{figure}[h]
\includegraphics[width=16.cm]{deltaD}
\caption{Upper Left panel: Difference between the (approximate) transfer function
$ {\tilde D}(k \; \eta_0) $ eqs.(\ref{dsr})-(\ref{dsr32}) for
$ \nu_\mathcal{R}=1.5182189\ldots $ and the numerical
(exact at least to a $ 10^{-7} $ relative error) transfer function
$ D(k) $, when $ N_{sr} = 63 $. Upper Right panel: Difference between the
(approximate) transfer function $ {\tilde D}(k \; \eta_0) $ eqs.(\ref{dsr})-(\ref{dsr32}) for
$ \nu_\mathcal{R}=3/2 $ (the scale-invariant value) and the numerical (exact)
transfer function. We see that the difference in the right panel [eq.(\ref{dsr32})] is
$ < 0.014 $ while in the left panel the difference of the analytic formula eq.(\ref{dsr})
is much smaller, $ < 0.0005 $.
Lower Left panel: difference between the exact (numerical) $ D(k) $ computed for
the fast-roll inflaton solution of table II and for the extreme slow--roll inflaton solution
of table I when BDic are imposed 63 efolds before the end of inflation. Lower Right
panel: difference between the numerical (exact) fast--roll $ D(k) $ and the approximate
$ {\tilde D}(k \; \eta_0) $ calculated with $ \nu_\mathcal{R}=3/2 $ and
$ \eta_0=-4.0169827\ldots $. We see that the differences are small in both cases.}
\label{fig:deltaD}
\end{figure}
We now find the exact (numerical) transfer
function $ D(k) $ for the initial conditions, by simply taking in eq.(\ref{powR})
the ratio of the two power spectra: $ P_{\mathcal{R}}(k) $ with BDic at time $ \tau_0 $
and $ P^{BD}_{\mathcal{R}}(k) $.
In the case of BDic at finite times the result is given in fig~\ref{fig:DBDC}.
At the largest value $ k/m=100 $ of the wavenumber interval considered, we have
$$
1+D(100 \, m)= 0.9996994\ldots , \; 1.0000061\ldots, \; 1.0000001\ldots
$$
for
$$
N_{sr}=61, \; 63, \; 65 \; {\rm and} \; 67 , \; {\rm respectively}.
$$
This provides a good check of the accuracy of the calculation.
In figs. \ref{fig:deltaD} we compare the numerically computed $ D(k) $
against $ {\tilde D}(k \; \eta_0) $ analytically computed for BDic imposed at time $ \eta_0 $
during slow--roll in eq.(\ref{dsr}), sec. \ref{bdicsr}.
The comparison is performed for BDic imposed when $ N_{sr} = 63 $ on the
extreme slow roll solution, which corresponds to $ \eta_0=-4.0202308\ldots $. We
consider two values of $ \nu_\mathcal{R} : \; \nu_\mathcal{R} = 2-n_s/2 = 1.5182189\ldots $,
$ n_s=0.9635620\ldots $ corresponding to slow--roll at leading $ 1/N $ order, and the
exactly scale-invariant case $ \nu_\mathcal{R}=3/2 $.
Notice that in the latter case
$ {\tilde D}(k \; \eta_0) $ has the explicit simple analytic form eq.(\ref{dsr32}).
The maximum of the numerical transfer function $ 1+D(k) $ is located at $ k/m=0.68755\ldots$
and has the value $ 1.13218\ldots $ The maximum of $ 1+ {\tilde D}(k \; \eta_0) $, when
$ \nu_\mathcal{R}=3/2 $ is in $ k/m=0.68755\ldots $ and has the value
$ 1.13009\ldots $. Recall that these values of $ k/m $ have the scale fixed
by the choice $ a=1 $ when $ N=60 $ efolds lack
before inflation ends.
\medskip
Let us now consider the fluctuations on the fast--roll solution of
Table II. Since $ \eta $ has a finite lower limit, the choice
$ A_\mathcal{R}(k)=1, \; B_\mathcal{R}(k) = 0 $ has little
meaning and BDic can be imposed only at a {\rm finite} time $ \tau_0 $ later than
the singularity time $ \tau_\ast $. If $ \tau_0 $ is exactly the transition time
$ \tau_{trans} $ when $ \epsilon_v=1/N $, fast-roll ends and slow--roll begins,
(to proceed for $ N_{sr}=63 $ efolds), then $ D(k) $ does not differ too much
from that computed with the extreme slow roll solution.
This comparison is performed in the lower left
panel of fig.~\ref{fig:deltaD}. In the right panel $ D(k) $ is compared
to the $ {\tilde D}(k \; \eta_0) $ for $ \nu_\mathcal{R}=3/2 $ and $ \eta_0=-4.0169827\ldots $,
which is the value of the conformal time at the onset of slow--roll
(see Table II).
\medskip
When the BDic are imposed during the fast--roll stage well {\bf before} it ends,
$ D(k) $ changes much more significantly than along the extreme slow roll solution.
This is due to two main effects: the
potential felt by the fluctuations is attractive during fast--roll and $ \eta_0 $, far from
being almost proportional to $ 1/a(\eta) $, tend to the constant value
$ \eta_\ast $ as $ \tau \to \tau_\ast^+ $ and $ a(\eta)\to 0 $. The numerical
transfer functions $ 1+ D(k) $ obtained from eqs.(\ref{curvapot}) and (\ref{powR})
are plotted in figs.~\ref{fig:DBDC}.
\medskip
The fact that choosing BDic leads to a primordial power and its respective
CMB multipoles which correctly {\bf reproduce} the observed spectrum
justifies the use of BDic for the scalar curvature fluctuations.
\subsection{The effect of the fast-roll stage on the low multipoles of the
CMB}\label{eiclm}
In the region of the Sachs-Wolfe plateau for $ l \lesssim 30 $, the
matter-radiation transfer function can be set equal to unity and the
CMB multipole coefficients $ C_l's $ are given by \cite{hu}
\begin{equation} \label{cl}
C_l =\frac{4\pi}{9} \int_0^\infty \frac{dk}{k}\, {P}_X(k) \left\{
j_l[k(\eta_0-\eta_{LSS})]\right\}^2 \; ,
\end{equation}
where $ P_X $ is the power spectrum of the corresponding perturbation, $
X=\mathcal{R} $ for curvature perturbations and $ X=T $ for tensor
perturbations, $ j_l(x) $ are spherical Bessel functions \cite{abra}
and $ \eta_0-\eta_{LSS} $ is the comoving
distance between today and the last scattering surface (LSS)
given by
\begin{equation} \label{etaLSS}
\eta_0-\eta_{LSS} = \frac1{H_0} \; \int^1_{\frac1{1+z_{LSS}}}
\frac{da}{\sqrt{\Omega_r + \Omega_M \; a + \Omega_{\Lambda} \; a^4}} \; ,
\end{equation}
where $ \Omega_r , \; \Omega_M $ and $ \Omega_{\Lambda} $ stand for the
fraction of radiation, matter and cosmological constant in today's
Universe. We find using $ z_{LSS} =1100 $,
\begin{equation}\label{329}
\eta_0-\eta_{LSS} = \frac{3.296}{H_0} \; .
\end{equation}
Notice that $ k/H_0 \sim d_H/\lambda_{phys}(t_0) $ is the ratio
between today's Hubble radius and the physical wavelength. The power
spectrum for curvature ($ \mathcal R $) perturbations
$ P_{\mathcal{R}}(k) $ is given by eqs.(\ref{powR})-(\ref{pbd}).
\begin{figure}[h]
\includegraphics[width=16.cm]{dcls.eps}
\caption{The change $ \Delta C_{\ell}/ C_{\ell} $ on the
CMB multipoles for $ \ell =1, \ldots, 5 $.
Upper plot: $ \Delta C_{\ell}/ C_{\ell} $ vs. $ \tau_0 - \tau_* $ for $ 0<\tau_0 - \tau_* <
0.2487963\ldots$. Lower plot: $ \Delta C_{\ell}/ C_{\ell} $ vs. $ \tau_0 - \tau_* $ for
$ 0.0193<\tau_0 - \tau_* < 0.2487963\ldots . \; \tau_0 $ is the time when the BDic
eq.(\ref{BD}) are imposed to the fluctuations. We choose $ \tau_0 $
inside the fast-roll stage. $ \Delta C_{\ell}/ C_{\ell} $ is {\bf positive} for
small $ \tau_0 - \tau_\ast $ and {\bf decreases} with $ \tau_0 $ becoming then
{\bf negative}. The CMB quadrupole observations indicate a large {\bf suppression}
thus indicating that $ \tau_0 - \tau_\ast \gtrsim 0.05 \simeq 10100 \; \tau_{Planck} $.
Our {\bf predictions} here for the quadrupole and octupole suppressions
are to be confronted with forthcoming CMB observations.
It will be extremely interesting to measure the primordial dipole and compare
with our predicted value.}
\label{dcl}
\end{figure}
Inserting eq.(\ref{powR}) into eq.(\ref{cl}) yields the $ C_l $ as the sum
of two terms
\begin{equation} \label{DelC}
C_l = C ^{BD}_l + \Delta C_l
\quad , \quad \frac{\Delta C_l}{C_l} = \frac{\int^\infty_0
D(\kappa \; x)~ f_l(x) \; dx}{\int^\infty_0 f_l(x) \; dx} \quad , \quad
x= k(\eta_0-\eta_{LSS}) = k/\kappa \; ,
\end{equation}
where from eq.(\ref{329}), $ \kappa \equiv H_0/3.296\ldots $,
\begin{equation} \label{fls}
f_l(x)= x^{n_s-2} \; [j_l(x)]^2 \; .
\end{equation}
and $ j_l(x) $ stand for the spherical Bessel functions.
The $ C ^{BD}_l $'s correspond to the standard BD power spectrum
$ P^{BD}_{\mathcal{R}}(k) $ eq.(\ref{pbd}) and the $ \Delta C_l $ exhibit
the effect of the transfer function $ D(k) $ on the $ C_l $.
\medskip
Using the transfer function $ D(k) $ obtained above eq.(\ref{DofkR}),
we computed the change on the
CMB multipoles $ \Delta C_{\ell}/ C_{\ell} $ for $ \ell =1,\ldots, 5 $ as
functions of the starting instant of the fluctuations $ \tau_0 $.
We plot $ \Delta C_{\ell}/ C_{\ell} $ for $ 1 \leq \ell \leq 5 $ vs. $ \tau_0 - \tau_\ast $
in fig. \ref{dcl}. We see that $ \Delta C_{\ell}/ C_{\ell} $ is {\bf positive} for
small $ \tau_0 - \tau_\ast $ and {\bf decreases} with $ \tau_0 $ becoming then
{\bf negative}. The CMB quadrupole observations indicate a large {\bf suppression}
thus indicating that $ \tau_0 - \tau_\ast \gtrsim 0.05 \simeq 10100 \; \tau_{Planck} $.
Being $ D(k) < 0 $ for low $ k $ as depicted in figs. \ref{fig:DBDC},
the primordial power at large scales is then suppresed and
the low $ C_{\ell} $ decrease as seen from eq.(\ref{DelC}).
$ \Delta C_{\ell}/ C_{\ell} $ mainly originates from the peak of $ D(k) $ displayed in figs.
\ref{fig:DBDC} whose position moves to smaller $ k $ for
decreasing $ \tau_0 $. Therefore, the primordial power suppression is
less important for decreasing $ \tau_0 $ and the CMB multipole suppression
$ \Delta C_{\ell}/ C_{\ell} $ less important as depicted in figs. \ref{dcl}.
For small $ \tau_0 - \tau_\ast \lesssim 0.05 $ the peak of $ D(k) $ grows
significantly and $ \Delta C_{\ell}/ C_{\ell} $ become positive, namely the low CMB multipoles
are enhanced.
\medskip
It should be recalled that the observation of a low CMB quadrupole
sparked many different proposals to explanain that suppression \cite{expla}.
\medskip
Besides finding a CMB quadrupole suppression in agreement with observations
\cite{biblia}-\cite{quamc},
we provide here {\bf predictions} for the dipole and octupole suppressions.
Forthcoming CMB observations can provide better data to confront our
quadrupole and octupole suppression predictions.
It will be extremely interesting to measure the primordial dipole and compare
with our predicted value.
\section{Analytic formulas for the transfer function $ D(k) $.}
It is very important to dispose of analytic formulas for the transfer
function $ D(k) $ in order to better understand the physical origin
of its oscillations and properties as well as in
the perspective of the MCMC data analysis.
However, the mode equations (\ref{fluces}) are not solvable in closed
form for $ k \neq 0 $ , not even for the approximated inflation solution
eq.(\ref{solfr}) which leads to the potential $ V_\mathcal{R}(\tau) $
eq.(\ref{vafr}).
The function $ D(k) $ must obey the general properties eq.(\ref{propg}).
\subsection{The primordial power spectrum vanishes for $ k \to 0 $
and becomes the BD power spectrum for $ k \to \infty $}\label{dcero}
The fluctuations equation (\ref{fluces}) can be solved explicitly for $ k = 0 $
\begin{equation}\label{S0}
s(\eta) = c_1 \; z(\eta) + c_2 \; z(\eta) \;
\int_{\eta_0}^{\eta} \frac{d\eta'}{z^2(\eta')} \; ,
\end{equation}
where $ c_1 $ and $ c_2 $ are arbitrary constants.
The BDic eq.(\ref{BD}) introduce for $ k \to 0 $ a $ 1/\sqrt{2 \, k} $ singularity
in the mode functions. Thus, the mode functions must have the behaviour
\begin{equation}\label{srk0}
S_\mathcal{R}(k;\eta)\buildrel{k \to 0 }\over= \frac{s(\eta)}{\sqrt{2 \, k}}\left[ 1 +
{\cal O}(k) \right]
\end{equation}
where $ s(\eta) $ is given by eq.(\ref{S0}).
Inserting eq.(\ref{srk0}) into the BDic eq.(\ref{BD}) yields for $ k \to 0 $,
$$
s(\eta_0) = 1 \quad , \quad \frac{ds(\eta_0)}{d\eta} = 0 \; ,
$$
which determines the coefficients $ c_1 $ and $ c_2 $ in eq.(\ref{S0}).
We finally obtain
\begin{equation}\label{sfin0}
s(\eta) = \frac{z(\eta)}{z(\eta_0)}- z'(\eta_0) \; z(\eta) \;
\int_{\eta_0}^{\eta} \frac{d\eta'}{z^2(\eta')}
\end{equation}
and using eq.(\ref{zeta2}) valid for $ \eta \to 0^- $ when slow--roll applies
\begin{equation}\label{sfin1}
{\displaystyle \lim_{\eta \to 0^-}}\frac{s(\eta)}{z(\eta)} = \frac1{z(\eta_0)} \; .
\end{equation}
The primordial power spectrum for $ k \to 0 $ follows by inserting
eq.(\ref{srk0}) and eq.(\ref{sfin1}) into the general expression eq.(\ref{curvapot}),
$$
P_\mathcal{R}(k) \buildrel{k \to 0 }\over= \left(\frac{m}{M_{PL}}\right)^2
\; \frac{k^3}{2 \; \pi^2} \; {\displaystyle \lim_{\eta \to 0^-}}
\left|\frac{S_\mathcal{R}(k;\eta)}{z(\eta)} \right|^2 \buildrel{k \to 0 }\over=
\left(\frac{m}{M_{PL}}\right)^2 \; \left( \frac{k}{2 \, \pi \; z(\eta_0)}\right)^2
$$
We thus find in general that the power spectrum vanishes as $ k^2 $ for $ k \to 0 $
and therefore
$$
1 + D(k) \buildrel{k \to 0 }\over= {\cal O}(k^{n_s+1})
$$
as stated in eq.(\ref{propg}). This property is generally true except for the extreme
slow-roll inflaton solution (sec. \ref{esr}) with BDic imposed at $ \eta_0 = -\infty $
in which case $ D(k) $ vanishes identically for all $ k $.
\bigskip
For growing $ k $ the modes exit the horizon later on,
during the slow--roll regime where eq.(\ref{bogo}) applies. For large $ k $
the mode functions $ S_\mathcal{R} $ as well as $ g_{\nu_\mathcal{R}} $
behave as plane waves [eqs.(\ref{BDk}) and (\ref{geta0})] and therefore
$$
A_\mathcal{R}(k)=1 \quad , \quad B_\mathcal{R}(k) = 0 \quad . \quad
{\rm Hence} \quad D(k) \buildrel{k \to \infty }\over= 0 \; .
$$.
\subsection{The transfer function $ D(k) $ when BDic are imposed during
slow--roll.}\label{bdicsr}
When the BDic eq.(\ref{BD}) are imposed during slow--roll at a finite time $ \eta_0 $
we can use eq.(\ref{bogo}) for the mode functions at $ \eta = \eta_0 $ and we obtain,
\begin{eqnarray}
\frac{e^{-i \; k \; \eta_0}}{\sqrt{2 \; k}} &=&
A_\mathcal{R}(k) \;
g_{\nu_\mathcal{R}}(k;\eta_0) + B_\mathcal{R}(k) \; g^*_{\nu_\mathcal{R}}(k;\eta_0) \cr \cr
-i \; k \; \frac{e^{-i \; k \; \eta_0}}{\sqrt{2 \; k}} &=&
A_\mathcal{R}(k) \;
g'_{\nu_\mathcal{R}}(k;\eta_0) + B_\mathcal{R}(k) \; g'^*_{\nu_\mathcal{R}}(k;\eta_0)
\end{eqnarray}
which determines
\begin{equation}\label{absr}
A_\mathcal{R}(k)=\frac{e^{-i \; k \; \eta_0}}{i \; \sqrt{2 \; k}}\left[
g'^*_{\nu_\mathcal{R}}(k;\eta_0) + i \; k \; g^*_{\nu_\mathcal{R}}(k;\eta_0) \right]
\quad , \quad
B_\mathcal{R}(k)=\frac{e^{-i \; k \; \eta_0}}{i \; \sqrt{2 \; k}}\left[
g_{\nu_\mathcal{R}}'(k;\eta_0) + i \; k \; g_{\nu_\mathcal{R}}(k;\eta_0) \right] \; .
\end{equation}
These coefficients satisfy eq.(\ref{abc}) and
$$
|A_\mathcal{R}(k)|^2 + |B_\mathcal{R}(k)|^2 = \frac1{k}\left[
|g_{\nu_\mathcal{R}}'(k;\eta_0)|^2 + k^2 \; |g_{\nu_\mathcal{R}}(k;\eta_0)|^2 \right]
$$
Notice that the function $ g_{\nu}(k;\eta) $ eq.(\ref{gnu}) and the $ k $ factors
in eq.(\ref{absr}) combine to produce functions $ A_\mathcal{R}(k) \equiv {
\tilde A}_\mathcal{R}(k \; \eta_0) $ and $ B_\mathcal{R}(k) \equiv
{\tilde B}_\mathcal{R}(k \; \eta_0) $ that only depend on the product $ k \; \eta_0 $.
We find from eqs.(\ref{DofkR}) and (\ref{absr}) the corresponding
transfer function which is a function of $ k \; \eta_0 $ too,
\begin{equation}\label{dsr}
1+{\tilde D}(k \; \eta_0) = \frac1{k} \left\{ |g_{\nu_\mathcal{R}}'(k;\eta_0)|^2 +
k^2 \; |g_{\nu_\mathcal{R}}(k;\eta_0)|^2- {\rm Re} \left[i^{3-2 \, \nu_\mathcal{R}}
\; \left( g'^2_{\nu_\mathcal{R}}(k;\eta_0) + k^2 \; g^2_{\nu_\mathcal{R}}(k;\eta_0)
\right)\right]\right\}
\end{equation}
The functional dependence on $ k \; \eta_0 $ confirms the assertion in sec. \ref{priflu}
that different initial times $ \tau_0 $ lead to a rescaling in $ k $.
\medskip
In the $ k \; \eta_0 \to \infty $ limit two types of vanishing terms show up in
$ {\tilde D}(k \; \eta_0) $: (a) terms that strongly oscillate as
$ e^{\pm 2 \, i \; k \; \eta_0} $ as they tend to zero and (b) non-oscillatory
decreasing terms. Under integrals on $ k $, the terms of type (a) yield convergent expressions.
We derive the non-oscillatory decreasing terms (b) by inserting the asymptotic behaviour of the
Hankel functions eq.(\ref{gnu}) \cite{abra} in eq.(\ref{dsr}) with the result
\begin{equation} \label{dkasi}
{\tilde D}(k \; \eta_0) \buildrel{k \to \infty}\over=
\frac{(\nu^2 - \frac14)^2}{8 \; (k \; \eta_0)^4 } +
{\rm terms ~ oscillating ~ as}~ e^{\pm 2 \, i \; k \; \eta_0} \; .
\end{equation}
However, this approximation will not be valid for large enough $ k $ since the
modes at small enough wavelength will exit the horizon after the end of slow--roll
where eq.(\ref{dkasi}) does not apply anymore. We recall that
the occupation number $ |B_\mathcal{R}(k)|^2 $ (and therefore $ D(k) $) must
decrease faster than $ 1/k^4 $ for
$ k \to \infty $ in order to ensure finite UV values for the expectation
value of the energy-momentum fluctuations \cite{biblia,motto2}.
\medskip
The case $ \nu_\mathcal{R} = 3/2 $ is a good approximation
which simplifies the expressions above. We obtain in this scale invariant case:
$$
A_\mathcal{R}(k) = 1 + \frac{i}{k \; \eta_0} - \frac1{2 \; k^2 \; \eta_0^2}
\quad , \quad B_\mathcal{R}(k) =- \frac{e^{-2 \, i \; k \; \eta_0}}{2 \; k^2 \; \eta_0^2}
\; .
$$
The transfer function is in this case,
\begin{equation}\label{dsr32}
{\tilde D}(x) = \frac{\cos 2x}{x^2} - \frac{\sin 2x}{x^3} + \frac{\sin^2 x}{x^4}
\quad , \quad \nu_\mathcal{R} = 3/2 \quad , \quad x \equiv k \; \eta_0 \; .
\end{equation}
Eq.(\ref{dkasi}) for $ \nu = 3/2 $ coincides with eq.(\ref{dsr32}) in the
$ x \to \infty $ limit, as it must be.
Notice that the simple formula eq.(\ref{dsr32}) obeys the general properties eq.(\ref{propg}).
In particular,
$$
{\tilde D}(x) \buildrel{x \to 0 }\over= -1 + \frac49 \; x^2 + {\cal O}(x^4) \; .
$$
\section{Fixing the Total Number of Inflation e-folds and the bound
from Entropy}
It is very useful to plot the comoving scales of the cosmological fluctuation wavenumbers
and the comoving Hubble radius together [see fig. \ref{hr}]. One sees in this way how and when
the cosmological fluctuations cross out and in the Hubble radius.
The comoving Hubble radius is defined by $ R_H \equiv 1/[a(\tau) \; H(\tau)] $. We
display in Table VI the dependence of $ R_H $ on the scale factor $ a $ for all
the relevant eras of the universe.
\begin{table}[h]
\begin{tabular}{|c|c|}
\hline
Expansion stage & Dependence of $ R_H $ on $ a $\\
\hline \hline
Extreme Fast-roll & $ a^2 $ \\ \hline
Fast-roll & $ a^2/\sqrt{a^6 + {\rm constant}} $ \\ \hline
Slow--Roll inflation & $ 1/a $ \\ \hline
Radiation Dominated & $ a $ \\ \hline
Matter Dominated & $ \sqrt{a} $ \\ \hline
\end{tabular}
\caption{Dependence of the comoving Hubble radius $ R_H = 1/[a \; H] $
on the scale factor $ a $ for the relevant eras of the universe.}
\end{table}
\begin{figure}[h]
\includegraphics[width=16.cm]{evol.eps}
\caption{The logarithm of the comoving scales and the logarithm
of the comoving Hubble radius $ R_H = 1/[a \; H] $ vs. $ \log a $.}
\label{hr}
\end{figure}
The observed CMB quadrupole suppression can be easily explained
if it exited the horizon by the end of fast-roll \cite{quadru,quamc}.
In that case, the modes which are horizon size today
had wavenumbers $ k_Q \simeq 11.5 \; m $ at horizon exit \cite{quamc}.
Combining this value of $ k_Q $ with the redshift since the pivot
wavenumber exited the horizon, eqs. (\ref{ka1}), (\ref{var}) and
(\ref{vk0}), determines the total redshift since the beginning of inflation to be
$$
z_{tot} = 0.9 \; 10^{56} \; \simeq e^{129} \; .
$$
Combining this value with the value of $ 1 + z_r \simeq 4 \; 10^{28} \simeq e^{66} $
by the end of inflation eq.(\ref{var}) yields a total number of $ N_{tot} = 63 $ inflation
efolds. This value is very close to the minimal number of inflation efolds required
to explain the entropy of the present universe due to photons and neutrinos
\cite{biblia}:
$$
N_{tot} \geq 62.4 \; .
$$
Namely, this is the minimum number of inflation efolds compatible
with the present entropy of the universe.
In summary, assuming that the CMB quadrupole is suppressed because
it exited the horizon by the end of fast-roll inflation {\bf fixes}
the total number of inflation efolds which turns to be
$$
N_{tot} \simeq 63 \; .
$$
\begin{acknowledgments}
We thank Anthony Lasenby for fruitful discussions and his interest in this work.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,712
|
\section{\label{sec:intro}Introduction}
Turbulence closure models are required in the dynamical cores of
global ocean-climate simulations. While grand challenge coupled
climate simulations can use an ocean resolution of 0.1\textdegree
($\sim10\,$km) to simulate timescales of decades, resolving the
turbulent cascade for submesocale, $O(1\,$km), eddies remains
computationally unachievable. For this reason, mesoscale ocean
large-eddy simulations (MOLES; \cite{FoKeMe2008}) are employed. {The
goal of a MOLES is to anticipate $1\,$km results at a much coarser
resolution. While such closures are sometimes compared subjectively
by visualizing the simulation results,} what is needed is a
{prescription} to objectively {and rigorously} compare {between} the
various proposed MOLES closures. Such a method is presented here:
\NEU{the computation of fluxes and comparison via the error-landscape measured against a high resolution benchmark.
Our application is to an idealized system, but the framework can be generalized}
for the
evaluation and development of closures applicable to World Ocean
simulations. \NEU{In \ref{sec:appendix}, we present the details for generalization to a 3D baroclinic zonally-reentrant channel.}
Often, the closure approach taken is {to set the
dissipation scale equal to the grid scale. This is equivalent to
setting the {appropriately-averaged} grid-scale Reynolds number to unity and is accomplished by simply
using a} constant viscosity, $\nu$, that is much larger than the
physical value ($\sim10^{-6}\,$m$^2$s$^{-1}$) so that a {numerically} resolved
simulation results. These large viscosities, however, also
result in unphysical damping of the large scales. To reduce this
effect while remaining in the paradigm of a linear dissipative model,
the order of the Laplacian, $\Delta=\nabla^2$, can be increased to
$\Delta^2=\nabla^4$ or higher. Such hyper-viscous models are more
scale-selective, applying dissipation concentrated near the grid scale
{(a new dissipation scale is derived from dimensional analysis of
the $\Delta^n$ dissipation and this scale is set equal to the grid scale).}
Turbulence is {far} more than a dissipative phenomenon, however, and purely
dissipative models cannot reproduce up-scale energy transfers due to
interactions between scales \NEU{(nor can they reproduce ``backscatter'' in the 3D case \cite{2000AnRFM..32....1M}).}
Another approach is to {use what is known about turbulent cascades and} apply dissipation only where it is required
with a spatio-temporally varying viscosity, e.g., the
{Smagorinsky \citep{Sm1963} and}
Leith
\citep{Le1996} models. In the {Smagorinsky} model, the global average {energy}
dissipation (due to a spatially uniform viscosity) is equated to the
local dissipation at the grid scale because the turbulence is assumed
homogeneous. The expression for $\nu_\ast(x,t)$ then follows from the
{3D turbulence} spectrum and dimensional analysis.
{For Leith, enstrophy dissipation and the 2D turbulence spectrum in
an enstrophy cascade are used {to derive the appropriate $\nu_\ast(x,t)$}. However,}
the assumption of
homogeneity is controversial \citep{LeMeCo2005} and there are also
issues with vorticity dissipation at the boundaries
\citep{FoKePe2004,FoKe2005}. Yet, the Leith model has been successful
in improving numerical stability in global eddy-permitting models
\citep{FoKeMe2008}.
{In 2D turbulent systems where enstrophy is clearly the quantity cascading to
unresolved scales, methods to}
{dissipate potential enstrophy while conserving energy} {have merit}.
{This {can be} accomplished by modifying the Coriolis force in the momentum equation such that the transport of potential vorticity is appropriately diffusive while still being energetically neutral \cite{SaBa1985}. The anticipated potential vorticity method (APVM)}
reproduces both the physical
{transfer of energy to larger scales} and the dissipation of small-scale
enstrophy \citep{VaHu1988}. APVM has also been extended to
variable-resolution grids \cite{ChGuRi2011}, {and it has been generalized to 3D rotating Boussinesq flows \cite{2012arXiv1206.2607G}.} However, it requires a
high-order Laplacian correction to concentrate the eddy viscosity to
the smallest scales \citep{VaHu1988}.
A {more} recent approach is to use a mathematical regularization {of the
underlying equations,} which ensures smooth {(hence, computable)}
solutions, as the closure model: e.g., the Lagrangian-averaged
$\alpha-$model
{\citep{HoMaRa1998,1998PhRvL..81.5338C,1999PhyD..133...49C,1999PhyD..133...66C,1999PhFl...11.2343C,2001PhyD..152..505F}.}
It is dispersive rather than dissipative: the transport is by a
spatially-smoothed velocity field (filter width $\sim\alpha$). For
three-dimensional (3D) incompressible, non-rotating, and
non-stratified flows the $\alpha-$model does not produce sizeable
computational gains because it {unphysically} develops rigid
bodies in the flow \citep{PiGrHoMi+2007}. This limitation {disappears when modelling systems that include} a body force. It has
been used successfully where there is a Lorentz force, in electrically
conducting fluids \citep{PiGrMiPo2009,PiGrMiPo2011}, and where there
is a Coriolis force, in rotating fluids, e.g., the two-dimensional
(2D) barotropic vorticity equation (BVE) on a $\beta-$plane
\citep{NaMa2001,HoNa2003}, the shallow water equations \citep{Wi2004},
a two-layer quasigeostrophic (QG) model \citep{HoWi2005}, {and the primitive equations \cite{2008JCoPh.227.5691H,2008JPhA...41H4009H}.}
For 2D
flows, relevant to this paper, the $\alpha-$model enhances the inverse
cascade of energy \citep{NaSh2001} and in the enstrophy cascade
regime, the rough kinetic energy and enstrophy spectra remain
unchanged ($k^{-3}$ and $k^{-1}$, respectively) in the limit
$\alpha\rightarrow\infty$ \citep{LuKuTa+2007}. With forcing {applied in the wavenumber shells $2<k<4$ with an amplitude proportional to $\alpha^2$,}
\cite{LuKuTa+2007} found that increasing $\alpha$ {led} to
increasing the amount of fine structure and, consequently, to the need
for increased resolution. They posited that with forcing unscaled,
computational gains (instead of losses) might be realized. We will
test whether or not this is so.
{The challenge in
evaluating the effectiveness of LES closures for MOLES should already be
clear. Not only do many possible closures exist, but these closures often
differ at the conceptual level of how unresolved turbulent motion should be
modeled. As such, we expect that the various possible closures will each
excel in some plausible evaluation metric. The challenge is then to determine
an approach, i.e {\it an evaluation framework}, that is both unbiased and fairly
measures the effectiveness of the various closures in mimicking the influence
of unresolved scales. The goal of this contribution is to do exactly that. }
Our approach here is to begin with the simplest system that we believe
might be applicable to MOLES, with the understanding that the results
obtained in such idealized systems will have to be reevaluated as the
system complexity and realism increases. With this caveat in mind, we
solve the 2D barotropic vorticity equation (2D BVE) in a
doubly-periodic domain. The motivation for using the 2D BVE is to
exploit the similarity of the the QG vorticity equation to the 2D BVE.
{(MOLES will be applied at grid resolutions near $5-10\,$km.)} The QG
vorticity equation has a potential enstrophy cascade of QG eddies
below the scale of the baroclinic instability. Similarly, the 2D BVE
has an enstrophy cascade below the forcing scale, which serves here as
an analog to the scale of the baroclinic instability. Furthermore, the
robust analysis of spectral fluxes of energy and enstrophy in 3D
systems, needed for more complex, realistic flows \NEU{(see
\ref{sec:appendix}),} is sufficiently complicated to warrant starting at a
lower spatial dimension. Since the 2D BVE system lacks the process of
baroclinic instability to initiate the turbulent mixing, we use
large-scale, slowly varying in time, wind stress to activate the
turbulence. As used in Ocean General Circulation Models (OGCMs),
quadratic bottom-drag is used to obtain realistic equilibrium
solutions.
Details of the enstrophy cascade process can be measured using {spectral}
enstrophy transfer analysis \citep{Kr1971,MaVa1993}. {The goal of
any LES is to anticipate higher resolution results. This is
accomplished by accurately modeling the interactions with the missing
scales. The statistics of these interactions, on a wavenumber
basis, are measured with spectral transfer analysis. If this
analysis shows an accurate reproduction, we can be sure we are
getting the right answer for the right reason.} The error-landscape
of enstrophy flux {is likely, then, the best measure of MOLES performance. We use it to} quantify the performance of the {six} popular
MOLES closures discussed above (the two linear dissipative models and
the {four} nonlinear models derived from hypotheses about turbulence)
employing a single, exponentially convergent, numerical model, the
Geophysical High Order Suite for Turbulence (GHOST;
\cite{MiRoRe+2011}).
To compare the models, we start by computing a fully-resolved
numerical solution of a flow with a fixed, physical viscosity as the
benchmark. This eliminates the possibility of any bias between the
parameterizations that could result from using any single MOLES at
higher resolution as the benchmark. {It also serves as our best
hope for the MOLES simulations: that they reproduce the benchmark.} In Section
\ref{sec:technique}, {spectral} enstrophy transfer analysis is reviewed: its
application to MOLES and how this will be combined with the
error-landscape is given. In Section \ref{sec:results}, the details
of the parameterizations are introduced and each parameterization is
optimized with the error-landscape technique in order to make a fair
{and objective}
comparison.
\section{\label{sec:technique}{Theory}}
\subsection{{2D turbulence}}
{For scales much smaller than the deformation radius, the quasi-geostrophic
potential vorticity equation reduces to the 2D-BVE (see, e.g., \cite{Va2006}).}
The 2D-BVE are
\begin{eqnarray}
\partial_t\zeta + \{\psi,\zeta\} = F+\nu\nabla^2\zeta-\frac{C_D}{h}\hat{\vec{z}}\cdot\nabla\times(|u|\vec{u})\nonumber\\
\zeta = \nabla^2 \psi\nonumber\\
\vec{u}=-\nabla\times(\psi\hat{\vec{z}})\,,
\label{eq:bve}
\end{eqnarray}
where $\zeta$ is the vorticity, $\psi$ the stream function, $\vec{u}$
the 2D velocity, $F$ an external time-varying forcing to mimic
wind stress, $\nu$ the viscosity,
$\hat{\vec{z}}$ the out-of-plane unit vector,
and $C_D/h$ the coefficient of
quadratic bottom drag.
As a constant Coriolis parameter has no effect
on 2D motion, Eqs. (\ref{eq:bve}) also describe the 2D-BVE on a
$f-$plane.
\begin{figure}[htbp]
\includegraphics[width=10cm]{2d.eps}\\
\includegraphics[width=10cm]{2dflux.eps}
\caption{{Cartoon depicting 2D turbulence theory: kinetic energy
spectrum ($E(k)$, top panel) and fluxes (bottom panel) of enstrophy ($\Pi_S(k)$, blue
solid line) and energy ($\Pi(k)$, red dashed line). Kinetic energy
undergoes an inverse cascade to large scales (negative flux) at
the kinetic energy injection rate, $\varepsilon$. Enstrophy undergoes a
direct cascade to small scales at the enstrophy injection rate,
$\eta$.}}
\label{fig:2dtheory}
\end{figure}
{A general overview of 2D turbulence theory (see, e.g., \cite{Va2006})
is presented {in} Fig. \ref{fig:2dtheory}.} Kinetic energy,
$|\vec{u}|^2/2$, and hence enstrophy, $|\zeta|^2/2$,
are injected into the fluid. Because both are quadratic ideal
invariants (conserved in the absence of forcing and viscosity) and
$\zeta=\hat{\vec{z}}\cdot\nabla\times\vec{u}$, enstrophy cascades to
smaller scales and energy undergoes an inverse cascade to larger
scales (Fjortoft's theorem). {(The central point \NEU{in deriving}
Fjortoft's theorem is \NEU{to realize that} energy, $E(k)$, and
enstrophy, $Z(k)$, spectra are related by $k^2E(k)=Z(k)$.)} Under
the assumption of spectral locality, forcing and dissipation
\NEU{cannot} affect the flow \NEU{except} over a finite range of
scales \NEU{near where they are prescribed: far from these ranges,} both cascades must \NEU{therefore} have a
constant flux (Fig. \ref{fig:2dtheory}, lower panel). The constant
flux cascade regimes are called inertial ranges because only the
inertial terms, $\vec{u}\cdot\nabla\vec{u}$ for energy and
$\vec{u}\cdot\nabla\zeta=\{\psi,\zeta\}$ for enstrophy, are
non-negligible. Dimensional analysis after equating constant fluxes
to the inertial terms for energy and enstrophy yields a $k^{-5/3}$
energy spectrum in the inverse energy cascade and a $k^{-3}$ energy
spectrum in the enstrophy cascade (Fig. \ref{fig:2dtheory}, upper
panel). {Fine theoretical details such as the logarithmic correction
to the $k^{-3}$ spectrum \citep{Kr1971} and arguments about locality
\citep{XiWaCh+2008} have here been omitted. }
\subsection{Transfer analysis}
{The $\{\psi,\zeta\}$ term is the only non-negligible term
in the enstrophy cascade regime. It will also be shown in Section
\ref{sec:LES} to be the term whose small-scale interactions we need to
parameterize. It is thus the focus of our comparison methodology.
Other terms in the analysis will heretofore be abbreviated as $\mathcal{F}$
for forcing, $\mathcal{D}$ for dissipation, and $\mathcal{Q}$ for large-scale
drag (where, e.g., $\mathcal{F}\equiv\zeta F$).}
The time evolution of enstrophy at any physical position is given by the
enstrophy-balance equation,
\begin{eqnarray}
\partial_t\frac{1}{2}\zeta^2=\zeta\partial_t\zeta=
-\zeta\{\psi,\zeta\}+\mathcal{F}+\mathcal{D}+\mathcal{Q}\,.
\end{eqnarray} The time evolution of the
enstrophy spectrum at wavenumber $k$, $Z(k)$, is similarly,
\begin{equation}
\partial_tZ(k) = \hat{\zeta}^\ast\partial_t\hat{\zeta}= S(k)
+\mathcal{F}(k) +\mathcal{D}(k) +\mathcal{Q}(k) \,,
\end{equation}
where $S(k)$ is the enstrophy transfer function (i.e., net enstrophy
received by wavenumber $k$ from all other wavenumbers),
\begin{equation}
S(k)=-\hat{\zeta}^\ast(k)\widehat{\{\psi,\zeta\}}(k)\,,
\end{equation}
and where the Fourier transform is represented by
$\hat{\cdot}$ and complex-conjugation by
$\cdot^\ast$.
The flux of enstrophy through wavenumber $k$, i.e., the sum
of {the rate of change of} enstrophy {leaving} all wavenumbers $\leq k$ {and going} to wavenumbers $>k$ {(i.e., moving to smaller scales)}, is
given by
\begin{equation}
\Pi_S(k) = -\int_\NEU{0}^k S(k') dk'\,,
\end{equation}
that is, the total {rate} of enstrophy flowing past wavenumber $k$ {to larger wavenumbers.}
The divergence of the flux is the transfer, {$S(k)$.}
{Because of the relation between energy and enstrophy spectra, the
transfer of energy is $T(k)=S(k)/k^2$.}
\subsection{\label{sec:LES}MOLES}
To reduce computational cost, MOLES solve only the largest scales of a
flow. The remaining {\sl unresolved} scales {from the anticipated higher-resolution simulation} are filtered out. The
filtering operation is indicated by $\bar{\cdot}$ and the resulting
equations are
\begin{equation}
\partial_t\bar{\zeta} + \{\bar{\psi},\bar{\zeta}\} =
\sigma + \bar{F}+\nu\nabla^2\bar{\zeta}
-\overline{\frac{C_D}{h}{\hat{\vec{z}}\cdot}\nabla\times(|u|\vec{u})}\,,
\label{eq:LES}
\end{equation}
where we have defined the
subgrid term $\sigma\equiv
-\overline{\{\psi,\zeta\}}+\{\bar{\psi},\bar{\zeta}\}$. The subgrid
term is the effects on the resolved scales by unresolved fluid
motions. {How well it is modeled is the measure of the success of the MOLES.} (Note that $\sigma = {\hat{\vec{z}}\cdot}\nabla\times\nabla\cdot\tau$ where
$\tau = -\overline{\vec{u}\vec{u}}+\bar{\vec{u}}\bar{\vec{u}}$, the
\NEU{momentum-equation} LES subgrid stress tensor.) The time evolution of the
enstrophy spectrum is now given by
\begin{equation}
\partial_tZ(k) = \bar{S}(k)
+ L(k)+{\mathcal{F}}(k) +\bar{\mathcal{D}}(k) +\mathcal{Q}(k)\,,
\end{equation}
where $\bar{S}(k)$ is {the rate of} enstrophy received by wavenumber $k$ from all other {\sl resolved} wavenumbers,
\begin{equation}
\bar{S}(k)=-\hat{\bar{\zeta}}^\ast\widehat{\{\bar{\psi},\bar{\zeta}\}}\,,
\end{equation}
and $L(k)$ is the {rate of} enstrophy received from all {\sl unresolved} wavenumbers,
\begin{equation}
L(k) = \hat{\bar{\zeta}}^\ast\hat{\sigma}\,.
\end{equation}
{Note that the rate of energy received from unresolved wavenumbers is $L(k)/k^2$.}
{How closely the sum of enstrophy transfer functions, from resolved and unresolved wavenumbers, approximate} the enstrophy transfer function from a
fully resolved system,
\begin{equation}
\bar{S}(k)+L(k) \approx S(k)\,,
\end{equation}
(for all wavenumbers smaller than the filter wavenumber) {is the spectral measure of the success of the model.}
\NEU{For $k$ in the inertial range, $S(k)=0$ and a successful model will produce $L(k)\approx-\bar{S}(k)$.}
The flux
of enstrophy through wavenumber $k$ due to resolved and modeled
interactions is given by
\begin{equation}
\Pi_T(k) = -\int_\NEU{0}^k {\big{[}}\bar{S}(k') + L(k'){\big{]}} dk'\,.
\end{equation}
\subsection{Objective method: error-landscape of enstrophy flux}
To objectively compare parameterizations, we make use of the
error-landscape assessment \citep{MeGeBa2003,MeSaGe2006,MeGeSa2007,Me2011}
{on the enstrophy flux.}
{We modify the method of \cite{Me2011} and employ $L^1$ instead of $L^2$
error norms,
\begin{equation}
D_p = \frac{\int_{1}^{k_{max}}\big{|}\Pi_S(k)-\Pi_T(k)\big{|}k^pdk}{\int_{1}^{k_{max}}\big{|}\Pi_S(k)\big{|}k^pdk}\,,
\end{equation}
where $k_{max}$ is determined from the MOLES resolution (see below).
We chose $p=0$ to obtain a good balance between the smaller resolved
scales and the largest, less model-sensitive, scales. The optimal
parameter value for each method is the point where this error norm
is minimized. (The term landscape is intuitive for two-parameter
models.) Inter-model comparisons are also made using the $D_0$ norm.}
\subsection{{Design of numerical experiments}}
We employ a well-tested parallelized pseudo-spectral code
\citep{MiRoRe+2011}. The computational box has size $[2\pi]^2$, and
wave numbers vary from $k_{min}=1$ to $k_{max}=N/3$ using a standard
2/3 de-aliasing rule, where $N$ is the number of grid points per
direction. To cast our results in meaningful units, the results are
dimensionalized by $l=l_0l'$, $t=t_0t'$ where $\cdot'$ indicates
non-dimensionalized pseudo-spectral result and $l_0 = 504 \times 10^4/
\pi\,$m and $t_0=1.2\times10^6$s. {To spin up our runs we begin with
a $1008^2$ simulation (dimensionalized grid spacing $\NEU{\Delta x=}10\,$km) initialized
with a few large-scale Fourier modes. The forcing is designed to mimic
wind-stress at $k=4$:
\begin{equation}
F = A(t)\bigg{[}\cos{(4y+\phi_y)} - \cos{(4x+\phi_x)}\bigg{]}\,,
\end{equation}
where $\phi_x=\pi\sin(1.2\times10^{-6}\,$s$^{-1} t)$
and $\phi_y=\pi\sin(1.2\times10^{-6} \pi\,$s$^{-1} t/3)$
so that the wind varies with a period of about 60 days.
The coefficient $A$ is dynamically controlled to hold a stead{y} enstrophy
injection rate of $1.75\times10^{-18}\,$s$^{-3}$ to reduce the amount of required statistics to measure a constant flux cascade, i.e.,
\begin{equation}
\frac{\int \zeta F dA}{ \int dA} = 1.75\times10^{-18}\,s^{-3}\,.
\end{equation}
Time step is $600\,$s, $\nu=88\,$m$^2$s$^{-1}$, and $C_D/h=1.25\times10^{-8}\,$m$^{-1}$.
The resulting root-mean-squared velocity is $v_{rms}=2.6\,$ms$^{-1}$
and the forcing scale ($k=4$) is $L_F=2520\,$km.
The corresponding forcing-scale turnover time is $11\,$days and the
Reynolds number is $Re\equiv v_{rms}L_F/\nu\approx75,000$.
{Simulations are integrated} for over $1300\,$days. The
final turbulent state of this run is used as initial conditions for
the benchmark and MOLES runs at $\nu=1.375\,$m$^2$s$^{-1}$. }
\section{\label{sec:results}Analysis of parameterizations}
The goal of MOLES is to anticipate higher resolution results at an
affordable resolution by representing the effects of the unresolved
eddies. To avoid any bias between the parameterizations, we use as
the benchmark a fully resolved direct numerical solution (DNS) at a
resolution of $8192^2$ of a flow with $\nu=1.375\,$m$^2$s$^{-1}$.
Each MOLES is then run at a resolution of $1008^2$ and tested for its
ability to reproduce the benchmark. This allows us to test the models'
representations against a known solution: a DNS flow. Accordingly,
the MOLES simulations also must use $\nu=1.375\,$m$^2$s$^{-1}$ in
addition to the subgrid term or they should be compared, instead, to a
$\nu=0$ benchmark which cannot be produced.
{The benchmark {is} run for 390 days, $v_{rms}=2.6\,$ms$^{-1}$ and
the corresponding forcing-scale turnover time is $11\,$days. The
Reynolds number is $\approx4.8\times10^6$.
A snapshot of the vorticity of the benchmark run is shown in the
Upper Left panel of Fig. \ref{fig:benchreal}. There are several
large vortices of both signs. Over time, vortices stretch and fold
vortex filaments into the fine-scale features as seen. This is the
enstrophy cascade process. This simulation is completely resolved
and this cascade is arrested at the smallest scales by dissipation
(Upper Right panel in Fig. \ref{fig:benchreal}). Energy is injected
by the forcing term (Lower Right panel in Fig. \ref{fig:benchreal})
at a constant injection rate: an inverse cascade of energy and
direct cascade of enstrophy result. The quadratic drag term serves
to arrest the inverse cascade of kinetic energy and primarily
removes energy (and enstrophy) at the largest scales. Though, it
does remove both from a wide range of scales (Lower Left panel in
Fig. \ref{fig:benchreal}). }
\begin{figure}[htbp]
\includegraphics[width=8cm]{fig_toddbench_w57}
\includegraphics[width=8cm]{fig_toddbench_diss57}\\
\hspace{0.01cm}\\
\includegraphics[width=8cm]{fig_toddbench_drag57}
\includegraphics[width=8cm]{fig_toddbench_force57}\\
\caption{{$8192^2$
benchmark; snapshot at 390 days for (Upper Left) vorticity,
$\zeta$, with thresholds $\pm1.5\times10^{-5}\,$s$^{-1}$
(counter-clockwise vorticity is shown in yellow; clockwise in
red); (Upper Right) absolute value of vorticity tendency due to dissipation,
$\nu\nabla^2\zeta$, black pixels are
$2.25\times10^{-7}\,$s$^{-2}$; (Lower Left) vorticity tendency
due to quadratic drag, $-\frac{C_D}{h}\nabla\times(|u|\vec{u})$,
with thresholds $\pm1.38\times10^{-6}\,$s$^{-2}$; (Lower Right)
vorticity tendency due to forcing, $F$, with thresholds
$\pm1\times10^{-4}\,$s$^{-2}$. }}
\label{fig:benchreal}
\end{figure}
The flux and resulting enstrophy spectrum for the benchmark are shown
in Fig. \ref{fig:bench}. A power-law spectrum, $Z(k)\sim k^{-1.2}$,
is observed in the enstrophy cascade inertial range. It is steeper
than the predicted $k^{-1}$ spectrum due to the quadratic drag which
acts at all scales of the flow: the difference between the enstrophy
flux (solid line) and a constant flux is exactly the cumulative drag
(dotted line). This steeper spectrum is similar to the result for
linear drag \citep{DaGu2001}. {Note that dissipation is not
significant for wavenumbers, $k<300$. Reproducing this flow at a
resolution of $1008^2$ ($k_{max}=336$) will thus be a {onerous test
for} the parameterizations.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{bench_fig_flux_.eps}\\
\includegraphics[width=10cm]{bench_fig_spec_.eps}
\caption{Benchmark run: (Top) Enstrophy flux
($\Pi_S(k)$, solid) and cumulative enstrophy injection
dash-triple-dotted), dissipation
dashed), and quadratic drag
dotted). As quadratic drag operates at all but the dissipative
scales, a constant enstrophy flux range is not seen. (Bottom) Compensated enstrophy spectrum,
$kZ(k)$, versus wavenumber, $k$, for $8192^2$ BVE benchmark. Quadratic drag acts at all scales
and precludes a pure $Z(k)\sim k^{-1}$ spectrum. }
\label{fig:bench}
\end{figure}
\NEU{The benchmark run contains all scales of motion at this $Re$. It
can be used to calculate the true transfers with scales that will be
unresolved at MOLES resolution by {spectral cut-off filtering}
the benchmark run down to a resolution of $1008^2$. These subgrid
transfers for energy and enstrophy are plotted in
Fig. \ref{fig:bench2}. The effects of the subgrid scales are to
remove enstrophy from a narrow band of wavenumbers near the
resolution limit and to generate a small amount of energy at the
very largest scales. These transfers can also been seen in Fig. 7
of \cite{VaHu1988}. The upscale energy transfer is a strong function
of the resolution, $\Delta x$: as can be seen by comparison with
\cite{VaHu1988}, the smaller $\Delta x$ is, the smaller in magnitude
is the upscale energy transfer. In fact, in the limit as $\Delta x$
approaches $\nu^{1/2}$ times some constant, both subgrid transfers
will tend to zero \cite{LuKuTa+2007}. However, at fixed $\Delta x$ both transfers will
tend to a non-zero function of $k$ that remains the same in the
limit of zero viscosity. This is due to spectral locality: only
those scales nearest $\Delta x$ will contribute to the transfers.
As $\nu$ decreases, and more and more scales are added, they will
contribute less and less to the transfers for $k<1/\Delta x$.}
\NEU{Given that an ideal MOLES will have $L(k)$ that exactly reproduces
Fig. \ref{fig:bench2}, we can anticipate the performance of the
proposed closures. None of the purely dissipative models (viscous,
hyper-viscous, Leith, or Smagorinsky) will be able to reproduce the
upscale transfer of energy. The hyper-viscous model should better
confine enstrophy dissipation to large wave numbers as its subgrid
term contains fourth-order derivatives compared to second-order for
the viscous model and first-order derivatives of the product of
first-order derivatives for Leith. Smagorinsky is derived for 3D
flow and is not expected to perform well in 2D. It has been
previously shown that AVM can produce the correct forms of the
transfers if high enough order viscosities and small enough
anticipation times are employed \cite{VaHu1988}. The $\alpha-$model
is non-dissipative, but could {\sl potentially} transport energy in
the correct direction \cite{NaSh2001}. }
\begin{figure}[htbp]
\includegraphics[width=10cm]{landscape_bench.eps}
\caption{\NEU{Benchmark run:
Transfers with what will be unresolved scales for MOLES simulations
for enstrophy, $S(k)-\bar{S}(k)$ (solid line),
and energy, $[S(k)-\bar{S}(k)]/k^2$ (dashed line). An ideal MOLES would
exactly reproduce these transfers with $L(k)=S(k)-\bar{S}(k)$.}}
\label{fig:bench2}
\end{figure}
\subsection{\label{sec:viscous}Linear viscous parameterizations and their performance}
The simplest parameterization is to assume the main
effect of subgrid turbulence is dissipative. Accordingly, the
viscosity is often increased until a numerically resolved solution is
possible. The subgrid term, $\sigma$, in the MOLES equation,
Eq. (\ref{eq:LES}) is then
\begin{equation}
\sigma = \big{(}\nu'-\nu\big{)}\nabla^2\zeta\,,
\end{equation}
with $\nu' \gg \nu$. A slightly more sophisticated
approach is to add higher-order dissipation, hyper-viscosity, e.g.
\begin{equation}
\sigma = \nu_4\nabla^4\zeta\,,
\end{equation}
or even higher order. We focus on $\nabla^2$ and $\nabla^4$
parameterizations here.
We apply the error-landscape of enstrophy flux technique to
optimize the viscous model. The modeled flux, $\Pi_T(k)$, for the
viscous model is shown in Fig. \ref{fig:viscoustrue}. Note that as
the viscosity is varied, the modeled flux brackets both sides of the
benchmark flux. This suggests an optimal $\nu'$ for the model should
be indicated by the enstrophy flux error-landscape. Indeed, {$D_0$} has its minimum for
${\nu'}=11\,$m$^2$s$^{-1}$. This is the optimal viscous model which we will
compare to the other parameterizations.
\begin{figure}[htbp]
\includegraphics[width=10cm]{viscous_ModelFlux.eps}\\
\includegraphics[width=10cm]{viscous_landscape_ModelFluxp.eps}
\caption{Viscous model: (Top) Modeled flux, $\Pi_T(k)$, for
${\nu'}=5.5\,$m$^2$s$^{-1}$ (red dotted), $11\,$m$^2$s$^{-1}$
(green dashed), $16.5\,$m$^2$s$^{-1}$ (blue dash-dotted),
$22\,$m$^2$s$^{-1}$ (pink dash-triple-dotted), and
$24.75\,$m$^2$s$^{-1}$ (cyan long-dashed) and $\Pi_S(k)$ for
$8192^2$ BVE benchmark (solid black). (Bottom) Flux error-landscape
{norm $D_0$.} The optimal value is
${\nu'}=11\,$m$^2$s$^{-1}$.}
\label{fig:viscoustrue}
\end{figure}
The approximate reproduction of the benchmark flux is accomplished by
the action of the subgrid enstrophy transfer $L(k)$
(Fig. \ref{fig:viscexp}). As expected, the action of the viscous
model is solely dissipative. The solid black line indicates what the
true transfer with the unresolved scales should be, $S(k)-\bar{S}(k)$ \NEU{(see Fig. \ref{fig:bench2}).} The viscous model dissipates enstrophy over a much larger
range of scales. Moreover, since energy is dissipated as
$\sim\nu'Z(k)\sim k^{-1.2}$, \NEU{eddy viscosity} is unphysically \NEU{positive} at
large scales. What the unresolved scales should be doing is
contributing to the {upscale transfer} of energy as shown by the solid,
black benchmark line. The enstrophy spectra are shown in
Fig. \ref{fig:viscous}. The result of too little dissipation is the
piling of small-scale thermal noise in the spectrum
\citep{CiBoDe2005}.
\begin{figure}[htbp]
\includegraphics[width=10cm]{viscous_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{viscous_landscape_ekSGS_.eps}
\caption{Viscous model: subgrid transfers for enstrophy ($L(k)$, Top)
and energy ($L(k)/k^2$, Bottom) {and
$S(k)-\bar{S}(k)$ for benchmark (solid black).} The model is solely dissipative of
enstrophy and energy. {Exact viscosities are denoted} in
Fig. \ref{fig:viscoustrue}.}
\label{fig:viscexp}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=10cm]{viscous_landscape_EnstrophySpectrum_.eps}
\caption{Viscous model: Compensated enstrophy spectrum;
{exact viscosities are denoted} in
Fig. \ref{fig:viscoustrue}.}
\label{fig:viscous}
\end{figure}
By looking at the hyper-viscous model's flux error-landscape norms
(Fig. \ref{fig:hypertrue}), we identify
$\nu_4=1.1\times10^9\,$m$^4$s$^{-1}$ as the optimal hyper-viscous
model. The hyper-viscous model much more closely models the
dissipation of enstrophy due to the unresolved scales than the viscous
model, see Fig. \ref{fig:hyperexp}. Additionally, as the energy
dissipation is $\sim k^2Z\sim k^{0.8}$, \NEU{the rate of energy dissipated at large
scales is insignificant (note} the difference in vertical scales
for energy transfer in Figs. \NEU{\ref{fig:bench2},} \ref{fig:viscexp} and \ref{fig:hyperexp}). This is a marked improvement, but no
solely-dissipative parameterization will model the {mechanism of upscale energy transfer.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{hyper_landscape_ModelFluxp.eps}
\caption{Hyper-viscous model: Flux error-landscape {norm $D_0$.} The optimal value is
$\nu_4=1.1\times10^9\,$m$^4$s$^{-1}$.}
\label{fig:hypertrue}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=10cm]{hyper_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{hyper_landscape_ekSGS_.eps}
\caption{Hyper-viscous model: Subgrid transfers for enstrophy ($L(k)$,
Top) and energy ($L(k)/k^2$, Bottom) for
$\nu_4=2.2\times10^8\,$m$^4$s$^{-1}$ (red dotted),
$3.3\times10^8\,$m$^4$s$^{-1}$ (green dashed),
$4.4\times10^8\,$m$^4$s$^{-1}$ (blue dash-dotted),
$5.5\times10^8\,$m$^4$s$^{-1}$ (pink dash-triple-dotted),
$1.1\times10^9\,$m$^4$s$^{-1}$ (cyan long-dashed), and
$S(k)-\bar{S}(k)$ for benchmark (solid black).}
\label{fig:hyperexp}
\end{figure}
\subsection{\label{sec:leith}Leith model}
The Leith model is derived by dimensional analysis \citep{Le1996}.
The local enstrophy dissipation rate is estimated as
\begin{equation}
\eta_\ast = \nu_\ast\big{|}\nabla_\ast\bar{\zeta}\big{|}^2\,,
\label{eq:localeta}
\end{equation}
and an enstrophy cascade spectrum is assumed,
\begin{equation}
Z(k) \propto \eta^{2/3}k^{-1}\,.
\end{equation}
The viscous
range, $k$, is when the viscous enstrophy losses in a given wavenumber
band, $\int\nu k^2 Z(k) dk$, are comparable to the enstrophy
injection, $\eta$, or
\begin{equation}
\eta\sim\nu^3k^6\,.
\label{eq:globaleta}
\end{equation}
Setting the global average dissipation, $\nu$, to the local, grid-scale
dissipation rate, $\nu_\ast$, and equating Eqs. (\ref{eq:localeta}) and
(\ref{eq:globaleta}), we find
\begin{equation}
\nu_\ast \propto \big{|}\nabla\bar{\zeta}\big{|} (\Delta x)^3\,.
\label{eq:leithnu}
\end{equation}
The BVE with Leith model, is \citep{Le1996,FoKeMe2008}
\begin{equation}
\partial_t\bar{\zeta} + \{\bar{\psi},\bar{\zeta}\} = \nabla\cdot\nu\nabla\bar{\zeta}+ \nabla\cdot\nu_\ast\nabla\bar{\zeta}{+\bar{F}+\bar{Q}}\,,
\label{eq:leith}
\end{equation}
where $\nu=0$ for an infinite Reynolds number flow.
The Leith subgrid term is then
\begin{equation}
\sigma = \nabla\cdot\bigg{[}\bigg{(}\frac{\Lambda\Delta x}{\pi}\bigg{)}^3\big{|}\nabla\bar{\zeta}\big{|}\nabla\bar{\zeta}\bigg{]}\,,
\end{equation}
{where $\Lambda$ is a free parameter.}
The subgrid transfers for the Leith model are very similar to the
viscous model results (see Fig. \ref{fig:leithexp}). This is to be
expected as the Leith model is also solely-dissipative. \NEU{Note} that there is strong enstrophy dissipation at
the forcing scale. This can be understood by looking at
Fig. \ref{fig:leitheye}. The Leith viscosity {$\nu_\ast$ is proportional to $|\nabla\bar{\zeta}|$} and, therefore, is concentrated along the
borders between oppositely-signed vortices. These large-scale
coherent structures of enhanced dissipation then project on the small
wavenumber Fourier-modes {(bottom left panel of
Fig. \ref{fig:leitheye}).} {Note that the optimal parameter value is found to be $\Lambda=1$.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{leith_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{leith_landscape_ekSGS_.eps}
\caption{Leith model: Subgrid transfers for enstrophy ($L(k)$, Top)
and energy ($L(k)/k^2$, Bottom) for {$\Lambda=0.5$ (red dotted),
$\Lambda=0.75$ (green dashed), $\Lambda=1$ (blue dash-dotted),
$\Lambda=1.25$ (pink dash-triple-dotted), $\Lambda=1.5$ (cyan
long-dashed), and benchmark (black solid). The optimal model is $\Lambda=1$.}}
\label{fig:leithexp}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=6cm]{fig_movieleith100_nustar9}
\includegraphics[width=6cm]{fig_smag1_nustar9}\\
\end{center}
\begin{center}
\includegraphics[width=7cm]{fig_movieleith100_fig_leith}
\includegraphics[width=6cm]{fig_movieleith100_w9}\\
\end{center}
\caption{Snapshots
of $\nu_\ast$ (Top Left) and Fourier power spectrum of
$\nu_\ast$ (Bottom Left) for Leith model,
{$\Lambda=1$,} of $\nu_\ast$ (Top Right) for {Smagorinsky, $\Lambda_S=1$,
} and of vorticity field (Bottom Right, shown for reference).
All snapshots are at $4\times10^4\,$min.}
\label{fig:leitheye}
\end{figure}
\subsection{\label{sec:smag}Smagorinsky model}
The Smagorinsky model \citep{Sm1963,Li67} is the 3D precursor of the
Leith model. It is derived with a similar dimensional analysis as in
Sec. \ref{sec:leith}, but assuming a 3D direct cascade of energy.
Consequently, the model for eddy-viscosity is
\begin{equation}
\nu_\ast=\bigg{(}\frac{{\Lambda_S}\Delta x}{\pi}\bigg{)}^2|S_{ij}|\,,
\end{equation}
where $S_{ij}=(\partial_jv_i+\partial_iv_j)/2$. For isotropic,
homogeneous 3D turbulence the Smagorinsky Constant,
$C_S\equiv{\Lambda_S}/\pi\approx0.2$ \citep{MeKa2000}. {It
should be noted that Smagorinsky was devised for 3D isotropic flow
and was not intended for 2D nor geostrophic flows, but has
been employed in global climate models \cite{2000MWRv..128.2935G,2012JCli...25.2755D}.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{smag_ModelFlux.eps}\\
\includegraphics[width=10cm]{smag_landscape_EnstrophySpectrum_.eps}
\caption{{Smagorinsky model: (Top) Modeled flux, $\Pi_T(k)$, for
$\Lambda_S=0.1$ (red dotted), $\Lambda_S=0.3$
(green dashed), $\Lambda_S=0.5$ (blue dash-dotted),
$\Lambda_S=1$ (pink dash-triple-dotted), and
$\Lambda_S=2$ (cyan long-dashed) and $\Pi_S(k)$ for
$8192^2$ BVE benchmark (solid black). (Bottom)
Compensated enstrophy spectrum.}}
\label{fig:smagtrue}
\end{figure}
{The enstrophy flux and enstrophy spectrum for Smagorinsky
(Fig. \ref{fig:smagtrue}), highlight the fact that good spectra can
be produced without necessarily reproducing the correct dynamics.
The best spectra are produced for $\Lambda_S=0.5$ (blue dash-dotted)
while the best flux is produced by $\Lambda_S=0.1$ (red dotted).
This is opposed to the case for the viscous model where the best
flux {\sl and} spectrum occur for {the same value of the model's
one free parameter, $\nu'$.} The reason for the disparity is that
the viscous parameterization captures the most important physical
process, small-scale enstrophy dissipation, while the Smagorinsky
model unphysically removes enstrophy and energy from the largest
scales (see Fig. \ref{fig:smagexp} and the real-space visualization
of $\nu_\ast$ in Fig. \ref{fig:leitheye}). Therefore, even when the
combination of modeling and numerical error produces a good
spectrum, the model is not capturing the correct physical dynamics.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{smag_landscape_SGS_.eps}
\includegraphics[width=10cm]{smag_landscape_ekSGS_.eps}
\caption{{Smagorinsky model: subgrid transfers {of enstrophy ($L(k)$, Top) and energy ($L(k)/k^2$,
Bottom).} The model dissipates
enstrophy and energy unphysically from the large scales. {Exact viscosities are denoted} in
Fig. \ref{fig:smagtrue}.}}
\label{fig:smagexp}
\end{figure}
\subsection{\label{sec:avm}Anticipated vorticity method (AVM)}
AVM (APVM when applied to potential vorticity, \cite{SaBa1985}) is
so-called because it can be seen as substituting the forward-in-time
vorticity in the BVE,
\begin{eqnarray}
\frac{\bar{\zeta}_{n+1}-\bar{\zeta}_n}{\theta} = - \{\bar{\psi},\bar{\zeta}_n\}\,,
\end{eqnarray}
where $\theta$ is the time step for the anticipation. Substituting
this anticipated value, $\bar{\zeta}_{n+1}$ {in Eq. (\ref{eq:bve})} results in the
lowest-order AVM,
\begin{eqnarray}
\partial_t\bar{\zeta} = -\{\bar{\psi},\bar{\zeta}_n\} +\theta\{\bar{\psi},\{\bar{\psi},\bar{\zeta}_n\}\}{+\bar{F}+\bar{D}+\bar{Q}}\,.
\end{eqnarray}
In practice, to weight the subgrid model to smaller scales,
\begin{eqnarray}
\sigma = -\frac{\theta}{k_{max}^{2m}}\{\bar{\psi},\nabla^{2m}\{\bar{\psi},\bar{\zeta}\}\}\,,
\end{eqnarray}
In this study we have used $m=1$ as even this order of diffusive
operator is not practical in finite-volume {and finite-difference
schemes typically used in global ocean modeling} {because of the
relationship between high-order derivative accuracy and stencil
size.} AVM is not Galilean invariant, i.e., it does not conserve
momentum, but it exactly conserves energy while dissipating enstrophy.
{Note that the subgrid term for the momentum equation is
$\nabla\cdot\tau=\big{[}(-1)^m\frac{\theta}{k_{max}^{2m}}\nabla^{2m}\big{(}\vec{u}\cdot\nabla(\zeta\hat{\vec{z}}\times\vec{u})\big{)}\big{]}\hat{\vec{z}}\times\vec{u}$
which is perpendicular to the velocity at every point in space.
AVM then exactly conserves energy even if $\theta$ varies
spatially and temporally.}
As AVM dissipates enstrophy at small scales, $L(k)<0$ for large $k$
(see Fig. \ref{fig:avmexp}), it must also remove some small-scale
energy, $k^{-2}L(k)<0$. Since AVM exactly conserves energy, this
energy shows up at large scales. AVM is the only parameterization
studied here that reproduces this signature of the correct transfer.
The physical effect, however, is over estimated by at least an order
of magnitude. This can be mitigated by reducing $\theta$. However,
too small $\theta$ (0.125$dt$ for our flow) results in an excess of
energy at all scales \citep{VaHu1988}. For $m=1$, as used here, AVM
is unable to mimic that eddy viscosity should only act in a small
range of wavenumbers near $k_{max}$ \citep{VaHu1988}. {Note that
setting the anticipation time equal to the time step, $\theta=1$,
very closely reproduces the low-wavenumber flux
(Fig. \ref{fig:avmtrue}). This large value for $\theta$, however,
makes the eddy viscosity act at even larger scales
(Fig. \ref{fig:avmexp}). If larger values of $m$ were practical in
actual ocean applications, a two parameter optimization might yield
a very robust model. Holding constant $m=1$,} the optimal value of
$\theta$ is {$0.16$.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{avm_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{avm_landscape_ekSGS_.eps}
\caption{AVM: Subgrid transfers of enstrophy ($L(k)$, Top) and energy
($L(k)/k^2$, Bottom) {for $8192^2$ BVE benchmark (solid black),}
for $\theta=0$ (NO MODEL, red dotted), $\theta=0.16$ (green
dashed), $0.25$ (blue dash-dotted), $0.5$ (pink dash-triple-dotted),
and $1$ (cyan long-dashed). The subgrid model transfer in AVM
changes sign so that the model dissipates no energy, sum of
$L(k)/k^2$ over all wavenumbers is $o(10^{-12})$, while enstrophy
dissipation (sum of $L(k)$) is $o(1)$. The negative energy
dissipation at large scales mimics the {upscale transfer} from
unresolved scales, though too strongly.}
\label{fig:avmexp}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=10cm]{avm_ModelFlux.eps}\\
\includegraphics[width=10cm]{avm_landscape_ModelFluxp.eps}
\caption{AVM: (Top) Modeled flux, $\Pi_T(k)$. {Exact values
of $\theta$ are given} in Fig. \ref{fig:avmexp}. (Bottom) Flux
error-landscape {norm. Optimal value is
$\theta=0.16$.} }
\label{fig:avmtrue}
\end{figure}
\subsection{\label{sec:alpha}$\alpha-$model}
The $\alpha-$model takes a different approach than the other
parameterizations. It is a non-dissipative, solely dispersive model
-- a mathematical regularization (smooth, and hence computable
solutions are ensured even in the limit $\nu\rightarrow0$) of the fluid equations
{\cite{HoMaRa1998,1998PhRvL..81.5338C,1999PhyD..133...49C,1999PhyD..133...66C,1999PhFl...11.2343C,2001PhyD..152..505F}.} The result is that the vorticity is advected by a
smoothed velocity,
$\overline{\vec{u_s}}=(1-\alpha^2\nabla^2)^{-1}\bar{\vec{u}}$, with
a filter scale $\sim\alpha$,
\begin{equation}
\partial_t\bar{\zeta} + \nabla\cdot\bigg{(}\overline{\vec{u_s}}\bar{\zeta}\bigg{)}
= \nu\nabla^2\bar{\zeta}+ {\bar{F}+\bar{D}+\bar{Q}}\,,
\label{eq:LANS}
\end{equation}
where
$\nabla\cdot\bigg{(}\overline{\vec{u_s}}\bar{\zeta}\bigg{)}=\{\bar{\psi_s},\bar{\zeta}\}$.
The alpha subgrid term is
\begin{equation}
\sigma = \{\bar{\psi},\bar{\zeta}\} - \{\bar{\psi_s},\bar{\zeta}\}\,.
\label{eq:lanssigma}
\end{equation}
{Note that the $\alpha-$model has complex conservation properties
in that the energy balance equation is in the $H^1_\alpha$ norm,
$\int \overline{\vec{u_s}}\cdot\bar{\vec{u}}dA$, and enstrophy is in
the $L^2$ norm, $\int \bar{\zeta}^2dA$. The subgrid energy transfer
is $L_\alpha(k)/k^2$ is related to the subgrid enstrophy
transfer by $L_\alpha(k)= L(k)/(1+\alpha^2k^2)$.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{alpha_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{alpha_landscape_ekSGS_.eps}
\caption{$\alpha-$model: subgrid transfers of enstrophy ($L(k)$, Top)
and energy {($L_\alpha(k)/k^2$,} Bottom) for $\alpha=\Delta x$ (red dotted),
$2\Delta x$ (green dashed), $9\Delta x$ (blue dash-dotted;
{vertical line shows wavenumber}), $16\Delta x$ (pink
dash-triple-dotted; {vertical line shows wavenumber}), and
benchmark (solid black, {nearly zero except for $k\gtrapprox300$ in $L(k)$).} Due to numerical cancellation noise in
Eq. (\ref{eq:lanssigma}), smoothing has been applied to the plots.}
\label{fig:alphaexp}
\end{figure}
The subgrid transfers, Fig. \ref{fig:alphaexp}, for the $\alpha-$model
are very large and in the wrong direction. As the model dissipates
neither energy nor enstrophy the transfers are conservative; they
remove energy and enstrophy from above the forcing scale and deposit
them below the forcing scale. As the filter width, $\alpha$, is increased so is the
amount of large-scale energy and enstrophy moved down-scale {to scales larger than $\alpha$ (vertical lines in Fig. \ref{fig:alphaexp}).}
\begin{figure}[htbp]
\includegraphics[width=4.425cm]{fig_movietrack_alpha2_w1}
\includegraphics[width=4.425cm]{fig_movietrack_alpha9_w1}
\includegraphics[width=4.425cm]{fig_movietrack_alpha16_w1}\\
\includegraphics[width=4.425cm]{fig_movietrack_alpha2_w2}
\includegraphics[width=4.425cm]{fig_movietrack_alpha9_w2}
\includegraphics[width=4.425cm]{fig_movietrack_alpha16_w2}\\
\includegraphics[width=4.425cm]{fig_movietrack_alpha2_w3}
\includegraphics[width=4.425cm]{fig_movietrack_alpha9_w3}
\includegraphics[width=4.425cm]{fig_movietrack_alpha16_w3}\\
\includegraphics[width=4.425cm]{fig_movietrack_alpha2_w4}
\includegraphics[width=4.425cm]{fig_movietrack_alpha9_w4}
\includegraphics[width=4.425cm]{fig_movietrack_alpha16_w4
\caption{Vorticity field, $\zeta$: Vortex merger event {(tracked
to center of field of view which is $1/5^2$ of the entire domain).} Time runs from top to bottom
starting 0min after initialization in steps of $10^4\,$min. {\sl
1st column} LANS $\alpha=2\Delta x$, {\sl 2nd column} $9\Delta x$,
{\sl 3rd column} $16\Delta x$. $\alpha=2\Delta x$ is the most
realistic result. {Cuts from 3rd row (black lines) are plotted in Fig. \ref{fig:darryl}.}}
\label{fig:alphatrace}
\end{figure}
{The physical effects of the $\alpha-$model are visualized in
Fig. \ref{fig:alphatrace}: small-scale vortical motions are removed
from the advecting field. As $\alpha$ is increased the rotation of
the central, yellow(light) V-shaped, vorticity feature is reduced.
This can be seen by viewing each row from left to right. To
visualize the effect on the vorticity filaments, 1D cuts are taken
as indicated by the black lines in the third row. The vorticity values are plotted
in Fig. \ref{fig:darryl}. There is a translation due to the removal
of small-scale vorticity from the advecting field. Disregarding
this, it is seen that the filaments are slightly larger as $\alpha$
is increased. The vorticity peaks are also taller. This indicates
that the dissipation of the filaments is reduced as $\alpha$ is
increased. The effect is also seen in the spectra: enstrophy is
removed from the largest (and smallest) scales and deposited at
scales bracketed by the forcing scale and $\alpha$. One
interpretation could be that the $\alpha-$model reduces both the
roll-up and the thinning of filamentation.
The reduced roll-up reduces spatially averaged vorticity gradients and,
hence, reduces dissipation. The reduction in thinning of the filaments
does not appear to be
large enough to be significant for the dissipation of individual,
small filaments. Also due to this, more vorticity and enstrophy
remains at super$-\alpha$ scales.}
{Note that our $\alpha-$model spectra do not compare to results
found by \cite{NaSh2001}: their forcing kept $Z_s(10)$ constant
rather than enstrophy injection constant, dissipated based on
$\zeta_s$ not $\zeta$, and plotted different quantities than we have
here. They studied $|u_s|^2$ and $|\zeta_s|^2$ which are not the
ideal invariants for the $\alpha-$model. Finally, unlike for the 3D
$\alpha-$model \cite{2001PhyD..152..505F}, no change in the scaling
of the dissipation scale with Reynolds number is expected for the 2D
$\alpha-$model \cite{LuKuTa+2007}. This suggests 2D$-\alpha$ will
not perform as a LES in the same regard as its 3D counterpart and,
perhaps, explains our results.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{alpha_fig_darryl3.eps}
\includegraphics[width=10cm]{darryl_landscape_EnstrophySpectrum_.eps}
\caption{{$\alpha-$model: (Top) Cut of vorticity field,
$2\times10^4\,$min into simulation for $\alpha=2\Delta x$ (green
dashed), $9\Delta x$ (blue dash-dotted), and $16\Delta x$ (pink
dash-triple-dotted). Section of cut is indicated in 3rd row of
Fig. \ref{fig:alphatrace}. (Bottom) Compensated enstrophy spectrum for same time.} }
\label{fig:darryl}
\end{figure}
\subsection{\label{sec:comparison}Comparison of parameterizations}
The subgrid transfers of the six parameterizations are compared in
Fig. \ref{fig:compareexp}. {Concentrating on the subgrid
enstrophy transfer, we can eliminate the $\alpha-$model because it
unphysically generates enstrophy for $100\lessapprox
k\lessapprox200$ and Smagorinsky can be eliminated because it
essentially eliminates zero small-scale enstrophy (the grey line is
flat an indistinguishable from zero on this vertical scale). Of the
remaining models, the hyper-viscous is closest to mimicking the true
subgrid transfers of both energy and enstrophy,} though for the largest
wavenumbers, $k>200$, the viscous and Leith parameterizations perform
similarly. The AVM is the only method that reproduces the correct
sign of the energy transfer, but it removes enstrophy preferentially
from {intermediate scales} instead of the smallest resolved
scales. This method would likely perform better for $m>1$
\citep{VaHu1988}.
\begin{figure}[htbp]
\includegraphics[width=10cm]{compare_landscape_SGS_.eps}\\
\includegraphics[width=10cm]{compare_landscape_ekSGS_.eps}
\caption{Subgrid transfers of enstrophy ($L(k)$, Top) and energy
($L(k)/k^2$, Bottom) for benchmark (solid black), hyper-viscous
$\nu_4=1.1\times10^{-9}m^4s^{-1}$ (red dotted), LANS $\alpha=2\Delta
x$ (green dashed), viscous $\nu=11m^2s^{-1}$ (blue dash-dotted),
Leith {$\Lambda=1$} (pink dash-triple-dotted), AVM {$\theta=0.16$}
(cyan long-dashed), {and Smagorinsky $\Lambda_S=0.1$
(solid grey).} }
\label{fig:compareexp}
\end{figure}
The enstrophy flux error landscape norms are given in
Fig. \ref{fig:comparetrue}. The $\alpha-$model {and Smagorinsky are} the obvious outliers
with {a factor of five poorer performance. The viscous, hyper-viscous
and Leith parameterizations have very
similar performance. The AVM is within a factor of two in performance.
Again, this could likely
be improved upon by using a larger value of $m$.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{compare_landscape_ModelFluxp.eps}
\includegraphics[width=10cm]{compare_landscape_ModelFlux_.eps}
\caption{{(Top)} Enstrophy flux error-landscape {norm $D_0$} for hyper-viscous
($\nu_4$), $\alpha-$model ($\alpha$), $\nabla^2$ viscosity ($\nu$),
Leith ($\Lambda$), AVM ($\theta$), and Smagorinsky ($\Lambda_S$).
{(Bottom) Modeled enstrophy flux, $\Pi_T$, for for benchmark
(solid black), hyper-viscous $\nu_4=1.1\times10^{-9}m^4s^{-1}$
(red dotted), LANS $\alpha=2\Delta x$ (green dashed), viscous
$\nu=11m^2s^{-1}$ (blue dash-dotted), Leith $\Lambda=1$ (pink
dash-triple-dotted), AVM {$\theta=0.16$} (cyan long-dashed), and
Smagorinsky $\Lambda_S=0.1$ (solid grey). } }
\label{fig:comparetrue}
\end{figure}
This similarity in performance between hyper-viscous, viscous, and
Leith parameterizations can also been seen in the resulting enstrophy
spectra, Fig. \ref{fig:compare}. All three methods have the same
spectra for $k\lessapprox100$. {Neither the $\alpha-$model
nor Smagorinsky reduces the pile-up of numerical thermalization
noise \cite{CiBoDe2005}} in the small scales. As seen in the previous results, the AVM
method with $m=1$ is dissipative at too large scales to perform as
well as the viscous, hyper-viscous, or Leith parameterizations.
{Smagorinsky performs poorly because it removes enstrophy from the
largest rather than the smallest resolved scales.}
\begin{figure}[htbp]
\includegraphics[width=10cm]{compare_landscape_EnstrophySpectrum_.eps}
\caption{Enstrophy spectra for benchmark (solid black), hyper-viscous
$\nu_4=1.1\times10^{-9}m^4s^{-1}$ (red dotted), LANS
$\alpha=2\Delta x$ (green dashed), viscous ${\nu'}=11m^2s^{-1}$ (blue
dash-dotted), Leith $\Lambda=1$ (pink dash-triple-dotted), AVM
{$\theta=0.16$} (cyan long-dashed), {and Smagorinsky $\Lambda_S=0.1$
(solid grey).}}
\label{fig:compare}
\end{figure}
\section{\label{sec:conclusions}Conclusions}
We have compared six popular turbulence parameterizations in the
enstrophy cascade regime of the barotropic vorticity equation on a
$f-$plane (equivalently, 2D Navier-Stokes) in forced-dissipative
simulations. The hyper-viscous, viscous, and Leith models all perform
well down to about $10\Delta x$.
The hyper-viscous model
reproduces the largest-resolved-scales ($1\le k\le100$) flux the best of the three and
the viscous model best reproduces the smallest-resolved-scales ($k\ge200$) flux.
The Leith model, {because its diffusion is anisotropic,} is expected to carry-over its performance to
anisotropic flows {(e.g., the 3D baroclinic ocean system)} which would
be challenging for the viscous and hyper-viscous models. {The
Smagorinsky model does not work in the enstrophy cascade regime--it
removes enstrophy from the largest rather than the smallest resolved
scales.} The anticipated vorticity method without a strong enough
weighting to small scales,
{larger values of $m$,}
does not perform as well as the prior three
parameterizations. As even this order of diffusive operator is not
practical in the finite-volume {and finite-difference schemes
typically used in global ocean modeling} (e.g.,
\cite{RiThKl+2010}), we chose not {to} investigate higher-orders.
We have confirmed \cite{LuKuTa+2007}'s suggestion that the
Lagrangian-averaged $\alpha-$model does not perform as a turbulence
model in this system {(see also \cite{NaSh2001}). Analytically,
one expects the numerical degrees of freedom to scale with Reynolds
number the same as unparameterized Navier-Stokes. The model reduces
rotation due to small-scale vorticity and, less dramatically, also
reduces the thinning of vortex filaments due to stretching. The
balance of the effect is a net reduction of dissipation of the
vorticity filaments and a piling of energy and enstrophy to
sub-forcing/super$-\alpha$ scales (enhancing the flux in this
spectral region).}
{One possible MOLES closure has not been scoped here,} the use of
monotone transport as the model for LES closure. These closures,
commonly referred to Monotone Implicit Large-Eddy Simulation (MILES),
require the evaluation of the nonlinear transport be carried out in
physical space, something that is not possible within the global
spectral model utilized for this study. Our future work, discussed
briefly below, will utilize a traditional finite-volume approach where
an evaluation of MILES will be possible. {Combined models have
also not been investigated here due to the enormous parameter space
that would entail.}
Subgrid transfers have been measured before, e.g., for the APVM
\citep{VaHu1988}, and again for the APVM, hyper-viscosity, and
implicit large-eddy simulations \citep{ThKeWo2011}. Error-landscapes
for LES have been produced for various quantities like spectra
\citep{MeGeBa2003,MeSaGe2006,MeGeSa2007,Me2011}. By combining these
two techniques, however, we have introduced a method for determining
the optimal turbulence parameterization also in flows different than
those considered here: the error-landscape of the enstrophy flux at
small-scales in a 2D flow can be replaced by the error-landscape of
the modeled {flux in a 3D baroclinic system \NEU{(see \ref{sec:appendix}).}
{We emphasize that MOLES comparisons based on spectra alone do
not ensure that the correct dynamics are being reproduced by a
parameterization. For example, consider the $\Lambda_S=0.5$
($C_S\approx0.16$) result for the Smagorinsky model (blue
dash-dotted line in Fig. \ref{fig:smagtrue}).
The spectrum is best approximated by this run, but for the wrong
reasons as the non-linear flux is more poorly reproduced than for
$\Lambda_S=0.1$. For the viscous model, however, which physically
correctly removes enstrophy from the small scales, both the
spectrum and the flux are best reproduced for
${\nu'}=11\,$m$^2$s$^{-1}$. In this latter case, the spectrum is
reproduced because the dynamics are reproduced.}
{For a 3D baroclinic system, at the scales on which the MOLES
acts \NEU{($5-10\,$km),} the system will be approximately \NEU{QG. Because of the similarities between QG and 2D \cite{FoKeMe2008}, we have some expectation} that our
results will hold: Smagorinsky, the $\alpha-$model, and APVM with
$m=1$ will not perform as well as viscosity, hyper-viscosity, and
Leith. In fact, because of the anisotropic diffusion offered by
Leith, it will likely perform the best. Our results {may} not extend
to the 3D system, however, if additional physics comes in to play,
like vertical mixing over small horizontal scales.}
Our next step is to move into an idealized, 3D baroclinic system, most
likely a re-entrant zonal channel that can serve as an idealized
Antarctic Circumpolar Current. While the move to three dimensions
allows for the direct simulation of baroclinic instability, it also
necessitates the development of analysis tools that can accurately
account for energy and enstrophy transfers between the disparate
horizontal and vertical scales of motion. Furthermore, the move to a
3D baroclinic system entail\NEU{s} the use of a height-based vertical
coordinate. Such a system requires the transport of one or more
tracers in order to close the system via an equation of state.
\NEU{The theoretical analysis of such a system is outlined in
\ref{sec:appendix}.} And finally, as we move to more realistic and,
thus, bounded domains, our ability to simulate the governing
equations, as well as analyze the fluxes, via global spectral
expansions will be increasingly cumbersome. As a result, we plan on
utilizing a traditional finite-volume global ocean model
\cite{Ringler:8qyuvoZk} in the next phase of this study. \\
{\bf Acknowledgments} We acknowledge discussions with B. Fox Kemper,
{D. Holm,}
and B. Wingate. This work was supported by the Earth System Modeling
and Regional and Global Climate Modeling programs of the Office of
Biological and Environmental Research within the US Department of
Energy's Office of Science.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 107
|
Southern is putting on extra trains as the RMT union gears-up for two further days of strike action this week on Friday 4 and Saturday 5 November.
Southern will be providing around 60 more trains between Haywards Heath and Victoria, with a half-hourly service running from 7am to 10pm. If the strike goes ahead, almost all 156 Southern stations will have either a train or bus service. There will be extra staff at stations to help passengers with arrangements in place for passengers to use alternative transport providers.
On both days, there will be a restricted service, with many routes having fewer trains, and on some routes there will be no train service at all. On some routes, a service will be provided through ticket acceptance with other train operators.
Southern Passenger Services Director Angie Doll said: "We're sorry to our passengers that we won't be able to provide a full service because of the RMT's pointless and wholly unjustified strike action.
"The union is causing yet further misery and disruption which is having a material impact on people's work and family lives.
Due to the strike action by the RMT union, it will not be possible for Southern to provide its usual service for the Lewes bonfire celebrations on Saturday November 5. As a result, no trains will stop at Lewes, London Road (Brighton), Moulsecoomb, Falmer and Glynde stations after midday on Saturday November 5 until normal services resume on Sunday November 6.
Passengers are urged to plan ahead and can find details of the service provision to help plan their journeys at www.southernrailway.com/rmtstrike. There is also a 'heat map' for passengers available on the page.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,020
|
Brioche Dorée is een Franse keten van bakkers/café-restaurants, opgericht in 1976 en oorspronkelijk begonnen in Brest. Het bedrijf werd opgericht door Louis Le Duff, de huidige president van Groupe Le Duff. De keten is wereldwijd actief en is de op een na grootste bakkers/ caféketen ter wereld. In de wereldwijd meer dan 500 café-restaurants komen dagelijks meer dan 200.000 klanten. De naam betekent "Gouden Brioche" in het Frans.
Het bedrijf heeft vestigingen in de Verenigde Staten, Zwitserland, Luxemburg, Egypte, Algerije, Canada, Costa Rica, Tsjechië, Denemarken, Verenigd Koninkrijk, Duitsland, Argentinië, Marokko, Saudi-Arabië, Syrië, China, Zuid-Korea, India, Japan, VAE, Oman, Koeweit, Qatar, Kameroen, Senegal en Bahrein.
Externe links
Officiële website
Franse winkelketen
Restaurantketen
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,623
|
Q: two coin (normal & both heads) toss probability I have two coins, one normal and one having both heads. Randomly picked one and tossed it 3 times. All three were HEADs.
What is the probability of seeing a tail on next toss?
Going with independent event approach,
normal: 1/2 x 1/2 = 1/4
biased: 1/2 x 0 = 0
total: 1/4 + 0 = 0.25
Is this correct? Or is there a different approach?
A: If the coin is fair then the number of heads in a row before you flip has no affect on the probability of another head so the the answer for a fair coin is $\frac{1}{2}$.
If this is a coin with both heads then you clearly has no chances to see tail at all.
Using Bayes' theorem you can say that
$$P(\mbox{your coin is double-headed}) = \frac{1\cdot \frac{1}{2}}{1\cdot \frac{1}{2} + \frac{1}{8}\cdot \frac{1}{2}} = \frac{8}{9}$$
Just to add more generality, using the same approach you can show that the probability that the coin is double-headed after $N$ heads in a row and no tails is
$$P = \frac{p}{p+(1-p)\cdot2^{-N}},$$
where $p$ is the probability you think the coin is double-headed before you receive any information via flips. After some rather big $N$ it will be very close to $1$ for any reasonable $p$.
A: $$\begin{align}
P(\text{T| HHH}) &= \dfrac{P(\text{T $\cap$ HHH})}{P(\text{HHH})} \\
&= \dfrac{P(\text{fair coin}) \cdot P_{\text{fair coin}}(\text{T $\cap$ HHH}) + P(\text{trick coin}) \cdot P_{\text{trick coin}}(\text{T $\cap$ HHH})}{P(\text{fair coin}) \cdot P_{\text{fair coin}}(\text{HHH}) + P(\text{trick coin}) \cdot P_{\text{trick coin}}(\text{HHH})} \\
&= \dfrac{\dfrac{1}{2} \cdot \dfrac{1}{2^4} + \dfrac{1}{2} \cdot(0)}{\dfrac{1}{2} \cdot \dfrac{1}{2^3} + \dfrac{1}{2} \cdot(1)} \\
&= \dfrac{1}{18}
\end{align}$$
A: This is not correct. If you didn't know what had happened on the three previous tosses, you would have equal chances of having either coin, so you would have a probability of $1/4$. However, after seeing three heads, the chance that the coin you are tossing is the double-headed one has increased.
You can work out the chance that you are tossing the normal coin using Bayes' rule:
$$P(\text{normal coin}\mid HHH)=\frac{P(\text{normal coin and }HHH)}{P(HHH)}$$
then use this to find the probability of a tail next.
A: You have a result of tossed three heads in three tries. What is the probability that the coin is fair?
*
*To obtain the result, you have either picked a fair coin and tossed two head, or else you have picked a double header and done so. $$\begin{align}\mathsf P(F\cap R)~&=~\tfrac 12\cdot\tfrac 12^3&&\text{Probability for Fair and Result}\\\mathsf P(F^\complement\cap R)~&=~\tfrac 12\cdot 1^3 &&\text{Probability for Unfair and Result}\end{align}$$
*
*So the (conditional) probability for having a fair coin given the results is:
$$\begin{align}\mathsf P(F\mid R) ~&=~ \dfrac{\mathsf P(F\cap R)}{\mathsf P(F\cap R)+\mathsf P(F^\complement\cap R)} \\[1ex] &=~ \dfrac{2^{-4}}{2^{-4}+2^{-1}} \\[1ex] &=~ \dfrac 19 \end{align}$$
*
*Then, by the law of total probability, the (conditional) probability for seeing a tail on the next toss given the result is: $$\begin{align}\mathsf P(T\mid R)~&=~\mathsf P(T\mid F)~\mathsf P(F\mid R)+\mathsf P(T\mid F^\complement)~\mathsf P(F^\complement\mid R)\\[1ex] &=~ \tfrac 12\cdot \tfrac 19+0\cdot \tfrac 89 \\[1ex] &=~ \dfrac 1{18}\end{align}$$
This is sensible, since seeing three heads in a row should make you more confident that you have selected the double headed coin.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,384
|
\section{Introduction}
In 5G and B5G era, to achieve better cellular performance, operators are
deploying more and more small-cell base stations (BSs) in the urban area
~\cite{rong20185g,liu2018millimeter}. This brings the backhaul challenge
for the dense small-cell deployment that huge amount of cellular data
need to be transmitted between BSs in such networks~\cite{feng2016millimetre,jaber20165g}.
However, most of small-cell BSs may not have wired connections to the
core network due to the construction limits and prohibitive cost in many
places such as North America, so that mmWave backhauling has been regarded
as a promising solution to the backhaul challenge in 5G cellular systems
since it is able to support multi-Gbps wireless backhaul links~\cite{taori2015point,shariat2015enabling,polese2020integrated}.
Currently, a lot of researches in this area focus on the mmWave integrated
access and backhaul (IAB) network architecture where the access tier and the
backhaul tier share the same frequency band~\cite{polese2020integrated,saha2018integrated,madapatha2020integrated}.
However, it could be challenging for the construction of line-of-sight (LOS)
mmWave backhaul links in such networks, since the location and deployment
of small-cell base stations are usually pre-fixed, which concerns more
from the perspective of increasing the cellular coverage and performance.
If the LOS path between a pair of BSs is blocked by the abundant obstacles
in the dense urban area, due to the blockage effect~\cite{liu2019analysis,gerasimenko2019capacity},
the capacity of the corresponding logical link will drop severely.
Therefore, we proposed a new relay-assisted mmWave backhaul network
architecture~\cite{hu2017relay,hu2018optimizing,hu2020feasibility,yan2021load},
where dedicated mmWave relays are introduced as additional
network elements in the backhaul network. With such simple relay devices,
not only the blockage can be addressed through flexible relay deployment
in the 3D space~\cite{liu2019joint,yan2018path,liu2018blockage}, but
also the link capacity of each physical link can be improved because of
the shorter physical link length.
In our previous work~\cite{hu2020feasibility},
we find that to support the ultra-high data rate in the 5G small-cell backhaul,
not only the location of relays need to be carefully selected, but also the
scheduling of the logical links between BSs have to be smartly coordinated.
Our recent work~\cite{hu2022maximum} focuses on optimizing the
throughput performance of the relay-assisted mmWave backhaul network with
a tree topology in either the downlink or uplink case.
However, the scheduling algorithm proposed in~\cite{hu2022maximum} only
aims to show the feasibility of finding a schedule that is able to satisfy
the set of schedule lengths of logical links obtained from solving the
optimization problems.
It is not an algorithm that can be directly deployed into the practical
mmWave backhaul network system. Moreover, in the performance optimization,
the link schedule length is a continuous value, but in the
realistic system, time resource is partitioned into small slots with the
same duration. Furthermore, in the practical cellular network, since the
aggregated traffic load may be larger than the network capacity, the backhaul
traffic has to be scheduled proportionally to the traffic demand of each
small-cell BS, so that a certain level of fairness can be achieved. Besides,
similar to the discussion in~\cite{hu2022maximum}, network
level intelligent scheduling has to come into play to handle the scenario
where mutual interference exists between logical links and limited number of
radio chains are available on BSs. Therefore, it is important to address the
scheduling problem in the practical backhaul system.
In this paper, we propose a novel distributed scheduling algorithm that
can be deployed to efficiently schedule the data traffic in the practical
relay-assisted mmWave backhaul network with a tree topology. Different
from the work in~\cite{hu2022maximum}, where either
downlink or uplink traffic is considered, the proposed distributed algorithm
is able to schedule both downlink and uplink traffic in every subframe. Since
the mutual interference relationship between a pair of logical links may vary
according to different transmission directions of both links, we assume
that two logical links are considered always interfering with each other if
there exists at least one combination of transmission directions which leads
to significant mutual interference when both logical links transmit concurrently.
Although applying this assumption sacrifices a certain amount of throughput
performance, it reduces the complexity of the algorithm design. In fact, since
there are only a few pairs of logical links interfering with each other in the
relay-assisted mmWave backhaul network in the dense urban area when the relay
locations are carefully selected, the throughput performance loss is likely to
be limited.
In the following sections, we first give an overview on the distributed
scheduling algorithm, which describes the general idea of the algorithm design.
Then, details on the system setting are provided. After that, three main
components of the algorithm, i.e., the handshaking procedure for exchanging
control information, the calculation of the local schedule of a small-cell BS,
and the determination of the final valid schedule of a BS, are elaborated one
after another. Simulation results show that the average throughput performance of
the distributed scheduling algorithm can reach closely to the calculated
maximum traffic demand of small-cell BSs (see~\cite{hu2022maximum})
in the tree-style mmWave backhaul network. Moreover, simulations are also
conducted to show that the distributed algorithm is able to adapt to the
dynamic traffic demand (with sharp changes) of BSs in the network as well.
\section{Related work}
Scheduling algorithm design for the mmWave backhaul network has attracted a
significant amount of interest in the research academia in the last decade.
The authors of~\cite{arribas2019optimizing} study the mmWave self-backhaul
scheduling problem and derive an MILP formulation for it as well as upper
and lower bounds. They prove that the problem is NP-hard and can be approximated,
but only if interference is negligible.
Given a set of mm-wave backhaul links, ~\cite{saad2019millimeter} addresses the problem of backhaul scheduling through considering the mutual interference and number of radio chains constraints. A succinct optimization-based formulation of the problem is provided and using reduction from the set-cover problem, the authors devise a provably good polynomial-time algorithm for the problem.
The authors of~\cite{niu2019relay} investigate the problem of optimal scheduling to maximize the number of flows satisfying their QoS requirements with relays exploited to overcome blockage, where relays refer to small-cell BSs. Both a relay selection algorithm and a transmission scheduling algorithm are proposed to increase the network throughput through addressing the blockage issue and exploiting concurrent transmissions.
~\cite{garncarek2019mmwave} studies the mmWave wireless backhaul scheduling problem in the mmWave IAB backhaul network, assuming stochastic arrival of packets at the macro-cell BS to be delivered to small-cell BSs. The authors present various results concerning system stability, defined as a bounded expected queue sizes of macro-cell BS and small-cell BSs, under different patterns of random traffic.
Note that all the works above consider the self-backhaul network where mmWave small-cell BSs serve as relays, and none of them take dedicated mmWave relays as network elements into account. In contrary, when dedicated relays are introduced in our network model, the scheduling problem becomes more sophisticated. Moreover, we also propose a new distributed scheduling algorithm for urban mmWave backhaul networks that takes several factors about practical deployment into account.
The authors in~\cite{fang2021joint} optimize the scheduling of access and backhaul links such that the minimum throughput of the access links is maximized based on the revised simplex method. ~\cite{ranjantwo} considers the radio resource scheduling problem and presents a QoS-based downlink scheduler designed explicitly for IAB networks. The scheduler is devised after considering multihop relaying topology, QoS requirements and backhaul constraints. These works are different from ours because they consider an IAB network structure whereas the frequency band used for backhaul in our research is dedicated to the backhaul usage.
In~\cite{yuan2018optimal}, the authors propose an efficient scheduling method, so-called schedule-oriented optimization, based on matching theory that optimizes QoS metrics jointly with routing in the mmWave cellular network. It is claimed to be capable of solving any scheduling problem that can be formulated as a linear program whose variables are link times and QoS metrics. As an example, they also show the optimal solution of the maximum throughput fair scheduling (MTFS).
Recently, the authors of~\cite{sha2022near} investigate two aspects of the near interference-free (NIF) space-time user scheduling in a multi-cell mmWave network with multi-RF-chain base stations. Firstly, they study the NIF user scheduling problem to minimize the unfulfilled user requirements; secondly, they study the joint NIF user scheduling and power allocation problem to minimize the total transmit power under the constraint of rate requirements.
The above works do not consider a relay-assisted backhaul network architecture, and focuses on the scheduling optimization in the mmWave access network.
~\cite{li2017practical} proposes Distributed Maximum QoS-aware (DMQ) scheduling algorithm for the mmWave backhaul network of small cells to maximize the system throughput while satisfying QoS requirements for each flow. Based on CSMA/CA, the proposed algorithm is the first that prioritizes MAC contention window to provide better concurrent transmission support while achieving QoS-aware capability. Different from our work, this paper consider a network architecture without dedicated mmWave relays deployed, and some practical factors such as the number of radio chains on BSs are not considered as well.
More recently, due to the fast development of artificial intelligence, several learning based scheduling methods for mmWave networks are also proposed. To achieve resilience to link and node failures, the authors in~\cite{dogan2021reinforcement} explore a state-of-the-art Soft Actor-Critic (SAC) deep reinforcement learning algorithm, that adapts the information flow through the network, without using knowledge of the link capacities or network topology. This work uses an N-relay Gaussian Full-Duplex (FD) 1-2-1 network model which is different from our network model. ~\cite{zhang2021resource} proposes a reinforcement learning approach based on column generation to address the resource allocation problem for mmWave IAB backhaul, as the method is claimed to be able to capture the environment dynamics such as moving obstacles in the network. ~\cite{gupta2020learning} is a work investigating the multihop link scheduling problem with the objective of minimizing the end-to-end delay experienced by a typical packet. The authors model the system as a network of queues and formulate it as a Markov decision process over a continuous action space. This allows them to leverage the deep deterministic policy gradient algorithm from reinforcement learning to learn the delay minimizing scheduling policy under two scenarios.
\section{Algorithm overview}
The basic idea of the proposed algorithm is that in the relay-assisted mmWave
backhaul network with a tree topology, every BS except the leaf small-cell BSs,
makes scheduling decisions for all the logical links between the BS and its child
BSs based on several types of information, including the traffic demand information,
the queue length information, the final valid schedule received from its parent BS,
and the set of local schedules received from its child BSs. All information is
collected through monitoring the BS's own status and exchanging control
messages between the BS and its one-hop neighbors via the handshaking
procedure introduced later.
The link scheduling happens on a per-subframe basis, and each subframe
contains a fixed number of time slots with the same duration. Therefore,
both the local schedule and the final valid schedule record the assignment
of time slots to a set of logical links for each subframe. In the distributed
scheduling algorithm, before a BS starts to calculate its final valid schedule
for a subframe, it has to know its parent BS's final valid schedule for the
same subframe in advance, because the schedule of the logical link between the
BS and its parent BS is specified therein. This indicates a strong ``happen-before"
relationship in the calculation of final valid schedule. Since the calculation and
propagation of the final valid schedules cannot complete instantly across
the entire network, every BS calculates its final valid schedule for a specific
subframe at least one subframe before the that subframe starts. Thus, in
a tree topology, the BSs closer to the macro-cell BS will always make the
scheduling decision for a future subframe earlier than those BSs farther
away from the macro-cell BS. However, the calculation of local schedule
does not need any information with a strict ``happen-before" relationship.
Therefore, a BS calculates its local schedule at the beginning of a
subframe, and send it to its parent BS immediately after the calculation
completes, so that the received local schedule can be used by the parent
BS to calculate its final valid schedule within the same subframe. Details
can be found in the section~\ref{sec:ch6-handshaking}.
Moreover, in the calculation of both types of schedules, the local fairness is
addressed, which means a BS assigns time resource to logical links
proportionally to their traffic demand or queue lengths. This will be
elaborated in later sections as well.
\section{System settings}
In a given relay-assisted mmWave backhaul network, there is one
macro-cell BS $M$, a set of small-cell BSs $\mathcal{B}=\{B_1, B_2, ...\}$,
and a set of mmWave relays. $M$, $\mathcal{B}$, and the set of logical
links $\mathcal{L}$ between them together form a tree topology, where
$M$ is the root node of the tree.
A logical link $L_i\in \mathcal{L}$ ends at the BS $B_i$, and it is either a
single-hop path or a multi-hop path going through several mmWave relays sequentially.
Every physical link within a logical link is a LoS. Every logical link
is ``intra-path" interference-minimal, which means the mutual interference between
the concurrent transmissions on the physical links within a logical link can be
ignored. Based on our previous work, the optimal scheduling of a interference-minimal
logical link can be computed. If a logical link $L_j$ is attached to a BS $B_i$ (i.e.,
$L_j$ is between $B_i$ and $B_j$), the physical link directly attached to $B_i$ within
$L_j$ is named as $l_{ji}$. Therefore, the portion of time used by $l_{ji}$ in the
optimal schedule length of $L_j$ is denoted as $P_{ji}$. The data rate of physical link
$l_{ji}$ is $r_{ji}$, which is a known system parameter.
It has to be clarified that the interference relationship between a pair of logical links
in the backhaul network may vary according to the combination of their transmission directions in a downlink and uplink hybrid case. If there are $x$ pairs of logical links that may interfere with each other, in total there could be $4^x$ different interference cases in the network, which introduces huge complexity to address them all. To address this issue, we assume that as long as two logical links with certain directions interfere with each other, they are considered always interfering with each other no matter which directions of their transmissions are. Under this assumption, despite that the throughput performance can not reach the optimal, we could first schedule both downlink and uplink traffic together, and then separate the traffic of different directions into different numbers of slots according to the traffic demand or queue length information.
The number of mmWave radio chains available on $B_i$ or $M$ is $N_{i}^R$ or $N_M^R$, respectively. There are 2 control time slots followed by $n_d$ data time slots in each subframe.
\section{Main components of the distributed scheduling algorithm}
In is section, we explain the distributed algorithm in details. The explanation
focuses on three main components of the algorithm: the handshaking of control messages,
the calculation of local schedule, and the determination of final valid
schedule.
\subsection{Handshaking of control messages}
\label{sec:ch6-handshaking}
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{figures/distributedAlgorithmHandshake.pdf}
\caption{Hand-shaking procedure of the distributed scheduling algorithm}
\label{fig:handshake}
\end{figure}
To elaborate the handshaking process, an example is provided in Figure~\ref{fig:handshake}. A small backhaul network with a tree topology is shown in the lower part of Figure~\ref{fig:handshake}, where BS 1 is the macro-cell BS, and BS 2-5 are all small-cell BSs.
It is noticed that some logical links between BSs are multi-hop relay paths. If control
messages are sent through the multi-hop logical links, the delay of these control information will be too long. To address this issue, each BS is equipped with a communication module working at a lower frequency band (e.g., sub-6 Ghz band), which uses an omni-directional antenna. That communication module takes responsibility of transmitting and receiving control messages in the first two control slots of each subframe. To avoid the collision of the control messages between a BS and its child BSs, each control slot is further partitioned into $n_{sub}$ sub-slots, and each child BS is assigned a unique sub-slot to exchange control messages. The assignment of sub-slots between a BS and its child BSs happens in the initial configuration phase when the relay-assisted backhaul network is constructed. During the initial configuration phase, a child BS is associated with its parent BS to form the backhaul logical link.
As we can see in Figure~\ref{fig:handshake}, at the first slot of each subframe, each small-cell BS calculates its local schedule based on the traffic demand information, which will be explained in detail later. The calculated local schedule and updated queue and traffic demand information of a small-cell BS will be sent immediately to its parent BS. Note that, since there is no parent BS of the macro-cell BS, it will skip this step.
After a BS receives the control messages from its child BSs, before it starts the calculation of a final valid schedule for a specific future subframe, it checks whether the final valid schedule from its parent BS for the same future subframe has been received or not. If it has not been received, the BS will not schedule transmissions in that future subframe, because it does not know which slots are assigned to the logical link between itself and its parent BS. Any transmission from the BS may potentially conflict with the transmission from its parent BS. On the other hand, if the interested final valid schedule from the parent BS has been received, the BS will starts the calculation of its final valid schedule.
As shown in the example, each BS will start its scheduling in a specific subframe according to its height in the tree topology. If the topology has a depth of $H$, which is the height of the macro-cell BS and each small-cell BS $B_i$'s height is denoted as $h_i$, the macro-cell BS will start its scheduling in subframe 1, while small-cell BS $B_i$ will start its scheduling in subframe ($H-h_i+1$). Note that, because the schedule of leaf BSs is determined by their parent BSs, they do not calculate their final valid schedule.
Moreover, in the first scheduling subframe, each non-leaf BS will calculate its schedule of the next several subframes in advance (i.e., from subframe $H-h_i+1$ to subframe $H$), because when a child BS calculates its final valid schedule of a subframe, it has to know its parent BS's final valid schedule of the same subframe in advance; however, the schedule needs time to propagate to its child BSs. After the first scheduling subframe, each BS only needs to calculate the final valid schedule of the next unscheduled subframe. As depicted in Figure~\ref{fig:handshake},
the initialization phase lasts $H-1$ subframe, and after that, the \textit{stable phase} starts.
\subsection{Calculate the local schedule of small-cell BSs}
\label{sec:local_schedule}
The purpose of calculating the local schedule on a small-cell BS $B_i$ is to obtain the ``desired" number of slots that $B_i$ wants its parent BS $B_{i}^p$ to assign for the data transmission on the logical link $L_i$ between $B_i$ and $B_i^p$. As a ``desired" local optimal, it is calculated based on the local traffic demand information rather than the queue information.
Traffic demand information is an upper layer (i.e., application) statistics which is not
typically used in the scheduling algorithm (i.e., MAC layer) in wireless ad hoc or mesh networks. However, in cellular networks starting from the 4G-LTE systems, the Quality-of-Service (QoS) has become a key performance metric.
QoS related policies such as traffic classification and prioritization have already been installed in BSs. Application or service data rate of each user accessing to a BS, as a crucial QoS parameter, is constantly collected by the BS. In the 5G system, as a wider range of applications are going to be supported, even finer-grained management of QoS is expected to be implemented, which means more accurate traffic demand information is likely available at each BS~\cite{ferrus20185g}. In fact, the local schedule aims to provide a ``desired" resource allocation requirement to the parent BS, the amount of which is usually much larger than the amount of resource that
the parent BS can offer in a backhaul network with the heavy traffic load. From this point of view, a small estimation error on the traffic demand of each BS is acceptable. From the discussion in the above section~\ref{sec:ch6-handshaking}, we know that each BS keeps updating its traffic demand information to its parent BS; thus, after a certain amount of time, BS $B_i$ can obtain the accurate total traffic demand $D_j$ from its child BS $B_j \in \mathcal{B}_{i}^c$, where $\mathcal{B}_i^c$ is the set of child BSs of $B_i$. Note that, first, the traffic demand $D_j$ of $B_j$ is in fact the
aggregated traffic demand of all small-cell BSs in the sub-tree rooted at $B_j$; second, as the proposed distributed algorithm schedules both uplink and downlink traffic, the traffic demand information contains both the downlink traffic demand and the uplink traffic demand information.
The local schedule of $B_i$ indicates the number of slots assigned for the data transmissions (both uplink and downlink) on all physical links attached to $B_i$. We denote the number of slots assigned to the attached physical link within the logical link $L_j$ as $n_j$.
Note that, the demand may exceed the maximum throughput capacity of a BS; therefore, we use a local continuous scale variable $S_i$ to indicate the fraction of traffic demand of each child BS actually being served at $B_i$ in one subframe. From this perspective, our proposed distributed scheduling algorithm addresses the local fairness through proportionally schedule the data traffic according to the amount of the traffic demand of each child BS.
Based on the above system setting, we can formulate a mixed integer programming problem to
address the calculation of the local scheduling on $B_i$, in which the local scale variable $S_i$
is to be maximized. In Equation~\ref{eq:local_schedule}, continuous variable $S_i$ is the
optimization objective, and integer variables $\{n_j|\ B_j \in B_i \bigcup \mathcal{B}_{i}^c\}$
are auxiliary variables.
\begin{equation}
\begin{split}
\max \quad & S_i \\
\mathrm{s.t.} \quad & n_i\cdot r_{ii} \geq S_i \cdot \sum_{B_j \in B_i \bigcup \mathcal{B}_{i}^c}{D_j} \\
& 0 \leq S_i \leq 1 \\
& n_j \cdot r_{ji} \geq S_i\cdot D_j,\ \forall\ B_j \in \mathcal{B}_{i}^c\\
& \frac{1}{P_{ji}} \leq \frac{n_j}{P_{ji}} \leq n_d,\ \mathrm{if}\ D_j > 0, \ \forall\ B_j \in B_i \bigcup \mathcal{B}_{i}^c \\
& {n_j} = 0, \ \mathrm{if}\ D_j = 0, \ \forall\ B_j \in B_i \bigcup \mathcal{B}_{i}^c \\
& \lceil\frac{n_j}{P_{ji}}\rceil + I_{jk}\cdot{\lceil\frac{n_k}{P_{ki}}\rceil} \leq n_d,\ \forall\ B_j, B_k \in B_i \bigcup \mathcal{B}_{i}^c \\
& \sum_{B_j \in B_i \bigcup \mathcal{B}_{i}^c}{n_j} \leq n_d\cdot N_i^R
\end{split}
\label{eq:local_schedule}
\end{equation}
As the distributed scheduling algorithm is expected to operate in the real relay-assisted mmWave backhaul system, in which every physical link has a length in the order of 200 m. Due to the use
of high-gain highly-directional antennas, the data transmissions on these LoS physical links are
likely to be able to use the highest level of modulation (e.g., 256 QAM) allowed in the system.
Therefore, the physical link data rate is the same, and we denote the unified physical link data
rate as $r$. Since the portion $P_{ji}$ of the transmission on the physical link within the logical link $L_j$ attached to $B_i$ is either 1 (i.e., single-hop) or 0.5 (i.e., multi-hop), we define a new integer parameter $\alpha_{ji}= \frac{1}{P_{ji}}$, whose value is either 1 or 2. Therefore, we can update the above mixed integer programming problem into a mixed integer linear programming problem.
\begin{equation}
\begin{split}
\max \quad & S_i \\
\mathrm{s.t.} \quad & n_i\cdot r \geq S_i \cdot \sum_{B_j \in B_i \bigcup \mathcal{B}_{i}^c}{D_j} \\
& 0 \leq S_i \leq 1 \\
& n_j \cdot r \geq S_i\cdot D_j,\ \forall\ B_j \in \mathcal{B}_{i}^c\\
& \alpha_{ji} \leq \alpha_{ji}\cdot {n_j} \leq n_d, \ \mathrm{if}\ D_j > 0, \ \forall\ B_j \in B_i \bigcup \mathcal{B}_{i}^c \\
& {n_j} = 0, \ \mathrm{if}\ D_j = 0, \ \forall\ B_j \in B_i \bigcup \mathcal{B}_{i}^c \\
& \alpha_{ji}\cdot {n_j} + I_{jk}\cdot{\alpha_{ki}\cdot {n_k}} \leq n_d,\ \forall\ B_j, B_k \in B_i \bigcup \mathcal{B}_{i}^c \\
& \sum_{B_j \in B_i \bigcup \mathcal{B}_{i}^c}{n_j} \leq n_d\cdot N_i^R
\end{split}
\label{eq:local_schedule_MILP}
\end{equation}
Since in the real system, $\mathcal{B}_i^c$ is a small set, $n_d$ is a pre-set small integer, the known interference relationship matrix $\{I_{jk}\}$ is sparse, the computation time for solving this mixed integer linear programming problem is very short. After solving the problem, the maximum $S_i$ is found, and correspondingly, we can calculate the minimum number of time slots $\widehat{n_i}$ to support the maximum achievable traffic demand for the physical link attached to $B_i$ within $L_i$. $\widehat{n_i}$ is the key parameter of the local schedule, which will be transmitted to $B_i$'s parent BS, as it indicates the maximum time slots that the parent BS shall allocate to the physical link within $L_i$. The parent BS could allocate more time slots to $L_i$; however, the extra traffic cannot be absorbed timely by $B_i$, as the total traffic on $L_i$ exceeds the maximum achievable traffic demand of $B_i$.
In the real deployment of the distributed algorithm, since the traffic demand of each small-cell BS
may be fluctuating within a small range in most of time or it changes gradually in a slow speed, the calculation of local schedule may fall into the trouble of tracking the demand change too sensitively, such that the final valid schedule may be changing too frequently as well. To address this issue, we modify the message exchange of local schedule that the BS $B_i$ only updates the value of $\widehat{n_i}$ to its parent BS when significant changes of $\widehat{n_i}$ happens. Specifically, $B_i$ keeps recording the
temporary $\widehat{n_i}$ value calculated in each subframe, and it applies a sliding window on the sequence of $\widehat{n_i}$ values to get the time-averaged value $\bar{\widehat{n_i}}$. When the difference between the current reporting $\widehat{n_i}$ value and the time-averaged value $\bar{\widehat{n_i}}$ is larger than a threshold of $T$ percentage, the reporting $\widehat{n_i}$ value is updated to $\bar{\widehat{n_i}}$; otherwise the reporting value does not change. In our simulations, we choose a relatively large threshold (i.e., 50\%), so that in a small backhaul network (e.g., 20 BSs), few small-cell BSs will experience sharp traffic demand change simultaneously.
\subsection{Determine the final valid schedule of a BS}
After a BS receives the queue, traffic and local schedule information from its child BSs, it will determine its schedule of a
future subframe based on its height in the topology, as mentioned in the above section on hand-shaking.
Different from the calculation of local schedule, the valid schedule is determined using not only the traffic demand information, but also the queue and local schedule information.
To realize the distributed algorithm, a non-leaf small-cell BS $B_i$ has to maintain queues
to store downlink and uplink packets, whose destinations are not $B_i$. We use $\mathcal{B}_i$
to denote the set of BSs located in the sub-tree rooted at $B_i$.
For each small-cell BS $B_k\in\mathcal{B}_i$, $B_i$ maintains a downlink queue and a
uplink queue for it, which temporarily store all packets with the same destination BS as $B_k$ and all
packets to be routed to the macro-cell BS $M$ with the same source BS as $B_k$, respectively.
At the beginning of a subframe on $B_i$, the total number of packets to $B_k$ in the corresponding
downlink queue is $q_{i,k}^D$; while the total number of packets from $B_k$ in the corresponding
uplink queue is $q_{i,k}^U$.
However, the set of queue lengths actually used in the final valid schedule calculation are
related to the routing, because we need to know the exact total number of uplink and downlink
packets that wait to be transmitted on each logical link $L_j$ between $B_i$ and $B_i$'s each
child BS $B_j$. Therefore, at BS $B_i$, the total number of downlink packets waiting to be
transmitted on $L_j$ is denoted as $Q_{i,j}^D$. Meanwhile, the uplink packets waiting to be
transmitted on $L_j$ are stored in the uplink queues at $B_j$, and the total number of them
is denoted as $Q_{j}^U$.
\begin{equation}
\begin{split}
Q_{i,j}^D = \sum_{B_k\in \mathcal{B}_j}{q_{i,k}^D}\\
Q_{j}^U = \sum_{B_k\in \mathcal{B}_j}{q_{j,k}^U}\\
\end{split}
\label{eq:queue_length}
\end{equation}
Based on Equation~\ref{eq:queue_length}, the total number of uplink and downlink packet waiting to be
transmitted on $L_j$ is $Q_j$, and
\begin{equation}
Q_j = Q_{i,j}^D + Q_{j}^U
\end{equation}
As mentioned in the ``handshaking" procedure, at the first time slot of each subframe, the child BS $B_j$
transmits the uplink queue information $Q_{j}^U$ to its parent BS $B_i$, so that $Q_j$ can be updated timely before the calculation of final schedule begins. Note that the queue maintenance on the macro-cell
BS $M$ and the leaf small-cell BS is similar to that on the small-cell BS, except that $M$ does not have uplink queues and leaf small-cell BSs do not have downlink queues. There is no parent BS of the macro-cell BS and no child BS of the leaf small-cell BSs. The process of generating the valid schedule
of a BS varies according to the type of that BS.
\subsubsection{Macro-cell BS}
To determine the final schedule at the macro-cell BS $M$, it has to first
find out the maximum number of packets it can transmit to each child BS
within a single subframe in the way local fairness is considered. Similar
to the procedure of calculating the local schedule, we first have to optimize
the local scaling variable $S_M$,
\begin{equation}
\begin{split}
\max \quad & S_M \\
\mathrm{s.t.} \quad & n_j \cdot r \geq S_M\cdot \min{\{Q_j,D_j\}},\ \forall\ B_j \in \mathcal{B}_{M}^c\\
& 0 \leq S_M \leq 1 \\
& 1 \leq {n_j} \leq \widehat{n_j}, \ \mathrm{if}\ Q_j > 0, \ \forall\ B_j \in \mathcal{B}_{M}^c \\
& n_j = 0, \ \mathrm{if}\ Q_j = 0, \ \forall\ B_j \in \mathcal{B}_M^c \\
& \alpha_{j}\cdot {n_j} + I_{jk}\cdot{\alpha_{k}\cdot {n_k}} \leq n_d,\ \forall\ B_j, B_k \in \mathcal{B}_{M}^c \\
& \sum_{B_j \in \mathcal{B}_{M}^c}{n_j} \leq n_d\cdot N_M^R
\end{split}
\label{eq:mbs_schedule}
\end{equation}
In Equation~\ref{eq:mbs_schedule}, the first constraint guarantees that
a logical link $L_j$ attached to the macro-cell BS will be allocated a
certain number of slots so that $S_M\cdot Q_j$ packets
in the queue of $L_j$ can be transmitted in the scheduled subframe,
where $Q_j$ is the total number of packets in both uplink and downlink
queues of $L_j$, while $\mathcal{B}_{M}^c$ is the set of child small-cell
BSs of the macro-cell BS $M$.
The third constraint indicates that for a logical link with non-empty
queue, at least 1 time slot will be assigned to it, and the macro-cell
BS will not assign more than $\widehat{n_j}$ to $L_j$, as this value
is from the local schedule of $B_j$, which is considered as the maximum
number of slots scheduled for $L_j$ expected by $B_j$.
The fourth constraint is the ``logical link interference constraint"
and the last constraint is the ``radio chain constraint", both explained
in~\cite{hu2022maximum}.
\subsubsection{Non-leaf small-cell BS}
As for a non-leaf small-cell BS $B_i$, its valid schedule of a subframe is calculated upon the received valid schedule of its parent BS, the received local schedule and uplink queue information of its child BSs, and its own downlink queue information. Similar to the case of macro-cell BS, the non-leaf small-cell BS also has to first maximize the local variable $S_i$ as well,
\begin{equation}
\begin{split}
\max \quad & S_i \\
\mathrm{s.t.} \quad & n_j \cdot r \geq S_i\cdot \min{\{Q_j,D_j\}},\ \forall\ B_j \in \mathcal{B}_{i}^c\\
& 0 \leq S_i \leq 1\\
& 1 \leq {n_j} \leq \widehat{n_j}, \ \mathrm{if}\ Q_j > 0, \ \forall\ B_j \in \mathcal{B}_{i}^c \\
& n_j = 0, \ \mathrm{if}\ Q_j = 0, \ \forall\ B_j \in \mathcal{B}_i^c \\
& I_{ik}\cdot{\alpha_{k}\cdot {n_k}} \leq n_d - \alpha_{i}\cdot {\widetilde{n_i}},\ \forall\ B_k \in \mathcal{B}_{i}^c \\
& \alpha_{j}\cdot {n_j} + I_{jk}\cdot{\alpha_{k}\cdot {n_k}} \leq n_d,\ \forall\ B_j, B_k \in \mathcal{B}_{i}^c \\
& \sum_{B_j \in \mathcal{B}_{i}^c}{n_j} \leq n_d\cdot N_M^R - \widetilde{n_i}
\end{split}
\label{eq:sbs_schedule}
\end{equation}
where $\widetilde{n_i}$ is the number of slots scheduled for the
physical link within $L_i$ attached to $B_i$, which is a known
information for $B_i$, because it is determined and sent to $B_i$
by its parent BS before the calculation occurs. All the constraints
are similar to the ones in the macro-cell BS case above, except that
the total number of available time slots should subtract the ones
(i.e., $\widetilde{n_i}$) already assigned to $L_i$.
\subsubsection{Leaf small-cell BS}
As leaf BSs do not have child BSs, their valid schedule is determined by their parent non-leaf BSs.
After obtaining the maximum $S_i$ or $S_M$, the number of slots assigned to a logical link $L_j$
attached to a BS $B_i$ or $M$ can be calculated. Based on the ratio between $Q_{i,j}^D$ and $Q_{j}^U$,
we can determine the number of slots for downlink and uplink traffic transmissions on $L_j$, respectively.
A similar scheduling procedure as the Algorithm in~\cite{hu2022maximum} can be used to actually schedule the set of slots assigned to each logical link within each subframe.
Moreover, we have to figure out how to fill packets into each slot when the schedule is executed. Based on the slot allocation, $B_i$ (i.e., the parent of $B_j$) can get the maximum numbers of downlink packets $N_{j}^{P,D}$ and $B_j$ can get the maximum number of uplink packets $N_{j}^{P,U}$ allowed to be transmitted on $L_i$ within a subframe. Then, $B_i$ selects the downlink packets from different queues proportionally according to $\{q_{i,k}^D\}$, and $B_j$ selects the uplink packets from different queues proportionally according to $\{q_{j,k}^U\}$.
\section{Numerical results and analysis}
\subsection{The throughput performance of distributed scheduling algorithm}
In this section, simulations are conducted to evaluate the throughput performance of the proposed distributed scheduling algorithm. In the simulations, the physical link data rate is set to 13.3 Gbps as discussed in section~\ref{sec:local_schedule}. There are in total 24 slots within a subframe which lasts for 0.1 ms, and the first two slots of each subframe is reserved to implement the operation of handshaking procedure described in section~\ref{sec:ch6-handshaking}. The rest 22 slots are data slots which can be used to transmit both uplink and downlink data traffic. All backhaul topologies used in the simulations are generated using a spanning-tree algorithm
introduced in the section 4.3 of~\cite{hu2019design}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{figures/distributed_throughput.pdf}
\caption{Throughput performance of distributed scheduling algorithm}
\label{fig:throughput_distributed}
\end{figure}
Figure~\ref{fig:throughput_distributed} shows the throughput performance of the distributed algorithm in different network settings (i.e., the interference condition and the number of radio chains available on each BS) under different traffic loads from 0.67 Gbps to 3.33 Gbps at each small-cell BS. Note that, all the simulated throughput values are averaged across 50 sets of individual simulations in each scenario. In each simulation, the ratio between the uplink and downlink input traffic demand of a small-cell BS is 1:2, and each small-cell BS has the same traffic demand. Each individual simulation runs for 1000 subframes. In Figure~\ref{fig:throughput_distributed}, we can see that the throughput performance of the ``MI-ER" scenario (i.e., interference-minimal and enough radio chains available) is much better than that of the ``LI-LR (2)" scenario (i.e., interference exists between a few pairs of logical links, and 2 radio chains on the macro-cell BS while 1 radio chain on each small-cell BS). This is because the maximum traffic demand achievable of each small-cell BS in the ``MI-ER" case is much higher than that of the ``LI-LR (2)" case. In fact, using the updated link capacity and $\{p_i^f\}$ values, we can get the updated maximum traffic demand of each small-cell BS in the ``MI-ER" case as 1.64 Gbps in average; while the value of the ``LI-LR (2)" case is 0.94 Gbps. As Figure~\ref{fig:throughput_distributed} shows, the maximum throughput values in the simulation are close to the calculated maximum traffic demand values in both scenarios, which means the distributed algorithm can schedule the transmissions efficiently in the mmWave backhaul networks.
\subsection{Enhance the aggregated throughput achieved using the distributed algorithm}
It is noticed that the determination of the final valid schedule on BSs strictly follows the idea of
scheduling the backhaul traffic proportionally according to the queue length of each flow bounded by
the traffic demand of each corresponding small-cell BS. Although it addresses the fairness issue, the
utilization of the network resource could be low when there exist several bottleneck routes in the
backhaul network. On those bottleneck routes, if the aggregated traffic demand is much larger than the
network capacity of the routes, it leads to very small values of the optimized scaling variable $S$.
As the local $S_M$ value is applied to all routes in the network at the macro-cell BS, a very small
$S_M$ will limit the amount of traffic transferred on those non-bottleneck routes, where plenty unused
network resource may exist.
To improve the low network resource utilization and increase the network aggregated throughput, we
add a step in the determination of the final valid schedule after the optimization of $S_M$ at the
macro-cell BS. In this new step, the optimized $\widehat{S_M}$ is used to bound smallest amount of
traffic demand of each small-cell BS that has to be serve in a subframe. In Equation~\ref{eq:mbs_schedule_modified}, we can see that the new optimization objective is to
maximize the number of data slots used in one subframe.
\begin{equation}
\begin{split}
\max \quad & \sum_{B_j\in \mathcal{B}_M^c}{n_j} \\
\mathrm{s.t.} \quad & n_j \cdot r \geq \widehat{S_M}\cdot \min{\{Q_j,D_j\}},\ \forall\ B_j \in \mathcal{B}_{M}^c\\
& 1 \leq {n_j} \leq \widehat{n_j}, \ \mathrm{if}\ Q_j > 0, \ \forall\ B_j \in \mathcal{B}_{M}^c \\
& n_j = 0, \ \mathrm{if}\ Q_j = 0, \ \forall\ B_j \in \mathcal{B}_M^c \\
& \alpha_{j}\cdot {n_j} + I_{jk}\cdot{\alpha_{k}\cdot {n_k}} \leq n_d,\ \forall\ B_j, B_k \in \mathcal{B}_{M}^c \\
& \sum_{B_j \in \mathcal{B}_{M}^c}{n_j} \leq n_d\cdot N_M^R
\end{split}
\label{eq:mbs_schedule_modified}
\end{equation}
We conduct simulations to compare the aggregated throughput achieved by using the modified distributed scheduling algorithm and the maximized aggregated traffic demand at the macro-cell BS that can be supported by the backhaul network, which is obtained in~\cite{hu2022maximum}. In our distributed scheduling simulations, the traffic demand of each small-cell BS is set as 3.33 Gbps, and the aggregated traffic demand surpasses the network capacity.
As we can see in Figure~\ref{fig:aggTD_dist_fair}, in both MIER and LILR(2) scenarios, the aggregated
throughput achieved using the modified algorithm is close to the maximized traffic demand that can be supported by the backhaul network. It shows that our proposed algorithm can schedule the backhaul traffic efficiently.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{figures/aggTD_dist_fair.pdf}
\caption{Comparison between the aggregated throughput achieved using the distributed algorithm and the maximized aggregated traffic demand}
\label{fig:aggTD_dist_fair}
\end{figure}
Moreover, as the input traffic demand is the same on each small-cell BS, we also calculate the Jain's fairness index of the set of throughput values of all small-cell BSs obtained in the distributed scheduling simulation. Figure~\ref{fig:fairIndex_dist_fair} shows that our algorithm can provide better fairness than the case in the theoretical analysis in~\cite{hu2022maximum} can. Because in~\cite{hu2022maximum}, the optimization does not address the fairness issue in the allocation of the extra to be supported traffic demand among small-cell BSs. In the solution to the optimization problem obtained in the theoretical analysis, usually the extra traffic demand is allocated to only a few BSs, which leads to an unfair situation. In contrary, using our distributed algorithm, the extra traffic demand is allocated locally fairly due to the procedure of determining final valid schedule at small-cell BSs.
\begin{figure}[h!]
\centering
\includegraphics[width=0.75\linewidth]{figures/fairIndex_dist_fair.pdf}
\caption{Comparison on the achieved fairness index values between applying the distributed algorithm and maximizing the aggregated traffic demand}
\label{fig:fairIndex_dist_fair}
\end{figure}
\subsection{Track the dynamic traffic demand of a small-cell BS}
In the practical backhaul system, the traffic demand of each small-cell BS may be fluctuating
within a small range in most of the time or it changes gradually with a slow speed; therefore,
during a relatively long period of time, only a few BSs are likely to experience sharp traffic
demand change. In this simulation, we aim to explore the feature of our proposed distributed
scheduling algorithm on tracking the dynamic traffic demand (with sharp demand change) of a
small-cell BS in the backhaul network. The topology used in the simulation is shown in
Figure~\ref{fig:backhaul_dynamic_traffic}, where the targeting BS is marked as $B^{*}$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\linewidth]{figures/network_topology_dynamic_traffic.pdf}
\caption{Backhaul topology (logical links) used in the simulation}
\label{fig:backhaul_dynamic_traffic}
\end{figure}
In the simulation, the traffic demand of all small-cell BSs is initialized as 0.67 Gbps for
downlink and 0.33 Gbps for uplink. When time arrives at the 250-th subframe, the downlink
traffic demand of $B^*$ is doubled to 1.34 Gbps; while its uplink traffic demand is unchanged
until the 400-th subframe. Starting from the 400-th subframe, the uplink traffic demand of
$B^*$ is doubled. The downlink and uplink traffic demands of $B^*$ are set back to their
initial values at the 600-th and 750-th subframe, respectively.
\begin{figure}[h!]
\centering
\includegraphics[width=0.8\linewidth]{figures/dynamic_traffic.pdf}
\caption{Dynamic throughput within each subframe at a BS}
\label{fig:dynamic_traffic}
\end{figure}
Figure~\ref{fig:dynamic_traffic} shows the dynamic throughput within each subframe of $B^*$.
From the figure, we can see that the proposed distributed scheduling algorithm can track the
sharp traffic demand change quickly and accurately. In fact, when the downlink traffic
demand doubles, it takes 8 subframes for the instantaneous downlink throughput to jump to the
new stage; while it takes 13 subframes for the instantaneous downlink throughput to drop back
to the old stage, after the downlink traffic demand sets back to the original value.
Correspondingly, the two values of number of subframes used to track the uplink traffic demand
changes are 4 and 8. The selected BS has a height of 4 in the topology. In the downlink traffic
demand change case, the traffic demand change has to first propagate to the macro-cell BS to
increase the ``desired" time slots for the logical links in the route. After that, more packets
can be transferred to the targeting BS. However, in the uplink case, the increased uplink
traffic flows together with the ``traffic increasing" message to the macro-cell BS, which is
a one-way trip intuitively. When the traffic demand drops, it takes a bit longer time for the throughput to drop, because the packets from the heavy traffic queued in the intermediate BSs need time to be absorbed by $B^*$.
\section{Conclusions}
In this paper, a novel distributed scheduling algorithm is created, which aims to efficiently schedule both the uplink and downlink backhaul traffic in the mmWave backhaul network with a tree topology. The handshaking of control messages, calculation of local schedules, and the determination of final valid schedule are all discussed. Simulation results show that the performance of the distributed algorithm can reach very close to the aforementioned maximum traffic demand of the backhaul network, and it can also adapt to the dynamic traffic with sharp traffic demand change of small-cell BSs quickly and accurately.
\bibliographystyle{ieeetr}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,575
|
Author: Raff Schreiner
Downtown Music Publishing's AVL Digital Group has acquired FUGA, an Amsterdam-based digital-music distributor and service provider.
The purchase, which was announced publicly this morning (January 22nd), is the latest in a series of aggressive, growth-minded investments from Downtown.
In March of 2019, the company finalized a $200 million deal and assumed ownership of AVL Digital Group, which, through its
subsidiaries—Sounddrop, AdRev, DashGo, and CD Baby, one of the largest digital distributors of independent music—helps independent artists monetize their work.
Per the deal's terms, FUGA will continue to operate independently.
Factoring for the Dutch company's 100 or so team members, Downtown now has more than 500 employees working on its behalf, as well as offices in New York, Los Angeles, Nashville, Amsterdam (separate from FUGA), and London.
Founded by Justin Kalifowitz in 2017, Downtown has quickly risen to the top of the artist-rights and royalty-collection sphere. The brand's ability to attract (and maintain) high-profile acts is without doubt; The Beatles, One Direction, Bruce Springsteen, and other extremely popular artists are part of the Downtown banner.
By acquiring FUGA, which will complement and benefit AVL's platforms, Downtown is signaling that its long-term goals aren't limited to North America—or even Europe.
FUGA debuted in 2014, as a small, ambitious startup with big ideas, limited funding, and only 12 employees. In less than half a decade, FUGA's team size shot up, and the company opened its own international offices. CEO Pieter van Rijn has long spoken of his intention to establish a foothold in the Asian and African markets, and doing so will be easier with the contacts and supplemental resources of Downtown at his disposal.
The financial terms of the Downtown-FUGA deal were not disclosed.
Author Raff SchreinerPosted on January 22, 2020 Categories MUSICTags ad, baby, eve, morning, Music, new, No, pop
Amazon Claims More Than 55 Million Music Streaming Users
Amazon says its Amazon Music streaming service has surpassed 55 million customers. There are six different pricing tiers for its streaming service, including a free ad-supported tier.
Amazon Music first launched in October 2016, but it still lags behind and . Just last summer, Apple Music announced 60 million subscribers. Last quarter, Spotify confirmed it has around 113 million paying subscribers and 248 million users globally.
There is no drill-down for how many users are in each of the Amazon Music tiers. Amazon does reveal that the growth of Amazon Music Unlimited is up 50% over the last year.
Amazon Music Unlimited is available in the United States, UK, Germany, Japan, and Brazil with HD-quality. France, Italy, Spain, and Mexico were also significant contributors to growth, though HD-quality music is not available yet.
Amazon's subscription tiers encourage Prime members to give its services a try. Amazon's dominance in the smart speaker market means there's an Amazon Music tier just for its Alexa devices. Prime users also get a discount on the Amazon Music Unlimited service, undercutting rivals.
Spotify is focusing on music and podcast exclusives to help differentiate. But during its last earnings call, Spotify gave some insights into Amazon's growth.
Amazon's music streaming business is picking up more ad-supported users than paying subscribers, according to Spotify.
"Our data also suggests that Amazon's user base skews significantly more to 'Ad-Supported' than 'Premium,' and that average engagement on our platform is approximately 3x," Spotify said during its .
Spotify says that Amazon's ad-supported growth leaves the company open to continue growing its paid streaming service. Spotify needs to be careful here, though, Amazon's ecosystem is attractive to people who want all services under one provider.
Amazon has its fingers in storage, video, reading, games, and now music. If Spotify isn't careful to diversify beyond just music, larger tech companies like Apple and Amazon could surpass it.
Author Raff SchreinerPosted on January 22, 2020 Categories MUSICTags ad, apple, blog, Music, No, podcast, summer, tech
Safaree & Erica Mena Get Roasted On Twitter After Tying The Knot On "L&HH"
On this week's episode Love & Hip Hop: New York, cast members Safaree Samuels and Erica Mena ficially tied the knot, but some Twitter users did not prove to be their biggest supporters. After Safaree proposed to Erica on Christmas Day in 2018, they got married the following year in October, a few months before they welcomed their first child together. Since then, the notorious couple have been causing some major drama on the show since the series' 10th season premiered in December. On Monday night, the world finally got to witness their wedding ceremony take place on their TV screens.
While Safaree Samuels and Erica Mena may have said "I do," the Internet highkey said "we don't." Plenty viewers were critical their union, pointing out how performative and fake their marriage looks considering how unhappy they've appeared to be together thus far.
The lack substance in their relationship have many feeling that the marriage is doomed and won't last long.
Still, some were moved by the ceremony and got emotional, expressing nothing but well wishes for the couple.
Author Raff SchreinerPosted on January 22, 2020 Categories C2C NewsTags christmas, happy, hip hop, Hiphop, love, News, No, party
Ja Morant Shows Off Early Look At The Nike Adapt BB 2.0
Ja Morant is one the most entertaining young stars in the entire NBA right now and continues to dominate on the fensive side the ball. He is currently under contract with Nike which makes him the perfect candidate to show f new models. A couple weeks ago, we reported on the Nike Adapt BB 2.0 and how some people overseas were getting their hands on some early pairs. Now, Morant is giving fans an ficial look at what this sneaker will resemble.
As you can see, the upper is black with a silver swoosh on the upper. From there, black and red are placed on the midsole. Overall, it's an interesting look that matches the futuristic vibe. For those who don't know, the Nike Adapt BB 2.0 features auto-lacing technology that can be controlled through an app. The original Nike Adapt BB was quite successful and we can only imagine what sort technological advances were made with this latest version.
For now, these are slated to drop on February 16th 2020 for $400 USD. In the meantime, let us know if you've tried auto-lacing sneakers yet and whether or not you enjoyed your experience.
Author Raff SchreinerPosted on January 22, 2020 Categories C2C NewsTags ad, Features, Hiphop, instagram, nba, News, No, tech
Nike Introduces New Sneaker: The Air Max 2090
Nike is celebrating the 30th anniversary the iconic Air Max 90 throughout 2020 with special edition colorways ranging from all-time classic to never before seen styles. As a further nod to the AM90, Nike will introduce an all-new silhouette, the Air Max 2090, billed as the company's current shoe the future.
Leveraging the same DNA that made the Air Max 90 so iconic and future-forward 30 years ago, the Air Max 2090 is designed to illustrate what the future holds. According to Nike, the kicks utilize colors typically seen in electric cars, and the overall design the AM2090 represents what Nike designers think cars will look like in 2090.
Other notable details include (H/T Nike):
Air Unit: With the aim all-day comfort, the Air Max 2090 employs a larger Air unit than its originator, with a 200 percent larger window than traditional side-visibility Air units seen 30 years ago.
Flexibility: Echoing the Air Max 90's waffle outsole, the Air Max 2090 also employs distinct grooves for flexibility and takes original tread lines and updates the pattern for performance in 2020.
Color Blocking: Color blocking was instrumental in the development the original Air Max 90, with darker colors underscoring the strength in the shoe's support structure. In that same vein, the 2090 relays this through levels opacity — the most supportive materials are 100 percent opaque, and the least supportive (and lighter weight) materials are see-through.
The Air Max 2090 will be launching at retailers this Spring. Continue scrolling for ficial images two upcoming colorways.
Author Raff SchreinerPosted on January 22, 2020 Categories C2C NewsTags ad, eve, new, News, nike, No, Sneakers, Twitter
Buffalo Wild Wings Announces 'Free Wings' Promotion For Super Bowl LIV
Buffalo Wild Wings is running back their overtime promotion for Super Bowl LIV, promising free wings for the whole country if the Kansas City Chiefs and San Francisco 49ers go to OT on February 2nd.
"Football fans love to have fun with our established connection to overtime, tweeting at us any time a game goes to extra time. That was taken to the next level last year when free wings were on the line, so we knew we had to bring this program back," said Seth Freeman, Chief Marketing Officer, Buffalo Wild Wings. "More football AND free wings? Now that's something all football fans can get behind and root for on February 2!"
According to BetMGM, the moneyline for the game going into overtime opened at +875, meaning a $100 bet would net $975 (including stake) if the game goes to OT. In other words, Vegas sportsbooks believe there is roughly a 10% chance winning free wings during Super Bowl LIV.
If the Chiefs and 49ers need an extra session to settle the score on February 2nd, fans can pickup a free order boneless or traditional snack-sized or 5/6 count chicken wings on February 17 between 4-7pm local time. Worth noting: No purchase is necessary to redeem the free wings, but the promotion is available for dine-in orders only.
Author Raff SchreinerPosted on January 22, 2020 Categories C2C NewsTags ad, eve, Football, love, new, No, Sports, Twitter
She ten tells people to mind their businesses when it comes to her personal life, but Apryl Jones has no problem sharing details about her intimate moments with the public. Her relationship with Fizz has been scrutinized for nearly a year after rumors circulated that she was dating her baby daddy Omarion's B2K groupmate. She and Fizz would later flaunt their romance for Love & Hip Hop Hollywood cameras, further placing them both at the forefront public criticism.
Emma McIntyre / Staff / Getty s
However, that's not the only topic that has caused Apryl to become a villain sorts. In a chat with Tiffany "New York" Pollard for VH1's Brunch With Tiffany webseries, Apryl also discussed the viral video that circulated where she explicitly named f celebrities she says wanted to sleep with her. Those men included including Kevin Durant, The Game, A$AP Rocky, Shaquille O'Neal, Paul Pierce, Shiggy Show, and last but certainly not least, Fizz.
Apryl explained that she was being accused sleeping with multiple men so she just listed their names in jest. "At this point, it's like, you're right, I f*cked every single one them because that's what you want to hear, so that's what they captured," she told Tiffany. "Not that I have had with those men]. I would have no problem when, sh*t, I should have. I mean, I should have f*cked The Game when I had the opportunity." She added that she never bedded any the other men she mentioned.
Elsewhere, Apryl spoke on co-parenting with Omarion and how it gets more difficult as time progresses. They haven't been in a relationship in over three years and she states the primary caregiver to their two children. "It's not been easy, that's the truth," she said. "I'm trying to figure it out on a daily. I'm prayed up that it's gonna get better, but I don't know."
Watch Apryl's Brunch With Tiffany episode below where she talks quite a bit about…well, …and check out her controversial viral video as well.
Author Raff SchreinerPosted on January 22, 2020 Categories C2C NewsTags hea, hip hop, Hiphop, love, No, personal, rock, sleep
CMT Pledges To Give 50% of All Video Plays to Female Artists
CMT says it will evenly split music video airtime between male and female artists. The former split favored male artists 40/60%.
CMT says it is stepping up to help ensure gender parity in the country music genre. A study released last year showcased what many female country stars already know – country music is a male-dominated genre.
The new policy will be standard on CMT and CMT Music channels. Male and female artists will split all 29-hour primetime video blocks equally.
Female country music star representation is something of a bugbear in the industry. Martina McBride brought the issue front and center with her last year. Now political comedian Samantha Bee is talking about the issue.
Last year, a study found that most country music stations are told to remove women. Some choice phrases in that study include, "women are not financially ble" and "country radio is a principally male format." The study also found that in 2000, women held 33.3% of songs on the year-end country airplay reports. By 2018, that number had dwindled to just 11.3%, or a decline of 66%.
That study is one of the first to use radio airplay data to investigate gender imbalances in country radio. The results not only confirm the gender imbalance but show its more than just listener preference behind country's male-dominated roster.
Unbalanced programming choices favoring men helps drive what seems like a preference for male artists. CMT is rising to the challenge of creating more opportunities for female country stars to create new fans.
The report released last year asked associations like CMA to set new standards for inclusion. Bringing attention to the programming imbalance means it can be corrected in the long run. Will the broader country music industry sit up and take notice, though? Decades of programming bias has enforced the belief that women aren't profitable.
That view isn't limited to country music either; the video game industry also dealt with issues of female representation. Back in 2012, games with male characters sold 75% better than those with only a female hero. Unsurprisingly, both country music and the video game industry are heavily male-dominant industries.
Author Raff SchreinerPosted on January 22, 2020 Categories MUSICTags ad, hea, Music, Music Video, nba, No, reports, study
Air Jordan 34 "Paris" Will Be The Most Limited Colorway Yet: Release Info
Jordan Brand is celebrating the NBA's first-ever regular season game in Paris with the release a super exclusive Air Jordan 34 this week.
The colorful kicks, limited to just 500 pairs worldwide, will reportedly release exclusively Nike SNKRS on Friday, January 24th, just before the Charlotte Hornets vs Milwaukee Bucks game at the AccorHotels Arena in Paris.
Each Air Jordan 34 features a unique heel logo and this particular colorway features an iridescent Jumpman logo on the left shoe, opposite Paris' '75' area code on the the right. The Air Jordan 34, billed as one the lightest basketball shoes ever created by Jordan Brand (size 9 weighs just 13.1 oz), includes the following features:
An Eclipse Plate, formed by two Pebax pieces, which helps reduce the overall stiffness and strips down the weight
A forefoot Zoom Air unit, with herringbone traction pattern – providing optimal explosion f the foot
Limited Upper – heel counter, tongue, eyestay and midfoot reinforcer – showcasing the purest form a basketball shoe
The number 23 is featured in Morse code on the vamp
Continue scrolling for ficial images the "Paris" colorway, and click here to preview the "MLK" Air Jordan 34 debuted by Zion Williamson.
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags basketball, Features, Hiphop, instagram, nba, News, No, Sneakers
Nipsey Hussle's 2020 Grammys Tribute Includes Performances By YG, Meek Mill & More
Los Angeles, CA – The 2020 Grammy Awards will include a special tribute to the late Nipsey Hussle. On Tuesday (January 21), Grammys' executive producer Ken Ehrlich revealed the tribute will entail performances by several of Nipsey's close friends and collaborators, including YG, Meek Mill, Roddy Rich, DJ Khaled, John Legend and Kirk Franklin.
"An activist, entrepreneur and rapper, Nipsey Hussle had a lasting impact on not just his community, but also the culture at large," Ehrlich said in a press release. "There is no denying the influence he had and his legacy will be felt for generations to come.
Th"We are honored to bring together this amazing group of artists to celebrate Nipsey's life and pay tribute to his many contributions to music. It's sure to be a memorable performance."
The Grammys will be broadcast live from the Los Angeles Staples Center on Sunday (January 26) with Alicia Keys serving as host. Additional performances include Tyler, The Creator, H.E.R. and Aerosmith with Run-DMC.
Nipsey earned a posthumous nomination in the Best Rap Performance and Best Rap Song categories for "Racks in the Middle" with Hit-Boy and Roddy Ricch. The John Legend and DJ Khaled collab "Higher" featuring Neighborhood Nip is also nominated for Best Rap/Sung Performance.
Nip was shot and killed in front of his Marathon Clothing store on March 31, 2019. He was just 33 years old.
Author Raff SchreinerPosted on January 21, 2020 Categories C2C News, MUSICTags ad, Hiphop, marathon, Meek Mill, Music, No, Rap, Tyler, The Creator
Pamela Anderson Secretly Marries 5th Husband Jon Peters In Malibu
Pamela Anderson and her ex-boyfriend Jon Peters have come back into each other's lives after they first dated and broke up 30 years ago. According to The Hollywood Reporter, Pamela and Jon secretly got married on January 20th in Malibu and the wedding marks their fifth time walking down the aisle for both them.
Pascal Le Segretain/Getty s
"Pamela has never seen her full potential as an artist. She has yet to shine in a real way. There is much more to her than meets the eye, or I wouldn't love her so much," Jon told the publication. "There are beautiful girls everywhere. I could have my pick, but — for 35 years — I've only wanted Pamela. She makes me wild — in a good way. She inspires me. I protect her and treat her the way she deserves to be treated." Jon is a famed film producer known for A Star Is Born (the original and the latest rendition) as well as The Witches Eastwick, Caddyshack and more.
Jon first met Pamela when she was just 19-years-old at the Playboy mansion. "I walked in and saw this little angel sitting at the bar. It was Pammy. She was like 19. I knew she would be a big star. We ended up living together. Of course, Hefner loved her and saw her beauty," Jon previously explained.
Pamela's famous exes have been Tommy Lee, Adil Rami, Rick Salomon and Kid Rock.
Vince Bucci/Getty s
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags ad, eve, husband, love, new, News, No, rock
T.I. Speaks On Honoring Nicki Minaj & Cardi B With Trap Music Museum Exhibit
Last week, we reported that the T.I.-founded Trap Music Museum in Atlanta would be unveiling an exhibit dedicated to Nicki Minaj and Cardi B. Despite Nicki and Cardi beefing in the past, paintings and other art pieces honoring the two artists' contributions to hip hop are placed side-by-side in this colorful installation. A dollhouse references Nicki's 2012 hit, "Beez In The Trap", and a stripper pole on a checkered platform references the aesthetic Cardi's debut album and her previous pression. T.I. spoke on how the lack women included in the museum since its 2018 inception contributed to the impetus behind this tribute.
"Ever since the Trap Music Museum opened, one the most common questions posed to us was, 'Why aren't there any exhibits showcasing women?' We were always committed to honoring women who contributed to the culture, but finally found a way to truly elevate and celebrate two the best," says T.I., per AllHipHop. "The Nicki Minaj and Cardi B exhibits lift two the greatest to their rightful place within the Trap Muzik Universe."
While the exhibit had its grand opening at the Trap Music Museum this week, a glimpse it was given at a pop-up in Los Angeles in July. T.I. faced some backlash for sanctioning one particular piece: a throne that was painted half pink and half checkered. The obvious symbolism is that Nicki and Cardi share the title being the "Queen Rap". While many argue that this a tired debate that unnecessarily pits women against one another, T.I. received complaints that Cardi's entrance in the game was too fresh to place her on the same plane as Nicki. He responded to this controversy in a 2019 interview: "Everybody was like], 'How could you? Cardi] hasn't been rapping long enough! She ain't put out enough! You're being disrespectful to Nicki!' When does celebrating two queens become disrespectful to one? I never understood that. This is an acknowledgment. Both these young ladies have struggled and overcame obstacles and defied the odds to put themselves in positions to represent themselves] and the next generation."
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags hip hop, Hiphop, Music, News, No, pop, Queen, Rap
Conor McGregor Claims Donald Trump Might Be The GOAT Of The USA
Twenty-twenty is a big year for politics in the United States America. Later this year, the Democratic party will have chosen their candidate who will go to battle against President Donald Trump in the general election. It remains to be seen how the election will go down, even with all the criticism surrounding Trump. While many speak out against the president, there are still plenty people who are ride or die for him. As it turns out, one those people is none other than UFC star Conor McGregor.
Yesterday, Trump celebrated Martin Luther King Jr. Day by talking about himself and his accomplishments as President. That's when McGregor jumped into the comments section to express his love for the 45th president the United States.
"Phenomenal President. Quite possibly the USA 🐐," McGregor wrote. "Most certainly one them anyway, as he sits atop the shoulders many amazing giants that came before him. No easy feet. Early stages term also. Incredible. Congrats and Happy Martin Luther King Jr. day America."
McGregor's comments were met with a mix support and strong disagreement. There is no denying both McGregor and Trump are polarizing people. Perhaps that's why they attract each other. Either way, we're sure some McGregor fans won't be too impressed with these comments.
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags ad, Conor McGregor, happy, Hiphop, love, No, party, Twitter
LeBron James Cops Diamond Lion Doorknocker Pendant With Red Lit Eyes
He's a modest, hard-working family man who is one the most respected basketball players on and f the court. Yet, just because he likes to be humble doesn't mean that LeBron James doesn't drop a few stacks now and then on some serious ice. Iceman Nick is well known in celebrity circles for creating one–a-kind pieces that can catch your attention from across a room, and TMZ reports that he recently helped craft a standout pendant for Bron.
Joe Scarnici / Stringer / Getty s
The Los Angeles Lakers star reportedly commissioned Iceman Nick for a large "Lion Gatekeeper" pendant that boasts "130 grams 14k gold, encrusted with 35 carats VS1 diamonds." Bron is a fan the King the Jungle because he's referenced lions in campaigns, logos, and apparel.
The crowned "Lion Gatekeeper" is also a fully functioning doorknocker with blazing red eyes that light up. TMZ didn't share how much this piece set Lebron back, but we're sure he spared no expense. Other clients Iceman Nick include Rap-a-Lot Records, Carmelo Anthony, Nelly, Lonzo Ball, Boosie Badazz, Megan Thee Stallion, Moneybagg Yo, Lil Baby, and many more. Check out Bron's new bling below along with a few other celebrity pieces Iceman Nick has crafted.
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags ad, Boosie Badazz, Hiphop, News, No, Rap, reports, tmz
LeBron James Praises Bronny James Jr. After Candy Was Thrown At Son During Game
When your father is arguably one the best NBA players to ever step in the paint, it's going to be difficult to follow in his footsteps. Nevertheless, Bronny James Jr. is carving out his own lane in the basketball world as the Sierra Canyon School sports star carries on the James name. LeBron is front and center at Bronny's games when he can make it, but even with the championship basketball player in attendance, his son is a target for abusive behavior.
On Monday (January 20) Sierra Canyon went up against Paul VI Catholic High School from Virginia in front a packed auditorium 4,000 people. LeBron reportedly entered the room to roaring applause, but even with all the support, someone thought it was appropriate to throw something at Bronny during the game.
According to Yahoo Sports, it was near the beginning the second half when someone in the stands tossed a piece candy at Bronny, causing the referee to interrupt gameplay. As LeBron shook his head in disgust, security escorted the person out the auditorium. The young James's teammate Ziaire Willaims told Yahoo Sports, "You'd be surprised by all the stuff Bronny has to go through. It's not fair, but he doesn't let it faze him at all. I'm learning how to be more like that from him and he's younger than me."
Later, LeBron took to Instagram to publicly declare how proud he is his son. "Know I got your back and front throughout it all! I'd die for you and your siblings," the Los Angeles Lakers star wrote. "Never waver no matter what disrespect, hate, envy, jealousy, etc comes your way! Only push forward and continue to be the great kid/young man/person you are! Love you kid!! ❤️ #JamesGang👑."
Author Raff SchreinerPosted on January 21, 2020 Categories C2C NewsTags ad, dance, hea, Hiphop, love, News, No, Sports
Eminem's "Music To Be Murdered By" Is Destined To Divide
Rihanna Can't Contain Herself While Giving Unintentionally Raunchy Makeup Tutorial
Air Jordan 1 High '85 "Varsity Red" Release Date Revealed: Stock Numbers
Lil Nas X Feels "Pressure" To Be A Role Model In LGBTQ Community
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,373
|
<?php
/**
* @namespace
*/
namespace Solarium\QueryType\Schema\Query\Command;
use Solarium\Core\ArrayableInterface;
use Solarium\Exception\RuntimeException;
use Solarium\QueryType\Schema\Query\Field\CopyField;
use Solarium\QueryType\Schema\Query\Query as SchemaQuery;
/**
* Class AddCopyField
* @author Beno!t POLASZEK
*/
class AddCopyField extends Command implements ArrayableInterface {
/**
* @var CopyField[]
*/
protected $fields = array();
/**
* Returns command type, for use in adapters
*
* @return string
*/
public function getType() {
return SchemaQuery::COMMAND_ADD_COPY_FIELD;
}
/**
* @return CopyField[]
*/
public function getFields() {
return $this->fields;
}
/**
* @param CopyField[] $fields
* @return $this - Provides Fluent Interface
*/
public function setFields(array $fields) {
$this->fields = array();
$this->addFields($fields);
return $this;
}
/**
* @param CopyField[] $fields
* @return $this - Provides Fluent Interface
*/
public function addFields(array $fields) {
foreach ($fields AS $field)
is_array($field) ? $this->createField($field) : $this->addField($field);
return $this;
}
/**
* @param CopyField $field
* @return $this
*/
public function addField(CopyField $field) {
if (!array_key_exists((string) $field, $this->getFields()))
$this->fields[(string) $field] = $field;
return $this;
}
/**
* @param array $attributes
* @return $this
*/
public function createField(array $attributes = array()) {
if (!array_key_exists('source', $attributes))
throw new RuntimeException("A copyField must have a source attribute.");
if (!array_key_exists('dest', $attributes))
throw new RuntimeException("A copyField must have a dest attribute.");
$copyField = new CopyField($attributes['source'], $attributes['dest'], isset($attributes['maxChars']) ? $attributes['maxChars'] : null);
$this->addField($copyField);
return $copyField;
}
/**
* @return array
*/
public function castAsArray() {
return array_values(array_map(function (CopyField $copyField) {
return $copyField->castAsArray();
}, $this->getFields()));
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,593
|
Until a whirlwind of NBA trades happened at the end of the week, it was another snoozer in the sports world.
But as always, there were those who impressed us and those who drove drunk in a Porsche with only three wheels.
And of course, one of our favorite couples won this week, basically for being awesome.
Pretty much every time a disgruntled superstar demands a trade, the fans turn against him.
Somehow, Chris Paul has avoided such scorn.
The NBA has bungled the Lakers-Hornet-Rockets trade so badly that the guy who wants to leave tiny New Orleans for Los Angeles comes out looking like the good guy.
$254 million over 10 years.
Shaq is just weeks away from returning to the national spotlight with his gig on TNT.
Hoopz, his tiny fiance, performed admirably on Jimmy Kimmel this week when she gave him a piggy-back ride.
Not only is Tim America's favorite dual-threat QB, he's also filthy rich.
He triggered a clause in his contract that gives him a $470,000 bonus at the end of the season.
Big Ben early snapped his leg in half in the first half against the Browns on Thursday night.
Yet somehow he came back in the second half and led the Steelers to a 14-3 win.
David Stern pulled a fast one on Chris Paul and said he couldn't be traded to the Lakers, after originally saying it was ok.
"We were all told by the league he was a trade-able player, and now they're saying that [Hornets general manager] Dell [Demps] doesn't have the authority to make the trade? Now, they're saying that Dell is an idiot, that he can't do it his job. [Expletive] this whole thing. David's drunk on power, and he doesn't give a [expletive] about the players, and he doesn't give a [expletive] about the hundreds of hours the teams put in to make that deal."
"I'm going to put my hands on you worse than that dude on those kids at Penn State."
After Nick Saban put Stanford at no.3 and Oklahoma State at no.4, Alabama snuck into the national championship game.
A lot of people think Oklahoma State was much more deserving.
Dottie Sandusky came out in support of her husband and said all of the alleged child molestation charges were not true.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,727
|
Baroness - Gold & Grey (2019)
Style: Progressive Rock
I started out my music reviews here in 2019 by alternating a new album with one from the previous year and I plan to continue in that vein ths January. Throughout this month, I'll be reviewing a new album alongside one from 2019 that I missed out on and has probably received strong critical acclaim. I'm kicking off with the new Baroness album, released last June, because that's been especially well received by critics and recommended to me personally as well.
I hadn't previously heard Baroness but they're a band from Savannah who have apparently evolved their sound over the years from sludge metal to a sort of alternative prog rock that still carries a metal crunch at points but sounds really big in other ways. It's how the songs are put together as much as how they're played, but it's as hard to imagine Baroness on a tiny club stage as it is to imagine Pink Floyd in anything but a vast stadium.
What felt odd to me was how they were at once inviting but deep. On a first listen, I found the songs trying to grab me but failing because they're not that immediate. Prog isn't always immediate, of course, but a more obscure prog album can often be a challenge, sounding awful or even ridiculous until we get on the right wavelength and suddenly realise what genius it is. This isn't that sort of challenging, because it's friendly hook-laden stuff from moment one but there are a lot of other things going on, on multiple layers, so the challenge is to move back far enough to see the big picture.
While the first track is in your face with its highly prominent bass and an odd fuzziness that feels uncomfortably close to static, the majority of the album is introspective, its songs constructed with care. Never mind sludge roots, the 2019 Baroness makes me think of an alternative rock band with a fondness for riffs mainlining on the Beatles until they end up as a combo of Radiohead and the Foo Fighters. These are big songs but they try not to be and, even four or five times through, I still think of this as an hour long piece of music rather than a collection of twelve songs and five interludes.
I first heard about Gold & Grey when a friend mentioned it after I reviewed the new Opeth album. I can see a lot of similarities in something this prog rock being released by a band who didn't start as prog rock in the slightest, but beyond the odd changes and contrary layers that link them, this is often livelier and more bombastic. Sure, there are inwardly looking numbers like Emmett - Radiating Light, but the power chords of Tourniquet just have to be accompanied by big light shows on a huge stage.
What impresses most is that they can follow a song like that with the calm, almost plainsong vocals and soothing strings of Anchor's Lament without us feeling jarred. Even the wild and experimental opening to Throw Me an Anchor doesn't seem abrasive, because by that point we've fully come to terms with the fact that Baroness play from as wide a colour palette as John Baizley in his role as cover artist. He's also the lead vocalist and guitarist and sole surviving founder member.
I have a feeling that I'll be discovering more new things in this album by a fiftieth listen. It's fundamentally rock music, but there's just as much pop in play as there is metal nowadays. I like that Cold-Blooded Angels can seem to be the former only to grow seamlessly into the latter. It even throws in some space rock sound effects for good measure. Other songs feature nods to folk, experimental, jazz, psych, ambient and even new wave. Broken Halo is punk at heart until it launches into a keyboard solo. Can Oscura and Assault on East Falls would feel at home on a krautrock album. Many songs highlight a Pink Floyd influence but, at one point, Borderlines suddenly becomes Thin Lizzy.
While my friend prefers this to the new Opeth album, I'd go the other way. I liked this a lot and expect to listen to it a lot too, but it didn't feel as consistent for me and that periodic fuzz turned me off every time it showed up. I like fuzz, I should add; I just don't like this fuzz. I'm now eager to travel backwards to see what Baroness used to sound like and there are four prior colour-themed albums for me to track down. Thanks, Mike!
By Hal C. F. Astell at January 06, 2020
Labels: 2019, progressive rock, USA
Dawn of Solace - Waves (2020)
The Wildhearts - Renaissance Men (2019)
Mortiis - Spirit of Rebellion (2020)
Wilderun - Veil of Imagination (2019)
Nektar - The Other Side (2020)
Algebra - Pulse? (2019)
Midnight - Rebirth by Blasphemy (2020)
Earth - Full Upon Her Burning Lips (2019)
ADX - Bestial (2020)
Revival Black - Step in Line (2019)
Decimator - Alienist (2020)
Diagonal - Arc (2019)
Val Tvoar - Today is Tomorrow's Yesterday (2020)
Smoulder - Times of Obscene Evil and Wild Daring (...
British Lion - The Burning (2020)
Spirit Adrift - Divided by Darkness (2019)
Magnum - The Serpent Rings (2020)
Inculter - Fatal Visions (2019)
Adrian Tăbăcaru - Lucifer: A Rock Opera (2020)
Waste of Space Orchestra - Syntheosis (2019)
Tortuga - Deities (2020)
Tom Keifer Band - Rise (2019)
Stabbing Westward - Dead & Gone (2020)
Avalanch - El secreto (2020)
Armadura - Nuevas tierras (2020)
Wolfmother - Rock 'n' Roll Baby (2019)
Rage - Wings of Rage (2020)
Lingua Ignota - Caligula (2019)
Andy's Game - Andy's Game (2020)
The Drowning - The Radiant Dark (2019)
Motorbooze - Motorbooze (2020)
White Ward - Love Exchange Failure (2019)
Desert Colossus - Eyes and Tongues (2020)
Blood Incantation - Hidden History of the Human Ra...
Burden of Ymir - Jötnar (2020)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,798
|
Q: Ajax queries in webview My problem is that I do not know if there is any listener that checks if ajax requests in my webview have finished. On page that I have opened in webview (it is not my page, so I cannot modify scripts) there are some ajax handlers that modify the HTML of the page. For example after clicking a link window titled "Loading..." shows up, and after a short moment this window changes title and fills itself with the result of ajax query. Is there any way to find out that page content has changed? I hope that there is a better way than to start new Thread that checks every few seconds if there is string "Loading..." in page source...
A: When Ajax sending request it receives success or error find it in your script.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,025
|
\section{Examples}
\label{sec.examples}
\subsection{Clemens complexes of toric varieties}
Let $T$ be an algebraic torus over a field~$F$.
Let $M$ and~$N$ denote the groups of characters and of cocharacters
of the torus~$T_{\bar F}$, endowed with the action of
the Galois group~$\Gamma=\Gal(\bar F/F)$
and the natural duality pairing $\langle\cdot,\cdot\rangle$.
Let $X$ be a smooth proper $F$-scheme
which is an equivariant compactification of~$T$.
By the theory of toric varieties, $X$ corresponds
to a fan~$\Sigma$ in the space~$N_{\mathbf R}$ which is invariant
under the action of~$\Gamma$. By definition, $\Sigma$
is a set of convex polyhedral rational cones in~$N_{\mathbf R}$
satisfying the following properties:
\begin{itemize}
\item for any cones $\sigma,\sigma'$ in~$\Sigma$, their intersection
is a face of both~$\sigma$ and~$\sigma'$;
\item all faces of each cone in~$\Sigma$ belong to~$\Sigma$;
\item for any $\gamma\in\Gamma_F$ and any cone~$\sigma\in\Sigma$,
$\gamma(\sigma)\in\Sigma$.
\end{itemize}
Assume for the moment that~$T$
is split, meaning that $\Gamma$ acts trivially on~$M$.
To each cone~$\sigma$ of~$\Sigma$ corresponds
a $T$-stable affine open subset $X_\sigma=\Spec F[\sigma^\vee \cap M]$
in~$X$,
where $\sigma^\vee$ is the cone in~$M_{\mathbf R}$ dual to~$\sigma$.
The variety~$X$ is glued from these affine charts~$X_\sigma$,
along the natural open immersions $X_\sigma\ra X_\tau$, for any two cones
$\sigma$ and~$\tau$ in~$\Sigma$ such that $\tau\supset \sigma$.
The zero-dimensional cone~$\{0\}$ corresponds to~$T$;
in particular, each $X_\sigma$ contains~$T$ and carries a
compatible action of~$T$.
Let $t$ denote the dimension of~$T$ and
let $\sigma$ be a cone of maximal dimension of~$\Sigma$.
By the theory of toric varieties, the smoothness assumption
on~$X$ implies that there exists a basis
of~$N$ which generates~$\sigma$ as a cone.
Consequently, the torus embedding $(T,X_\sigma)$
is isomorphic to $({\mathbf G}_m^t,{\mathbf A}^t)$.
The irreducible components of $X_\sigma\subset T$ then
correspond to the hyperplane coordinates in~${\mathbf A}^t$.
In particular, $X_\sigma\setminus T$ is a divisor
with strict normal crossings,
all of whose components of $X_\sigma\setminus T$
meet at the origin.
This shows that a set of components of~$X\setminus T$
has a non-trivial intersection whenever the corresponding
rays belong to a common cone in~$\Sigma$.
The converse holds since each point of~$X$ admits a $T$-invariant
affine neighborhood, hence of the form~$X_\sigma$.
In particular, what precedes holds over~$\bar F$
and implies the following description
of the geometric Clemens complex $\mathscr C(X,D)$ of~$(X,D)$.
There is a bijection between the set of orbits
of~$T_{\bar F}$ in~$X_{\bar F}$ and the set~$\Sigma$.
The irreducible components of~$D_{\bar F}$ are in bijection
with the set of one-dimensional cones (\emph{rays}) of~$\Sigma$
and a set of components has a non-trivial intersection
if and only if the corresponding rays
belong to a common cone in~$\Sigma$.
The cone~$\{0\}$ belongs to~$\Sigma$ and corresponds
to the open orbit~$T$.
Consequently, the Clemens complex $\mathscr C(X,D)$ is equal
to the set~$\Sigma\setminus\{\{0\}\}$, partially ordered by inclusion,
with the obvious action of the Galois group~$\Gamma_F$.
If $T$ is split, then $\Gamma_F$ acts trivially
on~$\mathscr C(X,D)$, hence $\mathscr C_F(X,D)=\mathscr C(X,D)$.
We have also observed in that case that the various strata of~$X$
even had $F$-rational points.
In other words, the $F$-analytic Clemens complex~$\mathscr C^{\text{\upshape an}}_F(X,D)$
is also equal to $\mathscr C(X,D)$.
Let us now treat the general case by proving
that $\mathscr C^{\text{\upshape an}}_F(X,D)=\mathscr C_F(X,D)$. By definition,
the $F$-rational Clemens complex
$\mathscr C_F(X,D)$ is the subcomplex of~$\mathscr C(X,D)$
consisting of $\Gamma_F$-invariant faces;
in other words, it corresponds to $\Gamma_F$-invariant
cones of positive dimension. We need to prove
that for any such cone~$\sigma$, the closure of
the corresponding orbit~$\mathscr O_\sigma$ in~$X$,
which is defined over~$F$, has an $F$-rational point.
For each ray~$r$ of~$\sigma$, let $n_r$ be the generator of $N\cap r$;
the group $\Gamma_F$ acts on the set of these~$n_r$,
and their sum~$n$ is fixed by $\Gamma_F$.
It corresponds to a cocharacter
$c_n\colon {\mathbf G}_m\ra T$ whose limit at~$0$ is an $F$-rational point
and this point belongs to~$\overline{\mathscr O_\sigma}$.
In fact, it even belongs to~$\mathscr O_\sigma$ because
the point~$n$ belongs to the relative interior of the cone~$\sigma$.
\begin{lemm}
Let $T_0$ be the maximal $F$-split torus in~$T$;
the group~$N_0$ of cocharacters of~$T_0$ is equal
to the subspace of~$N$ fixed by~$\Gamma_F$.
Let $X_0$ be the Zariski closure of~$T_0$ in~$X$.
It is a smooth equivariant compactification of~$T_0$
and its fan~$\Sigma_0$ in~$(N_0)_{\mathbf R}$
has as cones the intersections~$\sigma\cap (N_0)_{\mathbf R}$,
for all cones~$\sigma\in\Sigma^{\Gamma_F}$ which are invariant under~$\Gamma_F$.
\end{lemm}
\begin{proof}
It suffices to prove the desired result in each affine chart of~$X$.
Consider a cone~$\sigma\in\Sigma$ and
let $X_\sigma=\Spec \bar F[M\cap\sigma^\vee]$ be the corresponding
affine open subset of~$X_{\bar F}$.
A vector in~$M_0\cap \sigma$, being fixed under the action of~$\Gamma_F$,
belongs to all of the cones~$\gamma(\sigma)$, for $\gamma\in\Gamma_F$,
hence belongs to their intersection~$\tau$ which, by the definition
of a fan, is a cone in~$\Sigma$, obviously $\Gamma_F$-invariant.
As a consequence, the closure of~$T_0$ in~$X_\sigma$ is contained
in the toric open subvariety~$X_\tau$.
We thus can assume that $\sigma$ is a $\Gamma_F$-invariant cone,
and we may moreover assume that it is of maximal dimension.
Since $X$ is smooth, there is a basis $(e_1,\dots,e_d)$ of~$N$
and an integer~$s\in\{1,\dots,d\}$ such that $\sigma$
is generated by~$S=\{e_1,\dots,e_s\}$. We can also
assume that $e_{s+1},\dots,e_r$ belong to~$M_0$
and generate a complement to
the (saturated) subgroup generated by~$M_0\cap\sigma$.
Over~$\bar F$, this identifies
$T$ with~${\mathbf G}_m^d$ and $X_\sigma$ with ${\mathbf A}^s\times{\mathbf G}_m^{d-s}$.
Moreover, $\Gamma_F$ acts by permutations on~$S$. This implies that
$M_0\cap\sigma$ is generated by the vectors $\sum_{i\in O} e_i$,
where $O$ runs over the set $S/\Gamma_F$ of $\Gamma_F$-orbits in~$S$.
Consequently, $T_0\cap X_\sigma$ is the set of elements
$(x_1,\dots,x_d)$ in~${\mathbf G}_m^d$ such that $x_i$ is constant on
each orbit~$O$, and $x_i=1$ for $i>s$.
Its closure is the set of such elements
$(x_1,\dots,x_d)\in{\mathbf A}^r\times{\mathbf G}_m^{d-r}$
satisfying the same relations. This shows
that the closure of~$T_0$ in~$X_\sigma$ is
a isomorphic to the toric embedding of ${\mathbf G}_m^{\dim M_0}$
in ${\mathbf A}^{\mathop\# (S/\Gamma_F)}\times{\mathbf G}_m^{\dim M_0-\mathop\# (S/\Gamma_F)}$.
\end{proof}
\begin{coro}
By associating to a $T_0$-orbit in~$X_0$ the corresponding
$T$-orbit in~$X$, we obtain a bijection between the Clemens
complex of~$(X_0,X_0\setminus T_0)$ and the $F$-analytic
Clemens complex of~$(X,X\setminus T)$.
\end{coro}
\subsection{Clemens complexes of wonderful compactifications}
Let $G$ be a connected reductive group over a field~$F$.
Let $T_0$ denote a maximal split torus of~$G$ and
$T$ a maximal torus of~$G$ containing~$T_0$.
Let $X$ be a smooth proper~$F$-scheme which is
a biequivariant compactification of~$G$.
We denote by $Y$ and~$Y_0$ the closures
of the tori~$T$ and~$T_0$ in~$X$; these are toric varieties
defined over~$F$.
We assume that over a separable closure~$\bar F$ of~$F$,
the divisor $D=X\setminus G$
has strict normal crossings.
Observe that this assumption is satisfied if $X$ is
a wonderful compactification of~$G$.
\begin{prop}\label{prop.cartan}
With this notation,
one has: $X(F)=G(F)Y(F)G(F)$ and $Y(F)=T(F) Y_0(F)$;
in particular, $X(F)=G(F)Y_0(F)G(F)$.
\end{prop}
\begin{proof}
Let us first prove that $X(F)=G(F)Y(F)G(F)$.
It suffices to prove that $X(F)$ is contained in the latter set.
We use the arguments of~\cite{brion-kumar2005}, Lemma~6.1.4.
Let $x_0\in X(F)$; since $X$ is assumed to be smooth,
we can choose an arc $x\in X(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$ such that $x(0)=x_0$ and $x$
is not contained in $X\setminus G$.
We interpret $x$ as a $F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar$-point of~$G$;
by the Cartan decomposition
(\cite{bruhat-tits1972}, Proposition~4.4.3.
The fact that the group $G (F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra ) $ is good follows from the
property that the $F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar$-algebraic group $G_{F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar}$
is split over an unramified extension, namely $\bar F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar$,
and descent properties of the building)
\[ G( F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar ) = G( F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra ) T (F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar) G (F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t \rbra),
\]
we may write $x = g_1 y g_2$, where $g_1,g_2\in G(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra)$
and $y\in T(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar)$. Writing $y=g_1^{-1} x g_2^{-1}$,
we see that $y\in Y(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra)$.
Specializing at $t=0$, we now obtain $x_0=g_1(0) y(0) g_2(0)$,
as wanted.
Let us now prove the second equality, namely $Y(F)=T(F)Y_0(F)$.
Again, it suffices to prove that $Y(F)$ is contained in~$T(F)Y_0(F)$.
Let $x_0\in Y(F)$; as above, there
is an arc $x\in X(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )\cap G( F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar)$
such that $x(0)=x_0$. Using the Cartan decomposition,
we may replace $x$ by an arc of the form $y=g_1^{-1} x g_2$,
with $g_1,g_2\in G( F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra)$ satisfying
$y\in X(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra) \cap T(F \mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar)$.
Moreover, we may also assume that $g_1(0)=g_2(0)$ is
the neutral element of~$G(F)$. In particular,
$y\in Y(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra)$ and $y(0)=x_0$.
Let $S$ denote the anisotropic torus $T/T_0$
and $\pi\colon T\ra S$ the quotient map.
One has $\pi(y)\in S(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar)$.
Since $S$ is anisotropic,
one has $S(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )=S(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar )$
(Lemma~\ref{lemm.tore-anisotrope} below), hence $\pi(y)\in S(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$.
Looking at the exact sequence of tori
$$
1\ra T_0\ra T\ra S\ra 1
$$
and applying Hilbert's theorem~90 over the discrete valuation
ring $F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra $, it follows that there exists $z\in T(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$
such that $\pi(z)=\pi(y)$, hence $z^{-1}y\in T_0(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar )$.
Moreover, $z^{-1}y\in Y(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$.
Specializing~$t$ to~$0$, we see that $z(0)^{-1} y(0)\in Y_0(F)$,
that is $y(0)=x_0\in T(F) Y_0(F)$, as claimed.
The last equality follows immediately.
\end{proof}
The following lemma is well-known; we include a proof for the convenience
of the reader.
\begin{lemm}\label{lemm.tore-anisotrope}
Let $S$ be an anisotropic torus over a field~$F$.
Then, $S(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar )=S(F\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$.
\end{lemm}
\begin{proof}
Let $F'$ be a finite extension of~$F$ which splits~$S$; then,
for any $F$-algebra~$E$, $S(E)$ is the set
of morphisms from~$\mathfrak X^*(S)$ to~$E\otimes_F F'$
which commute with the actions of~$\Gamma_{F'/F}$ on both
sides. It follows that
a point~$P$ in $S(F\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar )$ corresponds to a
$\Gamma_F$-equivariant group morphisms~$\phi$
from $\mathfrak X^*(S)$ to~$F'\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar ^*$.
By composing~$\phi$ with the order map $F'\mathopen{(\!(}} \def\rpar{\mathclose{)\!)} t\rpar ^*\ra {\mathbf Z}$,
we obtain a $\Gamma_{F'/F}$-invariant morphism
$\operatorname{ord}\circ \phi\colon \mathfrak X^*(S)\ra{\mathbf Z}$,
which is necessarily~$0$ since $S$ is anisotropic.
Consequently, $\phi(c)=0$ for any $c\in\mathfrak X^*(S)$,
which means that $\phi(c)\in F'\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra ^*$.
In other words, $\phi$ is a $\Gamma_{F'/F}$-equivariant
morphism from $\mathfrak X^*(S)$ to~$F'\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra ^*$,
and the corresponding point~$P$ belongs
to $S(F'\mathopen{[\![}} \def\rbra{\mathclose{]\!]} t\rbra )$.
\end{proof}
Our goal now is to describe the various Clemens complexes
attached to the pair~$(X,D)$.
Let $W=\mathrm N_G(T)/T$ and $W_0 =\mathrm N_G(T_0)/\mathrm Z_G(T_0)$
be the Weyl groups of~$G$ relative to the tori~$T$ and~$T_0$.
Since the Weyl group~$W$ acts on~$G$ via the conjugation by an element
of~$G$, this extends to an action on~$X$. This action
induces the trivial action on~$\mathscr C(X,D)$.
Indeed, the group~$G$ being connected, each irreducible
component of $D_{\bar F}$ is preserved by
the actions of~$G_{\bar F}$.
\begin{prop} Over~$\bar F$, the open strata of the stratification of~$X_{\bar F}$ deduced from~$D_{\bar F}$ are exactly
the orbits of~$(G\times G)_{\bar F}$ in~$X_{\bar F}$.
Moreover, associating to an orbit its closure
defines a $\Gamma_F$-equivariant bijection from
$G(\bar F)\backslash X(\bar F)/G(\bar F)$ to~$\mathscr C(X,D)$.
\end{prop}
\begin{proof}
We may assume that the field~$F$ is separably closed.
We then have $T=T_0$.
By Proposition~\ref{prop.cartan}, the map
\[ Y(F)/T(F) \ra G(F)\backslash X(F)/ G(F) \]
which associates to the $T(F)$-orbit of a point~$x$ in~$Y(F)$
the orbit~$G(F)xG(F)$ in~$X(F)$
is surjective. By the theory of toric varieties,
$T(F)$ has only finitely many orbits in~$Y(F)$.
Consequently, $G\times G^{\mathrm {opp}}$ has only finitely many orbits in~$X$.
Let $Z$ be an element of~$\mathscr C(X,D)$, viewed as
an irreducible closed subvariety of~$D$. It is smooth, and possesses
a $G\times G^{\mathrm {opp}}$-action.
The group $G\times G^{\mathrm {opp}}$ acts on~$Z$ with only finitely
many orbits; necessarily, one of these orbits, say~$Z_0$,
is open in~$Z$. Since $Z$ is irreducible, it follows
that $Z=\overline{ Z_0}$,
and $Z$ is the closure of a $G\times G^{\mathrm {opp}}$-orbit.
By Theorem~2.1 of~\cite{knop1991}, there exists
an open affine subset~$X_0$ of~$X$
such that $Z_0$ is the unique closed orbit
of~$G\times G^{\mathrm {opp}}$ in~$GX_0G$.
Let us show that $Z\setminus Z_0$ is contained in $X\setminus GX_0G$
or, equivalently, that $Z\cap GX_0G=Z_0$.
Indeed, let $O$ be any orbit in~$Z$
distinct from~$Z_0$; we need to prove that $O\cap GX_0G=\emptyset$.
Let us denote by~$\overline O$ the closure of~$O$ in~$X$;
since $Z_0$ is open in~$Z$, $\overline O$ has empty interior in~$Z$,
hence is an irreducible subset of~$X$ whose dimension satisfies
$\dim \overline O < \dim Z$. However, any action of an algebraic
group on an algebraic variety has a closed orbit;
in particular $\overline O\cap GX_0G$ contains a closed orbit,
which implies that $\overline O\cap GX_0G\supset Z_0$,
and this contradicts the assumption $\dim\overline O<\dim Z_0$.
Since $X_0$ is open in~$X$, it contains at least one point of~$G$,
hence $GX_0G$ contains~$G$. Consequently, $X\setminus GX_0G$
is contained in~$X\setminus G$.
Since $GX_0G$ is an $G\times G^{\mathrm {opp}}$-invariant open subset of~$X$,
each irreducible component of~$Z\setminus Z_0$ is contained
in some irreducible component of~$X\setminus GX_0G$ which does not meet~$Z_0$.
Consequently, the orbit~$Z_0$ is an open stratum of the
stratification defined by the boundary divisor~$D$.
Conversely, let $O$ be an orbit of $G\times G^{\mathrm {opp}}$ in~$X$
and let $Z$ be the stratum of minimal dimension in~$\mathscr C(X,D)$
which contains~$O$.
By the preceding argument, $Z$ is the closure of
an orbit~$Z_0$ whose complement $Z\setminus Z_0$
is a union of lower dimensional strata.
By the minimality assumption on~$Z$, $Z_0=O$,
which proves that $O$ is an open stratum.
The rest of the proposition follows at once.
\end{proof}
The group of characters~$M_0$ of~$T_0$ is the group of coinvariants
of~$\Gamma$ in~$M$ (it is a quotient of~$M$),
and its group of cocharacters~$N_0$ is the group of invariants
of~$\Gamma$ in~$N$ (a subgroup of~$N$).
Let $t_0=\dim T_0$.
Let $\Sigma_0$ denote the fan of~$N_0$ induced by~$\Sigma$;
it is simplicial albeit not obviously smooth in general.
This fan defines a toric variety~$X_0$ which in fact is isomorphic
to the Zariski closure of~$T_0$ in~$X$.
\subsection{Volume estimates for compactifications of semi-simple groups}
\label{subsec.wonderful}
\subsubsection{Introduction}
Let $G$ be a semisimple algebraic group of adjoint type
over a field~$F$ of characteristic~$0$.
Let $\iota\colon G\ra {\operatorname{GL}}(V)$ be a
faithful algebraic representation of~$G$ in a
finite dimensional~$F$-vector space~$V$.
The natural map
\[
{\operatorname{GL}}(V)\ra{\operatorname{End}}(V)\setminus\{0\}\ra{\mathbf P}{\operatorname{End}}(V)
\]
induces a map $\bar\iota\colon G\ra{\mathbf P}{\operatorname{End}}(V)$. Let $X_\iota$ be
the Zariski closure of its image;
it is a bi-equivariant compactification
of~$G$. Let $\partial X_\iota$ be the complement to~$G$
in~$X_\iota$.
When $\iota$ is irreducible
with regular highest weight,
$X_\iota$ is the wonderful compactification defined
by De Concini and Procesi in~\cite[3.4]{deconcini-p83}.
In that case, $X_\iota$ is smooth and $\partial X_\iota$
is a divisor with strict normal crossings.
When $F$ is a local field, we endow
the vector space~${\operatorname{End}}(V)$ with a norm~$\norm{\cdot}$.
When $F$ is a number field, let us choose, for any place~$v$ of~$F$,
a $v$-adic norm on~${\operatorname{End}}(V)\otimes F_v$, so that
there is an $\mathfrak o_F$-lattice in~${\operatorname{End}}(V)$
inducing these norms for almost all finite places.
As it was explained in Sections~\ref{exam.projective-space.metric}
and~\ref{exam.projective-space.adelic.metric},
such choices give rise to a metric on the line bundle~$\mathscr O(1)$
on the projective space~${\mathbf P}{\operatorname{End}}(V)$, resp. an adelic metric,
when $F$ is a number field.
We claim that the line bundle~$\mathscr O(1)$ on~$X_\iota$
has a nonzero global section~$s_\iota$ which is invariant under~$G$.
Such a section~$s_\iota$
is unique up to multiplication by a scalar,
since the quotient of two of them is a $G$-invariant rational function
on~$X_\iota$. (See also~\cite{deconcini-p83}, 1.7, p.~9, proposition.)
To prove the existence, let us first observe that
the line in~${\operatorname{End}}(V)$ generated by $\operatorname {id}_V$ is $G$-invariant.
By semi-simplicity of~$G$, there exists a linear form~$\ell_\iota$
on~${\operatorname{End}}(V)$ which is invariant under~$G$ and maps~$\operatorname {id}_V$ to~$1$.
Let $s_\iota$ be the restriction
to~$G$ of the global section of~$\mathscr O(1)$ over~${\mathbf P}{\operatorname{End}}(V)$
defined by~$\ell_\iota$.
Now, we have
\[ \norm{s_\iota}(\bar\iota(g))=\frac{\abs{\ell_\iota(g)}}{\norm{\iota(g)}}
= \frac{\abs{\ell_\iota^g(e)}}{\norm{\iota(g))}} = {\norm{\iota(g)}}^{-1}, \]
for any $g\in G(F)$ when $F$ is a local field,
and similar equalities at all places of~$F$
in the number field case.
Consequently, for $F={\mathbf R}$ or~${\mathbf C}$,
the results of Section~\ref{subsec.geometric-volume}
imply, as a particular case, an asymptotic formula for the volume of
sets of $g\in G(F)$ with $\norm{\iota(g)}\leq B$, when $B\ra\infty$,
as well as similar (but weaker) estimates when $F$ is a $p$-adic field.
Similarly, when $F$ is a number field,
the volume estimates established in
Section~\ref{subsec.asymptotics.adelic} imply
estimates for the volume of adelic sets in~$G({\mathbb A}_F)$
consisting of adelic points of bounded height.
In the case of local fields,
it requires further computations for making these estimates explicit, in terms of the representation~$\iota$.
In particular, we will need to describe the analytic Clemens complex
of~$\partial X_\iota$
\subsubsection{The wonderful compactification of De Concini and Procesi}
For simplicity, we assume for the moment that $G$ is split.
We fix a maximal torus $T$ of~$G$ which is split,
as well as a Borel subgroup~$B$ of~$G$ containing~$T$.
We identify the groups of characters $\mathfrak X^*(T)$ and~$\mathfrak X^*(B)$,
as well as the groups of cocharacters $\mathfrak X_*(T)$ and~$\mathfrak X_*(B)$;
we also let $\mathfrak a=\mathfrak X_*(T)\otimes_{\mathbf Z}{\mathbf R}$ and
$\mathfrak a^*=\mathfrak X^*(T)\otimes_{\mathbf Z}{\mathbf R}$.
Let~$\Phi$, resp. $\Phi^+$, be the set of roots of~$G$, resp. of positive roots
in~$\mathfrak X^*(T)$; let $\beta$ denote the sum of all positive roots.
Let $\Delta\subset\Phi^+$ be the set of simple roots; they form a basis
of the real cone in~$\mathfrak a^*$
generated by positive roots,
hence we may write $\beta=\sum_{\alpha\in\Delta}m_\alpha\alpha$
for some positive integers~$m_\alpha$.
Let $\iota\colon G\ra{\operatorname{GL}}(V)$ be a representation as above;
we assume here that $\iota$ is irreducible
and that its highest weight~$\lambda$ is regular,
\emph{i.e.}, can be written as $\lambda=\sum_{\alpha\in\Delta} d_\alpha \alpha$,
for some positive integers~$d_\alpha$.
In that case, as recalled above, $X_\iota$
is the ``wonderful compactification of~$G$'' defined by De Concini and Procesi.
The variety~$X_\iota$ does not depend on the actual choice of~$\iota$,
but the projective embedding does.
The irreducible components of $X_\iota\setminus G$
are naturally indexed by the set~$\Delta$; we will write $D_\alpha$
for the divisor corresponding to a simple root~$\alpha$.
Let $D=\sum D_\alpha$.
The Clemens complex~$\mathscr C(\partial X_\iota)$ is simplicial,
and coincides with the $F$-analytic Clemens complex~$\mathscr C^{\text{\upshape an}}_F(\partial X_\iota)$
since $G$ is assumed to be split.
The line bundles~$\mathscr O(D_\alpha)$ form a basis of~$\Pic(X_\iota)$,
as well as generators of the cone of effective divisors
(which is therefore simplicial).
The restriction of~$\mathscr O_{{\mathbf P}}(1)$ to~$X$
corresponds precisely to~$\lambda$,
and there is a canonical isomorphism
\[ \mathscr O_{{\mathbf P}{\operatorname{End}}(V)} (1)|_{X_\iota} \simeq \mathscr O(\sum_{\alpha\in\Delta} d_\alpha D_\alpha). \]
According to~\cite{deconcini-p83}, the anticanonical line bundle of~$X_\iota$
is given by
\[ K_{X_\iota}^{-1} \simeq \mathscr O(\sum_{\alpha\in\Delta} (m_\alpha+1) D_\alpha).\]
Let $\mathscr C\subset\mathfrak a^*$ be the convex hull of the characters
of~$T$ appearing in the representation~$\iota$. This
is also the convex hull of the images of the highest weight~$\lambda$
under the action of the Weyl group. This is a convex
and compact polytope which contains~$0$ in its interior
by~\cite{maucourant2007}, Lemma 2.1.
\begin{lemm}
Let $\sigma=\max_{\alpha\in\Delta} (m_\alpha/d_\alpha)$,
let $t$ be the number of elements $\alpha\in\Delta$
where equality holds.
Then $\sigma$ is the smallest positive real number such
that $\beta/\sigma \in\mathscr C$ and
$t$ is the maximal codimension of a face of~$\mathscr C$
containing~$\beta/\sigma$.
\end{lemm}
\begin{proof}
For any simple root~$\alpha$, let~$\alpha^\vee$ be the corresponding
coroot, so that the simple reflexion~$s_\alpha$ associated to~$\alpha$
is given by $s_\alpha(x)=x-2\langle \alpha^\vee,x\rangle \alpha$,
for $x\in\mathfrak a^*$. We have $\langle\alpha^\vee,\alpha\rangle=1$
while $\langle\alpha^\vee,\alpha'\rangle=0$ for any other
simple root~$\alpha'$.
Within the Weyl chamber of~$\lambda$, the polytope~$\mathscr C$
is bounded by the affine hyperplanes orthogonal to the
simple roots~$\alpha$ and passing through~$\lambda$.
Consequently, a point~$x$ in this chamber
belongs to~$\mathscr C$
if and only $\langle \alpha^\vee,x\rangle\leq\langle\alpha^\vee,\lambda\rangle$.
Moreover, such a point~$x$ belongs to the boundary of~$\mathscr C$
if and only if equality is achieved for some simple root~$\alpha$;
then, the number of such~$\alpha$ is
the maximal codimension of a face of~$\mathscr C$ containing~$x$.
For $x=\beta/\sigma=\sum_{\alpha\in\Delta} (m_\alpha/\sigma) \alpha$,
we find
\[ \langle\alpha^\vee,x-\lambda\rangle=\frac{m_\alpha}\sigma-\lambda_\alpha,\]
hence the Lemma.
\end{proof}
Consequently, our geometric estimates (Theorem~\ref{theo.geometric-volume})
imply the following result of Maucourant~\cite{maucourant2007}
under the assumption that~$G$ is split and~$V$ has a unique
highest weight.
\begin{coro}\label{coro.maucourant}
Define $\sigma=\max_{\alpha\in\Delta} (m_\alpha/d_\alpha)$
and let $t$ be
be the number of~$\alpha\in\Delta$
where the equality holds.
Assume that $F={\mathbf R}$ or~${\mathbf C}$.
When $B\ra\infty$, the volume $V(B)$ of all $g\in G(F)$
such that $\norm{\iota(g)}\leq B$ satisfies an asymptotic formula
of the form:
\[ V(B)\sim c B^{\sigma}(\log B)^{t-1}. \]
\end{coro}
According to our theorem,
the positive constant~$c$ can be written as a product of
the combinatorial factor
\[ \frac1{\sigma(t-1)!} \prod_{\alpha\in A}\frac{1}{\lambda_\alpha}
\]
and an explicit integral on the stratum of~$X_\iota(F)$ defined by~$A$,
with respect to its normalized residue measure~$\tau_{D_A}$.
In the $p$-adic case, Corollary~\ref{coro.asymptotic.volume-ultrametric} similarly implies
the following result:
\begin{coro}\label{coro.maucourant.ultrametric}
Keep the same notation, assuming that $F$ is a $p$-adic field.
Then, when $B\ra\infty$,
\[ 0<\liminf \frac{V(B)}{ B^{\sigma}(\log B)^{t-1}} \leq
\limsup \frac{V(B)}{ B^{\sigma}(\log B)^{t-1}} <+ \infty. \]
\end{coro}
\subsubsection{The case of a general representations (split group)}
We now explain how to treat non-irreducible representations,
still assuming that the group~$G$ is split.
By Proposition~6.2.5 of~\cite{brion-kumar2005},
there is a diagram of equivariant compactifications
\[ \xymatrix{ & {\tilde X} \ar[ld]_{\pi} \ar[rd] \\
X_\iota && X_w } \]
where $X_w$ is the wonderful compactification previously studied
and $\tilde X$ is smooth. Since the boundary divisor of a
smooth toric variety is a divisor with strict normal crossings,
it then follows from~\cite{brion-kumar2005}, Proposition~6.2.3,
that the boundary~$\partial\tilde X$ of~$\tilde X$ is a divisor with
strict normal crossings in~$\tilde X$.
Moreover, the proof of this proposition and the local
description of toroidal $G$-embeddings in \emph{loc. cit.}, \S6.2,
show that $\tilde X$ is obtained from~$X_w$ by a sequence
of $T$-equivariant blow-ups.
Therefore, the boundary~$\partial\tilde X$
consists of the strict transforms~$\tilde D_\alpha$ of the divisors~$D_\alpha$,
indexed by the simple roots, and of the exceptional divisors~$E_i$
(for $i$ in some finite index set~$I$). These divisors form
a basis of the effective cone in the Picard group.
The anticanonical line bundle of~$\tilde X$ decomposes as a sum
\[ \sum_{\alpha\in\mathscr A} (m_\alpha+1) D_\alpha + \sum_{i\in I} E_i.\]
Let $\tilde L$ be the line bundle $\pi^*\mathscr O(1)$ on~$\tilde X$;
let us write it as $\tilde L=\sum \tilde\lambda_\alpha D_\alpha + \sum\tilde\lambda_i E_i$
in the above basis.
Since we can compute volumes on~$\tilde X$, our estimates
imply that $V(B)$ has an asymptotic expansion
of the form given in Corollary~\ref{coro.maucourant},
$\sigma$ being given by
\[ \sigma= \max \left( \max_{\alpha\in\mathscr A} \frac{m_\alpha}{\tilde\lambda_\alpha},
\max_{i\in I} \frac{0}{\tilde\lambda_i} \right)
= \max_{\alpha\in\mathscr A} \frac{m_\alpha}{\tilde\lambda_\alpha},
\]
and $t$ is again the number of indices~$\alpha\in\mathscr A$
where equality holds.
Observe that the definition of~$\sigma$
precisely means that the line bundle
$\sigma\pi^* L-(K_{\tilde X}+\partial \tilde X)$ belongs to the boundary
of the effective cone of~$\tilde X$; the integer~$t$
is then the codimension of the face of minimal dimension
containing that line bundle.
These properties can be checked on restriction to the toric
variety~$\tilde Y$ given by the closure of~$T$ in~$\tilde X$;
indeed the restriction map $\Pic(\tilde X)\ra\Pic(\tilde Y)$
is an isomorphism onto the part of~$\Pic(\tilde Y)$ invariant
under the Weyl group, and similarly for the effective cone.
To determine~$\sigma$,
it now suffices to test for the positivity of the piecewise linear
(\textsc {pl}\xspace) function on the vector space~$\mathfrak a^*$ associated to this
divisor.
The above formula for the anticanonical line bundle
of~$\tilde X$ implies that the \textsc {pl}\xspace function
for~$K_{\tilde X}+\partial\tilde X$
is that of $K_{X_w}+\partial {X_w}$.
On the other hand, the theory of heights on toric varieties
relates the \textsc {pl}\xspace function corresponding
to a divisor to the normalized local height function
(see~\cite{batyrev-t95b}).
The formula is as follows. Let $D$ be an effective $T$-invariant divisor,
the corresponding $T$-linearized line bundle has a canonical
$T$-invariant nonzero global section~$s_D$. For $t\in T(F)$,
let $\ell(t)$ be the linear form on~$\mathfrak a$
defined by $\chi\mapsto \log\abs{\chi(t)}$; dually, one
has $\ell(t)(\chi)=\langle\chi,\log(t)\rangle$.
Then, the norm of~$s_D$ with respect to the canonical normalization
is given by
\[ \log\norm{s_D(t)}^{-1} = \phi_D(\ell(t)). \]
If the local height function is not normalized, the previous
equality only holds up to the addition of a bounded term.
In our case, let $\Phi$
be the set of weights of~$T$ in the representation~$\iota$. Then,
for $t\in T(F)$,
\[ \norm{\iota(t)} \approx \max_{\chi\in\Phi} \abs{\chi(t)}, \]
hence
\[ \log\norm{s_\iota(t)}^{-1} = \log \norm{\iota(t)}
= \max_{\chi\in\Phi} \log\abs{\chi(t)} + \mathrm O(1). \]
In other words, the line bundle~$\pi^*L$
corresponds to the \textsc {pl}\xspace function
$\phi_\iota = \max_{\chi\in\Phi} \langle\chi, \cdot\rangle$
on~$\mathfrak a^*$.
As explained above, $\sigma$ is the least positive real
number such that
the \textsc {pl}\xspace function $ s\phi_\iota $ is greater than
the \textsc {pl}\xspace function~$\phi_\beta$ associated to~$K_{X_w}^{-1}(-\partial X)$.
Since the Weyl group acts trivially on the Picard group
of~$\tilde X$, these \textsc {pl}\xspace functions are invariant under the action
of the Weyl group and it is sufficient to test the inequality
on the positive Weyl chamber~$C$ in~$\mathfrak a$.
Let $(\varpi_\alpha)$ denote the basis of~$\mathfrak a^*$
dual to the basis~$(\alpha)$ --- up to the usual identification
of $\mathfrak a$ with its dual given by the Killing
form of~$G$, this is the basis of fundamental weights.
One has $C=\sum_\alpha{\mathbf R}_+\varpi_\alpha$.
Recall also that $\beta$ is the sum of the fundamental weights of~$G$,
hence belongs to the positive Weyl chamber in~$\mathfrak a^*$.
It follows that $w\beta\geq\beta$ for any element~$w$ in the Weyl group~$W$
(Bourbaki, LIE VI, \S1, Proposition~18, p.~158). It follows that
$\langle w\beta,\varpi_\alpha\rangle\leq\langle \beta,\varpi_\alpha\rangle$
for any $w\in W$. Consequently,
\[ \phi_\beta(y) = \max\langle w\beta,y\rangle = \langle\beta,y\rangle \]
for any $y\in C$.
Moreover, if $y=\sum y_\alpha\varpi_\alpha\in\mathfrak a$,
then
\[ \langle \beta,y\rangle=\langle \sum m_\alpha \alpha,\sum y_\alpha\varpi_\alpha\rangle = \sum m_\alpha y_\alpha. \]
Let $\Lambda$ be the set of dominant weights of~$\iota$
with respect to~$C$. For any $\lambda\in\Lambda$,
let us write $\lambda=\sum \lambda_\alpha \alpha$, for some
nonnegative $\lambda_\alpha\in{\mathbf Z}$.
Consequently, for any $y=\sum y_\alpha\varpi_\alpha\in C$,
with $y_\alpha\geq 0$ for all~$\alpha$, one has
\[ \phi_\iota(y) = \max_{\chi\in\Phi}\langle\chi,y\rangle
= \max_{\lambda\in\Lambda} \langle\lambda,y\rangle
= \max_{\lambda\in\Lambda} \left(\sum_\alpha \lambda_\alpha y_\alpha\right).
\]
The condition that $s\phi_\iota(y)\geq \phi_\beta$ on~$C$
therefore means that $s\max_{\lambda\in\Lambda}\lambda_\alpha\geq m_\alpha$
for any $\alpha\in\Phi$. In other words,
$ \sigma = \max (m_\alpha/\tilde\lambda_\alpha)$,
where we have set $\tilde\lambda_\alpha= \max_{\lambda\in\Lambda}\lambda_\alpha$.
Moreover, $t$ is the number of simple roots~$\alpha$ such that
$\tilde\lambda_\alpha \sigma=m_\alpha$.
As in~\cite{maucourant2007},
let $\mathscr C$ be the convex hull of~$\Phi$ in~$\mathfrak a^*$;
it is a compact polytope whose dual~$\mathscr C^*$
is defined by the inequality $\phi_\iota(\cdot)\leq 1$
in~$\mathfrak a^*$.
Let $s$ be a positive real number. By definition, $\beta/s$
belongs to~$\mathscr C$ if and only
if $\langle \beta/s, y\rangle\leq 1$ for any $y\in\mathfrak a^*$
such that $\phi_\iota(\mathfrak a)\leq 1$. This is precisely
equivalent to the fact that $s\phi_\iota-\langle\beta,\cdot\rangle$
is nonnegative on~$\mathfrak a^*$.
Let $y\in \mathfrak a^*$ and let $w\in W$ be such that $wy\in C$.
Since $\phi_\iota$ is invariant under~$W$,
\[ s\phi_\iota(y)-\langle\beta,y\rangle = s\phi_\iota(wy)-\langle w^{-1}\beta,wy\rangle
\leq s\phi_\iota(wy)-\langle \beta,wy \rangle, \]
with equality if and only if $w=e$.
We have thus shown that $\beta/s\in\mathscr C$ if and only
if the \textsc {pl}\xspace function $s\phi_\iota-\phi_\beta$ is nonnegative.
In other words, $\sigma$ is the least positive real number
such that $\beta/\sigma$ belongs to~$\mathscr C$,
as claimed by Maucourant in~\cite{maucourant2007}.
Let $\mathscr F$ be the face of~$\mathscr C$ containing~$\beta/\sigma$.
It is also explained in~\cite{maucourant2007}
that the dual face of~$\mathscr F$
is contained in the positive Weyl chamber~$C$ (Lemma~2.3)
and is given by
\[ \mathscr F^*=\{ y\in \mathscr C^* \,;\, \langle \beta,y\rangle=\sigma\}.\]
If again we decompose $y$ as $\sum y_\alpha\varpi_\alpha$,
it follows that $\mathscr F^*$ identifies as the set
of nonnegative $(y_\alpha)$ such that $\phi_\iota(y)\leq 1$
and $\langle \beta,y\rangle=\sigma$.
The first condition gives us $\sum\tilde\lambda_\alpha y_\alpha\leq 1$
and the second is equivalent to $\sum m_\alpha y_\alpha =\sigma$.
This implies $\sum y_\alpha=1$
and $y_\alpha=0$ if $\sigma \neq m_\alpha/\tilde\lambda_\alpha$.
Consequently, $\codim\mathscr F=\dim \mathscr F^*=t$,
showing the agreement of our general theorem with
the result obtained by Maucourant in~\cite{maucourant2007},
under the assumption that $G$ is split.
\subsubsection{The general case}
We now treat the general case of a possibly non-split group.
Let $T$ be be a maximal split torus in~$G$, so that
its Lie algebra~$\mathfrak a_{\mathbf R}$ is a Cartan subalgebra of
$\operatorname{Lie}(G)$.
We have already explained how the $F$-analytic Clemens complex
of~$X$ is related to the toric variety~$Y$ given by the closure of~$T$
in~$X$. As a consequence, all positivity conditions and dimensions
of faces which intervene in our geometric result rely only
on the divisors which are ``detected'' by~$Y$,
hence are expressed in terms of \textsc {pl}\xspace functions in~$\mathfrak a^*$.
The previous analysis now applies verbatim and allow us again
to recover Maucourant's theorem.
\subsubsection{Adelic volumes}
Let us now assume that $F$ is a number field.
Theorem~\ref{theo.volume.global}
describes the analytic behavior of the volume --- with
respect to the Haar measure --- of adelic
points in~$G({\mathbb A}_F)$ of height~$\leq B$, when $B\ra\infty$.
This allows to recover Theorem 4.13 in~\cite{gorodnik-m-o2009}.
In fact, that Theorem itself is proved as a corollary
of the analytic behavior of the associated Mellin transform
which had been previously shown in~\cite[Theorem 7.1]{shalika-tb-t2007}.
Similarly, this analytic behavior is a particular case of
our Proposition~\ref{prop.zeta.global}.
\subsection{Relation with the output of the circle method}
Let $X$ be a non-singular, geometrically irreducible, closed
subvariety of an affine space~$V$ of dimension~$n$ over a number field~$F$.
When $F={\mathbf Q}$, $W={\mathbf A}^n$ and $X$ is defined as the proper intersection of
$r$ hypersurfaces defined
Let $n$ be a positive integer and let $f_1,\dots,f_r\in{\mathbf Z}[X_1,\dots,X_n]$
be polynomials with integer coefficients.
The circle method furnishes eventually an estimate, when
the real number~$B$ grows to~$\infty$, of the number~$N(X,B)$
of solutions $x\in{\mathbf Z}^n$ of the system~$X$ given by $f_1(x)=\dots=f_r(x)=0$
whose ``height'' $\norm x$ is bounded by~$B$.
A set of conditions under which the circle method applies
is given by~\cite{birch62}; it suffices that
the hypersurfaces~$\{f_i=0\}$ meet properly and define
a non-singular codimension~$r$ subvariety of~${\mathbf A}^n$, and that
$n$ is very large in comparison to the degrees of the~$f_i$.
Let us assume that $X=V(f_1,\dots,f_r)$ is smooth and has codimension~$r$.
By the circle method, an approximation for~$N(B)$ is given
by a product of ``local densities'': for any prime number~$p$,
let
\[ \mu_p(X) = \lim_{k\ra\infty } \frac{\mathop\# X({\mathbf Z}/p^k{\mathbf Z})}{p^{k\dim X}};\]
for the infinite prime, let
\[ \mu_\infty (X,B)= \lim_{\eps\ra 0 } \eps^{-r} \vol \{x\in{\mathbf R}^n\,;\, \norm x\leq B, \ \abs{f_i(x)}<\eps/2\}, \]
where $\vol$ refers to the Euclidean volume in~${\mathbf R}^n$.
In some cases, one can indeed
prove that $N(X,B)\sim \mu_\infty(X,B)\prod_{p<\infty}\mu_p(X)$
when $B\ra\infty$.
As already observed by~\cite{borovoi-r1995} (Section~1.8)
we first want to recall that the right-hand-side $V(X,B)$
of this asymptotic expansion is really a volume.
Under the transversality assumption we have made on
the hypersurfaces defined by the~$f_i$,
there exists a differential form~$\tilde\omega$ on
a neighborhood of~$X$ in~${\mathbf A}^n$
such that
\[ \tilde\omega\wedge\mathrm df_1\wedge\dots\wedge\mathrm df_r
=\mathrm dx_1\wedge\dots\wedge\mathrm dx_n. \]
The restriction of~$\tilde\omega$ to~$X$ is a well-defined
gauge form~$\omega$ on~$X$ and
\[ \mu_\infty(X,B)=\int_{\substack{ X({\mathbf R}) \\ \norm x\leq B}} \mathrm d\abs\omega_\infty (x),
\qquad
\mu_p(X) = \int_{X({\mathbf Z}_p)} \abs\omega_p\quad\text{($p$ prime).} \]
Let us write $[t:x]$
for the homogeneous coordinates of a point in~$\mathbf P^n(K)$, so that $t\in K$
and $x=(x_1,\dots,x_n)\in K^n$.
One then defines an metric on $\mathscr O(1)$ by the formula
\[ \norm{s_F}([t:x])=\frac{\abs{F(t,x)}}{\max(\abs t,\norm x)^{\deg F}} \]
for
any point $[t:x]\in{\mathbf P}^n({\mathbf R})$
and
any homogeneous polynomial $F$ in ${\mathbf R}[t,x]$,
$s_F$ being the section of~$\mathscr O(\deg F)$ attached to~$F$.
Let $\mathsec f_0$ be the section corresponding to $F=t$;
for $B\geq 1$, the condition $\norm x\leq B$ can thus be
translated to $\norm{\mathsec f_0}([1:x])\geq 1/B$.
First assume that the Zariski closure of~$X$ in~$\mathbf P^n$
is smooth; let $D$ be the divisor $\mathbf V(\mathsec f_0)$ in~$X$.
Letting $d=\sum\deg(f_i)$, the divisor of~$\omega$ is equal to $d-n-1$.
In that case, the measure $\abs\omega$ coincides
with the measure $\tau_{(\bar X,(n+1-d)D}$ on~$X({\mathbf R})$, so
that $\mu_\infty(X,B)$ is an integral of the form studied
in this paper, namely
\[ \mu_\infty(X,B) = \tau_{(X,(n+1-d)D)}( \norm{\mathsec f_0}\geq 1/B ). \]
Assume $n> d$.
Then, when $B$ converges to infinity, Theorem~\ref{theo.geometric-volume}
implies that $\mu_\infty(X,B)\approx B^{n-d} (\log B)^{b-1}$,
where $b$ is the dimension of the Clemens complex of
the divisor~$D({\mathbf R})$, that is the maximal number of components
of $D({\mathbf R})$ that have a common intersection point.
(When $n=d$, a similar result holds except that $b-1$ has to be
replaced by~$b$.)
The paper \cite{duke-r-s1993}
showed that this asymptotic does not hold
for the specific example of a generic $\operatorname{SL}(2)$-orbit
of degree~$d$ binary forms. For example, when $d=3$,
this amounts to counting points on an hypersurface of degree~$4$
in~${\mathbf P}^4$ and the expected exponent~$0=4-4$ is replaced by~$2/3$.
This discrepancy is explained by~\cite{hassett-tschinkel2003}:
observing that $(\bar X,D)$ is not smooth in that case,
they computed a log-desingularization of $(\bar X,D)$
on which the behaviour of the integral $\mu_\infty(X,B)$
can be predicted.
In the general case,
let us thus consider a projective smooth compactification $\bar Y$ of~$X$
together with a projective, generically finite,
morphism $\pi\colon\bar Y\ra\bar X $
such that, over~$\bar{\mathbf Q}$, the complementary divisor~$E$ to~$X$ in~$\bar Y$
has strict normal crossings. Let $E_\alpha$ be the irreducible
components of~$E$.
It is customary to assume that $\pi$ induces an isomorphism over~$X$,
but this is not necessary in the following analysis;
it is in fact sufficient to assume that the degree of~$\pi$
is constant on the complement to a null set.
Let us pose $\eta=\pi^*\omega$; then, the divisor of~$\eta$
has the form $\div(\eta)=-\sum\rho_\alpha E_\alpha=-E'$, so that
\[ \mu_\infty(X,B) = \int_{\substack{\bar Y({\mathbf R}) \\\norm{ \pi^*\mathsec f_0}\geq 1/B}} \abs\eta = \tau_{(\bar Y,E')} (\{ \norm{\pi^*\mathsec f_0}\geq 1/B\}).
\]
Write also $\pi^*D=\sum\lambda_\alpha E_\alpha$.
According to Theorem~\ref{theo.geometric-volume},
$\mu_\infty(X,B)\approx B^a(\log B)^{b-1}$ where now,
$ a= \max (\rho_\alpha-1)/\lambda_\alpha$, the minimum
being restricted to those $\alpha$ such that $E_\alpha({\mathbf R})\neq\emptyset$;
the integer~$b$ is the dimension of the subcomplex
of the analytic Clemens complex
consisting of those~$E_\alpha$ achieving this minimum.
(When $a=0$, $b-1$ is replaced by~$b$.)
Let us give the specific example of an ${\operatorname{SL}}(2)$-orbit
in the affine space of binary forms of degree~$d$,
as treated in~\cite{hassett-tschinkel2003}.
That paper constructs a pair $(\bar Y,E)$ with an action of~$\mathfrak S_n$
such that $E=E_1+E_2$ has two irreducible components,
the divisors $E_1$ and~$E_2$ (denoted $A[n-1]$ and~$A[n]$ in that paper)
forming a basis of the $\mathfrak S_n$-invariant part of $\Pic(\bar Y)$,
Moreover, $K_{\bar Y}^{-1}=E_1+2E_2$.
They compute the inverse image of~$\mathscr O(1)$
and obtain $\frac{n-2}2 E_1+\frac n2 E_2$ (\emph{loccit.}~\ignorespaces, Lemma 3.3).
Consequently, $(\rho_1-1)/\lambda_1=0$ and
$(\rho_2-1)/\lambda_2=2/n$. Finally, $a=2/n$ and $b=1$,
implying that $\mu_\infty(X,B)\approx B^{2/n}$.
Let us return to the general case,
and assume that $K_{\bar X}$ is ${\mathbf Q}$-Cartier.
Then, the \emph{log-discrepancies} $(\eps_\alpha)$ are defined by the formula
\[ K_{\bar Y}(E) = \pi^*(\bar K_{\bar X}(D)) + \sum\eps_\alpha E_\alpha, \]
so that $1-\rho_\alpha=(d-n)\lambda_\alpha+\eps_\alpha$ for any~$\alpha$.
Then,
\[ a= n-d + \min_\alpha \frac{-\eps_\alpha}{\lambda_\alpha}, \]
where, again, the minimum is restricted to those~$\alpha$
such that $E_\alpha({\mathbf R})\neq\emptyset$.
When $(\bar X,D)$ has log-canonical singularities, $\eps_\alpha\geq 0$
for each~$\alpha$
and one obtains again $a\leq n-d$ since the log-discrepancy corresponding
to the strict transform of the components of~$D$ is~$0$.
However, in the case treated by~\cite{hassett-tschinkel2003},
$\eps_1=0$ and $\eps_2=-1$, leading to the
opposite inequality $a=\frac2n>0$.
\subsection{Matrices with given characteristic polynomial}
\subsubsection{}
Let $V_P$ be the ${\mathbf Z}$-scheme of matrices in the affine $4$-space
whose characteristic polynomial is $X^2+1$,
and let $B_T$ be the euclidean ball of radius~$T$ in ${\mathbf R}^4$.
Shah gives in~\cite{shah2000} the following
asymptotic expansion:
\[ \# (V_P({\mathbf Z})\cap B_T) \sim T \zeta_K^*(1) \dfrac{\pi^{1/2}}{\Gamma(3/2)}
\dfrac{\pi}{\Gamma(2/2)\zeta(2)}
= T \zeta_K^*(1) \frac{2\pi}{\zeta(2)} = C_{X^2+1}\, T , \]
where $\zeta_{{\mathbf Q}(i)}^*$ is the leading term at~$1$ of Dedekind's
zeta function relative to the number field ${\mathbf Q}(i)$.
A matrix in~$V_P$ has the form
$\left(\begin{smallmatrix} x & z \\ y & -x \end{smallmatrix}\right)$
with $x^2+yz+1=0$.
Let $X$ denote the subvariety of ${\mathbf P}^3$ defined by $X^2+YZ+T^2=0$. It
is smooth of dimension~$2$ over ${\mathbf Z}[1/2]$.
The scheme $V_P$ is exactly the open subset $U\subset X$ defined by $T\neq 0$.
The divisor at infinity $D=X\setminus U$ is defined by $T=0$;
it is smooth as well (still over ${\mathbf Z}[1/2]$).
\subsubsection{}
Let $p$ be an odd prime number.
One has $\# U({\mathbf F}_p)=p^2+\genfrac{(}{)}{}1{-1}{p} p$.
Indeed, $z\neq 0$ gives $p(p-1)$ points. If $z=0$, $y$ may be
arbitrary and $x$ has to be $\pm\sqrt{-1}$, hence
$2p$ points if $-1$ is a square and $0$ else.
Finally,
\[ \vol U({\mathbf Z}_p) = p^{-2} \# U({\mathbf F}_p) = 1 + \genfrac{(}{)}{}1{-1}{p} \frac1p. \]
For $p=2$, we split $U({\mathbf Z}_2)$ in two parts:
\begin{itemize}
\item $y$ odd, so that $z=-(x^2+1)/y$. This has measure $1/2$.
\item $y=2y'$ even, which implies that $y'$ and $x=2x'+1$ odd, so that
\[ z=-(x^2+1)/y= - (1+2x'+2(x')^2)/y'. \]
This has measure $1/4$. Indeed, it parametrizes as
\[
\left\{
\begin{aligned}
x& =1+2u \\ y& = \frac{2}{1+2v} \\ z& = (1+2u+2u^2)(1+2v) =
1+2u+2v+2u^2(1+2v)+4uv, \end{aligned}\right.
\]
with $(u,v)\in({\mathbf Z}_2)^2$.
The differential of this map is given by the matrix
\[ \begin{pmatrix} 2 & 0 \\
0 & -4(1+2v)^{-2} \\
2 + 4u(1+3v) & 2 + (4u^2+4u) \end{pmatrix}
= 2 \begin{pmatrix} 1 & 0 \\ 0& -2(1+2v)^{-2} \\ 1+2u(1+3v) & 1+2u(1+u)
\end{pmatrix}\]
and is twice a $2\times 3$ matrix with coefficients in~${\mathbf Z}_2$,
of which one of the $2\times 2$ is invertible.
Therefore, the measure
of its image is $\abs{2}_2^2=1/4$.
\end{itemize}
Finally, $\vol U({\mathbf Z}_2)=1/2+1/4=3/4$.
\subsubsection{}
The virtual character $\operatorname{EP} (U)$ is $-[\chi]$,
for $\chi=(-1/\cdot)$ the quadratic character corresponding to ${\mathbf Q}(i)$.
We will prove this in general below, however, this can be seen
directly as follows.
First, $U$ is an hypersurface of the affine~$3$-space,
so it has no non-constant invertible function.
Let us now study its Picard group.
Over ${\mathbf Q}(i)$, $X$ is the hypersurface of~${\mathbf P}^3$ defined
by the equation~$(x+it)(-x+it)=yz$, so is isomorphic to~${\mathbf P}^1\times{\mathbf P}^1$,
the two factors being interchanges by the complex conjugation.
Consequently, $\Pic(X_{\bar F})$, being generated by these two lines,
is the Galois module $\mathbf 1\oplus[\chi]$.
The result then follows from the fact that
$\Pic(X_{\bar F})$ maps surjectively to~$ \Pic(U_{\bar F})$,
its kernel being the submodule generated by the class
of the hyperplane of equation~$t=0$ in~${\mathbf P}^3$.
The ${\rm L}$-function of~$\operatorname{EP} (U)$ has local factors
${\rm L}_p(\operatorname{EP} (U),s)=1-\genfrac{(}{)}{}1{-1}p p^{-s}$.
The product ${\rm L}(\operatorname{EP} (U),s)$ of all ${\rm L}_p$ for $p>2$
is exactly ${\rm L}(\chi,s)^{-1}$. At $s=1$, it has neither a pole, nor a zero.
\subsubsection{}
One has
\[ \vol U({\mathbf Z}_2) \prod_{p>2} \vol U({\mathbf Z}_p) \mathrm L_p(1)
= \frac{3}4 \prod_{p>2} \left( 1 - p^{-2} \right)
= \frac {6}{\pi^2}
= \zeta(2)^{-1}.
\]
As ${\rm L}(\chi,s)\zeta(s)= \zeta_{{\mathbf Q}(i)}(s)$,
one has
${\rm L}(\chi,1) = \zeta_{{\mathbf Q}(i)}^*(1)$ and the normalized volume
of $U({\mathbb A}_f)$ is equal to
\[ \zeta_{{\mathbf Q}(i)}^*(1) \frac1{\zeta(2)}. \]
\subsubsection{}
Now we have to compute the volume of the divisor at infinity.
Due to the way matrices are counted in~\cite{shah2000}
(one wants $2x^2+y^2+z^2\leq T^2$)
the natural metric on the tautological bundle of ${\mathbf P}^3$
is given by the formula
\[ \norm{s_P}(x:y:z:t) = \frac{P(x,y,z,t)}{ \max\big(2\abs x^2+\abs y^2 + \abs
z^2 , \abs t^2\big)^{\deg P /2}}, \]
where $P$ is a homogeneous polynomial in $4$~variables
and $s_P$ the corresponding section of $\mathcal O(\deg P)$.
The divisor $D$ has equations $s_1=X^2+YZ+T^2=0$ and $s_2=T=0$ in~${\mathbf P}^3$.
We'll compute its volume using affine coordinates $(y,z,t)$ for ${\mathbf P}^3$
and $x$ for~$D$.
Affine equations of~$D$ corresponding to the two previous sections
are $1+yz+t^2=0$ and $t=0$.
One has
\[ \mathrm d(1+yz+t^2)\wedge \mathrm dz \wedge \mathrm dt
= z \mathrm dy\wedge\mathrm dz\wedge\mathrm dt, \]
and by definition,
\[
\norm{\mathrm dy\wedge\mathrm dz\wedge\mathrm dt}
= \max\big( 2+y^2+z^2,t^2\big)^2 . \]
Consequently, on~$D$, one obtains the following equalities
\[ \lim \frac{\norm{s_1}}{\abs{1+yz+t^2}}= \frac{1}{\max(2+y^2+z^2,t^2)}
=\frac1{2+y^2+z^2} \]
(since $t=0$ on~$D$) and
\[ \lim \frac{\norm{s_2}}{\abs t}= \frac{1}{\max(2+y^2+z^2,t^2)^{1/2}}
= \frac1{(2+y^2+z^2)^{1/2}}. \]
Finally,
\begin{align*}
\norm{\mathrm dz}
& = \norm{z \mathrm dy\wedge\mathrm dz\wedge\mathrm dt}
\lim \frac{\norm{s_1}}{\abs{1+yz+t^2}} \lim \frac{\norm{s_2}}{\abs t} \\
& = \abs{z} \max\big( 2+y^2+z^2,t^2\big)^{1/2}
= \abs z \big( 2+(1/z)^2+z^2\big)^{1/2} \\
& = \big( 1+2z^2+z^4 \big)^{1/2}
= 1+z^2, \end{align*}
so that the canonical measure on $D({\mathbf R})$ is given by
$\abs{\mathrm dz}/(1+z^2)$.
Hence, $\vol D({\mathbf R})=\pi$.
\subsubsection{}
Finally, we obtain that
Shah's constant $C_{X^2+1}$ satisfies
\[ C_{X^2+1} = 2 \vol D({\mathbf R}) \vol U({\mathbb A}_f), \]
compatibly with Theorem~\ref{theo.geometric-volume}.
\section{Metrics, heights, and Tamagawa measures}
\label{sec.general}
\subsection{Metrics and measures on local fields}
\subsubsection{Haar measures and absolute values}\label{subsubsec.local-fields}
Let $F$ be a local field of characteristic zero, \emph{i.e.}, either ${\mathbf R}$,
${\mathbf C}$, or a finite extension
of the field ${\mathbf Q}_p$ of $p$-adic numbers.
Fix a Haar measure $\mu$ on~$F$. Its ``modulus''
is an absolute value $\abs{\cdot}$ on~$F$, defined
by $\mu(a\Omega)=\abs a\mu(\Omega)$
for any $a\in F$ and any measurable subset $\Omega\subset F$.
For $F={\mathbf R}$, this is the usual absolute value, for $F={\mathbf C}$, it is its square.
For $F={\mathbf Q}_p$, it is given by $\abs{p}=1/p$ and if $F'/F$ is a
finite extension, one has $\abs{a}_{F'}=\abs{\mathrm N_{F'/F}(a)}_F$.
\subsubsection{Smooth functions}
If $F$ is ${\mathbf R}$ or ${\mathbf C}$, we say that a function $f$
defined on an open subset
of the $n$-dimensional affine space~$F^n$
is \emph{smooth} if it is $\mathrm C^\infty$.
If $F$ is non-archimedean, this will mean that $f$ is locally
constant. This notion is local and extends to functions defined
on open subsets of $F$-analytic manifolds. Observe moreover
than for any open subset~$U$ of~$F^n$ and any non-vanishing
analytic function~$f$ on~$U$, the function $x\mapsto \abs {f(x)}$
is smooth.
On a compact $F$-analytic manifold~$X$, a smooth function~$f$
has a sup-norm $\norm{f}=\sup_{x\in X}\abs{f(x)}$. In the
archimedean case, using charts (so, non-canonically),
we can also measure norms of derivatives
and define norms $\norm{f}_r$ (measuring the maximum of sup-norms
of all derivatives of~$f$ of orders~$\leq r$).
In the ultrametric case, using a distance~$d$,
we can define a norm $\norm{f}_1$
as follows:
\[ \norm{f}_1 = \norm{f}\big( 1 + \sup _{f(x)\neq f(y)} \frac
{1}{d(x,y)}\big) .\]
For $r> 1$, we define $\norm{f}_r=\norm{f}_1$.
\subsubsection{Metrics on line bundles}\label{subsubsec.metrics}
Let $X$ be an analytic variety over a locally compact valued field $F$
and let $\mathscr L$ be a line bundle on~$X$.
We define a metric on $\mathscr L$
to be a collection of functions $\mathscr L(x)\ra{\mathbf R}_+$,
for all $x\in X$,
denoted by $\ell\mapsto \norm{\ell}$
such that
\begin{itemize}
\item for $\ell\in\mathscr L(x)\setminus\{0\}$, $\norm{\ell}>0$;
\item
for any $a\in F$, $x\in X$ and any $\ell\in\mathscr L(x)$,
$\norm{a\ell}=\abs{a}\norm{\ell}$ ;
\item for any open subset $U\subset X$ and any section $\ell\in
\Gamma(U,\mathscr L)$, the function $x\mapsto \norm{\ell(x)}$
is continuous on~$U$.
\end{itemize}
We say that a metric on a line bundle $\mathscr L$
is smooth if for any non-vanishing local section
$\ell\in\Gamma(U,\mathcal L)$, the function $x\mapsto \norm{\ell(x)}$
is smooth on~$U$.
For a metric to be smooth, it suffices that there exists
an open cover~$(U_i)$ of~$X$, and, for each~$i$, a non-vanishing
section $\ell_i\in\Gamma(U_i,\mathcal L)$ such that the
function $x\mapsto\norm{\ell_i(x)}$ is smooth on~$U_i$.
Indeed, let $\ell\in\Gamma(U,\mathcal L)$ be a local non-vanishing
section of~$\mathcal L$; for each~$i$ there is a non-vanishing
regular function~$f_i\in\mathscr O_X(U_i\cap U)$ such
that $\ell=f_i\ell_i$ on~$U_i\cap U$, hence
$\norm{\ell}=\abs{f_i}\norm{\ell_i}$. Since the absolute value
of a non-vanishing regular function is smooth, $\norm{\ell}$
is a smooth function on~$U_i\cap U$. Since this holds for all~$i$,
$\norm{\ell}$ is smooth on~$U$.
The trivial line bundle $\mathscr O_X$ admits a canonical metric,
defined by $\norm{1}=1$.
If $\mathscr L$ and $\mathscr M$ are two metrized line bundles on~$X$,
there are metrics on $\mathscr L\otimes\mathscr M$ and on
$\mathscr L^\vee$ defined by
\[ \norm{\ell\otimes m}=\norm{\ell}\norm{m},
\qquad \norm{\phi}=\abs{\phi(\ell)} / {\norm{\ell}}, \]
with $x\in X$, $\ell\in\mathscr L(x)$, $m\in\mathscr M(x)$,
and $\phi\in\mathscr L^\vee(x)$.
\subsubsection{Divisors, line bundles and metrics}\label{subsubsec.Q-Cartier}
The theory of the preceding paragraph also applies if
$X$ is an analytic subspace of some $F$-analytic manifold, \emph{e.g.},
an algebraic variety, even if it possesses singularities.
Recall that by definition, a function on such a space~$X$ is
smooth if it extends to a smooth function
in a neighborhood of~$X$ in the ambient space.
Let $D$ be an effective Cartier divisor on~$X$ and let $\mathscr O_X(D)$
be the corresponding line bundle. It admits a canonical
section~$\mathsec f_D$, whose divisor
is equal to~$D$. If $\mathscr O_X(D)$ is endowed with a metric,
the function~$\norm{\mathsec f_D}$ is positive on~$X$ and
vanishes along~$D$.
More generally, let $D$ be an effective ${\mathbf Q}$-divisor,
that is, a linear combination of irreducible divisors
with rational coefficients such that a multiple~$nD$,
for some positive integer~$n$, is a Cartier divisor.
By a metric on~$\mathscr O_X(D)$ we mean
a metric on $\mathscr O_X(nD)$. By $\norm{\mathsec f_D}$,
we mean the function $\norm{\mathsec f_{nD}}^{1/n}$.
It does not depend on the choice of~$n$.
\subsubsection{Metrics defined by a model}
Here we assume that $F$ is non-archimedean, and let
$\mathfrak o_F$ be its ring of integers.
Let $X$ be a projective variety over~$F$ and $L$
a line bundle on~$X$. Choose
a projective flat $\mathfrak O_F$-scheme $\mathscr X$ and a line bundle
$\mathscr L$ on~$\mathscr X$ extending $X$ and $L$.
These choices determine a metric on the line bundle defined by $\mathscr
L$ on the analytic variety $X(F)$, by the following recipe:
for $x\in X(F)$ let $\tilde x\colon \Spec\mathfrak o_F\ra\mathscr
X$ be the unique morphism extending~$x$; by definition, the
set of $\ell\in L(x)$ such that $\norm{\ell}\leq 1$ is
equal to $\tilde x^*\mathscr L$, which is a lattice in $L(x)$.
Let~$\mathscr U$ be an open subset of~$\mathscr X$ over which
the line bundle~$\mathscr L$ is trivial and let $\eps\in\Gamma(\mathscr U,\mathscr L)$ be a trivialization of~$\mathscr L$ on~$\mathscr U$.
Then, for any point $x\in \mathscr U(F)$ such that $x$
extends to a morphism $\tilde x\colon\Spec\mathfrak o_F\ra\mathscr U$,
one has $\norm{\eps(x)}=1$. Indeed, $\tilde x^*\eps$ is a basis
of the free $\mathfrak o_F$-module~$\tilde x^*\mathscr L$.
Since $\mathscr X$ is projective, restriction to the
generic fiber identifies the set $\mathscr U(\mathfrak o_F)$ with
a compact open subset of~$\mathscr U_F(F)$,
still denoted by $\mathscr U(\mathfrak o_F)$.
Observe that these compact open subsets cover~$X(F)$.
Let $F'/F$ be a field extension. A model of $X$ determines
a model of $X_{F'}=X\otimes_F F'$ over the ring $\mathfrak o_{F'}$,
and thus metrics on the analytic variety $X(F')$.
Conversely, note that a model of $X_{F'}$ determines metrics
on $X(F')$ and hence, by restriction, also on $X(F)$.
\subsubsection{Example: projective space}\label{exam.projective-space.metric}
Let $F$ be a locally valued field and $V$ a finite-dimensional
vector space over~$F$. Let~$\abs\cdot$ be a norm on~$V$.
Let~${\mathbf P}(V^*)=\Proj{\operatorname{Sym}}^\bullet V^*$ be the projective space
of lines in~$V$; denote by~$[x]$ the point of~${\mathbf P}(V^*)$
associated to a non-zero element~$x\in V$, \emph{i.e.},
the lines it generates.
This projective space carries a tautological
ample line bundle, denoted $\mathscr O(1)$, whose
space of global sections is precisely~$V^*$.
The formula
\[ \norm{\ell([x])} = \frac {\abs{\ell(x)}}{\norm{x}}, \]
for $\ell\in V^*$ and $x\in V$,
defines a norm on the $F$-vector space~$\mathscr O(1)_{[x]}$.
These norms define a metric on~$\mathscr O(1)$.
Assume moreover that $F$ is non-archimedean and let~$\mathfrak o_F$
be its ring of integers. If the norm of~$V$ takes
its values in the set~$\abs{F^\times}$,
the unit ball~$\mathscr V$ of~$V$
is an~$\mathfrak o_F$-submodule of~$V$ which is free of rank~$\dim V$
(\cite{weil1967}, p.~28, Proposition~6 and p.~29).
Then, the $\mathfrak o_F$-scheme ${\mathbf P}(\mathscr V^\vee)$
is a model of~${\mathbf P}(V^\vee)$ and the metric on~$\mathscr O(1)$
is the one defined by this model and the canonical extension
of the line bundle~$\mathscr O(1)$ it carries.
\subsubsection{Volume forms}\label{subsubsec.volume-forms}
Let $X$ be an analytic manifold over a local field $F$.
To simplify the exposition, assume that $X$ is equidimensional
and let~$n$ be its dimension.
It is standard when $F={\mathbf R}$ or ${\mathbf C}$, and also true in general,
that any $n$-form $\omega$ on an open subset $U$ of~$X$ defines
a measure, usually denoted $\abs\omega$, on $U$, through the following
formula. Choose local coordinates $x_1,\dots,x_n$ on $U$ and write
\[ \omega= f(x_1,\dots,x_n) \,\mathrm dx_1\wedge \dots\wedge \mathrm dx_n.
\]
These coordinates allow to identify (a part of) $U$ with an open
subset of $F^n$ which we endow with the product measure $\mu^n$.
(Recall that a Haar measure has been fixed on~$F$
in~\ref{subsubsec.local-fields}.)
It pulls back to a measure which we denote
\[ \abs{\mathrm dx_1} \, \cdots \, \abs{\mathrm dx_n} \]
and by definition, we let
\begin{equation} \label{eq.abs.omega}
\abs\omega= \abs{f(x_1,\dots,x_n)}\,
\abs{\mathrm dx_1} \, \cdots \, \abs{\mathrm dx_n} .\end{equation}
By the change of variables formula in multiple integrals,
this is independent of the choice of local coordinates.
\subsubsection{Metrics and measures}\label{ss.metrics-measures}
Certain manifolds~$X$ possess a canonical
(up to scalar) non-vanishing~$n$-form,
sometimes called a \emph{gauge form}.
Examples are analytic groups, or Calabi-Yau varieties.
This property leads to the definition of a canonical measure on~$X$
(again, up to a scalar).
In the case of groups, this has been studied by Weil in the
context of Tamagawa numbers (\cite{weil82});
in the case of Calabi-Yau varieties, this measure has been used
by Batyrev to prove that smooth birational Calabi-Yau varieties
have equal Betti numbers (\cite{batyrev99}).
Even when $X$ has no global $n$-forms, it is still possible to define a measure
on $X$ provided the canonical line bundle $\omega_X=\bigwedge^n \Omega^1_X$
is endowed with a \emph{metric}. Indeed, we may then
attach to any local non-vanishing $n$-form
$\omega$ the local measure $\abs\omega/\norm\omega$. It
is immediate that these measures
patch and define a measure~$\tau_X$ on the whole of~$X$. This is a Radon
measure locally equivalent to any Lebesgue measure.
In particular, if $X$ is compact, its volume
with respect to this measure is a positive real number.
This construction is classical in differential geometry
as well as in arithmetic geometry (see~\cite{serre1981}, bottom
of page~146);
its introduction in the context of the counting problem
of points of bounded height on algebraic varieties over number
fields is due to E.~Peyre (\cite{peyre95}).
\subsubsection{Singular measures}
For the study of integral points,
we will have to consider several variants of this construction.
We assume here that $X$ is an algebraic variety over a local field~$F$
and that we are given an auxiliary effective Cartier ${\mathbf Q}$-divisor~$D$ in~$X$.
Let $U=X\setminus\abs D$ be the complement of the support of~$D$
in~$X$.
Let us first consider the case where $D$ is a Cartier divisor
and assume that the line bundle $\omega_X(D)$ is endowed with
a \emph{metric}.
If $\mathsec f_D$ is the canonical section of~$\mathcal O_X(D)$
and $\omega$ a local $n$-form, we can then consider the nonnegative
function~$\norm{\mathsec f_D\omega}$; it vanishes on~$\abs D$.
In the general case, we follow the conventions
of~Section\ref{subsubsec.Q-Cartier} and will freely
talk of the function~$\norm{\omega\mathsec f_D}$,
defined as $\norm{\omega^n\mathsec f_{nD}}$ where $n$ is any
positive integer such that~$nD$ is a Cartier divisor,
assuming the line bundle $\omega_X^{\otimes n}(nD)$ is
endowed with a metric.
Then, the measures $\abs\omega/\norm{\mathsec f_D\omega}$,
for $\omega$ any local non-vanishing $n$-form
$\omega$, patch and define a measure~$\tau_{(X,D)}$
on the open submanifold~$U(F)$.
This measure is a Radon measure on~$U(F)$, locally equivalent to the Lebesgue
measure.
However, the volume of~$U(F)$ is infinite;
this is in particular the case if $D$ is a divisor (with integer
coefficients) and the smooth locus of~$D$ has $F$-rational points.
We shall always identify this measure and the (generally ``singular'',
\emph{i.e.}, non-Radon) measure on~$X(F)$ obtained by push-out.
\subsubsection{Example: gauge forms}
Let us show how the definition of a measure using a gauge form
can be viewed as a particular case of this construction.
Let~$\omega$ be a meromorphic differential form on~$X$; let
$D$ be the \emph{opposite} of its divisor. One can write
$D=\sum d_\alpha D_\alpha$,
where the $D_\alpha$ are codimension~$1$ irreducible subvarieties in~$X$
and $d_\alpha$ are integers.
Let $U=X\setminus \abs D$ be the complement of the support of~$D$.
In other words, $\omega$
defines a trivialization of the line bundle $\omega_X(D)$.
There is a unique metrization of~$\omega_X(D)$ such that~$\omega$,
viewed as a global section of~$\omega_X(D)$,
has norm~$1$ at every point in~$X$.
The measure on the manifold~$U(F)$
defined by this metrization coincides with the measure~$\abs\omega$
defined by $\omega$ as a gauge form.
Moreover, if the line bundle $\mathcal O_X(D)$ is endowed
with a metric, then so is the line bundle~$\omega_X$.
In this case, the manifold~$X(F)$ is endowed with three measures,
$\tau_U$, $\tau_X$ and~$\tau_{(X,D)}$. Locally, one has:
\begin{itemize}
\item $\tau_X=\abs\omega/\norm{\omega}$;
\item the measure $\tau_U$ is its restriction to~$U$;
\item $\tau_{(X,D)}=\abs\omega=\abs\omega/\norm{\mathsec f_D\omega}
= \norm{\mathsec f_D}^{-1} \tau_X$.
\end{itemize}
Note that on each open, relatively compact subset of~$X(F)\setminus \abs D$,
the measures $\tau_X$ and~$\tau_{(X,D)}$ are comparable.
\subsubsection{Example: compactifications of algebraic groups}\label{subsubsec.haar-tamagawa}
We keep the notation of the previous paragraph,
assuming moreover that $U$ is an algebraic group~$G$ over~$F$
of which $X$ is an equivariant compactification,
and that the restriction to~$G$ of~$\omega$ is invariant.
If we consider~$\omega$ as a gauge form, it then defines
a invariant measure $\abs\omega$ on~$G(F)$, in other words,
a Haar measure on this locally compact group,
and also $\tau_{(X,D)}$.
\subsubsection{Residue measures}\label{subsubsec.residue-measures}
Let us return to the case of a general manifold~$X$.
Let $Z$ be a closed submanifold of $X$. To get a measure on~$Z$,
we need as before a metrization on $\omega_Z$. However, the ``adjunction
formula''
\[ \omega_X|_Z \simeq \omega_Z \otimes \det \mathcal N_Z (X) \]
implies that given a metric on $\omega_X$, it suffices
to endow the determinant of the normal bundle of $Z$ in $X$
with a metric.
The case where $Z=D$ is a divisor (\emph{i.e.},
locally in charts, $Z$ is defined by the vanishing of a coordinate)
is especially interesting.
In that case, one has
\[ \omega_D = \omega_X(D)|_D \]
and a metric on $\omega_X$ plus a metric on $\mathcal O_X(D)$
automatically define a metric on $\omega_D$, hence a measure on
$D$.
Let us give an explicit formula for this measure. Let
$\omega\in\Gamma(\omega_D)$ be a local $(n-1)$-form and $\tilde\omega$
any lift of $\omega$ to $\bigwedge^{n-1}\Omega^1_X$.
If $u\in\mathcal O(-D)$ is a local equation for $D$,
the image of $\omega$ in $\omega_X(D)|_D$ is nothing but
the restriction to~$D$ of the differential form with logarithmic
poles
$\tilde\omega\wedge u^{-1}\mathrm du$. Let $\mathsec f_D$ be
the canonical section of $\mathcal O_X(D)$. Then,
\[ \norm{\omega}=\norm{\tilde\omega\wedge u^{-1}\mathrm du}
= \norm{\tilde\omega\wedge \mathrm du}
\, \norm{u^{-1}} . \]
By definition, $\mathsec f_D$ corresponds locally to the function~1,
hence, for $x\not\in D$,
\[ \norm{u^{-1}(x)} = \frac{1}{\abs {u(x)}} \norm{\mathsec f_D(x)} \]
and this possesses a finite limit when $x$ approaches~$D$,
by the definition of a metric.
(As a section of $\mathscr O_X(D)$, $\mathsec f_D$ vanishes at order~$1$
on $D$). Moreover, the function $\lim \norm{\mathsec f_D}/\abs u$
on~$D$ defined by
\[ x\mapsto \lim_{\substack{y\ra x\\ y\not\in D}}
\norm{\mathsec f_D(y)}/\abs{u(y)} \]
is continuous and positive on~$D$.
By induction, if $Z$ is the transverse intersection of smooth divisors $D_j$
($1\leq j\leq m$),
with metrizations on all $\mathscr O_X(D_j)$, we have a similar formula:
\begin{equation}
\label{eq.abs.omega.D}
\norm{\omega} = \norm{\tilde\omega\wedge \mathrm du_1\wedge\cdots \wedge \mathrm du_m}
\lim \frac{\norm{\mathsec f_{D_1}}}{\abs{u_1}}\, \cdots
\lim \frac{\norm{\mathsec f_{D_m}}}{\abs{u_m}},
\end{equation}
where $u_1,\ldots,u_m$ are local equations for the divisors
$D_1,\ldots,D_m$.
\subsubsection{Residue measures in an algebraic context}
\label{sss.residue-algebraic}
Let $X$ be a smooth algebraic variety over~$F$; endow the
canonical line bundle $\omega_X$ on~$X(F)$ with a metric.
Let~$F'$ be a finite Galois extension of~$F$ and
let~$D_j$, for $1\leq j\leq m$, be smooth
irreducible divisors on~$X_{F'}$, whose union is
a divisor~$D$ with strict normal crossings which is defined over~$F$.
Then the intersection $Z=\bigcap_{j=1}^m D_j$ is defined over~$F$;
if $Z(F)\neq\emptyset$, then $Z(F)$ is a smooth $F$-analytic submanifold
of $X(F)$, of dimension~$\dim Z=\dim X-m$.
Moreover, the normal bundle of~$Z$ in~$X_{F'}$
is isomorphic to the restriction
to~$Z$ of the vector bundle $\bigoplus_{j=1}^m\mathscr O_{X'}(D_j)$,
hence the isomorphism $\det\mathscr N_Z(X)\simeq\mathscr O_X(D)$,
at least after extending the scalars to~$F'$.
Endow the line bundles $\mathscr O_{X_{F'}}(D_j)$
on the ${F'}$-analytic manifold~$X({F'})$ with metrics.
By the previous formulas, we obtain from the
metrics on~$\mathscr O_{X_{F'}}(D_j)$ a metric on
the determinant of the normal bundle of~$Z({F'})$ in the manifold~$X({F'})$.
The restriction of this metric to~$Z(F)$ gives us a metric
on the determinant of the normal bundle of~$Z(F)$ in~$X(F)$.
Accordingly,
we obtain a positive Radon measure~$\tau_Z$ on~$Z(F)$
which is locally comparable to any Lebesgue measure.
In particular, one has $\tau_Z(Z(F))=0$ if and only if $Z(F)=\emptyset$.
\subsection{Adeles of number fields: metrics and heights}
\subsubsection{Notation}
We specify some common notation concerning number fields
and adeles that we will use throughout this text.
Let $F$ be a number field and
$\Val(F)$ the set of places of~$F$.
For $v\in \Val(F)$, we let $F_v$ be the $v$-adic completion of~$F$.
This is a local
field; its absolute value, defined as in~\S\ref{subsubsec.local-fields},
is denoted by~$\abs\cdot_v$.
With these normalizations, the \emph{product formula} holds. Namely,
for all $a\in F^*$, one has
$\prod_{v\in\Val(F)} \abs a_v=1$,
where only finitely many factors differ from~$1$.
Let $v\in\Val(F)$. The absolute value $\abs{\cdot}_v$
is archimedean when $F_v={\mathbf R}$ or $F_v={\mathbf C}$; we will
say that $v$ is infinite, or archimedean.
Otherwise, the absolute value~$\abs\cdot_v$ is
ultrametric and the place $v$ is called finite, or non-archimedean.
Let $v$ be a finite place of~$F$.
The set~$\mathfrak o_v$ of all $a\in F_v$
such that $\abs a_v\leq 1$ is a subring of~$F_v$,
called the ring of $v$-adic integers. Its subset $\mathfrak m_v$
consisting of all $a\in F_v$ such that $\abs a_v<1$ is
its unique maximal ideal. The residue field $\mathfrak o_v/\mathfrak m_v$
is denoted by~$k_v$. It is a finite field and we write~$q_v$
for its cardinality. The ideal~$\mathfrak m_v$
is principal; a generator will be called a uniformizing
element at~$v$. For any such element~$\varpi$, one has
$\abs\varpi_v=q_v^{-1}$.
Fix an algebraic closure~$\bar F$ of~$F$. The group~$\Gamma_F$
of all $F$-automorphisms of~$\bar F$ is called the absolute
Galois group of~$F$.
For any finite place~$v\in\Val(F)$, fix an extension~$\abs{\cdot}_{\bar v}$ of
the absolute value~$\abs\cdot_v$ to~$\bar F$.
The subgroup of~$\Gamma_F$ consisting of all $\gamma\in\Gamma_F$
such that $\abs{\gamma(a)}_{\bar v}=\abs{a}_{\bar v}$
for all $a\in\bar F$ is called the decomposition subgroup of~$\Gamma_F$
at~$v$ and is denoted~$\Gamma_v$.
These data determine an algebraic closure~$\bar k_v$ of the residue field~$k_v$,
together with a surjective group homomorphism $\Gamma_v\ra\Gal(\bar k_v/k_v)$.
Its kernel~$\Gamma^0_v$ is called the inertia subgroup of~$\Gamma_F$ at~$v$.
Any element in~$\Gamma_v$
mapping to the Frobenius automorphism $x\mapsto x^{q_v}$
in~$\Gal(\bar k_v/k_v)$ is called an arithmetic Frobenius element at~$v$;
its inverse is called a geometric Frobenius element at~$v$.
The subgroups~$\Gamma_v$ and~$\Gamma_v^0$, and Frobenius elements, depend
on the choice of the chosen extension of the absolute value~$\abs\cdot_v$;
another choice changes them by conjugation in~$\Gamma_F$.
The ring of adeles~${\mathbb A}_F$ of the field~$F$ is the restricted product
of all local fields~$F_v$, for $v\in\Val(F)$,
with respect to the subrings $\mathfrak o_v$ for finite places~$v$.
It is a locally compact topological ring and carries
a Haar measure~$\mu$. The quotient space~${\mathbb A}_F/F$ is compact;
we shall often assume that $\mu$ is normalized so that
$\mu({\mathbb A}_F/F)=1$.
\subsubsection{The adelic space of an algebraic variety}
Let $U$ be an algebraic variety over~$F$.
The space $U({\mathbb A}_F)$ of adelic points of~$U$ has a natural locally
compact topology which we recall now.
Let $\mathscr U$ be a model of~$U$ over the integers of~$F$,
\emph{i.e.}, a scheme which is flat and of finite type over~$\Spec\mathfrak o_F$
together with an isomorphism of~$U$ with $\mathscr U\otimes F$.
The natural maps ${\mathbb A}_F\ra F_v$ induce a map from
$U({\mathbb A}_F)$ to $\prod_{v\in \Val(F)} U(F_v)$. It is not surjective
unless $U$ is proper over~$F$; indeed its image --- usually
called the restricted product --- can be described
as the set of all $(x_v)_v$ in the product such that
$x_v\in\mathscr U(\mathfrak o_v)$ for almost all finite places~$v$.
Since two models are isomorphic over a dense open subset
of~$\Spec\mathfrak o_F$,
observe also that this condition is independent of the choice of the
specific model~$\mathscr U$.
We endow each $U(F_v)$ with the natural $v$-adic topology;
notice that it is locally compact.
Moreover, for any finite place~$v$,
$\mathscr U(\mathfrak o_v)$ is open and compact in~$U(F_v)$ for this topology.
We then endow $U({\mathbb A}_F)$ with the ``restricted product topology'',
a basis of which is given by products $\prod_v \Omega_v$,
where for each~$v\in\Val(F)$, $\Omega_v$ is an open subset of~$U(F_v)$,
subject to the additional condition that $\Omega_v=\mathscr U(\mathfrak o_v)$
for almost all finite places~$v$.
Let $\Omega\subset U({\mathbb A}_F)$ be any subset of the form $\prod_v \Omega_v$,
with $\Omega_v\subset U(F_v)$, such that $\Omega_v=\mathscr U(\mathfrak o_v)$
for almost all finite places~$v$.
Then $\Omega$ is compact if and only each~$\Omega_v$ is compact.
It follows that the topology on~$U({\mathbb A}_F)$ is locally compact.
\subsubsection{Adelic metrics}\label{subsubsec.adelic-metrics}
Let $X$ be a proper variety over a number field~$F$
and
$\mathscr L$ a line bundle on~$X$.
An adelic metric on~$\mathscr L$ is a collection of $v$-adic metrics
on the associated line bundles on the $F_v$-analytic varieties $X(F_v)$,
for all places $v$ in~$F$,
which, except for a finite number of them, are defined by a single
model $(\mathscr X,\mathscr L)$ over the ring of integers of~$F$.
By standard properties of schemes of finite presentation, any two
flat proper models are isomorphic at almost all places
and will therefore define the same $v$-adic metrics at these places.
\subsubsection{Example: projective space}
\label{exam.projective-space.adelic.metric}
For any valued field~$F$, let us endow the vector-space~$F^{n+1}$
with the norm given by
$\abs{(x_0,\ldots,x_n)}=\max(\abs{x_0},\ldots,\abs{x_n})$.
When $F$ is a number field and $v\in\Val(F)$,
the construction~\ref{exam.projective-space.metric} on the field~$F_v$
furnishes
a metric on the line bundle~$\mathscr O(1)$ on~$\mathbf P^n$.
This collection of metrics is an adelic metric on~$\mathscr O(1)$,
which we will call the \emph{standard adelic metric}.
\subsubsection{Extension of the ground field}
If $F'$ is any finite extension of~$F$, observe that
an adelic metric on the line bundle~$\mathscr L\otimes F'$
over~$X\otimes F'$ induces by restriction an adelic metric on
the line bundle~$\mathscr L$ over~$X$. We only need
to check that the family of metrics defined by restriction
are induced at almost all places by a model of~$X$ on~$F$.
Indeed, fix a model~$(\mathscr X',\mathscr L')$ of~$(X\otimes F',
\mathscr L\otimes F')$ over~$\Spec\mathfrak o_{F'}$,
as well as a model~$(\mathscr X,\mathscr L)$ over~$\Spec\mathfrak o_F$.
There is a non-zero integer~$N$
such that the identity map $X\otimes F'\ra X\otimes F'$
extends to an isomorphism $\mathscr X\otimes \mathfrak o_{F'}[\frac1N]
\ra\mathscr X'\otimes\mathfrak o_{F'}[\frac1N]$.
Consequently, at all finite places of~$F'$ which do not divide~$N$,
both models define the same metric on~$L\otimes F'$.
It follows that the metrics of~$L$ are defined
by the model~$(\mathscr X,\mathscr L)$
at all but finitely many places of~$F$.
\subsubsection{Heights}\label{subsubsec.heights}
Let $X$ be a proper variety over a number field~$F$
and let $\mathscr L$ be a line bundle on~$X$ endowed
with an adelic metric.
Let $x\in X(F)$ and let $\ell$ be any non-zero element in $\mathscr L(x)$.
For almost all places~$v$, one has $\norm{\ell}_v=1$;
consequently, the product $\prod_v \norm{\ell}_v$
converges absolutely. It follows from the product formula
that its value does not depend on the choice of~$\ell$.
We denote it by $H_{\mathscr L}(x)$ and call it
the (exponential) \emph{height} of~$x$ with respect to the
metrized line bundle~$\mathscr L$.
As an example, assume that $X=\mathbf P^n$ and $\mathscr L=\mathscr O(1)$,
endowed with its standard adelic metric (\ref{exam.projective-space.adelic.metric}). Then for any point $x\in\mathbf P^n(F)$ with
homogeneous coordinates $[x_0\cdots:x_n]$ such that $x_0\neq 0$,
we can take the element~$\ell$ to be the value at~$x$ of
the global section~$X_0$. Consequently, the definition
of the standard adelic metric implies that
\[ H_{\mathscr L}(x)
= \prod_{v\in\Val(F)} \left(\frac{\abs{x_0}_v}{\max(\abs{x_0}_v,\ldots,\abs{x_n}_v)}\right)^{-1}
= \prod_{v\in\Val(F)} \max(\abs{x_0}_v,\ldots,\abs{x_n}_v), \]
in view of the product formula $\prod_{v\in\Val(F)}\abs{x_0}_v=1$.
We thus recover the classical (exponential) height of the point~$x$
in the projective space.
In general, the function
\[ \log H_{\mathscr L} \colon X(F)\ra {\mathbf R} \]
is a representative of the class of height functions
attached to the line bundle~$\mathscr L$
using Weil's classical ``height machine''
(see~\cite{hindry-silverman2000}, especially Part~B, \S10).
When $\mathscr L$ is ample, the set of points $x\in X(F)$
such that $H_{\mathscr L}(x)\leq B$ is finite for any real
number~$B$ (Northcott's theorem).
\subsection{Heights on adelic spaces}
\label{subsec.heights.adelic}
Let $\mathsec f$ be a non-zero global section of~$\mathscr L$
and let~$Z$ denote its divisor.
We extend the definition of $H_{\mathscr L}$
to the adelic space $X({\mathbb A}_F)$ by defining
\begin{equation}
\label{eq.height.f}
H_{\mathscr L,\mathsec f}(\mathbf x) = \big(\prod_v \norm{\mathsec f}_v(x)
\big)^{-1}
\end{equation}
for any $\mathbf x\in X({\mathbb A}_F)$ such that the infinite product makes sense.
To give an explicit example,
assume again that $X$ is the projective space~$\mathbf P^n$
and that $\mathscr L$ is the line bundle~$\mathscr O_{\mathbf P}(1)$ endowed
with its standard adelic metric;
let us choose $\mathsec f=x_0$.
Then, $U=X\setminus \div(s)$ can be identified to the set of points~$[1:\cdots:x_n]$ of~$\mathbf P^n$ whose first homogeneous coordinate
is equal to~$1$, \emph{i.e.}, to the affine space~$\mathbf A^n$.
For any $\mathbf x=(x_v)_\in U({\mathbb A}_F)$, given by $\mathbf x=[1:\mathbf x_1:\cdots:\mathbf x_n]$, with
$\mathbf x_i=(x_{i,v})_v\in {\mathbb A}_F$, one has
\[ H_{\mathscr L,\mathsec f}(x)
= \prod_{v\in\Val(F)} \max(1,\abs{x_{1,v}}_v,\ldots,\abs{x_{n,v}}_v). \]
\begin{lemm}\label{lemm.northcott.adelic}
Let $U$ be any open subset of~$X$.
\begin{enumerate} \def\alph{enumi}}\def\labelenumi{\theenumi){\roman{enumi}}
\item For any $\mathbf x\in X({\mathbb A}_F)$,
the infinite product defining~$H_{\mathscr L,\mathsec f}(\mathbf x)$
converges to an element of $(0,+\infty]$.
\item
The resulting function is lower semi-continuous and admits a positive
lower bound on the adelic space $X({\mathbb A}_F)$.
\item
If $\mathsec f$ does not vanish on~$U$,
then the restriction of~$H_{\mathscr L,\mathsec f}$
to $U({\mathbb A}_F)$ is continuous (for the topology of~$U({\mathbb A}_F)$).
\item
Assume that $X=U\cup Z$. Then, for any real number~$B$,
the set of points
$\mathbf x\in U({\mathbb A}_F)$ such that $H_{\mathscr L,\mathsec f}(\mathbf x)\leq
B$ is a compact subset of~$U({\mathbb A}_F)$.
\end{enumerate}
\end{lemm}
\begin{proof}
For any place~$v$, let us set $c_v=\max(1,\norm{\mathsec f}_v)$.
One has $c_v<\infty$ since $X(F_v)$ is compact and the
function $\norm{\mathsec f}_v$ on $X(F_v)$ is continuous.
Moreover, it follows from
the definition of an adelic metric that $\norm{\mathsec f}_v=1$
for almost all~$v$, so that $c_v=1$ for almost all places~$v$.
Consequently,
the infinite series $\sum_{v\in\Val(F)} \log\norm{\mathsec f_v}^{-1}$
has (almost all of) its terms nonnegative; it converges to a real
number or to~$+\infty$. This shows that the infinite
product $H_{\mathscr L,\mathsec f}(\mathbf x)$
converges to an element in~$(0,+\infty]$.
Letting $C=1/\prod_v c_v$,
one has $H_{\mathscr L,\mathsec f}(\mathbf x)\geq C$
for any~$\mathbf x\in X({\mathbb A}_F)$, which proves~(i) and
the second half of~(ii).
Let $V_1$ be the set of places $v\in\Val(F)$ such that $c_v>1$;
one has
\[ \log H_{\mathscr L,\mathsec f}(\mathbf x)
= \sup_{V_1\subset V\subset\Val(F)} \sum_{v\in V} \log\norm{\mathsec f_v}^{-1},
\]
where $V$ runs within the finite subsets of $\Val(F)$ containing~$V_1$.
This expression shows that
the function $\mathbf x\mapsto \log H_{\mathscr L,\mathsec f}(\mathbf x)$
is a supremum of lower semi-continuous functions, hence
is lower semi-continuous on~$X({\mathbb A}_F)$, hence the remaining
part of Assertion~(ii).
Let us now prove~(iii). Let $\mathscr X$
be a flat projective model of~$X$ over~$\mathfrak o_F$,
let $\mathscr Z$ be the Zariski closure of~$Z$
and $\mathscr V=\mathscr X\setminus\mathscr Z$,
let $\mathscr D$ be the Zariski closure of~$X\setminus U$
and let $\mathscr U=\mathscr X\setminus\mathscr D$.
By definition of an adelic metric,
for almost all finite places~$v\in\Val(F)$,
$\norm{f}_v$ is identically equal to~$1$ on~$\mathscr U(\mathfrak o_v)$.
By definition of the topology of $U({\mathbb A}_F)$, this implies that
the function~$H_{\mathscr L,\mathsec f}$
is locally given by a finite product of continuous functions;
it is therefore continuous.
Let us finally establish~(iv).
Consider integral models as above; since $U$ contains~$V$
by assumption, then $\mathscr U\supset \mathscr V$.
For any $\mathbf x=(x_v)_v\in X({\mathbb A}_F)$,
one has
\[ \norm{\mathsec f}(x_v) = H_{\mathscr L,\mathsec f}(\mathbf x)^{-1}
\prod_{w\neq v} \norm{\mathsec f}(x_v)^{-1}
\geq C^{-1} H_{\mathscr L,\mathsec f}(\mathbf x)^{-1} .
\]
If moreover $H_{\mathscr L,\mathsec f}(\mathbf x)\leq B$,
then $\norm{\mathsec f}(x_v)\geq (BC)^{-1}$.
The set of points~$x_v\in X(F_v)$ satisfying this inequality
is a closed subset of the compact space~$X(F_v)$, hence is compact.
Moreover, this set is contained in $U(F_v)$ because,
by assumption, $\mathsec f$ vanishes on $X\setminus U$.
Consequently, this set is a compact subset of $U(F_v)$.
If the cardinality~$q_v$ of the residue field~$k_v$
is big enough, so that the $v$-adic metric on~$\mathscr L$
is defined by the line bundle~$\mathscr O(\mathscr Z)$ at the place~$v$,
then $-\log\norm{\mathsec f}_v(x_v)$ is a non-negative integer
times $\log q_v$. The inequality $-\log\norm{\mathsec f}_v(x_v)\leq\log(BC)$
then implies that $-\log\norm{\mathsec f}_v(x_v)=0$.
Let $E_B$ denote the set of points $\mathbf x\in U({\mathbb A}_F)$
such that $H_{\mathscr L,\mathsec f}(\mathbf x)\leq B$.
By what we proved, there exists a finite set of places~$V$
(depending on~$B$) such that
$E_B$ is the product of a compact subset of $\prod_{v\in V} U(F_v)$
and of $\prod_{v\not\in V} \mathscr V(\mathfrak o_v)$.
Since $\mathscr V(\mathfrak o_v)\subset\mathscr U(\mathfrak o_v)$,
we see that $E_B$ is relatively compact in $U({\mathbb A}_F)$.
By lower semi-continuity, it is also closed in~$U({\mathbb A}_F)$,
hence compact.
\end{proof}
We observe that the hypotheses in~(iii) and~(iv) are actually necessary.
They hold in the important case where
$U=X\setminus Z$; then, $H_{\mathscr L,\mathsec f}$
defines a continuous exhaustion of $U({\mathbb A}_F)$ by compact subsets.
\smallbreak
\subsection{Convergence factors and Tamagawa measures on adelic spaces}\label{subsec.global-tamagawa}
\nobreak
\subsubsection{Volumes and local densities} \label{ss.local.densities}
Let $X$ be a smooth, proper, and geometrically integral
algebraic variety over a number field~$F$.
Fix an adelic metric on the canonical line bundle~$\omega_X$;
by the results recalled in~\ref{subsubsec.volume-forms},
for any place $v$ of~$F$, this induces a measure~$\tau_{X,v}$ on $X(F_v)$
and its restriction~$\tau_{U,v}$ to~$U(F_v)$.
Let $U$ be a Zariski open subset of~$X$ and let $Z=X\setminus U$.
Fix a model $\mathscr X$ of~$X$, and
let $\mathscr Z$ be the Zariski closure of~$Z$ in~$\mathscr X$
and $\mathscr U=\mathscr X\setminus \mathscr Z$.
By a well-known formula going back to Weil (\cite{weil82},
see also~\cite{salberger98}, Corollary~2.15),
the equality
\begin{equation} \label{eq.weil-formula}
\tau_{U,v}(\mathscr U(\mathfrak o_v)) = q_v^{-\dim X} \mathop\# \mathscr U(k_v) .
\end{equation}
holds for almost all non-archimedean places~$v$.
\subsubsection{Definition of an ${\rm L}$-function}
\begin{defi}\label{defi.P(u)}
We define $\operatorname{EP} (U)$ to be the following virtual
${\mathbf Q}[\Gamma_F]$-module:
\[ \left[\mathrm H^0(U_{\bar F},{\mathbf G}_m)/\bar F^* \right]_{\mathbf Q} -
\left[\mathrm H^1(U_{\bar F},{\mathbf G}_m)\right]_{\mathbf Q}.
\]
\end{defi}
It is an object of the Grothendieck group of the category
of finite dimensional ${\mathbf Q}$-vector spaces endowed with a continuous
action of~$\Gamma_F$, meaning that there is a finite extension~$F'$
of~$F$ such that the subgroup $\Gamma_{F'}$ acts trivially.
Such a virtual Galois-module
(we shall often skip the word ``virtual'')
has an Artin L-function, given by an Euler product
\[ \mathrm L(s,\operatorname{EP} (U))= \prod_{\text{$v$ finite}} \mathrm L_v(s,\operatorname{EP} (U)),
\qquad
\mathrm L_v(s,\operatorname{EP} (U))=\det\big(1-q_v^{-s}\Fr_v\big|\operatorname{EP}(U)^{\Gamma_v^0}\big)^{-1},
\]
where $\Fr_v$ is a geometric Frobenius element
and $\Gamma_v^0$ is an inertia subgroup at the place~$v$.
\begin{lemm}
For any finite place~$v$ of~$F$, $\mathrm L_v(s,\operatorname{EP} (U))$
is a positive real number.
\end{lemm}
\begin{proof}
We have $\mathrm L_v(s,\operatorname{EP} (U))=q_v^{s\rang \operatorname{EP} (U)}f(q_v^s)$,
where the rational function
\[
f(X)=\det(X-\Fr_v|\operatorname{EP} (U))
\]
is the virtual characteristic polynomial
of~$\Fr_v$ acting on~$\operatorname{EP} (U)$.
Since the action of~$\Gamma_F$ on~$\operatorname{EP} (U)$ factorizes
through a finite quotient, the rational function
has only roots of unity
for zeroes and poles.
By irreducibility
of the cyclotomic polynomials~$\Phi_n$ in~${\mathbf Q}[X]$, this
implies the existence of rational
integers $(a_n)_{n\geq 1}$, almost all zero,
such that
\[ \det(X-\Fr_v|\operatorname{EP} (U)) = \prod_{n\geq 1} \Phi_n(X)^{a_n}. \]
From the inductive definition of the cyclotomic polynomials~$\Phi_n$,
namely $\prod_{d|n}\Phi_n(X)=X^n-1$,
we see that $\Phi_n(x)>0$ for any real number~$x>1$
and any positive integer~$n$. In particular, $f(x)>0$
for any real number~$x>1$ and $\mathrm L_v(1,\operatorname{EP} (U))>0$,
as claimed.
\end{proof}
(To simplify notation,
we shall also put $\mathrm L_v=1$ for any archimedean place~$v$.)
More generally, if $S$ is an arbitrary set of places of~$F$,
we define
\[ \mathrm L^S(s,\operatorname{EP} (U))
= \prod_{ v\not\in S } \mathrm L_v(s,\operatorname{EP} (U))
.\]
This Euler product converges for $\Re(s)>1$ to a holomorphic function of~$s$
in that domain;
by Brauer's theorem~\cite{brauer1947},
it admits a meromorphic continuation to the whole complex plane.
We denote by
\[ \mathrm L^{S}_{*}(1,\operatorname{EP} (U))
= \lim_{s\ra 1} \mathrm L^S(s,\operatorname{EP} (U)) (s-1)^{-r},
\qquad r=\operatorname{ord}_{s=1} \mathrm L^S(s,\operatorname{EP} (U)), \]
its ``principal value'' at~$s=1$.
From the previous lemma, we deduce the following corollary.
\begin{coro}
For any finite set of places~$S$ of~$F$,
$\mathrm L^S_*(1,\operatorname{EP} (U))$ is a positive real number.
\end{coro}
We now show how this L-function furnishes renormalization factors
for the local measures~$\tau_{U,v}$.
\begin{theo}\label{prop.convergence}
In addition to the notation and assumptions of~\S\ref{ss.local.densities},
suppose that
\[ \mathrm H^1(X,\mathscr O_X)=\mathrm H^2(X,\mathscr O_X)=0. \]
Then
$\mathrm L_v(1,\operatorname{EP} (U))\tau_{U,v}(\mathcal U(\mathfrak o_v))
=1+\mathrm O(q_v^{-3/2})$. In particular,
the infinite product
\[ \prod_{v\not\in S} \mathrm L_v(1,\operatorname{EP} (U))
\tau_{U,v}(\mathcal U(\mathfrak o_v)) \]
converges absolutely.
\end{theo}
\begin{rema}
For a smooth projective variety~$X$ and any integer~$i$,
the vector-space $\mathrm H^0(X,\Omega^i_X)$
defines a birational invariant of~$X$.
Let us sketch the proof.
For any birational morphism $f\colon X\dashrightarrow X'$,
there are open subsets~$V\subset X$ and~$V'\subset X'$
whose complementary subsets have codimension at least~$2$
such that $f$ is defined on~$V$ and~$f^{-1}$ is defined on~$V'$.
The restriction morphism $\mathrm H^0(X,\Omega^i_X)\ra\mathrm H^0(V,\Omega^i_X)$ is then injective since $V$ is dense, and surjective by the
Hartogs principle; similarly, the restriction morphism
$\mathrm H^0(X',\Omega^i_{X'})\ra\mathrm H^0(V',\Omega^i_{X'})$ is an isomorphism.
Consequently, the corresponding regular maps
$g\colon V\ra X'$ and $g'\colon V'\ra X$
define morphisms $\mathrm H^0(X',\Omega^i_{X'})\ra\mathrm H^0(X,\Omega^i_X)$
and $\mathrm H^0(X,\Omega^i_X)\ra\mathrm H^0(X',\Omega^i_{X'})$
both compositions of which equal the identity map,
hence are isomorphisms.
When the ground field is the field of complex numbers, Hodge
theory identifies these vector-spaces with the conjugates
of the cohomology spaces $\mathrm H^i(X,\mathscr O_X)$.
Consequently, these spaces define birational invariants
of smooth complex projective varieties. Moreover,
by the Lefschetz principle, this extends to smooth projective
varieties defined over a field of characteristic~$0$.
In particular,
the assumptions of the theorem are therefore
conditions on the variety~$U$ and do not depend
on its smooth compactification~$X$.
\end{rema}
\begin{proof}
Blowing up the subscheme~$Z$ and resolving the singularities
of the resulting scheme, we obtain a smooth variety~$X'$, with
a proper morphism $\pi\colon X'\ra X$, which is an isomorphism
above~$U$, such that $Z'=\pi^{-1}(Z)$ is a divisor.
By the previous remark,
the variety~$X'$ satisfies
$\mathrm H^1(X',\mathscr O_{X'})=\mathrm H^2(X',\mathscr O_{X'})=0$.
Let $U'=\pi^{-1}(U)$; since~$\pi$ is an isomorphism,
any model $(\mathscr X',\mathscr Z')$ of~$(X',Z')$
will be such that $\mathscr U'=\mathscr X'\setminus\mathscr Z'$
is isomorphic to~$\mathscr U$, at least over a dense open
subset of~$\Spec\mathfrak o_F$. In particular, the
cardinalities of~$\mathscr U'(k_v)$ and~$\mathscr U(k_v)$
will be equal for almost all places~$v$.
In view of Weil's formula recalled as Equation~\eqref{eq.weil-formula},
this
implies that $\tau_{U',v}(\mathscr U'(\mathfrak o_v))=
\tau_{U,v}(\mathscr U(\mathfrak o_v))$ for almost all finite
places~$v$ of~$F$.
Consequently, assuming that~$Z$ is a divisor
does not reduce the generality of the argument which follows.
For almost all places~$v$, one has
\[ \tau_{U,v}(\mathcal U(\mathfrak o_v)) = q_v^{-\dim U} \mathop\# \mathcal U(k_v).\]
Evidently,
\[ \mathop\# \mathcal U(k_v) = \mathop\# \mathcal X(k_v) - \mathop\# \mathcal Z(k_v). \]
Let $\ell$ be a prime number.
By the smooth and proper base change theorems,
and by Poincaré duality,
the number of points of~$\mathscr X(k_v)$ can be computed,
for any finite place~$v$ of~$F$,
via the trace of a geometric Frobenius element~$\Fr_v\in\Gamma_v$
on the $\ell$-adic cohomology of~$Z_{\bar F}$,
as soon as
$\mathscr X$ is smooth over the local ring of~$\mathfrak o_F$ at~$v$
and the residual characteristic at~$v$ is not equal to~$\ell$.
Specifically, for any such finite place~$v$, one has
the equality
\[ q_v^{-\dim X}\mathop\# \mathcal X(k_v) = \sum_{i=0}^{2d} q_v^{-i/2}\tr(\Fr_v|\mathrm H^i(X_{\bar F},{\mathbf Q}_\ell)). \]
By Deligne's proof~\cite{deligne74} of the Weil conjectures (analogue of
the Riemann hypothesis) the eigenvalues of~$\Fr_v$
on $\mathrm H^i(X_{\bar F},{\mathbf Q}_\ell)$ are algebraic numbers with archimedean
eigenvalue~$q^{i/2}$. Therefore, the $i$th term of the sum above
is an algebraic number whose archimedean
absolute values are bounded by $\dim \mathrm H^i(X_{\bar F},{\mathbf Q}_\ell) q^{-i/2}$.
In particular, the sum of all terms corresponding to $i\geq 3$
is an algebraic number whose archimedean absolute values
are $\mathrm O(q_v^{-3/2})$ when $v$ varies through the
set of finite places of~$F$.
The term corresponding to $i=0$ is equal to~$1$ since $X$ is geometrically connected.
Moreover,
it follows from Peyre's arguments (\cite{peyre95}, proof of Lemma~2.1.1
and Proposition~2.2.2) that one has
\[ \mathrm H^1(X_{\bar F},{\mathbf Q}_\ell) = 0 \]
and that the cycle map induces an isomorphism of
${\mathbf Q}_\ell[\Gamma_F]$-modules
\[ \Pic(X_{\bar F}) \otimes{\mathbf Q}_\ell \xrightarrow\sim
\mathrm H^2(X_{\bar F},{\mathbf Q}_\ell(1)). \]
Let $\Pi(Z_{\bar F})$ denote the free ${\mathbf Z}$-module with basis
the set of irreducible components of $Z_{\bar F}$, endowed with
its Galois action.
\begin{lemm}
For almost all~$v$, one has the equality
\[ \mathop\# \mathcal Z(k_v) = n_v q_v^{\dim Z} + \mathrm O(q_v^{\dim Z-1/2}), \]
where $n_v=\tr(\Fr_v|\Pi(Z_{\bar F}))$
is the number of irreducible components of $\mathcal Z\otimes
k_v$ which are geometrically irreducible.
\end{lemm}
\begin{proof}[Proof of the lemma]\footnote{The proof
below uses Deligne's proof of Weil's conjecture but
the result can be deduced from the estimates
of Lang-Weil in~\cite{lang-w54}.}
In fact, we prove, for any scheme $\mathscr Z/\mathfrak o_F$
which is flat, of finite type, and
whose generic fibre~$Z$ has dimension~$d$,
the following equality
\[ \mathop\# \mathcal Z(k_v) = n_v q_v^d + \mathrm O(q_v^{d-1/2}), \]
where $n_v$ is the number of fixed points of~$\Fr_v$
in the set of $d$-dimensional irreducible components of~$Z_{\bar F}$.
First observe that there is a general upper bound
\[ \mathop\# \mathcal Z(k_v) \leq C q_v^d , \]
for some integer~$C>0$. When $\mathcal Z$ is quasi-projective,
this follows for example from Lemma~3.9
in~\cite{chambert-loir-t2000b}, where $C$ can be
taken to be the degree of the closure of~$Z$
in a projective compactification; the general case follows
from this since $\mathcal Z$ is assumed to be of finite type.
Consequently, adding or removing a subscheme of lower dimension
gives an equivalent inequality, so that
we may assume that $\mathcal Z$ is projective and equidimensional.
By resolution of singularities, we may also assume that
$Z$ is smooth.
Let $\ell$ be a fixed prime number. As above, we can compute
$\mathop\#\mathcal Z(k_v)$ using the $\ell$-adic cohomology
of~$Z_{\bar F}$: one has an upper bound
\[ q_v^{-d}\mathop\# \mathcal Z(k_v) = \tr(\Fr_v|\mathrm H^i(Z_{\bar F},{\mathbf Q}_\ell))+\mathrm O(q_v^{-1/2}). \]
Moreover,
$\mathrm H^{0}(Z_{\bar F},{\mathbf Q}_\ell) \simeq {\mathbf Q}_\ell^{\Pi(Z_{\bar F})}$,
where $\Pi(Z_{\bar F})$ denotes the set of connected
components of $Z_{\bar F}$, the Frobenius element~$\Fr_v$ acting
trivially on this ring.
Consequently,
\[ \tr(\Fr_v|\mathrm H^{0}(Z_{\bar F},{\mathbf Q}_\ell)) = \tr(\Fr_v|\Pi(Z_{\bar F}))
\]
and the lemma follows from this.
\end{proof}
Let us return to the proof of Theorem~\ref{prop.convergence}.
By the preceding lemma, we have
\[ q_v^{-\dim X} \mathop\# \mathcal Z(k_v) = q_v^{-1} \tr(\Fr_v|\Pi(Z_{\bar F}))
+ \mathrm O(q_v^{-3/2}), \]
hence
\[ q_v^{-\dim X} \mathop\# \mathcal U(k_v)
= 1 - \frac{1}{q_v} \tr\big(\Fr_v\big|\Pi(Z_{\bar F})\otimes{\mathbf Q}_\ell
- \mathrm H^2_{\text{\'et}}( X_{\bar F},{\mathbf Q}_\ell(1)) \big) + \mathrm O(q_v^{-3/2}). \]
Let us show that the trace appearing in the previous
formula is equal to $\tr(\Fr_v|\operatorname{EP} (U))$.
The order of vanishing/pole along any irreducible component
of $Z_{\bar F}$ defines an exact sequence of abelian sheaves
for the {\'e}tale site of $X_{\bar F}$:
\[ 1 \ra {\mathbf G}_m \ra i_*{\mathbf G}_m \xrightarrow{\operatorname{ord}}
\bigoplus j_{\alpha*} {\mathbf Z} \ra 0, \]
where $i\colon U_{\bar F}\ra X_{\bar F}$ is the inclusion
and $j_\alpha\colon Z_\alpha\ra X_{\bar F}$
are the inclusions of the irreducible components of $Z_{\bar F}$.
These sheaves are endowed with an action of $\Gamma_F$
for which the above exact sequence is equivariant.
Taking {\'e}tale cohomology gives an exact sequence of $\Gamma_F$-modules:
\[ 0 \ra \mathrm H^0(X_{\bar F},{\mathbf G}_m) \ra \mathrm H^0(U_{\bar F},{\mathbf G}_m)
\ra \bigoplus \mathrm H^0(Z_\alpha,{\mathbf Z})
\ra \mathrm H^1(X_{\bar F},{\mathbf G}_m) \ra \mathrm H^1(U_{\bar F},{\mathbf G}_m) . \]
Moreover, this last map is surjective: it can be identified with
the restriction map $\Pic(X_{\bar F})\ra \Pic(U_{\bar F})$
which is surjective because $X_{\bar F}$ is smooth and $U_{\bar F}$
open in~$X_{\bar F}$.
The scheme $X_{\bar F}$ is proper, smooth and connected
hence $\mathrm H^0(X_{\bar F},{\mathbf G}_m)={\bar F}^*$.
Moreover for any~$\alpha$, $Z_\alpha$ is connected which implies
$\mathrm H^0(Z_\alpha,{\mathbf Z})={\mathbf Z}$, that is $\bigoplus \mathrm H^0(Z_\alpha,{\mathbf Z})=\Pi(Z_{\bar
F})$.
Finally, one obtains an exact sequence of ${\mathbf Z}[\Gamma_F]$-modules
\[ 0 \ra \mathrm H^0(U_{\bar F},{\mathbf G}_m)/\bar F^* \ra \Pi(Z_{\bar F})
\ra \Pic(X_{\bar F})\ra \Pic(U_{\bar F})\ra 0 , \]
which after tensoring with ${\mathbf Q}_\ell$ gives an equality
of virtual representations
\begin{equation}\label{eq.virtual-eq}
\operatorname{EP} (U) = \Pi(Z_{\bar F})_{{\mathbf Q}_\ell}- \Pic(X_{\bar F})_{{\mathbf Q}_\ell} = \Pi(Z_{\bar F})_{{\mathbf Q}_\ell}- \mathrm H^2_{\text{\'et}}(X_{\bar F},{\mathbf Q}_\ell(1)).
\end{equation}
This implies that
\[ q_v^{-\dim X} \mathop\#\mathscr U(k_v)
= 1 - \dfrac1{q_v} \tr(\Fr_v|\operatorname{EP} (U)) + \mathrm O(q_v^{-3/2}). \]
Since the eigenvalues of~$\Fr_v$ on the two Galois modules
defining $\operatorname{EP} (U)$ are algebraic numbers
whose absolute values are bounded by $1$,
one has
\[ \det(1-q_v^{-1}\Fr_v|\operatorname{EP} (U))
= 1-\dfrac1{q_v} \tr(\Fr_v|\operatorname{EP} (U)) + \mathrm O(q_v^{-2}). \]
Now,
\[ \mathrm L_v(1,\operatorname{EP} (U))q_v^{-\dim X}\mathop\#\mathscr U(k_v)
=\det(1-q_v^{-1}\Fr_v|\operatorname{EP} (U))^{-1} q_v^{-\dim X}\mathop\#\mathscr U(k_v)
= 1+\mathrm O(q_v^{-3/2}), \]
and the asserted absolute convergence follows.
\end{proof}
\begin{defi}\label{defi.tamagawa-measure}
Let $F$ be a number field;
let $X$ be a smooth proper, geometrically integral variety over~$F$
such that $\mathrm H^1(X,\mathscr O_X)=\mathrm H^2(X,\mathscr O_X)=0$.
Let $Z$ be a Zariski closed
subset in~$X$, and let $U=X\setminus Z$.
The Tamagawa measure on the adelic space $U({\mathbb A}_F^S)$
is defined as the measure
\[ \tau_U^S = \mathrm L^{S}_{*}(1,\operatorname{EP} (U))^{-1}
\left(\prod_{v\not\in S} \mathrm L_v(1,\operatorname{EP} (U))\, \tau_{U,v} \right)
\]
\end{defi}
By Proposition~\ref{prop.convergence}, this infinite
product of measures converges;
moreover, non-empty, open and relatively compact,
subsets of $U({\mathbb A}_F^S)$ have a positive (finite) $\tau_U^S$-measure.
In particular,
compact subsets of~$U({\mathbb A}_F^S)$ have a finite~$\tau_U^S$-measure.
\begin{rems}
In the literature,
families $(\lambda_v)$ of positive real numbers such that
the product of measures $\prod_v(\lambda_v \tau_{U,v})$
converges absolutely are called sets of convergence factors
(see, \emph{e.g.}, \cite{weil82} or~\cite{salberger98}).
Our theorem can thus be stated as saying that for any smooth
geometrically integral algebraic variety~$U$ having
a smooth compactification~$X$ satisfying $\mathrm H^1(X,\mathscr O_X)=\mathrm H^2(X,\mathscr O_X)=0$, the family $(\mathrm L_v(1,\operatorname{EP} (U)))$
is a set of convergence factors. Let us compare our construction
with the previously known cases.
\emph a)
The Picard group of an affine connected algebraic group~$U$
is finite; in that case, $\operatorname{EP} (U)$ is therefore
the Galois module of characters of~$U$ and we recover
Weil's definition (\cite{weil82}).
For semi-simple groups, there are no non-trivial characters
and the product of the natural local densities converges
absolutely, as is well-known.
This is also the case for homogeneous varieties~$G/H$
studied by Borovoi and Rudnick in~\cite{borovoi-r1995},
where $G\supset H$ are semi-simple algebraic groups
without non-trivial rational characters.
\emph b)
On the opposite side, integral projective varieties have
no non-constant global functions, so that $\operatorname{EP} (U)=\Pic(U_{\bar F})$
if $U$ is projective. We thus recover Peyre's definition
of the Tamagawa measure of a projective variety~\cite{peyre95}.
\emph c)
Salberger~\cite{salberger98} has developed a theory of Tamagawa
measures on ``universal torsors''
introduced by Colliot-Thélène and Sansuc~\cite{colliot-thelene-sansuc1987}.
A universal torsor is a principal homogeneous space~$E$
over an algebraic variety~$V$ whose structure group is an algebraic
torus~$T$ dual to the Picard group of~$V$ such that the canonical
map $\Hom(T,\mathbf G_m)\ra \mathrm H^1(V,\mathbf G_m)$,
sending a character~$\chi$ to the $\mathbf G_m$-torsor
$\chi_*[E]$ deduced from~$E$ by push-out along~$\chi$,
is an isomorphism.
Such torsors have been successfully applied to the study of rational
points on the variety~$V$ (Hasse principle, weak approximation,
counting of rational points of bounded height);
see~\cite{colliot-thelene-sansuc1987,peyre98,breteche98b,skorobogatov2001}.
By a fundamental theorem of Colliot-Thélène and Sansuc\footnote
{Apply~\cite{colliot-thelene-sansuc1987}, Prop.~2.1.1
with $K=\bar k$, the map $\hat S\ra\Pic(X_{\bar k}))$
being an isomorphism by the very definition of a universal torsor.},
universal torsors have no non-constant invertible global functions
and their Picard group is trivial (at least up to torsion).
It follows that the virtual $\Gamma_F$-module $\operatorname{EP} (E)$
is trivial.
This gives a conceptual explanation for the discovery by Salberger
in~\cite{salberger98}
that the Tamagawa measures on universal torsors could be defined
by an absolutely convergent product of ``naive'' local measures,
without any regularizing factors.
\end{rems}
\subsubsection{}
Assume moreover that~$Z$ is the support
of a divisor~$D$ in~$X$ (more generally, of a Cartier ${\mathbf Q}$-divisor)
and that $\omega_X(D)$ is endowed with an adelic metric;
this induces a natural metric on~$\mathcal O_X(D)$.
For any place~$v$, the metric on~$\omega_X(D)$ on $X(F_v)$
gives rise to a measure~$\tau_{(X,D),v}$ on~$U(F_v)$,
which is related to the measure~$\tau_{X,v}$ by the formula
\[ \mathrm d\tau_{(X,D),v} (x) = \frac1{\norm{\mathsec f_D}(x)}
\,\mathrm d\tau_{X,v} (x), \]
where $\mathsec f_D$ is the canonical section of~$\mathscr O_X(D)$.
By definition of an adelic metric, the metric on~$\mathscr O_X(D)$
is induced, for almost all finite places~$v$, by the line
bundle~$\mathscr O_{\mathscr X}(\mathscr D)$ on the integral
model~$\mathscr X$, where $\mathscr D$ is a Cartier divisor
with generic fibre~$D$.
For such places~$v$,
one has $\norm{\mathsec f_D}(x)=1$ for any point~$x\in \mathscr U(\mathfrak o_v)$, so that
the measures $\tau_{(X,D),v}$ and $\tau_{X,v}$ coincide
on $\mathscr U(\mathfrak o_v)$.
This shows that the product
\[ \tau_{(X,D)}^S = \mathrm L^{S}_{*}(1,\operatorname{EP} (U))^{-1}
\left(\prod_{v\not\in S} \mathrm L_v(1,\operatorname{EP} (U))\, \tau_{(X,D),v} \right)
\]
also defines a Radon measure on $U({\mathbb A}_F^S)$.
In fact, one has
\[ \mathrm d\tau_{(X,D)}^S (x) = \big( \prod_{v} \norm{\mathsec f_D}(x_v)
\big)^{-1} \, \mathrm d\tau_U^S(x) = H_D(x) \,\mathrm d\tau_U^S(x) ,\]
and $H_D(x)$ is the height of~$x$ relative to the metrized
line bundle $\mathscr O_X(D)$.
\subsubsection{Example: compactifications of algebraic groups}
As in the case of local fields, we briefly explain
the case of algebraic groups and show how our theory of measures
interacts with the construction of a Haar measure on an adelic group.
We keep the notation of the previous paragraph,
and assume moreover that $U$ is an algebraic group~$G$ over~$F$
of which $X$ is an equivariant compactification.
Let us fix a invariant differential form of maximal degree~$\omega$
on~$G$; viewed as a meromorphic global section of~$\omega_X$,
it has poles on each component of~$X\setminus G$
(see~\cite{chambert-loir-t2002}, Lemma~2.4);
we therefore write $-D$ for its divisor.
For any place~$v\in\Val(F)$, viewing $\omega$ as a gauge
form on~$G(F_v)$ defines a Haar measure~$\abs\omega_v$
on $G(F_v)$.
For almost all finite places~$v$, $G(\mathfrak o_v)$ is a well-defined
compact subgroup of~$G(F_v)$.
We normalize the Haar measure $\mathrm dg_v$ on $G(F_v)$ be dividing
the measure~$\abs{\omega}_v$ by the quantity $ \mathop\# G(k_v)q_v^{-\dim G}$ for these places~$v$, and by~$1$ at other places.
By construction, for almost all~$v$,
$\mathrm dg_v$ assigns volume~1 to the compact
open subgroup $G(\mathfrak o_v)$, hence the product
$\prod_v \mathrm dg_v$ is a well-defined Haar measure on~$G({\mathbb A}_F)$.
By Proposition~\ref{prop.convergence}
we have the estimate
\[ \mathrm L_v(1,\operatorname{EP} (G)) \mathop\# G(k_v) q_v^{-\dim G} = 1+\mathrm O(q_v^{-3/2}), \]
from which we deduce that their infinite product
converges absolutely.
Consequently, $\prod_{v} \mathrm L_v(1,\operatorname{EP} (G))\abs{\omega}_v $
is also a Haar measure on $G({\mathbb A}_F)$.
Now, by the very definition of the adelic measures
$\tau_{(X,D)}$ and~$\tau_X$ on~$G({\mathbb A}_F)$, one has
\begin{align*}
\mathrm d\tau_{(X,D)}(g) &= H_D(g)\,\mathrm d \tau_X(g) \\
& = \prod_v \norm{\omega(g)}_v \quad\times\quad
\mathrm L_*(1,\operatorname{EP} (G))^{-1} \prod_v \left(
\mathrm L_v(1,\operatorname{EP} (G))
\frac{\mathrm d\abs{\omega}_v(g)}{\norm{\omega(g)}_v} \right)
\\
& = \mathrm L_*(1,\operatorname{EP} (G))^{-1} \prod_v \big(\mathrm L_v(1,\operatorname{EP} (G))
\,\mathrm d\abs\omega_v(g) ).
\end{align*}
This shows that,
in the particular case of an equivariant compactification
of an algebraic group, our general definition of the measure~$\tau_{(X,D)}$
on $X\setminus\abs D$ gives rise to a Haar measure on $G({\mathbb A}_F)$.
\subsection{An abstract equidistribution theorem}
In some cases, it is possible to count integral (resp.~rational) points
of bounded height
with respect to almost any normalization of the height,
\emph{i.e.}, with respect to any metrization on a given line bundle.
Analogously, we obtain below an asymptotic expansion
for the volume of height balls for almost any normalization of
the height.
In this subsection, we show how to extract from the obtained asymptotic
behavior
a measure-theoretic information
on the distribution of points of bounded height
or of height balls.
The following Proposition is an abstract general version
of a result going back to Peyre (\cite{peyre95}, Proposition~3.3).
\begin{prop}\label{prop.abstract.equi}
Let $X$ be a compact topological space, $U$ a subset of $X$
and $H\colon U\ra{\mathbf R}_+$ a function. Let $\nu$ be a positive measure
on~$X$ such that for any real number~$B$,
the set $\{x\in U\,;\, H(x)\leq B\}$ has finite~$\nu$-measure.
Let $\mathrm S$ be a dense subspace of the space $\mathrm C(X)$
of continuous functions on~$X$, endowed with the sup. norm.
Assume that there exist a function $\alpha\colon {\mathbf R}_+\ra{\mathbf R}_+^*$ and
a Radon measure $\mu$ on $X$ such that
for any positive function $\theta\in \mathrm S$,
\[ \nu \big( \{ x\in U\,;\, H(x)\leq \theta(x) B \}\big) \sim \alpha(B) \int_X
\theta(x)\,\mathrm d\mu(x) \]
for $B\ra+\infty$.
Then, for $B\ra +\infty$, the measures
\[ \nu_B = \frac{1}{\alpha(B)} \mathbf 1_{\{ H(x)\leq B\}} \,\mathrm d\nu(x)
\]
on $X$ converge vaguely to the measure $\mu$. In other words:
\begin{enumerate}\def\alph{enumi}}\def\labelenumi{\theenumi){\roman{enumi}}
\item
for any continuous function $f\in\mathrm C(X)$,
\[ \frac1{\alpha(B)} \int_X \mathbf 1_{\{ H(x)\leq B\}} f(x)
\,\mathrm d\nu(x)
\ra \int_X f(x)\,\mathrm d\mu(x), \qquad
\text{for $B\ra+\infty$;}\]
\item
for every open set $\Omega\subset X$ which is $\mu$-regular,
\[ \nu \big(\{x\in \Omega\cap U\,;\, H(x)\leq B \}\big) = \alpha(B) \mu(\Omega)
+ \mathrm o(\alpha(B)).
\]
\end{enumerate}
\end{prop}
Before entering the proof, let us introduce
one more notation.
For $\theta\in\mathrm C(X)$, $\theta>0$, let
\[ N(\theta;B) = \nu \big( \{x\in U\,;\, H(x)\leq \theta(x) B \}\big). \]
For any open subset $\Omega\subset X$, let
\[ N_\Omega(B) = \nu \big( \{x\in \Omega\cap U\,;\, H(x)\leq B \}\big).
\]
\begin{proof}
First remark that these hypotheses imply that for any $\lambda >0$,
$\alpha(\lambda B)/\alpha(B) \ra \lambda$ when $B\ra+\infty$.
Indeed, taking any positive~$\theta\in S$,
\[ \frac{\alpha(\lambda B)}{\alpha(B)}
= \frac{\alpha(\lambda B) \int_X \theta(x)\,\mathrm d\mu(x)}{\alpha(B) \int_X \theta(x)\,\mathrm d\mu(x)}
\sim \frac{ N(\theta; \lambda B)}{N(\theta;B)}
\sim \frac{ N(\lambda\theta; B)}{N(\theta;B)}
\sim \frac{\alpha(B) \int_X \lambda\theta(x)\,\mathrm d\mu(x)}{\alpha(B) \int_X \theta(x)\,\mathrm d\mu(x)} = \lambda. \]
Now, it is of course sufficient to prove that $N_\Omega(B)/
\alpha(B)\ra \mu(\Omega)$ for any $\mu$-regular open set $\Omega\subset
B$.
Fix $\eps>0$. Since $\Omega\subset X$
is $\mu$-regular, there exist continuous functions $f$ and $g$ on $X$
such that
\[ f \leq \mathbf 1_\Omega \leq g \]
and such that $\int(g-f)\mathrm d\mu\leq \eps$.
Since $\mathrm S$ is dense in $\mathrm C(X)$, we may also assume
that $f$ and $g$ belong to~$\mathrm S$ and that they are
non-negative.
Let $f_\eps=\eps+(1-\eps)f$, $\chi_\eps=\eps+(1-\eps)\mathbf 1_\Omega$
and $g_\eps=\eps+(1-\eps)g$.
The inequality
$ f_\eps\leq \chi_\eps\leq g_\eps $ implies that
\[ N(f_\eps;B)\leq N(\chi_\eps;B) \leq N(g_\eps;B). \]
Moreover, by definition,
\[ 0\leq N(\chi_\eps;B) -N_\Omega(B) \leq N_X(\eps B). \]
Therefore,
\begin{align*}
\liminf N_\Omega(B)\alpha(B)^{-1}
& \geq \liminf N(\chi_\eps;B)\alpha(B)^{-1}
- \limsup N_X(\eps B)\alpha(B)^{-1} \\
& \geq \liminf N(f_\eps;B) \alpha(B)^{-1} - \eps \\
& \geq \int f_\eps \,\mathrm d\mu - \eps = \int f\,\mathrm d\mu + \mathrm
O(\eps).
\end{align*}
Similarly,
\[ \limsup N_\Omega(B)\alpha(B)^{-1}
\leq \int g \,\mathrm d\mu + \mathrm O(\eps). \]
When $\eps\ra 0$, one thus finds
\[ \lim N_\Omega(B)\alpha(B)^{-1} = \mu(\Omega), \]
as claimed.
\end{proof}
\section{Geometric Igusa integrals: Preliminaries}
\label{s.igusa}
\subsection{Clemens complexes and variants}
\label{subsec.clemens}
In this section, we study analytic properties
of certain integrals of Igusa-type attached
to a divisor~$D$ in a variety~$X$.
This requires the introduction of a simplicial
set which encodes the intersections of the various components of~$D$,
and which we call the \emph{Clemens complex}.
Such a simplicial set has been used by Clemens in~\cite{clemens1969}
in his study of the Picard-Lefschetz transformation
(see also~\cite{gordon1980}).
\subsubsection{Simplicial sets} \label{sss.simplicial-sets}
We recall classical definitions concerning simplicial sets.
A partially ordered set, abridged as \emph{poset},
is a set endowed with a binary relation~$\prec$ which is transitive
($a\prec b$ and $b\prec c$ implies $a\prec c$), antisymmetric
($a\prec b$ and $b\prec a$ implies $a=b$) and reflexive ($a\prec a$).
If $a\prec b$, we say that $a$ is a face of~$b$;
in particular, the transitivity axiom for a poset means that a face of a face
is a face.
There are obvious definitions for morphisms of posets, sub-posets,
as well as actions of groups on posets.
As an example, if $\mathscr A$ is a set, the set $\mathscr P^*(\mathscr A)$
of non-empty subsets of~$\mathscr A$, together with the inclusion
relation, is a poset. We shall interpret it geometrically as
the simplex on the set of vertices~$\mathscr A$
and denote it by $\mathscr S_{\mathscr A}$.
More generally, a \emph{simplicial complex} on a set~$\mathscr A$
is a set of non-empty finite subsets of~$\mathscr A$, called \emph{simplices},
such that any non-empty subset of a simplex is itself a simplex.
The dimension of a simplex is defined as one less its cardinality:
points, edges,... are simplexes of dimension~$0$, $1$,...
The dimension of a simplicial complex is the supremum
of the dimensions of its faces.
The order relation endows a simplicial complex with the structure of poset.
In fact, simplicial complexes are
sub-posets of the simplex~$\mathscr S_{\mathscr A}$.
Let $S$ be a poset. The dimension of an element~$s\in S$,
denoted $\dim(s)$,
is defined as the supremum of the lengths~$n$ of chains
$s_0\precneq\cdots\precneq s_n$, with $s_n\prec s$.
Similarly, the codimension~$\codim(s)$ of an element~$s\in S$
is defined as the supremum of the lengths of such chains
with $s\prec s_0$.
Elements of dimension~$0$, $1$,\ldots, are called vertices,
edges,... The dimension~$\dim(S)$ of~$S$ is the supremum of all dimensions
of all of its elements.
It is important to observe that given a poset~$S$
and a sub-poset~$S'$, the dimension or the codimension of a face of~$S'$
may differ from their dimension or codimension as a face of~$S$.\footnote{This probably means that the terminology ``sub-poset'' is inappropriate;
sub-posets for which the dimension notion is compatible
are sometimes called ideals...}
Let $S$ be a poset. Categorically, an action of a group~$\Gamma$ on~$S$
is just a morphism of groups from~$\Gamma$ to the set $\Aut(S)$
of automorphisms of~$S$. In other words, it is the data, for any $\gamma\in\Gamma$, of a bijection $\gamma_*$ of~$S$ such that $\gamma_*s\prec \gamma_*s'$
is $s\prec s'$, subject to the compatibilities $(\gamma\gamma')_*=\gamma_*\gamma'_*$ for any~$\gamma$ and~$\gamma'\in\Gamma$.
Let there be given a poset~$S$ and an action of~$\Gamma$ on~$S$.
The set $S^\Gamma$ of fixed points of~$\Gamma$ in~$S$ is
a sub-poset of~$S$.
In the other direction, the set~$S/\Gamma$ of orbits of~$\Gamma$
in~$S$, can be endowed with the binary relation deduced
from~$\prec$ by passing to the quotient. Namely, for $s$ and~$s'$
in~$S$, with orbits $[s]$ and~$[s']$, we say that
$[s]\prec [s']$ if there exists $\gamma\in \Gamma$ such
that $s\prec \gamma_*s'$. (This condition does not depend
on the actual choice of elements~$s$ and~$s'$ in their
orbits~$[s]$ and~$[s']$.)
There are obvious morphisms of posets, $S^\Gamma\ra S$
and $S\ra S_\Gamma$; these morphisms are the universal
morphisms of posets, respectively to~$S$ and from~$S$,
which commute with the action of~$\Gamma$ on~$S$.
As an example,
assume that $\mathscr S=\mathscr S_{\mathscr A}$, the simplex with
vertices in a set~$\mathscr A$. Actions of a group~$\Gamma$
on~$\mathscr S_{\mathscr A}$ correspond to actions of~$\Gamma$
on~$\mathscr A$. The simplicial sets $\mathscr S^\Gamma$
and~$\mathscr S_\Gamma$ are respectively the simplices with vertices
in the fixed-points~$\mathscr A^\Gamma$ and the orbits~$\mathscr A/\Gamma$.
\def{\bar{\mathscr A}}{{\bar{\mathscr A}}}
\subsubsection{Incidence complexes}
Let $X$ be a (geometrically integral) variety
over a perfect field~$F$ and let $D$ be a divisor in~$X$.
Fix a separable closure~$\bar F$ of~$F$.
Let~${\bar{\mathscr A}}$ be the set of irreducible components of~$D_{\bar F}$;
for $\alpha\in{\bar{\mathscr A}}$, we denote by $D_\alpha$ the corresponding component
of~$D_{\bar F}$.
For any subset $A\subset {\bar{\mathscr A}}$, we let $D_A=\bigcap_{\alpha\in A}D_\alpha$;
in particular, $D_{\emptyset}=X_{\bar F}$.
We shall always make the assumption
that the divisor~$D_{\bar F}$ has simple normal crossings:
all irreducible components $D_\alpha$ of~$D_{\bar F}$ are supposed to be smooth
and to meet transversally.\footnote
{This is equivalent to
the weaker condition that $D$ has normal crossings together with
the smoothness of the geometric irreducible components of~$D$;
the condition that~$D$ has strict normal crossings is however
stronger since the smoothness of an irreducible component of~$D$
implies that its geometric irreducible components don't meet.}
In particular, for any subset $A\subset{\bar{\mathscr A}}$ such that $D_A\neq\emptyset$,
$D_A$ is a smooth subvariety of~$X_{\bar F}$ of codimension~$\mathop\# A$.
The closed subschemes~$D_A$,
for $A\subset{\bar{\mathscr A}}$, are the closed strata of a stratification
$(D_A^\circ)_{A\subset{\bar{\mathscr A}}}$, where, for any subset~$A\subset{\bar{\mathscr A}}$,
$D_A^\circ$ is defined by the formula:
\[ D_A^\circ = D_A \setminus \bigcup _{B\supsetneq A} D_B. \]
There are in fact several natural posets that enter the picture,
encoding in various ways the combinatorial data
of whether or not, for a given subset~$A$ of~$\mathscr A$,
the intersection~$D_A$ is empty.
The following observation is crucial for our definitions to make sense:
\begin{prop}
Let $A$ and~$A'$ be two subsets of~${\bar{\mathscr A}}$ such that $A'\subset A$.
For any irreducible component~$Z$ of~$D_A$, there is a unique
irreducible component~$Z'$ of~$D_{A'}$ which contains~$Z$.
\end{prop}
\begin{proof}
If an irreducible component~$Z$ were contained in two
distinct irreducible components of~$D_{A'}$,
these two components would meet along~$Z$, which contradicts
the simple normal crossings assumption on~$D_{\bar F}$.
Indeed, $D_A$ being smooth, its irreducible components must be disjoint.
\end{proof}
\subsubsection{The geometric Clemens complex}
The incidence complex~$\mathscr I_{\bar F}(D)$
defined by~$D$ is the sub-poset of~$\mathscr P^*({\bar{\mathscr A}})$
consisting of non-empty subsets~$A$ of~${\bar{\mathscr A}}$ such that the intersection
$D_A$ is not empty.
More precisely, we define
the geometric Clemens complex~$\mathscr C_{\bar F}(D)$ as the set
of all pairs $(A,Z)$, where $A\subset{\bar{\mathscr A}}$ is any non-empty subset, and
$Z$ is an irreducible component of the scheme~$D_{A}$,
together with the partial order relation
defined by $(A,Z)\prec (A',Z')$ if $A\subset A'$ and $Z\supset Z'$.
In other words, the set of vertices of~$\mathscr C_{\bar F}(D)$ is~${\bar{\mathscr A}}$,
there are edges corresponding to irreducible
component of each intersection $D_\alpha\cap D_{\alpha'}$, etc.
Mapping $(A,Z)$ to~$A$ induces a morphism of posets from
the geometric Clemens complex to the incidence complex.
In fact, we could have defined $\mathscr C_{\bar F}(D)$ without
any reference to~${\bar{\mathscr A}}$. Indeed,
thanks to the normal crossings condition, given an irreducible
component~$Z$ of a scheme~$D_A$, we may recover~$A$ as the set
of $\alpha\in {\bar{\mathscr A}}$ such that $D_\alpha$ contains the generic point of~$Z$.
\medskip
Since $X$ is defined over~$F$, $\Gamma_F$ acts naturally
on the set of integral subschemes of~$X_{\bar F}$.
Since $F$ is perfect, an integral subscheme~$Z$
of~$X_{\bar F}$ is defined over~$F$ if and only if
it is a fixed point for this action.
For any $\gamma\in\Gamma_F$ and any $\alpha\in{\bar{\mathscr A}}$, observe
that $\gamma_* D_{\alpha}$ is an irreducible component of~$D_{\bar F}$,
because $D$ is defined over~$F$;
this induces an action of~$\Gamma_F$ on~${\bar{\mathscr A}}$, defined
by $\gamma_* D_\alpha=D_{\gamma\alpha}$,
for $\alpha\in{\bar{\mathscr A}}$ and $\gamma\in\Gamma_F$.
Moreover, if $Z$ is an irreducible component of
an intersection $D_A$, for $A\subset{\bar{\mathscr A}}$, then
$\gamma_* Z$ is an irreducible component
of $\gamma_* D_A=D_{\gamma_* A}$.
Consequently, we have a natural action of~$\Gamma_F$ on
the geometric Clemens complex~$\mathscr C_{\bar F}(D)$,
given by $\gamma_*(A,Z)=(\gamma_* A,\gamma_* Z)$ for any element~$(A,Z)$
of~$\mathscr C_{\bar F}(D)$.
The natural morphism of posets
$\mathscr C_{\bar F}(D)\ra \mathscr I_{\bar F}(D)$ mapping $(A,Z)$ to~$A$
is $\Gamma_F$-equivariant.
\subsubsection{Rational Clemens complexes}
Let us denote by $\mathscr I_F(D)$ and~$\mathscr C_F(D)$
the sub-posets of~$\mathscr I_{\bar F}(D)$ and $\mathscr C_{\bar F}(D)$
consisting of $\Gamma_F$-fixed faces.
These posets correspond to the intersections of the
divisors~$D_\alpha$
and to the irreducible components of these intersections
which are defined over the base field~$F$.
Alternatively, they can be defined without
any reference to the algebraic closure~$\bar F$ by considering
irreducible components of the divisor~$D$, their intersections
and the irreducible components of these which are geometrically
irreducible.
More generally, let $E$ be any extension of~$F$,
together with an embedding of~$\bar F$ in an algebraic closure~$\bar E$.
As an example, one may take any extension of~$F$ contained in~$\bar F$.
In our study below, $F$ will be a number field and $E$ will be the completion
of~$F$ at a place~$v$; the choice of an embedding $\bar F\hra\bar E$
corresponds to the choice of a decomposition group at~$v$.
Under these conditions, there is a natural morphism
of groups from the Galois group~$\Gamma_E$ of~$\bar E/E$
to~$\Gamma_F$. In particular,
the posets~$\mathscr C_{\bar F}(D)$ and~$\mathscr I_{\bar F}(D)$
are endowed with an action of~$\Gamma_E$.
Let us define the $E$-rational Clemens complex, $\mathscr C_{E}(D)$, as
the sub-poset of~$\mathscr C_{\bar F}(D)$ fixed by~$\Gamma_E$.
In particular,
for any face $(A,Z)$ of~$\mathscr C_E(D)$, the subschemes~$(D_A)_{\overline E}$
and $Z_{\overline E}$ are defined over~$E$. We shall
denote $(D_A)_E$ and~$Z_E$, or even $D_A$ and~$Z$,
the corresponding subschemes of~$X_E$.
Observe that $Z_E$ is geometrically irreducible.
Conversely, let $A$ be any non-empty subset of~$\bar{\mathscr A}$
which is $\Gamma_E$-invariant; then, $(D_A)_{\overline E}$ is defined
over~$E$ and corresponds to some subscheme $(D_A)_E$ of~$X_E$.
Let $Z$ be any irreducible component of~$(D_A)_E$
which is geometrically irreducible. Then, $Z_{\bar E}$
is an irreducible component of~$(D_A)_{\bar E}$.
By EGA IV (4.5.1), the set of irreducible components doesn't change
when one extends the ground field from an algebraically
closed field to any extension;
consequently, $Z_{\bar E}$ is defined over~$\bar F$
and $(A,Z)$ corresponds to a face of~$\mathscr C_E(D)$.
\subsubsection{$E$-analytic Clemens complexes}
Let $E$ be any perfect extension of~$F$ together with an embedding
$\bar F\hra\bar E$ as above.
We define the \emph{$E$-analytic Clemens complex},
denoted $\mathscr C^{\text{\upshape an}}_E(D)$, as
the sub-poset of~$\mathscr C_E(D)$ whose faces are those faces
$(A,Z)$ such that $Z(E)\neq\emptyset$.
(When we write $Z(E)$, we identify~$Z$ with the unique
$E$-subscheme of~$X_E$ whose extension to~$\bar E$ is~$Z_{\bar E}$.)
\smallskip
This complex can also be defined as follows.
Let first $A$ be any non-empty subset of~$\bar{\mathscr A}$
which is $\Gamma_E$-invariant and let $Z$ be an irreducible
component of~$D_A$. While $(D_A)_{\bar E}$ is defined over~$E$,
for the moment, $Z$ is just a subscheme of~$X_{\bar F}$.
However, we can define $Z(E)$ as the intersection in $X(\bar E)$
of $X(E)$ and of $Z(\bar E)$; this is indeed what
one would get if $Z_{\bar E}$ where defined over~$E$.
Assume that $Z(E)\neq\emptyset$.
Let $Z'$ be the smallest subscheme of~$X_E$ such that $Z'_{\bar E}$
contains~$Z$. It is irreducible: if $Z'$ were
a union $Z'_1\cup Z'_2$,
one of them, say~$Z'_1$, would be such that
$(Z'_1)_{\bar E}$ contains $Z_{\bar E}$,
because $Z_{\bar E}$ is irreducible. Moreover, $Z'$ is contained
in~$(D_A)_E$, since $(D_A)_{\bar E}$ contains~$Z_{\bar E}$.
It follows that $Z'$ is an irreducible component of~$(D_A)_E$.
By assumption, $Z'(E)\neq\emptyset$ hence,
by EGA IV (4.5.17), $Z'$ is geometrically connected.
Since $D_A$ is smooth, so are $(D_A)_{\bar E}$ and $Z'_{\bar E}$,
this last subscheme being a union of irreducible components
of~$(D_A)_{\bar E}$. But a connected smooth scheme is irreducible,
hence $Z'$ is irreducible and $Z'_{\bar E}=Z_{\bar E}$.
It follows that $Z_{\bar E}$ is defined over~$E$
and $(A,Z)$ corresponds to a face of $\mathscr C^{\text{\upshape an}}_E(D)$.
\smallskip
Let us finally observe that the dimension of a face~$(A,Z)$
of $\mathscr C^{\text{\upshape an}}_E(D)$ is equal to $\mathop\#(A/\Gamma_E)-1$.
Indeed, let $n=\mathop\#(A/\Gamma_E)$ and consider
any sequence $A_1\subset\cdots\subset A_n=A$ of $\Gamma_E$-invariants
subsets of~$A$. Then, $D_{A_1}\supset D_{A_2}\supset\cdots\supset D_{A_n}$
and each of them has a unique irreducible component~$Z_i$ containing~$Z$.
Since $Z(E)\neq\emptyset$, $Z_i(E)$ is non-empty either,
and the $(A_i,Z_i)$ define a maximal increasing sequence
of faces of~$\mathscr C^{\text{\upshape an}}_E(D)$.
\smallskip
This notion will be of interest to us under the supplementary
assumption that $E$ is a locally compact valued field.
Indeed, such fields allow for a theory
of analytic manifolds as well as an implicit function theorem.
By the local description of~$X$ that
will be explained below, for any face $(A,Z)$ of $\mathscr C_E(D)$,
$Z(E)$ is either empty, or is an $E$-analytic submanifold of~$X(E)$
of codimension~$\mathop\#(A)$ which is Zariski-dense in~$Z$.
However, as a face of~$\mathscr C^{\text{\upshape an}}_E(D)$, $(A,Z)$ has dimension~$\mathop\#(A/\Gamma_E)-1$
and it is that invariant which will be relevant in our analysis below.
\subsubsection{Example}
Below we describe these Clemens complexes in special
cases, which are essentially governed by combinatorial data,
namely for toric varieties and equivariant compactifications
of semi-simple groups.
We want to make clear that for general varieties,
the three types of Clemens complexes
can be very different.
For example, let $X$ be the blow-up of the
projective space~${\mathbf P}^n$ along
a smooth subvariety~$V$ defined over the ground field,
let $D$ be the exceptional divisor in~$X$
and let $\pi\colon X\ra{\mathbf P}^n$ be the canonical morphism.
The map~$\pi$ induces
a bijection between the set of irreducible components of~$D_{\bar F}$
and the set of irreducible components of~$V_{\bar F}$,
as well as a bijection between the set of irreducible
components of~$D$ and the set of irreducible components of~$V$.
Moreover, a component~$Z$ of~$D$ is geometrically irreducible
if and only if the corresponding component~$\pi(Z)$ of~$V$ is;
indeed, $Z$ is isomorphic to the projectivized normal bundle
of~$V$ in~$X$.
This description shows also that $Z(F)=\emptyset$
if and only $\pi(Z)(F)=\emptyset$.
To give a specific example, take $V$ to be the disjoint union
of a geometrically irreducible smooth curve~$C_1$, without rational points,
of a geometrically irreducible smooth curve~$C_2$ with rational points
and of a smooth irreducible curve~$C_3$ with two geometrically components~$C'_3$ and~$C''_3$.
Then, $\mathscr C(X,D)$ consists of four points corresponding to~$C_1, C_2, C'_2, C'_3$
(there are no intersections),
the $F$-rational Clemens complex $\mathscr C_F(X,D)$ consists
of two points corresponding to~$C_1$ and~$C_2$,
and the $F$-analytic Clemens complex $\mathscr C^{\text{\upshape an}}_F(X,D)$
consists of the single point corresponding to~$C_1$.
We present further examples in Section~\ref{sec.examples}.
\subsection{Local description of a pair~$(X,D)$}
\subsubsection{Notation}
Let~$F$ be a perfect field, $\bar F$ an algebraic closure of~$F$,
$X$ a smooth algebraic variety over~$F$ and $D\subset X$ a reduced divisor
such that $D_{\bar F}$ has strict normal crossings in~$X_{\bar F}$.
Let us recall that
this means that $D_{\bar F}$ is the union of irreducible smooth divisors
in~$X_{\bar F}$ which meet transversally. In other words,
each point of~$X_{\bar F}$ has a neighborhood~$U$, together
with an étale map $U\ra{\mathbf A}^n_{\bar F}$ such that for each irreducible
component~$Z$ of~$D_{\bar F}\cap U$, $Z\cap U$ is the preimage of some
coordinate hyperplane in~${\mathbf A}^n_{\bar F}$.
Let~${\bar{\mathscr A}}$ be the set of irreducible components of~$D_{\bar F}$;
for $\alpha\in {\bar{\mathscr A}}$, let~$D_\alpha$ denote the corresponding divisor,
so that $D_{\bar F}=\sum_{\alpha\in \mathscr A}D_\alpha$.
For $\alpha\in {\bar{\mathscr A}}$, we denote by~$F_\alpha$ the field of definition
of~$D_\alpha$ in~$\bar F$; its Galois group~$\Gamma_{F_\alpha}$
is the stabilizer of~$D_\alpha$ in~$\Gamma_F$.
For $A\subset{\bar{\mathscr A}}$, we denote by~$D_A$ the intersection
of the divisors~$D_\alpha$, for $\alpha\in A$; by convention,
$D_\emptyset=X$.
Let $\alpha\in{\bar{\mathscr A}}$.
The union~$\bigcup_{a\in\Gamma_F\alpha} D_a$
of the conjugates of~$D_\alpha$ is a divisor in~$X_{\bar F}$
which is defined over~$F$; the corresponding divisor of~$X$
will be denoted by~$\Delta_\alpha$.
Similarly, the intersection of the divisors~$D_a$,
for $a\in\Gamma_F\alpha$ is a smooth subscheme of~$X_{\bar F}$
which is defined over~$X$; the corresponding subscheme of~$X$
will be denoted by~$E_\alpha$. If $E_\alpha\neq\emptyset$,
then its codimension~$r_\alpha$ is equal to~$[F_\alpha:F]$.
Moreover, $\Delta_\alpha(F)=E_\alpha(F)$.
The choice of another element~$\alpha'=\gamma_*\alpha$
in the orbit~$\Gamma_F\alpha$
doesn't change~$\Delta_\alpha$ nor~$E_\alpha$. However, it changes
the field~$F_\alpha$ to the conjugate~$F_{\alpha'}=\gamma_*F_\alpha$
and its stabilizer~$\Gamma_{F_\alpha}$ to its conjugate subgroup
$\Gamma_{F_{\alpha'}}=\gamma\Gamma_{F_\alpha}\gamma^{-1}$.
Observe that $r_{\alpha'}=r_\alpha$.
\subsubsection{Local equations for the étale topology}
Let us now give local equations of~$D$ around each rational point of~$X$.
Fix a point~$\xi\in X(F)$ and let $\alpha\in{\bar{\mathscr A}}$ be such that
$\xi\in D_\alpha$ (it is a $\Gamma_F$-invariant subset of~${\bar{\mathscr A}}$).
When $F_\alpha=F$, $D_\alpha=\Delta_\alpha\times\bar F$
is defined over~$F$ and the choice of a local equation for~$D_\alpha$
defines a smooth map from an open neighborhood~$U_\xi$ of~$\xi$ in~$X$
to the affine line which maps~$U_\xi\cap D_\alpha$ to~$\{0\}\subset{\mathbf A}^1$.
To explain the general case, let us introduce some more notation.
We write
${\mathbf A}_{F_\alpha}$ for the Weil restriction of scalars
$\Res_{F_\alpha/F}{\mathbf A}^1$
from~$F_\alpha$ to~$F$ of the affine line. It is endowed
with a canonical morphism ${\mathbf A}^1_{F_\alpha}\ra\left(\Res_{F_\alpha/F}{\mathbf A}^1\right)\times_F F_\alpha$ which induces, for any $F$-scheme~$V$,
a bijection
\[ \Hom(V,\Res_{F_\alpha/F}{\mathbf A}^1)=\Hom(V\times_F F_\alpha,{\mathbf A}^1). \]
In particular, the $F$-rational points of~${\mathbf A}_{F_\alpha}$
are in canonical bijection with~$F_\alpha$.
The $F$-scheme~${\mathbf A}_{F_\alpha}$
is an affine space of dimension~$r_\alpha=[F_\alpha:F]$.
Let us indeed choose a~$F$-linear basis $(u_1,\ldots,u_{r_\alpha})$
of~$F_\alpha$. Then, the morphism
${\mathbf A}^{r_\alpha}\ra {\mathbf A}_{F_\alpha}$ given by
$(x_1,\ldots,x_{r_\alpha})\mapsto \sum x_i u_i$
is an isomorphism.
In these coordinates, the norm-form~$\mathrm N_{F_\alpha/F}$
of~$F_\alpha$
is a homogeneous polynomial of degree~$[F_\alpha:F]$
which defines a hypersurface in~${\mathbf A}_{F_\alpha}$; it has a single
$F$-rational point~$0$.
Moreover, the $r_\alpha$ $F$-linear embeddings of~$F_\alpha$ into~$\bar F$
induce an isomorphism $F_\alpha\otimes_F \bar F\simeq
\bar F^{r_\alpha}$,
hence an identification of~${\mathbf A}_{F_\alpha}\times_F \bar F$
with the affine space~${\mathbf A}^{r_\alpha}_{\bar F}$.
Under this identification, the divisor of the norm-form
corresponds to the union of the coordinates hyperplanes.
\begin{lemm}
There is a Zariski open neighborhood~$U_\xi$ of~$\xi$ in~$X$
and a smooth map
\[
x_{\alpha}\colon U_\xi\ra{\mathbf A}_{F_\alpha}
\]
which maps~$\Delta_\alpha\cap U_\xi$ to the hypersurface of~${\mathbf A}_{F_\alpha}$
defined by the norm equation $\mathrm N_{F_\alpha/F}(x)=0$,
such that there is a diagram
\[ \xymatrix{ E_\alpha \ar@{^(->}[r] \ar[d] & \Delta_{\alpha} \ar@{^(->}[r] \ar[d] & **[r] U_\xi \subset X \ar[d] \\
{\{0\}} \ar@{^(->}[r] & {\{\mathrm N_{F_\alpha/F}(x)=0\}} \ar @{^(->}[r] & {\mathbf A}_{F\alpha}.} \]
\end{lemm}
\begin{proof}
Choose an element~$f$ in the local ring~$\mathscr O_{X_{\bar F},\xi}$
which is a generator of the ideal of~$D_\alpha$, so that
$f=0$ is a local equation of~$D_\alpha$.
For any~$\gamma\in \Gamma_F$, $f^\gamma=0$
is a local equation of~$\gamma_* D_\alpha$. Consequently,
$f^\gamma/f$ is a local unit at~$\xi$, for each
$\gamma\in\Gamma_{F_\alpha}$, and the map~$\gamma\mapsto f^\gamma/f$
is a $1$-cocycle for the Galois group~$\Gamma_{F_\alpha}$ with
values in the local ring~$\mathscr O_{X_{\bar F},\xi}^*$.
By Hilbert's Theorem~90 for the multiplicative group,
this cocycle is a coboundary
(see~\cite{milne80}, Proposition~4.9 and Lemma~4.10 for a non-elementary proof
in that context) and there exists a unit $u\in \mathscr O_{X_{\bar F},\xi}^*$
such that $f^\gamma/f=u^\gamma/u$ for any~$\gamma\in\Gamma_{F_\alpha}$.
Consequently, $fu^{-1}$ is an element of~$ \mathscr O_{X_{\bar F},\xi}$,
fixed by $\Gamma_{F_\alpha}$, which generates the ideal
of~$D_\alpha$.
In other words,
there is a Zariski open neighborhood~$V$ of~$\xi$ in~$X$,
and a function $g\in\Gamma(V\times_F {F_\alpha},\mathscr O_{V\times F_\alpha})$
such that~$g=0$ is an equation of~$D_\alpha$ in~$V\times_F {F_\alpha}$.
To the morphism $V\times_F F_\alpha \ra{\mathbf A}^1_{F_\alpha}$
corresponds by the universal property of the Weil restriction of scalars
a morphism, still denoted~$\theta$, $V\ra {\mathbf A}_{F_\alpha}$.
Through the canonical identification, over~$\bar F$, of~${\mathbf A}_{F_\alpha}$ with
the affine space~${\mathbf A}_{\bar F}^{r_\alpha}$, the map~$g$ induces an isomorphism
from~$D_\alpha$ to one of the coordinate hyperplanes,
from~$\Delta_\alpha$ to the union of the coordinate hyperplanes,
while $\xi$ maps to~$0$.
At the level of $F$-rational points, the diagram
$(V,\Delta_\alpha)\ra ({\mathbf A}_{F_\alpha},H)$ corresponds
to the map~$(V(F),D_\alpha(F))\ra (F_\alpha,0)$.
\end{proof}
\subsubsection{Local charts}
\label{subsubsec.local-charts}
Let $\xi\in X(F)$.
More generally, let $A_\xi$ be the set of~$\alpha\in\mathscr A$
such that $\xi\in D_\alpha$. Let $Z$ be the irreducible
component of~$D_{A_\xi}$ which contains~$\xi$. It is geometrically
irreducible and $(A_\xi,Z)$ is a face of~$\mathscr C^{\text{\upshape an}}_F(D)$.
For a sufficiently small neighborhood~$V$ of~$\xi$, there exists,
for each~$\alpha\in A_\xi$,
a smooth map~$x_\alpha\colon V\ra{\mathbf A}_{F_\alpha}$ as in the previous
paragraph.
Since the divisors~$D_\alpha$ meet transversally,
the map $(x_\alpha)\colon V\ra \prod_{\alpha\in A_\xi} {\mathbf A}_{F_\alpha}$
is smooth.
By choosing additional local coordinates, and shrinking~$V$ if needed,
we can extend it to
an \'etale map~$q_\xi\colon U_\xi\ra\prod_{\alpha\in A_\xi}{\mathbf A}_{F_\alpha}
\times {\mathbf A}^r$, with $r=\codim_\xi( D_{A_\xi},X)=\mathop\#(A_\xi)$.
We will also assume, as we may, that the open subset~$U_\xi$ is affine
and that $U_\xi\cap D_\alpha=\emptyset$ if $\alpha\not\in A_\xi$.
Assume moreover that $F$ is a local field.
It follows from the preceding discussion that
for each $\alpha\in\mathscr A$,
$\Delta_\alpha(F)=E_\alpha(F)$ is a smooth $F$-analytic subvariety
of~$X(F)$, either empty, or of codimension~$r_\alpha=[F:F_\alpha]$.
For any $ \Gamma_F$-invariant subset~$A$ of~$\mathscr A$, let us
denote by~$\Delta_A$ the intersection of all~$\Delta_\alpha$,
for $\alpha\in A$. Then, $\Delta_A(F)=E_A(F)$ is either empty,
or a smooth $F$-analytic subvariety
of~$X(F)$ of codimension $r_A = \mathop\#(A)$.
\subsubsection{Partitions of unity}
\label{subsubsec.partition-unity}
In this section, we assume in addition that $F$ is a local field
and we consider the situation from the analytic point of view.
Observe that
the maps~$q_\xi$ induce analytic \'etale maps
$U_\xi(F)\ra \prod_{\alpha\in A_\xi} F_\alpha\times F^r$.
The open sets $U_\xi(F)$ cover~$X(F)$ which is compact, since $X$ is projective.
Consequently, there is a finite partition of unity $(\theta_\xi)$ subordinate
to this covering: for any~$x$, $\theta_\xi\colon X(F)\ra{\mathbf R}$
is a smooth function whose support is contained in~$U_\xi(F)$,
the map $q_\xi$ is one-to-one on that support,
$\sum_{\xi\in X(F)} \theta_\xi=1$ and only a finite number of~$\theta_\xi$
are nonzero.
\subsubsection{Metrized line bundles, volume forms}
Let $\alpha\in{\bar{\mathscr A}}$ and
let us consider the line bundle~$\mathscr O_X(\Delta_\alpha)$.
It has a canonical section~$\mathsec f_\alpha$ the divisor of
which is~$\Delta_\alpha$.
Let us assume that $F$ is a local field. On each open set
$U_\xi$ of $X$, we have constructed
a regular map $x_\alpha\colon U_\xi \ra{\mathbf A}_{F_\alpha}$.
Composed with the norm ${\mathbf A}_{F_\alpha}\ra{\mathbf A}^1$, it gives
us a regular function $\mathrm N(x_\alpha)\in \Gamma(U_\xi,\mathscr O_X)$
which generates the ideal of~$\Delta_\alpha$ on~$U_\xi$.
This function induces therefore a trivialization
of $\mathscr O_X(-\Delta_\alpha)_{|U_\xi}$.
Consequently, a metric on
$\mathscr O_X(\Delta_\alpha)_{|U_\xi}$
takes the form $\norm{\mathsec f_\alpha}(x)=\abs{x_\alpha}_{F_\alpha}
h_\xi(x)$ on $U_\xi(F)$, where we view $x_\alpha$
as a local coordinate $U_\xi(F)\ra F_\alpha$ and
$\abs{x_\alpha}_{F_\alpha}=\abs{\mathrm N_{F_\alpha/F}(x_\alpha)}_F$,
and where $h_\xi\colon U_\xi(F)\ra{\mathbf R}_+^*$ is any continuous function.
Similarly, on such a chart~$U_\xi(F)$, we have a
measure in the Lebesgue class
defined by $\mathrm d\mathbf x=\prod_{\alpha\in A_\xi}\mathrm dx_\alpha\times
\mathrm dx_1\cdots\mathrm dx_r$. Any measure on $X(F)$ which
belongs to the Lebesgue class can therefore be expressed
in a chart $U_\xi(F)$ as the product of this measure $\mathrm d\mathbf x$
with a positive locally integrable function.
Let us also observe that there are canonical isomorphisms
\[ \mathscr O(\Delta_\alpha)|_{E_\alpha}\simeq
\mathscr N_{\Delta_\alpha}(X) |_{E_\alpha} \simeq
\det \mathscr N_{E_\alpha}(X). \]
These isomorphisms allow us to define residue measures on
$\Delta_\alpha(F)=E_\alpha(F)$, and on their intersections,
hence on all faces of the $F$-analytic Clemens complex~$\mathscr C^{\text{\upshape an}}_F(D)$.
\subsection{Mellin transformation over local fields}
\subsubsection{Local zeta functions}\label{subsubsec.local-zeta}
Let~$F$ be a local field of characteristic zero, with normalized
absolute value~$\abs{\cdot}_F$.
Define a constant $\mathrm c_F$ as follows:
\begin{equation}\label{eq.cF}
\mathrm c_F = \begin{cases}
\mu ( [-1;1]) & \text{if $F={\mathbf R}$;}\\
\mu (B(0;1)) & \text{if $F={\mathbf C}$;} \\
(1-q^{-1}) (\log q)^{-1} \mu (\mathfrak o_F) & \text{if $F\supset {\mathbf Q}_p$,
$q=\abs{\varpi_F}_F^{-1}$.}
\end{cases}
\end{equation}
(In the last formula, $\varpi_F$ denotes
a generator of the maximal ideal of~$\mathfrak o_F$.)
For any locally integrable function~$\phi\colon F\ra{\mathbf C}$, we let
\[ \mathscr M_F(\phi)(s)=\int_F \phi(x)\abs{x}_F^{s-1}\, \mathrm d\mu(x), \]
for any complex parameter~$s$ such that the integral
converges absolutely. We call it the Mellin transform of~$\phi$;
to shorten the notation and when no confusion can arise,
we will often write $\hat\phi$ instead
of~$\mathscr M_F(\phi)$,
We write~$\zeta_F$ for the Mellin transform~$\mathscr M_F(\phi_0)$ of
$\phi_0$, the characteristic function of the unit ball in~$F$.
The integral defining~$\zeta_F(s)$ converges if and only if~$\Res(s)>0$, and one has
\begin{align*}
\zeta_F(s) &= \mu([-1,1]) \int_0^1 x^{s-1}\, \mathrm dx = \frac{\mathrm c_F}{s} && \quad\text{for $F={\mathbf R}$}\\
&= \frac{\mu(B(0;1))}{\pi} 2\pi\int_0^1 r^{2s-1}\,\mathrm dr = \frac{\mathrm c_F}{s} && \quad\text{for $F={\mathbf C}$} \\
& = \sum_{n=0}^\infty q^{-n(s-1)} \mu(\varpi_F^n\mathfrak o_F^*)
= \mu(\mathfrak o_F) \big(1-\frac1q\big) \sum_{n=0}^\infty q^{-ns}
= \mu(\mathfrak o_F) \frac{1-q^{-1}}{1-q^{-s}} && \quad\text{otherwise}.
\end{align*}
In all cases, the function~$s\mapsto \zeta_F(s)$ is holomorphic
on the half-plane defined by~$\Re(s)>0$ and extends to a meromorphic
function on~${\mathbf C}$. It has a simple pole at~$s=0$, with residue~$\mathrm c_F$.
In the archimedean case, note that $s=0$ is the only pole of~$\zeta_F$
However, in the $p$-adic case, $\zeta_F$ is $(2i\pi/\log q)$-periodic
so that $s=2in\pi/\log q$ is also a pole for any integer~$n$;
this will play a role in the use of Tauberian arguments later.
\subsubsection{Mellin transform of smooth functions}
One of our goals in that paper is to establish
the meromorphic continuation, together with growth estimates,
for integrals on varieties that are defined locally like Mellin transforms.
To explain our arguments, we begin with the one-dimensional toy example.
We say that a function $\phi\colon F\ra{\mathbf C}$ is smooth if it
is either $\mathrm C^\infty$ when $F={\mathbf R}$ or~${\mathbf C}$, or
locally constant otherwise.
If $a$ and~$b$ are real numbers, we define
${\mathsf T}_{>a}$, resp.
${\mathsf T}_{(a,b)}$, as the set of $s\in{\mathbf C}$
such that $a<\Re(s)$, resp. $a < \Re(s) <b$.
\begin{lemm}
Let $F$ be a local field of characteristic zero, let $\phi$
be a measurable, bounded and compactly supported function on~$F$.
Then, the Mellin transform
\[ \mathscr M_F(\phi)(s) =\int_F \abs{x}_F^{s-1} \phi(x)\,\mathrm dx \]
converges for $s\in{\mathbf C}$ with $\Re(s)>0$ and defines
a holomorphic function on~${\mathsf T}_{>0}$. Moreover,
for any real numbers $a$ and $b$ such that $0<a<b$, the function
$\mathscr M_F(\phi(s))$ is bounded in~${\mathsf T}_{(a,b)}$.
Assume that $\phi$ is smooth.
Then there exists a holomorphic function $\phi_1$ on ${\mathsf T}_{>-1/2}$
such that $\phi_1(0)=\phi(0)$ and $\mathscr M_F(\phi)(s)=\zeta_F(s)\phi_1(s)$.
Moreover, for any real numbers~$a$ and~$b$ such that $-\frac12<a<b$,
$\phi_1$ is bounded in ${\mathsf T}_{(a,b)}$
if $F$ is ultrametric, while there is an upper bound
\[ \abs{\phi_1(s)}\ll 1+\abs{\Im(s)}, \qquad s\in{\mathsf T}_{(a,b)},\]
if $F$ is archimedean.
In particular,
the function
\[ s \mapsto \mathscr M_F(\phi)(s)-\phi(0)\zeta_F(s) \]
is holomorphic
and
\[ \lim\limits_{s\ra 0}s\mathscr M_F(\phi)(s)=\mathrm c_F\phi(0). \]
\end{lemm}
\begin{proof}
The absolute convergence of~$\mathscr M_F(\phi)(s)$ for $\Re(s)>0$
and its holomorphy in that domain immediately follow from the absolute
convergence of~$\zeta_F$ and the fact that~$\phi$
is bounded and has compact support. So does also its boundedness
in vertical strips ${\mathsf T}_{(a,b)}$ for $0<a<b$.
Assume that $\phi$ is smooth.
Let us prove the stated meromorphic continuation.
Again, let~$\phi_0$ denote the characteristic function of the unit ball in~$F$.
Replacing $\phi$ by $\phi-\phi(0)\phi_0$,
it suffices to show that $\mathscr M_F(\phi)$ is holomorphic
on the half-plane
$\Re(s)>-\frac12 $ whenever $\phi$ is a compactly supported function on~$F$
which is smooth in a neighborhood of~$0$ and satisfies $\phi(0)=0$.
In the real case, one can then write $\abs{\phi}(x)=\abs x_F\psi(x)$ for some
function~$\psi$ which is continuous at~$0$. Similarly, in the complex
case, one can write $\abs{\phi(z)}=\abs{z}_F^{1/2}\psi(z)$.
The required holomorphy of~$\mathscr M_F(\phi)(s)$ on the domain
${\mathsf T}_{>-\frac12}$ follows.
In the ultrametric case, $\phi$ vanishes in a neighborhood of~$0$
and $\mathscr M_F(\phi)$ even extends to a holomorphic function on~${\mathbf C}$.
In all cases, observe that $\mathscr M_F(\phi)(s)$ is bounded
in vertical strips ${\mathsf T}_{(\alpha,\beta)}$,
if $\alpha$ and~$\beta$ are real numbers such that $-\frac12<\alpha<\beta$.
Now, the existence of the function~$\phi_1$ and the asserted bound
on this function follow from the fact that $\zeta_F^{-1}$ is holomorphic
and satisfies this bound in these vertical strips.
\end{proof}
\subsubsection{Higher-dimensional Mellin transforms}
We consider a finite family $(F_j)_{1\leq j\leq n}$ of local
fields. For simplicity, we assume these to be finite extensions
of a single completion of~${\mathbf Q}$. Let us denote by~$V$
the space $\prod F_j$.
Adopting the terminology introduced for functions of one
variable, we say that a function on~$V$ is smooth if it is
either $\mathrm C^\infty$, in the case where the $F_j$ are archimedean,
or locally constant, when the~$F_j$ are ultrametric.
If $\alpha=(\alpha_1,\ldots,\alpha_n)$ and~$\beta=(\beta_1,\ldots,\beta_n)$
are families of real numbers, we define the sets
${\mathsf T}_{>\alpha}$ and ${\mathsf T}_{(\alpha,\beta)}$ as the open
subsets of~${\mathbf C}^n$ consisting of those $s\in{\mathbf C}^n$
such that $\alpha_j<\Re(s_j)$, resp. $\alpha_j < \Re(s_j) <\beta_j$,
for $j=1,\dots,n$.
When all $\alpha_j$ are equal to a single one $\alpha$,
and similarly for the~$\beta_j$, we will
also write ${\mathsf T}^n_{>\alpha}$
and ${\mathsf T}^n_{(\alpha,\beta)}$, or even ${\mathsf T}_{>\alpha}$
and~${\mathsf T}_{(\alpha,\beta)}$ when the dimension~$n$
is clear from the context.
Let $\mathscr F$ be the vector space generated by functions
on $V\times{\mathbf C}^n$ of the form
$(x;s)\mapsto u(x) v_1(x)^{s_1}\dots v_n(x)^{s_n}$,
where $u$ is smooth, compactly supported on~$V$,
and $v_1,\dots,v_n$ are smooth, positive, and equal to~$1$
outside of a compact subset of~$V$.
For any function $\phi\in\mathscr F$ and $s\in{\mathbf C}^n$, we let
\[\mathscr M_V(\phi) (s) = \int_V \abs{x_1}^{s_1-1}\dots\abs{x_n}^{s_n-1}
\phi(x;s)\, \mathrm dx_1\,\dots \mathrm dx_n, \]
whenever the integral converges.
\begin{prop}\label{prop.mellin-n}
The integral $\mathscr M_V(\phi) (s)$ converges for any $s\in{\mathbf C}^n$ such that
$\Re(s_j)>0$ for all~$j$, and defines an holomorphic function in
${\mathsf T}_{>0}$.
For any positive real numbers $\alpha$ and~$\beta$,
$\mathscr M_V(\phi)$ is bounded on~${\mathsf T}^n_{(\alpha,\beta)}$.
Moreover, there exists a unique holomorphic function $\phi_1$
on the domain~${\mathsf T}^n_{>-1/2}$
such that
\[ \mathscr M_V(\phi)(s) = \prod_{j=1}^n \zeta_{F_j}(s_j) \phi_1(s) \]
for $s\in{\mathsf T}^n_{>0}$.
Let $\alpha$ and~$\beta$ be real numbers such that
$-\frac12<\alpha<\beta$. Then, $\phi_1$ is bounded on~${\mathsf T}^n_{(\alpha,\beta)}$
if the $F_j$ are ultrametric, while there exists a real number~$c$
such that
\[ \abs{\phi_1(s)} \leq c \prod_{j=1}^n (1+\abs{s_j}), \qquad s\in{\mathsf T}_{(\alpha,\beta)}, \]
if the $F_j$ are archimedean.
Moreover, $\phi_1(0)=\phi(0;0)$.
\end{prop}
\section{Geometric Igusa integrals and volume estimates}
\label{sec.igusa-volumes}
\subsection{Igusa integrals over local fields}
\label{ss.geometric-igusa}
We return to our geometric situation: $X$ is a smooth quasi-projective
variety over a local field~$F$, $D$ is a divisor on~$X$ such
that $D_{\bar F}$ has simple normal crossings.
We let ${\bar{\mathscr A}}$ be the set of irreducible components
of~$D_{\bar F}$. For $\alpha\in{\bar{\mathscr A}}$, $D_\alpha$ denotes the corresponding
component, while $\Delta_\alpha$ and~$E_\alpha$ are respectively
the sum and the intersection of the conjugates of~$D_\alpha$.
For $\alpha\in{\bar{\mathscr A}}$,
we denote by $\mathsec f_{\alpha}$ the canonical section
of the line bundle~$\mathscr O(\Delta_\alpha)$.
Endow the line bundles $\mathscr O(\Delta_\alpha)$,
as well as the canonical bundle~$\omega_X$ with smooth metrics.
Let $\tau_X$ the corresponding measure on~$X$.
Following the definitions of~\ref{subsubsec.residue-measures},
the analytic variety $D_A(F)$, for any $\Gamma_F$-invariant subset
$A\subset{\bar{\mathscr A}}$ supports a natural ``residue measure''.
To simplify explicit formulas below, we define the
measure $\tau_{D_A}$ as the residue measure multiplied by
$\prod_{\alpha\in A} \mathrm c_{F_\alpha}$,
where the constants $\mathrm c_{F_\alpha}$ are defined in
Equation~\eqref{eq.cF}.
The integrals we are interested in
take the form
\[ \mathscr I(\Phi;(s_\alpha)_{\alpha \in \mathscr A}) =
\int_{X(F)} \prod_{\alpha\in \mathscr A}
\norm{\mathsec f_{D_\alpha}}(x)^{s_\alpha-1}\, \Phi(x)\, \mathrm d\tau_X(x), \]
where $\Phi$ is a smooth, compactly supported function on $X(F)$
and the~$s_\alpha$ are complex parameters.
Letting~$\Phi$ vary, it is convenient to view
these integrals as distributions
(of order~$0$) on~$X(F)$.
For any subset $A$ of the $F$-analytic Clemens complex
$\mathscr C^{\text{\upshape an}}_F(D)$, define also
\[ \mathscr I_A(\Phi;(s_\alpha)_{\alpha \not\in A})
= \int_{D_A(F)} \prod_{\alpha\not\in A} \norm{\mathsec f_{D_\alpha}}(x)^{s_\alpha-1}\, \Phi(x) \, \mathrm d\tau_{D_A}(x). \]
\begin{lemm}\label{lemm.local-convergence}
The integral $\mathscr I(\Phi;(s_\alpha)_{\alpha\in \mathscr A})$
converges for $s\in{\mathsf T}_{>0}^{\mathscr A}$ and the map
\[ (s_\alpha)_{\alpha\in\mathscr A} \mapsto \mathscr I(\Phi;(s_\alpha)_{\alpha\in \mathscr A}) \]
is holomorphic on ${\mathsf T}_{>0}^{\mathscr A}$.
Similarly, for any~$A\subset\mathscr A$,
the integral $\mathscr I_A(\Phi;(s_\alpha))$ converges for
$s\in{\mathsf T}_{>0}^{\mathscr A\setminus A}$ and
the function $s\mapsto \mathscr I_A(\Phi;s)$
is holomorphic on that domain.
\end{lemm}
We shall therefore call such integrals
``\emph{holomorphic distributions on~$X$}''.
\begin{proof}
For $\xi\in X(F)$, let~$\mathscr A_\xi$ be the set of~$\alpha\in\mathscr A$
such that $\xi\in D_\alpha$ and let
\[
q\colon U_\xi\ra\prod_{\alpha\in\mathscr A_\xi}{\mathbf A}_{F_\alpha}
\times {\mathbf A}^r
\]
be an \'etale chart around~$\xi$ adapted to~$D$,
as in Section~\ref{subsubsec.local-charts}.
Let~$x_\alpha\colon U_\xi\ra {\mathbf A}_{F_\alpha}$ be the composite of~$q$
followed by the projection to~${\mathbf A}_{F_\alpha}$,
and $y\colon U_\xi\ra{\mathbf A}^r$ be the composite of~$q$ with the projection
to~${\mathbf A}^r$.
These maps induce local coordinates in a neighborhood of~$U_\xi$,
valued in $\prod_{\alpha\in\mathscr A_\xi}F_\alpha\times F^r$
in which the measure~$\tau$ takes the form $\kappa((x_\alpha),y)
\prod \mathrm dx_\alpha\,\mathrm dy$.
By definition of a smooth metric,
there is,
for any $\alpha\in\mathscr A$,
a smooth non-vanishing function~$u_\alpha$ on~$U_\xi(F)$
such that $\norm{\mathsec f_{D_\alpha}}=\abs{x_\alpha}_{F_\alpha}u_\alpha$
if $\alpha\in\mathscr A_\xi$, and $\norm{\mathsec f_{D_\alpha}}=u_\alpha$
otherwise.
Further, introducing a partition of unity
(see Section~\ref{subsubsec.partition-unity}),
we see that it suffices to study integrals of the form
\[ \int_{\prod F_\alpha\times F^d} \prod_{\alpha\in\mathscr A_\xi} \abs{x_\alpha}_{F_\alpha}^{s_\alpha-1}
\Phi((x_\alpha),y)\prod_{\alpha\in\mathscr A} u_\alpha^{s_\alpha-1}
\kappa((x_\alpha),y) \prod \mathrm dx_{\alpha}\, \mathrm dy, \]
where $\Phi$ is a smooth function
with compact support on~$\prod F_\alpha\times F^r$.
The holomorphy of such integrals is precisely the object
of Proposition~\ref{prop.mellin-n} above.
The case of the integrals $\mathscr I_A(\Phi;(s_\alpha))$ is analogous.
\end{proof}
By the same arguments, but using the meromorphic continuation
of Mellin transforms and the estimate of their growth in vertical strips,
we obtain the following result.
\begin{prop}\label{prop.local-continuation}
The holomorphic function
\[ s\mapsto \prod_{\alpha\in\mathscr A}\zeta_{F_\alpha}(s_\alpha)^{-1}
\mathscr I(\Phi;(s_\alpha)) \]
on ${\mathsf T}_{>0}^{\mathscr A}$ extends to a
holomorphic function~$\mathscr M(\Phi;\cdot)$ on ${\mathsf T}_{>-1/2}^{\mathscr A}$.
Moreover, for any real numbers $a$ and~$b$ and for any
function~$\Phi$, there is a real number~$c$ such that
\[ \mathscr M(\Phi;s)\leq c\prod_{\alpha\in\mathscr A}(1+\abs{s_\alpha})
\qquad\text{ for any $s\in{\mathsf T}_{(a,b)}^{\mathscr A}$.} \]
\end{prop}
\subsubsection{Leading terms}
We will need to understand the leading terms at the poles
of these integrals after restricting the parameter $(s_\alpha)$
to an affine line in~${\mathbf C}^{\mathscr A}$.
For $\alpha \in \mathscr A$,
let $s\mapsto s_\alpha=-\rho_\alpha+\lambda_\alpha s$
be an increasing affine function with real coefficients.
To shorten notation, we write $\mathscr I(\Phi;s)$
instead of $\mathscr I(\Phi;(s_\alpha))$,
and similarly for the integrals~$\mathscr I_A$.
Define
\[ a(\lambda,\rho) = \max_{\alpha \in \mathscr A}
\frac{\rho_\alpha}{\lambda_\alpha}
\]
and let~$\mathscr A(\lambda,\rho)$ denote the set of all $\alpha\in \mathscr A$
where the maximum is obtained,
\emph{i.e.}, such that $a(\lambda,\rho)=\rho_\alpha/\lambda_\alpha$.
By the preceding lemma,
$\mathscr I(\Phi;s)=\mathscr I(\Phi,(-\rho_\alpha+\lambda_\alpha s))$
converges for $\Re(s)>a(\lambda,\rho)$ and defines a holomorphic
function there.
Similarly,
letting
\[ a_A(\lambda,\rho)= \max_{\alpha\not\in A} \frac{\rho_\alpha}{\lambda_\alpha}, \]
the function
$s\mapsto \mathscr I_A(\Phi;s):=\mathscr I_A(\Phi,(-\rho_\alpha+\lambda_\alpha s))$
converges for $\Re(s)>a_A(\lambda,\rho)$.
Notice that $a_A(\lambda,\rho)\leq a(\lambda,\rho)$
in general, and that the inequality may be strict,
\emph{e.g.}, if $\mathscr A(\rho,{\lambda})\subseteq A$.
Let~$\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}(D)$ be the intersection
of the Clemens complex~$\mathscr C^{\text{\upshape an}}_F(D)$ with the simplicial subset
$\mathscr P^+(\mathscr A(\lambda,\rho))$
of~$\mathscr P^+(\mathscr A)$. In other words, we remove
from~$\mathscr C^{\text{\upshape an}}_F(D)$ all faces containing a vertex~$\alpha$ such
that $\rho_\alpha< a(\lambda,\rho) \lambda_\alpha$.
\begin{prop}
\label{prop.igusa-limit}
With the above notation, there exists a positive real number~$\delta$,
and, for any face~$A$ of~$\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}(D)$ of
maximal dimension, a holomorphic function~$\mathscr J_A$ defined
on~${\mathsf T}_{>a(\lambda,\rho)-\delta}$ with polynomial growth
in vertical strips such that
\[ \mathscr J_A(\Phi;a(\lambda,\rho))
= \mathscr I_A(\Phi;(a(\lambda,\rho)\lambda_\alpha-\rho_\alpha)_{\alpha\not\in A}) \]
and such that
\[ \mathscr I(\Phi;s) = \sum_A \mathscr J_A(\Phi;s) \prod_{\alpha\in \mathscr A} \zeta_{F_\alpha}(s-a(\lambda,\rho)), \]
where the sum is restricted to the faces~$A$ of~$\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}(D)$
of maximal dimension.
In particular,
\[ \lim_{s\ra a(\lambda,\rho)}
\mathscr I(\Phi;s) (s-a(\lambda,\rho))^{\dim\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}}
= \sum_{A}
\mathscr I_A (\Phi;(a(\lambda,\rho)\lambda_\alpha-\rho_\alpha)_{\alpha\not\in A})
\prod_{\alpha\in A} \frac{1}{\lambda_\alpha} . \]
\end{prop}
\begin{proof}
As in the proof of Lemma~\ref{lemm.local-convergence},
we shall use a partition of unity and local coordinates.
Let $\xi\in X(F)$, let~$\mathscr A_\xi$ be the set of~$\alpha\in\mathscr A$
such that $\xi\in D_\alpha$.
In a neighborhood~$\Omega$ of~$\xi$, we have local coordinates
$x_\alpha\in F_\alpha$, for $\alpha\in\mathscr A_\xi$,
and other coordinates $(y_1,\dots,y_r)$ for some integer~$r$.
There exist smooth functions~$u_\alpha$ on~$\Omega$ such that
$\norm{\mathsec f_{D_\alpha}}=\abs{x_\alpha}_{F_\alpha} u_\alpha$
if $\alpha\in\mathscr A_\xi$,
and $\norm{\mathsec f_{D_\alpha}}=u_\alpha$ otherwise.
We may assume, after shrinking~$\Omega$,
that $D_\alpha\cap\Omega=\emptyset$ if $\alpha\not\in\mathscr A_\xi$;
then, $u_\alpha$ does not vanish on~$\Omega$, for any~$\alpha\in\mathscr A$.
Finally, the restriction to~$\Omega$ of the measure $\tau_X$ can be written
as $\kappa \mathrm dy_1\dots \mathrm dy_r \prod_{\alpha\in\mathscr A_x}\mathrm dx_\alpha $,
for some smooth function~$\kappa$ on~$\Omega$.
Let~$\theta_\xi$ be a smooth function with compact support in~$\Omega$.
The integral
\[ \mathscr I_\xi (s) = \int_\Omega \prod_{\alpha\in\mathscr A}
\norm{\mathsec f_{D_\alpha}}(x)^{s_\alpha-1}\Phi(x) \theta_\xi(x) \,\mathrm d\tau_X(x)\]
can be rewritten as
\[ \mathscr I_\xi(s)=
\int_\Omega
\prod_{\alpha\in\mathscr A_\xi}\abs{x_\alpha}_{F_\alpha}^{-\rho_\alpha+s\lambda_\alpha-1}
\times \prod_{\alpha\in\mathscr A} u_\alpha^{s_\alpha-1}(x)
\Phi(x)\theta_\xi(x)\kappa(x)\,
\prod_{\alpha\in\mathscr A_\xi}
\mathrm dx_\alpha \times \mathrm dy.
\]
Let $a=a(\lambda,\rho)$; when $s\ra a$, only the variables $s_\alpha$
such that $\rho_\alpha=a \lambda_\alpha$ and $\alpha\in\mathscr A_\xi$
contribute a pole.
Write $A =\mathscr A_\xi\cap\mathscr A(\lambda,\rho)$ for this subset.
Applying the regularization procedure which led to Prop.~\ref{prop.local-continuation},
but only to the variables~$x_\alpha$ with $\alpha\in A$,
furnishes an expression of the form
\[ \mathscr I_\xi (s)=\mathscr J_\xi(s) \prod_{\alpha\in A}\zeta_{F_\alpha}(-\rho_\alpha+s\lambda_\alpha) \]
for the integral~$\mathscr I_\xi(s)$,
where $\mathscr J_\xi$ is holomorphic for $\Re(s)>a(\lambda,\rho)-\delta$
and has polynomial growth in vertical strips.
Moreover,
\begin{align*}
\lim_{s\ra a} \mathscr J_\xi(s) & = \lim_{s\ra a} \mathscr I_\xi(s)
\prod_{\alpha\in A}
\zeta_{F_\alpha}(-\rho_\alpha+s\lambda_\alpha)^{-1} \\
& = \int_{\Omega\cap D_A(F)}
\prod_{\alpha\in\mathscr A} u_\alpha^{-\rho_\alpha+a\lambda _\alpha-1}
\prod_{\alpha\in\mathscr A_\xi\setminus A} \abs{x_\alpha}_{F_\alpha}^{-\rho_\alpha+a\lambda _\alpha-1}
\theta_\xi(x)\Phi(x)\kappa(x)\,
\prod_{\alpha\in\mathscr A_\xi\setminus A}\mathrm dx_\alpha\, \mathrm dy .
\end{align*}
By the definition of the residue measure on~$D_A(F)$
(Section~\ref{subsubsec.residue-measures})
and its normalization used here, one thus has
\[\lim_{s\ra a}\mathscr I_\xi(s)
\prod_{\alpha\in A}\zeta_{F_\alpha}(-\rho_\alpha+s\lambda_\alpha)^{-1}
= \prod_{\alpha\in A}\frac1{\mathrm c_{F_\alpha}} \int_{D_A(F)}
\prod_{\alpha\not\in A} \norm{\mathsec f_{D_\alpha}}^{-\rho_\alpha+ a \lambda_\alpha-1}
\theta_\xi(x)\Phi(x)\,\mathrm d\tau_{D_A}(x).\]
so that
\[ \lim_{s\ra a} (s-a)^{\mathop\# A} \mathscr I(\theta_\xi\Phi;s)
=\mathscr I_A(\theta_\xi\Phi;(-\rho_\alpha+a\lambda_\alpha))
\prod_{\alpha\in A} \lambda_\alpha^{-1}.\]
Observe that $A$ is a maximal face of~$\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}$,
though maybe not one of maximal dimension.
Now choose the functions~$\theta_\xi$
so that they form a finite partition of unity, \emph{i.e.}, $\sum\theta_\xi=1$,
and only finitely many~$\theta_\xi$ are not zero.
Then $\mathscr I(\Phi;s)=\sum\mathscr I_\xi(s)$; regrouping the non-zero terms
according to the minimal face of the Clemens complex~$\mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}$
to which they correspond furnishes
the desired expression for~$\mathscr I(\Phi;s)$.
Let~$b$ be the dimension of this complex
and let $\mathscr C^{\text{\upshape an,max}}_{F,(\lambda,\rho)}$
be the set of its faces of dimension~$b$.
Granted the previous limits computations, one has
\begin{align*}
\lim_{s\ra a} (s-a)^b\mathscr I(\Phi;s)
& = \sum_\xi \lim_{s\ra a} (s-a)^b \mathscr I(\theta_\xi\Phi;s) \\
& = \sum_{A\in\mathscr C^{\text{\upshape an,max}}_{F,(\lambda,\rho)}}
\prod_{\alpha\in A} \lambda_{\alpha}^{-1}
\mathscr I_{A}(\Phi;(-\rho_\alpha+a\lambda_\alpha)),
\end{align*}
as claimed.
\end{proof}
\begin{coro}
If $\Phi\equiv 1$ or, more generally, if the
restriction of~$\Phi$ to $D(F)$ is not identically~$0$,
then the order of the pole of~$\mathscr I(\Phi;s)$ at $s=a(\lambda,\rho)$
is equal to
\[ 1+\dim \mathscr C^{\text{\upshape an}}_{F,(\lambda,\rho)}(D). \]
\end{coro}
\subsubsection{The case of good reduction}
We recast in this geometric context a formula of J.~Denef
(\cite[Theorem 3.1]{denef87}, see also~\cite[Theorem 9.1
and Theorem 11.2]{chambert-loir-t2002}).
Assume that $F$ is non-archimedean and that our situation
comes from a smooth model $\mathscr X$ over $\mathfrak o_F$,
that the divisors $D_\alpha$ extend to divisors $\mathscr D_\alpha$
on~$\mathscr X$
whose sum becomes a relative Cartier divisor with strict normal crossings after
base change to a finite \'etale extension of~$\mathfrak o_F$,
and that all metrics are defined by this model.
The residue field of~$F$ is denoted by~$k$, its cardinality by~$q$.
For $\alpha\in\mathscr A$, the extension $F\subset F_\alpha$
is unramified by the good reduction hypothesis;
we denote by $f_\alpha$ its degree;
let $\mathfrak o_{F_\alpha}$ be the ring of integers of~$F_\alpha$
and $\mathfrak m_\alpha$ its maximal ideal.
The Zariski closure of the scheme $E_\alpha$
is a smooth subscheme $\mathscr E_\alpha$ of~$\mathscr X$,
of relative codimension $d_\alpha$, such that
$\mathscr E_\alpha(\mathfrak o_F)=\mathscr D_\alpha(\mathfrak o_F)$.
Similarly, the Zariski closure of~$E_A$ is a smooth geometrically connected
subscheme~$\mathscr E_A$ of~$\mathscr X$ of codimension~$d_A$
and $\mathscr E_A(\mathfrak o_F)=\mathscr D_A(\mathfrak o_F)$.
For any subset $A\subset\mathscr A$,
we let $\tau_{D_A}$ denote the Tamagawa measure on $D_A(F)=E_A(F)$.
Since the extensions $F\subset F_\alpha$ are unramified,
one also has $\mathscr D_\alpha(k)=\mathscr E_\alpha(k)$
for any $\alpha\in\mathscr A$, and $\mathscr D_A(k)=\mathscr E_A(k)$
for any $A\subset\mathscr A$.
Assume also that the function~$\Phi$ is constant
on residue classes; the induced function on $\mathscr X(k)$ will
still be denoted by~$\Phi$.
\begin{prop}\label{prop.denef}
Under the above conditions, one has
\[ \mathscr I(\Phi;(s_\alpha))
= \sum_{A\subset \mathscr A}
\big(q^{-1}\mu(\mathfrak o_{F})\big)^{\dim X}
\prod_{\alpha\in A}
\frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1}
\big(\sum_{\tilde\xi \in \mathscr D_A^\circ(k)} \Phi(\tilde\xi)\big)
.
\]
In particular, for $\Phi=1$,
one has
\[ \mathscr I(1;(s_\alpha))
= \sum_{A\subset \mathscr A}
\big(q^{-1}\mu(\mathfrak o_F)\big)^{\dim X}
\prod_{\alpha\in A}
\frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1}
\mathop\#(\mathscr D_A^\circ(k)).
\]
\end{prop}
\begin{proof}
Let $\tilde\xi\in\mathscr X(k)$, let $\mathscr A_{\tilde\xi}=\{\alpha\in\mathscr A\,;\, \tilde\xi\in \mathscr D_\alpha(k)\}$,
so that $\tilde\xi$ belongs to the open
stratum~$\mathscr D_{\mathscr A_{\tilde\xi}}^\circ$.
By the good reduction hypothesis, we can introduce local (étale)
coordinates $x_\alpha\in \mathfrak m_\alpha$
(for $\alpha\in \mathscr A_{\tilde\xi}$) and
$y_\beta\in\mathfrak m$ (for $\beta$
in a set~$\mathscr B_{\tilde\xi}$
of cardinality $\dim X-\sum_{\alpha\in \mathscr A_{\tilde\xi}}f_\alpha$)
on the residue class $\Omega_{\tilde\xi}$ of~$\tilde\xi$,
such that $\mathscr D_\alpha$ is defined by the equation $x_\alpha=0$
on $\Omega_{\tilde\xi}$. Then the local Tamagawa measure
identifies with the measure $\prod_{\alpha\in \mathscr A_{\tilde\xi}}
\mathrm dx_\alpha\times
\prod_{\beta\in \mathscr B_{\tilde\xi}}\mathrm dy_\beta$
on $\prod_{\alpha\in \mathscr A_{\tilde\xi}}\mathfrak m_\alpha
\times\prod_{\beta\in\mathscr B_{\tilde\xi}}\mathfrak m$.
Recall also (see~\S\ref{subsubsec.local-zeta})
that for any ultrametric local field~$F$,
with ring of integers~$\mathfrak o_F$ and maximal ideal~$\mathfrak m$,
and any complex number $s$ such that $\Re(s)>0$,
one has
\[ q \int_{\mathfrak m} \abs{x}_F^{s-1} \,\mathrm dx
= \frac{q-1}{q^{s}-1} \mu(\mathfrak o_F), \]
where $q$ is the cardinality of the residue field.
These formulas, applied to the fields~$F_\alpha$, and
the decomposition of the integral $\mathscr I$ as a sum of similar
integrals over the residue classes $\tilde\xi\in\mathscr X(k)$, give us
\begin{align*}
\mathscr I(\Phi;(s_\alpha))
&=
\sum_{\tilde\xi \in \mathscr X(k) }
\Phi(\tilde\xi)
\big(q^{-1}\mu(\mathfrak o_F)\big)^{\mathop\#\mathscr B_{\tilde\xi}}
\prod_{\alpha\in \mathscr A_{\tilde\xi} }
\int_{\mathfrak m_\alpha} \abs{x_\alpha}_{F_\alpha}^{s_\alpha-1}\,\mathrm dx_\alpha \\
&= \big(q^{-1} \mu(\mathfrak o_F)\big)^{\dim X}
\sum_{\tilde\xi \in\mathscr X(k)}
\Phi(\tilde\xi) \prod_{\alpha\in \mathscr A_{\tilde\xi}}
\frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1} ,
\end{align*}
since the residue field of~$F_\alpha$ has cardinality~$q^{f_\alpha}$
and $\mu(\mathfrak o_F)=\mu(\mathfrak o_{F_\alpha})$.
Let us interchange the order of summation: one gets
\[
\mathscr I(\Phi;(s_\alpha))
= \big(q^{-1} \mu(\mathfrak o_F)\big)^{\dim X}
\sum_{A\subset \mathscr A}
\prod_{\alpha\in A} \frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1}
\sum_{\tilde\xi \in \mathscr D_A^\circ(k)} \Phi(\tilde\xi) .
\]
In particular, if $\Phi$ is the constant function~$1$, one has
\[
\mathscr I(1,(s_\alpha))
= \sum_{A\subset \mathscr A}
\big(q^{-1}\mu(\mathfrak o_F)\big)^{\dim X}
\prod_{\alpha\in A}
\frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1}
\mathop\# (\mathscr D_A^\circ(k)).
\qedhere
\]
\end{proof}
By Weil's formula (\emph{cf.}~\ignorespaces Equation~\eqref{eq.weil-formula}), one then has
\[ \tau_{D_A}(\mathscr E_A^\circ (\mathfrak o_F))
= \big( q^{-1} \mu(\mathfrak o_F)\big)^{d_A} \big)
\mathop\# \mathscr E_A^\circ (k)
= \big( q^{-1} \mu(\mathfrak o_F)\big)^{d_A} \big)
\mathop\# \mathscr D_A^\circ (k)
. \]
Since moreover $D_A(F)=E_A(F)$,
the last formula of the Proposition can be rewritten as
\[
\mathscr I(1,(s_\alpha))
= \sum_{A\subset \mathscr A}
\big(q^{-1}\mu(\mathfrak o_F)\big)^{d_A}
\prod_{\alpha\in A}
\frac{q^{f_\alpha}-1}{q^{f_\alpha s_\alpha}-1}
\tau_{D_A} (\mathscr D_A^\circ(\mathfrak o_F)). \]
\subsection{Volume asymptotics over local fields}
\label{subsec.geometric-volume}
Let $X$ be a smooth projective variety over a local field~$F$.
Assume that $X$ is purely of dimension~$n$.
Let $D$ be an effective divisor in~$X$, denote by~$\mathscr A$
the set of its irreducible components, by $D_\alpha$ the
component corresponding to some $\alpha\in\mathscr A$ and
by $d_\alpha$ its multiplicity.
We have $d_\alpha>0$ for all~$\alpha$ and $D=\sum d_\alpha D_\alpha$.
Let $U=X\setminus D$, and assume that $\omega_X(D)$
is equipped with a metrization.
Let~$\tau_{(X,D)}$ denote the corresponding measure on~$U(F)$
(see \S\ref{ss.metrics-measures}).
Endow the line bundles $\mathscr O_X(D)$
and $\mathscr O_{X}(D_\alpha)$, for $\alpha\in\mathcal A$,
with metrics, in such a way that the natural isomorphism
$\mathscr O_X(D)\simeq \bigotimes \mathscr O_X(D_\alpha)^{d_\alpha}$
is an isometry.
For $\alpha\in\mathcal A$, denote by~$\mathsec f_\alpha$
the canonical section of~$\mathscr O_{X}(D_\alpha)$;
denote by~$\mathsec f_D$ the canonical section of~$\mathscr O_X(D)$.
One has
$\mathsec f_D=\prod_{\alpha\in\mathcal A}\mathsec f_{\alpha}^{d_\alpha}$.
We also endow $\omega_{X}$ with the metric which makes
the isomorphism $\omega_X(D)\simeq\omega_X\otimes\mathscr O_X(D)$
an isometry.
Letting~$\tau_{X}$ be the Tamagawa measure on~$X(F)$
defined by the metrized line bundle~$\omega_X$,
we have the following equalities:
\[ \mathrm d\tau_{(X,D)}(x) = \norm{\mathsec f_D(x)}^{-1}\,\mathrm d\tau_X(x)
= \prod_{\alpha} \norm{\mathsec f_\alpha(x)}^{-d_\alpha} \, \mathrm d\tau_{X}(x). \]
Let $L$ be an effective divisor in~$X$ whose support contains the support
of~$D$; assume that the corresponding line bundle $\mathscr O_X(L)$
is endowed with a metric. The norm of its canonical section~$\mathsec f_L$
vanishes on~$L$, hence on~$D$. Consequently, for any positive
real number~$B$,
the set of all $x\in X(F)$ such $\norm{\mathsec f_L(x)}\geq 1/B$
is a closed subset of~$X(F)$, which is contained in $U(F)$,
hence is compact in~$U(F)$. Consequently, its volume
with respect to the measure~$\tau_{(X,D)}$,
\begin{equation}
V(B) =
\int_{\norm{\mathsec f_L(x)}\geq 1/B}
\mathrm d\tau_{(X,D)}(x) ,
\end{equation}
is finite for any~$B>0$.
We are interested in its asymptotic behavior
when $B\ra\infty$.
Let us
introduce the Mellin transform of the function~$\norm{\mathsec f_L}$
with respect to the measure~$\tau_{(X,D)}$, namely:
\begin{equation}
Z(s) = \int_{U(F)} \norm{\mathsec f_L(x)}^s \mathrm d\tau_{(X,D)}(x).
\end{equation}
The analytic properties of $Z(s)$ and $V(B)$
strongly depend on the geometry of the pair~$(X,D)$.
We will assume throughout that over the algebraic closure~$\bar F$,
the divisor~$D$
has strict normal crossings in~$X$; in that case we will see
that the answer can be stated in terms of the analytic
Clemens complex of~$D$.
In principle, using resolution of singularities, we can reduce
to this situation, even if it may be difficult in explicit examples
(see~\cite{hassett-tschinkel2003} for a specific computation related
to the asymptotic behavior of integral points of bounded height
established in~\cite{duke-r-s1993}).
For $\alpha\in\mathcal A$,
let $\lambda_\alpha$ be the multiplicity of~$D_\alpha$ in~$L$;
the divisor $\Delta=L-\sum\lambda_\alpha D_\alpha$ is effective
and all of its irreducible components meet~$U$.
We denote by~$\mathsec f_\Delta$ the canonical section of
the line bundle~$\mathscr O_X(\Delta)$; we endow this line
bundle with a metric so that $\norm{\mathsec f_L}=\norm{\mathsec f_\Delta}
\prod\norm{\mathsec f_\alpha}^{\lambda_\alpha}$.
Following the conventions of Section~\ref{subsubsec.Q-Cartier},
the results extend to the case where~$D$
and~$L$ are ${\mathbf Q}$-Cartier divisors; in that case,
the coefficients~$\lambda_\alpha$ and~$d_\alpha$ are rational numbers.
Let $\sigma=\max (d_\alpha-1)/\lambda_\alpha$,
the maximum being over all $\alpha\in\mathcal A$
such that $D_\alpha(F)\neq\emptyset$.
If there is no such~$\alpha$, we let $\sigma=-\infty$ by convention;
this means precisely that $U(F)$ is compact.
Only the case $\sigma\geq 0$ will really matter.
Indeed, as we shall see below, the condition $\sigma<0$ is equivalent
to the fact that $U(F)$ has finite volume with respect to~$\tau_{(X,D)}$.
Let $\mathscr C^{\text{\upshape an}}_{F,(L,D)}(D)$ be the subcomplex
of the analytic Clemens complex~$\mathscr C^{\text{\upshape an}}_F(D)$ consisting
of all non-empty subsets~$A\subset{\bar{\mathscr A}}$ such that $E_A(F)\neq\emptyset$
and $d_\alpha=\lambda_\alpha\sigma+1$ for any $\alpha\in A$.
Let $b=\dim\mathscr C^{\text{\upshape an}}_{F,(L,D)}(D)$.
For any face~$A$ of maximal dimension~$b$ of~$\mathscr C^{\text{\upshape an}}_{F,(L,D)}(D)$,
let $D_A=\bigcap_{\alpha\in A}D_\alpha$ be the corresponding
stratum of $X$. The subset $D_A(F)$ carries a natural measure~$\mathrm d\tau_{D_A}$
and we define
\begin{equation}
Z_A(s) = \int_{D_A(F)} \norm{\mathsec f_\Delta(x)}^s
\prod_{\alpha\not\in A} \norm{\mathsec f_\alpha(x)}^{s\lambda_\alpha-d_\alpha} \, \mathrm d\tau_{A}(x).
\end{equation}
\begin{prop}\label{prop.igusa-volume}
Let $\sigma$, $\mathscr C^{\text{\upshape an}}_{F,(L,D)}(D)$ and~$b$ be defined as above. Then
the integral defining~$Z(s)$ converges for $\Re(s)>\sigma$
and defines a holomorphic function in that domain.
Assume that~$\sigma\neq-\infty$.
Then there is a positive real number~$\delta$
such that $Z$ has a meromorphic continuation to a half-plane
$\Re(s)>\sigma-\delta$, with
a pole of order~$b=\dim \mathscr C^{\text{\upshape an}}_{F,(L,D)}(D)$
at $s=\sigma$
with leading coefficient
\[ \lim_{s\ra\sigma} (s-\sigma)^bZ(s)
= \sum_{\substack{A\in \mathscr C^{\text{\upshape an}}_{F,(L,D)}(D) \\ \dim A=b}} Z_A (\sigma) \prod_{\alpha\in A}\frac1{\lambda_\alpha} \]
and moderate growth in vertical strips.
When $F={\mathbf R}$ or~${\mathbf C}$, $Z$ has no other pole
provided $\delta$ is chosen sufficiently small.
When $F$ is ultrametric, there is a positive integer~$f$
such that $(1-q^{f(\sigma-s)})^bZ(s)$ is holomorphic on the
half-plane $\{\Re(s)>\sigma-\delta\}$, again provided
$\delta$ is sufficiently small.
\end{prop}
\begin{proof}
By definition,
\[ Z(s)
= \int_{X(F)} \norm{\mathsec f_\Delta}^s
\prod_\alpha \norm{\mathsec f_\alpha(x)}^{s \lambda _\alpha -d_\alpha}
\mathrm d\tau_{X}(x), \]
an integral of the type studied in~\S\ref{s.igusa}.
Precisely, using the notations introduced in~\S\ref{ss.geometric-igusa},
we have
$Z(s)=\mathscr I(\mathbf 1,(s+1;s\lambda_\alpha -d_\alpha+1))$, where
the first parameter~$s$ refers to the divisor~$\Delta$,
while for each $\alpha\in\mathcal A$,
the parameter $s_\alpha=s\lambda_\alpha-d_\alpha$ corresponds
to the divisor~$D_\alpha$.
Similarly,
\[ Z_A(s)=\mathscr I_A(\mathbf 1,(s+1;s\lambda_\alpha-d_\alpha+1)).\]
By Lemma~\ref{lemm.local-convergence},
this integral converges and defines a holomorphic function
as long as $\Re(s)>0$ and $\Re(s)\lambda_\alpha>d_\alpha$.
This shows the holomorphy of~$Z(s)$ for $\Re(s)>\sigma$.
Assume that $\sigma\neq-\infty$.
By~Proposition~\ref{prop.local-continuation}, the function~$Z$
has a meromorphic continuation to the domain of~${\mathbf C}$
defined by the inequalities~$\Re(s_\alpha)>-\frac12$ and $\Re(s)>0$,
hence to some domain of the form~$\Re(s)>\sigma-\delta$.
In the ultrametric case, the existence of a positive integer~$f$
such that $(1-q^{(\sigma-s)f})^b Z(s)$ has no pole
on such a half-plane also follows directly
from Proposition~\ref{prop.local-continuation}
(one may take for~$f$ the l.c.m. of the~$f_\alpha$),
as well as the growth in vertical strips.
It remains to prove the asserted behavior at $s=\sigma$.
By Proposition~\ref{prop.igusa-limit},
\[ \lim_{s\ra \sigma} (s-\sigma)^b Z(s)=
\sum_{\substack{A\in\mathscr A_{L,D}^{\text{\upshape an}} \\ \dim A=b}}
\mathscr I_A (\mathbf 1;
(\sigma +1;\sigma \lambda_\alpha-d_\alpha+1)_{\alpha\not\in A})
\prod_{\alpha\in A} \frac1{\lambda_\alpha}
= \sum_{\substack{A\in\mathscr A_{L,D}^{\text{\upshape an}} \\ \dim A=b}}
Z_A(\sigma)
\prod_{\alpha\in A} \frac1{\lambda_\alpha}.
\]
\end{proof}
Using the Tauberian theorem~\ref{theo.tauber} recalled in the appendix,
we obtain the following estimate for the volume~$V(B)$
in the archimedean case.
\begin{theo}
\label{theo.geometric-volume}
Assume that $F={\mathbf R}$ or~${\mathbf C}$ and that $\sigma\geq 0$. This implies that $b\geq 1$.
There exists a polynomial~$P$
and a positive real number~$\delta$ such that
\[ V(B) = B^\sigma P(\log B) + \mathrm O(B^{\sigma-\delta}). \]
Moreover, if $\sigma>0$, then $P$ has degree~$b-1$ and leading
coefficient
\[ \operatorname{lcoeff}(P) = \frac1{\sigma(b-1)!}
\sum_{\substack{A\subset \mathscr C^{\text{\upshape an}}_{F,(L,D)}(D) \\ \dim A=b}}
Z_A(\sigma) \prod_{\alpha\in A} \frac{1}{\lambda_\alpha};\]
otherwise, if $\sigma=0$, then $P$ has degree~$b$ and its leading
coefficient satisfies
\[ \operatorname{lcoeff}(P) = \frac1{b!}
\sum_{\substack{A\subset \mathscr C^{\text{\upshape an}}_{F,(L,D)}(D) \\ \dim A=b}}
Z_A(\sigma) \prod_{\alpha\in A} \frac{1}{\lambda_\alpha}.\]
\end{theo}
Considering integrals of the form
\[ Z_\Phi(s) = \int_{U(F)} \Phi(x) \norm{\mathsec f_L(x)}^s\, \mathrm d\tau_{(X,D)}(x), \]
or applying the abstract equidistribution theorem
(Proposition~\ref{prop.abstract.equi}),
we deduce the following corollary (``equidistribution of height balls''):
\begin{coro}\label{coro.equidistribution.local}
Assume that $F={\mathbf R}$ or~${\mathbf C}$ and that~$\sigma>0$.
Then $b\geq 1$ and, when $B\ra+\infty$, the family of measures
\[ V(B)^{-1} \mathbf 1_{\{ \norm{\mathsec f_L(x)}\geq 1/B\}}
\,\mathrm d\tau_{(X,D)}(x) \]
converges tightly to the unique probability measure which
is proportional to
\[ \sum_{\substack{A\subset \mathscr A_{L,D}^\text{\upshape an} \\ \dim A=b}}
\norm{\mathsec f_\Delta(x)}^\sigma
\prod_{\alpha\in A} \frac{1}{\lambda_\alpha}
\prod_{\alpha\not\in A} \norm{\mathsec f_\alpha(x)}^{-1} \,\mathrm d\tau_{D_A}(x). \]
\end{coro}
In the case of ultrametric local fields,
our analysis leads to a good understanding
of the Igusa zeta functions but the output
of the corresponding Tauberian theorem,
\emph{i.e.}, the discussion leading to
Corollaries~\ref{coro.theo.tauber2}, \ref{coro.theo.tauber2b}
and \ref{coro.theo.tauber2c}
in the Appendix, is less convenient since
one can only obtain an asymptotic expansions for $V(q^N)$,
when $N$ belongs to a fixed congruence class modulo some positive integer~$f$.
In fact, when all irreducible components of~$D$ are geometrically irreducible,
one may take $f=1$ in Proposition~\ref{prop.igusa-volume}.
In that case, Corollary~\ref{coro.theo.tauber2b} even leads
to a precise asymptotic expansion in that case.
We leave the detailed statement to the interested reader
and we content ourselves with the following corollary.
\begin{coro}\label{coro.asymptotic.volume-ultrametric}
Assume that $F$ is ultrametric and that~$\sigma\geq 0$.
Let $b^*=b$ if $\sigma=0$, and $b^*=b-1$ if $\sigma>0$.
Then, when the integer~$N$ goes to infinity,
\[ 0< \liminf \frac{V(q^N)}{q^{N\sigma} N^{b^*}} \leq
\limsup \frac{V(q^N)}{q^{N\sigma} N^{b^*}} <\infty. \]
\end{coro}
\begin{proof}
Changing metrics modifies the volume forms and the height functions
by a factor which is lower- and upper-bounded; this does not
affect the result of the corollary.
Consequently, we may assume that all metrics are smooth
and that the function~$\norm{\mathsec f_L}$ is $q^{\mathbf Z}$-valued.
The Igusa zeta function~$Z(s)$ is then~$2i\pi/\log q$-periodic
and has a meromorphic continuation of the form
$\sum_A \Phi_A(s)\prod_{\alpha\in A}(1-q^{f_\alpha\lambda_\alpha(\sigma-s)})$,
for some functions~$\Phi_A$ which are holomorphic on an open half-plane
containing the closed half-plane given by $\{\Re(s)\geq\sigma\}$.
Let $f$ be any positive integer such that $f$ is
an integral multiple of $f_\alpha\lambda_\alpha$,
for any~$\alpha\in \mathscr A$;
we see that $Z(s)(1-q^{f(\sigma-s)})^b$ extends holomorphically
to this open half-plane and
it now suffices to apply Corollary~\ref{coro.theo.tauber2}.
\end{proof}
\subsection{Adelic Igusa integrals}
\label{sec.adelic-igusa}
\subsubsection{Geometric setup}
Let $F$ be a number field, let ${\mathbb A}_F$ be the ring of adeles of~$F$.
More generally, if $S$ is a finite set of places of~$F$,
let $\mathbf A_F^S$ be the restricted product of local fields~$F_v$,
for $v\not\in S$.
We fix an algebraic closure~$\bar F$ of~$F$;
for each place~$v$,
we also fix a decomposition group $\Gamma_v$ at~$v$
in the Galois group $\Gamma_F=\Gal(\bar F/F)$.
Let $\bar X$ be a smooth projective variety over~$F$,
let $(D_\alpha)_{\alpha\in \mathscr A}$
be a family of irreducible divisors in~$\bar X$ whose
sum $\Delta=\sum_{\alpha\in \mathscr A} D_\alpha$
is geometrically a divisor with strict normal crossings.
For $\alpha\in\mathscr A$,
let $\mathsf f_{D_\alpha}$ denote the canonical section
of the line bundle $\mathscr O_{\bar X}(D_\alpha)$
and let $F_\alpha$ be the algebraic closure of~$F$
in the function field of~$D_\alpha$.
Let $\bar{\mathscr A}$ be the set of irreducible
components of the divisor~$\Delta_{\bar F}$; it carries an action
of $\Gamma_F$, the set of orbits of which, $\bar{\mathscr A}/\Gamma_F$,
identifies canonically
with the set $\mathscr A$.
Endow all line bundles $\mathscr O_{\bar X}(D_\alpha)$, as well
as the canonical line bundle~$\omega_{\bar X}$, with adelic metrics.
Let $\mathscr B$ be any subset of~$\mathscr A$, corresponding
to a subset $\bar{\mathscr B}$ of~$\bar{\mathscr A}$
which is stable under the action of~$\Gamma_F$.
Let $Z=\bigcup_{\alpha\in\mathscr B} D_\alpha$
be the union of the corresponding divisors,
$D=\bigcup_{\alpha\not\in\mathscr B}D_\alpha$ the union
of the other divisors,
and let us define $X=\bar X\setminus Z$ and $U=X\setminus D$.
The local spaces $X(F_v)$, $U(F_v)$, for any place~$v$ of~$F$,
and the adelic spaces $X({\mathbb A}_F)$, $U({\mathbb A}_F)$
are locally compact and carry Tamagawa measures
$\tau_{X,v}$, $\tau_{U,v}$, $\tau_X$ and~$\tau_U$
(see Definition~\ref{defi.tamagawa-measure})
which are Radon measures, \emph{i.e.}, finite on compact subsets.
The set of irreducible components $\mathscr A_v$ of the divisor~$\Delta_{F_v}$
is in natural bijection with the set of $\Gamma_v$-orbits
in~$\bar{\mathscr A}$. For any such orbit~$\alpha$, we will
denote by $D_\alpha$ the corresponding divisor on $X_{F_v}$.
Our aim here is to establish analytic properties
of the adelic integral
\[ \mathscr I (\Phi; (s_\alpha)) =
\int_{U({\mathbb A}_F)} \prod_{\alpha\in \mathscr A}
\prod_{v} \norm{\mathsf f_{D_\alpha}}_v^{s_\alpha-1} \Phi(x)
\, \mathrm d\tau_U, \]
when $\Phi$ is the restriction to~$U({\mathbb A}_F)$
of a smooth function with compact support on $X({\mathbb A}_F)$.
It will be convenient to view the map $\alpha\mapsto s_\alpha$
as a $\Gamma_F$-equivariant map from $\bar{\mathscr A}$ to~${\mathbf C}$.
In other words, for each $\alpha\in\bar{\mathscr A}$, we
let $s_\alpha=s_{[\alpha]}$, where $[\alpha]\in\mathscr A$ is
the orbit of~$\alpha$ under~$\Gamma_F$. More generally,
if $\beta$ is any subset of such an orbit, we define $s_\beta=s_\alpha$,
for any $\alpha\in\bar{\mathscr A}$ belonging to~$\beta$
(it does not depend on the choice of~$\alpha$).
\subsubsection{Convergence of local integrals}
Assume that $\Phi=\prod_v \Phi_v$ is a product
of smooth functions and define, for any $v\in\Val(F)$,
\[ \mathscr I_v (\Phi_v; (s_\alpha)) =
\int_{U(F_v)}
\prod_{\alpha\in \mathscr A}
\norm{\mathsf f_{D_\alpha}}_v^{s_\alpha-1} \Phi_v(x) \,
\mathrm L_v(1,\operatorname{EP} (U))\mathrm d\tau_{X,v}. \]
When the local integrals $\mathscr I_v$ converge absolutely,
as well as the infinite product
$\prod_v \mathscr I_v(\Phi;(s_\alpha))$,
then the integral $\mathscr I(\Phi;(s_\alpha))$
exists and one has an equality
\[ \mathscr I(\Phi; (s_\alpha)) = \mathrm L_*(1,\operatorname{EP} (U))^{-1}
\prod_v \mathscr I_v(\Phi_v;(s_\alpha)). \]
Let $\alpha\in\mathscr A$.
Let us decompose the $\Gamma_F$-orbit~$\alpha$ as a union of disjoint
$\Gamma_v$-orbits $\alpha_1,\dots,\alpha_r$.
Then $D_\alpha=\sum_{i=1}^r D_{\alpha_i}$ and $\mathsec f_{D_\alpha}
=\prod_{i=1}^r \mathsec f_{D_{\alpha_i}}$.
It follows that the integral~$\mathscr I_v$ can be rewritten as
\[ \mathscr I_v (\Phi_v; (s_\alpha)) =
\int_{U(F_v)}
\prod_{\alpha\in \mathscr A_v}
\norm{\mathsf f_{D_\alpha}}_v^{s_\alpha-1} \Phi_v(x) \,
\mathrm L_v(1,\operatorname{EP} (U))\mathrm d\tau_{X,v}. \]
By Lemma~\ref{lemm.local-convergence}, $\mathscr I_v(\Phi_v;(s_\alpha))$
converges absolutely when $\Re(s_\alpha)>0$ for each $\alpha\in\mathscr A$.
If moreover $\Phi_v$ has compact support in~$X(F_v)$,
then the conditions $\Re(s_\alpha)>0$ for $\alpha\in\mathscr B$
are not necessary.
The decomposition $\alpha=\bigcup \alpha_i$ corresponds to
the decomposition $F_\alpha\otimes_F F_v =\prod_{i=1}^r F_{\alpha_i}$.
Observe that the local factor of Dedekind's zeta function~$\zeta_{F_\alpha}$
at~$v$ is given by the formula
\[ \zeta_{F_\alpha,v}(s) = \prod_{i=1}^r (1-q_v^{-f_{\alpha,i}s})^{-1}, \]
where for each~$i$, $f_{\alpha,i}=[F_{\alpha,i}:F_v]$.
Let $\zeta_{F_\alpha}^*(1)$ be the residue at $s=1$ of this
zeta function.
\subsubsection{Convergence of an Euler product}
To study the convergence of the product, we may ignore a finite
set of places.
Let $S$ be a finite set of places containing the archimedean places
so that, for all other places,
all metrics are defined by good integral models~$\overline{\mathscr X}$,
$\mathscr X$, $\mathscr U$, \dots, over $\Spec\mathfrak o_{F,S}$.
Assume moreover that for any $v\not\in S$, $\Phi_v$ is the characteristic
function of~$\mathscr X(\mathfrak o_v)$.
By Denef's formula (Prop.~\ref{prop.denef}),
one has
\[ \mathscr I_v(\Phi_v;(s_\alpha))
= \sum_{A\subset \mathscr A_v\setminus\mathscr B_v}
\big(q_v^{-1}\mu_v(\mathfrak o_v)\big)^{\dim X}
\mathop\# D_A^\circ (k_v)
\prod_{\alpha\in A} \frac{q_v^{f_\alpha}-1}{q_v^{f_\alpha s_\alpha}-1}
\]
for any place $v\not\in S$.
Combined with the estimate of Theorem~\ref{prop.convergence},
this relation implies that
\[ \mathscr I_v(\Phi_v;(s_\alpha))
\prod_{\alpha\in\mathscr A\setminus \mathscr B}
\zeta_{F_\alpha,v}(s_\alpha)^{-1} = 1 + \mathrm O(q_v^{-1-\eps})
\]
provided $\Re(s_\alpha)>\frac12+\eps$ for each $\alpha\not\in\mathscr B$.
(See Prop.~9.5 in~\cite{chambert-loir-t2002} for a similar computation.)
This asymptotic expansion will imply the desired convergence of the infinite
product.
\begin{prop}\label{prop.igusa-adelic-merom}
Assume that $\Phi$ is a smooth function with compact support on $X({\mathbb A}_F)$.
Then the integral $\mathscr I(\Phi;(s_\alpha))$ converges
for $\Re(s_\alpha)>1$, for each $\alpha\not\in\mathscr B$,
and defines a holomorphic function in this domain.
This function has a meromorphic continuation:
there is a holomorphic function~$\phi$ defined for
$\Re(s_\alpha)>\frac12$ if $\alpha\not\in\mathscr B$,
such that
\[ \mathscr I(\Phi;(s_\alpha)) =\phi(s)\prod_{\alpha\not\in\mathscr B} \zeta_{F_\alpha}(s_\alpha). \]
Moreover, if $s_\alpha=1$ for $\alpha\not\in\mathscr B$, then
\[ \phi(s) = \prod_{\alpha\not\in\mathscr B}\zeta_{F_\alpha}^*(1)^{-1}
\int_{X({\mathbb A}_F)} \Phi(x)\prod_{\beta\in\mathscr B}\prod_v \norm{\mathsec f_{D_\beta}}_v^{s_\beta-1}\,\mathrm d\tau_X(x) .\]
\end{prop}
(Note that, in the last formula,
the function under the integration sign has compact
support on $X({\mathbb A}_F)$, while $\mathrm d\tau_X$ is a Radon
measure on that space.)
\begin{proof}
The first two parts of the theorem follow from the estimates
that we have just derived and the absolute convergence for $\Re(s)>1$
of the Euler product defining the Dedekind zeta function of
a number field.
Let $s\in{\mathbf C}^{\mathscr A}$ be such that $s_\alpha=1$
for $\alpha\not\in\mathscr B$.
One has therefore
\[\mathscr I_v(\Phi_v;s) = \mathrm L_v(1,\operatorname{EP} (U))\int_{U(F_v)}
\prod_{\beta\in\mathscr B} \norm{\mathsec f_{D_\beta}}_v^{s_\beta-1}
\Phi_v(x)\,\mathrm d\tau_{X,v}(x). \]
Moreover, one has equalities of virtual representations of~$\Gamma_F$
(\emph{cf.}~\ignorespaces Equation~\eqref{eq.virtual-eq}),
\begin{align*}
\operatorname{EP} (U) & = \operatorname{EP} (\bar X) +\sum_{\alpha\in\mathscr A}\Ind_{\Gamma_{F_\alpha}}^{\Gamma_F}\mathbf 1\\
\operatorname{EP} (X) &= \operatorname{EP} (\bar X) +\sum_{\alpha\in\mathscr B}\Ind_{\Gamma_{F_\alpha}}^{\Gamma_F}\mathbf 1,
\end{align*}
from which it follows that
\[ \operatorname{EP} (X) = \operatorname{EP} (U) - \sum_{\alpha\not\in\mathscr B}\Ind_{\Gamma_{F_\alpha}}^{\Gamma_F}\mathbf 1. \]
In particular, for any finite place $v$ of~$F$,
\[ \mathrm L_v(1,\operatorname{EP} (X))
= \mathrm L_v(1,\operatorname{EP} (U)) \prod_{\alpha\not\in\mathscr B} \zeta_{F_\alpha,v}(1)^{-1} \]
and
\[ \mathrm L^*(1,\operatorname{EP} (X))
= \mathrm L^*(1,\operatorname{EP} (U)) \prod_{\alpha\not\in\mathscr B}\zeta_{F_\alpha}^*(1)^{-1}. \]
If $S_\infty$ is the set of archimedean places of~$F$, then
(recall that $s_\alpha=1$ for $\alpha\not\in\mathscr B$)
\begin{align*}
\phi(s) & = \mathrm L^*(1,\operatorname{EP} (U))^{-1}
\prod_{v\in S_\infty} \mathscr I_v(\Phi_v;s)
\prod_{v\not\in S_\infty} \left(\mathscr I_v(\Phi_v;s)
\prod_{\alpha\not\in\mathscr B}\zeta_{F_\alpha,v}(1)^{-1} \right) \\
&= \mathrm L^*(1,\operatorname{EP} (U))^{-1}
\prod_{v} \int_{X(F_v)} \Phi_v(x) \prod_{\beta\in\mathscr B}\norm{\mathsec f_{D_\beta}}_v^{s_\beta-1} \mathrm L_v(1,\operatorname{EP}(X)) \,\mathrm d\tau_{X,v}(x) \\
&= \mathrm L^*(1,\operatorname{EP} (U))^{-1}
\mathrm L^*(1,\operatorname{EP} (X))
\int_{X({\mathbb A}_F)} \Phi(x)\prod_{\beta\in\mathscr B}\prod_v
\norm{\mathsec f_{D_\beta}}_v^{s_\beta-1}\,\mathrm d\tau_X(x) \\
&= \prod_{\alpha\not\in\mathscr B}\zeta_{F_\alpha}^*(1)^{-1}
\int_{X({\mathbb A}_F)} \Phi(x)\prod_{\beta\in\mathscr B}\prod_v \norm{\mathsec f_{D_\beta}}_v^{s_\beta-1}\,\mathrm d\tau_X(x) .
\end{align*}
\end{proof}
In fact, it is possible to establish a more general theorem.
Let $S$ be a finite set of places of~$F$ containing the archimedean places.
For any place $v\in S$, let $\Phi_v$ be a smooth bounded function on $X(F_v)$;
let also $\Phi^S$ be a smooth function with compact support
on $X({\mathbb A}_F^S)$; let $\Phi$ be the function $\Phi^S\prod_{v\in S}\Phi_v$
on $X({\mathbb A}_F)$.
By the same arguments as above,
the integral $\mathscr I(\Phi;(s_\alpha))$ converges
provided $\Re(s_\alpha)>1$ for each $\alpha\not\in\mathscr B$
and
$\Re(s_\beta)>0$ for each $\beta\in\mathscr B$
and defines a holomorphic function in that domain.
\begin{prop}
Let $\Omega\subset{\mathbf C}^{\mathscr A}$ be the set of $(s_\alpha)$
such that $\Re(s_\alpha)>\frac12$ for $\alpha\not\in\mathscr B$
and $\Re(s_\beta)>-\frac12$ for $\beta\in\mathscr B$.
The function $\mathscr I(\Phi;(s_\alpha))$
admits a meromorphic continuation of the following form.
For any place $v\in S$ and any face $A$ of maximal
dimension of $\mathscr C^{\text{\upshape an}}_{F_v}(D)$, there is a holomorphic
function $\phi_A$ on~$\Omega$ such that
\[ \mathscr I(\Phi;(s_\alpha))=
\prod_{\alpha\not\in\mathscr B} \zeta_{F_\alpha}^S(s_\alpha)
\prod_{v\in S} \left( \sum_{A\in\mathscr C^{\text{\upshape an,max}}_{F_v}(D)} \phi_A(s)
\prod_{\alpha\in A}\zeta_{F_\alpha,v}(s_\alpha)\right) . \]
Moreover, the functions $\phi_A$ have moderate growth in vertical strips
in the sense that for any compact subset~$K$ of ${\mathbf R}^{\mathscr A}\cap\Omega$,
there are real numbers~$c$ and~$\kappa$ such that
\[ \abs{\phi_A(s)} \leq c \prod_{\alpha\in\mathscr A}(1+\abs{s_\alpha})^\kappa,\]
for $s\in{\mathbf C}^{\mathscr A}$ such that $\Re(s)\in K$.
\end{prop}
\subsection{Volume asymptotics over the adeles}
\label{subsec.asymptotics.adelic}
In this Subsection we derive asymptotic estimates
for volumes of height balls in adelic spaces,
similar to those we established above for height balls over local fields.
Let $F$ be a number field,
$X$ a smooth projective variety over~$F$, purely of dimension~$n$.
Let $D$ be an effective divisor in~$X$ and $\mathscr A$ its set of irreducible components; for $\alpha\in\mathscr A$, let $D_\alpha$ be the corresponding
component and $d_\alpha$ its multiplicity in~$D$. We have $D=\sum d_\alpha D_\alpha$.
For $\alpha\in\mathscr A$, let $\mathsec f_\alpha$ be the canonical
section of~$\mathscr O_X(D_\alpha)$; let $\mathsec f_D$
denote the canonical section of~$\mathscr O_X(D)$. We have
$\mathsec f_D=\prod \mathsec f_\alpha^{d_\alpha}$.
Let us endow these line bundles with adelic metrics, in such
a way that the isomorphism
$\mathscr O_X(D)\simeq \bigotimes \mathscr O_X(D_\alpha)^{d_\alpha}$
is an isometry.
Let $U=X\setminus D$ ; let us endow $\omega_X$ and $\omega_X(D)$
with adelic metrics in such a way that the isomorphism
$\omega_X(D)\simeq \omega_X\otimes \mathscr O_X(D)$
is an isometry.
By the constructions of Section~\ref{sec.general},
we obtain natural measures $\tau_X$, $\tau_{U}$
and $\tau_{(X,D)}$ on~$U({\mathbb A}_F)$ or $X({\mathbb A}_F)$, related
by the equalities
\[ \mathrm d\tau_{(X,D)}(x) = H_D(x) \mathrm d\tau_U(x), \]
where, for $x\in U({\mathbb A}_F)$, $H_D(x)=\prod_x \norm{\mathsec f_D(x)}_v^{-1}$.
Let $L$ be an effective divisor in~$X$
whose support is equal to the support of~$D$;
for $\alpha\in \mathscr A$, let $\lambda_\alpha$ be the multiplicity
of~$D_\alpha$ in~$L$, so that $L=\sum\lambda_\alpha D_\alpha$.
We endow the line bundle~$\mathscr O_X(L)$
with the natural adelic metric deduced from the
metrics of the line bundles~$\mathscr O_X(D_\alpha)$.
The canonical section of~$\mathscr O_X(L)$ vanishes on~$L$, hence on~$D$.
Let $H_L$ denote the corresponding height function
(denoted $H_{\mathscr O_X(L),\mathsec f_L}$
in Section~\ref{subsec.heights.adelic})
on the adelic space~$U({\mathbb A}_F)$; recall (Equation~\eqref{eq.height.f})
that it is defined by the formula
\[ H_L(\mathbf x) = \prod_{v\in\Val(F)} \norm{\mathsec f_L}(x_v)^{-1},
\qquad \text{for $\mathbf x = (x_v)_v$.} \]
By Lemma~\ref{lemm.northcott.adelic},
$H_L$ is bounded from below on~$U({\mathbb A}_F)$ and, for any real number~$B$,
the set of all $\mathbf x\in U({\mathbb A}_F)$ such that $H_L(\mathbf x)\leq B$
is compact in $(X\setminus L)({\mathbb A}_F)$.
Let $V(B)$ denote its volume with respect to the measure~$\tau_{(X,D)}$:
\begin{equation}
V(B) = \int_{\substack{U({\mathbb A}_F) \\ \{H_L(\mathbf x)\leq B\}}}
\,\mathrm d\tau_{(X,D)}(\mathbf x);
\end{equation}
it is a positive real number
(for $B$ large enough)
and we want to understand its asymptotic behavior
when $B\rightarrow \infty$.
We are also interested in the asymptotic behavior of the probability
measures
\[ \frac1{V(B)} \mathbf 1_{\{ H_L(\mathbf x)\leq B\}}
\mathrm d\tau_{(X,D)}(\mathbf x).
\]
As in the case of local fields, we introduce a geometric Igusa
zeta function, namely
\begin{equation}
Z(s) = \int_{U({\mathbb A}_F)} H_L(x)^{-s}\,\mathrm d\tau_{(X,D)}(x), \end{equation}
for any complex number~$s$ such that the integral converges
absolutely.
For any such~$s$, we thus have
\[ Z(s) = \int_{U({\mathbb A}_F)} \prod_v \norm{\mathsec f_\alpha(x_v)}^{s\lambda_\alpha-d\alpha} \, \mathrm d\tau_X(\mathbf x)=
\mathscr I(\mathbf 1; (s\lambda_\alpha-d_\alpha+1)) \]
with the notation of Section~\ref{sec.adelic-igusa}.
Let $\sigma=\max_{\alpha\in\mathscr A}(d_\alpha/\lambda_\alpha)$
and let $\mathscr A_{L,D}$ be the set of $\alpha\in\mathscr A$
such that $d_\alpha=\lambda_\alpha\sigma$.
Then, the integral defining~$Z(s)$ converges for any
complex number~$s$ such that $\Re(s)>\sigma$
(Proposition~\ref{prop.igusa-adelic-merom}).
Moreover, there exists a positive real number~$\delta$
such that function $s\mapsto Z(s)$ admits a meromorphic
continuation to the half-plane given by~$\Re(s)>\sigma-\delta$.
Namely, by Proposition~\ref{prop.igusa-adelic-merom} again,
there exists a holomorphic function~$\phi$ defined for $\Re(s)>\sigma-\delta$
such that
\[ Z(s) = \phi(s) \prod_{\alpha\in\mathscr A_{L,D}}
\zeta_{F_\alpha}(s\lambda_\alpha-d_\alpha+1) \]
and
\[ \phi(1) = \prod_{\alpha\in\mathscr A_{L,D}}
\zeta_{F_\alpha}^*(1)
\int_{X({\mathbb A}_F)} \prod_{\alpha\not\in\mathscr A_{L,D}}
H_{D_\alpha}(x)^{d_\alpha-\sigma\lambda_\alpha}
\,\mathrm d\tau_X(x). \]
In particular, the function~$Z$ has a pole at $s=1$
of order~$\mathop\#(\mathscr A_{L,D})$ and satisfies:
\[ \lim_{s\ra 1} (s-1)^{\mathop\#(\mathscr A_{L,D})} Z(z)
= \phi(1) \prod_{\alpha\in\mathscr A} \frac {\zeta_{F_\alpha}^*(1)}{\lambda_\alpha}
= \prod_{\alpha\in\mathscr A_{L,D}} \lambda_\alpha^{-1}
\int_{X({\mathbb A}_F)} \prod_{\alpha\not\in\mathscr A_{L,D}}
H_{D_\alpha}(x)^{d_\alpha-\sigma\lambda_\alpha}
\,\mathrm d\tau_X(x). \]
Let $E$ denote the ${\mathbf Q}$-divisor $\sigma L-D$.
We have
\[ E=\sigma L-D= \sum_{\alpha\in\mathscr A} (\sigma\lambda_\alpha-d_\alpha)D_\alpha = \sum_{\alpha\not\in\mathscr A_{L,D}} (\sigma\lambda_\alpha-d_\alpha)D_\alpha .\]
Consequently,
\begin{equation}\label{eqn.Z*(1)-adelic}
\lim_{s\ra \sigma} (s-\sigma)^{\mathop\#(\mathscr A_{L,D})} Z(\sigma)
= \prod_{\alpha\in\mathscr A_{L,D}} \lambda_\alpha^{-1}
\int_{X({\mathbb A}_F)} H_E(x)^{-1}\,\mathrm d\tau_X(x)
= \int_{X({\mathbb A}_F)} \,\mathrm d\tau_{(X,E)}(x).
\end{equation}
We summarize the results obtained in the following Proposition:
\begin{prop}\label{prop.zeta.global}
Let $\sigma$ and $\mathscr A_{L,D}$ be defined as above.
Then the integral defining $Z(s)$ converges for $\Re(s)>\sigma$
and defines a holomorphic function in that domain.
Moreover, there is a positive real number~$\delta$
such that $Z$ has a continuation to the half-plane $\Re(s)>\sigma-\delta$
as a meromorphic function with moderate growth in vertical strips,
whose only pole is at $s=\sigma$,
with order $b=\mathop\#(\mathscr A_{L,D})$
and leading coefficient given by Equation~\eqref{eqn.Z*(1)-adelic}.
\end{prop}
Similarly to what we did in the local case,
using the Tauberian theorem~\ref{theo.tauber} and the abstract
equidistribution theorem (Proposition~\ref{prop.abstract.equi}),
we obtain the following result.
\begin{theo}\label{theo.volume.global}
There exists a monic polynomial~$P$ of degree~$b$
and a positive real number~$\delta$ such that, when
$B\ra\infty$,
\[ V(B) =
\frac{\prod_{\alpha\in\mathscr A_{L,D}}\lambda_\alpha^{-1}}
{\sigma (b-1)!}
B^\sigma P(\log B)
\int_{X({\mathbb A}_F)} H_E(x)^{-1}\,\mathrm d\tau_X(x)
+ \mathrm O(B^{\sigma-\delta}). \]
Moreover, we have the tight convergence of probability measures
\[ \frac1{V(B)} \mathbf 1_{\{H_L(x)\leq B\}} \mathrm d\tau_{(X,D)}(x)
\ra \frac1{\int_{X({\mathbb A}_F)}H_E^{-1} \tau_X} H_E(x)^{-1} \,\mathrm d\tau_{X}(x). \]
\end{theo}
\begin{rema}
Let $S$ be a finite set of places of~$F$.
At least two variants of the preceding results
may be useful in $S$-integral contexts.
Let ${\mathbb A}_F^S$ be the ring of adeles ``outside~$S$''.
For the first variant, we consider points of $U({\mathbb A}_F^S)$
of bounded height, and their volume with respect to the Tamagawa
measure $\tau_{(X,D)}^S$ in which the local factors in~$S$.
In that context, all infinite products of the set~$\Val(F)$
of places of~$F$ are replaced by the products over the set
$\Val(F)\setminus S$ of places which do not belong to~$S$.
The modifications to be made to the statement and proof
of Theorem~\ref{theo.volume.global} are obvious
and lead to an asymptotic expansion for the
volumes of height balls in~$U({\mathbb A}_F^S)$.
A second variant is also possible which restricts
to the subset~$\Omega$ of~$U({\mathbb A}_F)$ consisting of
points $(x_v)_{v\in\Val(F)}$ which are ``integral''
at each place~$v\not\in S$. We do not need to be
specific about that condition; for all that matters
in our analysis, we understand that this subset
is relatively compact in~$U({\mathbb A}_F)$ and has non-empty
interior. A scheme-theoretic definition
would ask that we are given a projective and flat model of~$X$
over~$\mathfrak o_F$; then a point $(x_v)\in U({\mathbb A}_F)$
belongs to~$\Omega$ if and only if, for any finite
place $v\not\in S$, the reduction
mod~$v$ of~$x_v$ does not belong to $D({\mathbf F}_v)$.
An adelic definition would consider $\Omega$ to be
defined by a condition of the
form $\prod_{v\not\in S} \norm{\mathsec f_L}_v(x)\leq B_0$,
for some real number~$B_0$.
In that case, the analytic study of the adelic zeta
function involved is straightforward from what
has been done in Section~\ref{subsec.geometric-volume}.
Namely, it decomposes as an infinite product over
all places~$v\in\Val(F)$ of $v$-adic zeta functions.
The subproduct corresponding
to places~$v\not\in S$ extends holomorphically
to the whole complex plane, while each factor attached
to a place~$v\in S$ is the source of zeroes and poles
described by the $v$-adic analytic Clemens complex
as in Prop.~\ref{prop.igusa-volume}.
Although the procedure should be quite clear to the reader,
in any specific example,
a general statement would certainly be
too obfuscating to be of any help.
We would like the reader to remark than when all abscissae of convergence
at places in~$S$ are equal to a common real number~$\sigma$,
the order of the pole at~$\sigma$ will be the
sum of the orders~$b_v$,
and the leading coefficient the product of those
computed in Prop.~\ref{prop.igusa-volume}.
Moreover, if $S$ contains archimedean places,
the order of the pole at $\sigma$ will be
\emph{strictly} greater than the order of the other
possible poles on the line $\Re(s)=\sigma$.
In that case, the Tauberian theorem~\ref{theo.tauber.S}
gives a simple asymptotic formula for the volume
of points of bounded height.
With the notation of that Theorem, the $q_j$ are powers
of prime numbers and the non-Liouville property
of the quotients $\log q_j/\log q_{j'}$ follows
from Baker's theorem on linear forms in logarithms,
\cite[Theorem~3.1]{baker1975}.
We expect that these asymptotic expansions for volumes
can serve as a guide to understand the asymptotic number
of $S$-integral solutions of polynomial equations,
\emph{e.g.}, to a $S$-integral generalization of the circle method.
\end{rema}
\section{Introduction}
The study of rational and integral points on algebraic varieties
defined over a number field often leads to considerations
of volumes of real, $p$-adic or adelic spaces.
A typical problem in arithmetic geometry is to
establish asymptotic expansions, when $B\ra\infty$, for the number
$N_f(B)$ of solutions in rational integers smaller than~$B$
of a polynomial equation $f(\mathbf x)=0$.
When applicable,
the circle method gives an answer in terms of a ``singular integral''
and a ``singular series'', which itself can be viewed as a product
of $p$-adic densities.
The size condition is only reflected in a parameter in
the singular integral, whose asymptotic expansion
therefore governs that of $N_f(B)$.
More generally, one considers systems of polynomial equations,
\emph{i.e.}, algebraic varieties over a number field
or schemes of finite type over rings of integers,
together with embedding into a projective or affine space.
Such an embedding induces a \emph{height function} (see, \emph{e.g.},
\cite{lang1983,serre1997,hindry-silverman2000})
such that there are only finitely many solutions of bounded height,
in a fixed number field, resp. ring of integers.
A natural generalization of the problem above is to understand
the asymptotic behavior of the number of such solutions, as well
as their distribution in the ambient space
for the local or adelic topologies, when the bound grows to infinity.
Apart from applications of the circle method,
many other instances of this problem have been successfully
investigated in recent years, in particular, in the context
of linear algebraic groups and their homogeneous spaces.
For such varieties, techniques from ergodic theory and harmonic
analysis are very effective;
for integral points, see
\cite{duke-r-s1993}, \cite{eskin-mcmullen1993},
\cite{eskin-m-s1996}, \cite{borovoi-r1995},
\cite{maucourant2007}, \cite{gorodnik-o-s2006};
for rational points, see
\cite{batyrev-m90}, \cite{franke-m-t89},
\cite{peyre95}, \cite{batyrev-t96},
\cite{chambert-loir-t2002}.
In most cases, the proof subdivides into two parts:
\begin{enumerate}
\item comparison of the point counting with a volume asymptotic;
\item explicit computation of this volume asymptotic.
\end{enumerate}
In this paper, we develop a general geometric framework
for the second part, \emph{i.e.}, for the understanding
of densities and volumes occurring in the counting problems above.
We now explain the main results.
\subsection{Tamagawa measures for algebraic varieties}
Let $F$ be a number field. Let $\Val(F)$ be the set
of equivalence classes of absolute values of~$F$.
For $v\in\Val(F)$, we write $v\mid p$ if $v$ defines the $p$-adic
topology on~${\mathbf Q}$, and $v\mid \infty$ if it is archimedean.
For $v\in\Val(F)$, let $F_v$ be the corresponding completion of~$F$
and, if $v$ is ultrametric, let $\mathfrak o_v$ be its ring of integers.
We identify~$v$ with the specific absolute value~$\abs\cdot_v$
on~$F_v$ defined by the formula $\mu(a\Omega)=\abs a_v\mu(\Omega)$,
where $\mu$ is any Haar measure on the additive group~$F_v$,
$a\in F_v$ and $\Omega$ is a measurable subset of~$F_v$
of finite measure.
Let $X$ be a smooth projective algebraic variety over~$F$.
Fix an adelic metric on its canonical line bundle~$K_X$
(see Section~\ref{subsubsec.adelic-metrics}).
For any~$v\in\Val (F)$, the set $X(F_v)$ carries an analytic topology
and the chosen $v$-adic metric on~$K_X$
induces a Radon measure~$\tau_{X,v}$ on~$X(F_v)$
(see Section~\ref{subsubsec.volume-forms}).
Let $\mathscr X$ be a projective flat model of~$X$ over the ring
of integers of~$F$. Then, for almost all finite places~$v$,
the measure~$\tau_{X,v}$ coincides with the measure
on $\mathscr X(\mathfrak o_v)=X(F_v)$ defined by Weil in~\cite{weil82}.
Let $D$ be an effective divisor on~$X$ which, geometrically,
has strict normal crossings and set $U=X\setminus D$.
Let $\mathsec f_D$ denote the canonical section
of the line bundle~$\mathscr O_X(D)$ (corresponding to the regular
function~$1$ when $\mathscr O_X(D)$ is viewed as a sub-sheaf
of the sheaf of meromorphic functions); by construction,
its divisor is~$D$.
Let us also fix an adelic metric on this line bundle.
We let $\mathscr D$ be the Zariski closure of~$D$ in the model~$\mathscr X$
and $\mathscr U=\mathscr X\setminus\mathscr D$ be its complement.
For any place~$v\in\Val(F)$, we define a measure
\[ \tau_{(X,D),v}=\frac1{\norm{\mathsec f_D}_v} \tau_{X,v} \]
on $U(F_v)$. Note that it is still a Radon measure; however $U(F_v)$
has infinite volume. If $U$ is an algebraic group,
this construction allows to recover the Haar measure of~$U(F_v)$
(see~\ref{subsubsec.haar-tamagawa}).
Let ${\mathbb A}_F$ be the adele ring of~$F$, that is, the restricted
product of the fields~$F_v$ with respect to the subrings~$\mathfrak o_v$.
A nonzero Radon measure~$\tau$ on the adelic space~$U({\mathbb A}_F)$
induces measures~$\tau_v$ on any of the sets~$U(F_v)$,
which are well-defined up to a factor. Conversely, we can recover
the Radon measure~$\tau$ as the product of such measures~$\tau_v$
if the set of measures $(\tau_v)_{v\nmid\infty}$
satisfies the convergence condition: \emph{the infinite product
$\prod_{v\nmid\infty} \tau_v(\mathscr U(\mathfrak o_v))$
is absolutely convergent.}
A family of convergence factors for $(\tau_v)$
is a family $(\lambda_v)_{v\nmid\infty}$
of positive real numbers such that the family of
measures $(\lambda_v\tau_v)$ satisfies the above convergence condition.
Our first result in this article is a definition
of a measure on~$U({\mathbb A}_F)$ via an appropriate choice
of convergence factors.
Let $\bar F$ be an algebraic closure of~$F$ and let $\Gamma=\Gal(\bar F/F)$
be the absolute Galois group.
Let $M$ be a free ${\mathbf Z}$-module of finite rank endowed with a continuous
action of~$\Gamma$; we let $\mathrm L(s,M)$ be the corresponding
Artin L-function, and, for all finite places $v\in\Val(F)$,
$\mathrm L_v(s,M)$ its local factor at $v$.
The function $s\mapsto \mathrm L(s,M)$ is holomorphic
for $\Re(s)>1$ and admits a meromorphic continuation to~${\mathbf C}$;
let $\rho$ be its order at $s=1$ and define
\[ \mathrm L^*(1,M) = \lim_{s\ra 1} (s-1)^{-\rho} \mathrm L(s,M);\]
it is a positive real number.
\begin{theo}\label{theo.1.tamagawa}
Assume that $\mathrm H^1(X,\mathscr O_X)=\mathrm H^2(X,\mathscr O_X)=0$.
The abelian groups $M^0=\mathrm H^0(U_{\bar F},{\mathbf G}_m)/\bar F^*$
and $M^1=\mathrm H^1(U_{\bar F},{\mathbf G}_m)/\text{torsion}$ are free
${\mathbf Z}$-modules of finite rank with a continuous action of~$\Gamma$.
Moreover, the family $(\lambda_v)$
given by
\[ \lambda_v= \mathrm L_v(1,M^0)/\mathrm L_v(1,M^1) \]
is a family of convergence factors.
\end{theo}
Assume that the hypotheses
of the Theorem hold.
We then define a Radon measure on~$U({\mathbb A}_F)$ by the formula
\[ \tau_{(X,D)} = \frac{\mathrm L^*(1,M^1)}{\mathrm L^*(1,M^0)}
\prod_{v\in\Val(F)} \left(\lambda_v \tau_{(X,D),v}\right), \]
where $\lambda_v$ is given by the Theorem if $v\nmid\infty$
and $\lambda_v=1$ else.
We call it the \emph{Tamagawa measure on~$U$, or, more precisely,
on $U(\mathbb A_F)$.}
This generalizes
the construction of a Tamagawa measure on an algebraic group,
where there is no~$M^1$ (see, \emph{e.g.}, \cite{weil82})
or on a projective variety,
where there is no~$M^0$,
in~\cite{peyre95}.
\subsection{Volume asymptotics in analytic geometry}
Let $L$ be an effective divisor in~$X$ whose support contains
the support of~$D$; let again $\mathsec f_L$ be the canonical
section of the line bundle $\mathscr O_X(L)$. Fix an
adelic metric on~$\mathscr O_X(L)$ and a place~$v\in \Val(F)$.
For any positive real number~$B$, the set of all $x\in U(F_v)$
such that $\norm{\mathsec f_L(x)}_v\geq 1/B$ is a compact
subset in~$U(F_v)$. It has thus finite volume $V(B)$
with respect to the measure~$\tau_{(X,D),v}$.
Let us decompose the divisor~$D_v=D_{F_v}$ as a sum
of irreducible divisors:
\[ D_{v}=\sum_{\alpha\in\mathscr A_v} d_{\alpha,v} D_{\alpha,v}. \]
For $\alpha\in\mathscr A_v$,
let $\lambda_{\alpha,v}$ be the multiplicity of~$D_{\alpha,v}$
in~$L_v$; there exists an effective divisor~$E_v$ on~$X_{F_v}$
such that
\[ L_v= E_v+\sum_{\alpha\in\mathscr A} \lambda_{\alpha,v} D_{\alpha,v} .\]
For any subset $A\subseteq\mathscr A_v$, we let $D_{A,v}$ be the
intersections of the $D_{\alpha,v}$, for $\alpha\in A$.
Let now $a_v(L,D)$ be the least rational number such that
for any $\alpha\in\mathscr A_v$, with $D_{\alpha,v}(F_v)\neq\emptyset$,
one has
$a_v(L,D)\lambda_{\alpha,v}\geq d_{\alpha,v}-1$.
Let $\mathscr A_v(L,D)$ be the set of those $\alpha\in\mathscr A_v$
where equality holds and
$b_v(L,D)$ the maximal cardinality of subsets $A\in\mathscr A_v(L,D)$
such that $D_{A,v}(F_v)\neq\emptyset$.
To organize the combinatorial structure of these subsets,
we introduce variants of the simplicial complex considered, e.g.,
in~\cite{clemens1969} in the context of Hodge theory.
\begin{theo}\label{theo.2.volume-local}
Assume that $v$ is archimedean.
If $a_v(L,D)>0$, then $b_v(L,D)\geq 1$ and
there exists a positive real number~$c$ such that
\[ V(B) \sim c B^{a_v(L,D)} (\log B)^{b_v(L,D)-1}. \]
If $a_v(L,D)=0$, then there exists a positive real number~$c$
such that
\[ V(B) \sim c B^{a_v(L,D)} (\log B)^{b_v(L,D)}. \]
\end{theo}
With the notation above, we also give an explicit formula for the constant~$c$.
It involves integrals over the sets $D_{A,v}(F_v)$
such that $\mathop\#(A)=b_v(L,D)$, with respect to measures induced
from~$\tau_{(X,D),v}$ via the adjunction formula.
To prove this theorem, we introduce the Mellin transform
\[ Z(s) = \int_{U(F_v)} \norm{\mathsec f_L(x)}_v^s \,\mathrm d\tau_{(X,D),v}(x)\]
and establish its analytic properties. We regard $Z(s)$
as an integral over the compact analytic manifold~$X(F_v)$
of the function~$\norm{\mathsec f_L}_v^s$ with respect
to a singular measure, connecting the study of such zeta functions
with the theory of Igusa local zeta functions,
see~\cite{igusa1974a,igusa1974b}.
In particular, we show that $Z(s)$ is holomorphic
for $\Re(s)>a_v(L,D)$ and admits a meromorphic
continuation to some half-plane $\{\Re(s)>a_v(L,D)-\eps\}$,
with a pole at $s=a_v(L,D)$ of order~$b_v(L,D)$.
This part of the proof works over any local field.
If $v$ is archimedean (and $\eps>0$ is small enough), then
$Z(s)$ has no other pole in this half-plane.
Our volume estimate then follows from a standard Tauberian theorem.
When $v$ in non-archimedean, we can only deduce a weaker
estimate, \emph{i.e.}, upper and lower bounds of the stated
order of magnitude (Corollary~\ref{coro.asymptotic.volume-ultrametric}).
\subsection{Asymptotics of adelic volumes}
Assume that
$\mathrm H^1(X,\mathscr O_X)=\mathrm H^2(X,\mathscr O_X)=0$.
Theorem~\ref{theo.1.tamagawa} gives us Tamagawa measures~$\tau_{(X,D)}$
and~$\tau_X$ on the adelic spaces~$U({\mathbb A}_F)$ and $X({\mathbb A}_F)$ respectively.
Suppose furthermore that the supports of the divisors~$L$ and~$D$
are equal.
We then define a \emph{height function} $H_L$ on
the adelic space~$U({\mathbb A}_F)$ by the formula
\[ H_L ((x_v)_v)= \prod_{v\in\Val(F)} \norm{\mathsec f_L(x_v)}_v^{-1}. \]
This function $H_L\colon U({\mathbb A}_F)\ra{\mathbf R}_+$ is continuous and proper.
In particular, for any real number~$B$, the subset of~$U({\mathbb A}_F)$
defined by the inequality $H_L(\mathbf x)\leq B$ is compact,
hence has finite volume~$V(B)$ with respect to~$\tau_{(X,D)}$.
We are interested in the asymptotic behavior of $V(B)$
as $B\ra\infty$.
Let us decompose the divisors~$L$ and~$D$ as the sum
of their irreducible components (over~$F$).
Since $L$ and~$D$ have the same support, one can write
\[ D = \sum_{\alpha\in\mathscr A} d_\alpha D_\alpha, \qquad
L = \sum_{\alpha\in\mathscr A} \lambda_\alpha D_\alpha \]
for some positive integers~$d_\alpha$ and~$\lambda_\alpha$.
Let $a(L,D)$ be the least positive rational number such that
the~${\mathbf Q}$-divisor $E=a(L,D)L-D$ is effective;
in other words,
\[ a(L,D) = \max_{\alpha\in\mathscr A} {d_\alpha}/\lambda_\alpha. \]
Let moreover~$b(L,D)$ be the number of $\alpha\in\mathscr A$ for
which equality is achieved.
To the ${\mathbf Q}$-divisor~$E$, we can also attach a height function
on~$X({\mathbb A}_F)$ given by
\[ H_E(\mathbf x) = \prod_{v\in \Val(F)} \norm{\mathsec f_L(x_v)}_v^{-a(L,D)} \norm{\mathsec f_D(x,v)}_v \]
if $x_v\in U(F_v)$ for all~$v$, and by $H_E(\mathbf x)=+\infty$
else.
The product can diverge to~$+\infty$
but $H_E$ has a positive lower bound, reflecting the
effectivity of~$E$. In fact,
the function $H_E^{-1}$ is continuous on~$X({\mathbb A}_F)$.
\begin{theo}\label{theo.3.volume-global}
When $B\ra\infty$, one has the following asymptotic expansion
\[ V(B) \sim \frac1{a(L,D) (b(L,D)-1)!} B^{a(L,D)} (\log B)^{b(L,D)-1}
\int_{X({\mathbb A}_F)} H_E(\mathbf x)^{-1} \,\mathrm d\tau_X(\mathbf x). \]
\end{theo}
As a particular case, let us take $L=D$. We see that $a(L,D)=1$,
$b(L,D)$ is the number of irreducible components of~$D$
and the integral in the Theorem is equal
to the Tamagawa volume $\tau_X(X({\mathbb A}_F))$ defined by Peyre.
\medskip
In both local and adelic situations, our techniques
are valid for any metrization of the underlying line bundles.
As was explained by Peyre in~\cite{peyre95} in the context
of rational points,
this implies \emph{equidistribution theorems},
see Corollary~\ref{coro.equidistribution.local} in
the local case, and Theorem~\ref{theo.volume.global}
in the adelic case.
\subsection*{Roadmap of the paper}
Section~\ref{sec.general}
is concerned with heights and measures on adelic spaces.
We first recall notation and definitions
for adeles, adelic metrics and measures on analytic manifolds.
In Subsection~\ref{subsec.heights.adelic},
we then define height functions on adelic spaces
and establish their basic properties.
The construction of global Tamagawa measures
is done in Subsection~\ref{subsec.global-tamagawa}.
We conclude this Section by a general equidistribution theorem.
Section~\ref{s.igusa} is devoted to the theory
of geometric analogues of Igusa integrals,
both in the local and adelic settings.
These integrals define holomorphic functions in several variables
which admit meromorphic continuations. (In the adelic case,
these meromorphic continuations may have natural boundaries.)
To describe their first poles we
introduce in Subsection~\ref{subsec.clemens} the geometric, algebraic
and analytic Clemens complexes which encode the
incidence properties of divisors involved in the definition
of our geometric Igusa integrals.
We then apply this theory in Subsections~\ref{subsec.geometric-volume}
and~\ref{subsec.asymptotics.adelic},
where we establish
Theorems~\ref{theo.2.volume-local} and~\ref{theo.3.volume-global}
about volume asymptotics.
In Section~\ref{sec.examples}, we make explicit the main results of our article
in the case of wonderful compactifications
of semi-simple groups.
In particular, we explain how to recover
the volume estimates established in~\cite{maucourant2007}
for Lie groups,
and in~\cite{gorodnik-m-o2009,shalika-tb-t2007} for adelic groups.
\subsection*{Acknowledgments}
During the preparation of this work, we have benefited from conversations
with Michel Brion and François Loeser.
The first author would like to acknowledge the support of the Institut
universitaire de France, and to thank the organizers
of a conference at the Maxwell Institute,
Edinburgh, 2008, for having given him the opportunity to present
the results of this paper.
The second author
was partially supported by NSF grant DMS-0602333.
\section{Tauberian theorems}
\def\thesection.\arabic{subsubsection}{\thesection.\arabic{subsubsection}}
\def\thesection.\arabic{equation}{\thesection.\arabic{equation}}
Let $(X,\mu)$ be a measured space and $f$ a positive measurable
function on~$X$. Define
\[ Z(s) = \int_{X} f(x)^{-s}\, d\mu(x) \quad\text{and}\quad
V(B) = \mu(\{f(x)\leq B\}). \]
One has
\[ Z(s) = \int_0^{+\infty} B^{-s}\, dV(B). \]
\begin{theo}
\label{theo.tauber}
Let $a$ be a real number;
we assume that $Z(s)$ converges for~$\Re(s)>a$ and extends to a
meromorphic function in the neighborhood of
the closed half-plane~$\Re(s)\geq a-\delta$,
for some positive real number~$\delta$.
If $a<0$, then $V(B)$ has the limit $Z(0)$ when $B\ra\infty$.
Furthermore, assume that:
\begin{enumerate}
\item
$Z$ has a pole of order~$b$ at $s=a$ and no other pole in
the half-plane $\Re(s)\geq a-\delta$.
\item
$Z$ has moderate growth in vertical strips, \emph{i.e.},
there exists a positive real number~$\kappa$ such that for any~$\tau\in{\mathbf R}$,
\[ \abs{Z(a-\delta+i\tau)} \ll (1+\abs{\tau})^\kappa . \]
\end{enumerate}
Then, there exist a monic polynomial~$P$, a real number~$\Theta$
and a positive real number~$\eps$
such that, when $B\ra\infty$,
\[ V(B) = \begin{cases}
\Theta B^a P(\log B) + \mathrm O(B^{a-\eps}) & \text{if $a\geq 0$;}\\
Z(0)+\Theta B^aP(\log B) + \mathrm O(B^{a-\eps}) & \text{if $a<0$.} \end{cases}
\]
Moreover, if $a\neq 0$,
then
\[ \deg(P) = b-1 \quad\text{and}\quad \Theta \, a\, (b-1)! = \lim_{s\ra a}(s-a)^b Z(s)
\]
while
\[ \deg(P)=b \quad\text{and}\quad \Theta \,b! = \lim_{s\ra a}(s-a)^b Z(s
)
\]
if $a=0$.
\end{theo}
For any integer~$k\geq 0$, let us define
\[ V_k(B)= \frac1{k!} \int_{f(x)\leq B} \left( \log\frac B{f(x)}\right)^k \,d\mu(x)
= \frac1{k!} \int_X \left(\log^+ \frac B{f(x)}\right)^k\,d\mu(x), \]
where $\log^+(u)=\max(0,\log u)$ for any positive real number~$u$.
\begin{lemm}\label{lemm.tauber.k}
Let $k$ be an integer such that~$k>\kappa$. Then there exist
polynomials~$P$ and~$Q$ with real coefficients, a real number~$\delta>0$
such that
\[ V_k(B) = B^a P(\log B) + Q(\log B) + \mathrm O(B^{a-\delta}). \]
Moreover, the polynomials $P$ and $Q$ satisfy
\begin{gather}
P(T) = \begin{cases}
\frac{1}{a^{k+1}(b-1)!} \Theta T^{b-1}+\dots & \text{if $a\neq 0$,} \\
0 & \text{if $a=0$;} \end{cases} \\
Q(T)= \begin{cases}
0 & \text{if $a>0$;} \\
\frac{1}{(b+k)!} \Theta T^{b+k} + \dots &\text{if $a=0$;} \\
\frac1{k!} Z(0) T^{k}+\dots & \text{if $a<0$.}
\end{cases}
\end{gather}
\end{lemm}
\begin{proof}
We begin with the classical integral
\[ \int_{\sigma+i{\mathbf R}} \lambda^s \frac{\mathrm ds}{s^{k+1}} = \frac{2i\pi}{k!} (\log^+(\lambda))^k, \]
where $\sigma$ and~$\lambda$ are positive real numbers.
For $\sigma>\max(0,a)$, this implies that
\begin{align*}
V_k (B) & = \frac1{k!} \int_{X} \log^+(B/f(x))^k\,\mathrm d\mu(x) \\
& = \int_X \frac{1}{2i\pi} \int_{\sigma+i{\mathbf R}} (B/f(x))^s \frac{\mathrm ds}{s^{k+1}} \, \mathrm d\mu(x) \\
& = \frac{1}{2i\pi} \int_{\sigma+i{\mathbf R}} B^s Z(s) \,\frac{\mathrm ds}{s^{k+1}},
\end{align*}
where the written integrals converge absolutely.
We now move the contour of integration to the left of the pole $s=a$,
the estimates for $Z(s)$ in vertical strips allowing us to
apply the residue theorem. Only $s=a$ and $s=0$ may give a pole
within the two vertical lines $\Re(s)=\sigma$ and $\Re(s)=a-\delta$
and we obtain
\[ V_k(B) = \sum_{\substack{u\in\{0,a\}\\ a-\delta < u <\sigma}} \Res_{s=u}\left(\frac{B^sZ(s)}{s^{k+1}}\right)
+ \frac{1}{2i\pi} \int_{a-\delta+i{\mathbf R}} B^s Z(s)\,\frac{\mathrm ds}{s^{k+1}}. \]
If $a>0$ and $a-\delta>0$, or if $a<0$,
one checks that there exists a polynomial~$P$
of degree~$b-1$ and of leading coefficient $\Theta/a^{k+1}(b-1)!$
such that
\[ \Res_{s=a} \left( \frac{B^s Z(s)}{s^{k+1}}\right)
= B^a P(\log B). \]
In the case $a>0$, we moreover set $Q=0$.
If $a=0$, let us set $P=0$.
There exists a polynomial~$Q$ of degree~$b+k$
and of leading coefficient $\Theta/(b+k)!$
such that
\[ \Res_{s=0} \left( \frac{B^s Z(s)}{s^{k+1}}\right)
= Q(\log B). \]
Finally, if $a<0$, one verifies that there exists
a polynomial~$Q$ of degree~$k$ and of leading coefficient
$Z(0)/k!$ such that
\[ \Res_{s=0} \left( \frac{B^s Z(s)}{s^{k+1}}\right)
= Q(\log B). \]
This implies the lemma.
\end{proof}
\begin{lemm}\label{lemm.tauber.iteration}
Assume that there are polynomials~$P$ and~$Q$, and a positive
real number~$\delta$, such that
\[ V_k(B) = B^a P(\log B) + Q(\log B) + \mathrm O(B^{a-\delta}), \]
where we moreover assume that $P=0$ if $a=0$ and $Q=0$ if $a>0$.
Then for any positive real number~$\delta'$ such
that $\delta'<\delta/2$, one has the asymptotic expansion
\[ V_{k-1}(B) = B^a (aP(\log B)+P'(\log B)) + Q'(\log B) + \mathrm O(B^{a-\delta'}). \]
\end{lemm}
\begin{proof}
For any $u\in(-1,1)$, one has
\begin{align*}
V_k(B(1+u))-V_k(B) & = B^a\big(P(\log(B(1+u)))-P(\log B)\big) \\
& \qquad {} + Q(\log(B(1+u)))-Q(\log B) + \mathrm O(B^{a-\delta}) \\
&= B^a P(\log B) \big( (1+u)^a-1\big) \\
& \qquad {}+ B^a (1+u)^a \big( P(\log B+\log(1+u)) - P(\log B) \big) \\
& \qquad {} + \big( Q(\log B+\log(1+u))-Q(\log B) \big)
+ \mathrm O\big(B^{a-\delta}\big) \\
&= B^a P(\log B)\big(au +\mathrm O(u^2)\big)
+ B^a P'(\log B) u + B^a \mathrm O\big(\log B)^{\deg P-2}u^2\big)\\
& \qquad {}+ Q'(\log B) u + \mathrm O\big((\log B)^{\deg Q-2}u^2\big)
\qquad {}+ \mathrm O\big(B^{a-\delta}\big) .
\end{align*}
Let $\eps$ be a positive real number; if $u=\pm B^{-\eps}$,
we obtain
\begin{multline*} \frac{V_k (B(1+u))-V_k(B)}{\log(1+u)}
= B^a (aP(\log B)+P'(\log B)) +\tilde{\mathrm O}(B^{a-\eps})) \\
+ Q'(\log B) + \tilde{\mathrm O}\big( B^{-\eps}\big)
+ \mathrm O\big( B^{a-\delta+\eps}\big), \end{multline*}
where the~$\tilde{\mathrm O}$ notation indicates unspecified
powers of~$\log B$. If $\delta'<\delta/2$, we may choose
$\eps$ such that $\delta'<\eps<\delta/2$ and then,
\begin{equation}\label{eq.diffVk}
\frac{V_k (B(1+u))-V_k(B)}{\log(1+u)}
= B^a (aP(\log B)+P'(\log B))
+ Q'(\log B)
+ \mathrm O\big( B^{a-\delta'}\big). \end{equation}
On the other hand,
for any real number~$u$ such that $0<u<1$, and any positive
real number~$A$, one has
\[
\frac{ \log(A(1-u))^k-\log(A)^k}{\log(1-u)}
\leq k \log(A)^{k-1} \leq
\frac{ \log(A(1+u))^k-\log(A)^k}{\log(1+u)}, \]
which implies the inequality
\begin{equation}
\frac{V_k(B(1-u))-V_k(B)}{\log(1-u)} \leq V_{k-1}(B) \leq
\frac{V_k(B(1+u))-V_k(B)}{\log(1+u)} . \end{equation}
Plugging in the estimates of Equation~\eqref{eq.diffVk}, we deduce
the asymptotic expansion
\begin{equation}
V_{k-1}(B) = B^a (aP(\log B)+P'(\log B)) + Q'(\log B) +
+ \mathrm O\big( B^{a-\delta'}\big), \end{equation}
as claimed.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{theo.tauber}]
We now can prove our Tauberian theorem.
Let $k$ be any integer such that $k>\kappa$.
Lemma~\ref{lemm.tauber.k} implies an asymptotic expansion
for $V_k(B)$; let $P,Q,\delta$ be as in this Lemma.
Applying successively $k$ times
Lemma~\ref{lemm.tauber.iteration},
we obtain the existence of an asymptotic expansion for $V(B)$
of the form
\[ V(B) = B^a D_a^k P(\log B) + D_0^k Q(\log B) + \mathrm O(B^{a-\delta'}),\]
for some positive real number~$\delta'$
(any positive real number~$\delta'$ such that $\delta'<\delta/2^k$
is suitable),
where we have denoted by $D_a$ and $D_0$ the differential
operators $P\mapsto aP+P'$ and $P\mapsto P'$.
For $a\neq 0$, the operator~$D_a$ doesn't change the degree
but multiplies the leading coefficient by~$a$.
Consequently,
\[ D_a^k P (T)= \begin{cases}
\frac1{a(b-1)!} \Theta^{b-1}+\dots & \text{if $a\neq 0$,} \\
0 &\text{if $a=0$.} \end{cases}\]
Similarly, the operator~$D$ decreases the degree by~$1$
and multiplies the leading coefficient by the degree.
It follows that
\[ D_0^k Q(T) = \begin{cases}
Z(0) & \text{if $a<0$;}\\
\frac1{b!}\Theta T^b +\dots & \text{if $a=0$;} \\
0 & \text{if $a>0$.} \\
\end{cases} \]
Theorem~\ref{theo.tauber} now follows easily.
\end{proof}
\bigskip
In ultrametric contexts, the function~$f$ usually takes values
of the form~$q^n$, with $n\in{\mathbf Z}$, where $q$ is a real number
such that~$q>1$.
In that case, the function~$Z$ is $(2i\pi)/\log q$-periodic,
its poles form arithmetic progressions of common difference~$2i\pi/\log q$,
and Theorem~\ref{theo.tauber} doesn't apply.
Let $q$ be a real number with $q>1$ and
assume that $f(x)\in q^{\mathbf Z}$ for any $x\in X$.
Let $a$ be a nonnegative real number;
let us assume that $Z(s)$ converges for~$\Re(s)>a$ and extends to a
meromorphic function in the neighborhood of
the closed half-plane~$\Re(s)\geq a-\delta$,
for some positive real number~$\delta$.
Suppose furthermore that the poles of~$Z$ belong to
finitely many arithmetic progressions of the form
$a_j+\frac1{\log q} 2i\pi {\mathbf Z} $, where $a_1,\dots,a_t\in{\mathbf C}$
are complex numbers of real part~$a$, not two of them being congruent
mod $2i\pi/\log q$; let $b_j=\operatorname{ord}_{s=a_j} Z(s)$ and
let $c_j=\lim_{s\ra a_j} (s-a_j)^{b_j} Z(s)$.
Let us make the change of variable $u=q^{-s}$ and set $Z(s)=\Phi(u)$.
The function $\Phi$ is defined by
\[ \Phi(u)=\sum_{n\in{\mathbf Z}} Z_n u^n, \qquad \text{where}\quad
Z_n=\mu(\{ f(x)=q^{n} \}); \]
it is defined for $0<\abs u<q^{-a}$ and is holomorphic
in that domain; moreover, it extends to a meromorphic function
on the domain $0<\abs u<q^{\delta-a}$, with poles
at $q^{-a_j}$ (for $1\leq j\leq t$) such that
\[ \lim_{u\ra q^{-a_j}} (1-q^{a_j}u)^{b_j} \Phi(u)
= \lim_{s\ra a_j} (1-q^{a_j-s})^{b_j} Z(s)
= (\log q)^{b_j} c_j. \]
Consequently, there are polynomials $p_j$ of degree~$b_j$
and leading coefficient $(\log q)^{b_j} c_j$
such that
\[ Z(u) - \sum_{j=1}^t \frac {p_j(u)}{(1-q^{a_j}u)^{b_j}} \]
is holomorphic in the domain defined by $0<\abs u<q^{\delta-a}$.
Therefore, the Cauchy formula implies an asymptotic expansion of the form
\[ Z_n = \sum_{j=1}^t P_j(n) q^{a_j n} + \mathrm O(q^{(a-\delta')n}), \]
where $\delta'$ is any positive real number such that $0<\delta'<\delta$
and, for $1\leq j\leq t$, $P_j$ is a polynomial
of degree~$b_j-1$ and of leading coefficient $c_j (\log q)^{b_j} /(b_j-1)!$.
Since the Laurent series defining~$\Phi$ has nonnegative coefficients,
the inequality $\abs{Z(u)}\leq Z(\abs u)$ holds for any $u\in{\mathbf C}$
such that $0<\abs u<q^{-a}$.
In particular, we see, as is well known,
that $\Phi$ has a pole on its circle of convergence.
This means that, unless $t=0$, we can assume that $a_1=a$
and that $b_j\leq b_1$ for all~$j$.
We are not interested in the sequence~$(Z_n)$ itself,
but rather on the sums
$ V(q^n)=\sum_{m\leq n} Z_n$,
when $n\ra\infty$. (Their convergence
follows from the fact that $\Phi(u)$ converges for arbitrary
small nonzero complex numbers~$u$.)
We begin by observing that for any integer~$j$,
there exists a polynomial $Q_j$ such that
\[ P_j(m) = Q_j(m)- q^{-a_j}Q_j(m-1) \]
for any $m\in{\mathbf Z}$.
Moreover, if $q^{a_j}\neq 1$, then there
is only one polynomial satisfying these relations,
its degree satisfies $\deg(Q_j)=\deg(P_j)$
while its leading coefficient is equal to $\operatorname{lcoeff}(P_j)/(1-q^{-a_j})$;
however, if $q^{a_j}=1$, then $\deg(Q_j)=\deg(P_j)+1$
and $\operatorname{lcoeff}(Q_j)=\operatorname{lcoeff}(P_j)/(\deg(P_j)+1)$.
Let us now separate the discussion according to the value of~$a$.
\emph{Case $a<0$.} ---
Then, $V(q^n)$ has the limit $\Phi(1)$ when $n\ra+\infty$,
and
\[ V(q^n)=\Phi(1) - \sum_{m>n} Z_n
= \Phi(1) - \sum_{j=1}^t \sum_{m>n} q^{a_jm} P_j(m)+ \mathrm O(q^{(a-\delta')n}) , \]
provided $\delta'<\delta$ is chosen so that $a<a+\delta'<0$.
Since
\[ \sum_{m>n} q^{a_jm} P_j(m) = \sum_{m>n} \left(
q^{a_jm}Q_j(m)-q^{a_j(m-1)}Q_j(m-1)\right)
= - q^{a_j n} Q_j(n), \]
we obtain that
\begin{equation}
V(q^n)=\Phi(1) + \sum_{j=1}^t q^{a_j n}Q_j(n) + \mathrm O(q^{(a-\delta')n}).
\end{equation}
\emph{Case $a> 0$.} --- In that case, we have
\[ V(q^n)= \sum_{j=1}^t \sum_{m\leq n} \left(
q^{a_jm}Q_j(m)-q^{a_j(m-1)}Q_j(m-1)\right)+ \mathrm O(q^{a-\delta'n}), \]
so that
\begin{equation}\label{eq.q.a>0}
V(q^n)= \sum_{j=1}^t q^{a_jn} Q_j(n) + \mathrm O(q^{a-\delta'}n). \end{equation}
Moreover, $\deg(Q_j)=b_j$ for all~$j$.
\emph{Case $a=0$.} ---
Then $V(q^n)$ also satisfies the asymptotic expansion~\eqref{eq.q.a>0}.
However, $\deg Q_1=b_1+1$ and $\deg Q_j\leq b_j\leq b_1$ for $j\neq 1$,
and we obtain
\begin{equation} \lim_{n\ra\infty} V(q^n) n^{-b_1-1} \operatorname{lcoeff}(Q_1)
= \frac1{b_1+1}\operatorname{lcoeff}(P_1) = c_1 \frac{(\log q)^{b_1}}{b_1+1}.\end{equation}
\begin{coro}\label{coro.theo.tauber2}
Let us retain the previous hypotheses, assuming moreover that $a>0$.
Then, we have the following weak asymptotic behavior:
\[ 0< \liminf V(q^n) q^{-an} n^{-b_1} \leq \limsup V(q^n) q^{-an}n^{-b_1}<\infty. \]
\end{coro}
\begin{coro} \label{coro.theo.tauber2b}
Let us assume that for some positive integers~$b$ and~$d$, the function
$Z(s)(1-q^{(a-s)d})^b$ has a holomorphic expansion in
some neighborhood of the half-plane $\{\Re(s)\geq a-\delta\}$.
Then, where $n$ is restricted to belong to any arithmetic progression
mod.~$d$, the sequence $(V(q^n)q^{-na}\log(q^n)^{-b})$
has a limit.
\end{coro}
\begin{proof}
The assumptions allow us to set $t=d$
and $a_j=a+2i\pi(j-1)/d\log(q)$, for $1\leq j\leq d$, and $b_j=b$.
Let us fix $m\in{\mathbf N}$ and let us write $n=m+kd$, where $k\in{\mathbf N}$
goes to infinity.
For $n\in{\mathbf N}$, we may write
\[ V(q^n) q^{-na}\log(q^n)^{-b}
\sum_{j=1}^d q^{(a_j-a)n} \frac{Q_j(n)}{(n\log q)^b} + \mathrm O(q^{-\delta'n}).
\]
The asserted convergences follow from the fact the observation that
for any~$j$, $q^{(a_j-a)n}$ is a $d$th root of unity.
More precisely, we find
\[ \lim_{\substack{n\ra\infty \\ n\equiv n_0\pmod d}}
V(q^n) q^{-na} \log(q^n)^{-b}
=\frac1{(b-1)!}\sum_{j=1}^d \exp(2i\pi (j-1)/d) \frac{c_j}{1-q^{-a_j}}, \]
where
\[ c_j = \lim_{s\ra a_j} (s-a_j)^{b} Z(s). \]
\end{proof}
The following corollary has been inspired by work in progress
by Cluckers, Comte and Loeser.
\begin{coro}\label{coro.theo.tauber2c}
Let $V^*(q^n)$ be the Cesaro mean of $V(q^n)$, namely
\[ V^*(q^n)=\frac1{n+1}\sum_{m=0}^n V(q^m) .\]
If $a>0$, then $V^*$ satisfies
\[ \lim_{n\ra\infty} V^*(q^n) q^{-an} n^{-b_1}
= c_1 \frac{ (\log q)^{b_1}}{1-q^{-a}}. \]
If $a<0$, then
\[ \lim_{n\ra\infty} \left(V^*(q^n)-Z(0)\right)q^{-an} n^{-b_1}
=c_1 \frac{ (\log q)^{b_1}}{1-q^{-a}}.
\]
\end{coro}
\begin{proof}
This follows from the main result and the fact for any
complex number $z\in{\mathbf C}$ such that $\abs z=1$ but $z\neq 1$,
the sequence $(z^n)$ Ces\'aro-converges to~$0$.
\end{proof}
\bigskip
We conclude this Appendix by a Tauberian result which is
useful in $S$-integral contexts.
\begin{theo}
\label{theo.tauber.S}
Let $a$ be a real number;
we assume that $Z(s)$ converges to a holomorphic
function for~$\Re(s)>a$.
Let us furthermore assume that it has a meromorphic
continuation of the following form:
there exists a positive integer~$b\geq 1$
and a finite family $(q_j,b_j)_{j\in J}$
where $q_j$ is a real number such that~$q_j>1$ and $b_j$
is an integer satisfying $1\leq b_j\leq b-1$ such that,
setting $b_0=b-\sum_{j\in J}b_j$, the function~$Z_0$ defined by
\[
Z_0(s)=Z(s) \left(\frac {s-a}{s-a+1}\right)^{b_0}
\prod_{j\in J} \left(1-q_j^{s-a}\right)^{b_j}
\]
extends to a holomorphic function
with
moderate growth in vertical strips, \emph{i.e.},
there exists a positive real number~$\kappa$ such that for any~$\tau\in{\mathbf R}$,
\[ \abs{Z(a-\delta+\mathrm i\tau)} \ll (1+\abs{\tau})^\kappa . \]
Assume also that for any two $j,j'\in J$,
$\log q_j/\log q_{j'}$ is not a Liouville number.
Then, there exist a monic polynomial~$P$, a real number~$\Theta$
and a positive real number~$\eps$
such that, when $B\ra\infty$,
\[ V(B) = \begin{cases}
\Theta B^a P(\log B) + \mathrm O(B^{a}(\log B)^{\max(b_j)}) & \text{if $a\geq 0$;}\\
Z(0)+\Theta B^aP(\log B) + \mathrm O(B^{a}(\log B)^{\max(b_j)})
& \text{if $a<0$.} \end{cases}
\]
Moreover, if $a\neq 0$,
then
\[ \deg(P) = b-1 \quad\text{and}\quad \Theta \, a\, (b-1)! = \lim_{s\ra a}(s-a)^b Z(s)
\]
while
\[ \deg(P)=b \quad\text{and}\quad \Theta \,b! = \lim_{s\ra a}(s-a)^b Z(s)
\]
if $a=0$.
\end{theo}
\begin{proof}
For any integer~$k\geq 0$, let us define
\[ V_k(B)= \frac1{k!} \int_{f(x)\leq B} \left( \log\frac B{f(x)}\right)^k \,d\mu(x)
= \frac1{k!} \int_X \left(\log^+ \frac B{f(x)}\right)^k\,d\mu(x), \]
where $\log^+(u)=\max(0,\log u)$ for any positive real number~$u$.
As in Theorem~\ref{theo.tauber}, we begin by proving
an asymptotic expansion for $V_k(B)$, where $k$
is an integer satisfying $k>\kappa$.
As above, we have
\[
V_k (B) = \frac{1}{2i\pi} \int_{\sigma+i{\mathbf R}} B^s Z(s) \,\frac{\mathrm ds}{s^{k+1}},
\]
for $\sigma>a$, and we will move the line of integration to the left
of $s=a$,
the novelty being the presence of infinitely many poles
on the line $\Re(s)=a$, namely at any complex number
of the form $\alpha_{j,m}=a+2\mathrm i m\pi/\log q_j$, for some $j\in J$
and some integer $m\in{\mathbf Z}$.
Let $F_k$ be the holomorphic function given by $F_k(s)=B^s Z(s)/s^{k+1}$.
Let $\mu$ be any common irrationality measure
for the real numbers $\log q_j/\log q_{j'}$, namely
a real number such that for any two integers
$m$ and~$m'$ such that $m\log q_j + m'\log q_{j'}\neq 0$, we
have
\[ \abs{m \log q_j + m'\log q_{j'}} \geq \max(\abs m,\abs{m'})^{-\mu}. \]
A straightforward computation based on Leibniz and Cauchy formulae shows
the existence of a positive real number~$c$ such that
for any $j\in J$ and any nonzero~$m\in{\mathbf Z}$,
\[ \abs{\Res_{s=\alpha{j,m}} F_k(s) } \leq c (1+\abs{\Im{\alpha_{j,m}}})^{\kappa-k-1-\mu \beta} B^a (\log B)^{\max(b_j)} , \]
where $\beta=\sum_{j\in J} b_j$.
Moreover, as in the proof of Lemma~\ref{lemm.tauber.k},
there exists a polynomial $P$ of degree~$\deg(P)=b-1$ with leading
coefficient~$\Theta/a^{k+1}(b-1)!$ such that
\[ \Res_{s=a} F_k(s) = B^a P(\log B). \]
We choose $k$ such that $k>\kappa+\mu\beta$.
Since $b>\max(b_j)$, these upper bounds imply that the
sum of residues of~$F_k$ at all poles of real part~$a$
is dominated by the residue at $a=0$ and
satisfies the following estimate
\[ \sum_{\Re(s)=a} \Res F_k(s) = B^a P(\log B). \]
If $a>0$, we set $Q=0$; if $a\leq 0$, there exist a polynomial~$Q$
of
degree~$b+k$ and leading coefficient~$\Theta/(b+k)!$ such that
\[ \Res_{s=0} F_k(s)= Q(\log B).\]
We may then continue as in the proof of Lemma~\ref{lemm.tauber.k}
and conclude that
\[ V_k(B) = B^a P (\log B)+Q(\log B) + \mathrm O(B^a(\log B)^\beta). \]
An application of Lemma~\ref{lemm.tauber.iteration} similar
to that of Theorem~\ref{theo.tauber} then implies
Theorem~\ref{theo.tauber.S}.
\end{proof}
\normalsize
\section{\vskip 0pt plus 6\baselineskip\penalty-100
\theoremstyle{plain}
\newtheorem{prop}[subsubsection]{Proposition}
\newtheorem*{Theo}{Theorem}
\newtheorem{theo}[subsubsection]{Theorem}
\newtheorem{coro}[subsubsection]{Corollary}
\newtheorem{cond}[subsubsection]{Condition}
\newtheorem{lemm}[subsubsection]{Lemma}
\newtheorem{assu}[subsubsection]{Assumption}
\newtheorem{defi}[subsubsection]{Definition}
\theoremstyle{definition}
\newtheorem{prob}[subsubsection]{Problem}
\newtheorem{ques}[subsubsection]{Question}
\newtheorem{conj}[subsubsection]{Conjecture}
\theoremstyle{remark}
\newtheorem{rema}[subsubsection]{Remark}
\newtheorem{rems}[subsubsection]{Remarks}
\newtheorem{exam}[subsubsection]{Example}
\newtheorem{exer}[subsubsection]{Exercise}
\makeatletter
\def\subsection{\@startsection{subsection}{2}%
\z@{.7\linespacing\@plus.3\linespacing}{0.3\linespacing}
{\normalfont\bfseries\smf@boldmath}}
\let\c@equation\c@subsubsection
\let\cl@equation\cl@subsubsection
\let\thesection.\arabic{equation}\thesection.\arabic{subsubsection}
\def\arabic{section}.\arabic{subsection}.\arabic{equation}{\arabic{section}.\arabic{subsection}.\arabic{equation}}
\def\arabic{section}.\arabic{subsection}{\arabic{section}.\arabic{subsection}}
\let\toctable\tocsubsection
\def\l@table{\@tocline{0}{3pt plus2pt}{0pt}{}{\itshape}}
\makeatother
\def\,;\,{\,;\,}
\def\mathop\#{\mathop\#}
\def\mathscr C{\mathord{\mathscr C\!\ell}}
\def\mathscr C{\mathscr C}
\def\mathscr C^{\text{\upshape an}}{\mathscr C^{\text{\upshape an}}}
\def\mathscr C^{\text{\upshape an,max}}{\mathscr C^{\text{\upshape an,max}}}
\def{\mathbf A}{{\mathbf A}}
\def{\mathbb A}{{\mathbb A}}
\def{{\AD}_{\textrm{fin}}}{{{{\mathbb A}}_{\textrm{fin}}}}
\def{\mathbf C}{{\mathbf C}}
\def{\mathbf F}{{\mathbf F}}
\def{\mathbf G}{{\mathbf G}}
\def{\mathbf G}_a{{\mathbf G}_a}
\def{\mathbf G}_m{{\mathbf G}_m}
\def{\mathbf N}{{\mathbf N}}
\def{\mathbf P}{{\mathbf P}}
\def{\mathbf Q}{{\mathbf Q}}
\def{\mathbf R}{{\mathbf R}}
\def{\mathbf V}{{\mathbf V}}
\def{\mathbf Z}{{\mathbf Z}}
\def{\operatorname{GL}}{{\operatorname{GL}}}
\def{\operatorname{SL}}{{\operatorname{SL}}}
\def{\operatorname{End}}{{\operatorname{End}}}
\def{\operatorname{Sym}}{{\operatorname{Sym}}}
\def{\operatorname{Id}}{{\operatorname{Id}}}
\def{2\mathrm i\pi}{{2\mathrm i\pi}}
\def\operatorname{ord}{\operatorname{ord}}
\def{\mathrm Z}{{\mathrm Z}}
\def{\mathrm X}{{\mathrm X}}
\def{\mathsf T}{{\mathsf T}}
\def\operatorname{EP}{\operatorname{EP}}
\let\der\partial
\let\ra\rightarrow
\let\hra\hookrightarrow
\let\da\downarrow
\let\epsilon\varepsilon \let\eps\epsilon
\let\epsilon\varepsilon
\let\phi\varphi
\let\emptyset\varnothing
\let\leq\leqslant
\let\geq\geqslant
\let\le\leq
\let\ge\geq
\def\text{\upshape an}{\text{\upshape an}}
\def\operatorname {id}{\operatorname {id}}
\def\textsc {pl}\xspace{\textsc {pl}\xspace}
\def\operatorname{Lie}{\operatorname{Lie}}
\def{\text{\upshape eff}}{{\text{\upshape eff}}}
\def\abs#1{\left\lvert{#1}\right\rvert}
\def\norm#1{\left\|{#1}\right\|}
\def{\mathrm H}{{\mathrm H}}
\def{\mathrm{NS}}{{\mathrm{NS}}}
\def\DeclareMathOperator#1#2{\def #1{\operatorname{#2}}}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\Re}{Re}
\DeclareMathOperator{\Im}{Im}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\rang}{rank}
\DeclareMathOperator{\codim}{codim}
\DeclareMathOperator{\div}{div}
\DeclareMathOperator{\Val}{Val}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\res}{res}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\Val}{Val}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\Ind}{Ind}
\DeclareMathOperator{\Fr}{Fr}
\def\emph{loccit.}~\ignorespaces{\emph{loccit.}~\ignorespaces}
\def\emph{cf.}~\ignorespaces{\emph{cf.}~\ignorespaces}
\def\alph{enumi}}\def\labelenumi{\theenumi){\alph{enumi}}\def\labelenumi{\alph{enumi}}\def\labelenumi{\theenumi))}
\title{Igusa integrals and volume asymptotics
in analytic and adelic geometry}
\author{Antoine Chambert-Loir}
\address{Universit\'e de Rennes~1, IRMAR--UMR 6625 du CNRS, Campus de Beaulieu, 35042 Rennes Cedex, France}
\address{Institut universitaire de France}
\email{antoine.chambert-loir@univ-rennes1.fr}
\author{Yuri Tschinkel}
\address{Courant Institute, NYU, 251 Mercer St. New York, NY 10012, USA}
\email{tschinkel@cims.nyu.edu}
\begin{document}
\date{\today}
\begin{abstract}
We establish asymptotic formulas for volumes of
height balls in analytic varieties over local fields
and in adelic points of algebraic varieties over number fields,
relating the Mellin transforms of height functions
to Igusa integrals and to global geometric invariants
of the underlying variety.
In the adelic setting, this involves the construction
of general Tamagawa measures.
\end{abstract}
\ifsmfart
\begin{altabstract}
Nous établissons un développement asymptotique du volume
des boules de hauteur dans des variétés analytiques
sur des corps locaux et sur les points adéliques de variétés algébriques sur des corps de nombres. Pour cela, nous relions les transformées de Mellin
des fonctions hauteur à des intégrales de type Igusa et à des
invariants géométriques globaux de la variété sous-jacente.
Dans le cas adélique, nous construisons des mesures de Tamagawa
dans un cadre général.
\end{altabstract}
\fi
\keywords{Heights, Poisson formula, Manin's conjecture, Tamagawa measure}
\subjclass{11G50 (11G35, 14G05)}
\maketitle
\tableofcontents
\clearpage
\include{introduction}
\clearpage
\include{general}
\clearpage
\include{igusa}
\clearpage
\include{example}
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,892
|
You are here: Start / Themes / The Branobel Company / 1970 – a drawn-out liquidation process reaches its end
In 1970, 90 years after the foundation of the company and 50 years after its nationalisation, the end finally came. The remains of the huge oil empire had been managed, and reasonably protected, through two major wars. Ultimately, there were still assets worth close to SEK one billion.
The liquidation of the family of companies around Branobel took time. In 1934, the Branobel company had been sued by a group of bondholders in France for a loan that Emanuel Nobel took out with Deutsche Bank in 1904 and which should have been repaid in 1928. The plaintiffs were not demanding payment for their now worthless bonds; rather they intended to force the company into liquidation and a sale of assets that would be divided up between creditors and shareholders. Nobel responded that the company was Swedish, and therefore could not be brought before a French court. Swedish and French lawyers were engaged and the judgement was appealed against at a higher court in 1936.
An AGM was held on 22 February 1938, but the company's assets could not be sold during the world war. A lot were sensitive, such as wealth, and material was probably hidden away to avoid falling into the wrong hands.
At the end of the war in 1945, Gösta Nobel tried to convert the company's remaining assets into cash. A significant proportion were Finnish government bonds. At this point, Gösta became involved in an unpleasant process in London that had a severe effect on him. He died in 1955 without getting to see the end of the process, or the company's liquidation.
The company now had no board, as both Wilhelm Hagelin and Arthur Lessner had already died. Branobel's finance manager, accountant Oscar Kristoferson, and lawyer Harald Almer succeeded in initiating a settlement process in London, and organised a meeting of former shareholders in May 1959, which, by a sufficient majority, decided to liquidate the company. Harald Almer died, however, and it was left to Nils Oleinikoff to lead the meeting on 3 June 1964. Nils Oleinikoff, grandchild of Ludvig Nobel and son of his daughter, Marta, was the fourth generation of Nobels at the helm of what was left of the Brothers Nobel's oil company by its end.
A liquidation meeting was held in 1965, but due to formalities involving shareholders who did not have certificates or shares, the final meeting had to be postponed until 24 March 1970. At that point, the assets could be divided up, with a five-year appeal period. Funds were reserved for this. An initial division of the residue of SEK 9,253 million took place in Switzerland somewhat later. This had been determined on 3 June 1964 with the intention of executing it in 1969. In the same year, 1969, some of the Finnish bonds were redeemed luckily enough, and the money from that could be included in the residue.
The initial period in exile after the Bolsheviks seized power was difficult for the Russians who had been in the service of the company and the Nobel family. Most ended up in Paris. The Swedes and Finns who returned to their home countries found it difficult to find new work and the company's board offered them support and pensions as long as they could, ultimately offering them a one-off sum.
In Stockholm, the Swedish government set up the "Russian Investigation Commission" which, between 1922 and 1923, tried in vain to secure compensation from the Soviet regime for lost property in the old Russia. Some of the Swedes making claims were Edla, Gösta, Emil, Ludvig and his wife Louise, Mary, Rolf Nobel and Anna Sjögren. It was a question of compensation worth millions of roubles – for property, companies, bank funds, bonds and shares.
The Nobel family retained their summer residence, the Kirjola estate, outside Viborg in south-eastern Finland. During the Second World War, valuables were successfully removed from there and, in part, buried in part of the garden before the house was set alight by withdrawing Finnish troops who did not want the house to fall into Russian hands.
Many objects that the family acquired in Russia have since been sold at auction. Russia's new capitalists have bought "back" valuable art and jewellery, such as the Fabergé clock Emanuel Nobel ordered for the paraffin company's anniversary in 1906. One surprise was that Marta Nobel-Oleinikoff had bought Hungarian government bonds before the years of revolution and when these matured a few years ago, it came as a complete surprise. Erik Delin's and Gösta Nobel's silver objects still lie buried, somewhere in a plot of land in Baku.
The Fabegé clock made for Branobel for celebrating extraction of 1 billion pud crude oil. Foto Bukowskis Auktioner.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,419
|
\section{Summary}\label{summary}}
Gravitational waves in the sensitivity band of ground-based detectors
can be emitted by a number of astrophysical sources, including not only
binary coalescences, but also individual spinning neutron stars. The
most promising signals from such sources, although as of 2020 not yet
detected, are the long-lasting, quasi-monochromatic `Continuous Waves'
(CWs). Many search methods have been developed and applied on LIGO (Aasi
et al. 2015) and Virgo (Acernese et al. 2015) data. See Prix (2009),
Riles (2017), and Sieniawska and Bejger (2019) for reviews of the field.
The \texttt{PyFstat} package provides tools to perform a range of CW
data analysis tasks. It revolves around the \(\mathcal{F}\)-statistic,
first introduced by Jaranowski, Krolak, and Schutz (1998): a
matched-filter detection statistic for CW signals described by a set of
frequency evolution parameters and maximized over amplitude parameters.
This has been one of the standard methods for LIGO-Virgo CW searches for
two decades. \texttt{PyFstat} is built on top of established routines in
\texttt{LALSuite} (LIGO Scientific Collaboration 2018) but through its
more modern \texttt{Python} interface it enables a flexible approach to
designing new search strategies.
Classes for various search strategies and target signals are contained
in three main submodules:
\begin{itemize}
\tightlist
\item
\texttt{core}: The basic wrappers to \texttt{LALSuite}'s
\(\mathcal{F}\)-statistic algorithm. End-users should rarely need to
access these directly.
\item
\texttt{grid\_based\_searches}: Classes to search over regular
parameter-space grids.
\item
\texttt{mcmc\_based\_searches}: Classes to cover promising
parameter-space regions through stochastic template placement with the
Markov Chain Monte Carlo (MCMC) sampler \texttt{ptemcee} (Vousden,
Farr, and Mandel 2015).
\end{itemize}
Besides standard CWs from isolated neutron stars, \texttt{PyFstat} can
also be used to search for CWs from sources in binary systems (including
the additional orbital parameters), for CWs with a discontinuity at a
pulsar glitch, and for CW-like long-duration transient signals, e.g.,
from \emph{after} a pulsar glitch. Specialized versions of both
grid-based and MCMC-based search classes are provided for these
scenarios. Both fully-coherent and semi-coherent searches (where the
data is split into several segments for efficiency) are covered, and an
extension to the \(\mathcal{F}\)-statistic that is more robust against
single-detector noise artifacts (Keitel et al. 2014) is also supported.
While \texttt{PyFstat}'s grid-based searches do not compete with the
sophisticated grid setups and semi-coherent algorithms implemented in
various \texttt{LALSuite} programs, its main scientific use cases so far
are for the MCMC exploration of interesting parameter-space regions and
for the long-duration transient case.
\texttt{PyFstat} was first introduced in Ashton and Prix (2018), which
remains the main reference for the MCMC-based analysis implemented in
the package. The extension to transient signals, which uses
\texttt{PyCUDA} (Klöckner et al. 2012) for speedup, is discussed in
detail in Keitel and Ashton (2018), and the glitch-robust search
approaches in Ashton, Prix, and Jones (2018).
Additional helper classes, utility functions, and internals are included
for handling the common Short Fourier Transform (SFT) data format for
LIGO data, simulating artificial data with noise and signals in them,
and plotting results and diagnostics. Most of the underlying
\texttt{LALSuite} functionality is accessed through SWIG wrappings
(Wette 2020) though for some parts, such as the SFT handling, we still
(as of the writing of this paper) call stand-alone \texttt{lalapps}
executables. Completing the backend migration to pure SWIG usage is
planned for the future.
The source of \texttt{PyFstat} is hosted on
\href{https://github.com/PyFstat/PyFstat/}{GitHub}. The repository also
contains an automated test suite and a set of introductory example
scripts. Issues with the software can be submitted through GitHub and
pull requests are always welcome. \texttt{PyFstat} can be installed
through pip, conda or docker containers. Documentation in html and pdf
formats is available from
\href{https://readthedocs.org/projects/pyfstat/}{readthedocs.org} and
installation instructions can be found there or in the
\href{https://github.com/PyFstat/PyFstat/blob/master/README.md}{README}
file. PyFstat is also listed in the Astrophysics Source Code Library as
\href{https://ascl.net/2102.027}{ascl:2102.027}.
\hypertarget{statement-of-need}{%
\section{Statement of need}\label{statement-of-need}}
The sensitivity of searches for CWs and long-duration transient GWs is
generally limited by computational resources, as the required number of
matched-filter templates increases steeply for long observation times
and wide parameter spaces. The C-based \texttt{LALSuite} library (LIGO
Scientific Collaboration 2018) contains many sophisticated search
methods with a long development history and high level of optimization,
but is not very accessible for researchers new to the field or for
students; nor is it convenient for rapid development and integration
with modern technologies like GPUs or machine learning. Hence,
\texttt{PyFstat} serves a dual function of (i) making \texttt{LALSuite}
CW functionality more easily accessible through a \texttt{Python}
interface, thus facilitating the new user experience and, for
developers, the exploratory implementation of novel methods; and (ii)
providing a set of production-ready search classes for use cases not yet
covered by \texttt{LALSuite} itself, most notably for MCMC-based
followup of promising candidates from wide-parameter-space searches.
So far, \texttt{PyFstat} has been used for
\begin{itemize}
\tightlist
\item
the original proposal of MCMC followup for CW candidates (Ashton and
Prix 2018);
\item
developing glitch-robust CW search methods (Ashton, Prix, and Jones
2018);
\item
speeding up long-transient searches with GPUs (Keitel and Ashton
2018);
\item
followup of candidates from all-sky searches for CWs from sources in
binary systems, see Covas and Sintes (2020) and Abbott et al. (2021);
\item
studying the impact of neutron star proper motions on CW searches
(Covas 2020).
\end{itemize}
\hypertarget{acknowledgements}{%
\section{Acknowledgements}\label{acknowledgements}}
We acknowledge contributions to the package from Karl Wette, Sylvia Zhu
and Dan Foreman-Mackey; as well as helpful suggestions by John T.
Whelan, Luca Rei, and the LIGO-Virgo-KAGRA Continuous Wave working
group. D.K. and R.T. are supported by European Union FEDER funds; the
Spanish Ministerio de Ciencia, Innovación y Universidades and Agencia
Estatal de Investigación grants
PID2019-106416GB-I00/AEI/10.13039/501100011033, RED2018-102661-T,
RED2018-102573-E, FPA2017-90687-REDC, FPU 18/00694, and BEAGAL 18/00148
(cofinanced by the Universitat de les Illes Balears); the Comunitat
Autonoma de les Illes Balears through the Direcció General de Política
Universitaria i Recerca with funds from the Tourist Stay Tax Law ITS
2017-006 (PRD2018/24) and the Conselleria de Fons Europeus, Universitat
i Cultura; the Generalitat Valenciana (PROMETEO/2019/071); and EU COST
Actions CA18108, CA17137, CA16214, and CA16104. This paper has been
assigned document number LIGO-P2100008.
\hypertarget{references}{%
\section*{References}\label{references}}
\addcontentsline{toc}{section}{References}
\hypertarget{refs}{}
\leavevmode\hypertarget{ref-TheLIGOScientific:2014jea}{}%
Aasi, J., B. P. Abbott, R. Abbott, and others. 2015. ``Advanced LIGO.''
\emph{Class. Quant. Grav.} 32: 074001.
\url{https://doi.org/10.1088/0264-9381/32/7/074001}.
\leavevmode\hypertarget{ref-Abbott:2020mev}{}%
Abbott, R., T. D. Abbott, S. Abraham, and others. 2021. ``All-sky search
in early O3 LIGO data for continuous gravitational-wave signals from
unknown neutron stars in binary systems.'' \emph{Phys. Rev. D} 103 (6):
064017. \url{https://doi.org/10.1103/PhysRevD.103.064017}.
\leavevmode\hypertarget{ref-TheVirgo:2014hva}{}%
Acernese, F., M. Agathos, K. Agatsuma, and others. 2015. ``Advanced
Virgo: a second-generation interferometric gravitational wave
detector.'' \emph{Class. Quant. Grav.} 32 (2): 024001.
\url{https://doi.org/10.1088/0264-9381/32/2/024001}.
\leavevmode\hypertarget{ref-Ashton:2018ure}{}%
Ashton, Gregory, and Reinhard Prix. 2018. ``Hierarchical multistage MCMC
follow-up of continuous gravitational wave candidates.'' \emph{Phys.
Rev. D} 97 (10): 103020.
\url{https://doi.org/10.1103/PhysRevD.97.103020}.
\leavevmode\hypertarget{ref-Ashton:2018qth}{}%
Ashton, Gregory, Reinhard Prix, and D.I. Jones. 2018. ``A semicoherent
glitch-robust continuous-gravitational-wave search method.'' \emph{Phys.
Rev. D} 98 (6): 063011.
\url{https://doi.org/10.1103/PhysRevD.98.063011}.
\leavevmode\hypertarget{ref-Covas:2020hcy}{}%
Covas, P. B. 2020. ``Effects of proper motion of neutron stars on
continuous gravitational-wave searches.'' \emph{Mon. Not. Roy. Astron.
Soc.} 500 (4): 5167--76. \url{https://doi.org/10.1093/mnras/staa3624}.
\leavevmode\hypertarget{ref-Covas:2020nwy}{}%
Covas, P. B., and Alicia M. Sintes. 2020. ``First all-sky search for
continuous gravitational-wave signals from unknown neutron stars in
binary systems using Advanced LIGO data.'' \emph{Phys. Rev. Lett.} 124
(19): 191102. \url{https://doi.org/10.1103/PhysRevLett.124.191102}.
\leavevmode\hypertarget{ref-Jaranowski:1998qm}{}%
Jaranowski, Piotr, Andrzej Krolak, and Bernard F. Schutz. 1998. ``Data
analysis of gravitational - wave signals from spinning neutron stars. 1.
The Signal and its detection.'' \emph{Phys. Rev. D} 58: 063001.
\url{https://doi.org/10.1103/PhysRevD.58.063001}.
\leavevmode\hypertarget{ref-Keitel:2018pxz}{}%
Keitel, David, and Gregory Ashton. 2018. ``Faster search for long
gravitational-wave transients: GPU implementation of the transient
\(\mathcal F\)-statistic.'' \emph{Class. Quant. Grav.} 35 (20): 205003.
\url{https://doi.org/10.1088/1361-6382/aade34}.
\leavevmode\hypertarget{ref-Keitel:2013wga}{}%
Keitel, David, Reinhard Prix, Maria Alessandra Papa, Paola Leaci, and
Maham Siddiqi. 2014. ``Search for continuous gravitational waves:
Improving robustness versus instrumental artifacts.'' \emph{Phys. Rev.
D} 89 (6): 064023. \url{https://doi.org/10.1103/PhysRevD.89.064023}.
\leavevmode\hypertarget{ref-Kloeckner:2012pyc}{}%
Klöckner, Andreas, Nicolas Pinto, Yunsup Lee, B. Catanzaro, Paul Ivanov,
and Ahmed Fasih. 2012. ``PyCUDA and PyOpenCL: A Scripting-Based Approach
to GPU Run-Time Code Generation.'' \emph{Parallel Computing} 38 (3):
157--74. \url{https://doi.org/10.1016/j.parco.2011.09.001}.
\leavevmode\hypertarget{ref-lalsuite}{}%
LIGO Scientific Collaboration. 2018. ``LIGO Algorithm Library -
LALSuite.'' free software (GPL).
\url{https://doi.org/10.7935/GT1W-FZ16}.
\leavevmode\hypertarget{ref-Prix:2009oha}{}%
Prix, Reinhard. 2009. ``Gravitational Waves from Spinning Neutron
Stars.'' In \emph{Neutron Stars and Pulsars}, edited by Werner Becker,
357:651--85. Astrophys. Space Sci. Lib. Berlin Heidelberg: Springer.
\url{https://doi.org/10.1007/978-3-540-76965-1_24}.
\leavevmode\hypertarget{ref-Riles:2017evm}{}%
Riles, Keith. 2017. ``Recent searches for continuous gravitational
waves.'' \emph{Mod. Phys. Lett. A} 32 (39): 1730035.
\url{https://doi.org/10.1142/S021773231730035X}.
\leavevmode\hypertarget{ref-Sieniawska:2019hmd}{}%
Sieniawska, Magdalena, and Michał Bejger. 2019. ``Continuous
gravitational waves from neutron stars: current status and prospects.''
\emph{Universe} 5 (11): 217.
\url{https://doi.org/10.3390/universe5110217}.
\leavevmode\hypertarget{ref-Vousden:2015pte}{}%
Vousden, W. D., W. M. Farr, and I. Mandel. 2015. ``Dynamic temperature
selection for parallel tempering in Markov chain Monte Carlo
simulations.'' \emph{Mon. Not. Roy. Astron. Soc.} 455 (2): 1919--37.
\url{https://doi.org/10.1093/mnras/stv2422}.
\leavevmode\hypertarget{ref-Wette:2020air}{}%
Wette, Karl. 2020. ``SWIGLAL: Python and Octave interfaces to the
LALSuite gravitational-wave data analysis libraries.'' \emph{SoftwareX}
12: 100634. \url{https://doi.org/10.1016/j.softx.2020.100634}.
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,818
|
Global Resources (69 documents found)
Gender: a precursor for discriminating against women in paid employment in Nigeria
Authors: G. Owoyemi, O. Olusanya, American Journal of Business and Management, Jan 2014
When the UN declared 1975 - 1985 the 'decade for women', they did not know it was to serve as a precursor to several national and international constitutions that would be used to address and suppress unequal power relations between men... Find out more
Trading up, crowded out: ensuring economic diversification works for women
Authors: D. Jayasinghe, R. Noble, ActionAid International, Jun 2016
This ActionAid UK briefing explores the relationship between gender inequality and economic diversification in the context of export manufacturing in the developing world, drawing on the work of feminist economists and scholars to identify... Find out more
Small and medium scale enterprises in African setting: the place of women
Authors: I. C. Igbanugo, M. C. Uzonwanne, R. U. Ezenekwe, International Journal of Economics, Commerce and Management, Mar 2016
Small and medium scale enterprises (SMEs) are a major driver of economic growth in developing countries, and with between 31-38% of SMEs run by women, it is vital to recognise and support women entrepreneurs as a critical component of... Find out more
A review of the challenges militating against women entrepreneurship in developing nations
Authors: C. L. Moses, M. Olokundun, H. Falola, Mediterranean Journal of Social Sciences, Jan 2016
Particularly in developing nations, the development of women entrepreneurship has recently gained the attention of practitioners, policy makers, and even more importantly, academics. This is a trend consequent upon the fact women's... Find out more
Gender-based stereotypes and managerial careers in diverse socio-economic environments: the cases of Greece and Nigeria
Authors: D. M. Mihail, Christiana O. Ogbogu, International Journal of Business and Social Science, Feb 2016
In most parts of the world, the workplace has persistently reproduced gender segregation, wage inequality, sex discrimination, sexual harassment and under-representation of women in key management positions. As women have made gains in... Find out more
Gender influence on access to innovation resources in Nigeria
Authors: I. O. Salome, A. T. Damilola, E. K. Sunday, International Journal of Humanities and Social Science, Jan 2013
Previous studies have indicated that men are more innovative than women in every aspect of business, with reasons put forward ranging from innovation being relatively incompatible with female gender roles, to societal attitudes leading to... Find out more
Where are the women: inclusive boardrooms in Africa's top listed companies?
Authors: G. J. Fraser-Moleketi, S. Mizrahi, African Development Bank , Jan 2015
There was once an assumption that as women became more educated and represented in the workforce, they would naturally continue their advance and rise to the very top of corporate governance. Unfortunately, that dream never materialised,... Find out more
Gender, politics, and industrial development in Nigeria
Authors: U. M. Urim, D. O. Imhonopi, F. O. Ahmadu, Covenant University, Jan 2014
The interplay between gender, politics and industry in Nigeria over the last few decades has undermined the role and participation of women in industrial development in Nigeria. That is the argument put forth in this chapter of the book... Find out more
Gender inequality: Issues and challenges in basic Education in Nigeria
Author: Zainab Inuwa Adamu, The Business and Management Review, Nov 2014
Although every child has the right to education in Nigeria, the ratio of male and female students shows a significant bias favouring boys, with the rate of disparity being lower in primary schools than in secondary schools. This suggests... Find out more
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,920
|
As one of the world's biggest beef producers, Argentina has more than 55 million cows grazing in its famed Pampas grasslands.
Scientists are now carrying out trials of new diets designed to improve cows's digestion and hopefully reduce global warming. Silvia Valtorta, of the National Council of Scientific and Technical Investigations, said that by feeding cows clover and alfalfa instead of grain "you can reduce methane emissions by 25 percent".
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,408
|
@interface SimpleJSONIPResultParser : NSObject <PublicIPResultParser>
- (instancetype)initWithTargetKey:(NSString*)key;
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,977
|
A former social worker in Child Protective Services in Maryland Prince George's county and a criminal attorney with more than 20 years' experience. Rebecca has seenpeople at their lowest points. She is not easily shocked and does not judge you based on your circumstances. Rebecca has learned the value of discipline and a stellar work ethic at a young age. She applies that same discipline and work ethic to every case she handles. Rebecca understands that, to a certian extent, your freedom rests in her hands, and she works around the clock to ensure you receive the best defense possible.
When you hire the Law Offices of Rebecca A. Nitkin, P.C. to represent you, you get a compassionate, responsive legal team led by an experienced highly disciplined attorney.
Maryland State Bar Association, Inc.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,222
|
Q: SQL Server job to check 2 databases and act based on result? I need to check 2 tables on similar databases on different servers, if they contain the same information (or size, hash, whichever is the best way to check if they're the same) then one of them should have its information deleted, on a scheduled job every day.
Is this possible using only the Jobs interface from within SQL Server Management Studio?
A: As you are on SQL 2008, you could use the MERGE command to do the check (assuming you are still talking about the linked server table comparison from your other question here).
You would setup a job that runs under an account with the correct permissions to read the data from the linked server as the source for your MERGE command and then delete the entries in your target table that match it:
MERGE DBNAME.SCHEMANAME.TABLENAME t --this is your target (the local table)
using (SELECT * from SERVERB.DBNAME.SCHEMANAME.TABLENAME) s --this is your source (the linked server)
on (t.uniquefield = s.uniquefield) --join condition
when matched then delete -- when matched, then delete!
;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,322
|
Entertainer Quickie: Tom Cruise and Katie Holmes finalize divorce settlement; Kourtney Kardashian welcomes a beautiful baby girl
By Jacquelyne Yawn | July 11, 2012
Ton Cruise and Katie Holmes Divorce Settlement
After six years of marriage, Tom Cruise and Katie Holmes have split. The decision couldn't have been easy for Holmes, but she had planned it well when she started renting an apartment in secret two weeks before the file for divorce. Some odd days later, the divorce between the two stars has been settled and Holmes seemed happy with the results.
It is known that through the settlement Holmes has gained full custody of their only daughter, Suri, and Cruise will be allowed visitation, but under certain guidelines.
On Monday, a source spoke about the custody settlement, "Tom is really sad at the thought of not being able to see Suri. You can tell he really loves his kids." The child custody results contrast to Cruise's previous marriage to Nicole Kidman. When the former couple had split, Cruise had full custody of their two children Connor, 17, and Isabella, 19.
Katie and Tom had made their divorce settlement as quick and painless as possible. Both of them released a joint statement earlier to reaffirm their commitment to work peacefully together as parents for Suri:
"We are committed to working together as parents to accomplishing what is in our daughter Suri's best interests. We want to keep matters affecting our family private and express our respect for each other's commitment to each of our respective beliefs and support each other's roles as parents." (La Times)
The full details about the settlement haven't been revealed to the public yet. It is unknown what else Holmes has received from the settlement besides Suri.
Kourtney Kardashian Gives Birth
Speaking of family, Kourtney Kardashian gave birth last Sundy to a beautiful baby girl, Penelope Scotland Disick. The reality starlet, 33,and her boyfriend, Scott Disick, 29, welcomed the new member to the family at the Los Angeles Cedars-Sinai Medical center. The little girl was born at 7 pounds and 14 ounces.
The couple has been together for five years and already have a child, their two-year-old son Mason. Last November, the happy couple had confirmed Kourtney's second pregnancy and went to Mexico last February to celebrate.
During an episode of E!'s "Keeping Up with the Kardashians" the happy mother proudly talked about her new daughter. "I am excited to have a girl, to have a new experience. I'm really excited for Mason to not have to compete with a little brother!"
The starlet's mother, Kris Jenner, threw her daughter a baby shower in May. The guests to the shower included the two sisters Kim and Khloe, Kanye West, Lamar Odom, and other stars.
Khloe has stated, "I have bought Kourt's daughter so many tutus. All I want to do is dip my niece in glitter and put a tutu on her!"(US Magazine)
All of the family members have shown their excitement over the newest member. The family's reality show displayed many proud and happy faces.
Photos courtesy of Kathryn Decker-Krauth and South Florida Beach Photos via Flickr.
Related Itemskatie holmesKeeping up with the KardashiansKourtney KardashianSuri Cruisetom cruise
Jacquelyne Yawn
← Previous Story Katie Holmes and Tom Cruise marriage broken over Scientology
Next Story → "Fifty Shades of Grey" movie in 2013
Khloe Kardashian Opens up About Her Relationships in New Interview
Movie Review – Jack Reacher: Never Go Back
Movie Review – Mission Impossible: Rogue Nation
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,421
|
The principal owner of D&S Machine Service also owns and operates Kenosha Steel Castings, a full service steel foundry producing castings up to 10,000 lbs. in weight and flasks up to 10 ft. square. Alloys poured range from plain carbon steel to high grade steel and up to 150,000 – 165,000 psi tensile and yield strengths respectively. Custom designed alloys and heat treatments can be designed to meet special requirements.
Kenosha Steel Castings has full service capabilities including: pattern construction and rigging, molding in green sand or sodium silicate, core making in baked oil sand or sodium silicate and Phenolic Urethane no-bake, complete cleaning and finishing facilities, normalizing, annealing, stress relieving or quench and temper heat treating, inspection and painting.
Kenosha Steel Castings supplies castings for a wide range of industries in various sized parts.
Parts produced include gears, sheaves, pinions, bearings, bushings, slips/clamp assemblies, ball joints and crusher components.
We offer on-site spectrographic chemical analysis, magnetic particle inspection, Brinell hardness testing and dimensional verification.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,963
|
{"url":"http:\/\/community.boredofstudies.org\/1003\/maths\/352499\/parametrisation-curves.html","text":"2. ## Re: Parametrisation of Curves\n\n$\\noindent It is clear that the locus will be symmetric about the y-axis, so if a point (x,y) is on the locus, so is (-x,y). So let's just focus on when x\\geq 0 (i.e. when B lies on or to the right of the y-axis).$\n\n$\\noindent Note that the polar equation of the circle in equation is r=2a\\sin \\theta (standard polar equation for such a circle). Let the angle between the tangent L and OA be \\phi, 0<\\phi \\leq \\frac{\\pi}{2}. By this I am referring to \\angle OAX, where X is the point on the top of the circle, i.e. (0,2a). The y-value of P, y_P, is just the y-value of B. Since OB is of length 2a\\sin \\phi (from the polar equation of the circle), and the angle made by OB with the positive x-axis is \\phi (using alternate angles in parallel lines), we have that the y-value of B is r\\sin\\phi = 2a\\sin \\phi\\cdot \\sin \\phi = 2a\\sin^{2}\\phi. So y_P = 2a\\sin^{2}\\phi.$\n\n$\\noindent For the x-value of P, x_P, it is just the length XA. From right-angle trigonometry in triangle OXA, noting OX = 2a, we have XA = 2a\\cot \\phi. So x_P= 2a\\cot \\phi.$\n\n$\\noindent Thus using symmetry, to get the full locus, when B is on the left of the y-axis, we can treat the angle \\phi as negative (e.g. -30^\\circ), and this will negate x_P whilst leaving y_P unchanged, as we want. So the parametric equations of the locus in this setup are \\boxed{x=2a\\cot\\phi, y = 2a\\sin^{2}\\phi,\\text{ for }\\phi \\in \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right]\\backslash \\left\\{0\\right\\}}. Note that when \\phi = 90^\\circ = \\frac{\\pi}{2}, we have a degenerate' case with A and B both at (0,2a), and P also at (0,2a). The parametric equations still hold when \\phi = 90^\\circ though.$\n\n$\\noindent To get an implicit equation from x=2a\\cot\\phi, y = 2a\\sin^{2}\\phi for \\phi \\in \\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]\\backslash \\left\\{0\\right\\} we use trig. identities. Note y can never be 0 (both seen geometrically and from our restriction on \\phi). Therefore, we can write \\csc^{2}\\phi = \\frac{2a}{y} by rearranging y = 2a\\sin^{2}\\phi. Since \\cot^{2}\\phi+1 = \\csc^{2}\\phi, and \\cot\\phi = \\frac{x}{2a}, we have \\frac{x^2}{4a^2}+1 = \\frac{2a}{y} as an implicit Cartesian equation. But we can easily solve this for y, i.e. x^2 y + 4a^2 y = 8a^3, i.e. \\boxed{y = \\frac{8a^3}{x^2 + 4a^2}, -\\infty\n\nFor anyone curious, there's further info on the Witch of Agnesi (with an animation too) here: http:\/\/mathworld.wolfram.com\/WitchofAgnesi.html .\n\n3. ## Re: Parametrisation of Curves\n\nOriginally Posted by InteGrand\n$\\noindent It is clear that the locus will be symmetric about the y-axis, so if a point (x,y) is on the locus, so is (-x,y). So let's just focus on when x\\geq 0 (i.e. when B lies on or to the right of the y-axis).$\n\n$\\noindent Note that the polar equation of the circle in equation is r=2a\\sin \\theta (standard polar equation for such a circle). Let the angle between the tangent L and OA be \\phi, 0<\\phi \\leq \\frac{\\pi}{2}. The y-value of P, y_P, is just the y-value of B. Since OB is of length 2a\\sin \\phi (from the polar equation of the circle), and the angle made by OB with the positive x-axis is \\phi (using alternate angles in parallel lines), we have that the y-value of B is r\\sin\\phi = 2a\\sin \\phi\\cdot \\sin \\phi = 2a\\sin^{2}\\phi. So y_P = 2a\\sin^{2}\\phi.$\n\n$\\noindent For the x-value of P, x_P, it is just the length XA, where X is the point on the top of the circle, i.e. (0,2a). From right-angle trigonometry in triangle OXA, noting OX = 2a, we have XA = 2a\\cot \\phi. So x_P= 2a\\cot \\phi.$\n\n$\\noindent Thus using symmetry, to get the full locus, when B is on the left of the y-axis, we can treat the angle \\phi as negative (e.g. -30^\\circ), and this will negate x_P whilst leaving y_P unchanged, as we want. So the parametric equations of the locus in this setup are \\boxed{x=2a\\cot\\phi, y = 2a\\sin^{2}\\phi,\\text{ for }\\phi \\in \\left(-\\frac{\\pi}{2},\\right]\\backslash \\left\\{0\\right\\}}. Note that when \\phi = 90^\\circ = \\frac{\\pi}{2}, we have a degenerate' case with A and B both at (0,2a), and P also at (0,2a). The parametric equations still hold when \\phi = 90^\\circ though.$\n\n$\\noindent To get an implicit equation from x=2a\\cot\\phi, y = 2a\\sin^{2}\\phi for \\phi \\in \\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]\\backslash \\left\\{0\\right\\} we use trig. identities. Note y can never be 0 (both seen geometrically and from our restriction on \\phi). Therefore, we can write \\csc^{2}\\phi = \\frac{2a}{y} by rearranging y = 2a\\sin^{2}\\phi. Since \\cot^{2}\\phi+1 = \\csc^{2}\\phi, and \\cot\\phi = \\frac{x}{2a}, we have \\frac{x^2}{4a^2}+1 = \\frac{2a}{y} as an implicit Cartesian equation. But we can easily solve this for y, i.e. x^2 y + 4a^2 y = 8a^3, i.e. \\boxed{y = \\frac{8a^3}{x^2 + 4a^2}, -\\infty\n\nFor anyone curious, there's further info on the Witch of Agnesi (with an animation too) here: http:\/\/mathworld.wolfram.com\/WitchofAgnesi.html .\nvery impressive, ty\n\n4. ## Re: Parametrisation of Curves\n\nfound an old class test, how would one go about doing this, i can sorta see that the eqn will be something y=-x^2 + extras but otherwise am stuck\n\n5. ## Re: Parametrisation of Curves\n\nfound an old class test, how would one go about doing this, i can sorta see that the eqn will be something y=-x^2 + extras but otherwise am stuck\n\n$\\noindent Let B be the origin and C = (c,0) on the x-axis, where c is a fixed positive number. By a suitable choice of units and since whether the line L is above or below the x-axis clearly won't affect the \\emph{shape} of the locus, we can just take L to be the line y = 1. Now, let A be the point (t,1) on this line, where t\\in \\mathbb{R} will be allowed to vary.$\n\n$\\noindent Then the altitude from A to BC is just the vertical line through A, i.e. the line x = t. Note for t\\neq c, the slope of AC is \\frac{1-0}{t-c} = \\frac{1}{t-c}. So the slope of the altitude from B to AC is the negative reciprocal of this, i.e. c-t (this is also true for t=c, since then, A is directly above C, so the line BC \\emph{is} the altitude to AC, and has slope 0). So the equation of the altitude from B to AC is y = (c-t)x (since B is the origin).$\n\n$\\noindent So the intersection of these two altitudes (which is precisely the orthocentre) is at the point where x = t and y = (c-t)t. So there are the parametric equations of the orthocentre, which clearly gives us a parabola (which Cartesian equation y = x(c-x)).$\n\n6. ## Re: Parametrisation of Curves\n\nOriginally Posted by InteGrand\n$\\noindent Let B be the origin and C = (c,0) on the x-axis, where c is a fixed positive number. By a suitable choice of units and since whether the line L is above or below the x-axis clearly won't affect the \\emph{shape} of the locus, we can just take L to be the line y = 1. Now, let A be the point (t,1) on this line, where t\\in \\mathbb{R} will be allowed to vary.$\n\n$\\noindent Then the altitude from A to BC is just the vertical line through A, i.e. the line x = t. Note for t\\neq c, the slope of AC is \\frac{1-0}{t-c} = \\frac{1}{t-c}. So the slope of the altitude from B to AC is the negative reciprocal of this, i.e. c-t (this is also true for t=c, since then, A is directly above C, so the line BC \\emph{is} the altitude to AC, and has slope 0). So the equation of the altitude from B to AC is y = (c-t)x (since B is the origin).$\n\n$\\noindent So the intersection of these two altitudes (which is precisely the orthocentre) is at the point where x = t and y = (c-t)t. So there are the parametric equations of the orthocentre, which clearly gives us a parabola (which Cartesian equation y = x(c-x)).$\nahh forgot to realise altitude from A to BC will always have the same x eqn, anyways thx again\n\nThere are currently 1 users browsing this thread. (0 members and 1 guests)\n\n#### Posting Permissions\n\n\u2022 You may not post new threads\n\u2022 You may not post replies\n\u2022 You may not post attachments\n\u2022 You may not edit your posts\n\u2022","date":"2018-05-25 09:15:35","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\": 16, \"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\": 0.9673320651054382, \"perplexity\": 2217.1335852687635}, \"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-2018-22\/segments\/1526794867055.20\/warc\/CC-MAIN-20180525082822-20180525102822-00364.warc.gz\"}"}
| null | null |
\section{Introduction}
The experimental confirmation of oscillations of atmospheric and solar
neutrinos in Refs.~\cite{Fuk98,Ahm02} is the direct indication that neutrinos
have nonzero masses and mixing, which, in its turn, unambiguously
points out to the physics beyond the standard model. This experimental
success was followed by the determination of other parameters in the
neutrino mixing matrix, including $\theta_{13}$ (see, e.g., Ref.~\cite{An12})
and the CP violation phase (see, e.g., Ref.~\cite{Abe17}).
Although the great importance of neutrino flavor oscillations for the
experimental studies of properties of these particles, other channels of
neutrino oscillations are of interest for the evolution of astrophysical
and cosmological neutrinos~\cite{Raf96}. In the present work, we shall deal mainly
with spin-flavor oscillations of neutrinos, which imply the conversion
of the type $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$, where
both flavor, $\alpha,\beta = e,\mu,\tau,\dots$, and helicity, $\mathrm{L,R}$, change. This type of neutrino transitions implies
that these particles possess nonzero magnetic moments and interact with
a strong electromagnetic field. Neutrino electromagnetic properties
are reviewed in Ref.~\cite[pp.~461\textendash 479]{FukYan03}.
The majority of studies of neutrino spin-flavor oscillations
involve the neutrino interaction with a constant transverse magnetic
field. Other configurations of electromagnetic fields, including an
electromagnetic wave, were used in Refs.~\cite{EgoLobStu00,DvoStu01}
in the examination of spin-flavor oscillations. The neutrino interaction
with an electromagnetic wave is important for the studies of neutrino
propagation in strong laser pulses~\cite{MeuKeiPia15,Fom18}.
In the present work, we study neutrino spin and spin-flavor oscillations
in a plane electromagnetic wave on the basis of the exact solution
of the Dirac-Pauli equation for a massive neutrino in this external
electromagnetic field~\cite{BagGit90}. In our analysis, we suggest
that neutrinos are Dirac particles. Although some theoretical models
of the neutrino mass generation point out that neutrinos are likely
to be Majorana particles~\cite[pp.~387\textendash 460]{FukYan03}, the issue of the neutrino
nature is still open~\cite{EllFra15}.
Our work is organized as follows. In Sec.~\ref{sec:SPINOSC}, we start with the studies of neutrino
spin oscillations in a plane electromagnetic wave with circular polarization
within one neutrino mass eigenstate. Then, in Sec.~\ref{sec:TMM},
we generalize our treatment to account for neutrino spin-flavor oscillations
with the great transition magnetic moment. The description of spin-flavor
oscillations is based on the formulation of the initial condition
problem for flavor neutrinos~\cite{Dvo11}. The possibility of an astrophysical application
is also considered in Sec.~\ref{sec:TMM}. The influence of the
small diagonal magnetic moments on spin-flavor oscillations is studied in Sec.~\ref{sec:DMM}. Spin-flavor oscillations of neutrinos with great diagonal magnetic moments are considered in Sec.~\ref{sec:GDMM}. We summarize our results in Sec.~\ref{sec:CONCL}.
The matrix elements of the neutrino spin interaction are calculated in
Appendix~\ref{sec:MATREL}.
\section{Spin oscillations in an electromagnetic wave\label{sec:SPINOSC}}
In this section, we consider neutrino spin oscillations, within
one neutrino generation, in a plane electromagnetic wave with the circular
polarization. Our analysis is based on the exact solution of the Dirac-Pauli
equation for a massive neutral fermion found in Ref.~\cite{BagGit90}.
We shall take that a neutrino is a Dirac particle. In this section, we
shall neglect the mixing between different neutrino generations. Assuming
that the considered neutrino mass eigenstate has the nonzero mass
$m$ and the magnetic moment $\mu$, the Dirac equation for such a
neutrino, described by the bispinor $\psi$, in the external electromagnetic
field $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=(\mathbf{E},\mathbf{B})$
reads
\begin{equation}\label{eq:Direq}
\left[
\mathrm{i}\gamma^{\mu}\partial_{\mu}-m-\frac{\mu}{2}\sigma_{\mu\nu}F^{\mu\nu}
\right]
\psi=0,
\end{equation}
where $\gamma^{\mu}=(\gamma^{0},\bm{\gamma})$ and $\sigma_{\mu\nu}=\tfrac{\mathrm{i}}{2}\left[\gamma_{\mu},\gamma_{\nu}\right]_{-}$
are the Dirac matrices. In the following, we shall use the standard
representation for the Dirac matrices~\cite{ItzZub80}.
We shall take that the external electromagnetic field is in the form
of a plane electromagnetic wave propagating in the positive direction
of the $z$-axis. It is convenient to choose the following gauge for
the vector potential: $A^{\mu}=(0,\mathbf{A})$. Neglecting the dispersion
of the wave, we can take that $\mathbf{A}=\mathbf{A}(t-z)$. The electric
and magnetic fields are $\mathbf{E}=-\mathbf{A}'$ and $\mathbf{B}=(\mathbf{e}_{z}\times\mathbf{E})$,
where the prime means the derivative with respect to the whole argument
of $\mathbf{A}$.
The exact solution of Eq.~(\ref{eq:Direq}) was found in Ref.~\cite{BagGit90}
for the arbitrary propagation of the fermion with respect to
the electromagnetic wave. In the present work, we consider a special
situation when a neutrino propagates along the wave, i.e. $\psi=\psi(z,t)$.
In this case, the solution of Eq.~(\ref{eq:Direq}) has the form,
\begin{equation}\label{eq:psisol}
\psi=\frac{e^{-\mathrm{i}Et+\mathrm{i}pz}}{2\sqrt{E(E-p)}}
\left(
\begin{matrix
\left[
m+E-p
\right]
v\\
\left[
m-E+p
\right]
\sigma_{3}v
\end{matrix}
\right),
\end{equation}
where $v=v(t-z)$ is the two component spinor, $E=\sqrt{p^{2}+m^{2}}$
is the neutrino energy, $p$ is the neutrino momentum, and $\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$
are the Pauli matrices. It is interesting to notice that $|\psi|^{2}=1$
if $|v|^{2}=1$. The motivation to study neutrinos propagating along the electromagnetic wave results from Fig.~\ref{fig:stardet}.
\begin{figure}
\centering
\includegraphics[scale=.3]{fig1.eps}
\protect
\caption{The schematic illustration of the neutrino propagation in an electromagnetic wave.
If both a neutrino and a wave are produced by the same source, e.g., a neutron star
(see Sec.~\ref{sec:TMM}), they will propagate collinearly towards a detector.
Thus one studies oscillations of neutrinos, created in a star, under the influence of an
electromagnetic wave emitted by the same star.
It is the motivation to use the neutrino wave function in Eq.~\eqref{eq:psisol}.
\label{fig:stardet}}
\end{figure}
The spinor $v$ was found in Ref.~\cite{BagGit90} to obey the equation,
\begin{equation}\label{eq:veq}
\mathrm{i}v'=-\mu(\bm{\sigma}\mathbf{B})v.
\end{equation}
To proceed with the analysis of the solution of Eq.~(\ref{eq:veq})
we suppose that we have a circularly polarized electromagnetic wave,
i.e. $B_{x}=B_{0}\cos\left[\omega(t-z)\right]$, $B_{y}=B_{0}\sin\left[\omega(t-z)\right]$,
and $B_{z}=0$, where $B_{0}$ is the amplitude of the wave and $\omega$
is its frequency. In this case, the solution of Eq.~(\ref{eq:veq})
has the form,
\begin{align}\label{eq:vsol}
v(\phi)= & \exp(-\mathrm{i}\sigma_{3}\omega\phi/2)
\left\{
\cos
\left[
\Omega(\phi-\phi_{0})
\right]+
\frac{\mathrm{i}}{\Omega}
\left[
\mu B_{0}\sigma_{1}+\omega\sigma_{3}/2
\right]
\sin
\left[
\Omega(\phi-\phi_{0})
\right]
\right\}
\nonumber
\\
& \times
\exp(\mathrm{i}\sigma_{3}\omega\phi_{0}/2)v_{0},
\end{align}
where $\phi=t-z$ is the current phase, $\phi_{0}=t_{0}-z_{0}$ is
the initial phase, $v_{0}=v(\phi_{0})$ is the initial spinor, and
$\Omega=\sqrt{(\mu B_{0})^{2}+\omega^{2}/4}$. One can see in Eq.~(\ref{eq:vsol})
that $v=v_{0}$ if $\phi=\phi_{0}$. Moreover $|v|^{2}=|v_{0}|^{2}$.
Now we should specify the initial condition. Let us suppose that $v_{0}^{\mathrm{T}}=(0,1)$.
In this case, using Eq.~(\ref{eq:psisol}), one can check that $P_{-}\psi_{0}=\psi_{0}$,
where $\psi_{0}$ is the initial bispinor, corresponding to $v=v_0$, $P_{\pm}=(1\pm\Sigma_{z})/2$
are the helicity projection operators, $\bm{\Sigma}=\gamma^{5}\gamma^{0}\bm{\gamma}$
and $\gamma^{5}=\mathrm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$
are the Dirac matrices. It means that initially a neutrino is left-handed,
i.e. its spin is opposite to the particle momentum.
Neutrino spin oscillations correspond to the appearance of the nonzero component $\psi_{+}$,
i.e. the right-handed polarization, at $\phi>\phi_{0}$, where $\psi_{+}=P_{+}\psi$.
The probability of $\mathrm{L}\to\mathrm{R}$ transitions is $P_{\mathrm{L}\to\mathrm{R}}=\psi^{\dagger}P_{+}\psi$.
Using Eq.~(\ref{eq:psisol}), one can show that $P_{\mathrm{L}\to\mathrm{R}}=\left(1+v^{\dagger}\sigma_{3}v\right)/2$,
i.e. the transition probability is completely defined by the evolution
of the neutrino spin in Eq.~(\ref{eq:veq}).
After lengthy but straightforward calculations based on Eq.~(\ref{eq:vsol}),
we can represent $P_{\mathrm{L}\to\mathrm{R}}$ in the form,
\begin{equation}\label{eq:PLR}
P_{\mathrm{L}\to\mathrm{R}}=
\frac{\mu^{2}B_{0}^{2}}{\mu^{2}B_{0}^{2}+\omega^{2}/4}
\sin^{2}
\left[
\sqrt{\mu^{2}B_{0}^{2}+\omega^{2}/4}(\phi-\phi_{0})
\right].
\end{equation}
One can see in Eq.~(\ref{eq:PLR}) that the transition probability
is a function of $\Delta\phi=\phi-\phi_{0}$, i.e. it depends on both
$t$ and $z$. Thus neutrino spin oscillations can happen both in
time and in space. It is owing to the fact that the spinor $v$ in
Eqs.~(\ref{eq:psisol}) and~(\ref{eq:vsol}) depends on $\phi$.
It is the feature of the field theory based approach for the description
of neutrino oscillations.
The coordinate dependence of $P_{\mathrm{L}\to\mathrm{R}}$ in Eq.~(\ref{eq:PLR})
becomes important if a neutrino interacts with an electromagnetic wave for a
quite long time. In such a situation, the particle initial wave packet becomes broad
enough. It can happen for relatively low energy neutrinos. In practice,
a neutrino is an ultrarelativistic particle. We can localize it and
attribute a mean velocity $\beta=p/E$ of its wave packet. Thus $z=\beta t$
and $z_{0}=\beta t_{0}$.
In this way to modify the coordinate dependence of $P_{\mathrm{L}\to\mathrm{R}}$
in Eq.~(\ref{eq:PLR}), we rederive the transition probability of
neutrino spin oscillations in a circularly polarized electromagnetic
wave obtained in Ref.~\cite{EgoLobStu00} on the basis of the quasiclassical approach. For this purpose one should
set $\Delta m^{2}=0$ and neglect the interaction with matter in the
corresponding expressions in Ref.~\cite{EgoLobStu00} since, in this section, we study
neutrino spin oscillations induced by the electromagnetic wave only
within one neutrino generation. In the following section, we shall
generalize this formalism to describe neutrino spin-flavor oscillations.
\section{Spin-flavor oscillations: great transition magnetic moment\label{sec:TMM}}
In this section, we shall extend the formalism developed in Sec.~\ref{sec:SPINOSC}
to describe neutrino spin-flavor oscillations. For this purpose we
discuss the system of two flavor neutrinos and formulate the initial
condition problem for these particles. Here we will focus on the case of the great
transition magnetic moment.
For simplicity, we shall consider the system of two flavor neutrinos
$(\nu_{\alpha},\nu_{\beta})$. The generalization of the formalism for greater
number of neutrino flavors is straightforward. For instance, we can
take that $\nu_{\beta}=\nu_{e}$ and $\nu_{\alpha}=\nu_{\mu,\tau}$.
The Lagrangian for this system in the presence of the external electromagnetic
field reads
\begin{equation}\label{eq:Lagrfl}
\mathcal{L}=\sum_{\lambda=\alpha,\beta}
\bar{\nu}_{\lambda}\mathrm{i}\gamma^{\mu}\partial_{\mu}\nu_{\lambda} -
\sum_{\lambda,\lambda'=\alpha,\beta}\bar{\nu}_{\lambda}
\left(
m_{\lambda\lambda'}+\frac{1}{2}M_{\lambda\lambda'}\sigma_{\mu\nu}F^{\mu\nu}
\right)
\nu_{\lambda'},
\end{equation}
where $(m_{\lambda\lambda'})$ and $(M_{\lambda\lambda'})$ are the
matrices of masses and magnetic moments in the flavor eigenstates
basis. As in Sec.~\ref{sec:SPINOSC}, we shall suppose that only
an electromagnetic wave, propagating along the $z$-axis, is present
in the system and neutrinos move in the direction of the wave.
We shall define the initial conditions in the system described by
the Lagrangian in Eq.~(\ref{eq:Lagrfl}). For this purpose, we shall suppose that initially $\nu_{\alpha}(z,t=0)=0$ and $\nu_{\beta}(z,t=0)=f\nu_{i\mathrm{L}}$,
where $f=f(z)$ is the given coordinate dependence of the initial
wave function and $\nu_{i\mathrm{L}}$ is a constant bispinor. Let
us suppose that $\Sigma_{z}\nu_{i\mathrm{L}}=-\nu_{i\mathrm{L}}$.
For example, we can take $\nu_{i\mathrm{L}}^{\mathrm{T}}=(1/\sqrt{2})(0,1,0,-1)$.
Such a choice of the initial wave functions corresponds to the presence
of left-handed electron neutrinos (if $\nu_{\beta}\equiv\nu_{e}$)
and the absence of neutrinos of other flavors.
If only two neutrino flavors are present in the system, we can introduce
the neutrino mass eigenstates $\psi_{a}$, $a=1,2$, to diagonalize
the mass matrix in Eq.~(\ref{eq:Lagrfl}). The corresponding matrix
transformation reads
\begin{equation}\label{eq:mattrans}
\nu_{\lambda}=\sum_{a=1,2}U_{\lambda a}\psi_{a},
\quad
(U_{\lambda a})=
\left(
\begin{matrix
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{matrix}
\right),
\end{equation}
where $\theta$ is the vacuum mixing angle. Note that we can say whether
neutrinos are Dirac or Majorana only after the transition to the mass
eigenstates basis in Eq.~(\ref{eq:mattrans}).
The Dirac equation for $\psi_{a}$ has the form,
\begin{equation}\label{eq:psiaeq}
\mathrm{i}\dot{\psi}_{a}=H_{a}\psi_{a}+V\psi_{b},
\quad
a,b=1,2,
\quad
a\neq b,
\end{equation}
where
\begin{equation}\label{eq:HaV}
H_{a}=(\bm{\alpha}\mathbf{p})+\beta m_{a}-\mu_{a}\beta
\left[
(\bm{\Sigma}\mathbf{B})-\mathrm{i}(\bm{\alpha}\mathbf{E})
\right],
\quad
V=-\mu\beta
\left[
(\bm{\Sigma}\mathbf{B})-\mathrm{i}(\bm{\alpha}\mathbf{E})
\right].
\end{equation}
Here $\mu_{a}\equiv(\mu_{11},\mu_{22})$ and $\mu\equiv\mu_{12}$
are the diagonal and transition magnetic moments in the mass eigenstates
basis, $(\mu_{ab})=U^{\dagger}MU$, as well as $\bm{\alpha}=\gamma^{0}\bm{\gamma}$
and $\beta=\gamma^{0}$ are the Dirac matrices.
First, we shall discuss the situation when $\mu_{a}\ll\mu$. If we neglect
the antiparticle degrees of freedom, the general solution of Eq.~(\ref{eq:psiaeq})
has the form,
\begin{align}\label{eq:psiasol}
\psi_{a}(z,t) = & \int_{-\infty}^{+\infty}\frac{\mathrm{d}p}{2\pi}
\sum_{s=\pm}a_{as}e^{-\mathrm{i}E_{a}t+\mathrm{i}pz}u_{as},
\notag
\\
u_{as}= & \frac{1}{2\sqrt{E_{a}(E_{a}-p)}}
\left(
\begin{matrix
\left[
m_{a}+E_{a}-p
\right]
v_{s} \\
\left[
m_{a}-E_{a}+p
\right]
\sigma_{3}v_{s}
\end{matrix}
\right),
\end{align}
where $E_a = \sqrt{p^2+m_a^2}$ are the energies of different mass eigenstates, $v_{+}^{\mathrm{T}}=(1,0)$ corresponds to a right-handed neutrino
and $v_{-}^{\mathrm{T}}=(0,1)$ to a left-handed particle since $\sigma_{3}v_{\pm}=\pm v_{\pm}$.
Note that we omit the index $a$ in the spinors $v_{s}$ since we
neglect $\mu_{a}$. We also mention the orthogonality of the basis
bispinors, $u_{a\pm}^{\dagger}u_{a\mp}=0$.
The coefficients $a_{as}$ are supposed to be $c$-number functions
rather than operators acting on Fock states. If we assume that
they depend on neither $t$ nor $z$, the wave functions in Eq.~(\ref{eq:psiasol})
would satisfy the equations $\mathrm{i}\dot{\psi}_{a}=H_{a}\psi_{a}$.
To account for the potential $V$ in Eq.~(\ref{eq:HaV}), which mixes
different mass eigenstates in Eq.~(\ref{eq:psiaeq}), we have to
suppose that $a_{as}$ are no longer constant. Such an approach for
the description of the evolution of neutrino mass eigenstates in the
presence of external fields was used in Ref.~\cite{Dvo11}. However,
unlike Ref.~\cite{Dvo11}, where the corresponding coefficients depend
on time only, here we suppose that $a_{as}=a_{as}(t-z)$. Our main
goal is to study the behavior of these coefficients.
Substituting the ansatz in Eq.~(\ref{eq:psiasol}) to Eq.~(\ref{eq:psiaeq}),
we obtain the equation for $a_{as}$,
\begin{equation}\label{eq:aeqraw}
\mathrm{i}\sum_{s=\pm}a_{as}'u_{as'}^{\dagger}(1-\alpha_{z})u_{as} =
\sum_{s=\pm}u_{as'}^{\dagger}Vu_{2s}e^{\mathrm{i}(E_{a}-E_{b})t}a_{bs},
\quad
a\neq b.
\end{equation}
Accounting for the mean values
\begin{align}
u_{as'}^{\dagger}(1-\alpha_{z})u_{as} & =
\left(
1 - \frac{p}{E_{a}}
\right)\delta_{ss'},
\nonumber
\\
u_{1s'}^{\dagger}Vu_{2s} & =
-\mu v_{s'}^{\dagger}(\bm{\sigma}\mathbf{B})v_{s}
\sqrt{\frac{(E_{1}-p)(E_{2}-p)}{E_{1}E_{2}}},
\label{eq:meanu}
\\
v_{\pm}^{\dagger}(\bm{\sigma}\mathbf{B})v_{\mp} & =B_{0}\exp[\mp i\omega(t-z)],
\label{eq:meanv}
\end{align}
we can rewrite Eq.~(\ref{eq:aeqraw}) in the form,
\begin{equation}\label{eq:aa'}
\mathrm{i}a_{a\pm}' =-\mu B_{0}\sqrt{\frac{E_{a}(E_{b}-p)}{E_{b}(E_{a}-p)}}
\exp
\left[
\mathrm{i}(E_{a}-E_{b})t\mp\mathrm{i}\omega(t-z)
\right]
a_{b\mp}.
\end{equation}
In Eqs.~(\ref{eq:meanu}) and~(\ref{eq:meanv}), we suppose that
one deals with a circularly polarized wave.
Before we proceed with the analysis of the evolution of $a_{as}$
we should specify the initial conditions for them. As in Sec.~\ref{sec:SPINOSC},
we shall suppose that neutrinos are localized along their trajectories.
However, since different mass eigenstates have different masses, we
assume that $z=\bar{\beta}t$, where $\bar{\beta}$ is the center
of inertia velocity~\cite{LanLif94}, $\bar{\beta}=2p/(E_{1}+E_{2})$.
If neutrinos are ultrarelativistic, one gets that $\bar{\beta}\approx1-(m_{1}^{2}+m_{2}^{2})/4p^{2}$.
Therefore, if $t=0$, $a_{as}$ should be taken at zero argument:
$a_{as}(0)$.
Accounting for the initial condition for the flavor eigenstates specified
above, we obtain that
\begin{align}\label{eq:inicondaa}
a_{1-}(0) = & \sin\theta f(p)
\left(
u_{1-}^{\dagger}\nu_{i\mathrm{L}}
\right),
\quad
f(p)=\int_{-\infty}^{+\infty}f(z)e^{-\mathrm{i}pz}\mathrm{d}z
\notag
\\
a_{2-}(0) = & \cos\theta f(p)
\left(
u_{2-}^{\dagger}\nu_{i\mathrm{L}}
\right),
\end{align}
and $a_{a+}(0)=0$. Taking into account the explicit form of the initial
bispinor $\nu_{i\mathrm{L}}$ and $u_{a-}$ in Eq.~(\ref{eq:psiasol}),
we get that $\left(u_{a-}^{\dagger}\nu_{i\mathrm{L}}\right)=m_{a}/\sqrt{2E_{a}(E_{a}-p)}$.
If neutrinos are ultrarelativistic, $\left(u_{a-}^{\dagger}\nu_{i\mathrm{L}}\right)\to1$.
Taking into account the dependence of $a_{as}$ on $t$ in Eq.~(\ref{eq:aa'}),
$a_{as}=a_{as}(t[1-\bar{\beta}])$, we can rewrite Eq.~(\ref{eq:aa'})
as the effective Schr\"odinger equation,
\begin{align}\label{eq:SchPsi}
\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\Psi= & H_{\mathrm{eff}}\Psi,
\\
\notag
H_{\mathrm{eff}}= & -\mu B_{0}(1-\bar{\beta})
\left(
\begin{matrix
0 & 0 & 0 & \xi
\exp
\left(
\mathrm{i}\Phi_{-}t
\right)\\
0 & 0 & \xi
\exp
\left(
\mathrm{i}\Phi_{+}t
\right)
& 0\\
0 &
\exp
\left(
-\mathrm{i}\Phi_{+}t
\right) /
\xi & 0 & 0\\
\exp
\left(
-\mathrm{i}\Phi_{-}t
\right) /
\xi & 0 & 0 & 0
\end{matrix}
\right),
\end{align}
where $\Psi^{\mathrm{T}}=\left(a_{1+},a_{1-},a_{2+},a_{2-}\right)$
and
\begin{align}
\Phi_{\pm} & =E_{1}-E_{2}\pm\omega(1-\bar{\beta})
\approx
\frac{m_{1}^{2}-m_{2}^{2}}{2p}\pm\omega\frac{m_{1}^{2}+m_{2}^{2}}{4p^{2}},
\label{eq:Phipm}
\\
\xi & =\sqrt{\frac{E_{1}(E_{2}-p)}{E_{2}(E_{1}-p)}}
\approx
\frac{m_{2}}{m_{1}}.
\label{eq:xi}
\end{align}
In Eqs.~(\ref{eq:Phipm}) and~(\ref{eq:xi}), we use the approximation
of ultrarelativistic neutrinos.
One can see that $H_{\mathrm{eff}}$ in Eq.~(\ref{eq:SchPsi}) is
non-Hermitian since $\xi\neq1$ because $m_{1}\neq m_{2}$. This fact
results from the assumption that both mass eigenstates with different
masses propagate with the same velocity $\bar{\beta}$. Nevertheless
we can approximately set $\xi=1$. Indeed, let us assume that $m_{a}\sim1\,\text{eV}$~\cite{Ase11}.
Below, we shall be interested in the $\nu_{e}\nu_{\mu}$-oscillations
channel. In this situation, $\delta m^{2}=m_{1}^{2}-m_{2}^{2}\approx7.6\times10^{-5}\,\text{eV}^{2}$~\cite{Abe16}.
Therefore $|m_{1}-m_{2}|/m_{a}\sim10^{-2}$. Hence, we can take that
$\xi=1$ with a sufficient level of accuracy. Note that we should
keep $\delta m^{2}\neq0$ in $\Phi_{\pm}$ in Eq.~(\ref{eq:Phipm})
since these terms contribute to the phase of neutrino oscillations,
which is very sensitive to the change of parameters.
Let us change the variables in Eq.~(\ref{eq:SchPsi}),
\begin{equation}
\Psi=\mathcal{U}\tilde{\Psi},
\quad
\mathcal{U}=\text{diag}
\left(
e^{i\Phi_{-}t/2}, e^{i\Phi_{+}t/2},e^{-i\Phi_{+}t/2}, e^{-i\Phi_{-}t/2}
\right).
\end{equation}
The wave function $\tilde{\Psi}^{\mathrm{T}}=\left(\tilde{a}_{1+},\tilde{a}_{1-},\tilde{a}_{2+},\tilde{a}_{2-}\right)$
obeys the modified Schr\"odinger equation,
\begin{align}\label{eq:SchPsitilde}
\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\Psi}= &
\tilde{H}_{\mathrm{eff}}\tilde{\Psi},
\notag
\\
\tilde{H}_{\mathrm{eff}}= &
\left(
\begin{matrix
\Phi_{-}/2 & 0 & 0 & -\mu B_{0}(1-\bar{\beta})\\
0 & \Phi_{+}/2 & -\mu B_{0}(1-\bar{\beta}) & 0\\
0 & -\mu B_{0}(1-\bar{\beta}) & -\Phi_{+}/2 & 0\\
-\mu B_{0}(1-\bar{\beta}) & 0 & 0 & -\Phi_{-}/2
\end{matrix}
\right).
\end{align}
We recall that we set $\xi=1$ in the $\mu B_{0}$-terms in Eq.~(\ref{eq:SchPsitilde}).
The solution of Eq.~(\ref{eq:SchPsitilde}) has the form,
\begin{equation}\label{eq:tildePsisol}
\tilde{\Psi}(t)=\sum_{\zeta=\pm}
\left[
e^{-i\Omega_{\zeta}t}
\left(
U_{\zeta}\otimes U_{\zeta}^{\dagger}
\right)+
e^{i\Omega_{\zeta}t}
\left(
V_{\zeta}\otimes V_{\zeta}^{\dagger}
\right)
\right]
\tilde{\Psi}_{0},
\end{equation}
where $\Omega_{\pm}=\sqrt{\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2}+\Phi_{\pm}^{2}/4}$
and
\begin{align}\label{eq:UV}
U_{+} & =
\frac{\sqrt{\Omega_{+}+\Phi_{+}/2}}{\sqrt{2\Omega_{+}}}
\left(
\begin{matrix
0\\
1\\
-\dfrac{\mu B_{0}(1-\bar{\beta})}{\Omega_{+}+\Phi_{+}/2}\\
0
\end{matrix}
\right),
\quad
V_{+}=\frac{\sqrt{\Omega_{+}+\Phi_{+}/2}}{\sqrt{2\Omega_{+}}}
\left(
\begin{matrix
0\\
\dfrac{\mu B_{0}(1-\bar{\beta})}{\Omega_{+}+\Phi_{+}/2}\\
1\\
0
\end{matrix}
\right),
\nonumber
\\
U_{-} & =\frac{\sqrt{\Omega_{-}+\Phi_{-}/2}}{\sqrt{2\Omega_{-}}}
\left(
\begin{matrix
1\\
0\\
0\\
-\dfrac{\mu B_{0}(1-\bar{\beta})}{\Omega_{-}+\Phi_{-}/2}
\end{matrix}
\right),
\quad
V_{-}=\frac{\sqrt{\Omega_{-}+\Phi_{-}/2}}{\sqrt{2\Omega_{-}}}
\left(
\begin{matrix
\dfrac{\mu B_{0}(1-\bar{\beta})}{\Omega_{-}+\Phi_{-}/2}\\
0\\
0\\
1
\end{matrix}
\right),
\end{align}
are the eigenvectors of $\tilde{H}_{\mathrm{eff}}$: $\tilde{H}_{\mathrm{eff}}U_{\zeta}=\Omega_{\zeta}U_{\zeta}$
and $\tilde{H}_{\mathrm{eff}}V_{\zeta}=-\Omega_{\zeta}V_{\zeta}$.
Using Eqs.~(\ref{eq:inicondaa}), (\ref{eq:tildePsisol}) and~(\ref{eq:UV}),
we find that the coefficients $a_{a+}(t)$ have the form,
\begin{align}\label{eq:a12p}
a_{1+}(t) & =\mathrm{i}e^{\mathrm{i}\Phi_{-}t/2}
(1-\bar{\beta})\frac{\mu B_{0}}{\Omega_{-}}\sin(\Omega_{-}t)f(p)\cos\theta,
\nonumber
\\
a_{2+}(t) & =\mathrm{i}e^{-\mathrm{i}\Phi_{+}t/2}
(1-\bar{\beta})\frac{\mu B_{0}}{\Omega_{+}}\sin(\Omega_{+}t)f(p)\sin\theta.
\end{align}
The explicit form of $a_{a-}(t)$ is not important for our purposes
since these coefficients do not contribute to the evolution of the right-handed neutrino
states.
We are interested in the appearance of right-handed neutrinos of the
flavor $\alpha$ in a beam initially consisting of $\nu_{\beta\mathrm{L}}$.
We shall suppose that the initial wave packet is quite wide, i.e.
we take that $f(p)=2\pi\delta(p-p_{0})$, where $p_{0}$ is the initial
momentum. In the following, we shall omit the subscript 0 for brevity.
Using Eqs.~(\ref{eq:mattrans}), (\ref{eq:psiasol}), and~(\ref{eq:a12p}),
we obtain that the wave function $\nu_{\alpha\mathrm{R}}$ has the
form,
\begin{align}\label{eq:nualphaR}
\nu_{\alpha\mathrm{R}}(z,t)= &
\int_{-\infty}^{+\infty}\frac{\mathrm{d}p}{2\pi}
\left[
\cos\theta e^{-\mathrm{i}E_{1}t}a_{1+}(t)u_{1+}-
\sin\theta e^{-\mathrm{i}E_{2}t}a_{2+}(t)u_{2+}
\right]
\nonumber
\\
& =
\left[
\cos\theta e^{-\mathrm{i}E_{1}t}a_{1+}(t)
\left(
\nu_{f\mathrm{R}}^{\dagger}u_{1+}
\right) -
\sin\theta e^{-\mathrm{i}E_{2}t}a_{2+}(t)
\left(
\nu_{f\mathrm{R}}^{\dagger}u_{2+}
\right)
\right]
\nu_{f\mathrm{R}},
\end{align}
where $\nu_{f\mathrm{R}}^\mathrm{T}=(1/\sqrt{2})(1,0,1,0)$. Basing on Eq.~(\ref{eq:psiasol}),
one gets that $\left(\nu_{f\mathrm{R}}^{\dagger}u_{a+}\right)= m_{a} [2E_{a}(E_{a}-p)]^{-1/2}$.
If we study ultrarelativistic neutrinos, then $\left(\nu_{f\mathrm{R}}^{\dagger}u_{a+}\right)\to1$.
Using Eqs.~(\ref{eq:a12p}) and~(\ref{eq:nualphaR}), as well as
considering ultrarelativistic neutrinos, we obtain the transition
probability for $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$
oscillations,
\begin{equation}\label{eq:PbetaLalphaR}
P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}(t)=|\nu_{\alpha \mathrm{R}}|^{2}=
\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2}
\left[
\cos^{2}\theta\frac{\sin(\Omega_{-}t)}{\Omega_{-}}-
\sin^{2}\theta\frac{\sin(\Omega_{+}t)}{\Omega_{+}}
\right]^{2}.
\end{equation}
It is interesting to compare $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}$
in Eq.~(\ref{eq:PbetaLalphaR}) with the transition probability of
spin-flavor oscillations of Dirac neutrinos with large transition
magnetic moment in a constant transverse magnetic field studied in
Ref.~\cite{DvoMaa07}. For this purpose we should set $\omega=0$
and replace $(1-\bar{\beta})\to1$ in Eq.~(\ref{eq:PbetaLalphaR}).
In this case, $\Omega_{+}=\Omega_{-}=\sqrt{(\mu B_{0})^{2}+(\delta m^{2}/4p)^{2}}$.
Thus we reproduce the transition probability found in Ref.~\cite{DvoMaa07}.
Now we can compare $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}$
in Eq.~(\ref{eq:PbetaLalphaR}) with the corresponding transition
probability for neutrino spin-flavor oscillations in a plane electromagnetic
wave found in Ref.~\cite{EgoLobStu00}.
Using Eq.~(21) in Ref.~\cite{EgoLobStu00} and omitting the neutrino matter interaction there, since here we study the neutrino interaction with an electromagnetic wave only, one obtains the following transition probability for $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$ oscillations:
\begin{align}\label{eq:Pwrong}
P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}(t) = &
\frac{\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2}}
{\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2} +
[\delta m^2 A(\theta)/4p \pm \omega(1-\bar{\beta})/2]^2}
\notag
\\
& \times
\sin^2
\left(
\sqrt{\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2} +
[\delta m^2 A(\theta)/4p \pm \omega(1-\bar{\beta})/2]^2} t
\right).
\end{align}
If we consider $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ transitions (see below), the function $A(\theta)$ reads $A(\theta) = (1+\cos2\theta)/2$~\cite{LikStu95}. The signs $\pm$ in Eq.~\eqref{eq:Pwrong} correspond to different polarizations of the wave.
One can see that the result
of Ref.~\cite{EgoLobStu00}, shown in Eq.~\eqref{eq:Pwrong}, cannot be reproduced by Eq.~(\ref{eq:PbetaLalphaR}), except in the trivial case $\theta=0$.
However, the situation of a zero mixing angle between active neutrinos was recently excluded experimentally in Ref.~\cite{An12}, where it was shown that the remaining mixing angle $\theta_{13}$ is nonzero.
Moreover, basing on Eq.~(\ref{eq:PbetaLalphaR}), one cannot expect
the appearance of the resonant amplification of spin-flavor oscillations, which can result from Eq.~\eqref{eq:Pwrong} and was claimed in Ref.~\cite{EgoLobStu00}.
The mentioned discrepancy is because of the unsubstantiated account
of the neutrino vacuum oscillations phase $\delta m^{2}A(\theta)/4p$ in the
dynamics of the neutrino spin in Ref.~\cite{EgoLobStu00}. In Sec.~\ref{sec:SPINOSC},
using the method of the exact solution of the Dirac-Pauli equation, we
have confirmed that the results of Ref.~\cite{EgoLobStu00} are applicable
for the description of the neutrino spin evolution within one mass
eigenstate. Thus, in the present work, we have generalized the findings of Ref.~\cite{EgoLobStu00}
to correctly treat neutrino spin-flavor oscillations.
To analyze Eq.~(\ref{eq:PbetaLalphaR}) we suppose that $\delta\Omega\ll\Omega_{\pm}$,
where $\delta\Omega=\left(\Omega_{+}-\Omega_{-}\right)/2$. In this
situation, the function $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}$
becomes rapidly oscillating, being modulated by a slowly varying envelope
function. Therefore we can apply the method of the analysis of this
envelope function developed in Refs.~\cite{Dvo08,DvoMaa09}. Let
us represent $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}$
in the form,
\begin{equation}\label{eq:PLRenv}
P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}(t) =
A_{\mathrm{eff}}(t)\sin^{2}(\Omega_{0}t),
\quad
A_{\mathrm{eff}}(t)=A_{\mathrm{min}}+2\delta A\sin^{2}(\delta\Omega t),
\end{equation}
where
\begin{equation}\label{eq:AmindA}
A_{\mathrm{min}}=
\frac{
\left[
\mu B_{0}(1-\bar{\beta})\cos2\theta
\right]^{2}
}{\Omega_{0}^{2}}
\left(
1+\frac{2\delta\Omega}{\cos2\theta\Omega_{0}}
\right),
\quad
\delta A=
\frac{
\left[
\mu B_{0}(1-\bar{\beta})\sin2\theta
\right]^{2}
}{2\Omega_{0}^{2}},
\end{equation}
are the minimal value and the amplitude of the envelope function $A_{\mathrm{eff}}(t)$.
In Eqs.~(\ref{eq:PLRenv}) and~(\ref{eq:AmindA}), $\Omega_{0}=\sqrt{\mu^{2}B_{0}^{2}(1-\bar{\beta})^{2}+(\delta m^{2}/4p)^{2}}$.
The behavior of the transition probability for the $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$
oscillations channel for different $\omega$ is shown in Fig.~\ref{fig:PTMM}.
One can see that $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}$ is
a rapidly oscillating function. Therefore the averaged signal $\bar{P}(t)=A_{\mathrm{min}}/2+\delta A\sin^{2}(\delta\Omega t)$
will be detected. The maximal averaged transition probability reads
\begin{equation}\label{eq:Pavmax}
\bar{P}_{\mathrm{max}}=
\frac{
\left[
\mu B_{0}(1-\bar{\beta})
\right]^{2}
}{2\Omega_{0}^{2}}
\left(
1+2\cos2\theta\frac{\delta\Omega}{\Omega_{0}}
\right).
\end{equation}
The envelope function $A_{\mathrm{eff}}(t)$ and the averaged transition
probability $\bar{P}(t)$ are also shown in Fig.~\ref{fig:PTMM}.
\begin{figure}
\centering
\subfigure[]
{\label{1a}
\includegraphics[scale=.35]{fig2a.eps}}
\hskip-.6cm
\subfigure[]
{\label{1b}
\includegraphics[scale=.35]{fig2b.eps}}
\protect
\caption{The transition probability for the $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$
oscillations channel for different $\omega$ versus the propagation
distance $z=t$, based on Eq.~(\ref{eq:PLRenv}). The neutrino parameters
are $p=1\,\text{keV}$, $m_{1}\approx m_{2}=1\,\text{eV}$,
$\delta m^{2}=7.6\times10^{-5}\,\text{eV}^{2}$,
$\theta=0.6$~\cite{Abe16}, and $\mu=10^{-11}\mu_{\mathrm{B}}$~\cite{Bed13},
where $\mu_{\mathrm{B}}$ is the Bohr magneton. The amplitude of the
electromagnetic wave $B_{0}=10^{18}\,\text{G}$.
(a)~$\omega=10^{12}\,\text{s}^{-1}$;
(b)~$\omega=10^{13}\,\text{s}^{-1}$.
The envelope function $A_{\mathrm{eff}}(t)$
and the averaged transition probability $\bar{P}(t)$ are depicted
by the red and blue lines respectively.
We do not show the actual behavior of the transition probability in panel~(a) since it oscillates
very rapidly. Instead we represent it in the inset in panel~(a) at the small propagation
distance $z<500\,\text{m}$.\label{fig:PTMM}}
\end{figure}
The maximal averaged transition probability, calculated using Eq.~(\ref{eq:Pavmax})
with the parameters corresponding to Figs.~\ref{1a} and~\ref{1b}, is $\bar{P}_{\mathrm{max}}\approx0.35$.
This result is in the agreement with Figs.~\ref{1a} and~\ref{1b}. We can also estimate
the modulation length $L=\pi/\delta\Omega$. Using Eq.~(\ref{eq:PLRenv}),
one gets that
\begin{equation}\label{eq:dOmega}
\delta\Omega=\frac{\omega\delta m^{2}}{32\Omega_{0}p^{3}}
\left(
m_{1}^{2}+m_{2}^{2}
\right).
\end{equation}
Basing on the parameters corresponding to Fig.~\ref{1a} and Eq.~(\ref{eq:dOmega}),
we obtain that $L\approx7.2\,\text{km}$, whereas $L\approx720\,\text{m}$ for Fig.~\ref{1b}. These values of $L$ are in the agreement with Fig.~\ref{fig:PTMM}.
As an application of the obtained results, we shall consider the possibility of $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$
spin-flavor oscillations of astrophysical neutrinos in the vicinity
of a neutron star (NS). A magnetic field with the strength $\sim10^{18}\,\text{G}$,
chosen in Fig.~\ref{fig:PTMM}, is slightly weaker than the strongest
field allowed in NS~\cite{CarPraLat01}. The electromagnetic emission of pulsars ranges from radio to X-ray wavelengths~\cite{Man04}.
The frequencies adopted in Fig.~\ref{fig:PTMM} correspond to wavelengths $\lambda = 1.8\,\text{mm}$ in Fig.~\ref{1a} and $\lambda = 0.18\,\text{mm}$ in Fig.~\ref{1b}. These wavelengths lie in the infrared region. The electromagnetic emission of NS with such wavelengths was described in Ref.~\cite{Dan11}.
Note that further enhancement of the frequency is inexpedient since $\delta\Omega$ becomes comparable with $\Omega_{\pm}$ then. In principle, using our approach, we can also analyze spin-flavor oscillations in electromagnetic waves with wavelengths in the range: $(\text{several mm})<\lambda<(\text{several tens of cm})$, emitted by radio pulsars. For a radio pulsar, $L$ is greater than that shown in Fig.~\ref{fig:PTMM}.
An electromagnetic wave, which causes neutrino spin-flavor oscillations, can be considered as plane only in the near field zone. In the far field zone, a wave should be taken as spherical, with its amplitude decreasing as $1/z$. The size of the near field zone was estimated in Ref.~\cite{Rap02} as $l \sim D^2/\lambda \sim D^2\omega$, where $D$ is the length scale of the region where a wave is emitted. In case of a pulsar, we can take that $D \sim 10^4\,\text{cm}$~\cite{Bau15}. Thus, we get that $l \sim 3\times 10^4\,\text{km}$ for the case shown in Fig.~\ref{1a}, and $l \sim 3\times 10^5\,\text{km}$ for Fig.~\ref{1b}. The obtained values of $l$ are much greater than $L$ in Fig.~\ref{fig:PTMM}. Hence the approximation of a plane electromagnetic wave is valid.
We studied neutrino spin-flavor oscillations in constant in space magnetic fields in Refs.~\cite{DvoMaa07,Dvo11}. Although neutrino oscillations in a constant field are not suppressed by the factor $(1-\bar{\beta}) \ll 1$, cf. Eq.~\eqref{eq:PbetaLalphaR}, the analytical expressions for the transition probabilities, derived in Refs.~\cite{DvoMaa07,Dvo11}, cannot be applied for neutrinos propagating near NS at distances $R_\mathrm{NS} < z \lesssim l$, where $R_\mathrm{NS}\sim10\,\text{km}$ is the typical NS radius. Indeed, one cannot neglect the coordinate dependence of the magnetic field in NS at such distances, i.e. we have to take $B(z) \sim B_\text{surf} (R/z)^3$, where $B_\text{surf}$ is the magnetic field at the NS surface. Thus only a numerical transition probability
can be obtained by solving the effective Schr\"odinger equation with $B=B(z)$. On the contrary, the analytical expression in Eq.~\eqref{eq:PbetaLalphaR} is valid for the distances $z$ up to $l$.
The most powerful neutrino emission takes place during a supernova
explosion. In several minutes after an explosion, the neutrino flux
becomes rather weak to be detected. Nevertheless, when the NS temperature is above
$10^{7}\,\text{K}\sim1\,\text{keV}$, neutrinos carry away energy
from NS and are an effective tool for the NS cooling~\cite{HaePotYak07}.
That is why we take the neutrino energy $\sim1\,\text{keV}$ in Fig.~\ref{fig:PTMM}.
Thus such neutrinos, produced in modified Urca processes in the NS
core, are subject to the described spin-flavor oscillations. Basing
on $\bar{P}_{\mathrm{max}}$ in Fig.~\ref{fig:PTMM}, one gets that about
$35\%$ of $\nu_{e\mathrm{L}}$ will be converted to $\nu_{\mu\mathrm{R}}$.
It should be noted that the chosen parameters $B_{0}$ and $p$ correspond
to $\delta A\lesssim1$ in Eq.~(\ref{eq:AmindA}). It guarantees
that the transition probability is not suppressed significantly. We mention that the study of the electromagnetic emission of pulsars is a rapidly developing area of modern astrophysics. Although the recent advances in the description of the pulsars radiation~\cite{Bes17}, there is no self-consistent model of this process~\cite{Har17}. In realistic situation, the amplitude of an electromagnetic wave is likely to be smaller that $10^{18}\,\text{G}$. In the case, when $\mu B_{0}(1-\bar{\beta}) \ll \delta m^{2}/4p$, using Eq.~\eqref{eq:Pavmax}, the maximal averaged transition probability is
\begin{equation}\label{eq:PtrsmallB}
\bar{P}_\mathrm{max} \approx
2
\left[
\mu B_{0}(1-\bar{\beta}) \frac{2 p}{\delta m^2}
\right]^{2} < 0.35.
\end{equation}
Thus, the value $\bar{P}_\mathrm{max} = 0.35$, obtained above, is the upper bound on the maximal transition probability in NS since we supposed that $B_0$ equals to the strongest magnetic field, which can exist in such a system.
At the end of this section, we discuss the general issue on the approximations
used in the derivation of the main results. To obtain Eq.~(\ref{eq:inicondaa})
we supposed that the neutrino mass eigenstates are localized on their classical trajectories. However, to derive Eq.~(\ref{eq:nualphaR})
one has to assume that the neutrino wave packets are wide. The latter
assumption is necessary for a significant overlap of different
mass eigenstates for the oscillations process to occur. Formally we should
require that $\ell\gg\delta\Omega^{-1}$, where $\ell$ is the
width of the wave packet. However we can suggest that point-like neutrinos
with equal velocities are coherently emitted during the time interval
$\Delta t\gg\delta\Omega^{-1}$. In this case, one deals with an effectively
wide neutrino wave packet consisting of mass eigenstates propagating
with a certain velocity. Thus we can reconcile both assumptions necessary
to get Eqs.~(\ref{eq:inicondaa}) and~(\ref{eq:nualphaR}).
\section{Spin-flavor oscillations: influence of diagonal magnetic moments\label{sec:DMM}}
In this section, we discuss the influence of the diagonal magnetic
moments $\mu_{a}$ on the neutrino spin-flavor oscillations in a plane
electromagnetic wave. In our analysis, we shall still suppose that $\mu_{a}$ is small: $\mu_{a}\ll\mu$.
If one accounts for $\mu_{a}$, the main formalism, developed in Sec.~\ref{sec:TMM},
remains practically unchanged. First, Eq.~(\ref{eq:psiasol}) is
modified, so that the two component spinor $v_{s}$, entering to the bispinor $u_{as}$, becomes dependent on time and
$\mu_{a}$, i.e. $v_{s}\to v_{as}(t)$. The temporal dependence of
$v_{as}$ is given by Eq.~(\ref{eq:vsol}). We can choose two independent
spin states $v_{0s}$ in Eq.~(\ref{eq:vsol}) as $v_{0+}^{\mathrm{T}}=(1,0)$
and $v_{0-}^{\mathrm{T}}=(0,1)$. In this situation, both $v_{a\pm}(t)$
and $u_{a\pm}$, corresponding to the opposite spin states, are orthogonal.
Second, the matrix elements in Eq.~(\ref{eq:meanv}) become quite
cumbersome. The calculation of the matrix elements is provided in
Appendix~\ref{sec:MATREL} in the general form. Using Eq.~(\ref{eq:veps})
and making the calculations similar to those in Sec.~\ref{sec:TMM},
we derive the analogue of Eq.~(\ref{eq:SchPsi}),
\begin{align}\label{eq:SchPsiDMM}
\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\Psi = & H_{\mathrm{eff}}\Psi,
\notag
\\
H_{\mathrm{eff}}= & -\mu B_{0}(1-\bar{\beta})
\left(
\begin{matrix
0 & 0 & \epsilon_{+} &
\exp
\left(
\mathrm{i}\Phi_{-}t
\right) \\
0 & 0 &
\exp
\left(
\mathrm{i}\Phi_{+}t
\right) &
\epsilon_{-}\\
\epsilon_{+}^{*} &
\exp
\left(
-\mathrm{i}\Phi_{+}t
\right) &
0 & 0\\
\exp
\left(
-\mathrm{i}\Phi_{-}t
\right) &
\epsilon_{-}^{*} & 0 & 0
\end{matrix}
\right),
\end{align}
where
\begin{equation}\label{eq:epspm}
\epsilon_{\pm}=\frac{B_{0}}{\omega}
\left[
\pm
\left(
\mu_{1}+\mu_{2}
\right)
\mp\mu_{1}e^{\pm \mathrm{i}\omega t(1-\bar{\beta})}
\mp\mu_{2}e^{\mp \mathrm{i}\omega t(1-\bar{\beta})}
\right].
\end{equation}
In Eqs.~(\ref{eq:SchPsiDMM}) and~(\ref{eq:epspm}), we set $z=\bar{\beta}t$
and assume that $m_{1}\approx m_{2}$ in the off-diagonal terms $\sim\exp(\pm\mathrm{i}\Phi_{\pm}t)$
in $H_{\mathrm{eff}}$.
Since we discuss the situation when $m_{1}\approx m_{2}$, it is reasonable
to take that $\mu_{1}\approx\mu_{2}=\mu'$.
In this case, $\epsilon_{\pm}=\pm\epsilon$,
where $\epsilon=4\mu'B_{0}\sin^{2}\left[\omega t(1-\bar{\beta})/2\right]/\omega$.
Then, we can derive the analogue of Eq.~(\ref{eq:SchPsitilde}),
\begin{align}\label{eq:SchPsitildeDMM}
\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\tilde{\Psi}= &
\tilde{H}_{\mathrm{eff}}\tilde{\Psi},
\notag
\\
\tilde{H}_{\mathrm{eff}}= &
\left(
\begin{matrix
\Phi_{-}/2 & 0 & -\mu B_{0}(1-\bar{\beta})\epsilon & -\mu B_{0}(1-\bar{\beta})\\
0 & \Phi_{+}/2 & -\mu B_{0}(1-\bar{\beta}) & \mu B_{0}(1-\bar{\beta})\epsilon\\
-\mu B_{0}(1-\bar{\beta})\epsilon & -\mu B_{0}(1-\bar{\beta}) & -\Phi_{+}/2 & 0\\
-\mu B_{0}(1-\bar{\beta}) & \mu B_{0}(1-\bar{\beta})\epsilon & 0 & -\Phi_{-}/2
\end{matrix}
\right).
\end{align}
Should one have the solution of Eq.~(\ref{eq:SchPsitildeDMM}), the
transition probability for $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$
oscillations reads
\begin{equation}\label{eq:PLRtildea}
P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}(t)=
|\cos\theta\tilde{a}_{1+}(t)-\sin\theta\tilde{a}_{2+}(t)|^{2}.
\end{equation}
One can show that Eq.~(\ref{eq:PLRtildea}) is equivalent to $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}$
calculated using $a_{a+}(t)$, i.e. based on the solution of Eq.~(\ref{eq:SchPsiDMM}),
as $\nu_{\alpha\mathrm{R}}(z,t)$ in Eq.~(\ref{eq:nualphaR}).
For the illustration of the influence of the diagonal magnetic moments
on neutrino spin-flavor oscillations in a plane electromagnetic wave,
we consider $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ with the same
parameters as in Sec.~\ref{sec:TMM}.
In Figs.~\ref{fig:PDMM12} and~\ref{fig:PDMM13},
we show $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}$ based on the
numerical solution of Eq.~(\ref{eq:SchPsitildeDMM}) for different
$\mu'$ (or $\epsilon_{0}=\mu'B_{0}/\omega$) and $\omega$.
The upper and lower envelope functions, as well as the averaged transition
probability, are also depicted in Figs.~\ref{fig:PDMM12} and~\ref{fig:PDMM13}. To build these envelope functions we use the spline interpolation of the
maxima and minima of $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}$
respectively.
Note that, as in Fig.~\ref{1a}, we do not represent the evolution of $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}(z)$ in Fig.~\ref{fig:PDMM12} since this function is rapidly oscillating and, hence, is indistinguishable. Instead, we just show the envelope functions and the averaged transition probability in Fig.~\ref{fig:PDMM12}.
\begin{figure}
\centering
\subfigure[]
{\label{2a}
\includegraphics[scale=.35]{fig3a.eps}}
\hskip-.6cm
\subfigure[]
{\label{2b}
\includegraphics[scale=.35]{fig3b.eps}}
\\
\subfigure[]
{\label{2c}
\includegraphics[scale=.35]{fig3c.eps}}
\hskip-.6cm
\subfigure[]
{\label{2d}
\includegraphics[scale=.35]{fig3d.eps}}
\protect
\caption{The upper (red lines) and lower (green lines) envelope functions, as well as the averaged
transition probability (blue lines), corresponding to Eq.~(\ref{eq:PLRtildea}), for
$\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ oscillations in the electromagnetic
wave versus the distance $z$ passed by the neutrino beam for different
diagonal magnetic moment $\mu'$, based on the numerical solution
of Eq.~(\ref{eq:SchPsitildeDMM}). The parameters of the neutrino system
and $B_{0}$ are the same as in Fig.~\ref{fig:PTMM}. The frequency
of the electromagnetic wave is $\omega=10^{12}\,\text{s}^{-1}$.
(a)~$\mu'=0$ ($\epsilon_{0}=0$);
(b)~$\mu'=2.8\times10^{-3}\mu=2.8\times10^{-14}\mu_\mathrm{B}$
($\epsilon_{0}=0.1$);
(c)~$\mu'=5.5\times10^{-3}\mu=5.5\times10^{-14}\mu_\mathrm{B}$
($\epsilon_{0}=0.2$);
(d)~$\mu'=8.3\times10^{-3}\mu=8.3\times10^{-14}\mu_\mathrm{B}$
($\epsilon_{0}=0.3$).\label{fig:PDMM12}}
\end{figure}
\begin{figure}
\centering
\subfigure[]
{\label{3a}
\includegraphics[scale=.35]{fig4a.eps}}
\hskip-.6cm
\subfigure[]
{\label{3b}
\includegraphics[scale=.35]{fig4b.eps}}
\\
\subfigure[]
{\label{3c}
\includegraphics[scale=.35]{fig4c.eps}}
\hskip-.6cm
\subfigure[]
{\label{3d}
\includegraphics[scale=.35]{fig4d.eps}}
\protect
\caption{The transition probability (black lines) for
$\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$
oscillations, as well as the upper (red lines) and lower (green lines) envelope functions,
and the averaged
transition probability (blue lines) for this oscillations channel, based on the numerical solution of
Eq.~\eqref{eq:SchPsitildeDMM}. The parameters of the neutrino system and the
electromagnetic wave are the same as in Fig.~\ref{fig:PDMM12}, except for the frequency,
which is now equal to $\omega=10^{13}\,\text{s}^{-1}$.
\label{fig:PDMM13}}
\end{figure}
One can see that $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}$ and the envelope functions in
Figs.~\ref{2a} and~\ref{3a}, corresponding to $\mu'=0$, coincide with the curves shown in Figs.~\ref{1a} and~\ref{1b},
which were built on the basis of the exact solution of Eq.~(\ref{eq:SchPsitilde});
cf. Eqs.~(\ref{eq:tildePsisol}), (\ref{eq:UV}) and~(\ref{eq:PbetaLalphaR}).
It is also interesting to note that $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}}$ and the envelope functions
for spin-flavor oscillations of neutrinos with $\mu'\sim10^{-2}\mu\ll\mu$,
shown in Figs.~\ref{2d} and~\ref{3d}, significantly differs from the case $\mu'=0$ shown in Figs.~\ref{2a} and~\ref{3a}.
Basing on Figs.~\ref{2b}-\ref{2d} and~\ref{3b}-\ref{3d}, one can conclude that the averaged transition
probability diminishes if one accounts for $\mu'>0$ in the system.
Thus the situation when $\mu_{a} = 0$, studied in Sec.~\ref{sec:TMM},
is more preferable from the point of view of phenomenological applications.
Besides $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ oscillations, we can study other channels of neutrino oscillations on the basis of the numerical solution of Eq.~\eqref{eq:SchPsitildeDMM}. For example, in Fig.~\ref{fig:eLmuL13}, we demonstrate the transition probability, the upper and lower envelope functions, and the averaged transition probability for $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$ oscillations. These dependencies for $\nu_{e\mathrm{L}}\to\nu_{e\mathrm{R}}$ oscillations are shown in Fig.~\ref{fig:eLeR13}. The survival probability for oscillations $\nu_{e\mathrm{L}}\to\nu_{e\mathrm{L}}$, the upper and lower envelope functions, and the averaged survival probability are given in Fig.~\ref{fig:eLeL13}.
\begin{figure}
\centering
\subfigure[]
{\label{5a}
\includegraphics[scale=.35]{fig5a.eps}}
\hskip-.6cm
\subfigure[]
{\label{5b}
\includegraphics[scale=.35]{fig5b.eps}}
\\
\subfigure[]
{\label{5c}
\includegraphics[scale=.35]{fig5c.eps}}
\hskip-.6cm
\subfigure[]
{\label{5d}
\includegraphics[scale=.35]{fig5d.eps}}
\protect
\caption{The transition probability (black lines) for
$\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$
oscillations, as well as the upper (red lines) and lower (green lines) envelope functions,
and the averaged
transition probability (blue lines) for this oscillations channel, based on the numerical solution of
Eq.~\eqref{eq:SchPsitildeDMM}. The parameters of the neutrino system and the
electromagnetic wave are the same as in Fig.~\ref{fig:PDMM13}.
\label{fig:eLmuL13}}
\end{figure}
\begin{figure}
\centering
\subfigure[]
{\label{6a}
\includegraphics[scale=.35]{fig6a.eps}}
\hskip-.6cm
\subfigure[]
{\label{6b}
\includegraphics[scale=.35]{fig6b.eps}}
\\
\subfigure[]
{\label{6c}
\includegraphics[scale=.35]{fig6c.eps}}
\hskip-.6cm
\subfigure[]
{\label{6d}
\includegraphics[scale=.35]{fig6d.eps}}
\protect
\caption{The transition probability (black lines) for
$\nu_{e\mathrm{L}}\to\nu_{e\mathrm{R}}$
oscillations, as well as the upper (red lines) and lower (green lines) envelope functions,
and the averaged
transition probability (blue lines) for this oscillations channel, based on the numerical solution of
Eq.~\eqref{eq:SchPsitildeDMM}. The parameters of the neutrino system and the
electromagnetic wave are the same as in Fig.~\ref{fig:PDMM13}.
\label{fig:eLeR13}}
\end{figure}
\begin{figure}
\centering
\subfigure[]
{\label{7a}
\includegraphics[scale=.35]{fig7a.eps}}
\hskip-.6cm
\subfigure[]
{\label{7b}
\includegraphics[scale=.35]{fig7b.eps}}
\\
\subfigure[]
{\label{7c}
\includegraphics[scale=.35]{fig7c.eps}}
\hskip-.6cm
\subfigure[]
{\label{7d}
\includegraphics[scale=.35]{fig7d.eps}}
\protect
\caption{The survival probability (black lines) for
$\nu_{e\mathrm{L}}\to\nu_{e\mathrm{L}}$
oscillations, as well as the upper (red lines) and lower (green lines) envelope functions,
and the averaged
survival probability (blue lines) for this oscillations channel, based on the numerical solution of
Eq.~\eqref{eq:SchPsitildeDMM}. The parameters of the neutrino system and the
electromagnetic wave are the same as in Fig.~\ref{fig:PDMM13}.
\label{fig:eLeL13}}
\end{figure}
One can see in Figs.~\ref{fig:PDMM13}-\ref{fig:eLeR13} that $P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}} = P_{\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}} = P_{\nu_{e\mathrm{L}}\to\nu_{e\mathrm{R}}} = 0$ at $z=0$, whereas $P_{\nu_{e\mathrm{L}}\to\nu_{e\mathrm{L}}}(z=0) = 1$ in Fig.~\ref{fig:eLeL13}, i.e. the initial condition is fulfilled. Figs.~\ref{fig:PDMM13}-\ref{fig:eLeL13} are built for $\omega=10^{13}\,\text{s}^{-1}$. The analogues of these dependencies for $\omega=10^{12}\,\text{s}^{-1}$ are not presented here since the transition and survival probabilities are rapidly oscillating functions and, hence, are quite difficult for the analysis.
We also mention that, in this section, we study only the situation when $\omega \geq 10^{12}\,\text{s}^{-1}$.
The numerical solution of Eq.~(\ref{eq:SchPsitildeDMM}) for $\omega < 10^{12}\,\text{s}^{-1}$ (i.e. the situation corresponding to electromagnetic waves emitted by a radio pulsar; cf. Sec.~\ref{sec:TMM}) is quite problematic
since Eq.~(\ref{eq:SchPsitildeDMM}) becomes extremely stiff then.
\section{Spin-flavor oscillations: great diagonal magnetic moments\label{sec:GDMM}}
In this section, we study spin-flavor oscillations of neutrinos with great diagonal magnetic moments in a plane electromagnetic wave.
In some extensions of the standard model, e.g., in a minimally extended standard model, diagonal magnetic moments of neutrino are proportional to neutrino masses~\cite{DvoStu04}. The transition magnetic moment is much smaller than diagonal ones because of the Glashaw-Iliopoulos-Maiani (GIM) mechanism~\cite[pp.~461\textendash 479]{FukYan03}. Considering the situation of great diagonal magnetic moments, we can derive analytically the transition probability of neutrino spin-flavor oscillations in an electromagnetic wave for such neutrinos.
As in Sec.~\ref{sec:TMM}, we study the evolution of neutrino mass eigenstates. The Dirac equation for $\psi_a$ coincides with that in Eqs.~\eqref{eq:psiaeq} and~\eqref{eq:HaV}. However we should take that $V=0$ since we study the situation when $\mu_a \gg \mu$.
Since the transition magnetic moment is not taken into account, one has the following qualitative evolution of the neutrino spin and mass eigenstates. There will be independent spin precession within each mass eigenstate owing to the interaction of the neutrino magnetic moment $\mu_a$ with the electromagnetic field of the wave. The transition between different mass eigenstates is solely due to the presence of the nonzero vacuum mixing in Eq.~\eqref{eq:mattrans}.
Therefore, to find the transition probability for $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$ oscillations, we can use Eqs.~\eqref{eq:psisol}-\eqref{eq:vsol}, as well as the results of Ref.~\cite{DvoMaa07}, where spin-flavor oscillations of neutrinos with arbitrary magnetic moments were studied in the constant magnetic field. We assume that neutrinos are ultrarelativistic particles and adopt the quasiclassical approximation from the very beginning. Omitting some intermediate calculations, we just give the final result for the transition probability,
\begin{equation}\label{eq:Pgmua}
P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}}(t) =
\sin^2(2\theta)
\left[
\frac{1}{4}(A_1 - A_2)^2 + A_1 A_2 \sin^2(\Phi_\mathrm{vac}t)
\right],
\end{equation}
where
\begin{equation}\label{eq:A12}
A_{1,2}(t) =
\frac{\mu_{1,2} B_{0}}{\sqrt{\mu_{1,2}^{2}B_{0}^{2}+\omega^{2}/4}}
\sin
\left[
\sqrt{\mu_{1,2}^{2}B_{0}^{2}+\omega^{2}/4}(1-\beta_{1,2})t
\right],
\end{equation}
are the amplitudes of probabilities of spin oscillations within each mass eigenstate and $\Phi_\mathrm{vac} = \delta m^2/4p$ is the phase of flavor oscillations in vacuum.
If we set $\omega = 0$ and replace $(1-\beta_{1,2}) \to 1$ in Eq.~\eqref{eq:A12}, we get that $A_{1,2} = \sin(\mu_{1,2} B_0 t)$. Thus Eq.~\eqref{eq:Pgmua} reproduces the transition probability for $\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}$ oscillations in a constant transverse magnetic field derived in Ref.~\cite{DvoMaa07} for the case $\mu_a \gg \mu$.
Although the transition probability, obtained in Eqs.~\eqref{eq:Pgmua} and~\eqref{eq:A12} is analytical, the situation when $\mu_a \gg \mu$ has very limited applicability from the point of view of phenomenology. The fact is that $\mu_a \sim 10^{-19}\mu_\mathrm{B}$ in the minimally extended standard model for reasonable values of neutrino masses~\cite{Ase11}. Thus $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}} \ll 1$ for any $t$. That is why we do not provide the correction to $P_{\nu_{\beta\mathrm{L}}\to\nu_{\alpha\mathrm{R}}} \ll 1$ owing to $\mu\neq 0$ since such a quantity is really negligible.
\section{Conclusion\label{sec:CONCL}}
In conclusion, we mention that, in the present work, we have studied
spin and spin-flavor oscillations of Dirac neutrinos in a plane electromagnetic
wave with the circular polarization. Our analysis was based on solving
the initial condition problem for the system of flavor neutrinos interacting
with external fields. This method was reviewed in Ref.~\cite{Dvo11}.
The neutrino interaction with the electromagnetic wave is owing to
the presence of the nonzero neutrino magnetic moments. This interaction
was accounted for by using the exact solution of the Dirac-Pauli equation
for massive fermions found in Ref.~\cite{BagGit90}.
In Sec.~\ref{sec:SPINOSC}, we have considered neutrino spin oscillations
in a plane electromagnetic wave within one neutrino mass eigenstate.
We have obtained the exact expression for the transition probability
for $\nu_{\mathrm{L}}\to\nu_{\mathrm{R}}$ oscillations in Eq.~(\ref{eq:PLR}).
This transition probability turns out to depend on both time and the
spatial coordinate. However, for an ultrarelativistic neutrino, which
is localized along its trajectory, the spatial coordinate dependence
is transformed to the dependence on time only. In this case, we have
reproduced the transition probability obtained in Ref.~\cite{EgoLobStu00},
where the quasiclassical approach for the description of the neutrino
spin evolution in electromagnetic fields was developed.
Then, in Secs.~\ref{sec:TMM}-\ref{sec:GDMM}, we have generalized
the approach in Sec.~\ref{sec:SPINOSC} to treat neutrino spin-flavor
oscillations. In Sec.~\ref{sec:TMM}, we have started with the
analysis of the situation when only a great transition magnetic moment
is present in the mass eigenstates basis. In this situation, one can
find the analytic expression for the transition probability; cf. Eq.~(\ref{eq:PbetaLalphaR}).
Comparing the transition probabilities in Fig.~\ref{fig:PTMM} with
the results of Ref.~\cite{Dvo08}, one can see that spin-flavor oscillations
of Dirac neutrinos in an electromagnetic wave are analogous to oscillations
in a twisting magnetic field.
The expression for the transition probability in Eq.~(\ref{eq:PbetaLalphaR})
turns out to differ from the analogous result obtained in Ref.~\cite{EgoLobStu00}; cf. Eq.~\eqref{eq:Pwrong}.
This discrepancy is due to the incorrect treatment of transitions
between different neutrino flavor eigenstates in Ref.~\cite{EgoLobStu00}.
As we have mentioned above, the neutrino spin evolution within one
neutrino mass eigenstate was accounted for correctly in Ref.~\cite{EgoLobStu00}.
Thus, in the present work, we have provided the valid generalization
of the results of Ref.~\cite{EgoLobStu00} for the description of
neutrino spin-flavor oscillations in an electromagnetic wave.
Then, in Sec.~\ref{sec:TMM}, we have discussed a possibility of an astrophysical
application of the obtained results. We have studied $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$
oscillations in an electromagnetic wave emitted by a highly magnetized
NS. We found that the upper bound on the averaged transition probability can reach $35\%$; cf. Fig.~\ref{fig:PTMM}. If the amplitude of an electromagnetic wave is less than $10^{18}\,\text{G}$, which is likely to be the case in realistic astrophysical media, the maximal transition probability can be found in Eq.~\eqref{eq:PtrsmallB}.
In Sec.~\ref{sec:DMM}, we have briefly discussed the
influence of the diagonal magnetic moments on spin-flavor oscillations
in an electromagnetic wave. We have assumed that diagonal magnetic
moments, $\mu'\sim10^{-14}\mu_{\mathrm{B}}$, are much smaller than
the transition one, $\mu\sim10^{-11}\mu_{\mathrm{B}}$. This value
of $\mu'$ is in agreement with the upper bound on diagonal magnetic
moments of Dirac neutrinos revealed in Ref.~\cite{Bel05}. One can
see in Figs.~\ref{fig:PDMM12} and~\ref{fig:PDMM13} that, if diagonal magnetic moments
are taken into account, the averaged transition probability is less
than in the case when only a transition magnetic moment is accounted
for. Besides $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{R}}$ oscillations channel, we have studied other types of neutrino oscillations like $\nu_{e\mathrm{L}}\to\nu_{\mu\mathrm{L}}$, $\nu_{e\mathrm{L}}\to\nu_{e\mathrm{R}}$, and $\nu_{e\mathrm{L}}\to\nu_{e\mathrm{L}}$ in Sec.~\ref{sec:DMM}; cf. Figs.~\ref{fig:eLmuL13}-\ref{fig:eLeL13}.
Finally, in Sec.~\ref{sec:GDMM}, we have provided the analytical transition probability for neutrino spin-flavor oscillations in an electromagnetic wave in case when diagonal magnetic moments dominate over transition one; cf. Eqs.~\eqref{eq:Pgmua} and~\eqref{eq:A12}. This situation can take place, e.g., in the minimally extended standard model, where the transition magnetic moment is suppressed by the GIM mechanism~\cite[pp.~461\textendash 479]{FukYan03}. However, this our result is of limited phenomenological applicability since the diagonal magnetic moments are extremely small in this model.
\section*{Acknowledgments}
This work was partially supported by the Competitiveness Improvement
Program at the Tomsk State University, RFBR (Research Project No.
18-02-00149a), and the Foundation for the Advancement of Theoretical Physics and
Mathematics ``BASIS''. I am also thankful to S.~Dvornikov for the assistance
in numerical simulations.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,844
|
Media: Street lamps, magnets, aluminium, acetate, ink.
There are six street lamps between our studio/lodging, and the south wing of IMMA main building. We are turning them into revolving lanterns, where shadows of retired soldiers of the past who died here are projected around the entire space.
When night falls, we are gated in with the security guards. And the occasional visits from bygone residents, for whom the Royal Hospital was originally built. Surveillance cameras record fuzzy glimpses of shadows of the past and the present, meeting under the warm glow of this Six Lamp Divide.
Shadows come into existence only when light shines upon them. What a particular shadow looks like depends on where and how the light is shinning. May the light give them shape, and the winds bring them motion.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 733
|
{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/187654-find-min-sufficient-statistics-continuous-random-sample.html","text":"# Thread: Find min sufficient statistics for a continuous random sample\n\n1. ## Find min sufficient statistics for a continuous random sample\n\nLet $\\displaystyle X_1,...,X_n$ be a random variable from an absolutely continuous distribution with density:\n\n$\\displaystyle f_ \\theta (x) = \\left \\{ \\begin {array}{rcl} \\frac {2x}{ \\theta ^2} & x \\in (0, \\theta ) \\\\ 0 & \\mbox {otherwise} \\end {array} \\right.$\n\nFind the minimum sufficient statistics T and its density.\n\nThe solution is as of follows:\n\nLet $\\displaystyle T(x) = \\mbox {max} \\{ X_i \\} , M(x) = \\mbox {min} \\{ X_i \\}$\n\nThen $\\displaystyle f_ \\theta (x_1, \\ldots ,x_n) = f _ \\theta (x_1) \\cdots f_ \\theta (x_n)$\n\n$\\displaystyle = \\frac {2x_1}{ \\theta ^2} \\cdots \\frac {2x_n}{ \\theta ^2} = \\prod ^n_{i=1} \\frac {2x_i}{ \\theta ^2}$\n\n$\\displaystyle = \\frac {2^n}{ \\theta ^{2n}} \\cdot \\prod ^n_{i=1}x_i$\n\n$\\displaystyle = \\frac {2^n}{ \\theta ^{2n}} \\cdot \\prod ^n_{i=1} x_i \\cdot 1 _{ \\{ M(x)>0 \\} } \\cdot 1_{ \\{ T(x)< \\theta \\} }$, with $\\displaystyle T = T(X)$\n\nI understand that this density gives non zero values only when $\\displaystyle M(X)>0$ and $\\displaystyle T(X) < \\theta$ but I don't understand how that last line with the two indicator functions come into being. Thank you!\n\n2. ## Re: Find min sufficient statistics for a continuous random sample\n\nThe suff stat is the max, because it cannot be separated from $\\displaystyle \\theta$\nthat's the factorization theorem in use.\nThe marginal densities are ZERO if the X's are not in $\\displaystyle (0,\\theta)$\nHENCE, the joint density is ZERO if all the X's are not in $\\displaystyle (0,\\theta)$\nAnd, note that\n\n$\\displaystyle 0< X_1,...,X_n <\\theta$ is the same as $\\displaystyle 0< X_{(1)}<\\cdots <X_{(n)} <\\theta$\n\nis the same as $\\displaystyle 0< X_{(1)}$ and $\\displaystyle X_{(n)} <\\theta$\n\nYOUR teacher should have had indicator functions on the previous lines as well, let me show you\n\n3. ## Re: Find min sufficient statistics for a continuous random sample\n\nYour teacher left out the indicator functions on the first few steps\nThis should clear it up...\n\nLet $\\displaystyle X_1,...,X_n$ be a random variable from an absolutely continuous distribution with density:\n\n$\\displaystyle f_ \\theta (x) = \\left \\{ \\begin {array}{rcl} \\frac {2x}{ \\theta ^2} & x \\in (0, \\theta ) \\\\ 0 & \\mbox {otherwise} \\end {array} \\right.$\n\nFind the minimum sufficient statistics T and its density.\n\nThe solution is as of follows:\n\nLet $\\displaystyle T(x) = \\mbox {max} \\{ X_i \\} , M(x) = \\mbox {min} \\{ X_i \\}$\n\nThen $\\displaystyle f_ \\theta (x_1, \\ldots ,x_n) = f _ \\theta (x_1) \\cdots f_ \\theta (x_n)$\n\n$\\displaystyle = \\frac {2x_1}{ \\theta ^2} I(0<x_1<\\theta)\\cdots \\frac {2x_n}{ \\theta ^2} I(0<x_n<\\theta)$\n\n$\\displaystyle = \\prod ^n_{i=1} \\frac {2x_i}{ \\theta ^2}I(0<x_i<\\theta)$\n\n$\\displaystyle = \\frac {2^n}{ \\theta ^{2n}} \\cdot \\prod ^n_{i=1}x_i I(0<x_i<\\theta)$\n\n$\\displaystyle = \\frac {2^n}{ \\theta ^{2n}} \\cdot \\prod ^n_{i=1} x_i \\cdot 1 _{ \\{ M(x)>0 \\} } \\cdot 1_{ \\{ T(x)< \\theta \\} }$, with $\\displaystyle T = T(X)$\n\nI understand that this density gives non zero values only when $\\displaystyle M(X)>0$ and $\\displaystyle T(X) < \\theta$ but I don't understand how that last line with the two indicator functions come into being. Thank you!\nWhoever did this doesn't completely understand.\nThe indicators were there from the beginning.\nSticking them at the end means they don't understand functions as well.","date":"2018-04-22 19:19:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7858149409294128, \"perplexity\": 520.7647437673779}, \"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-2018-17\/segments\/1524125945637.51\/warc\/CC-MAIN-20180422174026-20180422194026-00043.warc.gz\"}"}
| null | null |
## DEDICATION
_To everyone who needs the encouragement_
_to fight for what your heart most desperately wants,_
_this book is for you._
## CONTENTS
1. Cover
2. Title Page
3. Dedication
4. Contents
5. Chapter 1
6. Chapter 2
7. Chapter 3
8. Chapter 4
9. Chapter 5
10. Chapter 6
11. Chapter 7
12. Chapter 8
13. Chapter 9
14. Chapter 10
15. Chapter 11
16. Chapter 12
17. Chapter 13
18. Chapter 14
19. Chapter 15
20. Chapter 16
21. Chapter 17
22. Chapter 18
23. Chapter 19
24. Chapter 20
25. Chapter 21
26. Chapter 22
27. Chapter 23
28. Chapter 24
29. Chapter 25
30. Chapter 26
31. Chapter 27
32. Chapter 28
33. Chapter 29
34. Chapter 30
35. Epilogue
36. Acknowledgments
37. Excerpt from _Ride Wild_
1. Chapter 1
2. Chapter 2
3. Chapter 3
4. Chapter 4
5. Chapter 5
6. Chapter 6
7. Chapter 7
8. Chapter 8
9. Chapter 9
10. Chapter 10
11. Chapter 11
12. Chapter 12
38. About the Author
39. Also by Laura Kaye
40. Copyright
41. About the Publisher
# Guide
1. Cover
2. Contents
3. Chapter 1
1. ii
2. iii
3. iv
4. v
5. vi
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
85.
86.
87.
88.
89.
90.
91.
92.
93.
94.
95.
96.
97.
98.
99.
100.
101.
102.
103.
104.
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
115.
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
129.
130.
131.
132.
133.
134.
135.
136.
137.
138.
139.
140.
141.
142.
143.
144.
145.
146.
147.
148.
149.
150.
151.
152.
153.
154.
155.
156.
157.
158.
159.
160.
161.
162.
163.
164.
165.
166.
167.
168.
169.
170.
171.
172.
173.
174.
175.
176.
177.
178.
179.
180.
181.
182.
183.
184.
185.
186.
187.
188.
189.
190.
191.
192.
193.
194.
195.
196.
197.
198.
199.
200.
201.
202.
203.
204.
205.
206.
207.
208.
209.
210.
211.
212.
213.
214.
215.
216.
217.
218.
219.
220.
221.
222.
223.
224.
225.
226.
227.
228.
229.
230.
231.
232.
233.
234.
235.
236.
237.
238.
239.
240.
241.
242.
243.
244.
245.
246.
247.
248.
249.
250.
251.
252.
253.
254.
255.
256.
257.
258.
259.
260.
261.
262.
263.
264.
265.
266.
267.
268.
269.
270.
271.
272.
273.
274.
275.
276.
277.
278.
279.
280.
281.
282.
283.
284.
285.
286.
287.
288.
289.
290.
291.
292.
293.
294.
295.
296.
297.
298.
299.
300.
301.
302.
303.
304.
305.
306.
307.
308.
309.
310.
311.
312.
313.
314.
315.
316.
317.
318.
319.
320.
321.
322.
323.
324.
325.
326.
327.
328.
329.
330.
331.
332.
333.
334.
335.
336.
337.
338.
339.
340.
341.
342.
343.
344.
345.
346.
347.
348.
349.
350.
351.
352.
353.
354.
355.
356.
357.
358.
359.
360.
361.
362.
363.
364.
365.
366.
367.
368.
369.
370.
371.
372.
373.
374.
375.
376.
377.
378.
379.
380.
381.
382.
383.
384.
385.
386.
387.
388.
389.
390.
391.
392.
393.
394.
395.
396.
397.
398.
399.
400.
401.
402.
403.
404.
405.
406.
407.
408.
409.
410.
411.
412.
413.
414.
415.
416.
417.
418.
419.
420.
421.
422.
423.
424.
425.
426.
427.
428.
429.
430.
431.
432.
433.
434.
435.
436.
437.
438.
439.
440.
441.
442.
443.
444.
445.
446.
447.
448.
449.
450.
451.
452.
453.
454.
455.
456.
457.
458.
459.
460.
461.
462.
463.
464.
465.
466.
467.
468.
469.
470.
471.
472.
473.
474.
475.
476.
477.
478.
479.
480.
481.
482.
483.
## CHAPTER 1
Alexa Harmon tore out of her car and ran into the house, her high heels clicking against the concrete of the three-car garage and then the travertine tiles of the hallway and kitchen. She was late getting home from work, and that meant she was going to be hard-pressed to get dinner on the table on time.
She beelined for the bedroom, already working at the buttons on her silk blouse. Despite being under the gun, she took the time to hang up her work clothes and put everything away in the walk-in closet that was nearly as big as her childhood bedroom had been.
Grant didn't like mess or clutter. Everything had to be in its place. Always.
Slipping into a pretty blue blouse, jeans, and her ballet flats, Alexa's gaze cut to the alarm clock on her nightstand. She had twenty-five minutes. Twenty-five minutes to make sure her lateness didn't ruin their whole evening.
_Damnit, Alexa_. _You should've kept your eyes on the time better_.
It was true. She'd just been elbows deep in materials arriving for the model home in Grant's newest development. This was the first time he was letting her take the lead on the interior design of a model, rather than hiring their usual outside contractor, and she wanted it to be good. Oh, who was she kidding? She wanted it to be perfect.
More than that, _she_ wanted to be good. Good enough. No, she wanted to be _perfect_. For Grant.
Grant _really_ liked perfection.
Alexa got it. Her fiancé Grant's perfectionist tendencies went a long way toward explaining how he'd built Grant Slater Enterprises, the biggest real estate development and management company in Western Maryland. Though he'd come into some kind of a trust fund when he was younger, he'd built most of his success with his own hard work and smart investments. Now, Frederick was almost a company town, at least where real estate was concerned. There were more developments in the area with the words _Grant_ or _Slater_ in their names than she could count. Their own neighborhood was a prime example—Slater Estates.
Running back out to the kitchen, a low pleading _Meow_ caught Alexa's attention.
"Come on, Lucy. Come with Mama," Alexa called, heading straight for the cat's bowl. She poured dry food into the dish, spilling a little in her haste. The hairless sphynx brushed against her leg in a show of affection. Alexa gave Lucy's sweater-covered blue-gray body a quick pet as she scooped up stray morsels of food with her other hand.
The clock on the microwave told her she had twenty-two minutes now.
She grabbed the package of two filet mignons from the fridge, along with a bag of fresh asparagus. Moving as fast as she could, she found the grill pan for the meat and the sauté pan for the asparagus, and got that much going. The baked potatoes she'd planned weren't going to be possible with this little time, and trying to boil water for corn on the cob would be pushing it. Her stomach knotted as her pulse raced. She buttered thick slices of Italian bread and seasoned them with garlic, then slid them into the warming oven to brown.
As soon as she turned the filets, she was back in the fridge. When her gaze settled on the container of chickpea salad from the weekend, relief flooded through her. She'd forgotten they had that. Finally, she threw together a green salad with chunky fresh vegetables.
Keeping a close eye on the time, she set the dining room table—Grant always preferred to eat in the formal dining room. She made sure to align the flatware just so, just as he liked. And then she was pouring the wine and plating the food with two minutes to spare.
Alexa might've fist-pumped if she wasn't so anxious about almost having been late. Her stomach was in so many knots she wasn't even sure she'd be able to eat. Though it was her own damn fault.
Six o'clock came and went. Six-oh-five. Six-ten. Sitting alone at the dining room table, Alexa frowned. Finally, her phone buzzed an incoming text message from Grant.
_I've got a dinner meeting tonight. Don't wait up._
Alexa stared at the screen for a long moment, then found herself blinking away threatening angry tears. She stuffed down all the things she wanted to say—and all the things she felt—and replied simply, _Okay xo._
_Look on the bright side_ , she thought _._ Okay. On the bright side, she'd now have time she hadn't expected to work on her final project for her senior seminar.
Still . . .
She let herself fume and wallow for several more minutes, and then she shook her head. "Stop it, Al," she said out loud. God, she really was overemotional lately, wasn't she? Just like Grant said she was.
Between her job, designing the model home, her class project, her recent move into Grant's house, and their upcoming wedding, there was just so much going on. She felt like she should be juggling it all with more grace and enthusiasm. Instead, what she _really_ felt scared her. Scared her bad.
Dread. Skin-crawling, stomach-dropping, run-while-you-can dread.
It was ridiculous.
Alexa was on the cusp of having everything she'd ever dreamed about. A beautiful home she could be proud of, a secure job that she loved, a man who wanted to be with her every moment he wasn't working, and more money than she'd ever be able to spend. She wasn't greedy; that wasn't where her interest in money and a nice house came from. Instead, it came from the way she'd grown up. Her father leaving her and her brother with nothing but a seriously ill mother, how _little_ she'd had as a kid, how terrible the conditions in the trailer she'd grown up in had been—against all of that, it was simply amazing to think about how much she had now.
And hard to believe. A lot of the time, she was sure she didn't deserve it. And a part of her couldn't quite accept that it would last. Grant was Armani suits and Ivy-League education and million-dollar bank accounts, while Alexa was mall clearance racks and part-time evening classes and life lived paycheck to paycheck. At least, that had been her before they'd gotten serious. She didn't need Grant to tell her how lucky she was that he'd wanted her, although he did sometimes tell her just that.
Mostly, she was grateful beyond imagination. Grateful to be safe and secure. Grateful not to be ashamed and embarrassed of where she lived. Grateful to be able to afford to take care of her mom, who suffered from an array of mental health problems and needed all the help Alexa could give her, which had been more and more since her older brother, Tyler, died five years before. Alexa was grateful to Grant for making so much possible that she never would've been able to accomplish on her own.
Which made the dread seriously ridiculous.
It was just wedding jitters. Totally normal.
Sighing, she surveyed the beautiful dinner she'd managed to throw together. Given how scarce food had been when she was younger, Alexa absolutely hated to waste anything. Problem was, her appetite had been all over the place lately. Either she couldn't stomach the thought of eating or she was binge-eating a bag of potato chips while Grant was at work.
_Knock, knock_.
The quick raps on the front door pulled Alexa from her thoughts. She crossed the dining room to the wide oval foyer framed by a grand curving staircase. A glittering chandelier hung from the ceiling, casting colorful prisms here and there from where it caught the late-day sun through the large picture window above the door. Out on the front porch, Alexa found a stack of packages. She gave a wave to the UPS driver as he pulled out of the end of their driveway.
With just over two weeks until the wedding, presents from the registry had been pouring in every day. Grant had so many friends, colleagues, and contacts that she'd never met, Alexa didn't know who most of the gifts were from.
She carried in two smaller ones, then two medium ones, and then found herself struggling to move the large square box on the bottom. It was too deep to get her arms around and not easily pushed. What the heck could it be? She crouched behind it to try to gain leverage, and was just about to give up when a strong breeze blew her hair across her face, and she heard a soft click.
Her gaze cut to the front door.
"Oh, shit," she said. Knowing what she was going to find, she tried the knob anyway. Locked.
She was locked out, and Grant was away until who knew what time. She couldn't easily go anywhere because her purse, car keys, and phone were all inside. And she didn't know her neighbors yet because she'd just moved in.
"Shit, shit, _shit_."
So much for getting work done tonight.
She sat heavily on the stupid box and dropped her head into her hands. And burst into tears.
Not because of being locked out. But because being . . . trapped with no easy way out of the situation? Suddenly, that felt like a crazy accurate metaphor for her life.
If she was being honest with herself.
Which she really, really didn't want to be.
"Stop it, Al," she said in a rasping voice. "You're not trapped. Stop thinking that." Except, just then, she leaned her left cheek too heavily against her hand. She sucked in a breath at the smarting of the healing bruise there.
The one from the fight she and Grant had last week. The fight that had started with Alexa leaving a big mess in the foyer from where she'd been unboxing another delivery of packages and had escalated into a huge argument, culminating in Grant saying Alexa was just like her mother—something Grant knew cut her deep on so many levels. The fight had ended when Alexa told him he was being mean and he'd kicked a box at her. When she'd tried to duck out of the way, she tripped over another box on the floor and fell, hitting her head against the leg of a console table in the foyer, giving her some nasty bruises.
Alexa had been totally and absolutely stunned, especially when Grant hadn't helped her. Instead, a bitter, humorless laugh had spilled out of him and he'd said, "Way to prove my point, Alexa. I don't know why I put up with your shit. If you can't show me and my house a little respect, you can leave," he'd said, and then he'd stormed out.
_His_ house? Granted, she'd only moved in a few weeks before, but she'd moved in for keeps.
Shocked, she'd lain there for long minutes, completely confused and overwhelmed by the pain and his cruel words. So she'd done what he said. She'd fled. To her past. To Maverick Rylan, her dead brother's best friend, and the man who'd once been her closest friend and lover. It had been pure instinct to seek him out at the clubhouse of the Raven Riders Motorcycle Club. Despite everything that'd happened between them, no part of her had doubted that he'd help.
And he had. Or, at least, he'd tried.
But Maverick represented a past she'd left behind for a whole lot of very good, logical, and well-thought-out reasons. So she hadn't stayed. And she hadn't answered his questions.
By the time she'd finally returned home, all she could see were the million mistakes she'd made—making that mess, overreacting, going to Maverick and opening that door to the past that she'd kept closed tight for so long. For years. She'd been prepared to plead for Grant's forgiveness, sure he was going to be done with her once and for all. But he'd surprised her. Because Grant had apologized so profusely he'd gotten down on his knees, his head buried in her lap.
Never before in the nearly five years they'd been together had Grant ever hurt her. At least, not physically. He could be short with her and more controlling when he was stressed, and occasionally his criticism bordered on the mean side. But the truth was Alexa _could_ be messy, which _was_ why she'd tripped, and she could be disorganized and she could be forgetful, all things that drove him crazy. At the same time, Grant could also be generous and sweet and he'd done so much for her and her mother, things Alexa wouldn't have been able to do on her own. Their lives were better because of Grant Slater.
The night of their fight, things had just gotten out of hand, for both of them. And it was behind them now, so there was no point in dwelling—
"Alexa?" came a deep voice.
Prickles ran up her spine as she pulled herself from the bad memories and lifted her head—and found herself staring at her past, into the dark blue eyes of Maverick Rylan.
Alexa jumped up off the box, her heart suddenly in her throat. She swiped at the wetness on her face half sure she was imagining this man. This man she'd done her best to avoid—for years—until last week.
With his longish sandy-blond hair and his square jaw and his ruthlessly masculine features and his Raven Riders cutoff jacket hanging on those broad shoulders, Maverick was the sexiest man she'd ever known. Had been when they were together five years before, still was even now. No, he was hotter now. More muscular. More rugged somehow. More self-possessed. Utterly desirable.
"You okay?" he asked, stunningly dark blue eyes looking deep into hers.
_Snap out of it, Al!_ Right. Because clearly she wasn't hallucinating. And that . . . that was a problem.
Releasing a shaky breath, Alexa met his gaze head-on. She had to know. "Maverick, what in the world are you doing here?"
## CHAPTER 2
Alexa glanced up and down the quiet street with its manicured lawns and huge colonials and didn't see any cars that didn't belong—or any motorcycles. "Where did you even come from?"
Maverick's gaze narrowed on her face. "You're crying," he said, like that explained anything.
"No, I'm not," she said, brazening it out despite the wetness clinging to her eyelashes. "Seriously, why are you here?" She stopped short of saying he _shouldn't_ be there, because she suspected that would make him dig in his heels and want to stay. Even though he really shouldn't be there. Grant wouldn't like it. Oh, God, why had she gone to Mav for help?
Maverick's head tilted the smallest amount, like he was assessing her, or challenging her. "You know why I'm here, Al."
_Al._ No one in her life called her by that nickname anymore. Tyler had almost _always_ called her Al, which was where Maverick had picked it up. Of course, very few people from her _before Grant_ life were still around either. Somehow, her relationship with Grant and the work they did together had taken over everything until she'd all but lost touch with her friends.
"No, I don't." She shook her head not just in answer, but against the old longing she felt for Maverick. In high school, Tyler and Maverick had been thick as thieves, which meant despite their four-year age difference, Alexa had known Maverick long enough to have crushed on him forever, pretty much. It seemed like some part of her had _always_ yearned for him—and always would.
But that didn't mean there hadn't been problems, too. For a woman who'd grown up wanting nothing more than stability and respectability, a guy who built his life around a motorcycle club engaged in at least some questionable activities didn't seem most likely to offer that. And then Tyler had become a prospective member of the Raven Riders, following Maverick as he always did—until her brother had wiped out on a rainy mountain road one night and died on the way to the hospital. Mixed in with her soul-deep grief over his loss was a red-hot rage that his recklessness had made him leave her when she needed him so much. When their mother needed him.
Tyler's death had thrown a stark clarity on all the reasons that being with Maverick was problematic. She couldn't be with someone who lived such a dangerous life and couldn't offer her the security her mother required. Because Alexa was the only one left to take care of her. And it wasn't like Mav had ever talked about settling down, despite being together for three years. They'd just been having fun, hanging out. But mired in her grief and fear, she'd needed more than that. So she'd broken up with him. Just went cold turkey. Not that Maverick accepted that.
He'd pursued her. Hard. Dropping by her house, coming to her work, calling, texting. Trying to convince her to change her mind. Until he'd found her at Tyler's grave on what would've been his twenty-eighth birthday, and they'd had it out.
_"Why are you doing this, Alexa? Why are you pushing me away?" Maverick asked, the two of them standing on either side of Tyler's burial plot as if it were a wall between them._
_Alexa drew on everything she had to hold back her tears. "My mother is a wreck, and taking care of her is all on me. I have to focus on her now, and I have to do what's best for her. She needs stability, and I need to figure out how to give it to her, and I . . ." She peered up at him. "I need it, too, Maverick."_
_He grasped her hand. "So I'll help."_
_"But for how long?" she blurted. He frowned like he was trying to make sense of her question. And then her fears poured out of her. "How long until you wreck? Or some club business goes bad and you get hurt? Or you don't want me because—"_
_"Don't want you? Where the hell did that come from?" he asked, anger flashing in his eyes._
_Grief was like a lead blanket wrapped around her shoulders, heavy and suffocating. And it hadn't escaped her notice that he hadn't addressed her fears about staying safe. Because he couldn't. "Maverick, losing Tyler was the last thing I ever imagined happening. I can't handle anything else. I need certainty right now, permanence, security." Forever. She needed forever. Once, she'd hoped Maverick would want it, too. With her, someday. But now what he did and who he was scared her. Especially standing there at Tyler's grave._
_"And you don't think I can give that to you?" he asked, anger sharpening the angles of his ruggedly handsome face._
_"Can you?" she whispered._
The silence that had followed had been her answer.
So Alexa had made her choices long ago.
"Yes, you do. You know exactly why I'm here." Maverick stalked toward her, his big body bearing down on hers in a way that set her heart to racing. With a gentleness that seemed impossible, given his size and all his rough edges, he grasped her chin and turned her face so that the side that had been bruised was toward him. "You came to me." He nailed her with a stare.
Hugging herself, she pulled her chin free from the heat of his fingers. "I'm sorry I bothered you last week. I shouldn't have gone to the clubhouse." She'd just been so shocked by the fight with Grant and his words and his storming out that she hadn't known what else to do. Going to her mother's would've just made things worse, and she felt too out of touch with all of her old friends to go to any of them. Not to mention too embarrassed.
She'd felt so alone.
She still wasn't sure why she'd driven out to the Raven Riders' compound on the edge of Frederick. Once there, she'd worried she'd unnecessarily complicated her life a whole lot by mixing the past with her present and future. The longer she'd stayed at the clubhouse, the more certain she'd been that Grant really was done with her. Scared that she'd ruined everything, she'd left without explaining what'd happened to her, but not before Maverick had seen her face.
Now, him standing on the front porch of the house she shared with Grant? That was all kinds of complicated.
"If you think coming to me was a bother, maybe you never knew me at all," he said, the muscle in his jaw clenching. He stepped away, his gaze fierce.
His words set off a pang in her chest. She _did_ know him. His protectiveness, his possessiveness, his goodness—despite his club not being totally aboveboard. Although, from what she understood, their less-than-legal activities funded the Ravens' quiet mission to defend those who couldn't defend themselves . . .
Wait. Was _that_ why she'd gone out there?
The question was like a sucker punch to the gut. Alexa wasn't someone who couldn't defend herself. She wasn't in an abusive relationship like the women the Ravens helped. It had only been that one time and Grant had apologized over and over again. And she'd been the one who tripped, after all.
"Maybe I . . . maybe I didn't," she said, her voice not much more than a whisper as the thoughts settled uncomfortably deep inside her.
"Fine. Have it your way," he said, his voice like gravel. Just when she thought he would leave, he stepped up to her front door and pulled something from his back pocket.
"What are you doing?" she asked, moving to his side.
"Letting you back into your house." He worked a silver tool that looked like a pocketknife at the door handle.
"Are you picking my lock?" She watched in fascination as he quickly manipulated the tools and turned the knob, opening her front door.
"I'm letting you back into your house." He flipped the little tool set closed and slipped it into his pocket.
"You carry lock picks on you?" she asked. Her night had gone from stressful to surreal.
"Yes. You're welcome." His right eyebrow arched, just the littlest bit. And it was so damn sexy. Her blood heated, her nipples hardened, her pulse raced. She and Maverick had always been good at fighting. And even better at making up afterward. Her body clearly remembered . . . and hungered.
She blinked as he pushed open the door. "Um. Okay. Thank you."
He stepped away from the opening, and Alexa slipped into the spot where he'd stood, the word _Good-bye_ on the tip of her tongue.
But he still didn't leave. He heaved the box off the porch and turned to her, the muscles of his biceps and in his neck popping out.
"You don't have to do that," she said, even as she backed into the house to get out of his way. As he passed her, she noticed a band of black fabric tied around one of his arms. She nearly gasped as memories again sucked her five years into the past. To Tyler's funeral. He'd been a prospective member of the Ravens, and they'd honored him by wearing strips of some of his cut-up T-shirts around their arms. Who'd died now?
Mav lugged the box inside. "Where do you want it?" he asked, pulling her from the memories.
"Maverick—"
"This thing is really fucking heavy, Al." He stared at her expectantly.
"Right around the corner in that room," she said. Her office had become the holding area for all the wedding gifts as they came in. Which suddenly made her uncomfortable about having sent Maverick in there. She followed him in and watched him settle the unwieldy box next to the long credenza filled with unwrapped gifts. China and picture frames and vases and luxurious linens, to name a few. "Thank you."
His gaze surveyed the wedding gifts, but he didn't make a comment, and she couldn't read his expression. At least, not until he looked at the project boards hanging over her desk, one filled with plans for the model home job and another filled with her work on a design project for her senior capstone. Then his expression became interested. "That for school or a job you're doing?" he asked, eyes still on her work.
"Both." What did he see when he looked at her design ideas? Maverick had always had an eye for design and an appreciation for aesthetics—it was one of the things that made him such a talented and sought-after custom bike builder.
He nodded. "Pretty cool," he said in a quiet voice. Before she'd started her interior design program, he'd been a big supporter of her going to school and pursuing her dreams. Now she was only her current course and one internship away from finishing after all these years of working full-time and pursuing her degree and accreditation part-time.
The sincerity of his words and in his expression did funny things to her chest. "Yeah?"
His gaze cut to her. "Yeah."
They stood looking at one another a long moment until Alexa's heart was thundering against her breastbone with an unnamed need. She shoved the desire back. She'd made her choice for a man and a life—and it hadn't been with Maverick. But being in his presence messed with her head . . . and her body. Clearly. Which was why she'd stayed away . . .
_God. I should've stayed away._
"Your arm," she said, staring at that black band and trying to distract herself from the way he made her feel. Still.
Maverick frowned, and for just a second his true feelings reached the surface of his eyes. Grief. Anguish. Rage. "A friend," he said.
"I'm sorry." When he didn't say anything more, Alexa hugged herself. Despite their long estrangement, she hated seeing him in pain, but it wasn't like she could comfort him either. Finally, something occurred to her. "Hey, how did you know I was locked out? And that I couldn't move the box?" Voicing the question gave rise to goose bumps on her arms.
He just looked at her, that little eyebrow arch making itself known again.
"No, Maverick." She shook her head as realization set in. "Oh my God. You can't do that. You can't be . . . hanging around and watching out for me."
"Someone has to—"
"Grant. Grant watches out for me," she said, the words falling uncomfortably from her tongue.
"Is that what he was doing when that happened?" He nodded toward her, his gaze locked on her face, and she knew he was talking about the bruise on her cheek. What wasn't faded was fairly well covered with makeup, but they both knew it was there.
"Maverick—"
"Look, I don't want to fight," he said, closing the distance between them.
He stood so close she had to tilt her head back to keep eye contact. So close that all she'd have to do was reach out her hand and he'd be hers again. The thought came entirely out of left field and nearly made her dizzy. She hadn't thought of Maverick that way in years.
What was wrong with her? She was over him. Had been for a long time. For a lifetime.
Shaking her head, she managed, "Good. Me, neither." But that didn't mean she wouldn't find a way to get him to stop watching out for her. If Grant found out, he'd know she'd gone to Maverick or suspect she was still seeing Maverick, and neither would be good for her. Or Maverick.
The universe of strings Grant could pull—or have pulled—was bigger and scarier than she liked to think about. She didn't know many specifics because he shielded her from them, but she'd heard rumors and rumblings. Payoffs. Threats. Intimidation. Though maybe none of that was true. Maybe such things were rumors spread by competitors trying to take Grant down a notch, just like he said when she once asked about the things she'd heard.
Looking down at the space between them, Maverick planted his hands on his hips. "Damnit Alexa, you deserve better—"
She gave in to the dangerous urge to touch him and laid a hand on his chest. She wasn't sure what else he'd planned to say, but suddenly she couldn't bear for him to finish that sentence. With a quick shake of her head, she said, "Don't, Maverick. Okay? If you say anymore, we'll just end up fighting."
Maverick huffed, his gaze absolutely on fire as it raked over her face and settled on her lips. He leaned toward her, just the littlest bit, and her heart lurched into her throat. She was suddenly sure he was going to kiss her. And, _oh, God_ , she wanted him to. She should push him away, say no, but she didn't know how to say no to him, never had, and especially not with five years of suppressed need roaring through her. She licked her lips, her breath catching, her mouth hungry for a taste . . .
_Meow._
Lucy wound her body around Alexa's legs, and Alexa jumped back, breaking the spell that had wrapped around them. God, what was she thinking? Needing a shield, Alexa scooped the hairless cat into her arms and hugged her tight.
Mav eyeballed the cat and made a face. "What the hell is that?" he asked, more than a little amusement in his voice. He stepped back from her, like maybe he needed the space from whatever had just happened between them.
"It's Lucy," Alexa said, kissing her soft head.
"But . . . what is it? Or . . . happened to it?"
Alexa rolled her eyes. So her cat was bald. And had wrinkly skin. And was wearing an argyle sweater. And had the world's disproportionately biggest ears. "It's a sphynx cat. They're hairless. And they're awesome. Coolest cat you will ever know."
"It's really—"
"Don't you dare say she's ugly." Alexa nailed him with a stare. The house she'd grown up in had been so cluttered and filthy that Alexa had developed terrible allergies and asthma, so hairless cats were the only kind she could have. She'd adopted Lucy from an animal shelter less than a month after Tyler died.
Maverick chuffed out a small laugh. "So ugly she's cute?"
Shaking her head, Alexa held back the smile that threatened. "I suppose that's acceptable. Just."
Giving Alexa a wink, Maverick held out his hand to the cat's nose. Alexa braced for Lucy to freak out and bolt, but though her little muscles did tense, she merely sniffed at Maverick and then turned away.
Whoa. Grant couldn't get within ten feet of Lucy. "Wow, she likes you."
"Of course she likes me. I'm awesome." He folded his arms, his expression full of challenge and humor.
Joking around with him felt all too comfortable, easy, relaxed. Which was really, really not good.
Alexa gave Lucy another kiss and turned away. Maverick needed to go—and he needed to stay gone—so that whatever had just nearly happened between them didn't happen again. "Well, thanks for your help," she said. "I'll walk you out." In the foyer, she dropped Lucy to the floor and opened the door.
All the easy affability faded from Maverick's expression, and the hard-ass biker was back in its place. "I'm worried about you, Alexa."
She ignored the pang in her chest caused by his concern. "There's no reason to be."
"Think Tyler would agree with that?" Maverick's gaze was filled with as much skepticism as his tone.
"Tyler would believe me," she said, knowing deep down that her brother would've flipped his shit if he'd seen her face. But he wasn't here now, was he? And she had responsibilities she had to take care of on her own. Still, Maverick's question set off an ache inside her—an ache for what used to be, and maybe even for what might've been. "Because I'm good. No, I'm great."
She would have to do better if she was going to keep Maverick from coming around.
Her stomach knotted as words came to mind, and then she let them fly. "But if you hang around and Grant finds out, you're going to mess things up for me. So please, don't come here again. I made a mistake coming to you last week, and I'm sorry. It won't happen again either."
Now Maverick's expression was rankly pissed off. He stared at her a long moment, like he was debating how to respond. "Fine," he finally said, his voice like it had been scoured with sandpaper. "Have a good life, Alexa. Hope it's what you really want." He wrenched open the door and stalked out.
The silence seemed loud all around her. She quietly closed and locked the door, then turned and leaned her back against it. "Me, too, Maverick. Me, too."
## CHAPTER 3
Sitting under the blue tent, his brothers gathered all around, Maverick couldn't hear the pastor's words. Mav's head was too full of churn and burn. Over the loss of one of their own. Over the attack on his club just last week. Over confronting Alexa an hour before. Damnit, why had he gone over there when coming to this funeral already had him so torn up?
_You know why. Something's not right. She's not safe._
The bruises she'd had—that she _still_ had—proved that. Whether she wanted to admit it or not. And he couldn't let it go, not when he might be able to do something about it.
Heaving a deep breath, he stared at the flower-draped casket and hoped this was the last funeral he and his brothers had to attend for a long damn time. This loss hit him and everyone in the Raven Riders Motorcycle Club particularly hard—because it had happened on their own turf. Inside their own clubhouse. And the victim—prospective member Jeb Fowler—had been too young and too good to get taken out in cold blood.
The proof of Jeb's goodness was sitting right beside Maverick. Bunny McKeon, Maverick's mother, whose face still bore the bruises and scratches from where she'd been struck and her mouth duct-taped. But she was alive because Jeb had taken the bullet intended for her when a group of low-life criminals had broken into the clubhouse looking for something—or _someone_ —that wasn't theirs.
Mav's gaze slid over to his right, where the club's president, Dare Kenyon, sat with his girlfriend, Haven. Three days out of the hospital, Dare's face was pale, and dark circles ringed his eyes. He'd been shot twice in the same attack that had hurt Bunny and killed Jeb. An attack by Haven's abusive father meant to kidnap her and force her back under his control by whatever means necessary. An attack that had ultimately failed, despite the losses the Ravens had sustained.
It all could've been so much worse.
A sentiment that had his thoughts returning to Alexa. Last week it had been a few bruises, a busted lip. What would happen the next time her bastard of a fiancé "took care" of her? That very question was why Mav had been shadowing her ever since she'd shown up bloodied and frightened at the clubhouse. Which was how he'd known she'd locked herself out. Seeing her looking so defeated had reached inside his chest and hauled him to her side. And seeing her cry made him not care that he'd just blown his cover. He couldn't have stayed away at that point if he'd tried.
The service concluded and people around Maverick rose to their feet. Despite the evening hour, the June air hung humid and gray. Almost oppressive. The weight of it was fitting. It was as if their collective grief had taken on a physical form.
Maverick held out his hand to Bunny. With her white-blond hair and dark blue eyes, she was still as pretty as she'd ever been. And every bit as feisty. Well, usually.
Accepting his help, she gave him a sad smile and rose. Her husband, Rodeo McKeon, steadied her from her other side.
"Thanks, Maverick," she said, stepping toward the casket. She pulled a long-stem red rose out of the arrangement and brought it to her nose. A moment later, she laid it on top of the lid by itself, her hand resting there for a moment.
When she turned away, her lips trembled, and when she made eye contact with Maverick, her whole expression crumpled.
Maverick pulled her into his arms, her tears like ice in his veins. "It's gonna be okay, Mom," he said, uttering a name for her he hadn't used regularly in years. _Everybody_ called his mother Bunny, and somewhere along the way it had stuck for him, too.
From behind her, Rodeo rubbed her back. Mav met the older man's gaze and saw reflected at him the same pain and regret Maverick felt. Normally, Bunny was the youngest sixty-something you'd ever meet, but the attack and Jeb's death had left her fragile. And Maverick fucking hated it. Not because he thought her shakiness wasn't warranted, but because it reminded him of another time. When Bunny's first husband—Maverick's father—had finally beaten her so badly she ended up in the hospital for days.
That had been seventeen years before when Maverick had just been a kid in high school, but not a day had gone by when he hadn't blamed himself for not realizing what was going on, not being there, not protecting her. _Oh, I just fell, hon. I just tripped/bumped my head/slipped in the shower_. He'd believed every lie, swallowed every excuse.
And damn if Alexa wasn't giving him déjà vu.
Sonofabitch.
Maverick's gaze slid over Rodeo's shoulder to the casket. _I will find a way to avenge you, Jeb. And you can believe I'll never forget_.
"I'm okay," Bunny said, wiping at her cheeks. She patted his chest. "Thanks, hon."
Rodeo gave Mav a nod that said he had her, and Maverick didn't doubt it. Not only was Rodeo one of his brothers in the Raven Riders, he was also the best thing to ever happen to Bunny.
"Are you coming to the clubhouse?" Maverick asked, trying like hell to keep the rage out of his voice. Bunny hadn't been back since the attack the week before, which was totally unlike her. Normally, she spent part of every day there, often cooking one or more meals for whichever Ravens happened to be around or drop by.
"Yeah," she said. "We'll see you there."
"Okay," he said. With a last look at Jeb's casket, Maverick turned for the drive that wound through the rolling hills of the cemetery. Motorcycles formed an unbroken wall of steel and chrome almost forty deep. The whole club had turned out to pay its respects. As it should be when a brother took his last ride.
As vice president, Maverick's bike was at the front. Normally, he'd be riding third position behind Road Captain Phoenix Creed and Dare, but the gunshots to Dare's side and arm meant that he'd only be driving four-wheel vehicles for the immediate future. So Maverick was second in line. Still standing, he brought the bike to life on a low growl. One by one, all the Ravens' bikes rumbled until the cemetery nearly vibrated with the sound. Turning to the brother beside him, Maverick waited for Phoenix's command.
Like the black bands they wore on their arms—made of thin strips torn from a couple of Jeb's Harley T-shirts—they had traditions they honored when one of their own died.
A few years younger than Maverick's almost thirty-five, Phoenix normally wore a mischievous, good-humored expression. Not today. Not when they were burying one of Phoenix's closest friends just a month after Phoenix had buried his cousin, too. Their road captain had taken a beating the past few weeks, and it showed in Phoenix's unusual frown and his lack of joking around.
When everyone else had started their engines, Phoenix finally started his own. Then he turned his throttle and revved his engine until it roared.
Every biker except one joined in.
_Roar, roar, roar, roar, roar_.
The five thunderous revs lodged a knot in Maverick's throat. Because the Last Rev was meant to alert heaven that a biker was on his way home.
And then all the bikes quieted to a low idle—except one. The one that had remained silent before now roared out. Ike Young, the Tail Gunner of the procession, revved his engine five times, as if Jeb was answering the club's call and saying his good-byes. One last time.
When the Last Rev ended, everyone mounted their bikes and the procession got underway.
Quietly and slowly, they made their way home—back to the Ravens' compound on the outskirts of Frederick, Maryland. Maverick knew he had to at least make an appearance at the reception, though his gut had him wanting to head back to Alexa's. Because instinct said the worst was yet to come, and experience had taught him that abusers didn't abuse just once.
Her brief reappearance into his life had reminded him of promises he hadn't been well tending and triggered every one of his protective instincts, not to mention stirring up all kinds of shit inside him he thought he'd boxed up tight.
And as if Maverick hadn't already been climbing out of his skin with worry over Alexa—whether she wanted him worrying about her or not—Bunny's attack whipped up all the old guilt inside him and made him _need_ to know that Alexa was okay.
Or, if need be, _ensure_ that she would be okay. Whatever that took.
After all, once, he and Alexa had been tight—not just lovers, but friends. Alexa had gone from the kid sister who clearly worshipped her big brother and his best friend, to the teenager with the smart mouth and the quick wit who loved to keep him and Tyler guessing what she'd get into next, to the ambitious woman who worked two jobs to put herself through school while bearing most of the burden of taking care of her sick mother. She was giving and tough and had always made him laugh and smile—even when shit at home had made that damn difficult. All of that _plus_ the fact that they were together as a couple for three years before it all fell apart, and Maverick couldn't let what'd happened to her go unaddressed.
Because she was the first and only woman he'd ever loved. The fact that she'd broken up with him and moved on without him? That didn't matter if her life was on the line.
At the very least, Maverick owed it to Tyler Harmon to take care of his little sister. After all, it wasn't like Tyler could do that job himself. And Mav had once promised Tyler he'd always look out for Alexa.
But even more than that, Maverick needed to watch out for Alexa because he'd once failed his mother when she'd been in a similar situation, and that failure ate at him a little bit every day, like a slow dripping leak of acid deep inside his veins. Even all these years later.
Then, Maverick had been young and naïve and weak. He hadn't realized all the kinds of evil that lurked in the world. But that wasn't him anymore. Now he knew. And he refused to ever make that same mistake again.
"WHAT THE HELL'S the matter with you?" Dare asked as he joined Maverick for a drink.
Maverick had been sitting at the far end of the bar in the Ravens' clubhouse for the past few hours, in the midst of the party celebrating Jeb's life but not really a part of it. The driving bass beat of a rock song blared out all around him, and people laughed and joked, but Maverick barely heard anything.
"Nothing," Mav said, staring down at his whiskey.
"I excel at moody motherfucker, remember? I know it when I see it." Dare wasn't just the club president, he was also Maverick's cousin, though for years they'd been as tight as brothers. Calling each other out on their shit went with the territory.
"The past is a pain in the ass, that's all," Maverick said.
"Well, that's the goddamned truth." Dare grimaced as he slid onto a bar stool and flagged down Blake for a drink. Jeb had been his best friend, and the prospect served up the whiskey with bleakness in his eyes. Prospects were probationary members who wore the club's cutoff jackets but without the club's colors and patch on them—those had to be earned through loyalty and dedication to the club and were a privilege of full membership.
"How you feeling?" Mav asked his cousin when Blake stepped away. Dare was only five days out from multiple gunshot wounds and surgery, but he'd thrown himself back into club business as quickly as he could. It showed in the darker-than-normal rings around his brown eyes.
"I'm feeling like I might kill the next sonofabitch who asks me how I'm feeling." Dare raked his dark hair back off his face and gave Mav a look.
And Maverick got it. He did. But that didn't mean he wasn't going to try to shoulder as much of the burden around here as he could until Dare was on his feet again. Which was why he'd already pushed off the delivery of his current custom bike order by two weeks. He'd offered the customer a twenty percent discount for his inconvenience, and that'd smoothed over any hard feelings. And Mav had enough cred built up in the business to take the hit. "Yeah, well, I won't ask how you're feeling if you don't ask me what bug crawled up my ass." He smirked and tilted out his glass.
Dare chuffed out a laugh and clinked. "I'll drink to that."
The hot bite of the whiskey felt good against the back of Maverick's throat. A couple more of these and maybe he could convince himself he didn't care about Alexa Harmon anymore. About the fact that her slick scumbag of a fiancé had hurt her. About the fact that she'd so easily dismissed Maverick earlier. About the fact that his fucking blood had been on fire in her presence. Just like old times.
Exactly. Just like old times, she'd made her choice today. And it wasn't him. Not that he'd expected anything else.
"I'd like you to call a special meeting of Church as fast as you can get it pulled together," Dare said, yanking Maverick from his thoughts. Church referred to the club's monthly membership meetings, though recent crises had them meeting more often than usual. "After the shooting at the track, we need to talk damage control where the races are concerned. And we need a plan to deal with the Iron Cross once and for all."
Maverick nodded. "Yeah, we do. A plan that involves burning their fucking world to the ground." Stock-car racing and betting were their biggest businesses, and part of the attack on the club last Friday night had happened at the Green Valley Race Track that they operated on the edge of the Ravens' huge tract of land. The Iron Cross was an up-and-coming gang in nearby Baltimore that was trying to take over the city in the wake of the recent destruction of what had been Baltimore's most powerful gang. And it was more than a little likely that they'd had a hand in helping Haven's father and his men attack the Ravens. That shit couldn't go unaddressed, despite the fact that the Iron Cross denied being involved.
What the fuck else were they going to say?
"I hear you, Maverick. I only regret that I was laid up in the damn hospital and couldn't act sooner. We're not letting this go. Trust me." Dare's gaze was ice cold.
"Not your fault, D," Maverick said.
The expression on his cousin's face said he didn't agree.
Maverick cleared his throat. "Since we're on the same page, I'll put in a call to Nick Rixey in the morning to see what else he and his men might've gathered on the Iron Cross's involvement." Nick was the leader of a team of former Army Special Ops guys who were opening a security services firm in the city. The Ravens and Nick's men had worked together on several occasions now and had become tight allies.
"Good. Do that," Dare said, throwing back a gulp of whiskey. "Because whether the Iron Cross actually told Haven's father where she was or not, they tried to blackmail us to keep her whereabouts quiet, and in doing so they risked her life. You threaten one of us, you threaten all of us. That's all I need to know."
Maverick nodded, the other man's words reflecting so much about who the Ravens were and what they stood for—family, brotherhood, loyalty. They didn't usually go on the offensive, but they sure as hell defended their own. On top of that, Dare had a protective streak a mile wide, one he'd come by painfully when his father had killed his mother and brother twenty-plus years before. Add to that what Mav's own father had done to Bunny, and it explained why the club was in the business of standing up for people who couldn't do it for themselves. But Maverick had never seen Dare as fiercely protective as he was of Haven Randall. Then again, Maverick had never seen Dare in a serious relationship before, either.
And Maverick was happy for him. He really was. The guy deserved a little slice of happiness, and after everything she'd been through, so did Haven.
Which had Maverick thinking about Alexa again, and about the fact that she deserved happiness, too. Despite her comfortable circumstances now, Alexa had grown up poor and with a mother who had issues—issues that had gotten worse when Tyler died five years ago. And Maverick felt at least some responsibility for that since Tyler had picked up his love of motorcycles from Maverick. Hell, Tyler had become a prospective member of the club at Maverick's encouragement.
On some level, Mav couldn't help but wonder if Alexa blamed him for Tyler's death. Maverick had always wondered if that hadn't been part of her reason for dumping his ass. He wouldn't blame her if she did feel that way, because even if he hadn't been responsible for Tyler's death, Maverick had sure as hell played a role in setting that particular chain of events into motion. A rock dropped into his gut as the old guilt dug its claws into his skin.
He was damn near an expert at failing those he cared about, wasn't he?
"Hey," Dare said, grimacing as he shifted on the stool. "Did you ever get around to informing Mike Renner that you were sponsoring his prospective membership?"
"Fuck," Maverick said. "No, but I'll call him in the morning." Mike Renner was a Hang-Around, or friend of the club, that the Ravens had approved for prospect status at one of their last meetings. "I know he's eager."
"Eager's good," Dare said with a smirk.
Chuckling, Maverick nodded. "Yeah." Prospects had to put up with a lot of bullshit from fully patched members and did a lot of the club's grunt work. It could be a pain in the ass sometimes, but the process helped determine if the guy was a good fit for the club. And they'd all been there.
Jeb had been a damn good prospect. Everything had rolled right off that guy's back. He worked hard, showed up, pitched in, and did it all with a good attitude and a fucking smile. He would've made a great addition to the club. Maverick looked at Dare as an idea came to mind.
"What?" Dare asked.
"D, we should make Jeb a fully patched member," Maverick said. "Like, as an honor. In protecting Bunny's life he more than earned it."
Dare's expression was thoughtful, and he nodded. "A posthumous award. That's a damn good idea, Maverick."
"Yeah. I have them every once in a while," he said, though the humor didn't quite reach his voice. Granting Jeb full membership didn't make up for much, but it was something, and it would be one more way to make sure Jeb—and what he'd sacrificed—was never forgotten.
"I'll put it to a vote at the meeting," Dare said.
"Hey," Haven said, coming to stand alongside Dare. Though she'd come out of her shell a lot since first arriving at the Ravens' compound over a month before, she was still soft-spoken and a little shy. But quiet didn't mean spineless, not by a long shot. Maverick could easily picture her from the night of the attack on the club—crouched over Dare's unconscious body, gun drawn against anyone else who might try to hurt him. She'd killed her own father to keep him from killing Dare and kidnapping her. She'd already been well liked by the club because she was a fucking amazing cook, but now she was damn near revered.
Dare's whole demeanor changed as he turned to her. Smiling, he sat up straighter, and then he slid his hand into Haven's light brown hair and pulled her in for a kiss.
As her arms slid around his cousin's neck, Maverick found himself needing to look away. Not because he was embarrassed by the display of public affection—that was kinda par for the course around here, especially on party nights. Instead, his discomfort came from the goddamned jealousy slinking through his blood.
Maverick wanted someone who looked at him with the kind of unconditional love he read in Haven's blue eyes when she looked at Dare. And he wanted the soul-deep solace he could see on Dare's face when the guy was in Haven's presence. Maverick wanted to fucking _belong_ to someone. Well, to someone who wanted to belong to him, too. Fact was, someone had owned him for years.
Alexa fucking Harmon.
"Can I ask you a question, Haven?" Maverick said, swirling the last of his whiskey in the bottom of his glass.
"Sure," she said.
Mav debated exactly what he wanted to ask and finally found the words. "If you thought someone was in trouble, in a relationship that was maybe abusive, but that person told you to keep your nose out of it, would you?" No doubt Haven would realize he was asking about Alexa since Haven had helped clean her up after Alexa had come to the clubhouse last week. Remembering the blood and bruises on Al's face still made Maverick's chest hot with rage.
"I can't speak for everyone," she finally said. "But sometimes people tell you to stay out of it because they're afraid of how hard it will be to get out of the situation. Or they're afraid that having asked for help will get back to the abuser. Or they believe that they're not worth the risk that helping them might pose to someone else. I know I felt all of those."
Dare's jaw ticked as he listened. He slid his arm around Haven's shoulders, the gesture full of reassurance.
After a moment, Dare's dark eyes cut his way. "Question is, are you okay backing off if there's a chance she's feeling even a little of that?" Dare didn't need to clarify who he was talking about. Despite how hard Maverick had worked to keep that shit buttoned up, his cousin knew what Maverick felt for Alexa.
And the answer was a total gut check. _Absofuckinglutely not._
What the fuck was he doing sitting there? Wallowing. Instead of doing what his instincts had been telling him for over a week to do—protect Alexa. Or, at least, be ready to do so if something else happened. That's why he'd started surveilling her after she'd left the clubhouse that day, insisting what'd happened had been a misunderstanding but refusing to actually explain what kind of misunderstanding led to a banged-up face.
Maverick pushed away the not-quite-empty glass of whiskey and slid off his stool. He needed to send out a meeting alert to the brotherhood, and then he needed to get his ass to Alexa's house. He clapped Dare on his good shoulder and pressed a quick kiss to Haven's temple. "Thank you," he said to her. She gave him a shy smile as pink filtered into her cheeks.
"Hey, where's my kiss, asshole?" Dare asked.
Maverick threw a look over his shoulder and flipped Dare the bird, but the approval he saw in the other man's gaze told him he was doing the right thing.
Because Dare was right—if there was any chance that Alexa was pushing him away out of fear, Maverick couldn't stay away. She could be pissed at him. Hell, she could hate him. As long as she was safe and sound, he could stand just about anything else.
## CHAPTER 4
The air was tense and weighted as everyone filed into the Ravens' meeting room at seven the next evening for Church. Maverick took the seat next to Dare's at the head of the long rectangular table. The room had probably been a lounge back during the inn's heyday, which explained the cabin-like floor-to-ceiling stone fireplace and the exposed wooden beams on the ceiling. The Raven's logo hung on a carved panel of wood above the mantel.
_Ride. Fight. Defend._
Dare banged an old, beat-up gavel against the table and eased into his seat. "I'm calling this meeting of Church to order." Everyone settled into chairs at the table or along the side walls. They had a decent-sized crowd here today.
Clearly, Maverick wasn't the only one who wanted revenge for the attack on their people and property.
"Dare, it's good to see you and Meat here and in one piece," Bear Lowry said. Maverick nodded in agreement with the Old Timer who also was one of the club's founders. Bear had served as the secretary/treasurer for years because Dare and Doc trusted him implicitly, not to mention he had a damn good head on his shoulder for investments.
The guys on either side of Meat, whose real name was Craig Miles, clapped him on the shoulder, and he gave a nod. The night of the attack, Haven's father had shot Meat point-blank in the abdomen to prove that he was serious about his intentions to take out innocents for every ten minutes that passed without the club delivering her up.
And the Iron Cross had been the ones to send that asshole their way.
Dare shifted uncomfortably in his chair, whether from his injuries or his unease at the attention, Maverick didn't know. Raking a hand through his brown hair, Dare looked down at the table, his brow furrowed. "What we faced at the track that night . . ." He shook his head. "It was unprecedented. And what you did"—he looked up, his dark gaze intense—"what _all_ of you did was above and beyond. It—"
"No, D," Meat said, looking the club's president in the eyes. "It was _exactly_ what we should've done. Exactly what we should _always_ do. Nothing more."
The visible emotion on Dare's face reached inside Maverick's chest. Because he couldn't agree with Meat more. Maverick only wished that he could've gotten to Dare to shield him from those bullets, or at the very least gotten to Haven before she'd been forced to kill her own father to protect them both. And God knew he wished he could've kept his mother from getting hurt. Mav knew that his cousin was shouldering a shitload of guilt over all the people who'd been hurt that night. They were alike that way. Not that Maverick thought any of it was Dare's fault, because it wasn't.
Only two groups were to blame. Haven's father's crew, who were all either dead or in jail. And the Iron Cross.
Sitting at the far end of the table, Doc nodded. "Meat's got it right. This is a family. And every man here is your brother. You can _always_ count on us having your back. And that goes for everyone here. That's what the Raven Riders are about. That's why we exist." Frank "Doc" Kenyon was Dare's grandfather, the club's founder, and co-owner with Dare of the Ravens' property, which the older man had inherited decades before. Doc was also Mav's uncle. With whitish hair and beard that made him a shoo-in for playing Santa for the kids among the club members' families, Doc was fiercely protective of the club and everyone in it.
So Dare came by that naturally. Mav did, too.
"That's right," Maverick said. "Now we focus on getting justice for Jeb and for everyone else who got hurt." Murmurs of agreement circled the room. And even though his focus needed to be right here on these men, he couldn't help but think of Alexa. Because she'd been hurt, too. And Maverick hadn't yet thought of a way to make that right for her. It was eating at him like an itch he couldn't reach.
"Which brings me to some news from Baltimore," Dare said, opening the folder of intel Maverick had received from Nick Rixey. "Off the record, Nick shared that the Feds are running some kind of undercover operation targeting Baltimore's gangs. Ongoing investigations into narcotics and especially heroin. Long story short, in the week since everything went down, the Feds have tightened the noose around the Iron Cross as the primary player in that market, so it's very possible the Feds are about to take them out all on their own."
"Well, that's . . . kinda fucking unsatisfying," Phoenix said.
"'Possible' isn't good enough for me," Maverick said. "And letting the Feds do it does nothing for avenging ourselves and defending our reputation."
"I agree on both counts," Dare said.
Doc sat forward in his seat. "Given everything this club's been through the past six weeks, first with the fight over at Hard Ink and then with the attack here, we do not need more good people put in harm's way." He'd been leery of getting involved with the Iron Cross in the first place. Well, they all had. Maverick sure knew he'd been. But those motherfuckers hadn't given them much choice if they didn't want to make an enemy of a new player in the region. Turned out they had anyway. Which pretty much proved that the road to hell was paved with good intentions. "I get that the Iron Cross is a problem that has to be resolved, but if someone else is willing to do the dirty work, I say we let them."
"All due respect, Doc," Maverick said, the anger inside of him hot and thick like the June night outside. "We sit back, even if the Feds do their job here, we pretty much invite every other fucker who wants to make a name for themselves to do it on our backs. They invaded our territory. Attacked our home. Hurt and killed our people."
"I agree with Maverick," Caine McKannon said, his ice-blue eyes slicing toward Dare. Low murmurs of agreement rumbled around the room. Wearing a black knit cap over black hair and with black gauges in both his ears, Caine served as the club's sergeant-at-arms, which put him in charge of rule enforcement. The guy radiated a quiet intensity you didn't want to cross. Maverick had _never_ heard the guy raise his voice, and could only imagine what kind of pissed off he'd have to be to do so. Of the six men who made up the club's board of directors—Dare, Maverick, Bear, Phoenix, their Race Captain Jagger Locke, and Caine—Caine was by far the one Maverick knew the least, despite having known him for _years_.
"So, what are we talking about here?" Phoenix asked, face set into an unusually serious frown.
Considering glances were traded around the table until Dare finally spoke. "Here's what I suggest. We're not typically into offensive assaults. But we have to assume that the Iron Cross will keep coming at us, because they made it clear they want to push us around until they push us out as a player in the power dynamics in this area. In my view, going after them would be a defensive preemptive strike." More murmurs of agreement. "I say we make a statement that hurts them where they live—we attack their clubhouse, clean them out until they're left with nothing, and then burn their world to the ground. And if that sonofabitch of a leader Dominic gets in the way, so much the better."
"And how do you think Haven will feel about you doing that?" Doc asked.
Dare's dark gaze cut to meet his grandfather's. "This is _club_ business, Doc. But if you really want to know, I told her this was the likely reality. Don't think for a minute she doesn't understand our world after the one she grew up in, nor that this is still her father's mess that needs to be cleaned up. She'd be on the back of a Harley helping us do it if she could. Haven Randall is a hundred percent behind every one of us. Always."
Maverick surveyed his brothers' faces and saw the bone-deep respect they held for Haven. By saving Dare's life, she'd more than earned it.
"So we're going after property, not bodies," Phoenix said, eyebrow arched.
"That's what I'm proposing." Dare looked from one member to the next. "Discussion?"
"It may not be a long-term solution," Caine said, eyes narrowed.
The concern echoed Maverick's own thoughts; then something else occurred to him. "But think about this. If the Feds take out the Iron Cross after we take them down, then maybe it's the best of both worlds. They're out of the picture _and_ we protect our rep and make it clear that fucking with us comes at a cost. A big one."
"Exactly," Dare said with a nod, his whole face set in a frown. "Anyone else?" When the room remained silent, Dare heaved a deep breath. "Let's put it to a vote. Raise your hand if you support going after the Iron Cross." His gaze sliced through each and every one of them, because they all knew he was asking them to cross some lines. But sometimes, that shit couldn't be helped.
"Goddamnit," Doc bit out, fisting his hands on the table.
Nearly every other hand went up. Dark satisfaction rolled through Maverick's gut. Maybe that made him a terrible person. He didn't know, but so be it. Because he'd seen two good friends shot and hospitalized, another murdered, and his own mother bound and gagged and stuffed in the back of a car. And that was all besides the harm done to the club's reputation and livelihood.
Dare let the weight of their decision hang in the air for a long moment before he spoke again. "We need to be the ones to do this, but I want the Hard Ink team to help strategize, especially since we're using intel they collected. We good with that?" Nods all around. Dare placed a call and let the ringing sound out on speakerphone.
Nick's voice mail answered, and Dare left a message. "Nick, I need your help ASAP. Gimme a call."
"Fuck, I hate waiting," Jagger said, his fingers moving in a perpetual progression of invisible guitar chords against the table top. Their race captain was brilliant with his hands—musical instructions, engines, it didn't matter—and was nearly single-handedly responsible for every good thing that happened down at the racetrack. A big fucking deal since that provided the club's main income stream.
"Well, we have other business to pass the time," Dare said. "Maverick came up with something else we can do to honor Jeb. Mav?"
Nodding, Mav glanced around at all his brothers. "I move that we posthumously vote Jeb Fowler in as a fully patched member. He gave his life for my mother, for Rodeo's wife. He more than earned his place among us."
The group's discussion was quick and in full agreement, and then Dare put it to a vote. "All in favor?" Dare asked.
Without a word, Maverick rose to his feet. Then Rodeo did. Then Meat and Bear and all the others until every man in the room stood in honor of their fallen brother. A knot lodged in Mav's throat—pride that these men were his family.
Dare stood, too. "It's unanimous. Jeb Fowler is a fully patched member of the Raven Riders Motorcycle Club."
Silence hung thick and laden with emotion for a long moment, because the words highlighted the fucking tragedy of it all, and then Bear's quiet voice broke the tension. "I'll get his picture up," he said, referring to the wall of honor in the clubhouse's lounge where photographs of every member hung.
"Yeah," Dare said, sitting again. "That's good." Everyone else sat, too, and for a moment, quiet conversations filled the room. Dare banged the gavel, his gaze latching onto his phone like he could will it to ring. "We've got two more issues to discuss. First, Phoenix and Caine, a heads-up that county social services has reached out to us with a new protective assignment. I e-mailed you all the information before the meeting. Assess and see what the situation is going to require."
As the road captain and sergeant-at-arms, respectively, Phoenix and Caine took point on organizing the club's protective efforts, whether the cases came the Ravens' way through social services, the sheriffs' office, or local people who knew what the Ravens were willing to do for those who couldn't protect themselves.
"What are the basics?" Phoenix asked.
"It's a child abuse situation," Dare said, his expression grim. "The mother's getting heat from several directions to keep it quiet."
"We'll handle it," Caine bit out, his tone like ice.
"Good." Dare slapped the folder closed in front of him and sighed. "Last thing. We've got a major fucking PR problem at the racetrack now following the shootings. Given that we missed five weeks of racing while we were tied up in Baltimore, and then a sixth week this weekend as the police wrap up their investigation, we can't allow our income stream to be further compromised. Ticket sales are down, and even though the club was cleared of wrongdoing, the press hasn't helped the situation with their continued coverage of what happened. So we need to stop the hemorrhage and restore the public's confidence in the safety of our events. To that end, Jagger has an idea he wants to propose."
Jagger raked his hands through the length of his brown hair, his gaze fixed on a sheet of paper in front of him. "So, this idea comes from the position of you gotta spend money to make money."
"How much money?" Bear asked, his voice skeptical.
Jagger shoved the paper in the treasurer's direction. "I'm proposing we do a big family weekend with half-price adult tickets, free tickets for kids under sixteen, and paid uniformed security. A carnival, a dunking booth, a fucking petting zoo. We get all the local businesses in here with their food trucks which motivates them to promote the event, too. With no vendor fees. Maybe we give away some prizes of season passes or photo ops with drivers. We go all out to get people here and make them see there's nothing to worry about. And, if we wanted to be really ambitious about it—"
"This number is pretty damn ambitious already," Bear said, studying Jagger's proposal.
Jagger shrugged. "Yeah, it is, but if we think of it as buying a publicity campaign and not giving a bunch of shit away for free, I think it could be worth it."
Nodding, Maverick agreed with Jagger's thinking. Having lived in Frederick his whole life, Mav knew that they were going to have to make amends with the community. Frankly, he didn't have a problem with that. The Ravens took care of the local community and the town took care of the Ravens in return. Well, most of the town did, anyway.
They had a couple of perpetual thorns in their side. Like fucking Slater, who hated them because he wanted to develop the prime real estate they held. And because the Ravens had once successfully campaigned against one of his developments on environmentally fragile land adjacent to their own that might've impacted the zoning for their races. And because Maverick had been balls deep inside his wife-to-be. Maverick wasn't sure which one of those made Slater hate them more. But, mostly, the Ravens had a symbiotic relationship with their neighbors, one that needed to be nurtured.
"So, what would be even more ambitious than all of that?" Mav asked.
"Donating all profits from the weekend to a local charity," Jagger said. Bear looked a little like he might have a heart attack.
"How quickly could all that be pulled together?" Dare asked. They couldn't wait too many weekends to get back up to their regular numbers, that much was clear.
"I've got a carnival company available the last Thursday through Saturday of the month, which would give us two weeks to plan."
The first thought that Maverick had was _perfect_.
The second? That Saturday was Alexa's wedding day. For fuck's sake. At least he'd have something to do besides sit around and resist driving to the church so he could raise his hand when the preacher asked if anyone had any reasons why the happy couple shouldn't marry.
Conversation pretty quickly fell in support of Jagger's idea. The club had a rainy-day fund for exactly this kind of thing. Protecting their livelihood was more than a good enough reason to dip into it. Without their activities at the Green Valley Race Track, things would get tight fast. Certainly, the funding for their protective mission would largely disappear.
"The only hitch is that the county wants to move up our annual inspection before the event. I actually think it's a good idea. Get our permits and paperwork in order. Have the county sign off that everything is as it should be," Jagger said.
"Sounds like one more thing in our favor," Dare said. "Let's vote." The vote was overwhelmingly in favor.
"I'll get on it," Jagger said.
"Talk to Haven," Dare said. "Bet she'd make a bunch of stuff for a bake sale."
Jagger winked. "Already did, and she already agreed." Dare grinned, pride clear on the man's face.
"Can I just buy everything she makes now?" Phoenix asked, feigning as if he was taking out his wallet. Maverick laughed, and everyone else did, too. You never had a better cookie or cinnamon bun than what Haven made, that was for sure. And damn, it was good to hear everyone just laughing and shooting the shit. With everything they'd been through, they hadn't had nearly enough of that lately.
"You can," Dare said. "But I have it on good authority that she, Bunny, and Cora are making dinner for us, so—"
A cell phone rang out, immediately silencing the ruckus that had risen up in the room. Dare's expression revealed who it was before he even answered. "Nick, thanks for calling back so fast."
"Of course," he said, his voice sounding out through the speaker. "Sounded urgent."
"It is," Dare said. "I need to know if your contact has specifics on the Iron Cross's whereabouts. Hangouts, headquarters, meeting places, that kind of thing."
"Marz actually just got some new intel from our guy this morning." Nick spoke to someone on his end for a minute. "He's emailing it to you right now. Why? What's going on?"
"I won't draw this out," Dare said. "We're going after the Iron Cross. To be clear, we don't want you involved, but we'd appreciate your team's thoughts on carrying it out."
Silence for a long moment. "Shit, okay," Nick said. "I can't say that I blame you, but this is complicated by the Feds. They have undercover agents on the inside."
"Then you need to talk to your contact and get them out of there before it goes down. We're not going for body count but that doesn't mean there won't be collateral," Dare said, echoing Maverick's own thoughts. "Just telling it straight up."
"I hear you. I'm just not sure how much sway I have, Dare." The concern was clear in Nick's voice.
"I know you understand honor, justice, and loyalty, Nick. I've seen you live by it. Fight for it. Be willing to die for it. We might not have worn a uniform, but our code's not all that different." Dare's expression was fierce, unyielding. Maverick couldn't help but nod.
"Fuck, all right. I'll see what I can do. And I'll talk to the team and get you some thoughts."
"Good. And Nick? Don't wait. We're riding on this as soon as possible."
They hung up, and Dare tossed his phone to the table. "Board members and anyone else who can stick around to plan this thing out, stay here. Everyone else is dismissed. But be prepared to ride within the next twenty-four to forty-eight. We'll stay in touch." Eight members remained after the others filed out, and Dare nodded to the door. "Let's head to my office and take a look at what Nick sent."
It turned out to be gold.
Standing behind Dare's desk chair while his cousin opened files revealing locations, layouts, numbers, and more, Maverick already felt the adrenaline of the coming fight surging through his blood.
Dare sat back in his chair, his gaze glued to the monitor. "Their headquarters is in the middle of a largely abandoned industrial area right on the water, and they're not going to be there tomorrow night because they've got some big deal going down that the Feds are focused on."
"That's fucking _perfect_ ," Maverick said. For what they had planned, it was exactly what they needed.
"Jesus, if I was a religious man I might almost say this was providence," Phoenix said. Everyone nodded.
"Moving this fast makes me itchy as hell," Caine said in a quiet voice. "But we might not get this kind of opening again."
"The faster we get it done, the better," Dare said. "Maximum impact with minimum possible risk."
"Unless everything goes tits up," Jagger said.
"As tits are wont to do," Phoenix agreed.
Hands clasped over his belly, Dare nodded. "Then we spend tonight and tomorrow working on contingencies. But otherwise, let's get the word out to all the brothers." He peered up at Maverick. "We ride tomorrow night."
Satisfaction was a tight coil in Mav's gut. "This is the right call, D."
His cousin nodded, and then he looked back to the open documents on the computer screen. "I know. I just hope it doesn't cost us too much, because I'm not sure how much more we can stand to lose."
## CHAPTER 5
Why are you up so early?" Grant asked, coming up behind Alexa where she stood at the bathroom mirror carefully brushing on mascara. His arms wrapped around her stomach.
"This is the morning I have to take my mom to her doctor's appointment," she said, capping the tube.
"Mmm, that's right," Grant said, something that sounded like disappointment in his tone. He was estranged from his parents and had been for way longer than she'd known him, though he wouldn't talk about why so she didn't know what'd happened. But it meant he didn't really get her relationship with her mother. Mostly, he tolerated it. He settled his chin on her shoulder, and it was possible that the phrase _tall, dark, and handsome_ had been coined just for him. Even just out of bed he was undeniably attractive, with his square jaw, his classical profile, his calculating brown eyes, and his stylishly cut short brown hair. "You came to bed too late."
"I know," she said, feeling the sleepy weight of having gone to sleep after two in the morning. "I'm sorry. I was working on my project. A rough draft is due two weeks after the wedding, so I'm trying to get ahead so most of it is done beforehand."
His hands roamed up to palm her breasts through her shirt. Pressing a kiss against the side of her throat, he said, "I need you."
Alexa watched him in the mirror, kissing her, grabbing her, rubbing against her. In another moment, it might've been sexy, but she didn't have time to have sex, which he would know if he remembered any of the three times she'd told him that she'd be late to work on Friday morning because of her mother's appointment. And now he was putting her in the position of turning him down, which he wouldn't like. Which he _never_ liked.
She leaned into his kiss and put on a smile. "I would like nothing more than to be with you, but I'll be late."
He hugged her harder, grinding himself into her rear, and sucked her earlobe into his mouth. "We'll be quick." His hands slid down to her skirt and grasped it. Slowly, he worked it up.
"Grant," she said, her tone full of regret, a niggle of dread stirring in her belly.
He rucked her skirt up around her waist. Pushed her panties down. Slipped his fingers between her thighs.
"Honey, this feels good, but I'm gonna be late," she said, gently grasping his wrist, trying to still his movements.
"You've got time," he said, twisting his wrist out of her hold. "Her appointment isn't for an hour and a half."
So he _did_ remember. "I know, but you know how my mother is." Her mother had anxiety about all kinds of things, including leaving the house and going to the doctor, which meant she was going to need lots of coaxing and reassuring just to get her out the door. There was no such thing as rushing Alexa's mother to do anything.
He pressed his fingers more firmly against her core, firm enough that the sensation went from arousing to uncomfortable. "Grant—"
"Fine," he said, stepping away so abruptly that Alexa stumbled back a step. Without another word, he opened the glass shower door and turned on the water.
Something that felt inexplicably like humiliation rolled through her as she awkwardly pulled her panties into place and smoothed down her skirt. Quickly, she bagged up the last of her makeup, knowing he didn't like anything out of place. "I'm sorry. Please don't be mad. It's just my schedule . . ."
"I know," he said, not sounding particularly convinced. Or appeased.
She walked to him and kissed his bare shoulder, torn between feeling guilty for upsetting him and feeling upset for having to apologize. "I promise to make it up to you tonight," she said, making her words upbeat and playful. Flirtatious. Sometimes stroking his ego was enough to make his mood rebound and smooth things over.
He looked at her, one eyebrow arched, his expression fierce, masculine, intense. "Yeah?"
She smiled, wanting so much to make him happy again, needing so much to be enough for him. Just as she was. "Yeah. Okay?"
For a long moment, he just stared at her. Then, in one quick motion, he turned, grasped the back of her neck, and hauled her into him. The kiss was aggressive, claiming, possessive. His hold was tight, his mouth hard, his tongue penetrating. Grant liked to be rough, and she usually found pleasure in it, too. Though lately, that's all their lovemaking ever was—rough, aggressive, quick. The slow, exploring, sensual sex they'd sometimes had earlier in their relationship had largely disappeared from his repertoire. Not that she'd complained. Because she couldn't begin to imagine how she'd ask for something different in bed. There was almost no chance he'd take it as anything but criticism.
Alexa barely had time to react to it before Grant released her again. "You _better._ " The corners of his mouth tilted upward, like he was teasing, but his eyes were dead serious.
She ignored his eyes and kept up with the flirtation. "You know I will."
He grinned, and it was the smile that had helped win her over to him years before, when he'd pursued her maybe harder than any man ever had. The more she'd held out—worrying over the fact that he owned the company she worked for, then as an administrative assistant for a sales manager several levels down the corporate chain—the more Grant seemed to want her. "I'll look forward to that, Alexa." He dropped his boxers and stepped into the shower, his gaze latched onto her.
"Good," she said, giving him a smile and a sultry look. Playfully, she blew him a kiss. And then she was out of the bathroom and closing the door between them.
And feeling the oddest sensation of . . . relief.
Ugh. It was just the stress of dealing with her mother and knowing it wasn't going to be easy, on top of being late to work and knowing how that was going to make the rest of her day a scramble to catch up. Nothing more. Besides, no one liked starting out their day in a tiff with their partner. She was just relieved that things seemed okay between them now.
In the foyer mirror, she fixed her lipstick, and then she grabbed her purse and made her way to the garage. The Acura SUV she drove was another of her many gifts from Grant. She'd never minded the old Toyota sedan she'd driven for years—in fact, she'd been proud of the fact that she'd bought it and paid it off entirely with her own money. But a year ago, she'd finally given into Grant's insistence that she needed something safer and nicer and accepted his gift of a car.
As she backed out of the driveway, her phone dinged an incoming message. She braked and grabbed the cell, then brought the screen to life.
_MAVERICK: Can't stop thinking of you, Al. Want you to be okay._
On a gasp, she did a scan of the neighborhood around her, but didn't see Maverick's motorcycle or the old pickup he had. Or anything else that looked out of place in the upscale Slater Estates.
Heart racing in her chest, she stared at the text unsure what to do. Because she'd pushed him away, and here he was still trying to take care of her. After what'd happened in the bathroom just now, the point of comparison wasn't a comfortable one. And it meant a lot. It meant . . . maybe even more than it should.
Which made it hard to know how to respond.
She _should_ just delete the message and pretend she never got it. But instead she found herself wanting to keep it, wanting to be able to reread it, wanting to know it was there. She slipped her phone back in her purse and put her car into reverse again.
Maybe Maverick would hear her silence and stay away. Just like she'd asked.
_Is that really what you want?_ a little voice whispered inside her.
It didn't matter. This was about what was right. Because if Grant ever found out that Maverick had been inside their house, let alone that he was texting her, she was certain there wouldn't be much she could do to smooth that over.
On a sigh, she deleted that message after all.
FROM OUTSIDE, THE house appeared perfectly normal. A brick-and-siding rancher on a quiet street in a quiet neighborhood. It was a house Grant had bought and renovated to flip—but when the trailer where Alexa had grown up got condemned as uninhabitable, Grant had done what he always did and came to their rescue after Alexa had promised she wouldn't let his house get so bad. Grant agreed in part because he said he couldn't stand to see the mother of his soon-to-be wife living in squalor.
So her mother had been living here, rent free, for almost two years. Though part of Alexa felt bad that Grant was supporting her mom this way, his help ensured that she lived in a nice, safe place and allowed Alexa to cover the food and medical expenses her mother's disability checks couldn't.
It would've been so much harder to support her mom alone, even with her salary and the remainder of Tyler's small life insurance policy. With that, Alexa covered what she needed for her mom, paid for her own tuition, and saved as much as she could in case of emergency.
At the front door, Alexa took a deep breath and braced herself for how bad it might've gotten since her last visit two weeks before—she hadn't come the previous week because of how fresh the marks from her fall had still been on her face. She knocked twice and opened the door. "Mom? I'm here."
"In the kitchen," her mother called.
Alexa closed the door behind her and stepped from the small, tidy foyer—the foyer was _always_ tidy to keep up appearances—into the living room. Her shoulders fell. The piles were . . . everywhere. Piles of stuff. _All kinds_ of stuff. Waist-high and worse. Clothing. Boxes of pictures and keepsakes. Used furniture. Lamps. Pictures and mirrors that weren't hung and would never get hung. Garage sale and flea market finds, some of them not even taken out of the bag—because it seemed to be the acquiring and possessing of stuff—rather than the actual items themselves—that her mother prized and needed.
Her mom called what she did collecting, and she considered herself a pack rat. But, really, Alexa's mother was a hoarder. Had been for as long as Alexa could remember. Her mother hoarded everything and anything to fill the empty spaces inside her—empty spaces caused by her mother dying when she was young, her husband leaving her, and her only son dying in a motorcycle accident.
Wearing an old stained house robe, Cynthia Harmon came into the living room from the kitchen on the opposite side of the room. "Don't look like that. It's not so bad," she said. She ran her fingers self-consciously over unkempt shoulder-length gray-brown hair. Wrinkles cut into her plump fifty-eight-year-old face. She walked a little stooped over, the result of being really overweight and having a bad back she always complained was sore. Anxiety and hoarding and irregular stints on antidepressants hadn't been kind to her.
Queasiness curled into Alexa's stomach. How had the piles in here grown _so much_ in just the last two weeks? _This_ was why Alexa tried to get over here for at least a few hours every Saturday . . . She shook her head, refusing to get sucked into an argument about the house the second she walked through the door. "Why aren't you dressed? We need to leave soon."
Carefully picking her steps, her mother made her way through a narrow path lined with stacks of newspapers, magazines, junk mail, and years and years' worth of photo albums, to her favorite recliner and sat down. Alexa couldn't remember a time when her mother didn't keep herself surrounded by those photographs. "I was thinking I might not feel up to going today." She sniffed and pushed herself back in the overstuffed chair, almost knocking over a full ashtray resting on the arm.
Alexa wasn't surprised by her mother's words—it was normal to have to convince her to do things she needed to do. But offer to take her to a yard sale or a flea market or an after-Christmas sale and she was dressed and ready to go faster than you could blink. "Mom, you need to go. We already canceled this appointment once, and I've scheduled off work this morning to take you. I'll choose an outfit for you if that will help," she said, moving farther into the room toward the hallway that ran to the bedrooms.
"No, I don't want you to do that," her mother said, shooting out of her chair. The ashtray toppled over onto a stack of newspapers. "I don't want you back there. I don't need you to pick my clothes."
Alexa frowned and gingerly stepped out of the way so her mom could get around her, and nearly tripped over the broken poles of a lamp as she did so. Making decisions as simple as what to wear sometimes caused Mom a great deal of difficulty, so her strident refusal to let Alexa help probably meant her bedroom was as bad as the living room. Or worse. Alexa had been so busy getting settled into Grant's house—well, _their_ house now—that she hadn't spent the time she usually did over here cleaning and trying to cull through the piles. It was only through Alexa's constant battles with her mother that the place had remained as livable as it was for as long as it had. Left to her own devices, her mother would've filled the place floor-to-ceiling by now. Just like when Alexa and Tyler were kids.
It was why her father had left them when Alexa was almost nine and Tyler was thirteen. Their dad hadn't been able to deal with the hoarding. Alexa could still remember them fighting about it. After each new loss, it just got worse.
"Have you eaten breakfast?" Alexa asked. Entering the kitchen, she cursed under her breath as a rancid smell hit her smack in the face. Dirty dishes filled the sink and spilled over onto the adjacent counter. She _hoped_ it was just days' old food causing the smell and not a dead mouse somewhere.
"Yes," her mother called back.
Alexa opened the refrigerator, which was emptier than she usually let it get. She mentally added a grocery trip to her to-do list for the weekend. "What did you eat?" she asked loud enough for her mother to hear.
"A frozen breakfast. Now get out of my fridge," came the agitated reply.
Rolling her eyes, Alexa closed the bottom door and opened the upper freezer door. A few frozen meals remained. She hated that her mother's diet consisted largely of microwavable food, but considering that the stove was often covered in crap—like today, for instance, when there was a huge bag of . . . something sitting on it, the microwave was often the only accessible means she had of cooking anyway.
Between her mother's habit of stacking things on the stove and her smoking, Alexa was terrified that her mom was going to accidentally start a fire and get trapped inside the blaze by the mountains of junk. A fire had broken out in their place when Alexa was fifteen. Luckily, Tyler and Maverick had been at the house that day and had been able to put it out before it damaged much more than the kitchen, but Alexa still sometimes had nightmares about it. Sighing, she lifted the bag off the stove and put it on the floor.
"Now it's in the way." Her mother appeared in the kitchen doorway wearing a floor-length black maxi dress and a pair of flip-flops. She'd pulled her hair into a neat, low ponytail and put on a pair of earrings and a matching necklace.
"It's not safe to put things on the stove like that," Alexa said.
Waving a hand, Mom shook her head. "It's not like the stove can magically turn itself on."
Alexa didn't take the bait. "Better safe than sorry, that's all. Unfortunately, I don't have time to do it today, but tomorrow I will come back with groceries and do a little cleaning for you." _Do a little cleaning_ was code word for _get rid of as much stuff as I can without causing you to have a panic attack_ , and her mother knew it.
"You don't have to do that," her mother said as she tried to stack a few of the dirty dishes from the counter into the too-full sink.
"You know I don't mind," Alexa said. It wasn't exactly the truth. There had been a time when she first moved out of her mother's house when she'd sworn to never deal with hoarding again. As a kid, not even her bedroom had been safe from storing the unneeded and unwanted things her mother brought home. At one point, she'd lost the use of her bedroom closet because she'd put all the stuff her mother kept dropping into her room in there until it was filled to the top.
"Still, I don't need you to take care of me. I'm fine on my own. Always have been." Mom shuffled to the kitchen table and retrieved her purse from the back of a chair. Grocery bags filled with things Alexa couldn't make out buried the table next to the chair.
Her mom's words were a lie and they both knew that, too. "Well, I like to take care of you, so it's no problem. You letting me help makes me happy."
For just a moment, her mother gave her the softest, most sincere smile. "You're a good girl."
"I try, Mom."
"I know." Her mother came right up to her and did something she didn't do often—she hugged Alexa. "My baby."
Closing her eyes, Alexa soaked in the unusual show of affection. After Tyler died, her mom had become anxious about being touched. She only seemed to tolerate it from people she knew well, and even then, she allowed it infrequently. Just another of the issues that had manifested as she'd tried to cope with Ty's loss. Unlike her mother, Alexa never had the luxury of falling apart because someone always had to hold Cynthia Harmon together. More often than not, even when Tyler had still lived, it had been her.
"You ready?" Alexa asked quietly.
"I hate doctors," Mom said, pulling away.
Alexa nodded. "I know you do."
"But whatever. I'm ready. The sooner we leave, the sooner we get back. I don't want to miss my shows." As she moved toward the kitchen doorway, her purse caught a cracker box stacked in a recycling bin on the floor, causing an avalanche. "Just leave it," she said when Alexa went to right the pile.
"Okay," Alexa said. She stepped over the mess. She could deal with it tomorrow. Besides, there were only so many battles she was willing to fight today.
## CHAPTER 6
The Ravens rolled into Baltimore under the cover of darkness on Friday night, the warm, humid air crackling with the electricity of an approaching summer storm. They'd ridden in six groups of five, each set of riders set to converge on one of two agreed-upon meeting points from different directions to hopefully make their approach less noteworthy. Maverick wasn't usually one to believe in fate or any of that bullshit, but it was at least lucky that the weather might offer them some additional cover.
Maverick pulled his Harley into an underground parking garage about four blocks from their target, the Iron Cross headquarters, joining two groups that had arrived before his. A box truck waited nearby to carry any acquisitions home. Their engines rumbled inside the subterranean space, but it was far enough away to be secure, to give them a place to wait, and to make sure their efforts were coordinated with the team meeting in the second location.
Their plan wasn't particularly sophisticated—they'd converge at one time on the location from two different directions. Infiltrate the building via multiple entrances, acquire any assets inside, and then set the place on fire, leaving the Iron Cross with nothing and pulling the rug out from underneath their efforts to take control of the city's underworld. More than that, their loss of power would encourage the city's other criminal elements to pick them off like sitting ducks—exactly what'd happened to the Church Gang. Hopefully the Feds would take care of the rest. And if they didn't, the Ravens just might have to come back for more. And put these fuckers to bed once and for all.
Maverick dismounted as Dare did. They tugged off their helmets and joined the guys milling about at one end of the lot.
The cold, hard press of his handgun in the small of his back felt too damn familiar. He usually only carried when they were actively involved in a protective situation with a client, but lately it'd been one shit storm after another that'd demanded they ride hot. It was hard as hell in Maryland to get a license to carry, but the Ravens had a friend and ally in the sheriff's department who'd helped a number of them get a permit over the years because of the protective work the club did. Not that a permit would've been important for a night like this, because carrying without one would be the least of the lines they were crossing.
"The storm will work in our favor," Mav said.
"I was thinking the same thing," Dare said, wincing and holding his arm close to his side. Maverick was worried about the guy. His injuries and surgery were too damn recent and he wasn't supposed to be riding, but Dare wouldn't hear of staying behind or driving his truck. Mav couldn't really blame him on either account. "Shit, hold on." On a grimace, Dare yanked his cell out of the pocket of his cut. "Nick, not a good time, man," he said by way of answering.
Maverick frowned.
" _What?_ " Dare whipped around to look at the entrance through which they'd all come. "Who?" Pause. "Sonofabitch." Another pause, and the adrenaline Maverick had been riding all day turned pause by pause into anger. What the fuck was Nick doing? "I'm not making any promises." Dare hung up, his hand going as if to tug at his hair, an old habit, despite the fact that it was covered by a black bandana.
"What the hell was that?" Maverick asked, the guys circling around them.
"We're about to have company," Dare said. The words were hardly out of his mouth before a nondescript sedan eased down the ramp and into the level where they were. Raindrops had pebbled on the car's surface, evidence that the storm was starting, which should've been perfect. Maverick bit out a curse and wondered what the hell was coming at them now.
The car parked nearby, and a single man got out. Maverick peered into the car, but didn't see anyone else.
"Chen," Dare said as the man approached. Navy pants, white button-down, short brown hair, the guy's looks were total Middle America.
"Who?" Maverick asked, heaving an impatient breath. "Dude, we don't have time for this."
"Well, you'll be making it, Mr. Rylan," the man said, his voice even, neutral, dispassionate.
Maverick's eyes narrowed. "Do I know you?"
"No," Dare said on a sigh. "But he knows us. Don't you, Mr. Chen?"
"Indeed. Dare, good to see you again." Chen gave a nod.
"Somebody want to explain what the fuck's going on right now?" Phoenix said.
"Meet the Hard Ink team's contact," Dare said, looking over his shoulder at the gathered Ravens. "I met him at the end of their investigation."
This guy was from the CIA? Well, fuck. This was going to be a pain in the ass, no doubt.
"So, what, did Nick tell you where to find us? Or did you strong-arm that information out of him?" Phoenix asked. Maverick nodded, his gut not yet sure who to be pissed at in all this. Thunder rumbled loud enough for the sound to reach them.
"Nick passed on your message about getting my men out, but we've been watching you since the assault on your place last week. Figured it was only a matter of time before you responded." Chen laced his hands in front of him. The guy didn't look like he could hurt a flea, but Maverick had a strange feeling that was a façade. He _wanted_ to be underestimated, to be thought little of, and maybe not to be thought of at all.
"And you're here now, why exactly?" Maverick asked, planting his hands on his hips.
"To ask you to stand down."
All hell broke loose behind him, and Maverick understood the rage and aggression washing off the other men. He felt it, too.
Chen held up a hand. "Not for good. Temporarily. Hopefully just a few hours."
Dare crossed his arms, his face set in a deep scowl. "Explain why we should even consider waiting."
Maverick nearly growled in frustration. "D—"
"Hold on, Maverick." Dare put a hand to his chest. "We should at least hear him out."
"Thank you," Chen said. "You're going to want to hear this since your intel is already outdated."
Fucking hell. Mav traded an agitated glance with Dare.
"Welcome to tits up," Phoenix bit out.
Chen ignored the comment. "The Iron Cross has moved the location of their deal to their headquarters. Everyone will be converging there within the next fifteen to thirty. We're not sure what's behind the switch because communication with our agents is limited. If you go charging in now, you risk an all-out gang war and we lose our shot at acquiring the intelligence our investigation needs. An investigation that would be near to your heart, Dare, I should think."
"Meaning?" Dare asked. A crack of thunder split the awkward silence.
"The Iron Cross hasn't just picked up the Church Gang's heroin activities. They're reviving their human trafficking trade, too. Tonight's deal is their first sale and our shot at blowing this ring apart. And we have good reason to believe we have what we need to do just that. You strike now, and we lose intel on the players involved and the chance to rescue those women."
"Fuck," Maverick bit out. That tugged just as hard as his conscience as it likely did at Dare's. Especially given that the now-destroyed Church Gang had kidnapped Haven and her best friend, Cora. Anything might've happened to those two, women who'd become an important part of the Ravens' community, if the Hard Ink team hadn't stumbled upon them during their investigation weeks ago. So this hit too close to home.
"Exactly," Chen said.
"Where are the women now?" Dare asked.
"Presumably in transport but we don't know from where," Chen said. "So here's what I propose. Wait until I get the signal from my men that they're clear and have the women, intelligence, and evidence we need. When the club empties out later tonight, you do your thing."
Maverick's mind chewed on this new plan, not bothering to wonder how this guy knew their intentions. He was clearly one step ahead of them. Waiting sucked, but if it ensured their own safety _and_ helping a group of innocents, what choice did they have? Though something about this bugged him . . . "Why are you willing to allow us to do anything here?"
Chen turned that too-observant gaze on Mav. "Shared liability. The more someone else does to take them out, the less my shop has to do."
"I thought the CIA wasn't supposed to do this shit inside the U.S. at all," Phoenix said, scowling.
Chen shrugged and gave him a droll stare. "I work for the government."
"And the government sanctions . . . this?" Phoenix asked, the arch of the eyebrow making the scar on the side of his face look more severe.
"The world's a complicated place, Mr. Creed. Given what happened to your cousin and your friend recently, I think you'd know that as well as anyone. These are bad men who want to be worse. And they're involved in more than I'm even sharing. They're on a crash course with fate one way or the other. Let's leave it at that."
"So, what? We're partners now?" Phoenix asked, arms crossed.
Chen shrugged with one shoulder. "In a manner of speaking."
Chen's words brought Maverick around more and more. The Ravens could help these women. They'd have the cover and assistance Chen and his men could provide. And it all still achieved their objectives. Mav turned to Dare. "I hate it, but I think we gotta wait."
Dare nodded and nailed Chen with a stare. "We'll wait. But this needs to happen tonight. We want to use the cover of the storm, and we want to be fast asleep in our beds well before dawn." He arched a brow. Thunder rumbled again as if to emphasize the point.
"I think that can be accomplished," Chen said with a nod.
"Given we're stretching this out now, you got a way to keep the police out of the situation until we're clear?" Maverick asked.
"Yes," Chen said, offering nothing more. Well. There ya go.
"Fine," Dare said. "We'll play it your way. For now. But if things don't go down like you said, we're still doing this. So you keep up your end of the bargain, and we'll keep up ours."
THE SIGNAL FINALLY came at quarter after one.
"We're about to be clear," Chen called from where he'd been sitting in the driver's seat of his car. "Get the word to your men. There might be a few Iron Cross left behind but it broke up about a half hour ago."
"Doing it right now," Dare called, his face already in his phone.
"Jesus," Maverick said, shaking off the lethargy that hours of waiting had caused. Not hard, because the imminence of the fight fired up his adrenaline again. A few Iron Cross was worse than they'd hoped but better than if they'd gone in there hours before with the wrong intelligence. All around him, last-minute plans were being confirmed, and men were mounting up, checking their weapons, and starting their engines. He turned to Dare, gingerly straddling his ride. "D, for real. Be careful." Maverick nailed him with a stare.
"I hear you," he said.
Maverick grasped his cousin's good shoulder. "I'm serious, Dare. It's storming like a motherfucker, we're about to ride into God knows what, and you have someone to go home to." The sentiment made him think of Alexa, even though she wasn't his. Still, he had promises to honor—promises he wouldn't be able to keep if he got his ass killed tonight. And the thought of never seeing Alexa made him fucking _ache_. How had she gotten so far beneath his skin again in so few days? Or maybe she'd always been there, and he'd just been ignoring it.
"I _hear_ you. And we all do. I'll be careful." Dare tugged his helmet on, the action clearly causing him pain.
Stubborn bastard. Family trait, though, wasn't it?
Cursing under his breath, Maverick made for his own ride. The Night Rod came to life underneath him, the rumble of the engine sounding like an old friend saying hello.
Phoenix and Dare led them out, headlights dark, the echo of their collective engines drowning out the storm raging outside. And then they were out in it, rain lashing at them and pretty much instantly soaking through Maverick's jeans and shirt. The visor on his helmet kept his eyes clear and hid his face, too, but the rain still limited his vision through the shield. His cut grew wet and heavy on his shoulders, but Mav didn't mind. He liked the reminder of who he was and why they were doing this tonight.
Within a few minutes, they were in position in hiding places along the derelict street and around the edge of the waterfront compound, surveying the site and preparing to converge. The storm would no doubt prove a lifesaver, because there was plenty of evidence of unfriendlies on site. Four cars sat between his position and the building, and two Hummers sat close to a side entrance. More problematic, three men hung by the water at the back door of the long two-story building covered in old white siding. The remains of ancient signage under an old metal lamp revealed that the place had once housed some sort of shipping facility. The deluge obscured everything else.
Mav's heart was a freight train in his chest as he positioned the semiautomatic he'd had stowed in his saddlebag. How ironic that this was one of the weapons they'd taken off the Church Gang weeks before and that the Iron Cross had more recently tried to twist their arms into selling to them. Fuckers. Time seemed to drag out as he glanced between his cell and the warehouse before him, waiting for Dare to give the signal.
His cell buzzed in his hand. Finally. Dare's command simply read, _MOVE IN_.
Maverick took off, moving fast and low, pausing for cover where he could—at a fence, behind one of the cars. He was aware of the movement of the others only because he knew they were there. Otherwise, the rain was doing what they'd hoped and cloaking their approach as they tightened the noose around the building.
This was the part that was going to suck. Crossing the big open stretch between the last of the cars and the building. Surveying the wide expanse again, Maverick double-checked the position of the men by the water and found them still seemingly oblivious. Which meant, fuck it, time to go.
He cut around the bumper and broke into a sprint, weapon in hand and at the ready.
_Boom._
Maverick paused, wondering—
_BoomboomBOOM!_
The shock wave of the explosion knocked Maverick back on his ass. He landed hard, his head cracking against the pavement and making his ears ring. A fireball rose to the sky as debris launched into the air, falling into the water and the parking lot all around him.
"Jesus Christ," Maverick said, dodging a flaming board that landed too fucking close for comfort. Heart in his throat, his brain slowly came back online. He found the weapon he'd dropped and surveyed the scene like he was stepping out of a fog, just able to make out some of his brothers' movement through the deluge. Oxygen-stealing heat roared off the blaze like an impenetrable wall, the fire engulfing nearly the whole structure, particularly the end closer to the water. Weapon in shaking hands, he retreated, wanting to return to the cover of the car until he could—
Automatic gunfire erupted, the sound just audible over the storm and the inferno. Maverick dove for the ground and caught a flash of motion near the Hummers. Survivors. Fleeing the building. And coming out shooting.
A hail of bullets sailed over his head, and Maverick soldier-crawled through the mud and grime and puddles to the closest car. Rounds ricocheted off the metal as he got closer, and he wasn't sure if they were shooting _at_ him or if it was just his own dumb bad luck. Fuck. This took _tits up_ to a whole new level. Where were the others? Had any of their men made it to the building before the explosion? How many of them were caught out in the open like he'd been?
Maverick didn't have time to worry about any of that just then, because the two Hummers lurched into drive, a piece of burning siding on the hood of the first. Time slowed to a dreamlike crawl as he slid under the car, using the tire for cover from the shots and the heat. He couldn't tell how many people were in the vehicles careening around the flaming debris. Mav only knew that the Iron Cross refugees were still spraying bullets like they thought they had to shoot their way through. And then the Ravens started to return that fire. The storm couldn't quite hide the sound, but it hardly mattered. Because everyone in the city would've heard that explosion, and there wasn't anything Chen could do to cover their asses now. They just needed to get out as fast as they fucking could. Setting up a shot, Mav debated what to do—
Another explosion erupted from inside the building. Maverick felt it in his chest against the pavement. The heat was searing.
"Fuck," he said, flinching and watching as the second Hummer spun out of control and crashed into the closest of the parked cars. The first one had been clear of the building when the new detonation occurred, but it suddenly spun out of control, too—and it was coming right for him.
Rolling hard and fast, Mav made for the opening on the far side of the undercarriage and had just about made it when the collision occurred. Metal scraped upon metal like a building was coming down on top of him. The Hummer pushed the sedan so hard that the far tire he'd just been hiding behind settled against his shoulder again.
He'd barely managed to swallow his heart when movement caught his eye and someone spilled out of the Hummer's driver's side maybe ten feet away. The man hit the ground hard, but Mav knew he remained lucid when he adjusted the grip on his weapon and scanned the scene with wide searching eyes. Pissed-off eyes. Then those eyes found Maverick through the darkness, and the man raised his gun.
Maverick got a shot off first because he'd had his weapon at the ready. It was the only thing that saved him. He knew the slug found its target when the man's head and arms slumped. Mav didn't feel great about that, but the situation was too hot to sit and spin on it. And no one made it through life without losing a few pieces of themselves along the way, did they?
He waited for another moment, making sure no more Iron Cross were going to come out that side door that the flames hadn't yet reached. But all was suddenly still. _Time to fly_.
He'd no more had the thought when his phone buzzed in his back pocket. Maverick looked at it to confirm the message.
_Pull back. Follow assigned route home._
No kidding.
Racing to his Harley, his boots hit deep puddles and slipped on the grass, but then he was astride the bike and underway. Hauling ass away from the blaze. Maverick looked back once more in time to see one of the walls collapsing inward in a loud crack of flying sparks and flame.
Despite Chen's assurances, Maverick expected the authorities to appear on their tails at any second. The only comfort was in seeing the groups of motorcycles forming up—each complete group of five a reassurance that their men had survived. It wasn't until they were out on the open highway and heading west toward Frederick that Mav believed that they were truly free and clear.
The Bluetooth inside his helmet picked up an incoming call. "What?" he answered. He heaved a breath, trying to shake off the stress of the night.
"You okay?" Dare asked.
"Fuck, I don't even know. Yeah. You?"
"I'm good," Dare said. "So, that was some shit."
"What the hell happened?" Maverick bit out, dark farmland flying by on both sides of the highway.
"I talked to Chen. He said there was some sort of power play inside the Iron Cross that tilted the whole fucking night on its axis, for starters. He's not sure why the initial blast occurred like that, though they had a meth lab and were building explosives. For what, he wouldn't say. They also had some kind of fail-safe destruct system hardwired in, which they apparently tripped," Dare said, his voice sounding tired. Or maybe that was just their connection.
Building explosives? So the Iron Cross were into even more bad shit than they knew, just like Chen said. That lightened a little of the darkness inside Maverick's chest over the man he'd taken out tonight. He sighed and shifted in the saddle, his back aching after being knocked on his ass.
"Are we whole?" Mav asked, still worrying about their men even though the bike count had been right on the way out of the city.
"Two not serious GSWs. Joker and Bandit. Otherwise, yes." Dare's voice didn't sound like they weren't serious. But the guy felt every hit the Ravens took personally and always had.
"Damnit," Mav said. "Did Chen mention the women?"
"His men got them, and took a number of other Iron Cross out in the process," Dare said. "They apparently got what they needed before it all went down."
"Good. That's good." Those women could've been Haven. Cora. Alexa. Goddamnit.
"Yeah. Tonight sure as fuck didn't go like we'd planned but hopefully it did what we needed it to do," Dare said.
Well, that was an understatement if Mav ever heard one, but his cousin was right. "I'm just glad things might be settled with the Iron Cross now," he said on a troubled sigh. "For Bunny, for Jeb, for all of us."
They hung up, and Maverick tried to embrace the peacefulness of the open road for the rest of the trip. All told, it wasn't long. Just about an hour door to door. Usually, having his fists and knees in the breeze was the surest path to chilling his ass out when life had him wound tight. Tonight was no exception, though perhaps he was wound a little tighter than normal.
Because of their own dead and injured. Because of what they'd just done. Because of his worry for Alexa.
He really was worried about her, and equally worried that there wasn't much he could do. Not if she wouldn't let him.
But at least with tonight behind him—with the Iron Cross finally out of the picture, Maverick could put his focus on watching out for her without feeling like he was letting the Ravens down.
He couldn't do more to help those women, but he could help Alexa. And he would.
If she needed him, he'd be ready. That was all he knew. And for now, that would be enough.
## CHAPTER 7
Dealing with her mother two mornings in a row was a special kind of hell. Guilt gripped Alexa for feeling that way, but she couldn't help it. Yesterday after the doctor's visit, her mother had broken down into hysterical tears that were, Alexa suspected, a ploy to get her to stay at the house longer. Which she'd done, making herself even later for work. Of course, Grant hadn't appreciated that.
This morning, Alexa had brought over an order of groceries just like she'd promised, but getting any cleaning done was proving harder and taking longer than it had in a long time.
So far, Alexa had washed and put away the dirty dishes and washed several loads of laundry, and every step had been filled with tears, negotiations, and protests. Her mother hated having things put away because then she couldn't see them, and she fiercely resisted any attempts to throw things away, even stuff that was obviously trash.
"Okay, Mom," Alexa said, coming into the living room again. "We need to make more of a walkway in here for you so you don't fall. Why don't you keep the last week of newspapers and the last month of magazines and catalogs and let me recycle the rest?"
Wearing a T-shirt and a pair of capris, her mother sat in her recliner wringing her hands. A pile of used Kleenex sat on the chair beside her. "I need my magazines. They have coupons and recipes and good stories, and I haven't read them all," she said, voice filled with threatening tears.
"Then how about I recycle the newspapers and catalogs and you hold on to the magazines for now?" That would get rid of three huge stacks of paper. A small victory, but a victory nonetheless.
"Oh," her mother said, frowning. "Oh, I don't know." She sighed as she grabbed a photo album and pulled it into her lap. She'd always found comfort in pictures of the past, and her needing to hold that album revealed her escalating stress level. "I guess."
"Okay, thank you," Alexa said, moving before her mother changed her mind. She grabbed a black trash bag and crouched next to one of the piles. Halfway through, a headline from last Saturday's local paper caught her eye.
**5 KILLED IN ATTEMPTED KIDNAPPING AT GREEN VALLEY RACE TRACK**
Alexa's gaze raced over the story. Green Valley belonged to the Raven Riders, and Alexa had spent many weekend nights there with Maverick watching stock-car races or the occasional demolition derbies the club hosted. Maverick's uncle had inherited the track and the huge piece of property that surrounded it, all of which the motorcycle club now called home.
According to the article, a gang of out-of-state criminals had tried to kidnap a woman under the club's protection, causing a shoot-out in the middle of a race.
The black band she'd seen tied around Maverick's arm . . .
_What if he'd been one of the five?_
She shuddered as goose bumps erupted over her arms. _But he wasn't. You saw him with your own eyes. He's fine._
"What are you looking at there?" her mother asked.
"Oh," Alexa said, holding up the paper. "The shoot-out at the race track. I hadn't seen this." Which made her wonder why. They got the newspaper at her house, but she hadn't seen one around in more than a week.
Her mom shook her head. "Terrible what happened. That Kenyon boy got shot and Maverick's mother got hurt, too."
"Oh, God. Are Dare and Bunny okay?" Alexa asked. Suddenly, she remembered the anguished look Maverick had worn that day in her office when she'd asked about the band he wore. An ache took up root in her chest for him. He'd lost a friend, and his mother and cousin had been hurt, too? And here Alexa had told him to stay away. Worse than that, she'd told him his presence could ruin things for her. Nausea curled through her belly.
"I don't know the specifics, but I think so," Mom said. "Bunny was treated and released. I think Dare was in the hospital for a few days. How have you not heard about this? It's been all over the news."
"I . . . I don't know," Alexa said. She didn't always read the newspaper, but Grant usually left it on her desk after he'd read it in the morning. Seeing this made her realize he hadn't done that all week, and in the midst of everything—including the aftermath of their fight, Alexa hadn't noticed. Was it possible he'd taken the papers so she _didn't_ see this? No. Surely not. Frowning, Alexa stuffed the paper she'd been reading in the trash.
Before long, the papers filled the bag, so she grabbed a second and turned to the stack of catalogs.
"I want my catalogs," her mother said in a small voice. "I like to look."
No, she liked to shop. "Mom, there are so many of them."
"I like my catalogs," she said, her voice rising. "I don't know why that's so much to ask."
"You don't need them. You order from the internet anyway," Alexa said, grabbing a handful.
"But I like to look at the bigger pictures in the catalogs first," her mom said.
_Don't lose your temper, don't lose your temper_. The minute Alexa did, her mother would burst into tears and refuse to make another concession for the rest of the day. And that would be totally counterproductive. Alexa needed to keep this place clean—not just for her mother's health and safety, but because Grant would flip if he ever saw exactly how bad the inside of this house sometimes got. It was why Alexa came over here most Saturdays to clean. He'd been inside this house exactly one time since she'd moved in, and he'd nearly had a panic attack at seeing it—and that had been early on, before it had really gotten bad. So he didn't help because he couldn't stand the mess and didn't like to spend time with her mother, anyway—he said she made him uncomfortable, and his discomfort made her mother nervous and anxious. It was just better to keep them separate, even though it made Alexa sad.
"Let me take these, Mom. You'll get more in the mail. By the end of the week, you'll have half this many right back again. You won't miss them. Please," she said, giving her mother a pleading look. "I worry you'll trip and fall with so much right around the foot of your chair." Especially since the pillow and folded blanket stacked next to her meant that she was sleeping in the chair again, too. More evidence that her bedroom had probably become unlivable.
"Why do you make me do this, Alexa? You know I like my things," her mother said. It wasn't a no. Progress.
"I know. But I love _you_. And I want you safe. It's not safe to have all this in here." She held up a handful of catalogs. "Okay?"
Her mom hugged herself and waved her hand dismissively.
It didn't take long to fill the second bag. Though getting rid of the newspapers and catalogs did clear some floor space, their removal hardly made a dent in the overall state of the room. "I'm going to take these outside. Be right back."
It was always best to remove things from the house after getting her mother to agree to part with them; otherwise Alexa would turn around and find her mom pulling stuff out of the trash. With more than a little difficulty, Alexa hefted the bags over her shoulders so they would clear the piles of junk she had to navigate on her way to the foyer.
Awkwardly, she made her way through the front door and then she walked the bags to the end of the driveway, her biceps burning. She set them down with a groan next to a row of other bags she'd brought out earlier. Catching her breath, she braced her hands on her bare knees and enjoyed the warm June breeze blowing across her neck. She'd been working for a little over three hours and she was beat.
She needed to get back into her routine of doing the elliptical a few times a week. Really, she had no excuse not to since Grant had a fully decked-out workout room in the basement. Well, no excuse except for the fact that her final project and all the last-minute wedding planning took up the time she might otherwise have devoted to the elliptical. Still, Alexa had a wedding dress to fit into in just two weeks. And Grant seemed to notice the minute her weight fluctuated more than a few pounds. Of course, she wasn't sure she knew someone more fit than Grant, who ran on the treadmill and lifted weights almost every morning.
"How 'bout I give you a hand?" came a voice from behind her.
Alexa whirled, heart in her throat. "Oh my God, Maverick. Don't sneak up on me."
"I wasn't sneaking," he said, looking sexier than any man had a right to look in a pair of jeans, a white T-shirt, and his Ravens cut hanging on those broad shoulders. And here she was, sweaty, grimy from cleaning her mother's house, and with her hair thrown up in a messy knot. Not that her appearance mattered, of course.
"Then where did you come from? Again?" She arched her brow and gave him a look. "I thought I said you shouldn't be keeping an eye on me?" Though she didn't manage to put the heat behind the words she probably should have.
"I have a problem with listening," he said with a smirk.
"No kidding." Alexa planted her hands on her hips.
"Besides, I like keeping my eyes on you," he said with a wink. His gaze swept over her in a slow, satisfied up and down. Heat rose into Alexa's cheeks, because he'd said the words jokingly but there was an intensity and a seriousness to his expression that made her think of dark rooms and frantic kisses and messed-up sheets. Molten-hot memories of the two of them together—memories she'd kept boxed up tight for the past five years—threatened to come roaring back.
She couldn't let them, so she changed the subject. "Hey, how's your mom? And Dare? I'm sorry I didn't ask the other day. I didn't know what'd happened."
The amusement melted off Maverick's rugged face. "Both doing better. They'll be okay."
Relief flooded through Alexa. "Oh, thank God. I was worried."
Lips pressed into a tight line, Maverick nodded as his gaze drifted toward the house. "How's your mom?"
"Oh. She's . . ." Alexa shrugged. She couldn't help but be struck by another comparison. Grant rarely asked after her mother, except to ask how long Alexa would be gone on Saturdays. "You know. Pretty much the same." As much time as Maverick had spent with Tyler and her over the years, he would know what she meant. In that moment, Alexa realized Maverick was the _only_ other person left who truly understood Cynthia Harmon's problems—and what it took to deal with them.
"Yeah. So let me help." The sincerity in those dark blue eyes reached inside her chest and played with long-buried needs and desires.
"Okay," she found herself saying. She shook her head, confusion swamping her even as she went with it. "Yeah. Sure. Um." She gestured toward the house. "Come say hi?"
Maverick smiled. And holy _wow_. It had been a long time since Alexa had seen him smile that way. It stole her breath. Because Maverick Rylan smiling was stunningly gorgeous. The smile brought out deep dimples on both sides of his mouth, and combined with the stubble covering his jaw and those blue, blue eyes—
"We going in?" he asked, one eyebrow cocked.
So busted. "Of course," she said, hugging herself and looking down at the blacktop of the driveway as she led him toward the house. "When we get inside, let me give Mom a heads-up that you're here in case she wants to put herself together a bit. Or something."
"Sure," Maverick said from beside her. Out of nowhere, Alexa was swimming in the most overwhelming sensation of déjà vu. How many times had Maverick come to visit her mother with Alexa? How many times had he helped drop off groceries or take out trash or sat and chatted with her mom while Alexa took care of something that otherwise would've caused her a lot of stress? Him being there was suddenly so familiar _and_ so foreign—because it had been a long time since she'd felt like she had any companionship in dealing with her mom. Going all the way back to Tyler's death.
A weird little knot of tension settled into Alexa's gut.
She opened the front door and stepped inside, Maverick right behind her. She held out a hand to him and moved to the living room doorway. "I have a surprise for you," she said to her mother. "Someone came to say hello."
"Who?" Her mom ran her hands over her hair. "I look a mess, Alexa."
"It's just me, Mrs. H," Maverick said.
"Oh!" Alexa hadn't seen her mother's face light up like that in a long time. "Oh, Maverick Rylan, you come in here and see me right now."
Grinning, Maverick slid past where Alexa stood in the doorway. Though it was soft and quick, the slight brush of his body against hers flashed heat through Alexa's body. Just that little touch. Just that promise of something more. And that knot in her belly got a lot bigger and a lot more uncomfortable. Because barely touching Maverick made her hot with want, but just last night she'd had to pep talk herself into keeping her promise to make her rushed morning up to Grant.
"Hi, Mrs. H," Maverick said, crossing the room to her.
"Oh, Maverick. What a sight for sore eyes you are," she said, standing up. She held her arms out to him, and Maverick didn't hesitate for a second. He not only hugged her, but he lifted her off her feet, making her mother laugh like a girl. "Put me down before you break me," she said, not meaning a word of it.
"It's good to see you." Maverick carefully put her back down. "How have you been?"
"Same old. You know." Mom sat again. "You here to help Alexa straighten me out, then?" she asked with a sniff.
Maverick laughed. "I'm here to help Alexa with whatever she needs help with," he said, looking over his shoulder at her. His expression revealed that he meant that in all kinds of ways Alexa didn't want to think about. _Shouldn't_ think about. Since she was engaged to another man and all. And had given Maverick up long ago.
Regret crept across her skin in a shiver, but Alexa refused to give it a second thought. "Well, if you wouldn't mind helping me in the kitchen, then?"
"You're the boss," he said.
"Ha. Says the man who doesn't listen to me." Alexa threaded her way through the room, glad that needing to be careful of her steps meant that she had a good reason not to make eye contact with Maverick.
"You be nice to him, Alexa," her mother said.
"Yeah, be nice to me," Mav whispered. The sexy bastard.
In the kitchen, Alexa pointed her finger at him and raised an eyebrow. A silent command to be good.
"What?" he asked, his mouth sliding into a cocky half grin. She wanted to kiss it off of him.
"What are you going to do in the kitchen?" her mother asked from the other room.
Alexa shook away the troubling impulse. What was wrong with her? "Just dealing with the trash and recycling, Mom. Don't worry," Alexa called. Even though Maverick had been in their trailer many times and knew just how bad their mother's hoarding could be, Alexa found herself really glad that she'd taken care of the dishes earlier. The room didn't smell nearly as offensive now. And luckily she hadn't found any buried critters—so far, at least. She looked at Maverick and spoke in a near whisper, "Sure you don't mind doing this?"
"I never have," he said.
She knew that was true, and it poked at something deep inside her. Alexa nodded. "Then we gotta work as fast as we can in here. She was getting pretty anxious earlier."
"I hear you," he said. "Just tell me what you want me to do."
## CHAPTER 8
Maverick had never been happier to do grunt work in his whole damn life. But he was doing it with Alexa, and that made a fucking difference. Not just because he was with her. But because she'd invited him in and welcomed his company. More than that, she seemed to be enjoying it.
It felt damn close to old times. The _good_ old times.
An hour passed in the blink of an eye as they worked together. The happiness flowing through him made him feel damn pathetic, truth be told. But he ignored it in favor of letting himself enjoy what might be one of the last days he'd be spending with her. Ever.
He followed Alexa out to the curb with what was going to be their last bags of trash. Mrs. H had blown up at them moments before and made them promise to stop, so she was clearly at her limit. Not that it was any skin off Maverick's nose. He'd seen it all before going years back. And, anyway, being a hoarder was a lot better than being an abuser like his father, so it wasn't like he had any room to criticize anyone else's parent.
Alexa dropped her bag into the big pile at the curb with a groan. She planted her hands on her hips, emphasizing her mouthwatering curves there. "I'm really sorry Mom yelled at you." She wiped at her brow on a long sigh. Even with little makeup, her hair thrown up in a messy bun, and being dressed down in a pair of jean shorts and a form-fitting black tank top, she was still the most beautiful woman Maverick had ever known. Damn it all to hell.
"You don't owe me any apologies, Al. I know how it is." He dropped his bags at the curb.
"I know you do," she said quietly. Her expression was thoughtful. Maybe even a little wistful. And it made him want to argue, convince, hell, even _plead_ if it had a chance of making things different between them—all shit he should've done more of five years before.
_"Can you?"_ she'd asked that day at Tyler's grave. He'd heard everything she'd asked without her needing to say the words. Can you be there for me? Can you give me what I need? Can you commit? Can you do forever? And he'd frozen. Like the damn fool, like the _damn immature kid_ , he'd been. It was a moment he'd replayed so many times, wishing he could do it over each and every time.
Alexa pulled her cell from her back pocket. "Maybe I can get one of Grant's guys to come haul these bags away. If I leave them here 'til Monday morning, Mom will end up going through them and taking stuff back inside."
The mention of Slater's name was like nails against a chalkboard and hauled Maverick right back to the present. He grasped her hand. "I'll take care of it," he said a little more harshly than he intended.
As if it wasn't bad enough that Maverick had lost Alexa to the wealthy asshole, Grant Slater was one of the few businessmen in Frederick who thought that the Ravens' racing activities at Green Valley weren't good for the town. The Ravens had allies among the restaurant, hotel, and retail store owners who benefited from the out-of-town visitors who came to see the races, and lots of locals supported the Ravens because of how many people they employed during racing season. But Slater argued that the races were detrimental to luring yuppies looking for an escape from city life into one of his developments of Mc-Mansions. So of course he'd jumped on last week's crisis at Green Valley to get face time with the local news that he used to criticize the club. Worse than that, his wealth gave him influence, including with the mayor, who was so far into Slater's pocket it wasn't funny.
So Maverick had all kinds of reasons to want Alexa away from him. Not that it was his choice, was it?
"I'll run to the house and get my truck. I can be back in twenty," he said.
"Oh. Really?" Alexa asked, those hazel eyes wide with surprise.
Maverick just looked at her. Because no way he was opening his mouth and chancing revealing how deep it cut when she doubted him. Especially since he knew that her not believing in him—in his ability to take care of her—was part of what made her choose the wealthy, seemingly respectable asshole over him. And if that wasn't enough, her doubts poked at his own fears about himself, born when he'd failed to take care of his mother years before.
"Okay. Wow. Um, are you sure? I can't imagine you intended to get hung up here all day."
"I don't say what I don't mean, Alexa," he said, nailing her with a stare.
"I . . . I know. It's not that. It's just . . ." She shook her head. "You know what? Never mind. Thank you for offering. If you got your truck, that would be great, actually."
Maverick nodded. "You got it." He made for the street, and then an idea had him stopping in his tracks and turning back to her. "What are you going to do?" He'd spent enough time with Alexa and her mom to know that Alexa could only take so much of her mother at a time.
Alexa shrugged and looked maybe just a little lost. It made Maverick want to hold her, claim her, tell her she was fucking found. Forever. Which was a problem given that she was engaged to be married in just two weeks. If his gut wasn't so sure she was in trouble, he'd cut himself off from seeing her again. Because on one level he was just torturing himself by coming around. "I don't know," she said with a rueful chuckle. "Fall down on the grass right here and take a nap maybe?"
"I got a better idea," he said, a yearning planting itself in his gut. _Could be the last time . . ._
"Oh, yeah? Do tell," Alexa said with a smirk and an arched eyebrow. And fuck it was sexy when she came at him with a little playful attitude like that. Always had been.
"Ride with me." The words flew out of his mouth, and he didn't want to take them back even though, _fuck_ , it put him out there to be smacked down again. But damn if it wouldn't be worth it to feel her holding him, pressed up tight against him, laughing in his ear.
She hesitated and maybe even looked a little nervous, but Maverick could see the interest in the liveliness of her eyes.
"Don't think. Just say yes," he said.
Alexa tilted her head and narrowed her gaze at him, and then she smiled. "Okay. Yes."
_That's my girl_. He kept the sentiment to himself, but he felt it down deep. _His_ Alexa was in there. The one who could throw caution to the wind and have fun doing absolutely nothing. The one who laughed freely and bantered relentlessly. Maverick gave a single nod.
"Let me tell Mom. Be right back," she said.
"I'll bring the bike around." Maverick took a step backward, but then paused when Alexa turned and hurried toward the house. His gaze traced her curves and lingered on the bare skin of her thighs. Thighs that were going to be wrapped around him in just a few short minutes. Thighs that _had been_ wrapped around him in all kinds of ways many times. _Many_ times. Fuck.
A few houses down, Maverick straddled his Harley Night Rod Special and brought the engine to life on a low rumble. He pulled the bike to the curb in front of the Harmon house and waited. He didn't have to wait long.
Alexa came flying out through the garage. She made eye contact with him and it fucking seared his blood. For some reason this ride felt weighted with a significance he couldn't put into words. And he wasn't trying. He was just going to enjoy the hell out of it. Because Maverick knew her agreeing this one time didn't change anything. Not anything important.
By the time she reached him, Alexa was smiling even though he could tell she was trying to hold that shit back. But she couldn't, and that got to him, too. For a moment, she hesitated, like she wasn't sure what to do. No way that was it, though, because even though it had been years, she'd been in his saddle as many times as his bed. He suspected the real issue was giving herself permission to do what she was gonna have to do—get close and hold tight. To him.
Something she hadn't done in years.
But then she did it.
She braced one hand on his shoulder and climbed onto the small leather seat behind him. Her thighs hugging his ass. The softness of her breasts against his back. Her breath against his neck.
Fuck. The good old times once more.
He slid on a pair of sunglasses, then handed her his helmet. "Put this on."
"What about you?" she asked as she took it.
"I'll be careful." He looked over his shoulder at her, and she gave a nod when she had it secured. "Ready?"
"Yeah," she said, and he didn't think he was imagining the anticipation in her voice.
The moment her hands came around him, Maverick pulled away from the curb. And the rightness of having her on his bike again made him want to drive up into the mountains for the rest of the day. Fists and knees in the breeze for a long ride. Find a quiet spot by one of the lakes. Fuck like a couple of kids against a tree or in the grass. They'd done it before. More than once.
But that wasn't who they were anymore, was it?
And he was working on accepting that once and for all.
But not today.
THIS WAS PROBABLY a big mistake. Huge. Gargantuan, even. But that didn't keep Alexa from getting on the back of Maverick's bike and holding him tight.
_One last time . . ._
And it was just a ride. For old time's sake. For _fun_ , something she hadn't prioritized in so long. And for her mom. Couldn't forget that, either.
So, yeah. It was just a ride.
A shiver ran through her that had nothing to do with the air rushing over her skin. It was Maverick. His heat, his muscles, his long hair that she'd always loved running her hands through. It was his rough masculinity and his daredevil attitude and his fierce loyalty.
It was Maverick that had her body getting hot and feeling needy. She found even the leather Raven Riders' patch on the back of his cut alluring, which made no sense since it was his membership in the club that had been one of her rationales for pushing Maverick away. But holding him as she was, all she could think about was how that same image was tattooed on his broad back. Watching the raven perched on a dagger move as his naked body did had _always_ attracted her to him, even if everything the tattoo represented had sorta scared her.
Alexa closed her eyes and heaved a deep breath. Why was she torturing herself like this? She'd made her choice. Her future was set. Her future was _Grant_.
This? This was just a ride with someone she used to know.
Alexa groaned to herself. She might not want to admit the truth out loud, but it wasn't like she could hide from herself. And if she was being honest, she hadn't accepted the ride just to get a break from hanging out with her mother, although that had been part of it. She'd also _wanted_ to ride with Maverick. Deep down, a part of her hadn't wanted to spend the minutes apart that it would take for him to get his truck and return. Because she wouldn't have many more—if any—opportunities like this again, not with her wedding just two weeks away. Now, experiencing all these feelings welling up inside her, it made her remember something she hadn't thought of in a really long time.
Being with Maverick . . . just made her want to be with him even more. It had _always_ been like that between them.
Which was one of the reasons, five years before, that she'd broken things off with him cold turkey. No friendship, even though they'd started out as friends. No hanging out casually or with groups of mutual friends, even though they'd done that all the time before they ever started dating. No asking for his help with her mom, even though he'd done it so many times and she could've used the help after Tyler died. She'd just been overwhelmed with grief and anger and responsibility, and it had been too hard for her to be around Maverick and not want him. Even if she'd been convinced he couldn't give her everything she and her mother needed.
It hadn't been easy. In fact, it'd hurt almost as much as Tyler's death had. And that'd made her feel maybe more guilty than she ever had in her life.
And then Grant had stepped in, offering her kindness and support and help. Even as Maverick made a few last attempts to bridge the gap between them.
At first, staying away from Maverick had been her idea—all self-preservation. So it hadn't mattered when Grant had made it clear early in their relationship that he didn't want Alexa seeing Mav—or any of the Ravens, really—anymore. In her grief for her brother, she was already avoiding Maverick. In her need to focus on taking care of her shattered mother, she'd already cut him off. So honoring Grant's preference had been easy because it went along with her own intentions. Either way, it'd been a long time since she'd spent any time with Maverick like this.
And staying away from him had worked. It had made it easier to avoid feeling what she most desired where Maverick was concerned. Which was _him_. Had always been him. Since she'd been a girl who was way too damn young to be crushing on a teenager, and then a man, who was four years older than her. A man whose lifestyle had scared her on so many levels and made her choose what seemed like the more rational—safer—choice.
Grant Slater.
But now . . .
Today was the third day in a little over a week that Alexa had seen Maverick. And _damn_ if that desire wasn't still there. Even after all this time.
And that was a total sucker punch because she'd convinced herself a long damn time ago that she was totally and completely over this man. Emotionally. Physically. In every way.
Maverick took a fast turn, tilting the Harley at what seemed to Alexa to be a dangerously low angle. It'd been years since she'd ridden a motorcycle, but when they'd been together she'd done it all the time with him. As if on instinct, her body leaned the way it needed to handle the turn. Still, she clutched her arms and legs around him tighter, and his laughter floated back to her on the wind. And even though her heart was racing and her belly was flipping from the unexpected move, Alexa had a big smile on her face, too.
She and Maverick had sometimes butted heads, but mostly things between them had been easy, natural, effortless. She hadn't realized just how true that was, but being with him again held a mirror up to her relationship with Grant. And maybe the comparisons were unfair since they were working on merging their lives as soon-to-be husband and wife, something she and Maverick had never broached during their time together, something Maverick had said he wasn't sure he'd ever even want. Not after how his parents' marriage had turned out. And it was far easier to be fun and lighthearted when forever wasn't on the line. Right?
Right. _Stop talking yourself into thinking there are problems where none exist, Alexa_. Maybe Grant was right. He said all the time Alexa worried and overthought things too much. This sure seemed like proof.
Gah. Why hadn't she just stayed at her mother's?
"You up for going fast?" Maverick's voice yanked her from her thoughts.
Without really thinking, the old answer she used to give when he asked that spilled out of her mouth. "Always."
Nodding, Maverick smiled over his shoulder, and then he turned off the busy commercial route that led most directly to the Raven Riders' compound and toward the more rural back way to his house. It didn't take long to reach the open country road, and then they were _flying_.
The roar of the engine. The rush of the wind. The sheer power and speed of the bike.
Joy bubbled up inside Alexa until she was smiling and laughing. She wished she could throw out her arms and tilt her head back and let herself just float on the wind. She settled instead for letting out a loud _woohoo_ that made Maverick laugh and had him giving the bike just a little more speed.
A nagging voice in the back of Alexa's head reminded her how dangerous a bike could be. Doctors didn't call them "donor cycles" for nothing. Motorcycles versus . . . just about _anything_ was more likely than not going to be worse for the motorcycle, which Tyler's death had proven. It had just been him versus a wet mountain road and a guardrail. And he hadn't survived it. Once, that had terrified Alexa for Maverick. Made his bike-riding a risk her heart couldn't tolerate. And there was still some fear there.
But there was more than that. There was just a sheer exhilaration, too. When was the last time she'd felt this alive, this free, this . . . happy?
Maverick slowed the bike as they approached the back driveway into the Raven Riders' compound, off of which his house sat. Maverick was one of the few members of the club who lived on the Ravens' compound, a privilege reserved for those related to the club's founder, who was also Maverick's uncle. Though she'd been young, Alexa could still remember the day Maverick and his mother moved into the old cottage on the club's land to get away from his abusive father. Bunny had moved out years ago into her new husband's house, but Maverick had stayed here, content to keep his life centered around the club and starting his then-new custom bike-building business.
The forest hung all around them as they approached a secured gate. Maverick swiped a card and they sailed through, then took a turnoff that went farther up the mountain. The trees opened up on Maverick's house, an old two-story white cottage with a killer view over a small pond and the valley below.
Alexa hadn't been there in years, which meant every change Maverick had made to the place jumped out at her. The house had a fresh coat of paint and a new roof. New landscaping lined the front of the house. The circular driveway had been paved. The detached garage had new doors. The place looked . . . fantastic. Homey and comfortable and charming.
The bike came to a stop in front of the garage as one of those doors rolled up, and then Maverick eased it into the empty space inside and killed the engine. Alexa pulled her helmet off and handed it to Maverick.
He smiled at her as he helped her off, his big hand warm and strong around hers. "Good time?"
She couldn't help but smile back. "Really good time."
"Good." He dismounted the bike and raked his hands through his hair, and everything about his actions was so freaking sexy. The way his big body moved. The way his T-shirt rode up, revealing a sliver of toned abs and a dark blond happy trail just above the waistband of his jeans. The way he grinned at her all smugly and annoyingly and knowingly when he caught her watching him.
Alexa turned toward the open garage door, needing a break from touching and admiring Maverick freaking Rylan. She scanned her gaze over his house and yard. "Your place looks great, Maverick. You've been doing a lot of work."
He came up beside her. "Checking things off my project list one at a time."
The house had always been charming, but the TLC he'd put into updating it gave it a curb appeal it never had before. "It really shows."
"Yeah?" he asked, pride clear in his tone. "You should see the inside."
Truth be told, the designer inside her was seriously tempted. As was the woman who'd spent many nights here in another lifetime. But Alexa had told her mother she'd be back in less than a half hour, and a part of her also wasn't sure it was a great idea to be alone with Maverick at his place. Not after what she'd been feeling while riding with him. What she was _still_ feeling . . . "I'd love to, but—"
"Right. I know," he said abruptly. "We gotta get back. Another time. Maybe." He turned toward the pickup.
A ribbon of guilt curled inside her belly. "Yeah," she said, not at all believing there'd be another time. Which was probably for the best.
Heading to the driver's side of his old truck, Maverick said, "Well, hop in and we'll take off. Not as fun as the Night Rod but it'll get the job done."
Alexa slid into the passenger seat and closed her door. "That's all that matters. If I don't keep things under control at the house, Mom will just fill it up. So it's good to get rid of what we took out right away. I can't afford for her to lose this place, too."
Maverick gave her a hard look. "Why would she lose it? I thought Slater owned her place."
Alexa didn't bother to ask how he knew that. The one thing Grant and Maverick had in common was that they were both networked into just about everything that happened in Frederick. Grant because of his business activities. And Maverick because of the Ravens' businesses, not to mention the club's long history in the area. "He does," she said. "But he doesn't know just how bad it can get. It makes me worry."
Maverick frowned. "Well, there's no way he'd ever put her out on the street. She's your mother. He has to know how much she means to you, so I can't imagine you need to worry. You're finally set where she's concerned, Alexa. I know that means a lot to you."
The words nearly stole Alexa's breath. Mav didn't like Grant. It didn't take a genius to figure that out or know why. So it was a really generous thing for him to say about her fiancé—and really . . . mature, too. She wasn't sure the old Maverick could've offered her that kind of understanding. It meant a lot to her. More than that, it almost sounded like he understood why she'd made the choices she'd made. Or maybe that was her wishful thinking reading into what he'd said. She hoped he did understand. Because Alexa had never wanted to hurt Maverick, even though she knew she had.
Hell. She'd hurt herself, too, hadn't she? She knew that. Now. Being around him again made her wonder how she'd ever forgotten just how much.
Alexa just hoped Maverick was right about Grant. She wished she knew for sure. The fact that she _didn't_ know? She refused to analyze too closely what that meant. So she just said, "Yeah. Thanks."
"Don't have to thank me," he said, turning the key. The engine echoed inside the old garage as he backed out of the spot. Then he was putting the truck in Drive and pulling around past the front of his house.
"I do," she said, studying his roughly handsome profile as he sat behind the wheel. "You didn't have to do all this today. I appreciate it." She hadn't realized how much she needed some support in taking care of her mother until she had Maverick working at her side.
Mav gave her a long sideways glance. "You know I'm always here for you and your mom, right? You and her and Ty were like family to me for a lot of years. That didn't end just because we didn't make it. Understand?"
After a moment, Alexa gave a tight nod, and then she had to look away, out her window at the passing trees. So that Maverick didn't see just how much his words meant to her. Or ask why she suddenly had tears in her eyes.
Because she wasn't sure she knew, either.
## CHAPTER 9
I have a surprise for you," Grant said almost as soon as Alexa returned home that afternoon. Wearing an expensive blue dress shirt with the sleeves rolled up and the collar open, he met her in the kitchen and pulled her into a hug and a long, lingering kiss.
Registering his good mood—for which she was grateful given how much later she was than usual—Alexa relaxed into his embrace. "You do?" she asked.
"Mmhmm." Finally, he eased away from the kiss. Tucking some loose strands of hair behind her ear, he smiled. "You're always so stressed when you come home from your mother's, so I'm whisking you away for an overnight getaway. We'll relax, eat great food, take a sail. Just you and me."
"Wow, really? That sounds amazing," she said. And so much like the kinds of things he planned for them early in their relationship. He used to take her away on trips and weekend getaways all the time the first year or two that they dated. Weekends in New York City, Atlantic City, and the Massanutten ski resort in the Shenandoahs, and surprise weeks in Mexico, Paris, Rome, to name a few. They still traveled sometimes, but now his business and her school and job responsibilities made it harder to get away as often.
"Having you all to myself is what sounds amazing," he said, nailing her with an intense stare. "No wedding planning, no jobs, no school, no mother, no life obligations at all. Just you and me."
A fleeting wave of stress passed through her, because she had been planning to spend Sunday running errands for the wedding. Picking up gifts for Grant and the bridesmaids, who were mostly colleagues at work. Dropping by the reception venue to finalize menu details. Finding a few last things to pack for their honeymoon in Cozumel, the first place they'd ever traveled out of the country together years before. Neither of them having parents who could help with the planning was a mixed blessing—on the one hand it meant they didn't have to try to please a bunch of people besides themselves, but it also meant she didn't have any help. And all of that was in addition to the reading she still had to do for Tuesday night's class. The list was never-ending.
But Grant was right. They could use some time in the midst of all the madness to just focus on each other. And it would help screw her head back on right, too. "That _does_ sound amazing. Thank you, Grant. This is so thoughtful." She pressed her lips to his. "I'll just pack a quick bag."
"Already done," he said, nodding to something behind her.
She turned to see two overnight bags in the doorway to the foyer. Alexa laughed. "My, someone _is_ eager, isn't he?"
He kissed her roughly, in a way that made her feel claimed. "Where you're concerned? Always." He swatted her on the butt. "Now go get pretty. I'll pack up the car."
After cleaning all morning, Alexa guessed she must look a mess. "Okay, I'll do that." Excitement and anticipation flitted through her. No doubt wherever he was taking her would be romantic and luxurious and sure to take all her cares away. With a quick last kiss, she pulled away.
He caught her wrist and gave her an intense look. "And wear that white sundress for me."
"You like that white one, huh?" she asked, batting her eyelashes.
"I like tearing that one off of you," he said. "I like playing with the knot at your neck and imagining undoing it whenever I want to. Wherever I want to." His gaze flashed hot, and her belly answered with a flutter of butterflies. Whether nerves or arousal, she wasn't sure. "Now, go."
In their room, she found the dress, a pair of panties, and a pair of tall wedges set out at the bottom of the bed, all folded and lined up with excruciating precision. It reminded her of his desk at work, where the few items he kept out on it had exact placements, everything all symmetrical and aligned. She smiled and shook her head. Pure Grant. Dotting every i and crossing every t, and not leaving anything to chance. In his personal life just as in business, he took charge and control of every detail.
She quickly showered, dressed, and styled her hair in a chignon that allowed her to get away without blowing it fully dry. Grant had left her makeup bag on the counter, so she put on her face and a pair of earrings, and then she was ready to go.
Alexa gave herself a once-over in the bathroom mirror. She looked summery and put-together. She wasn't a hundred percent comfortable going without a bra, but the halter top and low back on the dress didn't really allow for one. And Grant _really_ liked that about the dress, anyway. So she could be okay with it for the day.
"I'm ready," she announced when she returned to the kitchen.
Grant gestured with his finger for her to turn for him. She did. "Even more beautiful than I remembered you being in that dress," he said.
The ride was easy and the time passed quickly with Grant filling her in on how a couple of his projects were developing. The only weirdness happened when she brought up something she probably shouldn't have.
"Hey, I meant to ask you. What happened to all the newspapers this week? You never left them for me."
"What do you mean? They were on the desk in the kitchen," he said.
Alexa frowned. "Really? I didn't see them. You usually put them on my desk when you're done with them."
"Well, forgive me for being a little busy, Alexa." He gave her a look.
"No, no, of course. I'm sorry. I didn't mean anything by it. It just occurred to me that I hadn't seen them." She grasped his hand and gave it a squeeze.
"Why is it such a big deal, anyway?" he asked, gaze straight ahead.
"Oh, I just happened to see one of the papers at Mom's and I hadn't heard anything about that horrible shooting and attempted kidnapping incident that happened out at Green Valley. That's all. I was just surprised I hadn't seen anything about it."
He shrugged. "The Ravens are such bad fucking news for this town. Violence at their hands is hardly a surprise."
Schooling her reaction, Alexa swallowed. "I was just curious about it. That's all."
"Well, I can't help it if you didn't see the papers. It's not like I was hiding them from you," he said. His cell rang, and he picked it up on the car's Bluetooth. "Grant Slater here."
When he hung up a few minutes later, she apologized again, wanting his good humor back more than anything. Then she steered the conversation to safer subjects, and his mood rebounded.
Two hours later, they were driving through the quaint, small town of St. Michaels on the Eastern Shore of Maryland. They passed antique shops and small eateries on the main street and finally turned onto a long brick-paved and tree-lined driveway that eventually opened up onto a broad courtyard surrounded by a series of white buildings.
"Oh my God. Is this where they filmed _Wedding Crashers?_ " Alexa asked, peering out the window as Grant pulled the car up to the lobby.
Grant laughed and gave her a teasing look. "Leave it to you to know a pop culture reference for one of the nicest resorts in all of Maryland."
The valet opened her door. "Welcome to the Inn at Perry Cabin," a man wearing a gray-and-black uniform said.
"Thank you." She accepted his hand to rise out of the Mercedes.
Grant came around to guide her through the doors with a hand low on her bare back. As they walked inside, he leaned down and pressed his mouth close to her ear. "Every man is looking at this dress and wishing they could be the one to take it off of you. But that pleasure is mine alone. _You_ are mine alone."
"Of course," she said with a shiver as she peered up at him.
"Don't ever forget it," he said, his expression serious, his gaze piercing.
Before she could respond, they were standing at the reception desk and Grant was checking them in. His words echoed over her body along with a rush of goose bumps. His claiming words had been arousing, but that last line left her feeling . . . odd. Why had he felt the need to say that? She'd been with him for nearly five years. She worked for him, lived with him, wore his ring, and planned to marry him. There wasn't one part of her life that wasn't bound up with his. _Of course_ she was his.
Except, just then, images of another man flashed through her head. Maverick. Hugging her mother. Smiling over his shoulder at her on his bike. Raking his hands through his hair. _Oh, God. Does he know Maverick was at my mother's today?_
Fear and guilt sloshed through her belly. Although, why should she feel guilty? She hadn't invited Maverick to come, and certainly nothing had happened between them. _Then why aren't you volunteering that you saw him?_
"Alexa?" Grant asked.
She blinked up at him, and the look on his face told her he'd called her name more than once. "Sorry," she said. "Lost in a daydream."
Grant shook his head and gave her an indulgent smile. "You? Always. Let's check out our suite," he said. "I've reserved a private dining room for us and we have dinner in an hour."
"Wow, you didn't have to do all that, Grant. I'm sure the regular restaurant here is lovely," Alexa said, taking in the light and airy décor. It was a mix of colonial charm and nautical colors and accents. Large windows provided expansive water views throughout the whole back side of the building, and comfy sitting rooms and reading nooks appeared around every corner.
"Of course I didn't have to," Grant said as they made their way down a long hallway, "but tonight I don't want to share you. I just want you all to myself." Finally, he gestured toward a door. "This is us."
The suite was beautiful and spacious. The large sitting room with overstuffed couches and chairs arranged around a fireplace opened onto a porch that overlooked the water. A bowl piled high with fresh fruit sat on the coffee table. Italian marble covered every surface in the bathroom, and a massive four-poster king bed dominated the bedroom. "This is amazing, Grant."
He came up behind where she stood in the doorway to the bedroom. His hands circled her wrists. "If I wasn't so hungry, I'd have half a mind to handcuff you to that headboard for the rest of the night and have my way with you."
Alexa leaned back against him. "You don't need handcuffs to have your way with me," she said, putting a lightness in her tone she didn't quite feel. Grant had always made comments about not wanting to share her and wanting them to be alone, but suddenly it felt different. Less sexy than controlling. Less about desire and more about ownership.
"You know what? You're right." He grasped her by the shoulders and manhandled her into the bathroom.
Laughing, Alexa worked to keep from tripping in the high wedges. "What are we doing?"
"Getting what you wouldn't give me yesterday morning," he said, pushing her in front of one of the sinks in the long vanity. Confused, Alexa could hardly keep up with the rush of his movements. And then he was flipping up her skirt and tugging at her panties and pushing her upper body down with a palm planted in the center of her back.
She braced her hands on the marble. "Grant—"
"Been thinking about this," he gritted out. "I need it, Alexa. Just like this. Give it to me."
The blunt head of his erection pushed against her core, driving into her. Or trying to. She wasn't ready and her body wasn't opening for him. He pushed harder. As hot as his urgency and need were, the friction was uncomfortable and his words were confusing.
Grant smacked her ass, once, twice, and the sound reverberated within the room. "Let me in, goddamnit." He surged forward, penetrating her.
On a whimper, she pulled away, or tried to, but she couldn't move much with the way he had her pinned against the sink. If he'd just give her body a chance to respond. "Grant, wait—"
"Be still," he said roughly, his hand pushing her down harder. He withdrew, and she heard him spit. And then he was right back inside her again. "Don't make this harder than it has to be." Her fingers clenched and unclenched against the smooth surface of the counter, and she let out a moan at the burning friction. Her thoughts were a confusing, conflicting whirl, her heart raced in what felt a lot like fight or flight, and she almost perceived the moment as if she were watching what was happening rather than experiencing it.
What _was_ happening?
Finally, her body provided the slickness that eased the way, but the soreness didn't fade. "There it is," he groaned, palm still holding her down.
He tugged hard at the tie of her dress around her neck, the falling material exposing her breasts, and then his hand fisted in her hair, forcing her head back and her gaze to the mirror. "I put you together. I can mess you up, too," he said, meeting the reflection of her gaze. A cold sweat broke out across her body, making her dress cling. "So fucking hot." He slapped her ass again. Hard. The sting made her cry out.
But otherwise, Alexa was . . . frozen. She couldn't talk. Despite the emotions blowing up inside her, she couldn't cry. She couldn't even make sense of what they were doing, exactly. It had started out kinda hot, but it was also . . . scary. She liked rough sex and she loved a man who took command in the bedroom, but why was that the only way they ever came together anymore? And why did this feel like something else?
Like . . . punishment.
Like control.
Like he was showing her who had it. And that she didn't.
Except that was ridiculous, wasn't it? She liked it rough. She liked hair pulling and spanking and being restrained and having it hard and fast. Exactly what this was, and what they'd done many times. And Grant was her fiancé. They were getting married. She loved him.
Hand tight in her hair, he drove into her again and again, his expression harsh, his pace fast and frenzied, her hips and thighs knocking into the front of the counter each time. She could already tell that she'd be bruised. And sore. Despite the fact that she was wet, arousal wasn't a part of this for her. Shouldn't it have been?
Finally, Grant finished with a groan, his grip almost painfully tight on her hip. Panting, he held himself inside her for a long moment. Then he bent over, pressing her into the hard counter, and kissed her shoulder. "Dinner's in a half hour, so you have time to get cleaned up," he said. He pulled out, causing her to wince, and then he stepped to the toilet.
"Okay," Alexa whispered. She grabbed tissues from the fancy silver dispenser on the counter to clean herself up, avoiding her own gaze in the mirror as she did.
Because she wasn't sure what she'd see in her eyes, or how her expression might reflect the queasiness taking root in the pit of her stomach.
"Alexa?" Grant asked from behind her. She whirled, startled. "Everything all right?"
"Yeah," she said, clutching the counter behind her to hide her shaking hands. "Just catching my breath." _And trying not to lose my mind_.
He gave her a wolfish grin. "Okay," he said, and then he walked out into the sitting room.
Alexa stared at the empty place where he'd been standing. _Pull it together, Al. You're fine. Everything's fine_.
Except that feeling of dread she'd been flirting with for the past couple of weeks was stronger than it had ever been before.
## CHAPTER 10
The biker rolled into the big parking lot in front of the Raven Riders' clubhouse and brought his Harley to a stop. Maverick gave a wave and waited as their newest prospective member hung his helmet on his handlebars and dismounted.
"Maverick, thanks again for sponsoring me, man," Mike Renner said. Stocky, with reddish-brown hair and a close-shaved beard, the guy was a few years younger than Maverick's thirty-four and had been a club Hang Around for almost a year. Mav had met him about three years before when he'd bought a custom bike from him, which was when Mike had first become interested in the club.
"Glad to have you," Maverick said, shaking Mike's hand. "Do me proud."
Mike grinned. "I will."
Nodding toward the front steps of the big two-story clubhouse, Maverick led Mike inside. "The induction ceremony is pretty low-key. Some words from the prez, advice from some of the patched members, and then you get your cut." Maverick winked at the other man. "And then the hazing—I mean, the fun, begins."
Laughing, Mike nodded as they stepped into the big front lounge that had once served as the lobby and reception area to the inn when it had been the center of a mountain resort decades before. On the wall by the mess hall hung all the members' photos, including Jeb's. Above the old registration desk, foot-high words carved into the wood spelled out the club's motto, _Ride. Fight. Defend._
Tempting smells that had Mav's stomach growling came through the mess hall door—proof that Bunny, Haven, and Cora were putting together a fantastic, celebratory Sunday dinner—but Maverick led Mike in the other direction, toward the big rec room where about thirty Ravens were waiting.
A round of applause and cheers erupted as Maverick and Mike walked into the big rectangular room and made their way toward Dare, standing in front of the bar.
"Dare," Maverick said when the ruckus died down, "I'm proud to sponsor Mike Renner for prospective membership into the Raven Riders Motorcycle Club." After the insanity of Friday night's raid on the Iron Cross, it felt really fucking good to have something positive for the club to focus on.
Dare reached behind him and grabbed a denim cutoff jacket with black leather patches sewn on here and there. "Mike, are you interested in becoming a member of the Raven Riders?"
"Yes, I am." Wearing a black button-down shirt and jeans, he stood solemnly with his hands clasped in front of him.
"Are you committed to learning about the club, supporting its activities, being a brother in actions, not just name, protecting the club and its business, and putting your loyalty to the Ravens above all else?" Dare asked, nailing the younger man with his dark stare. The words sucked Maverick back into his own induction ceremony at the age of eighteen. He could still remember the pride he'd felt at standing next to Dare as Doc asked them these same questions and they became brand-new prospects together.
Mike nodded, his expression serious. "I am." An approving murmur rolled through the group.
"The Raven Riders is more than just a club. It's a brotherhood of men who share similar interests, interests that include standing up for those who can't stand up for themselves. It's a family of choice, made up of the brothers and their kin—a family that protects and takes care of its own. We ride, we fight, and we defend as one. If you want to be a part of the Raven Riders, declare your intentions and accept your prospective status by putting on this cut." Dare held up the jacket for all to see. Unlike the cuts of the fully patched members, a prospect's cut lacked the Raven Riders' patches and name and location rockers, or badges, on the back. Instead, both the back and the chest simply read, _Prospect_. The full patches had to be earned. "Wearing it proclaims your loyalty and membership until such time as we deem you fit for fully patched status."
Mike stepped forward, and Dare slid the cut onto his bulky frame. Maverick grinned as welcoming cheers broke out all around.
The prospect came to Maverick first. "Thanks, Maverick. Really proud to be here."
"We're glad to have you," Mav said, glad to have something to celebrate, something that let them all focus on the future. One by one, everyone shook Mike's hand and personally welcomed him, and grabbed one of the beers Blake had served up on the bar.
"Words of wisdom for our new brother?" Dare called out.
"The more you contribute, the easier it'll be to become a full member," Bear said. Words of agreement rang out.
"Don't be a fuckup," Phoenix said with a grin. He raised his beer in a salute and gave the man a wink. Guys raised their beers all around to that one, and Mike took it in good humor, laughing and nodding.
A man of few words, Caine shifted on his boots. His ice-blue eyes zeroed in on Mike. "Never discuss club business with anyone. Keeping your mouth shut is always the wise choice if you're not sure."
Doc rose out of his chair, no doubt the hip and knee he'd had replaced a few years before bothering him as they often did late in the day. He ran his hand over the white hair of his beard as he looked at Mike. "As Dare said, this is a family. So start by getting to know everyone. Learn their names, and the names of their wives and kids. Figure out who can use a helping hand, and offer it. If a brother breaks down at two in the morning, go help, no questions asked."
"Amen," someone said. Others echoed the sentiment.
"Always have your brothers' backs," Jagger said. "No matter what."
Maverick nodded, agreeing down deep. The loyalty and brotherhood he'd found in this club were just two of the reasons he'd built his life around it. He hadn't had a good relationship with his father, which just proved that you could create a family just as well as be born into one. No doubt many of the guys felt the same way because there were a lot of loners, outcasts, and men estranged from their families for all kinds of reasons standing in this room.
Mav turned to Mike. "Whenever you're wearing that cut, you're not just representing yourself. You're representing the club. And me. And every man in this room. Never forget that."
Mike nodded, taking it all in.
Dare raised his beer. "To brotherhood, club, and family."
"Brotherhood. Club. Family," everyone called out, following the toast. Maverick took a long pull off his beer and clapped Mike on the back. "Welcome to the Ravens."
The man smiled and nodded, and soon got pulled into a conversation with a couple other brothers. Music turned on. Laughter rang out. On the other side of the room, someone racked up the balls on one of the pool tables.
"The club needs this," Dare said, coming to stand beside Maverick. Was his face pale, or was Mav just imagining it? The guy hadn't seemed himself since Friday night's fight. "New blood. Especially with everything that's gone on."
"Yeah," Maverick said, eyeballing his cousin. "Fresh enthusiasm never hurts. Especially after the other night."
"Amen to that," Dare said. "At least it appears we put the Baltimore issue to bed once and for all. Nick emailed news from Chen this morning. They've confirmed that most of the Iron Cross are either in custody, dead, or scattered."
Mav sighed as a weight lifted off his shoulders. Having that threat gone? Knowing those who'd hurt his mother wouldn't be able to do it again? Yeah, he couldn't help but feel good about that. "Glad that's behind us."
Jagger joined them, his fingers tapping out a beat against the back of an iPad he held.
Maverick clasped hands with the guy. "What's up, Jag?"
He raked longish brown hair back from his face. "The carnival's what's up. And I'm nailing down volunteers for shit while everyone's here."
"Damn, Jagger. You work fast," Maverick said. The guy always went above and beyond for the club, which was why their operations at the racetrack were such a success, recent catastrophe notwithstanding. "What do you need?"
"Eh, it's not all me. I've had help," he said, tapping on the tablet. "Okay, I have tickets, parking, and meeting the race car drivers almost covered. But I need setup and tear-down and dunking booth victims." Jagger smirked.
"Count me out for that shit," Dare said, and then he rushed to add, "Wound. Very serious wound. Can't get it wet. Doctor's orders."
Maverick threw him a look. "Doctor also said to keep your stubborn ass off your ride. Expect me to push on that next time." Dare flipped him off, clearly pleased that he had a reason not to get wet. Shaking his head, Mav said, "Count me in for the damn dunking booth. And put both of us down for setting up." Dare gave a nod.
"Done," Jagger said, then he threw Dare a mischievous grin. "Oh, by the way, D."
"Yeah?"
"Haven volunteered for the dunking booth."
As Dare's expression visibly darkened, Jagger winked at Maverick and took off. Dare mumbled something under his breath about good hearts and wearing a parka that made Mav laugh.
Dare scrubbed at his face and released an agitated sigh. "You heading out?"
_To Alexa's_ hung unspoken in the air between them. Maverick tipped the bottle to his mouth and took a long draw. "Nah. Here with all you ugly motherfuckers is where I want to be for right now." Seeing her yesterday had been fantastic, but it had left him feeling hollower than he wanted to admit. And he wasn't leaving Dare when the guy looked like hell.
With a wry grin, Dare clapped him on the back. "Good. Then let's find a place to plant our asses, because my goddamned side is killing me."
The admission caught Maverick by surprise. "You—"
"Fine. Just overdid it. Which makes me sound like a fucking geriatric," Dare groused as they cut through the crowd to one of the big groupings of leather couches.
"A geriatric dating a twenty-two-year-old hottie. Not bad," Maverick said with a wink.
Dare glared at him. "Shut the fuck up." He dropped into the corner of one of the couches on a grimace.
Maverick looked at him, and the realization hit that Dare wasn't joking around like he had been. "Shit, D. You know I didn't mean anything by that." He sat facing Dare. "You gotta know I respect the hell out of Haven. As long as I live, I will _never_ be able to forget seeing her save your life and protect your body with hers. Damn. I'm sorry."
Waving him off, Dare shook his head. "Forget about it."
"You're really serious about her." Maverick knew he was, but he hadn't realized just _how_ serious. Clearly. Dare just eyeballed him. "How serious? Talk to me, man."
Dare nailed him with an assessing stare, then gave a little shrug with his uninjured shoulder that belied the significance of what came out of his mouth. "Forever serious."
The words impacted Maverick square in the chest. "Have you asked her—"
"No," Dare said, glancing around to see if anyone might be listening in. But the music and other conversations gave them cover in the midst of the chaos. "I'm in no rush. Neither is she. But when the time is right . . ." He shrugged again.
"Fuckin' A," Maverick said, managing a smile. "I'm damn happy for you. For both of you." He meant it, especially since not two weeks ago, Dare had convinced himself that he'd have to send Haven away—to keep both her and the club safe. Now, nothing stood in their way. Just as it should be.
Which just cast a big, ol' glaring spotlight on all the things Maverick didn't have, didn't it? Probably made him an asshole to spend one second of time feeling regret in this moment when his best friend admitted to having found everything he ever wanted. Maverick knew it did. But he couldn't help it.
"Thanks," Dare said. "But do me a favor and keep your trap shut about it. I don't want anyone teasing and making Haven feel uncomfortable. Or whatever." Mav nodded. "And besides, we got word on Friday that Haven may be due some inheritance from her father's estate, assuming it gets released by the Feds. So we got things to work through."
"Ain't that some shit," Maverick said. "What's she thinking?"
"She was stunned at first. Then said she wanted nothing to do with anything that'd been his. But then she asked me if the Ravens could use the money to help our protectees." Pride and affection came over Dare's expression.
"And then she was willing to consider it?" Mav asked. It was just like Haven. He still remembered watching her tend to Alexa's wounds the day she came here. He'd already liked Haven then, but that was the day she earned his undying loyalty.
"Yup," Dare said. "Cora told her she should use it to open her own bakery business, which I think is fucking brilliant, but Haven said she'd rather use it to help other people like herself. We'll see. Either way, it's up in the air for now."
"You got a good one, D. For real." Maverick took another pull from his beer.
"The best." On a sigh, he relaxed back into the leather.
"Hey, speaking of Cora, where has she been?" Maverick asked. "I haven't seen her around lately." Cora Campbell was Haven's best friend from back home in Georgia. The pair of them had run away together from some bad situations and landed with the Raven Riders last month. From what Maverick understood, Cora had played a big role in helping Haven escape from her father, and she was also sarcastic and funny as hell. What Maverick knew of her, he liked.
Dare peeled at the label on his beer bottle. "She's been at Slider's place watching the boys. He got in a jam because the lady who usually watches Sam and Ben has been sick."
"Huh. Kinda awesome of her to help out like that," Maverick said. Sam "Slider" Evans was a longtime member who'd lost his wife to breast cancer three years before. As far as Maverick could tell, the old Slider they'd all known—the one who'd wiped out on his bike without injuring himself or his ride and got up laughing about it—no longer existed and never would. The guy's pain was so tangible that he wore it like a shadow. He hadn't attended Church on Thursday or participated in their ride to Baltimore that next night, and Mav hadn't been the least surprised at either. His two boys seemed to be the only thing that kept him functioning.
"That's for damn sure." Dare nodded, his concern about Slider clear in his expression. "Haven said Cora's happy to do it. And I appreciate the hell out of anything that takes even a little weight off of Slider's shoulders."
"Damn straight," Maverick said. And the fact that Cora was doing just that was one more reason to like her.
Dare shifted and propped his boots up on a table, his ankles crossed. "I'm thirty-fucking-seven, Maverick. I didn't think anything like what I have with Haven would happen for me. In fact, I was sure it wouldn't."
Maverick sagged back, his hands clasped across his stomach. "Yeah."
"Hey," Dare said, tagging his arm. "You get what I'm saying?"
Frowning, Mav just stared at the guy. He wasn't the slightest bit interested in having this conversation.
Rolling his eyes, Dare nailed him with a pointed stare. "When you gonna fight for what you want?"
Like it was that easy. "Come on, D." Maverick looked away, not wanting his cousin to see the resentment he probably wore for even bringing it up. For making Maverick confront yet again the fact that he'd given his heart to someone who didn't want it. And he couldn't seem to get it back.
Dare heaved a belabored breath. "Fine. I'm just saying."
"Well, don't. Let's move on to religion or politics or something." Because hell if he wanted to keep pricking at all the sore places inside himself.
"All right," Dare said, and the concern and resignation in his voice was a total kick in the ass. But whatever. Mav's life was what it fucking was. And for the most part, it was good. Damnit, before Alexa had shown up the week before last, he probably would've said it was great. This place, these men, Bunny and Rodeo and Doc and Dare— _this_ was Maverick's home, his family. He didn't need a woman to give him a sense of belonging.
He really didn't.
And it made him resent her for stirring up all this shit for him again after so long. For reminding him of the feeling he'd battled five years before that a part of him had died and been put in the ground right alongside Tyler. Because he'd lost his friend. And then he'd lost his woman. And his love for her had been killing him ever since.
That resentment? That was good. He clutched onto it. Because it was a helluva lot easier to be angry than to be hurt. It was that anger that had him making a resolution, one that self-preservation meant he was gonna have to stick to once and for fucking all.
He'd do right by Tyler's memory and the history he shared with Alexa and keep an eye out for her as much as he could. Just in case his gut was reading things right. But in thirteen days, she'd be walking down an aisle and into another man's arms.
Which meant Maverick had just under two weeks to let Alexa Harmon go. For good. Because he couldn't keep wanting something he could never have.
## CHAPTER 11
The gown was quite possibly the most beautiful thing Alexa had ever seen. Standing on the raised step in front of the angled mirrors, Alexa couldn't stop admiring her wedding dress. It had a voluminous A-line skirt made of tulle, a long chapel train, a satin bodice with a flattering sweetheart neckline, and a wide champagne-colored ribbon at the waist. Classic. Romantic. Ungodly expensive, but Grant wanted to spare no expense on their wedding.
_Pick something beautiful. Something that will make me proud._
That's what he'd told her as she'd set out to shop for a dress, a personal shopper he'd hired at her side. The woman had steered Alexa away from what she'd deemed the _pedestrian_ dresses to more exclusive, one-of-a-kind designer gowns—following instructions Grant had apparently given her. He wanted their wedding to be the swankiest social event Western Maryland had ever seen. And since so many business associates would be there, she understood why, even though all of it was so much more than what she needed to be happy. Alexa had dreamed of having a beautiful wedding as much as any other woman, but it was the marriage that came after and the joy of building a life together that was most important to her. Because she wanted what her own parents never seemed to have had.
The seamstress inspected each alteration she'd made and frowned as she examined the hidden back zipper. "It is a little snug here," she said, running her fingers down Alexa's spine.
The French fries Alexa had gotten at the drive-thru on the way to the fitting sat like a rock in her stomach. Snug . . . because she'd been eating junk food nonstop the past three days. Ever since having sex with Grant in the bathroom at the inn had left her feeling so strange. So unsettled. So . . . unsure.
Even more than she'd already been. She just wasn't sure if she was making something out of nothing. _Or allowing Grant to walk all over me_ , a little voice whispered.
The older woman met Alexa's eyes in the mirror. "Hmm. Maybe I should—"
"Don't worry," Alexa said, shaking her head and putting on a smile despite the tendril of panic snaking through her veins. "We just did some celebrating this past weekend. I overindulged a little. But it'll be fine by the wedding." Alexa would make sure of it. Grant wanted everything to be perfect—including her. Maybe even especially her. And she certainly wouldn't make him proud if she was busting out the seams of her six-thousand-dollar gown.
So stupid. Why hadn't she thought more about the fitting today?
"Okay, then. If you're sure," the seamstress said with a smile. "Let's get this off of you. You're all set."
An hour later, Alexa was back at the office, that rock still heavy in her stomach.
"Hey, sweetie," Christina Lee said from behind the ornate reception desk as Alexa entered the building. Alexa and Christina had started working at Slater Enterprises at nearly the same time—about a month before Tyler's death, and they'd been friendly ever since. Friendly enough that Alexa had asked her to be her maid of honor, and Christina had agreed. They might've even become much closer, but Grant's pursuit of Alexa back then seemed to have scared some people off of befriending her, like they were afraid Alexa might report back on them to the big boss. But that was better than the people who resented her for their relationship. "How'd the fitting go?"
Alexa leaned against the high desk. "Good. It's so gorgeous I can barely stand it."
"Of course it is," Christina said, grinning. "Less than two weeks now."
Something which should've unleashed excitement inside of her, but . . . didn't. "I can hardly believe it," she said, forcing a smile of her own.
"What's left on the list that I can help with?" she asked.
"Honestly, not too much. I visited the venue yesterday and confirmed the menu and did a little shopping last night. So things are on track, but let me check with Grant about his schedule. We should get together for dinner one night this week." Assuming he had a late meeting so she wasn't taking time away from him.
"You know I'd love that," Christina said. A phone call came in through the switchboard. "Oh, duty beckons."
Alexa gave a little wave and made her way to her office on the second floor. She probably ought to search out Betsy, Maggie, and Ellen, other lunch friends at work who had agreed to be bridesmaids, to keep them in the loop of wedding goings-on, but she found that she wasn't as up to it as she wanted to be. There'd be other chances to tell them about her fun weekend and the amazing party favors that'd finally come in and just how gorgeous her dress really was another day this week.
Instead, she dropped into the chair at her desk and threw herself into reviewing the construction punch list from the model home she was decorating. The place was amazing—the kind of home she'd always dreamed of living in. Spacious and light and airy, comfortable and well designed, chic without being stuffy. She'd spent the morning on-site doing a walk-through with the foreman and compiling the list which noted incomplete installations that needed to be corrected and incidental damage that needed to be fixed. Furniture to stage the model would be arriving this Thursday, so she was hoping the guys would work through most of the list before the deliveries began. All of which was leading up to her deadline next Wednesday to have the place ready for Grant's stamp of approval. That gave her eight days. And today was half gone.
Her gaze slipped to the clock display at the bottom corner of her computer screen. _Maybe I should skip class tonight_.
She twisted her lips as she debated, and quickly ruled it out. It might only be her one-credit professional development course, but she was already going to miss one class during their honeymoon. And since the class only met once a week, not going would be the equivalent of missing a whole week's worth of material.
_Knock, knock_.
Grant stepped into her office wearing an exquisitely cut navy blue suit, his hand behind his back. "Alexa."
"Oh, hey," she said, smiling. "I wasn't sure if you were back yet." She'd known he had plans this morning to be out of the office inspecting an apartment complex he was considering buying and rehabbing. Alexa got up and came around her desk to him.
"I am, though I have another meeting off-site shortly. But I wanted to catch you before you left for class," he said. "And I brought you something." He handed her a single long-stemmed red rose.
The gesture reached right inside her chest. "Aw, Grant. This is so sweet," she said. It was just the pick-me-up she needed. She grasped the flower and brought the velvety petals to her nose. "Thank you."
"Anything and everything for you, babe. You know that." He pressed a kiss to her cheek, her jaw, the corner of her lips. And then his mouth came down on hers as his body boxed her in against the desk. Alexa's gaze strayed to her open office door and she ignored her discomfort over the possibility that someone might see them. It wasn't like everyone didn't know they were together, but Alexa still didn't like to flaunt it in front of their colleagues. It was one of the reasons she still drove herself to work, and definitely why she'd refused a bigger office in the executive suite. What she got here at work, she wanted to have _earned_. Maybe she'd feel differently after they were married. Or maybe not. Just when she was going to pull away, Grant did instead. "Any chance I can talk you into staying home tonight?" he asked, a mischievous smile on his face.
"I wish. But I don't want to get behind before we leave on our honeymoon. Then I'm all yours." She gave him a sweet peck on the cheek. "Thank you again for the rose. You made my day."
He stepped away and smoothed a hand down over his red silk tie. "Just as well, I suppose. The mayor's having a reception, and I should probably put in some face time."
"He's a good friend to you," Alexa said. "So go and enjoy. I won't stay after, though. I'll come right home."
"You do that," Grant said as he stepped to the door and made to leave. Then he peered back in, his smile sexy and playful. "I'll be waiting."
Alexa grinned as Grant left, pulling the door shut behind him. He could be so charming, so attentive, so thoughtful—qualities that had attracted her from the beginning. She hadn't realized how much she needed that side of him today. And it made her want to show him how grateful she was.
Which made her even more eager to have class over with.
But, fine, she'd go. At least now she had something to look forward to when she got home.
TIRED FROM CLASS and from driving home in the pounding rain, Alexa came in through the garage door to the dark kitchen and dumped her purse and messenger bag on the counter. Seeing Grant would make her feel better. Hands free, she flicked on the light.
"Grant? I'm home," she called. Her gaze snagged on the desk built into one wall of the kitchen because . . . it was totally reorganized. The mail had been sorted by size, shape, and color, the edges of the envelopes lined up exactly parallel to the edge of the desktop. The pins in the corkboard were all in neat rows, sorted by color. Grant did things like that every once in a while, but she'd learned to just take it in stride after asking about it had once caused an argument.
She turned to find Lucy curling her sleek body around the corner of the island, meowing a greeting at her.
"Hi, baby," Alexa said, crouching to give her little one some love. Sometimes it seemed that Lucy was the most dog-like cat Alexa had ever known. Lucy came when Alexa got home and when she called. She loved to play. Was generous with her affection. And pretty much always wanted to be where Alexa was.
So as Alexa went in search of Grant, Lucy padded behind her. The house felt unusually still. Quiet. The steady drumming of the rain on the roof was the only sound she heard. Grant wasn't in the living room, nor the media room, nor their bedroom. She knocked on his mostly open office door. "Grant?" she said as she peered into the darkness. His car was in the garage, so he had to be here somewhere. Her heart tripped into a sprint as worry lanced through her.
The chair at his carved mahogany desk sat empty.
Lucy hung back in the hallway as Alexa made her way across the room, turned on the desk lamp, and pulled her cell out of her back pocket.
"Just who is it you're calling?" came a voice from behind her.
Alexa nearly screamed. She jumped and whirled, her heart in her throat, the phone clutched tight in her hand. "Grant! Oh, my God. You scared me. Are you okay?" She rushed across the room to where he sat in the big wing-back leather chair in the corner. Leaning over him, she cupped his face in her hand. "Why didn't you answer me?"
He stared down at a tumbler in his hand that Alexa hadn't noticed he was holding. Bourbon, if she had to guess. A quick glance to the Chippendale table next to the chair revealed a mostly empty bottle. His lips pressed into a tight line and his brow slashed downward. He tilted the glass in his hand as if watching the amber liquid was somehow mesmerizing. When he finally peered up at her, it was as if he'd turned into another man.
Grant's expression was like a storm descending—dark, twisted, calculating. "Did you think I wouldn't find out?" he asked, the words gritty and harsh.
"Find out what?" she asked, her thoughts frozen, her stomach dropping.
"Don't play coy with me, Alexa," he said, batting her hand away from his face.
Stepping back, Alexa shook her head, dread a living thing inside her. "Grant, I don't—"
"Don't fucking deny it!" he roared, lunging up from his chair. He grabbed her by the biceps and got right in her face.
"Grant, stop. Deny what? What happened?" she asked as she curled in on herself. His fingers dug into her arms like hooks.
He shoved her free and brushed his hands down his shirt as if to straighten himself, and then he glared at her. "How do you think it felt to have one of the sheriffs tell me, in front of the mayor, that he saw you riding on the back of Maverick Rylan's motorcycle? A goddamned degenerate Raven Rider piece of shit." He spoke with a quiet reserve that was somehow scarier than when he'd raised his voice.
Alexa broke out in a cold sweat as nausea swept through her. "I . . . I can explain."
"I'm sure you can."
She shook her head and held out her hands. "It was innocent. He was helping me—"
"Oh, I'll bet. I know _exactly_ what Rylan wants to help you with." He picked up his glass off the carpet and poured just enough bourbon in it to finish in one big gulp. She hadn't even seen the glass fall, he'd moved so fast.
Alexa's mouth was so dry it was hard to talk. "He rode by my mom's house on Saturday morning and happened to see me hauling trash to the curb. I was doing some cleaning for her, and I wanted to get rid of the garbage instead of letting it sit there in a big pile. He offered to get his truck to take the stuff to the dump for me. That's all."
"What do you take me for, Alexa?" he asked.
"I don't . . . Nothing. It's true. I only went with him to get his truck because things had been really tense with Mom and I wanted a break before we got into an argument. You know how she is," Alexa pleaded, her head spinning, her heart thundering in her chest.
"Oh, I do. I live with her daughter, after all." He chuffed out a humorless laugh and shook his head. Paced back to the table and poured himself more liquor.
His words hit her like a body slam, knocking the breath out of her. Regret and guilt twisted inside her. "That's all that happened. He was just helping me."
Grant turned to her and glared, his face flushed. From anger or the alcohol, Alexa wasn't sure. "You made a fucking fool of me. Twice. First, by parading around town with a known criminal. And second, when I refuted the sheriff's words, saying, _oh, no, that must've been someone else because Alexa was at her mother's_. Only to have Davis insist he'd seen your face clearly while you'd been waiting at a red light on 15. Hanging all over another man."
"I'm sorry," she said, tears pricking at the backs of her eyes. "It didn't mean anything. He used to be very close to my mother because of how long he and Tyler had been friends. He was just helping. That's all it was." She heaved a breath, trying to calm herself down, trying to keep everything from falling apart.
"It's not how close he is to your _mother_ that worries me," Grant sneered.
Alexa's stomach dropped to the floor. "Seriously?" she asked, the thought voicing itself without her permission.
His head tilted as his eyes narrowed. "You don't agree that sneaking around behind my back and lying warrants suspicion?"
The air felt thick as she drew it into her lungs, as if her sensation of dread had taken on a physical form all around them. "I wasn't sneaking around. And we weren't doing anything wrong. We were in public. In broad daylight."
"And by your own admission, you went to his house," he said, his tone like he'd just produced the smoking gun.
Alexa put her hand to her forehead, absolutely at a loss for how to pull this back from the brink of disaster, and beyond stunned at how different this night was going from what she'd planned. She'd expected to come home and seduce Grant, to show him her gratitude for his thoughtfulness earlier, to make love to him slowly and thoroughly until they were both sated and sleepy. "I . . . I don't know what else to say. I'm sorry I didn't mention seeing him. I should have, and I'm sorry. Frankly, I didn't want it to cause a fight when seeing him had been completely random and unlikely to happen again. And honestly, I was so surprised by the trip you planned for us that all my thoughts were focused on being with you and us being together."
"At least you don't deny lying to me. I'm giving you all this"—he gestured with outspread arms at the room, the house, the physical proof of his wealth and power—"and lies and deceit and betrayal are what you give me in return." He braced his hands on his hips and shook his head, disgust pouring off of him.
The guilt inside her twisted, morphed, flashed hot. Anger took root in the center of her chest. "I don't need all this, Grant. I need a man who trusts me. Who doesn't assume the worst of me. Who knows I can be in the room with another man, even someone I used to date, and remain faithful. I give you that kind of trust, and what you're giving me in return is suspicion and accusation." The more she spoke, the faster the words spilled out of her, and the stronger the anger grew inside her. And, oh, man, the well of anger inside her was deep. So deep. _Scary_ deep. How had she not seen that before?
Color raised higher in his cheeks and his eyes blazed with anger and outrage. "Oh, you don't need all this? Is that it? You'd rather go back to the shithole of a life you had before me? Is that what you're saying? You have _nothing_ without me. You _are nothing_ without me."
_Oh, my God_.
"How could you say such a horrible thing?" she whispered, calling him out in a way she never had before. "How could you even think it?" Nausea rolled through her at his words. Was that what he really thought of her? _Why are you so surprised, Alexa? And how much more do you need to hear before you grow a spine?_
"Truth hurts, sweetheart." He glared at her. He stared at her like she was trash, which was exactly how he was trying to make her feel. But she wasn't feeding herself his running commentary this time. Not this time.
"Yeah, I guess it does," she said, staring at everything she thought she'd wanted. And seeing for the first time that it had all been a pretty charade. A pretty charade she'd let herself believe in with all her heart. "You know what? Let's take a break before one of us says something else we don't mean." Because she needed a moment. Just one. To decide whether to try to fix what was broken . . . or start packing her bags.
_Do you_ really _need to think about it?_ She suddenly felt like she was standing on the edge of a tall cliff and about to plunge into a free fall.
"I mean every word I'm saying, darling. Count on it."
The hollowness in her chest hurt so damn bad. What a fool she'd been. What a blind, stupid doormat. Maverick was right—Tyler would've killed her if he knew what she'd let Grant get away with. "Wow. Okay," she said, shaking her head. She made for the door.
"Don't walk away from me, Alexa," Grant growled.
She kept going. As she rounded the corner toward the kitchen, she nearly tripped on Lucy who was hiding in the shadows as if she'd been listening to them fight, but from a safe distance. Alexa caught herself on the wall and leaned down to scoop up the cat, needing her warmth and her unconditional love. Alexa's stomach was jiggly and her knees were weak and her heart hurt so damn bad she could barely breathe. Nothing felt real.
Fingers wrapped around Alexa's arm, squeezing a gasp out of her. The grip was hard. Painfully tight.
"Wha—"
Grant tugged and dragged her, not saying a word.
Alexa stumbled until her feet caught up with his pace. "Grant, stop. _Stop_. You're hurting me."
He pulled her through the hall and into the foyer. Lucy growled and her claws dug into Alexa's shoulder in response to being jostled.
When they got to the front door, Grant pushed her against it roughly, hand still like a manacle, her shoulder and the back of her head making contact with the hard wood. Lucy struggled in her arms, but Alexa was so stunned that she hung on.
"You don't need all this? You don't think I'm good enough for you?" His chuckle held absolutely no humor. "Then you're fucking free to go," he seethed, his face looming over hers.
"What that's suppo—"
"Enjoy seeing what life would be like without me, Alexa." In one fast movement, he yanked her to the side, pulled the door wide, and then pushed her through the opening onto the front porch, giving her a rough shove when he finally let go of her arm. Lucy did break free then, her back claws catching Alexa on the neck as she jumped.
Alexa was so shell-shocked that she barely felt it. "What in the world are you doing?"
"Nothing you didn't ask for." The door slammed in her face. A metallic click told her he'd thrown the dead bolt for good measure.
She stared at the door in disbelief. "Are you fucking kidding me?" she yelled, pounding on the door with her fist. "Grant! Open the damn door." She pounded again. "Grant!"
The porch light shut off, and then what lights had been on inside went out, too.
"Grant!" She pounded again. "You're being a lunatic!"
She blinked at the darkness, at the craziness of what was happening to her life. Of what she'd _let_ happen. Her fiancé had thrown her out of their house. Out of her own house. Over a motorcycle ride.
Alexa stepped back, the whole world feeling like a Tilt-A-Whirl. It wasn't until the thunder cracked that she even realized it was still raining. The porch roof protected her from the deluge, though the air was humid with it. Lucy sat as far away from the front door as she could get while still being under the porch's cover. Her tail flicked in agitation.
Alexa's fiancé had thrown her out of the house in a storm. At ten o'clock at night. Because she'd gone on a motorcycle ride.
It was so ridiculously unbelievable that for a moment she was frozen standing there, no idea what to do, unable to think. Remembering her cell phone, she fished it from her pocket and immediately found and pressed Grant's number. She hugged herself as she placed the cell to her ear and listened to the ringing.
Four rings. Voice mail. Thunder crashed above her. For a moment, lightning lit up the night. She hung up and redialed. Two and a half rings. Voice mail. Which meant he'd declined the call. _Asshole_.
Mind in a spiral, Alexa hugged herself and stared down at the floor. And was suddenly sucked back to Saturday afternoon in that fancy bathroom. Feeling sore and upset and _used_. Violated, if she really wanted to be honest. Why hadn't she been honest before?
How long had she been hiding from the truth?
And now this.
"Oh, my God," she said, pressing her palm against her forehead to ward off the headache blooming there.
Weeks' worth—hell, maybe even months' worth—of nagging doubts and quiet misgivings roared into her thoughts next. The way he always made her feel like she needed to apologize. The way he so often made her feel like she was crazy or irrational for asking a question or having a different point of view or even remembering something differently. How he so often made her feel guilty and like she wasn't good enough, and how he'd left her feeling in that bathroom.
The horrible things he'd said to her in his office. The way he'd manhandled her. The fact that he'd locked her out of her own house to prove a point, to make her grovel to be taken back, to punish her.
"I can't do this," she whispered, shaking her head. Tears welled until the porch went blurry. "I can't." She took one step back from the door, then another. "I'm _not_ doing this," she said louder, though the rain swallowed up the words. The weight of everything crashed over her like a wave, and a sob broke free. " _Fuck this_ ," she rasped. "I'm not begging to be taken back. I'm not standing out here waiting. I'm not . . . I'm not . . ." Her hands fisted. She bit back the tears, because she wasn't crying over this asshole either. She'd already given him enough pieces of herself as it was.
It was time to start taking them back.
Starting with her dignity.
Trembling with shock and anger, Alexa crouched and held out her hand toward Lucy. Slowly, the cat came to her and sniffed at her fingers, and Alexa scooped the sphynx into her arms.
Alexa made for the steps, and then something occurred to her. She paused. Considered. And then turned back to the door. Juggling the cat, she tugged the diamond off her ring finger, too numb to know if the relief that washed through her was real or confusion resulting from this insanity. Not that it mattered. She wasn't changing her mind.
She dropped the ring over the top of one of the pointed brass leaves on the pineapple door knocker.
Without even one more thought, she walked off the porch into the storm, not looking back even once.
The rain immediately drenched her straight through, but she didn't care. Especially given how fast her mind churned. Alexa had no idea where she was going, or what she was going to do, or how she was going to figure out what happened next.
All she knew was that she had to get away. Everything inside her was screaming for that. _Had been_ screaming for that for so long. Only, now, she was finally hearing it. She was finally _listening to it_. Where the hell had she been? The real Alexa. The _old_ Alexa.
She was four houses down the street when a flash of lights from behind her illuminated the sidewalk. Dread ignited inside her. If it was Grant, what would she do?
Panic nearly making her take flight, she looked over her shoulder. It was just a pickup, not Grant. Relief surged through her, and then an even bigger wave of it hit as she realized.
Maverick. It was Maverick.
The truck pulled to a stop beside her and the passenger door pushed open. "I've got you, Al. Get in."
## CHAPTER 12
Maverick was nearly shaking he was so pissed off.
From the moment he'd seen Alexa walk off her front porch into the rain, his body had gone on high alert. Because there hadn't been a damn thing normal about that. And when she'd rushed down the street in the darkness, in the middle of a goddamned storm, he'd known. Some kinda shit had gone down.
Thunder cracked so loud it rattled the old truck's windows. For a long moment, he concentrated on the roads, the traffic, just getting her away. "I know it's a stupid fucking question, but I gotta ask it anyway. Are you okay? Are you hurt?" He tried really hard to keep the rage out of his voice.
"He didn't hit me," she said, her voice barely audible over the rain and the windshield wipers. "Can we just . . . not? Not yet?" The anguish in her pretty eyes nearly slayed him.
He gave a tight nod. But that didn't mean he wasn't dying to know what the hell had happened. That Slater hadn't hit her was the only thing keeping the asshole above ground right now. But she hadn't said she was okay, had she? In the light that shone into the cab from oncoming cars, her skin appeared pale, her eyes unfocused, and there were angry scratches on her neck. Though given the constant low yowls coming from the cat curled in her lap, maybe the strange-looking thing had been responsible for those.
_All she left with were the clothes on her back and her cat._
What. The. Fuck.
When they came up along the outskirts of the Raven Riders' property, Alexa finally spoke again. "Where are we going?" she asked.
"I'm taking you home." Maverick felt the rightness of those words down deep, even if she wouldn't hear them the way he felt them. And even if he couldn't keep her.
"Home." She said the word with so much hollow despair that it made him need to touch her. He reached across the old bench seat and grasped her cold hand in his. She clutched him right back. Which was when he realized that she was shaking.
Everything inside him wanted to go ballistic. But he had to keep the urge to flip the fuck out under control. Whatever had happened hadn't been pretty, and he didn't need to make things worse for her. Because the only other time he'd seen Alexa Harmon look so fragile, so vulnerable, so lost, was when Tyler had died five years before.
Maverick had known Alexa Harmon long enough to know that Ty's death had been the worst moment of her life. And it had changed everything—about her life and about his own. Which meant whatever had happened tonight had been bad.
But Mav would have to let her open up at her own pace, because pushing her to explain what had happened hadn't worked worth shit two weeks before when she'd shown up at the clubhouse with a bloodied face.
Soon, he was parking on the circle as close to his front door as he could. "Sorry. We're gonna have to get wet to get in."
She shook her head. "God, Maverick, please don't apologize." For a moment, she stared out the rain-blurred window at his place, and then she hugged Lucy close and opened the door. She moved mechanically, like her brain wasn't fully connected to her body.
Marrow-deep concern lanced through him as he hauled ass around the truck, the rain pelting his face and drenching him straight through in seconds. Under the cover of the porch, he opened the screen door and unlocked the front, and then he pushed it open for Alexa and gestured for her to go first.
They stepped inside, both of them dripping all over the place. "I'll get some towels." He rushed into the hall bathroom and grabbed two, then returned to find Alexa standing right where he'd left her by the door, her nose pressed to the cat's head.
"You can put her down if you want," he said as he offered Alexa a towel. He was trying like hell to do what she needed, even though what he really wanted to do was haul her into his arms and make sure she was okay with his hands and his mouth and his body.
Alexa placed Lucy on the floor and grasped the terry cloth, pressing it to her face, squeezing it around her hair, and wrapping it around her shoulders like she was cold. Or like she needed the fabric to hold her together.
Fuck.
"Alexa—"
"Please." She gave a quick shake of her head, and those hazel eyes cut up to his.
Not knowing what she was asking for, Maverick just nodded as he gave his face and hair a quick swipe of the towel. "Do you want to, uh, sit? Or have something to drink? Or . . ." He shifted feet and dragged a hand through his wet hair.
"I'm soaked," she said, looking down at herself. When she looked up again, she had a strange expression on her face, one that quickly shifted into amusement. Laughter spilled out of her, and she slapped a hand over her mouth. "I don't—" More laughter, this time with a slightly hysterical tinge to it.
Maverick frowned.
"Oh, my God," she finally said. She cupped her hand to her forehead. "I don't have . . . anything. I don't have clothes or my purse or my keys or my schoolwork. I literally have nothing." She peered up at him, her expression full of incredulity.
Anger lanced hot and fast through Maverick's blood. "Anything you need, I will give you or help you get," he said, meaning in the short term, but he wasn't opposed to meaning it otherwise, too.
Alexa let go of the towel, dropping it to the floor, and pressed both of her hands to her mouth—and that was when he noticed. She wasn't wearing that big-ass rock. Her ring finger was bare.
Chaos erupted inside his head.
He couldn't help it. Maverick stepped closer and gently grabbed her left hand. For a long moment, he looked down at the pale indent where the ring had been, and something that felt dangerously like hope flared through him. Except he beat that shit back—hard—because he couldn't afford to make assumptions here. He wouldn't survive it. And that wasn't what she needed right now anyway.
Finally, he looked up at her, the air between them heavy with so much. Unanswered questions. History. Desire.
Alexa's grip tightened around his hand, her eyes shiny and bright as she looked into his. And then her whole face crumpled and she burst into tears.
Maverick finally gave in to what he'd been wanting to do and hauled her into his arms. "Fuck, Alexa," he said, stroking her wet hair with one hand while holding her tight to him with the other. The rightness of her against him made his body fucking sing. "Whatever it is, I'll help you figure it out." He wasn't even sure she heard him over her wracking sobs. Sobs for another man. Desire, his ass. On his part, maybe.
_Don't lose sight of what's going on here, Maverick. She didn't come to you. She didn't call you. You picked her up._
Which was all true. The only good thing about how upset she was right now was that it forced him to keep his feet planted squarely in reality. A reality where she was crying over her troubles with another man.
Standing there holding her, her body tucked tight against his, her hands fisted in his T-shirt under his Ravens cut, he pushed everything else away and focused on her. A million years ago, they'd started off as friends—good friends.
That's what he would be to her once again.
ALEXA HATED THAT she was crying, but there was so much noise inside her that she'd just needed to let it out. She wasn't crying over Grant, exactly, but over the loss of the life she thought she was building. And, even more, over the humiliation and soul-deep disappointment she felt in herself for tolerating all the things in their relationship that had made her feel so bad for so long. She was just so fucking _mad_.
Old, long-ingrained thoughts tried to sneak in around the anger. Thoughts that made excuses and put blame on herself and tore herself down. Thoughts she'd learned the past five years, from Grant. Thoughts she'd made a very bad habit of taking to heart and making her own.
How had that happened? Why had she let it? How long had it been going on?
Who the hell had she become?
"Whatever you need, Al. Just name it," Maverick said, his big hands giving her so much comfort.
She forced a deep breath. And another. Trying to rein herself in. She scrubbed at her face and leaned her forehead against his broad chest. She breathed him in, her heart and her body recognizing his scent, all leather and soap and Maverick. And she wondered if just a few of her tears weren't for the relationship she'd walked away from five years before out of grief over losing Tyler to the thing Maverick loved most, fear of getting hurt even worse if she lost Mav, too, and the crushing weight of responsibility for her mother.
Right there in that moment, Maverick was her rock. Dependable, reliable, and certain—all things she'd doubted about him five years before. All things she'd believed Grant absolutely guaranteed.
The weight of all her mistakes was almost too much to bear.
She looked up into Maverick's dark blue eyes, so filled with concern for her, and whispered a confession. "I've been so wrong about so much for so long."
He cupped her cheek in his hand and swiped at her tears with his thumb. His calluses were rough against her skin, but it was the most amazing thing she'd felt in a long, long time. She couldn't resist leaning in to his touch. "No fucking way this was your fault. Tell me what happened."
"Grant accused me of cheating on him with you," she said.
Rage flashed through his blue eyes, but his hands remained gentle, soothing, caring. It made her heart ache and soar at the same time. "Go on," he whispered, his voice raw.
"One of the sheriffs saw us on your bike and told Grant. Nothing I said mattered. No, it was worse than that. He twisted everything I said. Threw my words back in my face." Anger rose up inside her again and she shook her head as Grant's voice echoed inside her mind. _You have nothing without me. You are nothing without me._
"No doubt it was Curt Davis. He's so far up Slater's ass it's a surprise you don't hear Davis's voice when Slater talks." He chuffed out a humorless laugh. "All this over a fucking bike ride," Maverick said under his breath. "Damnit, I'm sorry. That's my fault."
Alexa braced her hands on his chest, his warmth seeping inside her where she was so cold, so lonely, so alone. "No, Maverick," she said, the anger coming through in her voice. "It's neither of our faults. That's the thing. There were _so many_ things I couldn't do, or shouldn't do, or he would get mad. So much that was better left unsaid because it would cause a fight. I knew he wouldn't like knowing I'd ridden with you, but we didn't do anything wrong. You were just helping me. And I needed you—"
She cut herself off, the words taking her by surprise. How long had it been since she'd last let herself think that thought? That she'd _needed_ Maverick Rylan.
He gave a tight nod, his eyes on fire with emotions she didn't dare to guess at.
"And then he threw me out." How humiliating was that to say out loud? Just voicing it made her stomach toss.
Maverick's eyes narrowed. "He kicked you out of the house?"
"He literally grabbed me and threw me out the front door," she whispered, barely able to give voice to such an ugly truth. "To punish me."
"Sonofabitch," he bit out, his gaze searching hers. "I'm going to kill this motherfucker. I swear to God."
" _No_. Promise me. You will _not_ go after him. You're right. He has friends in lots of places around town, including the sheriff's and mayor's offices. You lay one finger on him and he'll find a way to get you arrested. _Promise me_." She held his stare, all the while realizing she never would've talked this way to Grant. Well, not before tonight.
Maverick looked like he wanted to murder someone. Well, obviously, he did. "He threw you out of your own house, Alexa. In a goddamned storm. He hurt you."
"I know. But I refuse to let him hurt you. Promise me." She grasped Maverick by the edge of his cut and shook him. " _Promise_."
His eyes were blue fire. And, God, even rankly pissed off, he was so freaking hot. And it wasn't just his looks, though those were damn fine. It was his sense of honor. His protectiveness toward her. His basic decency. When had she stopped valuing that? "Fine. I promise. _For now_."
"Maverick—"
"Tell me what happened the day you came to the clubhouse." He arched a brow at her.
"He didn't hit me," she rushed to say.
The brow went up a little higher. "Hanging on by a very short thread here, Al."
"We got in a fight over a mess I made. He kicked a box at me and I jerked back out of the way and tripped. I fell and hit my head on a table. He told me it was my own fault and stormed out. I . . . I freaked out." The words spilled from her in a rush.
"Please tell me you realize that none of that is normal. None of that is the way normal people react to a mess or seeing someone with an ex or having a fight. Right? You get that?" The concern was back in his expression again. It only took a little of the edge off the anger, not that his anger bothered her. And it certainly didn't scare her. Because Maverick Rylan would never intentionally hurt her. It wasn't even a question.
"I know," she said on a sigh. Her shoulders fell and she ducked her chin. Unthinkingly, she rubbed her fingers over the indent where her ring used to be. "I _do_ know. I knew as it was happening, although he always made things feel like they were my fault. Or I always found a way to rationalize what'd happened. _I_ was the one who tripped on the box. _I_ was the one who made the mess in the hall. _I_ was the one who went on a bike ride with another man and didn't tell my fiancé." She shook her head. "I don't know when I started thinking that way. I was thinking about it on the ride here and I can't pinpoint exactly how it happened. Things were good between Grant and me at first. And then . . . it was like he changed, but only in bits and pieces that I learned to live with one at a time, until he'd become someone else and I'd accepted it all." She shook her head, so disappointed in herself. "What happened to me?"
Maverick caught her chin in his fingers and made her look up at him. "Abusers know they have to work up to the bad behavior. Suck you in. Make you feel special. Get you acclimated to the abuse a little at a time. Wear you down more and more until you not only can't fight back, you don't fight back," he said, voice gritty and raw. "I saw that shit with my mother, but I didn't recognize what it was. My father did that shit with me, too. Abuse is insidious, Alexa. That's why it's so effective at tearing you down and stripping you of your defenses, your support system, your independence."
Alexa nodded, surprised to hear Maverick talk about what had happened to Bunny. He'd never said much about it even when they'd been close. She often wondered if he'd said more to Tyler. But she knew it ate at him and always had. "Well, no more. I'm done. I left my ring and walked away, and it was the first smart thing I've done in so damn long even though I have no idea what it means for my life. But I can't be with someone who treats me that way. And I certainly can't marry him. I won't." She clutched her stomach as a wave of anxiety washed through her, the sensation almost as if she'd nearly been in a terrible accident. She fell back against the front door. "Oh, God, I can't believe I came so close to marrying him."
Maverick stepped in close. "He doesn't know that part, though, right?"
"Not unless he opened the door and saw I wasn't there, saw the ring." She blew out a breath. "But I don't care. I'm done."
"He's not going to let you go that easily," Maverick said, peering down at her.
Her stomach flipped again as she nodded. Maybe. Probably. This was likely just the beginning of some hard days. But it was also the beginning of something new—a new life. What exactly that would be, who exactly _she_ would be, she didn't know. "What am I going to do? My entire life is wrapped up in him. My job. Where I live. My car. Where Mom lives. _Everything_. Oh, God. This is going to be a nightmare, isn't it?"
Maverick grasped her hands. "One thing at a time. You're out. You can stay here as long as you need to. I have plenty of room. I'll help you through it. That'll give you time to figure out what it is _you_ want. There's no rush."
"Okay," she said. "Okay. God, thank you." A shiver raced over her skin.
"You don't have to thank me. I'm so relieved for you that I might fall the fuck over," he said, his thumb rubbing over her knuckles. She almost managed a little smile. "Your hands are like ice, Al. I shouldn't have made you stand here all this time when you're soaking wet."
Alexa shook her head. "It felt good to get it out, Maverick."
"Glad to hear it. Would a hot shower make you feel better? I can find you a T-shirt and a pair of boxers to change into while your clothes dry."
All the TLC and understanding he was giving her reached right inside her chest. When was the last time someone had taken care of her this way?
_The last time you were with Maverick_. The truth of that thought made her breath catch. "Yeah. I'd like that," she managed.
He gave a tight nod and stepped back, and she missed his heat immediately. "Uh, good. That's good. You remember your way around?"
"Of course," she said. How much time had she spent here over the years? Even before they dated? At one point, Maverick's place had practically been a second home. Though, as she'd finally calmed down enough to take note of her surroundings, all the changes he'd alluded to the other day jumped out. The hardwoods were refinished, new drywall replaced the old, dark paneling, and the fireplace had been redone with gorgeous stonework. The finishes, fixtures, and décor were a fantastic mix of modern and lakeside cottage.
"You okay?" he asked, watching her.
She pushed off the door. "I don't know yet," she said, coming to him. "But what I do know is how grateful I am to you. For everything. For being there when I told you not to. For listening. For just . . . being you."
"Always." He crossed his big arms and ducked his chin, like something about the exchange made him uncomfortable.
"I mean it," she said, and then she pressed onto tiptoes and kissed his cheek, his stubble tickling her lips. "You saved me tonight."
"No," he said, eyes flashing, one brow arched. "You saved yourself, Al. I was just your getaway driver."
Staring into one another's eyes, the tender moment stretched out, morphed, suddenly flashed hot. Being so close to him set her body on fire, and _finally_ she could actually consider acting on it. Need roared through her, a living, breathing thing. "Maverick," she whispered.
"Fuck, Alexa, I think you better go take that shower," he said, his jaw ticking. "Now."
She released a shaky breath and nodded. He was right. Of course, he was.
So, showering. She could handle that.
Afterward, God only knew what she was going to do.
## CHAPTER 13
Alexa woke up in the middle of the night and immediately knew where she was—and where she wasn't. She was in Maverick's guest room, not in her own bed at home. Though, she supposed, she actually didn't have a home right now, did she?
_Annnd_ that thought pretty much ensured she wasn't going back to sleep.
Part of her was surprised she'd fallen asleep in the first place, because her thoughts were a stressed-out, confused mess. But then she'd emerged from the shower feeling like she was carrying a lead blanket on her shoulders, and it had been all she could do to keep her eyes open. Maybe her brain had just needed to shut down. Maybe her sanity had just needed a break from this new reality. One where she had almost nothing.
The only saving grace in the whole situation was that she hadn't yet combined her savings account with Grant's, which meant she at least had some money to provide a cushion. Not that it was huge, but it was at least something.
She threw back the covers, got up, and made her way out to the kitchen in the dark. Lucy hopped off the end of the bed and padded after her. After a moment of fumbling for a light switch, Alexa found it, and then she was squinting against the brightness while her eyes adjusted. Standing at the sink, she filled a glass with water.
"You okay?" came a voice from behind her.
"Oh, hey," she said, the glass nearly slipping from her hand in surprise. Maverick stood at the edge of the room, sexy as all hell with his sleep-mussed hair and wearing an unbuttoned pair of jeans and nothing else. Which reminded her that he liked to sleep naked. Or he used to. His tattoo-covered muscles were lean and hard, and her brain unhelpfully supplied her with the memories—so many molten-hot memories—of how all that hardness felt against her. "Did I wake you up? I'm sorry."
"Wasn't asleep." He crossed his arms and leaned against the corner of the wall, the position emphasizing the bulge of his biceps and drawing her gaze to the maddening opening at his fly. Dark blond hair ran in a line down his abdomen and under the denim. And damn if her mind didn't helpfully supply her brain with all kinds of images of exactly what lay beneath. She swallowed hard. "You neither?"
"I slept for a little while." Sighing, she finished her drink and placed the glass in the sink. Everything inside her wanted to go to Maverick and burrow in against him. And at least a little of her wanted to go to him and push those jeans off his hips and down his thighs. Maybe reacquaint her mouth and fingers with every tattoo he had. The _Live Free/Ride Free_ and crossed wrenches tattoos on his chest, the tribal black motorcycle on his arm that morphed into black flames, the black-and-white checkered flag that wrapped around his ribs on one side, the big piece on his shoulder that looked like he was mechanical under his skin—that one was new. Well, at least since they'd been together. But she'd lost the right to do any of those things. Worse, she'd thrown that right away.
"You gonna try to sleep some more?" he asked, dragging a hand through his hair. Hair she knew was so soft.
What was wrong with her? The last thing she needed to be doing right now was drooling over another man, but she couldn't seem to stop noticing all the things that had always driven her crazy about Maverick Rylan.
"Al?"
"Huh?" Her gaze snapped to his, and found that dark, dark blue absolutely blazing. "Oh, uh. I'm not sure there's much point. Maybe I'll just watch some TV. Or something."
He nodded toward the living room. "Maybe there's something good on HGTV," he said with a wink. "Probably not, but . . ."
Alexa chuckled. "Don't say that like you don't like a lot of those shows, too. You know you do."
"Do not," he said, shuffling toward the living room and yawning.
Walking out of the kitchen, she came up behind him. And the sight of his back nearly took her breath away. She'd forgotten how sexy his massive tattoo there was. The words _Raven Riders_ arched across his shoulder blades, and below, a huge black raven clutched at the hilt of a blade stabbed through the eye of a skull. The club's logo. Every fully patched Raven had one like it. Mav's muscles rippled under the ink, making the big black bird seem alive.
"How much of the work around here did you do yourself?" she asked, glad when he dropped onto one end of the charcoal-gray couch, breaking her view of his impressive ink. She sat away from him, but not all the way at the other end either. She hugged her legs to her chest as Lucy perched on the rug on the other side of the coffee table and stared at them.
"I had some help, but I had a hand in all of it." Mav flipped on the TV and changed the channel to a show about fixer-uppers.
"Uh-huh. Don't worry, big bad biker. Your love of HGTV is safe with me." She gave him an innocent look, but inside this all felt so familiar that she could almost pretend that the last five years hadn't happened. She and Maverick had always shared an interest in design. Interiors were her first love and building custom bikes was his, but they'd watched more than their fair share of shows about rehabs, renovations, flips, and more. They both liked learning new things and neither was afraid to try to do something themselves. She'd always loved refinishing and repurposing old furniture and, as the updates to his place proved, he had more than a little handyman skill. She used to enjoy watching him work on his bike projects in the chop shop at the Ravens' clubhouse. She'd always found competence and confidence sexy as hell—and Maverick had both in spades.
"It's a good thing I like you," he said, yawning again. He stretched his legs out in front of him and settled his big body into the cushion behind him.
Alexa chuckled, but her smile slipped right back off her face. Because the past five years _had_ happened. She couldn't pretend they hadn't, after all. More than that, she was going to have to deal with a whole host of consequences for all the things she'd let get so out of control.
They watched the show for a couple minutes, but as soon as a commercial played, her thoughts raced. Laying her head against her knees, she peered at Maverick. "I don't know what to do about work in the morning."
Maverick gave her a serious look. "Do you have any kind of a contract?"
She shook her head. "I'm in the middle of a huge project, though. The deadline's next week. God, I don't even have any clothes to wear."
"I can run you home in the morning, Alexa. But if he's realized you left your ring there, that you intend to leave him for good, you gotta prepare for him to be vindictive. And that maybe means violent, too. He's already shown he's not above that. But don't worry. I can bring some of the guys if I need to." Maverick shifted toward her, crooking one of his legs up on the couch. The position pulled the gap at his waist open, and her gaze couldn't help but lock on the fascinating strip of skin it revealed.
"Yeah?" she said, looking back to his face. "Okay." The heat in his gaze said he'd caught her admiring him. She pressed her fingers into her neck and massaged a knot there. "You don't have to stay up with me, you know."
"I don't mind." His gaze flicked to her hand. "You hurting?"
She gave a little shrug. "I'm just a ball of stress."
"I'd say you came by that honestly, Al." He sat up, bringing his body closer to hers, and then he pushed her hand out of the way. "Lemme."
Maverick's big, warm hand gripped the back of her neck and massaged her sore muscles. She groaned and buried her face in her knees.
"Too much?" he asked, his voice gravelly.
"It's perfect," she said. And it was. Warm, firm, strong, his touch felt so damn good it was all she could do to keep from moaning.
"Turn your back to me."
She did, and then both of his hands fell on her skin where the opening to his too-big shirt hung wide on her shoulders. He kneaded at her neck, dragged his fingers against her scalp, and pressed delicious circles against her upper back. His big hands spanning her body, he pushed his thumbs into the muscles running down both sides of her spine.
Alexa did moan then. Heat licked over her body. From her embarrassment at the sound that had just spilled from her throat. From his warmth and closeness against her back. From his hands on her skin.
The quiet suddenly felt weighted, heady, full of anticipation. Was that her imagination? Or did Maverick feel it, too? She held stock-still, both because she didn't want him to stop _and_ because she felt like if she gave even the slightest indication that she wanted more, this moment would explode into something she maybe wasn't ready for. Certainly not tonight.
Which made Maverick massaging her the sweetest torture. Because his touch brought her body to life like she'd been hibernating for the past five years. Her skin became hypersensitive. Her nipples hardened. Her core ached with need.
She shivered from the intensity of the arousal suddenly flooding through her.
"Cold?" he asked, hands smoothing up her back again.
"No," she said.
"Tired?" The timbre of his voice was low, rough.
"I don't know." She peered over her shoulder. Maverick's eyes were hot and intense, though she wasn't sure what to make of that, or whether she was projecting her own desires onto him. "I don't want to sleep again, though. I'd rather not be alone with my thoughts right now."
He looked at her for a long moment, and then he pushed himself back into the corner of the couch. Regret at losing his touch rushed through Alexa until Maverick held open his arm to her. "Come here."
Alexa didn't let herself second-guess it. She moved into the space along the side of his body. Her head on his chest, her chest against his side, her legs pressed along his. The sensations were familiar and new at the same time.
His arm came around her shoulder. "Sleep, Alexa. Everything else we can figure out in the morning."
"I hope so," she said.
"I know it." He gave her a squeeze.
His warmth and his scent and the lulling grumble of his voice made her eyelids sag and then close altogether. Maybe everything would look better in the light of day. She could only hope.
IT WAS ONE of Maverick's favorite dreams.
He and Alexa had gone up to Swallow Falls in Western Maryland for a weekend getaway and were staying in one of the mini cabins at the state park. After a day of hiking and swimming and picnicking outside, they'd come back to their cabin tired and ready to crash, but getting naked for showers had sidetracked them for hours. And even once they finally fell asleep, Maverick was hard and ready every time he woke up, and he took her again and again, falling asleep still buried inside her . . .
And fuck if he wasn't ready right now.
He burrowed his face in her soft hair and banded his arm around her stomach. His hand filled with the soft mound of her breast. He pulled her back against his chest and ground his erection against the swell of her ass.
"Maverick," she moaned.
Hell, yeah. He nuzzled her neck, kissed her there, tasting and nipping and sucking. God, he needed in her. "Fuck, Alexa," he whispered.
Her hand gripped his. "Maverick."
He rolled her under him and crawled on top of her, his body falling into the cradle of her spread thighs.
"Uh, Maverick."
He frowned and kissed her jaw, her cheek, her mouth.
It was the kiss that did it. Something wasn't right. The memory playing out in his sleep-fogged mind didn't feel like the reality confronting his physical senses. His eyes blinked open.
And he found himself lying on top of Alexa. Not in the cabin at Swallow Falls years before. In the gray morning light of his house. Her wide hazel eyes stared up at him.
He reared off of her in an instant. "Fuck," he said, coming to stand by the couch. Alexa looked stunned—and so fucking sexy that Maverick barely resisted crawling back on top of her. She lay on her back in _his_ clothes, on _his_ couch, her knees drawn up and falling out, her hair sleep-mussed and sexy. Jesus. He adjusted himself, unable to hide his raging hard-on, and scrubbed at his face. "Goddamn dream. I'm sorry, Alexa. I didn't mean to do that."
"You . . . you were dreaming. Of me?" she whispered.
His gaze narrowed. "Don't ask a question you don't want the answer to."
She swallowed and licked her lips. "What if I do want it? The answer," she added.
Years of need and longing roared through Maverick like a drug he'd mainlined. His thoughts spilled out unfiltered. "Damnit, Al. You're laying there in my clothes in my house with your thighs spread after I've just woken up holding you. My skin smells like you. And I'm sporting an erection because of you. Don't fuck with me." The words came out harsher than he intended, but she couldn't play with him. Not on this. Not when he cared so much. Not when he _wanted_ so much.
Her mouth dropped open and her chest rose and fell a little quicker. "I'm not playing a game. I want to know."
Planting his hands on his hips, Maverick studied her. Her beautiful, languid body. Her pretty, open face. Her eyes, honest and free of pretense. He felt pulled in a million directions. Between wrong and right, between taking advantage and taking care, between giving in and opening himself up to a world of hurt. "I was dreaming of you. Of us. Up at Swallow Falls."
"That night we—"
"Yes," he growled.
"Maverick—"
" _Fuck_." He dropped his chin to his chest and closed his eyes. Trying to be bigger than his base needs. Trying to put her before himself. "Whatever is about to come out of your mouth is _not_ a good idea."
"Mav—"
"I mean it, Al—"
"Maverick!" she nearly yelled. "Listen to me."
His gaze cut up to hers in time to see her sit up a little and take off her shirt. Well, his shirt. Then she laid back again, her eyes on him, drinking him in, _inviting_ him in. "What if I do want it?" she whispered.
Something inside him snapped.
He was on her in a second. Body covering hers. Hands going to her warm skin. Mouth tasting her everywhere—her shoulder, her throat, her cheek. His chest pressed against her breasts, her hard nipples evident, her excitement palpable. And then his mouth found hers. On a triumphant groan, he claimed her, his lips sucking, his tongue penetrating. His big hand found her breast and kneaded at the soft mound.
And Alexa was right there with him. Moaning, kissing him, clutching on to him. Her thighs wrapped around his hips and her fingers twisted in his hair.
_Bad idea bad idea such a fucking bad idea_.
Why did bad ideas have to feel so good?
"Fuck, I want you," he said, grinding his cock against the soft, welcoming spot between her thighs.
"I want you, too. Feels like I've been wanting you my whole life." She peered up at him, her eyes so vibrant they were nearly green.
Maverick pulled back, her words hitting him in all kinds of places, some comfortable, some even healing, but some less so. "You can't toy with me, Alexa. Not when I've wanted you for so long. Not when I never wanted to let you go in the first place. This . . . this can just be fucking. But don't you dare say anything you don't mean."
She stroked his hair. "Truth?"
He gave a tight nod and prepared for the worst, even as he hoped for a shot.
"Truth is, I don't know what this means. Yet. I am so messed up right now. I'm not even going to hide that. But . . ." She tilted her head, and her expression was filled with something Mav didn't want to name, but it sure as hell looked like affection. "But I know I want this. I want _you_ , Maverick. Right now. And wanting you feels like one of the smartest decisions I've made in forever. Everything else be damned." Her voice was shaky and breathy and her words were so full of dangerous, dangerous hope that Maverick didn't dare move.
His muscles nearly shook with the force of his restraint, because his body was literally screaming at him. Screaming at him to just let go. "Fuck, Alexa."
"I don't want to hurt you. I _never_ wanted to hurt you. But I want you. I want you inside me right now more than I want my next breath."
Even though his mind was rolling out a list of _all the reasons why this was a no good what the fuck are you thinking very bad idea_ , Maverick's body grabbed the reins. On a groan, he went in for a kiss, rougher than he intended, but the moan she unleashed when his mouth crashed down on hers told him she didn't mind. They fumbled with clothes. Her boxers. His jeans. Until they were both naked and pressed tight, all hot skin on skin, and his cock ground against her soft folds until she was mewling and writhing and silently pleading with the rough, desperate grip of her hands on his shoulders, his back, his ass.
A big part of Maverick wanted to pick Alexa up and carry her to his bed—where his heart said she belonged—but this wasn't about that. At least not yet, and he wasn't giving either of them any more time to think than he already had.
He needed her too damn bad. Just this once. In case that's all it was.
Taking his cock in one hand, he dragged his head against her pussy, the contact sending a ferocious need through his veins. He braced his other hand next to her shoulder and met her bright eyes. "You sure?"
"Now," she said.
"What about prot—"
She shook her head. "I'm covered. And I want you just like this."
"Christ," he bit out, the trust inherent in her words kinda blowing his mind. His cock found her opening, penetrated her, slid slow and deep. It was perfection of the soul-deep, bone-bending, never-get-enough kind. He'd been with other women since they'd broken up, of course. But being with Alexa again reminded him just how much every one of those experiences had paled in comparison to being inside this woman. Damn if it wasn't like coming home.
It took everything Mav had not to rut against her like an animal, not to just let his hips fly and pound, because that was the kind of urgency firing through his blood.
But she deserved better than that. She deserved his _care_. And he didn't want to rush this. He wanted it to last forfuckingever.
He came down on one elbow as his cock bottomed out inside her. Her head wrenched back as she cried out, the look of pleasure on her face so damn beautiful. "Oh, God," she rasped. "Oh, Maverick."
His name from her mouth licked heat over his skin. He withdrew slowly, then hammered home again. Alexa's eyes went wide. Her mouth dropped open. Maverick did it again and again. Slow withdrawal, fast penetration. Until her core was fisting around him and driving him insane.
"You gonna come for me, Al?" Their gazes collided as his hips snapped against her clit. "Come for me." His hips hammered home again. Staying deep, he ground himself against her. That was all it took.
Alexa's whole body went taut as her mouth formed a silent cry and her eyes squeezed shut, her nails digging in to his arm and his side. Her core sucked at him over and over, and Maverick forced himself to move despite the utter fucking perfection of it, because he wanted it to be good for her. He wanted the goodness—the _rightness_ —of it to mark and change her on the inside. The way it was doing to him. Because this was shining a spotlight on everything he'd been wanting for so long.
Her whole body shook and she finally found her voice, crying out and calling his name. He kissed her, deep and claiming, keeping their faces close, and wrapped himself around her. And then he let his body off its leash. He fucked her in a fast grind that quickly had them hot and sweaty and panting. Their gazes locked, she raked at his hair and pulled him in for a kiss they couldn't hold. Small needful whimpers spilled out of her, the sounds driving him wild. The closeness, the honesty, the need—it was the most intimate moment he'd had in years. Maybe ever. And it gave his body a giant shove toward release.
"Come again for me," he said. "Show me how good I'm making you feel."
"Maverick," she rasped.
"That's right," he said, tilting his hips to concentrate on her clit. "Give it to me, Alexa."
"Shit," she whispered.
Taking the hair at the back of her head in hand, he forced her to open to him as he bore down on her in a claiming, penetrating kiss, his body still moving in hers. " _Come_ ," he growled.
The cry started low and then turned into a guttural moan in his ear. "Mav, baby." The spasming of her core around his—again—was all he could take. Braced up on one arm, he let his hips fly, snapping against her and concentrating sensation into his balls. And then he was coming and cursing and shooting inside her. He moved through it, trying to drag it out for both of them as long as he could.
Because he didn't know what this meant. Or whether it would happen again. And Alexa hadn't made any promises. Neither of them had.
"Jesus," he whispered, pressing a kiss to her forehead. They stared at one another for a long moment. Not speaking or hiding. Just looking. And he felt it—a bunch of things she was maybe thinking and maybe even feeling, but not saying. He knew because he was doing the same thing. "You okay?"
The small smile she gave him was so damn pretty. "I'm . . . definitely better than okay."
He chuckled and nodded, affecting a nonchalance he didn't feel. "Good. That's good."
Soft fingertips dragged down his face, then pushed the length of his hair back. "Thank you."
"Yeah," he said. Not sure what more to say when there was so much he couldn't. Or shouldn't. Easing out of her, he immediately missed her heat and her touch. "How 'bout I make some coffee?"
Alexa nodded, her smile still small but turning playful. "Coffee is life."
"Coffee is life," he said on a nod. He scooped his jeans off the floor and forced himself to act all chill, to walk away, to keep his mouth shut.
Because what else could he do?
SITTING ON THE couch, the sounds of Maverick puttering around in the kitchen coming from behind her, Alexa quickly pulled on her clothes and made her way to the bathroom. She shut herself inside and pressed her face to the cool wood of the door. And let the quiet tears flow.
That . . . that had been the single most amazing thing Alexa had experienced in almost five years.
Everything she'd been wanting. Everything she'd been missing. Everything she'd been _needing_. Maverick gave it to her, not even realizing what he was doing, and how earth-shattering it truly was for her. He hadn't just held a mirror up to her relationship with Grant—because, Jesus, when she thought of what'd happened in that bathroom at the inn, it was _nothing_ like this. Mav had taken a sledgehammer to whatever last pieces of the charade of her life that she'd been clinging to. And in the process, making her see how much she'd tolerated that she shouldn't have, how little she'd accepted for herself, and how much she'd convinced herself that something existed where it didn't. And maybe never had.
Where had she been?
Why the hell had she given up on Maverick the way she had? Given up on _herself?_ She'd had good reasons, right? Good, totally reasonable reasons. Except, what were they again? And why did they ring so hollow to her now?
God, she was a mess. And it was one problem too many for her to try to fix just then.
Shaking her head, she freshened up and met her reflection in the mirror. "Time to face reality, Al," she whispered to herself.
How Maverick fit into that reality, she didn't know.
Out in the hall, she found Lucy waiting for her and meowing repeatedly. The cat followed her to the kitchen, where Alexa found Maverick much as he'd been the night before—shirtless, jeans hanging on his hips, though buttoned this time, and sexy as hell. "Do you have some lunch meat or tuna I can feed Lucy? Just until I can buy her some cat food?" she asked.
"I have ham and cheese," he said, gesturing to the fridge. "Help yourself to whatever you need. I mean it, Al. Like it's your own house. Okay?"
She ducked inside the refrigerator, not wanting him to see just how much his words meant to her. "Okay, thanks." For the next few minutes, she diced lunch meat while Maverick made toast and fixed them both a cup of coffee. They worked in silence and the way they moved around one another reminded her of all the times they'd made meals here when they'd been together. So she really didn't need to ask how he knew how she liked her coffee.
When Lucy had her breakfast and Alexa had forced herself to eat a piece of toast, she sighed. "I should check my phone," she said, moving into the hall. In the guest room, she found her cell. "Holy shit," she whispered, her gaze latching onto the little notification numbers.
"What is it?" Maverick asked from behind her.
She turned, her stomach making a slow drop to the floor. "I have thirty missed calls and five messages."
Standing in the doorway, his big arms crossed, Maverick frowned. "Do you want me to listen to them first?"
"No." She had to face this. _She_ had to be in control. For once. Sinking onto the edge of the bed, she shook her head. "You're not going to be able to shield me from all this, Maverick. It's my mess. I have to fix it. I have to face it." She wasn't sure what would be harder to hear—Grant yelling at her or playing nice. And she honestly had no idea which to expect. That said a heck of a lot, didn't it? Taking a deep breath, she played the first voice mail on speaker.
"Alexa, where the hell are you? Where did you go? I'll come get you. Call me." The tone was urgent, clipped, not obviously angered, but not just concerned, either.
"One down," she said, playing the next message. Adrenaline made her hands tremble around the device.
"Damnit, Alexa. This is ridiculous now. Where are you? I'm . . . I'm worried about you, babe. Call me."
She made a face at the phone. Was he really concerned about her? And did she care if he was?
Maverick blew out a long breath. "Fuck, Alexa. You're killing me."
"Why?" Her eyes cut up to his.
"Because I'm terrified that you're going to give him another shot." He braced his hands on the doorframe and hung his head for a moment. "And this isn't about what we just did. It's about what he did to you last night. For starters."
She pulled her gaze away from the fascinating movement of the tattoos over his muscles. "I'm not. No matter what he says."
Blue eyes flashed to hers.
"I may not know a lot yet, but I know that," she said. "I won't go back to him." It was a promise. A vow. To herself more than anyone else. But to Maverick, too. Because even though he'd said what they'd done could just be fucking—and it had been some mind-blowing fucking at that—it had felt like so much more. What she should do with that just then, she didn't know. But it didn't mean nothing, either.
The third message played, the tone much more conciliatory. "Babe, please tell me where you are. I know I lost my temper. Things are just really stressed right now. Come home. Let's talk this out. Like you said."
She wasn't sure how to read the tone, but what she could see was that every one of his messages was filled with commands. _Tell me, call me, come home_. And she also heard all the things he _didn't_ say, too. No apology. No declaration of love. Or was she reading too much into it? God, after years of misreading Grant's words and intentions, she wasn't sure whether to trust her reactions.
Alexa sighed and dropped her forehead into her hand. "This whole thing is so crazy," she said almost to herself. "How is this my life?" Her fingers moved over the cell's screen to play the next.
"Goddamnit, Alexa. I'm calling the police. Come home." That one had the angriest tone of all.
"Great. You do that, I'll be happy to tell them you physically threw me out of the house," she whispered to herself. Although, if he called Davis, that guy wouldn't give two craps that Grant had hurt her. Maverick was right. Officer Davis was a total lackey. "Last one."
"Alexa, I love you. I'm . . . I'm sorry about last night. It won't ever happen again. Just let me know you're okay, babe. I need you so much."
She blew out a breath and dropped the phone onto the bed beside her.
Maverick crossed the room and sat next to her. "You okay?"
"Wanna know the sad thing?"
"Yeah," Maverick said, clearly trying like hell to keep his voice even.
"I'm not sure I know how to read those messages." She finally met his gaze. "I'm not sure I trust my own judgment about them."
Emotions rolled over Maverick's expression, and she watched him rein them in. "Okay. What do you think they might mean? What are you hearing?"
She swallowed hard and ducked her chin. "Well. I think he wants me back. But he didn't apologize or even say he loved me until the last message, so I think he's trying to tell me what I want to hear rather than what he really means. I think he wants to control me." She blinked up at him.
Mav tilted his head, and his hair fell across his eyes. "Sounds to me like you know more than you're giving yourself credit for, Al." He gave her a pointed stare, and she nodded. "Slater had two ways to play reaching out to you—attacking you and lashing out in anger, or appeasing you and luring you back in with what seems like worry and concern and remorse. What happened when you went back to him after the last fight?"
"He apologized. Said he hadn't meant for things to get so out of control. Said it would never happen again." She shook her head. "But it did."
"Yeah, it fucking did," he said, his voice like gravel. "Last night you said there are all kinds of things you're not supposed to say or do or he'll get mad. So, that means he likes to control you, yeah?"
"Yeah," she whispered, looking away.
"Fuck," he gritted out, huffing an angry breath. Mav grasped her chin and made her look at him again. "Well, he can't control you as easily if you're not living under his roof, so his first objective is to get you back home. No doubt he didn't expect you to leave, so he has to be on his best behavior now. Don't think for a second he doesn't know exactly what he's doing. Don't believe for even an instant he isn't calculating the best way to get what he wants. That's what guys like Slater do, Alexa. That's what he's been doing to you." He touched two fingers to her forehead. "Trust yourself in here." To her heart. "And in here." To her stomach. "And in here. You _know_."
Just then, her cell phone rang. Alexa flinched, then scooped it up off the mattress. "It's him."
"Of course it is," he bit out, anger giving his voice a harsh edge. "He won't give up 'til you give him what he wants."
She blew out a breath and her belly went on a loop-the-loop that left her a little queasy. "I guess I have to talk to him at some point, right?"
Maverick gave a tight nod, his face set in a scowl that was almost scary.
"Then here goes nothing," she said, answering the phone.
## CHAPTER 14
Grant," Alexa said, hating the shakiness in her voice. Part of her wished she'd let the call go to voice mail so she had more time to gather her thoughts and come up with a game plan. Grant had always been an expert at litigating arguments and disagreements, managing to turn things around so that what had started out as a legitimate concern on her part ended up feeling silly and unreasonable and inconsiderate.
"Oh, thank God, Alexa. Where are you? Are you okay?" His voice, filled with a convincing concern, sounded out on speaker. Letting Maverick hear her life fall apart was about as comfortable as eating glass, but he'd already _seen_ it fall apart. So what was the difference?
"No, I'm not okay," she said, her stomach tossing. But what more did she have to lose? If she wasn't going to speak the truth to him now, she never would. "You physically threw me outside in the middle of a storm late at night. I'm not okay."
"I know. I know. Things just got out of hand." In her mind's eye, she could almost see him pacing.
"Things didn't get out of hand, Grant. _You_ got out of hand. With your suspiciousness, you mistrusting me, and you physically hurting me. Do you know how humiliating it was to be locked out of my own house? You were supposed to take care of me." The anger she'd felt the night before, the anger that had given her the strength and determination to leave her ring and walk off that porch, pooled in her gut once again.
"I didn't hurt you, did I? I didn't mean it," he said, an odd tone to his voice.
"I just said that you did, so yeah." Her gaze cut to Maverick, who was studying her, that scowl getting deeper, the look in his eyes totally lethal. Not that she minded any of that. Because she meant what she'd admitted to him—she didn't trust her own judgment where Grant was concerned. She needed him to be a sounding board right now. "My arm hurts from how you grabbed me and the back of my head hurts from you pushing me into the door. And this is just a few weeks after you promised that nothing like our last fight would ever happen again." She blew out another shaky breath. Adrenaline coursed through her. She _never_ talked to Grant like this and her body was clearly preparing for the need to flee from the consequences.
"Yeah. Yeah. Uh, okay, Alexa. Can you please come home so I can make sure you're okay with my own eyes? I've been up all night worrying and imagining every nightmarish scenario. I need you to come home."
It didn't escape her notice that he'd brushed off her pain to list all the things he needed and had suffered. Had he always been this big of an asshole? "I don't think that's a good idea," she said, her voice less certain. Had he not found her ring?
"What? Why not?" he asked. The surprise in his voice sounded so damn genuine that Alexa believed it. Which meant that he either didn't expect her to draw a line in the sand where his behavior was concerned or that he really didn't think he'd done anything so wrong as to warrant this reaction. Both possibilities left her feeling sad and defeated. How had she not seen this sooner? How had she let him get away with it for so long?
"Something clearly isn't working between us, Grant. It's getting worse and I can't take it anymore," she said. She felt the weight of Maverick's gaze, but couldn't let herself look at him. Not just then.
"Alexa, I can fix this," Grant said. "It was just a fight."
"I don't think so," she said. "And it was more than that to me."
"Damnit, Alexa—"
"You told me I was _nothing_ , Grant," she said, shame making her face hot for having to say that in front of Maverick. "Who says that?"
"I know, babe. Things just got out of control."
Again with the _things_. He wasn't taking responsibility here, was he? She sighed and didn't answer. She'd already addressed this once.
"Well, uh." A long pause filtered down the line. "What about work? Are you planning to come in? The model home project won't get done without you. No one else could do it like you. And you love that project."
Relief lifted some of the weight pressing down on her shoulders. If he was asking her, then maybe he didn't see her being _with_ him and working _for_ him as being fundamentally linked. After all, she'd worked there initially before they'd become a couple. Maybe she'd be able to continue to work at Slater Enterprises and they'd be able to act like professionals and adults.
"I still have a job at Slater Enterprises?" she asked, figuring she might as well be direct about it.
"Of course you do, sweetheart. Yes."
She ignored the _sweetheart_. "Then I'll be in this morning, but I'll be late since I don't have any clothes with me."
"Whatever you need, Alexa. Of course. I'll see you later then?" he asked. Was that hope in his voice? Or something else?
"I'll see you at the office." A shiver rushed over her skin.
"I love you," he said.
For a moment, her tongue got tangled, because she couldn't say it—not just because of what had happened, but because she felt like she was talking to a stranger. Like she'd been loving and sleeping with a stranger. Not to mention that Maverick's burning blue eyes were focused so intently on her. "Okay," she finally managed. "Good-bye." She hung up without waiting for him to say anything else. Blowing out a breath, she scrubbed her hands over her face. "How'd I do? I know I didn't flip out on him," she said, chancing a glance at Maverick.
"You did fantastic, Al. I know that couldn't have been easy, but you stood up for yourself. You didn't let him get away with anything. And you set some ground rules for yourself." He nodded, his expression full of a pride that allowed her shoulders to relax and her lungs to inhale a deep breath. And God did it feel good to have someone in her corner. For once. She hadn't had that in so long. "I'm worried as fuck about you going back to work, though. I'm not going to lie. The idea of you being near him makes me want to smash things."
She sighed, as her belly did a little flip. "I know. I don't really want to be near him, either. But I have to work, especially given this situation. I have some savings, but I'm going to need more. Because almost everything I have is his. Including my mom's house. I don't even know if I'd get to keep my car because he convinced me to get rid of the one I had when he bought the new one for me, so it's in his name." Alexa hugged herself, sadness making her chest feel hollow. "I get that working for him long-term probably isn't realistic now. Even if we can manage a professional relationship, there's no way things won't be awkward. But I need to work there until I can find something else, at least. And seeing this project through to the end will give me a project for my portfolio I'll need to land a new job."
Mav bit out a curse under his breath and nailed her with a stare. "Fine. But I'm taking you and picking you up. At least until you get a better feel for how he's acting towards you. But I wouldn't trust good behavior for a fucking second."
Alexa thought about it for a moment and nodded. "I think that's probably a good idea. Thank you."
"You're welcome," he said, not sounding too happy about it.
The silence stretched out, and in that quiet, Alexa's mind returned to what had happened between them on the couch, to Maverick waking her up with his hands and his mouth and his body. Waking her up in so many ways.
Maverick Rylan was unquestionably masculine and unapologetically rough around the edges, and she found it even sexier now than she had when they'd been a couple. Maybe she'd been too young to appreciate him all those years ago. Maybe she'd been too focused on the wrong things. Maybe she'd been too distracted by her grief and her fear to see what was standing right in front of her.
Well, clearly all of that had been the case, hadn't it?
How could she ever ask him to forgive her?
_I never wanted to let you go in the first place._
The gravel of his voice played over again in her mind.
"When do you want to go?" Maverick asked. "It's eight-thirty."
His question pulled her from her thoughts. "Um, let's leave around nine? That'll make sure Grant's already left for work. I don't want to have to deal with him at the house. Oh, wait. I . . . I don't—" Alexa bit back her words and smiled.
"What?"
"I was going to say I don't have a way into the house, but I have you." She grinned. "Oh master of lock picking."
"Like that idea now, do ya?" He smirked at her.
"It's proven to come in handy," she said.
Maverick nodded and rose, and Alexa couldn't help but stare at his Ravens tattoo as he crossed the room. Why didn't the sight of it rattle her like it once had? "I'll grab some coffee and a quick shower, then we can go. Your clothes from last night should be dry if you want to change." He left, then quickly peered back into the room. "And I like it when you call me master." He winked and left before she could throw something at him.
Her grin faded as her thoughts got all tangled up in the past. After losing Tyler, Alexa had thought she'd learned everything she could know about being scared. She'd been scared of how she'd manage to take care of her mother by herself. She'd been scared of how her mother would handle Ty's death, and then of how her mom had fallen apart and gotten sicker. She'd been scared that she'd never get the life she'd always dreamed of having. And she'd been scared of what would happen—and how it would feel—if she lost Maverick the same way, too.
Now she knew there was a lot about fear she hadn't known.
She'd learned that living your life guided by fear meant that a scary amount of your life passed you by while you weren't doing the things or being with the people you most wanted. She'd learned that there was more than one kind of loss. Death was the most catastrophic, obviously. But losing your friends and your independence and your faith in your own judgment were pretty damn scary, too. And she'd learned that sometimes playing it safe was its own kind of loss—representing wasted chances, unrecognized opportunities, and unrealized possibility.
Maybe . . . maybe things crashing and burning with Grant didn't represent so much of a loss as an opportunity?
From the direction of Maverick's bedroom, the hum of the shower water sounded out, and her mind couldn't help but conjure up a series of old, secret images. The water sluicing over all Maverick's hard edges and ink. Him taking her against the tile wall. Her on her knees on the shower floor, his cock in her hands, her mouth. A shiver raced through her.
Maybe losing the life she thought she was going to have with Grant could give her a second chance with . . . other things?
Lucy curled around Alexa's ankles and let out a pleading meow. On a sigh, Alexa bent down to give the sphynx some love. "Don't worry, Luce. Everything's going to be all right. We'll figure it all out. I promise."
Though Alexa didn't know who she was trying to reassure most.
"WANT ME TO come in with you?" Maverick asked from the driver's seat of his truck.
Alexa looked out the window to the sleek glass building that housed Slater Enterprises. Would going inside feel as strange as going home had felt? Maverick had let her into her house so she could shower and change, and the first thing that had dropped a ball of nerves into her gut was that her ring hadn't been hanging on the door knocker. Which meant Grant had seen it after all. Which left her wondering even more about what he'd said—and what he hadn't said—when they'd spoken on the phone.
The second thing that had left her feeling uncertain was the question of whether she should take anything beyond Lucy's nearly empty food and kitty litter and her own clothes, toiletries, and things for school. She didn't even know whether to take her car. Maverick convinced her that she should, and he had a new guy who was joining the Ravens drive someone over who could take it back to Mav's house. Otherwise, she'd only taken what was definitely hers.
And Mav made sure she had _all_ of it so she didn't have to come back. He'd personally carted loads of hanging clothes and suitcases full of personal items out to his truck himself until the other Ravens showed up to help.
The third thing that had felt so weird about returning was how much she didn't feel like she belonged there. It was almost like poking around a stranger's house, her nerves telling her she was going to get caught at any moment. She could almost imagine it'd been months since she'd last been there—years, even. It was more than a little surreal.
Finally, Alexa shook her head, her eyes still glued to the Slater Enterprises building. Wondering. Worrying. Debating. But she had to do this. She had to be strong. It was time. "No. It'll be okay. If I get worried, I'll call you to come get me."
"Look at me," Maverick said. Alexa's gaze cut to him. "Promise me." He arched a brow over dark blue eyes filled with fierce concern, his expression so serious it emphasized all the hard angles of his face. It made him beautiful to her, although he'd probably think that was a ridiculous thing to think about him. But how could she find his protectiveness, concern, and support anything but beautiful? She knew too well what the opposite looked like.
"I promise," Alexa said, making her words strong and sure when her insides felt a whole lot like someone was taking a mixer to them. But she was going to have to face Grant some time. No sense in putting it off, and doing it in public had all kinds of advantages over doing it privately.
He gave a tight nod. "I'll be here when you come out at five."
"Okay." She resisted the urge to lean over and kiss Maverick, because despite _the couch_ , she had no idea what they were or weren't. She opened the door and climbed out. Hesitating, she looked back in. "Thanks, Maverick. You didn't have to do all this for me, and . . . yeah . . . just, thanks."
"That's bullshit," he said, eyes blazing with an intensity she didn't really understand. "You're family. It's what we do. But you're welcome." He winked.
For some reason his expression and his words made her ache with yearning—a yearning that there was more between them than just history. "Okay." With a little nod, she pushed the door shut and headed inside.
Alexa's nerves were on high alert as she entered the building, part of her sure some sort of trap would spring on her the moment she walked through the doors. But there was only Christina, smiling at her but her expression also a little concerned. "Hey. Your mom okay?"
Confused, Alexa nodded.
"It's just that you're never late unless your mom has an appointment," Christina said.
"Oh, uh, no. Mom's fine. Thank you, though." She debated what to say, and then decided it would be better not to say a word. Not yet. "I'm sorry to run, but I'm late for something." She thumbed over her shoulder as she moved away from the desk. "I'll see you later, though?"
"Okay," Christina said, clearly catching on to her weirdness. Not that Alexa could worry about it just then.
She made her way to her office, sure that everyone knew what'd happened, like everyone could just look at her and see that her world was falling apart. But she pasted on a smile, offered greetings as she passed her colleagues, and finally made it to her desk.
A gigantic over-the-top bouquet of fresh flowers in a thick crystal vase filled one whole end of her work space. Three dozen flowers in bold, bright colors, leafy foliage, and baby's breath made an arrangement so big it could've been for a wedding. Or a funeral.
She opened the card.
_Alexa—_
_Come home to me where you belong._
_Love, Grant_
She sighed. It almost didn't matter that he hadn't given the flowers as an apology, because him apologizing wouldn't change how she felt. The note bothered her for other reasons. She was worried that it revealed Grant's stubborn unwillingness to hear what she was saying. That he was ignoring the fact that she'd taken off her ring. And it bugged her that he'd put her in the position of having to keep the flowers in her office or else give her coworkers a reason to wonder why she wasn't keeping such an exquisite gift from her fiancé less than two weeks before their wedding. She moved them to the credenza to get them out of her way and then dove into all the work the model home and a few of her smaller projects required.
Maverick texted her throughout the day making sure she was okay, and who knew that the word _fuck_ could sound so sweet? _I'm fucking worried about you. He'll fucking pay if he hurts you. IS IT 5 OFUCKINGCLOCK YET?_ That one came complete with a winky face. And when a big tough biker dude sent you shouty caps and a winky face, it could make you smile.
Otherwise, no one disturbed Alexa all day. It was four o'clock before she knew it. And she was both relieved and anxious about not yet having seen Grant.
Of course, his knock finally came.
"Alexa," Grant said, stepping into the office wearing an exquisitely tailored blue suit. His gaze cut to the flowers off to the side before he closed the door. "Did you like the flowers?"
"They're beautiful, but you shouldn't have," she said, not trying to hide the edge in her voice. She clasped her hands on top of the desk.
Grant came closer until he stood right in front of her, forcing her to look up to meet his eyes. "Of course I should. I'd do anything to make things right."
Her stomach dropped. "Flowers can't fix what's broken between us, Grant."
For a moment, she would've sworn the narrowing of his eyes and the tightness around his mouth foreshadowed building anger, but then it was gone from his expression and he shook his head. "I know. I just wanted you to know I was thinking about you, wanting you."
Nerves made her knees feel like jelly, but Alexa stood up anyway, needing to be on a more even playing field while she said what she had to say. "You need to stop wanting me, Grant. I know you found the ring." She shook her head and gathered her thoughts. "We're over."
Grant's eyes did narrow then, and he pulled something out of his pocket. Her diamond. It glinted in the sunlight coming through the window behind her. "Damnit, Alexa, this is _yours_." He nailed her with an intimidating stare. It was hard not to duck her chin. "You can't just end it. We have five years of history. You can't throw that away. Don't you think we deserve some conversation, some time to cool down and talk this out, some effort to fix things?"
"Grant—"
He stepped forward and she could've sworn he meant to grasp her wrist, but then he put the ring down on her paperwork. "I'm not letting you give it back. This isn't over."
She crossed her arms and arched a brow at him. "Yes, it is."
He scrubbed his hands over his face, then held them up as if in surrender. "Please, Alexa. You're important to me."
Frowning, Alexa shook her head. For one of the first times ever, she seemed to be able to hear what he wasn't saying as clearly as what he was, and it solidified that she was doing the right thing. "Please take it, Grant. I know it's really valuable. If you don't take it I'm afraid it'll get lost or stolen, because I won't be taking it either."
"I don't care about the ring," he said, his voice quiet but intense. Just like it had been in his office the night before.
"You say you care about me and you want to fix things, but you won't even honor what I want right now," she said, working to keep her tone even. "Take the freaking ring."
She had no idea how she held out in the staring match that followed. Her heart raced. Her knees felt like they could give out. She fisted her hands to try to hide her shakiness. Finally, he heaved out a long breath and scooped the diamond up again. "This isn't over," he said, and then he turned and stalked out.
The moment the door closed behind him, Alexa nearly collapsed into her chair.
She was shaky and a little woozy from the adrenaline pumping through her, but it still felt like a damn victory. That was quite possibly the first disagreement with him she'd felt like she'd ever won.
Despite his parting words.
But they didn't matter. Because she and Grant _were_ over.
And there was absolutely nothing he could do to change that.
## CHAPTER 15
Maverick was crawling out of his skin waiting for Alexa to walk out of work. Leaning against the front of his truck, arms crossed and eyes trained on the doors, it took everything he had not to march in there and escort her out.
He'd been a mess of worst case scenarios all day. Slater hitting her. Slater nabbing her and hiding her away. Slater snapping and taking them both out in some melodramatic murder-suicide. Even as the thoughts had been rushing through his brain, he'd known they were fucking ridiculous. Grant Slater was nothing if not image-conscious, which was the only thing that'd kept Maverick sane. The guy wasn't going to do anything obvious, not in public at least.
That hadn't made it any easier for Maverick to concentrate on the 1950s Copper Hardtail he'd been working on reengineering for a client. And that didn't mean her ex wouldn't go all stalker. Because Maverick believed he had that in him. And he didn't trust Slater's concerned act on that call this morning in the least.
He just hoped Alexa didn't either.
Then, finally, there she was. A beautiful vision in a pair of form-fitting black pants, strappy black heels, and a silky pale pink blouse. A _sexy_ vision. Put together and polished and professional.
Unlike him. He was all ripped jeans, stained hands, and beat-up boots.
Did she care about that stuff?
Didn't she have to—at least on some level? Five years ago, she'd picked Slater over him, and he and Slater couldn't have been more different. Abusive-asshole factor aside, Slater was two-thousand-dollar business suits and high-end luxury cars and million-dollar mansions. From his own business and his cut from the Ravens, Maverick had money, but he didn't flash it. He was a jeans, T-shirt, bike kinda man.
Sonofabitch.
As she neared, Alexa smiled. And damn if that shit didn't light him up inside. "Hey," she said.
"Hey." He gave a nod as she came to stand in front of him. His gaze raced over her, checking for signs that anything bad had happened in the seven hours that they'd been apart. But she looked fine. She looked _perfect_. Need roared through him. "Everything okay?"
"Yep," she said, her voice cheerful. Too cheerful?
He eyeballed her. "Slater in there?"
Her smile slid away. "Yeah."
Maverick ground his teeth together as a fantasy flooded through his mind's eye. One he wanted that asshole to fucking see. "Know what I want to do?"
"You promised you wouldn't hurt—"
"No, that's not it. Well, it is, but it's not what I was thinking of right now," he said.
An uncertain smile crept back onto her pretty face. "Okay. Well, then, what do you want to do?"
One beat passed. And then another. And Maverick let the words that described his fantasy fly. "I want to pin you against my truck and kiss the fuck out of you. I want to put my hands on you. Everywhere. I want to push those sexy little pants off your even sexier ass and make you wet with my fingers until you're crying out my name. While all those eyes behind that dark glass are watching. While _Slater's_ watching."
Alexa let out a shaky breath and her mouth dropped open. Heat slid into her gaze, making it harder for him to resist turning his words into actions. "Probably . . . um, probably not a good idea."
He chuffed out a little laugh. "Probably not." He reached for the handle and held the door open for her.
With a little nod, she got up into the truck's cab.
Maverick stepped into the doorway. "You really okay?"
"Yes," she said, her voice breathy.
"He talk to you?" She nodded. "He _touch_ you?"
"No." Her eyes flashed to the building, making Mav's scalp prickle. He peered over his shoulder just to make sure Slater wasn't standing right behind him. "I told him we were over. He tried to convince me to at least talk it out. And then he gave me back my ring."
Maverick couldn't help it. His gaze dropped to her lap, where her hands were fisted against her thighs. No ring.
"I wouldn't accept it, though, and he finally had to take it back."
He met her eyes and what he saw there nearly sucker punched him. Pride. Strength. Courage. None of this was easy for her, and he knew that. It made him proud of her, too. And he wouldn't have been the only one. "Tyler would be proud of you."
She gave him a smile that was so sweet and so sad that it felt like she'd reached inside his chest and squeezed. "That . . . that means a lot, Maverick."
"Just calling it like I see it," he said in a low voice. "He say anything else?"
She rolled her eyes. "Just that it wasn't over."
Of course it wasn't. The fucker. "We'll see about that."
The ride home was quiet, and then they were back at his house again.
"Your stuff's all in your room," Maverick said. He dropped his keys on the kitchen counter.
"You didn't have to unload it all. I would've helped," she said, placing her purse next to his keys.
He shouldn't like the look of that so much. Or the way it felt to have company in here at the end of the day. Or the way the clicking of her heels against the hardwoods echoed through the house.
"No biggie, Al," he said, grabbing a soda from the fridge. Just to have something to do with his hands that didn't involve getting them wet with the slickness he could draw out from between her legs. Yeah, they'd fucked this morning. But that didn't mean she wanted it to happen again. He needed to rein his damn self in.
"Gonna change. Be right back."
When she was gone, he raked his hand through his hair and mentally ran through his favorite parts of the rebuild he was doing. It had a copper-forged gas tank and copper-coated lower shock arms, rear fender strut, bar risers, and a bunch of other copper accents. When it was done, it was going to have a sweet throwback vibe.
Nope. Didn't work. He still wanted to follow her back to her bedroom. Especially since his brain knew that she was stripping down and baring all that warm, warm skin.
Goddamnit. He needed to knock his head against a wall somewhere. Because the last thing she needed was him coming on too strong. Or, frankly, coming on at all.
"Hey, Mav?" she called. "Do you see Lucy anywhere?"
He pushed off the counter and made for the living room. "Not in here," he said. He grinned to himself because he had a few surprises for Alexa where Lucy was concerned. He checked the hall bathroom next. As little as the thing was, she could be hiding just about anywhere. When he'd returned earlier in the day with Alexa's stuff, the cat had been standoffish at first, but once her mama's things started coming in, Lucy came closer, poking around and sniffing at Alexa's clothes and bags, until she finally let Maverick pet her and even pick her up.
And then, naturally, he'd needed to win over the fucking cat all the way. So when he'd gone to the grocery store to make sure he had some of Alexa's favorites in the house, he'd just happened into the pet aisle. Not seeing much, he'd stopped at the pet store, where a shit-ton of stuff had fallen into his cart. For fuck sake.
"Maverick?"
He followed her voice to the back of the house and found her standing in his bedroom doorway. She'd changed into a light green tank top and a pair of beat-up jeans that framed her curves perfectly, and put up her wavy brown hair into a ponytail. Standing there in bare feet, the threads of the frayed denim just touching the floor, she was the sexiest woman he'd ever seen.
"I can't believe it," she said, peering inside the room.
Lucy was curled up asleep on his pillow. "Why not?"
Alexa's eyes stayed on her cat. "Because Lucy was terrified of Grant. Wouldn't go near him. Bolted out of any room he came into. The only exception was at night. She slept under my side of the bed to be near me." Al finally met his gaze. "I thought she had a problem with men."
Mav smirked, even though the awe in Alexa's tone was doing shit to his chest again. "Only the assholes, apparently."
She burst out laughing. "Only the assholes."
A meow rang out from the bed, where Lucy was engaged in a long, exaggerated stretch.
"Hey, baby," Alexa said, crossing the room. "What . . . what are you wearing?"
Grinning, Maverick waited.
"Oh my God." She laughed and scooped the cat into her arms. "You bought her a sweater?"
He finally let his laughter loose. "I made her a badass." The sweater covered the cat's thin body from neck to back legs and was black with white skulls and crossbones.
Alexa kissed the cat's head, and Maverick could hear Lucy purring from several feet away. "She didn't mind you putting it on?" More of that awe.
"I wouldn't go as far as saying that she loved it," he said, moving closer and holding his hand out for the cat. Lucy forced her head against his palm. "But we made friends. Plus, um, I might've bribed her with treats."
Chuckling, Alexa shook her head. "I'm sorry I missed that." Lucy hopped out of her arms. Alexa's expression slowly changed and sadness filled her hazel eyes. "I'm sorry . . . for so much."
The word stirred up all kinds of things inside him, but Maverick just shook his head. "You're all about the future now."
"Yeah. Yeah, I am."
His cell buzzed in his pocket. Looking at the screen, he found a text from Bunny.
_You coming to dinner tonight?_
He hadn't planned to, not with Alexa here, but it wasn't often that his mother asked that outright. Which had him feeling like she wanted him there at the clubhouse. Rodeo had confided that she still didn't feel entirely comfortable being there.
"Everything okay?" Alexa asked.
He shrugged. "Mom. Asking if I was coming up to the clubhouse for dinner. She's, uh, she's still having a hard time."
"I'm sorry, Maverick. I can't believe everything that happened to her. You should go."
"You, uh, you wanna come?" he asked.
"Oh. Uh. Really?"
"Is that a no?" he asked with a frown. He hadn't realized how much he wanted her to agree until she'd answered.
"Not a no," she said. "More of a 'are you sure.' I remember how Dare looked at me when I was there a couple of weeks ago. I know after everything that happened between us that he's not my biggest fan." She hugged herself. "I don't blame him for that. Just stating the facts."
"My cousin's a pain in the ass," he said. He couldn't deny her words, so there was no point doing so. It wasn't that Dare didn't like Alexa; he didn't like the way things had gone down between them five years before. But that wasn't any of Dare's business, and he'd get over it in five seconds if Maverick wanted him to.
"No, your cousin's loyal," Alexa said. The characterization earned her some points in Maverick's book, not that she needed them. Just that he respected the hell out of her saying something nice about someone he cared about. And it was true, too.
"Come," he said.
Her eyes rose to his, and a beautiful blush bloomed across her cheeks. What the hell? And then he thought about his choice of words.
_Come_.
A slow smile slid onto his face, and he dragged his knuckles down her cheek. "Come," he said again, just to see if the blush would deepen. It did.
On a laugh, she knocked his hand away. " _Now_ who's being an asshole? Lucy's going to take her sweater off." She pushed by him, and it felt a whole helluva lot like she was running from the innuendo, which kinda made him want to keep teasing.
"So are you coming yet?" he asked, laughter in his voice now. He was cracking himself up.
"Oh my God," she said as she went into her room up the hall.
"You know you want to," he said as he leaned in her door. He found her unzipping a suitcase full of shoes.
"You want me to come?" she asked, smiling up at him as she slid on a pair of sandals.
Grinning, he nodded, loving that she was playing along. Loving _seeing_ her happier. Because finding her in the rain the night before, her expression desolate, her voice anguished, had been a big part of the reason he hadn't been able to sleep. "Hell, yeah, I do."
She planted her hands on her hips and arched a brow, a blasé expression on her face. "Then why don't you _make me_ come?"
It took half a second for the challenge to sink in, and then Maverick was crossing the room and tossing her over his shoulder. "Oh, Alexa. Don't issue a challenge you don't want answered."
She screamed and laughed, her hands smacking his back. "Put me down, Maverick Rylan."
He swatted her butt, loving every squirm of her body against his. "No fucking way."
"You better put me down right now!" she said, laughing harder.
"Oh, yeah? Why's that?" he asked, enjoying the hell out of himself as he grabbed the keys for his Harley off the counter.
"Because . . . because . . . if you don't, I'll make you pay!"
He chuckled. "I think I like the sound of that."
"Oh, my God," she said, as he walked them toward the door. "What is . . . Maverick, wait."
The change in her voice made him stop. "What?"
"Is that . . ." She twisted her upper body. "Is that a new climber for Lucy?"
"Oh," he said, glancing at the carpeted three-level contraption sitting in front of the big window that looked out onto the pond behind the house. Lucy lay in the top level, her gaze fixed on something outside. "Uh, yeah."
For a long moment, she didn't say anything else. Fuck. He'd gone too far, hadn't he?
And then her arms came around his stomach and she laid her head against his back in an awkward but awesome embrace. "Take me to dinner, Mav," she said softly.
Satisfaction roared through him. "You bet."
## CHAPTER 16
The ride to the clubhouse wasn't long, but Alexa savored every minute on the back of Mav's Night Rod Special, which was quite possibly the sexiest bike ever made. The warm breeze across her skin felt like freedom, and riding gave her a totally legit reason to wrap herself around him. This man who was not only generously taking care of her, but also of her cat.
A cat climber was maybe a stupid thing to get choked up about, but she had anyway. Because it wasn't just a cat climber to her. It was Mav recognizing that something was vitally important to her, even if it wasn't to him, and treating it with care and respect. And that had touched her.
_When was the last time I had this much fun and felt this carefree after coming home from work?_
The easiness, the joking innuendos, him tossing her over his shoulder, them going for a ride. There was no stress about any of it. No right or wrong. No expectations to live up to. She had no doubt that she could've begged out of dinner, and Maverick would've been fine about it.
She could just be herself—wants, wishes, desires, needs, and all.
Which was when it occurred to her. How much acting she'd done in Grant's presence. The masks she'd donned and the roles she'd played. Ones he'd crafted for her slowly but surely over the past five years until she thought they'd been her own idea.
She released a deep breath, imagining as she did that she was letting go. Letting go of all that stuff that wasn't her true self. And just letting it blow away in the wind.
Before long, they pulled into the big parking lot in front of the Ravens' clubhouse. The building still had the charm of a mountain resort even with more than a dozen motorcycles parked in front of it. Maverick parked the NRS and the engine went quiet. As she dismounted and took off her helmet, her eyes strayed to the chop shop building across the way. Maverick did most of his bike-building work in there, and she'd spent many an hour watching him. Asking him questions about what he was doing and handing him tools.
"You working on anything fun right now?" she asked.
"Always," he said with a smile. "Although club business has been distracting me the past few weeks. I can show you after dinner if you want."
"Yeah," she said.
With a side nod, he gestured toward the door. "Ready?"
"Sure." Even though she was more nervous than she wanted to admit. The last time she'd come, she'd been so upset over the fight and her injuries that she really hadn't had time to think about how the guys might treat her. And even though she'd sensed a little wariness from Dare, he'd also been clearly concerned and ready to help her. Everyone had been kind. No one had been mean. In fact, a woman named Haven had not only tended to her wounds but made her feel welcomed, reassured, and so much calmer. So maybe she didn't have anything to worry about after all.
The thought didn't keep the butterflies from whipping around in her belly.
Inside, things looked pretty much like she remembered. Couches filled the big lounge in what had once been the inn's front lobby, and the registration desk still actually stood at the back of the room. Photos of the club members filled the wall by the mess hall door. And on the other side, she could just peer into the big rec room where the bar was and most of the partying happened.
Voices spilled out of the mess hall, and she followed Maverick in, those butterflies getting a lot more active.
A couple of the guys greeted Maverick right away. She hung back as he shook hands and gave a few of those one-armed hugs men did. He made introductions, or re-introductions in a few cases, and everyone acted totally normal towards her.
"Alexa?"
She turned to find a woman . . . who looked a lot different than she remembered. "Haven?"
Nodding, Haven laughed as she settled a big platter of grilled burgers, hot dogs, sausages, and chicken legs in the middle of the table. Beside her, a woman with shoulder-length blond hair was putting down an equally big platter of grilled vegetables. "The hair, right?" Haven asked, smiling.
"Yeah," Alexa said, remembering the hip-length pale blond hair she'd had when they'd first met. Now it was light brown and hung in soft waves to her shoulder blades. "It looks really great."
"Sometimes change is good," Haven said.
The words sure hit home for Alexa. "Yeah. Sometimes it really is."
"Alexa, this is my best friend, Cora," Haven said.
Cora's smile was immediate. "Hey, there. Nice to meet you." They shook hands.
"You, too. Can I help at all?" Alexa asked, feeling more and more comfortable with each passing minute.
"Sure," Haven said, waving her into the kitchen.
They passed through the swinging door, and then Alexa's gaze snagged on the chair where she'd sat that day, bruised and bloodied, while Haven knelt in front of her and cleaned her up. It felt like yesterday and years ago all at the same time.
"Alexa Harmon. How are you, honey?" Bunny asked.
"Hi, Bunny," she said, so happy to see Maverick's mother whom she'd always liked, even if she was also nervous about what the older woman thought of her. Despite having hair that was so pale blond it was nearly white, you'd never guess that Bunny McKeon was in her early sixties. She wore a stylish pair of dark wash jeans and a form-fitting black T-shirt. "It's really good to see you." Though it was equally hard to see the nearly healed but still visible scratches and bruises on Bunny's face.
"Well, come on over here and give me a hug, young lady. It's been too long," Bunny said, holding open her arms.
The welcoming gesture made the backs of her eyes prick with tears that Alexa blinked away. They hugged for a long moment. "I'm sorry for what happened," she said.
"It's a crazy world," Bunny said, her voice quiet.
Alexa nodded. "Yes, it is."
Bunny pulled back from the hug and held Alexa's face in her hands. Light blue eyes examined her. "You doing okay?"
"Much better now," Alexa said.
"Good. Well . . ." Bunny patted her shoulder. "Let's get these boys fed before there's an uprising."
They all laughed.
Dinner was a lively, raucous affair full of funny stories, snarky retorts, and at least one roll-throwing incident. Alexa hadn't laughed so much or so hard in a long, long time. God, she couldn't even remember when the last time was. Not that she'd ever had a big family, but that's what it felt like. And it made her even more certain that she'd misjudged things five years before. She wanted to go back in time and shake herself.
Emotion suddenly crashed over her, like a wave unexpectedly swamping her in the ocean. She excused herself from the table and rushed into the kitchen and then out onto the big back porch. Lounge chairs filled the long space, which had a stunning view of the mountain and the valley beyond. Bracing her hands on the railing, Alexa stared at the vista until it went blurry.
The screen door quietly closed behind her, and Alexa batted away the wetness on her face.
"You okay?" Haven asked.
Alexa released a long breath. "Mostly."
"Yeah," Haven said, coming to stand beside her. "Isn't this view the best? I love to sit out here."
"I've always loved it," Alexa said, her thoughts a jumbled mess. "I just wish it hadn't taken me so long to see it again." Not just talking about the view. Obviously. She could almost feel Haven's questions, and she finally turned to look at the other woman. Haven appeared younger than Alexa was, though there was something about her eyes—the brightest blue Alexa had ever seen—that read as older, like maybe she'd seen way more than she should've at her age. "Messing up really sucks."
Understatement of the century.
"It does," Haven said. "But sometimes it also makes what you should've done really crystal clear. And that at least gives you a direction to go in."
The words resonated with Alexa down deep and took a little of the weight off her shoulders. Nodding, she said, "Just got a little overwhelmed in there."
Haven laughed. "They're an overwhelming bunch."
The affection in the woman's words made Alexa smile. "That's for sure." She turned to Haven. "Do you mind if I ask how you're connected to the Ravens? I'm just curious." When Alexa had been with Maverick, Bunny; Bear's wife, Margie; and Slider's wife, Kim, were the women she most frequently saw around the clubhouse, with a few other wives and girlfriends pitching in for bigger events and on race nights. But Margie and Kim were both several years' gone now. Alexa had seen their obituaries in the paper.
"Well, it's kind of a long story," Haven said. "My father was a bad man, and I ran away from him with Cora's help. When our truck broke down in Baltimore, the tow truck driver turned us over to a gang in the city. A couple days later, we got rescued, and then the Ravens agreed to take us in until we figured out what was next."
Alexa's mouth dropped open. And she thought she'd been in a tough situation. "Oh, God, Haven. I'm so sorry. That's terrible."
"Best thing that ever happened to me," Haven said, a little humor in her tone, but mostly it was clear that she meant it. "It led me to this place and these people. To Bunny, who's become like a mom to me. And it brought me to Dare." Her face went pink at the admission, and Alexa was kinda blown away to learn that Dare had a girlfriend. Back in the day, she'd never known him to be remotely interested in settling down. "And to making new friends like you."
The words reached inside Alexa and made her see the Ravens through Haven's eyes. As protectors, as heroes, as _family_. And that made Alexa realize again that in her grief and fear she'd focused on the wrong things five years ago. Or, at least, it felt that way now. She shook the thoughts away. Maverick was right. She was all about the future. "I could use a new friend."
"Good. Then, would you like to see my trick to getting a big group of scary bikers to eat out of the palm of your hand?" Haven asked, a mischievous sparkle in her eyes.
Laughing, Alexa nodded. "You've got me curious, so, yes."
"Come on." Haven led her into the kitchen. The guys were bringing in their dirty dishes and stacking them in the sink. On the far counter, Haven pulled out two big containers. "My secret weapons," she whispered, taking off the lids.
The most glorious smells wafted into the air. "My God, what is that?"
"Are you getting dessert?" came a voice from behind them. Alexa turned to find Phoenix Creed grinning at them. With all the time she used to spend out in the chop shop, she'd once known him pretty well. He was a jokester and a hopeless flirt and always made people laugh. Wearing a charming smile and a playful expression, he didn't appear to have aged one day.
"You best stay back, Phoenix," Haven said, pointing at the guy as if to ward him off. "You try to get a peanut butter cookie before Dare and I'm not responsible for what happens."
"Aw, come on, Haven. Help a brother out," he said, pressing his hands together like he was praying.
"You know that's not an idle threat about Dare," Alexa said. "Better listen."
"Aw, you, too?" He clutched his heart as Cora joined them, a smile on her face and her eyebrow arched like she was trying to figure out what was going on. He put his arm around Cora's shoulders and held out a hand. "Why do all the women around here gang up on me?"
"Uh, because it's super fun?" Cora said, sliding out from under his arm and giving him a look.
"It really is." Haven nodded.
Expression roguish as he came up behind Phoenix, Dare asked, "You hassling my woman?" His voice was stern, clearly unhappy, and he schooled his expression as Phoenix turned to him.
"Dude. Me? No. Of course not. Was I, Haven? Just ask her. I totally was not hassling her." Alexa laughed as his words spilled out in a rush.
"You know, the longer you all chatter at me," Haven said, "the longer it takes me to get everything ready."
Phoenix held up his hands. "I'll be out here waiting patiently. Like a good biker."
"Sorry," Dare said with a wink as he kissed Haven on the cheek. And stealthily snatched a cookie from one of the tubs.
"Hey!" she said, grinning and shaking her head.
Smiling, he took a bite. And then his dark eyes turned on Alexa and scanned over her face. And she just knew he was looking for signs of the injuries she'd had. Gone now. "Good to have you here, Alexa."
"Thanks," she said. "For everything."
He nodded, his expression open, his tone genuine. "Anytime." Eating his cookie, he made his way back out to the mess hall.
"Okay," Haven said, so clearly at home among these men. Alexa found herself envying that. How long had it been since she'd really belonged somewhere? "So we have chocolate and caramel chip cookies. Peanut butter cookies. And mocha s'mores squares. Have one."
"I kinda want to go put on some comfy pajamas, find a good chick flick, and eat every single one of these," Alexa said, taking a chocolate and caramel chip cookie.
"That sounds like the best night ever," Cora said, grinning and snagging a cookie for herself.
"I agree. I could totally make that happen," Haven said, "but only if I'm invited."
"Deal." Alexa nibbled at her cookie, so good she had to close her eyes on the first bite, while she helped pile the treats on three big plates. "Is this your recipe? Because this is one of the best cookies I've ever had."
Cora chuckled and tucked her blond hair behind her ears. "And another one bites the dust."
Haven rolled her eyes. "Yeah, it's mine. Baking's kinda my thing."
"Clearly," Alexa said around her last bite, more than a little sad that her cookie was gone. Luckily, there appeared to be at least six dozen more left.
"It's not _kinda_ your thing," Cora said, nailing the other woman with the kind of calling-you-on-your-bullshit stare that only a best friend can get away with. "It _is_ your thing. Which is why you need to open your own bakery."
"Oh, are you thinking of doing that?" Alexa asked. Because she would buy the _crap_ out of these cookies.
Haven sighed and returned her friend's stare in a way that suggested there was a whole lot of something Alexa didn't know. "We'll see. But either way, I'm not doing it tonight, so you guys take those two and I'll take the peanut butter," Haven said, smiling at Alexa. "They'll love you forever."
Cheers rose up as she and the other girls walked into the mess hall, arms laden with sugar. Alexa couldn't help but smile as the guys thanked them and dove in like they'd never seen a cookie before. When Haven placed her plate in front of Dare, he pulled her down into his lap and gave her a big kiss that earned a round of catcalls. Dare flipped his middle finger at the room while he was still kissing her, and she finally pushed him away, laughing and shaking her head. Her face was bright red but her expression totally happy.
And Alexa . . . Alexa wanted that. So bad it made her ache.
Her eyes drifted to Maverick, who was looking at her like he was thinking the same thing. Her whole body flashed hot and everything else in the room faded away. God, what was happening between them? What _could_ happen between them, given what a mess her life was? Would Maverick even want something with her again after what she'd done to him five years ago? She had no doubt that he'd be a friend to her. And obviously he was still attracted to her—though that had never been their problem. And it also wasn't the same as having a relationship and trusting someone not to shut you out.
He'd given her that trust once and she'd thrown it away. And she knew him well enough to know that had to be huge to him.
"Tell me you're going to have some dessert," Phoenix said from the seat beside where she was standing. "Here, take a load off." He pushed out a chair for her.
"Thanks." Smiling, she sat and forced the thoughts away. And then she remembered something she'd seen a few weeks before. She turned to Phoenix and placed her hand on his arm. "Hey, I was really sorry to hear about your cousin." He'd apparently died in some kind of explosion. God, the Ravens had really been through it lately, hadn't they?
He clapped his hand atop hers. "Thanks, Al." His voice was softer, lacking the usual Phoenix playfulness. "I appreciate that." His brown eyes were sad for a moment, but then it was like he forced it away. Which reminded her that she wasn't the only one going through something. You never knew what a person was carrying inside them. She thought back to being at work earlier today. No one there realized that she and Grant were imploding, at least, she didn't think so. "Now have a cookie," he said. "They'll probably be gone in about thirty-five seconds."
Grinning, she looked down the table to the plate in front of Dare. "I want to try a peanut butter."
"Good luck with that," Phoenix said, sniggering.
"Hey, Dare?" she said, swallowing her nerves. "May I please have one of your cookies?"
The room got quieter in a hurry, and Alexa looked around to see what'd happened. Everyone was staring at her, then Dare. She looked back to him.
The guy's face was serious, almost stern, and he had one eyebrow cocked and his eyes locked on her. Like he wasn't happy. Like he was _pissed_. And she couldn't believe that he—
Dare grinned and laughed. "Only because you called them mine," he said, pushing the plate to the guy next to him to pass down. Everyone burst into laughter.
Shaking her head, Alexa had to laugh, too. Because she'd totally fallen for it.
As if he'd get pissed off over a cookie. These men had their rough edges, but they were good guys. They'd always been good to Tyler and her when she'd hung out with them years before. And everything she heard about what they did for people who needed help in the community said the same, too.
Grant _looked_ like Mr. Respectable, and yet, behind closed doors, he wasn't a nice person and he certainly hadn't been good for her the way she'd convinced herself he was. Meanwhile, sometimes Maverick looked like a total grease monkey and other times like a hard-ass biker who might kill you just as soon as ask your name, and yet he _was_ a good person, a fair person, generous and kind.
Damn. She'd totally fallen for the façade, hadn't she? For the stereotype. Not the truth.
In not much longer than thirty-five seconds, the cookies and s'mores squares were all gone, and Alexa pitched in with the dishes. Maverick and Dare hung out while they worked, helping by putting away the bigger things as the women dried them. Alexa was glad for the time to hang out with Bunny and the girls some more, because she already liked Haven a lot and Cora seemed pretty great, too. More than once, Alexa felt Maverick's eyes on her. Sure enough, when she gave into the urge to look, he was watching her work, his expression intense and approving. She didn't know what to make of that, except that it made her belly flip every time.
"Hey, uh, Cora?" came a deep voice from the doorway.
As the man exchanged nods with Dare and Maverick, it took Alexa a few seconds to realize who the new Raven was. Slider. Except he didn't look anything like the person she'd known five years before. This man seemed to carry the weight of the world on his rounded shoulders. His eyes were shuttered, his mouth was set in a grim line, and his hair was much longer and hung in a manner that made her think it'd been a long time since he'd had it cut.
"Hey," Cora said, grabbing a paper towel to dry her hands. "I can be ready in two minutes. Do you want some dinner? I can pack up something for you."
Not quite making eye contact, Slider shook his head.
"For the boys then?" she asked.
"They had pizza," he said, turning for the door. "I'll be in the truck."
Cora sighed and squeezed Haven's shoulder. "See you tomorrow," she said.
"Okay," Haven said, her gaze equal parts sad and worried.
They were all quiet for a moment after Cora left. Dare heaved a long troubled breath. Alexa's heart hurt for the changes she saw in Slider. She'd never known him that well, but he'd always seemed happy—good to Kim and a proud dad to their two little boys. Kim had died a few years before, but clearly he was still torn apart.
"Okay, youngins, I'm beat," Bunny said, folding a hand towel.
"Thanks for everything, Bunny," Dare said. "Dinner was great as usual."
"It's my pleasure. You boys know that. Besides, these girls did most of the work." She pulled her purse out of a cabinet in the corner and then gave a round of hugs and kisses before she left.
"I guess we'll head out, too," Maverick said, clasping hands with Dare. "You need anything?"
Dare shook his head. "Everything is quiet right now, which, knock on wood, will stay that way for a while." He rapped his fist against the closest cabinet.
"Amen to that," Mav said. "But just ask, okay? Don't be a fucking hero all the time." He turned to Haven. "Make him ask me for help."
She smiled and nodded. Dare gave him a shove. "Weren't you leaving?"
"Yeah, yeah," Maverick said, smiling at Alexa. "Still want to see the bike I'm working on?"
"Definitely," Alexa said. She turned to Haven and gave her a meaningful look. "Thanks for everything." She didn't just mean the awesome meal, but the pep talk out on the porch, the offer of friendship.
Haven seemed to understand. "You're welcome. Maverick has my cell phone number. Give me a call anytime." They hugged and said their good-byes.
And then it was just her and Maverick. Well, there were some other Ravens still around as they made their way through the clubhouse. But the moment Maverick placed his hand against her lower back, he was the only man Alexa could think about.
The only man she wanted.
## CHAPTER 17
As they walked across the darkening parking lot to the chop shop, the evening air still warm, Maverick felt a little like he'd traveled back in time. Alexa was here. Hanging with him and his brothers. And it felt good, natural, _right_.
Only tonight had actually been better and worse. Better because Maverick was older now, old enough to know what he actually wanted out of life, old enough to know that he didn't want to come home to an empty house forever, old enough to understand down to his marrow how important family was—and that he wanted one of his own. He hadn't understood all of that five years ago. Not really. Not enough to fight for it the way he should've. Not enough to man up when she'd all but asked him to.
Not enough to ask her what he should've asked her.
Damnit. Why hadn't he fought for her—for them—harder? Why hadn't he laid it all on the line and gotten down on one knee? Especially when he'd known that in her grief over losing Tyler she wasn't strong enough to do it herself. But then watching her head get turned by Slater had fucked with _his_ head. Maverick had let that shit happen. So that was on him.
At the same time, the night was worse because Mav had no fucking idea whether Alexa wanted any of that in return—with him. Sure, the chemistry between them was still off the charts. And the sex this morning had been phenomenal. Even better than he remembered it from before, and before had been fantastic. Worse because seeing Alexa at the clubhouse only made Maverick want her more, want her to see what he saw, want her for his, for keeps, forever.
And he knew it wasn't fair to even expect her to know what she wanted from or with him. Not when she was only a day into the implosion of the life she'd thought she wanted. There was no reason why she'd feel the same urgency he did. The urgency not to make the same mistake again, the urgency to hold tight and never let go. No matter how rough the ride got this time.
All of which meant it was going to fucking suck if he turned out to be her rebound man. And only her rebound man. And Maverick knew there was a chance that's all he was.
But, fuck, if that's all he could have of her, he'd take it. And deal with the fallout later.
The chop shop was a long building with four bay doors and an office at one end. He pushed in through the office door and hit the lights. The two desks in the room were primarily his and Phoenix's, though others used them sometimes. But this wasn't where Maverick did the lion's share of his nine-to-five. Instead, the last of the bays was his most regular domain. Kicking on lights as they went, he led Alexa through the shop past bikes other Ravens were working on to his space, separated from the rest by a cinder block wall. A sign over the door they walked through was the same as one that hung above the outside bay door and read, _Maverick Custom Cycles._
"Everything looks the same," Alexa said, slowly glancing around the room. She shook her head.
"What?" he asked, trying to see what she was seeing, trying to understand where her head was.
She gave a little shrug. "Standing here . . ." She cleared her throat and met his gaze. And damn if her eyes weren't a little glassy. "It feels like I was _just_ here. And it also feels like I've lived a whole lifetime since I was last here. I don't know." She looked away and moved closer to the bike he was working on.
So her head was kicking around some of the thoughts his was, then.
He watched her move around his space, liking seeing her there. How many hours had they spent here together? Her keeping him company when he was up late finishing something on deadline. Her singing along to the radio. Him teaching her about bikes and engines the way he'd been taught, and her being genuinely interested, asking a million and a half questions.
Them not being able to keep their hands off of each other, even when there were guys out in the rest of the shop working.
Heat rushed through his veins at the stream of memories that played against his mind's eye.
"I love the copper," she said, walking around the bike, running her fingers over the smooth, mirrored surface. "That really looks sharp. Kinda vintage."
She'd _always_ gotten his love for making something custom, unique, one-of-a-kind. Even when she hadn't known what the parts were called or what they even did, she'd had a great eye for what looked good.
Unlike her, he'd been around engines since he was a kid. He'd known the names of car parts before he'd been old enough to drive, courtesy of his mechanic father, who'd specialized in customizing trucks. Still did. Not that Maverick had seen him in a long time, because he avoided the sonofabitch as much as possible. It'd been the better part of a year since they'd last run into each other. Which was too recent for Mav's taste.
But the love of engines, of building and rebuilding, of putting your own custom spin on something? Yeah, he had to credit that to his old man. At least he had one positive thing that the guy did that he could point to.
Motorcycles, though . . . _that_ he'd learned from Doc. Maverick had spent a lot of time with his uncle growing up, and Doc didn't believe in idle hands. Never had.
"Yeah, vintage is the goal," Maverick said, moving closer. The bike stood between them. "The base was actually a 2004 Chevy, but it's been reengineered so heavily you'd never know. The goal is something similar to a 1950s hot rod style chopper."
She grinned, her eyes flashing.
"What?" he asked, her smile beckoning his.
"It's been a long time since I've heard you talk bike makes and models." Her smile got considerably sadder.
In a weird way, that gave him hope. "Too long?" he asked. Pushing her. Just a little.
"Too long," she said. "Definitely too long." She turned away like maybe the admission had been a lot to make. But that didn't take away from the satisfaction he felt. Not one bit.
She wandered around the space a little, looking at a couple of decorative ornaments he had laying on one counter. Leaning in close to some photographs he had pinned to a corkboard. And coming to—
Shit. Why hadn't he thought about this?
"What's this one?" she asked, her hand grabbing the dusty drop cloth covering the bike.
"Alexa, don't—"
The black cloth puddled on the floor around Tyler's repaired bike. In the months after her brother died, but before it was clear that Alexa wasn't going to take Maverick back, he'd fixed the bike. It had felt like a way for him to pay respect to his friend, and he'd thought Alexa could sell the bike if she didn't want the reminder. He knew she could use the money. At least, back then. But he never even got the chance to offer.
Shit.
"This is . . . is this . . ." She shook her head, her back still to him. "It's Tyler's." She said her brother's name so low Mav barely heard it.
"Yeah," he said, coming up behind her and aching for opening her up to this right now.
"You fixed it?"
He hated that he couldn't read the emotion in her voice, and finally gave in to the need to see her face. Taking her by the shoulders, he turned her to him. "Yeah," he said, leaning down to meet her eye to eye.
"Why?" she whispered, those pretty hazel eyes glassy—and that glassiness gutted him.
"For him," he said. A tear finally fell, and he caught it with his thumb. "And for you."
Her bottom lip trembled. "But I . . . I hurt you." More tears fell.
He wasn't going to deny it. "You did."
"I'm so sorry," she said, her face crumpling. "I was such a wreck when he died." She shook her head. "And I made so many mistakes. And pushing you away was the biggest of them all. And I'm so fucking sorry, Maverick." She clasped her hand over a sob.
He pulled her into his chest, wrapped one arm around her back, and cradled her head in his other hand. "I know, Al. I get it. And I did then, even though I hated it." And, _fuck_ , but hearing her words made him rage, too. Because he'd let himself be pushed away, hadn't he? He'd known she was a wreck, but hadn't done the hard work to stick it out and carry her through.
God, why hadn't he seen how much she needed him? Because hearing her now, it was so damn clear that he'd failed her.
And Tyler.
"I'm sorry, too," he said, his voice like gravel. Her tears scalded him with guilt and regret, but he'd take every one. For her. He stroked his hand through her soft hair, loving the way it felt. Loving the way _she_ felt pressed against him.
"No," she said, her voice no more than a rasp. She lifted her eyes to his. "All on me."
"Fuck, Al. Rarely is anything in a relationship all on one person. That day in the cemetery . . ." He shook his head. "You have no idea how many times I've wished that I'd given you what you needed. That . . . that was on me."
Her mouth dropped open, like the admission surprised her. And then they stared at one another a long moment, Maverick swiping away her falling tears until they slowed, then stopped.
"Wanna head home?" he finally asked.
She released a halting breath, her eyes searching his. "You're a good man, Maverick Rylan. Better than I gave you credit for. But I see you now." She paused, and he hung on her words like they were the oxygen he needed to breathe. "I know it may be too late. But, God, I see you."
His heart tripped into a thunderous beat, one that pounded through his veins. His cock hardened at the sentiment and the raw emotion on her face. "Not too late," he managed, need putting him on edge. "But, Jesus, you need to stop looking at me like that."
"Like what?"
"Like you want to take a bite out of me."
"Why?"
"Because my bite will be fucking bigger."
She shifted against him, the friction of her belly turning his cock to steel. His hands tightened on her, a silent command to be still. Alexa licked her lips, and he saw his hunger reflected back at him. Her gaze dropped to his mouth. Lingered there.
" _Alexa_ ," he growled.
"You sure? Because I don't want a bite. I want the whole meal."
The words hit him like a blowtorch licking over his skin. Without thought, he kissed her. Hard and claiming and rough. He absolutely plundered her mouth, his lips sucking, his tongue penetrating. The whimper she unleashed shot to his cock as her fingers twisted in his hair.
In his head, he was pulling down those sexy beat-up jeans and bending her over his bike . . .
Bike.
Maverick's eyes flashed open . . . and his gaze landed on the handlebars of Tyler's motorcycle.
He pulled back from the kiss, his fingers going immediately to his lips. "Fuck."
"What?"
"We're not doing this," he said, anger making his words come out sounding harsh. Anger at himself for thinking with his dick, for thinking of himself, for taking advantage of the emotional wreck she'd been just five goddamned minutes ago.
"Why?" she asked, a tinge of hurt in her voice.
He grabbed her hand and pressed it to the painfully hard ridge of his cock. "Not because I don't want you. Understand?" He nailed her with a stare. He didn't know all the shit Slater had filled her head with, but he'd never make her doubt his desire for her. Maverick didn't have the patience, tolerance, or disposition for game-playing. Never had.
Finally, she nodded, but she didn't remove her hand. "Then why?"
He shuddered out a breath and forced her touch away, but he slid his fingers through hers to try to take the sting out of the gesture. "Because you don't need anything else that messes with your head right now. Being inside you this morning was the best fucking thing I've felt in five years, but it also complicates the shit out of what you're going through. And me bending you over a bike in my garage isn't going to help you figure things out." He stepped away as what he was saying to her sank in to his own brain. Yeah. This was the right thing to do. Keep his damn hands off until she figured her life out. "So, yeah." He raked at his hair.
"Oh." She hugged herself, but nodded. "I guess, yeah. Makes sense. Sorry."
He stepped back into her space and grasped her chin. Forced her to meet his eyes. "Stop apologizing for things that aren't your fault, Alexa. That's _him_ talking and I won't fucking stand for it." He arched a brow until she agreed.
She did. "Okay. Should we go home, then?"
"Yeah," he said, his cock still rock hard. Traitorous bastard. "We should."
"ALEXA," GRANT SAID, walking into her office the next morning. Without knocking. He closed the door, then turned to her, his face set in a deep scowl. A shiver raced over her skin.
"Grant," she said, pulling her attention away from checking over the furniture deliveries scheduled to arrive at the model home beginning this afternoon. She forced strength and confidence into her voice, refusing to be cowed by the anger radiating off of him. "Can I help you?"
"Yes. By coming home. Today." He crossed his arms, his eyes set in a dark glare.
"We've already had this conversation," she said, her heart racing despite herself. He was back in that scary, quiet mode again. "Will there be anything else?"
He stalked toward the desk and braced his hands against it, and then he leaned down close. Too close. "Collect your belongings and _come home_."
Was that what this anger was over? He'd noticed she'd removed all her belongings? "I don't have a home right now," she said.
"Is that what this is still about? How many times would you like me to apologize?" he asked, no remorse in his tone whatsoever. So be it. Acting like this just confirmed that she'd made the right decision.
In fact, half of her wanted him to keep talking, to break into a full-out tirade, even. Because both of those were likely to further validate her choice to leave him. "One genuine apology would be nice. But there's a difference between apologies and forgiveness and forgetting or overlooking. I have no intention of doing the latter. So I won't be coming home. Or getting back together with you. Or marrying you. Which is why I returned your ring and packed my stuff." She clasped her hands on the desktop to hide how much they were shaking. "Perhaps we should talk about who is calling to cancel what. I could take care of the venue, the photographer, and the band if you'll cancel the honeymoon reservations, the florist, and the cake maker. And I assume you'd prefer to send out the email notifying the guests." At least they didn't have a slew of in-laws coming into town that they had to worry about disappointing. For once, neither of them having much in the way of family was an advantage.
His expression transformed, like he was confused and angry in equal measure. "Cancel the wedding? We'll do no such thing."
She blinked, because she'd been very clear on this already, yet he was acting almost surprised to hear it. "Grant, you're not listening to me. We're over."
The rolling changes to his expression were almost comical. She couldn't decide if he was trying to figure out how to play this or was actually this . . . emotionally out of control. It was so unlike him that all she could do was stare and wait for his reaction. Like watching to see which number the ball landed on in roulette.
"We are adults, Alexa. We will talk this out like adults," he finally said.
"There's nothing to talk about," she said, trying very hard to keep her frustration in check. But talking to him right now was a lot like talking to a brick wall.
"There's everything to talk about. There's _forever_ to talk about."
"Grant—"
"What do you think is going to happen here, Alexa? That I'm just going to let you go? That I'm going to allow you to walk away from the life we've started building, the one that we've invited three hundred people to come celebrate and _witness_ the start of in ten days? That you're going to shack up with a criminal biker gang and I'm just going to stand for it?" His volume escalated on those last few words, his anger finally coming through, and he leaned over the desk, invading her space until it was hard to breathe.
Needing distance, she pushed back in her chair, her scalp prickling and her hair standing up on end. She wasn't sure she'd _ever_ heard rage in his voice as scary or as lethal as what he'd just spoken. Whatever sass had been on the tip of her tongue melted away. Where had these mood swings come from? Or had she just never noticed because she'd always gone along to keep the peace?
"Surely, you realized that I would find out where you were."
She had. Of course she had. It had only been a matter of time, and Frederick wasn't that big—especially for Grant Slater. "If I was trying to hide where I was staying, I wouldn't have Maverick giving me rides to and from work," she said, hating the quivering in her voice. Maverick, who she'd had sex with. Maverick, who she wanted to have sex with again. Though she thought better than to share any of that with Grant. He probably wouldn't want her back if he knew, which could be good. Except she feared his reaction would be a whole lot worse—and more damaging—than that. And, anyway, she hadn't been with Maverick to get back at Grant, so she wasn't sharing that with him for anything.
He made a sound full of disgust. "You've made your point with this little stunt. I only have so much patience. And I'm not letting you go without a fight," he said, his tone seemingly calm but, to her, obviously razor sharp and ice cold.
"There's nothing left to fight about, Grant," she said, exasperated. How the heck was she going to get this through his head? "You said you wanted me to do this job for the model home, so would you please leave so I can do it? This conversation isn't getting us anywhere."
He jabbed his finger into the desk. "Agree to see me after work."
"No."
"Agree—"
"Grant, no."
" _Agree_ , Alexa." His stare was like sitting under the lights in an interrogation room. Hot and uncomfortable.
"I can't see you tonight. My mother has an appointment," she lied, terribly, but she was desperate.
"Fine. Tomorrow night." He arched a brow.
"Grant—"
"I'm hardly being unreasonable. You _owe_ me an hour of your time."
"No, I don't—"
"I swear to God." Anger washed off of him and over her. Despite the air-conditioning in the office, a trickle of sweat ran down the center of her back. He leaned closer and it seemed like his body vibrated with tension, like he might come right up over the desk at her. "Al—"
"Fine," she said, regretting it immediately. Especially when a smug smile spread over his face. "Just know that I will not be changing my mind. And there's nothing you can say to get me to. But out of respect for you and the time we shared, I will talk to you about our relationship. One last time. But please, stop doing this to me at work." The words rushed out as her heart hammered inside her chest.
"So be it. I won't bother you here again with personal conversation. But remember how persuasive I can be. I _will_ change your mind." The smugness overtook his whole expression in an attitude that was supposed to come off as sexy but now just struck her as smarmy and purposely obtuse. Who _was_ this man?
She took her seat and laced her hands together on the desk again. "Tomorrow night. Until then, good-bye."
His eyes flashed. If there was one thing Grant wasn't used to, it was being dismissed. But she was so angry at herself for agreeing to see him, and she'd had enough. "Tomorrow night," he said before finally turning on his heel and leaving.
On a heaving breath, Alexa bent over and rested her forehead on her hands. Why had she let him badger her into meeting? She didn't want to see him or spend time with him. She certainly didn't want to be alone with him. And it wasn't like talking was going to change anything anyway.
_Damnit. Way to cave, Al._
And if all that wasn't bad enough, Maverick was going to freaking kill her.
The thought made her remember that Grant knew she was staying with the Ravens. Maverick at least needed to know that much now.
_Grant knows I'm staying with you._ She shot off the text.
_Good,_ came Maverick's reply.
_What if he takes it out on you?_ She hit _Send_ , for the first time wondering about what the consequences could possibly be to the Ravens for offering her shelter.
I can handle Slater.
Her belly flipped at all the things that might possibly entail. _I don't want you to have to handle him._ Maverick didn't need Grant's kind of trouble, and after everything that Bunny and the club had been through recently, it was clear that none of them did.
Her cell rang. Maverick.
"Hey," she said. Her heart gave a little pang that he'd stopped what he was doing to call her.
"Hey. You okay?" he asked. And another pang over his concern. "What happened?"
"Yes, I'm okay, but it's a long story. Do you mind if I recount it all tonight?" she asked, not wanting to possibly start a second argument in five minutes. Plus, she really needed to get some paperwork finished so she could meet the general contractor over at the model home to go over room-by-room furniture layout.
"Fine," he said. "That's a plan. But listen to me, Al, and listen to me good. You're under my protection. You're under the protection of all of my brothers. Period. Something comes at you, it comes at all of us. We try not to go on the offensive if we can help it, but we sure as fuck defend our own. This is what we do. And that includes you."
His protectiveness chased away some of the anxiety that had settled into her muscles from the confrontation with Grant. "Okay," she said.
"Okay. You need me, you reach out. You hear?" he said, the fierce urgency of his voice sending comfort and strength down the line to her.
"I hear," she said. "Thanks, Maverick. It was . . . it was good to hear your voice."
"Any time." He paused like maybe he wanted to say something more, but then he just said good-bye.
Alexa couldn't help but compare how she felt after talking to Grant and Maverick. Grant left her stressed out, anxious, angry at him and herself, and just drained, while Maverick built her up, made her feel secure and less alone, and eased some of her concerns. Hammering home how much she'd misjudged the two men five years before.
Enough.
Enough personal bullshit for one day. It was only nine-thirty in the morning, for crap's sake. And here she thought her day couldn't get any more awkward after running into Maverick in the hall this morning wearing only a towel, water droplets running down his bare chest and back from the ends of his freshly washed hair. She'd wanted to drop to her knees and reacquaint her hands and mouth with what was under that towel, but she'd been too worried that he wouldn't want her to. Not after he'd put the brakes on the night before out in his garage.
God, she really was a mess right now, wasn't she? Her whole life. Which probably made Maverick right in wanting to put on those brakes.
She just needed to do her job, take care of her mom, and figure out how to put her life back on track. So that's what she'd stay focused on doing.
## CHAPTER 18
You did _what_?" Maverick asked, standing in the doorway to her bedroom that night. "Because I know I didn't just hear what I think I heard."
Alexa had been sitting on the edge of the bed trying to work up the nerve to tell him that she'd agreed to meet Grant when he'd popped in and asked what was wrong. "I told Grant I'd meet with him. Tomorrow after work." She forced herself to look Maverick in the eye.
He raked his hands through his dark blond hair, the movement raising his shirt and exposing a sliver of lean muscle and inked skin above his jeans. "Why? Why would you do that?"
"I didn't want to—"
"Did he threaten you?" Maverick asked, coming to stand in front of where she sat. Agitation rolled off of him, but it didn't scare Alexa. It made her worry that he'd be disappointed in her, but didn't scare her.
She thought back over the interaction with Grant. There hadn't been an overt threat, but she certainly hadn't felt entirely safe, either. He _had_ scared her, even though she hated it, so agreeing had seemed like the path of least resistance. "No, not really."
Mav held out his hands, still stained with grease around the tips of his fingers. "What does that mean, Al? Please, start from the beginning and tell me what happened."
She did, recounting everything she could remember. "So, no, he didn't threaten me. But he was really angry and I . . . I . . . just gave in. Okay? I hated it the minute I agreed, but now I'm stuck."
"Cancel. Tell him you're not meeting with him. It's bad enough you have to work with him. I don't trust him alone with you at all." Maverick paced in front of her.
"He won't let me cancel. I'm going to have to do this sometime. For closure. And stop pacing. You're making me even more nervous."
Oh a huff, Maverick crouched down in front of her, his hands on her knees. "Damnit, I don't mean to make you nervous. But my gut says this is bad news. Fucking Slater."
"Yeah." She shrugged and hugged herself. "But what's the worst he could do? It's not like he can lock me up in the basement."
Maverick's blue eyes flashed. "Don't even joke about that, Alexa. You'd be surprised what a control-freak with his reputation on the line might be willing to do to get what he wants. And I don't want to see that happen to you."
She let out a shaky breath. "So what do I do then?"
"Not you. We. You do this, we're going with you." He pulled out his cell phone and dialed.
Alexa's stomach flipped before the meaning of his words fully sank in, and by then, he already had Phoenix on the line.
"MAVERICK, I DON'T think this is a good idea," Alexa said when he picked her up after work at the model home she'd been designing. She was meeting with Slater in an hour, and Mav was kinda going out of his fucking mind about that.
Straddling his Harley parked in a driveway surrounded by newly laid sod, he handed her his helmet. "It's the _only_ idea." It was the first day he'd picked her up on the Night Rod instead of in his truck, but for what was happening tonight, he wanted the NRS. No, he _needed_ it. Needed Grant to see them two up on the back of the bike, Alexa with her arms and her thighs wrapped around Maverick.
Maybe it made him an asshole, but Maverick was worried about her being alone with Slater, and that had his protective instincts roaring to life. Being worried made Mav pissed off, too, and being pissed off made him feel all kinds of possessive and territorial. All of which meant he'd been a bear to be around the past twenty-four hours, something he'd apologize to Alexa for later. When all this was behind them and he knew she was safe. And this _would_ be behind them, one way or the other. Because by the time this night was over, Slater was going to know Alexa Harmon wasn't his. Once and for fucking all.
"Take the damn helmet, Alexa," he said, nailing her with a stare. Somehow it didn't help his mood that she looked stunning in a pair of slim black pants, strappy black heels, a silky tank top and a little black jacket with leather accents all over it. She had her hair up in a high ponytail that made Maverick want to suck on her neck. Hard enough to mark it.
She sighed and took it. "I don't need you to be bossy."
"Why can you stand up to me but not him? Ever think of that?" he asked.
She froze with the helmet halfway to her head. "Because . . ."
"Finish that sentence," he bit out, his boot scuffing the pavement.
She licked her lips, and her eyes appeared golden in the late-afternoon sunlight. "Because I know you won't hurt me."
He grasped her by the front of her jacket and hauled her into his side. Their bodies collided, and she threw her arms around him to steady herself. Maverick grasped her tightly, probably too damn tightly, but he had to touch her. Had to feel her. Had to—
He kissed her. Hard. Just one long, thorough, breathless kiss. Finally, he pulled his mouth away from hers. "You're right. I won't. But the fact that we both believe Slater would is why we're doing this my way. Got it?"
"Yeah," she said. "Okay."
He gave a nod. "Okay. Now get your pretty ass on my bike and let's do this."
The minute she was straddling his ride, her arms and legs wrapped around him, he felt about a million times better. Her embrace, her heat, her heartbeat against his back, it was all proof of life. Proof that she was safe. Proof that she was right there by his side—right where everything in him demanded she belonged.
He backed out of the driveway of the big brick McMansion and made his way through the yet-to-be-built development to the main road. From there, it was less than fifteen minutes to his rendezvous point—a downtown restaurant called Dutch's. Maverick pulled his Harley in line with seven others waiting there. Eight members of the Raven Riders Motorcycle Club ought to make the point—and be more than able to handle any shit should it go down.
Turning off the engine, Mav gave Alexa a hand off. "We have a few minutes to kill," he said. He raked a hand through his hair as he dismounted.
She just nodded as she peered at the big picture window that lined the street. His brothers were visible sitting at the counter just inside.
"Hey, com'ere." Maverick pulled her into his arms. More gently this time. "I know I'm being an insufferable prick, but just roll with it a bit longer. Please." He stroked his hand over the long, thick silk of her ponytail.
Alexa gave a little chuckle. "Can I remind you of that later?"
He bit back a grin. He liked Al when she was feisty. And he definitely liked her standing up for herself. "You really gonna listen if I say no?"
"Probably not." He felt her smile against his chest.
"That's my girl. Come on." He took her hand and led her into the diner. Dutch's sat on the corner, and had a long, narrow interior that filled the whole first floor of an old brick building. A Formica counter with spinning stools and red-and-white booths with juke boxes on the wall completed the old-timey look.
Somber greetings met them as they walked through the door, the bell jingling overhead. Phoenix, Caine, Jagger, Bear, Joker, Blake, and Mike Renner were all there—Maverick had insisted Dare keep his still-healing ass back at the clubhouse. His brothers were ready to stand with Maverick. Ready to fight with him, if it came to that.
Though the place used to be open later, because of Dutch's age, it now typically closed at five. But Dutch was a friend to the club and had agreed they could hang there for a bit after closing. The restaurant Slater had picked—after realizing he wasn't going to get Alexa to agree to come to his house—was only two blocks away.
"Maverick," Dutch said, extending his wrinkled brown hand across the counter. Despite owning a diner his whole life, Dutch Henderson was tall and thin. He had a friendly face and graying black hair, and he never forgot a name or a face. "Good to see you, son. How's your mother doing?"
"Better every day, Dutch. Thanks for asking," Maverick said. Dutch and his wife had been at the racetrack the night all hell broke loose, so he knew exactly what'd gone down.
"You tell her and Rodeo to come on in for some breakfast or lunch, and it'll be on me," he said.
"You bet," Maverick said. "She'll love that. But how are you? Dare said you've got a surgery coming up."
"Hip replacement," he said, patting his right hip. "Never get old, Maverick. Never get old."
"I'll remember that," Mav said with a grin. He turned to Alexa. "Dutch, do you know—"
"Alexa Harmon, of course I do. Though I don't think I've seen you in a whole lot of years."
"Hi, Dutch," Alexa said, giving him a smile. "Being here makes me remember how much I loved your milk shakes. Do you still make them with the whipped cream and the little cookies that slide over the straw?"
The question flashed a memory before Maverick's eyes. Him and Tyler and Alexa when she was seventeen or eighteen. Some asshole boy had spread a rumor around school that he'd scored with Al, and she'd come home upset but not wanting her mom to know why. So Mav and Tyler had brought her to Dutch's because she loved those damn milk shakes so much. They had the ice cream first, and then got dinner after. By the end, Alexa was smiling again. And the next day, Tyler put the fear of God into that kid. Best Mav knew, he never gave her another problem.
"I surely do," Dutch was saying. "I can make you one if you like."
"Oh, no," Alexa said, giving a quick shake of her head. "I don't think I can eat anything now. But another time."
"You know where to find me," he said with a wink.
"It's time," Caine said, icy blue eyes flashing.
"Let's do this," Phoenix said. "Fucking Slater."
"Fucking Slater," Jagger groused. Of anyone, Jag probably had to deal with Slater's bullshit the most. The wealthy prick hated the racetrack because it hurt his home-building business, or so he argued. He was constantly making noise—with the mayor, the city council, the sheriffs, the press—that the Green Valley Race Track was bad for Frederick. Occasionally, Slater managed to stir something up that would bring the sheriffs sniffing around. Once, he almost had the city council agree to debate zoning ordinances that would've seriously hampered the Ravens' business. And every time, Jagger had to deal with the brunt of the bullshit. Of all of them, the guy was probably the hardest to ruffle and the smoothest talker, and he'd memorized all the relevant rules, policies, and laws pertaining to the track like the brilliant motherfucker he was, so he thwarted Slater at pretty much every turn. It was a thing of beauty.
So Maverick nodded, joining in with the sentiment. "Fucking Slater." Everyone got up from their stools and moved toward the doors. Toward Alexa.
"Before we go," she said, bringing all the guys to a halt. Maverick eyed her, no idea what was about to come out of her mouth. "I just want to say thank you. For being here for me. And for Maverick. I know you're mostly here because he asked you, but I wanted to tell you that I appreciate it, too. I know I flaked after Tyler died"—she paused like it took something out of her to say her brother's name—"and I'm sorry for that. But that was more about me than it was about you. So, yeah, that's all I wanted to say."
"We're doing this for both of you," Phoenix said, giving her a flirty smile that under other circumstances Mav might've wanted to knock off his face. "Don't you worry none." The other men nodded.
Jagger rubbed her arm. "Losing Tyler was a damn shame, Alexa. We all regretted it. You were allowed to be messed up by that. No apologies needed here."
The men filed out, offering kind words that made Maverick proud to be one of them. Taking Alexa's hand, he squeezed. "Thank you for that."
"It was the least I could do," she said, her expression uncertain, maybe even a little overwhelmed. Had she expected them to do anything other than appreciate her gratitude?
He leaned down to look her in the eyes. "No, the least you could do was say nothing at all. Or, worse, disrespect them. Instead, you gave them your gratitude and respect. In our world, that means a helluva lot." When she nodded, he turned to Dutch, who was clearing the counter of a few soda glasses and coffee cups. Maverick put a hundred-dollar bill on the counter. "Thanks for always taking care of us, Dutch."
"Always," he said. It was just that simple. "Ride safe now."
Maverick gave a wave over his shoulder as he guided Alexa out the door.
And then they were on their way, riding through town in four sets of two. Maverick and Alexa rode at the front of the group with Phoenix at their side. They roared into parking spaces right in front of the restaurant's long windows. Eight Harleys. Eight bikers in full colors.
And what do you know? It was perfect timing. Because just then, Slater pulled up in his Mercedes and valeted the car. Alexa was standing at Mav's side, and he could feel anxiety rolling off of her. He wanted to take her into his arms and comfort her, reassure her, let her know that they'd be right there the whole time. But she knew that. They'd talked this through. And Maverick didn't want to do anything that might cause her dinner companion to get any more pissed off than he already looked.
Without a word, Alexa made her way up the sidewalk toward the front door where Slater stood waiting, scowling, looking like he wanted to break something with his hands. None of the Ravens moved. They just stood there watching. When Mav and Alexa's abusive prick of an ex made eye contact, Maverick arched a single brow. _Touch her, hurt her, and I'll fucking make you pay_.
The guy's face was red and his mouth was pressed into a tight line. He bit something out at Alexa that Mav couldn't hear, and it took everything he had to keep his feet planted and let this bullshit play out. They disappeared inside, but only for a second. Because then they appeared in the window, visible behind the pale, sheer curtains. They stood there long enough that it seemed like some sort of debate was going on, and then Maverick smiled. Because they sat at one of the tables in clear view of the street. Of the Ravens. Just like they'd planned.
_That's my girl, Al_.
For the next forty minutes, the Ravens waited. They got a lot of strange looks from passersby on the sidewalk and traffic slowed as it went by as if trying to figure out what was going on, but he and the guys didn't pay attention to any of it. Maverick couldn't tell a lot about the conversation taking place inside because the curtains obscured facial expressions, but he could see Alexa. And that was all he needed. For now.
The brief, high-pitched _whoopwhoop_ of a siren sounded out from just down the street. Maverick turned to see one of Frederick's finest rolling up behind the farthest of the bikes, lights flashing. And of course it was Davis behind the wheel. So Slater had called in his lapdog.
The guy got out, all five-foot-seven-inches of him, and glared at the lot of them. You never saw a man with a bigger fucking power trip than Curt Davis. The blowhard swiped his hand over his slicked-back brown hair and put on his cop hat. "Y'all need to move along."
Caine turned his lethal stare on the little weasel. "We're citizens peaceably inhabiting public property. What's the problem, officer?"
"The problem is _you_. All of you. We're getting complaints from business owners. You're loitering. And you're impeding the flow of traffic," Davis said, sounding a whole lot like he was trying to think of anything that might be relevant to harass them with. Maverick listened to the conversation, but mostly tried to keep his focus on Alexa. The cop's appearance wasn't really a surprise—it was a total Slater douche move—but it felt like a distraction. And distractions were dangerous.
Caine looked up and down the street, then back at the cop. "We all know who called you, Davis. You moonlighting as Slater's security detail while on duty these days?"
"That doesn't seem strictly kosher," Phoenix said.
"No, it doesn't," Caine said, arching a brow.
Davis planted his hands on his hips, though one fell on the butt of his holstered weapon. "Move on. Now. Or I'll add disturbing the peace and throw the lot of your sorry asses in jail."
"Remember when the football team stuffed his sorry ass in a locker?" Phoenix asked one of their brothers, just needling the cop.
As if Davis had no sense of self-preservation, he got up in Caine's face and spouted off with indignation and more threats.
"Go ahead. Try to arrest me," Caine said. " _Please_." If Davis couldn't hear the deadly intent in the Raven's tone, he was an idiot. And Maverick believed his brother meant it, which meant shit was about to escalate. Fast.
Inside, the curtains parted, briefly revealing Alexa's unhappy face, and then she pushed up from the table. Slater followed quickly after. As she spilled out onto the street, her ex grabbed her wrist. She stumbled as the asshole reeled her in against him.
Maverick saw fucking red, already moving to get her free.
"Oh, shit," someone said from behind him.
"Grant, let me go. Okay? Haven't you said enough?" she cried, her voice clearly upset.
"Don't forget what I said," he said. "And know that I'm dead serious."
"If you don't get your hands off of her, you'll be dead all right," Maverick said, coming up to them.
Alexa tugged her arm free and crashed into Mav's chest, pushing him back with both hands on his pecs. "No, Maverick. Don't."
He nailed the bastard with a murderous stare, letting Alexa hold him back. But just barely.
"Oh, please do, Maverick. By all means," Slater said. Arrogant goddamned sonofabitch.
Mav's blood fucking boiled with the need to take a swing at him. Which was exactly what he wanted.
"He's not worth it, Mav. Come on." She pushed him harder. "Come _on_. _This_ we're doing my way. It's over. Let's go."
He pointed at Slater over Alexa's shoulder. "Hurt her again, I rain down twice as much hurt on you." Then he let himself be pushed, not waiting to see or hear what baiting comeback the asshole would throw at him. He hadn't heard the other Ravens gather around him, but they were all right there on his six and walking with him back to their bikes.
"Have a nice evening, officer," Jagger said with a smile as they all got on and started their engines. Alexa held Maverick tight, way tighter than on the way there, and he didn't know whether she was still holding him back from Slater or just needing to feel him there. Whether her touch came from protectiveness or need, both sent heat through his blood.
One by one, they pulled out. Caine glared at Davis, while Phoenix flipped him off with a grin. But Maverick only had eyes for Alexa's ex. Because that was not a man who thought this was over. Not by a long shot.
## CHAPTER 19
The ride back to the Ravens' compound passed in a blur. Because Alexa had no idea how she was going to resolve the situation Slater had backed her into, or whether to believe he'd really do all the things he threatened to do.
As if he sensed her worry, Maverick squeezed her hand where it lay on his chest. She hugged him tighter, wishing she could get closer, needing him inside her, wanting him to just make all of this go away, even if for only a while.
His little gesture of caring escalated her worry on a whole other front—Grant was off-the-scale irate about what he deemed the Ravens' interference in his life and their relationship. In fact, she'd never seen Grant less in control than at that dinner. Obsessively arranging the flatware. Cutting the steak she hadn't stayed long enough for him to finish into precise, same-sized squares. At one point he'd actually stammered, his eyes blinking almost like a tic. The fact that he'd called in Davis at some point proved he wasn't going to let that interference go unaddressed.
It had been all she could do to keep Maverick from jumping Grant and beating him to a pulp right there in the middle of downtown Frederick. But then Maverick would've ended up arrested, shot, or dead. No doubt all things Grant would either like to see happen or maybe even was actively planning. Given what she could finally see about him now, Alexa wouldn't put such things past Grant. Not anymore.
If only she'd seen it sooner.
A nauseating flutter of nerves rushed through her when they returned to the clubhouse. The way Maverick had been when they'd left the restaurant, there was no way he wasn't going to be enraged at Grant's efforts to blackmail Alexa into giving in. And at the same time she hated to do anything that might pull the Ravens further into this mess.
But Maverick was like a dog with a bone. She knew he wasn't going to let this go, either.
"So what did he say?" Maverick asked when they got inside. The nine of them headed straight to the bar in the big rec room for a drink, and caught up with Dare, who was hanging there waiting for them to return.
"What happened?" Dare asked, rising from the couch where he'd been sitting with Haven and his grandfather.
"Exchanged some words with Slater and Frederick's very own Barney Fucking Fife," Mav said. "But nothing major. Except whatever happened inside. Alexa?"
Everyone looked to her.
She took a deep breath and tried to collect her thoughts. Alexa couldn't believe she had to give voice to Grant's wild demands. She was overwhelmed and worried and stunned. She should've been panicking about the implications of what Grant was trying to do. Instead, she was just . . . shocked. All the noise inside her was too much. She'd gone numb. "I . . . he . . ."
"What?" Maverick asked, worry and a banked rage emanating from him.
Finally, she summed it up as succinctly as she could. "If I don't do what he wants, he's going to destroy me." She crossed her arms and squeezed, trying to hold herself together. How was this her life? _How was this her life?_
"Sonofafuckingbitch," Maverick bit out, raking at his sandy blond hair.
"And he wants?" Dare asked, his voice low and lethal.
"Me. To have me back. To marry me. Why would he want to marry someone who doesn't want to marry him?" She shook her head, "bewildered" too small a word for what she felt. "He could have anyone. Why would he go to all this trouble . . . over me?"
Maverick was in front of her in an instant. "One, because you're you. You're smart, you're caring, you're talented, and you're beautiful, and any man would want you. Don't ever let me hear you ask such a bullshit question again." He arched an eyebrow and nailed her with a fierce stare until she nodded, his words piercing a hole through her numbness and letting in a little heat and light. "Two, because he's a control freak, and you're denying him something he wants. That makes him want it even more. Three, because he doesn't want to lose face by having it known that you called off the wedding and broke up with him. I'm sure he wants to avoid the questions that will naturally arise from that, not to mention the personal embarrassment at having been jilted, and he's willing to do whatever it takes to do so."
"And four," she said, having to swallow around the knot of emotion in her throat. She met Maverick's intense blue gaze. "Because you're involved—all of you. He's crazy pissed about that, and he sees this as a competition with the Raven Riders. And Grant doesn't like to lose."
Maverick gave a tight nod. "Sounds about right."
"But I don't want anything bad to happen to you," she told him; then, she looked around at the group. "To any of you."
"If Slater throws something at us, Alexa, we'll handle it," Dare said. A low rumble of agreement went around the room. She appreciated the sentiment and the support, more than she could say, even though it made her worry more.
"Why did you say he was going to destroy you, Alexa?" Haven asked, coming to stand right next to her. "What does that mean?" Her eyes radiated concern, and her expression was full of strength and resolution. Haven rubbed her back, making Alexa realize how tense her muscles were, like she was braced against an oncoming blow.
Alexa's shoulders dropped as she released a long breath. "He didn't say it this starkly, but he threatened to report my car as stolen and take it away and evict Mom from his house. And he hinted that my job isn't safe, either. And I . . . I have no idea what I'm going to do."
_I'd hate to see you lose everything you've worked so hard for, Alexa_.
The memory of Grant's smug voice made her stomach roll. He'd clearly spent a lot of time thinking about what would devastate her the most, and his attitude was full of confidence that she'd cave. Just like she always did. And who knows, maybe she would've if she'd sat there long enough enduring the brute force of his badgering and withering stare. But then Davis had shown up and Alexa had been able to see that things were getting tense between the sheriff and the Ravens. Her worry for them had sent her fleeing from the table, wanting to save them from any further trouble and needing to be free from Grant's threats.
Any one of which would be a huge blow, but all of them together would be catastrophic for Alexa.
And the issue wasn't just financial—although it was partly that, too. If Grant fired her, what would that do to her future prospects as an interior designer? She couldn't even fathom all the doors he could close to her. Even worse, her mother couldn't easily be moved from Grant's house. It wasn't just because she had a tractor trailer's worth of crap in her house to sort through, but because her mother wasn't going to want to go. Best case scenario, it would take some preparation and cajoling, and probably a little arm-twisting and bribing, too, to get her mother out of the house without sending her into a complete and total mental breakdown. And that was saying nothing of finding another place where a hoarder could live without good references or credit.
_Oh, my God, what am I going to do?_
"You're sure as shit not going back to him," Maverick said, as if she'd spoken the words aloud. Had she? The situation felt so surreal she wasn't sure.
Alexa met his gaze again and hated the doubt she saw there, and the little hidden sliver of fear. Maybe the others wouldn't have seen it, but she knew Maverick, and she did. "No, I'm not. I can't. Especially not now. But, Jesus, getting free of him means that my mom and me . . . we lose everything. How am I going to tell her? Where am I going to move her? Am I even going to be able to stay in this area if Grant ruins my career and reputation here?" She kneaded at the muscles in her neck. "What an arrogant, twisted asshole he is. I can't believe it's come to all this."
Maverick grasped her hand and leaned down until their eyes met. "We'll figure it out. Did he attach any sort of a deadline to his demands?"
She laced her fingers with his. "He wants me home—well, back to his place—immediately, but the real deadline is next Saturday."
"The wedding," Maverick said.
"Yeah. I don't show—"
"—and the shit hits the fan," Mav finished for her. She nodded.
"So we have a week to make arrangements for your mom," Dare said. "We _will_ figure something out. That's a guarantee."
"Shit," Maverick said, his eyes going wide. He turned to Phoenix. "I hate to ask, man."
"What?" Phoenix asked. "Name it."
Alexa loved that about the Ravens. Always had. How they were there for each other, no matter what. Like a family.
"Any chance you'd be willing to let Mrs. Harmon use Creed's house?" Maverick asked. Alexa gasped. Creed was Phoenix's cousin who'd been killed a month or two ago in Baltimore. He was older, and she hadn't known him well, but she knew Phoenix well enough to know that he'd idolized Creed. Which meant Maverick asking him such a thing for her was _huge_.
She shook her head. "Oh, no, I couldn't—"
"Hey, that's actually a damn good idea," Phoenix said, nodding, his expression thoughtful. "I haven't touched it since Creed died, so it'll take some work, and it's not very big, but it's just sitting there."
The kindness they extended to her disproved stereotype after stereotype. Her billionaire businessman fiancé was a manipulative, abusive control freak, and her law-breaking motorcycle gang friends made a habit out of helping people who needed it. "That's an amazingly generous offer, Phoenix, but my mother's not the easiest person. She's not likely to take the best care of the place even with my help, and I don't even know what I can afford yet."
He gave her a look. "I'm talking about a gift, Al. I don't want your money. Or your mother's. Once we get it cleaned and fixed up, she could stay there as long as she wants. Creed always liked Tyler. I think he'd like this."
The man's generosity pierced more holes into her numbness. "Okay, well. Wow. I don't even know what to say. Thank you isn't enough."
"Yeah it is." He winked. Nodding, her thoughts raced. That potentially alleviated one big problem, assuming the work could be done within the next week, and assuming that Alexa could get her mother and all her crap out of Grant's house within that time frame.
Maverick turned to Phoenix and offered his hand. "I know what this means. Thank you. We owe you. Anything, anytime. Just name it." The men clasped hands for a long moment.
_We_ owe you. We. Such a small word. Such a huge meaning. It shattered what was left of Alexa's numbness. Worry, fear, panic all came rushing in, but so, too, did courage, resolve, and—for the first time in a long time—the steadfast knowledge that she wasn't alone. We. She and Maverick. Whatever they were, he was in her corner. All of these guys were. After years of feeling so alone, that meant everything. She pressed a hand to her chest and nodded, emotion overwhelming her.
Maverick faced her again, his big hands going to her shoulders. And it was like his touch grounded her, shielded her, pulled all the disparate pieces of her together so she could stand strong. "The rest we'll figure out. I promise. But until then, no going out on your own anywhere. If you feel like you need to stick it out at work, I understand. But we keep with me dropping you off and picking you up." On their surface, his words seemed calm, but the agitation was clear in the tight clench of his jaw and the hot flash of his eyes.
"Knowing Slater, this will escalate before it gets resolved," Dare said matter-of-factly. In her heart, Alexa knew that was right. But what would that escalation mean? She couldn't stand the idea of anyone getting hurt because of all the mistakes she'd made.
Which meant she had to act. She had to do whatever she could to protect herself and her friends. "I need to get a lawyer first thing Monday morning. And maybe I should talk to Sheriff Martin this weekend, just to see what he thinks?" Alexa didn't know Henry Martin well, but she'd heard Grant grouse about a divide within Frederick's local police over the Ravens—and that Henry tended to fall on their side and even protect them. The details of those internal politics were a mystery to her, but certainly an enemy of Grant's could be a friend to her.
"Good. That's good," Maverick said, a gleam of pride in his gaze. "Touching base with Martin makes sense. Get you protected legally, too. Because this fucker's not going to get away with what he's trying to do. Just you wait."
No, he wasn't. At least, not without a fight.
Problem was, she feared into her very soul that she was going to have to pay in some way before everything was said and done. The only question was how great that cost would be.
## CHAPTER 20
As it got later, more Ravens arrived at the clubhouse. Bunny and Haven laid out the fixings for beef tacos, and everyone dove into building a plate. Afterward, the atmosphere clearly veered in the direction of partying. Truth be told, Maverick had some adrenaline and not a little anger burning through his blood that could use being worked out. But then he found Alexa sitting at the kitchen table talking to Bunny and Haven, her knees drawn up to her chest. And even though she put on a smile for him, it was clear that she wasn't in the partying mood.
"Why don't I take you home?" Maverick asked, something deep inside him needing to not just protect her, but take care of her, comfort her. However she needed, whatever she needed.
"It's okay," she said.
"What do you want to do?" he asked, crouching beside her. He was aware that his mom and Haven were listening in, but didn't mind, didn't care. There was no shame in wanting to take care of what was his—even if she was only that in his stubborn fucking heart.
"I'll do whatever you want," she said. Her hand fell on his where it rested on her knee, and the touch shot heat through his blood.
He arched an eyebrow. "I want to know what _you_ want."
She met his gaze for a long moment, then gave a little shrug. "I don't want to pull you away."
"I can give you a ride, hon," Bunny said.
Maverick shook his head. "Thanks, Bunny, but I got this. You want to go, Al, I want to take you."
Her expression went soft and sweet for him. "Yeah?"
His grip on her knee tightened. "Yeah."
"Well, whenever, then." Her fingers stroked over his.
Between the small touches and the way she was looking at him, all affectionate and even a little unguarded, his body came alive with need. He rose and held out his hand. "Let's go."
The smile she gave him told him it was the right call. She needed him more than he needed to blow off some steam. And that was that.
Maverick watched while Alexa hugged Bunny and Haven and said her good-byes, and he liked what he saw—Alexa being folded into the arms of his community, and embracing it right back. He liked it a lot. Fuck, it was too soon to be feeling this, to be _wanting_ this—too soon for her. Just today, Grant had driven that point home with a goddamned sledgehammer. She was right in the thick of dealing with her ex. The last thing she needed was to deal with Maverick, too. For fuck's sake.
_When you gonna fight for what you want?_ Dare's chiding voice echoed in his memory. The question—and the urgency to answer it—pulsed through his veins with every heartbeat.
"Ready?" she asked.
"Yeah," he said, shoving away the thoughts. He hugged Bunny and Haven.
"I make a mean cinnamon roll," Haven said to Alexa. "In case you want to come for breakfast. Maybe we could do that girls-only movie day."
"I just might take you up on that," Al said with a smile. And Maverick found himself grateful to Haven once again. Alexa needed friends and fun in her life right now, and he appreciated Haven making her feel like she belonged. He knew enough about how Al had grown up to know that mattered to her. So it mattered to him.
It took a couple of minutes of good-byes to make their way out of the clubhouse, but then Maverick was starting up the Night Rod and Alexa was sliding on behind him, her heat ramping up the arousal stirring in his body. Especially when she didn't just embrace him. Her hands gently rubbed his chest. The night air blew warm and inviting against them as they took the back way through the big Raven Riders tract to the pond and his house beyond.
Alexa didn't mean anything by the soft caresses against his chest, but it was driving him fucking crazy. His cock hardened against his thigh, setting off an ache that pounded in his blood. Finally, he stilled her hands against his sternum, but instead of that putting a stop to the rubbing as he'd intended, it set her hands off to _exploring_.
Up over his pecs, the heels of her palms dragging across his nipples. Down over his abs, turning his cock to steel. Down farther, onto his inner thighs. Her nails raked against the denim. And then her right hand found his hard length and tormented him with rubs and squeezes that had him spilling groans into the wind.
Christ, what was she doing to him? And should he let her do it?
By the time he pulled the NRS into his garage, he was wrecked by lust and need, the adrenaline that had been surging through him since the confrontation at the restaurant finding an outlet his body liked. A lot.
He didn't kill the engine or flip out the kickstand. His boots braced the bike upright, the engine's vibrations spilling through him and winding him tighter. And then Alexa's fingers made quick work of unbuttoning his jeans. Unzipping. And pulling out his cock.
"Fuck, Alexa," he groaned as her hand wrapped around the thick column of his flesh. She reached to hang her helmet on the handlebar, and he helped her, wanting to know what she wanted, needing to see where she wanted to take this, unable to do anything but give in to the steady jerking of her hand around his hard length.
"Your voice is so sexy when you're turned on," she whispered against his cheek. "Makes me wet." Her tongue traced his ear, and then she was kissing him there, her hand on his neck, trying to make him yield to her mouth, her tongue.
"What else makes you wet?" he rasped. He should be questioning this, questioning her, not egging her on. But his body had taken over this train and pulled it right on out of the station.
Her fist gripped him tighter, pumped harder. She hummed low in her throat, her tongue dragging up his neck. "How hard your cock is. How it fills my hand. How I know it could fill me."
"Jesus," he bit out, needing to hear every filthy secret she'd share with him. "What else?"
"Your long hair. Your eyes. Your ink." She punctuated each pronouncement with a jerk and twist of her hand. "The feel of this bike rumbling between my legs," she offered without him asking.
He swallowed thickly, his thoughts trying to break through, trying to rein in the beast that wanted to break free. "I wasn't going to start this."
"That's why I did," she whispered, swiping her thumb through the wetness on the head of his cock.
Each word, each touch hammered another nail into the coffin of his restraint. "I've been trying not to take advantage of you, Alexa. You've got a lot coming at you right now." His head dropped back on her shoulder. He felt connected to her by her touch, their words, the bike's vibrations.
"It's not taking advantage if I want to be with you, too, Maverick. Just because my life's a wreck doesn't mean I don't know what I want. And I see it now with a clarity I haven't had in years. Please, don't take that from me," she said, her voice strong, but with just a hint of pleading. Her words hit him square in the chest, unleashing need and hope in a one-two punch that buried the last of his concerns, at least for tonight. But then she released him. "Unless you don't want—"
"Oh, fuck, no. I want." Twisting, he wrapped an arm around Alexa's slim waist and hauled her in front of him so that she was straddling his thighs, her front to his.
She gasped and grinned. "That was impressive."
"Not nearly as impressive as what I'm about to do," he said, lust making him feel desperate and crazed as he roughly unfastened her thin dress pants.
"What's that?" she asked, bracing her hands on the gas tank behind her, reclining away from him a little as he worked.
He reached into his back pocket, urgency fueling his plans. "Is your pussy wet for me, Al?"
She squirmed against his thighs. "Yes."
"Good, then you'll forgive this." His flipped out the pocket knife, grasped the open bottom of the fly, and carefully sliced at the seam that ran between her legs. A quick precise slash separated the flimsy silk of her wet panties, and then she was exposed to him.
"Holy shit, Maverick!"
In a quick move, he pocketed the knife, ripped the ruined seam wider, and lined his cock up with her wet core. And then he was sinking deep, shifting her forward on the bike just a bit so he could bear down on her. Her moan lanced heat through his blood, her body arching over the curved tank, her hair spilling fucking beautifully over the matte blank with its red accents. When he was balls deep it felt so good that he had to grip the handlebars to steady himself.
"Oh, my God, this is so fucking hot," she cried. Her hands joined his grasping onto the handlebars. "Please, Mav."
"Yeah," he said, his voice like gravel. He withdrew and thrust, the bike rocking under them, the engine adding a hot full-body sensation. Her pussy was wet and already tightening around him. He grabbed the knife again and fisted her little silk tank top. "Need all of you," he growled, slicing it from the low neckline to the hem. It was easy work to cut through the satin holding together the cups of her bra, exposing the mounds of her tits.
"Yes," she said. "Take what you want."
"What about you?" he said through a clenched jaw as he penetrated faster, deeper, riding her without mercy. Not that she was asking for any. "You gonna take me?"
"Yeah," she cried. "Oh, yes."
And God help him, he needed her to, to take all the adrenaline and aggression inside him. With each mind-blowing thrust, it was like she was sucking the poison of darkness and loneliness from his soul. "Yeah, you are. Take me, Alexa. Fucking take me."
"Maverick."
Her moan twisted up the tension gathering low in his gut. "That's it. Take all of me." He bottomed out inside her, then ground himself even deeper.
"God!" She reached a hand between her legs, and watching her work her own pussy while he fucked her on his bike was the single hottest thing he'd ever seen in his life. She screamed her orgasm, her whole body bowing over the gas tank, her thighs bucking over his. He planted his hand on the flat plane of her belly just below her breasts, steadying her and pinning her in place.
And then he shifted his stance so that he was hammering down into her, his pelvis crashing into hers on every thrust. Sensation was barreling down his spine and stealing his breath. "Christ, you're gonna take me, Alexa. You're gonna take everything I've got to give."
She grabbed his wrists. "Everything," she rasped.
The word tripped his orgasm, just ripped it right out of him until he felt turned inside out, raw and exposed, on top of the world. His shout echoed over the bike's low rumble against the garage's concrete, his cock kicking inside her again and again and again. It went on until Maverick saw spots around the edge of his vision, like he'd taken a drug that had sent him on the best, purest high.
When his body finally calmed, he lifted her against him until he was cradling her in his arms, her legs wrapped around his hips, their fronts pressed together, his cock still buried to the hilt. When he finally killed the engine, her arms wrapped around him on a sigh so contented he could've dedicated his life to making her sound that way over and over. She stroked at his hair as her whole body went lax against him. The silence was loud all around them.
"So good with you," she whispered.
"Always," he said, not sure if she meant the sex or how she felt or, just, all of it—the sex and the closeness and the fun times and the hard times and the boring minutiae of daily life. Which was how he meant it.
"I can't even be mad that you ruined my clothes," she said with a little satisfied hum. "Because it was too damn hot."
He chuckled and nodded. Someday he'd be eighty and too arthritic to ride and too senile to remember what he was arguing about with Dare, but he'd sure as shit recall every detail of what'd just happened with her. It was seared into his very DNA. "I'll replace them."
She pulled back from the hug just enough to look him in the eye. "I don't care about the clothes, Maverick. I don't care about anything that's replaceable. Not anymore."
At just that moment, he realized that he'd made love to her without kissing her even once. His gaze dropped to her lips, and suddenly he was ravenous for a taste. He devoured her with a deep, slow, lingering kiss, full of tangling tongues and tugging lips. His senses were full of her, but not as full as his heart. So he kept right on kissing, because it ensured none of the words pinballing around in his head got voiced. Not yet.
Alexa shifted in his lap, emphasizing that his cock was hardening again inside of her. He gripped her hips and thrust himself as deep as he could go. Held himself there. His libido started making plans he wasn't sure either of their bodies was actually up to carrying out.
Just then, wind gusted a warm, humid breeze through the garage, and a thin spray of water blew over them.
Al squeaked and laughed. "When did it start raining?"
Maverick peered over his shoulder. "Hell if I know." A nuclear bomb could've exploded out there ten minutes before, and he wouldn't have realized it. She shivered, goose bumps rising up on her skin. "Let's get you inside." When she nodded, he eased out of her, helped her off the bike, and did up his jeans when he got off.
She tried to hold the pieces of her clothes together, but he'd literally ripped her to shreds. And he couldn't stop staring. "You are a wet fucking dream right now, Alexa. I'm not gonna lie."
"I am? I'm a disaster." Looking down at herself, she gave a little chuckle.
He shook his head. "A beautiful disaster, maybe. But all of us are sitting on some kind of disaster, Al. Some are just better at hiding it than others." He sure as hell was. The guilt over not protecting his mother all those years before still had the ability to steal his breath and trash his self-worth. He should've seen it. He should've acted sooner. He shouldn't have let it get so out of control. He shook away the thoughts and slipped out of his Ravens cut. "Here." He held it out for her to put on.
She put her arms through the holes and wrapped it around herself, and the denim and leather was so big on her that it hung to the tops of her thighs. And _fuck_ if seeing her wear his colors didn't threaten to rev him up all over again.
An image flashed before his mind's eye—a black-and-purple tribal raven with a red heart tattooed on Alexa's fair skin. Marking her as one of them, as claimed, as _his_. The imagining made him think of his conversation with Dare, and that his cousin might actually get the pleasure of seeing such a thing on Haven's skin in the not-too-distant future. And Maverick had maybe never been more jealous of another human being in his life until that very moment.
Not that his cousin didn't deserve every bit of happiness he found with Haven. Because he did. They both did. But that didn't keep Maverick from wanting a little slice of that for himself. Wanting it _bad_.
"Maverick? Are you sure it's okay for me to wear this?" Alexa asked, her fingers stroking over his name patch and impacting him like it was his cock instead.
"It'll be fine," he said in a gruff voice. "Just 'til we get you inside. Come here." He bent and scooped her into his arms.
"You don't have to carry me." She grinned and wound her arms around his neck.
"Yes, I do. Because then I get to touch you." With that, he walked out into the gentle summer rain.
She laid her head on his shoulder. "Yeah, okay. Touching is good."
Hell, after years of thinking it would never happen again, touching her was everything.
And he didn't want to stop.
Inside, he carried her to her bed and stripped them both down. "I want you in my arms tonight. You good with that?"
"Yeah, Mav." She fit herself into the nook along the side of his body until there wasn't a breath of space between them. Her hand settled on his chest, and he folded his around it. "I'm great with that."
And even though they lay there perfectly still, perfectly peaceful—at least for that one moment—Maverick realized that Alexa Harmon was the best fucking ride of his life. And he never wanted it to end.
## CHAPTER 21
Alexa first became aware of the delicious achiness of her body, an achiness that spoke of sex so hot just thinking about it made her sweat. Smiling, she kicked off the covers, her mind playing the highlights reel. The electrifying shock of Maverick cutting open her clothes. The deep satisfaction of him taking her in such a primal way. The rough need in his voice. But as raw as the sex had been, there'd been tenderness, too. The way he'd held her after, cradling her against him. How he'd covered her with his cut and carried her inside, his face nuzzling her hair. His tight hold all night long, his body molded to hers, his hand cupping her breast.
Except . . . her hands stretched out and found only emptiness.
Her eyes opened, confirming that she was alone. "Maverick? Hey, Maverick?" Pushing onto an elbow, a sheet of paper on his pillow caught her eye.
_Hey Al—_
_The club got an emergency call at 5:30 A.M. so I had to ride. Hated to leave your bed without saying good-bye. Come to the clubhouse when you get up?_
_—Mav_
Alexa pressed the paper to her chest as she fell back against her pillow. The man had spent last night protecting and helping her, and then he'd gotten up in the middle of the night to help a stranger. A warm pressure filled her chest, getting bigger and bigger until she knew.
She _knew_.
She still had feelings for Maverick—and maybe had never stopped having them in the first place. She still wanted Maverick. The clarity and growing certainty of her emotions was stunning and a little jarring after months of building questions and doubts.
Part of her worried that this was all happening too fast. Only days had passed since Grant had thrown her out. It _was_ fast.
But everything that had happened between her and her ex these past weeks had snapped Alexa into a level of awareness she hadn't had in such a long time, like she'd suddenly awakened from a drug-induced sleep. Now it was like Grant had simply been an interruption to something that'd existed since she first became aware of the opposite sex—her attraction to Maverick Rylan. Her longing for him. Her need for him in her life. Either way, she felt miles away from the woman who'd stared in that bathroom mirror last weekend and tried to convince herself that there was nothing wrong with the way Grant had taken her.
She wasn't perfect, she wasn't flawless, and she was far from being as strong as she wanted to be, but she wasn't _that_ Alexa anymore, either.
And now that she had that clarity, she wasn't wasting it. So once she'd dealt with the chaos of her life and finished fighting Grant, she was going to lay all her feelings and flaws out on the line and fight _for_ Maverick. Because she wasn't making the same mistake twice in letting him get away, not if he was willing to give her a second chance.
Lucy leapt onto the foot of the bed, meowing as she walked up Alexa's body until she was demanding attention by lying on Mav's note and pushing her head into Alexa's hand. "Well, there's my baby. How are you today?" Purring was her only answer, and it was a good one, full of contentment and lazy satisfaction. Lucy liked it here. "I do, too," Alexa whispered, her mind wandering, building hopes and dreams she didn't know whether she had any business wishing for. For long minutes, she stroked her fingers over the softness of Lucy's head and big ears. "You need a bath, little one. We'll do it tomorrow. Okay?" That wasn't the only thing she needed to do. Her final project was due on Tuesday, and Alexa hadn't done five minutes of work on it all week. But even that wasn't the most important thing she had to do. "I need to take care of my mama today."
The words chased away some of the happiness Alexa had felt upon awakening. Telling her mother about the move was going to be hard. So hard that Alexa realized she'd need help to do it. She had to get her mom's case worker out to keep her calm and offer some support. Which meant Alexa had things to do.
"Okay, little friend, I have to get up and start my day." She slid Lucy off of her and pushed out of bed. The screen on her phone said it was nearly seven-thirty.
She typed out a quick text to Maverick. _Hope everything's okay. Be to the clubhouse in 30._
_Be better when you get here_ , was the quick reply. Well, now she wanted to hurry.
So she did. She rushed through showering, dressing, and running some Google searches to make arrangements for moving her mom. She left a message with the case worker, hoping the lady could come out today. And then Alexa was heading out the door, eager to try Haven's mean cinnamon buns and see Maverick to make sure everything was okay.
When she arrived at the clubhouse, voices rang out from the big mess hall, indicating breakfast was in full swing. Alexa felt a little shy about walking in to a roomful of Ravens already sitting and eating, but she also wanted to be a part of it, too. A part of them. So she pushed through her anxiety and walked through the door.
She shouldn't have worried.
From among the dozen or so gathered Ravens, Phoenix and Jagger called out immediate hellos, and Bunny gave her a big smile and a wave. Haven rose from her seat next to Dare to give her a hug. "I'm so glad you came," she said.
"Thank you," Alexa said, feeling more comfortable by the second.
And Maverick . . . he came to her wearing a tired smile and eyes full of satisfaction. "I'm glad you came, too. Come, sit." He took her hand and guided her to the open seat next to him. A cinnamon bun already filled a big part of the plate. "I grabbed one for you because they never last long."
Aw, that was kind of crazy sweet. And thoughtful. And proof that he'd been thinking about her. All of which stirred up more of that warm pressure growing in her chest. For him.
"That's so sweet, Mav. Thank you," she said.
"Yeah, Maverick, that's soooo sweet. You're just the sweetest motherfucker I ever knew," Phoenix said in a singsong voice. He glared at Mav and eyeballed her cinnamon bun. Laughter rumbled around the table.
Maverick flipped his middle finger and gave Phoenix a big shit-eating grin, then he winked at her.
Smiling, she leaned a little closer to Mav. "Sorry," she said.
His hand fell on her bare thigh under the table. "Don't you worry about it," he said in a low voice full of humor.
Heat shot from his palm to her clit so forcefully her breath caught. And then that heat got amped up even further when Mav's gaze cut to her, his eyes full of blue fire, his expression smug and knowing. His fingers squeezed, his little finger slipping under the hem of her shorts.
"Yo, pass everything down for Alexa," he called out.
Warmth spilled into her cheeks as she accepted a platter of sausage and bacon from a younger guy with brown hair whom she didn't know. The patches on his cut indicated he was a prospective member. Like Tyler had been.
Thinking of Ty while sitting in the Ravens' clubhouse set off a small pang of longing in Alexa's heart, but it also did something she didn't expect—it made her feel a little closer to him. For the first time in a long time. Why hadn't she ever thought that could be the case? That being around the guys he'd so badly wanted to be a part of would allow her to be around a part of him, too? She was seeing all kinds of things in a new light lately, wasn't she?
"Thanks," she said to the prospect, taking a few slices of bacon. "I'm Alexa."
"I remember," he said, passing her a bowl. "Blake. Nice to meet you."
"Oh, have we met already? I'm sorry." She scooped scrambled eggs on to her plate next.
He shook his head. "Not officially. I was here when you—" He cut himself off and gave an awkward shrug. "A few weeks ago."
"Oh." So he'd been here when she'd shown up with the bloodied face. Awesome. "Right."
Maverick's hand squeezed again, softer this time, like he meant to reassure her.
Alexa dug into her food and listened to the banter that ran around the table. She remembered this, the rowdiness and camaraderie of meals at the Ravens' clubhouse. Her eyes scanned the room. It hadn't changed a bit. Two big long tables, one currently empty, filled the space in front of a large stone fireplace that harkened back to the building's past as an inn. A carving of the Ravens' raven/dagger/skull logo was mounted over the mantel, and POW/MIA flags hung from the rough-hewn exposed beams overhead.
She saved Haven's bun for last—a dessert for her breakfast. With as much icing as it had, it totally fit. "Oh, my God," she said around her first bite. Rich and creamy and sweet. She took another bite and moaned. She couldn't help it.
"Right?" Mav grinned at her, his gaze tracking her tongue as it swiped icing off her lip.
"This is amazing," Alexa said, leaning forward to see Haven at the far end of the table. She called the other woman's name. "Your cinnamon rolls are crazy. They're, like, happiness in food form."
Haven beamed as the men all joined in offering praise. "Why, thank you."
"You could totally sell these," Alexa said. "And those cookies you made the other day, too."
Sitting at the head of the table, Dare gave Haven a look, one eyebrow arched. "I agree," he said, his tone like they'd had this conversation before.
Just then, Alexa's cell buzzed in the back pocket of her shorts. She grabbed it. "It's my mom's case worker," she said, already rising. "Be right back." Putting the phone to her ear, she moved into the big, empty lounge so she could hear better. She explained the situation to the case worker, an older woman named Lillian Hite who Alexa hadn't had cause to call on in a while.
"I'm sorry to bother you on a Saturday," Alexa said.
"No, you were right to call me, dear. Cynthia's going to need some help adjusting to this news," Lillian said. "I could come around three, unless that's too late? Otherwise, I could be there Monday morning."
Relief flooded through Alexa. "Three today would be great. Thank you." The sooner she got things moving on that front, the better. She only had a week, so she couldn't afford to waste even one day.
Just as she hung up, the mess hall door swung open behind her. "Everything okay?" Maverick. Sweet, sexy Maverick.
"Yeah. I was able to get my mom's case worker to agree to come out this afternoon. So I guess I'm breaking the news about the move today."
Maverick rubbed her arms. "I know it sucks, Al. I'm sorry it's come to this."
"In the end, it'll be better to be completely free of Grant. It's just that Mom doesn't handle change well." The degree to which that was an understatement unleashed butterflies through her belly. This was not going to be an easy conversation.
"Can I come? Maybe another friendly face will help. Unless you think it would stress her out more for me to be there," he said, dropping his hands.
The offer reached into her chest and squeezed. "You're too good to me. I'd love to have you there."
Maverick's whole expression went stern as his hand cupped her face. "There's no such thing as too good for you, Alexa. When are you going to get that?"
She grasped his wrist like he was her anchor in the storm that was her life right now. And he was. "I . . . I'm sorry."
"I don't want you to apologize," he said, his eyes searching hers. "I just want you to believe I'm here for you. Whatever you need."
"God, Maverick. How could I not believe that? There's not even a question in my mind," she said, knowing the truth of his words to the bottom of her soul, and needing him to believe she knew it, too.
His eyes smoldered, all that sternness shifting into desire. In her mind's eye, she saw him staring down at her as he took her on his bike, trapped between the hardness of steel and chrome against her back and his body over her. "Good. That's good."
She nodded. "I really liked last night," she whispered.
He made a sound like a growl under his breath. "I fucking _loved_ last night. I can't ride the goddamned Night Rod now without picturing you laid out for me, hair all over the handlebars, legs spread, your clothes wrecked."
Need settled between Alexa's thighs as her heart pounded. "Good. Because I can't stop seeing you over me, _riding me_ , fully clothed while I was all exposed, the hard vibration of the bike rolling through me as—"
Two Ravens Alexa didn't know spilled out of the mess hall. Maverick's eyes held frustration and amusement as he clasped hands and said quick good-byes to them. And then his hand was back on her face again, the touch tender and possessive at the same time.
"Now back to me riding you," he said when the guys left.
Chuckling, she turned her face in his grasp and pressed a kiss against his palm. As the heat in his gaze reignited, the moment felt weighted, significant, full of promise. She sneaked her tongue out to teasingly trace his palm. Mav's jaw clenched, the look he was giving her like he was a breath from devouring her. Her heart raced and her nipples hardened.
His hand slid from her mouth to her hair. His fingers dug in and hauled her close, her breasts crushing against his chest. Maverick kissed her on a groan she greedily swallowed as her hands went around his neck.
The door to the mess pushed open and a couple more Ravens spilled into the lounge behind them. Alexa gasped into the kiss, instinctively trying to jump back, but Maverick held her fast. "Don't care about them. Not done with you," he whispered into her mouth.
A shiver ran through her whole body. The guys said good-bye to Maverick, but he paid them no mind, even when someone laughed good-naturedly at his expense. For her part, Alexa was hyperaware—of the hard press of Maverick's body against hers, of the shared sweetness of sugar and cinnamon on Mav's tongue, of being watched by the others.
When Maverick pulled his lips away, he wore the sexiest, most satisfied smile she'd maybe ever seen on his ruggedly handsome face. "Mmm, good morning."
She smiled and smirked at him. "Good morning to you."
"Yo, Mav," Jagger said, coming into the lounge. "I gotta head down to the track to check over a bunch of shit for Monday's license renewal inspection, but give me a call when you know when you need my help this week."
Maverick finally released Alexa from his embrace, though he kept her close with his arm around her shoulders. "Will do. We're heading over to Alexa's mom's today to start figuring things out so I should have more details in the next day or two."
"Good deal," Jagger said, his fingers tapping against his thigh. A memory came to her—listening to Jagger play his acoustic as a group of them sat at the fire pit out by Maverick's pond one fall night. The man was self-taught and absolutely amazing. She hadn't thought of that in years. "Just say the word. See ya, Alexa." He gave a wave as he headed out the front door.
"Bye," she said, trying to make sense of the exchange through the haze of lust and nostalgia clouding her brain. "Jagger's gonna help move my mom, too?"
Mav shrugged like it was obvious. "Of course. Everyone will as much as they can. Before you arrived, Jag reminded me that he's got a cousin who owns one of those three-guys-and-a-truck moving companies. He's going to call in a favor to see if he can do the move. We got to talking that if your mom will agree to it, maybe we get her and her most essential things moved as fast as we can, and then we take our time going through the rest. That gets her out of Grant's crosshairs sooner."
"Wow. Yeah, that all sounds great, assuming Mom can be brought around to the idea," Alexa said, the realization sinking in that the Ravens had spent time talking about how to help her and coming up with an actual plan to deal with Grant's blackmail attempt. It was so unusual to feel like she wasn't alone that she barely knew what to do with the emotion welling up inside her. "Thank you."
More Ravens came into the lounge, headed up by Phoenix. He clapped Maverick on the back and gave Alexa a smile. "Okay, we're heading over to Creed's to get it ready for your mom. Should I leave the furniture or does she have enough of her own?"
Alexa's gaze scanned over the small group that included Phoenix, Blake, and a Raven she didn't know well whose cut read _Bandit_. "She does have a lot, but it's not all in the best shape. Maybe leave it until we see what she actually wants to move and what she might need?"
Phoenix nodded and waved, already heading for the door with the others. "That's a plan. Okay, then, we're outie."
"Thank you, Phoenix," she said, overwhelmed by the generosity all of these men were showing her. Which made her remember . . . "Hey, what was the emergency you had to leave for this morning?"
A storm rolled in over Maverick's expression. "The situation for one of our clients escalated when her estranged husband tried to break in. The mother called nine-one-one. Sheriff Martin knew she was one of ours so he called us in on it, too. The husband had never done anything like that before so we didn't have a twenty-four/seven detail on her." He shook his head, clearly frustrated. "We moved them up here to one of the cabins for the weekend because the kid is so spooked. As if it wasn't bad enough that the sick fuck had abused her, now she's scared of him crawling through windows in the middle of the night."
"Oh, God," Alexa said, her heart aching for the little family. What would people like them do without people like Maverick and his brothers in the club? What would _she_ have done without them? Once, she'd thought she had nothing in common with the women and families the Ravens helped protect, but now Alexa knew that was just one more thing she'd denied because she hadn't _wanted_ to see the similarities. "That's so terrible. Was anyone hurt?"
"Just rattled. Mostly." His dark expression made it clear there was more he wasn't saying.
"I'm glad you were there for them," she said. "Is there anything I can do to help?"
His gaze ran over her face, soft and searching. "No, but thank you for offering."
She nodded and heaved a deep breath. "There are some things I should do before heading over to Mom's this afternoon."
"Then let's get to it," Maverick said. "Because the faster we sever your remaining ties to Slater, the easier I'll be able to rest."
"And the less ways he'll have to try to hurt or control me," Alexa said in a low voice.
"Hey." Maverick tipped up her chin. "Over my dead fucking body is anyone going to hurt you." He arched a brow.
As much as Alexa appreciated the sentiment, the words unleashed an icy chill all down her spine. "Please, don't say that, Maverick. I couldn't . . . it would . . ." She shook her head, the horror of imagining him dying—or even getting hurt—stealing her ability to pull together her thoughts.
"I'm not going anywhere, Al. Don't you worry." He took her hand. "Come on."
But as they headed out into the already hot summer morning, that chill wouldn't go away. Because she didn't know what she'd do if something happened to Maverick because of her.
And she hoped she never, ever had to find out.
## CHAPTER 22
Alexa was right.
Her mother took the news badly. She yelled, cried, and rocked in her chair, the nastiness of her rants and name-calling more a reflection of her stress than what she truly thought of her daughter. Intellectually, Alexa knew that. It still hurt to hear her mother accuse her of being selfish and stupid and uncaring. Especially when so much of what she'd tried to do these past five years had been about making sure her mother would be taken care of forever.
"How can you do this to me, Alexa? How can you do this to me?" her mother cried. It went on and on and on, no matter what she or Lillian said.
"I'm sorry, Mom. I wish this didn't have to happen—"
"It doesn't. You can just make up with Grant. Make things better," she said, ringing her hands against the stack of photo albums she'd hauled into her lap. The stacks of magazines and newspapers around her chair had regrown in height since Alexa had cleaned some out just last week. "Then I could keep this house and all my things."
Alexa pushed down the hurt that her mother was more worried about losing her junk than why she and Grant had broken up. Not that Alexa had offered the gory details, but her mother seemingly hadn't even thought to ask. "I can't, Mom. And I won't."
"You could if you wanted to. You could if you cared about me," she said, her voice nearly a shriek. Her sobs and sniffles filled the tense air.
Wearing a pair of khaki pants and a lavender top, Lillian sat on the end of the couch closest to Mom so she could pass her occasional tissues. "Now, Cynthia, you know your daughter's relationship has nothing to do with how she feels about you," she said, trying for the fiftieth time to interject a voice of reason.
"But it does, it does, if she cared about me at all . . ."
"Mom," Alexa said, guilt threatening to swamp her.
Her mother's hand stroked over an album with one of her brother's baby pictures on the cover. "Tyler would never have done this to me."
Alexa flinched, the words impacting her like she'd been struck.
"Enough!" Maverick yelled.
The room went eerily quiet.
He'd been standing just inside the living room by the door to the foyer for the whole conversation offering Alexa silent strength and the certainty that she wasn't alone in all this craziness. He hadn't said a thing through the entire hour-long conversation. Until now.
"Maverick Rylan, don't you raise your voice at me," her mother said with a sniff. But not yelling this time.
"I will raise my voice, because you're not hearing Alexa and she deserves to be heard. If Tyler were here, he'd be raising his voice, too. So I'm saying what he would say because I know you'd respect him enough to want to hear it." His agitation on her behalf eased some of the sting she felt, and his defense of her made her realize how long it'd been since she'd last had a champion, a defender, someone who'd always have her back. God, it felt good, and it meant everything.
"So, what is it you think he'd say?" Mom asked, dabbing at her eyes.
Maverick stepped up beside Alexa. Frustration rolled off of him, but he did a decent job of reining it in as he spoke. "He'd say Grant Slater is a coldhearted, controlling, abusive bastard. He'd say that Slater hurt your daughter and forcibly threw her out of the house. And he'd say that if those weren't reason enough, Alexa can't go back to Slater for the sheer fact that he's threatening to evict you from here to try to force her to bend to his will. Now I've known you most of my life and I know you've always loved your kids, so I know there's no way you'd want that kind of a man or that kind of a life for your daughter. And that because you're such a great fucking mom, you'll do whatever you can to protect Alexa, too, just like me. No matter how hard it is."
Alexa didn't quite know how to feel about the fact that Maverick had shared all her dirty secrets with her mother, but she had to admit it was the first time since they'd arrived that her mom had stopped yelling and crying. And was actually listening.
"Language, Maverick," her mother said, no heat to the admonishment at all.
"Yes, ma'am," he said.
Her mom looked at her. "Is all that true, baby? Grant hurt you? And he's blackmailing you, over me?"
Alexa swallowed hard, the heat of shame and embarrassment crawling up her face. She nodded and hugged herself, feeling raw and exposed. But at least it was for a good cause, because Alexa could see that Maverick had gotten through to her mother by appealing to her _as_ a mother. Smart man.
"Oh," Mom said on a gasp, her hand going to her mouth. "Oh, my poor girl. Why didn't you tell me?"
"I was ashamed for you to know," Alexa said, her gaze dropping to the floor. "For anyone to know." She felt Mav's gaze on her, but couldn't meet it. Not just then.
"Alexa," Lillian said, "how someone treats you is a reflection of them, not you."
"I know that," Alexa said, finally looking at the other woman. "But I also let him get away with an awful lot." She shook her head, refusing to dwell on her mistakes when there was so much she could be doing to try to fix them. "I'm not doing that anymore, though."
Lillian's expression was full of sympathy and approval. "Cynthia, what does learning about the details of Alexa's situation make you feel?"
"Well, I feel terrible, of course. And obviously she can't be with someone like that," Mom said. Regret and trepidation were clear in her voice, but it was as if Maverick's tirade had flipped some sort of reset switch inside her mom's mind. And it was closer to an apology than she usually got from her mom. So there was that.
"So how do you feel about giving this move a chance?" Lillian asked.
After a long moment, Mom nodded. "I guess we have to, don't we?" She nodded again. "So . . . we will." Alexa felt a weight lift off her shoulders, at least a little. And she had Maverick to thank for bringing her mother around.
"Good," Lillian said. "Then how about the four of us come up with a game plan that will make this as easy on you as it can be?"
"Fine," her mother said, a little cantankerousness sliding back into her tone. "But first I'd like to see this new place. Because I can't plan anything without knowing what kind of place I'm going to and how much space it has."
"That we can do," Maverick said. His hand went to Alexa's lower back, and his thumb slowly stroked like he knew she needed the comfort. She did.
Alexa finally looked at him, and the fierce protectiveness blazing from his dark blue eyes absolutely slayed her. Just sliced right open to the heart of her. Which wasn't so hard since her heart clearly beat for him. "Thank you," she mouthed.
He gave her a barely perceptible nod, the intensity of his expression communicating so much. Support. Concern. Need.
"Will you come, Lillian?" her mother asked as she stacked the photo albums on the floor again and rose.
"I'd love to see your new home, Cynthia. Now remember what Alexa said. There's a crew of men there working to fix it up. So it's not quite ready for you yet but it will be," Lillian said. Alexa appreciated the older woman offering the reminder.
"Of course," Mom said. "Let me just get my purse." Making her way through a path lined with boxes and stacks of every possible thing, she disappeared back down the hallway toward her bedroom.
Alexa blew out a long breath. "I can't believe you did that," she said to Maverick, "or that it worked so well."
"I couldn't listen to her continue to berate you when she didn't know all the facts," he said, gravel in his voice. "I'm sorry if you didn't want her to know. But I had to do something."
"I know that was hard, Alexa, but I think she needed to hear it," Lillian said. "It pushed her into a place of acceptance and cooperation, which is a start. But be prepared for there to be setbacks and resistance as all this progresses."
Alexa nodded. "I know." She turned to Maverick. "And don't worry. It's okay." He gave a tight nod just as her mother returned. Alexa donned a bright smile. "Okay, Mom. Let's go see your new house."
She'd only been by Creed's little rancher before, never gone inside, but no matter what it was like, it would be better than here. Better than something Grant could give—or take away. Creed's house would be safer and happier for the both of them.
And that was all Alexa had ever wanted for her mom. And herself.
"THIS WILL DO," Mrs. H said as she, Alexa, and Lillian walked through the rooms of Creed's house for the millionth time. "This will do just fine."
Maverick hung back with Phoenix, wanting to give the women privacy, and not wanting to take his anger out on Al's mom again. He hadn't really intended to interfere or lose his cool, but he could tell what her mother's tantrum was doing to Alexa—he could see it in her posture and hear it in her voice—and he couldn't stand for it to continue for another second, especially when she brought Tyler up that way.
Hanging in the mostly empty living room, he watched as the women went into the kitchen again.
"She took it bad then?" Phoenix asked in a low voice.
Maverick chuffed out a humorless laugh, his gaze tracking Alexa until he couldn't see her anymore. "You could say that."
"Seems like she's come around now, though," he said.
"Yeah," Mav said, his gaze scanning over the space. A dark blue couch, coffee table, and end tables were all that remained. The house was a ranch-style with all the rooms on one floor. It had a big combined den and eat-in kitchen that connected past a breakfast bar to the living room. Two decent-sized bedrooms and a bath sat down the hall. More than enough space for one lady, even one with a shit-ton of, well, shit. "You all worked fast here."
"Wanted to get it emptied out as fast as possible so I could see what work it might need. I called Renner in to help, and I paid the three boys who live next door fifty bucks apiece to haul stuff out to the garbage or the storage pod. Just have to do the master bedroom and I'm done." Phoenix gestured for Mav to follow him down the hall. "We probably don't have time to freshen the paint up for her, but other than cleaning the carpets, this is the main thing that I should probably address before she moves in." He flipped on the bathroom light and knelt by the toilet to pull back the old linoleum. Black mold covered the subflooring underneath. Well, hell. "It's all soft, so something's leaking."
Maverick nodded as his gaze scanned over the old fixtures and the dingy shower. "I got a contractor friend who might be able to help." Maybe he could get the guy to freshen up the whole room, in addition to fixing the water damage. "And I'll pay you back for hiring those kids and whatever else you're laying out."
"Fuck that," Phoenix said, giving him a look. "Like I can't do something nice for your woman?"
The words made desire explode inside Mav's chest. Desire for Alexa to be his, once and for all. "She's not mine. And you're giving her a goddamned house, Phoenix. I'd say that's already above and beyond."
"Coulda fooled me." He smirked and waved his hand. "Anyway, your money's no good to me, and neither is hers." A long moment of silence as Phoenix tugged back more of the ruined linoleum to reveal water stains and more spots of mold.
"Can't claim someone who's not ready to be claimed," Maverick finally said in a low voice. "Things are too up in the air."
"Maybe, maybe not. But she looks at you like you hung the fucking moon for her, so maybe she's more ready than you think." He washed his hands.
Mav couldn't entertain that possibility, even with everything that'd happened the past few days. Not yet. So he ignored Phoenix's observation even though it kinda made him want to cheer. "Well, how about I do those rival wheels you and I talked about for your bike. And maybe throw in some custom chrome accents?"
The grin made the deep scar on the side of Phoenix's eye seem less severe. The result of a knife fight back when he'd been a prospect. "Now you're just turning me on."
Chuckling, Mav rolled his eyes. "Bring your ride around any time."
"Hey, there you are," Alexa said, leaning against the doorjamb.
"How's she doing?" Maverick asked. He couldn't help noticing how stressed Al looked. It was in her eyes and the way she hugged herself.
"Honestly? Better than I would've expected given how things started out. She adores the big kitchen and den, so I think that won her over even more." She smiled at Phoenix. "I can't thank you enough."
"You already thanked me. We're good," he said, moving toward the door. "Actually, you can thank me." He slipped behind her and whispered something in her ear.
Alexa's cheeks went pink even as she nodded.
"What the hell was that?" Maverick asked. With Phoenix, you never knew what he might be up to. When she didn't answer right away, he stepped right up to her, pinning her back to the molding. "Tell me."
She peered up at him from under her lashes. "He said . . . that me taking care of you was the only thanks he needed."
Irritation flashed through Mav. "Nosy fucker. Ignore him. You don't need that kind of—"
"No, he's right," Alexa said, grabbing the edges of his cut in her fists. "You're doing so much for me. I want to be there for you, too. I just—"
"What?" he asked, needing to hear what she had to say like he needed his next breath.
"I just need more time."
He probably should've accepted it at that, but something made him push. "And then?"
She swallowed thickly. "And then . . . we'll see if . . . if you think you can forgive me and . . ." She gave a little, uncertain shrug. ". . . whether you maybe want to try again."
He cupped her face as triumph surged through him and then he leaned down so he could look at her eye to eye. "I want."
Her eyes went glassy as she whispered, "You do? After everything? Because I wouldn't blame you—"
" _I. Fucking. Want_." He didn't try to restrain a bit of the fierceness roughening up his voice. He wanted her to know it. To believe it. To feel it down deep. "You. Us. A future. Our history, even the fucked-up parts. All of it. Understand?"
"Yes," she said on a fast nod, her eyes suddenly brimming with unshed tears.
He caught the wetness with his thumbs. "Don't cry, baby. And don't think you're the only one who has things to apologize for. Because I told you I have regrets, too."
"God, Mav—"
"Alexa?" her mother called from the living room. "Are you ready to go?"
Looking into Alexa's eyes, Maverick loved the hope and affection he saw there. "We'll finish this another time."
"Okay," she said. And then she pushed up on her tiptoes and threw her arms around him, holding him so tight he could barely breathe. So much unspoken emotion rolled off of her, and it fucking slayed him.
And made him feel hopeful for the first time in a long damn time.
Alexa was going to be his again. Sooner or fucking later. And nothing was going to keep them apart this time.
## CHAPTER 23
Want to help me give the cat a bath?" Alexa asked, her face playfully hopeful. Wearing a long black shirt of his, her hair still mussed from sleep, she looked beautiful standing barefoot in the sunlight spilling into his kitchen the next morning.
Leaning against the counter with a cup of coffee in hand, Maverick smirked. "If that's code for getting between your legs, sign me up."
Throwing her head back, Alexa laughed out loud. "Like we need special code for that."
He grinned around another sip. "Good to know."
She rolled her eyes, but it was clear she was enjoying the conversation every bit as much as him. And he was glad after the day she'd had yesterday. After seemingly doing so well at her new place, her mother had fallen into another round of tears when they took her home. They'd ended up being there until nearly nine o'clock, and Alexa had been exhausted by the time they got back here.
"I mean, the _actual_ cat," Alexa said, her eyebrow arched. "Not the pussy. Maybe later, though . . ."
Maverick had to adjust himself. "You say shit like that you give me ideas of what to do with that mouth."
Laughing, she disappeared down the hall. The water came on in the bathroom. And then he heard the low murmur of her talking to Lucy. And it occurred to him how much he liked not being here alone. He loved the sounds of Alexa moving around his house. He loved seeing her things sitting with his on the counter. He loved her weird sweater-wearing cat greeting them at the door or jumping up in the bed when they woke up. And it made him realize how much he'd changed in the past five years. How much he valued things now that he'd never thought all that important before—big things like commitment and small things like the sounds of companionship.
Curious, he wandered down the hallway to find Alexa kneeling next to the tub, the cat in her arms, the skull sweater discarded on the counter. Lucy wasn't bolting but she also wasn't looking thrilled with the plan. "She likes baths?"
Alexa chuckled. "She likes it once she's in, but she always gets tense before."
"And why exactly are you torturing—I mean, bathing Lucy?"
Smiling up at him, Alexa shook her head. "Being hairless means her body has no way to absorb the oil from her skin or from when she grooms herself, so she has to be washed pretty regularly. It's worth it for how awesome she is, though. Plus, she's so cute when she plays in the water that I can barely stand it." She pressed her face to the cat's. "You ready, baby?" She leaned over the tub and slowly lowered Lucy in, her arms still cradling the thin, little body as it submerged.
The cat scrambled for a moment, and Maverick prepared to try to block her if she made a break for it, but Alexa just held her there until the animal visibly relaxed.
As he watched, Lucy pushed out of Al's hold and walked a circle through the tub's water. "I thought cats weren't supposed to like water?"
Alexa chuckled. "Just wait."
"For what?" he asked, just as Lucy plopped down on her butt, sat up, and pawed at the top of the water like she was playing the drums. Ten seconds, twenty, thirty, she pawed and pawed, sending up little splashes and shaking her head when they hit her in the face. Grinning, he crouched beside the tub. Alexa met his gaze for a moment, and the happiness in her eyes as she looked at him reached all the way inside him. And grasped hold.
Lucy paced the tub again, stopping to bat at the faucet, the wall-mounted soap dish, and the shower curtain. Alexa grabbed a bag at her side that Mav hadn't noticed and handed it to him. "Line these up on the edge."
He reached inside and found a rubber ducky, a plastic dog, a squeaky mouse that made Lucy turn and look, and a couple of other small toys. "You have bath toys for your cat?" he asked with an arched brow.
"She's my baby," she said, laughing.
One by one, he lined them up as Lucy watched. For a moment, the cat just looked at them. And then she attacked, batting them into the water one at a time like a slugger aiming for the back wall, until they were all floating in the water. And then she pounced on them there again and again.
Maverick couldn't help but chuckle. Because Attack Cat was pretty awesome.
Alexa fished them out once more and together they lined them up. Lucy took them all down a second time until they were both laughing at her antics.
"This is fucking hilarious," Maverick said, gathering the toys and lining them up again.
Lucy didn't wait. She swung for the rubber ducky, knocking it out of the tub and into Maverick's lap.
"This is the coolest cat that ever lived," he said, grinning like an idiot as he replaced the duck on the tub's edge.
"See? I told you." Alexa grinned as Lucy knocked the toy into the water and pounced on it, sending up a big spray of water that got both of them. Alexa let out a little shriek, a belly laugh spilling out of her.
Finally, Al actually got down to the business of washing Lucy, and then she was scooping the cat out and wrapping her in a big towel. "Will you hold her while I do her ears and nails?"
Mav took the bundle into his arms and peered down at the little wet head. This was not an attractive creature by any stretch of the imagination, but now he found that to be part of her appeal. Lucy was kinda like the Ravens—rough-looking on the outside but full of awesome. He grinned at the thought.
"What is that smile for?" Alexa asked as she swiped Q-tips through Lucy's giant ears and, one by one, clipped the cat's nails.
"Was just thinking that Lucy could totally be the club's mascot. You need a biker name, though," he said, hugging Lucy a little tighter to his chest as she tensed at the clipping. "'Lucy' is cute and all but it doesn't really fit our vibe. Or the sweater I bought you. Maybe Lucky or Bones or . . . Slugger. Right?" The cat purred and stared at him. "You like that idea, don't you? Yes, you do, kittykitty."
"You are totally cooing at my cat right now."
"Am not."
"It's turning me on."
"I'm totally cooing at your cat."
Alexa laughed, peering up at him with so much affection. "When you take care of something I care about, you take care of me. It means a lot, Mav."
He handed the cat back and watched as Alexa gave her a rub with the towel and finally set her free. Lucy curled against his leg before wandering out of the bathroom.
He hiked Alexa up to sit on the counter. "It's easy when I care about you so fucking much," he said in a low voice.
"Then I'm the lucky one."
Her words heated his blood and set off a warm, expansive pressure in his chest. His hand cupped her neck as he fit himself in tighter between her thighs. His phone buzzed in his back pocket, but Maverick ignored the fuck out of it as he leaned in and claimed her mouth. Once, twice. "Enjoyed washing your kitty."
She grinned into the kiss, her arms coming around him. "I enjoyed having my kitty washed."
He couldn't help smiling, too. His tongue traced her lips and slipped inside. Guiding her head, he took more, went deeper.
His phone buzzed again.
"Should you check that?" she asked, pulling back, her face so alive with happiness that he just wanted to hide from the world and hold them in this moment. And smash the phone into a million pieces.
"Probably." On a sigh, he pulled the cell from his back pocket to find two missed calls from Caine. Which meant something was up, because the guy never called anyone to shoot the shit. "It was Caine. Better see what—"
The phone rang in his hand, Caine's name on the screen.
"Caine, man. What's—"
"Cops are here. Arresting Dare and Jagger. Come now."
"On my way," Maverick said, anger and outrage welling up inside him. What the actual fuck?
"What's the matter?" Alexa said, all that happiness bleeding out of her expression.
"Dare and Jagger are being arrested," he said, already moving from the bathroom toward the kitchen table, where his cut hung over the back of a chair. "Don't know why. But I'm sure as hell going to find out because this is fucking bullshit."
"I'll be thirty seconds," Alexa said, running to her room.
Mav jammed his feet into his boots, his thoughts racing. What the hell could they be accused of? For being a Raven, Jagger tended to go out of the way to keep his nose clean because so much of the business at the track rode on his shoulders. And Dare was just a little over a week out of the hospital and had hardly left the clubhouse except for the ride to Baltimore . . .
Ice crawled through Mav's blood. Could that be it? But, if so, something didn't add up, because if the attack on the Iron Cross was at the bottom of this, Dare and Jagger wouldn't be the only ones getting hauled in. Maverick couldn't begin to figure it out, except his gut screamed that something wasn't right.
Alexa was good to her word, rushing back in a little navy blue cotton dress and a pair of sandals, no makeup, her hair in a messy knot. Beautiful, especially for the concern she wore on her face. For his brothers.
They jogged out to his bike, and she slipped her hand into his. "I'm sorry."
He gave a tight nod, glad to have her at his side. He drove more aggressively than he otherwise might've, but made it in time to see that goddamned motherfucker Davis shoving a cuffed Jagger into the back of his squad car. Sheriff Martin was there, too, clearly trying to keep Phoenix, Caine, and a few others from losing their shit and earning some one-way tickets to a holding cell.
"I'll be right behind you, D," Caine called out just as Davis grabbed his cousin's injured shoulder and shoved him toward the car. Dare grimaced, and Maverick saw red.
Mav charged, Martin catching him about the shoulders. "He's injured, you fucking asshole," Mav yelled. Davis smirked and pushed Dare into the backseat.
"Don't give him the satisfaction, Maverick. I need you to take care of shit here." Dare nailed him with a hard stare just before the door shut between them.
"What the hell happened?" Mav looked to Caine and the others, stepping back from the car as Davis started it up.
Caine's expression was set in a lethal scowl that was all rage and hard angles. "Someone dumped a shit-ton of tires and oil out by the drag strip. And then Davis supposedly got an anonymous tip reporting it. Now Dare and Jagger have been charged with multiple counts of illegal dumping."
Head spinning, Mav watched as Davis drove away. The Raven Riders hadn't used the quarter-mile strip where they did occasional car and bike drag races since last summer. This made no goddamned sense. He looked from Caine to Sheriff Martin and back again. "You know Jagger Locke. You know this is bullshit. We've got a fucking licensing inspection tomorrow, and the drag strip is part of that. If we'd dumped the shit, why would we leave it where the inspector might find it? And Dare's been laid up for over a week. Jesus Christ, Martin, you were the one who responded to his shooting. You know he hasn't been hauling anything around."
Martin held up his hands, already nodding. Mav and Dare went way back with Henry Martin, all the way back to high school. Though he'd been a hotshot football player and they were pretty much the same miscreants they were now. "I hear you, Maverick. But it's out of my hands. I'll help where I can but what Dare and Jagger need most now is a lawyer."
"And a private detective," Caine said, his voice like ice.
"Yeah," Maverick said. "Because someone knows who was behind this trumped-up bullshit."
"I'm sorry," Haven said. "But why would illegal dumping cause . . . all this anyway?"
Martin gave her a sympathetic look. "It carries big penalties in Maryland. Fines and jail time." Haven paled.
Maverick looked back to Martin. "Can you find anything out about this supposed anonymous tipper?"
"I'll see what I can do," Martin said, covering his short, wavy brown hair with a sheriff's office baseball cap. "It's odd that Davis didn't call for backup in investigating the tip until after he'd found the dump site. He wouldn't normally risk coming on to your property by himself."
"Yeah, well, I guess he was feeling confident of what he'd find. Which in and of itself is fucking fishy," Caine said.
"Like maybe he already knew," Phoenix said. Nods all around. "And the drag strip is far enough from the clubhouse and close enough to the highway for someone to get in and out quick without any of us noticing. Especially if they did it in the middle of the night."
Maverick nodded, Phoenix's words sounding more and more like the truth of it. Someone planted this shit to cause trouble for them, maybe even keep them from getting their license renewed . . .
Sonofafuckingbitch.
"The timing of this . . . someone knew about the inspection," Maverick said, his mind racing through the possibilities. And landing on one that made his blood boil.
"Oh, God," Alexa said from behind him.
Mav turned to find her holding a shaken-up Haven's hand, Cora on Haven's other side with an arm around her.
Alexa's face was sheet-white. "Grant," she whispered, giving voice to the same conclusion Maverick had just made. "Grant would be able to find that out. He would . . ." She shook her head, her eyes wide and panicked.
"Christ," Maverick said, his gaze swinging back to Martin. "She's right and you know it."
"Aw, hell," Martin said, planting his hands on his hips. "You can't go accusing Grant Slater of anything without proof. He's one of the mayor's best friends and one of the richest and most influential men in town. If he was behind it—and that's a big if—he certainly wouldn't have dirtied his own hands with it, so you'd be hard-pressed to connect it directly to him anyway."
"He's right," Caine said, palming the knit cap covering his shaved black hair. He shook his head. "He'd hire this out."
"Probably," Mav said, nailing Martin with a hard stare. "But what we can a hundred percent prove is that he's blackmailing Alexa to keep her from calling off their wedding next Saturday. Slater's an abusive, controlling prick, and there's no doubt in my damn mind that this bullshit is part of what he's trying to do to her."
Martin's gaze cut to Alexa, and she gave a nod to confirm what Mav had said. "Damnit," Martin said on a sigh. "Okay, I hear you. I'll dig into this as much as I can."
"Do it," Maverick said, his vision going red the more all the implications of this penetrated his thick skull. "Because if you don't get to the bottom of it, we'll have to. But I'm guessing the fucking body count will be lower if you take the lead."
The sheriff held up his hands and arched a brow. "Let me get out of this conversation before we all regret me being in it." The guy was a friend to them, but he drew the line at knowing anything that might cross a line. And _all kinds_ of lines were going to likely have to be crossed to right this mess. Because shit had gone seriously sideways. "I'll get down to the station and keep an eye on your guys. Get a lawyer and get down there." He rounded the back of his squad car.
"Take care of them, Martin," Maverick said, eyeballing the other man. "Anything happens to either of them and there's going to be hell to pay in this town and you know it."
Martin nodded as he opened his door. "Just keep your goddamned heads about you." He got in and drove off, passing a tow truck rumbling into the lot as he did. Slider rushed out, his expression making it clear that he'd heard the news. Phoenix started catching him up as Mav made a call.
James Walter was a personal attorney who Doc had known forever. When Walter answered, Mav quickly explained the situation. The guy agreed to meet them at the station without having to be asked. Next, Mav left a message with the business lawyer they used for matters related to the track.
When he hung up, he turned to the assembled group. The pissed-off expressions on the faces of Caine, Phoenix, Slider, Blake, and Bear looked like Maverick felt. And the worry and sadness he saw on Haven's and Alexa's faces ratcheted up his anger by a factor of about a thousand. "I want a detail up here to guard the women and people at the ready just in case something else happens. Assemble as many Ravens as you can. Caine and I will head down to that goddamned station."
"I'll let everyone know what's going on," Phoenix said, pulling out his phone. "But I want to come, too."
"No, Phoenix. We can't have people losing their shit and possibly making things worse," Maverick said. Because the last thing they needed was more men behind bars. Phoenix gave a tight nod.
"I want to come," Haven said.
"No," Maverick said, feeling bad for her but resolved. "Dare won't want you anywhere near the place. I can guarantee it."
"But—"
"Haven, if he even thinks you're there, it'll make him feel worse," he said, forcing his voice to gentle. "The best thing you can do 'til we get him home is stay here and stay safe."
"We don't know what else might be coming at us," Caine added. Mav nodded.
"Damnit, okay," she said, her voice cracking. "I just . . . if you get to see him . . ."
"He knows," Maverick said, remembering his cousin's words about how close they'd gotten. No doubt Dare was already going out of his mind worrying about how Haven would handle this situation.
"I'm so sorry," Alexa said, shaking fingers pressed to her lips. "This is my fault."
"No, it's not," Haven said at the same time that Maverick said, "No, it's fucking not."
"But Grant did this to get back at me," she said, her hazel eyes a little wild.
Maverick got right in her face. He couldn't tolerate her feeling one more ounce of guilt for what her bastard sleazeball of an ex had done. "That's right. _Slater_ did this. Not you. And he did it to get back at us for showing up at the restaurant. I'd put money on it."
"That's what my gut says, too," Caine said. "Look, I'm gonna get down to the station now."
"I'm ready, too," Mav said, nailing Alexa with a stare. "Stay strong for me."
A quick nod. "I will."
"We'll take care of each other," Haven said. "Go. Don't worry about us."
Mav kissed Alexa's forehead and made for his bike.
"I'll get things organized," Phoenix called, his cell pressed to his ear.
Waving, Mav mounted the NRS and brought it to life on a roar. He gave Alexa a look and she nodded. And then he took off.
Way he felt, he was ready to take on the whole goddamned world if he had to, so he'd handle Slater in a fucking heartbeat. One way or the other.
## CHAPTER 24
Stunned didn't begin to describe how Alexa felt. Shocked. Horrified. Those got a little closer. "I'm so sorry," she said again as she watched Maverick tear out of the parking lot like a demon made of flesh and steel.
"Hey," Haven said, tugging her hand. "Maverick was right. What your ex has done isn't your fault. In fact, to the extent you believe it is, you're falling into the trap he's trying to set for you. He wants you to feel guilty because then maybe you'll do what he wants. Giving in to his effort to manipulate you would be your fault, but this isn't."
Alexa blinked, surprised at the vehemence of Haven's words. She came across as soft-spoken and easygoing, but the younger woman had a fierceness about her that Alexa hadn't seen before. And that she really admired.
"Okay?" Haven asked.
"Okay," Alexa said with a nod.
"Can you take care of them?" came a gruff voice from behind Haven. She and Alexa turned to find Slider talking to Cora, two little boys standing close by. Boys who had been much, much smaller the last time Alexa had seen them. "I know I said I didn't need you today—"
"I've got them," Cora said. "They'll be fine. But just so you know, Maverick didn't want anyone else going to the station."
Slider's dark eyes flashed. "Well, he isn't here to stop me."
Cora nodded. "Then go do whatever you need for however long it takes. Right, guys?"
With their brown hair and hazel eyes, the two boys bore a striking resemblance to their father. "Yeah, Dad," the older one said, his forehead set in a wrinkle. "You gotta help Dare and Jagger."
Slider gave Cora a nod and a look, like maybe he wanted to say more, but then he just turned away, gave the boys' hair a ruffle on the way past, and jogged to the tow truck.
"Ahhh, oh, no, it looks like you two are stuck with a bunch of icky _girls_ today," Cora said in an exaggerated groan that made the littler boy laugh. "Sam, Ben, this is Maverick's friend, Alexa."
The boys said hello, and Alexa found herself glad for the distraction they might offer her and Haven. "You know, I've met you guys before. You wouldn't remember, because you"—Alexa pointed to Ben, who must've been six or seven—"were just starting to walk. And you"—she smiled at Sam—"were about his size."
"Then why haven't we seen you since?" Ben asked in that way little kids had of asking uncomfortably pointed questions.
"Maverick and I just stopped seeing as much of each other," Alexa said. The boy shrugged, satisfied.
"I think this day is going to require lots of cookies," Haven said on a troubled sigh.
"Awesome!" the boys yelled, barreling up the steps.
"Nothing like kids to offer perspective," Haven said, putting on a brave smile. If she could do it, Alexa would, too.
Phoenix came jogging out. "All right. Blake, Joker, and Meat are here. Mike Renner will be here in a few. Don't leave the clubhouse. Just stay put." He gave them all a look.
Standing next to him, Cora crossed her arms. "We got it. Go do whatever you need to do. And please keep Haven posted if you can."
"Will do," he said, taking off in a rush for his bike. And then he was gone, too.
A few minutes later, the women had gotten the boys set up with a snack and the TV and game console in the rec room, and the three of them took a plate of cookies and a pitcher of strawberry lemonade out onto the big back porch. It was a crystal clear, sunny day, the air fresh and clean, the sky bright blue. The beauty of the day seemed to mock the chaos that had just unfurled out front not a half an hour ago.
Settling into a group of cushioned lounge chairs, they just lay there quietly for a long moment.
"I'm glad you guys are both here," Haven said.
Alexa appreciated the sentiment so much. "Me too. Can I admit something else embarrassing?"
"Something else?" Cora asked, tucking her shoulder-length blond waves behind her ears.
"Something else besides the fact that my ex-fiancé is a controlling psychopath . . ."
"Ah, right. Honestly, there are too many of those running around," Cora said, grabbing a cookie.
"For real," Haven said, turning her bright blue eyes on Alexa. "What's your confession?"
"I haven't had any real friends in a long time." Saying that out loud made her heart race. It sounded so ridiculous, but it had been her reality for years now. She always thought of Christina and the other girls at the office as her _work friends_. It wasn't like she could talk about Grant with them, and since he'd become her whole world, that left her mostly with superficialities. Which defined most of her relationships until these past few days, didn't it? And she wasn't even sure exactly when or how it had happened, only that it had.
Haven reached between their chairs and squeezed her hand. "Cora's pretty much the only friend I've ever had."
"And I'm, like, totally awesome." Cora smirked.
Haven grinned, and so did Alexa. "And she's totally awesome, obviously. But I just mean to say that I'm very glad to have another."
"Me too," Cora said.
"Me three," Alexa said, taking a cookie of her own.
They sat there in silence for a while, enjoying the cookies and the breeze. And Alexa couldn't stop worrying about Dare and Jagger and what Grant might do next. If he was willing to frame the Ravens for something, what the hell else would he be willing to do? She blew out a long breath and tried to gather her spiraling thoughts. She looked from Haven to Cora, and blurted a question. "Cora, how did you come to babysit Sam and Ben?"
Cora and Haven exchanged a look, and Cora chuckled. "It just kinda happened. When we were first here, Slider was in a bind one day and we agreed to watch the boys for him because we didn't really have anything else to do."
"This was before we knew we were going to stay here," Haven said.
"And then once we knew we were staying, Slider asked a few more times and I agreed. So now he's hired me to work for him part-time when he needs. At some point I have to figure out what the heck I'm going to actually do with my life in the long term, but until then, I'm happy to help him out." Cora took another cookie.
"He's changed a lot since I last knew him," Alexa said, biting into a big chocolate chunk. "Not that I knew him that well, but he was always so outgoing and laid-back."
" _Slider?_ " Cora asked, her mouth dropping open. "Like, _my_ Slider? Well, not mine, but you know what I mean."
Alexa nodded, not at all blaming the guy, given the way grief had torn her life apart, too.
"Aw, that's even sadder," Haven said, stretching her legs out on the lounge chair. "Dare said his wife died of breast cancer and the poor guy was just devastated."
Cora shook her head. "He's definitely not outgoing and laid-back now."
Haven eyed her friend. "Is he too hard to work for? I don't want you feeling uncomfortable, and no one would expect—"
"No, no," Cora said. "He's fine. He can be gruff and non-communicative, but he's never mean. It's almost like he just doesn't have it in him to talk or interact. Except with his boys. He must save what he does have for them, because he can be really sweet with them."
"Yeah," Haven said. "Whatever he's going through, he's good with Sam and Ben."
Alexa listened as the friends went back and forth, admiring their closeness, wishing she had a friendship of her own that was as tight as theirs. And of course that made her think of Maverick, who'd been a friend before he became a lover, then more. She couldn't help wonder what was happening down at the police station. She just hoped Maverick managed to keep his temper reined in so he didn't end up sitting next to Dare in a cell.
"So, what are you thinking you might like to do in addition to babysitting?" Alexa asked Cora.
The blonde exchanged a glance with Haven and shrugged. "I'm not sure yet. I was a waitress back home and never expected to leave there, so I never let myself even imagine much before," Cora said, brushing the crumbs off her lap.
"But you've had time to imagine now," Haven said.
Cora nodded and stared off into the distance for a long time. "Okay. So, there's something I always thought would be cool, but I'd just ruled it out as an impossibility for me."
Haven sat forward in a rush. "Really? What is it? I can't believe you haven't said anything."
"It's not like it could happen tomorrow, Haven," Cora said, chuckling.
"So? Tell me." Haven hugged her knees to her chest and stared at her friend.
"I've always loved animals. If money and time weren't any object, I'd love to go to school to become a veterinarian. Even though that would take like a million years since I'd have to go to college first." Cora looked at them from under her hair, like she was afraid they'd laugh at her idea.
"I think that sounds awesome," Alexa said. Cora's plans made her think of the schoolwork she wasn't doing, but in the midst of everything going on, it suddenly seemed so damned unimportant. She pushed the thoughts away. "I love animals, too."
"Oh, Cora, I think that would be amazing," Haven said.
"We'll see," Cora said, clearly uncomfortable to be the focus of their attention. "I did find out that the local animal shelter takes volunteers, so I thought maybe I'd check that out. Just to get a feel for working with animals. See if I really like it. In the meantime, I can keep babysitting for Slider. Or get some other part-time job."
"You totally should volunteer." Haven took another cookie. "This is so exciting."
Cora laughed. "Simmer down. It's just an idea."
"It's a good one," Haven said, arching an eyebrow.
"So is you opening your own bakery," Cora shot back. They looked at each other for a long moment. "Okay, here's the deal. I explore the volunteer position and start looking at schools if you _seriously_ look into opening your own business. You have your dad's money coming, so you don't even have lack of start-up funds as an excuse."
Haven's mouth dropped open, and the woman looked to Alexa for help. "She's bribing me."
Al held up her hands, so glad she wasn't alone right now, so glad to have _friends_. "I'm not getting in the middle. Although I think you'd both be crazy not to do those things."
"Okay, then." Haven nodded slowly, like she was coming around to the idea. "Deal."
Grinning, Cora nodded, too. "Fine. Deal." They shook hands.
On a sigh, Haven flopped back against her lounge chair. "Now that we have our lives sorted, I just need my man out of jail and back by my side." The longing in the woman's voice made Alexa think of Maverick. She wanted him by her side, too. And so many things he'd said recently gave her hope that he felt the same exact way. His voice played in her ear.
_I. Fucking. Want. You. Us. A future. Our history, even the fucked-up parts. All of it._
Alexa's heart squeezed at the memory of the promising words, the rough determination in his voice, the hot fierceness of his expression.
"I like hearing you call him 'my man,'" Cora said, waggling her brows.
"I do, too," Alexa said. "Never thought I'd see Dare in a relationship. But he obviously adores you. It's really good to see."
Pink filled Haven's cheeks and she chuckled, but it was clear how much she enjoyed hearing it.
Suddenly, Alexa's cell buzzed where it laid on the seat next to her.
_Please, let it be good news from Maverick._
It was, quite possibly, the exact opposite.
_GRANT: Come home, Alexa._
The words were innocuous enough, but they still made her stomach drop. Because he just wasn't getting it, was he? And on top of what'd happened to Dare and Jagger today, that seemed . . . problematic. Purposely obtuse, at least. Disconnected from reality, at worst.
"Is that Maverick?" Haven asked, hope flaring in her bright blue eyes.
"No," Alexa said, blowing out a breath. "My ex. Asking me to come home again for like the millionth time."
"Home? What part of you having moved out doesn't he get?" Cora asked.
"For real," Haven said. "I thought this guy was supposed to be some high-powered businessman. Because the more I learn about him, the more he kinda seems a little unhinged."
Unhinged . . . suddenly seemed like a really good description of Grant. She thought back over her recent interactions with him. All the little manifestations of his compulsiveness, the mood swings, the paranoia, the refusal to accept reality, the anger. She recalled him stammering and the way his eye had twitched. Alexa shook her head. "He is, but I don't know. It's like me standing up to him pushed him off the deep end."
"Hey," Haven said, grasping her hand. "You're not alone, okay?"
Alexa nodded, appreciating the sentiment so much.
Just then, Alexa's cell rang. "Oh, thank God, now it's Maverick." She put it to her ear and answered. "Hey."
"Hey, Al." Frustration and anger were plain in his voice. "If Haven's there with you, can you put me on speaker?"
She pressed the button. "Yeah, you're on speaker now. What's happening?"
"Haven," Mav said, "we can't get bail hearings for Dare and Jagger until sometime tomorrow."
Haven gasped. "Tomorrow? They have to stay in overnight?"
His rough exhale came down the line. "Fucking judge is probably in Slater's pocket as well. He won't talk to the lawyer until tomorrow."
"Goddamn Grant," Alexa said, emotion welling up inside her. This was so unfair. For Dare and Jagger, for the Ravens, for Haven. For her, too. "I hate him so much I don't know what I'd do if I saw him."
"Tell me about it," Mav bit out.
The lethal tone made Alexa's heart pound with a sudden fear. "No, Maverick. Don't you do anything. Promise me."
"We'll talk about this later, Al. But you gotta know all bets are off. He blackmailed you, he threatened you, and now he's threatening the freedom of two of my brothers and the livelihood of the club. What happened today was a declaration of war."
She understood where he was coming from, she really did, but his words still sent a chill down her spine.
"Haven, the lawyer saw Dare. He said to tell you not to worry, that he's fine, and that he'll be home soon."
"Thank you," Haven said, those two words laden with so much more.
"Listen, I'm sorry to run," Maverick said, "but I want to keep an ear out and catch the lawyer before he goes." After a quick good-bye, they hung up.
"He'll be okay," Cora said, taking Haven's hand. "He's as tough as they come. You know that. And Jagger's there with him."
Haven gave a quick bob of her head. "I know. But he's still hurt, even though he refuses to admit it. And I just hate the idea of him in there."
"So do I," Alexa said. "But Cora's right. It'll just be the one night and then they'll make bail until we can get all this cleared up."
"One night," Haven said, taking a deep breath. "We can make it through one night."
That _we_ tugged at Alexa's heartstrings, because it was clear that Haven and Dare really had something special between them. And Alexa wanted it for herself.
With Maverick.
"One night," Cora said. "Easy peasy."
Alexa nodded and put on a brave smile for her new friend. But inside she couldn't help but wonder what other bombs Grant was preparing to launch at them the next day and the day after that—and how much Alexa was willing to let others be hurt and sacrificed for her so that she could escape him once and for all.
ALEXA KNEW SHE was alone the moment she woke up. "Maverick?" she called.
The sheets were cold beside her. Slipping out of bed, she wandered through the house looking for him—his room, the bathroom, the kitchen, the living room. She peered out the front door to check the garage, but it was dark. The LED on the microwave read a little after 4:00 A.M.
Worry sent a shiver over her skin, and Alexa grabbed the fleece throw off the back of the couch and wrapped it around her shoulders. Where could he be?
The Ravens had finally returned from the police station and guard duty around dinnertime, and the meal had been a somber, quiet, tension-filled affair. And it got even more tense when Alexa shared the text she'd received. She'd hated to pour salt into Mav's wounds, but also felt like she shouldn't keep anything from him. Not with so much on the line.
Most of the guys stayed around after dinner to drink and plot against her ex-fiancé, so Alexa had hung with Haven, Cora, and Bunny in the kitchen playing cards until Maverick had finally come in, bleary-eyed and a little unsteady on his feet, and said he was ready to go. When Bunny had said she'd drive them home, he thankfully hadn't argued.
But he also hadn't been interested in talking. When they'd returned to his house, he'd collapsed into her bed and fallen right to sleep, still wearing his Ravens cut and boots and everything. No matter what he'd said earlier in the day, Alexa hadn't been able to fight back the guilt she felt—guilt that he was so upset about his best friend being locked up in jail because of her ex—and she'd been unable to fall asleep.
She'd finally given up and gone out to the couch to work on her project for a couple hours, and part of her was glad that she had—because she was in better shape than she thought she was. A few more solid hours and it would be good enough to turn in. Maybe not her best work ever, but good enough. Right now, that felt like a victory.
A flicker of light out in the backyard caught her eye, and Alexa walked closer to the big picture window. A bright moon sparkled off the pond, and a low flash of red and yellow revealed a small fire.
In bare feet, Alexa crossed the back deck and padded down the sandy trail to the fire pit that sat at the little beach's edge. Maverick sat next to the fire in a white T-shirt and jeans, his feet bare, his elbows braced on drawn-up knees, his eyes focused out over the dark water. He was so beautiful to her that it made her chest ache.
"Hey," she said. The heat of the fire warmed her legs, the sensation a pleasant contrast against the night air.
His gaze cut to her and then dragged down her body. She felt the hungry look like a physical caress. "Hey. Why are you up?" His voice was flat, tired.
"Could ask you the same thing." She came to stand in front of him.
"Too much shit in my head," he said, raking his hands through his hair.
An apology sat on the tip of her tongue, but Alexa bit it back. That wasn't what he needed from her. He needed her to make it better, to make it all go away, even if only for a little while.
"Know what helps that?" She dropped the blanket by his feet, her heart doing a little flip inside her chest. Next, she shed her T-shirt. Then she stepped out of her panties until she finally stood naked in front of him.
The raw need she saw on his face was the greatest thrill and the biggest turn-on. But she wanted to ease his mind before she tended to his body.
She stepped backward, a smile growing on her face. "Midnight skinny-dipping helps that. Like when we were young. Not a care in the world. Just you and me against the world." Backward and backward, until her feet hit the cool water's edge. _Come on, Maverick_.
He tilted his head and his gaze lost some of the blankness from moments before.
Alexa kept going. The water hit her knees, her thighs, her waist. The soft bottom was cold against her feet. But she kept going until she turned her back on him and gently dove toward the streak of moonlight stretching out across the dark, glimmering surface. The temperature was a shock to Alexa's sleep-warmed skin, but it didn't take her long to adjust. And then it was like swimming through silk, soft and slick and tantalizing. The bottom was a little rocky in some places and muddy in others, so Alexa swam out until she could just barely touch . . . and then she floated.
Small splashes sounded out from the water's edge. She smiled up at the star-scattered sky, her heart exploding at the fact that Maverick was getting in, that he'd let her offer a distraction from everything weighing on his mind. Finally, she turned to find him thigh-deep, breathtaking in his nakedness, and heading her way. He dove, giving her a flash of the ink on his back before submerging for a long moment.
She scanned the surface, waiting for him to reemerge. Something grabbed her leg.
Alexa screamed and flailed. Maverick came up wearing a smug smirk, his arm around her waist, his hair appearing dark from the water.
She smacked his shoulder, laughing despite herself. "You scared me!"
"Sorry, baby," he said, his tone clearly not even a little sorry.
Even though he said it playfully, the term of endearment unleashed butterflies through her belly. Until this week, she hadn't heard it in years, and it was both familiar and achingly new after so long. "No you're not. You suck."
He held her tighter, their fronts pressed together from breast to thighs. "Is that an invitation?" His mouth dropped to her neck. Licked. Nipped. Sucked at the soft spot below her ear.
Alexa's chuckle turned into a moan. "A standing invitation."
Maverick's cock hardened between them, feeling deliciously thick. "Did you mean it?" he whispered against her skin.
"What?" she asked, his mouth and wandering hands making it hard for her to do anything but feel, especially when he lifted her to wrap her legs around his waist. The movement lined her core up with his hardness, and he didn't miss the chance to grind against her clit. Small, controlled thrusts that quickly had her panting.
"You and me against the world." Hips still rocking, he grasped her hair in an imitation of a ponytail and forced her head back. And then he licked her from between her breasts to her jaw in a slow, torturous line.
"I meant it," she whispered, her heart thundering. She locked her ankles behind his back to gain more leverage. As close as he was, he wasn't close enough. She needed more. She needed everything.
Maverick kissed her on a groan, his mouth tasting like mint and the earthy cleanness of the water and something that was all him. His hands were in her wet hair, on her breasts, squeezing her ass. He ground his cock against her, the wet slide over her clit driving her hard and fast toward an orgasm. And all Alexa could do was hold on and give herself over to the incredible feelings he was unleashing inside her. Desire. Wonder.
Love.
She loved Maverick Rylan.
She'd _always_ loved Maverick. And she knew now what she'd hidden from herself for years—she'd never stopped. Not when Tyler died. Not when Grant made it easier to shove down and ultimately hide from her grief with trips and presents and the excitement of the way he'd chased her. And not when Grant's touch had turned cold and controlling and hurtful.
Feeling the enormity of what she felt right now, she wasn't sure how she'd ever thought she was in love with another man.
Because underneath it all, Maverick had still been there. At the heart and soul of her. Waiting for her to see the truth. Waiting for her to _feel_ it.
Her truth.
_I love Maverick Rylan_.
The admission exploded her apart, destroying the last of her illusions and throwing light onto parts of her she'd ignored for so, so long. Emotion ballooned inside her until it was almost more than she could bear.
"I could get through anything if that was true, Alexa," he said, his forehead pressed to hers, his hips moving maddeningly between her legs.
"It is," she managed, her breath catching.
He pulled back, his eyes searching hers. "Hey, are you crying?"
"No," she said, on a fast shake, despite the fact that her voice was suddenly thick with restrained tears.
"Fuck, am I making you feel pressured?"
She held him tighter with both her arms and her legs. "Not even a little, Maverick. You're making me feel lo"—she swallowed back the word that'd almost spilled from her lips, not sure if it was too much to put out there so fast—"cared for in a way I haven't felt in so long. You're my silver lining in the mess that is my life right now. It all brought me back to you, and I've never been more grateful for something in my entire life."
"Jesus, Alexa," he said, claiming her mouth in a searing, exploring, lingering kiss. "I need you. I need you every way I can have you."
"Then have me," she whispered.
Still kissing her, he turned with her still in his arms and slowly waded back to the water's edge, and then he carried her out. She didn't feel the chill in the air for the heat pouring off Maverick's body and out of his dark blue eyes. The fire had died down but still flickered low among the red coals. Next to it he'd apparently spread out her blanket before he'd joined her in the water.
Maverick lowered her to the brown fleece and came down on top of her, his hips between her thighs, his cock so close to where she yearned for him. But he didn't take her. Not yet. He kissed and licked down her body, his tongue dragging over her sensitive skin until she was squirming and moaning and fisting her fingers in his long hair.
His big shoulders settled between her thighs, spreading them wider. He looked up her body. "I want you to say my name when you come on my tongue." And then that tongue was right there, licking hard and firm over her clit. Driving. Relentless. Mind-blowing.
Her heart racing, Alexa couldn't do anything but chase the orgasm barreling down on her. Hands tighter in his hair, she thrust her hips forward, seeking more of the delicious contact with his lips and tongue. He planted a hard palm against her lower belly and held her, his mouth sucking her clit deep and his tongue driving her over the edge.
"Maverick!" she cried out, her release crashing over her like a tidal wave. Her whole body spasmed at the goodness of it. "Mav . . . oh, God."
On a growl, he crawled up her body and claimed her mouth, his tongue immediately sliding deep. She reveled in the taste of herself on him because she couldn't see the pleasure he'd given her as anything other than a selfless gift. And that was something she hadn't received in such a long time.
"Ride me," he gritted out as he flipped them. He manhandled her until her thighs straddled his hips. "Ride me rough and hard and long, Alexa. Use me up."
"Yes." She grasped his cock and lined it up with her core. She sank down, taking him in inch by rock-hard inch. Her pulse beat so strong she felt it against her skin everywhere. "Oh, yes."
Mav's hands went to her hips. " _Fuuuck_." When he bottomed out inside her, he held her still, forcing her to take all of him. To keep all of him. She would. God, she would.
Bracing her hands on his chest, Alexa rode him in a slow up and down that had both of them groaning and straining and clutching tight. His eyes were intense blue fire peering up at her, and she couldn't look away. She didn't want to.
"This is where I'm supposed to be," he said. "Deep inside you."
"Yes," she said.
"And this is where _you're_ supposed to be." His hands pushed her down harder. "Claiming every inch of me."
" _Yes_ ," she moaned, grinding her clit against his pubic bone. "Oh, yes."
He pulled her down and hugged her against his chest, his arms banding around her, his hands in her hair. "Say it again," he said, kissing her as much as he could while they moved.
She didn't have to ask what he wanted to hear. "You and me against the world."
His hips moved faster. His arms held tighter. His eyes burned brighter. "Again."
Sensation spiraled hot and low in Alexa's belly. "You and me against the world."
He groaned. " _Again_."
Her orgasm detonated, stealing her breath. "You and me," she managed.
"That's fucking right," he yelled, his release pulsing inside her and drawing out her own. "You and me, baby. You and me."
When their bodies calmed, he eased out of her but didn't let her go. He settled her at his side, his arm around her shoulders holding their bodies tightly together. He kissed her forehead and sighed, the sound full of satisfaction and ease even as his heart pounded under her hand on his chest.
Or maybe that was her heart she felt, because he'd stolen it totally and completely.
## CHAPTER 25
I don't like this," Maverick said as he pulled up in front of the model home where Alexa was working on Monday morning. A number of work trucks filled the driveway of the otherwise empty development. His gaze cut to her, looking so pretty and put together in a pair of navy dress pants and flowy lavender shirt.
"I don't love it, either," she said, giving him a brave smile. "But Grant never comes out here, which is perfect because I can do what I have to do in peace. He won't be here until the walk-through on Wednesday, and then the whole team will be with us. It'll be fine."
Mav clenched his jaw, not wanting to open the argument they'd already had this morning even though everything inside of him demanded he push. For her own good.
Because he wouldn't be able to stand it if anything happened to her. Not when he knew she was in danger. And not when it was on his watch.
It would be him failing his mother all over again—only worse. This time, there'd be no excuses that could absolve him. Unlike then, Maverick knew fully that Alexa was in trouble.
"Look, your guys are nearby, right?" she asked, reaching over and putting her hand on his thigh. He gave a tight nod. Renner and Blake's presence just out of direct sight of the house was the only reason he could entertain the idea of her being out in the world without him all day. "If anything happens that even remotely worries me, I'll call or text them immediately."
_You and me against the world_.
Her words rang in between Maverick's ears, and he needed them to be true so fucking bad he could hardly stand it. But Alexa needed to be safe in his life and at his side for that to have a chance. For _them_ to have a chance.
"Don't even let it get to the point of being worried," he said, arching his brow and nailing her with a stare.
"I won't," she said. "I get it, Maverick. I won't take any chances."
He thought just going back to work was taking one, but they'd already been down that road. No matter how much Maverick wanted to lay down the law and tell her what to do, he couldn't. That was the controlling shit Slater had done and not even a little of what Alexa needed from Mav. He wanted to take care of her, not dictate to her. Damn it all to hell.
Sliding his hand behind her neck, he released a troubled breath. "I need you to come home to me. Safe and sound. You understand?"
Her expression went so soft and sweet. For him. "Aw, I think that's the nicest thing anyone's ever said to me."
He scowled. "I mean it."
"So do I," she whispered. And then she leaned across the big bench seat and kissed him.
He wanted to convince her with his mouth and hands and cock not to go. But he finally released her. "Have a good day," he managed.
She chuckled. "Thanks, even though it sounds like what you meant is, _I sure hope that house burns to the ground_."
He grinned and nodded. "Pretty much."
She shook her head, and then her expression grew serious. "Keep me posted on Dare and Jagger. I'm really nervous for them."
"I will," Maverick said, his gut in all kinds of knots over the fact that his cousin and one of his best friends had been sitting in a fucking jail cell overnight. At least their bail hearing was today. "We'll get them out on bail and go from there."
"Yeah," she said, sliding out of the truck. She turned and leaned back in. "You and me against the world?"
The words came out sounding like a question, and Maverick looked her right in the eye. "You better fucking believe it."
He watched her walk up the sidewalk through all the brand-new landscaping and disappear inside the big brick colonial. Pulling away and leaving her there was one of the hardest things he'd had to do in a long damn time. Besides Renner and Blake, the only thing that gave Mav any ease of mind was the belief that what Slater wanted more than anything else was to get her back under his control and walking up the aisle to the altar. So she was probably safe for now.
After she left him standing at that altar? All bets were probably off.
"WAITING FUCKING SUCKS," Phoenix said, expressing all of their views as they cooled their heels at the Ravens' clubhouse. Walter had thought that Dare and Jagger would have a better chance of a lower bail if the whole club didn't show up and emphasize the fact that, you know, they were a big fucking motorcycle club. So only Haven, Bunny, and Doc—wearing a suit instead of his colors—had gone.
"Christ, it really does," Maverick said, dropping his head back against the leather couch. This whole day sucked as far as he was concerned. Between his worry about Alexa being at work, the anticipation of hearing from Walter, and the tension ricocheting around this room, Mav felt like a rubber band pulled tight and ready to snap.
"It's gonna be fine," Cora said, sitting on a stool at the bar next to Phoenix. "The hearing will be over any minute and then the guys will be on their way home."
Murmurs of agreement went around the room.
"That's right," Rodeo said, the stress of the situation clearly weighing on the old man—Dare's great-uncle by marriage. It looked like Rodeo had aged five years since Maverick saw him here the night before. Doc had looked even worse.
Maverick heaved a breath. Crossed his ankles on the table. Got up and went to the bathroom. And still the fucking phone didn't ring.
"It's taking too long," Caine said. He'd been leaning up against one of the pool tables, standing statue still and not saying a word since they'd all gathered.
_Fuck._ The words struck a chord with the feeling of dread stirring in Mav's gut.
Sitting on the coffee table by his feet, Maverick's cell phone finally rang, the ringtone cutting through the thick tension like a knife. He wrenched forward and grabbed it, then held out a hand to quiet everyone as they moved closer. He answered and put the call on speaker. "Mr. Walter, it's about time."
"I'm sorry for the delay, Maverick. The hearing just concluded." Walter's tone deepened that sense of dread.
"And?" he asked, fearing he already knew the answer.
"I have good news and bad news."
"Just say what you have to say," Maverick said, working to restrain the anger welling up inside him.
"Dare's out on fifty thousand dollars bail, but Jagger was denied bail."
The room erupted in angry denunciations of the news.
Maverick held up his hands to quiet them again, his pulse a rushing roar in his ears. "He _what_? Why?"
"It's clear the district attorney has a stronger case against Jagger because he's the track manager and Dare's been laid up because of his injury. And the judge deemed Jagger a flight risk," Walter said.
"Jesus, this is utter fucking bullshit," Maverick said, his thoughts a jumbled mess. Jagger wouldn't run. The guy didn't have it in him. He wouldn't abandon his brothers in the club, his duties at the track that he loved, or his younger sister who lived in Baltimore. "How in the world does illegal dumping warrant all this?" Because fifty grand wasn't chump change. Not that they wouldn't spend it to free Dare—they'd do that in a heartbeat, and then some. But wasn't this all a little excessive?
"Each charge, of which there are two, carries a maximum penalty of five years in prison and a $25,000 fine. And it's clear that the recent shooting at the track played into the judge's decision," Walter added. "He was being a real hard-ass."
Hearing the possible jail time at stake was like getting hit in the gut. With a goddamned baseball bat. "Who's the judge?" Maverick asked, his instincts waving a red flag.
"His name's Harold Brennan."
Mav met Caine's ice-blue eyes and could almost see the same questions in the other man's eyes. What was the likelihood they could discover a connection between Brennan and Slater? What were the chances that Brennan was in Slater's pockets, just like the mayor and Davis? Better than average on both counts, Maverick would bet. Fuck.
They'd sure as shit find out.
Walter passed on a little more information about what would happen from here and they hung up.
Mav rose to his feet and scanned his gaze around the room. Every man's face was more angered and outraged than the last. "Here's what's gonna happen. Caine, you handle getting a private detective on this right fucking now."
"I'd placed some calls this morning," he said. "I'll nail one down right away."
Nodding, Maverick looked to Phoenix. "Hire out whatever work remains on Creed's house. I want you taking over the carnival preparations immediately. Jagger didn't go through all this trouble on the club's behalf so that everything could fall apart three days beforehand, so we can make sure it goes off without a hitch until we can get him home again. And call in whatever help you need. Nothing is a bigger priority."
"Consider it done," Phoenix said, a deep scowl in place of his normally playful, smart-ass expression.
"I can help," Cora said. Phoenix gave her a nod.
"What about the licensing inspection?" Bear asked. The inspector was set to arrive at two o'clock, and they'd learned he wouldn't be alone. An inspector from the environmental crimes unit would be joining him. Who'd have ever thought _that_ would be one of their problems? On a sigh, Mav looked at his phone. They had ninety minutes.
"You and I will handle that," Mav said to the older man. "Given what's happened, having two executive board members present might help smooth this situation over. And I want another pair of ears there to make sure we're clear on how to mitigate the environmental damage and begin the cleanup." Bear nodded.
Jesus, this was the last thing they needed when they were trying to repair their reputation in the community. Fucking Slater. Maverick wanted to wrap his hands around the other man's throat and watch the life slowly squeeze out of him.
"Rodeo, I want you to set up a perimeter watch . . ." Mav's words trailed off as the command reminded him of something he hadn't thought of before. "Fuckin' A."
"What is it?" Caine asked, his tone on edge.
"Good news. Maybe." He met his brother's intense gaze. "Remember how the Hard Ink team hacked in to the traffic cameras when we were keeping lookout for Haven's father?" Nods all around. "I wonder if they could go back through the footage and find anyone coming or going from the area of the drag strip?"
"Fuck, that would be good news," Phoenix said. "Worth asking."
"I'll make the call. The rest of you, get to it," Maverick said, meeting the gazes of each of his brothers. "And watch your backs. It's probably Slater, but maybe it's someone else. Either way, we know for sure that someone's gunning for us. And we can't afford another hit."
TWO LONG, TENSE, busy days later, Maverick found himself at the track with Phoenix and a bunch of other Ravens helping to organize the carnival. The company had arrived this morning to set up the rides, games, and food trucks. The weekend was supposed to be a scorcher, so they'd hired a tent company to put up a couple of big tents to throw shelter over rows of picnic tables, and those were going up, too.
As for their part, the Ravens were setting up cones to direct traffic in the big parking lot, hanging signage, stringing lighting, roping off areas inside the track where the public could meet and take pictures with the drivers, and otherwise answering a thousand phone calls and questions and putting out fires whenever they arose. Trying to take the heat off of Dare, Maverick had been fielding the phone which wouldn't stop ringing. Only good thing was that being busy made the day fly by. And helped distract him from the fact that Jagger had now been sitting in jail for three days. The injustice of it ate at him like a parasite.
Pausing in the shade under one of the tents, Maverick answered another call. "Maverick, it's Cynthia."
Surprised, he leaned back against a table. It had been years since Alexa's mother had last called him, and then it was usually only to track Tyler down because he wasn't answering his cell. "Mrs. H? You doing okay?"
"Yeah, hon. I was just wondering what time you all were coming over."
"Oh, uh." He swiped at the sweat on his forehead. The work at Creed's was being completed today so they weren't wasting any time—they'd planned to move Alexa's mom and the first load of her belongings tonight. "Alexa gets off around five and then she'll change and we'll be over." Which they'd told her several times this week, but that was a small thing compared to the fact that she hadn't had any more crying jags since Sunday night.
"Oh. Right. Okay. Just wanted to make sure nothing had changed," she said.
"No problem, Mrs. H." Phoenix flagged Mav with a wave and then cut across the grounds to him.
"Sorry to bother you, Maverick. Alexa didn't answer and I wanted to make sure I was ready when you got here."
Didn't answer? Alarm bells threatened to go off, but Maverick also knew Alexa was neck deep in last-minute details today. "You're no bother," he said. "We'll see you soon." They hung up. "Hey," he said to Phoenix.
"Thought you could use a drink," the guy said, handing him an icy bottle of Coke.
"Fuck, yeah. Hot as hell out here today." Maverick popped the top and took a long pull from the bottle. Cold and fizzy and sweet. Then he gave in to the urge clawing up his spine and texted Alexa. _You okay?_
His phone buzzed almost immediately. _Yes, almost done. Breaking for lunch and then that'll be a wrap._
See? She's fine. He shot off a reply. _Knock 'em dead, baby._ Appeased, Mav slid his phone back into his pocket.
"You did a great job stepping into Jagger's shoes and putting all of this together, Phoenix. He'll be grateful. And so am I."
Phoenix heaved a troubled breath. "I fucking hate that he's in there."
"Me too," Maverick said, the regret he felt a weight on his shoulders that wouldn't go away. And why should it go away when Jagger was locked up for a crime he didn't commit? _That_ was a weight.
They sat together for a long time and drank their sodas, both of them clearly lost in their own thoughts. Aside from Jagger sitting in jail, the week was going better than any of them had expected.
Dare was home and no worse for the wear—way more pissed off, but not harmed in any way. The Ravens already had the tires and oil barrels removed from the property, and an environmental firm was treating the small spills that had occurred. A new licensing inspection had been scheduled for tomorrow to review the property after the environmental violations had been addressed. The Hard Ink team had hacked into the traffic camera footage again and was reviewing it, although they said it could take a while. And the P.I. they'd hired had found a social connection between Slater and the judge, confirming their suspicions. Not great news, of course, but at least there was forward fucking motion. Alexa had even managed to finish and submit her schoolwork, and he knew she was feeling good about having that off her plate.
Even better, the illegal dumping story didn't seem to be hampering the carnival at all. Advance ticket sales were strong, local businesses had signed up in droves to sell their food or other products, and the word was getting out. The charity aspect of the carnival was the real selling factor, and it seemed to be getting more play in the press than the dumping story. Thank God for small favors.
Best of all? Grant Slater hadn't bothered Alexa at the model home even once.
"All right," Phoenix said, hopping off of the table. "No rest for the wicked. I'm gonna go find the carnival manager and see where everything stands."
Maverick nodded. "Grab me if you need me. I'll be around until about quarter 'til five." Assuming he could actually sit still for that long when he knew that Slater would be going to the model home today for Alexa's presentation. That fact had him absolutely itching to ride over there and park his bike on all that brand-new grass. They clasped hands, and then Mav sat for another minute and emptied his drink.
Everything was finally coming together—for the carnival. For Mrs. H's move. Even for him and Alexa. His thoughts drifted to the other night, making love to her in the moonlight by the pond. How her words had hit him square in the chest and made him _want_. Her. Them. Everything. More than ever before. They were so close to having her ex behind them that Maverick could taste it. And then he'd be free to lay it all on the line.
So all the news was as good as it could be.
Maverick's gaze scanned over the colorful rides, the bright food carts, and the festive strings of lights swaying in crisscrosses above the whole grounds. Music played over the track's sound system, and men talked and laughed as they worked. Everything was normal. Good. Exactly what it should be.
All of which made him wonder why his gut was still so tied up in knots.
## CHAPTER 26
Thanks Mike," Alexa said, hopping out of the prospect's car in front of the model home. She'd forgotten to pack a lunch this morning, so Mike had run her out to grab a sandwich before the walk-through with Grant, the general contractor, and the sales team. They would all be arriving soon.
"Any time, Alexa," the newest Raven said. "Call us if you need us. We're hanging out two minutes away."
She gave him a nod as he pulled a U-ey in front of the house. Holding her stomach, she frowned. The big rock sitting there at the thought of seeing Grant again said maybe she should've skipped lunch, but she wasn't sure when they were going to fit in eating dinner with moving her mother tonight.
_Everything will be fine. You won't be alone with him._
Right.
On a sigh, she made her way inside. The cool air wrapped around her, helping her stomach just a little, as did the calm, relaxing vibe of the home's design. The place made her want to sit down and stay awhile. And for the four-hundred-thousand-dollar price tag associated with this particular model, she hoped it made a lot of people feel that way.
She stowed her purse and washed her hands, then grabbed her clipboard from the sales office and made another pass through the house. Grant loved perfection. In fact, maybe it was the only thing he truly loved. And in this one instance, she was determined to give it to him. Everything had to be just right for the walk-through. Maybe then he'd see that she could be valuable to him without becoming his wife and just let her go.
She began in the basement, which featured a big rec room, an exercise room, and a media room that looked like a little theater. She checked the finishes, paint job, placement of furniture, and double-checked her punch sheet. Next, she made her way through the living spaces on the main floor. She fixed the plantation shutters on the library windows, making sure all the slats lined up just so because she knew that even one out of place would drive Grant crazy.
_Thump_.
She froze, her ears going on alert. People were arriving for the walk-through already? She'd thought she had another ten or fifteen minutes. She went to the front door and peered outside, but there weren't any cars in the driveway.
Damn nerves. They had her jumpy and anxious. Feeling time getting away from her, Alexa rushed up the steps to run through the bedrooms, bathrooms, and upstairs laundry room.
At the far end of the hallway, she pushed through the double doors of the master bedroom—and stopped short.
For a moment, her eyes couldn't make sense of what she was seeing. Rose petals on the bed. Candles flickering on the nightstands and dresser. The low strains of jazz music playing from the built-in stereo system.
What in the world? As it all started to make a crazy, surreal sense to her, she gasped and turned—and ran right into Grant's chest so hard she dropped the clipboard.
"Alexa," he said, his voice casual, happy even.
She reeled back, putting distance between them. "What are . . . what the hell are you doing? What is all this?"
"It's for you," he said, his body filling the doorway.
"For God's sake, Grant, the walk-through starts in ten minutes." Her hands fisted at her sides, hopefully hiding her shakiness.
He shook his head. "I moved it to tomorrow morning."
"What?" She pulled out her cell phone and checked her email. "I didn't get any message about rescheduling."
He grinned. He actually grinned and it was a sickening thing to her. "I wanted to surprise you."
Her mind was spinning, trying to reject the insanity of all this. "With what? Some kind of cheesy seduction scenario? This is my place of work."
"What if it was more than that?" He came closer, his body loose, his mannerisms relaxed.
Still, she retreated further into the big bedroom, her scalp prickling. "Meaning what?" she asked, not sure she really wanted to know.
"What if this place was yours, ours? I know you've always loved it. My house is ours, of course, but I lived there alone first. If we moved here, it could be ours together from the beginning. A fresh start." He said all this like it made perfect sense. And maybe a month ago, it would've. But too much had happened. Too much had changed, for her. Now it just felt . . . creepy.
Unhinged, she heard Haven's voice say.
_Get out of here!_ The thought shivered over her skin.
"I don't even know what to say to you right now." Shaking her head, she pressed her thumb to the fingerprint security button on her phone, bringing it to life. She opened her text messages to send the message already typed out there, the one she'd written just in case but never really thought she'd have to use. Her finger moved for the _Send_ button—
Grant swiped the phone from her hand. "Focus, Alexa," he said, slipping her cell into his pocket. "Focus on us. This is important." For the first time, anger slipped into his voice.
"Okay, okay," she said, her voice high and strained. She jumped back from his touch. Shit. _Shitshitshit._ Had her text gone through? If not, she was going to have to run for it. But where would she even go that he couldn't get to her? The office. She could lock herself in and use the business line. But that meant she had to get him farther into the room to create an open path to the door. "Um, I'm just . . . this is a lot."
"I know, babe. But I thought maybe a grand gesture would show you how serious I am, how much I want you back and in my life." He came closer. Her legs hit the mattress and she stumbled, coming to sit on the edge of the bed.
Her ears strained for the sound of a car or motorcycle, but the bedroom was at the back of the house. She didn't hear a thing. Oh, God, what if no one was coming to help?
"It's a thoughtful gesture, Grant," she said, her stomach queasy at sitting among the scattering of red petals. She tried to keep her words kind, her tone placating, unsure exactly what kind of man she was talking to and feeling like maybe she'd never known Grant Slater at all. "But too much has happened. You've scared me on multiple occasions. I can't live like that and I don't feel the same way anymore. You're handsome, successful, and talented. You deserve someone who can love you with her whole heart. Who can be everything you want in a woman. That's not me. And you don't need it to be—women will be lining up to date you—"
"I don't want another woman, Alexa. I want _you_." He stood right in front of her, his knees almost touching hers.
Everything inside her screamed to flee. "Why? I'm nothing, just like you said. I'm a poor girl who was a lowly administrative assistant when we met."
"You're _my_ fiancée. That's all there is to it," he said.
She was a possession to him, and that's all she was. One he didn't want to lose. Shaking her head, her thoughts swam. She had to get some space between them. "I, um . . ." She cleared her throat. "I could really use some water."
His expression transformed from angered to pleased, like he was glad that she was asking him for something, anything, and not just rejecting his overtures. "Of course," he said, making for the bathroom. "I knew you'd come around. I knew I had to try. Because I believe in making my own luck, securing my own destiny, not waiting for others to do it for me."
So would she.
She shot off the bed and bolted for the door, time suddenly going all slow-mo even as she raced. Down the hall. Down the stairs. Footsteps pounded after her.
"So help me, Alexa!" Grant bellowed.
_OhGodohGodohGod_.
Her flat sandal slipped on the tiles in the foyer, but she regained her footing and kept moving. Grant was closing in. She'd never make the office.
She grabbed the front door handle and pulled. It swung open.
Something yanked her hair and she wrenched back. She crashed into Grant, and then they both went down. Pain exploded through her as she crashed in a free fall against the hard floor.
"Ungrateful bitch," he groaned, flipping her over and crawling on top of her. "You'll learn to do what I say." He ground his erection against her thigh.
Nausea rolled through her as she pushed and twisted with her whole body. "No, Grant. Get off of me. This is crazy. _You're_ being crazy."
He pinned her wrists to the floor, squeezing tight. Pain shot through her fingers. "Don't call me that," he growled. "And don't make this harder than it has to be."
He kissed her. Hard. Their teeth knocked, making her mouth throb. He pressed in, trying to force her jaw open. She worked to draw up her knees, to brace the flats of her feet on the floor, to turn her head away. But he held her so tight.
_Click_.
Grant froze and Alexa's eyelids flipped open. The barrel of a gun dug into Grant's temple.
Alexa nearly cried in relief.
"Get. Off of her. Right now. Before I blow your fucking brains out." Maverick. Maverick was here. Oh, God, Maverick.
When Grant didn't move fast enough, he suddenly flew off of her.
On a groan, Alexa pushed up onto an elbow to see Mike and Blake holding Grant by the shoulders and arms in the doorway. She hurt everywhere.
Maverick crouched at her side and gently helped her stand, his gun still aimed at Grant. "You okay?" he asked her, maybe never looking more fierce in his entire life.
"Yeah. Um, yeah." She was shaking so badly it was hard to think. "I just um . . ." She swayed.
Maverick caught her. "Sit him down and keep him secured," he bit out at the other Ravens. The men manhandled Slater over to the bottom of the stairs and forced him down. "Let's get you outside." Mav helped her onto the little porch.
She sagged down onto the top brick step, unable to do anything else. "If you . . . if you hadn't . . ."
"I'm here. You're good. I've got you." His gaze ran over her face, his expression equal parts enraged and concerned. "Just rest for a minute, okay?" He rose.
"Don't hurt him," she said, her teeth chattering. "Don't do . . . anything that will get you in trouble. He's not worth it."
Maverick scowled but nodded. "Much as I'd love to beat the shit out of him, I have a better plan." He whipped out his cell, swiped his finger over the screen, and put it to his ear. "Yeah, I'd like to report that I've just interrupted an attack on a woman. I have the attacker restrained and need immediate police assistance. And an ambulance." He eyeballed her and moved the phone away from his mouth. "Don't you worry about a thing, Alexa. This time, Slater's fucking done."
Alexa gaped. As if it wasn't miracle enough that Maverick had showed up just when it mattered, his self-restraint in dealing with Grant was frankly a miracle, too. Maverick still had a temper, but five years ago, he'd have solved this problem with his fists first. Now, he was thinking strategically, thinking long-term, which meant he'd grown up a helluva lot in the past five years. But she actually had lots of evidence of that, didn't she? She saw it in the care he took of his house, the way he kept himself in check, the way he was taking care of her and her mom.
It made Alexa feel calmer. Safe. Secure.
Which was all she'd ever wanted.
MAVERICK'S BLOOD WAS boiling, but he wasn't giving Slater the satisfaction of seeing it.
Standing in the front doorway where he could keep an eye on both Slater and Alexa, Mav was doing everything he could to keep himself reined in. Blake and Renner stood on either side of Slater, keeping him from moving a muscle. Renner had ensured that by pulling down some sort of cord from the dining room curtains and tying Slater's hands to the bannister.
"I am going to sue you each individually and then take your little club for every cent it's got. And then I'm going to bulldoze every inch of the Raven Riders property until no one remembers you even existed." Slater chuffed out a chilling laugh. "I will ruin you and everything you ever cared about. You don't know who you messed with."
Maverick tuned him out. His words just made him angrier, but Mav was already savoring the sweet satisfaction of watching Grant Slater get arrested. That would have to be reward enough.
Instead of taking Slater's bait, Maverick kept his gaze on Alexa. Except for the occasional, random shudders, she sat shock-still on the stoop, her arms around herself.
Jesus, what if he hadn't gotten here when he did? What if he hadn't left the track way earlier than he had to? Knowing Alexa would be seeing Slater at her meeting, Maverick had come to hang with Blake and Renner until it was over, and then he'd planned to drop by here to check on her.
_I need your help at the model home right now._
The memory of all three of them receiving that message at the same time still made Maverick's blood run cold.
Sirens sounded in the distance. Closer. Until, finally, three cop cars turned onto the street and parked haphazardly in front of the house.
"Now you're going to pay," Slater bit out. "Home invasion. Carrying a weapon. Assault. False imprisonment. You're done, Rylan. All of you."
"Do you ever shut up?" Maverick muttered, and then he stowed his handgun in a holster against his lower back and went into the yard to meet Martin.
"What happened?" Martin asked, rushing up onto the grass. Davis came running up beside him, a glower covering his weasel face. A rookie cop joined them—Eckstein, his name tag read.
"First, I need to tell you that I have a concealed carry license and I'm carrying at the small of my back," Maverick said, wanting to play this a hundred percent by the book.
Martin nodded. "I know, but thank you. Start talking." Davis threw scowls at both of them.
As an ambulance pulled up, Maverick recounted Alexa's text message and what he'd seen when they'd arrived—Grant, pinning a crying Alexa to the floor of the model home, forcibly kissing her while she struggled and said "no." Saying the words brought the images back to life in his mind's eye, fueling the lethal anger vibrating inside him.
"We tied Slater up inside until you could arrive to arrest him," Maverick said. Davis blanched. They all moved toward the house, following a pair of paramedics who stopped in front of Alexa and started asking questions.
Slater began ranting louder, but Martin didn't pay him any mind. Staying out of the medics' way, the cop sat next to her. "Hey, Alexa. Can you tell me what happened?"
Maverick was torn—he needed to hear it. Every word of it. But no part of him _wanted_ to hear it. None of this was about him, though, was it? It was about taking care of Alexa, whatever she needed.
She blinked at them and frowned, her expression almost confused. "Um." She let the paramedics check her vitals, and then she peered at Martin again. "I came back from lunch and found Grant in the master bedroom upstairs. He'd put rose petals on the bed and lit candles." She licked her lips, the sound thick. "He said a bunch of stuff about giving me this house to get me back, and then he took my phone when I tried to text Maverick for help."
"We got the message," Maverick said, trying to send her encouragement through his eyes. And thank fuck that they had. If Mav had waited to come to the house until after he thought the meeting would be over . . . He couldn't even finish the thought.
"He refuses to accept my decision on calling off our engagement. He just kept trying to convince me that we could start over. The whole thing felt very creepy. When I had the chance, I ran from the room and down the stairs. I managed to get the front door open, but he caught me in the foyer and threw me to the floor." She shuddered.
Slater was a sick fuck, that much was for sure. Between planting the tires and oil on their property, blackmailing her, and now this, the guy was either losing touch with reality or was pure evil. Maverick wasn't sure which was worse. Jesus, this could've been so much worse than it was, and it was already damn bad. He could've lost her. God, he could've lost her.
Emotion welled inside him. Something bigger than the anger and the desire for vengeance. Something brighter. Something far more powerful.
Love.
But more than that. He didn't just love Alexa Harmon; he was head over fucking heels in love with her. He'd known he had feelings for her, of course—hell, they'd never gone away despite their years apart. But what he'd _thought_ he'd felt before this moment was the dim sliver of the moon compared to the burning midday sun that he knew for sure he felt for her now.
If he'd have lost her this time, she would've taken him with her.
"Did you hit your head, miss? Can you tell us where it hurts?" one of the paramedics asked.
"Here," she said, gingerly touching the left side of her head. "I fell flat and hard. Everything kinda hurts, but I don't think anything's broken."
"What's this?" the medic asked, taking her hand. Bruises. Her wrist was ringed with bruises.
Rage flashed like molten lava through Maverick's blood. "Sonofabitch," Mav bit out.
Martin held up a hand to him and said in a low voice, "Let's get a picture of that."
"I told him to get off of me but he pinned me to the floor and kissed me and told me not to make it harder than it had to be." Maverick tilted his head as she spoke, noticing how flat and monotone her voice had gone. God, why hadn't he killed Slater when he'd had the chance again? They all could've lied. Called it self-defense. No one would've known otherwise.
The paramedic checked Alexa's eyes, examined her head, and asked her more questions.
"I'm gonna check out the bedroom," Martin said, rising. Davis made for the door as well.
"I'm going, too," Maverick said.
Martin frowned. "Mav—"
"Slater has the mayor, that asshole"—he pointed at Davis—"and at least one judge in his pocket, so if you think I'm not going to see what happened up there with my own eyes before evidence starts disappearing right and left, you're out of your mind." He arched a brow.
Davis put a hand on Martin's chest. "Martin, he can't—"
"Can it, Davis. He's coming. Just don't touch anything, either of you."
Mav could do that. He nailed Davis with a stare that challenged him to do something about it until the fucker finally looked away. Then the three of them went upstairs. Eckstein stayed with Slater, who was getting more and more outraged by the minute because no one had yet spoken to him. Or untied him.
The master bedroom was just as Alexa had described. The rose petals were like blood droplets on the floor and bed, sending a chill over his skin. "Every single thing she said checks out," Maverick said. "She sure as fuck didn't smash her own head against the floor or bruise her own wrists. And Blake, Mike, and I saw him forcing himself on her when we got here, so you've got three witnesses. I don't care how goddamned prominent Slater is in this community, do your jobs." He nailed both cops with a hard stare.
Martin nodded and placed a call on the radio attached to his shoulder, then he looked at the two of them. "Out of the room. I need to take pictures."
After a few minutes, they all made their way back downstairs, and Martin paused in front of Slater. He patted the guy down and found a cell in his pocket.
"That's Alexa's," Maverick said, recognizing the watercolor graphic on the case.
Martin set it aside and nodded. "Grant Slater, you're under arrest for the sexual assault and battery of Alexa Harmon. You have the right to remain silent. Anything you say can and will be used against you in a court of law. You have the right to speak to an attorney, and to have an attorney present during any questioning. Do you understand your rights as I've presented them to you?"
Red-faced, Slater just shook his head like he couldn't believe this wasn't going his way. Martin cuffed Slater, then worked to untie the ropes.
Maverick had seen enough. "Can I get Alexa out of here?"
Martin nodded. "I can follow up with her if I need to."
Outside, Mav found the paramedics packing up their bags. "She's refusing transport to the hospital," one of the men said. "There's a possibility of a head injury, though. She may have a headache, sleepiness, or feel nauseous. Those would all be expected. But if she loses coordination and balance, gets dizzy, or starts vomiting, take her to the emergency department immediately."
"Why don't you go, Alexa?" Maverick asked, crouching down next to her. "Just to be sure."
Just then, Martin hauled Slater out and down the steps. Her ex glared at Alexa, and Maverick put his body between them until he was out of sight, working like hell to keep his own anger under wraps.
Alexa shuddered. "I just want to go home."
"Okay," Maverick said. He gave the medic a nod. "Then let's get you home. This is all over now. Slater can't hurt you anymore."
## CHAPTER 27
Sitting on the couch in Maverick's house, Alexa hung up the phone. Telling her mother what'd happened was the only way to make her understand why they couldn't move her tonight. But going through the story and answering all the questions again left Alexa feeling utterly drained.
Never in a million years would she have thought Grant would get that violent . . . or twisted. And who knew what more he would've done if Maverick and the guys hadn't shown up?
Maverick sat down beside her and held out a steaming mug and two little pills. "Here, Al. Try a little of this."
"What is it?" she asked. The ceramic was warm in her hands, and the heat felt good.
"Just chicken noodle soup and some ibuprofen. I thought . . ." He shrugged. "I don't know. It might help."
Aw, sweet, sweet man. She tossed back the medicine and washed it down with a big sip of the salty broth. "It's perfect, Mav." She took another sip, and another, hoping the soup would warm the icy chill that filled her chest. "Aren't you going to eat?"
"I'm not worried about me right now," he said.
"Well, I am."
"Don't be." He tucked her hair behind her ear. "As long as you're good, I'm good."
"I don't know if I'm good or not." Sip.
"You don't have to know right now. But you're here and you're safe and Slater's behind bars. So that's a start." His eyes blazed a dark blue intensity that seemed to peer into her soul.
The fierceness of his words chased off a little of that chill, like he was wielding a pickax and chipping away the pieces of Grant that her ex had left behind inside her. It was a terrifying, nauseating image and immediately chased away her appetite. She set the mug on the coffee table.
"How can I help?" he asked.
She tried to give him a smile. "You already are." Finding it hard to hold his gaze, she ducked her chin and stared at her lap, at the navy capris and sheer patterned blouse over a tank that she'd worn to work today. Knowing Grant had touched the clothing repulsed her. "I want to take a shower."
"Yeah. Okay. That sounds like a good idea." He gave her a hand up and walked her to the bathroom. Worry poured off of him like a waterfall, the force of it too intense and soothing at the same time.
"I won't be long," she said, peering at him around the door.
"However long you need."
Nodding, she pushed it closed. Avoiding her reflection, she undressed. Balled up her clothes and stuffed them in the little trash can. Turned on the water. Waited for it to heat. Got in.
She cleaned herself mechanically, barely feeling the spray against her skin. Her soapy hands found a bruise on her hip. One on her ribs. One on her shoulder. And each little jolt of pain chipped away more pieces of Grant. A bruise on her forehead. On her wrist. On her elbow. _Chip, chip, chip_.
Each piece of him that sloughed off exposed more of her, raw and vulnerable and unbearably sensitive.
It was suddenly like someone had plugged Alexa back into her body. Because all the pieces of _her_ snapped into place in one instant so that she felt _everything_. Every bit of it. The fear and the pain and the despair and the disbelief and the shock. Her head throbbed. Her aches were a pulsing agony. It was too much.
_Toomuchtoomuchtoomuch_.
A sob ripped out of her, and Alexa clapped a hand over her mouth as it folded her in half and finally forced her to her knees. She slid into a ball on the slick bottom of the tub, hugging her legs and burying her face into her knees as she cried. Grant was like a poison inside her she had to release, and every tear cleaned her, disinfected her, set her free.
So she gave herself over to it. One last time. Just until Grant was completely gone from her psyche the way he was from her life. And then she'd never let him have a piece of her again.
"I DON'T KNOW how I didn't kill him," Maverick said into his cell phone. "I probably fucking should've." Rage and regret were like viruses in his blood, running rampant through him and laying waste to everything in their paths. Damnit. He'd known Slater could do something like this. He should've pushed Alexa harder not to go to work. He should've stayed at the house himself. He should've done so many things that might've kept this from happening.
It was failing his mother all over again, only worse.
"I get that," Dare said, "but going after Slater only leads to one result that matters—you locked up. Do I have to worry about you staying smart?"
Pacing the living room, Maverick blew out a frustrated breath. "No. I called the cops and did it by the book. And now he's the one spending a night behind bars, the fucker." The drumming of the shower water sounded out, and Mav had to resist the urge to go to Alexa and reassure himself that she was really there, in his house, safe.
"Good. Because this is probably gonna get messier for her before it gets better." Dare's voice was gruff, tired.
"That much is clear," Maverick agreed. "I'm going to talk to her about filing a protective order in the morning." Though, given who Slater was, who knew how well local law enforcement would follow through on actually, you know, enforcing it. Still, it was better to have the legal paperwork in place. And it would be damn satisfying to imagine Slater getting served.
"Worth doing," Dare said. "But don't count on that to resolve the situation."
"I won't. Not even a little," Maverick said. He stopped in front of the big window that looked out over the backyard and the sunlit pond beyond. He couldn't quite make out the fire pit where he'd made love to Alexa—and where she'd made love to him—he believed that into his very bones. But he didn't have to see it to feel the impact of what they'd shared and what he felt for her.
_Everything_.
And damnit, he wanted to tell her. That he loved her, that he wanted her for his own, that he wanted forever. With her. But he also didn't want the memory of those declarations to be mired in the pain that Slater had caused. Fuck, he wasn't sure which was the right way to play it.
"Did you hear me?" Dare asked.
"Shit, no. Sorry."
"I asked how she's doing," Dare said.
Maverick shook his head. "I'm not sure. She's mostly holding it together, but I think she's in shock. He slammed her to the ground pretty good, so she's also hurting. Got bruises on the side of her face and some other places." _Afuckinggain_. "She'll be okay, but she's still processing."
"Yeah," Dare said. "Christ. I'm sorry, Maverick. And Haven is, too. She asked me to let Alexa know that she's here for her."
"Thanks. I'm sure she'll appreciate hearing that." And he did, too, the way everyone was embracing Alexa again. It meant a lot to him. A long pause, and everything building up inside him demanded some form of release. "Shit. I'm . . . I'm fighting for her, D. I'm all in. And I'm not letting anything stand in the way this time."
A satisfied sigh, like maybe Dare had been holding his breath, came down the line. "About goddamned time, my brother. About goddamned time." Someone called to Dare in the background. "Listen, I gotta go. Dinner's on. Assume you all aren't coming." It wasn't a question.
"No," Mav said. "And we shifted moving her mother until tomorrow morning, so I won't make it to help finish setting up until later in the day." Maverick wasn't sure whether to regret the carnival's poor timing or hope that it might provide Alexa with a welcome distraction. Either way, the Ravens needed it, and they needed it to be a success.
"We've got it covered. Phoenix pulled this shit together like a boss."
"Yeah, he did," Mav said. "Catch you tomorrow."
"Wait. Mav?"
"Yeah?" he asked.
"Don't you dare blame yourself for this." Dare's voice was gruff, stern, and Maverick could easily imagine the expression that went with it. He'd seen it on his cousin's face a million times. "I know Alexa's situation is probably screwing with your head given what Bunny went through, but this isn't your fault. Wasn't then and it isn't now."
But Maverick couldn't get there. Not yet. Not when Alexa wore fresh bruises he might've prevented. "I hear ya, D." They hung up. Maverick stood there for a long moment, his ears latching onto the sound of the running water once more. Was it just him being an anxious, impatient fucker or had Alexa been in there a long damn time?
Something in his gut twisted, pushing him in the direction of a long damn time.
He made for the bathroom. Knocked. No answer. Knocked again.
His gut went on a free fall.
Maverick opened the bathroom door. Steam hung thick in the air. "Alexa, you okay?"
And that was when he heard it. Crying.
The sound lashed at him and lanced urgency through his blood. He was at the tub's edge in an instant and pulled back the curtain. Seeing her in a tight ball on the floor nearly broke his fucking heart.
In a flash, Maverick turned off the water, grabbed a towel, and draped it around her shoulders. "Come here, baby." He slid his arms under her knees and around her back.
Crying harder, Alexa wound her arms around his neck, making it easier for him to lift her from the tub. "Maverick," she cried, her wet face burrowed into his neck.
"I got you. Don't you worry about a thing."
Water ran from her hair and body, soaking into his shirt and dripping down his arms, but he didn't care. He turned out of the bathroom and carried her down the hall, past the guest bedroom to his own room. Because fuck the guest bedroom. If the two of them were going to make a run at doing this thing again, they were doing it in _his_ bed. Their bed, now.
Alexa wasn't a guest to him. She would never be a guest to him. Because this house was where she belonged—and always had. They'd both just gotten lost for a while. But that was all fucking over.
In his room, Maverick settled her on the edge of the bed and pulled back the covers. He kicked off his boots and shrugged off his cut, then settled his gun on the nightstand. "Climb in, Al."
Her breath hitching, she squeezed at the length of her hair with the towel. "But I'm wet."
"You're perfect. Climb in."
She moved to the center and slipped her pretty, long legs under the covers. Masculine satisfaction roared through him at seeing her there, but he had to rein that shit in. Hard. After what'd happened to her today, he wasn't touching her until she indicated that she was ready. Anyway, despite his hard-on, what he needed more than anything was to help Alexa feel safe. Fuck, he needed that more than his next breath. To protect her, to comfort her, to be her shelter in the storm.
Maverick stretched out and pulled her in tight against the side of his body.
_Meow_.
He looked to the doorway to find Lucy standing there, eyeballing them in her skull sweater.
"Come on, Luce," he said. "Your mama needs us."
The cat padded across the room, hesitated at the side of the bed, then jumped up. She sniffed and stepped, sniffed and stepped, until she stood on his stomach within reach of Alexa's petting hand. And then Lucy curled up on his belly and fell asleep while Alexa stroked her ears.
And Christ, it was probably stupid, but protectiveness roared through Maverick like a demon. Because this felt like his little family. His woman. Their weird-ass cat. Their house.
"Fall asleep if you want to," Maverick said, running his fingers through the loose waves of her damp hair. "I'll wake you up every so often to make sure your head's okay."
"I don't know if I can sleep."
"You can."
She shook her head in a small movement that was just barely perceptible. "I'm afraid."
Her words were like getting sucker punched in the heart. "Don't be," he managed. "Slater's in jail. I've got my gun sitting right here. Nothing's going to hurt you, Alexa, I promise."
"You'll stay awake . . ."
He heard what she didn't say. _You'll stay awake just to be sure?_
"I'll stay awake. You sleep. I'll keep you safe. Count on it."
She drifted off in maybe five seconds, and that had satisfaction roaring through him, too. She was scared, she felt vulnerable, and yet his words and his body had given her the security she needed to let go. The trust and belief that represented staggered him. Just nearly knocked him stupid.
And it made him realize he'd do anything not to break his word.
The conversation with Dare came back to him. Yeah, Maverick would be smart. He wouldn't be the one to start shit. But if Slater came after her again, he sure as hell would be the one to finish it.
## CHAPTER 28
Despite being woken up over and over again, Alexa opened her eyes to the early morning sunlight and felt . . . better. Stronger. Freer. For the horrible things Grant had tried to do to her, he was now in jail. And the more she thought about what'd happened yesterday, the more she realized that she'd been stronger and braver than she had in a long, long time. She hadn't caved in to him. She hadn't cowered and fallen apart. She'd schemed. She'd run. She'd fought. And she'd recounted every bit of it to the authorities.
That was more than she ever would've done before.
"Hey. You awake?" Maverick's voice was low, gravelly, sexy.
She shifted to see him better. Dirty blond hair that he'd clearly been running his hands through. Scruff covering his jaw. A face full of hard angles. God, he was beautiful to her. "Yeah. You must be exhausted, though."
He shook his head. "I'm fine. How you feeling?"
Alexa still had some achiness, and she couldn't deny the small throb in the side of her head. But all of that would go away. And she'd still be free of Grant—and that made her freaking ecstatic. "Actually, I feel pretty decent. Nothing some ibuprofen can't fix."
He arched a brow, his gaze running over her face like he was analyzing her words. "Really? I was thinking you could rest here and I could take care of moving your mom. You'd be safe. Dare offered to come hang." He gestured to his cell sitting on the nightstand next to his gun.
She pushed up on an elbow. For a long moment, she just soaked him in. The thoughtfulness of his words shot right to her heart. The raw emotion in his dark blue eyes wrapped around her and made her feel so cared for and safe. And his scruff made her want to feel it against her thighs.
She cupped his face in her hand, settling for feeling the beginnings of his beard against her palm. For now. "That's not necessary, but thank you. I feel a lot better. And honestly? I'm done feeling bad over Grant Slater. Today is all about new beginnings, starting with moving Mom. And I want to be there for her."
His eyes searched hers. After a moment, he nodded and pressed a kiss to her palm. "In case it needs to be said, you handled yesterday like a champ. It about fucking killed me to see—" He shook his head. "But you called us and you ran and you were fighting. When we got there, we saw you. I was terrified but proud of you, too. And I know you're hurting right now but you survived, Alexa. You survived that and you survived him and even if there's other bullshit to deal with, you're out the other side."
His fierceness built her up and bolstered her resolve. "You make me stronger, Maverick."
"You're strong on your own." He kissed her palm again. "Don't forget it."
She swallowed hard, love and need for this man shooting through her fast and urgent. Alexa shifted her naked body on top of his, her knees settling on either side of his hips. "I won't. And you can remind me if I do."
"Better believe I will." As he peered up at her, his eyes flashed hot. His hands landed on the outsides of her thighs. "You sure?"
She ground against the bulge of his cock. "Of this? Of _you_? Always." Bracing her hands on his pillow, she leaned down and kissed him.
The groan he unleashed seemed connected to her clit, because desire suddenly had her aching there. Aching with need and lust and longing. She sucked on his tongue, and his hands went to her hips and ground her down harder against his erection. The rough denim rubbed maddeningly against her clit, making her gasp into the kiss.
"Want you," she whispered into his mouth. "Want you so much."
"Fuck, baby. Lay on your stomach." He helped her push the pillows and covers out of the way, and then she stretched out in the center of the bed. He climbed off and tore his shirt over his head. She watched as he undid his jeans and shoved them down his lean hips, his cock jutting out hard and thick. Looking at her like he wanted to make a meal of her, he wrapped his hand around the base and stroked, once, twice, three times. "Christ, you're beautiful."
"I need you inside of me," she said, licking her lips at the sight of him. Inked and rough and hard. For her.
He came over her, his knees on the outside of her thighs. "You're gonna be so fucking tight like this." His hand slipped between her legs, and she arched her back to give him better access. "Feel that," he said, his fingers finding her wetness, sinking deep. "So wet for me."
"Yes," she said, her voice breathy and needy. "Now, Maverick."
Holding her ass cheeks open, his head lined up with her core and penetrated her pussy inch by mind-blowing inch. A moan spilled out of her as he bottomed out, his hands bracing on her lower back, forcing her to arch further.
"Christ, Alexa, _feel that_." The arousal in his voice spiked her pulse.
"It's so good. But I need you to move. Oh, God. Fuck me. Please." Her words were almost shaky with need.
He withdrew and thrust, withdrew and thrust, slow, dragging movements that had her panting and pushing back against him and aching for him to go deeper, harder, faster. "My cock looks so goddamned good sliding in and out of you."
She felt herself get wetter around him, and it ratcheted up the urgency of her yearning. "More. Give me more."
"More of what, baby?" He came down on top of her, his whole weight covering her, his cock buried deep.
She looked over her shoulder. "More of you. More of everything. Don't hold back, Maverick."
"Al—"
" _Everything_."
"Fuck." He claimed her mouth in a short, searing kiss. "Everything," he gritted out. And then he did just as she asked. He wrapped his arms around her head and shoulders for leverage and took her like a man possessed. Like a man taking what was his. And that's what she wanted more than anything. His hips flew and slammed against her ass. His cock deliciously battered a spot inside her that had her moaning nonstop. His body covered and claimed every part of hers.
It was the most amazing thing she'd ever felt.
"Christ, I'm gonna come," he rasped.
"Come," she said, wanting to feel him pulsing inside of her. "Come in me, Maverick."
The roar he unleashed shoved her hard in the direction of her own orgasm, as did the way his cock jerked inside her again and again. And then he was off of her. Flipping her over. Pinning her hips down and planting his mouth right over her clit.
He sucked, _hard_. His tongue flicking and strumming until all she could do was fist her hands in his hair and hold her breath.
Alexa came so hard that her vision blurred around the edges and her heart threatened to pound out of her chest. But still, Maverick didn't let up. He growled against her and continued to tongue her until she was screaming another orgasm and trying to push him off before she lost her mind. She was so sensitive that when he finally released her, her whole body flinched and she couldn't help but laugh as she tried to roll away.
"Where do you think you're going?" he asked as he grabbed and crawled up her, wearing the most smug, satisfied expression perhaps any man had ever worn.
She couldn't help but grin. "Nowhere."
"That's right." He stroked her hair back off her face. "God it's good to hear you laugh."
They stared at each other a long moment, and suddenly their playfulness felt weighted, significant, like the ground was moving beneath their feet. For her, it was. Because emotion ballooned inside her chest until restraining it became simply impossible.
It was bigger than her, more than her. After everything, she had to tell him, and he deserved to know. "Maverick . . . I know we have a lot to figure out, and I know I'm a mess, but I also know that I love you so much that half the time I can barely breathe for holding it in. So I don't want to hold it in anymore. I love you. I'm in love with you." The hammering of her heart had nothing to do with the two incredible orgasms he'd just given her. "So, yeah. I just wanted you to know."
His eyes widened and his lips parted, and then his expression took on a softness she'd absolutely never seen before. "Aw, damn, Alexa. You have no idea what a fucking dream it is for me to hear you say that. I have never stopped loving you. Not one minute of one day. But what I feel now is so much deeper, just bigger, you know? Than anything I've ever felt before. I love you, too. Jesus."
His words—his love—healed every aching place inside her. Just snapped all the broken pieces back into place until she was whole, strong, the Alexa she was always meant to be. He kissed her, intense but gentle, claiming and healing, utter perfection.
"Are you mine, baby? Tell me you're mine," he rasped.
"I'm yours, Maverick." She ran her fingers through his hair, holding it back from his ruggedly handsome face. "I'm so yours."
"You own me. You gotta know that. Christ, I've wanted this for so long." He dropped his head to her shoulder.
The gravel in his voice reached into her chest and squeezed. She wrapped her arms around his big shoulders and held him.
Finally he looked at her again, his eyes bright and so full of love. For her. "This is our new beginning, Alexa. You and me."
"Against the world. I love you," she said, happier than she'd been in years. Happier than she ever thought she'd be again. And, God, she adored the happiness pouring off of him, too. She wasn't naïve enough to think that Grant was going to make her life easy, but there wasn't anything he could do to change this. Because Alexa had finally found herself and, through that, the man she truly needed.
MAVERICK WAS FLYING so goddamned high over the words he and Alexa had shared that not even spending the day with a very anxious Mrs. H schlepping her mountain of crap could touch how good he felt. Unbelievably, that mountain represented maybe a third of what the woman had in her house, but most of what they'd left behind was outright junk.
Only two things put a damper on how he was feeling. First, knowing Jagger was sitting in jail. It ate at Maverick like an itch he couldn't reach. He just had to trust that the Hard Ink guys and their private detective would find the evidence they needed to clear Jagger. It couldn't happen soon enough for Mav—for any of the Ravens. And second, the word that Grant had made bail. They knew it would happen, and they'd taken the time this morning to get the temporary protective order, but it was still a burr in his saddle to know that clearly guilty Grant got to walk while Jagger rotted. Justice was blind, Mav's ass.
At the new place now, they'd just finished unloading the moving truck. He, Alexa, and the movers had managed to get the boxes and furniture in the rooms where they needed to be, but the process of unpacking and really getting the house set up would take a while. At least Mrs. H was out of Slater's place for good.
Mav returned one of the dollies to the now-empty truck parked out in the driveway.
"That was a big job to get done," Alexa said, bringing him a glass of ice cold water. It was another hot one, but the weather was perfect—which boded well for the opening of the carnival tonight. "I'm so glad we got her out of there."
"Me too," he said, taking a big drink. The more ties Alexa severed with Grant, the better. So far that included the restraining order, moving her mother out of his house, and calling in sick to work. She hadn't pulled the trigger yet, but she'd pretty much come around to the conclusion that she was going to have to quit her job. Mav knew how much she liked what she did, and he hated that for her, hated that she was the one who was going to have to walk away when she'd done nothing wrong. "What else would you like to do here today?"
"I'm hoping that we can maybe set up her kitchen, bedroom, and enough of the living room so that she can watch TV. But I don't want to make us late for the carnival." She gingerly prodded the side of her head.
"You hurting?" he asked, his gut twisting.
"Just a little bit of a headache. Not too bad. Honestly, I feel too good from this morning to even care." Her smile was so damn pretty.
"From the orgasms or the _I love yous_?" He grinned and pulled her into his arms.
She chuckled. "Both were very nice."
"Nice? _Nice?_ You screamed so loud you almost shattered glass."
Laughing, she smacked him in the shoulder. "I did not."
"Yeah you did. That memory will play in my head for _years_." He arched a brow at her, loving the humor shining from her hazel eyes.
"Okay, I totally did." She ducked her face against his chest.
"That's right. Don't be denying my skills." He kissed her head and grinned again, having too much fun with her. What a difference a fucking day made.
"Oh, my God." She pushed against his chest and retreated, pointing at him, and trying not to smile. "You are too full of yourself."
"I call it like I see it." He followed her toward the house.
She rolled her eyes and turned around, but he could tell her shoulders were shaking. "So, why don't you try your skills on the mess that is the living room? And I'll tackle the kitchen with Mom?"
He winked. "Consider it done. Because, skills."
She threw up her hand and made for the kitchen. Maverick couldn't stop smiling like an idiot. Getting everything you ever wanted did that to a man. No shame in that.
Three hours later, they'd achieved most of what Alexa had hoped for. Everything was put away in the kitchen cabinets and drawers, her mom's bed was made and clothes were all put away, and Maverick got the TV set up, cable connected, and furniture in place so she'd be comfortable. He even found her box of photo albums and set them out on the end table next to her favorite chair.
When he was done, Maverick leaned against the arch to the kitchen. "How's it feeling, Mrs. H?"
"It's a little overwhelming, but it's coming together. You kids did a lot today," she said. "And I know I was a pain in the ass."
He chuckled and winked at Alexa. "A Harmon be a pain in the ass? Never."
Her mom smiled, just what he was going for. "I suppose we have our moments, don't we?"
"Who doesn't?" Mav asked. "What else can I do?"
"I'd love to get all the empty boxes and packing material outside. You're going to have so much space here, Mom. Just wait and see." Alexa tore down one of the boxes, making it flat.
"You got it." He headed for the other end of the house and cleared out the bedroom and hallway first. By the time he emptied the living room, Alexa had taken care of the kitchen.
Smiling, Alexa passed him with an armful of bubble wrap. "We make a good team."
He kissed her. "Damn right we do. Want me to take that out?"
She shook her head. "I got it."
The moment Alexa stepped out the front door, her mom came into the living room. "How's she doing? Really."
"She's strong, Mrs. H. She's okay. You should ask her, though," he said.
"I did, but of course she doesn't want to worry me." She waved a hand. "But I'd like to take that Grant over my knee."
Maverick might pay good money to see that. "I understand the sentiment."
Cynthia eyeballed him. "You're good for her. You always were. Even when you were more of a hellion."
The comment drew a smile from him, but also made him think. "She's good for me, too. And I'm serious about her. That okay with you?"
She nodded. "All I ever wanted was for her to be happy. You make her happy, Maverick. That's more than good enough for me."
Alexa came back in and stopped short. Looking between them, she asked, "What's the matter?"
He chuckled and shook his head. "Nothing's the matter. We're just talking." He managed to keep his voice casual, even though her mom's words added to the high he was riding. Knowing she approved of them—of him—took a load off his mind. Because Maverick was already thinking in terms of forever. What to do with that, he didn't know. But they had time to figure it out.
"You kids should go. I know you have that carnival. I'm pretty beat and think I might just like to put my feet up for a while." She made her way to her chair and sank down. "Aw, look at you putting my books out." She ran her hands over the album on top of the stack and gave him a smile.
Alexa squeezed his arm as she passed him, and then she crouched beside her mother. "You sure you'll be okay by yourself? New house and everything?"
"I'm an old pro at being by myself. You go and have fun. You deserve it after everything." She held out her arms. "Give your mama a hug."
Mrs. H was so much more relaxed than she'd been earlier in the day that it gave Maverick hope that she'd be happy here, and he knew that would make Alexa happy, too.
"Okay," Alexa said, rising. "I'll check on you later."
"You don't have to do that. I'm just gonna watch some shows and maybe work on unpacking some more." She pulled one of the photo albums into her lap which was something he'd seen her do when she was stressed or upset. So maybe she was hiding how she was feeling until they were gone. Fair enough. She was allowed to be stressed over such a big change—and on such short notice.
"All right then, Mrs. H, we'll get out of your hair for a while. But if you need anything, you call and we'll be here. Simple as." He nailed her with a stare so that she'd know he was serious.
"Don't worry." She pushed back in her recliner and stroked her hand over the album's cover.
Alexa gave her shoulder a squeeze. "I want you to be happy here, Mom."
"I think you did a good thing for me, Alexa. So thank you."
When Alexa turned toward him, the expression on her face nearly sucker punched him. Mrs. H wasn't much known for giving apologies or outwardly showing affection, and she'd given both to Alexa today in one form or another. Maverick knew what that meant and he was glad for her. Alexa deserved it, and she needed it right now, too.
She came right up to him. "So, do I get to dunk your butt tonight or what?"
He smirked. "Nope, I'm on tomorrow. But if you think you can take me down, bring it."
She grinned and arched a brow, her look full of sass. "Oh, I'm gonna take you down all right. But for tonight, I'll settle for you winning me lots of stuffed animals."
He nodded. "Challenge accepted. Let's go."
Outside, they made for his pickup truck. When she got in, Alexa turned to him. "You know, I'm glad the carnival's tonight. We need something happy and fun, a celebration."
He hauled her across the seat and pressed a careful kiss to the side of her head. "Yeah, we fucking do. We all do."
"I'm just sorry Jagger isn't here," she said, squeezing his thigh. It meant a lot that she thought of his brother, because he felt the same way.
"Me, too, Al. But he's why we have to do this and do it right. We need this to be the big success he knew it could be. Tonight, we'll celebrate brotherhood, club, and family in his honor until we can get him out of there."
She nodded. "Thank you for everything today, Maverick. You were such a huge help and you were so good with Mom. I appreciate it."
"No thanks required, Al. It's what you do for someone you love." He gave her a look.
She leaned her head against his shoulder. "I adore the way that sounds."
So did he. He started the truck. "Good. Get used to it, baby. Because nothing's going to change it. Ever."
## CHAPTER 29
I'd say we have a huge fucking success on our hands," Maverick said, toasting with Dare and Phoenix under one of the big tents. A bunch of the Ravens had claimed a few tables where they could take turns grabbing dinner. He and Alexa had just finished eating burgers and fries, which might've been a bad decision given she wanted to ride the Tilt-A-Whirl.
"We can use the good news. It's been a rough few weeks for everyone," Dare said, looking out over the crowds filling the carnival grounds. Mav's gaze followed, his brain half scanning for Slater, even though he knew he wasn't there. They'd had the P.I. put a tail on him the second he left the police station, and the guy had been directed to notify them immediately if Slater came anywhere near the Ravens' property.
Even though it wasn't even quite dark yet, more people had shown up than they'd expected, and they'd actually had to rent out the fallow cornfield from the farmer who owned the tract of land across the street from the racetrack for additional parking. That was saying something given the crowds they often entertained on busy race nights.
From the lines in Dare's face, it was clear that Jagger's absence weighed on him. Dare always tended to take the burden of the club's business—and all its members—on his shoulders. Always had. "There'll be more good news to come, too," Maverick said. "You know it."
"I know, but it can't come soon enough." He turned to Haven, sitting next to him on the picnic table bench. "Okay, what rides have you never ridden?"
She grinned like he hung the sun and the moon just for her, and Maverick loved seeing that. His cousin deserved it. And fuck if he hadn't seen a similar expression on Alexa's face when she looked at him. The realization made him feel ten feet tall. "The Tilt-A-Whirl and the Sea Dragon and the spinny thing that plays the music."
"The Gravitron?" he asked. She nodded. Did Dare turn a little green or did Mav imagine it? "Okay. It's a plan."
"Oh, let's ride the Tilt-A-Whirl together," Alexa said. The two women jumped up, excited by the idea.
"First one to hurl owes the other fifty bucks," Maverick said as he and Dare joined the women.
Dare chuckled. "I'm too fucking old for this shit," he said under his breath. "But I'll ride every goddamned thing here twice if it makes her happy."
Maverick understood the sentiment. He took Alexa's hand, linking their fingers. "Then let's do it. I'm curious to see what burgers and fries look like on the way back up."
Alexa and Haven laughed.
"That's disgusting, Maverick," Haven said. He winked at her.
People laughed, music played, bells and buzzers rang out, lights twinkled and flashed against the nearly dark sky. Standing in line, Mav almost felt like a kid again. Hanging at a carnival on a summer night with his best friend and his girl. Life was good. Anything was possible. He slid his fingers into Alexa's hair and kissed her, long and lingering.
"What was that for?" she asked, smiling.
"Do I need a reason to kiss you?"
"Never," she said.
"Oh, there's Cora," Haven said, waving her arms.
The blonde joined them with Slider's boys in tow.
"My main men," Maverick said, holding up his hand for high fives. Ben jumped to hit his palm, and Sam attempted to put every bit of his strength into it. Maverick laughed at their antics.
"Maverick," Ben said in a rush. "We sat all the way at the back of the Sea Dragon and it feels like you're going to get dumped out of your seat when it's all the way up. It's so cool."
"That sounds awesome, Ben. What else did you ride?" he asked. The six-year-old recounted all his adventures in a fast jumble of words. The boy could barely contain his excitement. Mav was glad to see them happy and having fun. They'd had a rough few years, just like their dad. Speaking of . . . "Is Slider here?" he asked Cora.
"He's working parking right now, so I offered to take the boys around. Who's better to go to a carnival with than some awesome kids?" She grinned at Sam and Ben, and both of them beamed up at her.
"When can we play games?" Sam asked.
"I'm following you two. Just lead the way," she said. As the boys ran off, she threw all of them a wave.
Mav and Dare exchanged a look, and he could read the general tenor of his cousin's thoughts—worry, for Slider. But at least the guy was here. He'd shown up and pitched in. Maverick had no doubt that said as much about his loyalty to Jagger as anything else. And he had to respect that.
"Tickets, please," the man at the head of the line said. They handed them over and found two cars near each other.
"This is going to be so fun." Alexa leaned over and kissed him.
"If you keep doing that, it'll be a fucking blast," he said.
She threw her head back and laughed.
Grinning, Maverick pointed at Dare. "Fifty bucks, brother."
Dare flipped him off. And then they started moving.
Maverick couldn't remember the last time he'd laughed so hard. He hadn't entirely ruled out the possibility of puking, but between the look on Dare's face, Alexa absolutely howling and screaming in delight, and the way centripetal force glued her to his side, he was having a great damn time.
"That was fucking awesome," he said when they all got off. He clapped a greenish Dare on the back. "What are we riding next?"
They rode the Sea Dragon—though Dare called bullshit on the very back, which Haven was only too happy to accommodate. They rode the spinning teacups, which was a funny sight with them in their Ravens colors sitting in dainty pastel cups. They rode the Gravitron, where the floor dropped out from under them, rock music blaring in their ears. They walked through the house of mirrors, where Haven got stuck in a little path of mirrors and couldn't find her way out—a fact that made her want to go through it a second time. And then they made their way to the bumper cars.
"Now this is more my speed," Dare said.
"But you were so cute in the teacups, see?" She brought up a selfie she'd taken of them on her phone.
"Aw, that is really cute, Dare." Alexa grinned at him.
"Baby, don't show that shit around," Dare said, no heat in his words at all. Maverick busted out laughing. "You have no room to talk, motherfucker." Dare arched a brow at him.
Mav held up his hands, grinning and shaking his head. "I didn't say a word."
They finally got to their cars, each of them taking their own, and then somehow it ended up three on one, everyone coming after Mav.
"What the hell?" he said, laughing and dodging Alexa as she beelined straight for him. "You're supposed to be on my side."
"All's fair in bumper cars, Maverick. Deal with it." She rammed him.
"Aw, you're gonna pay, little girl. Just you wait." He went to turn the car, but Dare hit him next. The guy grinned like an idiot. Mav just shook his head.
After, Alexa wrapped herself around him. "I'm sorry I crashed my car into you."
He feigned a scowl and nailed her with a hot stare. Her playfulness was fucking sexy. "I think I know how you can make it up to me."
"Hey, Haven wants to do the Tilt-A-Whirl again," Dare said. "Catch ya later?"
"You bet," Mav said.
When they were alone, Alexa pushed up on tiptoes and put her mouth to his ear. "Does it involve me sucking your cock?" She gave him an innocent look.
"Jesus Christ," he bit out, the teasing words sending blood rushing south. "You better believe it does now."
She bit her lip, her look so full of desire that he wanted to take her behind one of the rides right this second. "Mmm, I'll look forward to that." She tugged his hand. "But first we need games and stuffed animals. And maybe cotton candy. Or a funnel cake."
"Whatever you want," he said, meaning that in every way he could.
"Oh, I love this one," she said, pulling him toward the game where you shot water into the mouth of a clown to blow up and pop a balloon. They plunked down their money and took up the last two guns side by side.
Alexa lined up her shot like her life depended on it, and then the game began.
_Ding ding ding!_
"I won! Oh, my God, I won!" she cried.
"Yes, you did," Mav said. "What are you going to pick?"
The man pointed to her three options for an extra-large prize. "You get to pick from the black bear, the dinosaur, or the llama."
She turned to him. "Which one do you like?"
He peered up at them. "I don't know. That llama's ridiculous, though."
"Oh, yay. That's the one I like, too. I'll take the llama."
The man handed it to her, all two and a half feet of fucking weird-ass llama. To go with her cat. Mav chuckled. Who even knew they made llama stuffed animals?
"It's for you," she said.
"You won me a llama?" He looked at it skeptically as she pushed it against his chest. The eyeball on the one side of its head wouldn't stop looking at him.
"Yes! You totally do not have enough llamas in your life." She absolutely glowed with happiness.
"That is true, Al. Thank you. I think." He tucked the huge toy under his arm. People looked at him weird as he carried it around, the tall neck sagging this way and that. He just smiled.
"But now I have to win you something. Let's play that." He pointed to the dart game.
"Pop three in a row, win a large prize," the man yelled. "That's a mighty fine llama you got there."
Chuckling, Maverick laid out his cash. "Fucking llama." His first throw popped a balloon. Alexa cheered. The second one did it again.
"Go, baby!" she said.
Number three . . . And _pop_!
"Yay!" She threw her arms around his neck. "You did it."
"What's it gonna be, little lady?" the man asked, pointing to a row of toys smaller than the llama. Thank God.
"Holy crap, is that a hedgehog?" she asked, her eyes wide.
"Indeed it is," the man said, holding one out to her.
"Is it just me or does this carnival have the weirdest stuffed animals ever?" Maverick asked.
Alexa grinned. "I know. Isn't it amazing? I'll take it. Thanks," she said. And then she turned to him. "I'm so happy I could cry."
Maverick took her face in his hand and leaned in. The llama leaned in, too, which made her giggle. "I'll hold a carnival for you every day if it makes you this happy."
"Aw, silly man. It's not the carnival. It's you." She kissed him nice and deep, making him think of her plans for after. When she pulled back, she eyed the llama. "Okay, it's him a little bit, but mostly you."
Maverick laughed. "I guess I can live with that. Then what's next, baby? Because I can't get enough of hearing you laugh and seeing you smile."
"I think we need a funnel cake, Mav," she said, petting her hedgehog.
"Your wish," he said, seeing the food truck in the distance. "Let's go."
"I THINK THIS was my absolute favorite day ever," Alexa said as they made their way back to Maverick's truck. They had their llama, their hedgehog, and bellies full of sweets.
"I love you, Alexa," he said, pinning her to her door. He kissed her slowly, deeply, thoroughly, his tongue leaving no part of her mouth unexplored. Need and lust rose up inside her. She just couldn't get enough of him, and it was the most amazing thing to know that she didn't have to get enough, because he'd always be there. And they'd always be together.
She wound her arms around his neck. "Love you, too," she whispered. "I think we should go check on my mom as quickly as possible so we can get home."
"Mmm." He kissed and licked down her neck. "Yeah, we better, before I take you right fucking here." He pulled away wearing a mischievous grin, his eyes blazing.
She sighed, her body not feeling big enough to contain all the happiness overwhelming her. She wasn't sure she could ever remember a time when life seemed so full of promise, the future so bright. And Maverick had given that to her.
Mav started up the truck and made his way through the parking lot, the line of traffic to get out moving slow.
She thumbed on her phone. "Oh, shoot. I have a voice mail. I didn't hear it ring." She put it to her ear to listen.
Her mother's voice: "Hi, Alexa, it's your mom. I realized there were a few things I forgot so I took a cab over to my house. There's absolutely no rush because I can just take my time gathering some stuff, but drop by and grab me after the carnival. Okay? Bye."
Alexa groaned, frustration flooding through her. Getting her to leave things behind had been a real struggle, despite the fact that most of it was either trash, so old there was no way her mother even remembered owning it, or something nearly identical to what she was already taking. And now, the first minute they left her alone, she ran back for more. "Crap. Mom went back to her house to get more stuff."
"For real?" Maverick asked.
"Yeah." She huffed out a breath. "I'm sorry, but can we go get her?"
"Of course we can. How the hell else is she going to get home?" They finally got to the main road, and he turned in the direction of town.
"Honestly, I think she hoped we'd be a while so she'd have time to collect more things to bring back to the new place. Damnit. I'd really hoped she would see how nice it was without all the clutter and want to keep it that way. Who was I kidding?" Her stomach dropped. She'd been kidding herself, clearly.
"It is what it is, Al. You've given her a nice place, and you can help her keep it up if you want, but you can't force her to change her ways." He took her hand and held it against his thigh.
"I know you're right." And what a difference it was to have someone to help shoulder the burden of dealing with moments like these. She forced a deep breath. It was only some stuff. It would be fine.
Within fifteen minutes, they pulled up to the curb in front of her mom's old place. The driveway was largely blocked off with the most obvious and disgusting of the trash that they'd carried out during the move. Which was going to make schlepping whatever new boxes or bags she'd packed a pain in the butt, but it was too late to worry about that now.
"I'll go get her," Alexa said, opening her door. She caught the smoky scent of a wood fire or a cookout, making her thing of the yummy burgers they'd had at the carnival. God, that had been so much fun.
"You're not going alone, woman. Especially if she has a bunch of shit to carry." He joined her and they made their way up the driveway.
Looking at the house, Alexa couldn't help but think that it seemed dark, although there did seem to be some light—
Not light, a flicker. And a pale haze hung in the air. The scent of wood burning. And then she knew. Oh, God, she knew.
She gasped and bolted. "Maverick, the house is on fire!"
"Alexa, wait!" Maverick yelled.
But she couldn't. Her mother was in there. In that death trap of a house—even with everything they'd removed, it was still loaded down with crap all over the floor. Her heart pounded and her ears rang. The world closed in on her as she burst through the front door—
"Oh, Jesus," she moaned. Smoke hung thick in the dark air, a roiling, looming beast. She stopped short inside the living room. Fire engulfed the kitchen at the back of the house and was crawling into the living room, climbing up the walls, licking at the ceiling, and consuming the piles on the floor. "Mom!"
Maverick was right behind her. "Get out. I'll find her."
"No, I'm not leaving without her," she said, coughing. But at least she could call for help. With shaking hands, Alexa dialed nine-one-one.
"What's your emergency?" the operator answered.
"House fire with injured people inside at 825 Walnut Street. Hurry," she yelled, coughing again. The dispatcher asked a few more questions, but Alexa finally cut her off. "Just hurry." Because the heat was making it hard to breathe.
Maverick turned to her. "Alexa, I'll find her. Go."
"I'm not going without her and we don't have time to argue. This place is going to go up fast so we can't wait for help. Let's just find her and we'll all go together." She turned on the flashlight on her phone, but the light almost reflected off the smoke. "Mom? Answer me!"
"Cover your mouth and nose with your shirt," he said. She did, pulling it up over her face. "Cynthia!" he yelled, as they made their way farther in. He tripped over something he couldn't see, going down on a knee. God, the floor was a hidden debris field.
The fire popped, spraying an arc of embers as the fire found more fuel and spread surprisingly fast. Alexa screamed and ducked, but they had to keep going. Moving slow because they couldn't see what was underfoot, they made for the kitchen. God, this was her nightmare come to life. The thing she'd always most worried about with her mother's hoarding, ever since that fire when she was a kid. They got as close to the kitchen as they could with the fire consuming the doorway.
The heat threatened to sear her skin and wrung sweat from her body. Her chest was so tight that she couldn't get a deep breath, the asthma she hadn't dealt with in so long kicking in. "Mom!" she yelled, finding it harder to project her voice. "What if she's in there?"
He moved past her one more step, the flames stretching out above him now.
A giant _crack_ and a big piece of the ceiling caved in. Right above Maverick. The fiery debris rained down on him, taking him to the ground.
Alexa screamed and lurched toward him, thinking of nothing but needing to help him. She grabbed the flaming Sheetrock and lumber and pushed it off of him, the fire lashing at her skin and making her cry out.
"I'm okay," he said, crawling through the debris. Together they beat at his cut and shirt and jeans where they'd caught fire. "I'm okay." But nothing about his voice sounded okay.
"Mav'rick," she moaned, tears from the smoke streaming down her face. And from her realization. "We . . . have to go . . ."
He pushed onto hands and knees. Then stared down the hall. "Fuck, Al, look." He pointed.
And, oh, God, it was her mother, illuminated by the spreading flames. She was sprawled on the floor in front of her bedroom door.
"Stay back," he said. "I'll get her." Maverick rose but stumbled, clearly hurting. He used the wall for support until he finally got to her mom. As she watched, Maverick grabbed her mother by the legs and pulled, stumbling a little as he moved. It felt like forever until he returned to where Alexa waited.
Feeling a little light-headed, Alexa retreated toward the door, almost immediately tripping over something on the floor. She felt around with her hands, finding junk everywhere. She had to try to clear a wider path for her mom. Her chest felt like it could explode from the clenching tightness, but none of that mattered. She grabbed what turned out to be a broken metal table lamp and used the round base of it to shove and push at the piles.
"That's good, baby. Go out. Go on now." Maverick stumbled again.
"I'll help," she said, not sure if he could hear her. She came around and grabbed her mother's arms. Made it a few steps. But dizziness washed over her. She went down to her knees. Damn asthma. "Take her," she yelled as loud as she could. "And come back for me."
"Fuck, Alexa! Follow me, baby. Keep moving." It was like he found a new source of energy, because he was suddenly upright again and pulling her mother hard. "Come on, Alexa!"
She tried. Managed to get one foot underneath herself again. But her breaths were shallowing out and her chest hurt so bad.
"Goddamnit. I'll be right back, baby." He disappeared into the foyer with her mother. Relief flooded through Alexa, because they were getting out. Oh, God, they should be outside by now. The thought made her try to crawl again.
_Just keep moving, Al. Mav will be right back to help_. She got her other foot underneath her. There.
A voice. There he was, standing in the doorway to the foyer. She held out her hand and looked up, blinking. Her eyes stung like they were filled with crushed glass.
"Imagine my delight at seeing you two arrive. And here I'd thought I was only going to get the satisfaction of hurting your mother."
Grant?
Rage and hatred surged through her, jolting her to action. She got her feet under her, then finally stood up. "You are . . . a . . . sick . . . bastard," she managed.
"I've actually heard that before," he said with a shrug. "I didn't much like it back then, either."
She glared at him, the firelight throwing demonic shadows across his face. "Why are you doing this?" She forced herself upright, hoping he couldn't tell how unsteady she was. The heat was nearly unbearable.
He held up the neck of a broken bottle, then tossed it aside. Which made her wonder where Maverick was . . . and why he wasn't coming back. Dread snaked ice through her veins. "Because I can. Because I have to," Grant said, an odd flatness in his voice. She shook her head, trying to make sense of the man before her. But failing.
And the fact that Maverick still hadn't come back? Meant Grant had hurt him. She had to face that fact, didn't she? Fury unlike anything she'd ever felt before blasted through her until it became her blood, her cells, her very DNA. She was going to kill him. She was going to revel in it. Even if she died herself.
_Oh, God, Maverick! Please, be okay!_
Turning, she picked up the lamp again.
He shook his head. "You can barely breathe, let alone lift that thing, Alexa. Face it, I've won. I told you to just give in and do things my way. Now, _this_ is my way."
Maybe he had won. Probably. But she moved closer, closer, struggling for enough air to think. But what she was doing didn't require thought. Only instinct. To avenge. To survive.
Lights suddenly flashed outside, and Grant glanced to his side. Which was when Alexa swung the heavy metal lamp with all her might.
The lamp crashed into his right kneecap, and the crunch was a sickening thing. He let out a howl and fell forward, his knee at an unnatural angle. He tried and failed to catch himself, then tripped over the mess covering the floor and went down hard. He moaned and shifted, and she couldn't tell if he was conscious or not. Either way, he was down and her path was clear.
Gasping for breath, she looked at the fire quickly engulfing the living room and closing in on where he lay. And left him to it.
Stumbling to the foyer, Alexa could barely walk. And then she found Maverick lying face down across the threshold of the open front door, something wet darkening his hair. He'd come for her, just like he said. And Grant had hurt him. Oh, God, let him be okay.
But she couldn't focus on her worries just then. She had to get them both out of there.
Using every last piece of herself she could pull together, she stepped clumsily over him and grabbed his feet. She got him clear of the door, and then the world went wavy around her. Wavy and fuzzy. And then there was nothing at all.
## CHAPTER 30
Maverick came awake on a gasp that hurt like hell. His throat felt like it'd been scoured with coarse sandpaper, and his body felt like someone had dropped a house on it. He'd been in and out of it a hundred times, but never been able to keep his hold on reality.
_The fire. Something . . . hit me. Alexa!_
On a moan, he sat up, his hands clumsily pulling at tubes and wires.
"Hey, hey, none of that," a nurse in pink scrubs said, gently stilling his hands. "Just calm down, hon."
"Alexa," he rasped, his voice like gravel. "Alexa Harmon."
"She's right next to you, Mr. Rylan." She pushed the curtain back and revealed an unconscious Alexa propped up in a bed identical to his. Only she had a tube in her mouth and her hands were fully bandaged. "You two have got a lot of, uh, friends out in the waiting room and they pushed hard to get you in the same room. Miss Harmon is gonna be fine but her situation is a bit more serious right at the moment, so I need you to let her rest or I have to move her." She arched a brow at him.
More serious? The pain in the center of his chest had nothing to do with the smoke he'd inhaled. "What's wrong with her?" he asked, his heart fucking breaking. The nurse looked at him like she was debating, so Maverick pushed. He glanced to her name tag. "Erica, I'm going to marry that woman. Please, tell me."
"She had a severe asthma attack exacerbated by the smoke inhalation. Her lungs didn't sufficiently respond to anti-inflammatories, so we had to intubate her to stem the possibility of respiratory failure. Once her lungs are recovered and able to work on their own again, we'll be able to remove the tube. Probably in the morning. She also has burns on her hands and forearms.
Christ, from pulling the burning pieces of ceiling off of him, he bet. Sonofabitch. "But . . . but she's going to be okay?" he asked, needing to hear her say it again.
"She will. You both will. Your burns are minor and your CT scan was clear. You just have a pretty killer bump on your head."
He put his hand to his hair and flinched. Bump? That thing was a fucking mountain. "I don't know how that happened."
She frowned. "When you think you're ready, there's a police officer here who'd like to talk to you. He might be able to tell you more. Want me to send him in?"
Maverick nodded, and she turned to go. "Wait. Alexa's mom. Cynthia Harmon?"
Erica returned to his bedside. "I'm afraid her mother was in severe respiratory distress from prolonged smoke inhalation, and she also had a bump to the head. The doctors are hopeful for a recovery, but we'll know more in the next twelve to twenty-four hours."
"Fuck. Okay." Maverick shook his head, his thoughts a wreck. "Sorry for the language."
She smiled. "Don't worry about it, hon. You've had a rough night." She disappeared out the door. Martin came in shortly after.
"Christ, Maverick. You look like hell," Martin said.
"Fuck you, too," he said.
"We found you and Alexa lying together on the front porch of her mom's house. It looked like Alexa pulled you out before passing out herself."
His gaze cut to her. She'd pulled him out?
He shook his head. "I don't remember. Something hit me, and then—" He shook his head again. Flashes of disconnected images came together, but he couldn't piece them together to make any sense.
Martin sighed. "Grant Slater hit you. With a liquor bottle, as best we can tell. The house is a total loss, so it's going to take us a while to sift through all the evidence. But if I had to guess, I'd say he set the fire."
Rage was a living beast inside Maverick, stalking around and demanding release. "Goddamn Slater," he bit out. "I—"
A moan.
His gaze whipped back to Alexa. Her eyes fluttered, her wrapped hands struggling to move.
"I'm here, Al. I'm right here." He tore the oxygen mask off of his head and slid out of bed. His back and shoulders hurt like a motherfucker, so he grasped the IV pole and used it for support.
Martin reached out a hand. "You sure you should—"
"Try to keep me away from her." He limped to her bed and gently sat on the edge. "I'm here, Alexa. You're safe. I'm so fucking sorry, baby." If he'd taken Slater out at the house after he'd attacked her, this never would've happened. So much for playing it by the books. And trying to be the good guy.
Her eyes swam, then finally fixed on him. He could tell the moment her mind snapped to some semblance of awareness. She moaned again as panic cut lines into her face.
"Stay calm, Al. You're pretty banged up. You burned your hands helping me and your lungs need some time to recover from the severe asthma attack and smoke inhalation. The nurse said the tube can probably come out in the morning." His thoughts raced to what else she needed to know. "Your mom's here, too. She also had pretty bad smoke inhalation. That's all I know so far."
She pointed to his head, her expression full of questions.
"My head?" he asked. She nodded. "I'm good."
She frowned.
"Alexa," Martin said, "do you know how Maverick got hit on the head?"
Her face went white and she moaned, frustration clear in her helpless gestures.
Martin held up a hand. "How about this? I say something, and you just shake your head yes or no?" She nodded.
"Did you see Grant at the house?" She nodded. Surprise flooded through Maverick. Grant had been there the whole time? He looked to Alexa, confusion swamping him.
"Did you talk to him?" Another yes.
"Did he admit anything to you?" Another yes.
"Did he admit or did you see him hit Maverick?" She shrugged, but her expression said she had more to say on the topic. But Maverick knew—the fucker had totally jacked him up. Not only had Maverick not seen him, but he had no memory after the hit.
"Was he inside the house when you came out?" Martin asked.
She nodded. Did that mean . . .
"Did you find a body?" Maverick asked.
Martin nodded. "No positive ID yet. But one deceased in the house."
Slater was dead? Mav's mind reeled, shock and relief warring through him.
Erica pushed back into the room. "Mr. Rylan, what the heck do you think you're doing? Get back in bed and leave Miss Harmon alone."
"Ma'am—"
"That's an order. The rest of this can wait until the woman can fully breathe on her own." She nailed both of the men with a no-nonsense stare.
Maverick pressed a kiss to Alexa's forehead. "I'll be right over there, baby."
Al watched him move away, sadness clear on her face.
"What about this?" Erica said, gesturing at his bed. "Get in, Mr. Rylan."
"Maverick," he said, climbing back in but restless as hell over the ten feet separating them.
"Okay, Maverick. I see how it is, so I have an idea you might like." In that moment, he couldn't do anything but obey her; he was still too shell-shocked by the news that Slater was actually dead. He laid his ass down on a groan. Erica did some rearranging and pushed their beds closer together, leaving just enough narrow space for someone to walk between. "Better?"
"Yeah," he said. "As good as can be in the midst of all this."
"Sometimes that's all you have," Erica said, turning on Martin. "Officer, you're going to have to come back."
He nodded. "I'll let everyone know you're both awake. They're all here."
"Thanks," Maverick said, the comfort of having his brothers, his family, around easing some of the turmoil from the night.
"Now both of you get some rest," the nurse said, dimming the light.
He turned to Alexa, whose big, wide eyes were on him. "Christ, I love you, Alexa." She laid her bundled hands on her heart and looked at him, so much love shining from her eyes. Just then, that was all he needed. Because he had more to say. So much more. But he'd wait til she could speak, too.
Sleep came to him fitfully. His body ached, his worries focused on Alexa, and his mind kept wandering to any one of a million questions. Had Grant been at the house when Cynthia arrived? Or had he surprised her by arriving while she was packing her things? And why had Grant gone to the house in the first place? What had been the point? Maverick just didn't know. And maybe now they'd never find out.
When he finally did fall, he had dreams over and over again about getting hit in the head just as he'd been on the cusp of running back into the fire to bring Alexa out. He went down hard and stupid just inside the doorway every time, then the dream ended abruptly, making Mav want to scream his frustration.
Was that all that had happened? Was there more? Maybe he'd never know.
And with that, his brain finally shut down. Because his woman was alive and safe. And the man who wanted to take her away from him was gone for good. Just then, that was all Maverick could possibly ask for.
ALEXA DIDN'T THINK anything could surpass the relief of learning from Officer Martin that Grant had died, but then morning had come and they'd removed her breathing tube. And the sheer relief of being able to _breathe_ gave that news a good run for its money.
Her lack of grief or even remorse over Grant almost made her wonder if there was something wrong with her, but how could you grieve the loss of someone who only wanted to hurt you, to own you?
She couldn't. That was for sure. Not when her mother still lay unconscious, her prognosis unclear. Alexa felt like she stood at the edge of a tall cliff, her heart and soul set to shatter when it hit bottom, and she wouldn't know whether she'd get to stand or fall until her mother's situation was clearer. After everything she'd tried to do, she hadn't been able to take care of her mother in the end.
How would she ever forgive herself for that?
After getting some scans done, Alexa got wheeled in her bed back to their room, but voices from inside had her asking the orderly to wait a moment.
"I'm so fucking sorry, Mom." Maverick, the sound of his strained, smoked-scoured voice breaking Alexa's heart.
"Oh, son. You have nothing to apologize to me for," Bunny said, her voice strained, too.
"I do. I didn't protect you from my father. And now I've let Alexa get hurt. It's like I can't stop fucking failing the people I care about."
Alexa wanted nothing more than to go inside and set Maverick straight, but she didn't want to intrude on their conversation.
"You listen to me, Maverick Rylan," Bunny said, her words strong and full of certainty. "I _hid_ your father's abuse from you. On purpose. So you _wouldn't_ know. But the only one of the three of us to blame was your father. You couldn't have changed a thing until I was ready to get out. Tell me you hear me. Tell me you believe it."
A long pause.
"Tell me, hon."
"Fuck," Maverick said. "Okay." The conversation seemed to cut off, or the words they spoke were too quiet to share.
Alexa finally nodded to the orderly, who nodded in return and wheeled her into the room.
She found Dare in a chair beside Maverick's bed and Bunny standing on the other side holding Mav's hand. And, oh, seeing Maverick's incredible dark blue eyes, awake and aware and alive, brought a whole new wave of relief. When she'd dragged his dead weight out the door last night, she'd been afraid he might never look at her again.
"There she is," Bunny said, a big smile on her face.
"Hi," Alexa said, unable to take her gaze off Mav. But having his mother and cousin there was special, too. She never had a big family she could rely on before—and the Ravens were certainly that. Loyal, dependable, protective. Way more than just a club.
Now if her mother would wake up, everything would be as close to perfect as she could ever hope for.
"How are you?" she asked Maverick. She could see him sitting in front of her, of course—living and breathing. But she'd never forget the terror of learning that Grant had hurt Maverick, that he'd wanted to see him die. She'd never be able to unsee the picture of the blood streaking Mav's hair.
"Just some achiness. How about you?" Maverick asked, his eyes locked on her.
Her chest felt like an elephant had sat on it, and maybe was still there. "Same," she said, her voice sounding sultry and low, the result of all the smoke and the tube.
"You're both goddamned liars but I'm so fucking happy you're okay that I don't even care," Dare said. "What the hell happened?"
She and Mav exchanged a look, and then he launched into it.
When he stopped, she jumped in with the things he didn't know. "After you took my mom out, Grant came in and just started ranting. He admitted to intending to hurt my mom and us, too, once we showed up," she said, looking at Maverick, the memory of the terror and despair she'd felt upon realizing he must've been hurt boomeranging through her.
Pain flared in her chest. If he hadn't survived, she wasn't sure she could've, either. Not when she'd just found her way back to him after so long, and not when she loved him so damn much.
"I was so beside myself that he'd hurt you. I grabbed a metal lamp and hit him in the knee so hard it crunched and he fell." She shuddered, a confession welling up inside her, then spilling free. "I . . . I probably could've helped him, but I didn't. I left him."
"Good," Bunny said, that one word so filled with anger that it vibrated through the room.
"I agree," Dare said.
Alexa released a shaky breath and looked at Maverick, and the emotion shining from his eyes reached inside her chest and soothed so much of her pain.
"Shit, I don't remember anything after taking your mom out. But you were so strong, Alexa. You saved us," Maverick said.
Dare braced his elbows on his knees, his gaze going serious. "I hope you don't feel a second of guilt, Alexa. He wasn't behaving rationally. He would've kept coming at you, and he made it into a situation where it was you or him. And the P.I. learned something that I think maybe confirms that." His dark gaze moved to Maverick, then back to her again. "Did you know that Slater spent time in a psychiatric hospital when he was nineteen?"
Alexa gasped. "What? No." She shook her head, barely able to process the information.
Dare nodded. "All the P.I. discovered so far was that it was an involuntary admission. Lasted ten weeks. When Slater got out, he moved away from his hometown and, as best as can be told, never returned."
Shell-shocked, she glanced at Maverick. "That must be why he had no relationship with his parents. Wouldn't talk about it, either." A barrage of questions flooded through her that all boiled down to one—what had Grant been hiding from her? Because Dare was right, Grant had changed these past few weeks from a man always in tight, strict control to one completely out of it. Either way, it really was like she'd been with a stranger all those years. "I don't even know what to say."
"If the P.I. finds out more, I'm sure he'll tell us," Maverick said. "But the most important thing is that it's finally over." He shifted in his bed. "And damnit I want out of this bed."
Dare let loose a small laugh. "Welcome to my world."
Maverick scowled at his cousin, making Dare laugh harder.
"Why don't we let you all rest?" Bunny said, coming to Maverick's bedside. She leaned down and kissed his forehead, and it was the sweetest thing. "You're my favorite son. I don't know what I would've done . . . Well, I'm not even going to entertain that now."
Mav swallowed hard, Bunny's words clearly getting to him. "I'm your _only_ son, Bunny."
She managed a small smile and winked. "That's why you're my fave." Next, she came around to Alexa's bed and placed a hand on her shoulder. "I wish I would've had half your strength at your age. You hang in there and take care of each other."
"I will, Bunny." Now she was the one Bunny was getting choked up. "Thank you."
Dare said his good-byes next. Clasping Mav's hand, he said, "This was some fucking extreme lengths to go to in order to get out of your turn in the dunking booth."
Maverick barked out a laugh, and the sound was so amazing to her. Proof of life. Proof of truly living. "Silver linings, man. You gotta embrace 'em where you find 'em."
"Truer words," Dare said, then he came around to Alexa, and leaned down over her so he could look her in the eyes. "You did good, kid."
She blinked back tears and gave a quick nod. And then they were gone, and she and Maverick were finally alone.
"I'VE HAD ENOUGH of being apart from you," Maverick said, climbing out of bed and shuffling over to her, holding on to his IV stand. A pale blue hospital gown was all he wore, but he didn't care about any of it. About anything other than touching Alexa again.
"Are you naked under there?" Her expression was almost playful, and it made him want to distract her, to help her find a moment of light in all this dark.
He chuffed out a laugh. "You really feeling up to sex?"
Alexa grinned. "No, damnit. But I thought your ass might look cute sticking out."
He shook his head, and then he turned around for her and gave her his bare ass. "Happy?"
"Wow. That is . . . yeah. I'm super happy."
He chuckled and turned back to her.
"I feel _so much better_ now," she said. "I might need to see that every once in a while. You know, for purely medicinal purposes."
"You're so full of shit," he said, but he enjoyed her teasing, enjoyed seeing her come back to life, a little at a time. He put down the side railing to her bed. "Let me in." It was a little awkward with her bandaged hands, but she managed to scoot aside enough to make space for him. His back was one giant bruise from where the ceiling had collapsed on him, so he settled on his side facing her. Fuck, that felt good. Satisfying. Exactly where he belonged. "That's better."
"Yes, it is," she said, her breath catching as the smile slid off her face. "I feel happy and sad at the same time."
He cupped her cheek, his thumb brushing under her eye, but there were no tears. "I know, baby. Perfectly reasonable. I'm so sorry your mom is in such bad shape."
"I think I'm going to lose her, Mav. I feel it."
"Don't say that. The doctors don't know anything for sure yet," he said, even though none of the doctors sounded any more hopeful or positive about it than they'd been the night before. Mrs. H had just been in that smoke for so long. "And if the worst happens . . ." He swallowed around a lump. A lump caused by the fact that the first time he'd lost Alexa, it was because she'd lost her brother. What would losing her mother do? "We'll handle it together."
She nodded.
"Promise me," he said.
"Of course we will." She tilted her head. "What's the matter? You look upset."
"It's fucking selfish, given what we're talking about." But that didn't keep him from feeling the anxiety. Christ, he didn't want to have found her again just to lose her so quickly once more.
"We can say anything to each other. Remember?" She rested her bandaged hands against his chest.
"If the worst happens, Alexa, you have to promise to let me help you through it this time. Please." His heart thundered against his breastbone, the heavy beat from a mix of need and fear and worry.
Her breath caught. "Oh. Oh, God, Maverick. You're worried because my grief over Tyler drove me away from you before?" He nodded, a knot lodged in his throat. "I promise. I won't do that again. I'm not that girl anymore. And I've learned so damn much about what love is, and what it isn't."
Relief crashed over him so hard he shuddered. "Like what?"
Her expression went soft and thoughtful. "Love . . . love is the only thing that doesn't hurt. So many people are afraid of it. I think I was five years ago. Afraid that loving you would hurt me, one way or the other. But love doesn't hurt. Abuse hurts, and loneliness hurts, and jealousy hurts, and lies hurt. But love? Love heals. Love puts back together things that were broken before. Which is what you've done for me. So promising you, that's easy. I don't want to be anywhere else but with you. No matter what."
Damn if those words didn't resonate all the way to the bottom of his soul, and make him want things he never thought he'd have. Want them _right fucking now_.
He leaned his forehead against hers. "Marry me." The words almost took him by surprise, but they felt more right than maybe anything he'd ever said. Well, besides that he loved her. "It doesn't have to be right away, but marry me. Say you'll be mine. That we'll ride through this life together, you and me against the world, just like we said. Say you'll let me take care of you and that you'll take care of me, too. Laugh with me and joke with me and fight with me and share your weird cat with me, and _marry me_ , Alexa."
Tears made her eyes go glassy, but her smile was like the sun coming out after a storm. Bright and hopeful. "Yes, I'll marry you, Maverick. You are my strength and my home and my truth. Nothing would make me happier than being yours."
"Forever," he said, triumph making his blood sing.
"Forever," she whispered, the first tears slipping free of her pretty eyes.
He kissed her then, a soft pressing promise of lips on lips, and he managed to pull her into his arms. "I didn't exactly plan this today, and I know I need to get you a ring—"
"I don't care about a ring, Mav. I mean, I want to wear your ring, whatever it is, but your heart's the only thing I need." She kissed him again.
"Well, you have that. Because you are my friend and my love and the life I always wanted, Al. My everything." And, damn if that didn't give him an idea for a tattoo. Which made him think of the black-and-purple tribal raven with the red heart at its center. Seeing it on her skin someday was going to bring him to his knees. He already knew that was true.
"Oh, that's perfect, Maverick. Because you're my everything, too."
## EPILOGUE
Welcome home, Al," Maverick said, the next afternoon. He had his arm around her shoulders, supporting her weight and helping her across the threshold to his little lake house. They'd been released from the hospital an hour before, but had bowed out of the Saturday afternoon cookout being planned at the Ravens' clubhouse before the final night of the carnival began. Alexa's chest still hurt if she undertook much activity, so that was all more than she was up to just yet.
"I'm so glad to be home," she said. Home. God, that sounded nice. One of the things she'd always wanted but never really been able to find. Lucy came rushing right up to her. "Hi, baby. I bet you were lonely without us."
"Phoenix came by to feed her and keep her company. I told him to line the toys up on the tub to see what she did. He said we can ask him to Lucy-sit any time." He lowered her onto the couch.
She chuckled, her hand rubbing the cat's thin body when she jumped up. "Look at all the new friends you're making."
Maverick grinned. "What can I get you?"
"Nothing at all," she said. "Just be with me."
"My favorite thing." He grabbed the remote and put on HGTV, then settled in next to her.
"Ooh, I adore _Love It or List It_ , even though the couple almost always does the opposite of what I would do." She shifted against him, her head finding his shoulder. He put his arm around her, and she felt like she could've purred herself. Besides Jagger being freed, the only thing that could've made this moment better was knowing her mother was well, but there still hadn't been any change in her condition. All Alexa could do was hope and believe.
Mav propped his boots on the coffee table. "I enjoy the rebuild projects. Finding a new home doesn't feel as special as making the old thing you know and love grow with you and work for you in ways you never thought of before."
She nodded, loving how he phrased that. It felt like he was talking about them. And maybe he was. Honestly, what relationship wasn't a fixer-upper? At least the roughest parts of their ride were behind them.
She hoped.
Hours later, Maverick woke her up with a big tray of food. "I made us some dinner."
"You're going to spoil me."
"You're going to be my wife. There's no such thing." He held up a spoonful of mac and cheese to her mouth, since she still didn't have full use of her hands. Part of her was eager for the gauze to come off, but the other feared how bad all that damaged skin was going to hurt when it was exposed to the world. One day at a time, she guessed. "Want?"
She looked past the food to his face. "Want."
He chuckled and winked. "I'm not sure either of us is up to wanting."
Alexa grinned and took a bite. "Probably not. But mac and cheese hand-fed by my man is pretty good, too."
He shook his head and enjoyed a bite for himself.
Something pinged around in her brain that she'd been avoiding all day. She finally gave it voice. "Today . . . today would've been my wedding day."
"I know," he said, grabbing another spoonful to offer her.
"Know what I was thinking?"
His gaze cut to hers, the dark blue so intense. "Tell me."
"Maybe all the bad stuff was supposed to happen to bring me back to you."
He gave a tight nod. "I hate that bad stuff happened to you."
"I have no regrets, Maverick. I want you to know that. Except maybe walking away five years ago."
"Yeah? Well, I've been thinking about that. I've been thinking that neither of us was ready for us five years ago. You pushed me away and I let myself be pushed. Would you do that now?" He gave her a searching stare.
"Absolutely not." She shook her head.
"Me, neither."
They ate the rest of their dinner while watching _The Fast and the Furious_ , and then Alexa was too tired to stay awake any longer.
Mav helped her get ready for bed, and got changed himself. And then she walked into the bedroom—and burst out laughing. "This is perfect!" The llama and hedgehog sat on the bed.
"This is your home now. _Our_ home," Maverick said. He pointed around the room. "Our bed and our closet and our bathroom and our dresser. I want you to move in, rearrange the furniture, hang pictures, lay your stuff out on the bathroom counter, and leave Lucy's toys in the tub. Whatever makes you feel at home, I want you to do."
"I will, but I don't have to do any of that to feel at home. That's you, Maverick. Just you." She wound her arms around his neck, careful not to put too much pressure there.
"Fuck, I love you, baby." He kissed her on a groan. "Good, then climb into _our_ bed and let me hold you." It took a moment to get situated, but then she was all snuggled up to the long nook along the side of his lean body.
And that's when Alexa knew— _this_ was what being home really was. Finding love and belonging and family with one person, one person who could accept her mistakes and hold her tight and be her biggest cheerleader.
Maverick was that for her. She knew that for sure.
And that was everything.
## ACKNOWLEDGMENTS
I have a lot of people to thank for this one because sometimes books are hard, like this one was, and those are the ones that take the most support to finish. I have to start by thanking Lea Nolan for all the brainstorming, plotting, and hand-holding, and Christi Barth for reading the draft. I also have to thank my writing partner Stephanie Dray for her incredible patience, and Amanda Bergeron for the same! I also have to thank Amanda for the incredible encouragement she gave me by loving the book, and especially Maverick, as much as I do. It's an amazing experience to work with an editor who gets your worlds and your words and your characters so much, and I appreciate it to the bottom of my heart.
Next, I need to thank my wonderful agent, Kevan Lyon, for everything she did to help me get this one done. And my amazingly supportive husband and daughters for doing so much for me to give me the time I needed to write. I love you guys!
And I also need to thank my Heroes and my Reader Girls for the excitement you all had for the early excerpts I posted from the book—each and every one was a little boost of encouragement and motivation! It's amazing any time a reader mentions that a certain book of yours is their favorite, but a special thank you to every reader who told me _Ride Hard_ was your new favorite, because that provided me with such wonderful motivation when the going got tough on this one. I hope you find another story that grabs your heart within these pages.
Finally, a thanks as always to all the readers, who take my characters into their hearts and let them tell their stories over and over again.
~LK
## **EXCERPT FROM _RIDE WILD_**
**Don't miss the next Raven Riders novel**
RIDE WILD
**Coming Fall 2017!**
**Preorder it today.**
**Wild with grief over the death of his wife, Sam "Slider" Evans lives for his two sons. Nothing else holds his interest anymore, not riding his bike, not even his membership in the Raven Riders Motorcycle Club. And then he hires a new babysitter. . . .**
**Recently freed from a bad situation by the Ravens, Cora Campbell is determined to bury her past. When Slider offers her a nanny position, she accepts, needing the security and time to figure out what she wants from life. Cora adores his sweet boys, but she never expected the red-hot attraction to their brooding, sexy father that makes her wish he'd notice her. Just once.**
**Slider sees the beautiful, fun-loving woman he invited into his home. She makes him feel too much, and he hates it and yearns for it, not sure which he feels more, until Cora witnesses something she shouldn't have. Now, the new lives he and Cora have only just found are on the line, and he must claim and protect what's his before it's too late. . . .**
**And keep reading to enjoy the bonus Hard Ink novella**
HARD EVER AFTER
## CHAPTER 1
I don't want to let you go," Nick Rixey said, lying on his back in bed with Becca Merritt sprawled half on top of him. Dawn was already around the corner, but neither of them had slept much. At least he wasn't the only one who was a little stressed out about life returning to normal. You'd think it would be the opposite. But it's funny how your body and brain could get used to operating in crisis mode—and resist the reality that the crisis was over.
Becca traced a ticklish pattern across his chest with her fingertips. "I'm just going to work," she said. "I've been off for two months. It's time."
"I know," he said, pressing a kiss to her soft blond waves.
"But we might have time for a proper good-bye before I have to go," she said, a smile plain in her voice.
Nick couldn't help but grin, and Becca was a big part of why that was so much easier to do these days. "Might we now? And what would a proper good-bye look like?" Against his belly, his cock hardened.
"Mmm, a back rub might be nice," she said, her tone full of teasing.
"That's all?" He reached down to stroke his cock, and her head shifted, following the movement. And fuck if her watching him jack himself didn't get him even hotter. How the hell was he supposed to let her out of his sight for a twelve-hour shift at the hospital after they'd been around each other twenty-four/seven for most of the last two months?
"Maybe this, too." Becca slid down his body and joined her hand with his. Her tongue swirled around the head of his cock and her lips sucked at the tip as their hands moved together.
"Fuck, Sunshine. Take all of me," Nick rasped.
She pushed onto her knees, her face so filled with desire—for him—that it blew his mind. Every damn time. Looking him in the eye with those pretty baby blues, she leaned over and swallowed his length inch by maddening inch until he was buried in the back of her throat.
His hand went to her hair and his hips surged. "God, yes." For a few long moments, he savored the pleasure she was pulling out of his body with her lips and tongue and hands. And then he cupped her cheek in his palm. "Lay down," he said.
Smiling at him, Becca stretched out on her belly and laid her head on folded hands. Nick straddled her upper thighs, his cock rubbing against her ass as he smoothed his palms over her beautiful golden skin. To think that for most of the past year he'd been lonely, purposeless, angry, battling with demons so numerous it was hard to believe that they'd mostly been slayed. And it was Becca's walking through the door of his tattoo shop that had put him on the road to getting justice, having his honor restored, and clearing his name—not to mention finding love.
Becca Merritt had saved his life in every way that mattered.
Because of her, he had purpose again in a new business. His surviving Army Special Forces teammates—his family by choice if not by blood—were back working and fighting at his side. And he had hope for the future in a way he hadn't in a very long time, not since a roadside ambush in Afghanistan had killed half his team, sullied the name of his mentor and commander, and ended in the other-than-honorable discharge of Nick and his team. Not to mention injuring him and two of the other survivors. Although, shit, they'd all walked away with soul-deep wounds even if they hadn't been visible to the eye.
But that was all behind them now. After weeks of investigations, fighting, and off-the-books operations against a shit-ton of enemies, he and his team had come out on top and the bad guys were either dead or in custody. And now Nick could see the future stretching out in front of him. And this woman was at the very center of his vision.
"How's that feel?" he asked, kneading at Becca's shoulders. The position tugged at the lingering injuries he bore in his hip and lower back, where he'd taken two rounds during the ambush, but fuck if he was letting his body ruin this perfect moment of peace and closeness between them.
"Really good," she whispered.
He worked his way down her spine in a series of slow, deep massages. When he got to her ass, he pressed his cock between her cheeks and slowly stroked himself against her. Becca moaned. And Nick couldn't wait anymore. He had to get in her. "You wet for me, Sunshine?" He slipped his fingers between her legs and found her hot and slick and more than ready. "Aw, fuck, yeah you are."
"Always wet for you," she said, arching her back.
Straddling her thighs, Nick penetrated her, both of them moaning as he slid home.
"Oh, God, Nick," Becca rasped.
Grasping her cheeks, he began to move, his eyes trained on where he disappeared inside her. "So good together. Every damn time."
"Yes," she said, arching into him, urging him faster, harder.
Nick braced his arms against the bed and gave her everything he had—his body, his heart, his soul. The sounds of panting breaths and pleasured moans and slapping skin filled the gray-lit room. "Touch yourself," he said. "I want you to come on me."
Becca slid a hand between her legs, and Nick closed every bit of distance between them, his body coming down and covering her from head to toes. He wrapped one arm around her shoulder and another around her head, his fingers tangling in all that beautiful, soft blond hair. "Love you so fucking much," he rasped into her ear.
"Nick," she whispered, her tone tortured and needy. Her movements grew jerky and desperate beneath him, and then she was moaning and coming, her body fisting around him and stealing his sanity.
"Aw, God, Becca," he groaned as his orgasm nailed him in the back. On a series of punctuated thrusts, he sought to get as deep as he could as he poured himself into her.
When their bodies calmed, Nick shifted, but Becca grasped at his hip. "Don't leave yet."
"Don't want to crush you," he said, kissing her temple.
"I like the feeling of you on top of me."
Nick chuffed out a small laugh. "Keep talking like that, and I'm gonna get hard again."
Becca smiled. "I wouldn't complain."
His head resting against hers, Nick sighed. Contentment. Such a foreign feeling. And yet he found it in Becca's arms. And had, from the very beginning, even when he'd been too stubborn and too proud to see everything that she was.
And that was the moment Nick knew what he was doing during Becca's first day back to work. He wanted her in his life. He wanted that life to start now. Hell, to start yesterday. And he wanted it to be forever.
And that meant he needed a ring. The rightness of the idea settled bone-deep inside him. As much as anyone, he knew how life could change in a single unexpected instant. No way did he want to wait even a second more for their future to start.
"Nick?" she said, pulling him from the plans taking shape in his head.
"Yeah?"
"Will you do something for me tonight?"
He finally shifted off her, his body settling alongside hers so he could look at her while they talked. "You can always assume the answer to that will be yes, Sunshine," he said, brushing her hair back off her face.
"I've been thinking about it, and I'd like you to give me a tattoo tonight," she said, bright blue eyes looking up at him with so much warmth.
The request sent his heart beating a little faster. Nick was half owner of Hard Ink Tattoo, though he'd only been working as a tattoo artist on a part-time basis since he'd been discharged from the Army. "You know I've been dying to put my ink on you," he said with a smile.
She grinned. "Well, now's your chance." She pointed toward the drawer on the nightstand. "I printed something out to give you an idea."
Nick couldn't move fast enough. After all the times they'd talked about what she'd want if she ever got a tattoo, and after all the times he'd drawn on her body with skin markers just to put his mark on her—even if only temporarily—he couldn't wait to see what she'd finally decided she wanted on her skin. Forever. He sat on the edge of the bed and unfolded the sheet while Becca knelt behind him, her front pressed tight to his back, her arms wrapped around his stomach.
It was three intertwined cursive words.
_Only. Always. Forever._
"Fuck, Sunshine," Nick said, remembering the night he'd shown her the tattoo he'd gotten on his forearm for her and she'd written the word _YOURS_ over her heart with a marker. That had set off a raw, urgent lovemaking that had included them writing words of claiming and love and intention all over each other.
Only. Always. Forever.
"What do you think?" she asked.
"I think it's perfect," Nick said, staring at the page. Damn if his throat didn't get a little tight at the thought that she wanted to put their words on her body. "Do you know where you want it?"
She kissed the side of his neck and her breath caressed his ear. "On my right shoulder."
Nodding, Nick could already picture it—and it made him even more certain about what he needed to do today. "It'll look beautiful there, Becca." He shifted to the side so he could wrap her in his arms. "You make me fall in love with you a little more every day. You know that?"
"I think that's the sweetest thing anyone's ever said to me." Her kiss was slow and sweet and lingering. "I don't want to, but I should get moving."
"I know," Nick said, standing and giving her a hand off the bed. He watched her walk into the bathroom, his mind back on his plan for the day. Because he was giving her more than a tattoo tonight. And he couldn't fucking wait.
## CHAPTER 2
A few blocks away from the hospital, the nerves Becca had been shoving down all morning finally pushed through. Ridiculous to be nervous about returning to a place where she'd worked for years. But she was. Because the last time she'd been there, a man named Tyrell Woodson had grabbed her from behind, jabbed a knife into her ribs, and tried to abduct her from the staff lounge. Only her struggling—getting cut in the process—and Nick arriving, well, in the nick of time, had saved her from God only knew what horrible fate.
Even worse? The man had gotten away and tried to grab her again, though the team had caught him that time and made sure he wouldn't be a problem anymore. So Becca shouldn't be nervous. She shouldn't be worrying. And she certainly didn't want to let on to Nick that she was.
They caught the red light a block away from the hospital's downtown Baltimore campus, and Nick turned to her from the driver's seat. "I'll pick you up at seven, and then we can grab some dinner and head down to Hard Ink."
"Sounds like a plan," she said with a smile, looking forward so much to finally getting a tattoo—from Nick. He brought their joined hands to his mouth and kissed her knuckles. God, she loved this man. He'd insisted on driving her. Truth be told, she hadn't minded the extra time with him. It was going to be weird to go back to work in the emergency department after all these months off. All the people at Hard Ink had come to feel like her family now in addition to her brother, Charlie, who was in a relationship with Nick's younger brother, Jeremy. She was going to miss seeing them all the time.
Moments later, Nick pulled the car over to the curb in front of the hospital. "Have a great day, Sunshine," he said, leaning over the center console. His kisses made her want to stay. "Be safe."
"I will," she said, ignoring the butterflies in her belly. It really was ridiculous. "Miss you already." And then she was pushing out of the car and crossing the wide sidewalk plaza in front of the hospital's tall glass entrance. She'd purposely arrived close to the beginning of her shift so that she wouldn't have much time before she'd be busy, which she knew would be the perfect cure for her nerves.
A chorus of greetings rose up from the nurse's station of the emergency department. Becca made her way inside and gave a round of hugs. Luckily, things were busy enough with the shift change that no one had time to linger. She headed to the staff lounge to stow her belongings.
Alison Harding came out of the lounge just as Becca reached for the door. "Oh, Becca, it's so good to have you back," the woman said, a hint of sadness in her bright green eyes. Becca had been subbing for Alison the day the attempted abduction had occurred, and Alison had sent more than one guilt-ridden, apologetic text. Not that Becca blamed her. It was hardly Alison's fault that the undercover military investigation into narcotics smuggling that Becca's father had been investigating in Afghanistan had spilled over into the United States. Or that the bad guys had been selling their heroin to the Church Gang, headquartered just across the city in Baltimore. Or that somehow the bad guys had discovered that Charlie had stumbled onto his father's activities, leading them to grab him and attempt to grab Becca as well.
"It's good to be back," Becca said.
"How are you doing? Did the police ever catch the guy?" Alison asked, tucking a strand of light brown hair behind her ear.
"No, they didn't," Becca said, unable to share what she did know—that Nick's team had caught and interrogated Woodson, and that Nick had threatened the man within an inch of his life. "But I'm good. Really good."
Alison frowned. "God, it's scary that he's still out there, isn't it?"
Becca's belly did a little flip. "No, I really think he's long gone," she said. Marz had taken video of him spilling his guts about the Church Gang's secrets, which Nick had promised to put in the gang leader's hands should Woodson ever come near Becca again. Already beaten up for having failed to capture Becca, Woodson had tripped all over himself promising to stay away for good.
"Well, I hope so." Alison gave her an unconvincing smile. "All right. I'll see you out there." She squeezed Becca's arm and headed down the hall.
Taking a deep breath, Becca pushed into the room where she'd been attacked, worried that it was going to be filled with all kinds of ghosts. Instead, she found a big bouquet of balloons, a sheet cake that read, _Welcome back, Becca!_ and a plastic-wrapped bunch of flowers lying on one of the tables. The overhead lights and morning sunshine spilling through the window near the door—the door through which Woodson had tried to drag her—made the room bright and cheery, not the scary, dark place her nightmares sometimes depicted.
Shaking her head at herself, Becca crossed to her locker and ditched her purse. She made a small corner piece of cake with a big pink frosting flower her breakfast, then found herself so immersed in patients that it was noon before she knew it—and time for the other thing she wasn't looking forward to: an appointment with a hospital psychologist. It was standard operating procedure after the attack and the long leave of absence, but Becca wasn't relishing being asked to talk about what had happened. And she was well aware that medical personnel sometimes made the worst patients, herself included. She was way more comfortable taking care of others than being taken care of herself.
She waited in the fifth-floor mental health services suite. Finally, the door to the waiting room opened, and a tall, attractive woman in dress pants and a crisp blouse stepped out. "Becca Merritt?"
"Yes," Becca said, tossing the magazine she'd been skimming to the coffee table.
"I'm Dr. Parker," the woman said. "Please, come in."
Becca had seen her around the hospital a few times but didn't know her well. "Thanks," she said, slipping into the well-appointed office—all warm tones and relaxing landscape prints. She took a seat on the sofa.
The doctor grabbed a pen and folder from her desk, then sat in an armchair and smiled at Becca. "How has your first day back to work been?"
"Fine. Busy. But I'm right back in the swing of things," Becca said, lacing her fingers in her lap.
"Good, I'm glad to hear it. You know this meeting is routine. The hospital just needs to touch base, given the traumatic event that led to your leave of absence." Dr. Parker scanned a sheet inside Becca's folder.
Becca nodded. "I understand." Woodson had somehow managed to gain access to a set of hospital credentials and had posed as a maintenance man, so the hospital had been concerned that Becca would sue. But more than that, traumatic events could lead to bad decision making, which was never an acceptable risk when those decisions were of the life-and-death variety.
"So how are you doing? How are you finding being back in the hospital again?" The doctor's expression was carefully neutral, but Becca didn't doubt for a moment that her reactions were being scrutinized.
So she went for honesty. "I was a little nervous about coming in before I got here this morning, but once I was here, I was fine. As soon as the shift started, everything felt normal. So I think I'm doing pretty good."
Dr. Parker nodded. "I'm glad to hear it. Are you having any nightmares, anxiety, issues with panic, sleep or appetite problems?"
Becca clutched her hands tighter. "I've had occasional nightmares, and for a while I was jumpy if someone approached me from behind, but I haven't had any of the other issues." Frankly, given everything that Nick's team had faced during their investigation into the men who'd killed her father and abducted Charlie, Becca's issues had taken a total backseat. And she'd been fine with that. Because she _had been_ fine. And the last thing she'd wanted to do was distract or worry Nick by making him think she was anything but fine. Not when his life had been on the line—so many times. "The whole thing could've turned out a lot worse than it did, so I mostly feel lucky."
"That's a great way to look at it." The doctor scribbled something inside the folder. "Do you have any concerns about being back to work?"
"None," Becca said. "I'm glad to be back." And she was. She'd known she wanted to be a nurse since the age of thirteen, when her mother had died of an aneurysm. The feeling of helplessness Becca had experienced that night had made her determined to be able to help if something similar ever happened again to someone she cared about. She loved what she did.
After a few more questions, Dr. Parker handed her a form. "I'm happy to clear you to return," she said. "Just sign where it's highlighted."
"Great," Becca said. "Thank you." She signed and handed back the form, and then it was time for her lunch break.
Back in the staff lounge, she found a couple of people hanging out around the half-demolished cake. She was glad for the company and conversation as she settled down to the turkey sandwich, chips, and yogurt she'd brought from home, and she was equally glad to find that no one treated her weirdly despite the fact that everyone knew what had happened to her that day. Even if you could keep gossip that juicy under wraps around there, which you couldn't, the hospital had undertaken a security reevaluation and had implemented some new procedures and security mechanisms as a result. So her attack was no secret whatsoever.
Still, as the day progressed and patients were admitted in a nonstop stream, she found it easier and easier to relax. Finally, seven o'clock rolled around, and a bundle of anticipation took root in her belly. She couldn't wait to see Nick, and she really couldn't wait for him to do her first tattoo.
In the staff lounge, she collected her purse from her locker and gathered the flowers to take home. There wasn't much cake left, and she figured the night shift would easily finish it. The balloons made the otherwise plain blue-and-white lounge more cheery, so she decided to leave them there. They wouldn't fit in Nick's sports car anyway.
Not wanting to keep Nick waiting, Becca rushed across the room with her arms full. The door yanked open right in front of her, and a tall, bald man with dark brown skin stepped into the opening, looming over her.
Becca nearly choked on a gasp. Tyrell Woodson. For a moment, she was sucked back into the past so thoroughly that everything around her disappeared.
"Oh, sorry about that," the man said, his voice deep and friendly.
She blinked and swallowed hard. _Not Woodson. Holy shit, not Woodson. What's wrong with me?_ Becca forced a smile. "Oh, no. Not your fault," she managed. "I wasn't paying attention." He wore blue scrubs, not a maintenance uniform. And the identification tag clipped to his pocket read _Benton Tucker, Certified Nursing Assistant._ She stepped back to let him in.
He pointed at the flowers. "Are you Becca?"
"Uh, yeah," she said, her heart still racing in her chest. "How'd you know?"
"The cake. I had a piece earlier. When it still said your name," he said with a deep chuckle.
She smiled. "Right. Glad you got some, because it's almost gone. Free food never lasts long."
Another chuckle. "I guess that's right. I'm Ben," he said, extending his hand. "I've only been here for about a month."
"Nice to meet you, Ben," she said, returning the shake and feeling bad for the way she'd reacted to him, which had been not only ridiculous but also embarrassing. Not to mention a little concerning. For a moment, her brain had been entirely convinced that Woodson had been standing in front of her, despite the fact that Ben bore only a superficial resemblance to him. While both men were tall and dark skinned, Ben's head wasn't bald, like Woodson's, but was covered with closely trimmed hair. Ben didn't have any tattoos or scars, whereas Woodson had been covered in them. Ben's face was lean, and he wore a neatly trimmed goatee, where Woodson's face had been round, his cheeks full. And Ben radiated an easygoing good humor, not the menace she'd gotten from Woodson. "Well, hopefully our shifts will overlap soon. Hope you have a good night."
"You, too, Becca." He gave a wave and turned for the cake table.
Becca pushed out into the hall. After she'd successfully battled back her nerves all day, freaking out just because a man had stepped in front of her made her feel defeated and weak and stupid. And that pissed her off. She was stronger than this. And she refused to let a little anxiety get the better of her. Woodson was gone. The Church Gang had been largely destroyed. And Nick and his team had exposed the corruption that had led to her father's death and the team's being railroaded out of the Army. They'd also gotten the justice they deserved.
Everything was good now. The crises were all behind them.
Outside, July heat wrapped around her despite the evening hour, but the only thing Becca cared about was the man sitting in the black car idling at the curb. She rushed across the plaza, the smile on her face growing when Nick noticed her coming.
She couldn't get in the car fast enough. "Hi," she said.
"Sunshine," Nick said, the word filled with so much emotion it made tears prickle against the backs of her eyes. "Missed you." His hand found the back of her neck and pulled her in for a long kiss.
"Missed you, too," she said. His presence chased away the last of her nerves and allowed her to take a deep, cleansing breath. She was fine. No big deal.
He threw the car into gear and eased into traffic. "So, how was your day? Everything go okay?"
"Yeah," Becca said. "Everything went great."
## CHAPTER 3
You made Sloppy Joes," Becca said with a big grin when they got home. The rich, spicy smell of Nick's one and only specialty filled the whole loft apartment—and the gesture filled her heart with so much affection. He'd made her Sloppy Joes the very first night she'd spent there at Hard Ink, back when everything had seemed so uncertain, back when it had seemed like she might lose everyone she had left.
Nick grinned as he moved to the Crock-Pot on the counter and lifted the lid. "I did. Thought you might enjoy something homemade after a long shift."
Becca came up behind him and wrapped her arms around his firm stomach. "You're the sweetest."
He chuckled as he stirred the thick, meaty sauce. "Don't tell anyone."
"Too late," Jeremy said as he walked into the room, Charlie right behind him. The two of them were pretty much attached at the hip these days, which Becca found completely awesome. Her brother had been a loner for so much of his life. He deserved someone as special and fun and loving as Jeremy Rixey.
Eileen loped out after them. Becca had rescued the three-legged German shepherd puppy off the street near the hospital the first week she'd met Nick, back before she'd realized she'd never be returning to her own place again. At first, that was because it hadn't been safe—multiple break-ins had proven that. Now, even though all their mysteries had been solved and threats had been neutralized, it was because her home was here.
"Hey," Becca said, bending down to pet the monster-sized puppy. "You guys are just in time. We're getting ready to eat. Wanna join?"
Jeremy ran his inked hand over his short dark-brown hair, which was still growing out following brain surgery just over a month ago. None of them had emerged unscathed from the investigation into the corruption that had killed her father and blackened his team's reputations. Charlie had been abducted and maimed, two of his fingers cut off to try to make him talk. And Jeremy had been pistol-whipped by a fleeing bad guy who'd attacked them at a funeral. But both of them were doing so much better now. "We already had some," Jeremy said with a grin. "You know I wouldn't miss Nick's Sloppy Joes."
"Pretty much everyone else ate already," Nick said, pressing a kiss against Becca's hair. "Everyone" meaning the other four members of Nick's team and their respective girlfriends, all of whom were crashing in temporary digs here until a huge-scale rebuilding and renovation project was done, which would create six loft-style apartments in the other half of the L-shaped Hard Ink building.
Peering out from between long strands of blond hair, Charlie nodded. "We're gonna catch a movie."
After weeks of being on lockdown here, the idea of just going out to do something as casual and normal as seeing a movie still felt strange to Becca. "Oh, well, that sounds like fun."
"How was work today?" Charlie asked in that quiet way he had.
She came around the counter to him. Sometimes she was completely overwhelmed by her relief that they'd managed to rescue him from the Church Gang. And by her love for him, her only remaining family member. "It was good. Business as usual." Playfully, she pushed his hair back off his face. "I like it long, you know."
"Yeah?" he asked, his gaze a little shy. Even around her. "Me too."
"Me too," Jeremy said in a loaded tone, waggling his eyebrows as he planted a kiss on Charlie's cheek.
Becca laughed and held up her hands as Charlie's cheeks pinked, which was when she noticed the guys' T-shirts. Jeremy's was white with a headless stick figure. It read, _I need head_. Charlie's was blue and read, _I'm Getting Real Tired of Wearing Pants and Having Responsibilities._ Jeremy's innuendo-filled T-shirt collection was legendary around here, and Charlie had been borrowing Jer's clothes ever since he'd been rescued, although he usually picked the least dirty shirts Jeremy had. It was just another thing Becca loved about Jeremy, and about the way he loved and took care of her brother.
"Speaking of responsibilities, how did things go with the construction today?" she asked as Nick passed her two plates. She placed them on the breakfast bar, then grabbed some silverware and napkins.
Jeremy flicked his tongue against the piercing on his bottom lip and braced his hands on the counter. "Inspectors were out this morning and signed off on everything that's been done so far. Contractor's hoping to have the exterior shell totally done before winter. Fingers crossed."
"Considering a few weeks ago there was just a big hole out there, that sounds pretty good," Becca said.
"Yeah," Jeremy said, something dark momentarily passing through his gaze. And Becca didn't have to guess at what it was. There'd been a big hole because the arm of the building that had previously stood in that spot had been destroyed by a military-grade explosive device launched at the building in a predawn attack by the enemies of Nick's team. An attack that had resulted in the deaths of two of Jeremy's friends, members of the Raven Riders Motorcycle Club, which had been helping protect them. On some level, Becca knew Jeremy blamed himself for that. "Well, we better go."
Charlie nodded and made for the door, where he paused for a moment. "Hey, Becca?"
"Yeah?"
For a moment it seemed like he struggled for words. "Have a good night," he finally said, and then he ducked out, Jeremy right behind him.
"Thanks," she said, then turned to Nick. "Was that weird, or is it just me?"
Nick shrugged as he pulled buns out of a bag. "I think he was worried about you being at work today."
"Oh." The thought made her heart squeeze.
Soon, she and Nick were seated at the bar together with overflowing Sloppy Joe sandwiches, some of the pasta salad she'd made over the weekend, and chips. Eileen curled up on the floor next to Becca's tall chair.
"This is the best dinner ever," she said.
"That's because you're easy to please," he said with a smile that brought his dimple out to play. A man with so many rough edges . . . and a dimple. It slayed her every time.
"So how was your day?" she asked.
"Uh, good. Made a lot of progress on the new office," he said. They were turning the previously empty first-floor spot next to their tattoo shop into a high-tech suite of offices for the new security consulting company Nick and his team were opening. "Kinda funny that Jeremy bought this old warehouse because it was cheap, and now it's turned out to be the perfect space for all of us."
Becca smiled. "Yeah. I'm glad everyone is still going to be around here when all the work is done." It had seemed so empty around the building when, earlier in the summer, most of the team had cleared out to return to their homes and pack up their lives to relocate here permanently.
"Me too, Sunshine. This all feels right." Nick wiped up some sauce from his plate with the edge of his bun.
"Where is everyone anyway?" Becca asked. "It's so quiet." With six couples living out of two loft apartments, only one of which had a finished kitchen, it often felt a little like a college dorm around there.
"Shane, Sara, Easy, and Jenna went out to dinner earlier. And I think Beckett, Kat, Marz, and Emilie decided to finish up some painting downstairs."
Becca leaned in for a kiss. "It's weird to be alone."
Nick laughed. "Roger that."
"We could have sex on the counter," Becca said, giving him a seductive look.
He froze with a potato chip halfway to his mouth. "Is this something you've been thinking about?"
"Pretty much if it involves you and sex, you can bet I've thought about it," Becca said, grinning at the expression on his face, part dumbfounded, part aroused. "What can I say? You're very inspiring."
He wiped his mouth and slipped off his stool, then he spun her around to face him, his big body surrounding hers. He tilted up her chin. "Right back atcha, Becca. But nothing is sidetracking me from getting my ink on you tonight. You hear me?"
She rested her hands against his chest. "No sidetracking intended."
His fingers slid into her hair. "Uh-huh. Now, you ready for your tattoo? Because I'm dying to get my hands on you."
WEARING ONLY HER bra and jeans, Becca sat in a chair in the middle of Nick's tattoo room. Since the shop was closed while Jeremy focused on getting the construction on the other half of the building started, they were the only ones down there. The driving beat of a rock song played from the radio as Nick moved around the room getting everything ready.
Cabinets and a long counter filled one wall, which was otherwise decorated with drawings, tattoo designs, posters, and photographs of clients.
Becca had seen Nick work before and loved the dichotomy of this hard-edged, lethal soldier having a soft, artistic side. He was really freaking talented, too.
He handed her three sheets of paper. "I worked up a couple different fonts. What do you think?"
She shifted between the pages. "This one," she said, settling on the cursive design that best interweaved the letters in the words _Only, Always, Forever._
"That was my favorite, too," he said, giving her a wink. "How is this for size? Bigger? Smaller?"
The total design as he had it on the sheet was about four inches square, the words stacked atop one another. "This looks good to me. What do you think?"
Nick nodded and came behind her. He folded the sheet to focus on the design, then held it against the back of her right shoulder. "Yeah. This is a good size for the space. Gonna be fucking beautiful." He leaned down and pressed a kiss to her skin. "Let me go make the stencil, and we're ready to go."
A few minutes later, he cleaned her skin, affixed the stencil, and let her look at its placement before getting her settled into the chair again.
He pulled her bra strap off to the side. "Ready?"
"Very," she said, butterflies doing a small loop in her belly.
The tattoo machine came to life on a low buzz. "Just relax and let me know if you need a break, okay?" he said, dipping the tip into a little plastic cup of black ink.
"Okay." His gloved hands fell against her skin, and then the needles. Almost a scratching feeling, it didn't hurt nearly as bad as she thought it would. And just like when he'd drawn on her with skin markers, she was already dying to see what it looked like.
"How you doing?" he asked in a voice full of concentration she found utterly sexy. Just the thought that he was permanently altering her skin—just like he'd permanently altered her heart, her life, her very soul—sent a hot thrill through her blood.
"I'm good," she said, relaxing into the sensation of the bite moving across her skin. "Is it weird that I kinda like how it feels?"
He didn't answer right away as the needle moved in a long line. He pulled the machine away and wiped at her shoulder. "Not weird at all," he said, his voice a little gravelly. "Some people like the sensation and even find getting tattoos addictive."
"I can see that," she said. He worked without talking for a stretch, and the combination of the quiet intensity radiating off of him, the driving rock beat, and the buzz of the machine was heady and intoxicating. She found herself breathing a little faster and wanting so much more of him to be touching so much more of her. If she thought he was sexy putting ink on someone else, it was nothing compared to how she felt when he was doing it to her.
"What are you thinking about so hard?" Nick asked, his breath caressing her bare shoulder.
"Really want to know?" she asked, already smiling at what his reaction might be.
"Always," he said, wiping at her skin. He dipped the machine in the ink and leaned in again.
"How turned on this is making me." She really wanted to turn to see his expression but knew she wasn't supposed to move.
He pulled the machine away again. "Jesus, Becca. You're killing me here."
She grinned. "I asked if you really wanted to know."
Nick chuffed out a laugh. "Yeah, well, I've never had a fucking hard-on the entire time I've done a tattoo before, so you're not the only one."
Becca unleashed a small moan. "Now you've got me thinking about your cock, Nick." She couldn't help the hint of a whine in her voice.
"You'll never convince me that that's a bad thing, Sunshine."
"God, I really want to touch you right now," she said, heat spreading over her body.
"Be still," he said, his tone full of a stern command that made her smile.
"Yes, sir."
"Fucking _yes, sir,_ " he grumbled under his breath.
Another long stretch passed without them talking, but knowing that what they were doing was arousing Nick as much as it was her made her wet and needy and absolutely ready to jump him the minute she could.
Nearly ninety minutes had passed by the time Nick said, "There. It's done." He wiped at her skin and handed her a mirror. "Take a look."
Anticipation made her belly feel like she was looking over the edge of a tall cliff. She crossed to the mirror and turned her back to it, then lifted the hand mirror to see her first tattoo.
"Oh, Nick," she said, her gaze drinking it in. The way the stacked letters intertwined with one another was so beautifully done. "It's . . . gorgeous." Her heart squeezed in her chest. "You are so freaking talented." She looked from the mirror to where he still sat, his gaze glued to her face.
"I think it looks phenomenal on you. You really like it?" he asked.
She looked at her ink again. The stark crispness of the black lettering was so striking against her skin. She adored everything about it—the design, the words, their meaning. "I don't just like it. I love it, Nick. It's perfect. Everything I wanted." Her gaze cut back to him. "Just like you."
"Come here," he said, his voice a little rough. When she stood right in front of him, he pressed a kiss between her breasts. "It was an honor, you letting me do that."
She dragged her hands through his dark brown hair. "Sweet, sweet man," she said, leaning down to kiss him. It started off soft, full of gratitude and love, but quickly flashed hot until they were devouring one another, claiming, wanting, moaning.
"Fuck, Becca," he said, pulling back. "Let me take care of your tattoo."
Wiping the wetness from her lips, she smiled and nodded. "Okay." She sat back in the chair, and Nick cleaned the skin over her tattoo and taped a bandage to it.
"All done," he said. "Now, there's just one more thing I need to do."
## CHAPTER 4
Holy hell, had Nick ever been this nervous in his life? He'd faced down warlords, captured terrorists, survived IED explosions, and been shot on multiple occasions. Yet he'd never felt the kind of queasy, can't-quite-manage-a-deep-breath nerves he felt just then.
He retrieved the little black box from the drawer where he'd hidden it, fisted it in his palm, and came to stand in front of Becca. He gave her a hand to stand up, then slowly sank to one knee.
"What are you—" Becca gasped. "Nick?"
"Becca." Looking up at her beautiful face, he grasped her left hand. "When you walked through my door, you changed my whole life. You gave me purpose when I had none. You brought my family back together when I was so alone. You believed in me when no one else did, including myself. You fought for me and loved me and made me a better man." Glassy, bright blue eyes stared down at him with so much love. "You shined light on places inside me I thought would never emerge from the dark, and you helped me reclaim my integrity, my honor, and my life." A knot lodged in Nick's throat. "You saved me from becoming someone I didn't recognize, and because of you I have a life worth living. But only if you'll walk it with me."
"Nick," she rasped, her voice thick with unshed tears.
He flipped open the box and pulled out the diamond and platinum round-cut ring. A halo of smaller stones surrounded the center stone, creating what to him looked like a sun. More accent stones lined the band, giving it a classic, vintage look. He'd known it was the right ring as soon as he'd seen it. Slowly, he slid the diamond onto Becca's ring finger. "I love you with everything that I am, and everything I want to be. Please do me the greatest honor of my life and say you'll be my wife, my partner, my best friend, my companion. Becca Merritt, will you marry me?"
For a split second that felt like eternity, she looked down at her shaking hand. And then she sank to her knees in front of him and grasped his face. "Yes," she said, kissing him. "Only you. Always you. Forever you, Nick. Yes."
"Aw, Sunshine," he said, sliding his hands into her hair. "You make me happier than I ever thought I could be."
"I feel the exact same way," she said, tears finally leaking from the corners of her eyes. "I love you so much."
He kissed her on a groan, his spirit more buoyant and triumphant than it had ever been. Even more than when their records had been cleared and their honor had been restored. That had been exactly what he'd deserved, but this . . . this was more than he ever knew to want.
Becca's hands fisted in his shirt as she sucked hard on his tongue, a needy, desperate moan spilling from her throat. The sound shot right to his cock, making him rock hard in an instant and bringing back every bit of the aching lust he'd felt while he'd been doing her tattoo. Christ, if he thought it was arousing to mark her with his ink, it was nothing compared to what it did to him to know she'd just agreed to be his forever. He was fucking flying.
Nick tore open the button to her jeans. "Need in you."
"Yes," she said, tugging up his shirt. He helped her pull it over his head. For a moment, they were a whirl of shedding clothes and grasping hands and claiming kisses until they were both naked and panting and hot.
He sat in the chair where she'd been sitting and guided her down to his lap, her back to his front. "Take me inside of you, Becca. Ride me so fucking hard."
She took his cock in hand and sank down on him in one slow, slick stroke. "Oh, God," she rasped when he bottomed out inside her. "Needed you so much. All day."
He grasped her hips, his fingers digging into her soft flesh. "I'm here. Right here."
Hands braced on his thighs, Becca lifted herself up and down on his cock, riding him until they were both moaning, desperate, shaking. Her nails bit deliciously into his quads, making him do a double take at the big diamond on her left hand. And _fuck_ if that didn't escalate the urgent ache in his balls—to pour himself deep, deep inside her, and never let go.
She threw her head back, sending her long blond waves cascading down his chest. He fisted a thick column of her hair in his hand, forcing her back further against him until she was reclined against his chest and impaled on his cock. "Want you to come all over me," he said, reaching around to stroke her. She arched on contact, but Nick held her fast against him as his fingers pressed firm, quick circles against her clit.
The diamond caught the light as she grasped and kneaded her breasts, her movements growing desperate as she thrust forward against his fingers and back against his cock. On a guttural moan, she held her breath and her core fisted around his length until her muscles were pulsing, sucking, squeezing the sanity from him. A high-pitched moan wrenched out of her as she went slack on his lap, and the languid satisfaction of her body made him feel ten feet tall.
"Holy shit," she rasped.
He smacked her ass. "Kneel on the chair and bend over."
"Ooh. Yes, sir," she said as he helped her stand. He grinned as she got into position and looked back over her shoulder.
Gut instinct had him pulling off the bandage covering her tattoo. "I wanna see this ink while I fuck you." He penetrated her inch by maddening inch, his gaze glued to the words she would wear forever. For him.
Gripping the back of the chair, she peered at the design from the corner of her eye. "You're so good, Nick. All of you. I'm so lucky you're mine."
Her voice lit up places within him that once were so dark. Buried all the way inside her, he leaned over her back and braced his hand on the chair next to hers. Then his hips started to move in small, deep, punctuated thrusts that had her moaning with each stroke and his body screaming for release. As deep as it was, it wasn't deep enough. It would _never_ be deep enough. "Fuck, Becca. Just want you so goddamned much."
"You have me, Nick. Oh, God," she rasped as he banded an arm around her ribs and moved faster. The chair screeched against the floor and their skin slapped against the percussive beat of a grinding rock song.
He clutched her left hand on the backrest of the chair, and the diamond bit into his palm. It was the nail in the coffin of his remaining reserve. On a shout, he buried himself balls deep and came until he couldn't help but rest his weight against Becca's back. When his body finally stilled, he wrapped both arms around her and pressed a soft, open-mouthed kiss next to her tattoo. "I love you, Becca. And I always will."
NICK COULD BARELY keep the smile off his face as he opened the apartment door. Him, unable to hold back a smile. If that wasn't life doing a one-eighty, he didn't know what was.
"After you," he said to Becca. As she stepped inside the loft, he flicked on the lights to the main room.
"Surprise!" rang out in a great chorus of voices, along with a few barks. All their friends were there waiting for them—Jeremy and Charlie, Nick's sister, Kat, and all of Nick's teammates and their girlfriends. Baltimore police detective Kyler Vance, who'd been such an ally during their investigation, was there, too. And Nick had even managed to convince Walter and Louis Jackson to come. Walter had been Charlie's landlord and had taken a special interest in helping Becca, even calling in the assistance of his son, Louis, who'd turned out to be an amazing resource for the team as the coordinator of the city's task force on gangs.
"Oh, my God," Becca said with a huge smile on her face. She turned and threw her arms around Nick. "You planned all this?"
_Now_ he could grin. "I did good, huh?"
She laughed and hugged him tighter. "You did amazing."
When they broke apart, he and Becca were surrounded by their friends and family, although the distinction didn't mean much in this room. These people were all their family of choice. Everyone offered words of congratulations as Becca showed off her ring and recounted his proposal.
"Congrats, man. I couldn't be happier for you," Jeremy said, wearing the most unreserved smile Nick had seen on his brother in weeks. Not that Nick could blame him—between recovering from brain surgery, managing the construction on the other half of the Hard Ink building, and grappling with the death of two friends in the attack, the guy had a lot on his plate.
"Thanks, Jeremy. That means a lot," Nick said. He shook his brother's hand and tugged him in for a quick hug. "I'm happy for you, too. You and Charlie."
Becca arched a brow at Charlie. "So, going to a movie, huh?" Charlie's smile was a little sheepish, where Jer's was a total shit-eating number that said he was pleased with himself for pulling one over on her. "So you were in on all this?" she asked them.
Charlie nodded. "It's nice to have something else to celebrate."
"I couldn't agree more," Nick said as he watched Becca hug the blond-haired man who'd helped make tonight possible when he'd given Nick his blessing to propose to his sister.
Someone touched Nick's arm, and he turned to find Kat standing beside him. Short, with long brown hair, she matched Nick for stubbornness and guts, something she'd proven more than once during the recent investigation. "She's really good for you, Nick. You deserve this. And I'm really proud of you," Kat said. Given the way the two of them butted heads sometimes, the words meant a lot. They hugged, and Nick was reminded yet again just how much he had in his life now. Because of Becca.
"Thanks, Kat. Although all this happiness is really fucking weird," he said.
Rolling her bright green eyes, she shook her head. "Too damn bad. You'll just have to get used to it." She linked arms with Becca. "So when do we get to go dress shopping?"
"I'd love to help, too," Sara Dean said, brushing her red hair back from her face.
"I hadn't even thought about it yet," Becca said, looking between Kat and Sara. Of all the women here, Becca and Sara had known each other the longest. Nick knew that Becca held a special affection for Sara, who'd helped him and the team rescue Charlie from the basement of the strip club where Sara had been forced to work. "I'm off on Thursday and Friday, so maybe then? Jenna and Emilie can come, too. We'll do a whole girls' day."
"After everything that's happened, isn't it weird to think we can just go shopping?" Sara said. Words of agreement rose up all around, and Nick was glad that Becca had a close group of friends to share all this with.
Shane McCallan came up behind Sara and kissed her on the cheek. "Champagne?" He held out a tray of plastic flutes. Sara looked up at Shane with so much affection on her face, and Nick wondered if he and Becca were that blatant with their feelings. Hell, he guessed they probably were. But everyone in this room deserved a big old slice of happiness, so Nick couldn't begrudge a single one of them.
"You make a good waiter," Nick said to his best friend as he took a glass for Becca and himself. "In case this security business doesn't work out, and all."
"Don't make me tell you to fuck off at your engagement party," Shane said, a hint of his Southern accent coming through.
Nick laughed and shook the guy's free hand. "Thanks again for coming with me today."
"Wouldn't have been anywhere else," Shane said. And Nick knew that was true, despite the initial disbelief and subsequent ribbing Shane had dished out when Nick had told him what he planned to do.
When everyone had a glass of bubbly in their hands, Shane called out, "Hey everyone, gather 'round. I'd like to make a toast."
Standing in a big circle between the loft's open living room and kitchen, everyone quieted. Becca slipped her arm around Nick's back and leaned in tight against his side.
Shane held up his glass. "Nick Rixey is my best friend, my teammate, and my brother, and I know he'd lay down his life for me as quickly as I'd do the same for him."
"Fuckin' A," Nick said, giving Shane a nod. Their other teammates—Beckett Murda, Derek "Marz" DiMarzio, and Edward "Easy" Cantrell—sent up words of support, too.
"So I couldn't be happier," Shane continued, "to see him getting everything he deserves. Well, maybe even more than he deserves, given how amazing Becca Merritt is."
Against a round of laughter, Nick grinned and nodded, while Becca protested and hugged him.
Shane winked at her. "Becca, you went above and beyond in helping us clear our names, and I will forever be proud to call you my sister." Before things had a chance to turn serious, he added, "So if you ever need any help with this stubborn pain-in-the-ass man with whom you've chosen to spend your life, you just let any of us know." He gestured to the other guys in the room.
"Count me in for that, too," Jeremy said with a big smile. Everyone laughed, and damn, it felt good seeing their friends at such ease.
"So let's raise a glass to the couple who brought us all together. May love, peace, and happiness be your constant companions. To Nick and Becca." Shane raised his glass higher.
"To Nick and Becca," everyone called.
Grinning up at Nick, Becca clicked her plastic flute against his. "I love you," she said.
"Right back atcha, Sunshine," he said, his heart feeling two sizes too big for his chest. They drank.
"Is it time to eat the cake yet?" Marz said to more laughter as he leaned against the breakfast bar. Which was when Nick noticed a big cake with a figurine standing atop it next to Marz's elbow.
"Leave it to Marz to demand food," Beckett said with a smirk. Seeing Beckett relaxed and cutting it up was another big change, because for almost as long as they'd known one another, Beckett had been reserved and quiet, not one to shoot the shit or joke around. Before all this, only Marz had seemed to get behind the big guy's walls. Nick now knew that Kat had had a lot to do with how the man had changed, as much as their relationship had thrown Nick at first.
"Well, he did help me make it," Emilie said, planting a kiss on Marz's cheek. "It was all I could do to keep him from eating all the icing."
"Aw, you made this?" Becca said, stepping up to the counter. The square cake was two layers tall and read, _To Nick and Becca, The Best Is Yet To Come!_ Next to the words stood a porcelain figurine of a man in fatigues embracing a blond-haired woman in a wedding dress.
And _that_ was officially the first time Nick imagined seeing Becca in her wedding gown—about to get married to him. And hell if the thought of that didn't slay him.
"This is amazing, Emilie," Becca said as the women hugged.
"It really is. Thank you," Nick said, hugging Emilie next. After losing her brother—who'd been the team's enemy—her devotion to everyone in that room was truly amazing. Nick held out his hand to Marz next. "And thank you for not eating the cake before we had a chance to see it." Marz's appetite was pretty damn legendary.
Wearing his trademark grin, Marz nodded. "Dude, it was a close call." Marz could always be counted on to lighten a moment and make them laugh. Despite the fact that he'd borne the most serious injury from their ambush, he never let his amputation hold him back or get him down. Nick fucking admired that, he really did.
For a moment, surrounded by everyone he loved, Nick let himself bask in this moment of such fucking perfect contentment.
_The best is yet to come_.
Now that all the bullshit was behind them, Nick absolutely believed it. The future was theirs, and nothing could keep him from his happily ever after with Becca now.
"Well, what are we waiting for?" Nick asked. "Let's eat some damn cake."
## CHAPTER 5
I'm really excited to do this today," Becca said as Nick parked the car along the curb. She'd spent all her free time after work on Tuesday and Wednesday searching local venues, dress shops, caterers, florists, and other things related to planning a wedding. She already had her two days off booked with appointments, just so she could get an idea of what she'd want at her wedding.
Her wedding! Part of Becca could hardly believe it. She and Nick were getting married. She thought back to the stubborn, standoffish man she'd met that first day at Hard Ink all those months ago, and could only marvel at how far they'd come.
"Me too," Nick said. "Besides, whatever makes you happy makes me happy, you know that."
She gave him a look. "Smart man."
He grinned. "Come on. Let's go check it out."
They stepped out onto Front Street, a little cobblestone-paved gem right in the middle of downtown Baltimore. At one end of the block stood the towering 1840s Carrollton Inn and at the other, the historic Carroll Mansion, onetime home to Charles Carroll, Maryland's longest living signer of the Declaration of Independence. A brick-paved courtyard surrounded by lush gardens joined the two.
They'd barely stepped through the wrought-iron gatehouse entrance onto the property before Becca was head over heels in love—with the historic ambiance, the beautiful architecture, the whole romantic setting. A thrill of excitement shot through her. A few months ago, she'd been worried about losing her only remaining family member. Now, she'd gained a huge new family, had her brother firmly back in her life, and was about to become someone's wife. It was amazing how quickly life could change. For the good and for the bad.
"Hello," a tall woman in a smart pants suit said. With her long black hair and warm brown skin, she was strikingly pretty. "I'm Sonya Mayer, the assistant manager of the inn. Welcome." Nick and Becca introduced themselves, and Sonya gave them the tour of the inn's four interior levels and rooms, as well as its outdoor spaces.
"I can't believe I've never noticed this place before. It's just gorgeous," Becca said. The rooms each had their own unique atmosphere and decor, lending elegance to the inn's charming nineteenth-century architecture. And the courtyard would be gorgeous for an outdoor wedding, something she'd always wanted.
"We have our own little enclave back here," Sonya said. "A little oasis of calm and historical charm in the middle of the city."
Becca nodded, struggling to keep her outward cool when inside she was all _I want it I want it and I don't care how much it costs!_
Back downstairs, Sonya guided them to a table where they had a few flower and cake samples laid out. Menus, catalogs, and price lists sat on one corner of the table atop a shiny venue folder.
"Can I offer you a glass of champagne while we talk?" Sonya asked.
"I'd love that," Becca said. "Nick?"
"Sounds great," he said. When Sonya departed, he turned to her. "Okay, tell me how much you want this place."
Becca managed to hold in her enthusiasm for about five seconds. "It's so amazing. Don't you think so? It's pretty and charming and not too big and—"
Nick kissed her. "Done."
"It's pretty expensive, though," she said.
He shook his head, his pale green eyes locked on hers. "I don't care what it costs. If you want it, we're having it. I like it, too. And we deserve the best to start our new life."
"Really?" she asked. "How'd I get so lucky?"
The smile brought out his dimple. "That's my line."
"Here we go," Sonya said, settling two crystal champagne flutes down on the table. She had an iPad tucked under her arm.
"Cheers," Nick said, clinking glasses with Becca. The champagne was sweet and bubbly, absolutely delicious.
"So, have you picked a date?" Sonya asked, bringing the iPad to life in front of her.
Becca looked to Nick. "We only just got engaged," she said. "If we wanted to have the ceremony outside, I suppose we'd need to do it by the fall or wait until the spring?" Nick nodded and gave her hand a squeeze.
"Well, let me see," Sonya said, scrolling through her digital calendar. "We're actually booking a year out right now. I know all the weekend dates in the fall are completely booked. And I think the spring is, too." Becca's shoulders dropped. She shouldn't have been surprised, really, since most people had long engagements to allow them to make their plans. "If you're interested in a shorter engagement, it looks like . . ." The woman focused on the screen for a long moment. "It looks like I only have two options as of this moment. The third Saturday in December, which would preclude an outdoor ceremony. Or, oh, we had a cancellation on Saturday, August eighth. Though if you've only just gotten engaged, I imagine that's much too soon. Otherwise, our next weekend opening is next July."
Becca barely heard anything after the August date. That was three weeks from now. She looked up at Nick. "Is three weeks from now too crazy?"
The smile he gave her made her fall even more in love with him. "Only in the best possible ways, Sunshine."
Tears pricked the backs of her eyes. "I could be your wife in three weeks."
A heated masculine satisfaction slid into those pale green eyes, and Nick turned to Sonya. "We'll take August eighth."
"IT'S A GOOD thing you'd already made an appointment to look at dresses," Emilie said as the five women piled out of Shane's big truck the following day. Sara had sweet-talked him into letting them borrow it for their girls' day out.
"I know, right?" Becca said, still floating over the fact that she was getting married in three weeks. "But the inn was just too perfect to pass up, and neither of us wanted to wait a whole year to get married."
Sara elbowed her in the side. "And you're sure this isn't a shotgun wedding, right?"
Grinning, Becca shook her head. She and Nick had been answering this question ever since they'd returned from the inn yesterday after several hours of choosing food, cake, and flowers for their event. "Nope. There's no bun in this oven, I promise you." Everyone laughed. Becca walked up to the ornate carved desk and greeted the young woman standing there. "Hi, I'm Becca Merritt."
"Welcome, Becca. Please have a seat. Diana will be right out," the woman said.
The five friends sat on the overstuffed cream-colored sofas and nibbled at cookies and fancy wrapped chocolates displayed on plates covering the glass-topped end tables. Becca and Nick might be doing this fast, but it all still felt so special to her. She was glad they weren't waiting.
A woman with short strawberry-blond hair wearing a pretty teal wrap dress approached their group. Diana made quick introductions, then took them to a sitting room where Becca could show off the gowns she tried on. "Do you have a date in mind?" Diana asked.
"I do, and it's really short notice. August eighth," Becca said, not missing the woman's gaze flicker down to her belly and back. Becca almost laughed. "We fell in love with a venue that had a last-minute cancellation, so we went for it."
"Well, we can make this work, Becca. We've got quite a few gowns available in our annual sample sale, so let's see if we can't find something you love. This way, please." She led them into a long rectangular room filled with racks of gowns. "Do you have any preferences for color, silhouette, length?"
"I want to wear a gown that you couldn't wear any other time," Becca said. "Something romantic and full. Maybe sleeveless. And I think I prefer white to ivory."
"Let's start with ball gowns and A-lines, then," Diana said, already pulling a few things off the racks. "Ladies, feel free to pull anything fun that you see."
Becca searched through the gowns, pulling a couple of things that caught her eye. Before long, she was in the dressing room neck deep in satin and lace and tulle. Too much tulle in the case of the first dress, which she obligingly showed to her friends even though she knew it wasn't the one. "What do you think?" she asked as she spun on the dais in front of them.
Kat made a face. "I think it's too young for you."
Sara nodded. "Too froufrou."
Becca laughed. "I agree." She tried on another, this one in a mermaid cut that wasn't at all her style. "Who picked this one?" she asked, laughing.
"What? I thought it was cool," Sara's younger sister, Jenna, said. They both had matching red hair.
"Sadly, I don't have the hips to pull this off. Or the boobs. Or whatever else you need to make this work." Becca rolled her eyes as the women nodded and laughed. The third one was closer—a sleeveless ball-gown style with lots of lace and beading. "This is gorgeous," Becca said. "But it's so heavy I don't know how I'd dance in it."
"I don't know, Becca," Emilie said, tucking her brown hair behind her ears. "That one might be worth suffering for."
It took five more dresses before Becca fell in love. The white ball gown had a sleeveless sweetheart neckline and gorgeous beading at the waist, while the skirt fell in soft layers of satin, full but not poofy. It was two sizes too big and was missing a few buttons down the back, but Diana assured her it could be taken in and repaired in plenty of time. Staring in the mirror, Becca suddenly felt overwhelmed.
Kat was the first one to notice. She crossed from the sofa to stand beside Becca. "You okay?"
Afraid that trying to speak might hasten the threatening tears, Becca just nodded.
"This is the one, isn't it?" Kat met Becca's gaze in the mirror. Despite her petite stature, Kat looked so much like her brothers, with her chocolate brown hair and green eyes—and even a few shared facial expressions—that Becca immediately felt at ease. And she realized there was something she needed to ask Katherine.
"Yeah," Becca managed. "I don't need to look anymore." She turned to face Kat. "Will you be my maid of honor?" They might not have known each other very long, but in the few months since they'd met, they'd bonded hard and fast, not only over their love for the Rixey men but also because of the way Kat had taken care of Becca during the team's investigation.
Kat's eyes went wide. "You want me?"
"We're going to be sisters, right? I absolutely want you. If you'll do it," Becca said.
Kat hugged her. "I'd love to stand up for you and Nick. I'm so happy for both of you."
"Okay, now. Don't make me cry. I'm having a hard enough time with that as it is," Becca said. Everyone laughed. Becca gave herself one last look in the mirror. "Yeah, this is the one." She turned to see the back of it, all clean, soft lines of satin. On her shoulder, the healing tattoo peeked out through her hair. The design was as beautiful as the words were appropriate, so she had no qualms about the ink showing. She was proud of the gift Nick had given her.
"Well, then," Diana said. "Let's get the tailor to take a look at you, and then we'll get the ladies started on bridesmaid dresses. Any idea what color you'd like?"
"Yes," Becca said. "Kat, can you grab the picture from my purse?" Kat handed Diana the picture of the bridesmaids' bouquets Becca had chosen. "I figured it might be hard to get one bridesmaid dress that works for everyone in the short time we have," Becca continued. "So as long as the gown is a shade of purple that matches these flowers, I don't care what style or length it is. Whatever you guys like."
Diana studied the picture, which showed a bouquet with mauve roses, purple hydrangeas, dark purple irises, berry-colored orchids, burgundy dahlias, and light purple mini carnations. "Oh, yes, we can make this work. I'll grab the tailor for you and show the girls where to look."
In the time it took for Becca to get fitted, everyone found things to try on. And it didn't take long until all four friends decided on dresses that suited their taste and matched Becca's color scheme. Kat chose a sleek, sleeveless dark-purple gown that looked gorgeous with her long brown hair. Emilie picked a mauve V-neck gown with a satin belt at the waist. Sara chose a satin berry-colored sheath with cap sleeves and a sweetheart neckline, while Jenna went with a flirty lavender gown with a drop waist and a fuller skirt that accentuated her curves beautifully.
"You all look stunning," Becca said when they stood before her. "The guys aren't going to know what to do with themselves."
"Shane's never seen me in a dress like this before," Sara said, staring at herself in the mirror. "In fact, I don't think I've ever owned a dress like this before."
Jenna grasped her sister's hand. Despite being the youngest of the five of them, the Dean sisters had been through hell the past few years, especially Sara, who'd borne the burden of repaying her criminal father's debts after he'd died, sometimes in ways too horrible to imagine. "Well, it won't be the last one," Jenna said, smiling. "But, yeah, it's gonna be fun seeing their reactions."
"I think we're all pretty guaranteed to get laid at Becca's wedding," Kat said with a mischievous grin.
They all burst into laughter. "When do you _not_ get laid, Kat?" Emilie asked with an arched eyebrow.
"Aw, don't even talk to me, Miss Garza, because you and Marz are loud as hell. Not that I mind, because, _dude,_ does he have a mouth on him," Kat said with a big grin.
"Oh, my God," Sara said, her cheeks turning bright pink but her smile saying she was enjoying the teasing.
"He really does." Grinning, Emilie shrugged. "The hazards of sharing an unfinished apartment. Can't be helped." Kat and Beckett had been staying in the room Nick and Jeremy reserved for her in their apartment until one particularly loud session of lovemaking had apparently caught Jer's ear. His teasing had been relentless. Finally, Kat and Beckett relocated to an empty room upstairs. As much as they all enjoyed each other's company, everyone was going to be thrilled when the new building was done, that was for sure.
Becca could only laugh as the good-natured ribbing went on. "Well, I know I'm getting laid. The rest of you are on your own." By the time they'd all been fitted and had paid for their gowns, Becca was pretty sure they were on the verge of getting thrown out of the store.
They spilled out onto the street, laughing and hungry for lunch. Becca fell behind while she fished for her cell phone in her purse and paused to shoot off a quick text to Nick.
_All done dress shopping! I'm gonna knock your socks off! ;)_
Nick responded immediately. _Sunshine, you already do._
Grinning, she glanced down the block—and nearly gasped out loud. Tyrell Woodson stood at the corner, glaring at her.
"Hey, Becca, come on," Kat called. Becca blinked and the man was gone. Vanished. A figment of her imagination. Not that her body seemed to know the difference. Heart racing, she caught up with the group as they made their way to an Italian place they'd agreed on earlier. "You okay?" Kat asked.
"Yeah. Great." Her voice sounded flat to her own ears. She glanced back over her shoulder. Woodson wasn't there. Of course. She let out a long breath. She'd gotten through the whole workweek without another incident like the one she'd had with Ben at the end of her first shift back. Clearly, her subconscious wasn't done worrying about Woodson, though, however unnecessary—and unfounded—that worrying was.
Why was she freaking out about what'd happened to her now? For months, she'd been fine, just an occasional nightmare of being grabbed, being dragged away, being lost and never found. Then again, for most of that time she'd been shut up at Hard Ink with Nick.
They arrived at the restaurant, and Kat paused before she followed the others inside. "Are you sure you're okay?" she asked. "Because you know my brother will kill me if anything happens to you on my watch." Kat arched an eyebrow. Though Becca knew she was joking, there'd been a time not too long before when Kat had in fact been Becca's bodyguard, during a meeting with the man who'd turned out to be responsible for the death of Becca's dad. Nick had gotten angry at both of them when they'd had to deviate from the original plan to get Becca home safely.
"You just survived heart surgery, Kat," Becca said. Kat had gotten shot at the same funeral where Jeremy had been hurt. Watching Nick deal with both of his siblings fighting for their lives was one of the hardest things Becca had ever done. But they'd both pulled through. And now they all deserved a celebration. "Nick is so grateful you're okay that I'm pretty sure you'll be able to get away with absolutely anything for the rest of your life," Becca said with a smile. "And I'm fine, I promise."
"Good," Kat said, giving her a last look. "Then let's eat, because I'm starving."
## CHAPTER 6
It happened again the next week. More than once. Most recently, it occurred on her way back to the hospital after grabbing something for lunch. Becca saw Woodson lurking in a doorway farther down the street, but when she looked again, no one was there.
That night, as she and Nick lay in bed, Becca gave voice to the question whose answer she thought might best give her some peace of mind. "Do you know what happened to Tyrell Woodson after you interrogated him?"
Nick shifted to look at her, concern filling his pale green eyes. "Why?"
"I just wondered," she said, feeling a little bad for not sharing what she'd been experiencing. She didn't want to worry him, though, especially when they had so much going on—the construction, setting up his and the team's new business, her work at the ER, planning the wedding. During their off hours this week, they'd managed to send out invitations, find an officiant, hire a DJ, and apply for their marriage license. Luckily, all the guys already had their dress blues from when the Army had held a memorial service for the fallen members of their A-team a few weeks before.
Eyes narrowed, Nick studied her. "You know we sent the harbor police out to that island where we dropped him, but he was gone when they got there. So once Detective Vance joined our investigation, I had him search for Woodson. Just to be on the safe side."
Gone? Becca pushed up onto an elbow. Vance's inquiry was news to her. "So what did he find?"
Nick tucked her hair behind her ear. "Woodson's not in custody anywhere, but he's not here. Vance tracked him to South Carolina using credit card transactions. Apparently he has family down there. Looks like he left town right after everything went down. Marz took that video of him spilling all kinds of secrets. No doubt he wanted to get clear of the city in case we turned it over to Church like we threatened."
Relief flooded through Becca so fast that she sagged back down to her pillow. He wasn't in Baltimore. Hell, he wasn't in Maryland. So it really was her imagination at work. "Oh. That's good."
Leaning over her, Nick cupped Becca's face in his hand. "Shit, Becca. I'm sorry. Have you been worrying about this? I should've said something, but—"
"No, not worried, exactly. I guess I just needed to know. For closure." And it was true. Maybe she could stop seeing ghosts around every corner now.
Nick nodded. "Why don't I have Vance run another trace on him? Make sure he's still out of the picture? I can't imagine why he'd come back with how volatile Baltimore's gang scene is right now. Since the Church Gang imploded, Louis Jackson told us that the word on the street for former Churchmen is leave or die. It's been open season on them as other gangs fight to consolidate power and take over the heroin trade."
Becca nodded, Nick's words strengthening the bulwarks against her fear. "Yeah, that makes sense. But have Vance run the check. It can't hurt, right?"
"Absolutely. You know I'd do anything to keep you safe, Becca." He nailed her with a stare full of such fierce love that it stole her breath.
"Of course. And so would I," she said.
It only took the weekend before Vance had news. He called Monday night after Becca's shift while everyone was hanging out watching movies in Nick and Jeremy's living room.
"Vance," Nick said as he answered his cell and got up from the couch. "How are you?"
"Want me to pause it?" Jeremy asked, sitting in the corner of the other couch next to Charlie, Eileen attempting to fit her growing body on his lap.
Nick waved him off and headed back down the hall toward his office. Becca followed. "And what did you find?" Nick asked as he settled into the chair at his desk. As Becca looked at him sitting there, memories came rushing back. One of her first nights here. Giving Nick a massage after his back had been hurt when he and Beckett had gotten into a fistfight. How he'd kissed her. "Well, that's good news then."
The words pulled Becca from her thought. Good news had to mean Woodson was still far away, right?
"Okay, sounds like a plan. Thanks, man," Nick said.
"What did he say?" she asked when Nick hung up.
He tossed his cell to the desk and reached out for her, pulling her down to straddle his lap. "As recently as last Wednesday, Woodson got a speeding ticket in South Carolina. Vance is still working on recent credit card transactions. But that's pretty good evidence that he's stayed put." Nick stroked his fingers through Becca's hair. "Vance is going to put out a be-on-the-lookout for Woodson's car with local PD, on the off chance he returns to the area."
Becca nodded. "That all sounds good. Thank you for having him do that."
"A little extra peace of mind never hurts," he said. Hand slipping behind her neck, he gently pulled her down until their lips met. The kiss was slow and soft and exploring, full of the heat that always flared quickly to life between them. "You're gonna be my wife in less than two weeks," he whispered around the edge of the kiss.
The words made her smile. "I am." The kiss deepened. "Yours to do with whatever you want."
Nick's gaze flashed hot. "My thoughts exactly." He urged her to stand up, then he crossed to the door and closed it. When he turned to her, the expression on his face was predatory, so damn sexy she was immediately wet with anticipation. Without a word, he stood in front of her, unbuttoned and pushed down her jeans and panties, and sank to his knees in front of her. "Right now what I want is for you to come all over my tongue."
Becca clutched onto the desk behind her as Nick planted his mouth between her legs, his tongue immediately plunging into her folds and finding her clit. Together they worked her jeans the rest of the way off, then Nick pushed her stance wider and settled his big body in tight between her legs.
He was relentless, his fingers holding her open, his tongue alternating between lapping at her and hard, fast flicks, his mouth sucking.
"Oh, God, Nick," she rasped, her hand falling on his hair and clutching at it. He growled against her, not letting up for a second, and she thrust her hips forward, yearning and seeking and craving even more. She tugged his hair, pulling him in tighter, unable to restrain herself from demanding what she needed.
And it seemed to drive Nick harder, because he worked a thick finger inside her slickness and sucked her in a fast rhythm that had her panting and thrusting and straining. The orgasm hit her like a shock wave, not there one moment and then throwing her head over heels the next. She got light-headed and her knees went soft, forcing her to sag back against the desk.
"Holy shit," she rasped.
Nick eased his hand free of her and looked up, his gaze so full of smug satisfaction that it made her shake her head as she smiled. And then, eyes on hers, he sucked the wetness off his finger and licked his lips. "You taste fucking delicious, Sunshine."
Becca's heart did a little flip as she offered him a hand and he rose. "You know," she said, "if I hadn't already agreed to marry you, I would say yes just based on how good you are at that."
Nick threw his head back and laughed, the kind of free, joyful laughter she'd only heard from him a handful of times. His smile was freaking gorgeous, his dimple a mile deep, and his face so ruggedly beautiful that it made her whole chest ache with how much she loved him. "Good to know," he managed.
Grinning, she wrapped her arms around his neck. "Seriously. A man who can make the best Sloppy Joes on earth, helps total strangers, sings bad eighties anthems, loves his family fiercely, can kill in fifty-two ways if he has to, is smoking hot, _and_ eats pussy like he's starved for it is a keeper." She tried to keep a straight face, but laughter was already bubbling up inside her.
"I . . . am really freaking aroused now. Keep talking," he said, his eyebrow arched, that smug smile returning.
She reached for his jeans and tugged the button open. "About what? Should I talk about how much I love your cock? How amazing it feels inside me?"
Nick licked his lips, a kind of amused disbelief filling his gaze. "Yes, definitely talk about that. Like, a lot."
Grinning, Becca bit her lip and pushed off the desk. Walking backward, she hooked her fingers in the waistband of his jeans and pulled him along with her toward the bedroom. With her other hand, she grasped his hard length, squeezing just enough to wring a grunt out of him. "Come put this inside me, Nick, and I'll say absolutely anything you want."
NICK FELT LIKE he hadn't seen Becca in forever, despite the fact that they'd woken up together this morning. But that had been fifteen hours ago—before he'd spent all day working with the guys to install the carpet in their new offices. As much as possible, they were trying to do the work on the business suite to allow the contractor to stay focused on the much bigger project of the building. Given Jeremy's ability to figure out pretty much any home improvement project, Beckett's electrical know-how, and Marz and Charlie's expertise with all the wiring and secure computer hookups they needed done, they were in pretty good shape.
In the meantime, all of them had been working their contacts to get word out about the new security consulting services they'd be offering. Beckett, Marz, and Shane had already been working in various aspects of the field and had ready clients to reach out to. Having worked for years as a computer security consultant hacking into corporations' systems to find their weaknesses, Charlie turned out to be an amazing asset in making new contacts. All of which was keeping them on track for a targeted post–Labor Day opening date, which would give Nick and Becca two weeks for their honeymoon in Italy and another two for Nick to be back and helping with all the last-minute prep.
Nick's and Beckett's cell phones dinged an incoming message at about the same time, and both of them paused where they knelt on the floor to check their phones. Since Becca and Kat had gone shopping together after Becca's half-day shift, Nick didn't have to guess at who it was.
_On the way home. Be there in about twenty! xo_
"Aw, look at you two," Shane said, his tone full of sarcasm. "Stop chitchatting with the ladies and let's get this shit finished."
"You are as pussy-whipped as any of us, McCallan, so shut the fuck up," Beckett said as he shoved his phone in his back pocket. Ah, it was so nice that they were all getting along so good again. Just like old times. In all the best ways.
From the next room where Marz and Charlie were working on some wiring, Charlie called, "For the record, I am not as pussy-whipped as the rest of you."
Silence rang out for a moment, and then they all lost their shit. Just flat-out, tear-inducing laughter that had every one of them clutching their guts. Charlie fucking Merritt was coming out of his shell, that was for goddamn sure.
Wiping the tears from his eyes, Nick managed to get to two feet and make it to the doorway, where he found Marz red in the face with hilarity and Jeremy pressing a big kiss to his boyfriend's mouth. The affection his brother and soon-to-be brother-in-law bore for each other made Nick happy, it really did. Because Nick had never seen Jeremy settle down like this before—with a man or a woman, and he'd dated both. And no matter how much Nick admired so many other things about Becca and Charlie's father—the team's former commander—Nick would never understand the homophobic bullshit Frank Merritt had apparently rained down on Charlie from the moment he'd come out. So Charlie deserved this chance to be happy and loved and accepted. By all of them. And, amazingly, they all fit together like clockwork—even Jeremy and Charlie, who hadn't been part of their team.
They were now.
Beckett joined Nick in the doorway, his bright blue eyes gleaming. "Tou-fucking-che, Charlie."
Because he had his hair pulled back in some sort of a bun thing, the red covering Charlie's face was crystal clear, but the guy was smiling as Marz slapped him on the knee.
Everyone got back to work, shooting the shit as they finished up the last of the carpeting.
From where he knelt in the corner, Easy ran his dark hand over his short-trimmed hair and looked around the nearly finished space. "This is all really coming together."
"Yes, it is," Nick said. Right now, it was all plain white walls and industrial gray carpeting, but they'd come a helluva long way from the cement floor, exposed ceilings, and cinder-block walls that had stood there just a few weeks before. "Thanks to the hard work you all have been putting in."
"I'm glad to do it," Easy said. "It's good to be busy. And it's really fucking good to feel like I'm part of something again."
Nick, Easy, Beckett, and Shane all traded looks and nodded. Every one of them felt the same way after the ambush, the other-than-honorable discharge, having their reputations tarnished, and being scattered to the four winds once they'd returned stateside.
"Amen, brother," Shane said, clapping Easy on the shoulder.
When the other three men cleared out, Nick went into the next room to see how the wiring job was coming. "Almost done?" he asked Marz.
"Yeah, hoss. Maybe just another half hour," Marz said. Charlie nodded.
"Anything I can do to help?' Nick asked.
"Nope, we're good. If you see Em, just let her know I'll be up in a bit." Kneeling in front of some cables sticking out of the wall, Marz kneaded absentmindedly at his thigh.
Nick frowned. "You okay?" Sometimes you could almost forget about Marz's amputation because the guy never let it slow him down, even when he should.
"Hmm?" Marz looked up from the complicated jack he was working on. "Oh, yeah. No worries. Damn leg just gets cranky if I spend much time on the floor."
"I told you I'd do that." Charlie paused what he'd been coding on his laptop to look at Marz.
"I'm fine. And this is the last one anyway. Go see your bride-to-be." Marz looked up at Nick with a grin.
Nick nodded. "All right. Later." He stepped through the back door out into the warm July night and nearly walked into Beckett. "Sorry. What's the matter?" Nick asked, noticing the serious expression Beckett wore.
Beckett shook his head. "I just made Becca cry."
Frowning, Nick glanced around the otherwise empty parking lot. Kat's car was there, so they were back. "What?" he asked, heading toward the door to the main part of the Hard Ink building.
Beckett fell into step beside him. "I came up behind her and Kat out here, and when Becca noticed me, she freaked out. Nearly jumped out of her skin. At first she tried to play it off, but then she got upset and ran inside."
Nick punched the code into the keypad, and they stepped into the concrete-and-metal stairwell that led to Hard Ink on the first floor and their apartments and the gym on the upper floors. "Don't worry about it. I'm sure she's fine."
Beckett gave him a doubtful look but nodded. "Let me know if there's anything I can do."
Clapping him on the arm, Nick shook his head. "I'll take care of her." They parted ways as Nick let himself into his apartment and Beckett made his way upstairs. The big open kitchen and living room were empty. "Becca?" he called, an urgent need to see her and make sure she was okay flowing through him.
No answer. He went straight for their rooms at the back of the apartment.
"You need to tell him," someone said. Kat.
"I agree," came another voice. Emilie?
Nick rounded the corner into his office to see Becca sitting on his sofa, her face wet with tears. Kat and Emilie sat on either side of her. Nick's gut dropped to the floor. "Tell me what?"
## CHAPTER 7
Nick nailed Becca with a stare. As he watched, she made a valiant effort to button up just how upset she was, and that made him worry even more. He knelt down in front of her, his hands on her knees. "What's going on?"
She shook her head, and his heart fucking broke as he saw her struggle to find her voice and hold back the tears.
Kat grasped something from Becca's lap and held it out to him. "Someone left this in front of Becca's locker at work today."
Frowning, Nick accepted the small, floppy, black-and-brown stuffed animal into his hand. He guessed it was supposed to be a German shepherd, and the damn thing was missing a leg. It looked like someone had cut the back leg off and stapled the opening closed. What the hell?
"Look at the neck, Nick," Kat said, her voice serious.
He flipped it around, and that was when he noticed that the animal was so floppy because the head was connected to the body only by a thin strip of material along the back. He might not have thought much about that if someone hadn't gone to the trouble of spray painting the torn opening red. As if the neck had been slit. "Sonofabitch," he said, hot prickles running down his back. "Talk to me, Sunshine."
She heaved a deep breath. "I found it at the end of my shift," she said. "I went to get my stuff, and this was sitting upright on the floor in front of my locker. I thought someone had left me a present until . . ."
"This is fucking twisted," Kat bit out.
The words echoed Nick's own thoughts. Twisted and threatening. The removed leg was clearly meant to communicate that the person knew Becca's dog had only three legs, which made the threat personal and specific. "Did you talk to security about this?" he asked, making sure to keep his voice even.
Becca shook her head. "I was so freaked out, I just wanted to get out of there."
"I didn't know about this until just now, or I would've brought us home earlier," Kat said.
"I'm sorry," Becca said, turning to Nick's sister. "I just wanted to forget about it for a few hours."
"I know," Kat said. "You don't have to apologize, but I'm worried about this."
Nick nodded. "Can you think of anyone at the hospital who would do this? Who would have a problem with you?"
"I know some people are upset that I've asked for time off for the wedding and our honeymoon after being on leave for two months." Becca scrubbed her face with her hands. "I overheard some women talking at the nurses' station yesterday. But I can't imagine anyone doing something this cruel. And twisted. Kat's right."
"I don't like it, Becca," Nick said, his hackles all up. "I don't like it at all. I'd like to go with you tomorrow to talk to hospital security. They need to know."
She gave a quick nod. "Okay."
"Is this why you freaked out when Beckett came up behind you out back?" he asked.
Becca sagged back against the couch. "Partly."
Nick frowned, his instincts flaring. "What's the other part?"
The quick look Becca exchanged with Emilie had Nick's gut twisting with worry, especially when Em gave her a small nod. "I know it's ridiculous," Becca said in a small voice, "but I keep thinking I'm seeing Woodson."
The words hung there for a moment and rushed ice through Nick's veins.
Emilie got up and gestured for Nick to sit. He slid onto the couch next to Becca. He'd barely put his arm around her shoulders when she buried her face against his chest, her arm clutching his neck. Her shoulders shook with restrained tears. "Aw, Becca," he said, his heart absolutely aching. How had he not seen this?
"I'm sorry," she rasped.
He locked eyes with Kat, whose expression was every bit as concerned and upset as he felt. "You don't have anything to apologize for. You hear me?"
A quick nod against his chest.
Guilt flooded into his gut. Why hadn't he ever considered that the attempted abductions might have traumatized Becca? She'd been so strong through it all that he'd just assumed she was fine. No wonder she'd asked him what had happened to Woodson. Yet, once again, he hadn't probed deep enough. "Shit, I'm the one who should be apologizing." He stroked her hair.
"No," she said, shaking her head as she pulled away. Her face was red and wet and her eyes were puffy, but she was still the most beautiful thing he'd ever seen. "How could you have known? It just felt so ridiculous that I didn't want to say anything."
"Tell me what's been going on," he said, cupping her cheek in his palm and swiping at her tears with his thumb.
She gave a shy little shrug. "I keep thinking I see him. One minute he's there, and the next he's not. At random times. Around the hospital. On the street. Tonight at the mall I kept feeling like someone was watching me, but of course no one was there. Just like no one's ever there. I was just freaked out about the stuffed animal. After all that, Beckett just scared me and everything kinda crashed in on me." The words spilled out of her in a rush.
"It's not ridiculous, Becca," Emilie said. "It's PTSD." Before all this, Emilie had worked as a clinical psychologist at a local university. Given the shit all of them had been through, she'd been an incredible resource for the whole team these past months.
"But I was fine," Becca said, looking from Nick to Emilie to Kat. "I was fine after it happened."
"The crisis of the investigation probably kept your brain otherwise focused. But then you went back to the scene of the abduction, and you were out on your own for the first time in months." Emilie knelt where Nick had been. "Your nervous system is finally trying to process what happened to you. The anxiety, the reliving of the event, the spontaneous memories, the paranoia. These are all normal given what you went through."
The list of symptoms lashed at Nick's soul. He hated that Becca was hurting. Was the shit that had happened in Afghanistan never going to stop raining down on them? "She's right," Nick said, taking Becca's hand. "How frequently has this been happening?"
Becca frowned, and her gaze went distant. "Maybe a half dozen times since I started back to work."
"Aw, Sunshine," Nick said. He hated that she'd been carrying this all by herself. But no more. "What can I do to help?"
"I don't know," Becca said. "I know it's not real. But I can't seem to make it stop."
"It's gonna take time," Emilie said. "I can help you with some techniques to reduce and combat anxiety. Or you might consider seeing a therapist at the hospital."
"Okay," Becca said. "I'd like to talk to you, I think."
Emilie nodded.
"Maybe you should consider taking off even earlier than you planned," Kat said. Nick could've hugged her, because his thoughts were running in the same direction. But the last thing Becca needed was for him to be an overprotective asshole right now.
"I only have four and a half more shifts," Becca said. "I'd hate to bail on everyone last minute."
"Becca," Kat said, taking her other hand. "You're always taking care of everyone else. You have to let us take care of you, too."
"I know," Becca said in a small voice. "I think if we address this stuffed animal with the hospital and I'm talking to Emilie, I'll feel better."
"That's a good start," Nick said. "But until you take off, I'm walking you in and out of the hospital at the beginning and end of your shifts. If you have wedding errands you need to do, I want to be at your side. And, hey," he said, gently turning her face toward him. "Please talk to me. I can't be there for you if you don't let me know what you need."
Nodding, Becca gave him a look that nearly broke his fucking heart. "I just didn't want to worry you."
He lifted her left hand to his mouth and kissed her ring, then he pressed her hand to his heart. "Taking care of you is my job, Sunshine. For the rest of my life. In good times and in bad, remember?"
Glassiness filled her eyes. "Yeah."
Kat rose, and Emilie followed suit. "We'll give you guys some time alone," Kat said. She leaned over and pressed a kiss against the top of Becca's head. "I had fun shopping with you, sis."
It was the first smile he'd seen from Becca since he'd walked into the room. "I had fun with you, too, Kat," Becca said. "You're going to be the best sister ever."
"Hey, I already am," Kat said with a grin. She and Emilie left.
"Can we get ready for bed?" Becca asked in a small voice. "I'd really like to just lay down with you."
"Of course," Nick said, helping her up. They got changed without talking much, then he climbed into bed and held his arm open to her. Becca crawled in alongside his body and fitted herself tight against his side, like she always did. She fit so fucking perfectly against him. Nick stroked his fingers through her hair. "I don't want you to ever feel like you can't talk to me. About anything."
"I know. I do feel like I can," Becca said, shifting to meet his gaze. Her eyes were so blue. "I should've said something sooner. I'm sorry."
"I get it," Nick said. "I do. But the best thing about having a team is you get help carrying the load. You and me. We're a team now."
Becca smiled. "Always and forever."
"That's fucking right."
"I won't forget again," she said.
"You didn't forget, Sunshine. You're still just getting used to the idea. Me too." He hugged her in against him. "It's hard to lean on someone else when you've been so used to walking on your own."
She nodded. "I love you, Nick."
"I love you, too. There's nothing to worry about. I promise you," he said. And Nick was going to do whatever it took to make that the truth.
WHILE BECCA WAS in the shower the next morning, Nick let the guys know what was going on. Standing around the island in the kitchen, he said, "Becca is dealing with some PTSD from the abduction attempts. Probably triggered by returning to work, the scene of the first attack. And she keeps thinking she's seeing Woodson."
"Damn," Beckett said. "We all saw how roughly he treated her when he tried to grab her the second time. It's no wonder she's struggling."
Nick nodded, the memory souring the coffee in his gut. "Vance has given me some circumstantial evidence that places Woodson in South Carolina, where he's been since we interrogated him, but just keep your eyes open. Be on the lookout. I'm going to do a little more digging there to make sure Becca's not discounting something that's really there."
"Can't be too careful," Beckett said.
"No, not with Becca. That's for damn sure."
"This place is about as secure as we can make it," Marz said. "So we're good here. And the cameras from around the neighborhood are still up and running, so if you have Woodson's vehicle specs, I can keep an eye out. Make sure nothing's hanging around that shouldn't be."
"Do that," Nick said. "Thanks. I'm going to talk to hospital security this morning about this damn thing." He pointed to the stuffed animal, sitting in a plastic bag on the counter.
"Sick fuck," Shane said, glaring at it. "You should let Vance know about this, too."
"I will. I've got a whole to-do list on this today. I'm going to talk to Vance after I leave the hospital, and then I'm going to drop by the inn and work with them to beef up the event security they already provide. I don't want Becca thinking about anything besides having a good time on our wedding day."
Nods all around.
"What am I missing? Can you think of anything else?" Nick asked.
"Would it make her feel any better to carry a weapon?" Shane asked, his gaze serious.
"Good question. I'll talk to her about it. She doesn't have a concealed carry permit, though, and even though some of us flouted that law the past couple of months, I don't know if she'd be comfortable doing so." Though Nick and Beckett had had Maryland permits, the other guys were from out of state and hadn't. Carrying illegally was just one of the ways they'd had to work outside the law to clear their names. Now that they were opening a security firm of their own, all the guys were in compliance. Their business was going to have to be run completely by the book.
"Maybe you should get in contact with Chen," Beckett said. "If Vance can't track Woodson down, Chen sure as hell should be able to. He fucking owes us anyway."
Nick nodded. "True. I'll do that." Chen was the CIA operative who'd first assigned Frank Merritt to the undercover corruption investigation in Afghanistan that had snowballed into the shit storm of the last year. When Nick and his team had rescued Charlie and picked up the investigation, Chen had found them and offered the vital assistance that had finally allowed the team to take down the bad guys and clear their own names. Chen wanted them to work for the CIA from time to time, which meant he was predisposed to do them favors. And Nick wasn't above asking.
"Hey," Becca said, walking into the kitchen. Wearing a set of lavender scrubs, she looked fresh faced and beautiful. His sunshine.
"Hey," Nick said, hugging her in against him. "Want a bagel before we go?"
"Sure," she said.
Nick busied himself with the task, then turned around to find Easy wrapping her in his arms. "I'm here for you, Becca," he said in a quiet voice. One by one, the men repeated the action and the words. _This_ was what family looked like. It made Nick fucking proud. And, truth be told, it choked him up. Just a little.
An hour later, they were sitting across the desk from the hospital's chief security officer, a tall, wiry man with graying blond hair and a weather-beaten face. Barry Coleman had served for twenty years in the Marine Corps and worked in security for the past eight, facts that already made Nick feel a little better about leaving Becca there today.
Becca recounted when and how she'd found the stuffed animal, and Coleman asked a series of probing questions. Finally, he said, "Unfortunately, we don't have a security camera inside the staff break room. After Becca's attack, we secured and alarmed that external door, and we put cameras on all the main entrances into the ER, but I'll have one installed in there today. Just for some extra peace of mind. And I'll get my team on reviewing the personnel rosters and camera feeds from yesterday to see if we can pull together a list of people to talk to. This is harassment and intimidation, Becca, and we won't tolerate it for a second. I can promise you."
"Thank you," she said.
"In case it needs to be said, there's no chance Tyrell Woodson could get in here again. We have photographs of him posted at all the monitors. The whole security team knows what he looks like, including the BPD officers stationed in the waiting room," Coleman said.
"We have reason to believe he's out of the area anyway," Nick said. "We heard South Carolina."
Coleman nodded. "That's good to know. We'll get to the bottom of this, I promise."
Nick shook the man's hand, then he and Becca walked out through the back part of the ER. In the break room, Becca stowed her purse in her locker. "You sure you're okay being here?" Nick asked, his hands rubbing her shoulders. "No one would blame you for cutting out a few days early."
"I want to do this," Becca said. "I promise I'm okay. And I wouldn't hesitate to go to Coleman if something happened."
Nick nodded. "Okay. I'll be here at three to walk you out. Have a good day, Sunshine."
She smiled. "You, too. I can't wait to celebrate tonight."
"Me too," Nick said. Tonight all of them were having dinner together at a great local steak house before parting ways for their respective bachelor and bachelorette parties. Nick kissed her for a long moment, and he didn't want to admit how hard he found it to walk away and leave her.
But he had things he needed to do today to give them _both_ some peace of mind. Vance, Chen, the inn. Nick also wanted to drive by Woodson's last known address and make sure nothing was going on down there. Anything to help Becca feel better and get past the way his life had exploded all over hers.
It was the least he could do.
## CHAPTER 8
That night at dinner, Nick couldn't keep his hands off Becca. Despite the fantastic food, the great company of friends, and the well-deserved celebration, all he wanted was to get Becca alone somewhere so he could flip up the flirty skirt on the stunning little yellow dress she'd worn and get inside her any and every way he could.
Part of it was the top-shelf liquor flowing all around the table, and part of it was the relief Nick felt after all his efforts today had panned out in one way or another. Vance had found a parking ticket on Woodson's car from two days ago in South Carolina, and Chen had agreed to put his considerable resources into not only pinpointing the guy's location but also getting him off the street once and for all. The inn had agreed to additional security, and the head of the security company they used had even made the time to meet with Nick. The guy and his team seemed competent, smart, and savvy, so there was another thing on their side. And Woodson's last known address had not only been quiet as a grave but dust-covered to boot. No one had been there any time recently.
That still left the mystery of the stuffed animal, of course, but Coleman was on it, and Becca's day at work had been incident-free. They'd get to the bottom of that yet.
Becca had been visibly relieved when Nick had filled her in on his day. Now, she seemed so relaxed and happy that it made his fucking heart ache.
Sitting at the dinner table surrounded by their friends, Nick squeezed her thigh. She turned to him wearing a huge smile, a champagne glass in her hand. "Are you feeling frisky, Mr. Rixey?" she asked.
Nick leaned in close. "No, I'm fucking horny. I want to mess up your lipstick and tear off your panties and make my fingers and cock smell like you." He leaned back again, his face carefully neutral.
Her eyes were wide—and full of heat. "Holy shit. How am I supposed to be apart from you the rest of the night after that?"
"Welcome to my world, Sunshine." He threw back a gulp of whiskey.
"Come here. I want to taste that off your tongue," she said.
"Jesus," he gritted out, but it wasn't like he was turning down a kiss. She leaned in, giving him a great view of her cleavage down the front of her sequined strapless dress, and grasped his face in her hand. Her lips were warm, soft, and tasted like champagne and the chocolate mousse cake they'd shared for dessert. Fucking delicious. Her tongue slipped around his, and she pressed herself closer.
"Someone pull those two apart," one of the guys yelled.
Nick grinned even as they continued to kiss. He wasn't voluntarily giving up Becca's mouth, that was for goddamned sure.
"All right," Kat said from the other side of Nick. "We better get the rest of the night underway before we lose the bride and groom." Laughter all around as everyone got up from the table.
"Do you think they'd notice if we snuck away?" Becca asked, her face absolutely glowing.
Beckett grabbed Nick by the shoulders. "Get up, Rix. The tables are waiting for us."
"Apparently," Nick said. "You go have a good time, Sunshine. But you be ready for me later." He arched a brow.
"Oh, I will," Becca said, her tongue licking at her bottom lip.
Shit, he had it bad for her. And he fucking loved it. This woman was going to be his _wife_. How fantastically lucky was he? A man who just months ago would've said he didn't believe in luck, unless it was of the bad kind.
Outside, they found two massive stretch Hummer limousines waiting for them. Beckett had arranged their transportation for the evening through one of the companies he had experience working with—the cars were bulletproof and the drivers were prior military and armed. Nick appreciated the hell out of the gesture.
As the men headed for one vehicle and the women for the other, Nick pulled Becca into his arms. "Have fun, Sunshine. I love you."
"Love you, too, Nick." This kiss was softer, sweeter. Which was good, since all their friends started giving them shit.
"Yeah, yeah," Nick said, flipping the guys the finger. "Before you go, I have something for you," he murmured, then slipped a little wrapped box into Becca's hand. "Wear this and think of me."
"What is it?" she asked.
"I'm not telling," he said. Finding this present had been the other good thing he'd accomplished while she'd been at work.
With a little wave and a big grin, Becca turned to catch up with the women, her skirt twirling out and showing a dangerous amount of thigh. God, she looked gorgeous.
When she was safely tucked inside the Hummer and it pulled away from the curb, Nick got into his own limo and Shane pushed a fresh glass of whiskey into his hand. "Gentlemen, start your livers," Shane called out, loosening his tie and raising a glass of his own.
A round of laughter as everyone drank and the limo started moving. Colored lights ran around the tops of the leather seats, and a fully stocked bar filled one whole side.
Marz sat forward in his seat, a mischievous grin on his face. "If the ocean was vodka and I was a duck, I'd swim to the bottom and drink it all up. But the ocean's not vodka and I'm not a duck, so pass me the bottle and—"
"Let's get fucked up!" they all finished.
"Fuckin' A," Marz said with a laugh.
Beckett rolled up the sleeves of his dress shirt and held up his glass. With a sly grin, he said, "I'll keep mine short and sweet. May all your ups and downs be between the sheets."
"Hear fucking hear," Nick said, taking a drink and laughing at the blush filling Charlie's cheeks. He and Jer were in for a rude awakening—Nick's teammates were fucking fish, and it'd been a damn long time since they'd had a night like this to just cut loose. Hell, it'd been way more than a year since they'd last done it together.
"All right," Easy said, holding up his glass. "I'll play."
"Yes sir, E," Marz said, grinning.
The guy smiled, and it made Nick realize how much Easy had changed in the few months they'd all been back together. A few weeks into their investigation, he'd admitted to them that he'd been badly depressed and battling suicidal thoughts. They'd all been gutted to know how bad Easy had been silently struggling, but they'd banded together around him, and Shane and Emilie had made sure he'd gotten the medicine and therapy he'd needed to fight the demons in his head. "Here's to a long life, and a merry one. A quick death, and an easy one. A pretty girl," he said with a wink at Nick, "and a loyal one. A stiff drink, and another one." Another round of bottoms up. At this rate, they were going to lose every dollar they owned at the casino, and Nick didn't give a shit.
"I'm not good at this," Charlie said with a sheepish smile. "But I'll give it a go." Jeremy grinned at him as Marz clapped him on the back. "To Nick, if you hurt my sister, I'll kill you in your sleep."
For a moment, the words hung there, then everyone burst into guffaws. Yeah, Charlie Merritt fit in just fine.
"No worries. I'll fucking drink to that, Charlie," Nick said, laughing. He took a big gulp of whiskey, enjoying the bite as it went down. "Okay, I've got something to say. First, to nights and friends I'll never forget." Holding his glass high, he looked each man in the eye. "And second, to our enemies."
" _Fuck you!_ " they all called out.
"Amen," Nick said. But tonight wasn't a night to worry about enemies. Tonight was a night for celebrating the good things in life. And if Becca hadn't already done it, being with all these guys was making him realize exactly how much good Nick had.
"OOH, I HAVE a fun idea," Kat said, pulling out her phone. "Everyone take either a cleavage shot or an upskirt shot and text it to your guy. Make 'em remember what they're missing out on tonight." She tugged down the V-neck of her emerald green satin dress and took a picture of herself. A few flicks of her fingers, and she said, "There. Go on, now. Make 'em sweat."
Becca could only laugh as she lifted her skirt and took a picture of the virginal white panties she wore, complete with glittering sequins. They'd made her feel very bridal. "I love this idea," she said, shooting off a text to Nick. Then again, she'd already been three glasses of champagne into Happyville before they'd left the restaurant, and Kat had given her a fourth when she'd gotten into the limo. So she was prone to love just about any idea just then. "Ooh, I'm sending Nick one of each so he can see how beautiful this necklace is on me," she said, taking a shot down the top of her dress but making sure to get the incredible yellow diamond sun-shaped pendant he'd given her into the frame. She loved him so freaking much.
"Oh, my God," Sara said. "I have no cleavage to speak of, people. But Shane did like these red panties I have on." Awkwardly and with a lot of blushing, she managed to take a picture up the skirt of her little red dress. "You, on the other hand, have great boobs, Jenna," she said to her sister.
"I already sent mine," Jenna said, looking very pleased with herself. She'd worn a form-fitting black dress that gave her the most enviable hourglass shape. On Becca's last day off, they'd gone shopping for new dresses for tonight and the rehearsal dinner, and all the time Becca had gotten to spend with these women was making her fall in love with them even more. She had women she was friendly with at the hospital, but it hadn't been since nursing school that she'd last had truly close friends. Best friends. Now she had four of them.
"I need help with mine," Emilie said, grinning. She handed her phone to Kat, who sat next to her. Laughing, Emilie turned and got on all fours on the seat. She pulled up the bottom of her gold dress just enough to reveal a really tiny pair of satin black panties.
"I knew I liked you," Kat said, taking the picture. "And why am I not surprised that Derek is an ass man?"
"Oh, my. This is going to be a night of TMI, isn't it?" Sara asked, sipping at her champagne.
"Yes," Emilie said, sitting down again. "But if I've discovered anything, it's that life is too damn short and uncertain to hold back."
"I'll drink to that," Becca said, draining her glass.
And that's when all their phones started blowing up. Laughter filled the limo as they all read the guys' reactions to their selfies. Becca couldn't stop grinning—or fantasizing—about Nick's reply.
_I'm going to tear those fucking things off with my teeth. Count on it._
His reply to her second selfie made her all warm inside.
_You are so beautiful. My sunshine._
"Okay, as much as I want to get you drunk, I also don't want to see you sick." Kat handed Becca a bottle of water and grabbed one for herself. "Drink this before you have any more champagne."
"I will. But why aren't you drinking?" Becca asked. Kat hadn't touched her champagne at dinner, and she was the only one of them without a drink now.
"So I can take care of you," Kat said. "Besides, I'm naturally high on life. Runs in the family. Well, at least with Jeremy and me, anyway."
Becca laughed. Kat and Nick were alike in so many ways, and their stubbornness often had them butting heads. By the time the limo pulled to a stop, Becca had dutifully followed orders and emptied the bottle. "I'm so excited to see what we're doing," she said. The girls had insisted on keeping it a secret.
The driver, an older man named Tony, whose military bearing reminded Becca of her father, opened the door. They spilled out onto the street, and Kat wrapped her arm through Becca's as they walked up to the doors of a posh salon and spa. "We're getting completely pampered. Anything you want. The place is all ours for the night."
"Oh, my God," Becca said. "This is the coolest thing ever." And she didn't know the half of it until they were inside. There was more champagne, a table full of chocolate-covered strawberries and Godiva truffles, and a mountain of presents.
"You never got to have a shower," Emilie said. "So consider tonight your combination shower and bachelorette party."
Becca was completely overwhelmed by the thoughtfulness and perfection of the whole thing. "You shouldn't have done all this," she said. "But I'm really, really glad you did."
They all traded their dresses and heels for robes and slippers and settled in for pedicures. One of the ladies from the spa kept Becca in a continuous stream of drinks and treats and presents. So many presents. A gorgeous lingerie set for her wedding night. A big basket of body lotions and spa products and makeup. A set of pillow cases that read _Mr. Right_ and _Mrs. Always Right._ A trio of crystal picture frames. A happily ever after wish jar with their names on it to be put out for people to fill at their wedding. A pair of red satin panties with the words _You got lucky_ on the crotch. Sparkling, drop diamond earrings. A beautiful framed print that had Becca and Nick's names, their wedding date, and the words _And they lived happily ever after._
"I love everything so much," Becca said as the woman put the finishing touches on her purple toe polish. "Not that long ago, I had almost no one in my life. My parents were gone. Charlie had distanced himself. And I threw myself into work just to fill the void. Now, I can't believe everything I have. I'm so grateful for each and every one of you."
"I've never had friends like this before," Sara said, blinking fast. "Sometimes I'm afraid that I'm dreaming and I'll wake up."
"It's not a dream, Sara," Jenna said. "What happened is that you woke up from the nightmare. _This_ is your reality now."
"Oh, God, you guys are going to make me cry," Emilie said. "I think we need more champagne. And chocolate."
Kat's pedicure was already done, so she brought the bottle around and refilled everyone's glasses. In all, Becca was treated to a pedicure, a manicure, and a mini facial, and she had her eyebrows shaped. She hadn't felt so relaxed in forever. No doubt everything Nick had learned earlier in the day helped, and for the first time since she'd returned to work, she felt hopeful.
It was after eleven by the time they were all dressed again and had the presents and leftovers packed up to take home. Tony carried everything out to the Hummer for them, then Becca noticed him talking quietly with Kat.
"Is everything okay?" Becca asked, joining them at the front door of the spa.
"Yes, Miss Merritt," Tony said. "There's a man hanging around down the block who's drunk and belligerent. He threw a bottle at a passing car earlier. And he gave me a little bit of guff when I asked him to move away from the Hummer. He left without incident, but I don't want any of you stepping outside until we're ready to get in the vehicle and depart."
Through the haze of champagne and sugar, Becca's gut clenched. "Okay, of course."
"Don't worry," Kat said. "It's just a precaution."
When the other women emerged from the bathroom, Tony said, "Ladies, I'd like you to move directly into the limo once you're outside, please." He went out first, paused as he opened the door, and waved them out.
A half block down the street, a tall, thin man wearing too-big pants and an oversized black hoodie with the hood up skulked in a circle, his arms waving and his body gesticulating like he was having an argument. From this distance, Becca couldn't make out the man's face, but she couldn't deny the relief she felt at the fact that the man was way too thin to be Woodson, who'd been bulky and muscular. Not that she should be worrying about Woodson. Nick's research today really had made her feel a lot better.
Kat bustled Becca into the limo, then climbed in after. As the other women got in, Becca could just make out the man shouting.
"You think you so fucking better than me!" he yelled, his voice full of drunken slur. "Well, you not! You not! And I'm gonna show you! I'm gonna show you!"
The minute Emilie was in, Tony had the door secured behind them, cutting off the rest of the tirade. Almost immediately, the engine started and the Hummer eased away from the curb.
"Please don't let that tarnish your night," Kat said.
Becca smiled. "Not at all. Nothing could tarnish this night. It was fantastic. Perfect. One of the best ever." She meant it, too. And the whipped cream on her cake? In just a few minutes, it would be midnight. And that meant in just one week, Becca was going to be married to the love of her life.
And absolutely nothing could ruin the amazing miracle of that.
## CHAPTER 9
Nick," Chen said when he called on Wednesday morning. "I've got bad news."
"Shit, what is it?" Nick asked. When a guy like Chen said he had bad news, you knew it was _bad_.
From the driver's seat of his truck, Shane frowned, his expression full of questions. He parked the truck in front of the dry cleaner. They'd dropped Becca off at work a half hour before and were picking up their uniforms for the wedding.
"Woodson's in Baltimore. Has been for at least a week, maybe longer."
Nick's gut dropped to the floor, his mind racing. A week? That meant he'd been in town long enough to be responsible for the stuffed animal, for Becca's feeling of being watched at the mall, and maybe even for some of her sightings that they'd thought were impossible and chalked up to her PTSD. "Goddamnit. Are you sure? How do you know?"
"I put a guy on the ground in South Carolina. He learned from some locals that Woodson had left town and traded vehicles with his uncle. I managed to track the uncle's truck to a rest stop near Richmond, where another car had been reported stolen. That car was found abandoned in Baltimore County last week, which we just put together. Otherwise, the guy's been way off the grid. No credit cards. No known vehicles. I have two undercover agents in the city looking for him from within the gang scene. As soon as we locate him, we'll grab him."
"Fuck," Nick said, the weight of this new development crushing in on him. "Thanks for letting me know. Keep me posted." They hung up. "Head back to the hospital. Now," Nick said.
Shane had the truck in reverse and barreling out of the parking lot immediately. "Talk to me."
"Woodson's in town. Has been for over a week. We got fucking outfoxed." Nick dialed Becca. It went to voice mail. "Please call me as soon as you get this, Becca." Ice sloshed into his gut as Nick filled Shane in.
"No one stays off the grid like that unless they fear they're being hunted. Or they don't want to be noticed," Shane said, running the tail end of a yellow light.
Nick appreciated the hell out of his friend's aggressive driving. He really did. "Given the situation in the city, it's probably some of both in this case. But I'm a helluva lot more worried about the latter."
"Roger that," Shane said, darting around other cars as much as he could.
Nick tried Becca's cell again. Voice mail. Damnit. He was about ready to crawl out of his skin. Flipping through the contacts on his phone, he found the number for Barry Coleman at the hospital.
"Mr. Coleman, this is Nick Rixey, Becca Merritt's fiancé," Nick said, his knee bouncing as he scanned his gaze over the street as it flew by.
"Nick, what can I do for you?"
"I need you to find Becca and keep an eye on her until I get there. She's not answering her cell, but I know she might be with a patient. I just got word that Tyrell Woodson is back in town and has been for more than a week. Since we still don't know who pulled the stunt with the stuffed animal, I'd feel better if Becca left early today until we get to the bottom of this and know what Woodson's up to. It seems he took some pains to get back into the city unnoticed."
"I wish I had your connections for intel," Coleman said.
"Yeah, well I wish I didn't need them."
"I hear you," the other man said. "I'll find Becca and stay with her until you get here." They hung up.
In another five minutes, Nick and Shane made it back to the hospital. Nick barely waited for Shane to bring the truck to a stop before he was opening the door. "Pick us up near the ER entrance. It's more sheltered."
"You got it," Shane said.
Nick rushed across the plaza to the main entrance, his gaze scanning the streetscape, the crowd, the sea of faces. He let his guard down for five goddamned seconds, and this was what happened. Becca, potentially exposed to danger and completely unaware.
Inside, he made his way to Coleman's office. Relief flooded through him.
Becca. Sitting across from Coleman at his desk. Her face was a shade too pale, but otherwise she was safe, sound, a fucking sight for sore eyes.
"Nick," she said, rising as soon as she saw him. "He's back?"
Nick cupped her face in his hands. "Yes, but Chen's on it. Woodson won't be free for long. Don't worry, okay?"
She gave him a doubtful look that was like a knife to the gut.
Nick turned to Coleman. "Thanks for your help." They shook hands.
"Anything else you need, you just let me know," the man said.
Taking Becca's hand in his, Nick led her to the main ER entrance, keeping back from the glass until he saw Shane's big black truck pull into the drop-off lane. "That's our ride. Come on."
They jogged toward the truck, Nick's gaze doing a constant scanning circuit as they moved. He got Becca into the truck's backseat, shut her door, and moved to his own—which was when his eye caught it. A glint of morning sun off metal. There at the corner of the building.
Nick opened the passenger door just in time, the report of the gunfire reaching his ears only a second before the round pinged off the door. Close. Too damn close. He dove into the cab. "Go, go!"
"Fuck!" Shane punched the accelerator and pulled the truck into a hard U-ey. Another round hit the back quarter panel.
"Get down, Becca," Nick said as he reached for the Glock at the small of his back. But she was way ahead of him, tucked in a ball on the floor behind his seat. He didn't have a clear shot of anything, especially with chaos already breaking out around the ER's entrance as people dove for cover and the couple of on-site police officers rushed into defensive positions.
In what felt like long minutes but was only a few seconds, they were clear of the area. Shane ran a red light to get them away from the hospital altogether.
Twisting in his seat, Nick looked out the cab's rear window. "Watch for a tail."
"On it," Shane said. "You need to alert the team, Vance, Chen."
"Yeah," Nick said, but first he needed to check on Becca. Christ, every reassurance he'd offered the past week had just been blown to shit. His gut was a wreck, his mind unhelpfully crafting one horror story after another about what might've happened if Chen hadn't called. Or if Nick hadn't returned to the hospital immediately. Or if Coleman hadn't been able to pull Becca off the floor. "Becca, are you okay?"
"I don't know," she said, her voice shaky. When she looked up at him, her skin was ashen.
He reached back and clutched her hand as he dialed Marz.
"Yo, hoss, wassup?" Marz said.
"We've got a situation," Nick said, filling him in. "Let everyone know what's going on. And if you have a chance, scan the security feeds around our neighborhood looking for anything potentially suspicious. We've been keeping an eye out for the wrong damn car."
"You got it," Marz said.
Nick had just hung up when his cell rang again. Chen. Nick put it on speaker.
"I heard about the hospital. You okay?" Chen asked by way of a greeting. Nick wasn't surprised that Chen had information that was only minutes old.
"Yeah, we're in one piece," Nick said. "But it was fucking close. Too close."
"Damnit. Wanted to let you know we have a lead on where Woodson's been staying. Putting together a raid for tonight as we speak."
"Well there's a bit of good news," Nick said. "You need backup?"
"No, you stay hunkered down. I'll let you know when it's done."
"I need you to take this guy out," Nick said, anger lancing through the words. If it had just been him in danger, it would have been one thing. But now it was Becca. Now it was _his family_. And that was a whole other goddamned thing. "I need this situation to go the fuck away."
"I hear you," Chen said. "And I'm working on it." He clicked off.
"If anyone can take care of this, Chen can, right?" Becca asked from the backseat. A little color had returned to her cheeks.
"Yes," Nick and Shane both said at the same time.
When they got back to Hard Ink, Marz had everyone else assembled and briefed in the big unfinished space across from their apartment that was part gym, part war room. It was where they'd run the whole of their investigation against the Church Gang and the mercenaries who'd killed Becca's father and smuggled heroin from Afghanistan into Baltimore. Nick wasn't thrilled at all about the similarity of this meeting to the many they'd held during the investigation they'd thought was done. Closed. Behind them once and for all.
Except it wasn't. Because sometimes the past wouldn't fucking die.
"Chen's people think they've discovered where Woodson has been holing up. They're going after him tonight. His actions at the hospital demonstrate his intention to get revenge, so until we hear from Chen, we're back on lockdown again. I don't want anyone leaving the building today," Nick said.
"What if they don't get him?" Becca asked from where she sat on a folding chair near Marz's improvised desk.
The other team members traded looks with Nick. "I don't know the answer to that yet," Nick said.
Becca nodded. "Do we need to think about postponing the wedding?"
The question was like a punch to the gut, especially because he'd been asking himself the same thing. Fuck. "Not yet," Nick said. A bleak sadness filled Becca's baby blues, and the fact that this scumbag had managed to hurt her yet again lanced boiling hot rage through his blood. Enough was efuckingnough.
The meeting broke up, and the day crawled by like an inchworm moving in reverse.
Nick spent hours worshipping every inch of Becca's body, hoping to keep them both distracted from everything that was at stake as long as he could. The women made six batches of homemade chocolate chip cookies. They watched movies until they were all cross-eyed. And still it wasn't time for the raid.
Finally, a little after ten o'clock, Nick's cell rang with a call from Chen. The devastating news was that they'd apprehended a number of former Churchmen—but Woodson hadn't been among them.
"Well, what's next?" Nick barked into the phone. "This guy came after Becca three times. He's not going to stop."
"I know, Nick," Chen said. "We're interrogating the Churchmen we brought in. We'll find him."
But how fast would they find him? And would Chen find Woodson before Woodson found Becca again?
Because Nick would never survive if something happened to the only woman he'd ever loved.
## CHAPTER 10
Chen showed up at Hard Ink Friday morning. His people still hadn't caught Woodson. And Becca was beside herself. She couldn't believe . . . so many things. That Woodson was back. That they might have to cancel the wedding. That maybe she really had seen Woodson some of the times she'd chalked it up to her imagination.
The whole group gathered in the gym, and Chen sat in the middle of them, wearing his usual, nondescript khaki pants and light blue button-down. Chen wasn't his real name, but it was the only one they knew him by—the one that had been on the nametag fastened to the stolen doctor's coat Chen had been wearing the first time Nick had seen him. They'd been at the hospital where Jeremy and Kat had been treated after the funeral.
"I have a proposal for dealing with Woodson," Chen said, scanning the group and finally settling his gaze on Nick and Becca.
"Let's hear it," Nick said.
"I get the word out on the street that Becca is going to be at the restaurant tonight for her rehearsal dinner. We lure Woodson to us rather than wait for him to come at you." Chen's matter-of-fact words hung there for a long moment.
"No," Nick said. Just as matter-of-factly.
This wasn't the first time it had ever been proposed that they use Becca as bait. Nor was it the first time Nick had reacted negatively to the idea. "Nick, Becca said."
"No, Becca. We've been there, done that, and you got hurt," he said. The fierce protectiveness in his gaze made her love him even more.
"I know, but the last time, we also caught Woodson and got information out of him that saved Charlie's life. So it worked," she said. Tension hung so thick in the room you could cut it with a knife. "The alternative is that we cancel the wedding, stay shut up inside the building, and wait it out, right?"
Chen nodded. "We will get him. It's a matter of when, not if."
"I believe you," Becca said. "But when kinda matters a lot right now. The wedding can be rescheduled if we need to, of course, but none of us wants this hanging over our heads. If we can end it tonight, let's end it."
"I agree," Kat said, looking at Nick with sympathy in her eyes.
"So do I," Beckett said. "We'll all be there. We'll all be armed. Nothing's going to happen to Becca or anyone else."
"And my team will be there," Chen said.
"Is this what everyone thinks?" Nick asked, his voice like gravel. He crossed his arms and surveyed the group. Nods and affirmatives all around. "Fuck. Then what's the plan? Because I want it to be goddamned foolproof."
For the next hour, the guys strategized. Chen had brought plans for the Italian restaurant where they were scheduled to go, as well as a big map of the surrounding streets and alleys. He'd arranged to have surveillance on the restaurant starting immediately, to make sure no one arrived early and found a place to lay in wait. When they were done, Chen put in calls to his undercover contacts to get the word out. He was apparently confident enough in the way information moved within and between gangs to think that the word would make it to Woodson in time if in fact he was actively looking for her. Worst-case scenario, it didn't, and he didn't show. And then they were right back to square one, but no further behind.
After Chen left, Becca turned to Nick, where they were all still gathered in the gym. "I want to be armed tonight."
"Me too," Kat said.
Nick and Beckett traded a look, but then Nick nodded. "Everyone who's comfortable handling a weapon should be armed. I want redundancies on top of redundancies where safety is concerned. And for the record, I fucking hate this."
Becca wrapped her arms around Nick and laid her head against his chest. "I do, too. But I hate being scared more."
Nodding, Nick said, "I've got a few calls I want to make. But then I'll be wrapped up here."
"Okay," Becca said. "Maybe I'll go throw together some lunch for everyone." Staying busy was the best way to keep from going crazy. At least, it had worked for her during the team's investigation. No reason why it shouldn't now.
"I'll help," Emilie said.
"Me too," Sara said.
In the end, all four women made their way back to the apartment with Becca, and she appreciated the silent show of support. She really did. They decided on tacos, and everyone got to work chopping veggies and browning the meat. Emilie apparently made a mean spicy Spanish rice, so she took charge of that.
As Becca stood at the stove, the whole thing suddenly crashed over her like a tidal wave. The shooting at the hospital. Knowing everyone would be in danger tonight. The prospect of facing Woodson again—for real this time. "Can you watch this?" Becca asked Emilie, laying down the big spoon with which she'd been stirring the ground beef. "I'll be right back."
She rushed down the hallway and ducked into Kat's sometimes-room rather than her own—the one she shared with Nick. First, because she wanted to be alone in case Nick returned from making his calls. Second, because her wedding gown was in this room, hanging on the outside of the closet door. Luckily, they'd picked up their dresses from the bridal boutique before the lockdown had started, and now it was here waiting for her.
For whenever Nick and Becca were finally able to get married. Because her gut told her it wasn't going to be tomorrow.
Becca sagged down onto the edge of the mattress, her gaze drinking in the gleaming white satin and the sparkling beadwork at the waist. Her lip quivered and her eyes pricked, but she wasn't giving in to tears. She was done crying. Now she was just fucking angry.
_It doesn't matter, Becca. The wedding is just one day in a forever that lasts the rest of your life. It's just one day._
And it was. She knew it. But their love had overcome big obstacles—Nick's initial belief that her father had betrayed him, sophisticated and numerous enemies, multiple attempts on their lives. What they had was hard-fought and well-earned. They _deserved_ a day of celebration and happily ever afters.
Two soft knocks sounded against the door.
Becca straightened her back, took a deep breath, and said, "Come in."
"Hey," Kat said, leaning in the doorway. "Can I join you?"
"It's your room," Becca said with a small smile.
Kat shut the door and sat down on the bed next to Becca. "It really is an amazing gown. Nick is going to swallow his tongue when he sees you in it."
It might've been the first time all day Becca smiled. "I am looking forward to seeing him see me in it for the first time."
Grasping her hand, Kat nailed her with that Rixey stare. "You just have to hang in for a few more hours. This is almost over."
"I know," Becca said. "I know. But is it bad that I want to be the one to end this asshole's life once and for all? I just want to see the consciousness bleed out of his eyes so I can know it's over for good."
"Not even a little bad," Kat said. "You guys deserve a happily ever after."
Becca chuffed out a humorless laugh. "We all do. But sometimes I'm afraid all we're going to get is . . . a hard ever after. You know?"
Kat's gaze was full of determination. "This guy's going down one way or the other, Becca. And besides, a hard ever after sounds like it could be good to me. I mean, _you know,_ hard can be good."
That eked a smile out of Becca, and that smile turned into a chuckle. "Yeah, hard can be good."
"No," Kat said, grinning now. "Hard _is_ good. Really fucking good."
It was stupid and childish and ridiculous, but as the hard jokes started flowing out of them, they descended into outright crying giggles and really unattractive snorting that was a helluva better release than tears could ever be.
"Thank you," Becca finally managed. "I really needed that."
"Good. Now come on, let's go eat." Kat pulled her up from the bed.
"I'll catch up," Becca said. "Need to use the bathroom." She headed back toward her room, passing Shane and Sara's on the way. Their door was open, and something caught Becca's eye. She stepped back to the opening and peered in at Shane's big medic case just inside the doorway. Becca shut herself inside the room, knelt, and opened the lid to the case.
When her gaze finally landed on a bottle of injectable diazepam, an idea came to mind. Nick wanted redundancies upon redundancies; well, this certainly fit. And given that Woodson had gotten his hands on her twice before, she wanted a way to hurt him up close if it happened again. Without letting herself question what was probably a totally useless idea, she filled a syringe with a dose guaranteed to induce sleep in a man Woodson's size. Quickly, she put everything away and tucked the syringe into the pocket of her dress, which hung in Nick's closet. No one even had to know the syringe was there.
One thing was for sure, Becca Merritt was done feeling like a victim. She was ready to fight for this life she wanted, once and for freaking all.
THEY ARRIVED AT the inn early for the rehearsal. Before Becca even stepped foot out of the stretch Hummer, Nick wanted to take a look around and discuss a plan with the security he'd brought in extra for the rehearsal. This was the same team who'd be running things at the wedding, so they already understood the nature of the threat from when he'd met with them earlier. To add another layer of security, Chen had arranged to put agents from his detail at either end of the long block on which the inn sat, providing an effective roadblock and a defensive perimeter.
It was all likely overkill. Nick knew it was. The bait would lure Woodson to the restaurant, not here. Hell, they hadn't even published a wedding announcement, so the details of their arrangements could only be known by people with whom they'd shared invitations.
When he was satisfied, he opened the door to the ladies' limousine. "Let's practice us a wedding," he said with a smile. He might be tense as hell inside, but he didn't want to do anything—well, anything _more_ than what had to be done—to take away from the joy of the occasion. Becca deserved a happily ever after, and Nick was determined to give her one. No matter what it took.
Becca stepped out of the limo in a stunning long blue dress that made the color of her eyes almost glow. She adjusted the long decorative chain of her purse on her shoulder. He'd given her a small handgun that would fit the bag, and even though she knew how to use it—thanks to her father—Nick really fucking hoped she never had to. But he couldn't have agreed more that she should have the protection on her. It was _always_ wise to be prepared for a snafu, that was for damn sure.
Sonya greeted them at the gatehouse and guided the group of them inside. The first floor of the inn was where the cocktail party immediately after the ceremony would take place while they were taking wedding photographs. A tall round table filled with champagne flutes stood in the center of the floor. "Please help yourself while you wait for the minister to arrive."
"Do you want to see the upstairs where we'll hold the reception?" Becca asked the women.
Even though the security team had assured them that the house had been locked up tight all day, with no one coming or going until they and Sonya had arrived to open up for their rehearsal, Nick wasn't comfortable letting Becca go alone. In the end, everyone went along for the tour. It really was a nice place. The kind of place where happy memories were made. Nick wanted that for Becca. For both of them.
They didn't have to wait long for the chaplain to arrive. Nick didn't know the man personally, but Thomas McAdams was a military chaplain who was a friend of a friend, which was how they'd managed to book him on such short notice. Nick guessed that McAdams was not much older than he was, and the chaplain seemed eager and interested and kind. Nick immediately liked him.
McAdams greeted the whole wedding party, then Sonya gave him a general tour of the outdoor space where the ceremony would take place. They'd process up the brick-lined courtyard between chairs that were already set up for them, and the ceremony itself would take place on a raised brick patio against the backdrop of the rear of the gorgeous Carroll Mansion. A plain wide wooden arch stood there now. Tomorrow it would be decorated with flowers.
As Sonya, McAdams, and Becca spoke, Nick couldn't help but be on the alert. His eyes and ears were wide open, despite the fact that things were quiet. Exactly as they should be. And he wasn't the only one. Every one of his teammates had their game faces on. Nick was fucking glad.
"Okay, now that I have the lay of the land," McAdams said, "I'd like to go over what you all indicated you wanted for the ceremony. Then we'll walk through it as a group, and I'll show you where to walk and stand, all that sort of thing. And then we'll do an actual dry run, music and all if you have it."
"I have it on my phone," Becca said.
"That'll work," McAdams said. For a few moments, they discussed vows, wording for various parts of the ceremony, who was holding the rings, and how they wanted him to introduce them to the audience when the ceremony was complete. "Okay, let's walk through it all. Anyone should feel free to interrupt this time with questions or suggestions. In fact, _now_ is the main time to ask questions," he said. "Becca, where do you want to start the procession from?"
Sonya stepped forward. "If I might make a suggestion," she said, guiding them back into the first floor of the inn. In the back corner, there were three doors. "There are two private party rooms here we've set up for the bridal party to use before the ceremony." She let them into one and turned on the lights. The rectangular room had a long table that seated eight in the center, with old-fashioned leather armchairs clustered in groupings along one wall. A floor-to-ceiling mirror hung at the far end of the room. "Both rooms are identical. These usually work nicely to allow for last-minute preparations, privacy if the bride doesn't want to be seen by the groom beforehand, and staging for the ceremony itself."
"Yes, that's what we'd discussed. I don't want him to see me until I'm walking down the aisle," Becca said, smiling at him. Nick couldn't fucking wait. "So then we'd be processing from inside?"
"That's right," Sonya said, leading them back out to the main room. "The groomsmen will join Nick at the dais ahead of time. We'll keep these doors shut until it's time for the women to walk. When the music changes, we'll open the doors and the bridesmaids will go one by one. And then the music will change again to whatever song Becca has chosen to walk to, and she'll proceed as well. Is someone walking you down the aisle, Becca?"
"Oh." Without hesitation, she turned to Charlie. "Will you? Please?"
Her brother's face filled with emotion. "Yeah. Of course."
Becca pressed a kiss to Charlie's face. "Thank you. I'm sorry that in all the craziness I didn't think of this sooner." He shook his head, clearly moved by her request.
"Very good," McAdams said. "Let's head outside to see how we'll arrange things at the dais."
As they walked the length of the courtyard, Sonya said, "Remember not to walk too fast, bridesmaids. It's longer than usual, so everyone always wants to run up the aisle." The women all laughed. When they got to the raised brick porch, Sonya directed each of them to where they should stand for the ceremony itself.
After that, McAdams sped them through the words so they knew what would happen. And then Sonya was guiding them back down the aisle again to show them where to stand for the reception line they were going to form before returning to the porch for group pictures.
This all seemed pretty basic to Nick, and frankly he was a little impatient with it. But that was just the broader situation talking. He tried to block it out as much as he could, especially since Becca seemed to be enjoying herself. And given what they were yet to face at dinner, he didn't want to do anything to ruin that.
"Okay, let's do a full run-through," McAdams said. "Gentlemen, except Charlie, come with me."
Nick frowned and turned to Becca.
She smiled and winked. "Go ahead. I'll be right there."
He gave her a quick kiss. "You're already the prettiest bride there ever was."
"Aw, sweet man. I don't care what anyone says, I'm going to run up this aisle to you," she said as he headed for the door.
"I won't object to that at all," he called over his shoulder as he made his way toward the mansion.
And then Sonya guided the women back inside and closed the doors behind them.
## CHAPTER 11
This is so exciting, Becca," Sara said, looking around at the elegant room. "This place is so gorgeous."
"I know. I fell in love immediately," Becca said. The beautiful summer night, the sweet fragrance of the garden flowers, the ambiance provided by the historic architecture. Tomorrow was going to be amazing.
Assuming it happened.
No, it _was_ happening. Becca just had to believe it.
"Becca," Sonya said. "If you have your phone, I can hook it up to the sound system so it plays in the courtyard. Just show me what songs you want."
Becca opened her wedding playlist. "This is the music that can play while people are being seated," Becca said, pointing to a list of songs. "This one is for the bridesmaids' procession, and this one is for mine."
"Very good." Sonya hooked up the phone. Classical music immediately filtered in from outside. Sonya had Becca line the women up in the order in which she wanted them to march, and the music changed for the bridesmaids. "And now we're marching," Sonya said as she opened the doors. The warm twilight air spilled in. White lights twinkled in the trees and on lines strung across the courtyard. It was magical.
One by one, the women walked out the doors. Jenna, Sara, Emilie, then Kat. And then it was just Becca and Charlie left to walk down the aisle. She turned to him. "I'm so lucky to have you as my brother, Charlie. I just want you to know how grateful I am that all of this has brought us closer again." Next to finding Nick, that was one of the brightest silver linings in everything that had happened.
He gave her a small smile. "I feel the same exact way," he said.
"Okay," Sonya said from beside the doors. "Now I'll change to the wedding march, and it'll be your turn."
"Shut the fucking door," came a deep voice from behind them.
Becca whirled. She recognized the voice immediately. Woodson. For real this time.
"Who are you, sir?" Sonya asked. "You can't be in here."
He brandished a gun directly at her. "I _said_ , shut the fucking door. Now."
As the blood drained from her face, Sonya hastily pushed the doors closed.
Becca fumbled for her purse, but the decorative metal clasp stuck.
"Don't move, Becca. Don't even fucking breathe. In fact, toss that bag down. Now." Woodson stalked closer.
Which was when Becca realized he'd lost _a lot_ of weight since she'd seen him two months ago. His eyes and face had the sunken, haggard look of an addict. He'd let his hair grow back in, too. He was no longer bald. "It was you. On the street the other night."
"Told you I was gonna show you. And when I found out about your wedding, I thought, what better time. Was gonna wait 'til tomorrow. Do it up right for the big crowd. But I figure, I got you now for sure."
Charlie moved just the smallest amount to angle himself in front of her, and Woodson tracked the movement like a hawk. He trained his weapon at Charlie's head and arched an eyebrow. "Get down on the floor," he ordered Becca's brother. "You, too." He glared at Sonya, who rushed to comply. Charlie sank down more slowly, and Becca could feel the anger and frustration rolling off of him.
Becca needed to distract Woodson from whatever he planned to do. Delay him, at the very least. At some point Nick was going to realize something was wrong—she had faith in that into her very bones. He wouldn't let anything happen to her. So she just had to hold on for a short while.
"I told you to drop the bag. Do it _now,_ " Woodson barked out, punctuating his words by jabbing the gun at the air.
Her belly sinking, Becca dropped the purse next to her feet.
Woodson gave her a droll stare. "Don't fucking play me. Kick it away."
She did, a little of her hope going with it as it slid across the hardwood floor. "How did you know about my wedding?" she asked, wanting to keep him talking.
"Yeah, that was some lucky shit, wasn't it? For me, anyway." He stalked closer, slowly, like he was paranoid despite the fact that he was the one with the weapon. "Little sister of a Churchman who was killed works in housekeeping at the hospital. Guess someone left an invitation out in the staff break room."
"So she's how you got the stuffed animal in to me," she said, her voice shaky. She hadn't been freaking out for nothing after all. Woodson really had been lurking around the edges of her life. For how long she wasn't sure. But what a lesson to trust her instincts.
His grin was sadistic and cruel. "Enjoyed my little calling card, did ya?" He didn't give her the chance to answer, because as soon as he was close enough, he roughly grabbed her arm and tugged her against him. He spun her so her back was to his front, then he dragged her away from Charlie's reach. His forearm pressed savagely into her throat, choking her as she struggled to keep her heels under her. "Because of you," he hissed into her ear, "I lost everything I had. And now you're going to see how that feels, starting with him."
Woodson lifted the gun.
Becca had to do some—
The syringe!
In her rage and terror, it seemed to her that he moved in slow motion. And that she did, too. An eternity seemed to pass as he took aim and she reached into the pocket of her maxi dress. Uncapped the syringe. Jabbed it into Woodson's thigh.
The world froze for an eternity.
Woodson shouted.
The gun fired.
She stumbled as he did, his arm still squeezing her throat.
And then she was falling, falling backward with him, as the gun fired again.
NICK WATCHED AS the women made their way up the aisle. When Kat got closer, she grinned and made a face at him, but then her gaze shifted to his left. To Beckett. And hell if the look his sister was giving his friend wasn't how Becca sometimes looked at Nick. For as much as Nick had been thrown by Beckett's interest in Kat, the two of them had proven to be damn good for each other.
As Kat took her place, Nick searched for Becca. And found the doors to the first floor closed again. He frowned. Maybe it was to allow the bride to make a grand entrance once the wedding march began? Nick stretched his neck and rolled his shoulders. Probably made him an asshole, but he really wanted this rehearsal to be over.
He stared at the doors. The same music continued on. A prickle ran over his scalp. He looked to his teammates, standing at his left. But his gut had already decided. "I don't like this." No, more than that. "Something's fucking wrong."
The men took off as a unit. "Kat, get everyone in the limo and keep them there," Nick heard Beckett say. "Go."
_Becca. Jesus Christ, Becca. Where are you?_
Nick full out sprinted down the courtyard. Two security guards spilled out from the gatehouse and filled in behind them.
"Go around to the side and take the shot through a window if you have it," Nick called. Easy and Marz broke off. Guns in hand, the remaining three slowed as they approached the doors. Curtains covered the glass, keeping Nick from seeing inside. In a quiet jog, they hugged the building as they got closer, then Nick used hand signals to communicate the plan. Him on one side, Beckett and Marz on the other. Beckett would force entry, Marz would provide sweeping cover and fire, and Nick would take out the target—assuming he had a shot. There were three friendlies inside.
It was the only way he could think of Becca as his brain shifted to ice-cold operational mode.
Gunfire. One shot. Then another.
And it didn't fucking sound like it had come from the exterior of the building. It had come from inside.
_No! Not Becca! Not my sunshine!_
With a violent kick, Beckett exploded open the doors. A scream from inside.
Nick swept in to witness something he would never forget for the rest of his life. Becca on the floor on top of Woodson. She wrestled a gun away from him, then rose on stumbling, unsure feet, the gun trained right at the man's head.
Nick didn't know whether to be terrified, proud, or completely fucking dumbfounded.
Gun trained on Woodson, Nick slowly came around so he had a clear view of the man and of Becca's face.
"Becca, are you okay?" Nick asked, his heart a goddamned freight train in his chest. Seriously. The adrenaline coursing through him was strong enough to knock him off his feet, and as strong as the relief he felt at seeing Becca on hers. But she didn't seem to hear him. "Becca."
"I should kill you," she said, the tone of her voice something he'd never heard before. "I should." Despite the shudders racking her body, she gripped the weapon stably, competently. Her finger sat on the trigger.
Nick glanced to Woodson to find him unconscious, then all his focus narrowed in on her, even as his teammates moved around the room. Still vigilant, Nick moved closer. "Becca, it's me. It's over."
She shook her head. "I _should,_ " she said again, her face crumpling.
God, his heart was fucking breaking. "No, you shouldn't. No matter how much he deserves it, you don't want a death on your hands. Any death. You don't want that. And I don't want that for you."
Beckett moved around behind Becca, poised to disarm her if he needed to, but Nick gave a single shake of his head.
Nick crouched to force himself closer to her line of sight. "Becca. Sunshine. Look at me."
Shattered blue eyes cut to him, but her gun remained trained on the unconscious man who'd wreaked such havoc on their lives. "Is . . . is Charlie . . ."
"I'm okay," Charlie said, sitting up against the wall by the door. "I'm okay, Becca."
Nick spared a quick glance to her brother. Okay, but hit in the shoulder. Shane was taping gauze to the wound from a kit open on the floor beside him. Jeremy was crouched on Charlie's other side, his head against the guy's good shoulder.
"He's okay?" she asked, like she wasn't quite processing the information.
"Yeah. Charlie's okay. It's all over." Slowly, Nick reached out toward her, his hand gesturing for the gun. "You did so good, Becca. You took Woodson out. You saved Charlie. Let me take it from here." The _how_ of it all, Nick didn't yet know, but there was no doubting that Becca had saved this fucking day.
Nick's hand fell on the barrel of the gun. Exerted pressure. Forced it down and away.
Finally, she let it go.
It was like the gun had been holding her up.
Her legs went out beneath her. Beckett was right there and caught her as she sagged to the floor.
Nick was to her in an instant. He handed the gun off to Beckett and took Becca into his arms.
"Charlie," she rasped.
"He's right here," Nick said.
She turned within his embrace, a tortured gasp spilling out of her when she saw her brother. Blood had soaked a crimson circle through the gauze.
"It's just a scratch," Charlie said. "I'm fine."
Movement in the doorway. Chen and his team. "I got here as fast as I could," Chen said. "Are you all okay?"
Hell if Nick knew. "Can you stand?" he asked Becca. He wanted to get her out of there, away from Woodson.
"Yeah," she said as he helped her up. She needed the help. Adrenaline had her shaking like she was freezing, and her teeth were chattering.
Nick shrugged out of his suit coat and wrapped it around her.
"What's that?" Chen asked, pointing at the floor by Woodson's leg.
A syringe.
"Diazepam," Becca said in a weak voice.
"Smart," Chen said in that deadpan way he had.
Not smart. Fucking brilliant. When had she done that?
As they watched, one of Chen's men cuffed Woodson's hands.
"Can we go home?" Becca asked, her voice taking on a flat, odd quality. "I just want to go home."
His arm around her, Nick pulled her in against his chest and stroked her hair. He looked to Shane and Charlie.
"He needs stitches, but not surgery. Went clean through the meat above his collarbone," Shane said.
"Can you fix it up at home?" Charlie asked Shane.
Shane pressed his lips into a tight line. "The job will be neater and less painful if we take you to the hospital."
Shaking his head, Charlie looked from Shane to Nick. "I don't care about that. I want to take Becca home."
"Your call," Nick said to his best friend.
"Okay," Shane said after a moment. "Let's get you up." He and Jeremy both helped Charlie stand.
"He was gonna kill all of you. That was his plan," Becca said out of nowhere.
Chen's gaze swung to Becca, as did several of the other men's. "Did he tell you how he knew to come here?"
"Just that someone in housekeeping at the hospital found one of our wedding invitations and gave it to him," Becca said. Well, that answered some questions right there.
"He came from the basement," Sonya said from where she hovered with her security team at the door.
"We'll get to the bottom of it all," Chen said. "Go home. I'll touch base in a while."
"What's your plan with him?" Nick asked, giving Woodson one last glance.
"The less you know, the better." Chen gave him a pointed look.
Nick knew he didn't have to say anything more, so he just nodded and guided Becca toward the door. "Then let's go home."
## CHAPTER 12
Becca felt like she was trapped in the dark at the bottom of a well. Somewhere, she knew there was a way out, but as she felt around with blind hands, she couldn't find the ladder. She couldn't find the light.
Despite the fact that, physically, she was functioning.
She answered questions as if by rote. She watched Shane sew up Charlie's wound. She ate part of a piece of pizza that she didn't taste and couldn't finish. She let Nick change her into a pair of pajamas. She felt him touching her, but she couldn't reach him.
She couldn't find the ladder. She couldn't find the light.
But then her friends threw her a rope.
They'd all been sitting around on the couches in the living room for hours, just keeping one another company, just keeping _her_ company, when suddenly Kat shot to her feet, her hands fisted, her posture indicating she was waging some great internal debate.
And then she turned to Beckett, who was sitting with his hip resting on the back of the couch behind where she'd been. "I'm pregnant," Kat blurted.
The whole room froze.
Beckett's face was sculpture still. And then his eyes went wide. "Pregnant? Like . . . pregnant?"
"Pregnant like you're going to be a daddy," she said, her voice uncertain.
He came around the couch to her and grasped her face in his hands. And then he gave her a smile that lit up the entire room. "You're pregnant," he whispered, his voice absolutely reverent. One big hand dropped to her stomach. And then he wrapped her in his arms so tightly that it made Kat laugh.
"You're happy," she said.
"I'm terrified," Beckett said. "But I'm also fucking ecstatic."
The whole place erupted in laughter and cheers and words of congratulations, the energy in the room shifting like the planets had just realigned.
Light cracked through the fog clouding Becca's heart and soul.
"I thought I told you there better not be any goddamned children," Nick said.
Kat whirled. "Nick—"
"No, it's okay," Beckett said, his face going serious again.
And then Nick broke out in a deep belly laugh. "I had you. I totally had you." He grasped Beckett's hand and pulled him in for a back-slapping embrace. "Congratulations, man. I'm gonna be the coolest uncle ever. And you're going to be a kick-ass dad."
"Dude, you have no chance of being cooler than me, so give that shit up now," Jeremy said, hugging Kat in beside him.
"You're a fucking asshole," Beckett said as he gave Nick a playful shove. But that million-dollar smile was back on the guy's face again.
"You totally are," Kat said, reaching up to hug Nick next.
Woodenly, Becca rose from the couch to offer her congratulations.
"Well, shit," Marz said from where he stood next to Beckett. "If we're sharing good news, and why the hell not after this day, then I have to tell you that Emilie agreed to marry me this afternoon." He put his arm around her and kissed her on the temple. "And to be honest, I have no idea how I lasted this long without telling you."
"We were going to share it tomorrow at the reception," Emilie said, her voice cracking. "But I'm glad we did it now. We've been dying."
The smiles on the couple's faces were absolutely brilliant with joy. Another round of jubilation erupted.
The rope was in Becca's hands. The light was burning off the fog.
When everyone had a chance to offer congratulations, Sara called out, "I got accepted into college at Johns Hopkins for the fall. I'm going to finish my degree."
Jenna threw her arms around her sister's shoulders. "Oh, that's amazing, Sara. I'm so proud of you."
More celebrations. More light.
Becca could tell the moment her body finally plugged back in, because suddenly she felt _everything_.
"I'm so happy for all of you," she said, her voice strained. "So, so happy."
Tears exploded out of her. Hot, racking, full-body tears. Sobs that had been stored up since the beginning of time.
Someone wrapped her in their arms and sank down to the couch with her. A hand fell on her shoulder. Another on her back. Another on her knees. Someone grasped her hand.
_All_ she had was rope and light now, even though she couldn't rein herself in. The tears felt like they released a poison inside her that had to be purged, so she gave herself over to them. Not that she really had a choice.
Finally, _finally,_ she managed a deep, shuddering breath. Tears continued to leak from her eyes, but she could see enough to realize she'd literally soaked Easy with her tears. "I'm sorry," she croaked out.
He shook his head. "Don't you worry about a thing. I got you."
She pressed her hand to his heart. "You're a beautiful, beautiful soul, Easy. I'm so glad you're in my life."
The words visibly impacted him. He gave her hand a squeeze.
Becca turned to the others, who'd all formed a tight circle around her. "I feel that way about each and every one of you. I love you and I cherish you and you're just . . . you're all everything to me." At least she wasn't the only one crying, but she was pretty sure she saw only happy tears.
"We feel the same way about you, Becca," Kat said from where she knelt in front of her.
"Yeah," Marz said. "You're da bomb. Then again, I've thought so since you made a MacGyver reference, so . . ." The men groaned. "What?" Marz asked.
Her heart was so, so full. But something still needed to be said. Urgency flooded through her, and she whirled to face Nick, who'd been sitting behind her, holding her and stroking her back. "I want to marry you. Tomorrow," she said. "Just like we planned. I don't want to wait. I don't want to put it off another second." The words rushed out of her. She scrubbed at her face, despite the fact that her eyes seemed set to non-stop.
Nick grasped her hands. "Becca—"
"Please," she said, sensing that his concern for her was going to push him in the direction of caution, of taking things slow. "The best way to fight back the darkness is with love and light. The best way to cheat death is to live life with no regrets, holding nothing back, just throwing yourself into the messy, vibrant, unexpected beauty of it. _That's_ what I want. I don't care about what happened. I refuse to let that win. I don't want to wait to be your always and forever. And for you to be mine."
He studied her face for a long moment, his pale eyes shiny and searching hers. Anticipation hung over the room like a balloon about to burst. "I would love nothing more than to make you my wife. As soon as humanly possible. But tomorrow will do."
"BECCA, ARE YOU ready? It's five o'clock, so it's time," Kat said.
Staring at herself in the floor-to-ceiling mirror in the bridal room at the inn, Becca nodded. One by one, her friends formed a tight circle behind her. Kat, one of the bravest, fiercest women she knew. Emilie, one of the strongest and most generous. Sara, one of the most courageous and certainly the most resilient. Jenna, one of the most compassionate, with so much passion for life that she'd helped a broken man rediscover that within himself again.
How lucky was Becca to be surrounded by such extraordinary women? The only one missing was her own mother, and a little part of Becca's heart ached that her parents weren't there to see everything that she and Charlie had become. But that just proved that you had to love and cherish the ones you cared about while they were in your life and never waste a single minute.
"You look gorgeous," Sara said.
"We're all fucking hot," Kat said, making them all laugh.
"We are pretty stunning," Becca said. The gowns, the flowers in their hands and in their hair, the happiness radiating out from every one of their faces. "And I'm ready."
"Let me make sure the coast is clear. My brother has been a total crazy man about not seeing you." Kat winked at her as she made for the door. Becca hadn't seen Nick since they'd departed at noon for the salon. Truth be told, she was at her limit of missing him, too. "Okay, we're good."
They moved out into the main space, where Becca found Charlie waiting for her. The look he gave her was full of pride and affection.
"Are you feeling okay?" she asked him.
"Great," Charlie said. "I'm about to witness my favorite person on earth getting everything she ever wanted. How could I be anything but great?"
"I love you, Charlie," she said, gently hugging him.
"Bridesmaids, it's time to march," Sonya said. The lady had seemed a little rattled when they'd first arrived, but she'd really gone above and beyond in putting Becca at ease. But if Becca was honest, she didn't feel scarred by what had transpired there the day before. She felt freed.
One by one, her friends made their way to the dais at the far end of the courtyard.
"And now it's your turn," Sonya said as the music transitioned to the wedding march. "Congratulations, Becca."
"Thank you," she said as she slipped her arm through Charlie's good one. They stepped through the door and out into the evening sunlight. "I want this for you someday, Charlie. This happiness, this belonging. You deserve it."
He smiled. "I want it, too. And for the first time in my life, I believe I can have it."
"Jeremy is amazing," she said, giving him a little nudge with her elbow.
"Jeremy is everything," Charlie whispered as they neared the back row of chairs. The happiness in the words filled Becca's heart up to overflowing.
The audience all stood. It wasn't a huge gathering—a handful of Nick's friends from various stages of his life, some of Becca's hospital colleagues and spouses, friends they'd made along the course of the investigation—Detective Vance, Walter and Louis Jackson, Chen. Ike and Jess, who worked at Hard Ink Tattoo, sat to the far side holding Eileen by her leash—the puppy wore a collar with wedding bells hanging off it and a veil that ran down her back. And to top it off, nearly twenty members of the Raven Riders Motorcycle Club had come, some with dates, some without. Before all this, Becca had never before met a person in a motorcycle club, but without them, Nick and his teammates would never have been able to win all the fights that had ultimately allowed the guys to clear their names.
And finally put the past to rest.
Becca was looking forward to getting to know the Ravens more now that all the fighting was behind them.
Looking up, Becca found Nick straight ahead of her, looking as sexy as she'd _ever_ seen him. His dress uniform highlighted the strong width of his shoulders and the trim leanness of his waist. Metals hanging on one side of his chest spoke of a man of honor, loyalty, bravery, and so much more. But what she most noticed was the expression on his face. Total, abject, unrestrained love, unconditional devotion, incredible respect.
If you had those things with the person walking through life with you, what more could you possibly want? Becca certainly didn't know.
As she got closer, she could see the men standing up for Nick, standing by his side. Jeremy, who lived life with more pure delight than anyone she knew. Shane, within whom a fire to help and secure justice for others burned so bright. Marz, who was the most positive, loyal person she'd ever known. Easy, who would do anything for anyone, and who was one of the strongest people she'd ever met in her life, even if he didn't yet know it. And Beckett, fierce and self-sacrificing, the quintessential good guy hiding under a gruff exterior. She loved that about him. She loved all of them.
On a table in front of the men stood seven framed photographs. Her father and the six men from their A-team who hadn't survived that ambush on a dusty road in Afghanistan. They were all together again. Just as it should be.
Finally, she and Charlie reached the front. The chaplain asked, "Who presents this woman to be married to this man?"
"I do," Charlie said. He kissed her on the cheek, then turned to take a seat.
Except before he could do so, Kat came down the two brick steps, grabbed his hand, and led him to stand with her. "You should be here, too," she said.
It was the first moment all day that made Becca have to fight back tears.
Nick stepped forward and took her hand, and the glassiness in his eyes hammered the next nail into her effort to make it through the ceremony without smearing her makeup. As he guided her up the steps, he whispered, "I love you, Becca. I am absolutely the luckiest man alive."
"I love you, too, Nick. Always and forever," she whispered back.
And as they exchanged their vows and claimed one another with their rings and confirmed their commitment in front of everyone they loved, Becca knew this was what happily ever after felt like.
"Ladies and gentlemen," the chaplain called out when they'd said everything that needed to be said, "may I present the new mister and missus Nick and Becca Rixey. You may seal your union with a kiss."
As applause erupted, Nick slid his big hand behind her neck and slowly pulled her in. The kiss was deep and soulful and claiming, full of love and heat, the kind of kiss she would remember as an old lady, the kind of kiss that would make her remember that, once, she'd really lived.
But living was what she was all about now. And _this kiss_ was what their happily ever after was going to feel like all the time. Full of life and passion and love.
And if they ran into some hard times along the way, they'd fight through them together. Side by side. Surrounded by their friends.
After all, hard ever afters were good, too.
Because hard is good.
## ABOUT THE AUTHOR
**LAURA KAYE** is the _New York Times_ and _USA Today_ bestselling author of over twenty books in contemporary romance and romantic suspense. Laura grew up amid family lore involving angels, ghosts, and evil-eye curses, cementing her life-long fascination with storytelling and the supernatural. A former college history professor, Laura also writes historical women's fiction as Laura Kamoie. Laura lives in Maryland with her husband, two daughters, and cute-but-bad dog, and appreciates her view of the Chesapeake Bay every day.
www.laurakayeauthor.com
www.avonromance.com
www.facebook.com/avonromance
Discover great authors, exclusive offers, and more at hc.com.
## **ALSO BY LAURA KAYE**
_Raven Riders Series_
RIDE ROUGH
RIDE HARD
_Coming Soon_
RIDE WILD
_Hard Ink Novels_
HARD TO SERVE
HARD EVER AFTER (novella)
HARD AS STEEL
HARD TO LET GO
HARD TO BE GOOD (novella)
HARD TO COME BY
HARD TO HOLD ON TO (novella)
HARD AS YOU CAN
HARD AS IT GETS
## COPYRIGHT
This is a work of fiction. Names, characters, places, and incidents are products of the author's imagination or are used fictitiously and are not to be construed as real. Any resemblance to actual events, locales, organizations, or persons, living or dead, is entirely coincidental.
"Hard Ever After" was originally published as an e-book novella in February 2016 by Avon Impulse, an Imprint of HarperCollins Publishers.
"Hard Ever After" copyright © 2016 by Laura Kaye.
RIDE ROUGH. Copyright © 2017 by Laura Kaye. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, nontransferable right to access and read the text of this e-book on screen. No part of this text may be reproduced, transmitted, 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 hereafter invented, without the express written permission of HarperCollins e-books.
EPub Edition May 2017 ISBN: 9780062403391
Print Edition ISBN: 9780062403384
FIRST EDITION
## ABOUT THE PUBLISHER
**Australia**
HarperCollins Publishers Australia Pty. Ltd.
Level 13, 201 Elizabeth Street
Sydney, NSW 2000, Australia
www.harpercollins.com.au
**Canada**
HarperCollins Canada
2 Bloor Street East - 20th Floor
Toronto, ON M4W 1A8, Canada
www.harpercollins.ca
**New Zealand**
HarperCollins Publishers New Zealand
Unit D1, 63 Apollo Drive
Rosedale 0632
Auckland, New Zealand
www.harpercollins.co.nz
**United Kingdom**
HarperCollins Publishers Ltd.
1 London Bridge Street
London SE1 9GF, UK
www.harpercollins.co.uk
**United States**
HarperCollins Publishers Inc.
195 Broadway
New York, NY 10007
www.harpercollins.com
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 3,635
|
Produced by D Alexander, Peter Vickers, Juliet Sutherland
and the Online Distributed Proofreading Team at
http://www.pgdp.net
[Illustration: THREE TIMES THEY BROKE SPEARS]
TALES FROM TENNYSON
BY
MOLLY K. BELLEW
EDITOR OF
"TALES FROM LONGFELLOW"
"DICKENS' CHRISTMAS STORIES FOR CHILDREN"
ETC., ETC.
ILLUSTRATED BY H. S. CAMPBELL
NEW YORK AND BOSTON
H. M. CALDWELL CO.
PUBLISHERS
COPYRIGHT, 1902
BY
JAMIESON-HIGGINS CO.
CONTENTS.
The Coming of King Arthur 9
Gareth and Lynette 29
The Marriage of Geraint 46
Geraint's Quest of Honor 64
Merlin and Vivien 85
Balin and Balan 95
Lancelot and Elaine 104
The Holy Grail 119
Pelleas and Ettarre 132
The Last Tournament 142
The Passing of Arthur 150
To my Young Readers.
Alfred Lord Tennyson was the typically English poet, and none, perhaps
not even Shakespeare, has appealed so keenly to the human heart. No
other man's poems have caused as many readers to shed tears of sympathy
nor have awakened higher sentiments in the human heart. The critics
agree in pronouncing him the ideal poet laureate. In his "Idylls from
the King" are found the loftiest and proudest deeds of English history
and even in the retelling of these in prose the high spirit that is an
inspiration to the noblest deeds cannot fail to be preserved.
MOLLY K. BELLEW.
THE COMING OF KING ARTHUR.
Over a thousand years ago everybody was talking about the wonderful King
Arthur and his brilliant Knights of the Round Table, who everywhere were
pursuing bold quests, putting to rout the band of outlaws and robbers
which in those days infested every highway and by-way of the country,
going to war with tyrannical nobles, establishing law and order among
the rich, redressing the wrongs of women, the poor and the oppressed,
and winning glorious renown for their valor and their successes.
That was in England which at that time was not England as it is today,
all one kingdom under a single ruler, but was divided into many bits of
kingdoms each with its own king and all warring against each other.
Arthur's kingdom was the most unpeaceful of all. This was because for
twenty years or more, ever since the death of old King Uther, the
country had been without a ruler. Old King Uther had died about a score
of years before without leaving an heir to the throne, and all the
nobles of the realm had immediately gone to war with one another each
trying to get the most land and each trying to get the throne for
himself.
[Illustration: OLD MERLIN APPEARS.]
Suddenly, however, old Merlin, the wizard who had been King Uther's
magician, appeared one day in the royal council hall with a handsome
young man, Arthur, and declared him to be the king of the realm. Arthur
was crowned and for a time the nobles were quiet, for he ruled with a
strong hand of iron, put down all the evils in his kingdom and
everywhere gave it peace and order. People in every part of the island
sent for him and his knights, begging him to come to help them out of
their difficulties. But presently the nobles became troublesome again;
they said that Arthur was not the true king, that he was not the son of
Uther and that, therefore, he had no right to reign over them. So there
was fighting and unrest again, and in the midst of it Leodogran, the
king of the Land of Cameliard, asked Arthur to come with his knights and
drive away the enemies besetting him on every side. The country of
Cameliard had gone to waste and ruin, because of the continual warfare
that was waged with the kings that lived in the little neighboring
countries and a mass of wild-eyed foreign heathen peoples who invaded
the land. And so it happened that Cameliard was ravaged with battles,
its strong men were cut down with the sword and wild dogs, wolves, and
bears from the tangled weeds came rooting up the green fields and
wallowing into the palace gardens. Sometimes the wolves stole little
children from the villages and nursed them like their own cubs, until
finally these children grew up into a race of wolf-men who molested the
land worse than the wolves themselves. Then another king fought
Leodogran, and at last the heathen hordes came swarming from over the
seas and made all the earth red with his soldiers' blood, and they made
the sun red with the smoke of the burning homes of his people.
Leodogran simply did not know which way to turn for help until at last
he thought of young Arthur of the Round Table who recently had been
crowned king. So Leodogran sent for Arthur beseeching him to come and
help him, for between the men and the beasts his country was dying.
[Illustration: PRINCESS GUINEVERE.]
King Arthur and his men welcomed the chance and went at once into the
Land of Cameliard to drive away the heathen marauders. As he marched
with his men past the castle walls, pretty Princess Guinevere stood
outside to watch the glittering soldiers go by. Among so many richly
dressed knights she did not particularly notice Arthur, for he wore
nothing to show that he was king, although his kingly bearing and brave
forehead might suggest leadership. But no royal arms were engraved upon
his helmet or his shield, and he carried simple weapons not nearly so
gorgeously emblazoned as those of some of the others.
[Illustration: HE LED HIS WARRIORS BOLDLY.]
Although Guinevere did not see the fair young King, Arthur spied her
beside the castle wall; he felt the light of her beautiful eyes
glimmering out into his heart and setting it all aflame with a fire of
love for her.
He led his warriors boldly to the forests where they pitched their
tents, then fought all the heathen until they scampered away to their
own territories, he slew the frightful wild beasts that had plundered
the fields, cut down the forest trees so as to open out roads for the
people of Cameliard to pass over from one part of their land to the
other, then he traveled quietly away with his men, back to fight his own
battles in his own country. For there was fighting everywhere in those
days. But all the time in Arthur's heart, while he was doing those
wonderful things for Leodogran, he was thinking still, not of Leodogran,
but of the lovely Guinevere, and yearning for her.
If only she could be his queen he thought they two together could rule
on his throne as one strong, sweet, delicious life, and could exert a
mighty power over all his people to make them good and wise and happy.
Each day increased his love until he could not bear even to think for a
moment of living without her. So from the very field of battle, while
the swords were flashing and clashing about him, as he fought the barons
and great lords who had risen up against him, Arthur dispatched three
messengers to Leodogran, the King of Cameliard.
These three messengers were Ulfius, Brastias and Bedivere, the very
first knight Arthur had knighted upon his throne. They went to Leodogran
and said that if Arthur had been of any service to him in his recent
troubles with the heathen and the wild beasts, he should give the
Princess Guinevere to be Arthur's wife as a mark of his good will.
[Illustration: ARTHUR DISPATCHED THREE MESSENGERS TO LEODOGRAN.]
Well, when they had said this, Leodogran did not know what to do any
better than when the heathen and the beasts had come upon him. For while
he thought Arthur a very bold soldier and a very fine man, and, although
he felt very grateful indeed to him for all the great things he had
done, still he was not certain that Guinevere ought to marry him. For,
as Guinevere was the daughter of a king she should become the wife of
none but the son of a king. And Leodogran did not know precisely who
this King Arthur was; but he did know that the barons of Arthur's court
had burst out into this uproar against him because they said he was not
their true king and not the son of King Uther who had reigned before
him. Some of them declared him to be the child of Gerlois, and others
avowed that Sir Anton was his father.
As poor, puzzled Leodogran knew nothing about the matter himself, he
sent for his gray-headed trusty old chamberlain, who always had good
counsel to give him in any dilemma; and he asked the chamberlain whether
he had heard anything certainly as to Arthur's birth. The chamberlain
told him that there were just two men in all the world who knew the
truth with respect to Arthur and where he had come from, and that both
these men were twice as old as himself. One of them was Merlin the
wizard, the other was Bleys, Merlin's teacher in magic, who had written
a book of his renowned pupil's wonders, which probably related
everything regarding the secret of Arthur's birth.
"If King Arthur had done no more for me in my wars than you have just
now in my present trouble," the king answered the chamberlain, "I would
have died long ago from the wild beasts and the heathen. Send me in
Ulfius and Brastias and Bedivere again."
So the chamberlain went out and Arthur's three men came into Leodogran
who spoke to them this way: "I have often seen a big cuckoo chased by
little birds and understood why such tiny birds plagued him so, but why
are the nobles in your country rebelling against their king and saying
that he is not the son of a king. Tell me whether you yourselves think
he is the child of King Uther."
[Illustration: SIR KING, THERE ARE ALL SORTS OF STORIES ABOUT THAT.]
Ulfius and Brastias answered immediately "yes," but Bedivere, the first
of all Arthur's knights, became very bold when anyone slandered his
sovereign and he replied: "_Sir King, there are all sorts of stories
about that_; some of the nobles hate him just because he is good and
they are wicked; they cry out that he is no man because his ways are
gentler than their rough manners, while others again think he must be
an angel dropped from heaven. But I will tell you the facts as I know
them, King Uther and Gerlois were rivals long ago; they both loved
Ygerne. And she was the wife of Gerlois and had no sons, but three
daughters, one of them the Queen of Orkney who has clung to Arthur like
a sister. The two rivals, Gerlois and Uther went to war with each other
and Gerlois was killed in battle; then Uther quickly married the winsome
Ygerne, the widow of Gerlois, for he loved her dearly and impatiently.
In a few months Uther died, and on that very night of his death Arthur
was born. And as soon as he was born they carried him out by a secret
back gateway to Merlin the magician, to be brought up far away from the
court so that no one would hear about him until he was grown up ready to
sit upon Uther's, his father, throne.
"For those were wild lords in those years just like these of today,
always struggling for the rule, and they would have shattered the
helpless little prince to pieces had they known about him. So Merlin
took the baby and gave him over to old Sir Anton, a friend of Uther's,
and Sir Anton's wife tended Arthur with her own little ones so that
nobody knew who he was or where he had come from. But while the prince
was growing up the kingdom went to weed; the great lords and barons were
fighting all the time among themselves and nobody ruled. But during this
present year Arthur's time for ascending the throne had come, so Merlin
brought him from out of his hiding place, set him in the palace hall and
cried out to all the lords and ladies, 'This is Uther's heir, your
king!' Of course, none of them would have that. A hundred voices cried
back immediately: 'Away with him! he is no king of ours, that's the son
of Gerlois, or else the child of Anton, and no king.'
"In spite of this opposition Merlin was so crafty and clever he won the
day for the people, who were clamoring for a king and were glad to see
Arthur crowned. But after it all was over the lords banded together and
broke out in open war against Arthur. That is the whole story of this
war."
Although pleased with Bedivere's good account of Arthur, yet when it was
ended Leodogran scarcely felt satisfied. Was Bedivere right, he thought
to himself, or were the barons right? As he sat pondering over
everything in his palace, _three great visitors came to the castle_;
these were the Queen of Orkney, the daughter of Gerlois and Ygerne, with
her two sons, Gawain and Modred. Leodogran made a great feast for them
and while entertaining them at table remembered what Bedivere had said
about Arthur and this queen. So he turned to the queen and remarked:
[Illustration: THREE VISITORS TO THE CASTLE.]
"An insecure throne is no better than a mass of ice in a summer's sea;
it all melts away. You are from Arthur's court; tell me, do you think
this king with his few loyal Knights of the Round Table can triumph over
the rebellious lords, and keep his throne?"
"O King, they are few indeed," the Queen of Orkney cried, "but so bold
and true, and all of one mind with him. I was there at the coronation
when the savage yells of the nobles died away, and Arthur sat crowned
upon the dais with all his knights gathered round him to do his service
for him forever. Arthur in low, deep tones, with simple words of great
authority bound them to him with such wonderfully rigid vows that when
they rose from their knees one after the other, some of them looked as
pale as if a ghost had passed by them, others were flushed in their
faces, and yet others seemed dazed and blind with their awe as if not
fully awake. Then he spoke to them, cheering them with divine words that
are far more than my tongue can ever tell you, and while he spoke every
face flashed, for just a moment with his likeness, and from the crucifix
above, three rays in green, blue, scarlet, streamed across upon the
bright, sweet faces of the three tall fair queens, his friends who stood
silently beside his throne, and who will always be ready to help him if
he is in need.
"Merlin, the magician, came there too, with his hundred years of art
like so many hands of vassals to wait upon the young king. Near Merlin
stood the mystical, marvelous Lady of the Lake, who knows a deeper magic
than Merlin's own, dressed in white. A mist of incense curled all about
her and her face was fairly hidden in the dim gloom. But when the holy
hymns were sung a voice like flowing waters sounded through the music.
It was the voice of the Lady of the Lake who lives in the lowest waters
of the lake where it is always calm, no matter what storms may blow over
the earth and who when the waves tumble and roll above her can walk out
upon their crests just as our Lord did.
"_It was she who gave Arthur his remarkable sword_ Excalibur, with its
hilt like a cross wherewith he drove away the heathen for you. That
strange sword rose up from out the bosom of the lake, and Arthur rowed
over in a little boat and took it. The sword is incrusted with rich
jewels on the hilt, with a blade so bright that men are blinded by it.
On one side the words 'Take me' are graven upon it in the oldest
language of the world, while on the other side the words 'Cast me away'
are carved in the tongue that you speak.
[Illustration: SHE GAVE ARTHUR HIS REMARKABLE SWORD]
"Arthur became very sad when he saw the second inscription, but Merlin
advised him to take the beautiful blade and use it; he told him that now
was the time to strike and that the time to cast away was very, very far
off. So Arthur took the tremendous sword and with it he will beat down
his enemies, King Leodogran."
Leodogran was pleased with the queen's words, but he wished to test the
story Bedivere had told him, so he looked into her eyes narrowly as he
observed, with a question in his tones, "The swallow and the swift are
very near kin, but you are still closer to this noble prince as you are
his own dear sister."
"I am the daughter of Gerlois and Ygerne," she answered.
"Yes, that is why you are Arthur's sister," the king returned still
questioningly.
"These are secret things," the Queen of Orkney replied, and she motioned
with her hand for her two sons to leave her alone in the room with the
king.
Gawain immediately skipped away singing, his hair flying after and
frolicked outside like a frisky pony, _but cunning Modred laid his ear
close beside the door to listen_, so that he half heard all the strange
story his mother told the king. This is what the queen said in the
beginning to the king.
[Illustration: CUNNING MODRED BESIDE THE DOOR TO LISTEN]
"What should I know about it? For my mother's hair and eyes were dark,
and so were the eyes and hair of Gerlois, and Uther was dark too, almost
black, but the King Arthur is fairer than anyone else in Britain.
However, I remember how my mother used often to weep and say, 'O that
you had some brother, pretty little one, to guard you from the rough
ways of the world."
"Yes? She said that?" Leodogran rejoined, "but when did you see Arthur
first?"
"O king, I will tell you all about it," cried the Queen of Orkney. "Once
when I was a little bit of a girl and had been beaten for some childish
fault that I had not committed, I ran outside and flung myself on a
grassy bank and hated all the world and everything in it, and wished I
were dead. But all of a sudden little Arthur stood by my side. I don't
know how he came or anything about it. Perhaps Merlin brought him, for
Merlin, they say, can walk about and nobody see him, if he will, but any
rate, Arthur was there by my side, comforting me and drying my tears.
After that Arthur came very often without anybody knowing it and we were
children together, and in those golden days I felt sure he would be
king.
"But now I must tell you about Bleys, the old wizard who taught the
magician Merlin. You know they both served King Uther, and just a little
while ago when Bleys died he sent for me. He said he had something to
tell me that I must know before he left the world. He said that they
two, Merlin and he, sat beside the bed of King Uther on the night when
the king passed away, moaning and wailing because he left no heir to his
throne. After the king's death as Merlin and Bleys walked out from the
castle walls into the dismal misty night, they saw a wonderful
fairy-ship shaped like a winged dragon sailing the heavens, with shining
people collected on its decks; but in the twinkling of an eye the ship
was gone.
"Then Merlin and Bleys passed down into the cove by the seashore to
watch the billows, one after the other, as they lapped up against the
beach. And as they looked at last a great wave gathered up one-half of
the ocean and came full of voices, slowly rising and plunging, roaring
all the while. Then all the wave was in a flame; and down in the wave
and in the flame they saw lying a naked babe that was carried by the
water to Merlin's very feet.
"'The king!' cried Merlin. 'Here's an heir for Uther.'
"Then as old Merlin spoke the fringe of that terrible great flaming
breaker lashed at him as he held up the baby; it rose up round him in a
mantle of fire so that he and the child were clothed in fire. Then
suddenly there was a calm, the stars looked out and the sky was open.
"'And this same child,' Bleys whispered to me, 'is the young king who
reigns. And I could not die in peace unless the story had been told.'
Then Bleys passed away into the land where nobody can question him.
"So I came to Merlin to ask him whether that was all true about the
shining dragon-ship and the tiny bare baby floating down from heaven
over on the glory of the seas; but Merlin just laughed, as he always
does, and answered me in the riddles of the old song, this way:
"'Rain, rain and sun! a rainbow in the sky!
A young man will be wiser by and by;
An old man's wit may wander ere he die.
Rain, rain and sun! a rainbow on the lea!
And truth is this to me and that to thee;
And truth or clothed or naked let it be.
Rain, sun and rain! and the free blossom blows;
Sun, rain and sun! and where is he who knows.
From the great deep to the great deep he goes!'
"It vexed me dreadfully to have Merlin be so tantalizing; but you must
not be afraid, king, to give your only child Guinevere to this King
Arthur. For great poets will sing of his brave deeds in long years after
this; and Merlin has said, and not joking, either, that even although
Arthur's enemies may wound him in battle he will never, never die, but
will only pass away for a time, for a little while, and then will come
to us again. And Merlin says too, that sometime Arthur is going to
trample all the heathen kings under his feet until all the nations and
all the men will call him their king."
It pleased Leodogran tremendously to hear what the Queen of Orkney told
him of Arthur, and when she had ended he lay thinking over it all, still
puzzled as to whether he should say "yes" or "no" to the ambassadors
whom Arthur had sent. As he lay buried in his thoughts he grew very,
very drowsy and dreamy, and at last, he fell asleep. And while he slept
he saw a wonderful vision in a dream.
There was a strange, sloping land, rising before his eyes, that ascended
higher and higher, field after field, to a very great height and at the
top there was a lofty peak hidden in the heavy, hazy clouds; and on the
peak a phantom king stood. One moment the king was there, and the next
moment he was gone, while everything below him was in a frightful
confusion, a battle with swords, and the flocks of sheep and cattle
falling back, and all the villages burning and their smoke rolling up in
streams to the clouded pinnacle of the peak where the king stood in the
fog, hiding him the more. Now and then the king spoke out through the
haze, and some one here or there beneath would point upward toward him,
but the rest all went on fighting. They cried out, "He is no king of
ours, no son of Uther's, no king of ours." Then in a twinkling the dream
all changed; the mists had quite blown away, the solid earth below the
peak had vanished like a bubble and only the wonderful king remained,
crowned with his diadems, standing in the heavens.
Then Leodogran while still looking at him woke from his sleep. He called
for Ulfius and Brastias and Bedevere, and when they had come into this
presence he told them that Arthur should marry the fair Princess
Guinevere, and he sent them galloping back to Arthur's court.
That was a joyful day for King Arthur when the three knights delivered
King Leodogran's message. He made ready at once for his sweet queen. He
picked out Lancelot, his favorite Knight of the Round Table, whom he
loved better than any other man in all the world, to ride over into the
Land of Cameliard and bring back Guinevere for his bride. And as
Lancelot mounted his dancing steed and rode away _Arthur watched him
from the palace gates_, thinking of the lovely lady who would ride by
his side when he returned.
[Illustration: LANCELOT MOUNTED HIS DANCING STEED.]
Lancelot's horse trampled away among the flowers; for it was April when
he left the court of Arthur, and just one month later he came riding
back among the flowers of the May-time. Guinevere was with him on her
graceful palfrey.
Then Dubric, the head of the whole church in Britain, went out to meet
her. Happy Arthur was there too. They were married in the greatest and
noblest church in the land before the stately altar, with all the
Knights of the Round Table dressed in stainless white clothes, gathered
about them. And all the knights were as delighted as they could be
because their king was so glad. Holy Dubric spread out his hands above
the King and the lovely Queen to call down the blessings of heaven, and
he said:
[Illustration: KING ARTHUR AND THE LOVELY QUEEN.]
"Reign, King, and live and love, and make the world better, and may your
queen be one with you, and may all the Knights of the Order of the Round
Table fulfill the boundless purposes of their king."
There was spread a glorious marriage feast. Great lords came thither
from far away Rome, which once was the mistress of all the world, but
now was slowly fading away. These Roman lords called for the tribute
from Arthur that they had always received from Britain ever since Caesar
with his Roman legions had conquered it long years before.
But Arthur, the king and bridegroom, pointed to his snowy knights and
said: "These knights of mine have sworn to fight for me in all my wars
and to worship me as their king. The old order of things has passed away
and a new order will take its place. We are fighting for our fair father
Christ, while you have been growing so feeble and so weak and so old
that you cannot even drive away the heathen from your Roman walls any
more. So we will not pay tribute to you nor be your slaves. This is to
be our own free country which we will defend and maintain."
_The great lords from Rome drew back very angrily_ and went home and
told their king all about what Arthur had said. So Arthur had to battle
with Rome, but he won in the end.
Arthur trained his Knights of the Round Table so that they all felt like
one great, vast strong man, all of one will. Thus he became mightier
than any of the other kings in any part of Britain. And when he fought
with them he always conquered them. In that way he drew in all the
little kingdoms under him, so that he was the one king of the land, and
they all fought together for him.
There were twelve great battles against the heathen hordes that had
molested them from across the terrible seas, and each of these battles
he won. So he made one great realm and he reigned over it, the king.
[Illustration: THE GREAT LORDS FROM ROME DREW BACK.]
GARETH AND LYNETTE.
Old King Lot and good Queen Bellicent had three sons. Gawain and Modred
were Knights of the Round Table at Arthur's court, and young Gareth, who
was his mother's pet, sighed to think he had to stay home and be cuddled
and fondled like a baby boy instead of riding off like a venturesome
soldier fighting gloriously for the king and winning a great name.
"There!" he cried impatiently, one chilly spring day as he stood by the
brink of a rivulet and saw a bit of a pine tree caught from the bank by
the dashing, swollen waters of the stream and whirled madly away.
"That's the way the king's enemies would fall before my spear, if I had
a spear to use! That stream can do no more than I can, even although it
is merely icy water all cold with the snows while I'm tingling with hot
blood and have strong arms. When Gawain came home last summer and asked
me to tilt with him and Modred was the judge, didn't I shake him so in
his saddle that he said I had half overcome him? Humph! and mother
thinks I'm still a child!"
_Gareth went in to the queen_ and said: "Mother, if you love me listen
to a story I will tell. Once there was an egg which a great royal eagle
laid high above on the rocks somewhere almost out of sight and there was
a lad which saw the splendor sparkling from it, and the lightnings
playing around it and the little birds crying and clashing in the nest.
The boy thought if he could only reach that egg he would be richer than
a houseful of kings, and he was nearly driven from his sense with his
desire for it. But whenever he reached to clamber up for it some one who
loved him restrained him saying, 'If you love me do not climb, lest
you break your neck.' So the boy did not climb, mother, and he did not
break his neck, but he broke his heart pining for the glorious egg. How
can you keep me tethered here, Mother? Let me go!"
[Illustration: MOTHER, IF YOU LOVE ME LISTEN TO A STORY I WILL TELL.]
"Have you no pity for me?" Queen Bellicent asked. "Stay here by your
poor old father and me; chase the deer in our fir trees and marry some
lovely bride I will get for you. You're my best son and so young."
"Mother, a king once showed his son two brides and told him that he must
either win the beautiful one, or, if he failed, wed the other. The
pretty one was Fame and the other was Shame. Why should I follow the
deer when I can follow the king? Why was I born a man if I cannot do a
man's work?"
"But some of the barons say he isn't the true king."
"Hasn't he conquered the Romans and driven off the heathen and made all
the people free? Who has a right to be king if not the man who has done
that? He is the true king."
When Bellicent found that she could not turn Gareth from his purpose,
she said that if he was determined he must do one thing before he asked
the king to make him a knight.
"Anything," cried Gareth. "Give me a hundred proofs. Only be quick."
The queen looked at him very slowly and said: "You are a prince, Gareth,
but before you are fit to serve the king you must go into Arthur's court
disguised and hire yourself to serve his meats and drink among the
scullions and kitchen knaves. And you must not tell your name to anyone
and you must serve that way for a year and a day."
The queen made this condition, thinking that Gareth would be too proud
to play the slave. But he thought a moment, then answered: "A slave may
be free in his soul, and I can see the jousts there. You are my mother
so I must obey you and I will be a scullion in King Arthur's kitchen and
keep my name a secret from everyone, even the king."
So Bellicent grieved and watched Gareth every moment wherever he went,
dreading the time when he should leave. And he waited until one windy
night when she slept, then called two servants and slipped away with
them, all three dressed like poor peasants of the field.
They walked away towards the south and as they came to the plain
stretching to the mountain of Camelot, they saw the royal city upon its
brow. Sometimes its spires and towers flashed in the sunlight; sometimes
only the great gate shone out before their eyes, or again the whole fair
town vanished away. Then the servants said:
"Let us go no further, Lord. It's an enchanted city, and all a vision.
The people say anyway, that Arthur isn't the true king, but only a
changeling from fairyland, and that Merlin won his battles for him with
magic."
Gareth laughed and replied that he had magic enough in his blood and
hopes to plunge old Merlin into the Arabian sea. And he pushed them on
to the gate. There was no other gate like it under heaven. The Lady of
the Lake stood barefooted on the keystone and held up the cornice. Drops
of water fell from either hand and above were the three queens who were
Arthur's friends, and on each side Arthur's wars were pictured in weird
devices with dragons and elves so intertwined that they made men dizzy
to look at them. The servants cried out, "Lord, the gateway is alive!"
Then a blast of music pealed out of the city, and the three queens
stepped aside while an old man with a long beard came out and asked:
"Who are you, my sons?"
"We are peasants," answered Gareth, "who have come to see the glories of
your king, but the city looked so strange through the morning mist that
my men are wondering whether it is not a fairy city or perhaps no city
at all. So tell us the truth about it."
"Oh, it's a fairy city," the old man answered, "and a fairy king and
queen came out of the mountain cleft at sunrise with harps in their
hands and built it to music, which means it never was built at all, and
therefore built forever."
"Why do you mock me so?" Gareth cried angrily.
"I am not mocking you so much as you are mocking me and every one who
looks at you, for you are not what you seem, still I know what you truly
are."
Then the old man turned away and Gareth said to his men: "Our poor
little white lie stands like a ghost at the very beginning of our
enterprise. Blame my mother's love for it and not her nor me."
So they all laughed and came into the city of Camelot with its shadowy
and stately palaces. Here and there a knight passed in or out, his arms
clashing and the sound was good to Gareth's ears. Or out of a casement
window glanced the pure eyes of lovely women. But Gareth made at once
for the hall of the king where his heart fairly hammered into his ears
as he wondered whether Arthur would turn him aside because of the half
shadow of a lie he had told the old man by the gate about being a
peasant. There were many supplicants coming before the king to tell him
of some hurt done them by marauders or the wild beasts, and each one was
given a knight by the king to help them.
When Gareth's turn came, he rested his arms, one on each servant, and
stepped forward saying: "A boon, Sir King! Do you see how weak I seem,
leaning on these men? Pray let me go into your kitchen and serve there
for a year and a day, and do not ask me my name. After that I will fight
for you."
"You are a handsome youth," said the king, "and worth something better
from the king, but if that is what you wish, go and serve under the
seneschal, Sir Kay, Master of the Meats and Drinks."
Sir Kay thought the boy had probably run away from the farm belonging to
some Abbey where he had not had enough to eat, and he promised that if
Gareth would work well he would feed him until he was as plump as a
pigeon.
But Lancelot, the king's favorite, said to Kay: "You don't understand
boys as well as dogs and cattle. Can't you see by this lad's broad fair
forehead and fine hands that he is nobly born? Treat him well or he may
shame you."
"Fair and fine, forsooth," cried Kay. "If he had been a gentleman he
would have asked for a horse and armor."
So he hustled and harried Garreth, _set him to draw water_, _hew wood_
and labor harder than any of the grimy and smudgy kitchen knaves. Gareth
did all with a noble sort of ease and graced the lowliest act, and when
the knaves all gathered together of an evening to tell stories about
Arthur on the battlefields or of Lancelot in the tournament, Gareth
listened delightedly or made them all, with gaping mouths, listen
charmed, to some prodigious tale of his own about wonderful knights
cutting their scarlet way through twenty folds of twisted dragons. When
there was a Joust and Sir Kay let him attend it, he went half beside
himself in an ecstasy watching the warriors clash their springing
spears, and the sniffing chargers reel.
At the end of the first month, lonely Queen Bellicent felt sorry for her
poor, dear son, toiling and moiling among pots and pans, so she sent a
servant to Camelot with the beaming armor of a knight and freed him from
his vow. Gareth redder than any young girl and went alone in to
the king and told him all.
[Illustration: SET HIM TO DRAW WATER, HEW WOOD.]
"Make me your knight in secret," he begged Arthur, "and give me the very
next quest from your court!"
"Son," answered the king, "my knights are sworn to vows of utter
hardihood, of utter gentleness, of utter faithfulness in love and of
utter obedience to the king."
Gareth sprang lightly from his knees: "My king, I can promise you for my
hardihood; respecting my obedience, ask Sir Kay, and as for love I have
not loved yet, but God willing some day I will, and faithfully."
The reply so pleased the great king, he laid his hand on Gareth's arm
and smiled and knighted him.
A few days later _a noble maiden_ with a brow like a May-blossom and a
saucy nose _passed into the king's hall with her page_ and told Arthur
that her name was Lynette, and that her beautiful sister, the Lady
Lyonors lived in the Castle Perilous which was beset with bandit
knights.
[Illustration: A NOBLE MAIDEN WITH HER PAGE.]
"A river courses about the castle in three loops," said she, "each loop
has a bridge and every bridge is guarded by a wicked outlaw warrior, Sir
Morning-star, Sir Noon-sun and Sir Evening-star, while a fourth called
Death, a huge man-beast of boundless savageries, is besieging my sister
in her own castle so as to break her will and make her wed with him.
They are four fools," cried the maiden disdainfully, "but they are
mighty men so I have come to ask for Lancelot to ride away with me to
help us."
Gareth was up in a twinkling with kindled eyes. "A boon, Sir King, this
quest," he cried. "I am only a knave from your kitchen, but I can
topple over a hundred such fellows. Your promise, king."
"You are rough and sudden and worthy to be a knight. Therefore go," said
Arthur to the great amazement of the court.
"Fie on you, King!" exclaimed Lynette in a fury. "I asked you for your
best knight, Lancelot, and you give me a slave from your kitchen," and
she scampered down the aisle, leaped to her horse and flitted out of the
weird white gate. "A kitchen slave!" she sputtered as she flew. "Why
didn't the king send me a knight that fights for love and glory?"
Gareth in the meantime had strode to the side doorway of the royal hall
where he saw a war-horse awaiting him, the gift of Arthur and worth half
the price of a town. His two servants stood by with his shield and
helmet and spear. Dropping his coarse kitchen cloak to the floor, he
instantly harnessed himself in his armor, leaped to the back of his
beautiful steed and flashed out of the gateway while all his kitchen
mates threw up their caps and cried, "God bless the king and all his
fellowship!"
"Maiden, the quest is mine," he said to Lynette as he overtook her,
"Lead and I follow."
"Away with you!" she cried, nipping her slender nose. "You smell of
kitchen grease. See there, your master is coming!"
Indeed she told the truth, for Sir Kay, infuriated with Gareth's
boldness in the king's hall was hounding after them. "Don't you know
me?" he shouted.
"Yes, too well," returned Gareth. "I know you to be the most ungentle
knight in Arthur's court."
"Have at me, then," cried Kay, whereupon Gareth pounced upon him with
his gleaming lance and struck him instantly to the earth, then turned
for Lynette and said again, "Lead and I follow."
But Lynette had hurried her galloping palfrey away and would not stop
the beast until his heart had nearly burst with its violent throbbing.
Then she turned and eyed Gareth as scornfully as ever. As he pranced to
her side she observed:
"Do you suppose scullion, that I think any more of you now that by some
good luck you have overthrown your master. You dishwasher and
water-carrier, you smell of the kitchen quite as much as before."
"Maiden," Gareth rejoined gently, "Say what you will, but whatever you
say, I will not leave this quest until it is ended or I have died for
it."
"O, my, how the knave talks! But you'll soon meet with another knave
whom in spite of all the kitchen concoctions ever brewed, you'll not
dare look in the face."
"I'll try him," answered Gareth with a smile that maddened Lynette. And
away she darted again far into the strange avenues of the limitless
woods.
Gareth plunged on through the pine trees after her and a serving-man
came breaking through the black forest crying out, "They've bound my
master and are throwing him into the lake!"
"Lead and I follow," cried Gareth to Lynette, and she led, plunging into
the pine trees until they came upon a hollow sinking away into a lake,
where six tall men up to their thighs in reeds and bulrushes were
dragging a seventh man with a stone about his neck toward the water to
drown him.
Gareth sprang upon three and stilled them with his doughty blows, but
three scurried away through the trees; then Gareth loosened the stone
from the gentleman and set him on his feet. He proved to be a baron and
a friend of Arthur and asked Gareth what he could do to show his
gratitude for the saving of his life. Gareth said he would like a
night's shelter for the lady who was with him. So they rode over toward
the graceful manor house where the baron lived, and as they rode he said
to Gareth.
"I believe you are of the Table," meaning that Gareth was a Knight of
the Round Table.
"Yes, he is of the table after his own fashion," Lynette laughed, "for
he serves in Arthur's kitchen." And turning toward Gareth she added, "Do
not imagine that I admire you the more for having routed these miserable
cowardly foresters; any thresher with his flail could have done that."
And when they were seated at the baron's table, Gareth by Lynette's
side, she cried out to their host, "It seems dreadfully rude in you,
Lord Baron, to place this knave beside me. Listen to me: I went to King
Arthur's court to ask for Sir Lancelot to come to help my sister, and as
I ended my plea, up bawls this kitchen boy: 'Mine's the quest.' And
Arthur goes mad and sends me this fellow who was made to kill pigs and
not redress the wrongs of women."
So Gareth was seated at another table and the baron came to him and
asked him whether it might not be better for him to relinquish his
quest, but the lad replied that the king had given it to him and he
would carry it through. The next morning he said again to proud Lynette,
"Lead and I follow."
But the maiden responded, "We are almost at the place where one of the
knaves is stationed. Don't you want to go home? He will slay you and
then I'll go back to Arthur and shame him for giving me a knight from
his kitchen cinders."
"Just let me fight," cried Gareth, "and I'll have as good luck as little
Cinderella who married the prince."
So they came to the first coil of the river and on the other side saw a
rich white pavilion with a purple dome and a slender crimson flag
fluttering above. The lawless Sir Morning-star paced up and down
outside.
"Damsel, is this the knight you've brought me?" he shouted.
"Not a knight, but a knave. The king scorned you so he sent some one
from his kitchen."
"Come Daughters of the Dawn and arm me!" cried Sir Morning-star, and
three bare-footed, bare-headed maidens in pink and gold dresses brought
him a blue coat of mail and a blue shield.
"A kitchen knave in scorn of me!" roared the blue knight. "I won't fight
him. Go home, knave! It isn't proper for you to be riding abroad with a
lady."
"Dog, you lie! I'm sprung from nobler lineage than you," and saying
this, Gareth sprang fiercely at his adversary who met him in the middle
of the bridge. The two spears were hurled so harshly that both knights
were thrown from their horses like two stones but up they leaped
instantly. Gareth drew forth his sword and drove his enemy back down the
bridge and laid him at his feet.
"I yield," Sir Morning-star cried, "don't kill me."
"Your life is in the hands of this lady," Gareth replied. "If she asks
me to spare you I will."
"Scullion!" Lynette cried, reddening with shame. "Do you suppose I will
ask a favor of you?"
"Then he dies," and Gareth was about to slay the wounded knight when
Lynette screamed and told him he ought not to think of killing a man of
nobler birth than himself. So Gareth said, "Knight, your life is spared
at this lady's command. Go to King Arthur's court and tell him that his
kitchen knave sent you, and crave his pardon for breaking his laws."
"I thought the smells of the odors of the kitchen grew fainter while you
were fighting on the bridge," Lynette remarked to Gareth as he took his
place behind her and told her to lead, "but now they are as strong as
ever."
So they rode on until they arrived at the second loop of the river where
the knight of the Noonday-Sun flared with his burning shield that blazed
so violently that Gareth saw scarlet blots before his eyes as he turned
away from it.
"Here's a kitchen knave from Arthur's hall who has overthrown your
brother," Lynette called across the river to him.
"Ugh!" returned Sir Noonday-Sun, raising his visor to reveal his round
foolish face like a cipher, and with that he pushed his horse into the
foaming stream.
Gareth met him midway and struck him four blows of his sword. As he was
about to deal the fifth stroke the horse of the Noonday-Sun slipped and
the stream washed his dazzling master away. Gareth plucked him out of
the water and sent him back to King Arthur.
"Lead and I follow," he said to Lynette.
"Do not fancy," she rejoined, as she guided him toward the third passing
of the river, "that I thought you bold or brave when you overcame Sir
Noonday-Sun; he just slipped on the river-bed. Here we are at the third
fool in the allegory, Sir Evening-star. You see he looks naked but he is
only wrapped in hardened skins that fit him like his own. They will turn
the blade of your sword."
"Never mind," Gareth said, "the wind may turn again and the kitchen
odors grow faint."
Then Lynette called to the Evening-star:
"Both of your brothers have gone down before this youth and so will you.
Aren't you old?"
"Old with the strength of twenty boys," said Sir Evening-star.
"Old in boasting," Gareth cried, "but the same strength that slew your
brothers can slay you."
Then the Evening-star blew a deadly note upon his horn and a
storm-beaten, russet, grizzly old woman came out and armed him in a
quantity of dingy weapons. The two knights clashed together on the
bridge and Gareth brought the Evening-star groveling in a minute to his
feet on his knees. But the other vaulted up again so quickly that Gareth
panted and half despaired of winning the victory.
Then Lynette cried: "Well done, knave; you are as noble as any knight.
Now do not shame me; I said you would win. Strike! strike! and the wind
will change again."
Gareth struck harder, he hewed great pieces of armor from the old
knight, but clashed in vain with his sword against the hard skin, until
at last he lashed the Evening-star's sword and broke it at the hilt. "I
have you now!" he shouted, but the cowardly knight of the Evening-star
writhed his arms about the lad till Gareth was almost strangled. Yet
straining himself to the uttermost he finally _tossed his foe headlong
over the side of the bridge_ to sink or to swim as the waves allowed.
"Lead and I follow," Gareth said to Lynette.
"No, it is lead no longer," the maiden replied. "Ride beside me the
knightliest of all kitchen knaves. Sir I am ashamed that I have treated
you so. Pardon me. I do wonder who you are, you knave."
"You are not to blame for anything," Gareth said, "except for your
mistrusting of the king when he sent you some one to defend you. You
said what you thought and I answered by my actions."
At that moment he heard the hoofs of a horse clattering in the road
behind him. "Stay!" cried a knight with a veiled shield, "I have come to
avenge my friend, Sir Kay."
Gareth turned, and in a thrice had closed in upon the stranger, but when
he felt the touch of the stranger knight's magical spear, which was the
wonder of the world he fell to the earth. As he felt the grass in his
hands he burst into laughter.
[Illustration: TOSSED HIS FOE OVER THE SIDE OF THE BRIDGE.]
"Why do you laugh?" asked Lynette.
"Because here am I, the son of old King Lot and good Queen Bellicent,
the victor of the three bridges, and a knight of Arthur's thrown by no
one knows whom."
"I have come to help you and not harm you," said the strange knight,
revealing himself. It was Lancelot, whom King Arthur had sent to keep a
guardian eye upon young Gareth in this his first quest, to prevent him
from being killed or taken away.
"And why did you refuse to come when I wanted you, and now come just in
time to shame my poor defender just when I was beginning to feel proud
of him?" asked Lynette.
"But he isn't shamed," Lancelot answered. "What knight is not overthrown
sometimes? By being defeated we learn to overcome, so hail Prince and
Knight of our Round Table!" "You did well Gareth, only you and your
horse were a little weary."
[Illustration: SHE TENDED HIM AS GENTLY AS A MOTHER.]
Lynette led them into a glen and a cave where they found pleasant drinks
and meat, and where Gareth fell asleep.
"You have good reason to feel sleepy," cried Lynette. "Sleep soundly and
wake strong." _And she tended him as gently as a mother_, and watched
over him carefully as he slept.
When Gareth woke Lancelot gave him his own horse and shield to use in
fighting the last awful outlaw, but as they drew near Lynette clutched
at the shield and pleaded with him: "Give it back to Lancelot," said
she. "O curse my tongue that was reviling you so today. He must do the
fighting now. You have done wonders, but you cannot do miracles. You
have thrown three men today and that is glory enough. You will get all
maimed and mangled if you go on now when you are tired. There, I vow you
must not try the fourth."
But Gareth told her that her sharp words during the day had just spurred
him on to do his best and he said he must not now leave his quest until
he had finished. So Lancelot advised him how best to manage his horse
and his lance, his sword and his shield when meeting a foe that was
stouter than himself, winning with fineness and skill where he lacked in
strength.
But Gareth replied that he knew but one rule in fighting and that was to
dash against his foe and overcome him.
"Heaven help you," cried Lynette, and she made her palfrey halt.
"There!" They were facing the camp of the Knight of Death.
There was a huge black pavilion, a black banner and a black horn. Gareth
blew the horn and heard hollow tramplings to and fro and muffled voices.
Then on a night-black horse, in night-black arms rode forth the dread
warrior. A white breast-bone showed in front. He spoke not a word which
made him the more fearful.
"Fool!" shouted Gareth sturdily. "People say that you have the strength
of ten men; can't you trust to it without depending on these toggeries
and tricks?"
But the Knight of Death said nothing. Lady Lyonors at her castle window
wept, and one of her maids fainted away, and Gareth felt his head
prickling beneath his helmet and Lancelot felt his blood turning cold.
Every one stood aghast.
Then the chargers bounded forward and Gareth struck Death to the ground.
Drawing out his sword he split apart the vast skull; one half of it fell
to the right and one half to the left. Then he was about to strike at
the helmet when out of it peeped the face of a blooming young boy, as
fresh as a flower.
"O Knight!" cried the laddie. "Do not kill me. My three brothers made me
do it to make a horror all about the castle. They never dreamed that
anyone could pass the bridges."
Then Lady Lyonors with all her house had a great party of dancing and
revelry and song and making merry because the hideous Knight of Death
that had terrified them so was only a pretty little boy. And there was
mirth over Gareth's victorious quest.
And some people say that Gareth married Lynette, but others who tell the
story later say he wedded with Lyonors.
THE MARRIAGE OF GERAINT.
King Arthur had come to the old city of Caerleon on the River Usk to
hold his court, and was sitting high in his royal hall when a woodman,
all bedraggled with the mists of the forests came tripping up in haste
before his throne.
"O noble King," he cried, "today I saw a wonderful deer, a hart all
milky white running through among the trees, and, nothing like it has
ever been seen here before."
The king, who loved the chase, was very pleased and immediately gave
orders that the royal horns should be blown for all the court to go a
hunting after the beautiful white deer the following morning. Queen
Guinevere wished to go with them to watch the hounds and huntsmen and
dancing horses in the chase. She slept late, however, the next day with
her pleasant dreams, and Arthur with his Knights of the Round Table had
sped gloriously away on their snorting chargers when she arose, called
one of her maids to come with her, mounted her palfrey and forded the
River Usk to pass over by the forest.
[Illustration: A WOODMAN ALL BEDRAGGLED CAME IN HASTE BEFORE HIS
THRONE.]
There they climbed up on a little knoll and stood listening for the
hounds, but instead of the barking of the king's dogs they heard the
sound of a horse's hoofs trampling behind them. It was Prince Geraint's
charger as he flashed over the shallow ford of the river, then galloped
up the banks of the knoll to her side. He carried not a single weapon
except his golden-hilted sword and wore, not his hunting-dress, but gay
holiday silks with a purple scarf about him swinging an apple of gold at
either end and glancing like a dragon-fly. He bowed low to the sweet,
stately queen.
"You're late, very late, Sir Prince," said she, "later even than we."
"Yes, noble queen," replied Geraint, "I'm so late that I'm not going to
the hunt; I've come like you just to watch it."
"Then stay with me," the queen said, "for here on this little knoll, if
anywhere, you will have a good chance to see the hounds, often they dash
by at its very feet."
So Geraint stood by the queen, thinking he would catch particularly the
baying of Cavall, Arthur's loudest dog, which would tell him that the
hunters were coming. As they waited however, along the base of the
knoll, came a knight, a lady and a dwarf riding slowly by on their
horses. The knight wore his visor up showing his imperious and very
haughty young face. The dwarf lagged behind.
"That knight doesn't belong to the Round Table, does he?" asked the
queen. "I don't know him."
"No, nor I," replied Geraint.
So the queen sent her maid over to the dwarf to find out the name of his
master. But the dwarf was old and crotchety and would not tell her.
"Then I'll ask your master himself," cried the maid.
"No, indeed, you shall not!" cried the dwarf, "you are not fit even to
speak of him," and as the girl turned her horse to approach the proud
young knight, the misshapen little dwarf of a servant struck at her with
his whip, and she came scampering back indignantly to the queen.
[Illustration: HE STRUCK OUT HIS WHIP AND CUT THE PRINCE'S CHEEK.]
"I'll learn his name for you," Geraint exclaimed, and he rode off
sharply.
But the impudent dwarf answered just as before and when Prince Geraint
moved on toward his master he struck out his whip and cut the prince's
cheek so that the blood streamed upon the purple scarf dyeing it red.
Instantly Geraint reached for the hilt of his sword to strike down the
vicious little <DW40> but then remembering that he was a prince and
disdaining to fight with a dwarf, he did not even say a word, but
cantered back to Queen Guinevere's side.
"Noble Queen," he cried fiercely. "I am going to avenge this insult that
has been done you. I'll track these vermin to the earth. For even
although I am riding unarmed just now, as we go along I will come to
some place where I can borrow weapons or hire them. And then when I have
my man I'll fight him, and on the third day from today I'll be back
again unless I die in the fight. So good-bye, farewell."
"Farewell, handsome prince," the queen answered. "Good fortune in your
quest and may you live to marry your first love whoever that may be. But
whether she will be a princess or a beggar from the hedgerows, before
you wed with her bring her back to me and I will robe her for her
wedding day."
Prince Geraint bowed and with that he was off. One minute he thought he
heard the noble milk-white deer brought to bay by the dogs, the next he
thought he heard the hunter's horn far away and felt a little vexed to
think he must be following this stupid dwarf while all the others were
at the chase. But he had determined to avenge the queen and up and down
the grassy glades and valleys pursued the three enemies until at last at
sundown they emerged from the forest, climbed up on the ridge of a hill
where they looked like shadows against the dark sky, then sank again on
the other side.
Below on the other side of the ridge ran the long street of a clamoring
little town in a long valley, on one side a new white fortress and on
the other, across a ravine and a bridge, a fallen old castle in decay.
The knight, the lady and the dwarf rode on to the white fortress, then
vanished within its walls.
"There!" cried Geraint, "now I have him! I have tracked him to his hole,
and tomorrow when I'm rested I'll fight him."
Then he turned wearily down the long street of the noisy village to look
for his night's lodging, but he found every inn and tavern crowded, and
everywhere horses in the stables were being shod and young fellows were
busy burnishing their master's armor.
"What does all this hubbub mean?" asked Geraint of one of these youths.
The lad did not stop his work one instant, but went on scouring and
replied, "It's the sparrow-hawk."
As Prince Geraint did not know what was meant by the sparrow-hawk he
trotted a little farther along the street until he came to a quiet old
man trudging by with a sack of corn on his back.
"Why is your town so noisy and busy to-night, good old fellow?" he
cried.
"Ugh! the sparrow-hawk!" the old fellow said gruffly.
So the prince rode his horse yet a little farther until he saw an
armor-maker's shop. The armor-maker sat inside with his back turned, all
doubled over a helmet which he was riveting together upon his knee.
"Armorer," cried Geraint, "what is going on? Why is there such a din?"
The man did not pause in his riveting even to turn about and face the
stranger, but said quickly as if to finish speaking as rapidly as he
could, "Friend, the people who are working for the sparrow-hawk have no
time for idle questions."
At this Geraint flashed up angrily.
"A fig for your sparrow-hawk! I wish all the bits of birds of the air
would peck him dead. You imagine that this little cackle in your baby
town is all the noise and murmur of the great world. What do I care
about it? It is nothing to me. Listen to me, now, if you are not gone
hawk-mad like the rest, where can I get a lodging for the night, and
more than that, where can I get some arms, arms, arms, to fight my
enemy? Tell me."
The hurrying armor-maker looked about in amazement to see this gorgeous
cavalier in purple silks standing before his bit of a shop.
"O pardon me, stranger knight," said he very politely. "We are holding a
great tournament here tomorrow morning and there is hardly any time to
do one-half the work that has to be finished before then. Arms, did you
say? Indeed I cannot tell you where to get any; all that there are in
this town are needed for to-morrow in the lists. And as for lodging, I
don't know unless perhaps at Earl Yniol's in the old castle across the
bridge." Then he again picked up his helmet and turned his back to the
prince.
So Geraint, still a wee mite vexed, rode over the bridge that spanned
the ravine, to go to the ruined castle. There upon the farther side sat
the hoary-headed Earl Yniol, dressed in some magnificent shabby old
clothes which had been fit for a king's parties when they were new.
"Where are you going, son?" he queried of Geraint, waking from his
reveries and dreaminess.
"O friend, I'm looking for some shelter for the night," Geraint replied.
"Come in then," Yniol said, "and accept of my hospitality. Our house was
rich once and now it is poor, but it always keeps its door open to the
stranger."
"Oh, anything will do for me," cried Geraint. "If only you won't serve
me sparrow-hawks for my supper I'll eat with all the passion of a whole
day's fast."
The old earl smiled and sighed as he rejoined, "I have more serious
reason than you to curse this sparrow-hawk. But go in and we will not
have a word about him even jokingly unless you wish it."
Whereupon Geraint passed into the desolate castle court, where the
stones of the pavement were all broken and overgrown with wild plants,
and the turrets and walls were shattered. As he stood awaiting the Earl
Yniol, the voice of a young girl singing like a nightingale rang out
from one of the open castle windows.
It was the voice of Enid, Earl Yniol's daughter as she sang the song of
Fortune and her Wheel:
"Turn, Fortune, thy wheel with smile or frown,
With that wild wheel we go not up or down;
Our hoard is little, but our hearts are great."
"The song of that little bird describes the nest she lives in," cried
Earl Yniol approaching. "Enter."
Geraint alighted from his charger and stepped within the large dusky
cobwebbed hall, where an aged lady sat, with Enid moving about her, like
a little flower in a wilted sheath of a faded silk gown.
"Enid, the good knight's horse is standing in the court," cried the
earl. "Take him to the stall and give him some corn, then go to town and
buy us some meat and wine."
[Illustration: GERAINT STEPPED WITHIN THE DUSKY COBWEBBED HALL.]
Geraint wished that he might do this servant's work instead of this
pretty young lady, but as he started to follow her the old gray earl
stopped him.
"We're old and poor," he said, "but not so poor and old as to let our
guests wait upon themselves."
So Enid fetched the wine and the meat and the cakes and the bread; and
she served at the table while her mother, father and Geraint sat around.
Geraint wished that he might stoop to kiss her tender little thumb as it
held the platter when she laid it down.
[Illustration: ENID FETCHED THE WINE AND THE MEAT AND THE CAKES.]
"Fair host and Earl," he said after his refreshing supper, "who is this
sparrow-hawk that everybody in the town is talking about? And yet I do
not wish you to give me his name, for perhaps he is the knight I saw
riding into the new fortress the other side of the bridge at the other
end of the town. His name I am going to have from his own lips, for I
am Geraint of Devon. This morning when the queen sent her maid to find
out his name he struck at the girl with his whip, and I've sworn
vengeance for such a great insult done our queen, and have followed him
to his hold, and as soon as I can get arms I will fight him."
"And are you the renowned Geraint?" cried Earl Yniol beaming. "Well, as
soon as I saw you coming toward me on the bridge I knew that you were no
ordinary man. By the state and presence of your bearing I might have
guessed you to be one of Arthur's Knights of the Round Table at Camelot.
Pray do not suppose that I am flattering you foolishly. This dear child
of mine has often heard me telling glorious stories of all the famous
things you have done for the king and the people. And she has asked me
to repeat them again and again.
"Poor thing, there never has lived a woman with such miserable lovers as
she has had. The first was Limours, who did nothing but drink and brawl,
even when he was making love to her. And the second was the
'sparrow-hawk,' my nephew, my curse. I will not let his name slip from
me if I can help it. When I told him that he could not marry my daughter
he spread a false rumour all round here among the people that his father
had left him a great sum of money in my keeping and that I had never
passed it over to him but had retained it for myself. He bribed all my
servants with large promises and stirred up this whole little old town
of mine against me, my own town. That was the night of Enid's birthday
nearly three years ago. They sacked my house, ousted me from my earldom,
threw us into this dilapidated, dingy old place and built up that grand
new white fort. He would kill me if he did not despise me too much to
do so; and sometimes I believe I despise myself for letting him have his
way. I scarcely know whether I am very wise or very silly, very manly or
very base to suffer it all so patiently."
"Well said," cried Geraint eagerly. "But the arms, the arms, where can I
get arms for myself? Then if the sparrow-hawk will fight tomorrow in the
tourney I may be able to bring down his terrible pride a little."
"I have arms," said Yniol, "although they are old and rusty, Prince
Geraint, and you would be welcome to have them for the asking. But in
this tournament of tomorrow no knight is allowed to tilt unless the lady
he loves best come there too. The forks are fastened into the meadow
ground and over them is placed a silver wand, above that a golden
sparrow-hawk, the prize of beauty for the fairest woman there. And
whoever wins in the tourney presents this to the lady-love whom he has
brought with him. Since my nephew is a man of very large bone and is
clever with his lance he has always won it for his lady. That is how he
has earned his title of sparrow-hawk. But you have no lady so you will
not be able to fight."
Then Geraint leaned forward toward the earl.
"With your leave, noble Earl Yniol," he replied, "I will do battle for
your daughter. For although I have seen all the beauties of the day
never have I come upon anything so wonderfully lovely as she. If it
should happen that I prove victor, as true as heaven, I will make her my
wife!"
Yniol's heart danced in his bosom for joy, and he turned about for Enid,
but she had fluttered away as soon as her name had been mentioned, so
he tenderly grasped the hands of her mother in his own and said:
"Mother, young girls are shy little things and best understood by their
own mothers. Before you go to rest to night, find out what Enid will
think about this."
So the earl's wife passed out to speak with Enid, and Enid became so
glad and excited that she could not sleep the entire happy night long.
But very early the next morning, as soon as the pale sky began to redden
with the sun she arose, then called her mother, and hand in hand,
tripped over with her to the place of the tournament. There they awaited
for Yniol and Geraint. Geraint came wearing the Earl's rusty, worn old
arms, yet in spite of them looked stately and princely.
Many other knights in blazing armor gathered there for the jousts, with
many fine ladies, and by and by the whole town full of people flooded
in, settling in a circle around the lists. Then the two forks were fixed
into the earth, above them a wand of silver was laid, and over it the
golden sparrow-hawk. The trumpet was blown and Yniol's nephew rose and
spoke:
"Come forward, my lady," he cried to the maiden who had come with him.
"Fairest of the fair, take the prize of beauty which I have won for you
during the past two years."
"Stay!" Prince Geraint cried loudly. "There is a worthier beauty here."
The earl's nephew looked round with surprise and disdain to see his
uncle's family and the prince.
"Do battle for it then," he shouted angrily.
Geraint sprang forward and the tourney was begun. Three times the two
warriors clashed together. _Three times they broke their spears._ Then
both were thrown from their horses. They now drew their swords; and
with them lashed at one another so frequently and with such dreadfully
hard strokes that all the crowd wondered. Now and again from the distant
walls came the sounds of applause, like the clapping of phantom hands.
The perspiration and the blood flowed together down the strong bodies of
the combatants. Each was as sturdy as the other.
[Illustration]
"Remember the great insult done our queen!" Earl Yniol cried at last.
This so inflamed Geraint that he heaved his vast sword-blade aloft,
cracked through his enemy's helmet, bit into the bone of his head,
felled the haughty knight, and set his feet upon his breast.
"Your name!" demanded Geraint.
"Edryn, the son of Nudd," groaned the fallen warrior.
"Very well, then Edryn, the son of Nudd," returned Geraint, "you must do
these two things or else you will have to die. First, you with your lady
and your dwarf must ride to Arthur's court at Caerleon and crave their
pardon for the insult you did the queen yesterday morning, and you must
bide her decree in the punishment she awards you. Secondly, you must
give back the earldom to your uncle the Earl of Yniol. You will do these
two things or you die."
"I will do them," cried Edryn. "For never before was I ever overcome.
But now all of my pride is broken down, for Enid has seen me fall."
With that Edryn rose from the ground like a man, took his lady and the
dwarf on their horses to Arthur's court. There receiving the sweet
forgiveness of the queen, he became a true knight of the Round Table,
and at the last died in battle while he fought for his king.
But Geraint when the tourney was over and he had come back to the
castle, drew Enid aside to tell her that early the next morning he would
have to start for Caerleon and that she should be ready to ride away
with him to be married at the court with tremendous pomp. For that would
be three days after the King's chase, when the prince had promised Queen
Guinevere he would be back. But of that he did not speak to Enid, who
wondered why he was so bent on returning immediately, and why she could
not have time at home to prepare herself some pretty robes to wear.
Imagine, she thought, such a grand and frightful thing as a court, the
queen's court, with all the graceful ladies staring at her in that faded
old silk dress! And although she promised Geraint that she would go as
he wished, when she woke to the dread day for making her appearance at
court, she still yearned that he would only stay yet a little while so
that she could sew herself some clothes, that she had the flowered silk
which her mother had given her three years ago for her birthday and
which Edryn's men had robbed from her when they sacked the house and
scattered everything she ever owned to all the winds. How she wished
that handsome Geraint had known her then, those three years ago when she
wore so many pretty dresses and jewels!
But while she lay dreamily thinking, softly in trod her mother bearing
on her arm a gorgeous, delicate robe.
"Do you recognize it, child?" she cried.
It was that self-same birthday dress, three years old, but as beautiful
as new and never worn.
"Yesterday after the jousts your father went through all the town from
house to house and ordered that all sack and plunder which the men had
taken from us should be brought back, for he was again to be in his
earldom. So last evening while you were talking with the prince some one
came up from the town and placed this in my hands. I did not tell you
about it then for I wished to keep it as a sweet surprise for you this
morning. And it is a sweet surprise, isn't it? For although the prince
yesterday did say that you were the fairest of the fair there is no
handsome girl in the world but looks handsomer in new clothes than in
old. And it would have been a shame for you to go to the court in your
poor old faded silk which you have worn so long and so patiently. The
great ladies there might say that Prince Geraint had plucked up some
ragged robin from the hedges."
[Illustration: BEARING A GORGEOUS ROBE.]
So Enid was put into the fine flowered robe.
Her mother said that after she had gone to the queen's court, she, the
poor old mother at home, who was too feeble to journey so far with her
daughter, would think over and over again of her pretty princess at
Camelot. And the old gray Earl Yniol went in to tell Geraint of Enid's
fanciful apparel.
But Geraint was not delighted with the magnificence.
"Say to her," he answered the earl, "that by all my love for her,
although I give her no other reason, I entreat Enid to wear that faded
old silk dress of hers and no other."
This amazing and hard message from Geraint made poor little Enid's face
fall like a meadowful of corn blasted by a rainstorm. Still she
willingly laid aside her gold finery for his sake, slipped into the
faded silk, and pattered down the steps to meet Geraint. He scanned her
so eagerly from her tip to her toe that both her rosy cheeks burned like
flames. Then as he noted her mother's clouded face he said very kindly:
"My new mother don't be very angry, or grieved with your new son because
of what I have just asked Enid to do. I had a very good reason for it
and I will explain it all to you. The other day when I left the queen at
Caerleon to avenge the insult done her by Edryn, the son of Nudd, she
made me two wishes. The one was that I should be successful with my
quest and the other was that I should wed with my first love. Then she
promised that whoever my bride should be she herself with her own royal
hands would dress her for her wedding day, splendidly, like the very sun
in the skies. So when I found this lovely Enid of yours in her shabby
clothes I vowed that the queen's hands only should array her in handsome
new robes that befitted her grace and beauty. But never mind, dear
mother, some day you will come to see Enid and then she will wear the
golden, flowered birthday dress which you gave her three years ago."
Then the earl's wife smiled through her tears, wrapped Enid in a mantle,
kissed her gentle farewells, and in a moment saw her riding far, far
away beside Geraint.
The queen Guinevere that day had three times climbed the royal tower at
Caerleon to look far into the valley for some sign of Geraint, who had
promised to be back that day, if he did not fall in battle, and who
would certainly come now, since Edryn had been vanquished and had come
to the court. At last when evening had fallen she spied the prince's
charger pacing nobly along the road, and Enid's palfrey at his side.
Instantly Queen Guinevere sped down from the small window in the high
turret, tripped out to the gate to greet him and embrace the lovely Enid
as a long-loved friend.
The old City of Caerleon was gay for one whole week, over the wedding
week of Geraint and Enid. The queen herself dressed Enid for her
marriage like the very sunlight, Dubric, the highest saint of the
church, married them, and they lived for nearly a year at the court with
Arthur and sweet Guinevere.
And so the insult done the queen was avenged, and her two wishes were
fulfilled. For Geraint overcame his enemy and wedded with his
first-love, dressed for her marriage by the queen.
GERAINT'S QUEST OF HONOR.
One morning Prince Geraint went into Arthur's hall and said:
"O King, my princedom is in danger. It lies close to the territory which
is infested with bandits, earls and caitiff knights, assassins and all
sorts of outlaws. Give me your kind good leave and I will go there to
defend my lands."
The king said the prince might go, and sent fifty armed knights to
protect him and pretty Enid as they traveled away on their horses across
the Severn River into their own country, the Land of Devon.
After Geraint had come into Devon he forgot what he had said to the king
of ridding his princedom of outlawry, he forgot the chase where he had
always been so clever in tracking his game, forgot the tournament where
he had won victory after victory, forgot all his former glory and his
name, forgot his lands and their cares, forgot everything he ever did,
and did nothing at all but lie about at home and talk with Enid. At last
all his people began to gossip about their fine prince who once had been
illustrious everywhere and now had become an idle stay-at-home who spent
his time in making love to his wife.
[Illustration: ENID HEARD OF GERAINT FROM HER HAIR-DRESSER.]
Enid heard of the tattling about Geraint from her hair-dresser, and one
morning as he lay abed, she went over it all to herself, talking aloud.
She wished, that he would not abandon all his knightly pursuits but
would hunt and fight again and add to his lustre. She felt very bashful
about mentioning the matter to him as she was very shy by nature and
lived in a time when wives were altogether over-ruled by their husbands,
yet to say nothing she thought would not be showing herself a true wife
to Geraint. All this and more Enid went over to herself.
The drowsy prince, half awake, just half heard her and quite
misunderstood her meaning. When she said that in keeping quiet about the
gossip she was not a true wife to him he supposed she meant that she no
longer cared for him, that he was not a handsome and strong enough man
to suit her. This grieved him deeply and made him very angry with her,
for Geraint had really given up all the glory of the king's court just
to be alone with Enid, although no one knew it. And the thought that now
she looked down upon him infuriated all his heart. A word would have
made everything right but he didn't say it.
Springing up quickly from his bed he roused his squire and said, "Get
ready our horses, my charger and the princess' palfrey. And you,"
turning a frowning face to the princess, "put on the worst looking,
meanest, poorest dress you have and come away with me. We are going on a
quest of honor and then you will see what sort of soldier I am."
Enid wondered why her lord was so vexed with her and replied, "If I have
displeased you surely you will tell me why."
But Geraint would not say; he could not bear to speak of it. So Enid
hurried after her poor old faded silk gown with the summer flowers among
its folds, which she had worn to ride from her old home to Caerleon, and
hastily dressed.
"Do not ride at my side," Geraint said as they both mounted their horses
to start away. "Ride ahead of me, a good way ahead of me, and no matter
what may happen, do not speak a word to me, no not a word."
Enid listened, wondering what had come over her lord.
"There!" he cried as they were off, "we will make our way along with our
iron weapons, not with gold money." So saying, he loosed the great purse
which dangled from his belt and tossed it back to his squire who stood
on the marble threshold of the doorway where the golden coins flashed
and clattered as they scattered every which-way over the floor. "Now
then, Enid, to the wild woods!"
At that they made for the swampy, desolated forest lands that were
famous for their perilous paths and their bandits, Enid with a white
face going before, Geraint coming gloomily nearly a quarter of a mile
after.
The morning was only half begun when the white princess became aware
that behind a rock hiding in the shadow stood three tall knights on
horseback, armed from tip to toe, bandit outlaws lying in wait to fall
upon whoever should pass. She heard one saying to his comrades as he
pointed toward Geraint:
"Look here comes some lazy-bones who seems just about as bold as a dog
who has had the worst of it in a fight. Come, we will kill him, and then
we will take his horse and armor and his lady."
Enid thought, "I'll go back a little way to Geraint and tell him about
these ruffians, for even if it will madden him I should rather have him
kill me than to have him fall into their hands."
She guided her palfrey backward and bravely met the frowning face which
greeted her, saying timidly:
"My lord, there are three bandit knights behind a rock a little way
beyond us who are boasting that they will slay you and steal your horse
and armor and make me their captive."
"Did I tell you," cried Geraint angrily, "that you should warn me of any
danger. There was only one thing which I told you to do and that was to
keep quiet; and this is the way you have heeded me! a pretty way! But
win or lose, you shall see by these fellows that my vigor is not lost."
Then Enid stood back as the three outlaws flashed out of their ambush
and bore down upon the prince.
Geraint aimed first for the middle one, driving his long spear into the
bandit's breast and out on the other side. The two others in the
meanwhile had dashed upon him with their lances, but they had broken on
his magnificent armor like so many icicles. He now turned upon them with
his broadsword, swinging it first to the right and then to the left,
first stunning them with his blows, then slaying them outright. And when
all three had fallen he dismounted, and like a hunter skinning the wild
beasts he has shot, he stripped the three robber knights of their gay
suits of armor, and leaving the bodies lie, bound each man's sword,
spear and coat of arms to his horse, tied the three bridle reins of the
three empty horses together and cried to Enid.
"Drive these on before you."
Enid drove them on across the wastelands, Geraint following after. As
she passed into the first shallow shade of the forest she described
three more horsemen partly hidden in the gloom of three sturdy
oak-trees. All were armed and one was a veritable giant, so tall and
bulky, towering above his companions.
[Illustration: THE THREE OUTLAWS BORE DOWN UPON THE PRINCE.]
"See there, a prize!" bellowed the giant and set Enid's pulses in a
quiver. "Three horses and three suits of armor, and all in charge
of--whom? A girl! Isn't that simple? Lay on, my men!"
"No," cried the second, "behind is coming a knight. A coward and a fool,
for see how he hangs his head."
The giant thundered back gaily.
"Yes? Only one? Wait here and as he goes by make for him."
"I will go no farther until Geraint comes," Enid said to herself
stopping her horse. "And then I will tell him about these villains. He
must be so weary with his other fight and they will fall upon him
unawares. I shall have to disobey him again for his own sake. How could
I dare to obey him and let him be harmed? I must speak; if he kills me
for it I shall only have lost my own life to save a life that is dearer
to me than my own."
So she waited until the prince approached when she said with a timid
firmness, "Have I your leave to speak?"
"You take it without asking when you speak," he replied, and she
continued:
"There are three men lurking in the woods behind some oaks and one of
them is larger than you, a perfect giant. He told them to attack you as
you passed by them."
"If there were a hundred men in the wood and each of them a giant and if
they all made for me together I vow it would not anger me so as to have
you disobey me. Stand aside while we do battle and when we are done
stand by the victor."
At this, while Enid fell back breathing short fits of prayer but not
daring to watch, Geraint proceeded to meet his assailants. The giant was
the first to dash out for him aiming his lance at Geraint's helmet, but
the lance missed and went to one side. Geraint's spear had been a
little strained with his first encounter, but it struck through the
bulky giant's corselet and pierced his breast, then broke, one-half of
it still fast in the flesh as the giant knight fell to the earth. The
other two bandits now felt that their support and hero was gone, and
when Geraint darted rapidly on them, uttering his terrible warcry as if
there were a thousand men behind him to come to his aid, they flew into
the woods. But they were soon overtaken and pitilessly put to death.
Then Geraint, selecting the best lance, the brightest and strongest
among their spears to replace the one he had broken on the giant, he
plucked off the gaudy armor from each brigand's body, laid it on the
backs of the three horses, tied the bridle reins together and handed
them to Enid with the words, "Drive them on before you."
So Enid now followed the wild paths of the gloomy forest with two sets
of three horses, each horse laden with his master's jingling weapons and
coat of mail. Geraint came after. As they passed out of the wood into
the open sky they came to a little town with towers upon a rocky hill,
and beneath it a wide meadowland with mowers in it, mowing the hay. Down
a stony pathway from the town skipped a fair-haired lad carrying a
basket of lunch for the laborers in the field.
"Friend!" cried Geraint, as the lad trotted past him, for he saw that
Enid looked very white, "let my lady have something to eat. She is so
faint."
"Willingly," the youth answered, "and you too, my lord, even although
this feed is very coarse and only fit for the mowers."
He set down his basket and Enid and Geraint alighted and put all the
horses to graze, while they sat down on the green sward to have some
bread and barley. Enid felt too faint at heart, thinking of the
prince's strange conduct, to care a great deal for food, but Geraint was
hungry enough and had all the mowers' basket emptied almost before he
knew it.
"Boy," he cried half-ashamed, "everything is gone, which is a disgrace.
But take one of my horses and his arms by way of payment, choose the
very best."
The poor lad, who might as well have had a kingdom given him, reddened
with his extreme surprise and delight.
"My lord, you are over-paying me fifty times," he cried.
"You will be all the wealthier then," returned the prince, gaily.
"I'll take it as free gift, then," the lad answered. "The food is not
worth much. While your lady is resting here I can easily go back and
fetch more, some more for the earl's mowers. For all these mowers belong
to our great earl, and all these fields are his, and I am his, too. I'll
tell him what a fine man you are, and he will have you to his palace and
serve you with costly dinners."
"I wish no better fare than I have had," Geraint said, "I never ate
better in my life than just now when I left your poor mowers dinnerless.
And I will go into no earl's palace. If he desires to see me, let him
come to me. Now you go hire us some pleasant room in the town, stall our
horses and when you return with the food for these men tell us about
it."
"Yes, my kind lord," the glad youth cried, and he held his head high and
thought he was a gorgeous knight off to the wars as he disappeared up
the rocky path leading his handsome horse.
The prince turned himself sleepily to watch the lusty mowers laboring
under the sun as it blazed on their scythes, while Enid plucked the long
grass by the meadows' edge to weave it round and round her wedding
ring, until the boy returned and showed them the room he had got in the
town.
"If you wish anything, call the woman of the house," Prince Geraint said
to Enid as the door closed behind them. "Do not speak to me."
"Yes, my lord," returned Enid, still marvelling at his cold ways.
Silently they sat down, she at one end, he at the other, as quiet as
pictures. But suddenly a mass of voices sounded up the street, and heel
after heel echoing upon the pavement. In a twinkling the door to their
room was pushed back to the wall while a mob of boisterous young
gentlemen tumbled in led by the Earl of Limours, the wild lord of the
town, and Enid's old suitor whom her father had rejected long ago, a man
as beautiful as a woman and very graceful. He seized the prince's hand
warmly, welcomed him to the town and stealthily, out of the corner of
his eye, caught a glimpse of unhappy Enid nestled all alone at the
farther end of the room.
The prince immediately sent for every sort of delicious things to eat
and drink from the town, told the earl, to bid all his friends for a
feast and soon was gaily making merry with the men, drinking, laughing,
joking.
"May I have your leave, my lord," cried Earl Limours, "to cross the room
and speak a word with your lady who seems so lonely?"
"My free leave," cried the merry Prince Geraint, who did not know the
earl, "Get her to speak with you; she has nothing to say to me."
As Limours stepped to Enid's side he lifted his eyes adoringly, bowed at
her side and said in a whisper:
"Enid, you pilot star of my life, I see that Geraint is very unkind to
you and loves you no longer. What a laughing stock he is making of you
with that wretched old dress you have on! But I, I love you still as
always. Just say the word and I will have him put into the keep and you
will come with me. I will be kind to you forever."
The tears fluttered into the earl's eyes as he spoke.
"Earl," replied Enid, "if you love me as you used to do in the years
long ago, and are not joking now, come in the morning and take me by
force from the prince. But leave me tonight. I am wearied to death."
So the earl made a low bow, brandishing his plumes until they brushed
his very insteps, while the stout prince bade him a loud good night, and
he moved away talking to his men.
[Illustration: THE EARL MADE A LOW BOW.]
But as soon as he was gone Enid began to plan how she could escape with
Geraint before Earl Limours should come after her in the morning. She
was too afraid of Geraint to speak with him about it, but when he had
fallen asleep she stepped lightly about the room and gathered the pieces
of his armor together in one place ready for an early departure on the
morrow. Then she dropped off into slumber. But suddenly she heard a loud
sound, the earl with his wild following blowing his trumpet to call her
to come out, she thought. But it was only the great red cock in the
yard below crowing at the daylight which had begun to glimmer now across
the heap of Geraint's armor. She rose immediately in her fright to see
that all was well, went over to examine the weapons and unwittingly let
the casque fall jangling to the floor. This woke Geraint, who started up
and stared at her.
"My lord," began Enid, and then she told him all that Earl Limours had
said to her and how she had put him off by telling him to come this
morning.
"Call the woman of the house and tell her to bring the charger and the
palfrey," Geraint cried angrily. "Your sweet face makes fools of good
fellows." Geraint loved Enid still and he was in as great perplexity as
she, for after misunderstanding what she had said he no more knew
whether she cared for him truly than she knew what was troubling him and
making him act in this unaccountable manner.
Enid slipped through the sleeping household like a ghost to deliver the
prince's message to the landlord, hurried back to help Geraint with his
armor and came down with him to spring upon her palfrey.
"What do I owe you, friends?" the prince asked his host, but before the
man could reply he added "take those five horses and their burdens of
arms."
"My lord, I have scarcely spent the price of one of them on you!" cried
the landlord astonished.
"You'll have all the more riches then," the prince laughed, then turning
to Enid, "today I charge you more particularly than ever before that
whatever you may see, hear, fancy or imagine, do not speak to me, but
obey."
"Yes, my lord," answered Enid, "I know your wish and should like to
obey, but when I go riding ahead, I hear all the violent threats you do
not hear and see the danger you cannot see, and then not to give you
warning seems hard, almost beyond me. Yet, I wish to obey you."
"Do so, then," said he. "Do not be too wise, seeing that you are
married, not to a clown but a strong man with arms to guard his own head
and yours, too."
The broad beaten path which they now took passed through toward the
wasted lands bordering on the castle of Earl Doorm, the Bull, as his
people called him, because of his ferocity.
It was still early morning when Enid caught the sound of quantities of
hoofs galloping up the road. Turning round she saw cloudsful of dust and
the points of lances sparkling in it. Then, not to disobey the prince,
yet to give him warning, she held up her finger and pointed toward the
dust. Geraint was pleased at her cunning, and immediately stopped his
horse. The moment after, the Earl of Limours dashed in upon him on a
charger as black and as stormy as a thunder-cloud.
Geraint closed with the earl, bore down on him with his spear, and in a
minute brought him stunned or dead to the ground. Then he turned to the
next-comer after Limours, overthrew him and blindly rushed back upon all
the men behind. But they were so startled at the flash and movement of
the prince that they scrambled away in a panic, leaving their leader
lying on the public highway. The horses also of the fallen warriors
whisked off from their wounded masters and wildly flew away to mix with
the vanishing mob.
"Horse and man, all of one mind," remarked Geraint, smiling, "not a hoof
of them left. What do you say, Enid, shall we strip the earl and pay for
a dinner or shall we fast? Fast? Then go on and let us pray heaven to
send us some Earl of Doorm's men so that we can earn ourselves something
to eat."
Enid sadly eyed her bridle-reins and led the way, Geraint coming after,
scarcely knowing that he had been pricked by Limours in his side, and
that he was bleeding secretly beneath his armor. But at last his head
and helmet began to wag unsteadily, and at a sudden swerving of the road
he was tossed from his horse upon a bank of grass. Enid heard the
clashing of the fall, and too terrified to cry out, came back all pale.
Then she dismounted, loosed the fastenings of his armor and bound up his
wounds with her veil. Then she sat down desolately and began to cry,
wondering what ever she should do.
[Illustration: ENID SAT DOWN DESOLATELY AND BEGAN TO CRY.]
Many men passed by but no one took any notice of her. For in that
lawless, turbulent earldom no one minded a woman weeping for a murdered
lover than they now mind a summer shower. One man scurrying as fast as
ever he could travel toward the bandit earl's castle, drove the sand
sweeping into her poor eyes, and another coming in the opposite
direction from out the earl's castle park in seeming hot haste, turned
all the long dusty road into a column of smoke behind him, and
frightened her little palfrey so that it scoured off into the coppices
and was lost. But the prince's charger stood beside them and grieved
over the mishap like a man.
At noon a huge warrior with a big face and russet beard and eyes rolling
about in search of prey, came riding hard by with a hundred spearmen at
his back all bound for some foray. It was the frightful Earl Doorm.
"What, is he dead?" cried the earl loudly to Enid, as he spied her on
the wayside.
"No, no, not dead," she quickly answered. "Would some of your kind
people take him up and bear him off somewhere out of this cruel sun? I
am very sure, quite sure that he is not dead."
"Well, if he isn't dead, why should you cry for him so? Dead or not
dead, you just spoil your pretty face with idiotic tears. They will not
help him. But since it is a pretty face, come fellows, some of you, and
take him to our hall. If he lives he will be one of our band, and if
not, why there is earth enough to bury him in. See that you take his
charger, too, a noble one."
And so saying, the rude earl passed on, while two brawny horsemen came
forward growling to think they might lose their chance of booty from the
morning's raid all for this dead man. They raised the prince upon a
litter, laying him in the hollow of his shield, and brought him into
the barren hall of Doorm, while Enid and the gentle charger followed
after. They tossed him and his litter down on an oaken settle in the
hall, and then shot away for the woods.
Enid sat through long hours all alone with Geraint besides the oaken
settle, propping his head and chafing his hands, but in the late
afternoon she saw the huge Earl Doorm returning with his lusty spearmen
and their plunder. Each hurled down a heap of spoils on the floor, threw
aside his lance and doffed his helmet, while a tribe of brightly gowned
gentle-women fluttered into the hall and began to talk with them. Earl
Doorm struck his knife against the table and bellowed for meat, and
wine. In a moment the place fairly steamed and smoked with whole roast
hogs and oxen, and everybody sat down in a hodge-podge and ate like
cattle feeding in their stalls, while Enid shrank far back startled,
into her nook.
But suddenly, when Earl Doorm had eaten all he would, and all he could
for the moment, he revolved his eyes about the bare hall and caught a
glimpse of the fair little lady drooping in her niche. Then he
recollected how she had crouched weeping by the roadside for her fallen
lord that morning. A wild pity filled his gruff heart.
"Eat, eat!" he shouted. "I never before saw any thing so pale. Be
yourself. Isn't your lord lucky, for were I dead who is there in all the
world who would mourn for me? Sweet lady, never have I ever seen a lily
like you. If there were a bit of color living in your cheeks there is
not one among my gentle-women here who would be fit to wear your
slippers for gloves. But listen to me and you will share my earldom with
me, girl, and we will live like two birds in a nest and I will bring you
all sorts of finery from every part of the world to make you happy."
As the earl spoke his two cheeks bulged with the two tremendous morsels
of meat which he had tucked into his mouth.
Enid was more alarmed than ever.
"How can I be happy over anything," replied she, "until my lord is well
again?"
The earl laughed, then plucked her up out of the corner, carried her
over to the table, thrust a dish of food before her and held a horn of
wine to her lips.
"By all heaven," cried Enid, "I will not drink until my lord gets up and
drinks and eats with me. And if he will not rise again I will not drink
any wine until I die."
At this the earl turned perfectly red and paced up and down the hall,
gnawing first his upper and then his lower lip.
"Girl," shouted he, "why wail over a man who shames your beauty so, by
dressing it in that rag? Put off those beggar-woman's weeds and robe
yourself in this which my gentle-woman has brought you."
It was a gorgeous, wonderful dress, in the tints of a shallow
sea with the blue playing into the green, and gemmed with precious
stones all down the front of it as thick as dewdrops on the grass. But
Enid was harder to move than any cold tyrant on his throne, and said:
"Earl, in this poor gown my dear lord found me first and loved me while
I was living with my father; in this poor gown I rode with him to court
and was presented to the queen; in this poor gown he bade me ride as we
came out on this fatal quest of honor, and in this poor gown I am going
to stay until he gets up again, a live, strong man, and tells me to put
it away. I have griefs enough, pray be gentle with me, let me be. O God!
I beg of your gentleness, since he is as he is, to let me be."
Then the brutal earl strode up and down the hall and cried out:
"It is of no more use to be gentle with you than to be rough. So take my
salute," and with that he slapped her lightly on her white cheek.
Enid shrieked. Instantly the fallen Geraint was up on his feet with the
sword that had laid beside him in the hollow of the shield, making a
single bound for the earl, and with one sweep of it sheared through the
swarthy neck. The rolling eyes turned glassy, the russet-bearded head
tumbled over the floor like a ball, and all the bandit knights and the
gentle-women in the hall flitted, scampering pell-mell away, yelling as
if they had seen a ghoul. Enid and Geraint were left alone.
[Illustration: THE RUSSET-BEARDED HEAD TUMBLED OVER THE FLOOR LIKE A
BALL.]
Now Geraint had come out of his swoon before the earl had returned, and
he had lain perfectly silent and immovable because he wished to test
Enid and see what she would do when she thought he was sleeping or
fainted away, or perhaps dead. So he had listened to all that had taken
place and had heard everything that Earl Doorm had said to her and all
that Enid had replied, so now he knew that she loved him as ever and
that she stood steadfast by him. All his heart filled with pity and
remorse that he had brought her away on this hard, hard quest, and had
made her suffer so much and had been so rough and cold.
"Enid," said the prince tenderly, very tenderly. "I have used you worse
than that big dead brute of a man used you. I have done you more wrong
than he. I misunderstood you. Now, now you are three times mine."
Geraint's kindness burst upon Enid so abruptly and was so unforeseen
that she could not speak a word only this:
"Fly, Geraint, they will kill you, they will come back. Fly. Your horse
is outside, my poor little thing is lost."
"You shall ride behind me, then, Enid."
So they slipped quickly outside, found the stately charger and mounted
him, first Geraint, then Enid, climbing up the prince's feet, and
throwing her arms about him to hold herself firm as they bounded off.
But as the horse dashed outside of the earl's gateway there before them
in the highroad stood a knight of Arthur's court holding his lance as if
ready to spring upon Geraint.
"Stranger!" shrieked Enid, thinking of the prince's wound and loss of
blood, "do not kill a dead man!"
"The voice of Enid!" cried the stranger knight.
Then Enid saw that he was Edryn, the son of Nudd, and feeling the more
terrified as she remembered the jousts, cried out:
"O, cousin, this is the man who spared your life!"
[Illustration: BEFORE THEM IN THE HIGHROAD STOOD A KNIGHT OF ARTHUR'S
COURT.]
Edryn stepped forward. "My lord Geraint," he said, "I took you for some
bandit knight of Doorm's. Do not fear, Enid, that I will attack the
prince. I love him. When he overthrew me at the lists he threw me
higher. For now I have been made a Knight of the Round Table and am
altogether changed. But since I used to know Earl Doorm in the old days
when I was lawless and half a bandit myself, I have come as the
mouthpiece of our king to tell Doorm to disband all his men and become
subject to Arthur, who is now on his way hither."
"Doorm is now before the King of Kings," Geraint replied, "And his men
are already scattered," and the prince pointed to groups in the
thickets or still running off in their panic. Then back to the people
all aghast whom they could see huddling, he related fully to Edryn how
he had slain the huge earl in his own hall.
[Illustration: TO THE ROYAL CAMP WHERE ARTHUR CAME OUT TO GREET THEM.]
"Come with me to the king," astonished Edryn said.
So they all traveled off to the royal camp where Arthur himself came out
to greet them, lifted Enid from her saddle, kissed her and showed her a
tent where his own physician came in to attend to Geraint's wound. When
that was healed he rode away with them to Caerleon for a visit with
Queen Guinevere, who dressed Enid again in magnificent clothes. Then
fifty armed knights escorted Enid and the prince as far as the banks of
the Severn River, where they crossed over into the land of Devon. And
all their people welcomed them back.
Geraint after that never forgot his princedom or the tournament, but was
known through all the country round as the cleverest and bravest
warrior, while his princess was called Enid the Good.
MERLIN AND VIVIEN.
Vivien was a very clever, wily and wicked woman, who wanted to become a
greater magician than even the great Merlin, who was the most famous man
of all his times, who understood all the arts, who had built the king's
harbors, ships and halls, who was a fine poet and who could read the
future in the stars in the skies.
He had once told Vivien of a charm that he could work to make people
invisible. Whenever he worked it upon anyone that person would seem to
be imprisoned within the four walls of a tower and could not get out.
The person would seem dead, lost to every one, and could be seen only by
the person who worked the charm. Vivien yearned to know what the charm
was, for she wanted to cast its spell on Merlin so that no one would
know where he was and she could become a great enchantress in the realm,
as she foolishly thought. And she planned very cleverly so as to find
out the wise old man's secret.
She wanted him to think that she loved him dearly. At first she played
about him with lively, pretty talk, vivid smiles, and he watched and
laughed at her as if she were a playful kitten. Then as she saw that he
half disdained her she began to put on very grave and serious fits,
turned red and pale when he came near her, or sighed or gazed at him, so
silently and with such sweet devotion that he half believed that she
really loved him truly.
[Illustration: HE LAUGHED AT HER.]
But after a while a great melancholy fell over Merlin, he felt so
terribly sad that he passed away out of the kings' court and went down
to the beach. There he found a little boat and stepped into it. Vivien
had followed him without his knowing it. She sat down in the boat and
while he took the sail she seized the helm of the boat. They were driven
across the sea with a strong wind and came to the shores of Brittany.
Here Merlin got out and Vivien followed him all the way into the wild
woods of Broceliande. Every step of the way Merlin was perfectly quiet.
They sat down together, she lay beside him and kissed his feet as if in
the deepest reverence and love. A twist of gold was wound round her
hair, a priceless robe of satiny samite clung about her beautiful limbs.
As she kissed his feet she cried:
"Trample me down, dear feet which I have followed all through the world
and I will worship you. Tread me down and I will kiss you for it."
But Merlin still said not a word.
[Illustration: MERLIN FELT SO TERRIBLY SAD.]
"Merlin do you love me?" at last cried Vivien, with her face sadly
appealing to him. And again, "O, Merlin, do you love me?" "Great Master,
do you love me?" she cried for the third time.
And then when he was as quiet as ever she writhed up toward him, slid
upon his knee, twined her feet about his ankles, curved her arms about
his neck and used one of her hands as a white comb to run through his
long ashy beard which she drew all across her neck down to her knees.
"See! I'm clothing myself with wisdom," she cried. "I'm a golden summer
butterfly that's been caught in a great old tyrant spider's web that's
going to eat me up in this big wild wood without a word to me."
"What do you mean, Vivien, with these pretty tricks of yours?" cried
Merlin at last. "What do you want me to give you?"
"What!" said Vivien, smiling saucily, "have you found your tongue at
last? Now yesterday you didn't open your lips once except to drink. And
then I, with my own lady hands, made a pretty cup and offered you your
water kneeling before you and you drank it, but gave me not a word of
thanks. And when we stopped at the other spring when you lay with your
feet all golden with blossoms from the meadows we passed through you
know that I bathed your feet before I bathed my own. But yet no thanks
from you. And all through this wild wood, all through this morning when
I fondled you, still not a word of thanks."
Then Merlin locked her hand in his and said, "Vivien, have you never
seen a wave as it was coming up the beach ready to break? Well, I've
been seeing a wave that was ready to break on me. It seemed to me that
some dark, tremendous wave was going to come and sweep me away from my
hold on the world, away from my fame and my usefulness and my great
name. That's why I came away from Arthur's court to make me forget it
and feel better. And when I saw you coming after me it seemed to me that
you were that wave that was going to roll all over me. But pardon me,
now, child, your pretty ways have brightened everything again, and now
tell me what you would like to have from me. For I owe you something
three times over, once for neglecting you, twice for the thanks for your
goodness to me, and lastly for those dainty gambols of yours. So tell me
now, what will you have?"
Vivien smiled mournfully as she answered:
"I've always been afraid that you were not really mine, that you didn't
love me truly, that you didn't quite trust me, and now you yourself have
owned it. Don't you see, dear love, how this strange mood of yours must
make me feel it more than ever? must make me yearn still more to prove
that you are mine, must make me wish still more to know that great charm
of waving hands and woven footsteps that you told me about, just as a
proof that you trust me? If you told that to me I should know that you
are mine, and I should have the great proof of your love, because I
think that however wise you may be you do not know me yet."
"I never was less wise, you inquisitive Vivien," said Merlin, "than when
I told you about that charm. Why won't you ask me for another boon?"
Then Vivien, as if she were the tenderest hearted little maid that ever
lived, burst into tears and said:
"No, master, don't be angry at your little girl. Caress me, let me feel
myself forgiven, for I have not the heart to ask for another boon. I
don't suppose that you know the old rhyme, 'Trust not at all or all in
all?'"
Then Merlin looked at her and half believed what she said. Her voice was
so tender, her face was so fair, her eyes were so sweetly gleaming
behind her tears.
He locked her hand in his again and said, "If you should know this charm
you might sometimes in a wild moment of anger or a mood of overstrained
affection when you wanted me all to yourself or when you were jealous
in a sudden fit, you might work it on me."
"Good!" cried Vivien, as if she were angry, "I am not trusted. Well,
hide it away, hide it, and I shall find it out, and when I've found it
beware, look out for Vivien! When you use me so it's a wonder that I can
love you at all, and as for jealousy, it seems to me this wonderful
charm was invented just to make me jealous. I suppose you have a lot of
pretty girls whom you have caged here and there all over the world with
it."
Then the great master laughed merrily.
"Long, long years ago," he said, "there lived a King in the farthest
East of the East. A tawny pirate who had plundered twenty islands or
more anchored his boat in the King's port, and in the boat was a woman.
For, as he had passed one of the islands the pirates had seen two cities
full of men in boats fighting for a woman on the sea; he had pushed up
his black boat in among the rest, lightly scattered every one of them
and brought her off with half his people killed with arrows. She was a
maiden so smooth, so white, so wonderful that a light seemed to come
from her as she walked. When the pirate came upon the shore of the
Eastern King's island the King asked him for the woman, but he would not
give her up. So the King imprisoned the pirate and made the woman his
queen.
"All the people adored her, the King's councilmen and all his soldiers,
the beasts themselves. The camels knelt down before her unbidden, and
the black slaves of the mountains rang her golden ankle bells just to
see her smile. So little wonder that the King grew very jealous. He had
his horns blown through all the hundred under-kingdoms which he ruled,
telling the people that he wanted a wizard who would teach him some
charm to work upon the queen and make her all his own. To the wizard who
could do this he promised a league of mountain land full of golden
mines, a province with a hundred miles of coast, a palace and a
princess. But all the wizards who failed should be killed and their
heads would be hung on the city gates until they mouldered away.
"So there were many, many wizards all through the hundred kingdoms who
tried to work the charm, but failed; many wizard heads bleached on the
walls, and for weeks a troupe of carrion crows hung like a cloud above
the towers of the city gateways. But at last the king's men found a
little glassy headed, hairless man who lived alone in a great wilderness
and ate nothing but grass. He read only one book, and by always reading
had got grated down, filed away and lean, with monstrous eyes and his
skin clinging to his bones. But since he never tasted wine or flesh--the
wall that separates people from spirits became crystal to him. He could
see through it, perceive the spirits as they walked and hear them
talking; so he learned their secrets. Often he drew a cloud of rain
across a sunny sky, or when there was a wild storm and the pine woods
roared he made everything calm again.
"He was the man that was wanted. They dragged him to the king's court by
force, he didn't want to go. There he taught the king how to charm the
queen so that no one could see her again, and she could see no one
except the king as he passed about the palace. She lay as if quite dead
and lost to life. But when the king offered the magician his league of
golden mines, the province with a hundred miles of sea coast, the palace
and the princess, the old man turned away, went back to his wilderness
and lived on grass and vanished away. But his book came down to me."
"You have the book!" cried Vivian smiling saucily. "The charm is written
in it. Good, take my advice and let me know the secret at once, for if
you should hide it away like a puzzle in a chest, if you should put
chest upon chest, and lock and padlock each chest thirty times and bury
them all away under some vast mound like the heaps of soldiers on the
battle-field, still I should hit upon some way of digging it out, of
picking it, of opening it and reading the charm. And _then_ if I tried
it on you who would blame me?"
"You read the book, my pretty Vivien?" cried Merlin. "Well, it's only
twenty pages long, but such pages! Every page has a square of text that
looks like a blot, the letters no longer than fleas' legs written in a
language that has long gone by, and all the borders and margins
scribbled, crossed and crammed with notes. You read that book! No one,
not even I can read the text, and no one besides me can make out the
notes on the margins. I found the charm in the margin. Oh, it is simple
enough. Any child might work it and then not be able to undo it. Don't
ask me again for it, because even although you would love me too much to
try it on me, still you might try it on some of the knights of the Round
Table."
"O, you are crueller than any man ever told of in a story, or sung about
in song!" cried Vivien. She clapped her hands together and wailed out a
shriek. "I'm stabbed to the heart! I only wished that prove to you that
were wholly mine, that you loved me and now I'm killed with a word.
There's nothing left for me to do except crawl into some hole or cave,
and if the wolves won't tear me to pieces, just to weep my life away,
killed with unutterable unkindness!"
She paused, turned away, hung her head while the hair uncoiled itself.
Then she wept afresh.
The dark wood grew darker with a storm coming over the sky.
Merlin sat thinking quietly and half believed that she was true.
"Come out of the storm," he called over to her, "come here into the
hollow old oak tree."
Then since she didn't answer, he tried three times to calm her but quite
in vain. At last, however, she let herself be conquered, came back to
her old perch, and nestled there, half falling from his knees. Gentle
Merlin saw the slow tears still standing in her eyes and threw his arms
kindly about her. But Vivien unlinked herself at once, rose with her
arms crossed upon her bosom and fled away.
"No more love between us two," she cried, "for you do not trust me. Oh,
it would have been better if I had died three times over than to have
asked you once! Farewell, think gently of me and I will go. But before I
leave you let me swear once more that if I've been planning against you
in all this, may the dark heavens send one great flash from out the sky
to burn me to a cinder!"
Just as she ended a bolt of lightning darted across the sky, and sliced
the giant oak tree into a thousand splinters and spikes.
"Oh, Merlin, save me! save me!" cried Vivien, terrified lest the heavens
had heard her oath and were going to kill her. And she flew back to his
arms. She called him her dear protector, her lord and liege, her seer,
her bard, her silver star of evening, her God, her Merlin, the one
passionate love of her life, and hugged him close.
All the time overhead the tempest bellowed, the branches snapped above
them in the rushing rain. Her glittering eyes and neck seemed to come
and go before Merlin's eyes with the lightning. At last the storm had
spent its passion, the woodland was all in peace again, and Merlin,
overtalked and overworn had told all of the charm and had fallen asleep.
[Illustration: IN THE HOLLOW OF THE OLD OAK TREE LEFT HIM LYING DEAD.]
Then in a moment Vivien worked the charm with woven footsteps and waving
arms, and in the hollow of the old oak tree left him lying dead to all
life, use and fame and name.
"I have made his glory mine! O fool!" she shrieked, and she sprang down
through the great forest, the thicket closed about her behind her and
all the woods echoed, "Fool!"
BALIN AND BALAN.
King Pellam owed Arthur some tribute money so Arthur told three of his
knights to go see about it and collect it for him.
"Very well," said one of the knights, "but listen, on the way to King
Pellam's country, near Camelot, there are two strange knights sitting
beside a fountain. They challenge and overthrow every knight that
passes. Shall I stop to fight them as we go by and send them back to
you?"
Arthur laughed, "No, don't stop for anything; let them wait until they
can find some one stronger themselves."
With that the three men left. But after they had gone Arthur, who loved
a good fight himself, started away early one morning for the fountain
side of Camelot. On its right hand he saw the knight Balin sitting under
an alder tree, with his horse beside him, and on the left hand under a
poplar tree with his horse at his side sat the knight Balan.
"Fair sirs," cried Arthur, "why are you sitting here?"
"For the sake of glory," they answered. "We're stronger than all
Arthur's court. We've proved that because we easily overthrow every
knight that comes by here."
"Well, I'm of Arthur's court, too," replied the king, "although I've
never done so much in jousts as in real wars. But see whether you can
overthrow me so easily too."
So the two brothers came out boldly and fought with Arthur, but he
struck them both lightly down, then softly came away and nobody knew
anything about it.
But that evening while Balin and Balan sat very meekly by the bubbling
water a spangled messenger came riding by and cried out to them: "Sirs,
you are sent for by the King."
So they followed the man back to the court. "Tell me your names,"
demanded Arthur, "and why do you sit there by the fountain?"
[Illustration: TWO STRANGE KNIGHTS.]
"My name is Balin," answered one of the men, "and my brother's name is
Balan. Three years ago I struck down one of your slaves whom I heard had
spoken ill of me, and you sent me away for a three years' exile. Then I
thought that if we would sit by the well and would overcome every knight
who passed by you would be a more willing to take me back. But today
some man of yours came along and conquered us both. What do you wish
with me?"
"Be wiser for falling," Arthur said. "Your chair is in the hall vacant.
Take it again and be my knight once more."
So Balin went back into the old hall of the Knights of the Round Table,
and they all clashed their cups together drinking his welcome, and sang
until all of Arthur's banners of war hanging overhead began to stir as
they always did on the battlefield.
Meanwhile the men who had gone to collect the taxes from King Pellam
returned.
"Sir King," they cried to Arthur, "We scarcely could see Pellam for the
gloom in his hall. That man who used to be one of your roughest and most
riotous enemies is now living like a monk in his castle and has all
sorts of holy things about him, and says he has given up all matters of
the world. He wouldn't even talk about the tribute money and told us
that his heir Sir Garlon, attended to his business for him, so we went
to Garlon and after a struggle we got it. Then we came away, but as we
passed through the deep woods we found one of your knights lying dead,
killed by a spear. After we had buried him, we talked with an old
woodman who told us that there's a demon of the woods who had probably
slain the knight. This demon, he said, was once a man who lived all
alone and learned black magic. He hated people so much that when he died
he became a fiend. The woodman showed us the cave where he has seen the
demon go in and out and where he lives. We saw the print of a horse's
hoof, but no more."
"Foully and villainously slain!" cried Arthur thinking of his poor
killed knight in the woods. "Who will go hunt this demon of the woods
for me?"
"I!" exclaimed Balan, ready to dart instantly away, but first he
embraced Balin, saying, "Good brother, hear; don't let your angry
passions conquer you, fight them away. Remember how these knights of the
Round Table welcomed you back. Be a loving brother with them and don't
imagine that there is hatred among them here any more than there is in
heaven itself."
When bad Balan left, Balin set himself to learn how to curb his wildness
and become a courteous and manly knight. He always hovered about
Lancelot, the pattern knight of all the court, to see how he did, and
when he noticed Lancelot's sweet smiles and his little pleasant words
that gladdened every knight or churl or child that he passed, Balin
sighed like some lame boy who longed to scale a mountain top and could
scarcely limp up one hundred feet from the base.
"It's Lancelot's worship of the queen that helps to make him gentle,"
said he to himself. "If I want to be gentle I must serve and worship
lovely Queen Guinevere too. Suppose I ask the King to let me have some
token of hers on my shield instead of these pictures of wild beasts with
big teeth and grins. Then whenever I see it I'll forget my wild heats
and violences."
"What would you like to bear on your shield?" asked the king when Balin
spoke to him about his wish.
"The queen's own crown-royal," replied Balin.
Then the queen smiled and turned to Arthur. "The crown is only the
shadow of the king," she said, "and this crown is the shadow of that
shadow. But let him have it if it will help him out of his violences."
"It's no shadow to me, my queen," cried Balan, "no shadow to me, king.
It's a light for me."
So Balin was given the crown to bear on his shield and whenever he
looked at it, it seemed to make him feel gentle and patient.
But one morning as he heard Lancelot and the queen talking together on
the white walk of lilies that led to Queen Guinevere's bower, all his
old passions seemed to come back and filled him and he darted madly away
on his horse, not stopping until he had passed the fount where he had
sat with his brother Balan and had dived into the skyless woods beyond.
There the gray-headed woodman was hewing away wearily at a branch of a
tree.
[Illustration: BALIN WAS GIVEN THE CROWN TO WEAR ON HIS SHIELD.]
"Give me your axe, Churl," cried Balin, and with one sharp cut he struck
it down.
"Lord!" cried the woodman, "you could kill the devil of this woods if
any one can. Just yesterday I saw a flash of him. Some people say that
our Sir Garlon has learned black magic too and can ride armed unseen.
Just look into the demon's cave."
But Balin said the woodman was foolish, and rode off through the glades
with a drooping head. He did not notice that on his right a great cavern
chasm yawned out of the darkness. Once he heard the mosses beneath him
thud and tremble and then the shadow of a spear shot from behind him and
ran along the ground. The light of somebody's armor flashed by him and
vanished into the woods.
Balin dashed after this but he was so blinded by his rage that he
stumbled against a tree, breaking his lance and falling from his horse.
He sprang to his feet and darted off again not knowing where he was
going until the massy battlements of King Pellam's castle appeared.
"Why do you wear the crown royal on your shield?" Pellam's men asked him
as soon as they saw him.
"The fairest and best of ladies living gave it to me," Balin replied, as
he stalled his horse and strode across the court to the banquet hall.
"Why do you wear the royal crown?" Sir Garlon asked him as they sat at
table.
"The queen whom Lancelot and we all worship as the fairest, best and
purest gave it to me to wear," said Balin.
But Sir Garlon only hissed at him and made fun of what he said, and
Balin reached for a wonderful goblet embossed with a sacred picture to
hurl it at Garlon, but the thought of the gentle queen about whom he
was talking soothed his temper. The next morning, however, in the court
Sir Garlon mocked him again and Balin's face grew black with anger. He
tore out his sword from its shield and crying out fiercely, "Ha! I'll
make a ghost of you!" struck Garlon hard on the helmet.
The blade flew and splintered into six parts which clinked upon the
stones below while Garlon reeled slowly backward and fell. Balin dragged
him by the banneret of his helmet and struck again, but in a minute
twenty warriors with pointed lances were making for him from the castle.
Balin dashed his fist against the foremost face then dipped through a
low doorway out along a glimmering gallery until he saw the open portals
of King Pellam's chapel. He slipped inside this and crept behind the
door while the others howled past outside.
Before the golden altar he noticed lying the brightest lance he had ever
seen with its point painted red with blood. Seizing it he pushed it out
through an open casement, leaned on it and leaped in a half-circle to
the ground outside. Running along a path he found his horse, mounted him
and scudded away. An arrow whizzed to his right, another to his left and
a third over his head while he heard Pellam crying out feebly, "Catch
him, catch him! he mustn't pollute holy things!"
But Balin quickly dove beneath the tree boughs and raced through miles
of thick groves and open meadowland until his good horse, at last
wearied and uncertain in his footsteps, stumbled over a fallen oak and
threw Balin headlong.
As Balin rose to his feet he looked at the Queen's crown on his shield
and then drew the shield from off his neck. "I have shamed you," he
cried. "I won't carry you any more," and he hung it up on a branch and
threw himself on the ground in a passionate sleep.
While he slept there the beautiful wicked Vivien came riding by through
the woodland alleys with her squire, warbling a song.
"What is this?" she cried as she noticed the shield on the tree, "a
shield with a crown upon it. And there's a horse. Where's the rider? Oh!
there he is sleeping. Hail royal knight, I'm flying away from a bad king
and the knight I was riding with was hurt, and my poor squire isn't of
much use in helping me. But you, Sir Prince, will surely guide me to the
Warrior King Arthur, the Blameless, to get me some shelter."
"Oh, no, I'll never go to Arthur's court again," cried Balin. "I'm not a
prince any more, or a knight. I have brought the Queen's crown to
shame."
Then Vivien laughed shrilly, and told Balin a wicked story about the
Queen which she just imagined in her wicked mind. But she told it so
cunningly and smiled so sunnily as she talked that Balin believed her
and he flew into the more passionate rage because he thought he had been
deceived in the Queen whom he had worshipped.
He ground his teeth together, sprang up with a yell, tore the shield
from the branch and cast it on the ground, drove his heel _into the
royal crown_, stamped and trampled upon it until it was all spoiled,
then hurled the shield from him out among the forest weeds and cursed
the story, the queen and Vivien.
His weird yell had thrilled through the woods where Balan was lurking
for his foe. "There! that's the scream of the wood-devil I'm looking
for," he thought. "He has killed some knight and trampled on his shield
to show his loathing of our order and the queen. Devil or man,
whichever you are, take care of your head!"
[Illustration: HE DROVE HIS HEEL INTO THE ROYAL CROWN.]
With that he made swiftly for his poor brother whom he did not
recognize. Sir Balin spoke not a word but snatched the buckler from
Vivien's squire, vaulted on his horse and in a moment had clashed with
his brother's armor. King Pellam's holy spear reddened with blood as it
pricked through Balan's shield to his flesh. Then Balin's horse, wearied
to death, rolled back over his rider and crushed him inward and both men
fell and swooned away.
"The fools!" cried Vivien to her young squire. "Come, you Sir Chick,
loosen their casques and see who they are. They must be rivals for the
same woman to fight so hard."
"They are happy," her gentle squire answered, "if they died for love.
And Vivien, though you beat me like your dog I would die for you."
"Don't die, Sir Boy," cried Vivien, "I'd rather have a live dog than a
dead lion. Come away, I don't like to look at them," and she made her
palfrey leap off over the fallen oak tree.
Balin was the first to wake from his swoon. As soon as he saw his
brother's face he crawled over to his side moaning. Then Balan faintly
opened his eyes and seeing who was with him kissed Balin's forehead.
"O Balin," he cried, "why didn't you carry your own shield which I knew,
and why did you trample all over this one which bears the queen's own
crown which I know?"
So Balin slowly gasped out the whole story of his shield. Then they each
said good-night to the other and closed their eyes, locked in each
other's arms.
LANCELOT AND ELAINE.
Long before Arthur was crowned king while he was roving one night over
the trackless realms of Lyonesse he came upon a glen with a gray boulder
and a lake. As he rode up the highway in the misty moonshine he suddenly
stepped upon a white skeleton of a man with a crown of diamonds upon its
skull. The skull broke off from the body and rolled away into the lake.
Arthur alighted, reached down and picked up the crown and set it on his
head murmuring to himself, "_You too shall be king some day_," for the
skeleton was the bones of a king who had fought with his brother there
and been killed.
[Illustration: YOU TOO SHALL BE KING SOME DAY.]
When Arthur was crowned he plucked the nine gems out of the crown he had
found on the skeleton and showed them to his knights with the words:
"These jewels belong to the whole kingdom for everybody's use and not to
the king. Hereafter there is to be joust for one of them every year and
in that way in nine years time we will learn who is the mightiest in the
kingdom and we will race with each other to become skilful in the use
of arms until at last we shall be able to drive away the heathen horde
from the land."
Eight years had now passed and there had been eight jousts. Lancelot had
won the diamond every year and intended when he had been victorious in
all the jousts, to give the nine gems to the queen. When the ninth year
came Arthur proclaimed the tournament for the central and largest
diamond to be held at Camelot, where he was holding his court. But the
queen became ill as the time for the tour jousts drew near and he asked
her whether she was too feeble to go to see Lancelot in the lists.
"Yes, my lord," replied Guinevere, "and you know it," and she looked up
languidly to Lancelot who stood near.
Lancelot thinking that she would rather have him near while she was ill
than to receive all the diamonds of the crown, said:
"Sir King, that old wound of mine is not quite healed so I can hardly
ride in my saddle."
So the king went, excused Lancelot, and rode away alone to the lists
while Lancelot remained, but as soon as Arthur was gone the _queen told
Lancelot that he ought by all means go too and fight_.
"But how can I go now," replied Lancelot, "after what I have said to the
king."
"I will tell you what to do," said Guinevere. "Everybody says that men
go down before your spear just because of your great name. They are
afraid as soon as you appear and of course, they are conquered. Go in
today entirely unknown and win for yourself, then after all is over the
king will be pleased with you for being so clever."
[Illustration: THE QUEEN TOLD LANCELOT THAT HE OUGHT BY ALL MEANS
FIGHT.]
Lancelot quickly got his horse and leaving the beaten thoroughfare,
chose a green path among the downs to take him to the lists. It was a
new road to him however and he lost his way and did not know where to go
until at last he came upon a faintly traced pathway that led to the
castle of Astolat far away on a hill. He went thither, blew the horn at
the gate where a _dumb, wrinkled old man came to let him in_. In the
castle court he met the lord of Astolat with his two young sons, Sir
Torre and Sir Lavaine and behind them the lily maiden Elaine, Astolat's
daughter. They were jesting and laughing as they came.
[Illustration: A WRINKLED OLD MAN CAME AND LET HIM IN.]
"Where do you come from, my guest, and what is your name?" asked
Astolat. "By your state and presence I would guess you to be the chief
of Arthur's court, for I have seen him although the other knights of the
Round Table are strangers to me."
Lancelot, Arthur's chief knight replied, "I am of Arthur's court and I
am known, and my shield which I have happened to bring with me, is known
too. But as I am going to joust for the diamond at Camelot as a
stranger do not ask me my name. After it is over you shall know me and
my shield. If you have some blank shield around, or one with a strange
device, pray lend it to me."
"Here is Torre's," the Lord of Astolat replied. "He was hurt in his
first tilt and so his shield is blank enough, God knows. You can have
his."
"Yes," added Sir Torre simply, "since I can't use it you may have it."
His father laughed. "Fie, Churl, is that an answer for a noble knight?
You must pardon him, but Lavaine, my younger boy, is so full of life he
will ride in the lists, joust for the diamond, win and bring it in one
hour to set upon his sister's golden hair and make her three times as
wilful as before."
"Oh, no, good father! don't shame me before this noble knight. It was
all a joke. Elaine dreamed that some one had put the diamond into her
hand and it was so slippery it dropped into a pool of water. Then I told
her that if I fought and won it for her she must keep it safer than
that. But it was all in fun. However, if you'll give me your leave, I'll
ride to Camelot with this noble knight. I shall not win but I'll do my
best to win."
Lancelot smiled a moment. "If you'll give me the pleasure of your
company over the downs where I lost myself I'll be glad to have you as a
friend and guide. You shall win the diamond if you can and then give it
to your sister if you wish."
"Such diamonds are for queens and not for simple little girls," said Sir
Torre.
Elaine flushed at this and Lancelot said, "If beautiful things are for
beautiful people this maiden may wear as fine jewels as there are in the
world."
Then the lily maid lifted her eyes and thought that Lancelot was the
greatest man that had ever lived. She loved his bruised and bronzed face
seamed across with an old sword-cut.
They took the pet knight of Arthur's court into the rude hall of Astolat
where they entertained him with their best meats, wines and minstrel
melodies. They told him about the dumb old man at the gate, how ten
years ago he had warned Astolat of the heathen fighters coming, and how
they had all escaped to the woods and lived in a boatman's hut by the
river while the old man had been caught and had his tongue cut off.
"Those were dull days," said the Lord of Astolat, "until Arthur came and
drove the heathen away."
"O, great Lord!" cried Lavaine to Lancelot, "you fought in those
glorious wars with Arthur. Tell us about them!"
So Lancelot told him all about the fight all day long at the white mouth
of the river Glenn, the four loud battles on the shore of Duglas where
the glorious king wore on his cuirass an emerald carved into Our Lady's
head. "On the mount of Badon," he said, "I saw him charge at the head of
all of his Round Table and break the heathen hosts. Afterward he stood
on a heap of the killed, all red, from his spurs to the plumes of his
helmet, with their blood, and he cried to me: 'They are broken! they are
broken!' In this heathen war the fire of God filled him, I never saw
anyone like him, there is no greater leader."
"Except yourself," thought the lily maid Elaine. All through the night
she saw his dark, splendid face living before her eyes and early in the
morning she arose as if to bid goodbye to Lavaine, stole step after step
down the long tower stairs and passed out to the court where Lancelot
was smoothing the glossy shoulders of his horse. She drew nearer and
stood in the dewy light, studying his face as though it was a god. He
had never dreamed she was so beautiful.
[Illustration: "FAIR LORD," SAID ELAINE.]
"Fair lord," said Elaine, "I don't know your name but I believe it is
the noblest himself of them all. Will you wear a token of me at the
tournament today?"
"No, pretty lady," said he, "for I've never worn a token of any woman in
the lists; as every one who knows me knows."
"Then by wearing mine you'll be less likely to be found out this time."
"That's true, my child, well, I'll wear it. Fetch it out to me. What is
it?"
"A red sleeve bordered with pearls," replied Elaine, and she went in and
brought it out to him.
Then he wound it round his helmet and said he had never before done so
much for any girl in the world. The blood sprang to Elaine's face as he
said that, and filled her with delight, although she grew all the paler
as Lavaine came out and handed Sir Torre's shield to Lancelot. Lancelot
gave his own shield to Elaine saying, "Do me this favor, child, keep my
shield for me until I come back."
"It's a favor to me," she replied smiling, "I'll be your squire."
"Come, Lily Maid," cried Lavaine, "you'll be a lily maid in earnest if
you don't get to bed and have some sleep," and he kissed her good-bye.
Lancelot kissed her hand as they moved away. She watched them at the
gateway until their sparkling arms dipped below the downs, then climbed
up to her tower with the shield and there she studied it and mused over
it every day.
Meanwhile Lancelot and Lavaine passed far over the long downs until they
reached an old hermit who lived in a white rock. Here they spent the
night. The next morning as they rode away Lancelot said, "Listen to me,
but keep what I say a secret, you're riding with Lancelot of the Lake."
"The great Lancelot?" stammered Lavaine, catching his breath with
surprise. "There is only one other great man to see, and that is
Britain's king of kings, Arthur. And he's going to be at the tournament,
too."
As soon as they reached the lists in the meadows by Camelot, Lancelot
pointed out the king who, as he sat in the peopled gallery was very easy
to recognize because of his five dragons. A golden dragon clung to his
crown, another writhed down his robe while two others in gilded carved
wood-work formed the arms of his chair. The canopy above him blazed with
the last big diamond.
"You call me great," cried Lancelot, "I'm not great, there's the man."
Lavaine gaped at Arthur as if he were something miraculous. Then the
trumpets blew. The two sides, those who held the lists and those who
attacked them, set their lances in rest, then struck their spurs, moved
out suddenly and shocked in the center of the field. The ground shook
and there was a low thunder of arms. Lancelot waited a little until he
saw which was the weaker side, then sprang into the fight with them. In
those days of his glory, whomever he struck he overthrew, whether they
were kings, dukes, earls, counts or barons. But that day in the field
some of his relatives were holding the lists who did not know him and
who could not bear the idea that any stranger knight should out do the
feats of their own Lancelot.
"Who is this?" one of them asked, "Isn't it Lancelot?"
"When has Lancelot ever worn a lady's token?" the others replied.
"Who is it then?" they cried, furious to guard the name of Lancelot.
They pricked their steeds and moving all together bore down upon him
like a wild wave that upsets a ship. One spear lamed Lancelot's charger
and another pierced through Lancelot's side, snapped there and stuck.
Lavaine now did splendidly for he brought a famous old knight down by
Lancelot's side. Lancelot in the meantime rose to his feet in all his
agony and by a sort of miracle as it seemed to those who were on his
side, drove all his opponents back to the barrier. Then the trumpet blew
and proclaimed that the knight who wore the scarlet sleeve with pearls
was victor.
"Go up and get your diamond," his men said to him.
"Don't give me any diamonds," said Lancelot. "My prize is death, I'll
leave and don't follow."
Then he vanished into the poplar grove where he told Lavaine to draw out
the lance head.
"I'm afraid you'll die, if I do," cried Lavaine.
"I'm dying now with it," said Lancelot, so Lavaine drew it out and
Lancelot gave a wonderful shriek and swooned away.
Then the old hermit came out, carried him into the white rock and
stanched his wound.
Immediately after he had left the field the men of his side went to the
king and said that the knight who had won the day had left without
receiving his prize.
"Such a knight as that must not go uncared for," said the king. "Gawain,
ride out and find him and since he didn't come for his diamond we will
send it to him. Don't leave your quest until you have him."
Gawain the courteous was a good young knight but he didn't like it that
he had to leave the banquet and the king's side to look for a stranger
knight, so he mounted his horse rather crossly. He rode all round the
country to every place except the right one, poplar grove, and at last
very late reached the Castle of Astolat.
"What news from Camelot?" cried Elaine as soon as she saw him, "What
about the knight with the red sleeve?"
"He won."
"I knew it," she said.
"But he left the jousts wounded in his side."
Then Elaine almost swooned away. When the Lord of Astolat came out and
heard about Gawain's quest, "Stay with us, noble prince," said he. "For
the knight was here and left his shield with us, so he will certainly
come back or send for it. Besides my son is with him."
Gawain thought he would have a pleasant time with Elaine so he stayed.
But Elaine rebelled against his pretty love-making and asked him why he
neglected the king's quest and why he didn't ask to see the knight's
shield.
"I've lost my quest in the light of your blue eyes," said Gawain, "but
let me see the shield. Ah! the king was right!" he cried out when Elaine
showed it to him. "It was our Lancelot."
"I was right too," Elaine said merrily, "for I dreamed that my knight
was the greatest of them all."
"And suppose that I dreamed that you love this greatest knight?"
returned Gawain.
"What do I know?" Elaine answered simply. "I don't know whether I know
what love is, but I do know that if I do not love him there isn't
another man whom I can love."
"Yes, you love him well," said Gawain. "And I suppose you know just
where your greatest knight is hidden, so let me leave my quest with you.
If you love him it will be sweet to you to give him the diamond and if
he loves you it will be sweet to him to receive it from you, while even
if he doesn't love you, a diamond is always a diamond. Farewell a
thousand times. If he loves you I may see you at court after while."
Then Gawain lightly kissed her hand as he laid the diamond in it, and,
wearied of his quest, leaped on his horse and carrolling a love-ballad
airily rode away to the court where it was soon buzzed abroad that a
maid of Astolat loved Lancelot and that Lancelot loved a maid of
Astolat.
The maid meanwhile crept up to her father one day and received his leave
to take the diamond to Sir Lancelot. Sir Torre went with her to the
gates of Camelot where they saw Lavaine capering about on a horse.
"Lavaine!" she cried, "how is it with my lord Sir Lancelot?" and she
told him about the diamond. Then Sir Torre went on into the city while
Lavaine guided Elaine to the hermit's cave. As she saw her handsome
knight on the floor, a sort of skeleton of himself, she gave a little
tender dolorous cry.
"Your prize, the diamond, sent you by the king," said she, as she put it
into his hand and explained how she had received it from Gawain. Then he
kissed her as a father would kiss a dear little daughter and she went
back to the dim, rich city of Camelot for the night. But the next
morning she was back in the cave, and day after day she came, caring for
him more mildly, tenderly and kindly than any mother could with a child,
until at last the old hermit said she had nursed him back to life, then
all three rode back together one morning to Astolat where Lancelot asked
Elaine to tell him the dearest wish of her heart so that he could grant
it to her. Elaine turned as pale as a ghost when he first spoke but at
last one day she told him. She said she wanted him to love her, she
wanted to be his wife.
"If I had chosen to wed," Lancelot replied, slowly, "I would have been
married long before this. But now I shall never marry, sweet Elaine."
"No, no," cried Elaine, "it won't matter if I can't be your wife, if I
can only go with you always and go round the world with you and serve
you."
But Lancelot said that would be a poor way for him to requite the love
and kindness her father and brothers had shown him. "Noble maid," he
went on, "this is only the first flash of love with you. After awhile
you will smile at yourself about it when you find a knight who is fitter
for you to marry and not three times older than you as I am, and then I
will give you broad lands and territories even to a half of my kingdom
across the seas and I'll always be ready to fight for you in your
troubles. I'll do this, dear girl, but more I cannot."
"Of all this I care for nothing," Elaine said growing deathly pale and
falling in a swoon.
That evening Lancelot sent for his shield from the tower where Elaine
sat with it, and as his horse's hoofs clattered off upon the stone of
the highway she looked down from her tower, but he did not glance back.
After that Elaine dreamed her time sadly away in the tower and only
wished that she could die. She begged her father to send for the priest
to confess her and asked Lavaine to write a letter for her to Lancelot.
Then she arranged it that when she died the dumb old man at the gate was
to take her in the barge down the river to the king's palace. Eleven
days later this was done. Elaine was dressed like a little sleeping
queen and floated along the stream with her letter in one hand and a
lily in the other.
That day Lancelot was with the queen and as he looked out of the
casement upon the river he saw the barge hung with rich black samite,
the dumb old man and the lily maid of Astolat gliding up to the palace
door.
"What is it?" cried everybody streaming round. "A pale fairy queen come
to take Arthur to fairy land?"
Then the king bade meek Sir Percival and pure Sir Galahad carry her
reverently into the hall where the fine Gawain came and wondered at her
and Lancelot came and mused over her, and the queen came and pitied her.
But King Arthur spied a letter, opened it and read it aloud to all the
lords and ladies. It was Elaine's goodbye to Lancelot.
[Illustration: A PALE FAIRY QUEEN CAME TO TAKE ARTHUR TO FAIRY LAND.]
Then Sir Lancelot told them everything about Elaine and how he had
promised to give her his lands and riches when she should be ready to
marry some knight of her own age. The king said that he should see that
she was buried very grandly. So they had a procession with all the pomp
of a queen, with gorgeous ceremonies, mass and rolling music while all
the Order of the Round Table followed her to the tomb. Then they laid
the shield of Lancelot at her feet and put a lily in her hand.
THE HOLY GRAIL.
One day a new monk came into the abbey beyond Camelot. There was
something about him different from all the other monks there. He was so
polished and clever that old Ambrosious who had lived in the old
monastery for fifty years and had never seen a bit of the world guessed
in a minute that the new brother had come from King Arthur's court. And
one windy April morning as Ambrosious stood under the yew tree with this
gentle monk he asked him why he left the Knights of the Round Table.
Then Sir Percival answered:
"It was the sweet vision of the Holy Grail."
[Illustration: "THE HOLY GRAIL," CRIED AMBROSIOUS.]
"The Holy Grail," cried Ambrosious. "Heaven knows I don't know much, but
what is that, the phantom of a cup that comes and goes?"
"No, no," said Percival, "what phantom do you mean? It's the cup that
our Lord drank from at his sad last supper, and after he died Joseph of
Aramathea brought it to Glastonbury at Christmas time, and there it
stayed a while and every one who looked at it or touched it was healed
of their sicknesses. But the times grew so wicked that the cup was
caught up into heaven where nobody could see it."
"Yes, I remember reading in our old books," said Ambrosious, "how Joseph
built a lonely little church at Glastonbury on the marsh, but that was
long ago. Who first saw the vision of the Holy Grail to-day?"
"A woman," said Sir Percival, "a nun, my sister who was a holy maid if
ever there was one. The old man to whom she used to tell her sins (or
what she called her sins), often spoke to her about the legend of the
Holy Grail which had been handed down through six people, each of them a
hundred years old, from the Lord's time. And when Arthur made the order
of the Round Table and all hearts became clean and pure for a time this
old man thought surely the Holy Grail would come back again. 'O Christ!'
he used to say to my sister, 'if only it would come back and help all
the world of its wickedness!' And then my sister asked him whether it
might come to her by prayer and fasting.
"'Perhaps,' said the father, 'for your heart is as pure as snow.'
"So she prayed and fasted until the sun shone and the wind blew through
her and one day she sent for me. Her eyes were so beautiful with the
light of holiness that I did not know them.
"'Sweet Brother,' she said, 'I have seen the Holy Grail. I heard a sound
like a silver horn but sweeter than any music we can make, and then a
cold silver beam of light streamed in through my cell, and down the beam
stole the Holy Grail, rose red and throbbing as if it were alive. All
the walls of my cell grew rosy red with quivering rosy colors. Then the
music faded away, the Holy Grail vanished and the colors died out in
the darkness. So now we know the Holy Thing is here again, Brother fast,
too, and pray, and tell your brother-knights about it, then perhaps the
vision may be seen by you all, and the whole world will be healed.'
[Illustration: MY KNIGHT OF HEAVEN, GO FORTH.]
"So I told all the knights and we fasted and prayed for many weeks. Then
my sister cut off all her long streaming silken hair which used to fall
to her feet and out of it braided a strong sword belt and with silver
and crimson thread she wove into it a crimson grail in a silver beam.
Then she bound it on our beautiful boy knight, Sir Galahad, and said:
"'My knight of heaven, go forth, for you shall see what I have seen and
far in the spiritual city you will be crowned king.' Then she sent the
deathless passion of her eyes through him and he believed what she said.
"Then came a year of miracles. In our great hall there stood a chair
which Merlin had fashioned carved with strange figures like a serpent
and in and out among the strange figures ran a scroll of strange letters
in a language nobody knew like a serpent. Merlin called it the Seat
Perilous, because he said if any one sat in it he would get lost. And
Galahad said that if he got lost in it he would save himself. So one
summer night Sir Galahad sat down in the chair and all at once there was
a cracking of the roofs above us, and a blast and thunder, and in the
thunder there was a cry and in the blast there was a beam of light seven
times clearer than the daylight. Down the beam stole the Holy Grail all
covered over with a luminous cloud. Then it passed away but every knight
saw his brother knight's faces in a glory and we all rose and stared at
each other until at last I found my voice and swore a vow.
"I swore that because I had not seen the Holy Grail behind the cloud I
would ride away a year and a day in quest of it until I could see it as
my sister saw it. Galahad swore too, and good Sir Bors, and Lancelot and
many others, knights, and Gawain louder than all the rest.
"The king was not in the hall that day for he had gone out to help some
poor maiden, but as he came back over the plains beyond Camelot he saw
the roofs rolling in smoke and thought that his wonderfully dear,
beautiful hall which Merlin had built for him so wonderfully was afire.
So he rode fast and rushed into the tumult of knights and asked me what
it all meant.
"'Woe is me!' cried the king when I told him. 'Had I been here you would
not have sworn the vows.'
"'My king,' I answered boldly, had you been here you would have sworn
the vows yourself.'
"'Yes, yes,' said he, 'are you so bold when you didn't see the Grail?
You didn't see farther than the cloud, and what can you expect to see
now if you go out into the wilderness?'
"'No, no, Lord, I didn't see the Grail, I heard the sound, I saw the
light and since I didn't see the holy thing I swore the vow that I would
follow it until I did see.'
"'Then he asked us, knight by knight, whether we had seen it and each
one said, 'No, no, Lord, that was why we swore our vows,' but suddenly
Galahad called out, 'But I saw the Holy Grail, Sir Arthur, and heard the
cry, "O Galahad, follow me."'
"Ah, Galahad, Galahad,' said the king, 'the vision is for such as you
and for your holy nun but not for these. Are you all Galahads or all
Percivals? No, no, you are just men with the strength to right the
wrongs and violences of the land. But now since one has seen, all the
blind want to see. However, since you have made the vow, go. But oh, how
often the distressed people of the kingdom will come into the hall for
you to help them and all your chairs will be vacant while you are out
chasing a fire in the quagmire! Many of you, yes, most of you will never
come back again! But come to-morrow before you go, let us have one more
day of field sports so that before you go I can rejoice in the unbroken
strength of the Order I have made.'
"So the next day there was the greatest tournament that Camelot had ever
seen, and Galahad and I, with a strength which we had received from the
vision, overthrew so many knights that all the people cheered hotly for
Sir Galahad and Sir Percival. The next morning all the rich balconies
along the streets of Camelot were laden with ladies and showers of
flowers fell over us as we passed out and men and boys astride lions and
dragons, griffins and swans at the street corners, called us all by name
and cried, 'God Speed!' while many lords and ladies wept. Then we came
down to the gate of The Three Queens and there each one went on his own
way.
"I was feeling glad over my victories in the lists and thought the sky
never looked so blue nor the earth so green. All my blood danced within
me for I knew that I would see the Holy Grail. But after a while I
thought of the dark warning of the king. I looked about and saw that I
was quite alone in a sandy thorny place, and I thought I would die of
thirst. Then I came to a deep lawn with a flowing brook and apple trees
overhanging it. But while I was drinking of the water and eating of the
apples they all turned to dust, and I was alone and thirsty again in
among the sands and thorns. Next I saw a woman spinning beside a
beautiful house. She rose to greet me and stretched out her arms to
welcome me into her house to rest, but as soon as I touched her she fell
to dust, and the house turned into a shed with a dead baby inside, and
then it fell to dust too.
"Then I rode on and found a big hill and on the top was a walled city,
the spires with incredible pinnacles reaching up to the sky, and at the
gateway there was a crowd of people who cried out to me:
"Welcome, Percival, you mightiest and purest of men!"
"But when I reached the top there was no one there. I passed through to
the ruined old city and found only one person a very, very old man.
'Where is the crowd who called out to me?' I asked him.
"He could scarcely speak, but he gasped out, 'Where are you from and who
are you?' and then fell to dust.
[Illustration: NEXT I SAW A WOMAN SPINNING.]
"Then I was so unhappy I cried. I felt as though even if I should see
the Holy Grail itself and touched it it would crumble into dust. From
there I passed down into a deep valley, as low down as the city was
high up, where I found a chapel with a hermit in a hermitage near by. I
told him about all these phantoms.
"'You haven't true humility,' he said, 'which is the mother of all
virtue. You haven't lost yourself to find yourself as Galahad did.'
"Just as he ended suddenly Sir Galahad shone before us in silver armor.
He laid his lance beside the chapel door and we all went in and knelt in
prayer. Then my thirst was quenched. But when the mass was burned I saw
only the holy elements while Galahad saw the Holy Grail come down upon
the shrine.
"'The Holy Grail,' he said, 'has always been at my side ever since we
came away, fainter in the daytime, but blood-red at night. In its
strength I have overcome evil customs wherever I have gone, and have
passed through Pagan lands and clashed with Pagan hordes and broken them
down everywhere. But the time is very near now when I shall go into the
spiritual city far away where some one will crown me king. Come with me
for you will see the Holy Grail in a vision when I go.'
"At the close of the day I started away with him. We came to a hill
which only a man could climb, scarred all over with a hundred frozen
streams, and when we reached the top there was a wild storm. Galahad's
armor flashed and darkened again every instant with quick, thick
lightnings which struck the dead old tree trunks on every side until at
last they blazed into a fire. At the base was a great black swamp partly
whitened with bones of dead men. A chain of bridges lead across it to
the great sea, and Galahad crossed them, one after the other, but each
one burned away as soon as he had passed over so that I had to stay
behind. When he reached the great sea the Holy Grail hung over his head
in a brilliant cloud. Then a boat came swiftly by and when the sky
brightened again with the lightning I could see him floating away,
either in a boat with full sails or a winged creature which was flying,
I couldn't tell which. Above him hung the Holy Grail rosy red without
the cloud. I had seen the holy thing at last. When I saw Sir Galahad
again he looked like a silver star in the sky, and beyond the star was
the spiritual city with all her spires and gateways in a glory like one
pearl, no larger than a pearl. From the star a rosy red sparkle from the
Grail shot across to the city. But while I looked a flood of rain came
down in torrents, and how I ever came away I don't know, but anyway at
the dawn of the next day I had reached the little chapel again. There I
got my horse from the hermit and rode back to the gates of Camelot.
"Just once I met one of the other knights. That was one night when the
full moon was rising and the pelican of Sir Bors' casque made a shadow
on it. I spurred on my horse, hailed him and we were both very glad to
see each other.
"'Where is Sir Lancelot,' he asked. 'Have you seen him? Once he dashed
across me very madly, maddening his horse. When I asked him why he rode
so hotly on a holy quest he shouted, 'Don't keep me, I was a sluggard,
and now I'm going fast for there's a lion in the way.' Then he vanished.
When I saw how mad he was I felt very sad for I love him, and I cared no
more whether I saw the Holy Grail, or not; but I rode on until I came to
the loneliest parts of the country where some magicians told me I
followed a mocking fire. This vexed me and when the people saw that I
quarrelled with their priests they bound me and put me into a cell of
stones. I lay there for hours until one night a miracle happened. One
of the stones slipped away without any one touching it or any wind
blowing. Through the gap it made I saw the seven clear stars which we
have always called the stars of the Round Table and across the seven
stars the sweet Grail glided past. Close after a clap of thunder pealed.
Then a maiden came to me in secret and loosed me and let me go.'
[Illustration: ACROSS THE SEVEN STARS THE SWEET GRAIL GLIDED PAST.]
"Sir Bors and I rode along together and when we reached the city our
horses stumbled over heaps of ruined bits of houses that fell as they
trod along the streets. At last brought us to Arthur's hall.
"As we came in we saw Arthur sitting on his throne with just a tenth of
the knights who had gone out on the quest of the Holy Grail standing
before him, wasted and worn, also the knights who had stayed at home.
When he saw me he rose and said he was glad to see me back, that he had
been worrying about me because of the fierce gale that had made havoc
through the town and shaken even the new strong hall and half wrenched
the statue Merlin made for him.
"'But the quest,' the king went on, 'have you seen the cup that Joseph
brought long ago to Glastonbury?'
"Then when I told him all that you have been hearing just now and how I
was going to give up the tournament and tilt and pass into the quiet of
the life of the monk, he answered not a word, but turning quickly to
Gawain asked,
"'Gawain, was this quest for you?'
"'No, Lord,' replied Gawain, 'not for such as I. I talked with a saintly
old man about that and he made me very sure that it wasn't for me. I was
very tired of it. But I found a silk pavilion in the field with a lot of
merry girls in it, then this gale tore it off from the tenting pin and
blew my merry maidens all about with a great deal of discomfort. If it
hadn't been for that storm my twelve months and a day would have passed
very pleasantly for me.'
"Then Arthur turned to Sir Bors, who had pushed across the throng at
once to Lancelot's side, caught him by the hand and held it there half
hidden beside him until the king spied them.
"'Hail, Bors, if ever a true and loyal man could see the Grail you have
seen it,' cried Arthur.
"'Don't ask me about it,' replied Sir Bors with tears in his eyes 'I may
not speak about it; I saw it.'
"The others spoke only about the perils of their storm, and then it was
Lancelot's turn. Perhaps Arthur kept his best for the last.
"'My Lancelot,' said the king, 'our Strongest, has the quest availed for
you?'
"'Our strongest, O King!' groaned Lancelot and as he paused I thought I
saw a dying fire of madness in his eyes. 'O King, my friend, a sin lived
in me that was so strange that everything pure, noble and knightly in me
twined and clung around it until the good and the poisonous in me grew
together, and when your knights swore to make the quest I swore only in
the hope that could I see or touch the Holy Grail they might be pulled
apart. Then I spoke to a holy saint who said that if they could not be
plucked apart my quest would be all in vain. So I vowed to him that I
would do just as he told me, and while I was out trying to tear them
away from each other my old madness came back to me and whipped me off
into waste fields far away.
"There I was beaten down by little knights whom at one time I would have
frightened away just by the shadow of my spear. From there I rode over
to the sea-shore where such a blast of wind began to blow that you could
not hear the waves even although they were heaped up in mountains and
drove the sea like a cataract, while the sand on the beach swept by like
a river. A boat, half-swallowed by the seafoam, was moored to the shore
by a chain. I said to myself that I would embark in the boat and lose
myself and wash away my sin in the great sea.
"For seven days I rode around over the dreary water and on the seventh
night I felt the boat striking ground. In front of me rose the enchanted
towers of Carbonek, a castle like a rock upon a rock, with portals open
to the sea and steps that met the waves. A lion sat on each side of
them. I went up the steps and drew my sword. Suddenly flaring their
manes the lions stood up like men and gripped me on my shoulders. When I
was about to strike them a voice said to me, 'Don't be afraid, or the
beasts will tear you to pieces; go on.' Then my sword was dashed
violently from my hand and fell. Up into the sounding hall I passed but
saw not a bench, table, picture, shield or anything else except the moon
over the sea through the oriel window, but I heard a sweet voice as
clear as a lark singing in the topmost tower to the east. I climbed up a
thousand steps with great pain. It seemed as though I was climbing
forever but at last I reached a door with light shining through the
crannies and I heard voices singing 'Glory and joy and honor to our Lord
and the Holy Vessel, the Grail.'
"'Then I madly tried the door, it gave way and through a stormy glare of
heat that burned me and made me swoon away I thought I saw the Grail,
all veiled with crimson samite and around it great angels, awful shapes
and wings and eyes!'
"The long hall was silent after Lancelot was done, until airy Gawain
began with a sudden.
"'O King, my liege, my good friend Percival and your holy nun have
driven men mad. By my eyes and ears I swear I'll be deeper than a
blue-eyed cat and three times as blind as any owl at noon-time
hereafter to any holy virgins in their ecstasies.'
"'Gawain,' replied the king, 'don't try to become blinder; you're too
blind now to want to see. If a sign really came from heaven Bors,
Lancelot and Percival are blessed for they have each seen according to
their sight.'"
PELLEAS AND ETTARRE.
When his knights went after the Holy Grail Arthur made many new knights
to fill the gaps made by their absence. As he sat in his hall one day at
old Caerleon the high doors were softly parted and through these in came
a youth, and with him the outer sunshine and the sweet scent of meadows.
"Make me your knight, Sir King!" he cried, "because I know all about
everything that belongs to a knight and because I love a maiden."
This youth was Sir Pelleas-of-the-Isles who had heard that the king had
proclaimed a great tournament at Caerleon with a sword for the victor
and a golden crown for the victor's sweetheart as the prize. He longed
to win them, the circlet for his lady love, the sword for himself.
Just a few days before, while riding across the Forest of Dean to find
the king's palace hall at Caerleon, Pelleas had felt the sun beating on
his helmet so sharply that he reeled and almost fell from his horse.
Then, seeing a hillock near-by overgrown with stately beech trees and
flowers here and there beneath, he tied his horse to a tree, threw
himself down and was very soon lost in sweet dreams about a maiden, not
any particular maiden for he had no sweetheart at that time.
But suddenly he was wakened with a sound of chatter and laughing at the
outskirts of the grove, and glancing through fern he saw a party of
young girls in many colors like the clouds at sunset, all of them riding
on richly dressed horses. They were all talking together in a
hodgepodge, some pointing this way, some that, for they had lost their
way.
[Illustration: WAS VERY SOON LOST IN SWEET DREAMS ABOUT A MAIDEN.]
Pelleas sprang up, loosed his horse and led him into the light.
"Just in time!" cried the lady who seemed to be the leader of the party.
"See, our pilot-star! Youth, we are wandering damsels riding armed, as
you see, ready to tilt against the knights at Caerleon, but we've lost
our way. To the right? to the left? straight on? forward? backward?
which is it? tell us quickly."
Pelleas gazed at her and wondered to himself whether the famous Queen
Guinevere herself was as beautiful as this maiden. For her violet eyes,
scornful eyes, were large and the bloom on her cheeks was like the rosy
dawn. Her beauty made Pelleas timid and when she spoke to him he could
not answer but only stammered, for he had come from far away waste
islands where besides his sisters, he had scarcely known any women but
the tough wives of the islands who made fish nets.
With a slow smile the lady turned round to her companions the smile
spreading to them all. For she was Ettarre, a very great lady in her
land.
"O, wild man of the woods," she cried, "don't you understand our
language, or has heaven given you a beautiful face and no tongue?"
"Lady," he answered, "I just woke from my dreams, and coming out of the
gloomy woods I was dazzled by the sudden light, and beg your pardon. But
are you going to Caerleon? I'm going too. Shall I lead you to the king?"
"Lead," said she.
So through the woods they went together but his tender manner, his awe
of her and his bashfulness bothered her. "I've lighted on a fool," she
muttered to herself, "so raw and yet so stale!"
But since she wished to be crowned the Queen of Beauty in the king's
tournament, and since Pelleas looked strong she thought perhaps he would
fight for her, so she flattered him and was very pleasant and kind. Her
three knights and maidens were kind to him too, for she was a very
great lady and they had to do as she did. When they reached Caerleon
before she passed on to her lodgings she took Pelleas by the hand and
said:
[Illustration: SHE TOOK PELLEAS BY THE HAND.]
"O, how strong your hand is! See; look at my poor little weak one! Will
you fight for me and win me the crown, Pelleas, so that I may love you?"
Pelleas' heart danced. "Yes! Yes!" he cried, "and will you love me if I
win?"
"Yes, that I will," answered Ettarre laughing and flinging away his hand
as she peeped round to her knights and ladies until they all laughed
with her.
"O what a happy world!" thought glad Pelleas, "everybody seems happy and
I am the happiest of all."
He couldn't sleep that night for joy and on the next day when he was
knighted he swore to love one maiden only. As he came away from the
king's hall the men who met him all turned around to look at his face,
for it flamed with happiness, and at the great banquets which Arthur
gave to knights from all parts of the country Pelleas looked the noblest
of the noble. For he dreamed that his lady loved him and he knew that he
was loved by the king.
On the morning when the jousts began the first that was called was the
tournament of youth. Arthur wanted to keep the older, stronger men out
of it so that young Pelleas might win his lady's love as she had
promised, and be lord of the tourney. Down by the field along the river
Usk where it was held the gilded parapets were crowned with faces and
the great tower filled with eyes up to its top. Then the trumpets blew
for the tournament to begin.
All day long Sir Pelleas held the field. At the close a shout rang round
the galleries as Ettarre caught the gold crown from his lance and
crowned herself before all the people. Her eyes sparkled as she looked
at him, but that was the last time she was kind to her knight.
She lingered a few days at Caerleon, sunny to all the other people but
always frowning at him.
Still when she left for home with her knights and maidens Sir Pelleas
followed.
"Damsels," cried she as she saw him coming, "I ought to be ashamed to
say it and yet I can't bear that Sir Baby. Keep him back with
yourselves. I'd rather have some rough old knight who knows the ways of
the world to chatter and joke with; so don't let him come near me.
Tell him all sorts of baby fables that good mothers tell their little
boys, and if he runs off for us--it doesn't matter."
[Illustration: ETTARRE CROWNED HERSELF BEFORE ALL THE PEOPLE.]
So the young women didn't let him go near Ettarre but made him stay with
them, and as soon as they had all passed into Ettarre's castle gate up
sprang the drawbridge, down rang the iron grating, and Sir Pelleas was
left outside all alone.
"These are only the ways of ladies with their lovers when the ladies
want to find out whether the lovers are true or not. Well, she can try
me with anything, I'll be true through all."
So he stayed there until dark, then went to a priory not far off and the
next morning came back. Every day he did the same whether it rained or
shone, armed on his charger, and stayed all the day beneath the walls,
although nobody opened the gate for him.
This made Ettarre's scorn turn to anger. She told her three knights to
go out and drive him away. But when they came out Pelleas overthrew them
all as they dashed upon him one after the other. So they went back
inside and he kept his watch as before. This turned Ettarre's anger into
hate. As she walked on top of the walls with her three knights about a
week later she pointed down to Pelleas and said:
"He haunts me, look, he besieges me! I can't breathe. Strike him down,
put my hate into your blows and drive him away from my walls."
So down they went but Pelleas overthrew them all again so Ettarre called
down from the tower above, "Bind him and bring him in."
Pelleas heard her say this so he did not resist, but let the men bind
him and take him into his lady love. "See me, Lady," he said cheerily,
"your prisoner, and if you keep me in your dungeon here I'll be quite
content if you'll just let me see your face every day. For I've sworn my
vows and you've given me your promise and I know that when you've done
proving me you will give me your love and have me for your knight."
But she made fun of his vows and told her knights to put him outside
again and "if he isn't a fool to the middle of his bones," said she,
"he'll never come back." Then the three knights laughed and thrust him
out of the gates.
But a week later Ettarre called them again, "He's watching there yet. He
comes just like a dog that's been kicked out of his master's door. Don't
you hate him? Go after him, all of you at once, and if you don't kill
him bind him as you did before and bring him in."
So the three knights couched their spears all together, three against
one, ready to dash upon Pelleas, low down beneath the shadow of the
towers.
Gawain passing by on a lonely adventure saw them.
"The villains!" he shouted to Pelleas, "I'll strike for you!"
"No," cried Pelleas, "when one's doing a lady's will one doesn't need
any help."
Gawain stood by quivering to fight while the three knights sprang down
upon Pelleas, but Pelleas all alone beat the three of them together.
Then they rose to their feet, and he stood still while they bound him
and took him into their lady.
"You're scarcely fit to touch your victor, you dogs!" she cried to her
men, "far less bind him; but take him out as he is and let whoever wants
to untie him. Then if he comes again--"
She paused just a minute and Pelleas broke in at once with, "Lady, I
loved you and thought you very beautiful, but if you don't love me
don't trouble yourself about it; you won't see me again."
As soon as Pelleas was put outside the gate Gawain sprang forward,
loosed his bonds, flung them over the walls and cried out:
"My faith, and why did you let those wretches tie you up so when you
were victor of all the jousts?"
"O," said Pelleas, "they were just obeying the wishes of my lady, and
her wishes are mine."
Gawain laughed. "Lend me your horse and armor," he said, "and I'll tell
her I've killed you. Then she'll let me in just to hear all about it and
when I've made her listen I'll tell her all about you, what a great and
good fellow you are. Give me three days to melt her and on the third
evening I'll bring you golden news."
"Don't betray me," cried Pelleas, as he handed over his horse and all
his weapons except his sword. "Aren't you the knight they call
'Light-of-love?'"
"That is just because women are so light," Gawain rejoined, laughing.
Then he rode up to the castle gate, and blew the bugle so musically that
all the hidden echoes in the walls rang out.
"Away with you!" cried Ettarre's maidens, running up to the tower
window. "Our lady doesn't love you."
"I'm Gawain from Arthur's court," cried Gawain, lifting his vizor so
that they could see his face. "I've killed Pelleas whom you hate so.
Open the gates and I'll make you merry with my story."
The ladies ran down crying out to Ettarre, "Pelleas is dead! Sir Gawain
of Arthur's court has killed him and is blowing the bugle to come in to
tell us."
"Let him in," said Ettarre.
Then they opened the gates and Gawain rode inside.
For three days Pelleas wandered all about, doing nothing but thinking of
Gawain and Ettarre, and on the third night, when Gawain did not come, he
wondered why Gawain lingered with his golden news. At last he rode up to
Ettarre's castle, tied his horse outside and walked in through the wide
open gates. The court he found all dark and empty, not a light
glimmering from anywhere, so he passed out by the back gate, into the
large gardens beyond of red and white roses, where he saw three
pavilions. In one he found the three knights with their squires, all red
with revelling, and all asleep, in the second he saw the girls with
their scornful smiles frozen stiff in slumber, and in the third lay
Gawain with Ettarre, the golden crown he had won for her at the joust on
her forehead, both sleeping.
Pelleas drew back as if he had touched a snake.
"I'll kill them just as they lie," he cried in a passion. "O! to think
that any knight could be so false!"
But he was too manly to kill anyone in sleep, so he just laid his sword
across their throats and passed out to his horse, crushed his saddle
with his thighs, clenched his hands together and groaned.
"I loathe her now just as much as I loved her!" he cried, and dashing
his spurs into his horse he bounded out into the darkness and never came
back.
Meanwhile Ettarre, feeling the cold sword on her neck, awoke.
"Liar!" she cried to Gawain, as she saw that it was the sword of
Pelleas, "you haven't killed Pelleas, for he's been here and could have
killed us both just now."
And ever after that, as those who tell the story say, the proud and
scornful Ettarre sighed for Pelleas, the one true knight in the world,
her only faithful lover, and at last pined away because he never came
back.
THE LAST TOURNAMENT.
One day while King Arthur and Sir Lancelot were riding far, far beneath
a winding wall of rock they heard the wail of a child.
A half-dead oak tree climbed up the sides of the rock and up in mid-air
it held an eagle's nest. Through its branches rushed a rainy wind and
through the wind came the voice of a little child. Lancelot sprang up
the crag and from the nest at the tree-top he brought down a baby girl.
Round her neck was twined a necklace of rubies, wound round and round
three times.
Arthur took the baby and gave it to Queen Guinevere, who soon loved it
very tenderly and named her "Nestling." But Nestling had caught a
terrible cold in her strange little home in the wild eagle's nest and
died. And after that whenever the Queen looked at the ruby necklace it
made her very sad so she gave it to Arthur and said:
"Take these jewels of our Dead Innocence and make them a prize at a
tournament."
"Just as you wish," cried the King, "but why don't you wear the diamonds
that I found for you in the tarn, which Lancelot won for you at the
jousts?"
"Don't you know that they slipped out of my hands the very day that he
gave them to me, while I was leaning out of the window to see Elaine in
the barge on the river? But these rubies will bring better luck than
that to the lady who gets them, for they didn't come from a dead king's
skeleton, but from the body of a sweet baby girl. Perhaps, who knows,
the purest of your knights will win them at the jousts for the purest of
my ladies."
So the great jousts were proclaimed with trumpets that blew all along
the streets of Camelot and out across the faded fields to the farthest
towers, and everywhere the knights armed themselves for a day of glory
before the king.
But just the day before they were to be held, as King Arthur sat in his
great hall, a churl staggered in through the door; his face was all
striped with the lashes of a dog whip, his nose was broken, one eye was
out, a hand was off and the other hand dangled at his side with
shattered fingers.
"My poor Churl," cried the king, full of indignant pity, "what beast or
fiend has been after you? Or was it a man who hurt you so?"
"He took them all away," sputtered the churl, "a hundred good ones. It
was the Red Knight. He--Lord, I was tending sheep, my pigs, a hundred
good ones, and he drove them all off to his tower. And when I said that
you were always kind to poor churls like me as well as gentle lords and
ladies, he made for me and would have killed me outright if he didn't
want me to bring you message and made me swear that I would tell you.
"He said, 'Tell the king that I have made a Round Table of my own in the
North, and that whatever his knights swear not to do mine swear that
they will do; and tell him his hour has come, and that the heathen are
after him, and that his long lance is broken, and that his sword
Excalibur is a straw.'"
Then Arthur turned to Sir Kay the Seneschal and said: "Take this churl
of mine and tend him very carefully as if he were the son of a king
until all his hurts are healed," and as Sir Kay left the hall with the
churl the king went on to Lancelot: "The heathen have been quiet for a
long, long time, but now they are rising again in the North, and I will
go with my younger knights to put them down, so as to make the whole
island safe from one shore to the other. And while I go away, you, Sir
Lancelot, will sit in my chair to-morrow at the tournament and be the
judge there of the field. For why should you anyway care to go in again
yourself, when you've already won the nine diamonds for the queen?"
"Very well," replied Lancelot, "if you wish, although it would be better
if you would let me go off with the younger knights and you stay here
with the others and watch the tournament. But, if not, all is well?"
"Is all really well?" cried the king, "or have I just dreamed that our
knights are not quite so true and manly as they used to be and that my
noble realm which has been built up by noble deeds and noble vows is
going to fall back into beastly roughness and violence again?"
He gathered all the younger Knights of the Round Table together and
started away with them down the hilly streets of Camelot, and at the
gateway turned sharply North.
The next morning, the day of the Tournament, the Tournament of the Dead
Innocence they called it, a wet wind blew. But the streets were hung
with white samite, the fountains were filled with wine, and round each
fountain twelve little girls, all dressed in purest white sat with the
cups of gold and gave drinks to all that passed. The stately galleries
were filled with white-robed ladies. Lancelot mounted the steps to the
king's dragon-carved chair, the trumpets blew and the jousts began.
[Illustration: TWELVE LITTLE GIRLS GAVE DRINK TO ALL WHO PASSED.]
But Lancelot did not think of the sport before him, he was dreaming over
and over again the words of the king about the kingdom, and many rules
of the tournament were broken, and he didn't say a word. Once one of the
knights, who was overthrown cursed the little baby girl, the dead
innocence, and the king, and once one of the knight's helmets became
unlaced and the wicked face of Modred peeped through like a vermin, but
Lancelot didn't see.
After a while a roar of welcome shouted all round the galleries and
lists as a new knight came in dressed from his head to his feet in green
armor all trimmed with tiny silver deer, with holly berries on his
helmet crest. It was Sir Tristram of the Woods who had just crossed over
the seas from Brittany. Lancelot had fought with him long ago and
conquered him, and now he saw him and longed to fight him again. As
many, many knights of the Round Table fell down before the new knight
Lancelot gripped the golden dragons on each side of his throne to keep
himself in his seat, and groaned with passion. "Craven crests! oh,
shame!" he muttered, "the glory of the Round Table is gone."
So Tristram won the jousts and Sir Lancelot gave him the jewels.
"The hands with which you take these rubies are red," he said as he put
the necklace in Tristram's hands.
Then the thick rain began to fall, the plumes on the helmets of the
knights drooped and the dresses of the ladies were mussed. When they
went inside to feast the ladies took off their pure white gowns and
robed themselves in all the colors of the rainbow and field flowers,
like poppies, blue-bells, kingcups, and one said she was glad the time
to wear the pure innocent simple white was over. They grew so loud in
their frolics that at last the queen, who was angry that Sir Tristram
had won the prize and angry with the lawless youths, broke up the
banquet.
The next morning as Sir Tristram stood before the hall little Dagonet,
the fool, came dancing along and Sir Tristram threw his rubies round
the little fool's neck as he skipped about like a withered leaf, asking
him why he danced.
"It's stupid to dance without music," Tristram said, and picked up his
harp and began to twangle a tune on it; but as soon as Sir Tristram
began to play Dagonet stopped his dance. "And why don't you go on
skipping, Sir Fool?" asked Tristram.
"Because I'd rather skip twenty years to the music of my little brain
than skip a minute to the broken music you make."
"And what music have I broken?" cried Sir Tristram. "Arthur the King's
music," cried little Dagonet, skipping again and again as Sir Tristram
ceased. Then down the city he danced all the way, while Sir Tristram
passed out into the lonely avenues of the forests. He rode on toward
Lyonesse and the West, thinking of Isolt, the White, whom he loved, and
how he would put the rubies round her neck.
[Illustration: LITTLE DAGONET SKIPPING AGAIN AND AGAIN.]
Arthur, meanwhile, with his hundred spearmen had gone far, far away,
until at last over the countless reeds of marshes and islands he saw a
huge tower glaring in the wide-winged sunset of the West. As he drew
near he saw that the tower doors stood open and heard roars of rioting
and wicked songs of ruffian men and women.
"Look," cried one of his knights, for there high on a grim dead tree
before the tower, a brother of the Round Table was swinging by his neck,
his shield flowing with a shower of blood on a branch near by.
All the knights wanted to dash forward and blow the great horn that hung
beside the gate, but Arthur waved them back and went himself. He blew so
hard that the horn roared until all the grasses of the marshes flared
up, and out of the castle gate sallied a knight dressed from tip to toe
in blood-red arms, the Red Knight.
"Aren't you the king?" he bellowed, "the king that keeps us all with
such strict vows that we can't have any pleasures, a milky-hearted king?
Look to your life now!"
Arthur scorned to speak to so vile a man or to fight him with his sword.
He simply let the drunkard, stretching out from his horse to strike,
fall head-heavy, over from the castle causeway to the swamp below.
Then all the Round Table Knights roared and shouted, leaped down on the
fallen man, trampled out his face in the mire, sank his head so that it
could not be seen, and, still shouting, sprang through the open doors
among the people within. They hurled their swords right and left on men
and women, hurled over the tables and the wines and slew and slew until
all the rafters rang with yells and all the pavements streamed with
blood. Then they set the tower all afire and half the night through it
flushed the long low meadows and marshlands and lazily plunging sea with
its flames. That was how Arthur made the ways of the island safe from
one shore to the other.
Sir Tristram, not many nights after, reached Tintagil, where Isolt, the
White, lived in a crown of towers, where she now sat with the low
sea-sunset glorying her hair and glossy throat, thinking of him and of
Mark, her Cornish lord.
When Tristram's footsteps came grinding up the tower steps she flushed,
started out to meet him and threw her white arms about him.
"Not Mark, not Mark!" she cried. "At first your footsteps fluttered me,
for Mark steals into his own castle like a cat."
"No, it's I," said Sir Tristram, "and don't think about your Mark any
more, for he isn't yours any longer."
"But listen," she cried, "to-day he went away for a three days' hunt, he
said, and that means that he may be back in an hour for that's his way.
My God, my hate for him is as strong as my love for you. Let me tell you
how I sat here one evening thinking of you, one black midsummer night,
all alone, dreaming of you, and sometimes speaking your name aloud, when
suddenly there Mark stood behind me, for that's his way to steal behind
one in the dark.
"'Tristram has married her!' he hissed out and then this tower shook
with such a roar that I swooned away."
"Come," cried Sir Tristram, laughing, "never mind, I'm hungry, give me
some meat and wine."
So they ate and drank, talked and laughed about Mark with his long
crane-like legs, and Sir Tristram took a harp and sang a song. Then
while the last light of the day glimmered away he swung the ruby
necklace before Isolt.
"It's the fruit of a magical oak-tree that grew mid air," he cried, "and
was won by Sir Tristram as a tourney prize to bring to you."
Flinging the rubies round her neck he had just touched her jeweled
throat with his lips when behind him rose a shadow and a shriek.
"Mark's way!" cried Mark, the Cornish king, and he clove Tristram
through the brain.
* * * * *
That very night Arthur came back from the North, and as he climbed up
the tower steps to go to the queen, in the dark of the tower something
pulled at him. It was little Dagonet.
"Who are you?" said the king.
"I'm little Dagonet, your fool," sobbed the little jester, "and I cry
because I can never make you laugh again."
THE PASSING OF ARTHUR.
One night King Arthur saw Sir Gawain in a dream, and Gawain, who had
been killed, shrilly called out to him through the wind:
"Hail King! to-morrow you are going to pass away, and there's a land of
rest for you. Farewell!"
But when Arthur told his dream to Sir Bedivere, good old Sir Bedivere
replied, "Don't mind what dreams tell you, but get your knights together
and go out to the West to meet Sir Modred, who has stirred up against
you so many of the knights you love. They all know in their hearts that
you are king. Go and conquer them as of old."
So the king took his army by night and pushed upon Modred league after
league, until they reached the Western part of Lyonesse where the long
mountains ended in the moaning sea. There Modred's men could flee no
farther, so on the waste lands by the barren sea they began that last
dim weird battle of the West.
A white chill mist slept over all the land and water so that even Arthur
became confused since he could not see which were his friends and which
were his foes. Friends killed friends, some saw the faces of old ghosts
looking in upon the battle. Spears were splintered, shields were broken,
swords clashed, helmets were shattered, men shrieked and looked up to
heaven for help but saw only the white, white mists. There were cries
for light and moans.
At last toward the close of the day a hush fell over the whole shore; a
bitter wind from the North blew the mist aside and the pale king looked
across the battlefield. But no one was there only the waves breaking in
among the dead faces.
But bold Bedivere said: "My King! the man who hates you stands there,
Modred, the traitor of your house!"
"Don't call this traitor a person of my house," the king replied. "The
men of my house are not those who have lived under one roof with me, but
those who always call me their king."
With that, Arthur dashed after Modred. Modred struck at the king's
helmet, which had grown thin with all his heathen wars. Arthur with his
sword Excalibur struck Modred dead, then fell down himself almost killed
with the wound through his helmet.
Sir Bedivere lifted him up and carried him to a chapel near by.
"Take my sword, Excalibur," said the King, "and fling it out into the
middle of the sea, watch what happens to it and then come back at once
and tell me."
"It doesn't seem right to leave you all alone here," said Sir Bedivere,
"when you are wounded and ill, but since you wish me to go, I will, and
will do all that you have told me."
He slipped away by zigzag paths, points and jutting rock to the shining
level of the sea. There he drew out the sword Excalibur. The winter moon
sparkled against its hilt and made it twinkle with its diamond sparks,
with myriads of topaz lights and fine jewelry work. Bedivere gazed so
long at it that both his eyes were dazzled as he stood, and he wondered
whether he ought to throw away so beautiful a thing. At last he decided
to hide it away among the water-flags that grew along shore.
"Did you do as I said?" asked the king, when he saw him. "What did you
see?"
"I heard the ripple washing in the reeds," said Sir Bedivere, "and the
wild water lapping on the rock."
"You are not giving me a true answer," said the king, faint and pale.
"It's shameful for men to lie. Quickly go again and as you are true and
dear, do just as I bade you. Watch and bring me word."
Then Sir Bedivere went the second time and paced up and down beside the
pebbly water, counting the dewey pebbles, but when he saw the wonderful
sword he clapped his hands together and cried:
"If I threw that sword away, a precious thing will be lost forever. The
King is ill; he does not know what he is doing. His great sword ought to
be kept, then in long years hereafter people will look at it at the
tournament and they will say: 'This was the great Arthur's sword
Excalibur which was made by the lonely lady of the Lake, working in the
deep sea for nine years.'"
So the second time he hid Excalibur and strode very slowly back to the
king.
"What did you see or what did you hear?" asked Arthur breathing very
heavily.
"I heard the water lapping on the rock and the long ripples washing in
the reeds."
"Unkind! miserable! untrue! unknightly!" cried Arthur, filled with
anger. "I see what you are, for you are the only one left me of all the
knights, yet you would betray me for my sword, either to sell it or like
a girl, because you love its beauty. Go out now the third time and if
you do not throw out my sword Excalibur I'll get up and kill you with my
hands."
At this Sir Bedivere sprang up like a flash and ran down leaping lightly
over the ridges, plunged into the beds of bulrushes, clutched the sword,
wheeled it round strongly and threw it as far as he could.
Excalibur made lightning in the moonlight as it flashed round and round
and whirled in an arch, shooting far out to the water. But before it
quite dipped into the sea an arm robed in white samite, mystic and
wonderful, rose out of the waves, caught it by the hilt, brandished it
three times and drew it under.
"Now I can see by your eyes that you have done it!" cried the King.
"Speak out; what have you seen or heard?"
"Sir King," cried Sir Bedivere, "I closed my eyes when I picked it up so
that I would not be turned from my purpose of throwing it into the
water, for I could live three lives, Sir King, and I wouldn't again see
such a wonderful thing as your sword. Sir, I threw it out with both
hands, wheeling it round and when I looked an arm robed in white samite
reached up out of the water and caught it by the hilt, brandished it
three times and drew it under."
"Carry me to the shore," said the king.
[Illustration: AN ARM ROBED IN WHITE SAMITE.]
So Bedivere lifted him up and walked as swiftly as he could from the
ridge, heavily, heavily down to the beach. As they reached the shore
they saw a black barge beside the water filled with stately people all
dressed in black. Among the people were three queens wearing crowns of
gold.
"Put me into the barge," cried Arthur.
So they came to the barge and the three queens held out their hands and
took the king.
The tallest and fairest of them held his head upon her lap loosed his
shattered helmet and chafed his hands, and moaned tenderly over him.
"Ah, my lord Arthur," cried Sir Bedivere, "where shall I go now? For
the old times are past now and the whole Round Table is broken."
"Go and pray," cried the king. "Farewell, for I am going a very long way
to the lovely Island-valley of Avilion where it will never hail nor rain
nor snow, and where the loud winds never blow. It lies in deep meadows,
beautiful with lawns and fruit trees and flowery glens."
Then the barge set sail and oar, and moved away from the shore.
"The king is gone!" groaned Bedivere.
He walked away from the shore and climbed up to the highest peaks and
ridges about him and looked far, far away. And from far away out beyond
the world he thought he heard sounds from a beautiful city as if every
one in it all together were welcoming a great King who had just come
back from his wars.
END.
Transcriber's Note:
Minor spelling and typographical errors have been corrected without note.
There are inconsistencies with italicising text that refers to
illustrations. I have left these as in the original text.
Corrections made include the following:
p34. ecstacy => ecstasy
p37. meaintime => meantime
p52. magnificientn => magnificent
p66. Springly => Springing
p75. Geriant => Geraint
p90. jealously => jealousy
p100. though => through
p101. passed => past
p101. musn't => mustn't
p106. heathern => heathen
p106. Gunievere => Guinevere
p117. to => that
p146. Mordred => Modred
End of the Project Gutenberg EBook of Tales from Tennyson, by Molly K. Bellew
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 4,845
|
{"url":"http:\/\/mathhelpforum.com\/calculus\/216736-legendre-polynomial.html","text":"## Legendre Polynomial\n\nAssuming the result\n\n$P^{'}_{n+1}(z)-P^{'}_{n-1}(z)=(2n+1)P_{n}(z)$\n\nI need to show that\n\n$\\int^{1}_{-1}z^{n}P_{n}(z)dz={\\frac{n}{2n+1}}\\int^{1}_{-1}z^{n-1}P_{n-1}(z)dz$\n\nand then deduce that\n\n$\\int^{1}_{-1}z^{n}P_{n}(z)dz={\\frac{2^{n+1}(n!)^{2}}{(2n+1)!} }$","date":"2017-10-20 07:29:43","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\": 0.9419042468070984, \"perplexity\": 3914.1062591112063}, \"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-2017-43\/segments\/1508187823839.40\/warc\/CC-MAIN-20171020063725-20171020083725-00698.warc.gz\"}"}
| null | null |
La nazionale di calcio femminile di Panama è la rappresentativa calcistica femminile internazionale del Panama, gestita dalla locale federazione calcistica (FEPAFUT).
In base alla classifica emessa dalla FIFA il 9 dicembre 2022, la nazionale femminile occupa il 57º posto del FIFA/Coca-Cola Women's World Ranking.
Come membro della CONCACAF può partecipare a vari tornei di calcio internazionali, quali il campionato mondiale FIFA, la CONCACAF Women's Championship, i Giochi olimpici estivi, i Giochi panamericani e i tornei a invito.
Storia
La nazionale panamense ha disputato la sua prima partita ufficiale il 29 luglio 2002 con la partita d'esordio alle qualificazioni alla CONCACAF Women's Gold Cup 2002. Con due vittorie su quattro partite guadagnò l'accesso alla fase finale: inserita nel gruppo A, dopo aver battuto , ha perso contro e , venendo eliminata dalla competizione. Ottenne la qualificazione alla fase finale anche nel 2006, dalla quale venne subito eliminata al primo turno dalla .
Tornò a disputare la fase finale della CONCACAF Women's Championship, nuova denominazione della competizione, nel 2018. Nuovamente nel gruppo A con Stati Uniti, Messico e Trinidad e Tobago, diversamente dal 2002, con due vittorie su tre partite, le panamensi ottennero l'accesso alle semifinali. Qui giunse una netta sconfitta per 0-7 contro il e la successiva disputa della finale per il terzo posto contro la Giamaica, persa ai tiri di rigore, mancando così l'accesso diretto al campionato mondiale 2019. Grazie al quarto posto Panama giocò contro l' lo spareggio CONCACAF-CONMEBOL per un posto al campionato mondiale. Nella gara d'andata vinsero le argentine in casa per 4-0, mentre il ritorno si concluse in parità sull'1-1, così che fu l'Argentina a qualificarsi al mondiale.
La partecipazione alla CONCACAF Women's Championship 2022 si concluse con l'eliminazione al termine della fase a gruppi, ma il terzo posto nel proprio raggruppamento diede alle panamensi l'accesso ai play-off intercontinentali per la qualificazione al campionato mondiale 2023. Inserita nel raggruppamento C dei play-off, ha prima sconfitto in semifinale per 2-0 e poi battuto in finale il per 1-0 grazie a una rete di Lineth Cedeño, qualificandosi per la prima volta alla fase finale del campionato mondiale.
Partecipazioni ai tornei internazionali
Calciatrici
Tutte le rose
CONCACAF Women's Championship
Note
Altri progetti
Collegamenti esterni
Panama
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,299
|
Home Banking Banking Basics
Why and How to Create an Emergency Fund
Don't let an unexpected emergency clear out your household finances.
Syda Productions / Shutterstock
By Doug Whiteman
Amid the pandemic, many Americans have learned the hard way about the importance of putting money away, just in case.
Fewer than half of U.S. households (46%) had savings to cover three months of expenses during a financial emergency, like the COVID layoffs and business closures, according to research from the financial literacy nonprofit FINRA.
Some people who did have rainy day funds found they hadn't saved enough. In a survey done by CNBC and the investing app Acorns, 14% of Americans said the coronavirus cleaned out their emergency stashes.
Most consumers know they should be saving for unplanned expenses — they just don't know how to do it, or how much to squirrel away. It can all be a little overwhelming, especially if you're starting from scratch.
It's fine to start small, as long as you start somewhere. Here are a few tips to get the ball rolling.
First, define emergency
zoff / Shutterstock
A true emergency would be blown transmissions.
Since emergencies are sudden and unexpected, it may be easier to define what they're not, rather than what they are.
Your kid's senior prom, not having Super Bowl tickets or being behind on your holiday shopping are not emergencies. True emergencies come out of nowhere and can hit you with a major expense, or interfere with your ability to earn money.
Examples include a blown transmission, the loss of a job, or a burst appendix that leaves you with big out-of-pocket medical costs. For those, you want to set up an in-case-of-emergency-break-glass fund.
You'll need a separate account to save up for your next opportunity to go to the Super Bowl.
Choose the right kind of account
Proxima Studio / Shutterstock
Put your funds in a savings account with easy access.
Your emergency fund should be accessible enough that you can withdraw money on short notice, but not so accessible that you can just dip into it on a whim.
Consider opening a high-yield savings account or money market account. Compare interest rates and fees. Make sure it's convenient for transferring or withdrawing money in a hurry.
Eliminate the need for self-discipline or willpower. If your employer pays by direct deposit, divert a portion of your earnings to the savings account. If you deposit your paychecks into a checking account yourself, set up recurring transfers to the emergency fund.
You might choose to put your emergency money into certificates of deposit, or CDs, which pay higher interest. But there are pros and cons, because those are designed to be long-term investments.
Banks and credit unions typically charge penalties for cashing out CDs early. The penalties can encourage you to leave your emergency savings alone, but you'll take a financial hit if and when something happens and you need to tap into your money.
Find ways to beef up your fund
watermelontart / Shutterstock
Save a little. Make coffee at home, and resist the lattes.
One surefire way to boost the amount you can save is to stop spending money. Scrutinize your bank and credit accounts to find out where it's all going.
Consider downgrading or eliminating cable TV. Cancel the gym membership, and work out for free at the community center. Skip the daily Starbucks, carpool to work, install a programmable thermostat, and cook at home.
Teach art, music, swimming or carpentry in your spare time. You can also make extra money with any marketable skills you have by freelancing on an online marketplace like Fiverr. Sell excess furniture, electronics and housewares on eBay or Craigslist.
Install money-saving apps
Stash | learn.stash.com
A screenshot of the Stash app
There are dozens of free apps that can make your smartphone an important tool in helping you save.
Stash offers fractional investing, online banking, simple automation and personal guidance to get the most from your money.
Swagbucks gives you gift cards to over 100 of your favorite retailers in exchange for completing surveys and watching videos.
Truebill helps you keep track of your spending and checks whether you're paying for any subscriptions you no longer use.
Calculate how much you need to put away
NakoPhotography / Shutterstock
Crunch the numbers, and don't forget to include miscellanaeous expenses in the equation.
At a minimum, every household should save enough to cover expenses for three months. The target amount depends on factors such as the number of kids in the home, how much is owed on the mortgage, and each breadwinner's career prospects in the event of job loss.
Some experts recommend that you put aside enough money to get by for six months, while personal finance personality Suze Orman recently said the coronavirus crisis has been so bad that the new standard should be three-year emergency funds.
Once you've decided how many months (or years) your emergency cushion should cover, tally up your regular expenses — including rent or house payments, a car payment, car insurance, child care, utilities and groceries.
When you've determined your monthly expenses, determine your savings goal by multiplying by the number of months you want your fund to cover. A banker, accountant or credit counselor can take a look to make sure you're on the right track. There are also emergency fund calculators online.
Yes, you really need emergency savings
Nattakorn_Maneerat / Shutterstock
Don't just blow unexpected windfalls, contribute to your emergency fund!
Annual household spending in the U.S. is $63,036, on average, according to the U.S. Bureau of Labor Statistics, so a six-month emergency fund for average earners should be approximately $31,520. A three-month fund would be $15,760.
Five-figure savings goals can be intimidating. Just remember that saving even a small amount each month is better than saving nothing at all. Stick to your goal, whether it's $500, $200 or even $50 per month.
Jump-start your savings by shopping around for a bank that offers an introductory bonus for new customers. And resist the urge to blow the next big tax refund you receive. Each time you get an unexpected windfall, sock it away into the savings account.
If you build a big enough buffer, you'll never have to go into debt for unplanned expenses. You'll be able to afford your new transmission or appendix surgery, and you won't drain your retirement fund or be forced to sell stocks at a loss.
Doug Whiteman
Doug Whiteman is the editor-in-chief of MoneyWise. He has been quoted by The Wall Street Journal, USA Today and CNBC.com and has been interviewed on Fox Business, CBS Radio and the syndicated TV show "First Business."
How to Cash a Check
Here's how to get real money from those valuable slips of paper.
By Esther Trattner
APR vs. APY: Why You Need to Know the Difference
Don't ignore those initials next to interest rates. They're not the same thing.
What Is the Current Prime Rate?
The prime lending rate is a key interest rate that affects many other rates. See why it matters to you.
What Is a High-Yield Checking Account? How Do You Get One?
Some checking accounts allow you to earn decent money with your money.
By Richard Glunt
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,760
|
Harris Dirgantara Internasional
Act Locally, Think Globally, Contribute Universally
Indonesia certifies N219 regional turboprop
29 December 2020 harrisdian
An N219 test aircraft in 2017
State-owned Indonesian Aerospace (IAe), also known as Dirgantara Indonesia, has received certification for its N219 commuter aircraft. A type certificate was officially issued by the Directorate General of Civil Aviation on 18 December, the transport ministry said in a statement on 28 December.
The ministry plans to order the small regional turboprop for flight calibration purposes and for providing air transport services to remote regions, among others. It states: "The Minister of Transportation hopes that this achievement will motivate Dirgantara Indonesia to continue to innovate, because technical improvements are still needed in the next generation of aircraft so that they can compete with foreign-made aircraft and have high [commercial value]." The same statement indicates that the aircraft has been undergoing certification since February 2014 and the three-year validity on the certification period was extended twice, on 8 February 2017 and 11 February 2020.
According to the ministry, each aircraft is fitted with two Pratt & Whitney Canada's PT6A engines and can carry up to 19 passengers. Following the type certification, IAe has plans for the N219 to enter the commercialisation phase in 2021, it states in a 28 December evening update posted on its official Twitter account. English-language media Jakarta Globe reported on 11 December that the aircraft type completed its final test flight that day. According to the report, Indonesia's minister for research and technology Bambang Brodjonegoro said the final test flight would clear the way for a commercial aircraft licence for the N219 to fly commercial routes in the country.
Cirium fleets data shows that IAe has two prototypes. PK-XDT (MSN 001) was rolled out in November 2015 and launched its first flight in August 2017. PK-XDP (MSN 002) was rolled out in September 2018 and its first flight was in December that year. The local government of the semi-autonomous Aceh province most recently showed interest in the programme, placing in December 2019 an LOI to order four examples.
Domestic airline Aviastar Mandiri has by far shown the greatest support for the programme, placing in October 2018 an LOI for 20 orders, adding to an April 2015 LOI for 20 orders and 10 options. Other LOIs – for eight orders by national private charterer Air Born and 10 orders and options for five by for Jakarta-based Trigana Air – also date back to April 2015. Defunct airlines Merpati and Nusantara Buana Air had lapsed LOIs to order 20 examples each, placed in 2011 and 2012, respectively. Nusantara also had options for another 10.
Link: https://www.flightglobal.com/aerospace/indonesia-certifies-n219-regional-turboprop/141763.article
Commercial, English, Manufacturing, New Aircraft, Strategic Management N219 Leave a comment
For Airlines, Dry Ice in Vaccine Transport Demands Special Attention
9 December 2020 harrisdian
A worker with dry ice at a Pfizer plant in Puurs, Belgium. The Pfizer-BioNTech vaccine must be stored at minus-70 degrees Celsius.
PHOTO: OLIVIER MATTHYS/ASSOCIATED PRESS
The refrigerant needed to maintain some doses during distribution is regulated as a dangerous good for aviation transport
The large amounts of dry ice needed to speed Covid-19 vaccine candidates to pandemic-weary populations will call for special attention from airlines and safety regulators.
Dry ice, the solid form of carbon dioxide, is a critical part of plans to transport the vaccine developed by Pfizer Inc. and BioNTech SE, which must be kept at ultracold temperatures. Pfizer expects to ship 50 million doses world-wide by the end of the year. The vaccine was the first to be authorized in the West, receiving clearance for emergency use in the U.K. last week. It is under review by the Food and Drug Administration in the U.S.
Widely used as a refrigerant, dry ice is classified as a dangerous good by the International Civil Aviation Organization and the U.S. Department of Transportation because it changes to gas form as it breaks down, a process called sublimation. Shippers must use ventilated containers that allow the gas to release, to prevent pressure from building up and rupturing the packaging.
The gas can also displace oxygen in confined spaces with poor ventilation, creating a suffocation hazard, though the risk is minimal under normal cabin ventilation, according to the Federal Aviation Administration.
"If oxygen levels get down below 19%, that could cause a hazard to people and animals," said Delmer Billings, technical director for the Dangerous Goods Advisory Council, a nonprofit trade group that promotes safe transportation of hazardous materials. "If you deplete oxygen sufficiently, it could cause unconsciousness, even death," he added.
Air carriers involved in vaccine transport efforts are asking aviation regulators to increase the amount of dry ice they are allowed to carry on flights hauling vaccines as they work with drugmakers and governments to set up distribution channels. Restrictions on the amount of the material on planes are typically based on aircraft ventilation rates and factors such as the size of the plane and whether it is used for passenger or cargo flights, said Robert Coyle, senior vice president of pharma and healthcare strategy at freight forwarder Kuehne + Nagel International AG.
On Thursday, Delta Air Lines Inc. said it had received FAA approval to double the allowed load of dry ice on its Airbus A330 and A350 wide-body jets, and six times the prior allowed load for shipments using a special suitcase-sized storage container that Pfizer designed.
Delta has done trial runs with vaccine cargoes from Europe and to Latin America, and within the U.S., all on cargo-only flights.
United Airlines Holdings Inc. secured FAA approval last month to boost its dry-ice allowance to 15,000 pounds from 3,000 pounds, for chartered cargo flights between Brussels International Airport and Chicago O'Hare International Airport to support distribution of the Pfizer and BioNTech vaccine. A United spokeswoman said the airline "has effective procedures in place to ensure we safely handle all the hazardous materials we are permitted to carry on board our aircraft."
Extremely cold with a surface temperature of about minus-78 degrees Celsius, dry ice has long been used to ship medicine, pharmaceutical products and perishable food such as meat or ice cream.
"When packaged and stored properly, it poses no risk," said Rafael Teixeira, president of World Courier and ICS, a specialty logistics provider owned by drug distributor AmerisourceBergen Corp.
The scale of the Covid-19 vaccine distribution effort is unprecedented, involving billions of doses with strict temperature-control requirements that are expected to strain cold-chain shipping networks.
The Pfizer and BioNTech shots must be kept at minus-70 degrees Celsius—colder than the average annual temperature at the South Pole and lower than some other vaccine candidates require. Moderna Inc.'s shot, the other leading front-runner, must be shipped and stored at a below-freezing temperature that most home or medical freezers can accommodate.
Makers of dry ice are bracing for an expected demand surge. Logistics providers have been building "freezer farms" with hundreds of portable units that store pharmaceuticals at ultralow temperatures.
Plymouth, Minn.-based Pelican BioThermal LLC, which makes packaging that typically uses engineered materials to maintain temperatures, has tested and approved the use of dry ice in its systems to provide the sub-frozen temperatures needed to maintain the efficacy of Covid-19 vaccines. The company is also ramping up global production of its large shipping containers that can hold full pallets of goods on rising demand from pharmaceutical companies looking to ship vaccines.
"There are a lot of investments bhttps://www.wsj.com/articles/for-airlines-dry-ice-in-vaccine-transport-demands-special-attention-11607370720eing made right now to get this done," said Ira Smith, director of Pelican's rental program in the Americas.
Link: https://www.wsj.com/articles/for-airlines-dry-ice-in-vaccine-transport-demands-special-attention-11607370720
Cargo, Commercial, COVID-19, English, Environmental, Health, Safety COVID-19_Vaccine, Dangerous_Goods, Dry_Ice Leave a comment
Chuck Yeager, 1st to break sound barrier, dies at 97
GRASS VALLEY, Calif. (AP) — Retired Air Force Brig. Gen. Charles "Chuck" Yeager, the World War II fighter pilot ace and quintessential test pilot who showed he had the "right stuff" when in 1947 he became the first person to fly faster than sound, has died. He was 97.
Yeager died Monday (7 Dec 2020), his wife, Victoria Yeager, said on his Twitter account. "It is w/ profound sorrow, I must tell you that my life love General Chuck Yeager passed just before 9pm ET. An incredible life well lived, America's greatest Pilot, & a legacy of strength, adventure, & patriotism will be remembered forever."
Yeager's death is "a tremendous loss to our nation," NASA Administrator Jim Bridenstine said in a statement.
"Gen. Yeager's pioneering and innovative spirit advanced America's abilities in the sky and set our nation's dreams soaring into the jet age and the space age. He said, 'You don't concentrate on risks. You concentrate on results. No risk is too great to prevent the necessary job from getting done,'" Bridenstine said.
"In an age of media-made heroes, he is the real deal," Edwards Air Force Base historian Jim Young said in August 2006 at the unveiling of a bronze statue of Yeager.
He was "the most righteous of all those with the right stuff," said Maj. Gen. Curtis Bedke, commander of the Air Force Flight Test Center at Edwards.
Yeager, from a small town in the hills of West Virginia, flew for more than 60 years, including piloting an X-15 to near 1,000 mph (1,609 kph) at Edwards in October 2002 at age 79.
"Living to a ripe old age is not an end in itself. The trick is to enjoy the years remaining," he said in "Yeager: An Autobiography."
"I haven't yet done everything, but by the time I'm finished, I won't have missed much," he wrote. "If I auger in (crash) tomorrow, it won't be with a frown on my face. I've had a ball."
On Oct. 14, 1947, Yeager, then a 24-year-old captain, pushed an orange, bullet-shaped Bell X-1 rocket plane past 660 mph to break the sound barrier, at the time a daunting aviation milestone.
"Sure, I was apprehensive," he said in 1968. "When you're fooling around with something you don't know much about, there has to be apprehension. But you don't let that affect your job."
The modest Yeager said in 1947 he could have gone even faster had the plane carried more fuel. He said the ride "was nice, just like riding fast in a car."
Yeager nicknamed the rocket plane, and all his other aircraft, "Glamorous Glennis" for his wife, who died in 1990.
Yeager's feat was kept top secret for about a year when the world thought the British had broken the sound barrier first.
"It wasn't a matter of not having airplanes that would fly at speeds like this. It was a matter of keeping them from falling apart," Yeager said.
Sixty-five years later to the minute, on Oct. 14, 2012, Yeager commemorated the feat, flying in the back seat of an F-15 Eagle as it broke the sound barrier at more than 30,000 feet (9,144 meters) above California's Mojave Desert.
His exploits were told in Tom Wolfe's book "The Right Stuff," and the 1983 film it inspired.
Yeager was born Feb. 23, 1923, in Myra, a tiny community on the Mud River deep in an Appalachian hollow about 40 miles southwest of Charleston. The family later moved to Hamlin, the county seat. His father was an oil and gas driller and a farmer.
"What really strikes me looking over all those years is how lucky I was, how lucky, for example, to have been born in 1923 and not 1963 so that I came of age just as aviation itself was entering the modern era," Yeager said in a December 1985 speech at the Smithsonian Air and Space Museum.
"I was just a lucky kid who caught the right ride," he said.
Yeager enlisted in the Army Air Corps after graduating from high school in 1941. He later regretted that his lack of a college education prevented him from becoming an astronaut.
He started off as an aircraft mechanic and, despite becoming severely airsick during his first airplane ride, signed up for a program that allowed enlisted men to become pilots.
Yeager shot down 13 German planes on 64 missions during World War II, including five on a single mission. He was once shot down over German-held France but escaped with the help of French partisans.
After World War II, he became a test pilot beginning at Wright-Patterson Air Force Base in Dayton, Ohio.
Among the flights he made after breaking the sound barrier was one on Dec. 12. 1953, when he flew an X-1A to a record of more than 1,600 mph. He said he had gotten up at dawn that day and went hunting, bagging a goose before his flight. That night, he said, his family ate the goose for dinner.
He returned to combat during the Vietnam War, flying several missions a month in twin-engine B-57 Canberras making bombing and strafing runs over South Vietnam.
Yeager also commanded Air Force fighter squadrons and wings, and the Aerospace Research Pilot School for military astronauts.
"I've flown 341 types of military planes in every country in the world and logged about 18,000 hours," he said in an interview in the January 2009 issue of Men's Journal. "It might sound funny, but I've never owned an airplane in my life. If you're willing to bleed, Uncle Sam will give you all the planes you want."
When Yeager left Hamlin, he was already known as a daredevil. On later visits, he often buzzed the town.
"I live just down the street from his mother," said Gene Brewer, retired publisher of the weekly Lincoln Journal. "One day I climbed up on my roof with my 8 mm camera when he flew overhead. I thought he was going to take me off the roof. You can see the treetops in the bottom of the pictures."
Yeager flew an F-80 under a Charleston bridge at 450 mph on Oct. 10, 1948, according to newspaper accounts. When he was asked to repeat the feat for photographers, Yeager replied: "You should never strafe the same place twice 'cause the gunners will be waiting for you."
Yeager never forgot his roots and West Virginia named bridges, schools and Charleston's airport after him.
"My beginnings back in West Virginia tell who I am to this day," Yeager wrote. "My accomplishments as a test pilot tell more about luck, happenstance and a person's destiny. But the guy who broke the sound barrier was the kid who swam the Mud River with a swiped watermelon or shot the head off a squirrel before going to school."
Yeager was awarded the Silver Star, the Distinguished Flying Cross, the Bronze Star, the Air Medal and the Purple Heart.
President Harry S. Truman awarded him the Collier air trophy in December 1948 for his breaking the sound barrier. He also received the Presidential Medal of Freedom in 1985.
Yeager retired from the Air Force in 1975 and moved to a ranch in Cedar Ridge in Northern California where he continued working as a consultant to the Air Force and Northrop Corp. and became well known to younger generations as a television pitchman for automotive parts and heat pumps.
He married Glennis Dickhouse of Oroville, California, on Feb. 26, 1945. She died of ovarian cancer in December 1990. They had four children: Donald, Michael, Sharon and Susan.
Yeager married 45-year-old Victoria Scott D'Angelo in 2003.
On the Net:
Yeager: http://www.chuckyeager.com/
Link: https://www.yahoo.com/news/chuck-yeager-1st-break-sound-051713166.html
English, Technology Chuck_Yeager, Sound_Barrier Leave a comment
One year later: The crash of Flight 3296
23 October 2020 23 October 2020 harrisdian
A commuter airplane has crashed near the airport in a small Alaska community on the Bering Sea, Thursday, Oct. 17, 2019, in Unalaska, Alaska. Freelance photographer Jim Paulin says the crash at the Unalaska airport occurred Thursday after 5 p.m. Paulin says the Peninsula Airways flight from Anchorage to Dutch Harbor landed about 500 feet (152 meters) beyond the airport near the water. (Jim Paulin via AP)
Twelve months ago, on Oct. 17, 2019, PenAir Flight 3296 overran the runway while landing at the Dutch Harbor airport, resulting in one passenger killed and four others injured. Since then, Ravn Alaska, which owned PenAir along with sister companies Corvus Airlines and Hageland Aviation, declared bankruptcy and auctioned off or sold the bulk of its assets.
Company executives blamed Ravn's failure on the coronavirus, but on the Flight 3296 anniversary, it is worth considering just what happened to PenAir in the single year it was owned by Ravn, and what we have learned since the accident that exposes problems within the company in the months leading up to the tragedy.
Soon after the accident, the National Transportation Safety Board released an investigative update detailing the flight crew's minimal experience in the aircraft. Ravn stopped all flights of the Saab 2000 into Unalaska and Alaska Airlines dropped the lucrative Capacity Passenger Agreement (CPA) it had with Ravn. The loss of the CPA, which paid Ravn for the Unalaska flights at "predetermined rates plus a negotiated margin, regardless of the number of passengers on board or the revenue collected," had serious financial ramifications for the company. Questions raised by the NTSB's preliminary investigation, however, left Alaska Airlines with little choice.
According to the NTSB's initial report, and heavily covered in the media, the pilot in command (PIC) for Flight 3296 had an estimated 20,000 hours total flight time, but only 101 hours in the Saab 2000 (the co-pilot, with 1,446 hours total time, had 147 hours in the aircraft). Under PenAir's previous ownership by the Seybert family, PICs were required to have 300 hours minimum in the Saab 2000 before operating into Dutch Harbor. (Similar requirements have existed for other companies operating at the challenging airfield.)
Based on the PenAir Operations Manual, flight-time minimums could be waived if approved by the company Chief Pilot. While the existence of such a waiver has not been addressed publicly, one month after the accident, the Federal Aviation Administration confirmed that Chief Pilot Crystal Branchaud had been replaced and no longer held a position of operational control with PenAir. The extent to which she or any other management personnel played a role in assigning the PIC to Flight 3296 will likely receive serious attention in the accident's final report.
Another area of significant interest for investigators will be the flight crew's decision to land in turbulent weather conditions. In the report, the NTSB stated that when Flight 3296 first attempted to land on Runway 13, the winds were at 10 knots from 270 degrees. After initiating a go-around, the winds were reported at 16 knots, gusting to 30, from 290 degrees. While on final approach the second time, the winds were 24 knots from 300 degrees, providing almost a direct tailwind. The aircraft was configured for approach with 20 degrees of flaps both times.
Aircraft landing performance standards are based on multiple factors including weight and balance, wind and runway conditions. While Flight 3296′s weight and balance has not been released, it is possible to determine a conservative estimate of its total weight from available data. According to the manufacturer, the aircraft has a basic empty weight of about 30,500 pounds (this includes the three-member crew). Adding fuel for required reserves and Cold Bay as an alternate destination (about 2,000 pounds) and weight for 39 passengers at the FAA standard for summer adults (195 lbs x 39 = 7,605 pounds), a total weight of 40,105 pounds can be calculated. This excludes any baggage that may have been onboard.
For Runway 13 at Dutch Harbor, PenAir's company performance standards permitted a landing weight, with 20 degrees of flaps, of 40,628 pounds with zero wind, 35,402 pounds for 5 knots of tailwind and 29,955 pounds for 10 knots of tailwind. It recommended a reduction of 1,031 pounds for each additional knot of tailwind. There is thus no discernible calculation that would recommend landing on Runway 13 with the reported winds at the time of the crash at the aircraft's approximate weight.
According to the NTSB, the flight crew reported touching down about 1,000 feet down the runway, with skid marks first appearing at about 1,840 feet. From there, the marks continued 200 feet before the aircraft crossed a grassy area, impacted the airport's perimeter fence, crossed a ditch, hit a large rock and then crossed Ballyhoo Road. It was on the opposite shoulder of the road, over the rock seawall and nearly into the waters of Dutch Harbor, that Flight 3296 finally came to rest.
After the aircraft stopped and a desperate but ultimately unsuccessful effort was underway to save the life of passenger David Oltman, the flight crew waited with forward passengers for assistance in exiting. It was at that point, according to passenger Steve Ranney, that a brief verbal exchange occurred. "A passenger asked the captain why he landed," explained Ranney in an email, "and he calmly said the computer showed he was within the safety margin." According to Ranney, who was interviewed by NTSB investigators, neither the captain nor co-pilot spoke another word.
There is no onboard computer that calculates landing performance for the Saab 2000; the PIC could only have been referring to an app likely used on his company-issued iPad. "Electronic flight bags" are commonly utilized by pilots, but the use of any software for the purposes of formal flight planning in commercial operation would have to be approved by the FAA. When asked if PenAir had authorization to utilize performance calculation software, the FAA referred the question, as part of an ongoing investigation, to the NTSB. The NTSB would state only that "crew performance standards equipment procedures and a host of other factors" would be part of the investigation.
Decision-making is always an area of particular inquiry following a commercial crash, both on the part of the flight crew and company management. As investigators moved from the aircraft to the cockpit and back to the offices of PenAir, Ravn Air Group and even the FAA, there are other events in 2019 that may have garnered interest and point to further issues within the newly acquired company.
In February last year, PenAir Flight 3298 suffered an engine loss about an hour after departing King Salmon. In a statement to ADN at the time, FAA spokesman Allen Kenitzer said the aircraft "experienced engine trouble, so pilots shut it down." The flight crew then returned to the village. In a subsequent Service Difficulty Report (SDR), the company reported a "right engine auto shutdown in flight, did not attempt restart. Troubleshooting in progress." It is unknown what the final remedy was for that engine.
In July, PenAir Flight 2051 was en route from Anchorage to Dillingham when it suffered the loss of the right engine near its destination. As later detailed in a passenger complaint submitted to the FAA, the flight crew chose to turn around and fly all the way back to Anchorage on only one engine.
PenAir subsequently reported in an SDR that there was a "RT engine overtemp in cruise with auto shutdown" and that the engine was to be removed and replaced. Additionally, in a separate SDR the same day, the company reported a problem with the aircraft's left engine, which went to "0 PU's 5SEC." The remedy was for that engine also to be replaced.
No mention of the problems with the left engine nor the necessary replacement of both engines was passed on to the passenger who filed the complaint. Neither was an explanation provided for the flight crew's decision to forgo immediate landing at the nearest suitable airport (as required by federal regulation 121.565).
FAA Safety Inspector David Friend wrote to the passenger, a licensed pilot from the Bristol Bay region, that "it has been determined that the flight crew acted within the scope of all applicable Federal Aviation Regulations and associated PenAir Operations Specifications." In a subsequent Freedom of Information Act request I submitted for a deviation of 121.565 report, the FAA responded that nothing pertaining to my request existed.
Months later, in the days after the crash of Flight 3296, Ravn announced a shift to using Dash-8 aircraft on the route and company management initiated a concerted effort to deflect blame to the Saab 2000. In an October 25 town hall meeting, CEO Dave Pflieger said Ravn would "need to go through a multifaceted process to ensure it is safe to land Saabs in Unalaska before they can return to service there." This negative sentiment was echoed by Ravn's new management, which acquired the PenAir and Corvus Airlines certificates along with several Dash-8 aircraft in a private sale last summer. In a July interview with KUCB, that company's CEO, Rob McKinney, responded to questions about safe operations in rural Alaska by commenting on the crash of Flight 3296. "The Saab 2000 has a narrower margin of safety," he asserted, "so that… potentially was a contributory cause of that unfortunate accident last year."
Both Pfleiger and McKinney's assessments ran sharply counter to the more than two years of accident-free flying with the aircraft under the Seyberts' ownership, including thousands of flights into Unalaska. Further, from the time the Saab 2000s were acquired by the Seyberts and long before they were put into service, there was extensive flight testing, upgrades, modifications and certifications required for their transition to Part 121. All of this was heavily supervised by the FAA. By the time PenAir was purchased by Ravn in October 2018, there was nothing left for the Saab 2000 to prove; the aircraft simply needed the company to assign pilots who were trained how to fly it.
For now, Alaska Airlines flies scheduled service into Cold Bay, with continuing service to Unalaska provided by Grant Aviation. Alaska Central Express offers both regular cargo flights and passenger charter service and other operators, including Dena'ina Airtaxi, Alaska Air Transit, Resolve Aviation and Security Aviation also fly passenger charters. The Saab 2000s, which were leased by PenAir, have been parked at Anchorage International by their Florida-based owner since Ravn's collapse. They will likely be relocated to the Lower 48 for maintenance and storage in the near future.
The NTSB's final report on Flight 3296 should be released early next year. What it will reveal about problematic risk management assessments at all levels of the company is of great interest to anyone following aviation safety in Alaska. And while the detrimental fallout from the subsequent pandemic can not be ignored, it must be noted that Ravn was the only Alaska aviation company of significant size to file for bankruptcy after the virus. Further, although Ravn destroyed numerous financial, professional and customer relationships, many other companies shouldered the pieces it left behind while still continuing to navigate the current uncertain economic landscape.
The easiest thing in the world would be to dismiss PenAir's summer engine problems and the decisions leading up to the Unalaska crash, disregard how long Ravn's $90 million worth of unpaid bills were accruing, pay no attention to the likely sky-high fleet insurance the company was paying and simply blame everything that happened to it on the coronavirus. But just like the transparent attempt to shift responsibility of the Flight 3296 tragedy onto the aircraft, this would also require a determination to blindly ignore so many events leading up to Ravn's demise, including its 16 accidents and incidents over the previous ten years. It is worth noting the most recent of those was not Flight 3296, but rather a gear-up landing by Hageland Aviation in Fairbanks, four months before Ravn shut down. It was easy to miss that one when the company was so loudly insisting everything was COVID-19′s fault.
Link: https://www.adn.com/opinions/2020/10/21/one-year-later-the-crash-of-flight-3296/
Accident, English, Human_Error, Investigation, Safety, Unstabilized Approach Alaska, NTSB, Pen_Air, Pen_Air_3296, SAAB_2000, Unalaska Leave a comment
Tuesday's Market Minute: Southwest Asks Union Employees For Pay Cuts
8 October 2020 harrisdian
Southwest Airlines Co (NYSE: LUV) is asking its employees to agree to pay cuts for the first time through the end of next year. This notion is specific to union employees, and the airline has said is the trade-off to avoiding furloughs and future job cuts. Non-union salaries will already be cut by at least 10% until the beginning of 2022. And the Southwest CEO Gary Kelly is also already foregoing his salary entirely through 2021, with several senior-level executives taking a 20% pay cut.
While it is not news that the airlines as a whole have been one of the hardest-hit sectors throughout the pandemic, it remains unclear when they will see any degree of relief. Airline travel demand remains down 70% compared to a year ago despite receiving federal aid of $25 billion dollars earlier this year. Under the terms of the government package, all airlines were banned from laying off or furloughing any employees until October 1.
As October quickly crept up and federal aid restrictions lifted, the airlines began taking necessary measures to remain afloat. American Airlines Group Inc (NASDAQ: AAL) laid off roughly 19,000 employees and United Airlines laid off about 13,000 employees within the last week.
While another round of stimulus has been in talks, a formal package remains unapproved. U.S. House Speaker Nancy Pelosi even asked the U.S. airline executives on Monday to hold off on further job cuts, promising that relief is on the way. The Southwest Airlines' CEO has confirmed the company can avoid job losses if these terms are reached within the unions by the start of 2021, also noting "I feel like we have a moral obligation to them."
Southwest stock remains down about 28% this year, outperforming its airline peers who have on average lost about 50% of their value in the last 8 months.
Link: https://www.yahoo.com/news/tuesdays-market-minute-southwest-asks-142348948.html?.tsrc=daily_mail&uh_test=1_04
Commercial, English, Financial, Strategic Management Pay_Cuts, Southwest_Airlines Leave a comment
Covid-19 Vaccine Delivery Will Present Tough Challenge to Cargo Airlines
Inoculations for the new coronavirus will require thousands of extra flights, taxing stretched airlines
UPS is combining multiple refrigerators at its airport hubs to store vaccines in transit.
The pandemic has revealed shortcomings in global supply chains and forced business to make logistics a bigger strategic priority. Successfully delivering Covid-19 vaccines will test manufacturers and shippers on what lessons have been learned.
"If 50 million doses were available today, could we distribute them?" asked Glyn Hughes, head of cargo at the International Air Transport Association, a trade group. "The answer is almost certainly 'No', for every jurisdiction."
The air-cargo industry is making plans for delivering as many as 20 billion Covid-19 vaccination doses, even before regulators have approved any of the multiple treatments under development. Shippers say they are having to plan without knowing exactly how many vaccine doses they'll have to ship, where they will be manufactured and how cold they have to be kept.
Pharmaceutical companies and shippers say they expect the bulk of vaccine supplies to be transported by air. Cargo-airline executives are working on a delivery schedule that assumes initial batches become available during the traditional peak season for shipping that runs from fall through early February.
Carriers such as FedEx Corp. FDX 0.44% and the DHL arm of Deutsche Post AG DPSGY 0.22% have started preparations such as introducing new temperature-monitoring systems to track future vaccine shipments. United Parcel Service Inc. UPS -0.41% and Deutsche Lufthansa AG are building "freezer farms" combining multiple refrigerators at their airport hubs to store vaccines in transit.
Yet cargo flights are fast filling up through February with bookings for consumer electronics, apparel and industrial parts through the holiday season and new year, said airline executives. This year's peak season is expected to include a lift from the delayed iPhone 5G from Apple Inc. and Sony Corp.'s PlayStation 5.
"We're planning for the mother of all peaks," said Don Colleran, president of FedEx's express division, on an investor call last month.
Airlines said they would make room for essential supplies such as vaccines, just as they have for personal protective equipment throughout the pandemic.
Most of the potential vaccines have to be kept at a low constant temperature throughout the journey to prevent spoiling, according to cargo experts. These fall into two temperature ranges—around freezing and about minus 70 degrees Celsius—with very different transport and storage requirements.
Pharma executives said spoilage rates for other vaccines during transport range from 5% to as much as 20% because of poor temperature control.
"This is going to be one of the biggest challenges for the transportation industry," Michael Steen, chief operating officer at Atlas Air Worldwide Holdings Inc., said in an interview. Atlas is one of the largest cargo airlines, flying freight on behalf of customers including Amazon.com Inc. and DHL.
Cargo executives said they expect it will take two years for a vaccine to reach all of the world's population, with particular challenges in some emerging markets where infrastructure is limited.
The air-cargo industry isn't starting from scratch. Pharma products have been one of the fastest-growing and most profitable cargo types over the past decade. Shippers have developed increasingly sophisticated supply chains for vaccines in recent years, especially for the flu. Gene therapies, another booming area, already require transport and storage at very low temperatures.
Planning flu-season vaccine deliveries typically starts months ahead and includes analyzing which routes and airports carry the highest risk for delays and spoilage, said Larry St. Onge, president of DHL's life sciences and health care unit.
DHL is applying that analysis to potential Covid-19 vaccines, which will have more-urgent delivery needs and far larger volume.
IATA estimates transporting a single dose to the global population would require the equivalent of 8,000 fully-laden Boeing Co. 747 flights, based on dimensions for vials and packaging provided by pharma companies. A recent study by DHL and McKinsey & Co. pegged demand at 15,000 flights, while including syringes and protective equipment for medical staff would increase the cargo-space requirement.
Pharma shipments already account for around 1.9% of global air-cargo volume, said IATA, and adding Covid-19 vaccines could double that share. Not every freighter jet is able to handle very cold cargo because of regulatory restrictions on how much dry ice they can transport to cool them, said executives.
Air-cargo capacity is already tight, with flights flying fuller than before the pandemic started. International air-cargo capacity was down 32% in August from a year earlier while demand was only 14% lower.
The pandemic-driven travel downturn has removed from service hundreds of passenger jets and the belly space that once carried cargo. More freighters are joining the fleet, with Atlas returning stored 747s from the desert and passenger airlines converting around 100 planes to carry freight in their cabins.
Covid-19 vaccine makers such as Pfizer Inc. have already begun manufacturing doses to be ready for shipment should regulators authorize their use. However, the uncertainty over approval timing and shipping requirements has meant they have stopped short of booking space on cargo flights, said airline executives.
The U.S. government last month outlined its initial plans for distributing vaccines domestically under its Operation Warp Speed program run by the Department of Health and Human Services and the Pentagon, as well as the Centers for Disease Control and Prevention.
McKesson Corp. , one of the world's largest drug wholesalers, has been contracted by CDC to ship some vaccine types in the U.S. It hasn't detailed how they would be transported, and air-cargo executives said they haven't signed any Covid-19 vaccine-related deals yet. McKesson declined to comment.
President Trump said during the opening presidential debate last week that the military would support distribution of the vaccine. The Pentagon said it doesn't expect to have to use military transport aircraft, except to very remote areas. "Our best military assessment is that there is sufficient U.S. commercial-transportation capacity to fully support vaccine distribution," said a spokesman.
Cargo, Commercial, COVID-19, English, Strategic Management COVID-19_Vaccine Leave a comment
Virgin Atlantic to Test Cabin Crew and Pilots for COVID-19 at Least Once a Month
Photo Credit: Virgin Atlantic
Beleaguered airline Virgin Atlantic is to test all of its cabin crew and pilots for COVID-19 at least once a month as part of measures to "instil confidence" in the safety of air travel. The airline, which is majority-owned by Sir Richard Branson, however, warned the move would do little to increase ticket sales without a system to replace draconian quarantine rules with mass COVID-19 passenger testing.
Pilots and cabin crew have been taking part in a trial of rapid pre-flight COVID-19 testing since September 30 on flights to Hong Kong and Shanghai. Both of these destinations require crew to present a negative COVID-19 test certificate before being allowed entry.
The plan is now to extend rapid testing to other destinations that require negative tests for crew members, as well as those where such requirements don't currently exist. In effect, it would mean that all crew would be tested for COVID-19 at least once per month.
Testing would be mandatory for certain destinations.
The move follows a similar initiative developed by shareholder Delta Air Lines. The Atlanta-based airline has now tested every member of staff for the novel Coronavirus and plans to roll-out mass rapid testing in the near future.
Results from the mass-testing program, Delta claims, shows that cabin crew are less likely to succumb to COVID-19 than the general public. The airline put that down to safety measures like mandatory mask-wearing rules and enhanced aircraft cleaning.
Last week, Air Canada also announced plans to roll-out mass rapid COVID-19 testing for staff members on a voluntary basis. The airline is using the Abbott rapid ID NOW testing system that can produce results in as little as 15-minutes.
On Monday, the South African government had to clarify rules for flight crew entering the government after Emirates apparently cancelled some services over confusion with new COVID-19 travel restrictions. While passengers must present a negative test certificate dated within 72-hours of travel, this will no longer be a requirement for cabin crew.
They won't, however, now be allowed to leave their hotel rooms as part of preventative measures.
Link: https://www.paddleyourownkanoo.com/2020/10/05/virgin-atlantic-to-test-cabin-crew-and-pilots-for-covid-19-at-least-once-a-month/
COVID-19, English, Health, Regulation, Strategic Management Virgin_Atlantic Leave a comment
Sustainable Aviation Fuels Still The Immediate Future
The prevailing view at Air Transport Action Group's (ATAG) Global Sustainable Aviation Forum (29-30 September) was that the immediate future for decarbonization remains the adoption of sustainable aviation fuels (SAFs). Robin Hayes, CEO of US low-cost carrier JetBlue, says the airline chose SAFs as a focus for the next five to ten years, because new zero-emission aircraft will come later. JetBlue found that SAFs, which are already being used on flights from San Francisco, are a cost-effective way of reducing emissions, he adds.
Another way is having more efficient air traffic control in congested regions, he says, adding that JetBlue has noticed that fewer flight delays – in the current low traffic COVID-19 period – has resulted in lower fuel burn for each flight. JetBlue recently became the first US airline to achieve carbon neutrality for all domestic flights through the purchase of carbon offsets. Hayes say the airline did this because "customers, regulators and communities want [tangible] action now." During the forum's chief technology officers' (CTO) panel, Safran CTO Stéphane Cueille said ATAG's Waypoint 2050 target – of halving aviation's global emissions by 2050 as compared to 2005 – is achievable.
France's government has announced that it expects to see a regional aircraft – powered by either hybrid-electric or hydrogen technology – enter service around 2030 and see a more fuel-efficient Airbus A320 aircraft enter service from 2033-2035. Cueille says conventional turbine powered aircraft will still be the mainstay of the global commercial aircraft fleet in the early 2030s, so Safran is working to ensure the turbines are 100% compatible with alternative fuels like SAFs. Safran is also working on hydrogen-powered propulsion systems, Cueille says.
Paul Stein, Rolls-Royce CTO, says disruptive technologies are unlikely to arrive in the short term, and that it first aims for efficiency gains of about 10% using its upcoming UltraFan technology. Rolls-Royce also plans for its engines to be 100% compatible with SAFs, he adds. Rolls-Royce is also active in disruptive technologies including hydrogen, hybrid and battery-electric, Stein says. Early next year it will attempt to break the speed record for the world's fastest zero-emission aircraft using its electric aircraft demonstrator ACCEL, he adds.
GE Aviation general manager advanced technology, Arjan Hegeman, says GE is working on a wide range of technologies, including allowing engines to utilize SAFs and aiming for double-digit efficiency improvements in its open-rotor engine technology. He also says GE is developing megawatt-scale power generation for electric motors. Raytheon Technologies head of technology and global engineering, Mark Russell, says the company is working on a wide range of carbon-cutting technologies, through improved efficiency, SAFs and hybrid-electric propulsion.
Airbus CTO Grazia Vittadini says hydrogen fuel can be used for about 50% of the journey to a carbon neutral industry. She says Airbus' zero-emission hydrogen-powered aircraft will be available by 2035, but that SAFs will also play a part in the journey as well, she adds. ATR senior VP engineering and head of design organization, Stéphane Viala, says: "ATR has looked at battery technology, but it is not the way to go [as it is too heavy]. We are not carrying enough payload." Viala says the only technology they are considering are those using hydrogen. He says they are looking at hydrogen fuel cells and burning hydrogen in turbines. A challenge we face is producing hydrogen in a 'green way' and at affordable cost, says Viala, adding that the other issue is developing the infrastructure to support a hydrogen-powered aviation industry.
Link: http://www.smartaviation-apac.com/2020/10/sustainable-aviation-fuels-still-the-immediate-future/
English, Environmental, Strategic Management, Technology ATAG, Aviation_Fuel Leave a comment
Rolls-Royce completes ground-testing of technology set to power the world's fastest all-electric plane: ACCEL
2 October 2020 2 October 2020 harrisdian
Rolls-Royce has completed testing of the technology that will power the world's fastest all-electric plane. All the technology has been tested on a full-scale replica of the plane's core—called an 'ionBird'—including a 500hp electric powertrain powerful enough to set world speed records and a battery with enough energy to supply 250 homes. The plane is part of a Rolls-Royce initiative called ACCEL, short for "Accelerating the Electrification of Flight".
The ACCEL project team includes key partners YASA, the electric motor and controller manufacturer, and aviation start-up Electroflight. The team has been developing the technology while adhering to the UK Government's social distancing and other health guidelines; the systems will soon be integrated into Rolls-Royce's 'Spirit of Innovation' plane. There is a long history of iron-birds in aviation for testing propulsion systems ahead of flight; in this case Rolls-Royce named the test airframe "ionBird", after the zero-emission energy source propelling the aircraft.
The dedicated team have tested each and every component of the system including running the propeller up to full speed (approximately 2,400 rpm) using the most power-dense battery pack ever assembled for aircraft propulsion. When at full power during the flight-testing phase, it will propel the aircraft to more than 300 mph setting a new world speed record for electric flight.
More than 6,000 cells are packaged in the battery for maximum safety, minimum weight and full thermal protection. Since January, engineering and test pilots have spent many hours optimizing the system and developing operating procedures for electric flight. GBs of data—generated every hour of operation—are analyzed to improve performance wherever possible.
Rolls-Royce is committed to playing a leading role in reaching net zero carbon by 2050.The completion of ground-testing for the ACCEL project is a great achievement for the team and is another important step towards a world record attempt. This project is also helping to develop Rolls-Royce's capabilities and ensure that we remain a leader in delivering the electrification of flight, an important part of our sustainability strategy. —Rob Watson, Director – Rolls-Royce Electrical
Bremont will be the official timing partner for the all-electric speed record attempt. The British luxury watch maker has also helped develop the design of the plane's cockpit which will feature a stopwatch, while the company has machined canopy release parts at its Henley-on-Thames manufacturing facility.
The first flight is planned for later this year; the aim is to beat the current all-electric flight world record early next year. Half of the project's funding is provided by the Aerospace Technology Institute (ATI), in partnership with the Department for Business, Energy & Industrial Strategy and Innovate UK. The ACCEL project is the first Rolls-Royce project to use offsetting to make the whole program carbon neutral. The company also hopes to inspire young people with the ACCEL project to consider STEM careers (Science, Technology, Engineering and Math).
Link: https://www.greencarcongress.com/2020/10/20201001-accel.html
Airworthiness, English, Manufacturing, New Aircraft, Propulsion, Strategic Management, Technology ACCEL, Electric_Airplane, Rolls-Royce Leave a comment
How to Fix Plane Certification
In this Monday, June 29, 2020 file photo, a Boeing 737 MAX jet heads to a landing at Boeing Field following a test flight in Seattle, USA.
Photo credit: AP Photo/Elaine Thompson, file
After being grounded in March 2019 following two fatal crashes, the Boeing 737 MAX is expected to be certified by the Federal Aviation Administration to fly again later this fall. Investigations pointed to a problem with the aircraft's Maneuvering Characteristics Augmentation System, or MCAS. This automated control system was designed to stabilize the plane and compensate for the more powerful engines used on the 737 MAX compared to previous versions. The FAA's certification of the plane has come under fire because manufacturers can speed up the process by having only enhancements to a preapproved aircraft reviewed and certified. Ronnie R. Gipson Jr., an expert in aviation law and visiting professor at the University of Memphis Cecil C. Humphreys School of Law, whose work was cited in the House Committee on transportation and infrastructure's report on this issue, explains what happened and ways to improve these safety regulations.
What are the criticisms of the FAA certification process stemming from the 737 MAX crashes?
The process for the certification of a transport category aircraft is a very involved and costly process. The aircraft manufacturers that go down this path have to be committed to spending hundreds of millions of dollars. It starts with an initial design, and the aircraft that is produced is then subjected to dynamic flight testing for compliance with all of the Federal Aviation Administration regulations. Once the airplane satisfies all those requirements, the aircraft is given an original type certificate by the FAA. The aircraft manufacturer is then allowed to produce aircraft and sell them.
As time goes on, technology advances and the manufacturer identifies ways to improve on that original design. So the manufacturer goes back to the FAA and says, "We want to take this initial design that we have and amend it because we made some changes." At this point, the aircraft manufacturer files what's called an amended type certificate application for a derivative aircraft from the baseline aircraft. For example, the original type certificate for the first 737 design was submitted to the FAA in 1967. That original design has had multiple derivative aircraft approved by the FAA, with the 737 MAX being the 13th version. In the amended type certification process, the regulatory authority focuses only on what's changed.
Another thing to keep in mind is that the FAA just doesn't have the manpower to oversee all the tests that go with an amended type certificate approval. Therefore, the FAA reviews most of the critical changes related to safety and delegates the noncritical changes for review to the manufacturers – in this case to a body in Boeing which consists essentially of Boeing employees.
And that's what happened here. MCAS wasn't necessarily presented as a change in the design impacting control in flight. As a result, the MCAS was not a priority for the FAA in the amended certificate approval process. The MCAS capabilities and what it was supposed to control were never fully revealed. That's really where the problem started. It was with the narrative that was being presented to the FAA, and the lack of oversight in the amended type certificate process. The result was that the MCAS system that was initially presented to the FAA at the beginning of the amended type certificate process was not the same system that ended up in the aircraft (view chart in gallery).
How will the recent recertification for the 737 MAX ensure that the model is now safe?
The FAA has had to backtrack and give the MCAS system the intense level of scrutiny that it deserved. The FAA has required the manufacturer to go back and make significant adjustments to the software, in addition to changes to the operator's manual, which is what the pilots would see.
How can the certification process be improved?
I see two paths to take. First, for a transport category aircraft, regulations are changed so that the manufacturer can receive amended type certificates for only 20 years after the original type certificate has been issued by the FAA.Here's how that would work: An aircraft manufacturer designs an aircraft for certification in the transport category and applies for the original type certificate in 2020. Once the original type certification is awarded in, say, 2025, then the manufacturer should have 20 years. That means that the manufacturer would have until the year 2045 to seek an amendment to that original type certificate. Beginning in 2046, if the aircraft manufacturer wants to make subsequent design changes, they have to start over and get a new original type certificate.The second component to resolving this problem would be to step in and review what areas the FAA can delegate oversight authority for system changes in an amended aircraft certification application review.
What are the obstacles to making these changes?
One would be money. The FAA has a budget, and these are very costly measures because the FAA will need more engineers and administrators. And for that to happen, Congress has to be prepared to spend the money to make that happen by increasing the FAA's budget.
There's also going to be a cost to the industry. Implementing the proposal of a 20-year cap on the validity of that original type certificate is going to impose a greater financial cost on the aircraft manufacturers of transport category aircraft. They're not going to have as much time to get a return on their investment for the aircraft that they produce. So the aircraft are going to end up costing more, which means the airlines are going to end up paying more for those planes. And that cost is going to trickle down to the flying public in those seats.
Certification timeline of the Boeing 737 series of aircraft.US Department of Transportation, Office of Inspector General
Link: https://www.mbtmag.com/home/news/21196218/how-to-fix-plane-certification
Airworthiness, English, Manufacturing, New Aircraft, Regulation, Safety, Strategic Management, Technology B737_Max, Boeing Leave a comment
Archives Select Month December 2020 (3) October 2020 (20) September 2020 (24) August 2020 (107) July 2020 (125)
Pesawat Citilink yang Tabrak Layang-layang Mendarat Selamat, Kru dan Penumpang Aman
Maskapai Lion Air Digugat Pailit
13,326 hit
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,807
|
Q: Is it possible to run command on different thread inside other thread? So I want to check if nodes are in certain range and if so, draw them or not.
The nodes are checked inside a timer, so a thread is created. Only issue right now is that method setRenderable() have to be called from UI thread. So is there a way to bypass that ? I need this to update in certain interval that's why I create timer.
public void updateNodes()
{
Camera camera = arFragment.getArSceneView().getScene().getCamera();
timer = new Timer();
timer.schedule(new TimerTask(){
int last = 0;
int copy_last = 0;
@Override
public void run(){
Vector3 camera_position = camera.getWorldPosition();
// We're checking all nodes for distance between them
for(Node al : copied_nodes_list){
if(last<copy_last) last++;
else if(countDistanceOfNodes(al,camera_position)<3.0 && copy_last >= last) {
al.setRenderable(andyRenderable);
last++;
copy_last = last;
}
else {
al.setRenderable(andyRenderable);
}
}
last = 0;
}
},0,250);
}
A: *
*Create a variable in your class.
*Set the value when the object is initialized or write a setter against that variable.
*Within the run function, check against the condition against that variable and write the business rules accordingly.
A: Okay, so there's function in android called runOnUiThread(); Which basically doing whats it's called. I attach the working code.
public void updateNodes()
{
Camera camera = arFragment.getArSceneView().getScene().getCamera();
Timer timer = new Timer();
timer.schedule(new TimerTask(){
int last = 0;
int copy_last = 0;
@Override
public void run(){
Vector3 camera_position = camera.getWorldPosition();
// We're checking all nodes for distance between them
for(Node al : copied_nodes_list){
camera_position = camera.getWorldPosition();
if(last<copy_last) last++;
else if(countDistanceOfNodes(al,camera_position)<3.0 && copy_last >= last) {
runOnUiThread(new Runnable(){
@Override
public void run(){
al.setRenderable(andyRenderable);
}
});
last++;
copy_last = last;
}
else {
runOnUiThread(new Runnable(){
@Override
public void run(){
al.setRenderable(null);
}
});
}
}
last = 0;
}
},0,100);
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,828
|
cask 'oracle-jdk' do
version '12.0.1,12:69cfe15208a647278a19ef0990eea691'
sha256 'cf96dc630cd8efa60b2aa8b12d0092a09a2cd4d99567b330cbe0250206bf29ea'
url "https://download.oracle.com/otn-pub/java/jdk/#{version.before_comma}+#{version.after_comma.before_colon}/#{version.after_colon}/jdk-#{version.before_comma}_osx-x64_bin.dmg",
cookies: {
'oraclelicense' => 'accept-securebackup-cookie',
}
name 'Oracle Java Standard Edition Development Kit'
homepage 'https://www.oracle.com/technetwork/java/javase/overview/index.html'
depends_on macos: '>= :yosemite'
pkg "JDK #{version.before_comma}.pkg"
uninstall pkgutil: "com.oracle.jdk-#{version.before_comma}",
rmdir: '/Library/Java/JavaVirtualMachines'
caveats do
license 'https://www.oracle.com/technetwork/java/javase/terms/license/javase-license.html'
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,973
|
\section*{Introduction}\label{sec1_introduction}
It is well-known that accelerated gradient-type methods possess optimal complexity estimates \cite{NemirovskyYudin} for the class of convex smooth minimization problems. In many practical situations it makes sense to work with inexact gradient information (see e.g. \cite{DevolderThesis}, \cite{DevPaper}, \cite{Polyak1987}, \cite{Vasin2021}). For example, this is relevant for gradient-free optimization methods (when estimating the gradient by finite differences) in infinite dimensional spaces for inverse problems (see, e.g. \cite{Kabanikhin}).
However, this can lead to an accumulation of corresponding inexactness in the theoretical estimates of the rate of convergence. Let us consider minimization problems of convex and $L$-smooth function $f$ ($\|\cdot\|$ is a usual Euclidean norm)
\begin{equation}\label{Lsmothness}
\|\nabla f(x) - \nabla f(y)\| \leqslant L \|x - y\| \quad \forall x, y \in \mathbb{R}^n
\end{equation}
with an inexact gradient $g: \mathbb{R}^n \rightarrow \mathbb{R}^n$:
\begin{equation}\label{InexactGrad}
\|g(x) - \nabla f(x)\| \leqslant \delta,
\end{equation}
where $L>0$ and $\delta > 0$. For the considered class of problem, the following estimate for accelerated gradient-type methods:
$$
f(x_N) - \min_{x \in \mathbb{R}^n} f(x) = O\left(L\|x_0 - x^*\|^2N^{-2} + \delta\max_{k \leq N }{\|x_k - x^*\|}\right)
$$
is known \cite{Aspremont2008,Vasin2021} for each $x^*:$ $f(x^*) = \min_{x \in \mathbb{R}^n} f(x)$. It is clear that the quantity $\max_{k \leq N }{\|x_k - x^*\|}$ can be not small enough. In this paper, we propose one modification of the Sequential Subspace Optimization Method \cite{SESOP_2005} with an $\delta$-additive noise in gradient \eqref{InexactGrad} and prove
the following estimate:
$$f(x_N) - \min_{x \in \mathbb{R}^n} f(x) = O\left(L\|x_0 - x^*\|^2N^{-2} + \delta \|x_0 - x^*\|\right),$$
where $x_0$ is the starting point of algorithm. Thus, a certain solution to the problem of accumulating the gradient inexactness is proposed for a special accelerated gradient method. It is important that we also consider some type of non-convex problems \cite{Hardt,NearOpt}.
The article consists of an introduction, 3 main sections and conclusion.
In the first main section \ref{sect1}, we propose and analyze a new modification of Subspace Optimization Method (Algorithm \ref{alg:sesop}) for minimization problems with $\gamma$-quasar-convex functions with an inexact gradient. The use of such a specific method made it possible to obtain a significant result on the non-accumulation of the additive gradient inexactness in the estimate of the convergence rate (Theorem \ref{SESOP_theorem}).
However, this result for Algorithm \ref{alg:sesop} is essentially tied to the structure of this method, which is associated with auxiliary low-dimensional minimization problems. Therefore, it is important to investigate the influence of errors in solving such problems on the final estimate of the rate of convergence. Section \ref{sect2} is devoted to this question and Theorem \ref{SESOP_theorem_full} is obtained.
The last main section \ref{sect3} is devoted to numerical illustration of the obtained theoretical results for one example of a quadratic function minimization problem. Firstly, we show that the convergence may be significantly better than the theoretical estimates for Algorithm \ref{alg:sesop}. Secondly, we compare Algorithm \ref{alg:sesop} with another known accelerated Similar Triangles Method (STM) for the case of additive gradient noise \cite{Vasin2021}. The STM was chosen for comparison with Algorithm \ref{alg:sesop} for the following reasons:
\begin{itemize}
\item in the case of exact gradient information ($\delta = 0$ in \eqref{InexactGrad}) both the STM and the SESOP possess the optimal rate of convergence $O(N^{-2})$;
\item for STM with an inexact gradient \eqref{InexactGrad}, a theoretical estimate of the quality of the solution is known (\cite{Vasin2021}, Theorem 1 and Remark 3).
\end{itemize}
Let us introduce some auxiliary notations and definitions.
Throughout this paper $\langle \mathbf{x}, \mathbf{y} \rangle$ $\|\cdot\|$ means the inner product of vectors $\mathbf{x} = (x_1, x_2, ..., x_n), \, \mathbf{y} = (y_1, y_2, ..., y_n) \in\mathbb{R}^n$ and is given by the formula
$\langle \mathbf{x}, \mathbf{y} \rangle = \sum\limits_{k=1}^n x_ky_k$.
It turns out that it is possible to formulate the main results of the work for a certain class of not necessarily convex problems. Let us recall the definition of the class of $\gamma$-quasar-convex functions (see \cite{Hardt,NearOpt}).
\begin{definition}
Assume that $\gamma \in (0,1]$ and let $\mathbf{x}^*$ be a minimizer of the differentiable function $f:\mathbb{R}^n \rightarrow \mathbb{R}$. The function $f$ is $\gamma$-quasar-convex with respect to $\mathbf{x}^*$ if for all $\mathbf{x}\in\mathbb{R}^n$,
\begin{equation}
f(\mathbf{x}^*)\geq f(\mathbf{x}) + \frac{1}{\gamma}\langle\nabla f(\mathbf{x}), \mathbf{x}^*-\mathbf{x}\rangle.
\end{equation}
\end{definition}
For example, a non-convex function $f(x) = |x| (1 - e^{-|x|})$ is a $1$-quasar-convex \cite{AccWQC,NearOpt}. The class of $\gamma$-quasar-convex functions is also called $\gamma$-weakly quasi-convex functions (see \cite{AccWQC}). Clearly each convex function is also $1$-quasar-convex. So, all results of this paper are applicable to convex optimization problems with an inexact gradient information.
\section{Subspace Optimization Method with Inexact Gradient}\label{sect1}
In this section we present some variant of the SESOP (Sequential Subspace Optimization) method \cite{SESOP_2005} for $\gamma$-quasar-convex functions with an inexact gradient. We generalize the results \cite{AccWQC} for SESOP method with inexact gradient on the class of $\gamma$-quasar-convex functions. In other words, our modifications of the SESOP method works with some approximation $g(x)$ of gradient $\nabla f(x)$ at each point $x \in \mathbb{R}^n$.
Similarly to \cite{AccWQC,SESOP_2005} we start with a description of the investigated algorithm. Let $D_k=\|\mathbf{d}_k^0\,\mathbf{d}_k^1\,\mathbf{d}_k^2\|$ be an $n\times 3$ matrix, the columns of which are the following vector:
$$\mathbf{d}_k^0 = g(\mathbf{x}_k), \quad \mathbf{d}_k^1 = \mathbf{x}_k-\mathbf{x}_0,\quad\mathbf{d}_k^2 = \sum\limits_{i=0}^k \omega_i g(\mathbf{x}_i),$$
where $\omega_0=1, \omega_i = \frac{1}{2}+\sqrt{\frac{1}{4}+\omega_{i-1}^2}.$
For all $k\geq 1$ we have $\omega_k=\frac{1}{2}+\sqrt{\frac{1}{4}+\omega_{k-1}^2} \geq \frac{1}{2} + \omega_{k-1}$
and $\omega_k=\frac{1}{2}+\sqrt{\frac{1}{4}+\omega_{k-1}^2} \leq 1 + \omega_{k-1}.$
So, it holds that
\begin{equation}
\label{w_k_est}
k+1\geq \omega_k\geq \frac{k+1}{2}.
\end{equation}
The matrices $D_k$ will generate the subspaces over which we will minimize our objective function. With $D_k$ defined this way, the proposed algorithm takes the following form:
\begin{algorithm}
\caption{A modification of the SESOP method with an inexact gradient}
\label{alg:sesop}
\begin{algorithmic}[1]
\REQUIRE objective function $f$ with an inexact gradient $g$, initial point $\mathbf{x}_0$, number of iterations $T$.
\FOR{$k =0, \hdots, T-1$}
\STATE Find the optimal step
\begin{equation}\label{eqsubproblem}
\tau_k \leftarrow \arg\min_{\tau\in\mathbb{R}^3}f\left(\mathbf{x}_k + D_k\tau\right)
\end{equation}
\STATE $\mathbf{x}_{k+1} \leftarrow \mathbf{x}_k + D_k\tau_k$
\ENDFOR
\RETURN $\mathbf{x}_T$
\end{algorithmic}
\end{algorithm}
Let us show that the main advantage of the SESOP method with an inexact gradient is the absence in theoretical estimate the term $\max_k \|x_k - x^*\|$. Before the proof of this result, we need to estimate the error accumulation in $\mathbf{d}_k^3$ vector.
\begin{lemma}
\label{a_lemma_w}
Let the objective function $f$ be $L$-smooth and $\gamma$-quasar-convex with respect to $\mathbf{x}^*$. Suppose also for the inexact gradient $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ of function $f$ there is some constant $\delta_1\geq 0$ such that for all $\mathbf{x}\in\mathbb{R}^n$
\begin{equation}
\|g(\mathbf{x})-\nabla f(\mathbf{x})\|\leq\delta_1.
\end{equation}
Let $\{\mathbf{x}_j\}_j$ be a sequence of points generated by Algorithm \ref{alg:sesop}. Then the following inequality holds:
\begin{equation}
\left\|\sum\limits_{k=0}^{T} \omega_k g(\mathbf{x}_k)\right\|^2 \leq 2\sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2 + 72 T^4 \delta_1^2
\end{equation}
for each $T\in\mathbb{N}$.
\end{lemma}
\begin{proof}
Let us define $W_{T} = \left\|\sum\limits_{k=0}^{T} \omega_k g(\mathbf{x}_k)\right\|$. Note that $\mathbf{d}^2_j=\sum\limits_{k=0}^{j} \omega_k g(\mathbf{x}_k)$ and $\|d_j^2\| \leq W_j$ for each $j$. For $W_{T}$ we have equality for this value:
\begin{equation}
\label{W_T_2_expression}
W_T^2 = \omega_T^2 \|g(\mathbf{x}_T)\|^2 + 2\left\langle w_T g(\mathbf{x}_T), \mathbf{d}^2_{T-1}\right\rangle + W_{T-1}^2
\end{equation}
and we have that $$\nabla f(\mathbf{x}_T) \perp \mathbf{d}^2_{T-1},$$ because of optimizing on the subspace $\mathbf{x}_{T-1}~+~ \mathbf{d}^2_{j-1}$. Therefore we have the following inequality $\left(\left\langle g(\mathbf{x}_j), \ \mathbf{d}^2_{j-1}\right\rangle = 0\right)$:
\begin{equation}
\label{delta_ineq}
|\left\langle g(\mathbf{x}_j), \ \mathbf{d}^2_{j-1}\right\rangle| = |\left\langle
\nabla f(\mathbf{x}_j), \mathbf{d}^2_{j-1}\right\rangle + \left\langle g(\mathbf{x}_j) - \nabla f(\mathbf{x}_j), \mathbf{d}^2_{j-1}\right\rangle| \leq \delta_1 W_{j-1}
\end{equation}
for all $j\geq 1.$ So, we have the following correlations for $W_T$:
$$W_T^2 = \sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2 + 2\sum\limits_{j=1}^T w_j\left\langle g(\mathbf{x}_j), \mathbf{d}^2_{j-1}\right\rangle,$$
and
\begin{equation}
\label{up_W_est}
W_T^2 \leq \sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2 +2\delta_1 \sum\limits_{j=1}^T w_j W_{j-1}.
\end{equation}
On the other hand, we can estimate the inner product by the Cauchy–Schwarz inequality for $T=j$ in \eqref{W_T_2_expression} and to get the following estimate:
$$ W_j^2 \geq - 2w_j \delta_1 W_{j-1} + W_{j-1}^2.$$
Solving the previous inequality on $W_j$, we have ($\sqrt{a + b} \leq \sqrt{a} + \sqrt{b}$ for all $a, b \geq 0$)
$$W_{j-1} \leq w_j\delta_1 + \sqrt{w_j^2 \delta_1^2 + W_j^2} \leq 2w_j\delta_1 + W_j.$$
By induction we have the following estimate:
$$W_j \leq 2\delta_1 \sum\limits_{k=j+1}^T w_k + W_T\text{ for } j=\overline{0, T-1}.$$
From \eqref{w_k_j_T} we have that
$$W_j \leq \delta_1 T(T+3) + W_T\text{ for } j=\overline{0, T-1}.$$
The last inequality and \eqref{up_W_est} mean that
\begin{equation}
W_T^2 \leq \sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2 +2\delta_1^2 T (T+3) + 2\delta_1W_T \sum\limits_{j=1}^T w_j,
\end{equation}
\begin{equation}
W_T^2 \leq \sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2 +2\delta_1^2 T (T+3) + T(T + 3) \delta_1 W_T.
\end{equation}
The value $W_T \geq 0$ by definition. One of the roots of the previous quadratic inequality is always negative. So, for $W_T$ to meet this inequality, its value must be not more than the largest root of the corresponding quadratic function:
$$
W_T \leq \frac{1}{2}(T^2 + 3T)\delta_1 + \frac{1}{2}\sqrt{(T^2 + 3T)^2 \delta_1^2 + 4\sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j\|^2 +8\delta_1^2 T (T+3)}.$$
Taking into account $T\geq 1$, we have
$$W_T \leq 2T^2\delta_1 + \frac{1}{2}\sqrt{48 T^4 \delta_1^2 + 4\sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2};$$
Due to the inequality $\sqrt{a+b}\leq\sqrt{a}+\sqrt{b}$ (for all $a, b \geq 0$)
we have
$$W_T \leq 6 T^2\delta_1 + \sqrt{\sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2}.$$
Further, the inequality $(a+b)^2\leq 2a^2+2b^2$ (for all $a, b \in \mathbb{R}$) means that
$$W_T^2 \leq 72 T^4 \delta_1^2 + 2\sum\limits_{j=0}^T \omega_j^2 \|g(\mathbf{x}_j)\|^2.$$
\end{proof}
Using Lemma \ref{a_lemma_w}, we can prove the following main result of this section.
\begin{theorem}
\label{SESOP_theorem}
Let the objective function $f$ be $L$-smooth and $\gamma$-quasar-convex with respect to $\mathbf{x}^*$. Also, for the inexact gradient $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ there is some constant $\delta_1\geq 0$ such that for all $\mathbf{x}\in\mathbb{R}^n$
\begin{equation}
\label{g_cond_i}
\|g(\mathbf{x})-\nabla f(\mathbf{x})\|\leq\delta_1.
\end{equation}
Then the sequence $\{\mathbf{x}_k\}$ generated by Algorithm \ref{alg:sesop} satisfies
\begin{equation}
\label{est_SESOP}
f(\mathbf{x}_k)-f^*\leq \frac{8LR^2}{\gamma^2 k^2} + 4\left(\frac{R}{\gamma} + 17 \right)\delta_1,
\end{equation}
where $R=\|\mathbf{x}^*-\mathbf{x}_0\|$.
\end{theorem}
\begin{proof}
By constructing $\mathbf{x}_{k+1}$ we have the following inequality:
\begin{equation}
\label{k_min_ineq}
f(\mathbf{x}_{k+1}) = \min_{\mathbf{s}\in\mathbb{R}^3} f\left(\mathbf{x}_k + \sum\limits_{i=0}^2 s_i d_k^i\right)\leq f\left(\mathbf{x}_k + s_0 g(\mathbf{x}_k)\right).
\end{equation}
On the other hand, from $L$-smoothness we have that
$$f(\mathbf{y})\leq f(\mathbf{x}) + \langle\nabla f(\mathbf{x}), \mathbf{y} - \mathbf{x}\rangle + \frac{L}{2}\|\mathbf{x}-\mathbf{y}\|^2$$
for all $\mathbf{x}, \mathbf{y}\in\mathbb{R}^n$. Further, from $\eqref{g_cond_i}$ follows the corresponding inequality for an inexact gradient:
$$f(\mathbf{y})\leq f(\mathbf{x}) + \langle g(\mathbf{x}), \mathbf{y} - \mathbf{x}\rangle + \frac{L}{2}\|\mathbf{x}-\mathbf{y}\|^2 + \delta_1 \|\mathbf{x}-\mathbf{y}\| \quad \forall \mathbf{x}, \mathbf{y}\in\mathbb{R}^n.$$
Using the inequality $\delta_1 \|\mathbf{x}-\mathbf{y}\| = \left(\sqrt{\frac{1}{L}}\delta_1\right) \left(\sqrt{\frac{L}{1}}\|\mathbf{x}-\mathbf{y}\|\right)\leq \frac{\delta_1^2}{2L} + \frac{L}{2}\|\mathbf{x}-\mathbf{y}\|^2$, we have
\begin{equation}
\label{g_L_ineq}
f(\mathbf{y})\leq f(\mathbf{x}) + \langle g(\mathbf{x}), \mathbf{y} - \mathbf{x}\rangle + L\|\mathbf{x}-\mathbf{y}\|^2 + \frac{\delta_1^2}{2L} \quad \forall \mathbf{x}, \mathbf{y}\in\mathbb{R}^n.
\end{equation}
On the base of the last inequality and the right part of \eqref{k_min_ineq} for $\mathbf{y}:=\mathbf{x}_k + s_0 g(\mathbf{x}_k)$ and $\mathbf{x}=\mathbf{x}_k$ we can conclude that
\begin{equation}
f(\mathbf{x}_{k+1}) \leq f(\mathbf{x}_k) +\left(s_0 + s_0^2 L\right)\|g(\mathbf{x}_k)\|^2 + \frac{1}{2L}\delta_1^2
\end{equation}
for each $s_0\in\mathbb{R}$. Further,
\begin{equation}\label{eqineqs0}
-\left(s_0 + s_0^2 L\right) \|g(\mathbf{x}_k)\|^2 \leq f(\mathbf{x}_k) - f(\mathbf{x}_{k+1}) + \frac{1}{2L}\delta_1^2.
\end{equation}
Maximizing the left part of \eqref{eqineqs0} by $s_0$, we have the following estimate for the inexact gradient norm:
\begin{equation}
\|g(\mathbf{x}_k)\|^2 \leq 4L(f(\mathbf{x}_k) - f(\mathbf{x}_{k+1})) + 2\delta_1^2.
\end{equation}
So, $\nabla f(\mathbf{x}_k)\perp \mathbf{x}_k - \mathbf{x}_0$ for all $k>0$ because $\mathbf{x}_k$ is minimizer of $f$ on the subspace containing the directions $\mathbf{x}_k-\mathbf{x}_{k-1}$ and $\mathbf{x}_{k-1}-\mathbf{x}_0$. Because of it we can write
$$\langle \nabla f(\mathbf{x}_k), \mathbf{x}_k-\mathbf{x}^*\rangle = \langle \nabla f(\mathbf{x}_k), \mathbf{x}_0-\mathbf{x}^* \rangle.$$
From this equality and \eqref{g_cond_i} we have
$$f(\mathbf{x}_k)-f(\mathbf{x}^*)\leq \frac{1}{\gamma} \langle g(\mathbf{x}_k), \mathbf{x}_0-\mathbf{x}^* \rangle + \delta_1 \frac{R}{\gamma}.$$
Similarly to the proof of Theorem 3.1 from \cite{AccWQC} we have the following chain of correlations:
\begin{equation}
\label{eps_1}
\begin{aligned}
\sum\limits_{k=0}^{T-1}\omega_k(f(\mathbf{x}_k)-f^*)&\leq \frac{1}{\gamma} \left\langle \sum\limits_{k=0}^{T-1} \omega_k g(\mathbf{x}_k), \mathbf{x}_0-\mathbf{x}^* \right\rangle +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k\\
&\leq \frac{1}{\gamma} \left\|\sum\limits_{k=0}^{T-1} \omega_k g(\mathbf{x}_k)\right\|R +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k.
\end{aligned}
\end{equation}
Now we can estimate the multiplier $\left\|\sum\limits_{k=0}^{T-1} \omega_k g(\mathbf{x}_k)\right\|$. According to Lemma \ref{a_lemma_w} and \eqref{w_k_est}, we have the following estimates:
\begin{align*}
\left\|\sum\limits_{k=0}^{T-1} \omega_k g(\mathbf{x}_k)\right\|^2 &\leq 2\sum\limits_{j=0}^{T-1} \omega_j^2 \|g(\mathbf{x}_j)\|^2 + 256 T^4 \delta_1^2 \\
& \leq 8L \sum\limits_{k=0}^{T-1} \omega_k^2 (f(\mathbf{x}_k)-f(\mathbf{x}_{k+1})) + 260 T^4 \delta_1^2.
\end{align*}
Note that the choice of $\omega_k$ is equivalent to choosing the largest $\omega_k$ satisfying
$$\omega_k = \begin{cases}1, \text{if $k=0$},\\ \omega_k^2-\omega_{k-1}^2, \text{otherwise}.\end{cases}$$
Now we can estimate the left part of \eqref{eps_1} denoting $\varepsilon_k=f(\mathbf{x}_k)-f^*$ in the following way ($\sqrt{a+b} \leq \sqrt{a} + \sqrt{b}$ for all $a, b \geq 0$):
\begin{equation}
\begin{aligned}
S &= \sum\limits_{k=0}^{T-1}\omega_k \varepsilon_k \\
&\leq \frac{1}{\gamma} \left\|\sum\limits_{k=0}^{T-1} \omega_k g(\mathbf{x}_k)\right\|R +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k\\
&\leq\left(\frac{8LR^2}{\gamma^2} \sum\limits_{k=0}^{T-1} \omega_k^2 (\varepsilon_k-\varepsilon_{k+1}) + 260 T^4\delta_1^2\right)^{\frac{1}{2}} +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k\\
&\leq\left(\frac{8LR^2}{\gamma^2} \sum\limits_{k=0}^{T-1} \omega_k^2 (\varepsilon_k-\varepsilon_{k+1})\right)^{\frac{1}{2}} + 17 T^2 \delta_1 +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k\\
&=\sqrt{\frac{8LR^2}{\gamma^2}} \sqrt{S - \varepsilon_T\omega_{T-1}^2} + 17 T^2 \delta_1 +\delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k.
\end{aligned}
\end{equation}
From the inequality above we have
\begin{equation}\label{eqat26}
\omega_{T-1}^2 \varepsilon_T \leq S - \frac{\gamma^2}{8LR^2}\left(S - \delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k - 17 T^2 \delta_1 \right)^2.
\end{equation}
Maximizing the right part of \eqref{eqat26} on $S$ we get
\begin{equation}
\omega_{T-1}^2 \varepsilon_T \leq \frac{2LR^2}{\gamma^2} + \delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k + 17 T^2 \delta_1.
\end{equation}
Now from \eqref{w_k_est} we have
\begin{equation}
\begin{aligned}
\omega_{T-1}^2 \varepsilon_T &\leq \frac{2LR^2}{\gamma^2} + \delta_1 \frac{R}{\gamma} \sum\limits_{k=0}^{T-1} \omega_k + 17 T^2 \delta_1\\
&\leq\frac{2LR^2}{\gamma^2} + T^2\left( \delta_1 \frac{R}{\gamma} +17 \delta_1\right).
\end{aligned}
\end{equation}
Dividing both parts of this inequality by $w_{T-1}^2$ and using the lower estimate for it \eqref{w_k_est} we get
\begin{equation}
\begin{aligned}
\varepsilon_T \leq \frac{8LR^2}{\gamma^2 T^2} + 4\delta_1 \frac{R}{\gamma} + 68\delta_1 =
\frac{8LR^2}{\gamma^2 T^2} + 4\left(\frac{R}{\gamma} + 17 \right)\delta_1,
\end{aligned}
\end{equation}
Q.E.D.
\end{proof}
\section{Subspace Optimization Method with Inexact Solutions of Auxiliary Subproblems}\label{sect2}
The result of the previous section shows that the SESOP algorithm can work with additive noise in a gradient. It is essential that the method leads to the need to solve auxiliary low-dimensional optimization problems.
So, there is an interesting case when the auxiliary problem \eqref{eqsubproblem} cannot be solved exactly. We consider this case in the following theorem.
\begin{theorem}
\label{SESOP_theorem_full}
Let the objective function $f$ be $L$-smooth and $\gamma$-quasar-convex with respect to $\mathbf{x}^*$. Let $\tau_k$ be the step value obtained with the inexact solution of the auxiliary problem \eqref{eqsubproblem} on step 2 in Algorithm \ref{alg:sesop} on the $k$-th iteration. Namely, the following conditions for inexactness hold:
\begin{itemize}
\item[(i)] For the inexact gradient $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$ there is some constant $\delta_1\geq 0$ such that for all points $\mathbf{x}\in\mathbb{R}^n$ condition \eqref{g_cond_i} holds.
\item[(ii)] The inexact solution $\tau_k$ meets the following condition:
\begin{equation}
\label{grad_orth_i}
|\left\langle \nabla f(\mathbf{x}_k), \mathbf{d}_{k-1}^2 \right\rangle |\leq k^2 \delta_2
\end{equation}
for some constant $\delta_2 \geq 0$ and each $k\in\mathbb{N}$. Note that $\mathbf{x}_k=\mathbf{x}_{k-1}+D_{k-1}\tau_{k-1}$.
\item[(iii)] The inexact solution $\tau_k$ meets the following condition for some constant $\delta_3 \geq 0$:
\begin{equation}
\label{grad_orth_ii}
|\left\langle \nabla f(\mathbf{x}_k), \mathbf{x}_k - \mathbf{x}_0\right\rangle| \leq \delta_3.
\end{equation}
\item[(iv)] The problem from step 2 in Algorithm \ref{alg:sesop} is solved with accuracy $\delta_4 \geq 0$ on function on each iteration, i.e.
$f(\mathbf{x}_k) - \min_{\tau\in\mathbb{R}^n}f(\mathbf{x}_{k-1} + D_{k-1}\tau) \leq \delta_4$.
\end{itemize}
Then the sequence $\{\mathbf{x}_k\}$ generated by Algorithm \ref{alg:sesop} satisfies
\begin{equation}
f(\mathbf{x}_k)-f^* \leq \frac{8LR^2}{\gamma^2 k^2} + \left(\frac{R}{\gamma}+ 10\right)\delta_1 + 4\sqrt{\delta_2} + \delta_3 + 5\sqrt{\frac{L\delta_4}{k}}
\end{equation}
for each $k\geq 8$, where $R=\|\mathbf{x}^*-\mathbf{x}_0\|$.
\end{theorem}
\begin{proof}
The proof of this theorem is somewhat similar to the proof of Theorem \ref{SESOP_theorem} and was moved to the appendix C.
\end{proof}
\begin{remark}
The obtained estimate of the rate of convergence for Algorithm \ref{alg:sesop} does not depend on the value $\max_k\|\mathbf{x}_k-\mathbf{x}^*\|$ and it depends only on $R, L$ and $\gamma > 0$.
\end{remark}
\begin{remark}
It is clear that when the auxiliary problem \eqref{eqsubproblem} in Algorithm \ref{alg:sesop} has exact solution then $\delta_2=\delta_3=\delta_4=0$. The constant before $\delta_1$ was improved in comparison with the result from Theorem \ref{SESOP_theorem} because of more accurate work with constants in proofs (see Lemmas \ref{a_lemma_w} and \ref{a_lemma_w_full}).
\end{remark}
According to Theorem \ref{SESOP_theorem_full}, the SESOP method for a $\gamma$-convex function can find solution with quality $\varepsilon$ by function after $N=\sqrt{\frac{16LR^2}{\gamma^2 \varepsilon}}$ iterations when the following condition holds:
$$\left(\frac{R}{\gamma}+ 10\right)\delta_1 + 4\sqrt{\delta_2} + \delta_3 + 5\sqrt{\frac{L\delta_4}{k}}\leq\frac{\varepsilon}{2}.$$
In particular, the SESOP method finds solution with this quality after $N=\sqrt{\frac{16LR^2}{\varepsilon}}$ iterations for convex functions.
Now we want to discuss the relationship between conditions $(ii)$, $(iii)$ and $(iv)$ from Theorem \ref{SESOP_theorem_full}. The condition on the accuracy of the subproblem $(iv)$ is natural enough for such methods. Conditions $(ii)$ and $(iii)$ are caused by the form of the method and provide almost orthogonality of the gradient and vectors $\mathbf{d}_k^j, j=1,2$. We can prove the following simple result.
\begin{theorem}
\label{delta_link}
If condition $(iv)$ from Theorem \ref{SESOP_theorem_full} holds, then we can choose $\delta_2, \delta_3 \geq 0$ according to the following estimates:
$$\delta_3 \leq \sqrt{2L \delta_4}\left(\sqrt{\max_{k}(\|D_k\| \|\tau_k\|)} + \sqrt{\|\max_{k}\mathbf{d}^1_{k-1}\|}\right)$$
and
$$\delta_2 \leq \frac{1}{k^2}\sqrt{2L \max_{k}\|\mathbf{d}_k^3\|\delta_4}.$$
\end{theorem}
\begin{proof}
Now we want to express conditions \eqref{grad_orth_i} and \eqref{grad_orth_ii} through accuracy of subproblem solution \eqref{eqsubproblem} $\delta_4$. We need to introduce the following auxiliary function:
\begin{equation}
\label{f_k}
f_k(\tau) = f(\mathbf{x}_k + D_k \tau).
\end{equation}
Note that $f_k: \mathbb{R} \rightarrow \mathbb{R}$ and its gradient is a one-dimension derivative. Let function $f_k$ have a Lipschitz continuous gradient with constants $L_k^j$, $j={1,3}$. We can derive these constants from $L$ and the norms of directions $d_k^j$:
$$\left|\frac{d}{d\tau_j} f_k(\tau+\alpha e_j)- \frac{d}{d\tau_j} f_k(\tau)\right| = \left|\left\langle d_k^j, \nabla f(\mathbf{x}^1) - \nabla f(\mathbf{x}^2)\right\rangle\right| \leq L \|d_k^j\| |\alpha|.$$
where $e_j\in\mathbb{R}^3$ is the $j$-th vector in the standard basis, $\alpha\in\mathbb{R}$ is some constant, $\mathbf{x}^1=\mathbf{x}_k + D_k \tau$ and $\mathbf{x}^2=\mathbf{x}_k + D_k (\tau+\alpha e_j)$. So we have the following expression for Lipschitz constant of a gradient for $f_k$ with respect to the $j$-th component:
\begin{equation}
\label{L_k_j}
L_k^j = L\|d_k^j\|.
\end{equation}
It is easy to see that
$$\left|\frac{d}{d\tau_j} f_k(\tau)\right|^2 \leq 2L_k^j \left(f_k(\tau) - \min_{\tau_j} f_k(\tau)\right)\leq 2L_k^j \left(f_k(\tau) - \min_{\tau} f_k(\tau)\right)=2L_k^j \delta_4$$
for all $\tau_j$. From \eqref{L_k_j}, the inequality above and the definition of $f_k$ \eqref{f_k}, we have the following expression:
\begin{equation}
\label{g_orth_delta_4}
|\langle\nabla f(\mathbf{x}_{k+1}), d_k^j\rangle|\leq \sqrt{2L \|\mathbf{d}_k^j\| \delta_4}.
\end{equation}
It means if we choose $\delta_2>0$ in the following way:
$$\delta_2 \leq \frac{1}{k^2}\sqrt{2L \max_{k}\|\mathbf{d}_k^3\|\delta_4}.$$
then condition $(ii)$ in Theorem \ref{SESOP_theorem_full} meets.
In a similar way we can obtain that $f_k$ has Lipschitz continuous gradient with constant $L_k$:
$$L_k = \|D_k\| L$$
and
$$\left|\nabla_\tau f_k(\tau)\right|^2 \leq 2L_k \left(f_k(\tau) - \min_{\tau} f_k(\tau)\right)=2L_k \delta_4.$$
Note that
$$\mathbf{x}_{k}-\mathbf{x}_0 = D_{k-1} \tau_{k-1} + \mathbf{d}^1_{k-1} $$
Finally, we can choose $\delta_3$ in the following way:
$$\delta_3 \leq \sqrt{2L \delta_4}\left(\sqrt{\max_{k}(\|D_k\| \|\tau_k\|)} + \sqrt{\|\max_{k}\mathbf{d}^1_{k-1}\|}\right).$$
\end{proof}
\section{Numerical Experiments}\label{sect3}
In the current section we provide the results of numerical experiments. All experiments were carried out on Python 3.7.3 on computer Acer Swift 5 SF514-55TA-56B6 with processor Intel(R) Core(TM) i5-8250U @ CPU 1.60GHz, 1800 MHz.
All experiments were carried out in the assumption that we can solve the subspace optimization problem at each iteration with some accuracy on function. For this, we used the quadratic test function
$$f(\mathbf{x})= \mathbf{x}^\top A \mathbf{x} + 2\mathbf{b}^\top x$$
with $A\in \mathbb{S}^n_+$ ($A$ is a symmetric positive semidefinite matrix), $\mathbf{b}\in\mathbf{R}^n$. Obviously, this function is convex and consequently $1$-quasar-convex. The components of parameter $\mathbf{b}$ were generated randomly i.i.d. from uniform distribituion $\mathcal{U}([-1,1])$. The matrix $A=B^\top B$ where components $B\in\mathbb{R}^{n\times n}$ were generated by the same way as for vector $\mathbf{b}$.
The shift $\tau_k$ can be found as a solution of convex quadratic optimization problem:
$$\min_{\tau\in\mathbf{R}^3} \tau^\top D_k^\top A D_k\tau - 2 \left( \mathbf{b} + A \mathbf{x}_k\right)^\top D_k \tau $$
with any accuracy that we will vary in our experiments (see details below). The Lipschitz constant $L$ of $\nabla f$ is also known and equals the maximal singular value of matrix $A$. For all experiments we take dimension $n=500$.
\begin{figure}[ht!]
\vspace{-4ex} \centering \subfigure{
\includegraphics[width=0.45\linewidth]{images/grad_exp_f_N.png} \label{fig:grad_exp_f_N} }
\hspace{4ex}
\subfigure{
\includegraphics[width=0.45\linewidth]{images/grad_exp_f_delta.png} \label{fig:grad_exp_f_delta} }
\caption{The dependencies of convergence on gradient inexactness in the case of an exact solution of the subspace optimization problem: \subref{fig:grad_exp_f_N} convergence for different $\delta_1$; \subref{fig:grad_exp_f_delta} minimal values found for different $\delta_1$.} \label{fig:res_1}
\end{figure}
The first experiment compares the theoretical estimation from Theorem \ref{SESOP_theorem} and the real experiment in the case of inexactness in the gradient only. It means that we solve quadratic optimization problem with machine accuracy that is significantly less than inexactness in the gradient. The inexact gradient will be given as an usual gradient with some noise
$g(\mathbf{x})= \nabla f(\mathbf{x}) + \delta \xi(\mathbf{x}),$
where $\xi(\mathbf{x})\sim\mathcal{U}\left(S_1(0)\right)$ is a random vector from the unit sphere with uniform distribution. Obviously such a vector meets the conditions of Theorem \ref{SESOP_theorem}. The results of this experiment are presented in Figure \ref{fig:res_1}.
We can see in Figure \ref{fig:grad_exp_f_delta} that the convergence of the proposed variant of the SESOP method (Algorithm \ref{alg:sesop}) at the first 100000 iterations is better than the theoretical convergence (the line $\frac{LR^2}{k^2}$ on graph) without noise for any gradient inexactness for $\delta\in\left[10^{-4}, 10\right]$. Moreover, in Figure \ref{fig:grad_exp_f_delta} the dependence of the function residual on the gradient inexactness shows that there is no significant error accumulation for $\delta<1$ at the first 100000 iterations. Such an optimistic result was obtained by Algorithm \ref{alg:sesop} due to the exact solution of the low-dimensional optimization subproblems \eqref{eqsubproblem}.
\begin{figure}[ht!]
\vspace{-4ex} \centering \subfigure{
\includegraphics[width=0.45\linewidth]{images/argmin_exp_f_N.png} \label{fig:argmin_exp_f_N} }
\hspace{4ex}
\subfigure{
\includegraphics[width=0.45\linewidth]{images/argmin_exp_f_delta.png} \label{fig:argmin_exp_f_delta} }
\caption{The dependencies of convergence on the inexactness of the subspace optimization problem in Algorithm \ref{alg:sesop}: \subref{fig:argmin_exp_f_N} convergence for different $\delta_4$; \subref{fig:argmin_exp_f_delta} minimal values found for different $\delta_4$.} \label{fig:res_2}
\end{figure}
In the second experiment we studied the practical convergence rate for different inexactness $\delta_j,j=\overline{2,4}$ when $\delta_1$ is fixed. In this experiments we take $\delta_1=10^{-3}$. Even in the ideal case, we cannot estimate the dependence of convergence on these parameters independently because when inexactness on the function of the subspace optimization problem \eqref{eqsubproblem} solution is small enough $\delta_4\rightarrow 0$, then other inexactness also tends to zero. We varied the inexactness of subspace optimization solution $\delta_4$. The results of the second experiment are shown in Figure \ref{fig:res_2}.
In this case in Figure \ref{fig:argmin_exp_f_N} we can see that the convergence is significantly better than the theoretical estimation only for accuracy values $\delta=10^{-7},10^{-6},10^{-5}$. For values $10^{-2},10^{-3}$ there is no improvement after 20000 iterations and the theoretical estimation obtains better convergence. For value $10^{-4}$ the convergence stopped after 20000 iterations too but the theoretical convergence is not better due to a small number of iterations. In the figure \ref{fig:argmin_exp_f_delta} we can see that approached function value degrades with the linear rate depending on $\delta\geq10^{-6}$, which corresponds to the results of Theorem \ref{SESOP_theorem_full}. So, the proposed modification of the SESOP method is more sensitive to the accuracy of subproblem solution \eqref{eqsubproblem} than to the inexactness of the gradient.
Finally, we want to compare Algorithm \ref{alg:sesop} with an inexact gradient with another method that can work with gradient inexactness. We choice the known Similar Triangles Method (STM) with gradient inexactness from \cite{Vasin2021}. Similar to the previous experiment, we will consider two cases: the case of inexactness only in the gradient and the case of fixed additive gradient inexactness when subspace optimization is being solved inexactly too.
\begin{figure}[ht!]
\vspace{-4ex} \centering \subfigure{
\includegraphics[width=0.45\linewidth]{images/grad_SESOP_STM_0.1.png} \label{fig:grad_SESOP_STM_1} }
\hspace{4ex}
\subfigure{
\includegraphics[width=0.45\linewidth]{images/grad_SESOP_STM_0.001.png} \label{fig:grad_SESOP_STM_2} }
\vspace{4ex}
\subfigure{
\includegraphics[width=0.45\linewidth]{images/grad_SESOP_STM_1e-05.png} \label{fig:grad_SESOP_STM_3} }
\caption{The convergence of SESOP and STM for different additive noise with an exact solution of the subspace optimization problem: \subref{fig:grad_SESOP_STM_1} $\delta_1=0.001$; \subref{fig:grad_SESOP_STM_2} $\delta_1=0.00001$; \subref{fig:grad_SESOP_STM_3} $\delta_1=0.1$.} \label{fig:res_3}
\end{figure}
The results for the first case for different values $\delta_1$ are presented in Figure \ref{fig:res_3}. We can see that because of the exact solution of the subspace optimization problem Algorithm \ref{alg:sesop} is almost everywhere better than the STM \cite{Vasin2021} with inexact gradient.
\begin{figure}[H]
\centering
\includegraphics[width=0.90\linewidth]{images/argmin_SESOP_STM.png}
\caption{The convergence of Algorithm \ref{alg:sesop} with an inexact solution of the subspace optimization problem and STM for additive noise $\delta_1=10^{-3}$.}
\label{fig:res_4}
\end{figure}
The results for the second case for different accuracy of the subspace problem solution are presented in Figure \ref{fig:res_4}. There is a natural result that for enough exact solution at each iteration, Algorithm \ref{alg:sesop} stays better than the STM. Nevertheless, for the inexactness in the low-dimensional subproblems solution larger or equal to $10^{-4}$ the STM becomes better than the provided method (Algorithm \ref{alg:sesop}).
\section*{Conclusion}
The contributions of the paper can be summarized as follows:
\begin{itemize}
\item We propose one modification of the Sequential Subspace Optimization Method \cite{SESOP_2005} with a $\delta$-additive noise in the gradient \eqref{InexactGrad}. For the first time, the result was obtained describing the influence of this inexactness on the estimate of the convergence rate, whereby the quantity $O(\delta \max_k \|x_k - x^*\|)$ is replaced by the constant $O(\delta \|x_0 - x^*\|)$, $\|x_0 - x^*\| \leq \max_k \|x_k - x^*\|$ .
\item The influence of inexactness in solving auxiliary minimization problems \eqref{eqsubproblem} to the general theoretical estimate for Algorithm \eqref{alg:sesop} is investigated.
\item We provide numerical experiments which demonstrate the effectiveness of the proposed approach in this paper. Algorithm \ref{alg:sesop} is compared with another known Similar Triangles Method (STM) with an additive gradient noise.
\end{itemize}
In the further works, we plan to continue the analysis of error accumulation in other methods for non-convex $\gamma$-quasar-convex functions. It is planned to develop some methods with auxiliary subproblems of dimension less than 3. In particular, we are going to consider the Conjugate Gradients method considered in \cite{NemirovskyYudin,AccWQC} and near-optimal methods from work \cite{NearOpt}.
The authors are grateful to Alexander Gasnikov and Mohammad Alkousa for very useful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,155
|
Mills Field
a random object
our weblog
Flights coming and going from Gate F76 See just the arrivals or only the departures at Gate F76 / or see all the airports that flights departing this gate have traveled to or the airports that flights arriving at Gate F76 have come from / or see the airlines or the different aircraft that have flown in or out of Gate F76
NZ9390 (SFO-PIT)
This Air New Zealand flight between San Francisco International Airport and Greater Pittsburgh International Airport, on July 14, 2019, was operated by another airline. It departed from gate F76. See all the flights for NZ9390.
CA7334 (SFO-PIT)
This Air China flight between San Francisco International Airport and Greater Pittsburgh International Airport, on July 14, 2019, was operated by another airline. It departed from gate F76. See all the flights for CA7334.
UA1839 (SFO-PIT)
This United Airlines flight traveled from San Francisco International Airport to Greater Pittsburgh International Airport, on July 14, 2019. It departed from gate F76. See all the flights for UA1839.
UA214 (SEA-SFO)
This United Airlines flight traveled from Tacoma International Airport to San Francisco International Airport, on July 14, 2019. It arrived at gate F76. See all the flights for UA214.
NZ9422 (SEA-SFO)
This Air New Zealand flight between Tacoma International Airport and San Francisco International Airport, on July 14, 2019, was operated by another airline. It arrived at gate F76. See all the flights for NZ9422.
CM1446 (SEA-SFO)
This Copa Airlines flight between Tacoma International Airport and San Francisco International Airport, on July 14, 2019, was operated by another airline. It arrived at gate F76. See all the flights for CM1446.
NZ9099 (SFO-ABQ)
This Air New Zealand flight between San Francisco International Airport and Albuquerque International Airport, on July 14, 2019, was operated by another airline. It departed from gate F76. See all the flights for NZ9099.
AC4032 (SFO-ABQ)
This Air Canada flight between San Francisco International Airport and Albuquerque International Airport, on July 14, 2019, was operated by another airline. It departed from gate F76. See all the flights for AC4032.
UA5648 (SFO-ABQ)
This United Airlines flight traveled from San Francisco International Airport to Albuquerque International Airport, on July 14, 2019. It departed from gate F76. See all the flights for UA5648.
UA5255 (MCI-SFO)
This United Airlines flight traveled from Kansas City International Airport to San Francisco International Airport, on July 14, 2019. It arrived at gate F76. See all the flights for UA5255.
NZ9502 (MCI-SFO)
This Air New Zealand flight between Kansas City International Airport and San Francisco International Airport, on July 14, 2019, was operated by another airline. It arrived at gate F76. See all the flights for NZ9502.
OZ6318 (MCI-SFO)
This Asiana Airlines flight between Kansas City International Airport and San Francisco International Airport, on July 14, 2019, was operated by another airline. It arrived at gate F76. See all the flights for OZ6318.
Visit FlySFO
© San Francisco International Airport
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,668
|
Australian rangers find three-eyed snake
Katie Fitzgerald
Watch: Are UFOs and psychics real? Credits: Image - Facebook/ Northern Territory Parks and Wildlife; Video - Newshub.
Australian rangers have discovered a three-eyed snake living in the northern territories.
The snake was discovered on the Arnhem Highway near the town of Humpty Doo, 30 minutes north of Darwin on Wednesday.
Snake-free New Zealand? Slithery intruder found in ute at South Island town
Nightmarish video of gigantic snake hoisting bird into sky
Cane toads hitch ride on the back of a python
"An x-ray revealed it was not two separate heads forged together, rather it appeared to be one skull with an additional eye socket and three functioning eyes," the Northern Territory Parks and Wildlife said in a post on Facebook.
"It was generally agreed that the eye likely developed very early during the embryonic stage of development.
"It is extremely unlikely that this is from environmental factors and is almost certainly a natural occurrence as malformed reptiles are relatively common."
A spokesperson for Parks and Wildlife told NT News the snake was around 40cm long when it was found, and believed to be around three-weeks-old.
The rangers named it Monty Python before it passed away.
Facebook users were delighted by Monty's different look and many joked about it with eye related puns.
"Was going to write a pun , but eye see it's already been done," one person said.
"I tried to come up with a few jokes but they just got cornea and cornea," another wrote.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 592
|
Perfect Bid: The Contestant Who Knew Too Much (2017)
I've often thought about which game show I'd most like to be on. It would probably be Jeopardy!, mostly because it pays cash and there's no limit on how often you can come back. Looking at shows like The Price Is Right, you can only win non-cash prizes and, if you do, you have to pay taxes out of your own pocket. Personally, I'd rather have the money.
Theodore Slauson, on the other hand, really liked The Price Is Right. He and his brother even noticed that the prices were fairly consistent. In fact, if you account for varying features, they were always the same. If a particular brand of a pack of gum was 69¢ one week, that same pack of gum would be 69¢ a few weeks later. It makes sense.
Most people would have left it at that. Not Ted. It would be a while before he would turn 18 and be eligible to participate in the show. That would give him plenty of time to memorize all of the prices. There were some products which couldn't be memorized. Vacations, for instance, had a lot of variables, such as where you flew from and which airline was used. There were a lot of products, like cars, that had maybe three or four different models (read: three or four different prices) at most. This gave Ted a leg up on the other contestants.
Ted made it to the show dozens of times as a member of the studio audience. Over the years, he helped many people in Contestants' Row get the right bid. On his 24th visit, Ted had the opportunity to bid himself. And yes, he managed to get an exact bid. A few more correct bids let him go to the Big Wheel, where he actually lost to someone else. (For those that don't watch the show, contestants have to spin a big wheel with dollar values and get as close to $1 without going over. Ted didn't get the highest amount.)
You'd think that this would be the end of the story, but it's not. After all, this is called Perfect Bid. It's about someone that made a perfect bid in the actual showcase. Again, for those not in the know, two contestants are each given a showcase consisting of several prizes. It might include a car or an RV. It might even be a vacation. Whoever bids closer wins their showcase. If they bid within a certain amount, they win both prizes. The Price Is Right has debuted in 1972. In all those years, there has been only one perfect bid on a showcase. This documentary is about that bid.
Ted might have been content with his showing. He would have had to have been, as at the time, there was a rule that you could only compete once. Then, the changed the rule so that you could try again after ten years. Ted made several more trips to the studio audience. While he wasn't selected, Terry Kniess was. Terry made it to the Showcase. While the details differ, the important fact is that he bid $23,743.. The value of his showcase? $23,743.
In the 40+ years that the show has been on the air, this remains the only time that has ever happened. People have come close. A site I found even has someone being $2 off. To put that in perspective, Bob Barker is credited on IMDb as having appeared on 6,719 episodes. Drew Carey is credited at 1,768 as I'm writing this. That's almost 8,500 episodes between them. Only once has there been an exact bid.
Now, you might wonder what the big deal is. I wouldn't blame you if you hadn't heard about this. This is something that you'd put in the trivia section on IMDb and forget about it. There was some scandal, as Roger Dobkowitz, a producer with the show, had recently been fired. Was this payback? Those producing the episode weren't really clear on what to do. While it was possible for this to happen, I don't think anyone expected it. In retrospect, it was kind of the show's fault, as they should have varied the prices more.
The runtime is a little long at 72 minutes. A lot of it seemed like filler. I think this could have been done in a much shorter time. Maybe have it as a segment on a TV show about game-show history or something. While it is an interesting footnote in the history of daytime television, that's really all it's going to be to a lot of people. If you were to watch it, I'd save it for your next layover at an airport.
Labels: Bob Barker , C.J. Wallis , Drew Carey , Kevin Pollak , Roger Dobkowitz , Theodore Slauson
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,140
|
Das 16. 24-Stunden-Rennen auf dem Nürburgring fand vom 18. auf den 19. Juni 1988 auf dem Nürburgring statt.
Rennergebnis
Das 24-Stunden-Rennen 1988 gewannen zwei Porsche 911 RSR vor einem Opel Manta 400. Im ersten Porsche vom Team Original ATE-Bremsen saßen Edgar Dören, Gerhard Holup und Peter Faubel, im zweiten vom Team Motortuning Horst Derkum Heinz Dorr, Günter Doebler und Jürgen Barth. Den Manta des Deutschen Opel Teams pilotierten Norbert Haug, Karl Mauer und Joachim Winkelhock.
Die Sieger fuhren 140 Runden und erreichten damit einen neuen Distanzrekord von 3549,00 km. Von den 183 gestarteten Fahrzeugen wurden 110 gewertet und eines disqualifiziert.
Rennverlauf
1988 schlug die Stunde der Privatteams, nachdem zur Halbzeit des Rennens bereits alle Werks-M3 nach Motorschäden, Unfällen und gebrochenen Kardanwellen ausgeschieden waren und auch die werksseitig eingesetzten Sierra RS Cosworth mit Motorschaden ausfielen. Die beiden Vorjahressieger Klaus Ludwig und Steve Soper im Texaco-Sierra von Eggenberger, die Frank Biela anstatt Klaus Niedzwiedz nun Frank Biela als dritten Fahrer an ihrer Seite haben, müssen ihren Cossie bereits nach elf Runden abstellen, der zweite Texaco-Sierra mit Soper, Pierre Dieudonné und Gianfranco Brancatelli am Steuer hält 121 Runden durch und ist das letzte Fahrzeug, das es nicht in die Wertung schafft.
Streckenführung
Seit dem Umbau der Strecke im Jahr 1983 wurde das 24-Stunden-Rennen am Nürburgring auf einer Kombination aus Nordschleife und Grand-Prix-Strecke ausgetragen.
Besonderheiten
Die drei Gentlemandriver im Siegerfahrzeug waren zum Zeitpunkt des Rennens zusammen 165 Jahre alt und gewannen dieses in einem bereits 13 Jahre alten Porsche 911 RSR. Den dritten Platz belegte mit dem Opel Manta 400 ebenfalls ein älteres Fahrzeug, das bereits 1981 vorgestellt wurde.
Das Rennen 1988 markierte zudem in dreifacher Hinsicht das Ende einer Ära. Zum letzten Mal zählte das 24-Stunden-Rennen zum Langstreckenpokal, zum letzten Mal wurde die Gruppe H zum Start zugelassen und der Escort RS2000 der zweiten Baureihe, der das Siegfahrzeug in den Jahren 1979 und 1980 war und mehrere Podiumsplatzierungen und Klassensiege einfuhr, war zum letzten Mal am Start.
Weblinks
www.24h-rennen.de – Offizielle Website zum 24-Stunden-Rennen auf dem Nürburgring
24-Stunden-Rennen auf nuerburgring.de
24h-Rennen auf veedol-freunde-der-nordschleife.de
Einzelnachweise
1988
Motorsportveranstaltung 1988
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 23
|
Miriam Barr is a New Zealand page and performance poet.
She is one of the original members of The Literatti, an Auckland-based performance poetry group started in 2006. Barr became creative director for the group in 2007.
Her poetry has appeared in Landfall, Black Mail Press, Tongue in Your Ear, JAAM, De-Formed Paper, Enamel, Live Lines, Debate, The Wild Goose Poetry Review, Magazine, Brief, Takahe, and Poetry New Zealand (now Poetry NZ Yearbook).
She has self-published two volumes of poetry, Tangents, 2006 (now out of print) and Observations from the Poetry Factory, 2007. Both of these books were released with CDs of Barr's poetry set to music. Her collection, Bullet Hole Riddle was published by Steele Roberts Aotearoa in 2014.
In 2007 Barr created the free, independent poetry zine, Side Stream, which was distributed by a network of poets from around the world until its demise in 2011.
Barr has been one of the coordinators of Poetry Live since 2006. Poetry Live has been a gathering point for the Auckland poetry community since the early 1980s.
Barr was the winner of the 2007 Auckland Writers and Readers Festival Poetry Idol slam. She has performed widely, including at AK07, the 2007 Ubud Readers and Writers Festival, the Dunedin Fringe Festival, the Parihaka International Peace Festival, Montana Poetry Day, the Parnell Festival of Roses, the Auckland Fringe Festival and the Whangarei Midwinter Arts Festival. She has also collaborated with many visiting international poets, including acclaimed American slam poet Nikki Patin.
Published poetry
References
External links
1982 births
Living people
New Zealand poets
New Zealand women poets
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,752
|
\section*{Acknowledgement}
Thanks to A. Samanta, IIT Kanpur for sharing the kineme code.
\subsection{Datasets}\label{sec:datasets}
\begin{sloppypar}
\noindent \textbf{FICS:} The FICS dataset~\cite{escalante2020modeling} comprises 10K ($\approx$15s long) \emph{YouTube} self-presentation videos, and is labeled via crowdworkers for the OCEAN personality traits. As Neuroticism is a negative trait, N scores encode inverse of Neuroticism scores in this dataset. FICS contains 6K \emph{training}, 2K \emph{validation} and 2K \emph{test} videos. \\
\end{sloppypar}
\noindent \textbf{MIT:} The MIT interview~\cite{Naim18} dataset comprises 138 mock internship interview videos, with an average length of 4.7 minutes. These videos were rated for 16 traits by crowdworkers; along with the four traits considered in Sec.~\ref{Sec:UC}, we also evaluate prediction performance on the \emph{overall} interview score (Ov) in this section.
\subsection{Experimental settings}\label{Sec:ES}
\textbf{Prediction Type
:} For our experiments, we modeled the personality and interview trait scores as either continuous or discrete variables (upon thresholding trait scores at their median value), and present regression (Tables~\ref{tab:FICS_reg},~\ref{tab:MIT_reg}) and classification (Tables~\ref{tab:FICS_class},~\ref{tab:MIT_class}) results. As the FICS dataset is already partitioned into the train, validation and test sets, we employed the validation set for fine-tuning model parameters, or for early stopping of model training for LSTM. On the other hand, as the MIT dataset only comprises 138 videos, we report performance in the form of $\mu \pm \sigma$ over five repetitions of 10-fold video cross validation (total of 50 runs) in Tables~\ref{tab:MIT_reg} and~\ref{tab:MIT_class}. Validation sets were obtained by randomly holding out 10\% of the training data during each run. \\
\noindent \textbf{Chunk vs video-level predictions:} In line with the thin-slice approach for behavioral trait prediction, we segmented the original videos into smaller chunks, 3, 5 and 7s for the FICS data and 5--60s chunks for the MIT data (see Fig.~\ref{fig:Chunk_vs_Vid_MIT}) and repeated the video label for all chunks. We then computed metrics over a) all chunks (chunk-level performance), and b) over all videos by assigning the majority label over all chunks to each video (video-level performance).
\subsection{Performance Metrics} For regression, we considered the accuracy (Acc) and Pearson Correlation Coefficient (PCC) metrics. As in~\cite{Gucluturk2018}, Acc is measured as 1-MAE, the mean absolute error in the predictions with respect to groud-truth scores; the PCC measures how well the predictions correlate with the ground-truth scores (PCC=1 for precise predictions). For classification, we employed the accuracy (Acc) and F1-score metrics. Given the imbalanced class distributions for some traits (see Table~\ref{tab:class-dist}), the F1-score denoting the harmonic mean of precision and recall is more suited for performance evaluation.
\subsection{Models}
\noindent \textbf{Linear regression with principal components (PCA-Lin-Reg):} We collated the yaw, pitch and roll values over each chunk, and then performed principal component analysis (PCA) over the training data. As many principal components explaining 90\% data variance were preserved, and linear regression was performed on the same. This method ignores the temporal head and facial motion dynamics. \\
\noindent \textbf{Hidden Markov Model for classification (HMM Kin):} To learn trait-wise temporal head motion patterns, we employed a HMM to iteratively deduce model parameters and class labels from kineme sequences via the Baum-Welch algorithm. \\
\noindent \textbf{Long short-term memory for regression and classification:} LSTMs denote another efficient methodology to learn latent temporal head and facial motion dynamics. We trained LSTMs with the kineme sequences \textbf{(LSTM Kin)}, AU sequences \textbf{(LSTM AU)} and their combination \textbf{(LSTM Kin+AU FF)}. The kineme sequences were input in the one-hot encoding form, where the kineme corresponding to a given time-window is coded to 1 and the other 15 coded to 0. AUs on the other hand were input in a coded vector form, where all AUs deemed dominant upon thresholding within a time-window were set to 1 and others set to 0. To fuse the kineme and AU-based representations (denoted as feature fusion or FF), we employed the architecture shown in Fig.~\ref{fig:Kin_AU_arch}. Kineme features form a 3D matrix of 16 (one-hot-kineme-vector) $\times$ data-points $\times$ chunk-length, while the AU matrix is of size 17 (vector of active AUs coded to 1) $\times$ data-points $\times$ chunk-length. A single hidden LSTM layer with 32 neurons was employed for kinemes and AUs, while these outputs were merged to obtain 64 neurons for FF. The LSTM layer is followed by a dense layer involving two neurons with sigmoidal activation for classification, and one neuron with linear activation for regression. In both cases, a dropout value of 0.2 was employed for the LSTM layer to prevent overfitting, and an Adam optimizer with learning rate of 0.01 was utilized for training. Binary-cross entropy was defined as the loss function for classification, and mean absolute error for regression. \\
\noindent \textbf{2D CNN for regression and classification:} A 19 layered VGG model, which processes images (video frames) was used. Upon removing the output layer, two hidden dense layers with 512 and 64 neurons respectively were added along with output layer involving 5 neurons (one neuron
each for the OCEAN/Interview traits). Mean squared error (MSE) for regression, and binary cross-entropy (BCE) loss for classification were used to train on a random frame from each video, with learning rate of 1e-4 and a batch size of 64. \\
\begin{sloppypar}
\noindent \textbf{Decision fusion for regression and classification {(LSTM Kin+AU (DF))}:} Apart from LSTM-based kineme AU fusion, we attempted fusion of the unimodal LSTM predictions. We adopted the fusion weight estimation approach proposed in~\cite{KOELSTRA2013164}. Assuming that the test sample score is $\alpha p_{Kin} + (1-\alpha) p_{AU}, \alpha \in [0,1]$, where $p_{Kin}$ and $p_{AU}$ are the individual classifier (regressor) scores, we performed grid-search incrementing $\alpha$ in steps of 0.01 to estimate the optimal $\alpha^*$ maximizing PCC for regression and F1-score for classification; $\alpha^*$ was then applied to compute test sample scores.
\end{sloppypar}
\begin{figure}[!htb]
\centering
\includegraphics[width=\linewidth]{Images/LSTM_arch}\vspace{-2mm}
\caption{Kineme$+$AU LSTM classification architecture. The dense layer output involves a single neuron with linear activation for regression. Best viewed in color and zoom.} \label{fig:Kin_AU_arch}\vspace{-4mm}
\end{figure}
\begin{table*}[!htbp]
\centering
\small
\caption{Trait-wise train (Tr) and test (Te) class distributions for the FICS and MIT datasets obtained for classification experiments. MIT class distributions correspond to 1-minute video samples employed for analysis.}\label{tab:class-dist} \vspace{-2mm}
\begin{tabular}{|c|cc|cc|cc|cc|cc|cc|cc|cc|cc|cc|}
\toprule
& \multicolumn{10}{|c|}{\textbf{FICS}} & \multicolumn{10}{|c|}{\textbf{MIT}} \\ \hline
& \multicolumn{2}{|c|}{\textbf{O}} & \multicolumn{2}{|c|}{\textbf{C}} & \multicolumn{2}{|c|}{\textbf{E}} & \multicolumn{2}{|c|}{\textbf{A}} & \multicolumn{2}{|c|}{\textbf{N}} & \multicolumn{2}{|c|}{\textbf{Ov}} & \multicolumn{2}{|c|}{\textbf{RH}} & \multicolumn{2}{|c|}{\textbf{Ex}} & \multicolumn{2}{|c|}{\textbf{EC}} & \multicolumn{2}{|c|}{\textbf{Fr}}\\ \hline
\textbf{Label} & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te & \bf Tr & \bf Te \\
\textbf{-ve} & 0.53 & 0.52 & 0.51 & 0.51 & 0.51 & 0.51 & 0.53 & 0.53 & 0.52 & 0.52 & 0.50 & 0.50 & 0.59 & 0.59 & 0.50 & 0.50 & 0.43 & 0.43 & 0.39 & 0.39\\
\textbf{+ve} & 0.47 & 0.48 & 0.49 & 0.49 & 0.49 & 0.49 & 0.47 & 0.47 & 0.48 & 0.48 & 0.50 & 0.50 & 0.41 & 0.41 & 0.50 & 0.50 & 0.57 & 0.57 & 0.61 & 0.61\\
\bottomrule
\end{tabular}
\vspace{-2mm}
\end{table*}
\begin{table*}[!htbp]
\centering
\small
\caption{FICS Regression results: Accuracy and PCC values for different methods are tabulated.} \vspace{-2mm}
\begin{tabular}{|l|cc|cc|cc|cc|cc|cc|}
\toprule
\bf Trait & \multicolumn{2}{|c|}{\textbf{PCA Lin-Reg}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin}} & \multicolumn{2}{|c|}{\textbf{LSTM AU}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (FF)}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (DF)}} & \multicolumn{2}{|c|}{\textbf{2D-CNN}}\\
& \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC}\\
\hline
\textbf{Open} & 0.884 & 0.085 & 0.872 & 0.060 & 0.889 & 0.370 & 0.892 & 0.368 & 0.893 & 0.382 &\textbf{0.906} & \textbf{0.392}\\
\textbf{Con} & 0.875 & 0.086 & 0.864 & 0.027 & 0.882 & 0.317 & 0.880 & 0.304 & 0.882 & 0.282 &\textbf{0.908} & \textbf{0.295}\\
\textbf{Extra} & 0.877 & 0.060 & 0.869 & 0.048 & 0.891 & 0.491 & 0.893 & 0.474 & 0.891 & 0.485 &\textbf{0.907} & \textbf{0.492}\\
\textbf{Agree} & 0.892 & 0.035 & 0.885 & 0.046 & 0.897 & 0.251 & 0.892 & 0.253 & 0.896 & 0.275 &\textbf{0.906} & \textbf{0.283}\\
\textbf{Neuro} & 0.877 & 0.071 & 0.867 & 0.051 & 0.885 & 0.370 & 0.884 & 0.365 & 0.887 & 0.387 &\textbf{0.903} & \textbf{0.395}\\
\bottomrule
\end{tabular}
\label{tab:FICS_reg}\vspace{-2mm}
\end{table*}
\begin{table*}[!htbp]
\centering
\fontsize{7}{7}\selectfont
\renewcommand{\arraystretch}{1.5}
\caption{MIT Regression results: Accuracy and PCC values are tabulated.} \vspace{-2mm}
\begin{tabular}{|l|cc|cc|cc|cc|cc|cc|}
\toprule
\bf Trait & \multicolumn{2}{|c|}{\textbf{PCA Lin-Reg}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin}} & \multicolumn{2}{|c|}{\textbf{LSTM AU}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (FF)}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (DF)}} & \multicolumn{2}{|c|}{\textbf{2D-CNN}}\\
& \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC} & \textbf{Acc} & \textbf{PCC}\\
\hline
\textbf{Ov} & 0.86$\pm$0.00 & 0.15$\pm$0.26 & 0.93$\pm$0.04 & 0.84$\pm$0.26 & 0.93$\pm$0.04 & 0.84$\pm$0.26 & \textbf{0.97$\pm$0.03} &\textbf{ 0.92$\pm$0.17} & 0.95$\pm$0.04 & 0.89$\pm$0.21 & 0.86$\pm$0.03 & 0.47±0.25\\
\textbf{RH} & 0.83$\pm$0.03 & 0.13$\pm$0.26 & \textbf{0.95$\pm$0.03} & \textbf{0.93$\pm$0.10} & \textbf{0.95$\pm$0.03} & \textbf{0.93$\pm$0.10} & 0.96$\pm$0.04 & 0.91$\pm$0.21 & 0.94$\pm$0.04 & 0.90$\pm$0.19 &0.85$\pm$0.03 & 0.45$\pm$0.24\\
\textbf{Ex} & 0.82$\pm$0.04 & 0.05$\pm$0.20 & 0.94$\pm$0.04 & 0.89$\pm$0.20 & 0.94$\pm$0.04 & 0.89$\pm$0.20 & \textbf{0.96$\pm$0.05} & \textbf{0.91$\pm$0.20} & 0.94$\pm$0.04 & 0.91$\pm$0.17 & 0.85$\pm$0.03 & 0.57$\pm$0.18\\
\textbf{EC} & 0.83$\pm$0.03 & 0.10$\pm$0.21 & 0.94$\pm$0.04 & 0.89$\pm$0.13 & 0.94$\pm$0.04 & 0.89$\pm$0.22 & \textbf{0.96$\pm$0.04} & \textbf{0.94$\pm$0.13} & 0.95$\pm$0.04 & 0.91$\pm$0.16 & 0.84$\pm$0.03 & 0.44$\pm$0.22\\
\textbf{Fr} & 0.82$\pm$0.04 & 0.02$\pm$0.28 & 0.95$\pm$0.03 & 0.93$\pm$0.10 & 0.95$\pm$0.03 & 0.93$\pm$0.10 & \textbf{0.97$\pm$0.03} & \textbf{0.96$\pm$0.08} & 0.95$\pm$0.03 & 0.94$\pm$0.08 & 0.86$\pm$0.03 & 0.58 $\pm$0.20\\
\bottomrule
\end{tabular}
\label{tab:MIT_reg}
\vspace{-2mm}
\end{table*}
\begin{table*}[!htbp]
\centering
\small
\caption{FICS Classification results: Accuracy and F1 values are tabulated.} \vspace{-2mm}
\begin{tabular}{|l|cc|cc|cc|cc|cc|cc|}
\toprule
\bf Trait & \multicolumn{2}{|c|}{\textbf{HMM Kin}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin}} & \multicolumn{2}{|c|}{\textbf{LSTM AU}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (FF)}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (DF)}} & \multicolumn{2}{|c|}{\textbf{2D-CNN}}\\ \hline
& \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1}\\
\textbf{Open} & 0.529 & 0.505 & 0.519 & 0.516 & 0.635 & 0.634 & 0.629 & 0.628 & 0.632 & 0.632 & 0.697 & \textbf{0.697}\\
\textbf{Con} & 0.524 & 0.515 & 0.513 & 0.513 & 0.618 & 0.618 & 0.604 & 0.604 & 0.599 & 0.599 & 0.705 & \textbf{0.705}\\
\textbf{Extra} & 0.514 & 0.509 & 0.505 & 0.505 & 0.651 & 0.651 & 0.648 & 0.648 & 0.657 & 0.653 & 0.712 & \textbf{0.712} \\
\textbf{Agree} & 0.528 & 0.497 & 0.481 & 0.479 & 0.580 & 0.580 & 0.593 & 0.586 & 0.584 & 0.583 & 0.656 & \textbf{0.656}\\
\textbf{Neuro} & 0.524 & 0.506 & 0.523 & 0.518 & 0.627 & 0.624 & 0.626 & 0.623 & 0.620 & 0.616 & 0.717 & \textbf{0.717}\\
\bottomrule
\end{tabular}
\label{tab:FICS_class}\vspace{-2mm}
\end{table*}
\begin{table*}[!htbp]
\centering
\fontsize{7}{7}\selectfont
\renewcommand{\arraystretch}{1.5}
\caption{MIT Classification results: Accuracy and F1 values are tabulated.} \vspace{-2mm}
\begin{tabular}{|l|cc|cc|cc|cc|cc|cc|}
\toprule
\bf Trait & \multicolumn{2}{|c|}{\textbf{HMM}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin}} & \multicolumn{2}{|c|}{\textbf{LSTM AU}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (FF)}} & \multicolumn{2}{|c|}{\textbf{LSTM Kin+AU (DF)}} & \multicolumn{2}{|c|}{\textbf{2D-CNN}}\\
& \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1} & \textbf{Acc} & \textbf{F1}\\
\hline
\textbf{Ov} & 0.56$\pm$0.17 & 0.48$\pm$0.18 & 0.83$\pm$0.11 & 0.82$\pm$0.13 & 0.82$\pm$0.14 & 0.81$\pm$0.15 & 0.80$\pm$0.14 & 0.80$\pm$0.14 & 0.85$\pm$0.13 & \textbf{0.85$\pm$0.14} & 0.65$\pm$0.09 & 0.64$\pm$0.08\\
\textbf{RH} & 0.55$\pm$0.14 & 0.34$\pm$0.27 & 0.79$\pm$0.12 & 0.79$\pm$0.12 & 0.83$\pm$0.13 & 0.83$\pm$0.14 & 0.81$\pm$0.12 & 0.80$\pm$0.12 & 0.84$\pm$0.11 & \textbf{0.83$\pm$0.12} & 0.59$\pm$0.07 & 0.58$\pm$0.07\\
\textbf{Ex} & 0.57$\pm$0.14 & 0.51$\pm$0.10 & 0.82$\pm$0.13 & 0.82$\pm$0.13 & {0.82$\pm$0.12} & 0.82$\pm$0.10 & 0.79$\pm$0.13 & 0.79$\pm$0.13 & 0.83$\pm$0.11 & \textbf{0.82$\pm$0.12} & 0.72$\pm$0.08 & 0.71$\pm$0.08\\
\textbf{EC} & 0.53$\pm$0.17 & 0.44$\pm$0.24 & 0.79$\pm$0.13 & 0.79$\pm$0.13 & 0.81$\pm$0.12 & 0.80$\pm$0.13 & 0.78$\pm$0.12 & 0.76$\pm$0.13 & 0.84$\pm$0.13 & \textbf{0.83$\pm$0.14} & 0.57$\pm$0.07 & 0.55$\pm$0.09\\
\textbf{Fr} & 0.57$\pm$0.18 & 0.54$\pm$0.12 & 0.80$\pm$0.15 & 0.80$\pm$0.16 & 0.86$\pm$0.09 & 0.85$\pm$0.09 & 0.84$\pm$0.10 & 0.84$\pm$0.11 & 0.87$\pm$0.11 & \textbf{0.86$\pm$0.12} & 0.61$\pm$0.08 & 0.61$\pm$0.08\\
\bottomrule
\end{tabular}
\label{tab:MIT_class}\vspace{-4mm}
\end{table*}
\begin{figure}[!htbp]
\centering
\includegraphics[width=\linewidth]{Images/GradCam_ex}
\vspace{-5mm}\caption{Exemplar Gradcam~\cite{Selvaraju_2019} maps for CNN predictions.}\label{fig:Gradcam_out}
\vspace{-6mm}
\end{figure}
\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.48\linewidth]{Images/plot_FICS_Kineme}\hspace{0.05cm}\includegraphics[width=0.48\linewidth]{Images/plot_FICS_AU}
\vspace{-2mm}\caption{Chunk vs video-level predictions with kinemes (left) and AUs (right) for the FICS dataset.}\label{fig:Chunk_vs_Vid_FICS}
\vspace{.5mm}
\centering
\includegraphics[width=0.48\linewidth,height=5cm]{Images/plot_MIT_Kineme}\hspace{0.05cm}\includegraphics[width=0.48\linewidth,height=5cm]{Images/plot_MIT_AU}
\vspace{-2mm}\caption{Chunk vs video-level predictions with kinemes (left) and AUs (right) for the MIT dataset.}\label{fig:Chunk_vs_Vid_MIT}\vspace{-4mm}
\end{figure*}
\subsection{Results and Discussion}
Tables~\ref{tab:FICS_reg} and~\ref{tab:MIT_reg} present regression results, while Tables~\ref{tab:FICS_class},~\ref{tab:MIT_class} tabulate classification results. We make the following remarks therefrom:
\begin{itemize}
\item One can note from Tables~\ref{tab:FICS_reg} and~\ref{tab:MIT_reg} that very low PCC values are noted even for relatively high accuracies for the FICS dataset; these results convey that larger variability in prediction scores is observed for the FICS videos, and that PCC is more suited for MIT performance evaluation. Also, comparing Tables~\ref{tab:FICS_reg},~\ref{tab:MIT_reg} with Tables~\ref{tab:FICS_class},~\ref{tab:MIT_class}, much higher performance is achieved for regression than classification. This can be attributed to the Gaussian-distributed FICS OCEAN trait scores around 0.5 (see Figs 12 and 13 in~\cite{escalante2020modeling}), and the MIT Interview trait scores about the median.
\item Focusing on Table~\ref{tab:FICS_reg}, PCA Lin-Reg which ignores temporal head-motion dynamics, performs better than LSTM Kin; this implies that the extracted kinemes cannot adequately describe the FICS OCEAN scores. Very poor PCC scores are obtained with both PCA Lin-Reg and LSTM Kin.
\item AUs achieve superior LSTM-based prediction than kinemes. However, complementarity of the kineme and AU encodings in general enables slightly superior performance with both feature and decision fusion, with both fusion schemes performing comparably. 2D-CNN achieves optimal predictions on the FICS videos; the explanatory power of CNNs is however limited. Grad-cam visualizations (Fig.~\ref{fig:Gradcam_out}) which highlight regions deemed critical for model prediction focus on the eye, mouth and nasal regions as in~\cite{Ventura17}. Unlike kinemes however, trait-characteristic temporal dynamics of these regions are not conveyed by these saliency maps.
\item Kinemes however demonstrate superior predictive power for interview traits (Table~\ref{tab:MIT_reg}), considerably outperforming Lin-Reg and performing identical to AUs. Fusing unimodal features/decisions is beneficial for the MIT dataset, with feature fusion outperforming decision fusion.
\item Comparing against the Resnet-based trait prediction model proposed in~\cite{Gucluturk2018}, we note that the 2D-CNN achieves similar or superior performance, while the Kineme/AU-based LSTM performs inferiorly.
\item On the smaller MIT dataset however, LSTM Kin considerably outperforms 2D-CNN, revealing the need for large training data to effectively tune VGG-type networks.
\item In Table~\ref{tab:FICS_reg}, 2D-CNN achieves highest accuracy for Conscientiousness, while Agreeableness is predicted best via all other methods. Highest PCC values are achieved for Friendliness with all methods excepting PCA Lin-Reg in Table~\ref{tab:MIT_reg}.
\item Focusing on Table~\ref{tab:FICS_class}, HMM Kin and LSTM Kin perform worst for OCEAN trait prediction. LSTM AU performs considerably better than LSTM Kin and similar to the feature and decision fusion schemes, implying that multimodal fusion is not very beneficial for the FICS dataset. The 2D-CNN achieves optimal trait classification, substantially outperforming other methods. With all schemes, least classification performance is observed for Agreeableness, while Extraversion is best classified by all schemes excepting HMM Kin and LSTM Kin.
\item With respect to Table~\ref{tab:MIT_class}, the LSTM network employing kineme features achieves considerably higher classification performance than HMM Kin. This trend again confirms that kinemes are effective predictors of interview traits. LSTM AU performs slightly better than LSTM Kin, while the decision fusion framework achieves optimal classification, marginally outperforming LSTM AU.
\item As with regression, LSTM Kin achieves substantially better F1-scores than 2D-CNN, confirming the efficacy of the kineme representation with fewer training data. The Friendliness trait is best isolated by four of the six classifiers.
\end{itemize}
The above results are achieved by considering 15s chunks for the FICS videos and 1-minute chunks for the MIT videos, and assigning the video label as the majority label over all chunk predictions. As mentioned in Sec.~\ref{Sec:ES}, we attempted trait prediction from thin behavioral slices (chunks) of varying lengths and evaluated prediction performance at the chunk and video levels.
Figures~\ref{fig:Chunk_vs_Vid_FICS} and ~\ref{fig:Chunk_vs_Vid_MIT} present corresponding results. In both figures, higher PCC values are achieved with video-level labels than chunk-level labels, implying that while episodic behaviors may be inconsistent with one another, trait-specific behaviors are indeed homogeneous over a long time-span. From Fig.~\ref{fig:Chunk_vs_Vid_FICS}, we note that kineme-based trait prediction performance generally decreases with larger time-slices, while AU-based predictions become more accurate with larger time-slices. Conversely, one can note a general increase in video and chunk-level PCC values with both Kineme and AU features for the MIT dataset. Interestingly, even with 5s slices, we obtain a PCC > 0.3 at the chunk-level and PCC > 0.65 at the video level with kineme features, and still higher values are obtained with AUs. These trends suggest that AUs describing facial behavior, encode more trait-specific information than kinemes characterizing head motion, consistent with one's expectation.
\vspace{-4mm}
\subsection{Head motion as kineme sequence}\label{HMtoKin}
Given the head-motion time-series extracted from a set of videos, we divide each time-series $\boldsymbol{\theta}$ into short overlapping segments of length $\ell$ (overlapping segments enable shift-invariance, are empirically found to generate better representations in~\cite{samanta2021emotion}). The $i^{th}$ segment is denoted by a vector $\mathbf{h}^{(i)} = [\theta_p^{i:i+\ell}\, \theta_y^{i:i+\ell}\, \theta_r^{i:i+\ell}]^\intercal$. The characterization matrix $\mathbf{H}_{\boldsymbol\theta}$ is defined as
$ \mathbf{H}_{\boldsymbol\theta} = [\mathbf{h}^{(1)}, \mathbf{h}^{(2)},\cdots, \mathbf{h}^{(s)}]$, where $s$ is the total number of segments in $\boldsymbol{\theta}$. Given $N$ training samples, a head motion matrix $\mathbf{H}\in\mathbb{R}_+^{3\ell\times Ns}$ is created as $\mathbf{H} = [\mathbf{H}_{\boldsymbol\theta_1}|\mathbf{H}_{\boldsymbol\theta_2}|\cdots|\mathbf{H}_{\boldsymbol\theta_N}]$. $\mathbf{H}$ is then subject to a Non-negative Matrix Factorization (NMF), yielding a basis matrix $\mathbf{B}$ and a coefficient matrix $\mathbf{C}$. We cluster head motion segments in the transformed space by grouping the coefficient vectors (columns in $\mathbf{C}$) via a Gaussian Mixture Model (GMM) into $k<<Ns$ clusters. This produces a $k$ column matrix ${\mathbf{C}^*}$. Kinemes are transformed back to the original \emph{yaw}-\emph{pitch}-\emph{roll} space via $\mathbf{H}^*=\mathbf{B}\mathbf{C}^*$, whose columns yield the set of $k$ kinemes $\mathcal{K}$.
Upon learning kinemes for the input video set, we can represent any head motion time-series as a sequence of kinemes by associating each $\ell$-long segment to one of the $K$ kinemes. Consider the $i^{th}$ segment in $\boldsymbol{\theta}$ characterized by $\mathbf{h}^{(i)}$. We project $\mathbf{h}^{(i)}$ onto the learned subspace to obtain $\mathbf{c}^{(i)}$:
\begin{equation*}
\hat{\mathbf{c}} = \underset{\mathbf{c}^{(i)} \geq 0}{\text{arg min}} \lVert{\mathbf{h}^{(i)} - \mathbf{B}\mathbf{c}^{(i)}}\rVert_F^2
\end{equation*}
The corresponding kineme $K^{(i)}$ for the $i^{th}$ segment is given by maximizing the posterior probability $P({K}|\hat{\mathbf{c}})$, over all $K\in\mathcal{K}$. On mapping each head motion time-series segment to a kineme, we get the corresponding kineme sequence $\boldsymbol\theta: \{K^{(1)} \cdots K^{(s)}\}, K^{(j)}\in \mathcal{K}$.
\subsection{Trait prediction from kinemes}
Prior studies have shown that human-centered traits are characterized by specific non-verbal (and specifically, head-motion) behaviors; \textit{e.g.}, an upright posture maintaining eye-contact conveys high Conscientiousness, while gaze avoidance is indicative of low Conscientiousness~\cite{Hoppe18,researchdigest220714}. Likewise, frequent head nodding is seen as courteous and agreeable behavior, while head shaking and frowning indicates a cold demeanor~\cite{Ishii2020,Kawahara18}. Kinemes inherently enable discovery of temporal head-motion patterns characteristic of a given trait, and sequence learning methods such as Hidden Markov Models (HMMs) and Long-short term memory (LSTM) networks can be employed to learn these latent temporal signatures for continuous or categorical trait prediction
\subsection{Personality and Interview Trait Prediction}
It is well known that personality drives human behavior, and the big-five or OCEAN personality trait model~\cite{Costa1992} describes human personality in terms of the Openness (curious vs cautious), Conscientiousness (diligent vs insincere), Extraversion (outgoing vs reserved), Agreeableness (sympathetic vs dispassionate) and Neuroticism (nervous vs emotionally stable) dimensions. Numerous studies have attempted personality prediction from behavioral, and specifically non-verbal cues such as proxemics~\cite{Takayama09}, speaking and head movement behavior~\cite{Jayagopi09,Subramanian13}, facial characteristics~\cite{escalante2020modeling,Gucluturk2018} and physiological responses to emotional scenes~\cite{Subramanian18}.
Many studies have also examined the relationship between personality traits and job/job-interview performance. \emph{E.g.}, Mount~\textit{et al.}~\cite{Murray1998} observe that Conscientiousness, Agreeableness and Emotional stability positively impact job performance involving interpersonal interactions. Authors of~\cite{Moy03} observe that the OCEAN traits are among the major attributes influencing hiring decisions. Two recent studies that examine the relation between personality traits and human factors in job interviews are~\cite{Gucluturk2018,escalante2020modeling}. Naim~\textit{et al.}~\cite{Naim18} perform multimodal analyses of interview videos and conclude that prosodic features (speaking style) critically influence impressions of interview-specific traits.\\
\noindent \textbf{Explainable personality and interview trait prediction:} While deep learning architectures such as CNNs and LSTMs have achieved excellent performance on multiple pattern recognition problems, their predictions are often not interpretable~\cite{Ventura17}. A deep residual network (ResNet) is proposed in~\cite{Gucluturk2018} to predict personality trait impressions, and linear regression is employed to predict interview scores from personality trait annotations; regression coefficients are employed to assess the influence of the OCEAN traits on interview scores. Face visualizations are also presented to demonstrate similarities among individuals achieving high and low trait scores. CNN-based trait prediction is explored in~\cite{Ventura17}, and analyses show that CNNs primarily analyze key facial regions such as eyes and mouth for prediction. The FICS dataset developed for a candidate screening challenge is presented in~\cite{escalante2020modeling}. Decision trees and visual-plus-verbal explanations are presented to convey relationships among personality and interview scores. Facial action units, which are typically used to describe emotions are shown to effectively predict personality traits in~\cite{Gavrilescu15}, and the correlations among AUs and learned CNN features is demonstrated in~\cite{Ventura17}. \vspace{-2mm}
\subsection{Head Motion for Behavioral Analytics} A majority of existing works on head-motion rely on extracting low-level features from head motion data. For example, Ding~\textit{et al.}~\cite{ding2018low} use amplitude of representative Fourier components. Samanta and Guha~\cite{samanta2017role} propose to extract energy of displacement, velocity and acceleration of the Euler rotation angles, while Gunes and Pantic~\cite{gunes2010dimensional} employ magnitude and direction of the 2D head motion. The use of semantically meaningful head gestures has been limited to extracting nods and shakes~\cite{gunes2010dimensional}. However, head motion generated during dyadic interactions and self-presentation videos is complex; therefore high-level head gestures may not be limited to only nods and shakes. To this end, Yang and Narayanan~\cite{yang2017modeling} propose to extract arbitrary head gesture segments. These head gesture segments are abstract, and do not have physical interpretation. \\
\noindent\textbf{Head motion in human-centered traits:} In the context of affect analysis, head motion patterns have been used to study coordination between mothers and infants~\cite{hammal2015head,hammal2015can}, emotion recognition~\cite{samanta2021emotion}, measuring engagement levels of dementia patients~\cite{Parekh18} and for analyzing interpersonal coordination in couple therapy \cite{hammal2014interpersonal, xiao2015head}.
\subsection{Inference Summary}
Analysis of the literature reveals the following shortcomings: (1) While many works examine prediction of personality and interview traits, and explain predictions via statistical analysis or visualizations, these explanations are limited to discovering salient facial features or examining connections between interview performance and personality traits. (2) While head motion patterns have been identified as critical non-verbal behavioral cues in interactive scenarios, they have nevertheless not been employed for personality or interview trait prediction.
Differently, we novelly attempt explanations for personality and interview traits from kinemes which are inherently explanatory; apart from generic explanations such as head-nodding evoking positive and head-shaking eliciting negative impressions, trait-specific characteristics such as Openness and emotional stability associating with persisting head movements, and Conscientiousness being impacted by head tilting to maintain/avoid eye-contact are also evident. More intuitive explanations are achieved on combining kinemes and facial action units such as nodding and expressive facial behavior achieving high interview scores, and head shaking and frowning being seen as less friendly and sociable. In addition to their explanatory characteristics, kinemes and AUs are also found to effectively predict the targeted traits.
\section{Introduction}\label{Sec:Intro}
\input{Intro}
\section{Related Work}\label{Sec:RW}
\input{RW}
\section{Kineme Formulation}\label{Sec:KF}
\input{KF}
\section{Explainable Trait Prediction}\label{Sec:UC}
\input{UC}
\section{Experiments and Results}\label{Sec:ER}
\input{ER}
\section{Conclusion}\label{Sec:DC}
\input{DisCon}
\bibliographystyle{ACM-Reference-Format}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 388
|
\section{Introduction} \label{sec:I}
The Large Area Telescope (LAT) onboard the {\em Fermi}\xspace space telescope
has proven to be a powerful tool for studying the extragalactic component
of the gamma-ray sky. It already has resolved nearly 90\% of the
extragalactic gamma-ray flux into discrete point sources. These are
overwhelmingly dominated by active galactic nuclei (specifically
blazars), with small populations of dimmer, though potentially much
more numerous, sources
\citep[e.g., starburst and radio galaxies, see Table 5 of][]{2LAC}.
These observations have produced the strongest constraints to date
upon the gamma-ray bright blazar population and its evolution.
Likewise, the unresolved component of the extragalactic gamma-ray sky
constrains the nature and number
of point sources that lie below the current detection threshold. At
present this arises in two contexts: 1.{\ }the isotropic extragalactic
gamma-ray background (EGRB) spectrum, which sets the total flux that
must be accounted for, and 2.{\ }the angular structure in the EGRB,
characterized by an anisotropy coefficient (see below), that limits
the fraction of the EGRB that can be produced by bright point sources.
Of these, the second has most severely constrained models that produce
the EGRB from extensions of the gamma-ray point source populations to
low fluxes and high redshifts.
Attempts to model the EGRB by extending existing point source
populations receive strong theoretical support from our current
understanding of the history of baryon accretion into dark matter
halos, and its associated observational tracers. Such models have been
generally successful in reproducing the observed EGRB spectrum, both
via phenomenological and physically motivated extensions
\citep[see, e.g.,][]{SPA12,PaperI,PaperV,Cavadini+2011}. They also receive
qualitative support from the structure of the measured EGRB angular
power spectrum, $C_\ell$, within a variety of energy bands. Between
$\ell=150$ and 500 the $C_\ell$ are constant, consistent with the EGRB
being produced by a population of discrete sources with angular scales
smaller than the width of the {\em Fermi}\xspace point spread function
\citep{Fermi_aniso}.
Quantitatively, however, the EGRB power spectrum is {\em not} consistent with
point source populations that are extensions of the known gamma-ray
bright source populations \citep{Cuoco}. The origin of the limit is
easy to understand: if the EGRB is produced by few sources just below
the current detection thresholds, it should exhibit a correspondingly
large degree of angular variability. From this, it has been argued
that no more than 20\% of the isotropic EGRB can be due to gamma-ray
blazars without violating the EGRB anisotropy constraints, and thus
motivated renewed interest in alternative gamma-ray sources
\citep{Cuoco}. However, this conclusion is predicated upon the proper
normalization of the angular power.
The general success of extensions of the known blazar population to
low fluxes provides a natural reason to revisit the EGRB anisotropy
normalization. However, more disturbing is the precipitous decline
in blazar numbers below the {\em Fermi}\xspace detection threshold required for
consistency. This is difficult to envision, even in principle,
requiring a pathological redshift evolution and luminosity
function for which there is currently no other evidence. Motivated by
this, here we present an external, empirical assessment of the
normalization of the reported EGRB anisotropy.
\begin{deluxetable*}{lccccc}[th!]
\tablecaption{Estimated Lower Limits upon the EGRB Anisotropy\tablenotemark{a} \label{tab:CPs}}
\tablehead{
& & \multicolumn{4}{c}{$\Delta C_P$ within various energy bands ($10^{-19}~{\rm ph^2 cm^{-4} s^{-2} sr^{-1}}$)}
\\
&&&&&\\
Method\tablenotemark{b}
&
Section
&
1.04--1.99~GeV
&
1.99--5.00~GeV
&
5.00--10.4~GeV
&
10.4--50~GeV
}
\startdata
\citet{Fermi_aniso}, {\sc data:cleaned} & --- & $46.2\pm11.1$ & $13.0\pm2.2$ & $0.85\pm0.25$ & $0.211\pm0.086$\\
\hline\\
Unmasked 2FGL detected in each energy band & 3.1 & $27.5\pm~~5.0$ & $~~7.0\pm1.3$ & $0.63\pm0.13$ & $0.168\pm0.063$\\
Unmasked 1FHL detected above 10~GeV & 3.1 & --- & --- & --- & $0.238\pm0.092$\\
Unmasked 2FGL K-corrected $\mathcal{F}_{35}$ & 3.2 & $26.9\pm~~3.7$ & $~~8.6\pm1.1$ & $0.86\pm0.11$ & $0.54~~\pm0.09~~$\\
\hline\\
Statistical 2FGL, w/ 2FG & 4.3 & $38.5\pm~~2.8$ & $12.5\pm0.9$ & $1.35\pm0.11$ & $0.98~~\pm0.12~~$\\
Statistical 2FGL, w/ 2FGL-EB & 4.3 & $16.0\pm~~2.7$ & $~~5.2\pm0.9$ & $0.61\pm0.11$ & $0.52~~\pm0.10~~$\\
Statistical 2FGL, w/ PL & 4.3 & $49.8\pm~~3.4$ & $16.1\pm1.0$ & $1.71\pm0.13$ & $1.20~~\pm0.13~~$\\
Statistical 2FGL, w/ PL-EB & 4.3 & $36.7\pm~~2.0$ & $11.9\pm0.6$ & $1.29\pm0.08$ & $0.94~~\pm0.10~~$\\
Statistical 2FGL, hard sources w/ 2FGL & 4.3 & $~~6.9\pm~~0.9$ & $~~4.1\pm0.5$ & $0.73\pm0.09$ & $0.76~~\pm0.11~~$\\
Statistical 2FGL, hard sources w/ 2FGL-EB & 4.3 & $~~3.5\pm~~0.7$ & $~~2.1\pm0.5$ & $0.39\pm0.09$ & $0.45~~\pm0.10~~$\\
Statistical 2FGL, hard sources w/ PL & 4.3 & $~~8.9\pm~~1.1$ & $~~5.1\pm0.6$ & $0.90\pm0.10$ & $0.91~~\pm0.13~~$\\
Statistical 2FGL, hard sources w/ PL-EB & 4.3 & $~~7.0\pm~~0.8$ & $~~4.0\pm0.4$ & $0.71\pm0.07$ & $0.73~~\pm0.10~~$\\
\hline\\
Statistical power law extension w/ $\mathcal{F}_{35}>3\times10^{-10}~{\rm ph~cm^{-2} s^{-1}}$ \& PL & 4.4 & $84.9\pm~~9.2$ & $27.2\pm2.8$ & $2.80\pm0.34$ & $1.73~~\pm0.30~~$\\
Statistical power law extension w/ $\mathcal{F}_{35}>3\times10^{-10}~{\rm ph~cm^{-2} s^{-1}}$ \& PL-EB & 4.4 & $68.5\pm~~5.9$ & $21.9\pm1.8$ & $2.26\pm0.22$ & $1.40~~\pm0.19~~$\\
Statistical power law extension w/ $\mathcal{F}_{35}>10^{-10}~{\rm ph~cm^{-2} s^{-1}}$ \& PL & 4.4 & $145.0\pm~~9.6~~$ & $46.4\pm3.0$ & $4.78\pm0.35$ & $2.96~~\pm0.31~~$\\
Statistical power law extension w/ $\mathcal{F}_{35}>10^{-10}~{\rm ph~cm^{-2} s^{-1}}$ \& PL-EB & 4.4 & $128.6\pm~~6.6~~$ & $41.2\pm2.0$ & $4.24\pm0.24$ & $2.63~~\pm0.21~~$\\
Statistical power law extension w/ $\mathcal{F}_{35}>10^{-12}~{\rm ph~cm^{-2} s^{-1}}$ \& PL & 4.4 & $187.4\pm~~9.6~~$ & $60.0\pm3.0$ & $6.18\pm0.35$ & $3.83~~\pm0.31~~$\\
Statistical power law extension w/ $\mathcal{F}_{35}>10^{-12}~{\rm ph~cm^{-2} s^{-1}}$ \& PL-EB & 4.4 & $171.1\pm~~6.6~~$ & $54.8\pm2.0$ & $5.64\pm0.24$ & $3.50~~\pm0.22~~$
\enddata
\tablenotetext{a}{The $1\sigma$ uncertainty intervals indicate the cosmic variance and include the propagation of flux uncertainties.}
\tablenotetext{b}{See Table \ref{tab:wfits} for the definitions of the threshold functions 2FGL, 2FGL-EB, PL, and PL-EB.}
\end{deluxetable*}
We do this by exploiting the sensitivity difference between the
First and 2 Year {\em Fermi}\xspace-LAT Source Catalogs
\citep[1FGL and 2FGL, respectively,][]{1FGL,2FGL} and the more
recently published First {\em Fermi}\xspace-LAT catalog of $>10$~GeV sources
\citep[1FHL,][]{1FHL}. The EGRB anisotropy measurements reported by
\citet{Fermi_aniso} employed the 1FGL for point source identification
and removal. Despite this, the {\em Fermi}\xspace data used for the anisotropy
measurement was collected over a 22 month period, coincident with the
24 month period used to construct the more sensitive 2FGL and 36 month
period used to construct the 1FHL. Thus,
subsets of the 2FGL and 1FHL necessarily will have contributed to the observed
EGRB anisotropy. By comparing their contributions to the observed
EGRB anisotropy with the reported values, we obtain
independent limits on the EGRB anisotropy magnitude.
Note that this is quite different than the discussion of the
dependence upon the catalog (1FGL or 2FGL) used to construct the
point-source mask in \citet{Fermi_aniso}. There the fraction of the
EGRB anisotropy associated with the 2FGL was assessed by comparing
the anisotropy signal in residual sky maps obtained by masking on
the 2FGL sources instead of the 1FGL (which comprised their main
analysis). Here, we are concerned with verifying the absolute
normalization of the anisotropy signal using an {\em independent}
and potentially more robust method which is based directly on the
properties of the unmasked point sources listed in the 2FGL rather
than on the angular power spectra of the residual sky maps.
In the interests of completeness we present a variety of estimates of
the expected anisotropy. These may be broadly placed into two
categories: estimates that explicitly use the sources in the 2FGL and
1FHL that are unmasked by the 1FGL (which we call ``direct
estimates''), and estimates that use the entire 2FGL and 1FHL as
statistical measures of the unmasked source population (``statistical
estimates''). The former set is necessarily much more susceptible to
cosmic variance, though places hard lower limits upon the measured
EGRB anisotropy spectrum. The latter set is statistically more
robust, more appropriate for comparison to theoretically motivated
source populations, and more easily extended beyond the 2FGL and 1FHL
catalogs. The assumption made is that an accurate statistical
description of the sources that remain after masking on the 1FGL can
be obtained from the 2FGL source population (which we will justify
empirically). {\em In all cases we find qualitatively similar
results: at high energies the reported EGRB anisotropy is
significantly below what may already be accounted for from the
observed point source populations.} A summary of the lower limits
upon the anisotropy within the energy bands reported in
\citet{Fermi_aniso} is collected in Table \ref{tab:CPs}, together with
a brief description of the method employed and the section where it is
described.
We begin in Section \ref{sec:CP} with a short derivation of the
contribution to the EGRB anisotropy spectrum from a population of
point sources. We present direct estimates for various
sub-populations of the 2FGL and 1FHL in Section \ref{sec:direct}.
Statistical estimates are collected in Section \ref{sec:statistical}.
In Section \ref{sec:discussion} we discuss our results in the context
of the measured EGRB and possible reasons for differences. Finally,
conclusions are collected in Section \ref{sec:conclusions}.
\section{Definition of $C_P$} \label{sec:CP}
Above $\ell\simeq150$, the EGRB angular power
spectrum is well characterized by a constant, i.e., $C_\ell = C_P$ \citep{Fermi_aniso}.
In the flat-sky limit\footnote{
Expanding point sources sky maps obtained in Section \ref{sec:direct}
into spherical harmonics, we measure the angular power spectrum and
verify that it becomes flat for multipoles $\ell>15$ and
consistent with a Poisson power spectrum. This provides an explicit
validation of the flat-sky limit at multipoles of interest.}, this is
related to the gamma-ray flux map via
\begin{equation}
C_\ell
=
\int d^2\theta e^{i\bmath{\ell}\cdot\bmath{\theta}}
\int \frac{d^2\theta'}{4\pi} \mathcal{F}(\bmath{\theta}'+\bmath{\theta}) \mathcal{F}(\bmath{\theta}')\,,
\end{equation}
where for a given ensemble of unresolved point sources with
fluxes $\{\mathcal{F}_j\}$ and positions $\bmath{\theta}_j$ the flux per unit
solid angle is,
\begin{equation}
\mathcal{F}(\bmath{\theta}) = \sum_j \mathcal{F}_j \delta^2(\bmath{\theta}-\bmath{\theta}_j)\,.
\end{equation}
Inserting this into the above, and averaging over ensembles (including
source sky positions) yields
\begin{equation}
\label{eq:Cell}
\begin{aligned}
\left< C_\ell \right>
&=
\left<
\int d^2\theta e^{i\bmath{\ell}\cdot\bmath{\theta}}
\int \frac{d^2\theta'}{4\pi} \sum_{j,k}\mathcal{F}_j\mathcal{F}_k
\delta^2(\bmath{\theta}'+\bmath{\theta}-\bmath{\theta}_j)
\delta^2(\bmath{\theta}'-\bmath{\theta}_k)
\right>\\
&=
\left<
\sum_{j,k}
\frac{\mathcal{F}_j \mathcal{F}_k}{4\pi}
e^{i\bmath{\ell}\cdot\left(\bmath{\theta}_j-\bmath{\theta}_k\right)}
\right>\\
&=
\left< \sum_j \frac{\mathcal{F}_j^2}{4\pi} \right>\,,
\end{aligned}
\end{equation}
where we identify the final expression with $C_P$. Note that
larger individual fluxes or an enhanced number of sources increases
this dimensional measure of the anisotropy coefficient $C_P$.
If the flux distribution of {\em unresolved} sources is a function of
$\mathcal{F}$ only, this reduces to the standard expression:
$C_P = \int d\mathcal{F}\,\mathcal{F}^2\,d\mathcal{N}/d\mathcal{F}$, where $d\mathcal{N}/d\mathcal{F}$ is the number of
sources {\em unresolved} in the 1FGL per steradian per unit flux.
Generally, this can be quite complicated, with the result that
\begin{equation}
C_P
=
\int d^n\!p\,\mathcal{F}^2\frac{d\mathcal{N}}{d^n\!p}\,,
\label{eq:CPdef}
\end{equation}
where $\bmath{p}$ are some set of $n$ parameters needed to describe
the flux distribution (e.g., flux and spectral index, or in our case
here, the 1--100~GeV band flux $\mathcal{F}_{35}$ and spectral index).
Further progress requires an explicit estimate for $d\mathcal{N}/d^n\!p$.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig1.eps}
\end{center}
\caption{ Contribution to the EGRB anisotropy $\Delta C_P$ by unmasked
point sources employing two different latitude cuts of
$|b|>20^\circ$ (red) and $|b|>30^\circ$ (green). We compare the
reported measurements of $C_P$ \citep[blue,][]{Fermi_aniso} to the
contribution of individual unmasked 2FGL sources, which are detected
in the respective energy bins (triangles) and the unmasked 1FHL
sources above 10 GeV (squares). The contribution of all unmasked
2FGL sources (detected above 1 GeV) that are K-corrected to the
respective energy band and corrected for spectral curvature
(circles) already overproduce the {\em Fermi}\xspace measurement by $3.2\sigma$
($3.4\sigma$) in the highest energy band for $|b|>30^\circ$
($|b|>20^\circ$). The error bars indicate the cosmic variance and
include the propagation of flux uncertainties. The points have been
shifted horizontally within their respective energy bin centers for
clarity.}
\label{fig:direct_main}
\end{figure}
Here we will use the 2FGL and 1FHL to provide an approximation of the
source population that is (statistically) not detected in the
1FGL. We do this primarily via distributions of the form
\begin{equation}
\frac{d\mathcal{N}}{d\mathcal{F} d\Gamma} \simeq \sum_j \frac{w_j}{\Omega_{\rm sky}}
\delta(\mathcal{F}-\mathcal{F}_j) \delta(\Gamma-\Gamma_j)\,,\label{eq:dNdFdGamma}
\end{equation}
where $\Omega_{\rm sky}$ is the fraction of the sky being included,
$F_j$ are the band-specific fluxes, and the $w_j$ are source-dependent
weights describing the likelihood that a given source was not
observed in the 1FGL\footnote{Modulo the weights, $w_j$, this is simply the
approximation of the probability distribution often used for
bootstrap methods.}.
Effectively, in Equation (\ref{eq:dNdFdGamma}), we are constructing
the source distribution directly from the 2FGL, as opposed to
utilizing fits to previously selected forms of $d\mathcal{N}/d\mathcal{F} d\Gamma$, as
is has been the case elsewhere \citep[see, e.g.,][]{Cuoco}.
These empirical estimates of the source distribution correspond to
conservative estimates of the lower limit on the EGRB anisotropy from
unresolved blazars, with objects unresolved in the 2FGL necessarily
enhancing the anisotropy further.
For comparison, we also consider a simple
extrapolation of the 2FGL $d\mathcal{N}/d\mathcal{F} d\Gamma$ to assess the
fraction of the contribution to the EGRB anisotropy arising from
sources in the 2FGL alone.
Inserting Equation (\ref{eq:dNdFdGamma}) into the estimate for $C_P$,
Equation (\ref{eq:CPdef}), gives our conservative estimate of the 2FGL
contribution to the measured EGRB anisotropy:
\begin{equation}
C_P = \frac{1}{\Omega_{\rm sky}} \sum_j \mathcal{F}_j^2~w_j\,.
\label{eq:CP}
\end{equation}
This procedure is then repeated for each energy band reported in
\citet{Fermi_aniso}, yielding a construction of the $C_P$ spectrum.
While a number of potential sources of uncertainty are present in
principle, after the detection efficiency, chief among them is cosmic
variance. Via a procedure similar to that employed above, we estimate
this to be
\begin{equation}
\label{eq:sigma_Cp}
\sigma_{C_P}
=
C_P
\frac{
\sqrt{\sum_j \mathcal{F}_j^4 w_j}}{\sum_j \mathcal{F}_j^2 w_j}
\,.
\end{equation}
Typical values of $\sigma_{C_P}/C_P$ are 6\%--10\%. By comparison,
the error induced by the intrinsic uncertainty in the K-corrected
flux, ignoring any uncertainty associated with the spectral shape
correction (see Appendix \ref{sec:K-shape-corr}), is estimated to be
roughly 2-7\%. Where relevant, both are included in the error
estimates shown. Note that because the 2FGL contribution to the EGRB
anisotropy is constructed directly from the point sources, there is no
photon noise term (beyond that associated with the point-source
flux uncertainties themselves).
To make further progress, we must necessarily describe how the
weights, $w_j$, are determined and how estimates for the band-specific
fluxes, $\mathcal{F}_j$, are obtained. The direct (Section (\ref{sec:direct})
and statistical (Section \ref{sec:statistical}) approaches differ
essentially in how the weights are obtained. In the case of the
former, these are found using the particular realization of point
sources found in the 2FGL/1FHL and 1FGL catalogs, while in the latter
the 2FGL is used as a statistical measure of the point source
population.
\section{Direct Estimates of the EGRB Anisotropy} \label{sec:direct}
The direct methods are characterized by how the $w_j$ in Equation
(\ref{eq:dNdFdGamma}) are chosen: set either to zero or unity
depending on whether or not the source in question would have been
masked out via the procedure followed in \citet{Fermi_aniso}.
We reproduce the mask employed by \citet{Fermi_aniso} by first
excluding sources within $2^\circ$ circular regions centered on all
source reported in the 1FGL, and then applying a Galactic latitude
cut, e.g., $|b|>30^\circ$. The resulting mask is shown in Figure \ref{fig:mask}.
Unlike \citet{Fermi_aniso}, since we are estimating the contribution
from resolved point sources directly, we are not limited by the
contaminating diffuse Galactic emission component. Thus, we are able
to consider alternative Galactic latitude cuts, providing some measure
of the role cosmic variance plays. Upon comparing the point source
populations at various potential Galactic latitude cuts, we find a
strong similarity in the point source populations for $|b|>15^\circ$
(see Appendix \ref{sec:CV}). Hence, in what follows we show both the
$|b|>30^\circ$ mask employed
by \citet{Fermi_aniso} as well as results associated with a somewhat
conservative, but nonetheless more complete, $|b|>20^\circ$ cut.
Following the implementation of the mask, the resulting value of $C_P$
is then obtained directly from Equation (\ref{eq:CP}), giving,
\begin{equation}
\label{eq:direct}
\Delta C_P = \frac{1}{\Omega_{\mathrm{sky}}}\,\sum_j \mathcal{F}_j^2,
\end{equation}
where the sum extends over all unmasked sources and
$\Omega_{\mathrm{sky}} =4\pi f_{\mathrm{sky}}$. Values of the unmasked sky
fraction, $f_{\mathrm{sky}}$, are obtained explicitly via a Monte-Carlo
integration of the respective sky masks (see Appendix~\ref{sec:CV}). Note that this is
necessarily a lower limit; it both fails to include any potential
diffuse component and the contributions from sources below the
2FGL/1FHL detection thresholds. For this reason, we denote the EGRB
anisotropy spectrum estimates obtained here by $\Delta C_P$, with the
understanding that they can represent only a fraction of the total
values.
Here we describe a variety of limits of the form described above,
distinguished by the point source catalog (2FGL and 1FHL) and estimate
of the fluxes, $\mathcal{F}_j$, used.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig2.eps}
\end{center}
\caption{Aitoff projection of the 1FGL sky mask employed (black
region). The low Galactic-latitude region ($|b|\le30^\circ$) is
shown by the hatched area and the Galactic plane ($|b|\le15^\circ$)
is shown in back. Masked 2FGL sources are shown by the
grey dots. Unmasked 2FGL sources detected above 10~GeV are denoted
by red (orange) squares at high (low) latitudes. Unmasked 2FGL
sources with only upper limits in the 2FGL above 10~GeV are denoted
by blue (light blue) circles at high (low) latitudes.
}\label{fig:mask}
\end{figure}
\subsection{$C_P$ from Unmasked Sources with Detections in each Energy Band}
In our most conservative approach, we {\em only} use sources with detections
in the respective energy bands. Since the energy bands reported in the
catalogs do not correspond to the energy band of the reported $C_P$,
the flux computation requires some spectral correction, which
necessarily depends upon the spectral shape. Here we only use those
point sources that are well fit by a power-law spectrum with spectral
index $\Gamma$ (and are classified as such in the catalogs), making the
standard K-correction for each source individually:
\begin{equation}
\mathcal{F}_{mM,j} = K(\Gamma_j) \mathcal{F}_{xy,j}
\quad\text{where}\quad
K(\Gamma) \equiv
\frac{E_m^{1-\Gamma}-E_M^{1-\Gamma}}{E_x^{1-\Gamma}-E_y^{1-\Gamma}}\,,
\end{equation}
where $E_{m,M}$ are the lower and upper energy limits of the target
band in GeV and $E_{x,y}$ are the lower and upper energy limits of the
bands reported in the catalogs. We apply the following K-corrections
to 2FGL sources:
\begin{equation}
\label{eq:Kcorr1}
\nonumber
\begin{array}{llll}
1-3 \mbox{ GeV} &\to& 1.04-1.99 \mbox{ GeV,} & \mbox{for 1-3 GeV sources}, \\
1-10 \mbox{ GeV} &\to& 1.99-5 \mbox{ GeV,} & \mbox{for 3-10 GeV sources}, \\
3-10 \mbox{ GeV} &\to& 5-10.4 \mbox{ GeV,} & \mbox{for 3-10 GeV sources}, \\
10-100 \mbox{ GeV} &\to& 10.4-50 \mbox{ GeV,} & \mbox{for 10-100 GeV sources}. \\
\end{array}
\end{equation}
In the case of the 1FHL catalog, we K-correct the $10-30$~GeV fluxes
to $10.4-50$~GeV. As a result, we find 15 (21) unmasked sources for
$|b|>30^\circ$ in the highest energy band of the 2FGL catalog
(respectively the $10-30$~GeV band of the 1FHL catalog) that
contribute to $\Delta C_P$. As shown in Figure~\ref{fig:direct_main}, in the
case of the 1FHL catalog, these 21 sources already explain the
measured anisotropy in the high-energy band.
Note that this can only be a lower limit to the true anisotropy since,
in addition to the restrictions mentioned at the end of the preceding
section, here it is further assumed that sources detected at lower
energies in the 2FGL---which are not significantly detected at the
high-energy band due to the decreasing {\em Fermi}\xspace-LAT sensitivity
there---would not emit any flux at $E>10$~GeV.
This latter assumption is even more severe than assuming that the 2FGL
is complete.
\subsection{$C_P$ from all Unmasked 2FGL Sources}
Within the 2FGL we find 139 unmasked power-law sources with
$|b|>30^\circ$, many more than the 15 that have explicit detections
above 10~GeV. In principle, all of these should contribute to the
$\Delta C_P$. Here we assess the point source contribution to the EGRB
anisotropy spectrum using fluxes K-corrected from the 1--100~GeV
band ($\mathcal{F}_{35}$) for all sources in the 2FGL.
This is complicated by the possibility that despite their
characterization as power-law sources some spectral curvature exists
at high energies. Such a deviation necessarily introduces a
systematic uncertainty in the K-corrected fluxes. Using sources well
above the flux limit, we compute a spectral correction factor to the
standard K-correction (see Appendix~\ref{sec:K-shape-corr}).
Typically this makes $\lesssim10\%$ correction to the K-corrected
fluxes and $\lesssim15\%$ difference in their contributions to
$\Delta C_P$ at the energies of interest.
In Figure~\ref{fig:direct_main}, we show the contribution of all
unmasked 2FGL power-law sources (i.e., detected above 1 GeV) that are
K-corrected to the respective energy band and corrected for spectral
curvature. At the highest energy band, the inferred $\Delta C_P$
values {\rm exceed} the {\em Fermi}\xspace-LAT measurement by
$3.2\sigma$ ($3.4\sigma$) for $|b|>30^\circ$ ($|b|>20^\circ$). That is using
only the unmasked 1FHL and 2FGL sources, the lower limit upon the
anisotropy either fully accounts for, or significantly exceeds the
reported values!
\section{Statistical Estimates of the EGRB Anisotropy} \label{sec:statistical}
The preceding direct estimates employed the realization of gamma-ray
point sources present in the unmasked 2FGL/1FHL. However, both to
reduce cosmic variance, which appears to produce an anomalously low
anisotropy signal near latitude cuts of $30^\circ$ (see Appendix
\ref{sec:CV}), and to provide a better comparison to theoretical
models of the gamma-ray point source population, we also provide a set
of statistical estimates of the EGRB anisotropy.
These employ the entire 2FGL as a statistical representation of the
point source population. Key to this is the assumption of the
isotropy (i.e., the masked and unmasked 2FGL sources are statistically
similar) and a characterization of the 1FGL detection threshold,
corresponding to the computation of the $w_j$ in Equation
(\ref{eq:CP}). We treat both of these here before describing the
corresponding estimate for the EGRB anisotropy spectrum.
\subsection{Isotropy of the 2FGL}
Clustering of the 2FGL sources would presumably decrease their
representation in the masked sample. However, evidence for this
isotropy may be found in the investigation of mask dependence in
\citet{Fermi_aniso}\footnote{We do not appeal directly to the
EGRB power spectrum since to do so we would necessarily beg the
question, having assumed that the background is due nearly
exclusively to point sources. Nonetheless, that we do find this to
be the case, in retrospect the EGRB power spectrum provides powerful
confirmation of this assumption.}. Assuming Poisson statistics, with 154
more high-latitude sources ($|b|\ge30^\circ$) found in the 2FGL than
in the 1FGL, the anticipated ratio of unmasked sky fractions is
\begin{equation}
\frac{f_{\rm sky}^{\rm 2FGL}}{f_{\rm sky}^{\rm 1FGL}}
\simeq
e^{-(\Omega_{\rm psm}/\Omega)\Delta N}
=
0.910\,,
\end{equation}
where $\Omega_{\rm psm}=\pi(2^\circ)^2=3.83\times10^{-3}~{\rm sr}$ is
the solid angle of the point source mask. This agrees nearly exactly
with the stated ratio in \citet{Fermi_aniso}, $0.295/0.325 = 0.907$,
and our own estimate (see Appendix \ref{sec:CV}), $0.310/0.338=0.917$,
providing some confidence that clustering may be neglected. A more
complete comparison of source properties with various latitude cuts is
presented in Appendix \ref{sec:CV}, where it was found that at
Galactic latitudes above $15^\circ$ the source populations are
statistically similar (i.e., have similar flux and spectral
properties) and have nearly identical numbers of point sources per
unit solid angle at different Galactic latitudes
(i.e., once the Galactic component no longer contributes
significantly, the number of objects per square degree
is fixed within the expected Poisson fluctuations).
\subsection{The 1FGL Detection Threshold}\label{sec:threshold}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig3a.eps}
\includegraphics[width=0.9\columnwidth]{fig3b.eps}
\end{center}
\caption{Photon spectral index vs flux for the 1FGL (top) and 2FGL
(bottom). The vertical dotted lines show very roughly the flux limits
of the 1FGL (dashed) and 2FGL (dotted). Red points in the
latter indicate sources with spectra that are better fit with
non-power law models. These are excluded in the estimate of the
EGRB anisotropy limits.}\label{fig:FGL_GvF}
\end{figure}
In the direct measures the 1FGL detection threshold entered implicitly
through the mask. Here we use the observed 1FGL source distribution
to estimate the detection threshold explicitly, and therefore assign
explicit values to the $w_j$.
In practice, this appears to be consistent with a
condition solely upon the photon flux from 1--100~GeV ($\mathcal{F}_{35}$).
Importantly, there does not appear to be any systematic correlation
between $\mathcal{F}_{35}$ and the photon spectral index, $\Gamma$ (see the
top panel of Figure \ref{fig:FGL_GvF}). It is not, however,
consistent with a simple cut-off, and in the interest of completeness,
we characterize the threshold function here.
\begin{figure}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig4.eps}
\end{center}
\caption{ Bottom: Distribution of 1FGL sources with 1~GeV--100~GeV
flux (black), in comparison to two models for the true source
distribution. At high fluxes, the distribution is well fit by a
single power-law (dashed blue line), which provides an upper limit
upon the inferred source population at low fluxes. An absolute
lower limit comes from the distribution of sources in the 2FGL,
normalized to be consistent with the 1FGL for
$\mathcal{F}_{35}>10^{-9}~{\rm ph~cm^{-2}~s^{-1}}$ (dotted red line).
Middle and Top: Inferred detection efficiencies, $1-w(\mathcal{F}_{35})$, for
the two approximations of the true source distribution shown in the
bottom panel. In both cases we show a best fit non-negative
$w(\mathcal{F}_{35})$ of the form described in the text, and a version that
allows $w(\mathcal{F}_{35})$ to extend below zero (detection efficiencies
greater than unity) to describe the potential impact of Eddington
bias upon the low-flux tail of the 1FGL source distribution.
These bound the plausible range of detection efficiencies.
}\label{fig:cut}
\end{figure}
It is not possible to estimate the detection threshold from the 1FGL
alone; necessarily some assumption must be made regarding the
undetected source population.
The flux distribution of the 1FGL is well approximated by a power law
at high fluxes,
\begin{equation}
\frac{d\mathcal{N}}{d\log_{10}\mathcal{F}_{35}}
\simeq
1.8\times10^3\left(\frac{\mathcal{F}_{35}}{10^{-9}~{\rm ph~cm^{-2}~s^{-1}}}\right)^{-1.25}\,,
\label{eq:dNdlogF}
\end{equation}
(see Figure \ref{fig:cut}; note the above is corrected for sky
fraction), providing a plausible upper bound upon the number of
undetected sources.
Also shown in Figures \ref{fig:FGL_GvF} and \ref{fig:cut} is the 2FGL
population. As anticipated, it extends to marginally lower fluxes,
resolving a portion of the gamma-ray background unresolved by 1FGL,
roughly consistent with the decrease expected from the increase in
exposure time. At high fluxes, the normalized 1FGL and 2FGL
are consistent with being drawn from the same population, i.e., above
the 1FGL detection threshold, the source population flux distributions
are statistically similar.
Making the conservative
assumption that the 2FGL is complete, it provides an absolute lower
bound upon the number of undetected sources. More complicated source
populations discussed in the literature fall between these two limits
\citep[see, e.g.,][]{Fermi_lNlS,SPA12}.
\begin{deluxetable}{llccc}
\tablecaption{Weight function parameters for the 1FGL.\label{tab:wfits}}
\tablehead{ Threshold Model & Source Distribution & $\mathcal{F}_{\rm max}$\tablenotemark{a} & $m$ & $y$ }
\startdata
2FGL & 2FGL & 1.00 & 7 & 0\\
2FGL-EB & 2FGL w/ Eddington bias & 1.18 & 6 & 0.176\\
PL & Power Law & 1.12 & 7 & 0\\
PL-EB & Power Law w/ Eddington bias & 1.19 & 6.5 & 0.097
\enddata
\tablenotetext{a}{Fluxes are in units of $(10^{-9} {\rm ph~cm^{-2}s^{-1}})$.}
\end{deluxetable}
In practice, due to its abrupt nature, the detection threshold is only
weakly sensitive to the form of the low-flux extension assumed. In
both cases, it is well modeled by a log-normal cutoff. That is, the
probability of {\em non-detection} is
\begin{equation}
w(\mathcal{F}_{35}) \simeq
\begin{cases}
1-10^{-m [\log_{10}(\mathcal{F}_{35}/\mathcal{F}_{\rm max})]^2 + y} & \mathcal{F}_{35}<\mathcal{F}_{\rm max}\\
0 & \text{otherwise}\,,
\end{cases}
\label{eq:w1FGL}
\end{equation}
where the values of the various constants are listed in Table \ref{tab:wfits}.
The threshold flux, $\mathcal{F}_{\rm max}$ is consistently near
$10^{-9}~{\rm ph~cm^{-2}s^{-1}}$, and the steepness parameter, $m$, is
near $6.5$. While this consistently over-predicts the detection
efficiency at very low fluxes, it provides a good approximation near
the threshold, and therefore at the fluxes that dominate the
contribution to the EGRB anisotropy. The
characteristic value of $5\times10^{-10}~{\rm cm^{-2}s^{-1}}$ employed
by \citet{Cuoco} produces a similar threshold, if not slightly lower.
Some care should be taken in interpreting the distribution of sources
near the detection threshold due to the potential for a significant
Eddington bias \citep{Eddi:1913,Eddi:1940}. That this is present in the 1FGL is clear from the
fact that the number of 1FGL sources right near the threshold
{\em exceeds} that in the presumably more complete 2FGL. Deconvolving
the Eddington bias is non-trivial, and depends upon both the
instrumental and intrinsic fluctuations in the measured source
fluxes. Here we take a simplified, if somewhat unphysical approach,
of allowing the detection efficiency to exceed unity immediately at
the threshold, describing the detection of false positives
corresponding to lower-flux sources that are temporally above the
threshold due to statistical and intrinsic fluctuations \citep[a
similar approach is adopted in][see Figure 30 and the surrounding
discussion]{1FHL}. This corresponds to a negative $w(\mathcal{F}_{35})$, and
{\em lowers} the EGRB anisotropy.
Many bright sources, well above the putative 1FGL flux limit, appear
in the 2FGL but not the 1FGL. Unlike the high-flux behavior, this is
presumably due to variability, i.e., sources that were bright only
following the initial 9 months included in the 1FGL. Similarly, in
the 2FGL many 1FGL sources lie below the approximate 1FGL flux
threshold, likely due to Eddington bias (both due to intrinsic
variability and statistical fluctuations).
\subsection{Estimating $C_P$ from the 2FGL} \label{sec:2FGL} We
supplement the weights obtained above with a Galactic latitude cut of
$|b|>30^\circ$ to more directly represent the population of relevance
to the results reported in \citet{Fermi_aniso}. As previously
mentioned, this guarantees that any Galactic point source contribution
has been eliminated. Inserting these conditions into Equation
(\ref{eq:CP}), i.e., only accounting for objects with $|b|>30^\circ$
in the summation and employing the weights in Equation
\ref{eq:w1FGL}, then provides an estimate for the $\Delta C_P$.
Note that in this case the source distribution is expressed in terms
of $\mathcal{F}_{35}$, and not the flux over the energy range for which $C_P$
is being constructed, $\mathcal{F}$. This is necessary since our detection
threshold is specified in terms $\mathcal{F}_{35}$; fluxes over other energy
ranges exhibit substantial spectral-index induced biases. That is we
specify
\begin{equation}
\frac{d\mathcal{N}}{d\mathcal{F}_{\rm 35} d\Gamma} \simeq \sum_j \frac{w_j}{\Omega_{\rm sky}}
\delta(\mathcal{F}_{35}-\mathcal{F}_{35,j}) \delta(\Gamma-\Gamma_j)\,,\label{eq:dNdFdGamma2}
\end{equation}
where $w_j=w(\mathcal{F}_{35,j})$.
In accordance with the particular form of the K-correction used, and
given the importance of the spectral shape to the reconstruction of
the EGRB anisotropy, we restrict ourselves to the subset of the 2FGL
that is best fit by power law spectra, comprising roughly 78\% of the
total sample. Since the sources for which non-power-law fits are
statistically favored tend to be bright, this restriction makes an
insignificant difference in our EGRB anisotropy estimates in
practice. In addition we make the modest spectral-shape correction
described in Appendix \ref{sec:K-shape-corr}, producing a roughly 13\%
enhancement of the $\Delta C_P$ estimate.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig5.eps}
\end{center}
\caption{Reported EGRB anisotropy spectrum \citep[blue error bars,
taken from][]{Fermi_aniso} in comparison to the implied
contributions from high-latitude ($|b|>30^\circ$) 2FGL sources
(triangles). Downward pointing triangles assume the best fit
non-negative $w(\mathcal{F}_{35})$ of the form described in the text, and
upward pointing triangles allow $w(\mathcal{F}_{35})$ to extend below zero
(detection efficiencies greater than unity) to describe the
potential impact of Eddington bias upon the low-flux tail of the
1FGL source distribution. Orange triangles show the EGRB anisotropy
spectrum inferred assuming a power-law low-flux extension of the
1FGL, and the red triangles present the conservative lower limit
obtained by assuming the 2FGL is complete. For reference, the
contributions from only hard sources (i.e., $\Gamma\le2$) are also
shown (purple and magenta). The error bars indicate the cosmic
variance and include the propagation of flux uncertainties. The
points are offset horizontally within their respective energy bin
centers for clarity.}\label{fig:EAlim}
\end{figure}
The estimated contribution to the EGRB anisotropy from the low-flux,
high-latitude sources in the 2FGL is shown by the orange
triangles in Figure \ref{fig:EAlim}, with the
conservative lower-limits arising from assuming the 2FGL is complete
shown by the red triangles. The estimates using
$w_j$ that account for the Eddington bias in the 1FGL (upwards
pointing triangles for each method respectively) are consistent
with the direct estimates obtained in Section \ref{sec:direct},
yielding some confidence in the statistical approach. When $w_j$
constructed assuming a power-law extrapolation of the low-flux source
population are used, the lower limits rise further.
Nevertheless, the anisotropy due to sources resolved by the 2FGL alone
are sufficient to produce the entirety of the EGRB anisotropy from
2--5~GeV, and {\em exceed} the reported EGRB anisotropy above 5~GeV by
many $\sigma$. At high energies, this is due primarily to the
presence of hard gamma-ray sources, i.e., objects for which
$\Gamma\le2$, shown by the magenta and purple
triangles\footnote{At all energies, however, the 2FGL contribution
to the isotropic component of the EGRB is well below the reported
limits.}. These are dominated by BL Lacs, and are presumably
responsible for some fraction of the isotropic EGRB component.
However, these are not solely responsible for the anisotropy excess,
playing a sub-dominant role below 10.4~GeV.
\subsection{Estimating $C_P$ from power-law extensions of the 2FGL} \label{sec:statpl}
The assumption made above that only sources that are detected in the
2FGL contribute to the EGRB anisotropy is itself extreme. While
contributions to the $C_P$ is
heavily biased towards high-flux objects, the range about the flux
threshold for which significant contributions are found extends for at
least a decade. By comparison, the reduction in the detection
threshold between the 1FGL and 2FGL is roughly 0.2 dex, considerably
smaller. Thus, the assumption that the 2FGL is complete corresponds
to a dramatic suppression in the number of sources immediately below
the 2FGL detection threshold. Evidence that this is not the case may
be found in the recently published catalog of hard Fermi sources, the
1FHL, which does not exhibit any notable features at the 2FGL flux
limit. Hence, the $\Delta C_P$ obtained in the previous sections can
at most represent a modest contribution to the anticipated value.
To provide a reasonable upper limit upon the expected contribution to
the EGRB anisotropy due to gamma-ray point sources, we consider the
$C_P$ obtained from a single power-law extrapolation of the 2FGL
population, given in Equation (\ref{eq:dNdlogF}). This is necessarily
highly uncertain. To guarantee a finite contribution to the isotropic
EGRB component a break in the $d\mathcal{N}/d\mathcal{F}_{35}$ relation must necessarily
exist at sufficiently low fluxes. Similarly, to guarantee a finite
source population, a cutoff must necessarily exist as well (though
this is much less well constrained). To assess the sensitivity to
these putative features, we produce anticipated EGRB anisotropy
spectra for a variety of lower flux cutoffs.
We must supplement the extrapolated $d\mathcal{N}/d\mathcal{F}_{35}$ relation with a
spectral index distribution. Based upon Figure \ref{fig:FGL_GvF} we
assume the distribution in $\mathcal{F}_{35}$ and $\Gamma$ is separable, i.e.,
we choose
\begin{equation}
\frac{d\mathcal{N}}{d\mathcal{F}_{35}d\Gamma}
=
\frac{d\mathcal{N}}{d\mathcal{F}_{35}}
\frac{e^{-(\Gamma-\bar{\Gamma})^2/2\sigma_\Gamma^2}}{\sqrt{2\pi} \sigma_\Gamma}\,,
\end{equation}
where $\bar{\Gamma}=2.2$ and $\sigma_\Gamma=0.3$, measured from the
2FGL directly. While the intrinsic photon spectral index distribution
is clearly skewed, the above does an especially good job of
reproducing the hard component, critical for the high-energy EGRB
anisotropy estimates.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig6.eps}
\end{center}
\caption{Reported EGRB anisotropy spectrum \citep[blue error
bars, taken from][]{Fermi_aniso} in comparison to that inferred
from a power-law extrapolation of the 2FGL to lower fluxes. The
green triangles show the EGRB spectrum for various low-flux
cutoffs, $3\times10^{-10}~{\rm cm^{-2}s^{-1}}$, $10^{-10}~{\rm
cm^{-2}s^{-1}}$, and $10^{-12}~{\rm cm^{-2}s^{-1}}$ (effectively
zero) from left to right within each energy bin. Downward
pointing triangles assume the best fit non-negative $w(\mathcal{F}_{35})$
of the form described in the text, and upward pointing triangles
allow $w(\mathcal{F}_{35})$ to extend below zero (detection efficiencies
greater than unity) to describe the potential impact of Eddington
bias upon the low-flux tail of the 1FGL source distribution. For
reference, the red triangles show the limits obtained under the
assumption that the 2FGL is complete (see Figure \ref{fig:EAlim}).
The error bars indicate the cosmic variance and include the
propagation of flux uncertainties. The points are offset
horizontally within their respective energy bin centers for
clarity.}\label{fig:EApl}
\end{figure}
Figure \ref{fig:EApl} shows the associated EGRB spectrum for a handful
of lower-flux cutoffs. From this it is apparent that the $C_P$
receive a substantial contribution from sources that lie below the
2FGL flux threshold. In the absence of a dramatic suppression of the
source population above $\mathcal{F}_{35}\simeq10^{-10}~{\rm cm^{-2}s^{-1}}$,
these unresolved sources comprises more than half of the anticipated
contribution.
\newpage
\section{Discussion} \label{sec:discussion}
Even in the most conservative cases our lower limit upon the EGRB
anisotropy spectrum significantly exceeds the values reported by
\citet{Fermi_aniso}. The reason for this is not immediately clear.
Here we consider handful possibilities.
The first is that sources without detections above 10~GeV in the 2FGL
really do have vanishing flux at that energy, i.e., K-correcting their
low-energy emission substantially over-estimates their true
high-energy fluxes. This is argued against by the detection of
high energy emission from a number of additional sources reported in
the 1FHL, though presumably this may arise from variability. Perhaps
more conclusive is the evidence that the K-corrected $\mathcal{F}_{35}$
provides a good estimate of the 1FHL measured fluxes (see Appendix
\ref{sec:K-shape-corr}), strongly suggesting that this will be true
for sources falling immediately below the relevant detection
thresholds. For this reason we believe this is highly disfavored.
The second is that the deficit in anisotropy is due to a surfeit of
dim sources in the 1FGL resulting from Eddington bias. While we have
not made any attempt to debias the 1FGL (which is well beyond the
scope of this paper), this effect alone is incapable of producing the
the factor of two reduction necessary to resolve the discrepancy.
The third is the point-spread function (PSF) deconvolution procedure
employed by \citet{Fermi_aniso}. The most recent point spread
function estimates are much broader than those used, implying a
corresponding underestimate of the deconvolved $C_P$. As shown in
Appendix \ref{sec:PSF}, this can amount to a factor of $\sim2$
correction under reasonable assumptions. When coupled with the cosmic
variance, this can boost the reported anisotropy to the minimum limits
implied by the 2FGL. However, it is unable to increase it to the
values suggested by smooth extrapolations of the 2FGL population to
significantly lower fluxes.
\section{Conclusions} \label{sec:conclusions} Estimates of the EGRB
anisotropy resulting from the now resolved point sources in the 2FGL
and 1FHL alone significantly exceed the reported measurements above
5~GeV in \citet{Fermi_aniso}, where the BL Lacs dominate the resolved
extragalactic gamma-ray sky and the EGRB anisotropy proved most
constraining. This represents a purely empirical estimate of the
anisotropy associated with the known point source population. In
particular, we do not appeal to a parametrized source distribution,
and hence our estimate represents the most conservative lower limit
possible to the EGRB anisotropy. Given 2FGL and the EGRB anisotropy
observations are largely coincident, reconciling the apparent
disparity between the estimates presented here and that of
\citet{Fermi_aniso} of the EGRB anisotropy is difficult.
For estimates that make reasonable assumptions regarding the
high-energy fluxes of 2FGL sources, the discrepancy above 10.4~GeV
ranges from 2.6$\sigma$ (direct) to 5.2$\sigma$ (statistical, no
Eddington bias), with the 2FGL contribution alone exceeding that
reported in \citet{Fermi_aniso} by a factor of $2.6$ to $4.6$, respectively.
The 2FGL estimates are quite robust, independent of many of the
complications that plague estimating the EGRB anisotropy
directly\footnote{Of course, this comes at the price of resolving the
background, and thus is not a reasonable alternative.}. In
particular, it is insensitive to the particulars of the low-flux
source population (which can only increase the anisotropy), and the
dominant remaining source of potential uncertainty, cosmic variance,
is small in comparison to the uncertainties on the reported anisotropy
values.
Given the lack of a precipitous decline in the number of sources
immediately below the 1FGL threshold observed in the 2FGL, and more
recently in the 1FHL below the 2FGL threshold, the observed
discrepancy is unsurprising; the 2FGL over-produces the EGRB
anisotropy for precisely the same reason that models of the gamma-ray
point source distributions do.
The limited flux range below the 1FGL threshold probed by the 2FGL
implies that extrapolations of the 2FGL population to fluxes below the
2FGL threshold exceed the 2FGL limits by a substantial amount.
Barring a cutoff immediately below the 2FGL limit, our estimates of
the point source contribution to the EGRB is an underestimate by at
least a factor of two at high energies. As a consequence of the 2FGL
detection threshold the 2FGL contribution to the EGRB anisotropy
probes primarily the nearby point source population. That is, unlike
the EGRB anisotropy generally, it does not provide an independent
measure of the high-redshift source distribution. Thus, it is not
surprising that models which reproduce the characteristics of the 2FGL
are consistent with our implied limits upon the EGRB anisotropy
\citep[e.g.,][]{PaperV}.
Similarly, in the absence of a better understanding of the discrepancy
between estimates of the EGRB anisotropy at high energies, it is
unclear that it provides an independent constraint upon the
high-redshift gamma-ray universe at present.
\begin{appendix}
\section{K Correction}
\subsection{K-correcting $C_P$}
Computing the flux within the specific energy bands relevant for
comparison to the measured EGRB anisotropy generally requires some
spectral correction from fluxes measured in bands for which data is
readily available. This necessarily depends upon the spectral
shape. Here we use the power-law fit to the point source spectrum,
making the standard K-correction for each source individually:
\begin{equation}
\mathcal{F}_j = K(\Gamma_j) \mathcal{F}_{35,j}
\quad\text{where}\quad
K(\Gamma) \equiv
\frac{E_m^{1-\Gamma}-E_M^{1-\Gamma}}{1-100^{1-\Gamma}}\,,
\end{equation}
where $E_{m,M}$ are the lower and upper energy limits of the band in
GeV, and $\mathcal{F}_{35}$ is our standard flux reference.
Note that while the above provides the K-correction for
{\em individual} sources, since their individual spectra differ it is
not equivalent to a single K-correction, evaluated at some
effective spectral index. At high energies the population
near the 1FGL detection threshold is dominated by increasingly hard
sources, imparting an energy dependence to the typical spectral index
and generally enhancing the anisotropy
\citep[see, e.g., section VI of][]{Cuoco}.
\subsection{Spectral Shape Corrections}
\label{sec:K-shape-corr}
Applying the K-correction described above necessarily assumes that the
source spectra are well approximated by power laws. Despite their
characterization as such at lower energies, it is not clear that at
high energies this remains the case. In particular, above 100~GeV
considerable softening is anticipated for sources with redshifts
$\gtrsim0.5$. Thus, prior to using the K-correction, here we assess
its applicability and estimate the relevant spectral corrections. We
do this by comparing the K-corrected $\mathcal{F}_{35}$ to the fluxes measured
in individual energy bands near those of interest. For this purpose,
we make use of the 1FHL sources with counterparts in the 2FGL due to
the formers higher sensitivity and better reported energy resolution
(especially at high energies). To exclude any putative Galactic
component we impose a cut on Galactic latitude of $|b|>30^\circ$.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig7.eps}
\end{center}
\caption{Left: Distribution of the spectral corrections to the K-corrected
$\mathcal{F}_{35}$ for sources in the 1FHL and 2FGL catalogs for various
energy bands. The green line shows a log-normal distribution fit.
Sources beyond the vertical dashed lines are excluded so as not to
bias the width towards larger values. Right: Measured versus
K-corrected $\mathcal{F}_{35}$ within each energy band. Red and blue dots denote
detections and upper limits, respectively. The dotted line shows
the equal case, and the shaded area corresponds to the region
excluded by Equation (\ref{eq:kex}).
}\label{fig:kcorr}
\end{figure}
\begin{deluxetable}{ccccc}
\tablecaption{Spectral Corrections to the K-correction for {\em Fermi}\xspace
Point Sources. \label{tab:kcorr}}
\tablehead{
$E_m$\tablenotemark{a}
&
$E_M$\tablenotemark{a}
&
$F_{\rm min}$\tablenotemark{b}
&
$\left<k\right>$
&
$\left<k^2\right>$
}
\startdata
0.3 & 1 & $2\times10^{-9}$ & 1.01 & 1.05 \\
1 & 3 & $5\times10^{-10}$ & 1.06 & 1.16 \\
3 & 10 & $1.5\times10^{-10}$ & 1.04 & 1.13 \\
10 & 30 & $3\times10^{-11}$ & 1.01 & 1.22 \\
30 & 100 & $2\times10^{-11}$ & 0.88 & 1.08 \\
100 & 500 & $2\times10^{-11}$ & 0.47 & 0.29
\enddata
\tablenotetext{a}{In units of GeV}
\tablenotetext{b}{In units of ${\rm cm^{-2}s^{-1}}$}
\end{deluxetable}
The measured and K-corrected $\mathcal{F}_{35}$ are clearly strongly
correlated, exhibiting larger scatter at low fluxes, eventually
developing a plateau, typically dominated by upper limits (see the
right-hand panels of Figure \ref{fig:kcorr}). To avoid the Eddington
bias at low measured fluxes induced by the detection threshold, we
institute a lower flux limit of the form
\begin{equation}
\mathcal{F}_{mM} \times (K \mathcal{F}_{35}) > \mathcal{F}_{\rm min}^2\,,
\label{eq:kex}
\end{equation}
where $\mathcal{F}_{\rm min}$ is an energy-band dependent flux limit. The
excluded region is denoted by the shaded regions in the right-hand
panels of Figure \ref{fig:kcorr}, and the boundary is orthogonal to
the expected proportionality relation (and therefore should not induce
any Eddington bias). Values for the band-specific $\mathcal{F}_{\rm min}$
adopted here are listed in Table \ref{tab:kcorr}.
We define the source-specific spectral correction by
\begin{equation}
k \equiv \frac{\mathcal{F}_{mM}}{K \mathcal{F}_{35}}\,,
\end{equation}
which is simply the multiplicative correction to the K-correction.
Within each energy band the distribution of $k$, shown in Figure
\ref{fig:kcorr}, is well approximated by a log-normal distribution.
The parameters of the distributions are relatively independent of the
particular value of $F_{\rm min}$ employed. However, among bands the
distributions vary substantially. In the characterization of the
properties of the $k$ distributions we have excluded outliers, which
we define as objects with spectral corrections located more than three
standard deviations from the mean. This has the effect of reducing
the width of the distributions and biasing our estimated corrections
to the EGRB anisotropy spectrum towards lower values, making it a
conservative assumption.
At energies less than 30~GeV the K-corrected fluxes are quite
accurate, with mean spectral corrections, $\left< k \right>$, all of
order unity. At high energies, $\left< k \right>$ is less than unity,
indicating the anticipated softening of the spectra. Moreover, there
is some tentative evidence that this softening becomes more severe at
higher energies.
However, since it is $\mathcal{F}^2$ that enters into the estimate of $C_P$
employed here, of more importance is the mean-{\em square} spectral
correction, i.e., $\left< k^2 \right>$. This is impacted by not only
the movement in the centroid of the $k$ distribution, but also by its
width. Because of this, $\left< k^2 \right>$ is larger than unity for
all but the highest energy bin. For the 30--100~GeV bin this is
despite $\left<k\right>\simeq0.88$. The estimates of
$\left< k^2 \right>$ vary about 1.13.
Repeating the above analysis with various other latitude cuts (e.g.,
$|b|>20^\circ$, $|b|>40^\circ$) yields nearly identical results,
implying that the spectral corrections are indeed intrinsic to the
extragalactic source population and not associated with any
contaminating Galactic subpopulation.
Therefore, assuming that the distribution in $k$ is uncorrelated with
$\Gamma$ or $\mathcal{F}_{35}$, we adopt a uniform spectral correction of 1.13
below 100~GeV, corresponding to the average value across bins at these
energies.
\begin{deluxetable}{ccc}[th!]
\tablecaption{Kolmogorov-Smirnov Comparison of Latitude Cuts\label{tab:bKS}}
\tablehead{
$b_{\rm min}$
&
$P_{KS}^{\mathcal{F}_{35}}$
&
$P_{KS}^\Gamma$
}
\startdata
$0^\circ$ & $9.2\times10^{-6}$ & 0.13 \\
$5^\circ$ & $3.2\times10^{-4}$ & 0.43 \\
$10^\circ$ & $9.7\times10^{-3}$ & 0.65 \\
$15^\circ$ & 0.15 & 0.70 \\
$20^\circ$ & 0.60 & 0.74 \\
$25^\circ$ & 0.97 & 1.0 \\
$30^\circ$ & 1.0 & 1.0 \\
$35^\circ$ & 0.96 & 1.0 \\
$40^\circ$ & 0.30 & 0.98 \\
$45^\circ$ & 0.23 & 0.48
\enddata
\end{deluxetable}
\begin{figure*}
\begin{center}
\includegraphics[width=0.9\columnwidth]{fig8a.eps}\hspace{2em}
\includegraphics[width=0.9\columnwidth]{fig8b.eps}
\end{center}
\caption{Contribution to the EGRB anisotropy $\Delta C_P$ by unmasked
point sources. We contrast the cumulative (left) and and
differential (right) contribution of galactic latitude (color coded
for different cuts). We compare the reported measurements of $C_P$
\citep[blue,][]{Fermi_aniso} to the contribution of individual
unmasked 2FGL sources, which are detected in the respective energy
bins (triangles) and the unmasked 1FHL sources above 10 GeV
(squares). We show the contribution of all unmasked 2FGL sources
(detected above 1 GeV) that are K-corrected to the respective energy
band and corrected for spectral curvature (circles). The error bars
account for cosmic variance and the data points have been shifted
horizontally within their respective energy bin centers for
clarity.}\label{fig:direct_app}
\end{figure*}
\section{Cosmic Variance} \label{sec:CV}
Our method of summing over the unmasked 2FGL sources to obtain a lower
limit on $C_P$ (in either our conservative direct or statistical
approaches) allows a less stringent latitude cut than that applied by
the {\em Fermi}\xspace-LAT collaboration to increase the statistics of the source
number. Of course, this is only justified if there is no
contaminating Galactic population of point sources, which would bias
the inferred values of $C_P$. Here we demonstrate that this is indeed
the case: above $15^\circ$ the source populations are statistically
indistinguishable from that above $30^\circ$. Following this, we
estimate the cosmic variance in the measured $C_P$ via the cumulative
and differential contributions to the anisotropy from various Galactic
latitude cuts.
\subsection{Excluding the Galactic Point Source Population}
Extending the 2FGL to lower latitudes than the $30^\circ$ cut made by
\citet{Fermi_aniso} results in a considerably larger sample size, and
correspondingly lower cosmic variance. More over, it provides
additional realizations with which to estimate the cosmic variance
expected for the measured anisotropy spectrum. Here we verify that
less stringent latitude cuts are allowed by the 2FGL (though perhaps
not by the diffuse Galactic emission) by comparing the distribution of
source properties at different latitudes via the Kolmogorov-Smirnov
test.
Since the KS-test is one-dimensional, we explicitly compare the source
flux ($\mathcal{F}_{35}$) distribution and photon spectra index ($\Gamma$)
distribution separately. A cursory comparison by eye of the joint
distribution does not reveal any noticeable correlation, suggesting
that the comparison of the projected distributions is sufficient.
The KS-test returns a probability that may be loosely interpreted as
the likelihood that two samples drawn from the same distribution
differ as much as the two being compared. These are listed for a
variety of latitude cuts, encompassing the $30^\circ$ employed by
\citet{Fermi_aniso}, for the flux and photon spectral index
distributions in Table \ref{tab:bKS}.
(The inclusion of $b_{\rm min}=30^\circ$ is gratuitous, and returns
the expected probability of unity.) In these, very small values
indicate differing source populations.
Fluctuations become large at
very high latitude cuts due to the small number of remaining sources.
However, at latitudes below $10^\circ$ the disparity can only be
explained by the presence of an
additional, Galactic population. Above latitudes of $15^\circ$, this
population is subdominant, and above $20^\circ$ it may be ignored
altogether. These corresponds to an increase of nearly 50\% and 30\%
in the total source count, respectively.
\subsection{Direct Estimates of the EGRB Anisotropy for Varying Galactic Latitude}
In Figure~\ref{fig:direct_app}, we show the cumulative and and
differential contribution to the EGRB anisotropy $\Delta C_P$ by unmasked
point sources for varying galactic latitude. We perform a Monte-Carlo
integration of the respective sky masks to obtain the unmasked sky
fractions
$f_{\mathrm{sky}}=\{0.613,0.556,0.446,0.338,0.240,0.158,0.093\}$ for
$|b|>\{5^\circ,10^\circ,20^\circ,30^\circ,40^\circ,50^\circ,60^\circ\}$
(a similar computation using the 2FGL to construct the mask yields
$f^{\rm 2FGL}_{\mathrm{sky}}=\{0.556,0.507,0.406,0.310,0.222,0.145,0.083\}$
for the same latitude cuts). For
the differential $\Delta C_P$, our Monte-Carlo integration in
galactic area rings of equal area (indicated in
Figure~\ref{fig:direct_app}) yield sky fractions that scatter around
0.09.
At low energies, there is a systematic trend of an increasing
cumulative $\Delta C_P$ for decreasing galactic latitude, which could be due
to two reasons. 1.~Since the different values for $\Delta C_P$ are not
statistically independent (a less stringent galactic latitude cut
contains the point sources with a more conservative cut as a
subsample), this sequence of $\Delta C_P$ could be due to a regression to a
(high) mean with increasing sample size or 2.~this could sign a
population of soft-spectrum galactic point sources out to
$|b|<30^\circ$. At energies $E>10$~GeV, the differential contribution
to $C_P$ of equal-sky-area galactic latitude rings shows no systematic
trend in the (more complete) 1FHL sample and in the 2FGL sample for
$|b|>20^\circ$. The data is inconclusive whether the low-energy data
points at $|b|<30^\circ$ signal positive outliers or the hint of a
population of soft-spectrum galactic point sources. Our
Kolmogorov-Smirnov test suggest the first possibility and that it is
safer to use point sources with $|b|>20^\circ$ (at least at higher
energies).
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=\columnwidth]{fig9.eps}
\end{center}
\caption{Point spread functions (left) for the front (top) and back
(bottom) LAT detectors for the energy bands relevant for the EGRB
anisotropy measurements for the Pass 6v3 (red) and Pass 7v6 (blue)
IRFs, and their associated window functions (right). The point
spread function/window function for the 1.04--1.99~GeV,
1.99--5.00~GeV, 5.00--10.4~GeV, and 10.4--50.0~GeV energy bands are
shown by the solid, long-dash, short-dash, and dotted lines,
respectively.}\label{fig:PSF}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.75\columnwidth]{fig10.eps}
\end{center}
\caption{Correction factor relating the EGRB anisotropy spectrum
using the Pass 6v3 and Pass 7v6 IRF to deconvolve the beam. The
correction for the front and back detector in the LAT are shown by
the blue circles and red squares, respectively.}\label{fig:fPSF}
\end{figure}
\section{{\em Fermi}\xspace Point Spread Function} \label{sec:PSF}
Here we present an estimate of the impact of the recent update of the
point-spread function (PSF) deconvolution procedure employed by
\citet{Fermi_aniso}.
The PSF used was from the Pass 6v3
instrument response function (IRF), and based upon Monte Carlo
simulations of the detector. Subsequent on-orbit calibrations, first
reported in Pass 6v11 IRF and used above 1~GeV in the current IRF,
Pass 7v6, exhibit much broader PSFs at high energies, shown in Figure
\ref{fig:PSF}. While this effect was discussed in
\citet{Fermi_aniso}, estimates using the most recent IRF suggest that
the impact of the PSF on the window function, related to the PSF by
\begin{equation}
W_l = 2\pi \int_{-1}^1 d\mu P_l(\mu) {\rm PSF}(\theta)\,,
\end{equation}
where $\mu\equiv\cos\theta$ and $P_l$ are Legendre polynomials, is
considerably larger than reported there.
The Pass 6v3 and Pass 7v6 PSFs and window functions are compared in
Figure \ref{fig:PSF}. Since the PSF deconvolved anisotropy signal is
$\propto |W_l|^{-2}$ \citep[see Equation (4) of][]{Fermi_aniso},
underestimating the width of the PSF (and thus overestimating
$|W_l|^2$) results in a corresponding underestimate of the anisotropy.
The magnitude of the correction to the window function depends upon
the multipole at which the $C_l$ is determined. The reported $C_P$
correspond to the $C_l$ averaged over $150\le l\le500$, and thus the
relevant correction factor is
\begin{equation}
f_{\rm PSF}
=
\sum_{l=150}^{500} \frac{1}{|W_{{\rm 7v6},l}|^{2}}
\bigg/
\sum_{l=150}^{500} \frac{1}{|W_{{\rm 6v3},l}|^{2}}\,,
\end{equation}\\
which is most heavily weighted towards large $l$, where $|W_l|^2$ is
smallest. This is shown for both the back and front detectors in
Figure \ref{fig:fPSF}, with a typical value at high energies ranging
from 1.5 to 2.8.
In practice, the correction factor depends upon the fraction of the
anisotropy signal associated with the front and back detectors.
Given the similarity in the effective area of the two LAT detectors,
adopting an equal anisotropy signal in each gives an effective
correction factor of roughly 2.2.
\end{appendix}
\acknowledgments The authors thank Niayesh Afshordi, Alessandro Cuoco,
Jennifer Siegal-Gaskins, and the {\em Fermi}\xspace~collaboration for helpful
discussions and Volker Springel for careful reading of the
manuscript. A.E.B.~receives financial support from the Perimeter
Institute for Theoretical Physics and the Natural Sciences and
Engineering Research Council of Canada through a Discovery Grant.
Research at Perimeter Institute is supported by the Government of
Canada through Industry Canada and by the Province of Ontario through
the Ministry of Research and Innovation. C.P.~gratefully acknowledges
financial support of the Klaus Tschira Foundation. E.P. acknowledges
support by the DFG through Transregio 33. P.C. gratefully acknowledges
support from the UWM Research Growth Initiative, from {\em Fermi}\xspace Cycle 5
through NASA grant NNX12AP24G, from the NASA ATP program through NASA
grant NNX13AH43G, and NSF grant AST-1255469.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,727
|
On May 3-5, the biggest paramotor event of 2019 in Latin America: read everything here!
Our journey to discover the Atom 80 engine continues, keep on following us!
From August 30th to September 1st do not miss the largest paramotor rally in Italy, near Vittorazi headquarters!
We start today a long journey inside our Atom 80, to let you know its most important technical features.
Follow us to learn all the secrets of our little, big engine!
Vittorazi will organize in May the first Maintenance Masterclass in South America. The course will be held in Intanhaém, São Paulo, Brazil, between May 3/5 during the big event 'Break the World Record - More pilots in flight'.
A year full of events and competitions, to be lived together. Discover the calendar here and mark the appointments of the season, not to miss anyone! Among the most important events, there will be the Vittorazi Day, in memory of Vittoriano Orazi.
Finnaly it has arrived! Our new Moster 185 MY '19 Handy box it's a small supply of spare parts to use for ordinary or extraordinary maintenance of the engine, thanks to a ready-to-use kit that allows you to intervene for the small necessities.
A unique Festival, combining various events from flight and entertainment to concerts and shows, together with other typical festival activities: Parafest, where Vittorazi will participate together with our partner Parajet.
A chat with one of America's most renowned paramotor pilots, who has chosen Vittorazi products for his endeavours, and who was given a special, custom-made engine emblazoned with the colours and graphics of Old Glory.
Vittorazi is taking part at one of the most important competitions of the international paramotoristic sports season: the French National Championship, held for a week in Blois.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,084
|
Nov. 3, 2020—A selection of recent accolades awarded to faculty and alumni
Medieval Mindset: Kress Foundation grant allows for expansion of access to medieval and Renaissance works
Nov. 3, 2020—The Vanderbilt University Fine Arts Gallery has been selected among spring 2020 applicants to receive support from the Samuel H. Kress Foundation for an exhibition of medieval and Renaissance artworks.
From the Dean, Fall 2020 Peabody Reflector
Oct. 6, 2020—A message from Dean Camilla P. Benbow for the Fall 2020 issue of the Peabody Reflector
Ideas in Action – The Conversation
Oct. 5, 2020—Peabody researchers regularly contribute to The Conversation, an independent, not-for-profit media outlet that uses content sourced from academics and researchers. The research-based stories are often republished by large media outlets.
Ideas in Action – Thought Leaders
Oct. 5, 2020—Peabody faculty members frequently contribute ideas to public discourse about education and human development. Here is a selection of recent faculty appearances in popular media.
Ideas in Action – Additional Faculty Honors
Oct. 5, 2020—Listing of additional faculty honors from the Fall 2020 Peabody Reflector
Recent books by Peabody faculty
Oct. 2, 2020—Books on specific strategies and practices for using digital tools to reduce inequities in educational opportunities and improve student outcomes and the core ingredients of leadership have been recently published by Vanderbilt faculty.
Vanderbilt alumnus' persistence helped propel 19th Amendment ratification
Aug. 26, 2020—At a time when Americans are voicing their opinions at the ballot box and in other ways, Vanderbilt University is joining people across Tennessee and the nation in commemorating the centennial of the long-fought battle to secure a woman's constitutional right to vote.
"Ignore Boundaries" Dean Mark Wait reflects on his decades at the Blair School of Music
Jul. 17, 2020—by John Pitcher It's the end of an era at Blair. Mark Wait, the Martha Rivers Ingram Dean and professor of music, retired June 30. Blair was a relatively small institution when Wait took the helm in 1993. It had 19 full-time faculty who taught and performed in a 31,000-square-foot facility. Today the school boasts...
Blair Faculty Notes
Jul. 17, 2020—Kirsten Agresta Copely, adjunct associate professor of harp, released Around The Sun, an all-original new age harp album in January. It takes listeners on a journey through a 365-day cycle from "Daybreak" to "Winter's Bone" with introspective melodies and ambient soundscapes. This marks her third solo album and first as composer. Molly Barth, assistant...
Blair Student Accolades
Jul. 17, 2020—Undergraduate competition winners Max Randal, a rising senior piano performance major and computer science minor, was named the Tennessee Music Teachers Association Young Artist Collegiate Competition state winner in November 2019. Kaila Geisinger, a rising senior harp performance major, took second prize representing the Southeastern Region in the Advanced Division at the American Harp Society...
Meet Lorenzo F. Candelaria, new dean at the Blair School of Music
Jul. 6, 2020—The Blair School of Music begins a new leadership era as it welcomes Lorenzo F. Candelaria as its new dean.
Page 2 of 24«12345678...»Last »
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,025
|
Messages from a Master Teacher
Empress of Everything–a film by Wendy Galson and Greg Windle
A Public Screening of the Empress of Everything–Messages from a Master Teacher
On May 8, 2018 May 9, 2018 By wendygalsonIn Uncategorized
Please join me in supporting Science Leadership Academy at Beeber (SLA@B), which is adding its first 5th grade class in School Year 2018-2019! I have been the School Psychologist at SLA@B since it began in 2013. I admire the staff and am amazed by the students. Adding project-based middle years is their next adventure.
Please join me at the first School District of Philadelphia screening of Empress of Everything-Messages from a Master Teacher at 4:00 PM on Wednesday May 23 at 5925 Malvern Ave, Philadelphia, PA. (This is school with a parking lot.) Suggested donation to the Home and School Association is $5.00.
I began working on this documentary film before I worked for the School District of Philadelphia, and when I retire from the School District in June, I will be traveling to conferences and festivals to promote it. The film shows the inner workings of a school where young children love learning. I hope that the film starts conversations about the importance of the relationship between student and teacher.
For background and updates, please follow my blog Messages from a Master Teacher and my Facebook page Messages from a Master Teacher
Empress of Everything-Messages from a Master Teacher is available for purchase!
Responsive Classroom®
Talking about Learning
I am Dr. Wendy Galson, school psychologist and first time filmmaker. With our film Empress of Everything–Messages from a Master Teacher, and with this blog, I want to encourage conversation about RELATIONSHIP–knowing and being known–between teacher and learner. I want to encourage conversation about how skilled attention to children's interactions can help build a supportive learning community where everyone is valued.
Pizza Day!
Children Learn from People They Love
DVD of Empress of Everything now available for sale!
Copyright Wendy Galson 2015
View doctorwendy's profile on Twitter
View wrgalson's profile on Instagram
View wendy-galson's profile on LinkedIn
View uwendygalson's profile on Vimeo
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,922
|
Der Arroyo de las Víboras ist ein Fluss in Uruguay.
Der Arroyo de las Víboras verläuft auf dem Gebiet des Departamentos Treinta y Tres und gleicht mehr einer sumpfigen Tallandschaft als einem Fluss. Er mündet in den Arroyo de Avestruz Chico.
Einzelnachweise
Fluss im Departamento Treinta y Tres
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,147
|
Brazil's Costa Establishes New Streak at Dramatic Walt Disney World Marathon
2012 Walt Disney World Marathon
@Brazilian runner wins second straight Disney Marathon after fierce battle with perennial winner Adriano Bastos; Virginia Beach's Renee High easily wins women's title; field of more than 16,500 pushes weekend total to record 56,000 runners
LAKE BUENA VISTA, Fla. (Jan. 8, 2012) – Brazilian runner Fredison Costa did it again, holding off his heavily-favored countryman Adriano Bastos on Sunday to win his second straight Walt Disney World Marathon presented by Cigna.
Costa waged a fierce battle with the 7-time champion Bastos throughout the 26.2 mile race through Disney's four theme parks – Magic Kingdom, Epcot, Animal Kingdom and Hollywood Studios – before narrowly winning. Costa crossed the finish line in 2:19:01 – just 14 seconds ahead of Bastos (2:19:15), who had won six Disney Marathons in a row before Costa's back-to-back feat.
Renee High of Virginia Beach, VA., was the first woman to cross the line. High, who is scheduled to run the marathon at the US Olympic Trials next week in Houston, completed the race Sunday in 2:48:33 just ahead of Emily Kroshus (2:53:11) of Boston and Christa Stephens (2:55:10) of St. Petersburg, Fla.
Costa, 34, and Bastos 33, ran virtually shoulder to shoulder through much of the race. But with about two miles to go, Costa inched ahead slightly and maintained that slim lead until the end top become the first person other than Bastos to win back-to-back Disney Marathon titles since Brazil's Santiago de Araujo did it in 1998 and 1999. Costa's time was even quicker than a year ago when he won with a time of 2:21:15. Matt Hensley of Port St. Lucie, Fla. was third (2:23:11).
"I think a victory is a victory and they are always very special," Costa said through an interpreter. "But I think this was such a difficult time overcoming some things for me. I've been through so many things that this race was very, very special to me."
Douglas Morton of Sebring, Fla. won the men's Masters Division (runners age 40 and over) with a time of 2:35:52 and Terri Rejimbal of Tampa won the women's Masters Division (3:09:33). In the wheelchair division, Krige Schabort of Cedartown, Ga. (1:46:16) won for the fourth straight year.
On Saturday, Jose De Morais of Brazil won the 13.1-mile Disney Half Marathon with a time of 1:10.12. Rosa Chacha of Ecuador was the top woman finisher, posting a time of 1:16.43.
In all, runners from all 50 states and 60 other countries took part in a variety of Disney Marathon Weekend events, such as the Family Fiesta 5K, Kids Races and the Disney Half-Marathon in addition to the Disney Marathon. And once again more than 6,500 runners put themselves to the ultimate test – Goofy's Race and a Half Challenge – by running the half-marathon on Saturday and the marathon on Sunday.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,484
|
Fort Carlton (initialement Carlton House) fut construit en 1810 comme poste de ravitaillement pour la Compagnie de la Baie d'Hudson. Situé au sud de la rivière Saskatchewan Nord, le Fort reste un dépôt de peaux important presque jusqu'à la fin du commerce de la fourrure dans la région. Le fort a été désigné lieu historique national du Canada en 1976 et est un parc provincial de la Saskatchewan depuis 1987.
C'est dans ses parages que sera signé le Traité 6 de 1876 entre les Cris et le gouvernement du Canada. Le fort est détruit par un incendie en 1885, lors de la Rébellion du Nord-Ouest.
Histoire
En 1976, le fort Carlton a été désigné Lieu historique national. Il est devenu un parc provincial le .
Notes et références
Annexes
Liens externes
Parc provincial en Saskatchewan
Rébellion du Nord-Ouest
Compagnie de la Baie d'Hudson
Fort au Canada
Lieu historique national en Saskatchewan
Aire protégée créée en 1986
Lieu patrimonial en Saskatchewan
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,114
|
{"url":"https:\/\/www.scipedia.com\/public\/Guti%C3%A9rrez_Romero_et_al_2017a","text":"## Abstract\n\nThe complexity of the dynamic behaviour of offshore marine structures requires advanced simulations tools for the accurate assessment of the seakeeping behaviour of these devices. The aim of this work is to present a time-domain model for solving the dynamics of floating marine devices, subjected to non-linear environmental loads and paying special attention on the mooring dynamics. First, the formulation of the hydrodynamic approach for solving the wave-floater interaction is introduced. Second, the solver of the mooring dynamics, based on a non-linear Finite Element Method approach, is presented. Third, a procedure for coupling the hydrodynamic along with other external loads, with the floating structure and mooring dynamics is described. Fourth, some validation examples and comparisons among different mooring approaches are presented. Fifth, an analysis of the OC3 floating wind turbine concept is performed to study the influence of different mooring models, the effects of non-linear waves on the platform, and the tension in the mooring system. The dynamic mooring model along with the second-order wave model produce realistic simulations of the floating wind turbine performance.\n\n## 1 Introduction\n\nResearch trends in Marine Renewable Energies (MRE) are mainly focused in offshore wind energy due to the high expectations raised by this technology. The technology for marine wind turbines is currently well-developed, but limited for fixed installations in shallow-water areas. The next horizon is focused on deep-water technology [1], but different challenges for Floating Offshore Wind Turbines (FOWT) are not solved yet [2], such as the dynamic stability in the presence of non-linear ambient loads [3]. In fact, an accurate prediction of the dynamic response of a FOWT, considering the interaction among the hydrodynamics, mooring, and aerodynamics of the turbine, is identified as one of the key challenges for the simulation tools required to design the future FOWTs [4,5,6].\n\nStandard design procedures and simulation tools for marine structures come from the existing technology and experience of the offshore oil and gas industry. For instance, the classic simulation approaches are based on uncoupled formulations, where the hydrodynamic response of the floater is linearised and can be decoupled from the mooring [7,8]. Recently, coupled simulations have been adopted to solve the seakeeping of FOWT devices, since the dynamic of FOWT offers a high complexity due to the variety of loads and non-linear effects. Anyhow, the interaction among different components, such as the wind turbine structure, the rotor dynamics, the mooring arrangements and the floating structure must be taken into account in a more accurate way [9,10].\n\nThe analysis of FOWT should be carried out with simulation codes capable to include the physics governing the dynamic response of these devices. With regards to marine structures, Low and Langley [11] showed that the dynamic response of a floating production system in a random sea can be split in two timescales: low frequency and wave frequency responses. Moreover the seakeeping of the floating device and the dynamics of the mooring lines are coupled. Then, the analysis should take into account the interaction between them, and the time-domain analysis seems to be the right way to simulate this sort of coupled problems. In fact, the American Bureau of Shipping [32] considers that the global seakeeping analysis of a FOWT should take into account: the unsteady wind loads, wind turbine control systems, wind turbine-platform interaction, wave actions over the platform, currents, mooring loads, and any other types of external actions. It will be presented later on that the time-domain approach allows to handle these actions in a natural manner.\n\nThe first works proposing coupled formulations for floating production systems were developed for TLP-type platforms [12,13,14], where it was found that the sum-frequency effects are important. Also other authors investigated and developed models to couple different effects and loads in floating devices [15,16,17,18]. More recently, Cordle and Jonkman [5] made a review of the state of art of the simulation tools for FOWT. Bae and Kim [9] recently presented a research on the effects of second-order wave excitation in a mono-column-TLP type FOWT comparing with the uncoupled simulations in time-domain. Karimirad and Moan [19] developed a simplified method for the coupled dynamics of a offshore wind turbine aiming at saving computational time. They also analysed a spar-type FOWT, and compared different hydro-elasticity codes [2]. In [3], the dynamic response of the spar-type platform subject to wave induced excitation was investigated numerically, including catenary mooring cables. Matha et al. [6] investigated a platform-tower coupled with a TLP-type FOWT. Jonkman and Sclavounos [20] presented a tool integrating an aeroelastic model for onshore wind turbines, and a hydrodynamic load model, together with a Quasi-Static (QS) catenary model for mooring lines. More sophisticated mooring models, which are usually based on Finite Element Methods (FEM) have been investigated; for instance, Kim et al. [21] presented a comparison between a linear spring and a non-linear FEM mooring for the analysis of the dynamic response when they are coupled with a floating structure. Several coupled seakeeping analyses with FEM mooring line models can also be found in the literature [22,23,24,25,26,27].\n\nFrequency-domain and time-domain approaches are used for seakeeping analyses of marine structures. However, the frequency-domain approach has difficulties to accurately handle non-linearities such as those arising from the mooring lines, and the low frequency components of the wave-body interaction, as appointed by Low [29], while the time-domain analysis can straightforwardly include any non-linearity within each time step in a natural manner. However, the main drawback of the time-domain is an increase of the computational time required for the simulations. To overcome this limitation, Low and Langley [11] and Low [29] presented a hybrid time-frequency domain model to simulate low and wave frequency response of coupled problems.\n\nIn this work, an extension of the time-domain method proposed by Serv\u00e1n-Camas and Garc\u00eda-Espinosa [30,31] is used. This method obtains a relevant reduction of the computational time thanks to the use of deflation techniques and High Performance Computation based on Graphical Processor Units (GPU). The tool is based on the Finite Element Method (FEM) and is able to solve multi-body seakeeping problems on unstructured meshes including any type of external load. Other examples of FEM solvers for seakeeping analysis can be found in the literature. For instance, Hong and Nam [33], who presented a second-order analysis of wave forces using FEM in the time-domain, and investigated the interactions with multi-body devices.\n\nThis work presents the development of a coupled FEM seakeeping and mooring models for the analysis of floating offshore structures. The model is able to take into account first and the second-order irregular waves, wind loads, and mooring loads. A novel iterative scheme for coupling the wave diffraction-radiation solver developed by Serv\u00e1n-Camas and Garc\u00eda-Espinosa [28,30,31] with non-linear aerodynamics and mooring loads is also described. The paper is organized as follows. First, the governing equations of the body dynamics are introduced. Second, the second-order diffraction-radiation wave problem is described. Third, the non-linear FEM model for mooring lines is introduced in detailed. Fourth, the coupling between the dynamics of the floater with non-linear loads, such as mooring lines forces, is explained. Fifth, some validations with experimental results are shown, as well as a comparison of the non-linear FEM model with the Quasi-Static catenary model. Sixth, we analyse an operational case of a spar buoy FOWT based on the OC3 concept and NREL 5 MW turbine. This case study aims at evaluating the influence of the type of mooring model, as well as the effects of non-linear wave on the dynamics of the platform. Finally, some relevant conclusions from the obtained results are presented.\n\n## 2 Problem statement\n\n### 2.1 General dynamics of floater\n\nThe motion of a floater subject to ambient loads can be modelled using the rigid body dynamics. Let OXYZ be a fixed global frame of reference, and let Gxyz be a local frame of reference located at the center of gravity, and whose axis are parallel to the axis of the global frame (see Fig. 1). Assuming small rotations, for each time step, the body accelerations can be obtained respect to the local frame from the rigid-body dynamic equations:\n\n ${\\displaystyle \\mathbf {M} _{b}\\cdot {\\ddot {\\mathbf {x} }}=\\mathbf {f} \\left(t\\right),}$ (1) ${\\displaystyle {\\bar {\\bar {\\mathbf {I} }}}_{b}\\cdot {\\ddot {\\mathbf {r} }}_{b}=\\mathbf {m} \\left(t\\right),}$ (2)\n\nwhere ${\\textstyle \\mathbf {M} _{b}}$ is the mass matrix, ${\\textstyle {\\ddot {\\mathbf {x} }}}$ is the linear acceleration vector , ${\\textstyle {\\bar {\\bar {\\mathbf {I} }}}_{b}}$ is the instantaneous inertia tensor, ${\\textstyle \\mathbf {f} \\left(t\\right)}$ and ${\\textstyle \\mathbf {m} \\left(t\\right)}$ are the external loads vector (hydrostatics, wave loading, mooring, wind, etc.) and the external moments respectively; finally ${\\textstyle {\\ddot {\\mathbf {r} }}_{b}}$ is the angular acceleration vector.\n\n Figure 1: Global and local frame of reference used in computation of floater dynamics.\n\nThe calculation of the external forces ${\\textstyle \\mathbf {f} \\left(t\\right)}$ over the body constitutes an essential part of the dynamics, and will be described in more detailed later on.\n\n### 2.2 Flow equation and boundary value problem\n\nAssuming incompressible and irrotational flow in a domain ${\\textstyle \\Omega }$, being ${\\textstyle \\Gamma _{b}}$ the wetted surface of the floating device, the wave problem governing equations are [28]:\n\n ${\\displaystyle \\Delta \\varphi =0{\\mbox{ on}}\\;\\Omega ,}$ (3) ${\\displaystyle {\\frac {\\partial \\xi }{\\partial {t}}}+{\\frac {\\partial {\\varphi }}{\\partial {x}}}{\\frac {\\partial {\\xi }}{\\partial {x}}}+{\\frac {\\partial {\\varphi }}{\\partial {y}}}{\\frac {\\partial {\\xi }}{\\partial {y}}}-{\\frac {\\partial {\\xi }}{\\partial {z}}}=0{\\mbox{ on}}\\;z=\\zeta ,}$ (4) ${\\displaystyle {\\frac {\\partial {\\varphi }}{\\partial {t}}}+{\\frac {1}{2}}\\left|\\nabla ^{2}\\varphi \\right|+g\\xi +{\\frac {{\\mbox{P}}_{fs}}{\\rho }}=0{\\mbox{ on}}\\;z=\\zeta ,}$ (5) ${\\displaystyle \\mathbf {\\mbox{v}} _{\\varphi }\\cdot \\mathbf {n} +\\mathbf {\\mbox{v}} _{b}\\cdot \\mathbf {n} =0{\\mbox{ on}}\\;\\Gamma _{b},}$ (6) ${\\displaystyle {\\mbox{ P}}_{b}=-\\rho {\\frac {\\partial {\\varphi }}{\\partial {t}}}-{\\frac {1}{2}}\\rho \\left|\\nabla ^{2}\\varphi \\right|-\\rho gz_{b}{\\mbox{ on}}\\;\\Gamma _{b},}$ (7)\n\nwhere ${\\textstyle \\varphi }$ is the velocity potential, ${\\textstyle \\xi }$ is the free surface elevation, ${\\textstyle g}$ is the gravity acceleration, ${\\textstyle {\\mbox{P}}_{fs}}$ is the pressure on free surface, ${\\textstyle \\rho }$ is the fluid density, ${\\textstyle \\xi }$ is the wave elevation, ${\\textstyle {\\mbox{P}}_{b}}$ is a pressure at a point over body, ${\\textstyle \\mathbf {v} _{\\varphi }}$ is the fluid velocity, ${\\textstyle \\mathbf {v} _{b}}$ is the local velocity at a point on the wetted body surface, ${\\textstyle \\mathbf {n} }$ is the vector normal to the body wetted surface (pointing upward this surface), and ${\\textstyle z_{b}}$ is the vertical coordinate of any point of the body.\n\n### 2.3 Second-order diffraction-radiation governing equations\n\nThe governing equations for the second-order diffraction-radiation wave problem are obtained applying Taylor expansion on the boundary surfaces of a time-independent domain. This approach allows to approximate the free surface on ${\\textstyle z=\\zeta }$ and the mean body surface ${\\textstyle \\Gamma _{b}^{0}}$ at time ${\\textstyle t}$. Then, a perturbed solutions based on Stokes expansion procedure is applied to the velocity potential, free surface elevation, and body motion. More details can be found in [28].\n\nThe solution can be decomposed as\n\n ${\\displaystyle \\varphi =\\psi +\\theta ,}$ (8) ${\\displaystyle \\xi =\\eta +\\zeta ,}$ (9)\n\nwhere ${\\textstyle \\psi }$ is the incident wave velocity potential, ${\\textstyle \\theta }$ is the diffraction-radiation wave velocity potential, ${\\textstyle \\eta }$ is the incident wave elevation, and ${\\textstyle \\zeta }$ is the diffraction-radiation wave elevation. Then, the wave diffraction-radiation governing equations up to second-order are [28]:\n\n ${\\displaystyle \\Delta \\theta ^{1+2}=0{\\mbox{ in}}\\,\\Omega ,}$ (10) ${\\displaystyle {\\frac {\\partial \\eta ^{1+2}}{\\partial {t}}}-{\\frac {\\partial {\\theta ^{1+2}}}{\\partial {z}}}=-{\\frac {\\partial {\\theta ^{1}}}{\\partial {x}}}{\\frac {\\partial {\\eta ^{1}}}{\\partial {x}}}-{\\frac {\\partial {\\theta ^{1}}}{\\partial {y}}}{\\frac {\\partial {\\eta ^{1}}}{\\partial {y}}}}$ ${\\displaystyle -{\\frac {\\partial {\\theta ^{1}}}{\\partial {x}}}{\\frac {\\partial {\\zeta ^{1}}}{\\partial {x}}}-{\\frac {\\partial {\\theta ^{1}}}{\\partial {y}}}{\\frac {\\partial {\\zeta ^{1}}}{\\partial {y}}}-{\\frac {\\partial {\\psi ^{1}}}{\\partial {x}}}{\\frac {\\partial {\\eta ^{1}}}{\\partial {x}}}-{\\frac {\\partial {\\psi ^{1}}}{\\partial {y}}}{\\frac {\\partial {\\eta ^{1}}}{\\partial {y}}}{\\mbox{ on }}z=0}$ (11) ${\\displaystyle {\\frac {\\partial {\\theta ^{1+2}}}{\\partial {t}}}+{\\frac {{\\mbox{P}}_{fs}}{\\rho }}+g\\eta ^{1+2}=-\\eta ^{1}{\\frac {\\partial {}}{\\partial {z}}}\\left({\\frac {\\partial {\\theta ^{1}}}{\\partial {t}}}\\right)-\\zeta ^{1}{\\frac {\\partial {}}{\\partial {z}}}\\left({\\frac {\\partial {\\theta ^{1}}}{\\partial {t}}}\\right)}$ ${\\displaystyle -\\eta ^{1}{\\frac {\\partial {}}{\\partial {z}}}\\left({\\frac {\\partial {\\psi ^{1}}}{\\partial {t}}}\\right)-{\\frac {1}{2}}\\nabla \\theta ^{1}\\cdot \\nabla \\theta ^{1}-\\nabla \\psi ^{1}\\cdot \\nabla \\theta ^{1}{\\mbox{ on }}z=0,}$ (12) ${\\displaystyle \\mathbf {\\mbox{v}} _{\\theta }^{1+2}\\cdot \\mathbf {\\mbox{n}} ^{1}=-\\left(\\mathbf {\\mbox{v}} _{b}^{1+2}+\\mathbf {\\mbox{v}} _{\\psi }^{1+2}+\\mathbf {r} _{b}^{1}\\cdot \\left(\\nabla \\mathbf {\\mbox{v}} _{\\theta }^{1}+\\nabla \\mathbf {\\mbox{v}} _{\\psi }^{1}\\right)\\right)\\cdot \\mathbf {\\mbox{n}} ^{1}{\\mbox{ on}}\\,\\Gamma _{b},}$ (13) ${\\displaystyle {\\frac {{\\mbox{P}}_{b}^{1+2}}{\\rho }}=-gz_{b}-g\\mathbf {r} _{b}^{1+2}-{\\frac {\\partial {\\theta }^{1+2}}{\\partial {t}}}-\\mathbf {r} _{b}^{1}\\cdot \\nabla \\left({\\frac {\\partial {\\theta ^{1}}}{\\partial {t}}}\\right)}$ ${\\displaystyle -{\\frac {1}{2}}\\nabla \\theta ^{1}\\cdot \\nabla \\theta ^{1}-\\nabla \\psi ^{1}\\cdot \\nabla \\theta ^{1}{\\mbox{ on}}\\,\\Gamma _{b}\\,\\,,}$ (14)\n\nwhere superscripts 1 and ${\\textstyle 1+2}$ denote the components at the first-order and up to second-order solution, and ${\\textstyle r_{b}}$ is the displacement vector at a point over body.\n\nWith regards to the ambient loads applied over wetted body surface, hydrodynamic wave forces and moments can be obtained directly from pressure integration over the wetted body surface. Thus, it can be written that\n\n ${\\displaystyle \\mathbf {F} ^{1+2}=\\mathbf {F} _{H}^{0}+\\mathbf {F} _{H}^{1+2}+\\mathbf {F} _{D}^{1+2},}$ (15) ${\\displaystyle \\mathbf {M} ^{1+2}=\\mathbf {M} _{H}^{0}+\\mathbf {M} _{H}^{1+2}+\\mathbf {M} _{D}^{1+2},}$ (16)\n\nwhere subscript ${\\textstyle H}$ denotes the hydrostatic loads and ${\\textstyle D}$ denotes the dynamic loads. ${\\textstyle \\mathbf {F} ^{0}}$ and ${\\textstyle \\mathbf {M} ^{0}}$ are the initial hydrostatic forces and moments over the body, ${\\textstyle \\mathbf {F} _{H}^{1+2}=\\mathbf {F} _{H}^{1}+\\mathbf {F} _{H}^{2}}$ and ${\\textstyle \\mathbf {M} _{H}^{1+2}=\\mathbf {M} _{H}^{1}+\\mathbf {M} _{H}^{2}}$ are the up to second-order hydrostatic loads and moments, and ${\\textstyle \\mathbf {F} _{D}^{1+2}}$ and ${\\textstyle \\mathbf {M} _{D}^{1+2}}$ are the up to second-order dynamic loads and moments. Details of each component can be found in [28].\n\n## 3 Non-linear FEM model for mooring dynamics\n\n### 3.1 Equations for cable dynamics\n\nThe dynamic equations for a mooring cable with length ${\\textstyle L}$ with negligible bending and torsional stiffness can be formulated as [34,35]\n\n ${\\displaystyle \\left(\\rho _{w}C_{m}A_{0}+\\rho _{0}\\right){\\frac {\\partial ^{2}\\mathbf {r} _{l}}{\\partial t^{2}}}={\\frac {\\partial }{\\partial l}}\\left(EA_{0}{\\frac {e}{e+1}}{\\frac {\\partial \\mathbf {r} _{l}}{\\partial l}}\\right)+\\mathbf {f} \\left(t\\right)\\left(1+e\\right),}$\n(17)\n\nwhere ${\\textstyle \\rho _{w}}$ is the water density, ${\\textstyle C_{m}}$ is the added mass coefficient, ${\\textstyle \\rho _{0}}$ is the mass per unit length of the unstretched cable, ${\\textstyle \\mathbf {r} _{l}}$ is the position vector, ${\\textstyle E}$ is the Young's modulus, ${\\textstyle A_{0}}$ is the cross-sectional area of the cable, ${\\textstyle e}$ is the strain, ${\\textstyle \\mathbf {f} }$ are the external loads applied on the cable, and ${\\textstyle l}$ is the length along the unstretched cable.\n\nThe boundary conditions are given by\n\n ${\\displaystyle {\\frac {\\partial ^{2}\\mathbf {r} _{l}}{\\partial t^{2}}}=0l=0\\;\\;\\;\\;\\left({\\mbox{ at the anchor connection point}}\\right),}$ (18) ${\\displaystyle {\\frac {\\partial ^{2}\\mathbf {r} _{l}}{\\partial t^{2}}}={\\ddot {\\mathbf {r} }}_{fl}l=L\\;\\;\\;\\;\\left({\\mbox{ at the fairlead connection point}}\\right),}$ (19)\n\nwhere ${\\textstyle {\\ddot {\\mathbf {r} }}_{fl}}$ is the second derivative of the position vector at the fairlead connection point. The external loads acting on the cable, considered in this work, are the self-weight of the cable, hydrostatic loads, drag forces [36], and seabed interaction.\n\n### 3.2 Seabed interaction\n\nWhen the cable undergoes an excitation, the part which lies between the anchor and the touchdown point interacts with the seabed. This interaction is a complex non-linear effect. In this work, the seabed-mooring interaction is modelled as a spring-damping system [37,38], and is implemented as follows:\n\n1. The vertical forces due to the stiffness of the seabed in each time step are calculated as\n2. ${\\displaystyle {\\begin{array}{l}c_{l}w\\,l_{l}&&{\\mbox{ if}}\\;G_{s}A_{c}\\left(z_{r}-z^{t+\\Delta t}\\right)>c_{l}wl_{l},\\\\G_{s}A_{c}\\left(z^{t}-z_{r}+\\Delta t{\\dot {z}}^{t}+\\Delta t^{2}\\left({\\frac {1}{2}}-\\beta \\right){\\ddot {z}}^{t}\\right)&&{\\mbox{ otherwise}},\\end{array}}}$\n(21)\n\nwhere ${\\textstyle c_{l}>1}$ is a coefficient limiting the seabed reactions, ${\\textstyle l_{l}}$ is the portion of the cable resting on the seabed, ${\\textstyle w}$ is the weight per unit of length of the line, ${\\textstyle w}$ is the apparent weight of the cable, ${\\textstyle \\beta }$ is the term related to the numerical integration of the dynamic FEM model, ${\\textstyle z}$ is the vertical coordinate of the cable, ${\\textstyle G_{s}}$ is the seabed stiffness, ${\\textstyle A_{c}}$ is the contact area of the line, and ${\\textstyle \\Delta t}$ is the time step. The term ${\\textstyle z_{r}}$ is defined as\n\n ${\\displaystyle z_{r}=z+{\\frac {wl_{l}}{G_{s}D}},}$\n(22)\n\nwhere ${\\textstyle D}$ is the diameter of line.\n\nThe term ${\\textstyle z_{r}}$ can be defined as a limit of the line subsidence.\n\n3. The vertical forces due to the damping system are modelled as\n4. ${\\displaystyle d_{l}\\left(z^{t}+\\left(1-\\gamma \\right)\\Delta t{\\ddot {z}}^{t}\\right){\\mbox{ if}}\\;\\left(z^{t+\\Delta t}\\right) (23) ${\\displaystyle 0{\\mbox{ otherwise}}}$ (24)\n\nbeing ${\\textstyle \\gamma }$ a term related to the numerical scheme adopted. In addition ${\\textstyle d_{l}}$ is expressed as\n\n ${\\displaystyle d_{l}=2\\varsigma _{l}{\\sqrt {G_{s}\\left({\\frac {wl_{l}}{g}}\\right)Dl_{l}}},}$\n(25)\n\nwhere ${\\textstyle \\varsigma _{l}}$ is the seabed damping Palm.\n\n5. Finally, the horizontal forces due to the seabed interaction are modelled as\n6. ${\\displaystyle d_{l}\\left(x^{t}+[1-\\gamma ]\\Delta t\\,{\\ddot {x}}^{t}\\right),\\;\\;\\;\\;(x-{\\mbox{direction}}),}$\n(26)\n ${\\displaystyle d_{l}\\left(y^{t}+[1-\\gamma ]\\Delta t\\,{\\ddot {y}}^{t}\\right),\\;\\;\\;\\;(y-{\\mbox{direction}}).}$\n(27)\n\nThe algorithm developed to solve the mooring dynamics is described next.\n\n### 3.3 Non-linear FEM algorithm solution\n\nTruss elements, with just three translational degrees of freedom per node, are used for modelling the mooring lines, since bending effects can be neglected in most of cases [7,40]. In this work, a FEM approach combined with an updated Lagrange formulation is used for describing the dynamics of the mooring cable. Some authors showed that the corotational formulation can get a better performance than the updated formulation. However some instabilities and lack of convergence can appear. Figure 2 shows a scheme of the discretization adopted in this work.\n\n Figure 2: Scheme showing the general approach adopted for the spatial discretization of the cable mooring.\n\nApplying the standard FEM formulation for the non-linear elastodynamics problems [41], the equations for the dynamic equilibrium of the forces on a cable for each time step ${\\textstyle \\Delta t+t}$ can be written as\n\n ${\\displaystyle {\\mathbf {f} }^{t+\\Delta t}={\\mathbf {M} }^{t+\\Delta t}\\cdot {\\ddot {\\mathbf {x} }}+{\\mathbf {C} }^{t+\\Delta t}\\cdot {\\dot {\\mathbf {x} }}+{\\mathbf {P} }^{0}+{\\mathbf {R} }^{t+\\Delta t},}$\n(28)\n\nwhere ${\\textstyle {\\mathbf {f} }}$ are the external loads vector, ${\\textstyle {\\mathbf {M} }}$ is a consistent mass matrix of the line, considering inertia and added mass, ${\\textstyle {\\mathbf {P} }^{0}}$ is the pretension vector in the initial configuration, and ${\\textstyle {\\mathbf {R} }}$ is the internal forces vector of the cable. Damping effects of the mooring cable are introduced through a Rayleigh proportional damping matrix of ${\\textstyle {\\mathbf {C} }}$. The application of the non-linear FEM approach to the cable equation can be found in [41,42].\n\nConsidering the linearization of the current configuration, Eq. (28) can be expressed as\n\n ${\\displaystyle {\\mathbf {f} }^{t+\\Delta t}={\\mathbf {M} }^{t+\\Delta t}\\cdot {\\ddot {\\mathbf {x} }}+{\\mathbf {C} }^{t+\\Delta t}\\cdot {\\dot {\\mathbf {x} }}+\\left({\\mathbf {K} }_{L}^{t+\\Delta t}+{\\mathbf {K} }_{NL}^{t+\\Delta t}\\right)\\cdot \\delta \\mathbf {x} +{\\mathbf {P} }^{0}+{\\mathbf {R} }^{t},}$\n(29)\n\nwhere ${\\textstyle {\\mathbf {K} }_{L}}$ and ${\\textstyle {\\mathbf {K} }_{NL}}$ are called the material stiffness matrix and the geometric stiffness matrix, respectively, in the FEM literature [41,42]. The sum of both matrices is the so-called tangent stiffness matrix. The geometric part of the stiffness matrix is considered as non-linear since no material properties appear, and it depends of the Second-Piola stress tensor of the current configuration. Incremental displacement of the cable nodes is ${\\textstyle \\delta \\mathbf {x} }$.\n\nAn implicit Bossak-Newmark [43] time integration scheme is applied for the time-integration of the Eq. (29). This provides a set of algebraic equations to be solved iteratively,\n\n ${\\displaystyle \\left[\\left(1-\\alpha \\right){\\mathbf {M} }^{t+\\Delta t,i}+\\Delta t\\gamma \\,{\\mathbf {C} }^{t+\\Delta t,i}+\\Delta t^{2}\\beta \\,{\\mathbf {K} }^{t+\\Delta t,i}\\right]\\cdot {\\ddot {\\mathbf {x} }}^{t+\\Delta t,i+1}=}$ (30) ${\\displaystyle \\Delta t^{2}\\beta \\left({\\mathbf {K} }_{L}^{t+\\Delta t,i}+{\\mathbf {K} }_{NL}^{t+\\Delta t,i}\\right)\\cdot {\\ddot {\\mathbf {x} }}^{t+\\Delta t,i}+{\\mathbf {F} }^{t+\\Delta t,i}-{\\mathbf {P} }^{0}-{\\mathbf {R} }^{t+\\Delta t,i}}$ ${\\displaystyle -{\\mathbf {C} }^{t+\\Delta t,i}\\left[{\\dot {\\mathbf {x} }}^{t}-\\Delta t\\left(\\gamma +1\\right){\\ddot {\\mathbf {x} }}^{t}\\right]-\\alpha \\,{\\mathbf {M} }^{t+\\Delta t,i}\\cdot {\\ddot {\\mathbf {x} }}^{t},}$\n\nwhere ${\\textstyle \\Delta t}$ is the time step, ${\\textstyle i}$ denotes the ith iteration, ${\\textstyle \\alpha }$ is a parameter concerned with the Bossak-Newmark implicit method, and ${\\textstyle \\gamma }$ and ${\\textstyle \\beta }$ are parameters related to the Newmark time integration scheme.\n\nThe new position and velocity of each node in each time step can be expressed as (see Fig. 3)\n\n ${\\displaystyle \\mathbf {x} ^{t+\\Delta t}=\\mathbf {x} ^{t}+\\Delta t{\\dot {\\mathbf {x} }}^{t}+{\\frac {\\Delta t^{2}}{2}}\\left[\\left(1+2\\beta \\right){\\ddot {\\mathbf {x} }}^{t}+2\\beta \\,{\\ddot {\\mathbf {x} }}^{t+\\Delta t}\\right],}$ (31) ${\\displaystyle {\\dot {\\mathbf {x} }}^{t+\\Delta t}={\\dot {\\mathbf {x} }}^{t}+\\Delta t\\left[\\left(1-\\gamma \\right){\\ddot {\\mathbf {x} }}^{t}+\\gamma \\,{\\ddot {\\mathbf {x} }}^{t+\\Delta t}\\right].}$ (32)\n\nIt can be highlighted that multi-segmented mooring systems can be solved with this approach, and the equilibrium between two or more mooring cables is determined using the Newton-Raphson method. A criterion based on the maximum difference between the position reached by the nodes in two consecutive iterations is applied to evaluate the convergence of the algorithm.\n\n Figure 3: General approach to algorithm for solving the floater dynamics.\n\nThe iterative procedure for solving the cable dynamics can be summarized as\n\n1. Setting the problem statement and boundary conditions, as well as the initial equilibrium configuration, of the mooring line. For instance, initial position based on a static catenary with touchdown can be chosen. Then, the tension obtained in this preliminary calculation is used as an initial pretension of the line ${\\textstyle {\\mathbf {P} }^{0}}$.\n2. Evaluation of mass, damping, and tangent stiffness matrices. The procedure to calculate the tangent stiffness matrix can be found in [44].\n3. Estimation of forces in each node of the cable at ${\\textstyle t+\\Delta t}$, including drag, seabed interaction, hydrostatic, and gravity loads.\n4. Solving the cable dynamic Eqs. (31-32) in order to obtain the new position of the cable nodes.\n5. Iteration from the step 2 through 4 until convergence is reached for each element of the line. In this work an Aitken's ${\\textstyle \\Delta ^{2}}$ method is used to accelerate the iterative algorithm [45].\n\n## 4 Algorithm to solve the floater dynamics\n\nMost of the models found in the literature to solve the coupled mooring-body dynamics use a frequency domain analysis of the floater together with a convolution integral [46] for solving the motions of the coupled system [9,20,19]. The most common approach is based on a frequency-domain diffraction-radiation solver based on the Boundary Element Method (BEM) combined with a FEM model to solve the mooring dynamics. In this work, a different approach based on a time-domain FEM seakeeping model recently presented [30,31] is proposed. This approach allows to straightforwardly couple the seakeeping of the floater with non-linear loads.\n\nThe algorithm used in this work to solve the dynamics of the coupled system is presented in Fig. 3. In order to accelerate the solution of the non-linear solver, it includes two nested loops; the external loop iterates on the wave diffraction-radiation problem, while the internal loop takes into account the remaining external forces vector acting on the body. In this figure ${\\textstyle {\\mathbf {F} }^{0}}$ are the external forces different from the hydrodynamic and the mooring forces, and ${\\textstyle {\\mathbf {K} }_{M}}$ is the matrix of the linearised mooring. The procedure to carry out the coupled analysis is as follows:\n\n1. For each iteration ${\\textstyle k}$ of the solver loop, the wave diffraction-radiation problem is solved using the body velocities obtained previously, within the ${\\textstyle l}$ iteration, in the body dynamics loop (see Fig. 4).\n2. On the one hand, once the velocity potential ${\\textstyle \\varphi }$ is known, hydrodynamic loads are computed. On the other hand, the stiffness matrix of the mooring lines are obtained by means of the Jacobian matrix for the current position of the fairlead points. Those matrixes are used to estimate the mooring forces within the inner loop of the dynamic solver.\n3. The hydrodynamic loads and the mooring stiffness matrix are sent to the body dynamic loop ${\\textstyle l}$.\n4. For each iteration ${\\textstyle l}$ of the body dynamics loop, the body dynamic equations are solved.\n5. ${\\displaystyle {\\mathbf {M} }_{b}\\left[\\left(1-\\alpha \\right){\\mathbf {{\\ddot {x}}^{t+\\Delta t}} }+\\alpha \\,{\\mathbf {{\\ddot {x}}^{t}} }\\right]={\\mathbf {f} }^{t+\\Delta t}+{\\mathbf {f} }_{mooring}^{t+\\Delta t},}$\n(33)\n\nwhere ${\\textstyle {\\mathbf {f} }_{mooring}}$ is the restoring force vector of the mooring cables.\n\n6. Then, the body position and velocity are sent to the solver loop and introduced in the boundary conditions for the wave diffraction-radiation problem.\n7. The solver loops iterate until convergence is reached.\n8. Once the solver loop converges, it is proceeded to the calculation of the next time step.\n\n### 4.1 Coupling between rigid body motions and non-linear FEM mooring\n\nAs mentioned above, the cable dynamics is formulated in term of the acceleration, and this is also valid for the boundary condition of the fairlead connection point. A linear approximation to evaluate the end node acceleration based on the prescribed displacement is proposed in this work. Each time step is divided in ${\\textstyle n}$ sub-interval, that is to say, ${\\textstyle \\Delta t\/n}$. So, in the ${\\textstyle i}$-th time step, the acceleration of end node fulfills the following compatibility relationship\n\n ${\\displaystyle {\\ddot {\\mathbf {x} }}^{t+i{\\frac {\\Delta t}{n}}}=\\mathbf {x} ^{t}+i{\\frac {\\Delta t}{n}}{\\dot {\\mathbf {x} }}^{t}+{\\frac {1}{2}}\\left(i{\\frac {\\Delta t}{n}}\\right)^{2}\\left[{\\ddot {\\mathbf {x} }}^{t}+2\\beta c^{i}\\,i{\\frac {\\Delta t}{n}}\\right],}$\n(34)\n\nwhere ${\\textstyle c^{i}}$ is a coefficient to be determined in each time step. Applying an appropriate boundary condition, this coefficient can be formulated as\n\n ${\\displaystyle c^{i}={\\frac {1}{\\beta {\\frac {\\Delta t}{n}}}}\\left[{\\frac {\\mathbf {x} ^{t+i{\\frac {\\Delta t}{n}}}-\\mathbf {x} ^{t+\\left(i-1\\right){\\frac {\\Delta t}{n}}}}{\\left({\\frac {\\Delta t}{n}}\\right)^{2}}}-{\\frac {{\\dot {\\mathbf {x} }}^{t+\\left(i-1\\right){\\frac {\\Delta t}{n}}}}{\\frac {\\Delta t}{n}}}-{\\frac {{\\ddot {\\mathbf {{}x} }}^{t+\\left(i-1\\right){\\frac {\\Delta t}{n}}}}{2}}\\right].}$\n(35)\n\nAbove formulation for the boundary condition at the fairlead point has shown a better stability for the dynamics solver than higher order approximations.\n\nAs it was presented in the previous section, in the first ${\\textstyle l}$ iteration of the dynamic loop, the stiffness matrix of the cable ${\\textstyle {\\mathbf {K} }_{M}}$ is estimated by obtaining the Jacobian of the reaction forces using numerical differentiation,\n\n ${\\displaystyle {\\mathbf {K} }_{M}=\\left[{\\begin{array}{ccc}\\partial R_{x}\/\\partial x&\\partial R_{x}\/\\partial y&\\partial R_{x}\/\\partial z\\\\\\partial R_{y}\/\\partial x&\\partial R_{y}\/\\partial y&\\partial R_{y}\/\\partial z\\\\\\partial R_{z}\/\\partial x&\\partial R_{z}\/\\partial y&\\partial R_{z}\/\\partial z\\end{array}}\\right],}$\n(36)\n\nwhere ${\\textstyle R}$ is the reaction of the mooring cable at the fairlead point. Thus, the cable response is linearised within each time step ${\\textstyle \\Delta t}$, by estimating the mooring restoring forces as\n\n ${\\displaystyle \\mathbf {f} _{x}=R_{x}+k_{11}\\Delta x+k_{12}\\Delta y+k_{13}\\Delta z,}$ (37) ${\\displaystyle \\mathbf {f} _{y}=R_{y}+k_{21}\\Delta x+k_{22}\\Delta y+k_{23}\\Delta z,}$ ${\\displaystyle \\mathbf {f} _{z}=R_{z}+k_{31}\\Delta x+k_{32}\\Delta y+k_{33}\\Delta z,}$\n\nwhere the terms ${\\textstyle k_{ij}}$ correspond to a linear stiffness matrix, and ${\\textstyle \\Delta x}$, ${\\textstyle \\Delta y}$ and ${\\textstyle \\Delta z}$ are the displacements of the fairlead point from the position at the beginning of the current time step.\n\n## 5 Validation of the non-linear FEM mooring model\n\nIn this section some validation examples of the non-linear FEM model are presented.\n\n## Case 1: Cable under its self weight\n\nThe first case is based on that presented in [47]. It consists of an isolated cable, with fixed ends, subjected to its own weight (see Fig. 4). Initially the cable has a flat form. The expected deformation is a U form, and the reactions at the ends must be equal to the cable weight. The properties of the cable are: the stiffness ${\\textstyle EA=}$ 50 N, the weight per unit length ${\\textstyle w=}$ ${\\textstyle 0.4}$ N\/m, and ${\\textstyle 14.1421}$ m of span length. The cable is discretized into twenty-two bar elements, and the time step adopted is ${\\textstyle 0.014}$ s. Figure 5 shows different positions of the cable until it gets the expected U form. Note that a damping increment leads to a faster convergence of the stationary solution. The reactions at end points are ${\\textstyle 3.4681}$, and the maximum deflection (0.715 m) is reached at ${\\textstyle X=7.07105}$ m.\n\n Figure 4: Validation of the non-linear FEM mooring model. Case 1: cable under its self weight.\n Figure 5: Validation of the non-linear FEM mooring model. Case 1: different stages of time evolution of cable under its self weight.\n\nThe Ms Excel file below includes the results of the time evolution of the cable under its self weight (corresponding to Figure 5).\n\n Case_1.xlsx\n\n\n## Case 2: Free vibration cable\n\nThe second case is based on that proposed by [48]. The experiment consists of a cable initially in a horizontal position, and letting one end free and leaving the cable evolve under gravitational loads. The computed results are compared with the experimental results taken from [48], as well as with the data obtained from simulation with the FEM structural solver RamSeries (www.compassis.com). The properties of the cable are: length ${\\textstyle L=}$ ${\\textstyle 1.0}$ m, stiffness ${\\textstyle EA=}$ 50 N, weight per unit of length ${\\textstyle w=}$ ${\\textstyle 0.98}$ N\/m, and ${\\textstyle 0.881}$ m distance between the cable end points. The cable was divided into forty four bar elements, and the time step adopted was ${\\textstyle 0.001}$ s. This time step must be low enough for obtaining an accurate description of the cable motion. The obtained numerical results show a good agreement between our results, and those experimentally and numerically obtained by other authors. The cable position at different times obtained from computed results can be observed in Fig. 6.\n\n Figure 6: Validation of the non-linear FEM mooring model. Case 2: different time step positions of free vibration cable obtained from computed results.\n\n Figure 7: Validation of the non-linear FEM mooring model. Case 2: comparison between computed results of the path of free end of the cable obtained and experimental results of [48] and numerical computed with RamSeries.\n\nSlight differences can be appreciated at the lower end in Fig. 7. This fact can be explained by the different numerical parameters chosen to carry out the simulation.\n\nThe Ms Excel file below includes the results of the time evolution of the free vibration cable (corresponding to figures 6, 7).\n\n Case_2.xlsx\n\n\n## Case 3: Cable attached to a circular rotating plate\n\nNow, results obtained by the developed FEM solver are compared for the model test proposed by Lindhal and Sjoberg [38,39]. The experimental set up is shown in Fig. 8. The lower end was attached to the concrete floor and the upper end was attached to a circular plate with a fixed rotation speed. The radius of the circular motion was ${\\textstyle 0.2}$ m. Two cases with different rotational periods of ${\\textstyle T_{r}}$ = 1.25 s and ${\\textstyle T_{r}=3.5}$ s respectively, were investigated. The reaction forces at the top end of the cable are measured and compared with those computed by the proposed numerical model.\n\n Figure 8: Validation of the non-linear FEM mooring model. Case 3: the geometrical set-up of the experimental tests by [38].\n\n Figure 9: Validation of the non-linear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period ${\\displaystyle T_{r}}$ = 3.5 s.\n\nThe properties of the cable are: length ${\\textstyle L=}$ ${\\textstyle 33}$ m, stiffness ${\\textstyle EA=}$ ${\\textstyle 10^{4}}$ N, weight per unit length ${\\textstyle w=}$ ${\\textstyle 0.18}$ N\/m, and diameter ${\\textstyle D=10^{-3}}$ m. In this validation case, the cable is discretized into 200 bar elements, and the time step adopted is ${\\textstyle 0.001}$ s. The motion of the top end is defined by the following expressions (including an initialization period to build up the spinning of the plate):\n\n ${\\displaystyle x\\left(t\\right)=0.2\\cdot {\\mbox{ tanh}}\\left({\\frac {t}{2}}\\right)\\left({\\mbox{ cos}}\\left[-{\\frac {2\\pi }{T_{r}}}t+\\delta \\right]-{\\mbox{ cos}}\\delta \\right)\\;,}$ (38) ${\\displaystyle y\\left(t\\right)=0.0\\;,}$ (39) ${\\displaystyle z\\left(t\\right)=0.2\\cdot {\\mbox{ tanh}}\\left({\\frac {t}{2}}\\right)\\left({\\mbox{ sin}}\\left[-{\\frac {2\\pi }{T_{r}}}t+\\delta \\right]-{\\mbox{ sin}}\\delta \\right)\\;}$ (40)\n\nThe results are compared with the experimental results taken from [38]. A good agreement between the computed reaction forces and the experimental results [38] can be observed in Fig. 9 (for ${\\textstyle T_{r}=3.5}$ s), and 10 (for ${\\textstyle T_{r}=1.25}$ s) for both, the maximum values and the time evolution. Only slight differences are appreciated due to the uncertainty of the experimental data. The entire cable loses stiffness at some instants in time, and the numerical oscillations after the slack are larger in the case of the larger excitation frequency, as it was observed by [39].\n\n Figure 10: Validation of the nonl-inear FEM mooring model. Case 3: comparison between the experimental and numerical cable top end reaction forces. Rotational period ${\\displaystyle T_{r}}$ = 1.25 s.\n\nThe Ms Excel file below includes the results of the circular rotating plate case (corresponding to figures 9, 10).\n\n Case_3.xlsx\n\n\n## 6 Demonstration of the coupled seakeeping-mooring solver\n\nA fully coupled analysis of the OC3 spar buoy offshore wind turbine, called Hywind (www.ieawind.org\/task23\/) is presented next. Main particulars of the spar buoy FOWT are presented in [52,53]. A general view of the buoy concept can be observed in Fig. 11. In this section, several analysis are made:\n\n Figure 11: General view of the spar buoy wind turbine concept (OC3-Hywind concept).\n\u2022 The Response Amplitude Operator (RAO) are compared for a rigid wind turbine with no wind loads, with those obtained by other authors [49].\n\u2022 The analysis of a spar wind turbine subjected to different monochromatic waves with periods close to the pitch resonance. The effects of the linear, the quasi-static, and the dynamic mooring models are compared.\n\u2022 A simulation in real operational conditions of the spar FOWT OC3 is performed. First and second-order irregular waves, real wind flow, and mooring loads (with three different types of mooring models) are taken into account. The influence of the mooring type model, the effects of second-order wave, and mooring tensions are studied.\n\n### 6.1 RAO analysis and comparison\n\nFirst, a RAO analysis is performed in the absence of wind. The frequency results of the seakeeping solver are obtained after applying a Fourier transform to the time history generated. Figure 12 shows an inter-code comparison. Note that a good agreement among the different solvers is found [49].\n\n Figure 12: OC3-Hywind concept. Comparison between the computed result by SeaFEM with those taken from other authors [49] for a rigid wind turbine with no wind.\n\nThe Ms Excel file below includes the RAO curves obtained in the present work and the data from HydroDyn and WAMIT used as reference (see Figure 12).\n\n RAOS.xlsx\n\n\n### 6.2 Mooring analysis around pitch resonance\n\nBelow, the influence of three different mooring models on the OC3 spar buoy FOWT is analysed. These models are: the linear model cable (that behaves as springs and is represented by a linearised mooring matrix [53]); the Quasi-Static model, similar to the one presented in [54]; and the dynamic model developed in this work. Six first-order monochromatic waves are used in the analysis. Key parameters of the mooring layout and cable properties can be found in Table 1. Fig. 13 compares the pitch motion for each mooring model.\n\n Item Value Unit Wave Amplitude 1.0 m Wave Period analysed 10; 25; 20; 35; 40; 55 s Number of Mooring Lines 3 Angle Between Adjacent Lines 120 deg Depth to Anchors Below SWL 320 m Depth to Fairleads Below SWL 70 m Radius to Fairleads from Platform Centerline 853.9 m Unstretched Mooring Line Length 902.2 m Mooring Line Diameter 0.9 m Mooring Line Mass Density 77.71 kg\/m Mooring Line Extensional Stiffness 3.84${\\textstyle \\times }$10${\\textstyle ^{8}}$ N\n\nIt can be observed that there are big differences in the results in those cases close to the pitch resonance (about ${\\textstyle T_{w}=}$ 30 s), while the results are quite similar for the cases with ${\\textstyle T_{w}=}$ 10 s and ${\\textstyle T_{w}=}$ 55 s. These results suggest that the use of simpler mooring models can lead to big errors near the resonance frequency, and as a consequence to magnify safety factors, contributing to increase the cost of the FOWT.\n\n### 6.3 Simulation in operational conditions\n\nFinally, four analyses of the OC3 spar buoy in operational conditions are presented. The different analyses are carried out in similar environmental conditions, but using first and second-order irregular waves. Furthermore, additional studies including Quasi-Static and dynamic mooring models are performed. The goal of these analyses is to evaluate the effects of the mooring model and the wave order on the dynamics of the system, as well as to estimate the tension in the mooring lines. On the one hand, the wind turbine system is assumed to be operating at an average wind speed of 11.4 m\/s, which generates the maximum thrust and torque. FASTLognoter [50,51] has been used to linearise with FAST [50] the behaviour of the wind turbine around the operating wind speed. Restoration and damping matrices resulting from the linearisation of the wind turbine system are included into the global dynamics. It should be remarked that the rotational and periodicity effects are considered in the calculation of the steady state matrix. On the other hand, the wind loads are estimated considering non-uniform wind flow, with an average wind speed of 11.4 m\/s. The wind flow profile is obtained using Turbsim [55], and the wind loads on the wind turbine are obtained from FAST\/AeroDyn [50].\n\n Case 1 Case 2 Case 3 Case 4 Average Wind velocity ${\\displaystyle W_{rel}}$ (m\/s) 11.4 11.4 11.4 11.4 Wind direction ${\\displaystyle \\theta _{rel}}$ (deg) 0.0 0.0 0.0 0.0 Wave Spectrum (-) JONSWAP 1${\\textstyle ^{st}}$ JONSWAP 1${\\textstyle ^{st}}$ JONSWAP 2${\\textstyle ^{nd}}$ JONSWAP 2${\\textstyle ^{nd}}$ Significant wave height ${\\displaystyle H_{s}}$ (m) 6.0 6.0 6.0 6.0 Peak period ${\\displaystyle T_{p}}$ (s) 12.0 12.0 12.0 12.0 Mean wave direction ${\\displaystyle \\theta _{w}}$ (deg) 0 0 0 0 Mooring model (-) Quasi-static Nonlinear Quasi-static Nonlinear Number of mooring lines (-) 3 3 3 3 Mooring lines elements (-) 200 200 50 200\n\nA JONSWAP spectrum with a mean wave period ${\\textstyle T_{m}}$= 12.0 s, and significant wave height ${\\textstyle H_{s}}$= 6.0 m, is simulated. The key parameters of the different case studies are presented in Table 2. As stated above, two different types of mooring models are analysed; one based on the Quasi-Static catenary model [54], and the other based on the dynamic (NFEM) cable model, presented in this work. For the dynamic cable analysis each mooring line is divided into 200 bar elements.\n\nFigures 14, 15 and 16 show the computed heave, roll and pitch motions from 600 s to 900 s of simulation time. Noticeable differences are found between the first and the second-order movements, while the QS and NFEM mooring models offer quite similar results.\n\n Figure 13: Results obtained for mooring analysis around pitch resonance of OC3-Hywind.\n\nTable 3 shows the mean and RMS values, as well as the motion amplitude for the first and second-order movements. When comparing the QS and the NFEM models, only slight differences are observed. In particular, the second-order pitch motion is higher when using the NFEM model, compared to the QS model, while the other values remain with similar trends for both models.\n\n Case 1 Surge (m) Sway (m) Heave (m) Roll (deg) Pitch (deg) Yaw (deg) Mean 1${\\textstyle ^{st}}$ 0.04 0.00 0.00 0.29 0.41 0.21 Mean 1${\\textstyle ^{st}}$ 0.04 0.00 0.00 0.29 0.41 0.21 Amplitude 1${\\textstyle ^{st}}$ 14.96 1.34 2.75 1.91 6.12 10.38 Amplitude 2${\\textstyle ^{nd}}$ 14.96 1.34 2.75 1.91 6.12 10.38 RMS 1${\\textstyle ^{st}}$ 2.76 0.21 0.53 0.52 1.09 1.92 RMS 2${\\textstyle ^{nd}}$ 2.73 0.24 0.45 0.48 1.22 1.91 Case 2 Surge (m) Sway (m) Heave (m) Roll (deg) Pitch (deg) Yaw (deg) Mean 1${\\textstyle ^{st}}$ 0.00 0.00 -0.01 0.29 0.40 0.21 Mean 1${\\textstyle ^{st}}$ 0.10 0.00 -0.03 0.30 0.42 0.28 Amplitude 1${\\textstyle ^{st}}$ 13.82 1.12 2.61 1.92 6.03 10.31 Amplitude 2${\\textstyle ^{nd}}$ 15.19 1.55 2.59 2.02 8.00 10.83 RMS 1${\\textstyle ^{st}}$ 2.54 0.18 0.50 0.52 1.09 1.92 RMS 2${\\textstyle ^{nd}}$ 2.73 0.24 0.45 0.48 1.22 1.91\n\nNext, a video of the analysis case 1 is presented.\n\n Video 1. OC3 spar buoy wind turbine simulation (analysis case 1).\n\nFigure 17 shows the tension for each mooring line at the fairlead point. The NFEM model recorded larger amplitude oscillations (in the range of 5 s to 10 s period) compared with QS model. The differences observed in Fig. 17 in the tension amplitude reach up to 24 ${\\textstyle \\%}$. This fact suggests that using a QS model for fatigue assessment of the mooring lines could overestimate their fatigue life.\n\n Figure 14: Comparison between heave motion for first and second-order wave environment for Case 1-4 (described in Table 2).\n Figure 15: Comparison between roll motion for first and second-order wave environment for Case 1-4 (described in Table 2).\n Figure 16: Comparison between pitch motion for first and second-order wave environment for Case 1-4 (described in Table 2).\n Figure 17: Comparison of fairlead tension of each mooring line for Case 3, and 4.\n\nTable 4 compares the maximum, minimum, average, and RMS tension values at the fairlead points, obtaining similar values for both mooring models. Furthermore, based on Figure 17, the second-order simulation provided larger tension values than the first-order. This result suggests that a first-order approximation can underestimate the fatigue loads and might lead to a wrong mooring design.\n\n Line Case Max. (N) Min. (N) Mean (N) RMS (N) Line 1 1 1.435${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.073${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.250${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.252${\\textstyle \\times }$10${\\textstyle ^{6}}$ Line 2 1 1.446${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.072${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.240${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.242${\\textstyle \\times }$10${\\textstyle ^{6}}$ Line 3 1 7.411${\\textstyle \\times }$10${\\textstyle ^{5}}$ 5.209${\\textstyle \\times }$10${\\textstyle ^{5}}$ 5.991${\\textstyle \\times }$10${\\textstyle ^{5}}$ 6.003${\\textstyle \\times }$10${\\textstyle ^{5}}$ Line 1 2 1.457${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.052${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.246${\\textstyle \\times }$10${\\textstyle ^{5}}$ 1.248${\\textstyle \\times }$10${\\textstyle ^{6}}$ Line 2 2 1.471${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.040${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.236${\\textstyle \\times }$10${\\textstyle ^{6}}$ 1.238${\\textstyle \\times }$10${\\textstyle ^{6}}$ Line 3 2 7.421${\\textstyle \\times }$10${\\textstyle ^{5}}$ 4.510${\\textstyle \\times }$10${\\textstyle ^{5}}$ 5.923${\\textstyle \\times }$10${\\textstyle ^{5}}$ 5.936${\\textstyle \\times }$10${\\textstyle ^{5}}$\n\nIt is emphasized that the NFEM mooring model allows to consider non-linear dynamic effects which cannot be taken into account by QS models. However, in deep water, the dynamics of catenary mooring lines are negligible as reported in Low.\n\n## 7 Conclusions\n\nA FEM coupled seakeeping and mooring model for the analysis of floating wind turbines is presented. Based on the results obtained in this work, the following conclusions are made:\n\n\u2022 A second-order time-domain seakeeping model based on FEM and capable of using unstructured meshes [28,30,31] is successfully applied to simulate floating wind turbines.\n\u2022 A dynamic cable model based on Non-linear FEM is presented. A Lagrangian formulation is used to describe the dynamics of the mooring line. Several non-linear effects are considered, such as seabed interaction and drag forces. In order to accelerate the body dynamics convergence, the restoring mooring forces on the floating structure are linearised for each iteration within each time step. Therefore, a stiffness matrix has to be evaluated for each iteration.\n\u2022 An algorithm is developed to couple the non-linear cable dynamics with the body dynamics solver within the time-domain seakeeping solver [28].\n\u2022 The dynamic cable model presented in this work is validated against several experimental results, obtaining a good agreement between the numerical and experimental ones.\n\u2022 Different analyses of a FOWT based on the Hywind concept [52,53] are performed. RAO curves are compared with those obtained by other authors, showing a good agreement. Moreover, a comparison among the linear, the QS, and the NFEM models are carried out for frequencies around pitch resonance, obtaining large differences among mooring models.\n\u2022 The Hywind FOWT is analysed under a realistic operational condition. Second-order coupled simulations are carried out considering a real wind profile, and with two types of mooring models; the QS and the NFEM. Results from both cases are compared. The NFEM mooring model combined with second-order waves and real wind provides a realistic simulation of the performance of the floating wind turbine. The comparison suggests that using a QS model for fatigue assessment of the mooring lines could overestimate their fatigue life. Furthermore, first-order approximation can underestimate tension values on the mooring systems.\n\n## References\n\n[1] Breton SP, Moe H. Status, plans and technologies for offshore wind turbines in Europe and North America. Renew Energy. 2009;34:646\u201354.\n\n[2] Karimirad M, Meissonnier Q, Gao Z, Moan T. Hydroelastic code-to-code comparison for a tension leg spar-type floating wind turbine. Marine Struct. 2010;24:412\u2013435.\n\n[3] Jeon SH, Cho YU, Seo MW, Cho JR, Jeong WB. Dynamic response of floating substructure of spar-type offshore wind turbine with catenary mooring cables. Ocean Eng. 2013;72:356\u2013364.\n\n[4] Sclavounos P, Butterfield S, Musial W, Jonkman JM. Engineering challenges for floating offshore wind turbines. In Proc: Copenhagen Offshore Wind Conference, 2005.\n\n[5] Cordle A, Jonkman JM. State of the art in floating wind turbine design tools. In Proc: 21st International Offshore and Polar Engineering Conference (ISOPE 2011). 2011.\n\n[6] Matha D, Schlif M, Cordle A, Pereira R, Jonkman JM. Challenges in simulation of aerodynamics, hydrodynamics, and mooring-line dynamics of floating offshore wind turbines. In Proc: 21st International Offshore and Polar Engineering Conference (ISOPE 2011), 2011.\n\n[7] Jacob PB, Bahiense RA, Correa FN, Jacovazzo BM. Parallel implementations of coupled formulations for the analysis of floating production systems, part I: Coupling formulations. Ocean Eng. 2012;55:206\u2013218.\n\n[8] Jacob PB, Franco LD, Rodrigues MV, Correa FN, Jacovazzo BM. Parallel implementations of coupled formulations for the analysis of floating production systems, part II: Domain decomposition strategies. Ocean Eng. 2012;55:219\u2013234.\n\n[9] Bae YH, Kim MH. Rotor-floater-theter coupled dynamics including second-order sum-frequency wave loads for a mono-column-TLP-type FOWT (Floating Offshore Wind Turbine). Ocean Eng. 2013;61:109\u2013122.\n\n[10] Bachinsky EE, Moan T. Design considerations for tension leg platform wind turbines. Marine Struc. 2012;29:89\u2013114.\n\n[11] Low YM, Langley RS. Time and frequency domain coupled analysis of deep-water floating production systems. Appl. Ocean Res. 2006;28:371\u2013385.\n\n[12] Paulling JR, Webster WC. Large-amplitude analysis of the coupled response of a TLP and tendon system. In Proc: International Conference on Ocean, Offshore and Arctic Engineering (OMAE'86). 1986, p. 86.\n\n[13] Sircar S, Kleinhans JW, Prasad J.Impact of coupled analysis on global performance of deep-water TLP's. In Proc: Offshore Technology Conference (OTC 1993), 1993. Paper No. 7145.\n\n[14] Schott WE, Rodenbusch G, Mercier RS, Webb CM. Global Design and Analysis of the Auger Tension Leg Platform. In Proc: of the Offshore Technology Conference (OTC 1994), Paper No. 7621, 1994. p. 541\u2013552.\n\n[15] Hong SY, Hong S. Motion simulation of a floating structure coupled with mooring lines. ISOPE 1996. In Proc: Sixth International Offshore and Polar Engineering Conference (ISOPE 1996). 1996.\n\n[16] Jacob BP, Masetti IQ. PROSIM Program:Coupled Numerical Simulation of the Behavior of Moored Floating Units (in Portuguese). LAMCSO\/COPPE-Petrobras Internal Report. 1998.\n\n[17] Kim S, Sclavounos P. Fully coupled response simulations of theme offshore structures in water depths to up to 10,000 feet. In Proc: 11th International Offshore and Polar Engineering Conference (ISOPE 2001). 2001.\n\n[18] Senra SF, Correa FN, Jacob BP, Mourelle MM, Masetti IQ. Towards the integration of analysis and design of mooring systems and risers, part I: studies on a semisubmersible platform. In Proc: 21st International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2002). 2002.\n\n[19] Karimirad M, Moan T. A simplified method for coupled analysis of floating offshore wind turbines. Marine Struc. 2012;27:45\u201363.\n\n[20] Jonkman JM, Sclavounos P. Development a fully coupled aeroelastic and hydrodynamic models for offshore wind turbines. ASME 2006 Wind Engineering Symposium. 2006.\n\n[21] Kim BW, Sung G, Kim JA, Hong SY. Comparison of linear spring and nonlinear FEM methods in dynamic coupled analysis of floating structure and mooring system. J. Fluid Struct. 2013;42:205\u2013227.\n\n[22] Tahar A, Kim MH. Coupled-dynamic analysis of floating structures with polyester mooring lines. Ocean Eng. 2008;35:1676\u20131685.\n\n[23] Kim BW, Sung HG, Hong SY, Jung HJ. Finite element non linear analysis for catenary structure considering elastic deformation. Comput. Modeling Eng. Sci. 2010;63:29\u201345.\n\n[24] Yang M, Teg B, Ning D, Shi Z. Coupled dynamic analysis for wave interaction with a truss spar and its mooring line\/riser system in time domain. Ocean Eng. 2012;39:72\u201387.\n\n[25] Cunff CL, Heurtier J-M, Piriou L, Berhault C, Perdrizet T, Teixeira D, Ferrer G, Gilloteaux J-C. Fully coupled floating wind turbine simulator based on nonlinear finite element method: Part I - Methodology. In Proc: ASME 32th International Conference on Ocean, Offshore and Arctic Engineering. 2013.\n\n[26] Perdrizet T, Gilloteaux J-C, Teixeira D, Ferrer G, Piriou L, Cadiou D, Heurtier J-M, Cunff CL. Coupled floating wind turbine simulator based on nonlinear finite element method: Part II - Validation results. In Proc: ASME 32th International Conference on Ocean, Offshore and Arctic Engineering. 2013.\n\n[27] Kvittem MI, Moan T. Time domain analysis procedures for fatigue assessment of a semi-submersible wind turbine. Marine Struc. 2015:40;38\u201359.\n\n[29] Low YM. Prediction of extreme responses of floating structures using a hybrid time\/frequency domain coupled analysis approach. Ocean Eng. 2008;35:1416\u20131428.\n\n[30] Serv\u00e1n B, Garc\u00eda-Espinosa J. Accelerated 3D multi-body seakeeping simulations using unstructured finite elements. J. Compu. Phys. 2013;252:382\u2013403.\n\n[31] Garc\u00eda-Espinosa J, Di-Capua D, Ubach, PA, O\u00f1ate E. A FEM fluid-structure interaction algorithm for analysis of the seal dynamics of a Surface-Effect Ship. Accepted for publication in Comp. Meth. in App. Mech. and Eng.2015.\n\n[32] ABS. Floating Wind Turbines. American Bureau of Shipping. Bureau of Safety and Environmental Enforcement. Retrieved from http:\/\/www.offshorewindhub.org\/. 2012.\n\n[33] Hong SY, Nam B-W. Analysis of second-order wave force of floating bodies using FEM in time-domain. In Proc: 21th International Offshore and Polar Engineering Conference (ISOPE 2010). 2010.\n\n[34] Aamo OM, Fossen TI. Finite element modelling of mooring lines. Math. Comput. Simul. 2000;53:415\u2013422.\n\n[35] Gobat J, Grosenbaugh M. 2006. Time-domain numerical simulation of ocean cable structures. Ocean Eng. 2006;33:1373\u20131400.\n\n[36] Morison JR, O'Brien MP, Johnson JW, Schaaf SA. The force exerted by surface wave on piles. Journal of Petroleum Transaction 1950;189:149\u201354.\n\n[38] Lindahl L, Sjoberg A. Dynamic analysis of mooring cables. In Proc: 2nd International symposium on ocean engineering and ship handling. 1983. pp. 281\u2013319.\n\n[39] Palm J, Paredes GM, Eskilsson C, Pinto FT, Bergahl L. Simulation of mooring cable dynamics using a discontinuous Galerkin Method. In Proc: of the V International Conference on Computational Methods in Marine Engineering (MARINE 2013). 2013.\n\n[40] Rodrigues MV, Correa FN, Jacob PB. Implicit domain decomposition methods for coupled analysis of offshore platforms. Commun. Numer. Meth. Engng. 2007;23:599\u2013621.\n\n[41] Felippa CA. Nonlinear Finite Element Method, Department of Aerospace Engineering Sciences. University of Colorado at Boulder. 2014.\n\n[42] Bathe KH. Finite element procedures. Prentice Hall. 1996.\n\n[43] Wood W, Bossak M, Zienkiewicz O. An alpha modification of Newmark's method. Int. J. Num. Meth. Engng. 1980;1:1562\u20131566.\n\n[44] Thai H, Kim SE. Nonlinear static and dynamic analysis of cable structures. Fin. Elem. Analysis Design. 2011;47:237\u2013246.\n\n[45] Irons BM, Tuck RC. A version of the Aitken accelerator for computer iteration. International J. Num. Meth. Engng. 1969;1:275\u2013277.\n\n[46] Cummins W. The impulse response function and ship motions. Symposium on Ship Theory 1962. Institut fur Schiffbauder Universitat. 1962.\n\n[47] Ortigosa I, Development of a decision support system for the design and adjustment of sailboat rigging. PhD Thesis, Universitat Polit\u00e9cnica de Catalunya, Spain. 2011.\n\n[48] Lazzari M, Saetta AV, Vitaliani RV. Nonlinear dynamic analysis of cable-suspended structures subjected to wind actions. Comput. Struct. 2001;79:953\u2013969.\n\n[49] Ramanchandran GCV, Robertson A, Jonkman JM, Masciola MD. Investigation of Response Amplitude Operators for Floating Offshore Wind Turbines. In Proc: 23rd International Ocean, Offshore and Polar Engineering Conference (ISOPE 2013). 2013.\n\n[50] Jonkman JM, Buhl Jr ML. FAST's Users Guide. Technical Report NREL\/EL-500-38230. National Renewable Energy Laboratory, www.nrel.gov, Colorado (USA). 2005.\n\n[51] Guti\u00e9rrez-Romero JE, Zamora B, Garc\u00eda\u2013Espinosa J. Peyrau R. Tool development based on FAST for performing design optimization of offshore wind turbines: FASTLognoter. Renew. Energy. 2013;55:69\u201378.\n\n[52] Jonkman JM, Butterfield S, Musial W, Scott G. Definition of a 5-MW reference wind turbine for offshore system development. Technical Report NREL\/TP-500-38060. National Renewable Energy Laboratory, www.nrel.gov, Colorado (USA). 2009.\n\n[53] Jonkman JM. Definition of the floating system for Phase IV of OC3. Technical Report NREL\/TP-500-47535. National Renewable Energy Laboratory, www.nrel.gov, Colorado (USA). 2010.\n\n[54] Jonkman JM. Dynamics Modeling and Loads Analysis of an Offshore Floating Wind Turbine. Technical Report NREL\/TP-500-41958. National Renewable Energy Laboratory, www.nrel.gov, Colorado (USA). 2007.\n\n[55] Jonkman BJ, Kilcher, L.Turbsim User's Guide. Version 1.06.00. Technical Report NREL\/TP-xxx-xxxxx. National Renewable Energy Laboratory, www.nrel.gov, Colorado (USA). 2012.\n\n### Document information\n\nPublished on 01\/01\/2016\n\nDOI: 10.1016\/j.marstruc.2016.05.002\n\n### Document Score\n\n5\n\nTimes cited: 15\nViews 8\nRecommendations 1","date":"2020-08-09 11:40:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 252, \"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\": 0.8628291487693787, \"perplexity\": 9054.949894138788}, \"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-2020-34\/segments\/1596439738552.17\/warc\/CC-MAIN-20200809102845-20200809132845-00188.warc.gz\"}"}
| null | null |
\section{Introduction} \label{introduction}
In recent years nanoelectromechanical
systems\cite{Blencowe05,Schwab05} (NEMs), nanoscale mechanical
oscillators coupled to electronic devices of comparable
dimensions, have attracted substantial research effort. A major
motivation for this effort is the ability to observe quantum
behavior in a macroscopic system under realizable experimental
conditions\cite{Leggett86,Leggett02}. Indeed, NEMs today can be
fabricated with vibrational mode frequencies of 1 MHz-10 GHz and
quality factors in the range of $10^3-10^5$, allowing the quantum
regime to be reached at milli-Kelvin temperatures for high
frequency oscillators\cite{LaHaye04,OConnell10}. Possible quantum
effects in NEMs under such conditions include quantized energy
levels, superposition of states, entanglement and
squeezing\cite{Rabl04,Ruskov05,Zhou06}. In addition, NEMs are
applied to high-sensitive detection of
mass\cite{Li04,Buks06-2,Jensen08}, force\cite{Blencowe07} and
displacement\cite{Stampfer06}, electrometers\cite{Blick00}, and
also to classical memory elements\cite{Blick07,Mahboob08}.
Observing or changing the state of NEMs requires some type of
transducer which couples to them. Optical
coupling\cite{Schliesser09} can be performed, e.g., by a microwave
cavity\cite{Regal08}, but is difficult to integrate in circuits
and suffers from the diffraction limit and heating of the NEMS.
Non-optical coupling methods are therefore more common in
experiments today. With magnetomotive coupling\cite{Huang05}, the
magnetic force on a thin metallic layer on the NEMS is measured.
Capacitive coupling can take many forms, one of which uses a
normal or superconducting single electron transistor
(SET)\cite{Mozyrsky04,Ruskov05}. The NEMS changes the island
charging energy in the SET and hence the tunneling rates, which
can be read electronically. Other forms of capacitive coupling use
Cooper pair boxes\cite{Irish03,Suh10}, flux qubits\cite{Xue07-2},
quantum point contacts\cite{Benatov12} and quantum
dots\cite{Lambert08}.
An inductive coupling scheme with a potential for displacement
precision greater than the standard quantum limit is obtained by
integrating a doubly clamped micron-scale beam within a
superconducting quantum interference device (SQUID). In a dc SQUID
the motion of the resonator changes the area of the SQUID loop and
hence the magnetic flux and the current through it, which is then
measured. This system was only recently
implemented\cite{Poot08,Poot10}. A more sophisticated design,
where the dc SQUID, and hence the resonator, is coupled to a
charge qubit, was also proposed\cite{Zhou06}. For an rf SQUID it
was found\cite{Buks06} that the change in the magnetic flux due to
the motion of the beam affects the visibility of Rabi oscillations
in the SQUID levels. The detection of discrete Fock states in a
resonator integrated with an rf SQUID was suggested in another
work\cite{Buks08}.
Squeezed states, originally introduced in quantum
optics\cite{Walls83}, are defined as minimum-uncertainty states
with less noise in one field quadrature than a coherent
state\cite{Walls08}. Several methods to generate squeezing in NEMs
were suggested. Coupling to a charge qubit\cite{Rabl04,Zhou06} as
means of generating squeezing was proposed, while another work
described squeezing by periodic position measurement with a
weakly-coupled detector\cite{Ruskov05}. Squeezing in nanoresonators can
be applied to decrease the noise in force or displacement
measurements to below the standard quantum limit, greatly
improving the sensitivity of the device\cite{Rabl04,Teufel09}.
In this work, we present a scheme to create quantum entanglement
and squeezing in two nanoresonators integrated in a dc SQUID. A
previous study\cite{Xue07} analyzed a similar system but
introduced many approximations that are difficult to implement
experimentally, whereas our present study is closer to an
experimentally realizable system. For instance, we do not overlook
the generally non-negligible self-inductance of the SQUID as done
in previous work\cite{Xue07}, and we assume mega-Hertz frequency
nanomechanical oscillators rather than giga-Hertz frequency
resonators, which are difficult to integrate with a SQUID. Lastly,
we do not require the SQUID to be prepared in a high-$|\alpha|$
coherent state in order to have squeezing, as the previous study
does\cite{Xue07}, and require instead a thermal equilibrium state,
which is easier to accomplish. Finally, we consider different
aspects of the system and draw conclusions, e.g., on the reading
and writing processes, that were not advanced in previous
literature.
The paper is organized as follows. In Sec. \ref{section1} we
present the system model and its classical Lagrangian and
Hamiltonian formulations. We then proceed to quantize the
Hamiltonian for the non-dissipative case and derive the effective
Hamiltonian. Next, in Sec. \ref{section2} we treat the system as a
quantum memory and explain how one can read its quantum state or
write on it. In Sec. \ref{section3} we put forward a scheme for
generating quadrature-squeezed states of the nanomechanical beams
when dissipation is neglected, whereas in Sec. \ref{section4} we
use a quantum master equation approach to test the range of
validity of our results in the the presence of dissipation.
Lastly, we discuss our results and present the conclusions in Sec.
\ref{conclusions}.
\section{System Model and Hamiltonian} \label{section1}
Our system consists of a dc SQUID, shown schematically in Fig.
\ref{fig01}, in which each arm includes a Josephson junction and
an integrated doubly clamped beam of length $l_i$ and mass $m_i$
that can oscillate mechanically in the plane of the SQUID with an
angular frequency $\widetilde{\omega}_i$ ($i=1,2$). The notation
we use is similar to the one in Ref. \onlinecite{Pugnetti10}. A
uniform magnetic field $B$ is applied perpendicularly to the plane
of the loop, and a dc bias current $I_b$ flows through it after
splitting to $I_1$ and $I_2$ in the lower and upper arms,
respectively. Two current-carrying wires with currents $I_3$ and
$I_4$ create additional magnetic fields $B_1$ and $B_2$ at the
positions of the first and second beams, respectively. The beam
amplitudes are much smaller than the beam-wire distance, allowing
these fields to be approximated as being spatially uniform. For
simplicity, the two Josephson junctions on the SQUID arms are
taken to be identical and their gauge-invariant phase changes are
denoted by $\gamma_i$. The critical current and shunting
capacitance of each junction are taken as $I_c$ and $C$,
respectively, and are used to define the characteristic junction
energy scales: the Josephson energy $E_J=\hbar I_c/2e$ and the
charging energy $E_C=e^2/2C$. The plasma frequency
$\omega_{pl}=\sqrt{2E_JE_C}/\hbar$ sets the typical time scale for
the SQUID dynamics.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.50cm]{fig01.eps}
\caption{\label{fig01} Schematic of the device we study: Two nanomechanical beams oscillating in plane are embedded in a dc SQUID with area $A$ when beams are at rest. Uniform magnetic field $B$ threads the SQUID, and bias current $I_b$ is assumed. The two identical Josephson junctions on the SQUID have phase drops $\gamma_i$ ($i=1,2$). The currents $I_3$ and $I_4$ create additional magnetic fields $B_1$ and $B_2$ at the resonators.}
\end{center}
\end{figure}
The area of the SQUID loop depends on the center of mass positions
of the nanomechanical resonators, denoted by $x_i$ and defined as
zero when the beam is at rest, and positive when it is inside the
loop. Since the superconducting order parameter is single valued,
we must have
\begin{eqnarray}
&\gamma_1-\gamma_2-\frac{2\pi\Phi}{\Phi_0}=2\pi p, \label{eq01-1}\\
&\Phi=BA-\sum\limits_i(B+B_i)l_ix_i+L(I_1-I_2)/2, \label{eq01-2}
\end{eqnarray}
where $p$ is an integer, $\Phi$ is the total magnetic flux
threading the loop, $A$ is the loop area when the beams are at
rest, $\Phi_0=h/2e$ is the flux quantum, $L$ is the
self-inductance of the loop, and $I_i$ is the current in its $i$th
arm. The first term in Eq. (\ref{eq01-2}) comes from the external
magnetic field and the second, responsible for the coupling of the
mechanical and magnetic degrees of freedom, from the oscillation
of the beams. The difference between $l_ix_i$ and the actual area
enclosed by the $i$th beam is negligible, being of third order in
the ratio of the beam amplitude to its length. Lastly, the third
term originates from the magnetic flux induced by the circulating
current in the SQUID.
The kinetic and potential energies of the system are functions of
four dimensionless variables defined by
$\gamma=(\gamma_1+\gamma_2)/2$, $\phi=\Phi/\Phi_0$ and
$\xi_i=(B+B_i)l_ix_i/\Phi_0$. They are
\begin{eqnarray}
&T=\sum\limits_i(\frac{\hbar^2}{4E_C}\frac{1}{\Omega_i^2}\frac{1}{\mathcal{A}_i^2}\dot{\xi}_i^2)+\frac{\hbar^2}{2E_C}\dot{\gamma}^2+\frac{\pi^2\hbar^2}{2E_C}\dot{\phi}^2, \label{eq02-1}\\
&U=E_{J} [-2\cos\gamma \cos(\pi\phi) -
\frac{I_b}{I_c}\gamma+\sum\limits_i ((-1)^i\pi \frac{I_b}{I_c}\xi_i + \frac{\xi_i^2}{2\mathcal{A}_i^2})\nonumber \\
&+\frac{2\pi}{\beta_L}(\phi-\xi_1-\xi_2-\phi_e)^2], \label{eq02-2}
\end{eqnarray}
where mechanical dissipation was assumed to be negligible, and
where we define the screening parameter $\beta_L=2LI_c/\Phi_0$ and
external flux $\phi_e=BA$ for the SQUID, while the dimensionless
magnetic field
\begin{equation}
\mathcal{A}_i=\sqrt{\frac{E_J}{m_i}}\frac{(B+B_i)l_i}{\widetilde{\omega}_i\Phi_0}
\label{eq03}
\end{equation}
and oscillation frequencies
$\Omega_i=\widetilde{\omega}_i/\omega_{pl}$ are defined for each
of the beams. The first term in Eq. (\ref{eq02-1}) corresponds to
the kinetic energy of the beams, while the second and third terms
to the capacitive energy of the junctions. The first term in Eq.
(\ref{eq02-2}) relates to the Josephson junctions energy, while
the second term is the washboard potential term\cite{Poole07}. The
third term corresponds to the Lorentz force on the beams in the
classical equations of motion (EOMs), and the fourth term to the
beams' elastic potential, taken to be harmonic, as nonlinear terms
are negligible at the amplitudes concerned\cite{Aldridge05}.
Lastly, the fifth term corresponds to the inductive energy of the
SQUID.
The classical EOMs for the four variables $\gamma,\phi,\xi_1$ and
$\xi_2$ are the Euler-Lagrange equations for the system Lagrangian
$\mathcal{L}=T-U$. Before writing the Hamiltonian, we expand the
potential in series about a minimum
$(\overline{\phi},\overline{\gamma},\overline{\xi}_1,\overline{\xi}_2)$,
around which the system oscillates. Under current experimental
conditions\cite{Poot08,Poot10} such a minimum exists, as the
Hessian matrix for $U$ there, proportional to the one in Eq.
\ref{eq06}, is positive definite. The well containing the minimum
can accommodate $\sim 20$ states in $\gamma$ and $\sim 900$ in
$\phi$. If we take these two parameters to be ``frozen'' at their
respective ground states, as we will later assume, we find the
well to be infinitely deep for the $\xi_i$ parameters. This
assumption also allows us to neglect in the series expansion of
$U$ terms higher than quadratic ones in $\phi-\overline{\phi}$ and
$\gamma-\overline{\gamma}$. With these approximations in mind the
Hamiltonian $H$ reads
\begin{equation}
H=T+U=\sum\limits_i
\frac{E_C}{2\hbar^2}p_i^2+\sum\limits_{i,j}E_JV_{ij}q_iq_j,\label{eq04}
\end{equation}
where the coordinates $q_i$ are given by
\begin{equation}
q_1=\gamma-\overline{\gamma},\ \ q_2=\pi(\phi-\overline{\phi}),\ \
\
q_{2+j}=\frac{1}{\sqrt{2}\Omega_j\mathcal{A}_j}(\xi_j-\overline{\xi}_j)\label{eq05}
\end{equation}
($j=1,2$), and the canonically conjugate momenta $p_i$ are
$p_i=(\hbar^2/E_C)\dot{q}_i$. In addition,
\begin{equation}
V= \left(
\begin{array}{cccc}
r & - s & 0 & 0 \\
- s & r+\frac{2}{\pi\beta _L} & -\frac{2\sqrt{2}\Omega_1\mathcal{A}_1}{\beta _L} & -\frac{2\sqrt{2}\Omega_2\mathcal{A}_2}{\beta _L} \\
0 & -\frac{2\sqrt{2}\Omega_1\mathcal{A}_1}{\beta _L} & \Omega_1^2(1+\frac{4 \pi \mathcal{A}_1^2}{\beta _L}) & \frac{4 \pi \Omega_1\mathcal{A}_1 \Omega_2\mathcal{A}_2}{\beta _L} \\
0 & -\frac{2\sqrt{2}\Omega_2\mathcal{A}_2}{\beta _L} & \frac{4 \pi \Omega_1\mathcal{A}_1 \Omega_2\mathcal{A}_2}{\beta _L} & \Omega_2^2(1+\frac{4 \pi \mathcal{A}_2^2}{\beta _L})
\end{array}
\right),\label{eq06}
\end{equation}
where $r=\cos \overline{\gamma} \cos \left(\pi
\overline{\phi}\right)$ and $s=\sin \overline{\gamma} \sin
\left(\pi \overline{\phi}\right)$ were introduced. We see that
the beam oscillations are coupled inductively via the $V_{34}$
term. This coupling can be used to generate squeezed states in the
beams as we will show below.
The Hamiltonian is quantized in the standard way by converting the
coordinates $q_i$ and their canonically conjugate momenta $p_i$ to
operators and postulating the canonical commutation relation
$[\widehat{q}_i,\widehat{p}_j]=i\hbar\delta_{ij}$. In terms of
creation and annihilation operators $\widehat{q}_i$ and
$\widehat{p}_i$ are
\begin{eqnarray}
&\hat{q}_i=\frac{1}{2}(\frac{2E_C}{E_JV_{ii}})^{1/4}(a_i^\dagger+a_i), \label{eq07-1}\\
&\hat{p}_i=i\hbar(\frac{E_JV_{ii}}{2E_C})^{1/4}(a_i^\dagger-a_i),
\label{eq07-2}
\end{eqnarray}
and the quantized Hamiltonian is given by
\begin{eqnarray}
&H=\sum\limits_{i} \hbar\omega_i(a_i^\dagger
a_i+\frac{1}{2})\nonumber\\
&+\frac{1}{4}\hbar\omega_{pl}\sum\limits_{i\neq j}
\frac{\omega_{pl}}{\sqrt{\omega_i\omega_j}}V_{ij}(a_i+a_i^\dagger)(a_j+a_j^\dagger),\label{eq08}
\end{eqnarray}
where $\omega_i=\omega_{pl}\sqrt{V_{ii}}$. We note the frequencies
$\omega_3$ and $\omega_4$ are the same as the resonators
frequencies, $\widetilde{\omega}_1$ and $\widetilde{\omega}_2$,
apart from each having a factor due to the magnetic field at the
resonator. Thus we see that the Lagrangian classical memory
variables $\xi_1$ and $\xi_2$, in complete analogy with memory
variables in electronic circuits\cite{Cohen12}, become memory
quanta in the Hamiltonian.
Taking the same experimental conditions\cite{Poot08,Poot10} and
tuning $B$ to make $r$ of order unity results in $\hbar\omega_1\gg
k_BT$ and $\hbar\omega_2\gg k_BT$. Consequently, the first and
second harmonic oscillators are ``frozen'' at their respective
ground states. Moreover, since $\omega_1,\omega_2\gg
\omega_3,\omega_4$, exciting the nanomechanical oscillators will
not budge them from their ground state. Removing constant terms,
we are then left with the effective Hamiltonian
\begin{equation}
H=\hbar\omega_3 a_3^\dagger a_3+\hbar\omega_4 a_4^\dagger
a_4+\widetilde{V}(a_3+a_3^\dagger)(a_4+a_4^\dagger),\label{eq09}
\end{equation}
where the interaction coefficient reads
\begin{equation}
\widetilde{V}=\frac{2\pi\hbar\sqrt{\widetilde{\omega}_1\widetilde{\omega}_2}\mathcal{A}_1\mathcal{A}_2}{\beta_L(1+\frac{4\pi
\mathcal{A}_1^2}{\beta_L})^{1/4}(1+\frac{4\pi
\mathcal{A}_2^2}{\beta_L})^{1/4}}.\label{eq10}
\end{equation}
\section{Reading and Writing Quantum Information}\label{section2}
We now wish to employ this system to create entangled
nanomechanical quantum memory that can be read from and written
on. We assume the beams are cooled to a temperature low enough so
as to reduce the equilibrium state to the ground state for each of
the beams. This is possible today, e.g. by coupling to a
superconducting microwave resonator\cite{Teufel08}, even if the
environment of the beams, which includes the SQUID, has a higher
temperature.
If the interaction term in Eq. (\ref{eq09}) is small relative to
the other two terms, perturbation theory gives first-order energy
corrections in $\widetilde{V}$ only when $|\omega_3-\omega_4|\ll
\widetilde{V}/\hbar$. Thus, we will henceforth assume the beams
are identical. The Hamiltonian (\ref{eq09}) is quadratic in the
ladder operators and is thus amenable to an exact solution at all
interaction strengths\cite{Bruus04}. This solution is found by
moving to the differential representation and then diagonalizing
the quadratic form of the potential by a canonical transformation
to even and odd coordinates,
\begin{equation}
x_{e,o}=\frac{1}{\sqrt{2}}(x_1\pm x_2),\qquad
p_{e,o}=\frac{1}{\sqrt{2}}(p_1\pm p_2),\label{eq11-2}
\end{equation}
where $x_i$ and $p_i$ are the position and momentum coordinates of
the $i$th beam.
Applying Eq. (\ref{eq11-2}) on the ladder operators, we find
\begin{widetext}
\begin{eqnarray}
a_5=\frac{1}{2\sqrt{2}}\left[(\sqrt{\frac{\omega_5}{\omega_3}}+\sqrt{\frac{\omega_3}{\omega_5}})(a_3+a_4)+(\sqrt{\frac{\omega_5}{\omega_3}}-\sqrt{\frac{\omega_3}{\omega_5}})(a_3^\dagger+a_4^\dagger)\right],\label{eq11-3}\\
a_6=\frac{1}{2\sqrt{2}}\left[(\sqrt{\frac{\omega_6}{\omega_3}}+\sqrt{\frac{\omega_3}{\omega_6}})(a_3-a_4)+(\sqrt{\frac{\omega_6}{\omega_3}}-\sqrt{\frac{\omega_3}{\omega_6}})(a_3^\dagger-a_4^\dagger)\right],\label{eq11-4}
\end{eqnarray}
\end{widetext}
where $a_5^\dagger$ corresponds to creation of an even mode
quantum in which both beams oscillate in phase, and $a_6^\dagger$
to creation of an odd mode quantum, where the beams oscillate in
anti-phase. The even and odd oscillation frequencies are given by
\begin{equation}
\omega_{5,6}=\sqrt{\omega_3^2\pm2\widetilde{V}\omega_3/\hbar}.\label{eq11-5}
\end{equation}
Using this transformation and omitting constant terms, the
Hamiltonian (\ref{eq09}) is reduced to
\begin{equation}
H=\hbar\omega_5 a_5^\dagger a_5+\hbar\omega_6 a_6^\dagger
a_6.\label{eq12}
\end{equation}
We see that the even mode is decoupled from the odd mode in this
Hamiltonian, which is thus separable to an even and an odd part.
The energy spectrum of this Hamiltonian is given by
\begin{equation}
E_{nm}=n\hbar\omega_5 +m\hbar\omega_6 ,\label{eq13}
\end{equation}
while the eigenstates are
\begin{eqnarray}
|nm\rangle=\frac{1}{\sqrt{n!m!}}(a_5^\dagger)^n
(a_6^\dagger)^m|00\rangle,\label{eq14}
\end{eqnarray}
which, upon substitution of Eqs. (\ref{eq11-3}) and
(\ref{eq11-4}), are seen to be highly entangled states of the two
beams.
The quantum state of the system is read by measuring the magnetic
flux threading through the SQUID, which is done by a current
measurement in the standard way\cite{Poole07}. The operator for
this observable is
\begin{equation}
\widehat{\Phi}=-(B+B_1)l_1x_1-(B+B_2)l_2x_2,\label{eq16}
\end{equation}
where the constant term $BA$ was omitted. With the assumption of
$B_1=B_2$, we have
\begin{equation}
\widehat{\Phi}=-\sqrt{2}(B+B_1)l_1\lambda_3(a_5+a_5^\dagger),\label{eq16.5}
\end{equation}
where the zero-point fluctuation, the resonator displacement
uncertainty at the ground state, is defined as
$\lambda_i=\sqrt{\hbar/2m_1\omega_{i+2}}$ with the definition
extended also for $i=3,4$.
We thus see, as expected, that the measurement of the magnetic
flux cannot detect the odd mode, since oscillations in this mode
do not amount to a change in the area of the SQUID loop. We
therefore set to read and write quantum information only in the
$n$ quantum number in the state $|nm\rangle$. Moreover, we note
that for the eigenstates of the Hamiltonian we have $\langle
\widehat{\Phi}\rangle=0$, implying that a better observable would
be the standard deviation $\langle \Delta \widehat{\Phi}\rangle$.
This is indeed true, with the values of this observable on the
eigenstates being
\begin{equation}
\langle \Delta
\widehat{\Phi}\rangle=\sqrt{2}(B+B_1)l_1\lambda_3\sqrt{1+2n},\label{eq17}
\end{equation}
enabling us to measure the value of $n$.
Having established the reading process, we now set to describe how
to write quantum information on this system. It would seem the
best way to excite the system is via a resonant ac current of
frequency $\omega_5$ in the external wires that, according to the
Hamiltonian (\ref{eq12}), will pump the beams to their excited
state. However, such a current will also pump the beams to even
higher excited states, since the energy level difference is fixed
in this system. A better method would be to use constant currents
in the external wires. The addition to the potential
(\ref{eq02-2}) due to such currents, keeping only first-order
terms in $B_1/B$ and $B_2/B$, is
\begin{eqnarray}
H_1=\frac{\pi E_J l_1}{I_c
\beta_L\Phi_0}\{-B_1x_1[4I_c(\overline{\phi}-\phi_e)+I_b\beta_L]\nonumber\\
+B_2x_2[I_b\beta_L-4I_c(\overline{\phi}-\phi_e)]\}\label{eq18}
\end{eqnarray}
where constant terms were omitted and only linear terms in $x_i$
were kept, owing to the quadratic terms being smaller by several
orders of magnitude.
Since reading can be done only for the even mode, and the
Hamiltonian (\ref{eq12}) is separable into odd and even
components, we do not consider the odd part in Eq. (\ref{eq18}),
and by choosing also $B_1=B_2$, the even part of Eq. (\ref{eq18})
is
\begin{eqnarray}
H_{1,e}&=&-4\sqrt{2}\pi\frac{E_J l_1 \lambda_3(\overline{\phi}-\phi_e)}{\beta_L\Phi_0}B_1(a_5+a_5^\dagger)\nonumber\\
&\equiv&f(B_1)(a_5+a_5^\dagger),\label{eq18.5}
\end{eqnarray}
where constant terms were again omitted. The even part of the
total Hamiltonian $H+H_1$ is therefore
\begin{equation}
H_e=\hbar\omega_5 a_5^\dagger
a_5+f(B_1)(a_5+a_5^\dagger).\label{eq19}
\end{equation}
The idea of writing is then the following: We create a constant
magnetic field $B_1$, which shifts the harmonic potential and then
let the system relax to its new ground state. We then suddenly
revert $B_1$ to zero thereby obtaining an excited state of the
original system. The Hamiltonian in the differential
representation is
\begin{equation}
H_e=-\frac{\hbar^2}{2m_1}\frac{d^2}{d
x^2}+\frac{1}{2}m_1\omega_5^2x^2+\lambda_3^{-1}f(B_1)x,\label{eq20}
\end{equation}
where the mass $m_1$ is the beam mass. Apart from a constant, the
Hamiltonian (\ref{eq20}) is
\begin{eqnarray}
H_e&=&-\frac{\hbar^2}{2m_1}\frac{d^2}{d
x^2}+\frac{1}{2}m_1\omega_5^2(x+\Delta
x)^2,\label{eq21}\\
\Delta x&=&\frac{f(B_1)}{m_1 \lambda_3\omega_5^2}.\label{eq22}
\end{eqnarray}
According to the scheme described above, we wish to maximize the
probability $P_{nn}=|\langle \psi_0(x+\Delta
x)|\psi_n(x)\rangle|^2$, with $\psi_n(x)$ being the $n$th harmonic
oscillator wavefunction, to obtain the desired state $|n\rangle$
by tuning $B_1$ and with it $\Delta x$. Using standard results of
the quantum harmonic oscillator to write the integral $\langle
\psi_0(x+\Delta x)|\psi_n(x)\rangle$ and then using the generating
function of the Hermite polynomials to find its value for every
$n$, we find the maximum value of $P_{nn}$ is reached when $\Delta
x=2\sqrt{n}\lambda_3$ with the probability then to measure $k$
phonons after $B_1$ is removed given by
\begin{equation}
P_{kn}=(k!)^{-1}e^{-n}n^k,\label{eq23}
\end{equation}
which is a Poisson distribution with mean $n$. The $P_{nn}$
function is plotted in Fig. \ref{fig02}. We see that the probability
drops sharply for small $n$ and evens out for larger values. The
limit at $n\rightarrow\infty$ is 0. Although the writing process
does not create a pure number state $|n\rangle$, the standard
deviation in the number of phonons, by the properties of the
Poisson distribution, is $\sqrt{n}$, which is reasonably low.
\begin{figure}[t]
\begin{center}
\includegraphics[width=9.00cm]{fig02.eps}
\caption{\label{fig02} Graph of the maximum probability of the ground state of the shifted harmonic potential having a $|n\rangle$ component in the unshifted potential. We have $P_{00}=1$ and $P_{11}=e^{-1}$. The values for non-integer $n$ were interpolated by $n!=\Gamma(n+1)$.}
\end{center}
\end{figure}
\section{Squeezed states} \label{section3}
Having shown how to read and write quantum information in this
system, we now wish to demonstrate the possibility of creating
squeezed states. In an effort to mimic the Hamiltonian of a
degenerate parametric amplifier from quantum optics\cite{Walls08},
we set the external wires magnetic fields to oscillate at double
the frequency of the even mode, namely,
\begin{eqnarray}
B_1&=&B_{1,0}e^{2i\langle\omega_5\rangle t}+\mathrm{c.c},\nonumber\\
B_2&=&B_{2,0}e^{2i\langle\omega_5\rangle
t}+\mathrm{c.c},\label{eq24}
\end{eqnarray}
where $\langle \omega_5\rangle$ denotes the time average of
$\omega_5$, and the oscillations of $\omega_5$ about this average
are small since $B_i\ll B$.
The non-interacting Hamiltonian is found from Eq. (\ref{eq12}) to
be
\begin{eqnarray}
H_0=\hbar \langle\omega_5\rangle a_5^\dagger a_5+\hbar
\langle\omega_6\rangle a_6^\dagger a_6.\label{eq25}
\end{eqnarray}
Keeping only terms to first order in $B_i/B$, the addition to the
potential (\ref{eq02-2}) due to the oscillating magnetic field in
the rotating wave approximation (RWA), valid here due to the weak
damping, is
\begin{eqnarray}
\hspace{-2em} H_1=\frac{4 \pi E_J l_1^2 B}{\Phi _0^2 \beta
_L}\{(B_1+B_2)x_1x_2+B_1 x_1^2+B_2 x_2^2\}.\label{eq26}
\end{eqnarray}
We write this addition in terms of ladder operators, take
$B_{1,0}=B_{2,0}$ to be pure imaginary and eliminate constant
terms to find the interaction picture Hamiltonian in the RWA to be
\begin{eqnarray}
&H_I=\frac{8\pi i\langle\lambda_3\rangle^2 E_J l_1^2 B
|B_{1,0}|}{\Phi_0^2\beta_L}\{a_5^2-(a_5^\dagger)^2\},\label{eq27}
\end{eqnarray}
which is the squeezing Hamiltonian. In writing Eq. (\ref{eq27}) we
neglected the time-dependent term $(\omega_5-\langle
\omega_5\rangle)a_5^\dagger a_5+(\omega_6-\langle
\omega_6\rangle)a_6^\dagger a_6$, since it is negligible relative
to the squeezing term under the assumed experimental conditions.
It is interesting to note that squeezing of the odd mode is not
possible using this scheme, even if $\omega_5$ is replaced with
$\omega_6$ in Eq. (\ref{eq24}). The fundamental reason for this is
Lenz law, which makes the coefficient of $x_1x_2$ in Eq.
(\ref{eq26}) positive and thus excludes terms proportional to
$a_6^2$ in Eq. (\ref{eq27}). The coefficient is positive because
moving both beams in the positive direction costs energy, since
both movements decrease the magnetic flux.
We now consider the effect of squeezing in this system and devise
means to observe it. We assume dissipation is weak, and both beams
are initially in the ground state. Conforming to standard
notation, the squeezing parameter is
\begin{eqnarray}
g=\frac{16\pi \langle\lambda_3\rangle^2 E_J l_1^2 B
|B_{1,0}|t}{\hbar\Phi_0^2\beta_L},\label{eq28}
\end{eqnarray}
which is real, and the squeezing operator reads
\begin{eqnarray}
S(g)=\exp[g(a_5^2-(a_5^\dagger)^2)/2].\label{eq29}
\end{eqnarray}
The time evolution of the $a_5$ operator in the interaction
picture is given by
\begin{eqnarray}
S^\dagger(g)a_5 S(g)=a_5\cosh g-a_5^\dagger \sinh g.\label{eq30}
\end{eqnarray}
In the rotating frame the uncertainty in the positions and momenta
of the beams are
\begin{eqnarray}
\langle\Delta x_i\rangle&=&\lambda_4(1+\tanh (g+\ln(\omega_5/\omega_6)/2))^{-1/2},\label{eq31}\\
\langle\Delta p_i\rangle&=& \frac{\hbar}{2} \lambda_4^{-1}(1-\tanh
(g+\ln(\omega_5/\omega_6)/2))^{-1/2},\label{eq32}
\end{eqnarray}
where Eqs. (\ref{eq05}), (\ref{eq07-1}), (\ref{eq11-3}),
(\ref{eq11-4}) and (\ref{eq30}) were used. We see that we have
limited squeezing to below the standard quantum limit in the
positions and unlimited anti-squeezing in the momenta, as the even
mode is squeezed, while the odd mode is not. The interaction
modifies the squeezing by adding the positive term of
$\ln(\omega_5/\omega_6)/2$ to the squeezing parameter. In
addition, with the product of the uncertainties being
\begin{eqnarray}
\langle \Delta x_i \rangle \langle\Delta
p_i\rangle=\frac{\hbar}{2}\cosh(g+\ln(\omega_5/\omega_6)/2),\label{eq33}
\end{eqnarray}
we see that due to the interaction the minimum uncertainty is no
longer attained before squeezing takes place.
It is interesting to write the wavefunction for the beams in the
differential representation in the presence of squeezing. This
wavefunction can be found by using the relation between the
two-photon coherent states and the squeezed states\cite{Walls08}
to write for the squeezed state $|g\rangle$
\begin{eqnarray}
(\cosh g a_5+\sinh g a_5^\dagger)a_6|g\rangle=0.\label{eq34}
\end{eqnarray}
After moving to the differential representation, Eq. (\ref{eq34})
is solved to find a wavefunction of the form
\begin{equation}
\psi(x_e,x_o)=C_1e^{-\frac{m_1\omega_5}{2\hbar}e^{2g}x_e^2}e^{-\frac{m_1\omega_6}{2\hbar}x_o^2},\label{eq34-2}
\end{equation}
with $C_1$ being a normalizing constant. When we transform this
wavefunction to the beam coordinates via Eq. (\ref{eq11-2}) we
find that the new wavefunction is in a jointly Gaussian form,
namely
\begin{equation}
\psi(x_1,x_2)=C_1\exp(-\frac{\frac{x_1^2}{\sigma_1^2}+\frac{x_2^2}{\sigma_2^2}-\frac{2
r x_1 x_2}{\sigma_1\sigma_2}}{2(1-r^2)}),\label{eq35}
\end{equation}
where $\sigma_1=\sigma_2=\sqrt{2}\langle\Delta x_1\rangle$, with
$\langle\Delta x_1\rangle$ given by Eq. (\ref{eq31}), and the
correlation coefficient given by
\begin{equation}
r=-\tanh (g+\ln(\omega_5/\omega_6)/2).\label{eq36}
\end{equation}
We note that the factor of $\sqrt{2}$ in $\sigma_i$ comes from
$|\psi(x_1,x_2)|^2$, rather than $\psi(x_1,x_2)$, being the
probability distribution. In addition, we see that, as before, the
beam interaction results in an addition to the squeezing
parameter, which gives negative correlation even at $t=0$. The
correlation due to the squeezing is negative, because the
influence of the odd mode, which is not squeezed, increases with
time, producing perfect anti-correlation when the squeezing
parameter goes to infinity.
Lastly, we consider the effect of the squeezing on the measurement
of the magnetic flux in the SQUID. Using Eq. (\ref{eq16.5}), we
find the standard deviation of $\widehat{\Phi}$ in the rotating
frame to be
\begin{eqnarray}
\langle \Delta\widehat{\Phi}\rangle=\sqrt{2}(B+B_1)l_1\lambda_3
e^{-g},\label{eq37}
\end{eqnarray}
which is fully squeezed, while in the lab frame we have
\begin{eqnarray}
\langle
\Delta\widehat{\Phi}\rangle=\sqrt{2}(B+B_1)l_1\lambda_3\nonumber\\
\times\sqrt{\cosh(2g)-\sinh(2g)\cos(2\omega_5 t)},\label{eq38}
\end{eqnarray}
which characteristically oscillates between fully squeezed values
at $t=(\pi/\omega_5)p$, corresponding to Eq. (\ref{eq37}), and
fully anti-squeezed values at $t=(\pi/\omega_5)(p+1/2)$, where $p$
is an integer. We conclude that the squeezing effect is
measurable, and that the squeezing parameter can be found from the
measurements.
\section{Effect of Mechanical Damping} \label{section4}
In reality, the damping of the beam oscillations is weak but
nonzero. With regard to reading and writing quantum information,
this is not a problem, so long as the reading or writing is
performed within a period much shorter than the characteristic
decay time. The squeezed states, however, are measurably degraded even by very weak dissipation, as we show in this
section.
Many models were devised for describing dissipation in quantum
systems\cite{Gardiner04}. We choose here to work with the quantum
master equation. In the interaction picture with the Hamiltonian
(\ref{eq27}), the quantum master equation takes the
form\cite{Chiao08}
\begin{eqnarray}
&\frac{\partial}{\partial
t}\rho(t)=\mathcal{L}_S\rho(t)+\mathcal{L}_{dis}\rho(t),\label{eq39}\\
&\mathcal{L}_S\rho(t)=\frac{1}{2} \zeta[a^2-(a^\dagger)^2,\rho(t)],\label{eq40}\\
&\mathcal{L}_{dis}\rho(t)=-\frac{\gamma}{2}(n_{cav}+1)\{[a^\dagger,a\rho(t)]+[\rho(t)a^\dagger,a]\}\nonumber\\
&-\frac{\gamma}{2}n_{cav}\{[a,a^\dagger\rho(t)]+[\rho(t)a,a^\dagger]\},\label{eq41}
\end{eqnarray}
where $\zeta=g/t$ is the squeezing rate and for brevity we write
$a$ instead of $a_5$ and $\omega$ instead of $\omega_5$. In Eqs.
(\ref{eq39}-\ref{eq41}) $\rho(t)$ is the statistical operator for
the system, $\mathcal{L}_S$ and $\mathcal{L}_{dis}$ are the
Liouville operators for the squeezing and dissipation,
respectively, $\gamma=\omega/Q$ is the damping rate of the even
mode, where $Q$ is the beam quality factor, and
$n_{cav}=(e^{\hbar\omega/k_BT}-1)^{-1}$ is the average phonon
occupation in the even mode.
The system can be equivalently described by the Wigner
quasi-probability distribution $W(\alpha,\alpha^*)$ instead of by
the statistical operator $\rho(t)$, where we omit the explicit
time dependence in $W(\alpha,\alpha^*)$ to make the notation
concise. The parameter $\alpha=X_1+iX_2$ is a complex number that
is related to the phase space coordinates via
$\alpha=\frac{1}{2\lambda}x+\frac{i\lambda}{\hbar}p$, where $x$
and $p$ are the even mode position and momentum coordinates,
respectively, and $\lambda=\sqrt{\hbar/2m_1\omega}$ as before. We
convert\cite{Gardiner04,Rabl04} Eq. (\ref{eq39}) to an equation
for the Wigner distribution to find
\begin{eqnarray}
&\frac{\partial W(X_1,X_2)}{\partial
t}=[\zeta(X_1\frac{\partial}{\partial
X_1}-X_2\frac{\partial}{\partial
X_2})+\frac{\gamma}{2}(\frac{\partial}{\partial X_1}
X_1+\frac{\partial}{\partial X_2} X_2)\nonumber\\
&+\frac{1}{4}\gamma (n_{cav}+\frac{1}{2})(\frac{\partial^2}{\partial
X_1^2}+\frac{\partial^2}{\partial X_2^2})]W(X_1,X_2),\label{eq42}
\end{eqnarray}
where we note that the original Wigner function
$\overline{W}(x,p)$ is related to the one used here by
$W(\alpha,\alpha^*)=W(X_1,X_2)=2\hbar\overline{W}(x,p)$.
Equation (\ref{eq42}) is seen to be a special case of the
Fokker-Planck equation with $W(\mathbf{u})$ corresponding to the
probability distribution $P(\mathbf{u};t)$, where
$\mathbf{u}=(X_1,X_2)$. Put in this form, the equation can be
formally written as
\begin{eqnarray}
\hspace{-2em}\frac{\partial W(\mathbf{u})}{\partial
t}=-\nabla\cdot
[\mathbf{F}(\mathbf{u})W(\mathbf{u})]+\frac{D_0}{2}\nabla^2
W(\mathbf{u}),\label{eq43}
\end{eqnarray}
where
$\mathbf{F}=(-(\zeta+\frac{\gamma}{2})X_1,(\zeta-\frac{\gamma}{2})X_2)$
and $D_0=\frac{1}{2}\gamma (n_{cav}+\frac{1}{2})$ are the force and
diffusion constant, respectively. Due to the form of the force in
Eq. (\ref{eq43}), we can use separation of variables to break this
equation into two one-dimensional Fokker-Planck equations with
solutions $W_1(X_1)$ and $W_2(X_2)$, where
$W(X_1,X_2)=W_1(X_1)W_2(X_2)$. These solutions are given by
($i=1,2$)
\begin{eqnarray}
W_i(X_i)&=&\frac{1}{\sqrt{2\pi} \sigma_i(t)} \exp
(-\frac{X_i^2}{2\sigma_i^2(t)}),\label{eq44}\\
\sigma_i(t)&=&\sqrt{(\frac{1}{4}-\frac{D_0}{k_i})e^{-k_it}+\frac{D_0}{k_i}},\label{eq45}\\
k_1&=&2\zeta+\gamma,\label{eq46}\\
k_2&=&\gamma-2\zeta,\label{eq47}
\end{eqnarray}
where $k_i$ are the decay rates.
Equations (\ref{eq44}-\ref{eq47}) indicate that a steady-state
solution always exists for $W_1(X_1)$ and is given by Eq.
(\ref{eq44}) with $\sigma_1(t)=\sqrt{D_0/k_1}$. This finite
distribution width corresponds to a saturation in the squeezing in
contrast with the dissipationless case, when the field quadrature
$X_1$ is squeezed without limit.\cite{Walls08} For $W_2(X_2)$ on
the other hand, we have a steady-state solution only at the strong
damping regime, $\gamma>2\zeta$, and this solution exhibits
$\sigma_2(t)=\sqrt{D_0/k_2}$. When the strong damping condition is
not satisfied, $k_2$ is negative, there is no steady state, and
$X_2$ is anti-squeezed as in the dissipationless
case\cite{Walls08}, but at a slower pace since the leading
behavior in $\Delta X_2$ is $e^{(\zeta-\gamma/2)t}$ instead of
$e^{\zeta t}$ as in the dissipationless case.
The knowledge of the Wigner function in Eq. (\ref{eq44}) enables
us to calculate of system properties via the
relation\cite{Gardiner04}
\begin{eqnarray}
\langle \{a^r(a^\dagger)^s\}_{\mathrm{sym}}\rangle=\int d^2\alpha
\alpha^r(\alpha^*)^s W(\alpha,\alpha^*),\label{eq48}
\end{eqnarray}
where $\{\cdot\}_{\mathrm{sym}}$ indicates the average of all the
permutations of the ladder operators, and $d^2\alpha=dX_1dX_2$.
Working in the rotating frame, the resulting uncertainties in the
positions and momenta of the beams read $(i=1,2)$
\begin{eqnarray}
\langle \Delta
x_i\rangle&=&\sqrt{2}\lambda_3\sqrt{\frac{1}{4}\frac{\omega_5}{\omega_6}+\sigma_1(t)^2},\label{eq49}\\
\langle \Delta
p_i\rangle&=&\frac{1}{\sqrt{2}}\hbar\lambda_3^{-1}\sqrt{\frac{1}{4}\frac{\omega_6}{\omega_5}+\sigma_2(t)^2},\label{eq50}
\end{eqnarray}
which reduce to Eqs. (\ref{eq31}-\ref{eq32}) when $\gamma=0$.
We see that the squeezing in the position coordinates, already
limited to $\sqrt{\omega_3/2\omega_6}$ of the standard quantum
limit, $\lambda_1$, in the dissipationless case of Eq.
(\ref{eq31}), is limited here as well with the same limit, where
we take $n_{cav}= 0$ due to the previous assumption of
$\hbar\omega\gg k_BT$. The momenta uncertainties, in comparison,
are anti-squeezed only in the weak damping regime,
$\gamma<2\zeta$, compared with being always anti-squeezed in Eq.
(\ref{eq32}), when there is no damping. As with the quadrature
field $X_2$, the momenta anti-squeezing in the weak damping regime
has a slower rate relative to the dissipationless case with a
leading behavior of $e^{(\zeta-\gamma/2)t}$ vs. $e^{\zeta t}$ for
the dissipationless case. The product of the position and momentum
uncertainties in Eqs. (\ref{eq49}-\ref{eq50}) gives the lowest
uncertainty at $t=0$ and higher values afterwards.
\begin{figure}[t]
\begin{center}
\large\flushleft{\mbox{(a)}}\\
\vspace{-2.5em}\includegraphics[width=9.00cm]{fig03-1.eps}\\
\vspace{-1.1em}\flushleft{\mbox{(b)}}\\
\vspace{-2.5em}\includegraphics[width=9.00cm]{fig03-2.eps}\\
\vspace{-1.1em}\flushleft{\mbox{(c)}}\\
\vspace{-2.5em}\includegraphics[width=9.00cm]{fig03-3.eps}\\
\caption{\label{fig03} Normalized deviation in the SQUID magnetic flux $\Delta \Phi(t)/\Delta \Phi(t=0)$ as a function of $t/T_1$, where $n_{cav}=0$, $T_1=2\pi/\omega$ and
$\zeta=0.1/T_1$. (a) No damping ($\gamma=0$): Lower and upper limits are $0$ and $\infty$. (b) Weak damping ($\gamma=\zeta$): Lower and upper limits are $1/\sqrt{3}$ and $\infty$. (c)
Strong damping ($\gamma=3\zeta$): Lower and upper limits are $\sqrt{3}/2$ and $\sqrt{3}$.}
\end{center}
\end{figure}
As in Sec. \ref{section3}, we wish to find here the effect of
squeezing on the measurement of the magnetic flux. Using Eq.
(\ref{eq16.5}), we find the standard deviation in the rotating
frame to be
\begin{eqnarray}
\langle \Delta\widehat{\Phi}\rangle=2\sqrt{2}(B+B_1)l_1\lambda_3
\sigma_1(t),\label{eq51}
\end{eqnarray}
which is squeezed, though only to a finite extent unlike the
dissipationless case in Eq. (\ref{eq37}), where it is fully
squeezed. In the lab frame we have
\begin{eqnarray}
\langle
\Delta\widehat{\Phi}\rangle=2\sqrt{2}(B+B_1)l_1\lambda_3\nonumber\\
\times\sqrt{\cos^2(\omega t)\sigma_1^2(t)+\sin^2(\omega
t)\sigma_2^2(t)},\label{eq52}
\end{eqnarray}
which oscillates between squeezed values at $t=(\pi/\omega)p$,
corresponding to Eq. (\ref{eq51}), and squeezed/anti-squeezed
values, depending on the damping regime, at
$t=(\pi/\omega)(p+1/2)$, where $p$ is an integer. In the
dissipationless limit of $\gamma=0$ Eq. (\ref{eq52}) reduces to
Eq. (\ref{eq38}). Eq. (\ref{eq52}) is plotted in Fig. \ref{fig03}
with normalized units for the no-damping, weak-damping and
strong-damping cases. We conclude again that the squeezing effect
is detectible and leads to reduced variation in the measured
magnetic flux in the SQUID.
\section{Conclusions} \label{conclusions}
In this work we have demonstrated that a system composed of two
nanomechanical resonators embedded in a dc SQUID can be used as
two units of quantum memory and that only the even mode in these
two units is readable by the SQUID. We showed how the state of the
beams can be altered, corresponding to writing quantum
information, and proved the amplitude distribution of the number
states in the resulting state is Poisson distributed. We then
proposed a scheme to squeeze the even mode of the resonators and
thus decrease the noise in the SQUID magnetic flux. Taking
dissipation into account, we found a criterion that separates the
weak damping regime, where a steady state exists only in one field
quadrature, from the strong damping one, where both field
quadratures exhibit steady states. We then predicted the form of
the fluctuations in the magnetic flux in the SQUID, by which
squeezing can be observed.
The approximations and assumptions made during our derivations
hold well for reasonable experimental values. For instance, for
two identical 8 MHz resonators of length 25 $\mu$m and quality
factor $Q=2\cdot 10^4$, an external magnetic field of 10 T, beam
temperature of $0.1$ mK, SQUID temperature of 20 mK and other
parameter values similar to the ones in Refs.
\onlinecite{Poot08,Poot10}, we find the energy level differences
in the Hamiltonian (\ref{eq12}) to be much larger than both $k_BT$
and the level widths. Moreover, for the reading process, Eq.
(\ref{eq17}) gives a required SQUID sensitivity of $1.3\cdot
10^{-5}\frac{\Phi_0}{\sqrt{\mathrm{Hz}}}$ for $n\sim 1$ and
sensitivity of $1.3\cdot
10^{-5}\frac{1}{\sqrt{2n}}\frac{\Phi_0}{\sqrt{\mathrm{Hz}}}$ for
$n\gg 1$. A typical SQUID with a flux sensitivity of
$10^{-6}\Phi_0/\sqrt{\mathrm{Hz}}$ satisfies these conditions for
$n<80$.
Regarding the squeezing, a major question is whether substantial
squeezing can be achieved within the decoherence time for the
states. The decoherence time for the resonators here can be made
to be at least 5 $\mu$s\cite{Regal08,Groblacher09,Verhagen12},
while substituting the parameters above in Eq. (\ref{eq28}) gives
a characteristic squeezing time of $\tau_{sq}\sim$ 2 $\mu$s. We
therefore conclude that substantial squeezing is achievable within
the dephasing time.
The experimental realization of this system will be an important
demonstration of macroscopic quantum behavior and squeezing in a
nanomechanical system. In addition it can be used for detecting
the position of the embedded nanomechanical beams with accuracy
higher than the standard quantum limit. Stacking such SQUIDS in
series, with the upper arm of the lower SQUID being also the lower
arm of the upper one, can form a quantum data
bus\cite{Li05,Sillanpaa07}, lead to a multi-mode entangled
state\cite{Nielsen10}, and possibly multi-mode
squeezing\cite{Walls08}. Another application of this system or a
close variant of it is that of a quantum gate\cite{Nielsen10}
acting on the two states by means of currents in the external
wires. A series of such quantum gates can form the basis of a
nanomechanical quantum computer\cite{Cleland04,Ladd10}. We leave
the development of these ideas for future studies.
\section*{Acknowledgments}
This work has been
partially funded by the NSF grant No. DMR-0802830. One of us (MD) is grateful to the Scuola Normale Superiore of Pisa for the hospitality during a visit where part of this
work has been initiated, and to S. Pugnetti and R. Fazio for useful discussions.
\section*{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,579
|
Q: How Do I Put My Bootloader And Kernel On A USB I've written a bootloader and Basic kernel as a fun side project while i'm learning 2 stage bootloaders, I want to load my bootloader at sector 1 (or the MBR) of the USB and the Kernel at sector 2. I've compiled both into Bootloader.bin & Kernel.bin using NASM. I just need a little help on actually writing them onto the USB. I have access to both Windows and Linux so any answers a appreciated.
Bootloader.asm
[BITS 16]
[ORG 0x7C00]
ResetDisk:
XOR AH, AH
INT 0x13
JC ResetDisk
ReadDisk:
MOV BX, 0x8000
MOV ES, BX
MOV BX, 0x0000
MOV AH, 0x02
MOV AL, 1
MOV CH, 0x00
MOV CL, 0x02
MOV DH, 0x00
INT 0x13
JC ResetDisk
JMP 0x8000:0x0000
TIMES 510-($-$$) DB 0
DW 0xAA55
Kernel.asm
[BITS 16]
[ORG 0x8000]
MOV SI, HelloString
CALL PrintString
JMP $
PrintChar:
MOV AH, 0x0E
MOV BH, 0x00
MOV BL, 0x0F
RET
PrintString:
MOV AL, [SI]
INC SI
OR AL, AL
JZ Exit
CALL PrintChar
JMP PrintString
Exit:
RET
HelloString DB 'Hello World!',0
TIMES 512-($-$$) DB 0
A: Figure out which device represents the USB key in /dev (the key itself, NOT a partition on it), then you can simply use dd or a similar tool to copy your data.
Example, where /dev/xxx is your USB key:
cat Bootloader.bin Kernel.bin > image.bin
sudo dd if=image.bin of=/dev/xxx bs=4k
Note that you may have to umount any mounted partition that comes from the USB key first. Also note that, but this goes without saying, make sure there is nothing important on the USB key first.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 80
|
\section{Submodules of nest modules}
In this section we characterize submodules of $\mathcal{T}_{(M, N)}$ in terms of certain finite sequences of left row reduced echelon or right column reduced echelon matrices with entries from $D$. First we need the following useful lemma.
\begin{lem}\label{2.1}
{\rm (i)} {\it If $ J \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big) $, then ${\rm Row}_i (J) \in \mathcal{LS}_{m_i \times n}$ for all $ 1 \leq i \leq r$. }
{\rm (ii)} {\it If $ J \in \mathcal{RS}\big(\mathcal{T}_{(M, N)}\big) $, then ${\rm Col}_j (J) \in \mathcal{RS}_{m \times n_j}$ for all $ 1 \leq j \leq s$.}
\end{lem}
\bigskip
\noindent {\bf Proof.} It suffices to prove (i). Part (ii) can be proved analogously. Let $ J \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big) $ and $ 1 \leq i \leq r$ be given. Clearly, ${\rm Row}_i (J)$ is additive. So it remains to show that $ B {\rm Row}_i (J) \subseteq {\rm Row}_i (J)$ for all $ B \in M_{m_i} $. To see this, given $ B \in M_{m_i}$, we can write
$$ B {\rm Row}_i (J) = {\rm Row}_i ({\bf B}_{ii}J) \subseteq {\rm Row}_i (J),$$
where ${\bf B}_{ii} \in \mathcal{T}_M$ is the block matrix with $B$ in the $(i, i)$ place and zero elsewhere. This completes the proof.
\hfill\qed
\bigskip
Let $ N = (n_1, \ldots, n_s)$, where $ n_i \in \mathbb{N}$ and $ n_1 + \cdots + n_s= n$. Define $\mathcal{R}_N(r; D)$ or simply $\mathcal{R}_N(r)$ as follows
$$ \mathcal{R}_N(r ;D)= \mathcal{R}_N(r)$$
$$:= \Big\{ (R_1, \ldots, R_r) \in \mathcal{R}_n^r: R_1 \geq \cdots \geq R_r, {\rm Col}_j (R_i) = 0\ \forall j < i, 1 \leq i \leq r, 1 \leq j \leq s \Big\}.$$
Define $ \vee$ and $\wedge$ on $\mathcal{R}_N(r)$ componentwise, e.g., for $ R = ( R_1, \ldots, R_r)$ and $R'= (R'_1, \ldots, R'_r)$ in $ \mathcal{R}_N(r; D)$, we write $ R \vee R' := ( R_1 \vee R'_1, \ldots, R_r \vee R'_r)$. Also when $ M = N$, define $\ast$ on $ \mathcal{R}_M (r)$ componentwise via the mapping $\phi$. Note that the operations $ \vee$ and $\wedge$, we just defined on $\mathcal{R}_N(r)$, and the operation $\ast$ defined on $ \mathcal{R}_M (r)$, are well-defined in the sense that $ R \vee R' , R \wedge R'\in \mathcal{R}_N(r)$ whenever $ R , R' \in \mathcal{R}_N(r)$, and that $R \ast R'\in \mathcal{R}_M(r)$ whenever $ R , R' \in \mathcal{R}_M(r)$. Likewise, for $M= (m_1, \ldots, m_r)$ with $ m_i \in \mathbb{N}$ and $ m_1 + \cdots + m_r= m$, we define
$$ \mathcal{C}_M(s; D) = \mathcal{C}_M(s)$$
$$ := \Big\{ (C_1, \ldots, C_s) \in \mathcal{C}_m^s:C_1 \leq \cdots \leq C_s, {\rm Row}_i (C_j) = 0\ \forall j < i, 1 \leq i \leq r , 1 \leq j \leq s\Big\}.$$
Again define $ \vee$, $\wedge$ on $\mathcal{C}_M(s)$ and, when $M=N$, define the operation $\ast$ on $\mathcal{C}_M(r)$ componentwise via $\phi$ in a similar fashion. It is easily checked that $(\mathcal{R}_N(r), \vee, \wedge)$ and
$(\mathcal{C}_M(s), \vee, \wedge)$ are modular lattices, and that if $M=N$, then $(\mathcal{R}_M(r), \vee, \ast)$ and
$(\mathcal{C}_M(r), \vee, \ast)$ are hemirings. Also, it is readily verified that $\big(\mathcal{LS}(\mathcal{T}_{(M, N)}), +, \cap \big) $ and $ \big( \mathcal{RS}(\mathcal{T}_{(M, N)}), +, \cap \big )$ are modular lattices and that $\big(\mathcal{LS}(\mathcal{T}_{N}), +, . \big) $ and $ \big( \mathcal{RS}(\mathcal{T}_{N}), +, . \big )$ are hemirings. Our next result shows that these modular lattices are pairwise isomorphic via a natural map exhibited in the theorem below.
\bigskip
\begin{thm}\label{2.2}
{\rm (i)} {\it There exists an isomorphism of lattices
$$ \Phi : \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big) \longrightarrow \mathcal{R}_{N}(r)$$
with the property that
$$ I = \left(\begin{array}{cccc}
M_{m_1 \times n} R_1\\
M_{m_2 \times n} R_2\\
\vdots \\
M_{m_r \times n} R_r\\
\end{array}\right), $$
where $ \Phi(I) = R = (R_1 , \ldots , R_r)$ and $R_i = \phi\big({\rm Row}_i(I)\big)$
for all $ I \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big)$ and $ 1 \leq i \leq r$.}
{\rm (ii)} {\it There exists an isomorphism of lattices
$$ \Phi : \mathcal{RS}\big(\mathcal{T}_{(M, N)}\big) \longrightarrow \mathcal{C}_{M }(s)$$
with the property that
$$ I = \left(\begin{array}{cccc}
C_1 M_{m \times n_1} &
C_2 M_{m \times n_2} & \cdots & C_s M_{m \times n_s}
\end{array}\right), $$
where $ \Phi(I) = C = (C_1 , \ldots , C_s)$ with $ C_j = \phi\big({\rm Col}_j(I)\big)$
for all $ I \in \mathcal{RS}\big(\mathcal{T}_{(M, N)}\big)$ and $ 1 \leq j \leq s$.}
\end{thm}
\bigskip
\noindent {\bf Proof.} We prove (i). Part (ii) can be proved analogously. Let $ I \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big)$ be given. In view of Lemma \ref{2.1} and the useful observation we made prior to Theorem \ref{1.2}, we see that, for each $ 1 \leq i \leq r$, there exists a unique $ R_i \in M_n$ such that $ {\rm Row}_i (I) = M_{m_i \times n} R_i$, i.e., $ \phi\big({\rm Row}_i(I)\big)= R_i$, and moreover
$$ I = \left(\begin{array}{cccc}
M_{m_1 \times n} R_1\\
M_{m_2 \times n} R_2\\
\vdots \\
M_{m_r \times n} R_r\\
\end{array}\right).$$
It remains to show that $(R_1, \ldots, R_r ) \in \mathcal{R}_{N}(r)$, and hence $ \Phi(I) = R = (R_1 , \ldots , R_r)$ is well-defined and that $ \Phi$ is an isomorphism of modular lattices. First, we show that $ R_q \leq R_p$ whenever $ 1 \leq p, q \leq r$ and $ p < q$. Recall that
$ {\rm Row}_q (I) = M_{m_q \times n} R_q$. In particular, for a given and arbitrary $ 1 \leq i \leq n$, we have $E_{1i} R_q \in M_{m_q \times n} R_q= {\rm Row}_q (I)$, where $E_{1i} \in M_{m_q \times n} $ is a standard matrix. Thus, there exists an $A_i \in I$ such that $E_{1i} R_q = {\rm Row}_q (A_i)$. Consequently, the $i$th row of $R_q$, which is the first row of $E_{1i} R_q$, occurs as the $i_q$th row of $A_i$, where $ i_q = m_1 + \cdots + m_{q-1} + 1$. Let $ i_p= m_1 + \cdots + m_{p-1} + 1$ if $ p > 1$ and $i_p = 1$ if $p=1$. Clearly, $ i_p < i_q$. Now note that $ E_{i_p i_q} \in \mathcal{T}_M$ because $ p < q$, and that $ E_{i_p i_q} A_i$ takes the $i_q$th row of $A_i$, which is in fact the $i$th row of $R_q$, to the $i_p$th row of $A_i$. This means
${\rm row}_i (R_q) = {\rm row}_1 \Big( {\rm Row}_p ( E_{i_p i_q}A_i) \Big) \in {\rm row}_1 \Big({\rm Row}_p (I)\Big)$, for $ E_{i_p i_q} \in \mathcal{T}_M$. That is, ${\rm row}_i (R_q) \in {\rm row}_1 \Big(M_{m_p \times n} R_p\Big)$, which easily implies ${\rm row}_i (R_q) \in {\rm LRS} (R_p)$. But $ 1 \leq i \leq n$ was arbitrary. Therefore $ {\rm LRS} (R_q) \leq {\rm LRS} (R_p)$, which means $ R_q \leq R_p$. Next, we need to show that ${\rm Col}_j (R_i) = 0$ for all $ j < i $ with $ 1 \leq i \leq r$ and $ 1 \leq j \leq s$. But this is obvious because for each $ 1 \leq i \leq r$, every row of $R_i$ occurs as a row of ${\rm Row}_i(A) $ for some $ A \in I \subseteq \mathcal{T}_{(M, N)}$. In view of the useful observation made regarding the block rows of the left submodules of $\mathcal{T}_{(M, N)}$ prior to Theorem \ref{1.2}, a proof almost identical to that presented in Theorem \ref{1.2} shows that
$$ {\rm Row}_i (I_1 + I_2) = {\rm Row}_i (I_1) +{\rm Row}_i ( I_2) , {\rm Row}_i(I_1 \cap I_2) = {\rm Row}_i(I_1) \cap {\rm Row}_i( I_2),$$
for all $ I_1 , I_2 \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big)$ and $ 1 \leq i \leq r$. This clearly implies that the mapping $\Phi$ is a homomorphism of lattices. That $\Phi$ is one-to-one follows from the fact that the $\phi$'s are all one-to-one by Theorem \ref{1.2} and that $I_1 = I_2$ if and only if $ {\rm Row}_i (I_1) = {\rm Row}_i (I_2)$ for each $ 1 \leq i \leq r$. To see that $\Phi$ is onto, for a given $ R = (R_1, \ldots, R_r) \in \mathcal{R}_{N}(r)$, it is readily checked that $ \Phi(I) = R$, where
$$ I = \left(\begin{array}{cccc}
M_{m_1 \times n} R_1\\
M_{m_2 \times n} R_2\\
\vdots \\
M_{m_r \times n} R_r\\
\end{array}\right).$$
This completes the proof.
\hfill\qed
\bigskip
\noindent {\bf Remarks.} (i) In the special case when $ m= r$, $ s =n$, $ m_i = 1$, $n_j = 1$ for all $ 1 \leq i \leq m$ and $ 1 \leq j \leq n$, the theorem characterizes one-sided submodules of upper triangular rectangular matrices in terms of the elements of $\mathcal{R}_{e_n}(m)$
or $ \mathcal{C}_{e_m}(n)$, where $ e_n \in \mathbb{N}^n$ and $ e_m \in \mathbb{N}^m$ are the elements whose components are all $1$. In the more special case when $m = r = s = n$ and $ m_i = n_j = 1$ for all $ 1 \leq i, j \leq n$, the theorem characterizes all one-sided ideals of the ring of all upper triangular square matrices with entries from $D$ in terms of the elements of $\mathcal{R}_{e_n}(n)$
or $ \mathcal{C}_{e_n}(n)$.
(ii) It is clear that the mapping $\Phi$ is an isomorphism of hemirings whenever $ M= N$ and $\big(\mathcal{LS}(\mathcal{T}_{N}), +, . \big) $, $ \big( \mathcal{RS}(\mathcal{T}_{N}), +, . \big )$, $(\mathcal{R}_N(r), \vee, \ast)$, and
$(\mathcal{C}_N(r), \vee, \ast)$ are viewed as hemirings.
\bigskip
\begin{lem}\label{2.3}
{\rm (i)} {\it Let $ I \in \mathcal{LS}_{m \times n}$ with $ \phi(I) = R$. Then $I = M_m A$ for some $ A \in M_{m \times n} $ if and only if $ {\rm LRS} (R) = {\rm LRS} (A)$. }
{\rm (ii)} {\it Let $ I \in \mathcal{RS}_{m \times n} $ with $ \phi(I) = C$. Then $I = AM_n$ for some $ A \in M_{m \times n} $ if and only if $ {\rm RCS} (C) = {\rm RCS} (A)$.}
\end{lem}
\bigskip
\noindent {\bf Proof.} We prove (i). Part (ii) can be proved in a similar fashion. First let $ I \in \mathcal{LS}_{m \times n} $ with $ \phi(I) = R \in {\rm R}_n$ and $I = M_m A$ for some $ A \in M_{m \times n} $. We need to show that $ {\rm LRS} (R) = {\rm LRS} (A)$. By Theorem \ref{1.2}, $I = M_{m\times n} R$. Thus $ I = M_m A= M_{m\times n} R$. As $ A = I_m A \in I$, we have $ A = X R $ for some $ X \in M_{m\times n}$. So by the exercise we pointed out preceding Theorem \ref{1.1}, we have $ {\rm LRS} (A) \subseteq {\rm LRS} (R)$. To see the reverse inclusion, let $ 1 \leq i \leq n$ be given. It follows that $ E_{1i} R \in I = M_m(D) A$, where $ E_{1i} \in M_{m\times n}$ is a standard matrix. Thus $ {\rm LRS} (E_{1i} R) \subseteq {\rm LRS} (A)$. Consequently, $ {\rm row}_i (R) \in {\rm LRS} (A)$. This yields $ {\rm LRS} ( R) \subseteq {\rm LRS} (A)$, for $ 1 \leq i \leq n$ is arbitrary. Therefore, $ {\rm LRS} (A) = {\rm LRS} (R)$. Next suppose $ {\rm LRS} (A) = {\rm LRS} (R)$. We need to show that $I = M_mA$. Note that $ A \in I$ because $I = M_{m\times n} R$ and $ A = X R$ for some $ X \in M_{m\times n}$, for $ {\rm LRS} (A) \subseteq {\rm LRS} (R)$. This yields $ M_m A \subseteq I$. For the reverse inclusion, again since $ {\rm LRS} (R) \subseteq {\rm LRS} (A)$, we obtain $ R = P A$ for some $ P \in M_{n \times m}$. But
$$ I = M_{m\times n} R = M_{m\times n} PA \subseteq M_m A.$$
This completes the proof.
\hfill\qed
\bigskip
We need the following proposition to characterize principal submodules of $ \mathcal{T}_{(M, N)}$.
\bigskip
\begin{prop}\label{2.4}
{\rm (i)} {\it Let $ 0\not= J \in \mathcal{LS}\big(\mathcal{T}_{(M, N)}\big) $ with $ \Phi(J) = (R_1, \ldots, R_k, 0_n , \ldots, 0_n)$, where $1\leq k \leq r$ is the largest index for which $ R_k \not= 0$, and $ A \in \mathcal{T}_{(M, N)}$. Then $J = \mathcal{T}_{M} A$ if and only if
$${\rm Row}_i (J) = M_{m_i \times m_i} {\rm Row}_i (A) + \cdots + M_{m_i \times m_k } {\rm Row}_k (A)$$ for all $ 1 \leq i \leq k$ if and only if
$${\rm LRS}(R_i) = {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big),$$
for all $ 1 \leq i \leq k$.
}
{\rm (ii)} {\it Let $ 0\not= J \in \mathcal{RS}\big(\mathcal{T}_{(M, N)}\big) $ with $ \Phi(J) = ( 0_m, \ldots, 0_m, C_k, \ldots, C_s)$, where $ 1 \leq k \leq s$ is the smallest index for which $ C_k \not= 0$, and $ A \in \mathcal{T}_{(M, N)}$. Then $J = A \mathcal{T}_{N} $ if and only if
$$ {\rm Col}_j(J) = {\rm Col}_k (A) M_{n_k \times n_j } + \cdots + {\rm Col}_j (A)M_{ n_j \times n_j } $$
for all $ k \leq j \leq s$ if and only if
$${\rm RCS}(C_j) = {\rm RCS}\big( {\rm Col}_k (A) \big) + \cdots + {\rm RCS}\big( {\rm Col}_j (A) \big)$$
for all $ k \leq j \leq s$.
}
\end{prop}
\bigskip
\noindent {\bf Proof.} The assertion easily follows from the useful observations we have already made, namely $J = \mathcal{T}_{M} A$ if and only if $M_{m_i \times n } R_i = {\rm Row}_i (J) = {\rm Row}_i (\mathcal{T}_{M} A)$ for all $ 1 \leq i \leq r$, and that $J = A \mathcal{T}_{N} $ if and only if $C_j M_{m \times n_j }= {\rm Col}_j(J) = {\rm Col}_j (A \mathcal{T}_{N})$
for all $ 1 \leq j \leq s$.
\hfill\qed
\bigskip
\begin{cor}\label{2.5}
{\it Let $ A \in \mathcal{T}_{m \times n}$. Then }
\noindent {\rm (i)} {\it given $ J \in \mathcal{LS}( \mathcal{T}_{m \times n} )$, $ J =\mathcal{T}_{m} A$ if and only if $ {\rm row}_i (J)= \langle \{{\rm row}_k (A): i \leq k \leq m \}\rangle $ for each $ 1 \leq i \leq m$. }
\noindent {\rm (ii)} {\it given $ J \in \mathcal{RS}( \mathcal{T}_{m \times n} )$, $ J =A \mathcal{T}_{n} $ if and only if $ {\rm col}_j (J)= \langle \{{\rm col}_k (A): 1 \leq k \leq j \}\rangle $ for each $ 1 \leq j \leq n$. }
\end{cor}
\bigskip
\noindent {\bf Proof.} This corollary is a special case of the preceding proposition.
\hfill\qed
\bigskip
The following characterizes the principal submodules of $ \mathcal{T}_{(M, N)}$.
\bigskip
\begin{thm}\label{2.6}
{\rm (i)} {\it Let $0\not= J \in \mathcal{LS}( \mathcal{T}_{(M, N)} )$ with $ \Phi(J) = ( R_1, \ldots, R_k, 0_n, \ldots, 0_n)$, where $1\leq k \leq r$ is the largest index for which $ R_k \not= 0_n$. Then $J$ is principal if and only if $ {\rm rank}(R_{i}) - {\rm rank}(R_{i+1}) \leq m_i$ for each $ 1 \leq i \leq k$. By convention, $ R_{r+1}:= 0_n$.}
{\rm (ii)} {\it Let $ 0\not= J \in \mathcal{RS}( \mathcal{T}_{(M, N)} )$ with $ \Phi(J) = ( 0_m, \ldots, 0_m, C_k, \ldots, C_s)$, where $ 1 \leq k \leq s$ is the smallest index for which $ C_k \not= 0_m$. Then $J$ is principal if and only if $ {\rm rank}(C_{j}) - {\rm rank}(C_{j-1}) \leq n_j$ for each $ k \leq j \leq s$. By convention, $ C_{0}:= 0_m$.}
\end{thm}
\bigskip
\noindent {\bf Proof.} We prove (i). Part (ii) can be proved analogously. First, let $ J = \mathcal{T}_{M} A$ for some $ A \in \mathcal{T}_{(M, N)}$. It follows from Proposition \ref{2.4} that
$${\rm LRS}(R_i) = {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big).$$
It is plain that
\begin{eqnarray*}
{\rm rank}(R_{i}) & =& \dim {\rm LRS}(R_i) \\
& \leq& \dim {\rm LRS}\big( {\rm Row}_i (A)\big) + \dim\Big( {\rm LRS}\big( {\rm Row}_{i+1} (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big) \Big)\\
& \leq & m_i + \dim {\rm LRS}(R_{i+1}),
\end{eqnarray*}
and hence $ {\rm rank}(R_{i}) - {\rm rank}(R_{i+1}) \leq m_i$ for each $ 1 \leq i \leq k$, as desired. Next, let $ {\rm rank}(R_{i}) - {\rm rank}(R_{i+1}) \leq m_i$ for each $ 1 \leq i \leq k$. In view of Proposition \ref{2.4}, it suffices to find an $A \in \mathcal{T}_{(M, N)}$ such that
$${\rm LRS}(R_i) = {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big),$$
for all $ 1 \leq i \leq k$. We find $A$ by finding its block rows, i.e., ${\rm Row}_i (A)$'s where $ 1 \leq i \leq r$. To this end, set ${\rm Row}_i (A) = 0_{m_i \times n}$ for each $ k < i \leq r$. By hypothesis ${\rm rank} (R_k) \leq m_k$. Thus there exists an $A_k \in M_{m_k \times n}$, whose rows are chosen from those of $ {\rm LRS}(R_k)$ or are zero, such that $ {\rm LRS}(A_k) ={\rm LRS}(R_k)$. Set $ {\rm Row}_k (A) =A_k$. Now as $ {\rm rank} (R_{k-1}) - {\rm rank} (R_k) \leq m_{k-1}$ and $ {\rm LRS}(R_k) \subseteq {\rm LRS}(R_{k-1}) $,
we can find $A_{k-1} \in M_{m_{k-1} \times n}$, whose rows are chosen from those of ${\rm LRS}( R_{k-1})$ or are zero, such that $ {\rm rank}(A_{k-1}) = {\rm rank} (R_{k-1}) - {\rm rank} (R_k) \leq m_{k-1}$. Continuing in this way, we obtain an $A \in \mathcal{T}_{(M, N)}$ such that
$$ {\rm rank}({\rm Row}_i (A)) ={\rm rank}(A_i)= {\rm rank} (R_{i-1}) - {\rm rank} (R_i) \leq m_i $$
for all $ 1 \leq i \leq k$. It remains to show that $${\rm LRS}(R_i) = {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big),$$
for all $ 1 \leq i \leq k$. Fix $ 1 \leq i \leq k$ and let $ \alpha $ be an arbitrary nonzero row of $R_i$. As $ R_i \geq \cdots \geq R_k$ there exists a largest $i \leq i' \leq k $ such that $ \alpha $ is in ${\rm LRS}(R_i')$, and hence not in ${\rm LRS}(R_{i'+1})$. But this means $ \alpha$ is in ${\rm LRS}(A_{i'})= {\rm LRS}\big({\rm Row}_{i'} (A)\big)$. Consequently,
$$ \alpha \in {\rm LRS}\big( {\rm Row}_{i'} (A)\big) \subseteq {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big).$$
This implies
$${\rm LRS}(R_i) \subseteq {\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big),$$
for $ \alpha $ was an arbitrary nonzero row of $R_i$. The reverse inclusion is trivial because by the way we constructed $A_i$'s, we have
$$ {\rm LRS}\big( {\rm Row}_p (A)\big) \subseteq {\rm LRS}(R_p) \subseteq {\rm LRS}(R_i)$$
for each $ i \leq p \leq k$. This yields
$${\rm LRS}\big( {\rm Row}_i (A)\big) + \cdots + {\rm LRS}\big( {\rm Row}_k (A)\big) \subseteq {\rm LRS}(R_i),$$
which is what we want. This completes the proof.
\hfill\qed
\bigskip
\bigskip
\begin{cor}\label{2.7}
{\rm (i)} {\it Let $0\not= J \in \mathcal{LS}( \mathcal{T}_{m \times n} )$ with $ \Phi(J) = ( R_1, \ldots, R_k, 0_n \ldots, 0_n)$, where $1\leq k \leq m$ is the largest index for which $ R_k \not= 0_n$. Then $J$ is principal if and only if $ {\rm rank}(R_{i}) - {\rm rank}(R_{i+1}) \in \{0, 1\}$ for each $ 1 \leq i \leq k$. By convention, $ R_{r+1}:= 0_n$.}
{\rm (ii)} {\it Let $0\not= J \in \mathcal{RS}( \mathcal{T}_{m \times n} )$ with $ \Phi(J) = ( 0_m, \ldots, 0_m, C_k, \ldots, C_n)$, where $ 1 \leq k \leq n$ is the smallest index for which $ C_k \not= 0_m$. Then $J$ is principal if and only if $ {\rm rank}(C_{j}) - {\rm rank}(C_{j-1}) \in \{0, 1\}$ for each $ k \leq j \leq n$. By convention, $ C_{0}:= 0_m$.}
\end{cor}
\bigskip
\noindent {\bf Proof.} This is a quick consequence of the preceding theorem.
\hfill\qed
\bigskip
\section{Subbimodules of nest modules}
\bigskip
We now intend to characterize subbimodules of nest modules. As it turns out the only subbimodules of $M_{m \times n}$ are $ 0$ and $M_{m \times n}$ itself. A proof can be given based on \cite[Corollary 1.10]{RY2}. Here, we present a proof based on one of the facts we left as an exercise preceding Theorem \ref{1.1}.
\bigskip
\begin{lem}\label{3.1}
{\it The only subbimodules of $M_{m \times n}$ are the trivial ones, namely $ 0$ and $M_{m \times n}$. }
\end{lem}
\bigskip
\noindent {\bf Proof.} Let $I$ be a nonzero subbimodule of $M_{m \times n}$. We show that $ I = M_{m \times n}$. As $ I \not= 0$, there is a nonzero element $A=(a_{ij}) \in I$. Thus there are $ 1 \leq i_0 \leq m $ and $ 1 \leq j_0 \leq n$ such that $ a_{i_0 j_0} \not= 0$. Let $ B= (b_{ij}) \in M_{m \times n}$ be arbitrary. Set $ K := \{(i, j ) : 1 \leq i \leq m, 1 \leq j \leq n, b_{ij} \not= 0\}$. We can write
$ B = \sum_{(i, j) \in K} B_{ij}$, where $ B_{ij} = E_{ii} B E_{jj}$, which is the matrix whose entries are all zero except for its $(i, j)$ entry which is the same as that of $B$. Note that $ E_{ii} \in M_m $ and $ E_{jj} \in M_n $ are standard matrices. Clearly,
$${\rm LRS}(B_{ij}) \leq \langle e_j \rangle = {\rm LRS}( E_{ii_0} A E_{jj_0}),$$
where $ e_j \in D_n$ is the row vector whose components are all zero except for its $i$th component which is $1$.
Again $ E_{ii_0} \in M_m $ and $ E_{jj_0} \in M_n $ are standard matrices.
Thus there is a $ C_{ij} \in M_m$ such that $ B_{ij} = C_{ij} E_{ii_0} A E_{jj_0}$, implying that $ B_{ij} \in I$ for each $(i, j) \in K$. This yields $ B \in I$. Therefore, $ I = M_{m \times n}$, as desired.
\hfill\qed
\bigskip
\noindent {\bf Remark.} It is possible to use Theorem \ref{1.2}, to present a short proof of this lemma. To this end, viewing the given nonzero subbimodule $I$ as a left submodule, it suffices to show that ${\rm rank}(R) = n$, where $ R = \phi(I)$. As $I$ is a nonzero subbimodule of $M_{m \times n}$, we have $ IP = I$ for any invertible $ P \in M_n$. So we must have $R \in \mathcal{R}_n \cap \mathcal{C}_n$, implying that $R = \left(\begin{array}{cc}
I_r & 0 \\
0 & 0
\end{array} \right)$, where $ r = {\rm rank}(R)$. But $r = n$, for otherwise multiplying $I$ from the right by the permutation matrix $P$ obtained by exchanging the $r$th column and the $n$th column of the identity matrix, we see that $ \phi(I) $ could be both $R, RP \in \mathcal{R}_n$, which is impossible as $ R \not= RP$.
\bigskip
The following characterizes all subbimodules of nest modules of matrices.
\bigskip
\begin{thm}\label{3.2}
{\it Let $J$ be a nonzero subbimodule of $ \mathcal{T}_{(M, N)}$. Then there exists a unique $ 1 \leq i \leq \min (r, s)$, which depends on $J$, a unique increasing sequence $(j_1, \ldots, j_i)$ such that $k \leq j_k \leq s$ and $ j_k < j_{k+1} $ if $ j_k = k< i$ for all $ 1 \leq k \leq i$, and a unique sequence $(R_1 , \ldots, R_i)$ with
$ R_1 = I_n$ if $j_1 = 1$, $R_1 =\left(\begin{array}{ll}
0_{r_1 \times (n-r_1)} & I_{r_1} \\
0_{n-r_1} & 0_{(n-r_1) \times r_1}
\end{array} \right) \in M_n $ if $ j_1 > 1$, and $R_k =\left(\begin{array}{ll}
0_{r_k \times (n-r_k)} & I_{r_k} \\
0_{n-r_k} & 0_{(n-r_k) \times r_k}
\end{array} \right) \in M_n$ if $ k\geq 2$, where
$ r_k = \sum_{l= j_k}^s n_l$ provided $j_k > 1$, such that
$$ J = \left(\begin{array}{ccccc}
M_{m_1 \times n} R_1\\
\vdots \\
M_{m_i \times n} R_i\\
0_{m_{i+1} \times n} \\
\vdots \\
0_{m_r \times n}
\end{array}\right).$$
Moreover, every subbimodule of $ \mathcal{T}_{(M, N)}$ is principal. }
\end{thm}
\bigskip
\noindent {\bf Proof.} Let $J$ be a nonzero subbimodule of $ \mathcal{T}_{(M, N)}$ and $1 \leq i \leq r$ be the largest integer for which $ {\rm Row}_i (J) \not= 0$. If $ r \leq s$, then $ i \leq r = \min (r, s)$. If $ r > s$, then $ {\rm Row}_k (J) = 0 $ for each $ k > r$ because $ I \subseteq \mathcal{T}_{(M, N)}$. Thus $ i \leq s= \min(r, s)$. Now we show that $J_{kl}$ is a subbimodule of $M_{m_k \times n_l}$ for each $ 1 \leq k \leq r$ and $ 1 \leq l \leq s$. Let $ X \in J_{kl}$ and $ B \in M_{m_k}$ and $ C \in M_{n_l}$ be arbitrary. It follows that $ X = A_{kl}$ for some $ A \in J$. We can write $ \widehat{B}_{kk} A \widehat{C}_{ll} = \widehat{BXC}_{kl}$. But $ A \in J$ and $ J$ is a subbimodule. Thus $BXC \in J_{kl}$, and hence $J_{kl}$ is a subbimodule of $M_{m_k \times n_l}$. It follows from the preceding lemma that $ J_{kl} = 0_{m_k \times n_l}$ or $ J_{kl} =M_{m_k \times n_l}$. For $ 1 \leq k \leq i$, let $ k \leq j_k \leq s$ be the smallest integer for which $ J_{kj_k} \not= 0_{m_k \times n_{j_k}}$, or equivalently, $ J_{kj_k}= M_{m_k \times n_{j_k}}$. So there exists an $A \in J$ such that $A _{kj_k} = E_{11} \in M_{m_k \times n_{j_k}}$. For $ 1 \leq u \leq i$, let $ p_u =1 + \sum_{t=1}^{u-1}m_t $. Clearly, $ E_{p_k p_l} \in M_{m \times n} \cap \mathcal{T}_M$ for each $ 1\leq k < l < i$. Thus $ E_{p_k p_l} A \in J$. On the other hand, the $p_k$th row of $ E_{p_k p_l} A \in J$ is the same as its $ p_l$th row, and hence $A _{kj_l} \not= 0$. Therefore, $ J_{kj_l} \not= 0_{m_k \times n_{j_l}}$. This yields $ j_k \leq j_l$. That is, the sequence $(j_1, \ldots, j_i)$ is increasing. Note that if $ j_k = k< i$, then $ j_k < j_{k+1} $, for $ J \subseteq \mathcal{T}_{(M, N)}$. To prove the equality, note first that
$$ J = \left(\begin{array}{ccccc}
{\rm Row}_1 (J)\\
\vdots \\
{\rm Row}_i (J)\\
0_{m_{i+1} \times n} \\
\vdots \\
0_{m_r \times n}
\end{array}\right), $$
for $ {\rm Row}_k(J) = 0_{m_k \times n}$ for each $ k > i$. But $ {\rm Row}_k (J)= \left(\begin{array}{ccc} 0_{m_k \times x_k} & M_{m_k \times y_k} \end{array}\right)= M_{m_1 \times n} (D) R_k$, where $ x_k = \sum_{j=1}^{j_{k}-1} n_j$ and $y_k = \sum_{j=j_k}^{s} n_j$. Note that if $ j_k = 1$, then $ k= 1$, $x_1= 0$, $y_1= n$, and $R_1= I_n$, and hence $ {\rm Row}_1 (J)= M_{m_1 \times n} = M_{m_1 \times n} (D) R_1$. Conversely, with an $ 1 \leq i \leq \min (r, s)$, an increasing sequence $(j_1, \ldots, j_i)$ with $k \leq j_k \leq s$ and $ j_k < j_{k+1} $ if $ j_k = k< i$ for all $ 1 \leq k \leq i$, and a sequence $(R_1 , \ldots, R_i)$ as in the statement of the theorem, a simple calculation with block matrices shows that
$$ \left(\begin{array}{ccccc}
M_{m_1 \times n} R_1\\
\vdots \\
M_{m_i \times n} R_i\\
0_{m_{i+1} \times n} \\
\vdots \\
0_{m_r \times n}
\end{array}\right)$$
is in fact a subbimodule of $ \mathcal{T}_{(M, N)}$. Clearly the subbimodule $\{0_{m \times n}\}$ is principal. So let $$ J = \left(\begin{array}{ccccc}
M_{m_1 \times n} R_1\\
\vdots \\
M_{m_i \times n} R_i\\
0_{m_{i+1} \times n} \\
\vdots \\
0_{m_r \times n}
\end{array}\right)$$
be a nonzero subbimodule of $ \mathcal{T}_{(M, N)}$. Then $J$ is generated by any matrix $ A \in J$ for which $ A_{kj_k} \not= 0$ for all $ 1 \leq k \leq i$, e.g., for
$$ A = \left(\begin{array}{ccccc}
A_1\\
\vdots \\
A_i\\
0_{m_{i+1} \times n} \\
\vdots \\
0_{m_r \times n}
\end{array}\right), $$
where $A_k = \left(\begin{array}{ccc} 0_{m_k \times x_k} & E_{m_k 1} \end{array}\right) \in M_{m_1 \times n} R_k$, where $ E_{m_k 1} \in M_{m_k \times y_k}$ for each $ 1 \leq k \leq i$. Clearly, the subbimodule generated by $A$ shares the same $ 1 \leq i \leq \min (r, s)$ and $j_k$'s ($ 1 \leq k \leq i$) as $J$. This implies that $J$ is generated by $A$. So the proof is complete.
\hfill\qed
\bigskip
\noindent {\bf Remarks.} (i) In the special case $M=N$, the theorem characterizes two-sided ideals of the nest algebra $ \mathcal{T}_{M}$. Moreover any two-sided ideal of $ \mathcal{T}_{M}$ is principal.
(ii) In the special case when $ m= r$, $ s =n$, $ m_i = 1$, $n_j = 1$ for all $ 1 \leq i \leq m$ and $ 1 \leq j \leq n$, the theorem characterizes subbimodules of upper triangular rectangular matrices and in the more special case when $m = r = s = n$ and $ m_i = n_j = 1$ for all $ 1 \leq i, j \leq n$, the theorem characterizes all two-sided ideals of the ring of all upper triangular square matrices with entries from $D$.
\bigskip
\noindent {\bf Acknowledgment.} This paper was submitted while the second named author was on sabbatical leave at the University of Waterloo. He would like to thank Department of Pure Mathematics of the University of Waterloo, and in particular Professors Laurent Marcoux and Heydar Radjavi, for their support.
\vspace{2cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,906
|
Home > Catalogue > Chopin Complete Edition
Chopin Complete Edition
Composer Frédéric Chopin
Artist various artists
Format 17 CD
Cat. number 94660
Download & streams
Other buying options
- select country -AustraliaAustriaBelgiumDenmarkFinlandFranceGermanyGreeceItalyNorwaySwedenSwitzerlandUnited KingdomUnited States
Buy in Australia
Fish Fine Music
Level 2, Shop 40-42
www.fishfinemusic.com.au
Thomas Music
31 Bourke Street
Melbourne, Victoria, 3000
www.thomasmusic.net.au
Readings: Carlton
309 Lygon St
Readings: Hawthorn
Readings: St Kilda
112 Acland St
Readings: Malvern
Malvern VIC 3144
Readings: State Library of Victoria
Cnr La Trobe & Swanston Str
Readings: Online
www.readings.com.au
Buy in Austria
www.gramola.at
Buy in Belgium
Coudenberg 74
www.laboiteamusique.eu
Buy in Denmark
Danacord Butik
www.danacordbutik.dk
Naxos Direct DK
www.naxosdirect.dk
Buy in Finland
Naxos Direct Finland
www.naxosdirect.fi
Buy in France
UVM Distribution
www.uvmdistribution.com
Fnac France
www.fnac.com
www.amazon.fr
www.gibertjoseph.com
Qobuz Digital & Streaming
www.qobuz.com
Buy in Germany
www.jpc.de
Zweitauseneins
www.zweitausendeins.de
Buy in Greece
Opera Greece
www.operacd.gr
Buy in Italy
Ducale Music
www.ducalemusic.it
www.ibs.it
www.amazon.it
Discoland Mail
www.discolandmail.com
www.lafeltrinelli.it
Buy in Norway
Torggt. 9A
http://www.eldo.no/
Skei Musikkhandel
Gjesdalveien 25
Naxos Direct Norway
www.naxosdirect.no
Buy in Sweden
Naxos Direct Sweden
www.naxosdirect.se
CDON.COM Sweden
www.cdon.se
Ginza.se
www.ginza.se
Buy in Switzerland
Musikvertrieb AG
www.musikvertrieb.ch
Buy in United Kingdom
www.amazon.co.uk
www.mdt.co.uk
Presto Classical
www.prestoclassical.co.uk
www.europadisc.co.uk
Buy in United States
HBDirect Classical
www.hbdirect.com
www.arkivmusic.com
About this release
The master of the piano must surely be Frédéric Chopin. Every one of his compositions includes the piano in some form, yet within his oeuvre there is still an exciting variety of music to be found. Chopin made the solo piano form into an art, extensively developing various styles – including the piano sonata, waltz, polonaise and impromptu – and heavily popularising others for the first time, such as the Polish mazurka, the form he often chose to express his nationalism. Chopin straddled the divide between the Classical and Romantic worlds, and the changes that took place in his writing is clear to see on an authoritative edition such as this one. His masterful études remain less virtuosic than those of his contemporaries, such as Liszt; the emphasis is still on the technical aspect, and they reveal Chopin's supreme talent for the piano, as a performer as well as a composer. His waltzes were the first to move away from pieces composed for dancing; rather, they were designed to be performed in salons, and therefore gave the potential for a faster tempo and freer sense of rhythm. Never fading from popularity, Chopin has long been regarded as a symbol of Romanticism, appreciated both in his own lifetime and in the years since his death.
This release features a mixture of consummate Chopin performers and new artists. Rem Urasin, the 1992 winner of the Moscow Chopin Competition, plays Chopin's complete mazurkas. He is one of a number of well-known pianists featured on this collection; Earl Wild, who performs the nocturnes, was previously praised for conveying a "convincing dark foreboding" by MusicWeb for his performance of the Nocturne in C sharp minor KK Anh.la No.6. EwaKupiecis joined by the Saarbrücken Radio Symphony Orchestra for Chopin's piano concertos; the Polish pianist feels a special connection with Chopin, and celebrated his 200th birthday with three specially selected Chopin recital programmes. Also shining a light on Chopin's orchestral music is renowned American pianist Abbey Simon, whose 1999 all-Chopin recital in Carnegie Hall was praised by the New York Times for mixing "architectural clarity" with a "free-spirited quality". These experienced Chopin performers are joined by newer artists, such as Alessandro Deljavan, whose prize-winning performance of Chopin's études were praised by the New York Concert Review as 'beautiful'.
Brilliant Classics is proud to offer a new release to add to its growing collection of authoritative composer editions. This box set is excellent value for money. An excellent gift idea for any lover of Romantic piano music: it spans the length of Chopin's illustrious career, and celebrates the best of his innovative and enchanting music.
This 17CD set is the third Chopin Edition by Brilliant Classics, an all-time favourite of one of the most popular and universally loved composers of all times. The piano works of the Polish master touch the heart chords of every listener and music lover around the world, they speak the universal language of beauty, melancholy, tenderness and passion.
This third Edition is also the best: it contains no less than 7 new recordings and several new licensed recordings. New recordings are: Etudes (Alessandro Deljavan, one of the most extraordinary performances ever recorded), Waltzes (Alessandro Deljavan), Piano Sonatas 2 & 3, Preludes, Ballades, Impromptus (Wolfram Schmitt-Leonardy, wonderfully idiomatic, "Fanciful, illuminating, stimulating and compelling" Classics today), Mazurkas (Rem Urasin). The piano concertos are played by Ewa Kupiec, the other works for piano and orchestra by Abbey Simon. The Nocturnes are played by Earl Wild ("silvery, deeply felt"), the Scherzi by Ivan Moravec, one of the best Chopin recordings by the Czech master.
An important and high quality set, not to be missed at bargain price!
Frédéric Chopin: Piano Concerto No. 1 in E Minor, Op. 11: I. Allegro maestoso
Frédéric Chopin: Piano Concerto No. 1 in E Minor, Op. 11: II. Larghetto
Frédéric Chopin: Piano Concerto No. 1 in E Minor, Op. 11: III. Rondo. Vivace
Frédéric Chopin: Piano Concerto No. 2 in F Minor, Op. 21: I. Maestoso
Frédéric Chopin: Piano Concerto No. 2 in F Minor, Op. 21: II. Larghetto
Frédéric Chopin: Piano Concerto No. 2 in F Minor, Op. 21: III. Allegro vivace
Frédéric Chopin: Variations on "Là ci darem la mano" in B-Flat Major, Op. 2
Frédéric Chopin: Grande fantaisie sur des airs polonais in A Major, Op. 13
Frédéric Chopin: Krakowiak in F Major, Op. 14
Frédéric Chopin: Andante spianato and grande polonaise in E-Flat Major, Op. 22
Frédéric Chopin: Cello sonata in G Minor, Op. 65: I. Allegro moderato
Frédéric Chopin: Cello sonata in G Minor, Op. 65: II. Scherzo. Allegro con brio
Frédéric Chopin: Cello sonata in G Minor, Op. 65: III. Largo
Frédéric Chopin: Cello sonata in G Minor, Op. 65: IV. Finale. Allegro
Frédéric Chopin: Grand duo concertant sur des thèmes de Robert Le Diable in E Major, B. 70
Frédéric Chopin: Introduction and polonaise brillante in C Major, Op. 3
Frédéric Chopin: Piano Trio in G Minor, Op. 8: I. Allegro con fuoco
Frédéric Chopin: Piano Trio in G Minor, Op. 8: II. Scherzo. Con moto, ma non troppo
Frédéric Chopin: Piano Trio in G Minor, Op. 8: III. Adagio sostenuto
Frédéric Chopin: Piano Trio in G Minor, Op. 8: IV. Finale. Allegretto
Frédéric Chopin: Polish Songs Op. 74: I. Zyczenie (The Wish)
Frédéric Chopin: Polish Songs Op. 74: II. Wiosna (Spring)
Frédéric Chopin: Polish Songs Op. 74: III. Smutna rzeka (The Sad Stream)
Frédéric Chopin: Polish Songs Op. 74: IV.Hulanka (Merrymaking)
Frédéric Chopin: Polish Songs Op. 74: V. Gdzie lubi (There where she loves)
Frédéric Chopin: Polish Songs Op. 74: VI. Precz z moich oczu! (Out of my sight!)
Frédéric Chopin: Polish Songs Op. 74: VII. Poseł (The Envoy)
Frédéric Chopin: Polish Songs Op. 74: VIII. S´liczny chłopiec (Handsome Lad)
Frédéric Chopin: Polish Songs Op. 74: IX. Melodya (Elegy)
Frédéric Chopin: Polish Songs Op. 74: X. Wojak (The Warrior)
Frédéric Chopin: Polish Songs Op. 74: XI. Dwojaki koniec (The Double End)
Frédéric Chopin: Polish Songs Op. 74: XII. Moja pieszczotka (My Darling)
Frédéric Chopin: Polish Songs Op. 74: XIII. Nie ma czego trzeba (There is no need)
Frédéric Chopin: Polish Songs Op. 74: XIV. Piers´cien´ (The Ring)
Frédéric Chopin: Polish Songs Op. 74: XV. Narzeczony (The Bridegroom)
Frédéric Chopin: Polish Songs Op. 74: XVI. Piosnka litewska (Lithuanian Song)
Frédéric Chopin: Polish Songs Op. 74: XVII. S´piew grobowy (Hymn from the tomb)
Frédéric Chopin: Polish Songs Op. 74: XVIII. Czary (Charms)
Frédéric Chopin: Polish Songs Op. 74: XIX. Dumka (Reverie)
Frédéric Chopin: Études, Op. 10: I. Etude in C Major "Waterfall". Allegro
Frédéric Chopin: Études, Op. 10: II. Etude in A Minor. Allegro
Frédéric Chopin: Études, Op. 10: III.Etude in E Major "Tristesse. Lento ma non troppo
Frédéric Chopin: Études, Op. 10: IV. Etude in C-Sharp Minor. Presto
Frédéric Chopin: Études, Op. 10: V. Etude in G-Flat Major "Black keys". Vivace
Frédéric Chopin: Études, Op. 10: VI. Etude in E-Flat Minor. Andante
Frédéric Chopin: Études, Op. 10: VII. Etude in C Major. Vivace
Frédéric Chopin: Études, Op. 10: VIII. Etude in F Major. Allegro
Frédéric Chopin: Études, Op. 10: IX. Etude in F Minor. Allegro molto agitato
Frédéric Chopin: Études, Op. 10: X. Etude in A-Flat Major. Vivace assai
Frédéric Chopin: Études, Op. 10: XI. Etude in E-Flat Major. Allegretto
Frédéric Chopin: Études, Op. 10: XII. Etude in C Minor "Revolutionary". Allegro con fuoco
Frédéric Chopin: Études, Op. 25: I. Etude in A-Flat Major "Aeolian Harp". Allegro sostenuto
Frédéric Chopin: Études, Op. 25: II. Etude in F Minor. Presto
Frédéric Chopin: Études, Op. 25: III. Etude in F Major. Allegro
Frédéric Chopin: Études, Op. 25: IV. Etude in A Minor. Agitato
Frédéric Chopin: Études, Op. 25: V. Etude in E Minor. Vivace
Frédéric Chopin: Études, Op. 25: VI. Etude in G-Sharp Minor. Allegro
Frédéric Chopin: Études, Op. 25: VII. Etude in C-Sharp Minor. Lento
Frédéric Chopin: Études, Op. 25: VIII. Etude in D-Flat Major. Vivace 1'13
Frédéric Chopin: Études, Op. 25: IX. Etude in G-Flat Major "Butterfly". Allegro assai
Frédéric Chopin: Études, Op. 25: X. Etude in B Minor. Allegro con fuoco
Frédéric Chopin: Études, Op. 25: XI. Etude in A Minor "Winter Wind". Lento
Frédéric Chopin: Études, Op. 25: XII. Etude in C Minor. Allegro molto con fuoco
Frédéric Chopin: 3 Nouvelles études, B. 130: I. Andantino in F Minor
Frédéric Chopin: 3 Nouvelles études, B. 130: II. Allegretto in D-Flat Major
Frédéric Chopin: 3 Nouvelles études, B. 130: III. Allegretto in A-Flat Major
Frédéric Chopin: Ballade No. 1 in G Minor, Op. 23
Frédéric Chopin: Ballade No. 2 in F Major, Op. 38
Frédéric Chopin: Ballade No. 3 in A-Flat, Op. 47
Frédéric Chopin: Ballade No. 4 in F Minor, Op. 52
Frédéric Chopin: Impromptu No. 1 in A-Flat Major, Op. 29
Frédéric Chopin: Impromptu No. 2 in F-Sharp, Op. 36
Frédéric Chopin: Impromptu No. 3 in G-Flat Major, Op. 51
Frédéric Chopin: Fantasy-Impromptu in C-Sharp Minor, Op. 66
Frédéric Chopin: Andante spianato and grande polonaise in G Major, Op. 22
Frédéric Chopin: Polonaise in C-Sharp Minor, Op. 26
Frédéric Chopin: Polonaise in E-Flat Minor, Op. 26
Frédéric Chopin: Polonaise in A Major, Op. 40: I. "Military"
Frédéric Chopin: Polonaise in C Minor, Op. 40
Frédéric Chopin: Polonaise in F-Sharp Minor, Op. 44
Frédéric Chopin: Polonaise in A-Flat Major, Op. 53 "Heroic"
Frédéric Chopin: Polonaise-fantaisie in A-Flat Major, Op. 61
Frédéric Chopin: 3 Polonaises, Op. 71: Polonaise in D Minor
Frédéric Chopin: 3 Polonaises, Op. 71: Polonaise in B-Flat Major
Frédéric Chopin: 3 Polonaises, Op. 71: Polonaise in F Minor
Frédéric Chopin: Polonaise in G Minor, Op. S 1, B. 1
Frédéric Chopin: Polonaise in B-Flat Minor, B. 13, KKIVa
Frédéric Chopin: Polonaise in A-Flat Major, B. 5, KKIVa
Frédéric Chopin: Polonaise in G-Sharp Minor, B. 6, KKIVa
Frédéric Chopin: Polonaise in G-Flat Major, KKIVa
Frédéric Chopin: Bourrée No. 1, Op. D2 in G Major
Frédéric Chopin: Bourrée No. 2, Op. D2 in A Major
Frédéric Chopin: Moderato in E Major, B. 151: No. 12 "Albumblatt"
Frédéric Chopin: Largo in E-Flat Major , B. 109
Frédéric Chopin: Sostenuto in E-Flat Major
Frédéric Chopin: Hexaméron: Variation No. 6
Frédéric Chopin: Fugue in A Minor, B. 144
Frédéric Chopin: Canon in F Minor, No. 1
Frédéric Chopin: Piano Sonata No. 1 in C Minor, Op. 4: I. Allegro maestoso
Frédéric Chopin: Piano Sonata No. 1 in C Minor, Op. 4: II. Menuetto
Frédéric Chopin: Piano Sonata No. 1 in C Minor, Op. 4: III. Larghetto
Frédéric Chopin: Piano sonata No. 1 in C Minor, Op. 4: IV. Finale. Presto
Frédéric Chopin: Piano Sonata No. 2 in B-Flat Minor, Op. 35: I. Grave – Doppio movimento
Frédéric Chopin: Piano Sonata No. 2 in B-Flat Minor, Op. 35: II. Scherzo
Frédéric Chopin: Piano Sonata No. 2 in B-Flat Minor, Op. 35: III. Marche funèbre. Lento
Frédéric Chopin: Piano Sonata No. 2 in B-Flat Minor, Op. 35: IV. Finale. Presto
Frédéric Chopin: Piano Sonata No. 3 in B Minor, Op. 58: I. Allegro maestoso
Frédéric Chopin: Piano Sonata No. 3 in B Minor, Op. 58: II. Scherzo. Molto vivace
Frédéric Chopin: Piano Sonata No. 3 in B Minor, Op. 58: III. Largo
Frédéric Chopin: Piano Sonata No. 3 in B Minor, Op. 58: IV. Finale. Presto non tanto
Disk 10
Frédéric Chopin: Scherzo No. 1 in B Minor, Op. 20
Frédéric Chopin: Scherzo No. 2 in B-Flat Minor, Op. 31
Frédéric Chopin: Scherzo No. 3 in C-Sharp Minor, Op. 39
Frédéric Chopin: Scherzo No. 4 in E Major, Op. 54
Frédéric Chopin: Fantaisie in F Minor, Op. 49
Frédéric Chopin: Berceuse in D-Flat Major, Op. 57
Frédéric Chopin: Barcarolle in F-Sharp Major, Op. 60
Frédéric Chopin: Preludes, Op. 28: I. Prelude in C Major. Agitato
Frédéric Chopin: Preludes, Op. 28: I. Prelude in A Minor. Lento
Frédéric Chopin: Preludes, Op. 28: I. Prelude in G Major. Vivace
Frédéric Chopin: Preludes, Op. 28: I. Prelude in E Minor. Largo
Frédéric Chopin: Preludes, Op. 28: I. Prelude in D Major. Allegro molto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in B Minor. Lento assai
Frédéric Chopin: Preludes, Op. 28: I. Prelude in A Major. Andantino
Frédéric Chopin: Preludes, Op. 28: I. Prelude in F-Sharp Minor. Molto agitato
Frédéric Chopin: Preludes, Op. 28: I. Prelude in E Major. Largo
Frédéric Chopin: Preludes, Op. 28: I. Prelude in C-Sharp Minor. Allegro molto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in B Major. Vivace
Frédéric Chopin: Preludes, Op. 28: I. Prelude in G-Sharp Minor. Presto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in F-Sharp. Lento
Frédéric Chopin: Preludes, Op. 28: I. Prelude in E-Flat Minor. Allegro
Frédéric Chopin: Preludes, Op. 28: I. Prelude in D-Flat Major "Raindrop". Sostenuto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in B-Flat Minor. Presto con fuoco
Frédéric Chopin: Preludes, Op. 28: I. Prelude in A-Flat Major. Allegretto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in F Minor. Allegro molto
Frédéric Chopin: Preludes, Op. 28: I. Prelude in E-Flat Major. Vivace
Frédéric Chopin: Preludes, Op. 28: I. Prelude in C Minor. Largo
Frédéric Chopin: Preludes, Op. 28: I. Prelude in B-Flat Major. Cantabile
Frédéric Chopin: Preludes, Op. 28: I. Prelude in G Minor. Molto agitato
Frédéric Chopin: Preludes, Op. 28: I. Prelude in F Major. Moderato
Frédéric Chopin: Preludes, Op. 28: I. Prelude in D Minor. Allegro appassionato
Frédéric Chopin: Prelude in C-Sharp Minor, Op. 45
Frédéric Chopin: 3 Écossaises, Op. 72 No. 3: I. Écossaise in D Major. Vivace
Frédéric Chopin: 3 Écossaises, Op. 72 No. 3: Écossaise in G Major
Frédéric Chopin: 3 Écossaises, Op. 72 No. 3: Écossaise in D-Flat Major
Frédéric Chopin: Waltz in F Minor, Op. Posth. 70 No. 2 (for Elise Gavard, original version)
Frédéric Chopin: Waltz in F Minor, Op. Posth. 70 No. 2(after Fontana)
Frédéric Chopin: Marche funèbre in C Minor, Op. 72 No. 2 (after Fontana
Frédéric Chopin: Marche funèbre in C Minor, Op. 72 No. 2 (Oxford Edition)
Frédéric Chopin: Cantabile in B-Flat Major, B. 84
Frédéric Chopin: Presto con leggerezza in A-Flat Major
Frédéric Chopin: 2 Mazurkas, B. 16: I. Mazurka in G Major
Frédéric Chopin: 2 Mazurkas, B. 16: II. Mazurka in B-Flat Major
Frédéric Chopin: Mazurka in A Minor, Op. 68 No. 2
Frédéric Chopin: Mazurka in C Major, Op. 68 No. 1
Frédéric Chopin: Mazurka in F Major, Op. 68 No. 3
Frédéric Chopin: Mazurkas Op. 6: I. Mazurka in F-Sharp Minor
Frédéric Chopin: Mazurkas Op. 6: II. Sotto voce in C--Sharp Minor
Frédéric Chopin: Mazurkas Op. 6: III. Vivace in E Major
Frédéric Chopin: Mazurkas Op. 6: IV. Presto ma non troppo in E-Flat Minor
Frédéric Chopin: Mazurkas, Op. 7: I. Vivace in B-Flat Major
Frédéric Chopin: Mazurkas, Op. 7: II. Vivo, ma non troppo in A Minor
Frédéric Chopin: Mazurkas, Op. 7: III. Mazurka in F Minor
Frédéric Chopin: Mazurkas, Op. 7: IV. Presto in A-Flat Major
Frédéric Chopin: Mazurkas, Op. 7: V. Vivo in C Major
Frédéric Chopin: Mazurka in B-Flat Major, Op.P2 No. 1 "Wolowska"
Frédéric Chopin: Mazurka in D Major, B. 71
Frédéric Chopin: Mazurkas Op. 17: I. Vivo e risoluto in B-Flat Major
Frédéric Chopin: Mazurkas Op. 17: II. Lento, ma non troppo in E Minor
Frédéric Chopin: Mazurkas Op. 17: III. Legato assai in A-Flat Major
Frédéric Chopin: Mazurkas Op. 17: IV. Lento, ma non troppo in A Minor
Frédéric Chopin: Mazurka in C Major, Op. 68
Frédéric Chopin: Mazurka in A-Flat Major, B. 85
Frédéric Chopin: Mazurkas, Op. 24: I. Lento in G Minor 3'24
Frédéric Chopin: Mazurkas, Op. 24: II. Allegro non troppo in C Major
Frédéric Chopin: Mazurkas, Op. 24: III. Moderato con anima in A-Flat Major
Frédéric Chopin: Mazurkas, Op. 24: IV. Moderato in B-Fflat Minor
Frédéric Chopin: Mazurkas, Op. 67: III. Sllegretto in C Major
Frédéric Chopin: Mazurkas, Op. 67: I. Vivace in G Major
Frédéric Chopin: Mazurkas, Op. 30: I. Allegretto non tanto in C Minor
Frédéric Chopin: Mazurkas, Op. 30: II. Vivace in B Minor
Frédéric Chopin: Mazurkas, Op. 30: III. Allegro non troppo in D-Flat Major
Frédéric Chopin: Mazurkas, Op. 30: IV. Allegretto in C-Sharp Minor
Frédéric Chopin: Mazurka in A Minor, B. 140 (ded. Émile Gaillard)
Frédéric Chopin: Mazurkas, Op. 33: I. Mesto in G-Sharp Minor
Frédéric Chopin: Mazurkas, Op. 33: II. Vivacee in D Major
Frédéric Chopin: Mazurkas, Op. 33: III. Semplice in C Major
Frédéric Chopin: Mazurkas, Op. 33: IV. Mesto in B Minor
Frédéric Chopin: Mazurkas, Op. 41: I. Maestoso in C-Sharp Minor
Frédéric Chopin: Mazurkas, Op. 41: II. Andantino in E Minor
Frédéric Chopin: Mazurkas, Op. 41: III. Animato in B Major
Frédéric Chopin: Mazurkas, Op. 41: IV. Allegretto in A-Flat Major
Frédéric Chopin: Mazurka in A Minor, B. 140: "Notre temps"
Frédéric Chopin: Mazurkas, Op. 50: II. Allegretto in A-Flat Major
Frédéric Chopin: Mazurkas, Op. 50: III. Moderato in C-Sharp Minor
Frédéric Chopin: Mazurkas, Op. 56: I. Allegro non tanto in B Major
Frédéric Chopin: Mazurkas, Op. 56: II. Vivace in C Major
Frédéric Chopin: Mazurkas, Op. 56: III. Moderato in C Minor
Frédéric Chopin: Mazurkas, Op. 59: I. Moderato in A Minor
Frédéric Chopin: Mazurkas, Op. 59: III. Vivace in F-Sharp Minor
Frédéric Chopin: Mazurkas, Op .63: I. Vivace in B Major
Frédéric Chopin: Mazurkas, Op .63: II. Lento in F Minor
Frédéric Chopin: Mazurkas, Op .63: III. Allegretto in C-Sharp Minor
Frédéric Chopin: Mazurkas, Op. 67: IV. Moderato animato in A Minor
Frédéric Chopin: Mazurkas, Op. 67: II. Cantabile in G Minor
Frédéric Chopin: Mazurkas, Op. 68: No. 4 in F Minor
Frédéric Chopin: Nocturnes, Op. 48: I. Lento in C Minor
Frédéric Chopin: Nocturne in C-Sharp Minor, B. 49 No. 6
Frédéric Chopin: Nocturnes, Op. 15: II. Larghetto in F-Sharp Major
Frédéric Chopin: Nocturnes, Op. 32: I. Nocturne in B Major
Frédéric Chopin: Nocturne in E Minor, Op. 72 No. 1
Frédéric Chopin: Nocturnes, Op. 62: II. Nocturne in E Major
Frédéric Chopin: Nocturnes, Op. 9: III. Allegretto in B Major
Frédéric Chopin: Nocturnes, Op. 37: II. Andantino in G Major
Frédéric Chopin: Nocturnes, Op. 32: II. Nocturne in A-Flat Major
Frédéric Chopin: Nocturnes, Op. 27: II. Lento sostenuto in D-Flat Major
Frédéric Chopin: Nocturnes, Op. 9: I. Larghetto in B-Flat Minor
Frédéric Chopin: Nocturnes, Op. 15: III. Lento in G Minor
Frédéric Chopin: Nocturnes, Op. 48: II. Andantino in F-Sharp Minor
Frédéric Chopin: Nocturnes, Op. 9: II. Andante in E-Flat Major
Frédéric Chopin: Nocturnes, Op. 55: I. Andantie in F Minor
Frédéric Chopin: Nocturnes, Op. 55: II. Lento sostenuto in E-Flat Majro
Frédéric Chopin: Nocturnes, Op. 27: I. Larghetto in C-Sharp Minor
Frédéric Chopin: Nocturnes, Op. 15: I. Andante cantabile in F Major
Frédéric Chopin: Nocturnes, Op. 37: I. Andante sostenuto in G Minor
Frédéric Chopin: Nocturne in C Minor, B. 108
Frédéric Chopin: Rondo in C Major, Op. 73
Frédéric Chopin: Variations sur un air national de Moore, B. 12a
Frédéric Chopin: Grande valse brillante in E-Flat Major, Op. 18
Frédéric Chopin: Waltzes, Op. 34: I. Vivace in A-Flat Major
Frédéric Chopin: Waltzes, Op. 34: II. Lento in A Minor
Frédéric Chopin: Waltzes, Op. 34: III. Vivace in F Major
Frédéric Chopin: Waltz in A-Flat Major, Op. 42
Frédéric Chopin: Waltzes, Op. 64: I. Waltz in D-Flat Major "Minute Waltz"
Frédéric Chopin: Waltzes, Op. 64: II. Waltz in C-Sharp Minor
Frédéric Chopin: Waltzes, Op. 64: III. Waltz in A-Flat Major
Frédéric Chopin: Waltzes, Op. 69: I. Waltz in A-Flat Major
Frédéric Chopin: Waltzes, Op. 69: II. Waltz in B Minor
Frédéric Chopin: Waltzes, Op. 70: I. Waltz in G-Flat Major
Frédéric Chopin: Waltzes, Op. 70: II. Waltz in F Minor
Frédéric Chopin: Waltzes, Op. 70: III. Waltz in D-Flat Major
Frédéric Chopin: Waltz in A-Flat Major, B. 21
Frédéric Chopin: Waltz in E Major, B. 44
Frédéric Chopin: Waltz in E Minor, B. 56
Frédéric Chopin: Waltz in A Minor, B. 150
Frédéric Chopin: Waltz in E-Flat Major: No. 10. Sostenuto
Frédéric Chopin: Waltz in E-Flat Major: No. 14
Frédéric Chopin: Variations brillantes in B-Flat Major, Op. 12
Frédéric Chopin: Rondo in C Minor, Op. 1
Frédéric Chopin: Variations sur un air national allemand in E Major, B. 14
Frédéric Chopin: Rondo à la mazur in F Major, Op. 5
Frédéric Chopin: Allegro de concert in A Major, Op. 46
Frédéric Chopin: Bolero in A Minor, Op. 19
Frédéric Chopin: Rondo in E-Flat Major, Op. 16
Frédéric Chopin: Tarantella in A-Flat Major, Op. 43
Frédéric Chopin: Variations in A Major, B. 37
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,125
|
Kollumer Museum Mr. Andreae is een streekmuseum in Kollum in de Nederlandse provincie Friesland.
Beschrijving
Het museum is vernoemd naar notaris en historicus Arnoldus Johannes Andreae. Het was vanaf 1980 gevestigd in het voormalige postkantoor. Sinds 2017 is het ondergebracht in multi functioneel centrum 'Oostenburg', een voormalig werkhuis voor armen uit 1838. De vaste collectie bevat objecten van inwoners uit Kollumerland omstreeks 1900. Onder het museum valt sinds 2009 het Archeologisch steunpunt Kollum dat in 2016 een plaats kreeg in de hal van het gemeentehuis van Kollumerland.
Collectie
Oude ambachten
Dienstbodekamer
Stijlkamer
Fotoatelier
Schoolklas
Kruidenierswinkeltje
Kostuumafdeling
Bibliotheek
Zie ook
Lijst van musea in Friesland
Externe link
Website museum
Museum in Friesland
Cultuur in Noardeast-Fryslân
Rijksmonument in Noardeast-Fryslân
Cultuurhistorisch museum
Streekmuseum in Nederland
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,887
|
Q: Excel copy paste in multiple sheets at once I have Button(Form control) assigned to a macro which returns me to the first sheet.
i want to paste this button to all sheets at once.
Process which i tried and failed
*
*Copied the button > then selected all my sheets > paste
excel gives error
*Used the option to Fill across all worksheet, but nothing happened
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 88
|
Q: Getting "AttributeError: 'NoneType' object has no attribute 'groupdict'" when executing an Excel file in Python using the Formulas library My problem is to save data into an excel file and then make it run its calculations and then read it using python without having to open it. For this I have tried using the formulas library. It worked perfectly fine on a test excel sheet that I tries that had a few calculations. However when I try it on a larger excel sheet with a lot more and complex calculations I get this error message.
AttributeError: 'NoneType' object has no attribute 'groupdict
I will also post the entire error message here:
AttributeError: 'NoneType' object has no attribute 'groupdict'
---------------------------------------------------------------------------
AttributeError Traceback (most recent call last)
/tmp/ipykernel_113/60233469.py in <module>
1 fpath = 'MTOOutput.xlsx'
2 dirname = 'MTOOutput'
----> 3 xl_model = formulas.ExcelModel().loads(fpath).finish(circular=True)
4 xl_model.calculate()
5 xl_model.write(dirpath=dirname)
~/.local/lib/python3.8/site-packages/formulas/excel/__init__.py in loads(self, *file_names)
93 def loads(self, *file_names):
94 for filename in file_names:
---> 95 self.load(filename)
96 return self
97
~/.local/lib/python3.8/site-packages/formulas/excel/__init__.py in load(self, filename)
98 def load(self, filename):
99 book, context = self.add_book(filename)
--> 100 self.pushes(*book.worksheets, context=context)
101 return self
102
~/.local/lib/python3.8/site-packages/formulas/excel/__init__.py in pushes(self, context, *worksheets)
106 def pushes(self, *worksheets, context=None):
107 for ws in worksheets:
--> 108 self.push(ws, context=context)
109 return self
110
~/.local/lib/python3.8/site-packages/formulas/excel/__init__.py in push(self, worksheet, context)
119 for c in row:
120 if hasattr(c, 'value'):
--> 121 self.add_cell(
122 c, context, references=references,
123 formula_references=formula_references,
~/.local/lib/python3.8/site-packages/formulas/excel/__init__.py in add_cell(self, cell, context, references, formula_references, formula_ranges, external_links)
232 val = cell.data_type == 'f' and val[:2] == '==' and val[1:] or val
233 check_formula = cell.data_type != 's'
--> 234 cell = Cell(crd, val, context=ctx, check_formula=check_formula).compile(
235 references=references, context=ctx
236 )
~/.local/lib/python3.8/site-packages/formulas/cell.py in compile(self, references, context)
88 def compile(self, references=None, context=None):
89 if self.builder:
---> 90 func = self.builder.compile(
91 references=references, context=context, **{CELL: self.range}
92 )
~/.local/lib/python3.8/site-packages/formulas/builder.py in compile(self, references, context, **inputs)
127 else:
128 try:
--> 129 i[k] = Ranges().push(k, context=context)
130 except ValueError:
131 i[k] = None
~/.local/lib/python3.8/site-packages/formulas/ranges.py in push(self, ref, value, context)
167
168 def push(self, ref, value=sh.EMPTY, context=None):
--> 169 rng = self.get_range(self.format_range, ref, context)
170 return self.set_value(rng, value)
171
~/.local/lib/python3.8/site-packages/formulas/ranges.py in get_range(format_range, ref, context)
159 def get_range(format_range, ref, context=None):
160 ctx = (context or {}).copy()
--> 161 for k, v in _re_range.match(ref).groupdict().items():
162 if v is not None:
163 if k == 'ref':
AttributeError: 'NoneType' object has no attribute 'groupdict'
I initially had not used the circular=True argument inside finish. But after a bit of trouble shooting I realized that could be the problem. I say that could because I did not create the excel sheet and hence do not know much about what is in there.
I also tried installing the formulas[all] version again after a bit of troubleshooting before getting the same error. I tried including the use of range references in my test file to check if that was causing the issue. However the test file worked fine with range references. I was wondering if anyone who had faced an issue like this with formulas had found a solution or an alternative.
PS: It would not be possible for me to attach the excel file as it is work related.
Thanks and Regards,
Yadhu
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,044
|
El Priorato de Santa Clara (en sueco: Sankta Klara kloster) fue un convento católico de la orden de Santa Clara de Estocolmo en Suecia, que funcionó desde 1289 hasta la reforma protestante de 1527.
El convento recibió grandes donaciones y tierras en su fundación por el rey Magnus III de Suecia. el Rey Magnus también dio a su hija la princesa Richeza Magnusdotter de Suecia al convento, y ella sirvió como abadesa entre 1335 y 1347.
Durante la guerra sueca de Liberación de 1521-23, la Estocolmo danesa fue sitiada por los suecos. La abadesa del convento de monjas, Anna Leuhusen, ofreció permitir a los suecos que vivían en Estocolmo escapar de manera segura de los daneses y salir de la ciudad a través del convento.
Véase también
Monasterio
Abadía
Priorato
Referencias
Prioratos de Suecia
Edificios y estructuras de Estocolmo
Arquitectura de Suecia del siglo XIII
Iglesia católica previa a la reforma luterana en los países nórdicos
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,948
|
\section{Introduction}
Two methods are mainly used at LEP2 to extract the $W$ mass:
the threshold method and the direct reconstruction technique \cite{lep2}.
In the first case the total $W^+ W^-$ cross section is measured near
threshold (161 GeV), where the sensitivity to $M_W$ is stronger, and
plotted as a function of the $W$ mass.
Conversely, the direct reconstruction method is applied at higher energies
and requires two steps:\\
\noindent 1. From the experimental data the invariant mass distribution
$\frac{d\sigma}{dM}$ is reconstructed. To improve the mass
resolution, a constrained fit is usually performed event by
event, assuming no initial-state radiation (ISR) and equality between
the invariant masses coming from different $W$'s.\\
\noindent 2. The experimental distribution is compared with the
theoretical prediction for $\frac{d\sigma}{dM}$
and, after Monte Carlo corrections, a reconstructed $W$ mass
$M_R$ is extracted, with an error $\Delta M_R$.\\
Recently, a new method has been proposed \cite{plb}
(direct fit method), in which only the best
measured quantities are used to extract the $W$ mass. The idea is simple.
Given a set of well measured quantities $\{\Phi\}$ one
computes, event by event, the theoretical probability ${\cal P}_i$ of
getting the observed set of values $\{\Phi_i\}$ for $\{\Phi\}$.
Since this is a function of $M_W$,
given $N$ observed events, the logarithm of the likelihood function
($ L= \prod_i {\cal P}_i$)
is distributed, for large $N$, as a quadratic function of $M_W$.
A parabola can then be fitted, from which the reconstructed $W$ mass
$M_R$ is obtained with an error $\Delta M_R$.
Although one can always consider more sets $\{\Phi\}$,
the following choices seem reasonable, in practice, for different
four-fermion final states \cite{plb,oxf}:
\begin{itemize}
\item[] Semileptonic case: $\ell_3 \nu_4 q_5 q_6 $
\begin{itemize}
\item[1.] $\{\Phi\}= \{E_3, \Omega_3, \Omega_5, \Omega_6,
E_h\}$, where $E_h$ is the total energy of the two jets.
\item[2.] $\{\Phi\}= \{E_3, \Omega_3, \Omega_5, \Omega_6 \}$.
\end{itemize}
\item[] Purely hadronic case: $q_3 q_4 q_5 q_6$
\begin{itemize}
\item[3.] $\{\Phi\}= \{\Omega_3, \Omega_4, \Omega_5, \Omega_6 \}$.
\end{itemize}
\item[] Purely leptonic case: $\ell_3 \nu_4 \nu_5 \ell_6 $
\begin{itemize}
\item[4.] $\{\Phi\}= \{E_3, \Omega_3,
E_6, \Omega_6 \}$.
\end{itemize}
\end{itemize}
In this paper, we explicitly give all the formulae needed to compute
the probabilities referring to the above four cases and present two
{\tt FORTRAN} programs ({\tt WEXTER} and {\tt ERAFITTER}) to extract
$M_R$ from the LEP2 data using the direct fit method.
The two programs have been developed in parallel and continuously cross
checked. For this reason we chose to present them in a common paper.
Also, we decided to put emphasis on the description of the common
algorithms and on a general illustration of the codes,
skipping the most technical details. Further information
is available directly from the authors.
\section{Theory}
Let us start from the case when the analysed events
belong to only one particular class of four-fermion final states
(for example $\mu^- \bar \nu_\mu q_1 \bar q_2$).
The main problem is computing, for each given event $i$, the probability
${\cal P}(\{\Phi_i\},M_W)$ of measuring the observed values $\{\Phi_i\}$
for the set $\{\Phi\}$. This is a function of $M_W$ and reads
\begin{eqnarray}
{\cal P}(\{\Phi_i\},M_W)= \frac{1}{\sigma_{tot}}\,
\frac{d \sigma}{d \{\Phi_i\}}\,,
\end{eqnarray}
where $\frac{d \sigma}{d \{\Phi_i\}}$ and $\sigma_{tot}$
are the differential and the total (namely integrated over the
whole fiducial volume) cross sections for the process
under study, respectively.
Once $\sigma_{tot}$ is known, the computation of
$\frac{d \sigma}{d \{\Phi_i\}}\,$ for the four cases
listed in the introduction is needed. Then, a likelihood function may be
constructed as follows:
\begin{eqnarray}
L= \prod_{i} {\cal P}(\{\Phi_i\},M_W)\,.
\end{eqnarray}
For large numbers of events, $L$ is distributed as a Gaussian. Therefore,
a quadratic fit of the form $Y= a\,X^2+b\,X+c$,
with $Y= \log L$ and $X=M_W$, can be performed.
The maximum of the fitted parabola ($-\frac{b}{2 a}$) gives
$M_R$, while the statistical error given by the set of
analysed events is $\sqrt{-\frac{1}{2 a}}$.
The above procedure can be easily generalized to the case when
the analysed events belong to different processes. In principle,
this allows us to extract $M_R$, using together all LEP2 events
\footnote{Notice that, since the differential
cross sections differ, also CP-conjugate processes such as
$\mu^- \bar \nu_\mu q_1 \bar q_2$
and $\mu^+ \nu_\mu \bar q_1 q_2$ must be considered separately.}.
If the events refer to $m$ different processes, the likelihood
function is given by
\begin{eqnarray}
&& L = \prod_{i,j} {\cal P}^{(j)}(\{\Phi_i\},M_W) \nonumber \\
&& {\cal P}^{(j)}(\{\Phi_i\},M_W) = \frac{1}{\sigma^{(j)}_{tot}}\,
\frac{d \sigma^{(j)}}{d \{\Phi_i\}}\,,
\end{eqnarray}
where $j= 1:m$ labels the $m$ different processes.
Therefore, the most general formula for $\log L$ reads
\begin{eqnarray}
\log L= \sum_{j=1}^{m} \left( \sum_{i=1}^{N_j}
\log \left( \frac{d \sigma^{(j)}}{d \{\Phi_i\}} \right)
-N_j \log \sigma^{(j)}_{tot} \right)\,,
\label{eqp}
\end{eqnarray}
where $N_j$ is the number of analysed events for the $j^{th}$ process.
In the following, we list the algorithms needed to compute
$\frac{d \sigma^{(j)}}{d \{\Phi_i\}}$ for the four cases given
in the introduction. Due to the ISR, the sum of the two
incoming momenta reads
\begin{eqnarray}
P = p_1+p_2= (E,p,0,0)\,,~~\,{\rm with}~~E= \frac{\sqrt{s}}{2} (x_1+x_2)\,,
~p= \frac{\sqrt{s}}{2} (x_1-x_2)\,,
\label{Ep}
\end{eqnarray}
where $x_{1,2}$ are the fractions of energy left to $e^{\pm}$
after QED radiation, and
\begin{eqnarray}
(p_1+p_2)^2 \equiv \hat{s}= x_1 x_2 s\,.
\end{eqnarray}
\paragraph{Purely hadronic case\\}
This is the simplest case because
$\{\Phi\}=\{\Omega_3, \Omega_4, \Omega_5, \Omega_6 \}$
is 8-dimensional. Then, when neglecting ISR, the kinematics
of the event is completely determined.
The inclusion of ISR would instead imply two integrations over $x_1$ and $x_2$.
The algorithm is as follows.
First one generates $x_1$ and $x_2$ between $0$ and $1$, and computes
$E$ and $p$ in eq. (\ref{Ep}).
Then, after parametrizing the final-state momenta as
\begin{eqnarray}
&& p_i= E_i\,(1,c_{\theta_i},s_{\theta_i} c_{\phi_i}
,s_{\theta_i} s_{\phi_i}) \nonumber \\
&& i= 3:6\,,~~~~c_{\theta_i} \equiv \cos{\theta_i}\,,
~~~~s_{\theta_i} \equiv \sin{\theta_i}\,,~~~{\rm etc.}\,,
\end{eqnarray}
the four unknown energies $E_i$
are found by using the energy-momentum conservation constraints:
\begin{eqnarray}
&&\Delta
\cdot
\left( \begin{array}{l}
E_3 \\
E_4 \\
E_5 \\
E_6
\end{array}
\right) =
\left( \begin{array}{c}
E \\
p \\
0 \\
0
\end{array}
\right)\,\nonumber \\
&&\Delta=
\left( \begin{array}{cccc}
1 &1 &1 & 1 \\
c_{\theta_3}&c_{\theta_4}&c_{\theta_5}&c_{\theta_6} \\
s_{\theta_3}c_{\phi_3} &s_{\theta_4}c_{\phi_4} &
s_{\theta_5}c_{\phi_5} &s_{\theta_6}c_{\phi_6} \\
s_{\theta_3}s_{\phi_3} &s_{\theta_4}s_{\phi_4} &
s_{\theta_5}s_{\phi_5} &s_{\theta_6}s_{\phi_6}
\end{array}
\right) \,.
\end{eqnarray}
For some values of $x_1$ and $x_2$, the above system may give
unphysical negative $E_i$. Such configurations must off course be discarded.
The matrix element squared $|M|^2$ can then be computed by
using the reconstructed momenta, so that the kernel cross section,
to be convoluted with the ISR structure functions, is
\begin{eqnarray}
d \hat{\sigma}(x_1,x_2)= \frac{|M|^2}{16} \delta^4(P-p_3-p_4-p_5-p_6)
\prod_{i=3}^{6} E_i\,dE_i\,d\Omega_i \,,
\end{eqnarray}
where $(2 \pi)^8$ and the flux factor have been included in the definition
of $|M|^2$.
Since
\begin{eqnarray}
\int dE_3\,dE_4\,dE_5\,dE_6\, \delta^4(P-p_3-p_4-p_5-p_6)=
\frac{1}{{\rm det} \Delta}\,,
\end{eqnarray}
the final answer reads
\begin{eqnarray}
\frac{d \hat{\sigma}(x_1,x_2)}{d \{\Phi_i\}} \equiv
\frac{d \hat{\sigma}(x_1,x_2)}{d \Omega_3 d \Omega_4 d \Omega_5 d \Omega_6}=
\frac{E_3 E_4 E_5 E_6}{16\,{\rm det}\Delta}\,|M|^2\,.
\end{eqnarray}
\paragraph{Purely leptonic case\\}
The measured quantities are now
$\{\Phi\}= \{E_3, \Omega_3, E_6, \Omega_6 \}$.
An additional integration over $\Omega_4$ is required,
besides that over $x_1$ and $x_2$.
Therefore one first generates
$x_1,\,x_2,\,\Omega_4$ and then, from the on-shell condition
$0= (P-p_3-p_4-p_6)^2$, one gets $E_4$:
\begin{eqnarray}
&& E_4= \frac{1}{2}\,\frac{\hat{s}-2\,E_3\,(E-p\,c_3)
-2\,E_6\,(E-p\,c_6)+2\,E_3\,E_6(1-\,c_{36})}
{E-p\,c_4-E_3\,(1-c_{34})-E_6\,(1-c_{46})} \nonumber \\
&& c_i= c_{\theta_i}\,,~~~~~~~c_{ij}= \cos \angle {(p_i,p_j)}\,.
\end{eqnarray}
The four-vector $p_5$ is given by
$p_5= P-p_3-p_4-p_6$; then, for any value of
$x_1,\,x_2\,{\rm and}\,\,\Omega_4$, the final-state momenta are known
(as in the previous case, unphysical solutions must be explicitly discarded).
The kernel multidifferential cross section is therefore
\begin{eqnarray}
&& \frac{d \hat{\sigma}(x_1,x_2)}{d \{\Phi_i\}} \equiv
\frac{d \hat{\sigma}(x_1,x_2)}{dE_3 d\Omega_3 dE_6 d\Omega_6} \nonumber \\
&& = \int d\Omega_4
\frac{E_3 E_4 E_6}{16\,|E-p\,c_4-E_3\,(1-c_{34})-E_6\,(1-c_{46})|}\,|M|^2\,.
\end{eqnarray}
\paragraph{Semileptonic case without hadronic energy\\}
Now $\{\Phi\}= \{E_3, \Omega_3, \Omega_5, \Omega_6 \}$ and one integration
is needed to compute the kernel differential cross section. We chose to
integrate over $E_5$.
As usual, one first generates $x_1$ and $x_2$. Then, a bound for $E_5$
is found from the condition $(P-p_3-p_5)^2 \ge 0$:
\begin{eqnarray}
0 \le E_5 \le \frac{1}{2}
\frac{\hat{s}-2\,E_3\,(E-p\,c_3)}{E-p\,c_5-E_3\,(1-c_{35})}\,.
\end{eqnarray}
By generating $E_5$ in the above interval, computing $E_6$ from
the on-shell condition $(P-p_3-p_5-p_6)^2 = 0$:
\begin{eqnarray}
&& \!\!\!\!\!E_6= \frac{1}{2}\,\frac{\hat{s}-2\,E_3\,(E-p\,c_3)
-2\,E_5\,(E-p\,c_5)+2\,E_3\,E_5(1-\,c_{35})}
{E-p\,c_6-E_3\,(1-c_{36})-E_5\,(1-c_{56})}
\end{eqnarray}
and discarding the unphysical solutions, the final-state momenta
are reconstructed.
Finally, the kernel cross section reads
\begin{eqnarray}
&& \frac{d \hat{\sigma}(x_1,x_2)}{d \{\Phi_i\}} \equiv
\frac{d \hat{\sigma}(x_1,x_2)}{dE_3 d\Omega_3 d\Omega_5 d\Omega_6} \nonumber \\
&& = \int dE_5
\frac{E_3 E_5 E_6}{16\,|E-p\,c_6-E_3\,(1-c_{36})-E_5\,(1-c_{56})|}\,|M|^2\,.
\end{eqnarray}
\paragraph{Semileptonic case with hadronic energy\\}
For this case, $\{\Phi\}= \{E_3, \Omega_3, \Omega_5, \Omega_6,
E_h\}$, where $E_h= E_5 + E_6$.
The set $\{\Phi\}$ is 8-dimensional; therefore, by giving $x_1$
and $x_2$, the kinematics of the event is completely fixed.
The algorithm is as follows.
First one generates $x_1$ and $x_2$, then one computes $E_5$ and $E_6$
from the system
\begin{eqnarray}
\left\{ \begin{array}{l}
E_h= E_5+E_6 \\
(P-p_3-p_5-p_6)^2=0\,.
\end{array}
\right.
\end{eqnarray}
By imposing $E_{5,6}= \frac{E_h}{2} \pm \delta$, one finds the
following two solutions:
\begin{eqnarray}
&& \delta^{\pm}= \frac{-\beta \pm \sqrt{D}}{2\,\alpha}\,, \nonumber \\
&& D= \beta^2-4\,\alpha\,\gamma~~~~~~~~\alpha= -2\,(1- c_{56})\,, \nonumber \\
&& \beta= 2\,\left[ p\,(c_5-c_6)+E_3\,(c_{36}-c_{35}) \right]\,, \nonumber \\
&& \gamma= \hat{s}-2E_3\,(E-p\,c_3)- E_h(2E-p\,(c_5+c_6)) \nonumber \\
&&~~+E_3 E_h\,(2-c_{35}-c_{36})+\frac{E_h^2}{2}\,(1-c_{56})\,.
\end{eqnarray}
Then $p_4= P-p_3-p_5-p_6$. Not always are both solutions physical.
The conditions to be fulfilled are
\begin{eqnarray}
\left\{ \begin{array}{l}
| \delta | \le \frac{E_h}{2} \\
D \ge 0 \\
E_4 \ge 0\,.
\end{array}
\right.
\end{eqnarray}
Finally, the kernel cross section is
\begin{eqnarray}
&& \frac{d \hat{\sigma}(x_1,x_2)}{d \{\Phi_i\}} \equiv
\frac{d \hat{\sigma}(x_1,x_2)}{dE_3 dE_h d\Omega_3 d\Omega_5 d\Omega_6} \nonumber \\
&& = \int d\delta\,
\frac{|M(\delta)|^2}{8}\,E_3\,\left( \frac{E_h^2}{4}-\delta^2 \right)
\, \delta\left((P-p_3-p_5-p_6)^2 \right) \nonumber \\
&& = F(\delta^+)+F(\delta^-)\,, \nonumber \\
&& F(\delta)= \frac{|M(\delta)|^2}{8}\,E_3\,
\left( \frac{E_h^2}{4}-\delta^2 \right)\,\frac{1}{|2\alpha\delta+\beta|}\,.
\end{eqnarray}
\section{The program {\tt WEXTER}}
In this section, we present the first of the two programs that use
the described probabilities to extract $M_R$ from the LEP2 data.
The program {\tt WEXTER} consists of three parts: the evaluation
of the matrix element, the computation of the relevant
differential cross sections, by integration over the unobserved
variables, and the fit to the likelihood curve, to extract $M_R$.
The knowledge of the total cross sections
$\sigma^{(j)}_{tot}$ in eq. (\ref{eqp})
is required as an input. The needed $\sigma^{(j)}_{tot}$
can be computed once for all, for each value of $M_W$ used in the fit,
using, for example, {\tt EXCALIBUR} \cite{exca}.
After an initialization in {\tt SUBROUTINE SETPRO}, the matrix
element evaluation is performed in {\tt SUBROUTINE MATRIX}
and {\tt SUBROUTINE DIAGA}.
The differential cross sections are evaluated in {\tt SUBROUTINE DIFF},
while all needed integrations and the fit are performed in the
{\tt MAIN} of the program.
\subsection{The {\tt MAIN}}
In the {\tt MAIN}, the input file is read
(see later for a detailed discussion). The input file must include
the name of the data file ({\tt DATANAME}) containing the events to be
fitted. In {\tt DATANAME}, all events must be given in terms of a complete set
of four-momenta readable with the following format:
\begin{verbatim}
open (unit=2,file=DATANAME,status='old')
read (2,60) lp,(p(0,kl),p(1,kl),p(2,kl),p(3,kl),kl= 3,6)
60 format(i4/,(4d19.10))
\end{verbatim}
where {\tt lp} is a flag that defines the process and
the array {\tt p(0:3,1:6)} contains the four-momenta.
The first index in {\tt p(0:3,1:6)} refers to the component
while the second one labels
the particles: $1$ is the incoming $e^+$, $2$ the incoming $e^-$, while
$3$, $4$, $5$ and $6$ are the four outgoing fermions.
By convention, the beam is along the $x$-axis (component 1), with the incoming
$e^{+}$ along the positive values.
\par From the $i^{th}$ event given in the above form, the program reconstructs the
values $\{\Phi_i\}$ for the set $\{\Phi\}$ of best measured variables
(see section 2 and ref. \cite{plb}), namely energies and
angles for charged leptons and solid angles for the quarks.
The set $\{\Phi\}$ is automatically determined according to the value
of {\tt LFLAG} returned by {\tt SUBROUTINE SETPRO} (see next section).
Of course, the above input format is not suitable when analysing
real data: in that case the values of the measured variables
$\{\Phi_i\}$ should be directly
given as an input. This requires a trivial change in the reading
format of the program.
However, we chose to feed the program directly with the four-momenta
in order to facilitate Monte Carlo studies.
Then, the probability ${\cal P}(\{\Phi_i\},M_W)$ of measuring the observed
values $\{\Phi_i\}$ for the set $\{\Phi\}$ is computed,
by a Monte Carlo integration over the initial-state
QED radiation - implemented as in ref. \cite{iqed} -
and, when necessary, over the unobserved quantities.
Finally, in the last part of the {\tt MAIN}, the quadratic fit
$Y= a\,X^2+b\,X+c$ described in the previous section is performed
to extract $M_R$ from the analysed events. Also the correlation matrix
is computed, to estimate the error on the fit due to the Monte Carlo
integration.
\subsection{The subroutines {\tt SETPRO}, {\tt MATRIX} and {\tt DIAGA}}
These three subroutines and the whole strategy for the computation
of the matrix element are taken from {\tt EXCALIBUR} \cite{exca},
to which we refer for further details.
In the first part of {\tt SUBROUTINE SETPRO} the strong and the electroweak
parameters used in the program are set.
They are $M_Z$ ({\tt ZMI}), $\Gamma_Z$ ({\tt WZI}),
$\sin^2 \theta_W$ ({\tt STH2}), $\alpha_{e.m.}$ ({\tt ALPHA}) and
$\Gamma_W$ ({\tt WWI}). Also $\alpha_s$ is an input. In the program,
two different $\alpha_s$'s are used. The first one ({\tt ALS}) controls the
coupling of the additional gluonic diagrams appearing
in four-quark final states \cite{qcd} (setting {\tt ALS= 0} switches
off such diagrams). The second one ({\tt ALSN}) is used in the computation
of the so-called ``naive'' QCD factor \cite{qcdn}.
By default, the widths of the gauge bosons (set in {\tt FUNCTION CM2})
are taken to be fixed and different from zero also in the $t$-channel.
This ensures QED gauge invariance of the results \cite{pask},
but induces a small shift in $M_W$. To compensate for it, the variable
{\tt LSHI} is introduced. When {\tt LSHI} is chosen to be 1 in the
input file, masses and widths are redefined as described in ref. \cite{shift}.
If {\tt LSHI = 0} such a redefinition is not performed.
Then, the four-fermion processes, chosen in the input file, are read and
the corresponding Feynman diagrams built up and printed in the output
file. In {\tt SUBROUTINE SETPRO}, the chosen processes are also classified
according to four categories.
The variable that controls the classification is {\tt LFLAG}:
{\tt LFLAG= I (I= 1:4)} corresponds to the {\tt I}-th choice for
$\{\Phi\}$ described in the introduction.
Finally, the right permutation of the four particles in the final state
is assigned to the variables {\tt N3, N4, N5} and {\tt N6}, for later
use in {\tt SUBROUTINE DIFF}. This is relevant for leptonic and
semileptonic final states only. In fact, all four particles
in fully hadronic final states are equivalent.
In {\tt SUBROUTINE MATRIX} and {\tt DIAGA} the matrix element
squared is computed, using the helicity techniques
described in ref. \cite{exca}.
\subsection{The subroutine {\tt DIFF}}
In {\tt SUBROUTINE DIFF} the differential cross section is computed
according to the value of {\tt LFLAG} given by the event.
The algorithms for the computation have already been described in
section 2. {\tt SQJAC} is the value returned by the subroutine. It
is the product of the matrix element times the relevant Jacobian.
{\tt SUBROUTINE DIFF} is called from the {\tt MAIN} of the program,
where the numerical integration over ISR and not measured variables
is performed.
\subsection{The input}
The meaning of the input parameters to be specified in
order to run {\tt WEXTER} is the following:
\begin{itemize}
\item {\tt OUTPUTNAME (CHARACTER*15)}\\
The name of the output file.
\item {\tt DATANAME (CHARACTER*15)}\\
The name of the file containing the events.
\item {\tt NLP (INTEGER)}\\
The number of different processes contained in the file {\tt DATANAME}.
\item {\tt PAR(3,I) (CHARACTER*2)}\\
Produced fermion with label 3 for process {\tt I}
( to be chosen among
'{\tt EL}', '{\tt NE}', '{\tt MU}', '{\tt NM}', '{\tt TA}',
'{\tt NT}', '{\tt DQ}', '{\tt UQ}', '{\tt SQ}', '{\tt CQ}',
'{\tt BQ}', '{\tt TQ}).
\item {\tt PAR(4,I) (CHARACTER*2)}\\
Produced antifermion with label 4 for process {\tt I}.
\item {\tt PAR(5,I) (CHARACTER*2)}\\
Produced fermion with label 5 for process {\tt I}.
\item {\tt PAR(6,I) (CHARACTER*2)}\\
Produced antifermion with label 6 for process {\tt I}.
\end{itemize}
The block of the previous four entries, specifying the
processes, should be repeated {\tt NLP} times ({\tt I= 1:NLP}).
\begin{itemize}
\item {\tt NEV (INTEGER)}\\
The number of events in the file {\tt DATANAME} to be analysed for the fit.
\item {\tt NPF (INTEGER)}\\
The number of different values of $M_W$ used to
fit the likelihood curve.
\item {\tt N (INTEGER)}\\
The total number of integration points.
\item {\tt KREL (INTEGER)}\\
It selects the Feynman diagrams. If {\tt KREL = 0} all possible Feynman diagrams
contributing to the chosen processes are taken into account.
If {\tt KREL= 1}, only the {\tt CC03} diagrams leading to the reaction
$ e^+ e^- \to W^+ W^- $.
\item {\tt LQED (INTEGER)}\\
It includes ({\tt LQED= 1}) or excludes ({\tt LQED = 0}) ISR.
\item {\tt LCOUL (INTEGER)}\\
It includes ({\tt LCOUL= 1}) or excludes ({\tt LCOUL = 0}) the
Coulomb factor described in ref. \cite{coul}.
\item {\tt LQCD (INTEGER)}\\
It includes ({\tt LQCD= 1}) or excludes ({\tt LQCD = 0}) the
``naive'' QCD factor \cite{qcdn}.
\item {\tt LENER (INTEGER)}\\
For semileptonic processes, the sum of the hadronic energies
is used ({\tt LENER= 1}) or not used ({\tt LENER = 0})
as an input. In other words, if {\tt LENER= 1},
a semileptonic process is classified with {\tt LFLAG= 1}.
Otherwise with {\tt LFLAG= 2}.
\item {\tt LFOLD (INTEGER)}\\
For semileptonic (hadronic) processes a 2-folding (24-folding)
over all possible jet assignments is performed
({\tt LFOLD= 1}) or not performed ({\tt LFOLD= 0}).
\item {\tt LSHI (INTEGER)}\\
A shift of masses and widths as described in ref. \cite{shift}
is performed ({\tt LSHI= 1}) or not performed ({\tt LSHI= 0}).
\item {\tt ROOTS (REAL*8)}\\
The total energy (in GeV) of the colliding $e^+$ and $e^-$.
\item {\tt X(I) (REAL*8)}\\
{\tt NPF} values of $M_W$ (in GeV) used to fit the likelihood curve
({\tt I= 1:NPF}).
\item {\tt SIG(I,J) (REAL*8)}\\
{\tt NPF} values ({\tt I= 1:NPF}) of the cross section integrated
over the fiducial volume for the process labelled with {\tt J}.
The {\tt I}-th cross section must be computed with
the {\tt I}-th value for $M_W$ ({\tt X(I)}).
These and the following quantities are an input
for {\tt WEXTER} and should be computed
once for all using, for example, {\tt EXCALIBUR} \cite{exca}.
\item {\tt DSIG(I,J) (REAL*8)}\\
The {\tt NPF} values ({\tt I= 1:NPF}) of the error
corresponding to the above quantities.
\end{itemize}
The block of the previous two entries, specifying the
cross sections and their errors, should be repeated
{\tt NLP} times ({\tt J= 1:NLP}).
\subsection{The output}
Since presenting a complete test run output
would require the specification of too long a list of numbers,
such as the events contained
in the file {\tt DATANAME}, we decided not to include it here.
We just describe what a typical output file looks like.
After printing out information about the process and the parameters
given in the input file, the program writes down,
for each value of $M_W$ chosen for the fit, the following quantities:
\begin{eqnarray}
\sum_{j=1}^{m}\sum_{i=1}^{N_j}
\log \left( \frac{d \sigma^{(j)}}{d \{\Phi_i\}} \right)\,,~~~
\sigma_{tot}^{(j)}\,,~~~X= M_W\,,~~~Y= \log L\,.
\end{eqnarray}
Then the final result of the fit is reported in the following form:
\begin{verbatim}
FITTED MW WITH A 3 PARAM FIT (Y= a*X^2+b*X+c) :
wm= 0.803187D+02 +/- 0.503275D-01 +/- 0.750389D-02
Chi^2/d.o.f. = 0.129242D+01
a = -0.197406D+03
b = 0.317107D+05
c = -0.128700D+07
\end{verbatim}
where {\tt wm} is the reconstructed $W$ mass ($M_R$) and the first and
second errors are the statistical and the Monte Carlo errors,
respectively. The previous result has been obtained by analysing a
set of 1600 unweighted {\tt CC03} semileptonic events, produced by
{\tt EXCALIBUR}, with an input mass $M_W= 80.35$ GeV at $\sqrt{s}= 190$ GeV.
This set of unweighted events, together with the input
file used to get the above output, are available, upon request,
from the authors.
\section{The program {\tt ERAFITTER}}
The program provides all necessary elements to fit the $W$ mass and
it is based on {\tt ERATO} \cite{erato}. Below we give a brief
description of the program.
\subsection{The Computational Tree}
The program {\tt main.f} evaluates the differential cross section for
the different cases described above. Phase-space generation proceeds
through the appropriate algorithms {\tt ALGO01} to {\tt ALGO04}. Then the
momentum assignment is used to evaluate the matrix element through
the routine {\tt MASTER}, which is extracted from {\tt ERATO}, and it is
specific for each selected channel, i.e. leptonic, semileptonic and hadronic.
The total cross sections, needed for the evaluation of the likelihood function,
are calculated in the programs {\tt evud\_mass.f}, {\tt llll\_c2\_mass.f} and
{\tt qqqq\_mass.f}. Finally {\tt fit2.f} collects all information and
does the actual fit to extract the $W$ mass by using standard
{\tt MINUIT} \cite{minuit} calling sequences.
In {\tt main.f}, the input data are read and all physical
constants needed for the computation are defined. The output is the
value of $\sum_{i=1}^{N} \log(d\sigma/d\{\Phi_i\})$ for the various values
of $M_W$ specified by the user.
In the subroutines
{\tt setc1}-{\tt setc4} the transformation from the
Cartesian representation of the four momenta to the polar one, needed
to define the well-measured variables, is performed.
In the subroutines {\tt ALGO01}-{\tt ALGO04}, the generation of the
full phase space, including, if specified, the ISR,
is performed, according to the discussion in section 2. Each subroutine
corresponds to the four cases defined in the introduction.
Then the matrix element
is computed by the standard call of {\tt ERATO}. All needed routines
are incorporated in the file {\tt comrou.f}.
As far as the common blocks used in this code are concerned,
they are identical to the ones used by {\tt ERATO},
with the addition of {\tt VARI01}-{\tt VARI04}, which give
the values of the well measured variables in each case,
and {\tt masses}, where {\tt ndim} is the total number of $M_W$ values
used in the computation and {\tt dwmas} is an array containing these
values. Finally, {\tt iaprox} defines the approximation scheme
according to which the $M_W$ dependence is taken into account.
If {\tt iaprox= 1} the $M_W$ dependence is computed
by only considering the Breit-Wigner functions,
\[ 1/((s_{12}-M_W^2)^2+M_W^2 \Gamma_W^2)\times
1/((s_{34}-M_W^2)^2+M_W^2 \Gamma_W^2)
\,,\]
which is exact in the {\tt CC03} class of Feynman diagrams, and constitutes
a good approximation for {\tt CC10}, {\tt CC11} and, depending on the cuts,
for {\tt CC20} as well. On the other hand, if {\tt iaprox= 0} the
exact computation is performed.
\subsection{Input description}
The input needed to run the {\tt main.f} code looks as follows:
{\tt
\begin{verbatim}
3 !case 1=1b 2=1a 3=2 4=3
190 !energy
1 1600 !first and last generated event read
0.3 10000 !derr,maxmc
9 79.73 79.93 80.03 80.13 80.23 80.33 80.43 80.53 80.73
!nim,dwmas
1 !isr
0 0 1 !ipro icoulomb iaprox
'/users/papadopo/Fortr/tmp/newfit/gen/run/evud/unw190100.data'
'lili100f'
'mass100f'
128.07 0.2310309 91.1888 2.4974 80.23 2.033 !input parameters
24 !# assignments (foldings) default =1 -> no folding
\end{verbatim}
}
\noindent The {\tt FORTRAN} variables corresponding to the above
input file are:
\begin{itemize}
\item {\tt ial}\\ The flag defining the choice of the set $\{\Phi\}$,
according to the order given in the introduction.
\item {\tt e0} \\ The collision energy.
\item {\tt nev1, nev2}\\ The first and last event read for analysis.
\item {\tt derr, maxmc}\\ The relative error required in the computation
of $d\sigma/d\Omega$ and the maximum number of MC iterations.
\item {\tt nim,dwmas(20)}\\ The number of and the actual values of $M_W$ used
in the calculation.
\item {\tt isr, ipro, icoulomb, iaprox} \\The flags for the ISR, the
actual process (i.e. $e^+e^-\to q_1\, \bar q_2\, q_3\, \bar q_4 $),
Coulomb correction and the approximation scheme as defined above.
\item {\tt pupu}\\ The input file where the events are stored.
\item {\tt pupu}\\ The output file containing the log-likelihood values
for each $M_W$.
\item {\tt pupu}\\ The output file containing the invariant masses of the
$W$ decay products.
\item {\tt ALPHA1, SINW2, ZMAS, ZGAMA, WMAS, WGAMA}\\ The physical constants
needed for the computation.
\item {\tt nfol}\\ The parameter controlling the number of foldings to be done,
{\tt nfol=2} for semileptonic and {\tt nfol=24} for hadronic. By folding we
mean all possible different jet assignments.
\end{itemize}
\noindent For {\tt fit2.f} the first input file is:
\begin{verbatim}
inf_min_1:
'../total/qqqq100' !input file (tot_cs or lili)
'tot2' !output file (tot_cs_coef)
80.23 !mass_0
0 !dnorm
0 !events
1,5 !ndat1-ndat2
0,1 !option 0(2) ,ifirst 1(2)
\end{verbatim}
\noindent with the following correspondence to the {\tt FORTRAN} variables:
\begin{itemize}
\item{\tt FILNAM}\\
The input file including the total cross section for different values
of the $W$ mass.
\item{\tt FILNAM}\\
The output file containing the coefficients of the fit of the total
cross section:
\[
\sigma=\mbox{par(1)}+\mbox{par(2)}\, M_W+\mbox{par(3)}\, M_W^2\,.
\]
\item{\tt rmax}\\
The assumed central value for the measured $M_W$, usually the input value
in a Monte Carlo simulation.
\item{\tt dnorm}\\
A normalization parameter (default = 0).
\item{\tt nev}\\
Not used.
\item{\tt ndat1, ndat2}\\
The first and the last rows to be read by the code from the
input file.
\item{\tt ioption, ifirst}\\
{\tt ioption=0} and {\tt ifirst=1} in order to perform the
total cross section fit.
\end{itemize}
\noindent The second input file reads
\begin{verbatim}
inf_min_2:
'lili100f' !input file (tot_cs or lili)
'tot2' !output file (tot_cs_coef)
80.23 !mass_0
0 !dnorm
1600 !events
145 153 !ndat1-ndat2
2 2 !option 0(2) ,ifirst 1(2)
\end{verbatim}
\noindent with
\begin{itemize}
\item{\tt FILNAM}\\
The input file including the log-likelihood values for each event and
for the different values of the $W$ mass required.
\item{\tt FILNAM}\\
The output file containing the coefficients of the fit and the fitted $M_W$.
\item{\tt rmax}\\
The assumed central value for the measured $M_W$.
\item{\tt dnorm}\\
A normalization parameter (default = 0).
\item{\tt nev}\\
The number of events read by the {\tt main.f} code.
\item{\tt ndat1, ndat2}\\
The first and the last rows to be read by the code from the
input file.
\item{\tt ioption, ifirst}\\
{\tt ioption=2} and {\tt ifirst=2} in order to perform the
fit to the logarithm of the likelihood function.
\end{itemize}
\noindent The first run performs the fit to the total cross section,
whereas the second one uses the previous results to extract $M_R$,
including the information from the differential
cross-section.
\subsection{The output}
Using the input files described above, we performed the fit to
a selection of 1600 {\tt CC03} unweighted events produced by {\tt ERATO}
with ISR and $M_W=80.23$~GeV. Then a 24-folding was performed
over all possible jet assignments. Moreover, the total cross section
has been evaluated using the same settings ({\tt CC03}, ISR).
The result of the {\tt MINUIT} program is as follows:
\begin{verbatim}
**********
** 3 **MIGRAD
**********
MIGRAD MINIMIZATION HAS CONVERGED.
FCN=.3168413E-01 FROM MIGRAD STATUS=CONVERGED
13 CALLS 137 TOTAL
EDM= .57E-21 STRATEGY=1 ERROR MATRIX UNCERTAINTY= .0%
EXT PARAMETER STEP FIRST
NO. NAME VALUE ERROR SIZE DERIVATIVE
1 A0 2278.3 2.8190 -.10390E-09 .19360E-04
2 A1 23.469 7.0501 .16109E-09 .25818E-05
3 A2 -503.17 22.433 .11872E-08 .25067E-05
**********
\end{verbatim}
In this output {\tt A0}, {\tt A1} and {\tt A2} are the
coefficients of the quadratic form $Y= {\tt A2}\,X^2+{\tt A1}\,X+{\tt
A0}$, with $Y= \log L$ and $X= (M_R-80.23)$,
from which one can obtain $M_R$ in GeV,
\[
M_R-80.23={\tt 2.33E-02}\, \pm\, {\tt 3.15E-02}\, \pm\, {\tt 7.08E-03}\,,
\]
where the first is the statistical and the second
the Monte Carlo error respectively.
\section{Conclusions}
We have introduced the relevant formalism and described two
{\tt FORTRAN} programs to reconstruct the $W$ mass
at LEP2 using the direct fit method introduced in ref. \cite{plb}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,508
|
Teaching – And Defending – American Democracy
by Leo Casey -- August 27, 2018
If recent history demonstrates anything, it is the old truth that American democracy is a work in progress, and that it can suffer reversals as well as advances. The teaching of civics in our schools should convey the complex and fluid character of American government, and the concurrent responsibility of citizens to be actively involved in politics in order to defend and expand the rights and freedoms of American democracy. At a moment of great risk for democracy, both in the United States and abroad, it is especially important for young people to understand that the moral arc of history does not bend on its own, but only by the active intervention of ordinary people. We may still have a republic, even in this moment of dangerous turmoil, but—as Benjamin Franklin famously opined—only if the citizenry can keep it.
Seen in this light, the crash course on how to teach civics offered by the Fordham Institute's Checker Finn is an exemplar of what NOT to do. In an age of the rise of authoritarian and racist populisms of the far right, including that found at the pinnacle of American government, Finn is exercised about the emergence of an embryonic democratic socialist current in American politics. Of particular concern is what he sees as an "appalling" New York Times op-ed by two young editors of the socialist journal Jacobin, which argued that "subversion of democracy was the explicit intent of the framers" of the Constitution, and advocated constitutional reform to make the American system more democratic.
The idea that the 1789 Constitution contained significant anti-democratic elements seems to be anathema to Finn. Armed with an exegesis of Federalist Paper 10 which misses the essence of James Madison's argument, he asserts that the purpose of the Constitution was the promotion and defense of democracy, full stop, and that is how it must be taught in civics courses.
In rebuttal, I offer a quick refresher course in U.S. history: The 1789 Constitution had the enslavement of African-Americans written into it in both the "three-fifths" and "fugitive slave" clauses. It delegated the power to determine who could vote to states, where the franchise was generally restricted to white men of property—or about 6% of the entire U.S. population. It created a federal government in which only one part of one of the three branches (the House of Representatives) was elected directly by this tiny electorate; the President, the Senate and the Supreme Court were all to be chosen by intermediary bodies. In Federalist Paper 68, Alexander Hamilton made an argument for having an Electoral College choose the President and Vice President, which was generally applicable to all of the systems of indirect election in the 1789 Constitution: the Electoral College would be populated by the nation's elite, who would be "most likely to have the information and discernment" to avoid the election of a President "not in an eminent degree endowed with the requisite qualifications." Ordinary Americans, even propertied white men, were not to be trusted with these tasks. Reading these passages today, it is hard to miss the irony that it was precisely this method of indirect election by the Electoral College that gave the United States a President who is a reckless demagogue and manifestly unfit for the office, when the popular vote would have spared us this result. That irony appears lost on Finn.
The reasoning found in Federalist Paper 10, incorrectly rendered in Finn's commentary, is at the heart of a number of the anti-democratic features of the 1789 Constitution. Madison's argument against factions was actually focused on one particular faction—the economic strata of small farmers, mechanics and craftsmen—which he saw as a threat to the wealthy. Looming behind this argument is a late 18th century version of the clash between the 99% and 1%, with Madison taking the side of the 1%. Since majority rule is a fundamental element of democracy, Madison was worried that the working classes, which constituted the majority, might be able to exercise political power on behalf of their economic interests. Organized as a faction, they could impose "an abolition of debts, an equal division of property, or any other improper or wicked project." The solution, Madison argued, was to make it difficult for a majority to pass legislation in this vein by dividing up the power of government: the different branches of the federal government would check and balance each other and the national government could counter state governments. Limits on the franchise and the indirect election of the President, the Senate and the Supreme Court would also keep the working classes at bay, ensuring the hegemony of the small class of very wealthy property owners.
The story of American democracy is, in no small measure, the tale of reforming the anti-democratic features of the 1789 Constitution. It is, for example, the story of the struggle of abolitionists and of a bloody Civil War to end slavery, a struggle that concluded with passage of the 13th, 14th and 15th Amendments to the Constitution. These were the amendments that excised the "three-fifths" and "fugitive slave" clauses from the Constitution and overturned the infamous Supreme Court ruling in Dred Scott v. Sanford, which had used those clauses to declare that African-Americans "had no rights a white man was bound to respect."
But this is an ongoing story, with defeats as well as victories. The right of all American men to vote, which was won in the 15th Amendment, was whittled away in the states through poll taxes, literacy tests and Grandfather clauses. Universal suffrage, expanded and reestablished through the 19th and 24th Amendments, as well as the Voting Rights Act, is now subject to a new series of assaults from voter suppression measures. The birthright citizenship rights of the 13th Amendment that overturned the Dred Scott decision are under attack by a Trumpian argument that is an analogue of Chief Justice Taney's majority opinion. The struggle for American democracy is never complete, never fully attained; this is an essential lesson in civics education that all young people should learn.
In their New York Times op-ed, the two Jacobin editors argue that American politics would be far more democratic if we had a "strong federal government powered by a proportionally elected unicameral legislature." Finn can hardly contain himself at such a suggestion: it is anti-democratic, ahistorical "madness." Yes, Checker, if the United States government had a single house legislature with representatives apportioned on the principle of one person, one vote, it might resemble those "dictatorial regimes" in Denmark, Norway and Sweden and the "tyrannies" of Canada and the United Kingdom.*
Or even worse, it might look that home-grown, red state version of 'authoritarian rule,' the Nebraska state legislature. It might even be akin to the United States Congress, minus the gerrymandering of House districts and a Senate in which a voter in Wyoming has 68 times the weight of a voter in California in terms of representation.
Seriously, one could have a pragmatic argument about whether it would be most efficacious to focus the democratic reform of American government on a one-house legislature with one-person-one vote representation; I would concentrate instead on the role of unlimited, dark money in the political process. But the implication that it is a wild, revolutionary scheme says more about the commentator's misapprehensions about the different forms of democratic governance in the world than about the proposal itself.
Finn's horror at the emergence of democratic socialist politics in the United States betrays a similar ignorance. Across the globe, parties that that describe themselves, somewhat interchangeably, as democratic socialist, social democratic and labor have been integral parts of democratic politics for a century or more, and have often participated in and led democratic governments. In the Nordic countries, for example, there are well-established, long-lasting models of social democracy built by such parties. These countries are characterized by low rates of poverty, unemployment and income inequality, high rates of unionization, strong protections for the rights of workers and consumers and egalitarian wage policies. The Nordic model provides a comprehensive social welfare safety net, with publicly provided free and universal education (pre-K-university), health care, day care, elder care and pensions. It is also typified by higher rates of voter participation and citizen satisfaction than in the United States.
One can reasonably debate the virtues of such a social democratic system versus the laissez-faire market system that has dominated the United States since the Reagan years. But what is indisputable is that the alarmist rhetoric of Finn that socialism brings "repression, famine, political prisoners, and a more or less total collapse of human rights" along with "corruption and tyranny" has no conceivable basis in the historical record of democratic socialism in government. Instead, this evidence-free rhetorical association disingenuously imputes the despotism of regimes such as Stalin's Soviet Union and Mao's China to the democratic socialists who have been among Communism's strongest political opponents. Throughout American history, democratic socialists have been both the strongest champions of social justice and the strongest foes of authoritarianism, from A. Philip Randolph, Bayard Rustin, Martin Luther King, Ella Baker and Dolores Huerta to John Dewey, Norman Thomas, Michael Harrington, and Walter Reuther. That roster includes a long line of presidents of the American Federation of Teachers, including Charlie Cogen, Dave Selden, Sandy Feldman and Al Shanker, after whom this blog is named.
In his younger years, Checker Finn knew many a democratic socialist, and worked closely with a number of them. The historical record of democratic socialist opposition to authoritarianism, both abroad and amongst America's leading democratic socialists, is not exactly news to him. That is why it is beyond odd that Finn, who never tires of insisting on the primacy of teaching civic knowledge over civic skills, would make an argument that so egregiously misrepresents basic information about democratic governance and democratic socialism.
While advancing years can lead to forgetfulness, it's doubtful that age provides an explanation for Finn's political amnesia. Something else is at work here. In his essay, Finn calls out by name three elected officials—Bernie Sanders, Elizabeth Warren and Alexandria Ocasio-Cortez—as if their social democratic calls for an economy that serves the many, and not just the few, constitutes a clear and present danger to American democracy. By contrast, the name of Donald Trump never appears in Finn's essay, and the names of Trump's enablers in the U.S. Congress and on the Supreme Court are also nowhere mentioned. One finds only the mildest of allusions to Trump as "the occasional bad apple (who) might slip into the governmental barrel, even get elected to high office." A reader might be excused for missing this reference altogether, since it is embedded in the argument that one cannot concede the anti-democratic features of the 1789 Constitution, as that document provided "the surest hedge against this happening very often—and to limiting the damage when it did."
But without an educated and politically active citizenry prepared to defend democracy and freedom, a constitution is nothing but script on a piece of parchment. No matter how many times universal suffrage is written into our Constitution and law, it becomes a 'dead letter' when states engage in voter suppression measures, racial gerrymandering and extreme partisan gerrymandering, and when the Supreme Court allows these practices, as it did in its most recent session. In the face of the evisceration of bedrock democratic principles and practices such as universal suffrage, silence is tantamount to consent. And I fear that the current and growing assault on the foundations of democracy—along with the silence and inaction in the face of these attacks by too many conservatives who have made their quiet peace with Trump and Trumpism—constitutes the clear and present danger to American democracy.
To be a credible teacher of democracy, one must be willing to defend it, openly and without apology.
* Denmark, Norway and Sweden have pure unicameral legislatures. Great Britain and Canada have vestigial Houses of Lords, but unlike the American Senate that was modeled after the British House of Lords, these houses no longer have a meaningful role in actual governance: both nations are governed by their House of Commons.
Issues Areas
Leo Casey
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,365
|
Q: error: 'TRUE' was not declared in this scope pad.pvx*= -1; } void move_paddle(PADDLE pad, bool alongX)
{
if(alongX!=TRUE)
{
if((pad.py+pad.length/2)>=B || (pad.py-pad.length/2)<=BB)
pad.pvx*= -1;
}
else if((pad.px+pad.length/2)>=A || (pad.py-pad.length/2)<=AA)
pad.pvx*= -1;
}
What is the actual error ? M unable to get through.
A: There is no TRUE keyword in standard C language. Most probably, this is a macro declaration that you are missing. Where to get it depends on what compiler and libraries you are using. If you cannot find its definition, putting this code before the usage of TRUE (in the beginning of the file, but after all includes) will fix the problem:
#ifndef FALSE
#define FALSE (0)
#endif
#ifndef TRUE
#define TRUE (!FALSE)
#endif
A: Since you are using bool data type, looks like you are using stdbool.h in C99. If that is the case then you should change TRUE to true which expands to 1.
A: I'm guessing that if the compiler says your error is that TRUE is not declared in this scope, then that's the actual error. Maybe you should define TRUE. Also, as I pointed out in my comment (before I had looked at the title) you're probably better off not even using TRUE in this snippet anyway. Comparing vs. TRUE in C is likely to lead to subtle errors that are hard to debug. if (!alongX) means what you wanted to say, and I find it clearer than if (alongX == FALSE).
A: You should use something like an int instead of a bool in c, and as long as it's value ain't 0 it's true, I would change this to
void move_paddle(PADDLE pad, int alongX)
{
if(!alongX)
{
if((pad.py+pad.length/2)>=B || (pad.py-pad.length/2)<=BB)
pad.pvx*= -1;
}
else if((pad.px+pad.length/2)>=A || (pad.py-pad.length/2)<=AA)
pad.pvx*= -1;
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,730
|
Klaus Seibel (* 1959) ist ein deutscher Schriftsteller mit den Schwerpunkten Science-Fiction und Thriller.
Leben
Seibel studierte Theologie und arbeitete als Pastor. Später war er Manager in einem Softwarehaus. Seit 2014 ist er hauptberuflich Schriftsteller. 2009 erhielt er den Krimipreis der Frankfurter Neuen Presse. Nach zwei Buchveröffentlichungen bei Verlagen konzentriert er sich auf Veröffentlichungen als Selbstpublikation. Die meisten Bücher sind als Book-on-demand auch in gedruckter Form erhältlich.
Werke
Romane
Krieg um den Mond., Kriftel 2012, ISBN 978-3-7431-0164-7
Englische Ausgabe: The Screw on the Moon. SeibelDigital 2013. Übersetzung ins Englische durch Alan. N. Shapiro, ISBN 978-1-4825-6667-3
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2018.
Schwarze Energie., Kriftel 2012, ISBN 978-3-7431-0134-0
Zehntausend Augen. Kriminalroman. Emons, Köln 2012, ISBN 978-3-7431-3877-3
Zehntausend Fallen. Der zweite Fall für Ellen Faber. Kriminalroman. Emons, Köln 2013, ISBN 978-3-7431-3923-7
Das Erbe der ersten Menschheit. (Die erste Menschheit Band 1), Kriftel 2014, ISBN 978-3-7412-7975-1
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2019.
Die erste Menschheit lebt. (Die erste Menschheit Band 2), Kriftel 2014, ISBN 978-3-7412-8350-5
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2019.
Die dunkle Seite des Erbes. (Die erste Menschheit Band 3), erschienen am 29. März 2015, ISBN 978-3-7412-9153-1
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2019.
Lantis (Sammelband der 3 Bücher zur ersten Menschheit), erschienen am 24. November 2015, ISBN 978-3-7393-2698-6
Strafe – Alte Sünden., Thriller, erschienen am 10. August 2015, ISBN 978-3-7386-2348-2
Strafe – Kein Vergessen., Thriller, erschienen am 4. Januar 2016, ISBN 978-3-7392-1948-6
Meister der Gene, (Die erste Menschheit, Band 4), erschienen Mai 2016, ISBN 978-3-7412-9782-3
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2019.
Spuren der ersten Menschheit (Die erste Menschheit, Band 5), erschienen am 12. Oktober 2016, ISBN 978-3-7412-8851-7
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2020.
Hoffnung Atlantis (Die erste Menschheit, Band 6), erschienen am 23. März 2017, ISBN 978-3-7431-6216-7
Hörbuch (Sprecher Mark Bremer), Rubikon Audioverlag 2020.
Chimären (Science Force 1) (der 3. Fall für Ellen Faber), erschienen Oktober 2017, ISBN 978-3-7393-9710-8
Hörbuch (Sprecher Florian Hoffmann), Rubikon Audioverlag 2022
Aufstand der Mikroben (Science Force 2) (der 4. Fall für Ellen Faber), erschienen März 2018, ISBN 978-3-7460-0600-0
Das Gehirn in der Tiefe (Science Force 3) (der 5. Fall für Ellen Faber), erschienen August 2018, ISBN 978-3-7528-8685-6
Aufbruch aus Atlantis (Aufbruch aus Atlantis 1), erschienen am 1. Januar 2019, ISBN 978-3-7481-4045-0
Flucht der Saurier (Aufbruch aus Atlantis 2), erschienen am 1. April 2019, ISBN 978-3-7494-3377-3
Der Präsident - SF-Thriller (Aufbruch aus Atlantis 3), erschienen am 1. Juli 2019, ISBN 978-3-7460-2967-2
Xeeh (Aufbruch aus Atlantis 4), erschienen im Oktober 2019, ISBN 978-3-7481-3177-9
Für eine bessere Welt (Sammelband der Bücher 2 – 4 der Staffel Aufbruch aus Atlantis), erschienen am 11. November 2019, ISBN 978-3-7460-9957-6
Aufmarsch der Klone (Aufbruch aus Atlantis 5), erschienen am 26. Februar 2020, ISBN 978-3-7504-6875-7
Angriff auf Atlantis (Aufbruch aus Atlantis 6), erschienen am 12. Mai 2020, ISBN 978-3-7519-3291-2
Der große Rat des Lebens (Aufbruch aus Atlantis 7), erschienen am 3. August 2020, ISBN 978-3-7519-2073-5
Endspiel um die Zukunft (Sammelband der Bücher 5 – 7 der Staffel Aufbruch aus Atlantis), erschienen am 22. September 2020, ISBN 978-3-7519-9009-7
Endlich! - Wir sind nicht allein, erschienen am 5. November 2020, ISBN 978-3-7526-4296-4
Hörbuch (Sprecher Florian Hoffmann), Rubikon Audioverlag 2021
Endlich 2 - Die Körperlosen, 2021, ISBN 978-3-7534-0370-0
Hörbuch (Sprecher Florian Hoffmann), Rubikon Audioverlag 2021
Endlich 3 - Agenten der KI, erschienen am 12. Juli 2021, ISBN 978-3-7543-2338-0
Hörbuch (Sprecher Florian Hoffmann), Rubikon Audioverlag 2021
Das unmögliche Foto, erschienen am 6. Dezember 2021, ISBN 978-3-7557-5485-5
Menschendruckerei, erschienen am 27. Mai 2022, ISBN 978-3-7562-1025-1
Jagd auf die Abtrünnigen, erschienen am 9. November 2022, ISBN 978-3-7568-7442-2
Kurzgeschichten
Die falsche Nadel (2011 veröffentlicht als Hörbuch durch die Navarra-Verlagsgesellschaft)
Casting zum Tod (2011 veröffentlicht als Hörbuch durch die Navarra-Verlagsgesellschaft)
Sein letzter Ton
Weblinks
Homepage des Autors
Oliver Heil: Digitaler Selfmade-Schriftsteller. In: Frankfurter Rundschau Lokalausgabe Main-Taunus vom 8. November 2012
Autor
Literatur (21. Jahrhundert)
Literatur (Deutsch)
Literatur (Deutschland)
Kriminalliteratur
Science-Fiction-Literatur
Roman, Epik
Kurzgeschichte
Deutscher
Geboren 1959
Mann
Selfpublisher
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,427
|
\section{Proof of \eqref{eq:xpx_sum}}
\label{sec:proof-of-xpx-sum}
\begin{proof}
Let $f(t):=\Paren{\frac c t}^t$ be defined on $\R^+$. Remark that $f$ is decreasing for $t\geq
c/e$. Therefore, if $k\in\N, k> c/e$, the sum can be bounded by
\begin{align*}
\sum_{m>k} \Paren{\frac c m}^m \leq \int_k^{+\infty} f(t) dt = I.
\end{align*}
On the other hand, for any $x>1$, we define the function $g(t):=f(t)x^t$ which attains at
$t=cx/e$ its maximum $e^{x(c/e)} = a^x$. Then,
\begin{align*}
I = \int_k^{+\infty} g(t) x^{-t} dt \leq a^x \int_k^{+\infty} x^{-t} dt = a^x \Paren{\frac{x^{-k}}{\ln x}}.
\end{align*}
Since $x>1$ is arbitrary, we deduce
\begin{align*}
I \leq \inf_{x>1} a^x \Paren{\frac{x^{-k}}{\ln x}} \leq \inf_{x\geq ke/c} a^x \Paren{\frac{x^{-k}}{\ln
x}} \leq \Paren{\inf_{x\geq ke/c} a^x x^{-k}} \Paren{\frac{1}{\ln(ke/c)}}.
\end{align*}
The function $a^x x^{-k}$ is convex on $x>1$ and attains at $x=ke/c>1$ its minimum $(c/k)^k$. Putting everything together, we finally obtain
\begin{align*}
I \leq \Paren{\frac c k}^k \Paren{\frac{1}{1+\ln(k/c)}}
\end{align*}
as desired.
\end{proof}
\section{Introduction}
\label{sec:intro}
Echolocation is a form of acoustics that uses active sonar to locate and identify targets. Many animals, such as bats and dolphins, use this method to avoid collisions, to select, identify, and attack preys,
and to navigate by emitting sounds and then analyzing the reflected waves \cite{nature2,nature1}. Animals with the ability of echolocation rely on multiple receivers to allow a better perception of the targets' distance and direction. By noting a difference in sound level and the delay in arrival time of the reflected sound, the animal determines the size and the location of the target. Moreover, the echolocating animal is able to learn how
to identify certain targets and discriminate them from all other
targets. As far as we know, shape identification and classification in echolocation is not well understood yet. It is the purpose of this paper to develop the first shape identification algorithm in echolocation.
We explain how to identify
and classify a target, knowing by advance that the latter belongs
to a certain collection of shapes. The model relies on
differential imaging, {\it i.e.}, by forming an image from the
echoes due to targets at multiple frequencies, and physics-based classification.
Considering a multistatic configuration, we first locate the target
using a specific location search algorithm. Then, we extract frequency-dependent scattering coefficients of the target from the perturbation of the echoes using a least-squares algorithm. Computing the invariants under rigid motions and scaling from the extracted features yields shape descriptors. Finally, a target is classified by comparing its invariants with those of a
set of learned shapes at multiple frequencies.
The main application we have in mind is an improved autonomous navigation using acoustic waves. It is challenging to equip autonomous robots with
echolocation perception and provide them, by mimicking echolocating animals, with acoustic imaging and classification capabilities \cite{nature3}.
The present paper extends our recent results on shape description and perception in electrosensing \cite{ammari_modeling_2012, ammari_target_2012}. For the conductivity equation, the concept of generalized polarization tensors \cite{ammari_boundary_2004, ammari_reconstruction_2004, AKLZ12, yves} has been successfully used for shape identification and classfication \cite{ammari_target_2012, ammari_shape_2013, ammari_generalized_2011}. Invariants with respect to rigid transformations and scaling for the generalized polarization tensors in two and three dimensions have been derived \cite{ammari_target_2012, ammari_invariance_2012}.
The paper is organized as follows. In Section 2 we first recall
the small-volume framework \cite{ammari_reconstruction_2004,capdeboscq_review_2004, CMV98, seo,VV} for imaging acoustic targets. In Section 3 we recover the scattering coefficients from multistatic measurements
using a least-squares method and estimate the resolving power of the reconstruction.
In Section 4
we construct shape descriptors for multiple frequencies based on
scaling, rotation, and translation properties of
the scattering coefficients. In
Section 5, we illustrate numerically on test examples the perfomance of the proposed classification algorithm and its stability with respect to measurement noise and the limited-view aspect. A few concluding remarks are given in the last
section.
\section{Problem formulation}
Let $D \subset \mathbb{R}^2$ be a bounded domain with Lipschitz boundary $\partial D$. We denote by
$\ve_*$ the electric permittivity and $\mu_*$ the magnetic permeability of $D$, and $\ve_0,
\mu_0$ the electric permittivity and the magnetic permeability of the background medium $\R^2
\setminus \bar D$, respectively. Then we introduce
\begin{align}
\ve = \left\{ \begin{array}{ll}
\ve_0&x \in \mathbb{R}^2 \setminus \bar D, \\
\ve_*&x \in D,
\end{array} \right. , \ \
\mu = \left\{ \begin{array}{ll}
\mu_0&x \in \mathbb{R}^2 \setminus \bar D, \\
\mu_*&x \in D,
\end{array} \right. \label{defepsmu}
\end{align}
and set $k = \omega\sqrt{\ve_*\mu_*}$ and $k_0 =
\omega\sqrt{\ve_0\mu_0}$, where $\omega$ is the operating
frequency.
\subsection{Helmholtz equation}
Let $\Gamma_k$ be the fundamental solution to the Helmholtz equation:
\begin{equation}
\label{eq:Helm}
(\Lap+k^2) \Gamma_k(x) = \delta_0
\end{equation}
subject to the Sommerfield outgoing radiation condition. In two
dimensions, $\Gamma_k$ is given by
\begin{equation}
\Gamma_k(x) = -\frac{i}{4}H_0^{(1)}(k\abs{x})
\end{equation}
with $H_0^{(1)}$ being the Hankel function of the first kind of
order 0.
We consider the scattered wave $u$, that is, the solution to
following equation:
\begin{equation}
\label{eq:uUsol}
\begin{cases}
\Grad \cdot \displaystyle{\frac{1}{\mu}} \Grad u + \omega^2 \ve u = 0 \qquad {\rm in}\,\, \R^2,\\
(u-U)\,\,\, {\rm satisfies \,\, the \,\, outgoing\,\, radiation\,\,
condition},
\end{cases}
\end{equation}
where $U$ is a solution to $(\Lap+k_0^2)U = 0$.
\subsubsection{Layer potentials and representation of the solution}
\label{sec:layer-potent-repr}
The solution $u$ to \eqref{eq:uUsol} admits an integral
representation. For this, we introduce the single layer potential
as follows:
\begin{equation}
\label{SDk}
\Sglkf D \phi (x) = \int_{\partial D} {\Gamma_k(x,y)\phi(y)d\sigma(y)}, \qquad x\in
\mathbb{R}^2,
\end{equation}
where $\phi \in L^2(\partial D)$. The following jump formula holds \cite{ammari_reconstruction_2004}:
\begin{equation}
\label{eq:SDk_nderiv}
\displaystyle{\left. \frac{\partial \mathcal{S}_D^{k}[\phi] }{\partial \nu} \right|_\pm} (x) =
\left( \pm\displaystyle{\frac{1}{2}}I + (\mathcal{K}^k_D)^* \right)[\phi](x) \qquad {\rm a.e.}
\quad x \in \partial D,
\end{equation}
where $f|_+$ and $f|_-$ denote respectively the limit on $\partial D$ from the outside and the
inside of $D$ along the normal direction, and $\p/\p \nu$ denotes the normal
derivative. Here, $\Kstar D$ is the Neumann-Poincar\'e operator defined by
\begin{equation}
\label{eq:KDk*}
(\mathcal{K}^k_D)^*[\phi](x) = \int_{\partial D} {\displaystyle{\frac{\partial
\Gamma_k(x,y)}{\partial \nu_x}}\phi(y)d\sigma(y)}
\end{equation}
with $\nu_x$ being the outward normal vectors of $\partial D$ at
$x$. Then the following system admits a unique solution
$(\varphi,\psi) \in L^2(\partial D) \times L^2(\partial D)$
\cite{ammari_boundary_2004}:
\begin{equation}
\label{eq:psiphisys}
\begin{cases}
\Sglkf D \varphi - \Sglknf D \psi = U & \text{ on } \p D, \\
\displaystyle{ \frac{1}{\mu_*} \left.{\ddn{\Sglkf D \varphi}}\right|_{-} - \frac{1}{\mu_0}
\left.{\ddn{\Sglknf D \psi}}\right|_{+} = \frac{1}{\mu_0} \ddn {U}} & \text{ on } \p
D,
\end{cases}
\end{equation}
provided that $k_0^2$ is not a Dirichlet eigenvalue for $-\Delta$ on $D$.
Moreover, the solution $u$ to \eqref{eq:uUsol} can be written as
\begin{equation}
\label{eq:solu}
u(x) =
\begin{cases}
U(x) + \Sglknf D \psi (x), \qquad x \in \mathbb{R}^2 \setminus \bar D, \\
\Sglkf D \varphi (x), \qquad x \in D.
\end{cases}
\end{equation}
\subsubsection{Asymptotic expansion}
\label{sec:asymptotic-expansion} We first recall Graf's formula
\cite{abramowitz_handbook_1964}:
\begin{equation}
\label{eq:graf}
H_0^{(1)}(k|x-y|) = \sum\limits_{n \in \mathbb{Z}}
{H_n^{(1)}(k|x|)e^{in\theta_x}J_n(k|y|)e^{-in\theta_y} } \ \text{ for }
|x|>|y|,
\end{equation}
where $H_n^{(1)}$ is the Hankel function of the first kind of
order $n$ defined by
\begin{align}
\label{eq:Hankel_n_def}
H_n^{(1)}(x) = J_n(x) + i Y_n(x),
\end{align}
and $J_n, Y_n$ are the Bessel function of the first and the second kind of order $n$, respectively.
Then by plugging Graf's formula into \eqref{eq:solu}, we obtain an
asymptotic form of $u-U$ as $|x| \to \infty$:
\begin{equation}
\label{eq:umU}
u(x) - U(x) = -\displaystyle{\frac{i}{4}}\sum\limits_{n \in \mathbb{Z}}
{H_n^{(1)}(k_0|x|)e^{in\theta_x}\int_{\partial D}J_n(k_0|y|)e^{-in\theta_y}\psi(y)d\sigma(y)}.
\end{equation}
In the following, we use $\cwv_m$ to denote the cylindrical wave
of index $m\in\Z$ and of wave number $k_0$, which is defined by
\begin{align}
\label{eq:cylind_wav}
\cwv_m(x)=\cwv_{m,k_0}(x):=J_m(k_0|x|)e^{im\theta_{x}}.
\end{align}
\subsection{Scattering coefficients}
\subsubsection{Definition}
Let $(\varphi_m, \psi_m)$ be the solution to \eqref{eq:psiphisys} with the cylindrical wave
$\cwv_{m}$ as the source term $U$. We introduce the \emph{scattering coefficients} as follows
\cite{ammari_enhancement_2011-2}.
\begin{dfn}
The scattering coefficients $W_{mn}$ ($m,n \in \mathbb{Z}$) associated with the permittivity and
permeability distributions $\ve, \mu$ given by (\ref{defepsmu}) and the frequency $\omega$ are defined by
\begin{equation}
\label{eq:sct}
W_{mn} = W_{mn}[D] = W_{mn}[D, \ve, \mu, \omega] := \int_{\partial D}
{\widebar{\cwv_n(y)} \psi_m(y)d\sigma(y)}.
\end{equation}
\end{dfn}
The coefficient $W_{mn}$ decays very fast as the orders $m,n$
increase. In fact, it has been proved in
\cite{ammari_enhancement_2011-2} that there is a constant $\CW$
depending on $(D, \ve, \mu, \omega)$ such that
\begin{align}
\label{eq:control_SCT}
\abs{W_{mn}[D, \ve,\mu,\omega]} \leq \frac{\CW^{\abs m + \abs n}}{\abs{m}^{\abs m} \abs{n}^{\abs
n}}\,\, \text{ for all } m,n\in\Z.
\end{align}
\subsubsection{Transformation formulas}
\label{sec:SCT_transform_formulae} We introduce the notation for
translation, scaling and rotation of a vector $x$ as $x^z:=x+z,
x^s:=sx, x^\theta:=R_\theta x$, and those of a shape as $$D^z:=D+z,
D^s:=sD, D^\theta:=R_\theta D,$$ and define new functions $f^z,
f^\theta, f^s$ so that
$$f^z(x^z)=f^\theta(x^\theta)=f^s(x^s)=f(x).$$
Explicit relations exist between the scattering coefficients
of $D$ and $D^z, D^s, D^\theta$. We will need the following
technical lemma:
\begin{lem}
\label{lem:psi_Dz_Uz}
We denote by $\psif{U,D}=\psif{U,D,\omega}$ the solution to \eqref{eq:psiphisys} given the domain
$D$, the source term $U$ and the frequency $\omega$. Then for any $z\in\R^2$, $\theta\in\R$ and
$s>0$:
\begin{align}
\label{eq:psi_Dz_Uz}
\psif{U,D}(x) = \psif{U^z, D^z}(x^z) = \psif{U^\theta, D^\theta}(x^\theta) = s \psif{U^s, D^s,
\omega/s}(x^s), \, \forall x\in\p D.
\end{align}
\end{lem}
\begin{proof}
We first prove $\psif{U,D}(x) = \psif{U^z, D^z}(x^z)$. The following equalities are easily
verified:
\begin{align}
\label{eq:Sgl_Kstar_Dz}
\Sglkf{D^z}{\phi^z}(x^z) &= \Sglkf{D}{\phi}(x) \, \text{ for } x \in \R^2 ,\\
\Kstarf{D^z}{\phi^z}(x^z) &= \Kstarf{D}{\phi}(x) \, \text{ for } x\in \p D
\notag,
\end{align}
which imply, using the jump formula \eqref{eq:SDk_nderiv}, that if $(\varphi, \psi)$ is the
solution to \eqref{eq:psiphisys}, $(\varphi^z, \psi^z)$ will be the solution to the following
system:
\begin{equation}
\label{eq:psiphisys_Dz}
\begin{cases}
\Sglkf{D^z}{\varphi} - \Sglknf{D^z}{\psi} = U^z & \text{ on } \p D^z, \\
\displaystyle{ \frac{1}{\mu_*} \left.{\ddn{\Sglkf{D^z}{\varphi}}}\right|_{-} - \frac{1}{\mu_0}
\left.{\ddn{\Sglknf{D^z}{\psi}}}\right|_{+} = \frac{1}{\mu_0} \ddn {U^z}} & \text{ on } \p D^z.
\end{cases}
\end{equation}
Hence we obtain $\psif{U^z, D^z}(x) = \psi^z(x)$ and so forth the first identity in
\eqref{eq:psi_Dz_Uz}. The proof of the identity $\psif{U,D}(x) = \psif{U^\theta,
D^\theta}(x^\theta)$ follows analogous arguments and is skipped for brevity.
We prove now $\psif{U,D}(x) = s \psif{U^s, D^s, \omega/s}(x^s)$. Similarly, the following
equalities can be verified:
\begin{align*}
s^{-1} \Sglf{D^s}{k/s}{\phi^s}(x^s) &= \Sglkf{D}{\phi}(x) \, \text{ for } x \in \R^2, \ \\
\frac{\p \Sglf{D^s}{k/s}{\phi^s}(x^s)}{\p \nu} \bigg|_{\pm} &= \frac{\p \Sglkf{D}{\phi}(x)}{\partial \nu} \bigg|_{\pm} \, \text{ for }
x\in \p D,
\end{align*}
which implies that if $(\varphi, \psi)$ is the solution to \eqref{eq:psiphisys}, then $(\varphi^s,
\psi^s)$ will be the solution to the following system:
\begin{equation}
\label{eq:psiphisys_Ds}
\begin{cases}
\Sglf{D^s}{k/s}{\varphi} - \Sglf{D^s}{k_0/s}{\psi} = sU^s & \text{ on } \p D^s, \\
\displaystyle{ \frac{1}{\mu_*} \left.{\ddn{\Sglf{D^s}{k/s}{\varphi}}}\right|_{-} -
\frac{1}{\mu_0} \left.{\ddn{\Sglf{D^s}{k_0/s}{\psi}}}\right|_{+} = \frac{1}{\mu_0} \ddn
{\Paren{sU^s}}} & \text{ on } \p D^s.
\end{cases}
\end{equation}
Hence, we obtain $\psif{sU^s, D^s, \omega/s}(x) = \psi^s(x)$, and so forth the last identity in
\eqref{eq:psi_Dz_Uz} by noticing that $\psif{sU^s, D^s, \omega/s}(x) = s\psif{U^s, D^s,
\omega/s}(x)$.
\end{proof}
\begin{prop}
\label{prop:SCT-tsr}
For any $z\in\R^2, \theta\in[0, 2\pi), s>0$, the following relations hold:
\begin{subequations}
\label{eq:SCT-tsr}
\begin{align}
W_{mn}[D^z] &= \sum_{a,b\in\Z} {\cwv_{a}(z)} \widebar{\cwv_{b}(z)} W_{m-a,n-b}[D] ,\label{eq:SCT-trl}\\
W_{mn}[D^\theta] &= e^{i(m-n)\theta} W_{mn}[D] ,\label{eq:SCT-rot}\\
W_{mn}[D^s, \ve,\mu,\omega] &= W_{mn}[D, \ve,\mu,s\omega] . \label{eq:SCT-scl}
\end{align}
\end{subequations}
\end{prop}
\begin{proof}
We establish only \eqref{eq:SCT-trl} here, and the identities \eqref{eq:SCT-rot} and
\eqref{eq:SCT-scl} can be proved in a similar way using Lemma \ref{lem:psi_Dz_Uz} and change of
variables.
We write $\psif{\cwv_m, D}(y):=\psi_m(y)$ in order to highlight the dependence on $\cwv_m$ and
$D$, and recall the identity \cite{han_book}:
\begin{align}
\label{eq:cylwav_sum}
\cwv_{m}(x+y) = \sum_{l\in\Z} \cwv_{l+m}(x) \widebar{\cwv_l(-y)}.
\end{align}
Then from Lemma \ref{lem:psi_Dz_Uz} and by the principle of superposition, it follows that
\begin{align}
\label{eq:psi_cwv_sum}
\psif{\cwv_m, D}(y) = \psif{\cwv^{z}_m, D^{z}}(y^{z}) = \sum_{l\in\Z} \widebar{\cwv_l(z)}
\psif{\cwv_{l+m}, D^{z}}(y^{z}).
\end{align}
By the change of variables $y\rightarrow y-z$ in \eqref{eq:sct} and by substituting \eqref{eq:psi_cwv_sum}, we get
\begin{align*}
W_{mn}[D] &= \int_{\p D} \widebar{\cwv_n(y)} \psif{\cwv_m, D}(y) d\sigma(y)
= \sum_{a,b\in\Z} \widebar{\cwv_{a}(z)} {\cwv_{b}(z)} \int_{\p D^z}
\widebar{\cwv_{b+n}(y)} \psif{\cwv_{a+m}, D^z}(y) d\sigma(y)\\
&= \sum_{a,b\in\Z} \widebar{\cwv_{a}(z)} {\cwv_{b}(z)}
W_{a+m,b+n}[D^z],
\end{align*}
which is equivalent to
\begin{align*}
W_{mn}[D^z] = \sum_{a,b\in\Z} \widebar{\cwv_{a}(-z)} {\cwv_{b}(-z)}
W_{a+m,b+n}[D].
\end{align*}
Therefore, we obtain \eqref{eq:SCT-trl} by the fact
$\cwv_a(-z)=\widebar{\cwv_{-a}(z)}$. Finally, we
remark that the sum in \eqref{eq:SCT-tsr} converges absolutely, due to the decay
\eqref{eq:control_SCT} of $W_{mn}$ and the decay \eqref{eq:Bessel_bound_JY} of the Bessel
functions.
\end{proof}
Using \eqref{eq:psi_Dz_Uz} and \eqref{eq:Sgl_Kstar_Dz}, we prove easily the identity:
\begin{align*}
\Sglf{D}{k_0}{\psif{U, D}}(x) = \Sglf{D^{z}}{k_0}{\psif{U^{z}, D^{z}}}(x^{z}) \, \text{ for } x \in
\R^2.
\end{align*}
Consequently, by the integral representation formula
\eqref{eq:solu} this yields the following result.
\begin{cor}
\label{prop:umU_uzmUz}
Let $u_{\sss{U,D}}$ be the solution to~\eqref{eq:psiphisys} given the domain $D$ and the source
term $U$. Then for any $z\in\R^2$:
\begin{align}
\label{eq:uDzmUz_eq_umU}
u_{\sss{U,D}}(x) - U(x) = u_{\sss{U^z,D^z}}(x^z) - U^z(x^z), \ \forall\, x\in \R^2 \setminus \bar
D.
\end{align}
\end{cor}
\section{Reconstruction of scattering coefficients and stability analysis}
\label{sec:reconstr-form-scts}
In this section we investigate the reconstruction of scattering coefficients from the measurements, and
analyze the stability of the reconstruction. We consider a multistatic configuration \cite{han_book}. Formally, let $u_s$ be the solution to \eqref{eq:uUsol}
corresponding to the source term $U_s$ for $s=1,2\ldots N_s$, and $\{x_r \}$ for $r = 1,2,...,N_r$
be the set of receivers. The multistatic response matrix (MSR) $\V:=(V_{sr})_{sr}$ is defined by
\begin{equation}
\label{eq:Vrs}
V_{sr} := u_s(x_r) - U_s(x_r) \quad {\rm for} \quad r = 1\ldots N_r, \text{ and } s = 1\ldots N_s.
\end{equation}
The entry $V_{sr}$ is the measurement recorded by the $r$-th receiver when the $s$-th source $U_s$ is in use.
\subsection{Full aperture acquisition}
\label{sec:full-angle-view}
In the following, we adopt a circular acquisition system using plane waves, which refers to the
situation that the receivers $x_r$ are uniformly distributed on a circle of radius $R$ and centered
at $z_0$ (typically, $z_0$ can be obtained using some localization algorithm and we assume that
$z_0$ is close to the center of $D$), and the sources are plane waves with equally distributed wave
direction; see Figure~\ref{fig:acqsys} (a). More specifically, for the $r$-th receiver we have
$\abs{x_r-z_0}=R$ and the angle $\theta_r:=\theta_{x_r-z_0} = 2\pi r/N_r$, where as the $s$-th plane wave source is given by
\begin{align}
\label{eq:Us}
U_s(x)=e^{ik_0 \xi_s\cdot x}
\end{align}
with the unit vector $\xi_s$ satisfying $\theta_s:=\theta_{\xi_s}=2\pi s/N_s$. Thanks to
\eqref{eq:uDzmUz_eq_umU}, we have
\begin{align}
\label{eq:Vsr_umU_uzmUz}
V_{sr} = u_{\sss{U_s,D}}(x_r) - U_s(x_r) = u_{\sss{U_s^{-z_0},D^{-z_0}}}(x_r-z_0) -
U_s^{-z_0}(x_r-z_0),
\end{align}
which means that the measurement recorded at $x_r$ given the inclusion $D$ and the source $U_s$ is the
same as that recorded at $x_r-z_0$ given the inclusion $D^{-z_0}$ and the source $U_s^{-z_0}$.
\subsection{Linear system of scattering coefficients}
\label{sec:linear-system-sct}
We deduce from \eqref{eq:Vsr_umU_uzmUz} a linear system involving $W_{mn}$.
Recall first the Jacobi-Anger decomposition of plane waves:
\begin{equation}
\label{eq:JA_plw}
U_s^{-z_0} (x) = e^{ik_0 \xi_s \cdot(x+z_0)} = e^{i k_0 \xi_s \cdot z_0}
\sum\limits_{m \in \mathbb{Z}} {e^{im(\frac{\pi}{2} -
\theta_{s})}\cwv_{m}(x)},
\end{equation}
and by the principle of superposition, the solution $\psif{U_s^{-z_0}, D^{-z_0}}$ to
\eqref{eq:psiphisys} given the inclusion $D^{-z_0}$ and the source $U_s^{-z_0}$ reads:
\begin{align}
\label{eq:psipsim}
\psif{U_s^{-z_0}, D^{-z_0}}(x) = e^{i k_0 \xi_s\cdot z_0} \sum_{m \in \Z} {e^{im(\frac{\pi}{2} - \theta_{s})}
\psi_{\cwv_m, D^{-z_0}}(x)}.
\end{align}
On the other hand, we assume $R$ large enough so that $R > \abs y$
for $y\in\p D^{-z_0}$ and Graf's formula \eqref{eq:graf} holds
with $x$ replaced by $x_r-z_0$. Then we apply the asymptotic
expansion \eqref{eq:umU} on the last term of
\eqref{eq:Vsr_umU_uzmUz}, by substituting \eqref{eq:psipsim}, to
finally obtain
\begin{align}
\label{eq:Vrs_ex}
V_{sr} &= -\displaystyle{\frac{i}{4}} \sum\limits_{n \in \mathbb{Z}} H_n^{(1)}(k_0 R)e^{in\theta_r}
\sum_{m \in \Z} e^{ik_0\xi_s\cdot z_0} {e^{im(\frac{\pi}{2} - \theta_{s})} \int_{\partial
D^{-z_0}} \widebar{\cwv_n(y)} \psi_{\cwv_m, D^{-z_0}}(y)} d\sigma(y) \notag \\
&= -\displaystyle{\frac{i}{4}} \sum\limits_{n \in \mathbb{Z}} H_n^{(1)}(k_0 R)e^{in\theta_r}
\sum_{m \in \Z} e^{ik_0\xi_s\cdot z_0} e^{im(\frac{\pi}{2} - \theta_{s})}
W_{mn}[D^{-z_0}].
\end{align}
This motivates us to introduce the constants $A_{sm}$ and $B_{rn}$:
\begin{equation}
\label{eq:AsmBrn}
A_{sm} = e^{ik_0\xi_s\cdot z_0} e^{im(\frac{\pi}{2}-\theta_s)} ; \quad B_{rn} =
\displaystyle{\frac{i}{4}}\widebar{H_n^{(1)}(k_0 R)}e^{-in\theta_r}, \quad m,n \in \mathbb{Z},
\end{equation}
and rewrite \eqref{eq:Vrs_ex} as
\begin{align}
\label{eq:Vrs_ex2}
V_{sr} = \sum_{m,n\in\Z} A_{sm} {W}_{mn}[D^{-z_0}] {(B^H)}_{nr},
\end{align}
where $B^H$ denotes the Hermitian conjugate transpose of $B$.
\subsubsection{Connection with the far-field pattern}\label{sec:far-field-pattern}
The \emph{far-field pattern} (the scattering amplitude) when the incident field is the plane wave
$U(x)=e^{i k_0\xi\cdot x}, \abs\xi=1$ at the frequency $\omega$, is a two-dimensional
$2\pi$-periodic function $\Af_D(\cdot;\omega)$ defined to be \cite{ammari_enhancement_2011-2}
\begin{align}
\label{eq:far_field_pattern_def}
\Af_D({(\theta_\xi, \theta_x)}^\top; \omega) = \int_{\p D} e^{-ik_0 \abs y \cos(\theta_x -
\theta_y)} \psif{U,D}(y) d\sigma(y)
\end{align}
with $\theta_\xi,\theta_x\in[0,2\pi]$ being respectively the angle
of $\xi$ and $x$ in polar coordinates.
In particular, for the circular acquisition system mentioned in
Section \ref{sec:full-angle-view} the following relation holds
between the MSR and the far-field pattern.
\begin{prop}
For a circular acquisition system entered at $z_0$ and the radius of the measurement circle $R\to +\infty$:
\begin{align}
\label{eq:Vsr_far_field}
\abs{V_{sr}} \simeq \frac{1}{\sqrt{8 \pi k_0 R}} \Abs{\Af_{D^{-z_0}}({(\theta_s, \theta_r)}^\top; \omega)}.
\end{align}
Moreover, there exists a constant $C=C(k_0, k, D^{-z_0})$ such that
\begin{align}
\label{eq:Vsr_bound}
\abs{V_{sr}} \leq C\Paren{\frac{\abs{\p D}}{\sqrt R}} \ \text{ when } R \rightarrow +\infty.
\end{align}
\end{prop}
\begin{proof}
Recall the following asymptotic of the Hankel function
\begin{align}
\label{eq:H0_asymp_t_inf}
H_0^{(1)}(t) = \sqrt{\frac 2{\pi t}} e^{i\Paren{t -\frac \pi 4}} + O(1/t) \ \text{ as }
t\rightarrow \infty.
\end{align}
Substituting in the integral representation of \eqref{eq:Vsr_umU_uzmUz}, we find
\begin{align}
\label{eq:Vsr_far_field_intexpr}
V_{sr} &= \Sglf{D^{-z_0}}{k_0}{\psif{U_s^{-z_0}, D^{-z_0}}}(x_r) \notag \\
&\simeq - \frac i 4 e^{-\frac{\pi i} 4}e^{i k_0 R} \sqrt{\frac{2}{\pi k_0 R}} \int_{\p
D^{-z_0}} e^{-ik_0 \abs y \cos(\theta_x-\theta_y)} \psif{U_s^{-z_0}, D^{-z_0}}(y) d\sigma(y),
\end{align}
which gives \eqref{eq:Vsr_far_field} by definition of the far-field pattern after taking the
modulus.
It is proved in \cite{ammari_boundary_2004} that there exists a constant $C=C(k, k_0, D)$
such that the solution $(\phi, \psi)$ of \eqref{eq:psiphisys} satisfies:
\begin{align}
\label{eq:phipsi_energy_control}
\norm{\varphi}_{L^2(\p D)} + \norm{\psi}_{L^2(\p D)} \leq C( \norm{U}_{L^2(\p D)} + \norm{\Grad
U}_{L^2(\p D)}).
\end{align}
For the plane wave source $U_s$, this implies:
\begin{align*}
\norm{\psif{U_s, D}}_{L^2(\p D)} \leq C (1+k_0) \abs{\p D}^{1/2}.
\end{align*}
Substituting back in \eqref{eq:Vsr_far_field_intexpr} and applying Cauchy-Schwartz inequality, we
obtain \eqref{eq:Vsr_bound}.
\end{proof}
\subsubsection{Truncated linear system}
\label{sec:trunc-line-syst}
A truncated version of \eqref{eq:Vrs_ex2} using the first $\abs m, \abs n \leq K$ terms reads
\begin{equation}
\label{eq:V_AWB_Ers}
V_{sr} = \sum_{m=-K}^{K}\sum_{n=-K}^{K} {A_{sm} {W}_{mn}[D^{-z_0}] {(B^H)}_{nr}} +
E_{sr},
\end{equation}
where $E_{sr}$ denotes the truncation error. For the fixed
expansion-order $K$, we introduce the matrix $\W =
({W}_{mn}[D^{-z_0}])_{mn}$ of dimension $(2K+1)\times(2K+1)$, the
source matrix $\A=(A_{sm})_{sm}$ of dimension $N_s\times (2K+1)$
and the receiver matrix $\B=(B_{rn})_{rn}$ of dimension $N_r\times
(2K+1)$, as well as the truncation error matrix $\E=(E_{sr})_{sr}$
of dimension $N_s\times N_r$. Then \eqref{eq:V_AWB_Ers} can be put
into a matrix form
\begin{align}
\label{eq:AWBE_mat}
\V = \A \W \B^H + \E = \bL(\W) + \E,
\end{align}
where the linear operator $\bL: \mathbb{R}^{(2K+1)\times(2K+1)}
\rightarrow \mathbb{R}^{N_s \times
N_r}$ is given by
\begin{equation}
\label{eq:opL}
\bL(\mbf X) = \A \mbf X \B^H.
\end{equation}
In practical situations, the data $\V$ may be contaminated by
some noise $\Wnoise$, thereby modifing \eqref{eq:AWBE_mat} to
\begin{equation}
\label{eq:AWBE_mat_noise}
\V = \bL(\W) + \E + \Wnoise .
\end{equation}
We suppose that $\Wnoise = \snoise \bN_0$, where $\snoise>0$ and $\bN_0$ is a $N_s$-by-$N_r$ complex
random matrix with independent and identically distributed $\normallaw 1$ entries.
From the bound \eqref{eq:Vsr_bound}, one can see the size of the measurement $V_{sr}$ is of order
${\abs{\p D}}/{\sqrt R}$, hence we define signal-to-noise ratio (SNR) to be
\begin{align}
\label{eq:def_SNR}
\SNR = \frac{\abs{\p D}/{\sqrt R}}{\snoise}.
\end{align}
\subsection{Bound of the truncation error $E_{sr}$}
\label{sec:trunc_err_E}
In this section we estimate the truncation error $E_{sr}$ which will be used later to determine the
maximal resolving order $K$ that one can achieve in a noisy environment. The following results will
be useful:
\begin{itemize}
\item For a fixed $t$ when $n$ is large, we have the following asymptotic behaviors of the Bessel functions \cite{abramowitz_handbook_1964}:
\begin{align}
\label{eq:Bessel_bound_JY}
J_{n}(t) \simeq \displaystyle{\sqrt{\frac{1}{2\pi|n|}}\left(\frac{et}{2|n|} \right)^{|n|}}, \ \
Y_{n}(t) \simeq \displaystyle{-\sqrt{\frac{2}{\pi|n|}}\left(\frac{et}{2|n|}
\right)^{-|n|}},
\end{align}
which implies an upper bound of the Hankel function:
\begin{align}
\label{eq:Hankel_bound}
\Abs{H_{n}^{(1)}(k_0R)}
\lesssim {\Paren{C_R\abs n}^{\abs n} + \Paren{C_R\abs n}^{-\abs n}
},
\end{align}
where $C_R:= 2/(ek_0 R)$.
\item For $c>0$ and $k\in\N$ such that $k> c/e$, the following bound holds:
\begin{align}
\label{eq:xpx_sum}
\sum_{m>k}\Paren{\frac{c}{m}}^m \leq \Paren{\frac c k}^{k}
\Paren{\frac{1}{1+\ln(k/c)}}.
\end{align}
The proof is given in Appendix \ref{sec:proof-of-xpx-sum}.
\end{itemize}
\begin{prop}
\label{prop:Esr_bound_K}
Let $C_R$ be the constant defined in \eqref{eq:Hankel_bound}, and $\CW>1$ be the constant in
\eqref{eq:control_SCT} with the shape $D^{-z_0}$. Suppose the radius of measurement circle $R$
satisfies $\CW^2C_R<1$. Then there exists a sufficiently large truncation order $K$ satisfying
$K>\CW/(C_R e)$, such that the truncation error $E_{sr}$ at the order $K$ decays as
\begin{align}
\label{eq:Esr_bound_K}
\abs{E_{sr}} = O(\rho^{-K})
\end{align}
with $\rho=\Paren{\CW^2 C_R}^{-1}>1$ a constant depending only on $D^{-z_0},\ve, \mu, \omega,$ and $R$.
\end{prop}
\begin{proof}
We separate $E_{sr}$ in three terms as follows:
\begin{align*}
{E_{sr}} = \Paren{\sum_{\substack{|m| \le K\\|n| > K}} + \sum_{\substack{|n| \le K\\|m| >
K}}+\sum_{\substack{|m| > K\\|n| > K}}} {A_{sm} W_{mn} (B^H)_{rn}} = I_1 + I_2 +
I_3.
\end{align*}
Let the radius $R$ be sufficiently large so that $\CW^2C_R <1$. Then by \eqref{eq:control_SCT} and
\eqref{eq:Hankel_bound}, we can bound $\abs{I_1}$ by
\begin{align*}
\abs{I_1} &\leq \sum_{|m|\leq K} \frac{\CW^{\abs m}}{\abs{m}^{\abs m}} \sum_{|n| > K}
\Abs{H^{(1)}_n(k_0 R)} \frac{\CW^{\abs n}}{\abs{n}^{\abs n}}\\
&\lesssim \Paren{\sum_{|m|\leq K} \frac{\CW^{\abs m}}{\abs{m}^{\abs m}}} \Paren{\sum_{\abs n > K}
{\Paren{\CW C_R}^{\abs n}} + \sum_{\abs n > K} \Paren{\frac{\CW/C_R} {\abs{n}^2}}^{\abs n}}.
\end{align*}
Up to some factors independent of $K$, we can bound the first sum in the last expression by
$\CW^K$, and the second sum by $(\CW C_R)^K$ (since $\CW C_R<\CW^2 C_R<1$). Furthermore, as a consequence of
\eqref{eq:xpx_sum} the last sum can be neglected by choosing
\begin{align*}
K > \CW / (C_R e)
\end{align*}
sufficiently large. Therefore, we obtain $\abs{I_1} \lesssim (\CW^2C_R)^{K}$. Proceeding in a similar
way one can establish for $I_2$ and $I_3$:
\begin{align*}
\abs{I_2}\lesssim (\CW^2C_R)^{K}, \ \text{ and } \abs{I_3}\lesssim (\CW^2C_R)^{K} K^{-K}.
\end{align*}
Putting these bounds together gives \eqref{eq:Esr_bound_K}.
\end{proof}
\subsection{Stability of the linear operator L}
\label{sec:stab-line-oper}
Here we show that the operator $\bL$ is ill-conditioned for the
circular acquisition system mentioned in section
\ref{sec:linear-system-sct}. If $N_s, N_r \ge 2K+1$, then it can
be verified easily that the matrices $\A,\B$ defined in
\eqref{eq:AsmBrn} are orthogonal:
\begin{align}
\label{eq:As_Ar_ortho}
\A^H \A = N_s \I, \ \text{ and } \B^H \B = N_r \bD,
\end{align}
where $\I$ is the identity matrix of dimension $\Paren{2K+1} \times \Paren{2K+1}$ and the $\Paren{2K+1} \times \Paren{2K+1}$ diagonal matrix
\begin{align}
\label{eq:P_matrix}
\bD =
\begin{pmatrix}
|d_{-K}|^2 & 0 & \ldots & 0\\
0 & |d_{-K+1}|^2 & \ldots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 &\ldots & |d_K|^2
\end{pmatrix}\
\ \text{ with } d_n = \displaystyle{\frac{i}{4}}{H_n^{(1)}(k_0 R)}.
\end{align}
\begin{prop}
\label{prop:SVD_L}
Suppose $N_s, N_r \ge 2K+1$. The right singular vector of $\bL$ is the canonical basis of
$\R^{(2K+1) \times (2K+1)}$, and the $(m,n)$-th singular value of operator $\bL$ is
\begin{equation}
\label{eq:sing_vl}
\lambda_{mn} = \sqrt{N_sN_r}|d_n|, \ \forall\, m,n=-K,\ldots, K
.
\end{equation}
\end{prop}
\begin{proof}
\label{prf:sing_vl}
For $m,n=-K\ldots K$, let $\bv_{mn}$ be the $\Paren{2K+1} \times \Paren{2K+1}$ square matrix
\begin{equation}
(\bv_{mn})_{ij} = \delta_{mi}\delta_{nj}, \ \forall\, i,j=-K,\ldots, K.
\end{equation}
We define the inner product of two complex matrices as $\seq{A, B}:=\sum_{m,n} (A^H)_{mn}
B_{mn}$. A simple computation using \eqref{eq:As_Ar_ortho} shows
\begin{align*}
\seq{\bL(\bv_{mn}),\bL(\bv_{m'n'})} = N_s N_r \seq{\bv_{mn}, \bv_{m'n'}\bD} =
\delta_{mm'}\delta_{nn'} N_s\,N_r\,\abs{d_n}^2.
\end{align*}
This proves that the canonical basis $\set{\bv_{mn}}_{m,n=-K\ldots K}$ constitutes the right
singular vectors of $\bL$. The $(m,n)$-th singular value of $\bL$ is hence given by
\begin{equation}
\lambda_{mn} = ||\bL(\bv_{mn})||_F = \sqrt{N_sN_r}|d_n|, \quad m,n= -K, \ldots, K,
\end{equation}
where $\norm{\cdot}_F$ denotes the Frobenius norm of matrices. The left singular vectors are given by
$\bu_{mn}:= \bL(\bv_{mn})/\lambda_{mn}$.
This completes the proof.
\end{proof}
\subsubsection{Condition number of $\bL$}
\label{sec:condition-number-bL} Remark that as $K\rightarrow
+\infty$, $\abs{d_K}$ diverges, and hence $\bL$ is an unbounded
operator and the following bound holds for the condition number of
$\bL$:
\begin{cor}
\label{cor:cond_L}
Under the same assumptions of Proposition \ref{prop:SVD_L}, we have
\begin{align}
\label{eq:condnum_L_bound}
\mathrm{\cond}\,(\bL) \lesssim (C_R K)^{K} \ \text{ as } K \rightarrow +\infty
\end{align}
with the constant $C_R$ defined in \eqref{eq:Hankel_bound}.
\end{cor}
\begin{proof}
We denote respectively by $\dmax$ and $\dmin$ the maximum and the minimum value of $\abs{d_n}$ for
$n=-K\ldots K$. Using the asymptotic expansion in \eqref{eq:Bessel_bound_JY}, we obtain
\begin{align}
\label{eq:dn_asymptotic}
\abs{d_n} \propto \Abs{H^{(1)}_n(k_0 R)} \simeq \sqrt{\frac {2}{\pi \abs n}} \Paren{C_R \abs
n}^{\abs n} \ \text{ as } \abs n\rightarrow +\infty.
\end{align}
Hence, when $K$ is large, $\dmax\simeq d_K$ and $\dmin$ is bounded from below. Therefore, the condition
number is bounded by
\begin{align*}
\cond(\bL) = \frac\dmax \dmin \lesssim (C_R K)^{K},
\end{align*}
where the underlying constant depends only on $\dmin$.
\end{proof}
\subsection{Least-squares reconstruction}\label{sec:least-square-reconst}
We can reconstruct $\W$ by solving the least-squares problem
\begin{align}
\label{eq:least_square}
\West = \argmin_{\W\in\ker\bL^\bot} \Norm{\bL(\W)-\V}_F,
\end{align}
where $\ker\bL$ denotes the kernel of $\bL$.
The solution of (\ref{eq:least_square}) is given by $\West=\bL^\dagger(\V)$ with $\bL^\dagger$ being the pseudo-inverse of
$\bL$. Since the order $K$ satisfies $2K+1 <N_s, N_r$, the matrices $\A$ and $\B$ have full columns rank and both of them have left pseudo-inverses denoted by $\A^\dagger$ and $\B^\dagger$, respectively. Then we have the pseudo-inverse formulas
\begin{align}
\label{addeq}
\A^\dagger = (\A^H \A)^{-1} \A^H,\quad \B^\dagger = (\B^H \B)^{-1} \B^H.
\end{align}
Using the orthogonal property \eqref{eq:As_Ar_ortho} of $\A$ and $\B$ together with (\ref{addeq}), we verify that
\begin{align}
\label{eq:L_pseudoinv}
\bL^\dagger(\V) = \frac{1}{N_s N_r} \A^H \V \B \D^{-1},
\end{align}
which provides an analytical reconstruction formula for $\W$.
\subsection{Maximal resolving order}
\label{sec:maxim-resolv-order}
Here we give an estimation of the maximal resolving order $K$ that can be reconstructed by
\eqref{eq:least_square} as a function of the SNR.
Let $\rho$ be the constant in Proposition \ref{prop:Esr_bound_K}. We work under the assumption
\begin{equation}
\label{eq:meas_noise}
\rho^{-K} \ll \snoise \ll {\abs{\p D}}/{\sqrt R},
\end{equation}
which is the regime that the measurement noise is much larger than the truncation error but much
smaller than the signal (or $\SNR \gg 1$).
Since $\bL$ is injective, $\bL^\dagger \bL = \I$ and from \eqref{eq:AWBE_mat_noise} we have
\begin{align*}
\Exp{\Abs{\Paren{\West - \W}_{mn}}^2}^{1/2} = \Exp{\abs{{\bL^\dagger(\E+\Wnoise)}_{mn}}^2}^{1/2}
\leq \abs{\bL^\dagger(\E)_{mn}} + \snoise\lambda_{mn}^{-1}\sqrt{N_sN_r}.
\end{align*}
On one hand, by Cauchy-Schwartz inequality and the bound \eqref{eq:Esr_bound_K}:
\begin{align*}
\abs{\bL^\dagger(\E)_{mn}} \leq \lambda_{mn}^{-1} \norm{\E}_F \lesssim \lambda_{mn}^{-1}
\sqrt{N_sN_r} \rho^{-K},
\end{align*}
which can be neglected thanks to the left inequality of
\eqref{eq:meas_noise}. Therefore, we obtain
\begin{align}
\label{eq:recon_error_bound_1}
\Exp{\Abs{\Paren{\West - \W}_{mn}}^2}^{1/2} \lesssim \snoise\lambda_{mn}^{-1}\sqrt{N_sN_r} =
\snoise \abs{d_n}^{-1},
\end{align}
which decreases to zero for all $m,n\geq K$ as $K\rightarrow
+\infty$, due to \eqref{eq:dn_asymptotic}. Apparently, this seems
to imply that the maximal resolving order is infinity. However,
from the decay behavior \eqref{eq:control_SCT} of $W_{mn}$, it is
reasonable to require the reconstruction error to be smaller than
the signal level:
\begin{align*}
\Exp{\Abs{\Paren{\West - \W}_{mn}}^2}^{1/2} \leq \tau_0 \frac{\CW^{\abs m + \abs n}}{\abs{m}^{\abs m}
\abs{n}^{\abs n}}
\end{align*}
with $\tau_0>0$ being a tolerance number. Using
\eqref{eq:recon_error_bound_1} at the order $m=n=K$, this
implies
\begin{align*}
\snoise \frac{1}{\abs{d_K}} \lesssim \tau_0 \Paren{\frac{\CW}{K}}^{2K},
\end{align*}
which yields $\snoise K^{K+1/2} \lesssim \tau_0 \rho^{-K}$ by \eqref{eq:dn_asymptotic}.
Finally, according to the regime \eqref{eq:meas_noise} and by definition of the SNR, we find
\begin{align}
\label{eq:max_resolving_order_bound}
K^{K+1/2} \lesssim \tau_0 \SNR,
\end{align}
and the maximal resolving order is defined as the largest $K$ satisfying
\eqref{eq:max_resolving_order_bound}.
\section{Shape descriptor and identification in a dictionary}
\label{sec:Shape_Descriptor}
In this section we construct the shape descriptors based on the
scattering coefficients which are invariant to rigid transform.
For a collection of standard shapes, we build a
frequency-dependent dictionary of shape descriptors and develop a
shape identification algorithm in order to identify a shape from
the dictionary up to some translation, rotation and scaling. In
the following, we denote by $B$ a reference shape of size $1$
centered at the origin, so that the unknown inclusion $D$ is
obtained from $B$ by a rotation with angle $\theta$, a scaling
$s>0$ and a translation $z\in \R^2$ as $D = z+ s R_\theta B$.
\subsection{Translation and rotation invariant shape descriptor}
\label{sec:shape-descriptor}
The construction of the shape descriptor starts from the formula \eqref{eq:SCT-trl}, where the
convolution relationship suggests us to pass to the Fourier domain. It turns out that the Fourier
series of $W_{mn}[D]$ is just the far-field pattern $\Af_D$ \cite{ammari_enhancement_2011-2}. The following proposition holds.
\begin{prop}
\label{prop:farfieldpat_SCT}
Let $\xi= {(\xi_1,\xi_2)}^\top \in [0, 2\pi]^2$. Then, we have
\begin{align}
\label{eq:Fourier_SCT}
\Af_D(\xi; \omega) = \sum_{m,n\in\Z} W_{mn}[D, \ve, \mu, \omega] e^{i m(\frac\pi 2
-\xi_1)} e^{-i n(\frac\pi 2 - \xi_2)}.
\end{align}
\end{prop}
\begin{rmk}
For the circular acquisition system mentioned in Section \ref{sec:full-angle-view}, the modulus
of the far-field pattern can be read off directly from the measurement when the radius $R$ and the
number of sources $N_s$ and receivers $N_r$ are large, thanks to
\eqref{eq:Vsr_far_field}. See Figure~\ref{fig:ffpattern_Vsr}.
\end{rmk}
\graphicspath{{./figures/}}
\def7.5cm{7cm}
\begin{figure}[htp]
\centering
\subfigure[$\Abs{V_{sr}}$]{\includegraphics[width=7.5cm]{Vsr.eps}}
\subfigure[$\Abs{\Af_D(\cdot;2\pi)}/\sqrt{8\pi k_0
R}$]{\includegraphics[width=7.5cm]{ffpattern.eps}}
\caption{(a) Modulus of the measurement of a flower-shaped object of size 1 with $N_s=512$ sources
and $N_r=512$ receivers at the frequency $\omega=2\pi$ and the radius $R=10$, (b)
$\Abs{\Af_D(\cdot;2\pi)}/\sqrt{8\pi k_0 R}$ computed via \eqref{eq:Fourier_SCT}. The relative
difference between (a) and (b) is about $10^{-2}$.}
\label{fig:ffpattern_Vsr}
\end{figure}
We emphasize that the Fourier series in \eqref{eq:Fourier_SCT} converges thanks to the decay
\eqref{eq:control_SCT} of $W_{mn}$. Then using the translation formula \eqref{eq:SCT-trl}, we
establish the relations between the far-field pattern of $D$ and $B$.
\begin{prop}
\label{prop:Fourier_SCT_DB}
Let $D=z+s R_\theta B$. We denote by $\theta_z$ the angle of $z$ in polar coordinate, and define
$\phi_z(\xi):=e^{ik_0\abs z \cos(\xi_1-\theta_z)} e^{-ik_0\abs z \cos(\xi_2-\theta_z)}$. Then, we have
\begin{align}
\label{eq:Fourier_SCT_DB}
\Af_D(\xi; \omega) = \phi_z(\xi)\, \Af_B(\xi_\theta; s\omega) \ \text{ with } \xi_\theta:=\xi
- {(\theta,\theta)}^\top.
\end{align}
\end{prop}
\begin{proof}
By the transformation formulas \eqref{eq:SCT-tsr}, we have
\begin{align}
\label{eq:Wmn_DB}
W_{mn}[D, \ve, \mu, \omega] &= \sum_{a,b\in\Z} \cwv_a(z) \widebar{\cwv_b(z)}
W_{m-a,n-b}[s R_\theta B,\ve, \mu, \omega] \notag \\
&= \sum_{a,b\in\Z} \cwv_a(z) \widebar{\cwv_b(z)} e^{i(m-a)\theta}e^{-i(n-b)\theta}
W_{m-a,n-b}[B, \ve, \mu, s\omega].
\end{align}
Using the Jacobi-Anger expansion, the Fourier series of the cylindrical wave reads:
\begin{align*}
\sum_{a\in\Z} \cwv_a(z) e^{i a (\frac \pi 2 - \xi_1)} = \sum_{a\in\Z} J_a(k_0\abs z) e^{i a
(\frac \pi 2 - (\xi_1-\theta_z))} = e^{i k_0\abs z
\cos(\xi_1-\theta_z)}.
\end{align*}
Therefore, computing the Fourier series on the both sides of \eqref{eq:Wmn_DB} yields
\eqref{eq:Fourier_SCT_DB}.
\end{proof}
Hereafter, we illustrate the descriptor construction based on the scattering amplitude.
We define the following quantity:
\begin{align}
\label{eq:Invar_S}
\mathcal S_D(v; \omega) := \int_{[0,2\pi]^2} \Abs{\Af_D(\xi; \omega){\Af_D(\xi-v; \omega)}} d\xi
.
\end{align}
More precisely, taking the modulus on $\Af_D$ removes the effect of the translation, and the
auto-correlation in $\xi$ removes that of the rotation. In fact, using \eqref{eq:Fourier_SCT_DB}:
\begin{align}
\label{eq:Shapedescrp_DB}
\mathcal S_D(v; \omega) = \int_{[0,2\pi]^2} \abs{\Af_B(\xi_\theta; s\omega)
\Af_B(\xi_\theta-v; s\omega)} d\xi = \mathcal S_B(v; s\omega),
\end{align}
where the last identity comes from the periodicity in $\xi$ of $\Af_B(\xi; \omega)$. We name
\eqref{eq:Invar_S} the \emph{shape descriptor}. The shape descriptor $\mcl S_D(v;\omega)$ is invariant to any translation and rotation.
Remark that for fixed $\omega$, it carries the information of the far-field
pattern of the shape, while for fixed $v\in[0,2\pi]^2$, it shows how the far-field pattern varies as a function of the frequency.
\subsection{Scale estimation}
\label{sec:estim-scal}
From $\mcl S_D(v;\omega)$ we could hopefully construct, at least formally, some quantities $\mcl
I_D(v)$ being further invariant to the scaling, and obtain in that way shape descriptors easily
adapted for the purpose of identification like in \cite{ammari_target_2012}. For example, one may
consider the Hilbert transform in $\omega$ of $\mcl S_D$:
\begin{align}
\label{eq:Hilbert_t}
\mathcal{I}_D(v) := \int_0^{+\infty} {\frac{\mcl S_D(v; \omega)}{\omega}}
d\omega,
\end{align}
which is invariant to any translation, rotation and scaling, since
\begin{align}
\label{eq:Invar_tsr_Hilbert}
\mathcal{I}_D(v) = \int_0^{+\infty} {\frac{\mcl S_B(v; s\omega)}{\omega}} d\omega = \int_0^{+\infty}
{\frac{\mcl S_B(v; \omega)}{\omega}} d\omega = \mcl I_B(v).
\end{align}
Not to mention the well-definedness of \eqref{eq:Hilbert_t}, let
us point out immediately the physical infeasibility of such kind
of quantities, whatever the method in use. In fact, the frequency
$\omega$ is coupled with the scaling factor $s$ in the relation
\eqref{eq:Shapedescrp_DB}, hence if the knowledge of $\mcl S_D(v;
\omega)$ in the frequency $\omega$ is bounded away from $0$ or
infinity, then it is not possible to construct an $\mcl I_D$ being
invariant to arbitrary scaling factor $s\in\R^+$, which would
require the frequency at 0 or infinity.
\subsubsection{Determine $s$ by lookup table}
\label{sec:determ-lookup-table}
We assume that the physical operating frequency $0<\omin\leq \omega \leq \omax<\infty$, and $0<\smin
\leq s \leq \smax<\infty$ which means the inclusions that we are interested in should not be too
small nor to large. Then we can estimate $s$ by solving
\begin{align}
\label{eq:optim_scaling_matching}
\sest = \argmin_{s\in[\smin, \smax]} \Set{\int_\omin^\omax\Paren{\int_{[0,2\pi]^2} [\mcl S_D(v;\omega)-\mcl
S_B(v;s\omega)]\, dv}^2 d\omega},
\end{align}
which is a lookup table technique since we are seeking the best value of $s$ in the table
$\int_{\sss{[0,2\pi]^2}} \mcl S_B(v; s\omega) dv$ indexed by $s$, by comparing with
$\int_{\sss{[0,2\pi]^2}} \mcl S_D(v; \omega) dv$. This idea is illustrated in Figure~\ref{fig:SDSB}.
Remark that the functional in \eqref{eq:optim_scaling_matching} is minimized at the value of the
true scaling factor, although the minimizer might not be unique.
\graphicspath{{./figures/}}
\def7.5cm{7cm}
\begin{figure}[htp]
\centering
\subfigure[]{\includegraphics[width=7.5cm]{SB.eps}}
\subfigure[]{\includegraphics[width=7.5cm]{SD.eps}}
\caption{Shape descriptors of a flower-shaped object $B$ and of $D=1.5\times B$, as functions of
the frequency $\omega$. (a) $\int_{\sss{[0,2\pi]^2}} \mcl S_B(v; \omega) dv$ with
$\omega\in[0.5\pi, 4\pi]$. (b) $\int_{\sss{[0,2\pi]^2}} \mcl S_D(v; \omega) dv$ with
$\omega\in[\pi, 2\pi]$. The red part in (a) corresponds to the frequency $\omega\in[1.5\pi,
3\pi]$.}
\label{fig:SDSB}
\end{figure}
\subsubsection{Numerical implementation of the scaling estimation}
\label{sec:numer-impl}
Numerically, \eqref{eq:optim_scaling_matching} can be solved approximately by sampling. Let $\Nwdic,
\Nw, \Nv, N_s$ be positive integers. We define $\set{\omegar_l, l=0\ldots \Nwdic}$ uniformly
distributed points on $[\ormin, \ormax]$, with $\ormin:=\omin\smin, \ormax:=\omax\smax$. Similarly,
let $\set{\omega_k, k=0\ldots \Nw}$ be uniformly distributed on $[\omin, \omax]$, and
$\set{(v^1_i,v^2_j)^\top, i,j=1\ldots \Nv}$ be uniformly distributed in $[0,2\pi]^2$. Then we sample
the functions $\mcl S_B$ and $\mcl S_D$ at discrete positions as follows:
\begin{align}
\label{eq:SD_ijk_SB_ijl}
\mcl S^D_{ijk}:= \mcl S_D({(v^1_i, v^2_j)}^\top; \omega_k), \ \mcl S^B_{ijl}:= \mcl S_B({(v^1_i,
v^2_j)}^\top; \omegar_l).
\end{align}
For $\set{s_t, t=0\ldots, N_\delta}$ uniformly distributed in $[\smin, \smax]$, we can approximate the
functional inside the argmin in \eqref{eq:optim_scaling_matching} by a discrete version:
\begin{align}
\label{eq:optim_scaling_discrete}
\mcl J(t; D, B) = \sum_{k=0}^{\Nw} \sum_{l\in
I_k(s_t)} \Paren{\sum_{i,j=1}^\Nv \Paren{\mcl S^D_{ijk} - \mcl S^B_{ijl}}}^2
\end{align}
with the index set $I_k(s):=\set{1\leq l \leq \Nwdic, \text{ s.t. } \omegar_{l-1}\leq s\omega_k \leq
\omegar_l}$. Finally, the scaling is estimated through
\begin{align}
\label{eq:err_DB}
\ve(D,B) = \min_{t=0\ldots N_\delta} \mcl J(t; D, B).
\end{align}
\subsection{Frequency-dependent dictionary and identification algorithm}
\label{sec:freq-depend-dict}
For a collection of standard shapes $\set{B_n}_{n=1\ldots N}$, we
pre-compute the discrete samples $\Paren{\mcl
S^{B_n}_{ijl}}_{ijl}$ of the shape descriptor $\mcl
S_{B_n}(v;\omega)$ as in \eqref{eq:SD_ijk_SB_ijl} for
$v\in[0,2\pi]^2$ and $\omega\in[\ormin, \ormax]$. This constitutes
a frequency-dependent dictionary of shape descriptors.
Suppose $D$ is the realization of a shape from the dictionary
$\set{B_n}_n$, up to some unknown translation, rotation and
scaling, and the scaling factor satisfies $\smin\leq s\leq \smax$.
In order to identify $D$ from the dictionary, we compute the
discrete samples $\Paren{\mcl
S^{D}_{ijk}}_{ijk}$ of the shape descriptor $\mcl S_D(v;\omega)$ as in \eqref{eq:SD_ijk_SB_ijl},
and calculate $\ve(D, B_n)$ in \eqref{eq:err_DB} for all shapes of the dictionary. The true shape is expected to give the best estimation of scaling and to minimize the error $\ve(D,B_n)$ among all the
dictionary elements, thus we take the minimizer of $\set{\ve(D,B_n)}_n$ as the identified shape. This
procedure is described in Algorithm \ref{alg:shape_identify}.
\begin{algorithm}[H]
\caption{Shape identification algorithm}
\label{alg:shape_identify}
\begin{algorithmic}
\STATE Input: {$\Paren{\mcl S^{D}_{ijk}}_{ijk}$ of unknown shape $D$, $\dicoSB$ of the whole dictionary}
\FOR{$B_n$ in the dictionary}
\STATE $\ve_n \leftarrow \ve(D, B_n)$;
\STATE $n \leftarrow n+1$;
\ENDFOR
\STATE Output: {The true dictionary element $n^* \leftarrow \argmin_n \ve_n$.}
\end{algorithmic}
\end{algorithm}
\subsection{Shape descriptor with partial far-field pattern}
\label{sec:shape-descr-partial}
Let us mention an interesting property of $\mcl S_D$ at the end of this section. Given $0<\alpha\leq
2\pi$, we denote the partial far-field pattern $\Af_D$
\begin{align}
\label{eq:partial_far_field}
\wt{\Af_D}(\xi;\omega):={\Af_D}(\xi;\omega)\mathbbm{1}(\abs{\xi_1-\xi_2}\leq \alpha),
\end{align}
\ie, only the band diagonal of width $\alpha$ in $\Af_D(\cdot; \omega)$ is available. Since
\begin{align*}
\mathbbm{1}(\abs{\xi_1-\xi_2}\leq \alpha) = \mathbbm{1}(\abs{(\xi_1-\theta)-(\xi_2-\theta)}\leq \alpha),
\end{align*}
so for the same reason as in \eqref{eq:Shapedescrp_DB} we deduce that
\begin{align*}
\int_{[0,2\pi]^2} \Abs{\wt{\Af_D}(\xi; \omega) \wt{\Af_D}(\xi-v; \omega)} d\xi =
\int_{[0,2\pi]^2} \Abs{\wt{\Af_B}(\xi_\theta; s\omega) \wt{\Af_B}(\xi_\theta-v; s\omega)} d\xi.
\end{align*}
Therefore, the shape descriptor $\mcl S_D$ computed with $\wt{\Af_D}$ is still invariant to
translation and rotation. Assuming that the dictionary $\dicoSBn$ is also computed using partial far
field pattern $\wt{\Af_{B_n}}$ of the same constant $\alpha$, then one can always apply
Algorithm~\ref{alg:shape_identify} for shape identification.
\section{Numerical experiments}
\label{sec:numer-exper}
In the rest of this paper, we present numerical results
demonstrating the theoretical framework presented in previous
sections. Given a shape $D$, the overall procedure of a numerical
experiment consists of the following steps:
\begin{enumerate}
\item Data simulation. The MSR matrix $\V$ is simulated for a range of frequency $[\omin, \omax]$ by
evaluating the integral representation \eqref{eq:solu}. Then the white noise is of level
\begin{align*}
\snoise = \sigma_0 \norm{\V}_F/\sqrt{N_sN_r}
\end{align*}
is added as in \eqref{eq:AWBE_mat_noise}. $\sigma_0$ is the percentage of the noise in the
measurement and it is proportional to $\SNR^{-1}$.
\item Reconstruction of scattering coefficients. For each frequency, we reconstruct $\W$ either by
solving the least-squares problem \eqref{eq:least_square} (or directly by the formula
\eqref{eq:L_pseudoinv} in case of the full aperture of view). This step is skipped in the case of
limited aperture of view, see Section~\ref{sec:shape-ident-with-lim-view}.
\item Shape identification. We calculate the shape descriptors and follow the procedure described in
Algorithm~\ref{alg:shape_identify} for identification.
\end{enumerate}
Our dictionary of shapes consists of eight elements as shown in
Figure~\ref{fig:8shapes}. All shapes share the same permittivity
$\ve_*=3$ and the permeability $\mu_*=3$, while the values of the
background are $\ve_0=1, \mu_0=1$. The frequency-dependent
dictionary of shape descriptors $\dicoSB$ is computed for the
frequency range $[\ormin,\ormax]=[0.5\pi,4\pi]$ with $\Nwdic=219$
and $\Nv=512$. The data are simulated for the operating frequency
range $[\omin,\omax]=[\pi, 2\pi]$ with $\Nw=109$ and the valid
scaling range is $[\smin, \smax]=[0.5, 2]$ with $N_\delta=751$.
\graphicspath{{./figures/dico/}}
\def7.5cm{3cm}
\begin{figure}[htp]
\centering
\subfigure[Ellipse]{\includegraphics[width=7.5cm]{Ellipse.eps}}
\subfigure[Flower]{\includegraphics[width=7.5cm]{Flower.eps}}
\subfigure[Letter A]{\includegraphics[width=7.5cm]{A.eps}}
\subfigure[Square]{\includegraphics[width=7.5cm]{Square.eps}}
\subfigure[Letter E]{\includegraphics[width=7.5cm]{E.eps}}
\subfigure[Rectangle]{\includegraphics[width=7.5cm]{Rectangle.eps}}
\subfigure[Disk]{\includegraphics[width=7.5cm]{Circle.eps}}
\subfigure[Triangle]{\includegraphics[width=7.5cm]{Triangle.eps}}
\caption{A small dictionary of shapes.}
\label{fig:8shapes}
\end{figure}
\subsection{Experiment and parameter settings}
\label{sec:exp-parm-settings}
We generate a circular acquisition system like in Section~\ref{sec:full-angle-view} with the radius
$R=3$, and $N_s=91$ plane waves as sources and $N_r=91$ receivers. For reason of simplicity, we
always choose the center $z_0=[0,0]^\top$. Figure~\ref{fig:acqsys} (a) illustrates this acquisition
system.
Another possibility is shown in Figure~\ref{fig:acqsys} (b), where the sources and receivers are
divided into different groups of aperture angle $\alpha$ such that no intercommunication exists between
groups. Such an acquisition is close to that used by the bat. Each group in Figure~\ref{fig:acqsys}
(b) represents the spatial position of a flying bat's body, which sends plane waves with limited
aperture of wave direction and receives the scattered field via receptors on its body. By flying
around the target and taking measurement at many positions, the bat actually acquires data
corresponding to the band diagonal part of the MSR matrix of a full aperture of view system, and the
width of the band diagonal is the number of the receivers (or proportional to $\alpha$).
\graphicspath{{./figures/}}
\def7.5cm{6.5cm}
\begin{figure}[htp]
\centering
\subfigure[Full view]{\includegraphics[width=7.5cm]{Acqsys_fullview.eps}}
\subfigure[Limited view]{\includegraphics[width=7.5cm]{Acqsys_limview_group.eps}}
\caption{Circular acquisition systems. (a) Full aperture with $N_s=91$ plane wave sources (with
the angular position marked by 'o') and $N_r=91$ receivers (marked by 'x'). (b) Limited aperture
with the sources and receivers divided into 5 groups, and the receivers of one group can only
see the sources of the same group. The measurement center $z_0$ in both cases is marked by '*'.}
\label{fig:acqsys}
\end{figure}
Figure~\ref{fig:svd_acqsys} shows the singular values of the
associated operators $\bL$ at the operating frequency
$\omega=2\pi$. The stairwise distribution and the increase of
singular values with the order $K$ in Figure~\ref{fig:svd_acqsys}
(a) confirm the theoretical result in \eqref{eq:sing_vl}. It can
be seen that at the low-order (\eg $K\leq 30$) the operator $\bL$
is numerically well-conditioned. On the contrary, as shown in
Figure~\ref{fig:svd_acqsys} (b) the situation in the limited
aperture of view is dramatically different: the operator is
ill-conditioned even for very low-order (\eg, $K=5$), which means
the reconstruction of scattering coefficients is highly unstable
with such an acquisition
system.
\def7.5cm{7cm}
\begin{figure}[htp]
\centering
\subfigure[Full view]{\includegraphics[width=7.5cm]{SVD_fullview.eps}}
\subfigure[Limited view]{\includegraphics[width=7.5cm]{SVD_limview_group.eps}}
\caption{Singular values of the operator $\bL$ of the acquisition systems of Figure~\ref{fig:acqsys}
(a) and (b) at the frequency $\omega=2\pi$. The curves from top to bottom correspond to the
operators $\bL$ of order $K=40, 30, 20$ in (a), and $K=15, 10, 5$ in (b), respectively. }
\label{fig:svd_acqsys}
\end{figure}
\subsubsection{Resolving order $K$}
The maximal resolving order $K$ given by
\eqref{eq:max_resolving_order_bound} is an asymptotic bound which
holds when both the radius $R$ and $K$ are large, and it might be
too pessimistic for the numerical range that we are interested in.
In practice, the maximal resolving order can be determined by a
tuning procedure. For the system in Figure~\ref{fig:acqsys} (a), we
vary the noise level and reconstruct the matrix $\West$ at
different truncation order $K$. Figure~\ref{fig:resolv_ord} plots
the relative error of the least-squares reconstruction
$\norm{\West-\W}_F/\norm{\W}_F$ as a function of $K$. It can be
seen that in the full aperture case, the reconstruction is rather
robust and with $20\%$ of noise the resolving order $K$ can go
beyond $30$ by bearing only $10\%$ of error. On the contrary, the
limited aperture of view can not provide any stable
reconstructions even at very low noise level and small truncation
order.
\def7.5cm{7cm}
\begin{figure}[htp]
\centering
\subfigure[Full view]{\includegraphics[width=7.5cm]{res_ord_fullview.eps}}
\subfigure[Limited view]{\includegraphics[width=7.5cm]{res_ord_limview.eps}}
\caption{Relative error of the least-squares reconstruction $\norm{\West-\W}_F/\norm{\W}_F$ for the
systems in Figure~\ref{fig:acqsys} at different truncation order $K$. The curves from bottom to
top correspond to respectively: (a) percentage of noise $\sigma_0=20\%, 40\%, 60\%, 80\%$ and
$100\%$, (b): $\sigma_0=2.5\%$ and $5\%$.}
\label{fig:resolv_ord}
\end{figure}
\subsection{Shape identification with full aperture of view}
\label{sec:shape-ident-with-full-view} Here we present results of
shape identification obtained using the full aperture of view
setting Figure~\ref{fig:acqsys} (a). For each shape $B_n$ of the
dictionary, we take the three steps mentioned in the beginning of
Section~\ref{sec:numer-exper} on a target $D$ obtained by
transform $B_n$ with the parameters $z=[-0.5,0.5]^\top, s=1.5,
\theta=\pi/3$. The order $K$ is set to 30.
The computation of the error $\ve(D,B_n)$ is represented in Figure~\ref{fig:ident_full_view} by error bars, where the
$m$-th error bar in the $n$-th group corresponds to the error $\ve(D,B_m)$ of the identification
experiment using the generating shape $B_n$. The error bars are arranged in the same way as in
Figure \ref{fig:8shapes}. The shortest bar in each group is the identified shape
and is marked in green, while the true shape is marked in red in case that the identification
fails. It can be seen that the identification succeeded for all shapes with $20\%$ of noise, and it
failed only for the rectangle with $40\%$ of noise. The estimated scaling factor computed as in (\ref{eq:err_DB}) is close to the
true value $1.5$. The discrepancies between them are displayed in Figure~\ref{fig:sclest_full_view}.
\def7.5cm{7.5cm}
\begin{figure}[htp]
\centering
\subfigure[$\sigma_0=20\%$]{\includegraphics[width=7.5cm]{ident_nlvl_0p2_fullview.eps}}
\subfigure[$\sigma_0=40\%$]{\includegraphics[width=7.5cm]{ident_nlvl_0p4_fullview.eps}}
\caption{Results of identification $\ve(D,B_n)$ for all shapes of the dictionary using the full
aperture of view setting with (a) $\sigma_0=20\%$ and (b) $\sigma_0=40\%$ of noise. All shapes
were correctly identified in (a), and the rectangle was identified as the square in (b).}
\label{fig:ident_full_view}
\end{figure}
\def7.5cm{6.5cm}
\begin{figure}[htp]
\centering
\subfigure[$\sigma_0=20\%$]{\includegraphics[width=7.5cm]{sclest_nlvl_0p2_fullview.eps}}
\subfigure[$\sigma_0=40\%$]{\includegraphics[width=7.5cm]{sclest_nlvl_0p4_fullview.eps}}
\caption{Difference between the estimated scaling factor and the true one ($s=1.5$) for the
identified shapes in Figure~\ref{fig:ident_full_view}.}
\label{fig:sclest_full_view}
\end{figure}
\subsection{Shape identification with limited aperture of view}
\label{sec:shape-ident-with-lim-view}
We use a limited view system as Figure~\ref{fig:acqsys} (b) of small aperture angle $\alpha$, with 512
source/receiver groups (the number of sources/receivers in each group is about $512\times
\alpha/2\pi$) uniformly distributed on a circle of radius $R=10$ around the target. Visually
speaking, the modulus of the measurement $\abs{V_{sr}}$ is similar to that shown in
Figure~\ref{fig:ffpattern_Vsr} with all entries set to zero except the band diagonal (in red).
As analyzed in Section~\ref{sec:exp-parm-settings}, in this case one can not expect to reconstruct
the scattering coefficients with high precision, and consequently compute neither the full far-field
pattern via \eqref{eq:Fourier_SCT} nor the shape descriptor \eqref{eq:Invar_S}. However, thanks to
the relation \eqref{eq:Vsr_far_field} the measurement $\abs{V_{sr}}$ gives directly the band
diagonal part of the far-field pattern $\abs{\Af_D}$, and one can compute the shape descriptors from
the partial far-field pattern hence apply the identification algorithm as described in
Section~\ref{sec:shape-descr-partial}.
The results of identification with the limited aperture are shown in Figure~\ref{fig:ident_lim_view},
where all shapes were identified correctly with $\alpha=\pi/3$ and only the square was missed with
$\alpha=\pi/6$. The performance may deteriorate when the dictionary contains more shapes. In that
case one should increase the aperture angle and broaden the range of operating frequency $[\omin,
\omax]$ in order to compensate the lose of information in the incomplete far-field pattern.
\def7.5cm{7.5cm}
\begin{figure}[htp]
\centering
\subfigure[$\alpha=\pi/6$]{\includegraphics[width=7.5cm]{ident_nlvl_0p2_limview_30deg.eps}}
\subfigure[$\alpha=\pi/3$]{\includegraphics[width=7.5cm]{ident_nlvl_0p2_limview_60deg.eps}}
\caption{Results of identification in the limited angle of view setting with angular aperture
(a) $\alpha=\pi/6$ and (b) $\alpha=\pi/3$. $\sigma_0=20\%$ of noise is added in both cases.}
\label{fig:ident_lim_view}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this paper we presented a framework of shape identification
using the frequency-dependent dictionary of shape descriptors,
which is invariant to rigid transforms and allows to handle the
scaling within certain ranges. The shape descriptor is based on
the far-field pattern which can be either computed from the
scattering coefficients or read off directly from the measurement.
We analyzed the stability of the reconstruction and presented
results of identification with both full and limited aperture of
view. Our approach in this paper can be used for tracking the location and the orientation of a target from acoustic echoes. The generalization of our work on target tracking in electrosensing \cite{ammari_tracking_2012} to echolocation will be the subject of a forthcoming paper.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,997
|
Home Research > Members > 1386-1421 > SHERARD, Robert
SHERARD, Robert, of Stapleford, Leics.
Published in The History of Parliament: the House of Commons 1386-1421, ed. J.S. Roskell, L. Clark, C. Rawcliffe., 1993
?s. of George Sherard by Joan, da. of Richard Berners. m. bet. Mich. 1396, Agnes, da. and h. of Sir Laurence Hauberk† (d.1381) of Stapleford by Margaret, da. and coh. of Roger Cheyne of 'Bukenhalle', Salop, wid. of (Sir) Hugh Calveley* (d.1393) of Calveley and Mottram St. Andrew, Cheshire, 1s.
Duchy of Lancaster bailiff of Stapleford c. Mich. 1404-c. Mar. 1413.1
No evidence has been found either to support or disprove Nichols's statement that Sherard came from an old Cheshire family, and he is not known to have possessed landed holdings in that county other than those he acquired by marriage. He is first recorded in February 1394 while accusing a man from Northampton of mayhem and a breach of the peace.2 Two years later he made his important match with Agnes, the widow of (Sir) Hugh Calveley, who, having earlier inherited from her father, Sir Laurence Hauberk, sizeable estates at Stapleford and Saxby (Leicestershire) and Barrow in Cottesmore (Rutland), had recently managed with the assistance of her stepfather, Sir John Calveley*, to establish her title to the manor of Scalford against the claims of her uncle, John Hauberk. In the autumn of 1396, Stapleford and Scalford were formally settled on Robert and Agnes Sherard, with remainder to the lady's issue, although a few years later Robert persuaded his wife to accept new entails of these and other Hauberk estates, giving preference to his son before her older Calveley offspring, with the outcome that after her death only the property at Saxby was divided between the Calveleys and the Sherards, the latter acquiring all the rest. Agnes held for life as dower from her previous marriage a third part of the Cheshire manor of Mottram St. Andrew.3
Stapleford, Sherard's principal manor, formed part of a large estate pertaining to the duchy of Lancaster, the profits of which, from December 1400 until his death at the battle of Shrewsbury, were paid to Sir John Calveley for his loyal service as one of the King's knights-bachelor. Thereafter, from 1404 for the rest of Henry IV's reign, Sherard himself enjoyed an annuity of £20 charged on the estate at Stapleford, which he farmed on an eight-year lease and administered on behalf of the duchy as a bailiff. While so doing, he was elected to Parliament for the only recorded time. Under Henry V, perhaps because he had ceased to hold office, Sherard's annuity was reduced to 20 marks. His service to the Crown in other respects was of little account, although he did contract on 29 Apr. 1415 to join the King's first expedition to France, albeit with a contingent of no more than two archers.4
Sherard died at an unknown date before Easter 1422, and was probably buried in Stapleford church, where he and his wife were to be depicted on windows inserted in the 17th century.5 Their son, Laurence, who became a prominent figure in royal local administration as sheriff of Rutland in 1437-8 and of Warwickshire and Leicestershire in 1443-4, also served James, Lord Audley, as steward of certain of his estates.6
Author: L. S. Woodger
1. DL29/212/3249.
2. J. Nichols, Leics. ii. 333, 343; CCR, 1392-6, p. 266.
3. Nichols, ii. 309, 314, 333; Early Lincoln Wills ed. Gibbons, 48-49; Leics. Village Notes ed. Farnham, ii. 344; iv. 31; v. 370; VCH Rutland, ii. 123; CP25(1)192/8/3; DKR, xxxvi. 80-81.
4. DL28/27/3, 6; DL29/212/3249, 738/12100; DL42/15, f. 17d, 17, f. 24; E101/69/5/419.
5. Leics. Village Notes, iv. 32, 130; Nichols, ii. pl. opposite p. 339. Sherard's widow was called Johan in a lawsuit (CP40/645 m. 336d), though this was apparently an error, for Agnes was still alive at Michaelmas 1421.
6. Nichols, ii. 348.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,633
|
{"url":"https:\/\/physics.stackexchange.com\/questions\/350588\/measuring-through-meter-scale","text":"Measuring through meter scale\n\nWhat if the measuring thing is found to have length between 6mm and 7mm. Like u r measuring the length of an object which has the length between 5.6cm and 5.7 cm. The millimeter distance was between somewhere of 6 mm and 7 mm. How can I deduce the distance?\n\nNote: I am just using meter scale. Not Vernier Caliper.\n\n\u2022 You cannot. Metre scale cannot give that accuracy. \u2013\u00a0Wrichik Basu Aug 6 '17 at 13:24\n\n1. In a single measurement with a ruler, you can not tell $0.1~mm$ apart, but you can certainly do better than $1~mm$. By eyeballing, I would say that a standard deviation $\\sigma~\\sim~0.2~mm$ is not unrealistic.\n\n2. If you repeat the measurement sufficiently many times, you are able to statistically deduce one extra digit beyond the millimeters.\n\n3. The trick is to not put one end of the object at $0~mm$ on the ruler: Place instead the ruler arbitrary along the object, and write down the readings $XX.X~mm$ and $YY.Y~mm$ at each end, eyeballing the last digit. The length is then $(YY.Y-XX.X)~mm \\pm \\sigma\\sqrt{2}$.\n\n4. Slide the ruler arbitrarily along the object. Repeat measurement $n$ times. Take the average. The resulting standard deviation $\\sim \\sigma\\sqrt{2\/n}$ is now well into the sub-millimeter range.","date":"2019-10-22 16:16:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7782237529754639, \"perplexity\": 792.9355817522336}, \"config\": {\"markdown_headings\": false, \"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-2019-43\/segments\/1570987822458.91\/warc\/CC-MAIN-20191022155241-20191022182741-00190.warc.gz\"}"}
| null | null |
LAST MAN STANDING
For every copy of this book sold in the UK, the publisher will make a donation of 20 pence to UNICEF UK (registered charity no. 1072612)
First published in Great Britain in 2014 by
Michael O'Mara Books Limited
9 Lion Yard
Tremadoc Road
London SW4 7NQ
Copyright © Sir Roger Moore 2014
All rights reserved. You may not copy, store, distribute, transmit, reproduce or otherwise make available this publication (or any part of it) in any form, or by any means (electronic, digital, optical, mechanical, photocopying, recording or otherwise), without the prior written permission of the publisher. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.
A CIP catalogue record for this book is available from the British Library.
ISBN: 978-1-78243-207-4 in hardback print format
ISBN: 978-1-78243-267-8 in ebook format
ISBN: 978-1-78243-286-9 in trade paperback format
www.mombooks.com
Designed and typeset by D23
CONTENTS
Foreword
Introduction
CHAPTER 1
_The Fun – and Feisty – Leading Ladies_
CHAPTER 2
_The Pinewood Years_
CHAPTER 3
_Stage-struck_
CHAPTER 4
_On-set Tales_
CHAPTER 5
_The Good Guys (and a Few Rascals)_
CHAPTER 6
_The Rat Pack_
CHAPTER 7
_The Creative Geniuses_
CHAPTER 8
_The Producers_
_Postscript_
_In Closing ..._
_Acknowledgements_
_Picture Credits_
_Index_
_List of Illustrations_
_Plates_
FOREWORD
WHEN I STARTED OUT WRITING THIS TOME, I HAD THE idea of calling it _One Lucky Bastard_ because that's what I feel I certainly am. But the 'b-word' was thought to be a little too risqué and wouldn't look good on the bookshop shelves, so I thought I'd better come up with another title that would describe, perhaps more accurately, what I hope you will find to be an interesting, amusing and moving collection of memories and stories about friends, colleagues and loved ones I've encountered in my eighty-odd years.
Lana Turner, whom I had the greatest pleasure of working with in Hollywood, told me her pet hatred was another actress named Linda Christian, namely because when Lana was engaged to Tyrone Power, Linda found out where he was going to stay in Rome while working on a film and booked herself into a room next to his ... and the rest was history.
Why am I telling you this? Well, a while later, Linda and Edmund Purdom – who was under contract at MGM at the same time as me – started a big affair and to complicate matters further, Linda found herself in the centre of a rather sticky situation regarding _another_ past affair, this time with a wealthy industrialist who had presented her with expensive jewels and precious diamonds that his family now wanted back. Linda felt she should have some recompense for her trouble, and when the day for a changeover of cash for jewels was set, she asked me to accompany her and Edmund, feeling that because I was a fairly athletic and fit young man, I would 'scare off' any unwanted intervention.
A year or two later, I was offered a TV play with Linda, and it was quite the worst script I'd ever read. Though the stage directions made it very clear why Linda was so interested: 'In the first scene, Linda makes her entrance and her beautiful hair is held back behind her ears ...'
Scene two: 'Linda comes in with her beautiful hair and dress hanging over her shoulder and looks even more lovely than before ...'
This went on, and on. Vanity was obviously in play.
But the one thing I remember from the script was the description and explanation of death: 'When one dies one has actually just gone into another room; we know you're in there but don't have the key to get in.'
That line has always stuck in my mind, and now being one of the last men standing I'm finding that a great many of my friends are in the next room. I don't wish to be morbid, nor want to write a collection of obituaries, but I do write about quite a few of my friends in the past tense ... but don't feel depressed, dear reader, feel happy that we've had these wonderful characters in our lives, as I certainly do. Frank Sinatra used to say, 'Who's going to be left to turn the light off?'
Hopefully, it'll be me!
INTRODUCTION
DUE TO THE PHENOMENAL WORLDWIDE SUCCESS OF MY first published autobiography, _My Word is My Bond_ , namely sales of two softback copies and one hardback in Burkina Faso, my publishers – poor misguided people with big hearts but short purse strings – have commissioned me to attempt to pen another pack of near truths.
By the time I deliver this manuscript I will have arrived at the ripe old age of eighty-six – I hope – four score years and six, and I'm very much reminded of dear old Bette Davis saying, 'Old age ain't no place for sissies' as my creaking knees and aching back certainly attest. But where did these eighty-six years go? Many things have happened yet they seem to have flashed by in eighty-six minutes; I must have met hundreds of thousands of people, but can I remember them all, some of them ... a few? Well, I'll try.
I have always imagined that somewhere in space a recording machine has documented every word, every image, and even more terrifyingly, every thought I have been involved with. I wonder what they'd think in Heaven if they tuned in to the lascivious thoughts that crossed my mind, aged thirteen, on seeing the girls at school with gymslips tucked into their dark blue bloomers as they performed in the hall or playground during PT? I know these are hardly the ideal reflections for a future UNICEF Ambassador and I apologize for this momentary lapse into early teenage indiscretions, but at my age these matters come to mind much more readily than others, such as what I had for breakfast this morning.
I have been very fortunate to spend most of my life in the business we call 'show'. It's always interesting, often challenging and if Lady Luck favours us, and benevolent producers take pity on us, then it's quite possible to make a living out of doing something really enjoyable. I've always maintained that any modicum of success I have savoured has been primarily down to good luck; yes, it helps if you look like a hero, if you can remember lines and if you work cheaply; but ultimately, if you're not in the right place at the right time then you could still be an eighty-six-year-old 'extra' carrying a spear in a crowd scene.
While fame, success and good fortune affect people differently, we are, of course, all equal underneath; some like to think they are more equal than others I grant you. However, proving that beyond the glitz, glamour and flashbulbs, actors are still human, is a story that was told to me by Honor Blackman, who is perhaps most fondly remembered by Bond aficionados as the delightfully named Pussy Galore in _Goldfinger_. Honor had been attending a function in Birmingham and, prior to making her departure for the drive home, took the opportunity to powder her nose. Her friend-cum-driver was standing near the door awaiting her re-emergence when two elderly ladies exited ahead of Honor and were heard by him to say,
Number 1: 'Did you see who that was?'
Number 2: 'Yes, it was her, wasn't it? Honor Blackman.'
Number 1: 'Yeah and just think, she goes to the toilet like the rest of us!'
This marks my third literary effort, and this time I want to share with you some of the fun I've experienced with showbiz folk in my long and illustrious career, along with stories and tales that I've been told. In the pages ahead, while I would like to take the opportunity of updating you on the exciting six years since the publication of my first tome (and, perhaps most importantly for any fellow hypochondriacs, to share with you all my latest ailments, accidents and surgeries), I realize – and my wife Kristina often reminds me – that tales of my kidney stones, pacemaker, accidents and the like might not fascinate you, dear reader, as much as they do me and my doctors (my proctologist, congratulating me after the publication of my first volume, did say that he'd seen me from a whole different angle ...). Therefore I shall limit the bulk of what follows to more of a mixture of adventures and anecdotes drawn from the deepest recesses of my mind – or failing that, ones I've just made up.
What I will say is that, in between paying jobs and book tours, Kristina and I split our time between Switzerland and Monaco. I've had a home in the South of France since the 1970s, the first being in St Paul de Vence – in fact that's where I met Kristina; as neighbours we used to play tennis. Now we just watch.
It's a wonderful part of the world, and very peaceful save for the odd private jet flying over. When you think of the Cote d'Azur, images of fancy yachts, golden beaches and sun-kissed restaurant and cafe terraces where patrons take shade with a glass of something pink come to mind. The tranquillity is only ever upset by the annual invasion for the Cannes Film Festival and Monaco Grand Prix – events we happily try and dodge these days.
Today at the Film Festival there are more stars than you can shake a stick at, all grappling for publicity on the red carpet. It's not an experience I enjoy anymore and my last foray to the festival was as a guest of Tom Hanks for _The Ladykillers_ premiere in 2004. That night, the festival organizers kindly arranged for a car to take us, and a PR person to guide us along the line of waiting photographers and into the Grand Auditorium. However, when we emerged after the film and our job had been done there was no such PR person to assist us, nor was there a car – there was, however, a bus to the party in Juan-les-Pins.
The story of my life – I arrive by limo and get sent home on a bus!
I should warn you that there might possibly be the odd rude word in the pages ahead, not uttered by my lips I hasten to add, but by those of other people I have associated with; and while I know the majority of broadminded readers will take it in their stride, I am conscious of receiving a letter following the publication of my earlier effort, from a lady who said she'd never read so much filth in her life and my continual mentioning of four-letter words disgusted her to the point she'd never watch one of my films again. I have to admit that the immediate halving of my fan base is something that has weighed heavily on my mind ever since.
So, _Last Man Standing_ , an odd title for a book of tales from Hollywood and beyond, I grant you, but it works for me as I write these stories. Many of the folk you'll encounter in these pages have shuffled off this mortal coil now – some great stars, some legendary directors, all great friends of mine from a career that spans almost seven decades. Where did that time go? How did that happen? And where on earth does one start with a book like this ...?
Well, ladies first, obviously ...
Lana Turner as Diane de Poitiers curtsies to me, Prince Henri, in our 1955 film _Diane_. Lana was always full of surprises.
CHAPTER 1
The Fun – and Feisty – Leading Ladies
WHEN I FIRST ARRIVED IN HOLLYWOOD IN 1954, reporting for duty at MGM, Grace Kelly was also under contract to the studio. I remember sitting in publicist Dore Friedman's office one morning when the door burst open, and in came Grace, fuming that she'd just seen the posters for _Green Fire_ – a film she'd made with Stewart Granger – and that the studio had superimposed Grace's head onto Ava Gardner's body.
'I do not have tits like that!' she shouted.
The studio liked to 'sex up' their posters, in order to sell the films, all in the very best possible taste, of course.
Dore Friedman, incidentally, told me he was invited to accompany someone to one of the 'pledge luncheons' that used to be held at Romanov's restaurant in LA. All the studio heads were there – Zukor of Paramount, Jack Warner, Zanuck from Fox etc. – and it suddenly dawned on poor Dore what was happening.
Zanuck stood up and said, 'I pledge $250,000!'
'Then I will pledge $300,000!' shouted Adolph Zukor ... and so it went on, with them all trying to outbid one another.
As they came around to Dore, he wasn't sure quite how far his $65 weekly pay cheque would stretch and, thinking on his feet, he declared, 'The same as last year!', which gained a great round of applause.
When asked about Grace Kelly, all the male directors and executives at MGM would tell you how much they fantasized about doing things only men and women can do together with her. She was unquestionably one of the most desirable women in Hollywood. William Holden and Ray Milland, two of the film world's most unrepentant lotharios, were said to be 'out of their minds' with passion for her.
I found myself seated next to Grace at dinner one evening at Hollywood hairdresser Sydney Guilaroff's house. The conversation started turning to politics, of which, as a young Brit, I knew very little, and Grace said to me, 'You know, Roosevelt sold us down the river.' I'm afraid I had no idea what she was talking about, and for some time after that I often kicked myself for not being able to continue the conversation.
Some years later, when I became a regular visitor to the South of France and she had become Princess Grace of Monaco, she invited me up to the Grimaldi family's farm retreat, Roccagele, in the hills high above Monte Carlo, and that's where I first met Prince Albert who I guess was eleven or twelve. He struck me as being a very quiet and shy young man, who took great pleasure in showing me the many animals around the farm.
Grace wasn't at all stuffy as her royal status would have entitled her to be had she wished. Far from it, she had a mischievous sense of humour, a glint of naughtiness in her eye and a great passion for limericks – especially saucy ones.
I may not have known a great deal about US politics in the early days, but I _did_ know that Grace Kelly was one of the most desirable women in Hollywood.
Grace was a very precious gift to Monaco, albeit for too short a time.
Talking of Ava Gardner, she was one of the biggest stars in Hollywood when I arrived at MGM in the 1950s. In fact MGM's publicity department was reportedly sending out three thousand photos of Ava each week. A decade earlier, Louis B. Mayer himself had signed Ava, reportedly saying after viewing her screen test, 'She can't act. She can't talk. She's terrific!'
After a few years of fairly nondescript roles, it was her part in the 1946 film _The Killers_ that really launched her as a star. The studio gave her Norma Shearer's old dressing-room suite, the largest on site, with a bedroom, bathroom, kitchen and the actual dressing room itself, lined with mirrors, light bulbs and wardrobes – it was certainly befitting of her new standing.
Ava was a very funny and pithy lady, though was, perhaps, equally well known for her sexual conquests and husbands as much as her films. She was married three times in all, to Mickey Rooney (himself an MGM contract artist when they met), Artie Shaw and Frank Sinatra; and her high-profile affairs included those with Clark Gable, Robert Taylor, George C. Scott and Robert Mitchum. In fact, legend has it that it was while filming _My Forbidden Past_ in 1951, that she was first attracted to co-star Mitchum, who was himself under contract to Howard Hughes, with whom Ava had been romantically linked.
Mitchum telephoned his boss. 'Do you mind if I go to bed with Ava?' he asked.
'If you don't,' Hughes replied, 'they'll think you're a pansy.'
In her autobiography, though, Ava stated that Sinatra was the real love of her life. They'd actually met when Ava was an eighteen-year-old starlet, newly arrived in Hollywood, but, despite describing her as 'smoulderingly sexy', Frank thought she was just too young at the time. Five years later – by which time she was not only divorced from Rooney but also from her second husband, bandleader Artie Shaw – they met again and there was a huge mutual attraction. Soon after, Frank left his wife, Nancy, for her.
The whole story caused a huge scandal among the Hollywood establishment, and the scandal was happily fuelled by gossip columnists Hedda Hopper and Louella Parsons, not to mention within the Catholic Church, and among Frank's fans. Ava was portrayed as the _femme fatale_ who had stolen Frank away from his family.
Frank's career suffered both critically and commercially, but Ava used her considerable influence to get him cast in what was to be his Oscar-winning role in _From Here to Eternity_ in 1953. That film, and the award that followed it, revitalized both Frank's acting and singing careers. He was soon re-established as the world's top recording artist.
During their six-year marriage, Ava became pregnant twice, but had abortions. 'MGM had all sorts of penalty clauses about their stars having babies,' she later said. Sadly, the marriage didn't last, as Ava pursued other, younger, lovers while on filming locations in Europe when Frank was working in Hollywood. It broke his heart, it really did.
In the early 1990s, Tina Sinatra, Frank's daughter from his first marriage, produced a TV movie about her father. There was obviously still a feeling of great bitterness over her parents' split, as Tina chose the most beautiful actress she could find to play Nancy but when it came to casting someone to play Ava – the greatest Hollywood beauty of all – the part went to a rather plain-looking actress.
A very dear friend from my earliest acting days was Dinah Sheridan. Dinah's parents ran a photographic studio, Studio Lisa, in Welwyn, where I used to do some of my modelling work, and I'll forever remember her for giving me a lift back to London in her car after a photographic assignment – and saving me the valuable train fare. Sadly, I never had the chance to actually work with her, as in the 1950s she married John Davis, the feared head of the Rank Organisation, and he forbade her ever to act again as, 'no wife of his should work'.
Dinah told me that on their wedding day Davis said to her, 'I can't remember if you're the third or fourth, but I'm sure you won't be the last of my wives.' It surprised few of us that she later filed for divorce – and was granted one, incredibly swiftly – on the grounds of 'cruelty'. Happily, she later returned to acting and made one of my favourite films, _The Railway Children_ , for Bryan Forbes at ABPC in Elstree in 1970.
It was Lionel Jeffries who brought E. Nesbit's acclaimed book to Bryan's attention and said he'd adapted it as a screenplay; Bryan read it and said he'd love to make a film version. 'But who should we approach to direct?' he asked.
'Well, actually ...' offered Lionel, 'I really rather fancied directing it myself.'
Bryan readily agreed and the result was one of the finest British films ever made.
Dinah's daughter, Jenny Hanley, followed in her footsteps as an actress and in fact became a Bond Girl in _On Her Majesty's Secret Service_ with the other fella. My favourite story from Jenny's career is actually from when she co-starred with my old sparring partner Christopher Lee in _Scars of Dracula_ , which at one point called for a blood-sucking bat to swoop down and remove her crucifix necklace in order that the evil Count could sink his teeth into her neck. Of course, back in 1970, there wasn't any CGI and so the bat was brought to life as a model, which was operated by two prop men who were very much an item – and extremely camp with it.
For the wide shots the bat was suspended on a wire and had to swoop down towards Jenny but the problems started when it swooped a little too low and bounced off her rather ample bosom. After a couple more 'bouncing takes' the whole cast and crew fell about in a terrible fit of giggles.
My modelling days took me from Dinah Sheridan's parents' photographic studio and a selection of tank tops to the 'hotspots' of the British seaside. You may not be able to see them, but I bet I'm covered in goosebumps.
'This prompted Sir Christopher Lee to walk on set and tell everyone they ought to take the film more seriously as, after all, Vlad the Impaler, upon whom Dracula was based, had been a real person and a little more gravitas – as befitting this noble figure – was called for,' said Jenny.
Of course, Christopher's intervention only served to make everyone start giggling further, though they eventually managed to get the take, and moved on to the close-ups.
'This meant the two prop men – who were dressed identically by the way – bringing the bat down to my eye level,' Jenny told me. 'One operated the wings and the other, with his arms wrapped around his partner, reached inside it to operate the mouth. They then started bickering about how fast the wings should move versus how fast the mouth should snap, and I fell about in a fit of uncontrollable hysterics.
"That's far too fast on the wings, dear!" said one.
"No it's not! You're too slow on the mouth, dear. Move it faster – and squirt the blood!" suggested the other.
"I'm not squirting _any_ blood until you get the wings at the right speed."
"God you're _such_ a bloody diva!" It was impossible to take any of it seriously, let alone look terrified as a damsel in distress,' Jenny concluded.
Towards the end of my tenure as Simon Templar at ABPC Studios in Borehamwood in 1968, Hammer Films moved in on an adjacent stage with a film version of the hit West End play _The Anniversary_. It starred Hollywood _grand dame_ Bette Davis, along with Sheila Hancock and James Cossins in support.
Bette Davis was a formidable actress and a formidable force in the movie business. In the 1930s, for example, she took on the Hollywood Studios, accusing them of 'slavery' and saying they only ever offered her mediocre films to star in. She wasn't to be messed with, and producers feared crossing her.
A very talented, award-winning young director named Alvin Rakoff was signed (with whom I happily worked myself the following year on _Crossplot_ ) to helm this new Hammer production, and the British cast were told that Miss Davis was to command their utmost respect, but they were not to approach her directly on set. Furthermore, on her first day they were given instructions to gather around and applaud the star as she made her entrance.
Within a few days it was clear Bette was not only standoffish with her co-stars but was even unwilling to engage in any form of dialogue with Alvin. 'She was above taking or talking about direction of any kind,' he said.
She had the producers fire Alvin, very unceremoniously, after a week and hired in Roy Ward Baker to replace him. Shortly afterwards, the Director of Photography was fired after Miss Davis accused him of not lighting her properly, and she subsequently gave her own specific instructions on where lights should be placed.
Sheila Hancock had the dressing room next door to Miss Davis' and she was able to hear the conversations through the radiator pipes and – almost on a daily basis – heard whom Miss Davis demanded be sacked next.
A decade later, Bette Davis landed at Pinewood Studios to make _Death on the Nile_ and her reputation certainly went before her. When a call was placed to production designer Peter Murton (who happily designed _The Man with the GoldenGun_ starring yours truly) saying, 'Miss Davis would like to see him', it was an ashen-faced Peter who turned to his assistant, Terry Ackland-Snow, and said, 'Terry, you've worked with Bette Davis before ...' Terry took a step backwards, wary of what his boss was going to say next, '... so be a dear and go see what she wants.'
'Well, she did ask for you, Peter,' Terry reasoned.
'Yes, but tell her I'm out looking at locations. Go on, you go.'
A nervous young Terry walked to the set and reported to the star.
'Are you Peter Murton?' she asked. Terry apologized that he wasn't and that Peter was out on location.
'They tell me he designed this set?'
'Yes, he did, Ma'am.'
'It is quite the most wonderful set I've ever been on, so please thank him.'
Terry returned to the Art Department, where his nervous boss asked, 'Well? What did she want?'
'She loves the set!' exclaimed Terry. Before he could say another word, Peter had grabbed his jacket and as he was halfway out the door he turned back to say, 'I'd best get over there in that case!'
But Bette had always been feisty – it wasn't something that came with great age or experience. In her very first film, back in 1931, Bette starred with Humphrey Bogart in _Bad Sister_. They were on set one day when someone screamed out, 'Move that broad to the other side of the room!'
Well, our Bette wasn't having any of that! 'Don't you ever call me a broad again!' she declared, deeply insulted. Alas it turned out that a 'broad' is one of the biggest lights on set and they hadn't been referring to her at all. But you can bet all the crew knew not to mess with Miss Davis!
I remember they'd just opened a new restaurant at ABPC Studios in around 1964-5 and Bette Davis, who was filming _The Nanny_ at the studio, came in one lunchtime and, to my great surprise, made a beeline for my table.
'I'm Bette Davis,' she said, as though she needed any introduction, and went on to tell me how she and her daughter loved watching _The Saint_ on TV. My head swelled greatly, so much so that I've not got enough hair to cover it now, and a friendship was formed.
I invited Bette to join Robert Wagner and I to the dog races at White City Stadium in west London – and she loved it. The idea of being able to dine, place bets and watch races every fifteen minutes, from our table, was like manna from heaven to Bette. She did rather well financially too. The dining room was in 'sections', with tables cordoned off from each other, and everybody was so preoccupied with looking at their racing forms, eating and placing bets that they never paid much attention to us, which I think Bette enjoyed hugely. She could be anonymous for once.
Though as sweet and scintillating a dinner guest as she was, I also saw the other side of her character when she spoke to her assistant, snapping orders and instructions as though he were some lesser breed of mortal. I certainly wouldn't have wished to get on the wrong side of her.
For as long as I can remember, rumours of an intense rivalry between Bette Davis and Joan Crawford abounded in Hollywood. I, like many others, put it down to their brilliant acting in _What Ever Happened to Baby Jane?_ , but no, there seems to have been a more sinister loathing that extended beyond the screen, although both denied it.
During the filming of _Baby Jane_ , Bette, the antagonist in the movie, was said to have actually made contact – in no uncertain terms – with Crawford during fight scenes, after which medical attention and stitches were required.
It turned out that the reason for their lifelong hatred was the fact that Crawford's second husband, Franchot Tone, was revealed to be the one true love of Bette Davis's life. Davis had worked with Tone in 1935 on the film _Dangerous_. Complicating matters a bit further was Crawford's bisexuality, and her declaration that although her husband wasn't interested in Bette, she 'wouldn't mind giving her a poke'.
They constantly tried to upstage and upset one another, from making back-handed compliments at The Oscars, 'Dear Bette, what a lovely frock', said Crawford when Bette was announced the winner of the best actress award, to when gossip columnist Louella Parsons first rumoured that they might star in a film together, 'When hell freezes over', said Bette.
Years later, when Tone was struck down by cancer, Crawford – though by then divorced from him – took him in to her New York apartment and nursed him until his death.
'Even when the poor bastard was dying, that bitch wouldn't let him go,' Bette said to the press. 'She had to monopolize him even in death.'
After Crawford died, Bette continued to rant about her. When asked why, she replied, 'Just because a person is dead, doesn't mean they've changed.'
Bette gave one of her last TV interviews to Gloria Hunniford, while she was in London to promote a new book. Gloria – quite naturally – spent the first part of the interview paying homage to the actress, her films, roles and co-stars.
Bette listened for a while before berating Gloria with, 'When are you going to ask about my book, that's the reason I'm here, isn't it?'
We've all thought it, but she said it. What a lady!
Another wonderful actress and feisty lady was Lana Turner. I wrote about Lana in my autobiography, about the time in the early 1950s when she taught me how to kiss on the set of the movie _Diane._ I actually thought my technique was pretty good – I had already been married twice and hadn't had many complaints in that department – but Lana taught me the new technique of 'passion without pressure' – what a lady she was! Of course, when she came to make _Diane_ , Lana was already a huge Hollywood star with lots of classic films to her name – not to mention several husbands and lovers. However, I will also forever remember her for the day she told our producer on the film, Edwin Knopf, to 'fuck off', after a seemingly trivial difference of opinion on set. In fact, Eddie was so upset that he stormed directly off the stage and into my trailer, where he was sitting, pink-eyed, when I returned a short while later.
'I've known Lana since she first walked onto this lot as a young girl,' he said to me. 'And now she speaks to me like that, in front of the whole cast and crew!'
I returned to the set and asked Lana why she'd been so rotten to Eddie who was, as everyone who knew him will attest, a lovely guy. He'd also overcome disability, leaving him with only one arm, which endeared him to everyone even more.
'Sweetheart,' she replied, matter-of-factly. 'When I first came on this lot all the producers fucked me. So now _I'm_ fucking _them_.'
In 1962, Lana was making a film at MGM British Studios in Borehamwood. It was at the time of the really dense fogs – known as pea-soupers – that used to descend over London. Lana was married to Lex Barker at the time and he was visiting her on set one day when one of these fogs came down. The production called time early to enable everyone to get home safely.
Lana and Lex left the studio in a chauffeur-driven car to head back to the Savoy Hotel in London, but a few miles down the road – at Apex Corner – the driver could barely see the front of the car, let alone anything else on the road ahead of him, and Lex, being the concerned husband he was, got out and said he would walk in front of the car until either the fog lifted or they reached the hotel.
It must be about thirteen miles from there to the Savoy, but Lex dutifully paced it out and lead his wife to safety – or at least he thought so, until he went to open the door of the car and discovered it was another car all together that had somehow started following him and Lana was nowhere to be seen!
And speaking of feisty ladies, they don't come much feistier – or more fun – than Joan Collins. Some time ago in Hollywood, Joan was having a romance with Arthur Loew Jr of the cinematic dynasty (not of the _Dad's Army_ dynasty, as some newspapers mistakenly reported in saying she was the girlfriend of the former Captain Mainwaring).
Anyhow, one night Joan was late for a ball in Hollywood, which she was attending with Arthur. Arthur had been brooding about her punctuality for some time, and this time he snapped, 'You are fucking boring!'
Two Mavericks and a Dynasty. With James Garner and Joan Collins – a veritable rose between two thorns – at an LA premiere in 1983.
'And you are a boring fuck!' snapped Joan, without a blink.
Joan has a lovely turn of phrase, as does her sister Jackie, and I love reading their books and articles. Their father, Joe Collins, was a big theatrical agent in the 1950s and was, in fact, my wife Dorothy Squires' agent, and I got to know him well. He was a very dashing, handsome man-about-town and Elsa, his wife, was a very graceful, classically beautiful woman who believed implicitly in Joan's innocence. When Joan announced to Elsa that she wanted to marry Maxwell Reed, the man to whom she'd lost her virginity, Elsa said, 'Darling, he's an _actor_ – and a spivvy sort of actor at that.'
Maxwell Reed was six foot four and wore jackets with enormous shoulders in them, a trilby pulled down over his eyes and looked every bit the gangster. He boasted of connections in the London underworld too. But in total contrast to his large frame was his high pitched and squeaky voice.
He had been offered a film contract with Alexander Korda in the mid-40s but was sent for voice coaching first as they wanted him to sound more like he looked – big, gruff and mean. When he returned with his squeaky shrill intact, Korda dropped him. It was then someone suggested he see a lady named Elsie who taught voice production, and she told Max that if he spoke with his chin pointing downwards, then he'd produce deep, round sounds.
I was in the Army at the time I next bumped into Max in London. He was wearing an over-sized camel-hair coat, with a script under his arm and with his chin facing downwards he said, 'Hello, old man, how are you?'
He told me – in deep, rounded tones – that he was now under contract to Sydney Box and was on his way to a script conference, and just then he lifted his hand up – along with his head – to call for a taxi ... and all of a sudden his deep, rounded tones became a very high-pitched shriek again.
I digress. Joan told her mother that she planned to marry Max and Elsa said, 'But Daddy won't allow that.'
'Then I will live with him,' Joan replied, with a flourish.
Elsa told me that she never doubted it, and so, reluctantly, that's why they agreed to the marriage.
Joan told me years later that one day, Max called out from the bathroom of the flat they'd rented, 'Joan! Have you been using my fucking mascara?'
'No, I haven't!' came the reply.
'Yes, you have!'
'How do you know?' she asked.
'Because you always spit in yours, whereas I put it under the tap!'
Sadly, the marriage was doomed from the start and ended in bitter divorce a few years later; ironically just as Joan's career was taking off and her earning power was on the up. Max demanded a hefty settlement, claiming he had 'discovered' Joan. Having now been married five times, Joan says that she's kissed enough frogs to have finally found her true prince in Percy Gibson, whom she married in 2002. They're incredibly happy together and complement one another perfectly.
Joan was aged just twenty-one when she was offered a seven-year contract with 20th Century Fox in California. Darryl F. Zanuck, the rather tiny though incredibly randy boss of the studio, had seen her in _Land of the Pharaohs_ , an epic set in the land of the pyramids, in which British stalwart Jack Hawkins played Pharaoh Khufu – inspired casting!
Zanuck had a reputation for propositioning virtually every actress who crossed his path and he was struck by the image of a semi-clad Joan sporting a diamond in her naval. Sure enough, when he bumped into Joan in the corridor one day he pressed her up against the wall.
'You haven't had anyone till you've had me,' he said. 'I've got the biggest and best and I can go all night.'
Joan sensibly declined his kind invitation, though she did catch sight of what she had missed when she visited his office, as Zanuck had a life-sized mould of his manhood – in solid gold – in pride of place on his desk. I've never asked Joan whether the sight of it impressed her or otherwise, but do know that another famous Joan – Joan Crawford – was in his office on one occasion and Zanuck – gesturing to his mould – said, 'Impressive, huh?'
Without missing a beat, Joan Crawford replied, 'I've seen bigger things crawl out of cabbages.'
Zsa Zsa Gabor is perhaps better known for the number of her marriages rather than anything else, and I was once coupled with her by MGM – albeit platonically. It was the studio's habit of partnering their contract artists with each other to attend events, premieres and dinners, purely for publicity purposes. I accompanied Zsa Zsa to one such premiere, and on to dinner afterwards. She was exquisitely beautiful, if a little large in the lower rear region I felt – well, not literally felt, you understand.
At one point Zsa Zsa was married to George Sanders, he was husband number three of nine I think, but she was also having a great affair with Rubirosa (aka Mr Ever Ready). Porfirio Rubirosa was a Dominican diplomat whose reputation as a playboy far exceeded any political accomplishments and was only matched by stories of his sexual prowess. His larger-than-average penis actually inspired restaurant waiters to name the gigantic pepper mills 'Rubirosas'. Many women, and some men, have assured me he was indeed built like a stallion, and his penchant for rich women saw him marry heiresses Doris Duke and Barbara Hutton among three other wives.
George was obviously aware of something going on between his wife and the playboy and returned home one day, just before Christmas, propped a ladder up against the bedroom window and caught the duo in mid-service. The ensuing flash of a camera bulb quite put Rubirosa off his stroke, and there was a mad scramble out of the bed as George gently descended his ladder, and let himself in through the front door to wait at the foot of the stairs.
Zsa Zsa and Rubirosa sheepishly descended.
'Merry Christmas, Zsa Zsa ... and to you Rubi,' he said in his deliciously wonderful sardonic voice, before leaving. They divorced the following April, and Rubirosa continued his womanizing ways elsewhere.
Zsa Zsa had followed her younger sister Eva to Hollywood, and it was Eva I knew better, having worked with her in _The Last Time I Saw Paris_. I was having a cup of coffee with her in her trailer one day, between set-ups, and Bill Shanks the First Assistant Director appeared and said, 'Eva, you're in the next shot.'
'Oh my goodness,' she said, leaping up and taking off a diamond ring the size of a baseball. 'I didn't have this on in the last shot, Bill. Would you look after it for me?'
'I'll put it in my trouser pocket,' Bill suggested. 'Is it worth much?'
'About $50,000,' replied Eva.
'Oh my god!' shouted Bill. 'Someone will cut my goddamn leg off for it!'
'Don't worry, dahlin',' she replied. 'It was only two nights' hard work.'
That was the difference between her and her sister – Zsa Zsa would have said it was 'only one night's hard work'.
Eva was very down to earth and nothing really fazed her. One day, while on the road publicizing a film, Eva was staying in a fairly grand hotel suite that had an interconnecting door with her publicist's room. They were due to appear at a television studio, so the publicist knocked on said door at the designated hour, entered and stood patiently waiting for Eva in her sitting room. Moments later, Eva, having thought she heard something, walked into the room absolutely naked apart from multiple layers of jewels.
Without missing a beat she spread her arms, gave a twirl and said, 'Well, Jeffrey? How do I look?'
Shelley Winters always had a great reputation for being good fun on set. We made a film together called _That Lucky Touch_ in Belgium in 1975, and late one evening in the depths of winter we were preparing for a night shoot. The set-up was that I was to be filmed hanging around outside on a window ledge and then had to go off to a field somewhere – the details escape me but it was bloody cold. Consequently, my wardrobe man had procured all manner of thermal underwear for me. At one point, Shelley walked in to my dressing room to discuss something, and noticed all my long johns and vests hanging over the chair.
'What are all those?' she asked.
'My warm underwear,' I replied.
'If it's going to be cold, then I want some as well,' she said, and picked up a selection of mine.
In the film, Shelley was playing Diana, the brassy wife of the American General (played by Lee J. Cobb) and she certainly stole every scene she was in – along with making an indelible imprint on my memory when she alighted, in character, from the General's car on our location wearing a lovely, warm fur coat. Just as she stepped out of the car she flashed at us – wearing nothing underneath apart from rolls of ample flesh all held in place by my white thermals.
Shelley Winters eyes up my socks – after all, she'd already taken my long johns ... At the Garrick Club in London for a party after filming _That Lucky Touch_ in 1975, with ( _front row, l to r_ ) Shelley, Lee J. Cobb, Susannah York; ( _back row_ ) Jean-Pierre Cassel, Sydne Rome and Raf Vallone.
Shelley had a great sense of humour. One evening we were shooting in a chateau and while we were waiting for everything to be set up, Lee and half the crew – and yours truly – were playing poker. Lee was a great card enthusiast. Enter Shelley Winters, sweet-faced and innocent.
'Oh! What's this game?' she asked, in wonderment.
'Poker,' Lee replied.
'Oh, I think I played that once. May I join in?' she asked with bashful sweetness.
Lee beckoned her to pull up a chair, and within thirty minutes she'd cleaned us all out! We knew never to play with her again.
A few years later, an up-and-coming young director, no doubt still wet behind the ears, was considering Shelley for a role in a film and asked her to audition. Now, you don't ask stars of Shelley's calibre to audition: you _invite_ them to lunch to _discuss_ a role, but you _don't_ ask them to come in and read! If anybody had suggested that to me, I'd have told them where to shove their script. But Shelley loved to work and – somewhat surprisingly to those around her – agreed to meet the director at his office and run through some lines. She duly reported, but arrived carrying an enormous bag over one shoulder.
The director gave the usual flannel about being delighted she had come in to read, and how he'd heard nothing but great things about her. He suggested they go through a scene but as Shelley sat down, she opened her bag, rummaged around in it for a bit, pulled out an Oscar statuette and put it down on the desk. Then she rummaged around again, and pulled out a second Oscar statuette.
'So,' she asked. 'Do I still need to audition?'
Diana Dors was perhaps the Rank Organisation's most glamorous blonde bombshell in the 1950s and 60s – and often regarded as the British Marilyn Monroe. She told the most hilarious story of returning to her hometown of Swindon to open a local fair, where the Mayor was due to introduce her to the gathered crowds but was conscious of not messing up his welcoming speech, in which he intended to refer to Diana by her birth name of Diana Fluck. He didn't want to fluck it up, you see.
The fair was about to commence and the rather nervous dignitary took to the podium. 'Ladies and gentlemen,' he began. 'Today we are joined by a star of the big screen – and someone we are very proud to say was born in Swindon. You know her today as Diana Dors, but Swindon knows her better as – Diana Clunt!'
Diana wet herself with laughter.
Oh, and another story about Diana Dors was related to me by my old neighbour from Denham, Jess Conrad. Occasionally Jess would accompany Di to cabaret functions, as she liked to have someone to present her with a big bouquet after her act – as well as help ensure the money was paid up front. 'Always get the money when you arrive,' she told Jess, 'as afterwards you're introduced to friends, family and the champagne comes out ... and everyone forgets about the business.'
Anyhow, this one particular evening they arrived at a club and were shown into the manager's office and after the usual, 'Hello ... what a thrill it is ...' etc., the manager showed them his prized plant. Well it wasn't so much a plant, Jess said, as a triffid-like vine, and he proudly described how rare it was, how unusual that one should survive in such a climate and so on.
'Lovely,' said Di, feigning interest. 'But shall we do the business side of the deal, darling?'
After paying the money over, the manager said his office was to be Di's dressing room and that she should come and go as she liked. There was just one small snag, which Di hadn't realized until a few minutes before she was due to go on stage: there was no en suite bathroom. Come the time that she _did_ realize, in her full outfit, made-up and looking a million dollars, she suddenly also realized that she was desperate to gain some relief, but didn't want to have to walk through the assembled crowd to go to the loo – what would that do to her big entrance a couple of minutes later?
Ah, dear Diana Dors, a bundle of fun and a force to be reckoned with. I look a little concerned that the moose is going to take off, while Carol Hawkins looks on, bemused.
'Well, what can we do?' asked Jess, in a panic. 'They're all standing outside the door waiting for you.'
Di looked around the room and spotted the plant. I won't go into the detail but I'm sure you know where this story is going ...
'And you thought a horse could pee!' laughed Jess to me some time later.
After her cabaret, Di went back into the manager's office for a glass of champagne but there was a bit of a kerfuffle as she found the manager almost in tears, leaning over his prized plant, which was no longer growing vertically but was lying, lifeless, horizontal across the floor.
'Oh, we won't stay, Jess,' said Di matter-of-factly. 'We've got a long drive home.' And with that they made their escape – they were in hysterics all the way!
I can't resist a toilet story, if you'll forgive me for dwelling in the smallest room for a moment more, and this one involves Tallulah Bankhead, a hugely successful American actress whose fame was such that, for example, she was the first choice to play Scarlett O'Hara in _Gone with the Wind_ (only for her thirty-six years to appear a few too many when the decision was made to switch from black-and-white to glorious Technicolor, and instead the role went to Vivien Leigh, a mere decade younger ...).
Anyhow, fame aside, Tallulah was notorious for being mean with money and the story goes that she was in a lavatory in Hollywood and discovered there was no paper. With that she knocked on the wall of the next cubicle and pushed under a $10 bill.
'Can you split that for two fives?' she asked her neighbouring occupant.
Actually, Tallulah was deported from Britain in the early 1930s, reportedly after having worked her way through most of the boys – and many of the Masters – at Eton public school, and Scotland Yard declared her a menace.
Incidentally, when the opening night of the London musical _Gone with the Wind_ (which starred June Ritchie and not Miss Bankhead, but don't let that get in the way of me telling a good story) was marred by an obnoxious young actress and a horse that relieved itself onstage, Noel Coward was in the audience and was heard to say, 'If they'd stuffed the child's head up the horse's arse, they would have solved two problems at once.' He did have such a way with words!
As I started this chapter with a feisty – and fun – princess, I think it only right that I should end it with another one ... And you know how I like to drop the odd royal name here and there, when I can.
I first met Princess Lilian of Sweden on a visit to Stockholm for UNICEF, my first visit to the country, in fact, and Ingvar Hjartso, my liaison and contact in Sweden, had arranged a visit to the Royal Palace. The King and Queen were away at the time, so left Princess Lilian to meet with me. I discovered that she had in fact been born in Wales and had been a model, at one point married to actor Ivan Craig. She met Prince Bertil of Sweden when she was in her twenties and they fell in love, but it was many years before they were given permission to marry. Prince Bertil's elder brother, the future King, had died very young, when his son and heir was only one year old, meaning if the reigning monarch died before the child, Carl Gustav, came of age, Bertil would have to assume the role of Prince Regent – and him being married to a commoner, and a divorcee, was not something the constitution would allow.
With Princess Lilian, the Duchess of Halland, a wonderful lady and a great friend.
However, when Carl Gustav did come of age and ascended directly to the throne, he granted Bertil and Lilian permission to marry in 1976.
Princess Lilian was greatly loved by the Swedish nation and deservedly so as she had a wonderful sense of fun, as well as duty, as I discovered when we went to lunch at a restaurant in the old town. I must admit that I sat rather stiffly for the first ten minutes, until the Princess pointed at my wine glass and said, 'Will you hurry up and bloody well skol me as a lady can't drink in this country until she is skoled!'
We became firm friends, and my wife Kristina also knew Princess Lilian through her oldest friend Ewa Wretman, who was married to the great Swedish cook Tore Wretman. Ewa, Tore and the Princess spent many happy times together at their holiday homes in the South of France.
And while I'm 'princess name-dropping', with Princess Anne at a Wildlife Fund Gala in 1970.
Tore Wretman, by the way, became a great friend – and was another person with a fascinating backstory. He began his career in the kitchen at the age of sixteen as an apprentice at the Hotel Continental in Stockholm; he swiftly moved to positions at the Opera Bar in Stockholm and then Maxim's in Paris, where he learned all about French cuisine under legendary chef Louis Barth.
When war came Tore spent a few years in the United States where, in 1941, he signed on a Finnish cargo ship for the return trip home. However, the ship was boarded by the British fleet near Iceland and Tore was taken to the Orkney Islands then on to London until 1943, when he was finally able to return to Sweden and a job as head waiter at Operakällaren – without doubt the finest, and my favourite, restaurant in Stockholm. In 1945, aged only twenty-nine, Tore was able to buy his own restaurant and, later, went on to take over Operakällaren. He became the favoured chef of the Royal Family and, in particular, Princess Lilian.
Whenever Kristina and I were in Stockholm we would meet the Princess for tea and would also enjoy many dinners together. Sadly, the last few years of her ninety-seven-year life were complicated by illness and she was forced to withdraw from public life. We were unable to attend Princess Lilian's funeral in 2013, despite the Swedish press reporting we were there, but we were able to attend her memorial service in the September, where we all shared our many immensely happy and fond memories of our times with the Princess.
With my two bodyguards on my spiritual – and literal – home turf: Pinewood Studios, where I've kept an office for over forty years.
CHAPTER 2
The Pinewood Years
LONG BEFORE HOLLYWOOD BECKONED ME, I FOUND MYSELF auditioning for a film at London's Pinewood Studios. Little did I realize that it would be my home three decades later for a major TV series and seven Bond films. But before I get to that, I thought it might be an idea to start where I started – and that was at the very epicentre of the British film industry: Wardour Street in London's Soho district, though admittedly my first cartoon filler-in job was on D'Arblay Street just around the corner.
It was such an exciting experience for a cinema buff like me to walk along seeing all the familiar logos of the major film companies, including ABPC (at Film House, 142 Wardour Street), Rank (at 127), British Lion, Paramount, Hammer (at Hammer House 113-117), Columbia, Warner-Pathé and others that were all congregated on this one magical road. Wardour Street was named after Sir Archibald Wardour, the architect of many of its buildings, though along with all the famous film interests it also had its share of, shall we say, more 'dubious' operators in the area and this caused the street to be known – even on the sunniest of days – as 'shady on both sides'.
It was to here that hopeful producers ventured with scripts firmly tucked under their arms, would-be directors wooed film chiefs over lunch, and some aspiring actors even attended auditions.
Many of the aforementioned companies also, at one point or another, controlled the film studios where yours truly hankered to work – Rank owned Pinewood and Denham, ABPC owned Elstree, British Lion controlled Shepperton, and Hammer were out at Bray. While there were smaller concerns at Beaconsfield, Ealing and Southall, it was the larger studios that offered the most tantalizing prizes – Pinewood being the largest of all.
In 1947 the view across the fields on the approach road to Pinewood was broken only by a cluster of tall pine trees, and then, as if from nowhere, appeared the mock-Tudor double-lodge entrance, and a friendly commissionaire. It was just like arriving at a stately home.
At that time I was a rather green twenty-year-old lieutenant serving in the Combined Services Entertainment Unit and being tested for the male lead in _The Blue Lagoon_. It marked the beginning of my long association with the studio and now aged eighty-six I am Pinewood's oldest (and longest-serving) resident, as I moved in to my office there during 1970, when I began work on the TV series _The Persuaders!_ and I've been paying rent ever since.
An early publicity shot from MGM.
British film mogul J. Arthur Rank opened his Pinewood Studios in 1936 as his dream rival to Hollywood; the final syllable of which, plus the abundance of pine trees on the 100-acre site, gave him the name. When I first turned up, the studio hadn't long reopened after being used during the war as a base for the army, RAF and Crown Film Units making documentaries. One stage was also requisitioned by the Royal Mint – some say that was the first time that Pinewood had made any money.
Just before my arrival, great film-makers were hard at work – names such as David Lean, Michael Powell and Emeric Pressburger, Ronald Neame, Frank Launder and Sidney Gilliat – making some of the country's greatest films, including _Great Expectations, The Red Shoes, Oliver Twist_ and _Black Narcissus._ Even as a young studio, Pinewood had an outstanding reputation.
With Bob Baker and Monty Berman, producers on _The Saint_. Flat feet, boys? A likely story!
During the war years, my two future _Saint_ producers, Monty Berman and Bob Baker, were both sergeants in the Army Film Unit stationed at the studio. Being the young tearaways they undoubtedly were, they'd found a way in under the wire, and used it for coming back after late-night shenanigans in the hotspots of Iver Heath. However, one night, just before breakfast, actually, they were caught midway under the wire, and were hauled up in front of the adjutant.
'What's your story?' he asked Monty.
'Well, I missed the last bus, sir,' Monty replied, 'and had to wait for the first one this morning.'
'Why didn't you walk?'
'I have flat feet, sir, so I can't,' Monty added.
Bob was then brought in.
'Why were you back so late?'
'I missed the last bus, sir.'
'Why didn't you walk? Have you got flat feet too?'
'No, sir, but my friend Monty has ... and I couldn't leave him on his own.'
They were both stopped a week's pay.
Bob, I should add, was the first allied cameraman in the ruined Reich Chancellery after Berlin fell and was a formidable cameraman as well as a hugely talented producer and director.
For my first screen test, I was led from the grandeur of Heatherden Hall, which formed the centre of the studio lot, through long clinical corridors across to one of the five stages – a huge, dark, soundproofed room with a smell of greasepaint, make-up and burning filters on the giant lamps. Soon it was my turn to step under the lights and in front of the cameras. Even though I didn't get the part, I was thrilled just to be there.
Later, I learned that I had been recommended as a possible 'contract artiste' for the studio's Company of Youth, more often referred to as 'the Rank Charm School'. Now unheard of in the modern industry, the studio had then established a stable of aspiring talent, producing its own stars of the future: Christopher Lee, Joan Collins, Anthony Steel, Diana Dors, Donald Sinden, Kenneth More and Petula Clark were all under contract.
Sadly, for me, it was at a time when John Davis, the much-feared company MD, was dealing with a £16 million overdraft. They consequently weren't interested in a young Roger Moore being added to the roster and ever-increasing wage bills. So while I mixed socially with my Rank contemporaries, I had to slip off to earn a crust elsewhere, but I always dreamed of returning to the wonderful film factory in the Buckinghamshire countryside.
Meanwhile, over at Shepperton Studios, where I occasionally auditioned for bit parts, Hungarian filmmaker Alexander Korda was busy building his empire, and while prudence was the watchword at Pinewood, extravagance was the order of the day at the rival studio, where the charming movie mogul began an impressive production programme: _The Third Man, The Fallen Idol, Anna Karenina, The Wooden Horse_ being a few of the films I marvelled at over in the Odeon Streatham. Korda – unlike his Methodist rival J. Arthur Rank, who was a reserved and very unlikely film industry magnate – was a great showman who loved and courted publicity. He was also an astute and talented filmmaker in his own right and made his opinions known.
What can I say? Another early publicity pose, this time from Warner Bros.
Guy Hamilton, who directed two of my 007 outings, told me one of his favourite memories of Korda, when he worked as an assistant director to the great man. Korda summoned Guy one Saturday morning, together with one editor, one cameraman and one assistant art director, to view the rough cut of Emeric Pressburger's first and only solo directorial effort, _Twice Upon a Time_. It was obvious that retakes were on the cards. The three Korda brothers (Vincent, Alex and Zoltan) walked in. The lights went out and they watched in silence until the end at which point Alex lit a cigar and addressed the assembled group.
'Boys, I could eat a tin of film trims and shit a better picture.'
Although Korda was based in Piccadilly and rarely ever visited the studios, there was always the 'threat' that he might descend at any time and in the studio restaurant a large round table in the corner was constantly reserved for him.
Just before one of his rare planned visits, studio manager Lew Thorburn had stained the wood that ran the length of the long interconnecting corridors of the main house a lovely shade of red. It really was immaculate. Korda arrived and walked partway down the corridor before turning to Lew and in his thick Hungarian accent said, 'Lew, this corridor smells of cats' piss. Do something about it.'
One of the aforementioned great films backed by Korda, and one of my favourites, was _The Fallen Idol_ directed by Carol Reed, on which Guy Hamilton was his First Assistant Director. The house used in the film, by the way, was the Spanish Embassy in Belgrave Square. One of the supporting cast members was Dora Bryan, who told me she received a call to audition at Shepperton to play the part of a prostitute. Very excited, she took the train out from Waterloo and on arriving at Shepperton Station, realized that the studio was actually a couple of miles away. So she walked across the fields, arriving rather the worse for wear.
Looking exhausted and a little flustered, but in her very best red dress and poshest shoes, she read the lines and Mr Reed offered her the part and told her to report to the studio at eight o'clock on the Friday morning.
'What should I wear, Mr Reed?' asked Dora, wondering just how tarty he envisaged the character.
He looked her up and down and said, 'What you're wearing today is fine.'
Poor Dora! She didn't know whether to feel insulted or not!
Another inimitable actress at that time was Dame Thora Hird who had, by the time Dora and I took our first steps inside a studio, been making a name for herself in films at Ealing Studios – a much smaller, but equally prolific facility as Pinewood. Dame Thora later told a wonderful tale about the mealtimes there.
Prior to the lunch break each day, one of the carpenters or electricians would usually go down to the canteen to get a copy of the typed menu for the day and bring it back to the set for the crew to place their orders. On one particular day, an electrician produced the menu which offered: fried Spam and chips, cold Spam and salad, Rissoles and a couple of other items ... only the capital 'R' on the old typewriter wasn't working correctly and instead printed as a 'P'.
'OK, then,' said the spark to the formidable canteen manageress. 'We'll have three Spam and chips, four Pissoles and chips ...'
'What did you say?' snapped the dinner lady.
'Four Pissholes ...'
'That is an R! An _R_ – did you hear me?' she screamed.
'Oh, sorry,' replied our trusty spark and, without missing a beat, continued, 'We'll have three Spam and chips and four R-soles and chips, please!'
All of the British studios remained busy through the 1940s, but when television became a real threat the government introduced a tax on box-office receipts, which was to be reinvested in British films. Called the Eady Levy, it helped to attract many overseas producers to the British studios, including Walt Disney and my friend Albert R. 'Cubby' Broccoli. Once here, they stayed because, quite frankly, they fell in love with our studios and technicians. That love led to millions of pounds being injected into the UK economy and employment for many, many actors and creative personnel.
As I write, Pinewood is buzzing with activity and the big news is that Disney have moved in again, but this time on a ten-year rental deal bringing with them _Star Wars_ , and the first film in the new series is directed by my friend J.J. Abrams.
I say 'again' because it's not the first time Disney has set up at the studio, as back in 1952 they became the first 'renters' to move in.
Whereas back then British studios had a regular tea trolley visit the stages twice a day, part of the American way of working was to have continual refreshments on set, and Mr Disney was adamant – he wanted hot and cold running drinks all day.
Keen to avoid what he thought would be a mass daily invasion from surrounding stages, Pinewood's Managing Director Kip Herren suggested one of the old brigade of tea ladies, Margaret, would man the station and thereby, after a few days, would recognize the Disney crew, despatching any interlopers with a flea in their ear. All was well and good until one day a tall moustached gentleman in a raincoat asked Margaret for a cup of coffee.
'No you don't,' said Margaret. 'You're not on this production.'
'Oh, but I am, I assure you.'
'I don't think so. I know everyone on this set and you've not been here before,' Margaret continued, as she picked up a copy of the unit list. 'So, come on then, what's your name?'
'I'm Walt Disney,' the man replied with a big smile.
Margaret melted into a corner, but Disney was apparently delighted that his pennies were being looked after so diligently.
Disney's newest employee, J.J. Abrams, came to my aid recently (and I'm so delighted he's achieved great success since giving this old English actor a job as a British spymaster on the long-running ABC TV series _Alias_ ) when Lucasfilm (a division of Disney) moved into the corridor just down from my office. The next thing we knew, their part of the corridor was sealed by security doors through which access could be gained only via a swipe card. Ordinarily I wouldn't have raised an eyebrow, as different productions have all had different security arrangements over the years, but the problem in this instance is that the kitchen and gents' loo are all situated in the inaccessible part of the corridor – meaning no tea, and perhaps even no pee.
Pinewood staff shrugged their shoulders saying 'that's what the client wanted' and didn't offer any real alternative save for using the workmen's lavvy in the other direction, which, to be honest, wasn't somewhere I'd have sent a workman – let alone an international megastar such as myself – to fill a kettle. (They've since refurbished it, I'm pleased to say.)
I dropped a line to J.J. – who was still in LA – asking if we could use the kitchen, and promising that we wouldn't spill any of the secrets of _Star Wars_. The next thing we knew, not only was access granted but an apology came from Pinewood for inconveniencing us. Ah, what it is to have friends in high places.
My book on _The Secrets of Star Wars_ , meanwhile, will be in shops later in the year...
Television became hugely important to me in my career, and in the late 1940s my first, and a very handy, means of earning an extra few quid through the medium came when my agent Gordon Harbud suggested me for some assistant stage management work (as well as acting gigs) at Alexandra Palace.
In doing a little bit of research for this book and _googling_ myself, a certain well-known reference website states that my first TV appearance was in 1950 for _The Drawing Room Detective._
That's not correct, dear readers!
My first tentative steps as a TV actor were taken a full year earlier in 1949 at Alexandra Palace, more fondly known as Ally Pally. The BBC produced most of its early TV programmes at Ally Pally in north London and as such it's often referred to as being the 'birthplace of television'. While I wasn't old enough to be there for the birth itself, back in 1935 when the Corporation leased the building, I do vaguely remember the following year as an excited nine-year-old when it started its broadcast trials; up until then our only mass entertainment was cinema or radio – one of my favourite radio shows being _Educating Archie_ , which was actually a ventriloquist act. A vent act on radio – work that one out!
I saw my first TV pictures on a tiny box with a fuzzy little screen, introduced by Elizabeth Cowell with the words, 'This is direct television from Alexandra Palace...' The local baker was the only person we knew who owned a TV set and it was so exciting to gather around it waiting for the valves to warm up and seeing the picture emerge. Little did I realize that, a decade later, I would be starring in my own show on the box.
The momentous day I turned from viewing to being viewed was 27 March 1949, in a production of _The Governess_ by Patrick Hamilton. It was transmitted live at 8.30 p.m., and I received the grand sum of twenty-three guineas to play the part of 'Bob Drew'. According to the BBC files I was allowed to study recently, the story all took place in Drew's house 'outside London, in the middle of the Victorian epoch'. The plot centred on the kidnapping of my sister, and I had to come in to the drawing room where the police had gathered and say, 'Hello, mother! What's going forward here?' I never understood the line, and am sure viewers were equally perplexed by the Victorian turn of phrase.
Other cast members included Clive Morton, Betty Ann Davies, Joan Harben, Jean Anderson, Dorothy Gordon and Willoughby Gray, whom I vividly remember describing his interest in restoring model soldiers to me, and that he had a whole battalion of them. Almost forty years later, Willoughby starred as Dr Carl Mortner in my last Bond film _A View to a Kill_. You can't keep a good pairing down!
Almost everything was transmitted live because TV budgets didn't extend to the luxury of recording on expensive tape or film, but thankfully we had a couple of weeks' rehearsal to get everything spot on and in this instance we all decamped to the cold and draughty Methodist Hall in Thayer Street, London. Our producer/director was Stephen Harrison, who guided us through the text and explained the various set-ups the camera would move through, stressing how careful we had to be so as not to get in its way, nor to be on the wrong set at the wrong time, as it would simply have spelled disaster for the whole production. No pressure then.
The possibility of an actor drying, a scene shift not working or a camera breakdown was something of which we were all aware, but tried not to think about. Electrical equipment wasn't anywhere near as reliable as it is today and in fact during the technical rehearsal on the morning of transmission we had to wait thirty minutes for a camera fault to be remedied. Mercifully it was all right on the night.
I unearthed a couple of interesting production memos from those BBC files. One was from Evelyn Moore's agent saying that in the _Radio Times_ listing for the show, 'Miss Moore's credit should also state she is now appearing in _The Dark of the Moon_ at the Lyric Hammersmith'. There's nothing like an unashamed plug for one's current project!
Another memo was from the head of programming to our producer, stating, 'Running time is 100 minutes including a one-minute opening and three-minute interval. The budget will be a maximum of £660 for one performance and should include all costs of wardrobe, design, film, sound, artists, script copyright, orchestration, transport, hospitality and photos.'
Even in the hands of the most prudent BBC accountants, I don't think £660 would go very far nowadays.
Of course, beaming into people's living rooms made you 'real characters' in the viewers' eyes, and many blurred that reality with drama. For example, one of my live TV contemporaries was Leonard Rossiter who was later – and most fondly – remembered for playing the miserly landlord Rigsby in _Rising Damp_. In one particular drama he was being examined by a doctor and, while fully trousered, had his shirt off.
'You can get dressed again now,' said the doctor and the dialogue continued while Len buttoned up his shirt and moved on to the next set-up.
Before the programme had ended word came through that somehow his mother, Mrs Rossiter in Liverpool, had been out to a phone box, got through to the BBC – which was no easy task in itself – and then miraculously to the production office to leave a message, 'Len, you never put your vest on!'
Mind you, that blurring continues today, with soap opera characters often being mistaken for the actors who portray them. Mark Eden, who played villain Alan Bradley in _Coronation Street_ , was innocently doing his grocery shopping at a local supermarket when he felt a sudden searing pain across the back of his head. He turned around to see an elderly lady swinging her handbag. 'That's for what you did to Rita!' she exclaimed.
I was in great demand at the BBC and on 24 April 1950 appeared in _The House on the Square_ as 'John'. In fact, there were two performances, the second (or repeat) being four days later. Again, it was all staged at Ally Pally and this time for director Harold Clayton. An hour into the first performance and an electrical breakdown on the stage meant it was time for the infamous _Potter's Wheel_ interlude film to appear on TV screens, along with the words 'Please bear with us while we try to restore your programme'. There were quite a few technical breakdowns in those days and so the _Potter's Wheel_ was pretty well known in its day. Thankfully this time it was only on for a minute and we were able to resume; just as well we were the consummate professionals.
_Drawing Room Detective_ was, in fact, my third BBC drama opus, broadcast on 27 May 1950. In it, I played a part as well as performed the duties of Assistant Stage Manager (ASM) for the grand fee of fifteen guineas. It was a sort of whodunnit, hosted by Leslie Mitchell, in which viewers were invited to guess the person responsible for a crime.
With no further drama casting in the pipeline, I accepted an ASM role on a few episodes of _Lucky Dip_ in June 1950, which was described as being a 'Variety Hopscotch'. I was paid less at seven guineas this time, but it only involved a couple of days' rehearsal at Lime Grove, followed by the live transmission from Ally Pally. It was actually rather fascinating to be part of a variety programme as the thirty minutes featured some regular comics – Duggie Wakefield and Archie Glen – along with a terrific line-up of guest artists including Julie Andrews, George Moon, Benny Lee, Lynette Rae, Jenny Lee, The Great Gingalee and a host of extras. There was also a new TV segment that excited BBC bosses, in which a member of the public chose a tune and Nat Allen's band in the studio had to see if they could play it... If they could rise to the challenge the lucky punter received a prize of two BBC TV show tickets of their choice.
I also lent my ASM credentials to a Caribbean Miscellany called _Bal Creole_ in which Boscoe Holder – brother of Geoffrey Holder, with whom I starred in _Live and Let Die_ – was brought in from New York with his steel band, which, I believe, was the first time the metal dustbin-lid-type drums had ever been seen on British screens.
_The Man Who Haunted Himself_ was one of the first films made at Elstree Studios under the leadership of Bryan Forbes. It's now attracted a bit of a cult following, I'm told.
Being a jobbing actor I was happy to accept anything, but when the mention of a film was made I thought I'd hit the big time. The Automobile Association (AA) made a number of training films and semi-documentaries, and I was drafted in to play a patrolman, along with my old friend Leslie Phillips. I wasn't sure it would lead to my name in lights over the entrance to the Odeon, but it was a start. The exotic location for the twenty-minute epic was a road somewhere on the outskirts of Guildford and while the crew set up the cameras behind a hedgerow on one side, I was to be found across the way, happily leaning on my motorcycle combination, dragging on a cigarette while awaiting my call to duty.
Just then, an old Austin 7 with two rather antiquated ladies in the front pulled up.
'I say, patrolman! Patrolman!' called the driver, with a rather strained upper-crust accent. It took a few moments for it to sink into my thick skull that she meant me and so in my full AA uniform, puffing on my fag, I ambled over.
'Is this the road to Guildford?' she asked.
'Guildford?' I pondered. 'Well, back there I did see a sign that had Guildford on it, but I can't remember in which direction it was pointing ...'
At that moment it obviously dawned on the old dear that there was something rather peculiar about this particular patrolman, as not only did I slouch and not salute (as a real AA man at that time would have), I was also smeared in make-up. Without taking her eyes off me, she feigned a smile, reached down with her left hand and, after a few attempts, ground the gear stick into first and kangarooed off down the road. Sixty-four years later they're probably still circling the outskirts somewhere looking for Guildford.
Meanwhile, back at Pinewood, things were ticking over nicely with British crowd-pleaser films such as the Norman Wisdom comedies, the _Doctor_ series, starring Rank's biggest star, Dirk Bogarde, and (a little later) the _Carry Ons_.
Dirk Bogarde made most of his Pinewood films with Betty Box and Ralph Thomas. Betty, his producer, was married to _Carry On_ producer Peter Rogers and was one of only a couple of women who had held such a role within the film industry, while Ralph – the director – was the brother of _Carry On_ director Gerald Thomas. They were often described as the 'royal family' of the studio. One of the duo's films with Bogarde was _The Wind Cannot Read_ , in which he played a flying officer in World War II who fell in love with a Japanese language instructor. Not a lot of people know that Bogarde had a false tooth in his upper set of gnashers and wore it throughout all of his films, only ever removing it at night when he turned in and placed it on his bedside table. Every morning the first thing he did was slip the tooth back in.
On location, Bogarde was having terrible trouble sleeping and consequently, day by day, looked increasingly more haggard on set. Ralph Thomas was becoming concerned that his leading man was going to look anything but his best, and suggested Bogarde might take a sleeping pill. Bogarde wasn't keen on the idea, but Ralph nevertheless left a couple on his bedside table and suggested if he couldn't get off to sleep after an hour or so, he should take one.
After an hour of tossing and turning, the star finally reached over and took a pill. At three o'clock in the morning, Ralph Thomas was woken by Bogarde banging furiously on his hotel room door. The star told his director that he'd taken his advice, but after getting up to use the bathroom he had noticed that both pills were still next to the bed and, yes you've guessed it, the false tooth wasn't. The next day's schedule called for a number of close-ups of Bogarde and, understandably, panic set in. Half a dozen bottles of castor oil and a jug of disinfectant were sent up to the room ... and yes, they got their close-ups.
Which is your favourite, Simon Templar's Volvo P1800 coupe in _The Saint_...
However, it wasn't until 1961, when Cubby Broccoli and his new producing partner Harry Saltzman wanted to set up a series of spy adventures based on Ian Fleming's hero James Bond, that Pinewood really hit the big time. Typically, I missed out on all that fun as I didn't return to Pinewood until 1970, after hanging up my halo on _The Saint_ at Elstree, when Bob Baker, Johnny Goodman and I set up a new series called _The Persuaders!._
When Tony Curtis's name was mentioned to me by Lew Grade as being one of the three possible co-stars for the 1971 series _The Persuaders!_ (the other two being Glenn Ford and Rock Hudson) I was immediately grabbed by the idea. I thought Tony was a brilliant actor in films like _The Sweet Smell of Success_ , _Trapeze, The Boston Strangler_ and, of course, he showed his comedic skills in _Some Like It Hot_ in which he based his English accent on Cary Grant. Incidentally, when director Billy Wilder later told Cary, he said, 'But I don't talk like that!' in _exactly_ the same way in which Tony had taken him off.
... or Brett Sinclair's Aston Martin DBS in _The Persuaders!_?
Tony had worked with an impressive roster of directors, and while he rarely spoke ill of anyone, he did tell me he had a tough time on _Spartacus_ with director Stanley Kubrick. The whole cast had endured an agonizingly long shoot, and one day Tony turned to co-star Jean Simmons and asked, 'Who do you have to fuck to get off this picture?'
Danny Wilde and Sir Brett Sinclair doing what they did best on _The Persuaders!_. As you might imagine, Tony and I had a great deal of fun on and off set.
As part of the package of luring Tony to make a TV series, Lew Grade bought him a house in Chester Square in London's fashionable Belgravia for an astonishing £49,000 – and it was Tony's to keep. A few years later he sold it for £250,000 and thought he'd made a pretty good deal, but a couple of years after that, when he returned to England and looked at buying a similar property, he discovered the asking price was nearer £2 million.
Throughout the fifteen-month shoot, or at least most of it, Tony's wife Leslie was pregnant. She was already 'well endowed', but was now even more ample of bosom. Tony had always been what you'd call a 'boob man', and so he liked Leslie to wear low-cut dresses that exposed as much as possible. One evening I was at his house for a party and, as his wife walked into the room, Tony said to the assembled throng, 'Look at my darling Leslie, doesn't she have the most wonderful tits?'
In between set-ups one day on set Tony said, 'Dear sweet Roger, Burt Lancaster once told me that if you're in your dressing room at the studio with a young lady and your wife should walk in, continue with what you're doing and when you get home deny it, and say, "But they have people who look like me".'
I was never tempted into such a situation, though a year or two later I found myself filming in South Africa, and in one scene my character was to bed a rather attractive young lady in her apartment, demonstrating his Bond-like masculinity no doubt, and in a very thick Afrikaans accent the young actress said to me, 'I really wish there weren't all these people around', referring, of course, to the crew.
'Oh, why?' I asked innocently.
'Because I could show you a really good time!' I thought about Tony's words for a split second, but that thought soon turned to my (then) Italian wife, who was sitting downstairs and who knew my stunt double was off that day!
_The Persuaders!_ was scheduled for a twelve-month shoot but ended up taking fifteen. You see, not only did Tony like to wander off script and improvise on occasions, meaning we found ourselves taking a little longer to shoot a particular sequence, he had a total aversion to overtime.
The British Trade Unions were all powerful at this time, and the ACTT (Association of Cinematograph Television and Allied Technicians) had very strict rules about working hours, with the only exception being you could 'call the quarter' (an extra fifteen minutes) if the red 'shooting' light was on at 5.30 p.m. and you needed to finish a certain scene.
Tony got wise to this, and would never start a scene after 5.15 p.m.!
We also found it particularly difficult to persuade him to come in for looping. This is the process undertaken at the end of all movies to make sure the soundtrack is consistent in each scene or to add music.
'But why do we need to do it?' Tony – a veteran of twenty years in movies – asked.
'Tony, in the last set-up in the gardens there was a close-up of me talking, and then it cut to you but an aeroplane was flying over at that point. So when we edit it together it'll go from no background noise on me, to noise on you, to no noise on me again ... we need to re-record your dialogue,' I explained.
'Audiences are sophisticated,' he replied in all seriousness. 'They understand these things.'
Knowing he didn't want to stay back after 5.30 p.m., when it was usual to loop any such scenes, I suggested we might do it during our lunchtime for half an hour.
'Okay, okay,' he conceded. 'But I want champagne and smoked salmon in the theatre.'
The next day, he arrived at the theatre in Pinewood and a few moments later Johnny Goodman, our Associate Producer, just happened to walk in.
'Tony!' Johnny shouted. 'Smoked salmon and champagne! You've never had it so good!'
'Goddam son of a bitch!' Tony shouted. 'You've just blown your half hour!' and he stormed out.
Tony was what you might call 'very careful' with his money, to the point you might have actually wondered if his trouser pockets were sewn up. One Christmas, we were shooting _The Persuaders!_ over the Festive period and he invited my (then) wife and I to join him and Leslie for dinner one night. I'll never forget it: he produced the tiniest roast chicken, which, we discovered, was not only to serve us four, but his two household staff as well. Talk about trying to pick the meat off the bones!
Just then a group of carol singers arrived outside the front door and piped up with 'Once In Royal David's City'. At the end of it, they rang the bell in expectation of receiving a little Christmas offering. Tony started waving his arms and screamed like a banshee, 'Get away, get away or I'll call the cops!'
Leslie, who obviously felt pleased at having picked up a little English, said, 'No, Tony! It's Bobbies, honey, _Bobbies_!'
His meanness was demonstrated further at the end of filming, when it is customary for the leading actors to be offered some of the items of clothing they had worn in the production. Tony took absolutely everything, and then held a sale for the crew to come over and buy it all off him!
He next called Johnny Goodman to his dressing room.
'Well, Johnny, we've been working together for fifteen months and I'd like to give you something in appreciation of everything you've done for me.'
Johnny, being somewhat surprised at this out-of-character generosity, thanked Tony profusely – and was duly presented with a bottle of the cheapest sherry you could buy from the local supermarket.
'Now, what shall I write on it?' Tony pondered aloud. 'I know ...' and he scribbled 'Best wishes, Tony'.
Johnny has never opened it, and often stares at it ... in disbelief.
But you couldn't help but love Tony, he was a terrific character and a gift as a co-star.
Meanness is not an attractive trait in actors, and I remember a production accountant telling me he had been working with a certain thespian of Scottish origin and had arranged for his _per diem_ (agreed daily out-of-pocket expenses) to be dropped in to the actor's dressing room each morning.
'Keep hold of it until the end of the week, would you?' the actor said each morning. Then at the end of the week he said, 'I don't need it at present, so keep hold of it until I come and see you.'
A couple of months later, with filming complete, there was a stack of these little brown envelopes in the safe, and together they amounted to a considerable sum. The actor came to collect them with a large briefcase.
The following week bills started arriving from restaurants, theatres, car companies – the actor had charged everything he should have paid with his _per diem_ to the production and pocketed the cash!
Whatever his eccentricities, though, Tony ensured that there was never a dull day when we were making _The Persuaders!_. Over our fifteen-month schedule we filmed in every nook and cranny of Pinewood and the adjoining Black Park.
Cubby Broccoli was a regular in the Pinewood dining room and always made a point of introducing me to his guests. He sat at the large round table – the same one that had been reserved for Emeric Pressburger all those years earlier – where he entertained backers, sponsors, royalty and visiting journalists over sumptuous lunches. It was a magical environment in which to impress visitors and inveigle finance. Stars such as Bette Davis, James Caan, Peter Ustinov, Katharine Hepburn, David Niven, Gregory Peck, Stewart Granger, Richard Burton and Elizabeth Taylor could all be spotted at the tables. Liz would be showing off her latest jewel, and they'd talk about who was doing what next and for how much, or what offers they'd refused, or gossip about who was sleeping with whom, all often punctuated by the unmistakable laugh of Sid James and the _Carry On_ gang on neighbouring tables.
'How many set-ups did you get in this morning?' Sid would shout across, inducing a sort of friendly rivalry to anyone in earshot (he'd no doubt taken side bets on it).Tony Curtis was once prompted to boast that we'd managed 'five'.
The _Carry On_ films were in their early days when this shot was taken in the 1960s, but Barbara Windsor and the gang were always around at Pinewood, and we enjoyed a friendly rivalry in the dining room.
'Oh, we slipped in eight,' Barbara Windsor chuckled back, much to Tony's chagrin.
Meanwhile, the _Carry On_ producer Peter Rogers locked the stage doors at lunchtime to prevent any of his artistes or crew claiming overtime.
Kenneth Williams starred in more of the _Carry On_ films than any other actor, though he never stopped complaining about what he felt were poor scripts, terrible money and co-stars with whom he didn't get along. However, his moaning and bitching aside, he was without doubt one of the funniest raconteurs you could ever meet. When I started on my first book tour in 2008, I was reminded of the story Kenneth told about attending a store signing in Australia – though actually he stole the story from Monica Dickens, to whom it actually happened, but we won't let that detract from my tale.
'Who's it for?' he asked the lady at the front of the queue.
'Emma Chisset.'
He duly signed and handed her the book.
'What's this?' asked the lady, looking at the inscription.
'Your name!' exclaimed Williams.
'No, I didn't say " _Emma Chisset_ ", I asked " _Ha much issit_?".'
Aside from his acting work, and borrowing other people's stories, Kenneth also took great pride in another job within the industry, which not a lot of people know about.
When director Kevin Connor and casting director Allan Foenander set out in search of a young leading lady for _Arabian Adventure_ in the late 1970s, they scoured the length and breadth of Britain without success, but then someone suggested they go and see a young girl at drama school by the name of Emma Samuelson. They were absolutely bowled over by the young actress and Kevin was in no doubt that he'd found his young lead. However, there was a problem: being still in drama school, Emma was not yet a member of the actors' union Equity and as such wouldn't be allowed to work on a British film. Ironically, the rules at the time stated that she would not be able to get her Equity ticket until she had a certain number of paid acting jobs under her belt. It was quite a ridiculous situation, but the union was all-powerful back then. All was not lost though as, in extraordinary circumstances, Equity might consider waiving the rule provided they could be convinced there was no one else suitable for the part.
Kevin, Allan and producer John Dark convened a meeting at Pinewood for the visiting Equity representative, in the knowledge it was really make or break for young Emma's film career – and the picture. They sat solemnly waiting for the rep to arrive ... and in walked Kenneth Williams, with his trademark nostrils flaring.
The trio argued the case for casting Emma, and a stern and very serious Kenneth asked, 'And there is no member of Equity who could play this role?'
'No, we've cast extensively and Emma really is the only one suitable,' replied Kevin Connor.
'Then, in the circumstances, we shall offer Miss Samuelson membership and permission to be contracted.'
Despite taking his union role very seriously, he couldn't help but then let his guard down over tea and biscuits and revert to his outrageous self, telling everyone about his recent 'bum trouble'.
Incidentally, producer John Dark said that the name 'Emma Samuelson' would be too long to appear on cinema marquees and so shortened it to Samms. And so began the career of a wonderful actress.
The first Bond girl, Eunice Gayson, had a similar discussion with her employers. She was born 'Eunice Sargaison' and when she secured her first West End play, the producer – who paid for signage out in front of the theatre by the letter – said she must shorten her name. Funny how these things happen, isn't it? And as for my old mate Maurice Micklewhite, his agent suggested it wasn't the sort of name that tripped off the tongue easily, so young Maurice looked across from the phone box at Leicester Square where he was calling from and saw _The Caine Mutiny_ was playing at the Odeon. Henceforth Michael Caine, film star, was born.
Kenneth Williams, meanwhile, had a habit of flashing his manhood around the Pinewood sets. 'Oh no, Kenny, put it away!' his female co-stars would moan. I think he did it more in an attempt to shock than anything else, although I did hear on one occasion that he complained to producer Peter Rogers that he had 'grazed his penis' on set and was sent off to see the nurse. Twenty minutes later, and with his director and co-stars waiting, there was still no sign of Williams, so Peter marched down to the nurse's office, opened the door and saw him lying on the table naked, sighing gently, while the nurse was massaging his tool with a handful of ointment.
'I'll be there in a few minutes,' replied the star, as Peter almost wet himself with laughter.
On another occasion, at a party thrown by comedy actress Betty Marsden, the hostess said to Williams, 'Now, Kenneth, are you behaving yourself?'
'Is my cock hanging out?' he asked.
'No,' she said, cautiously.
'Well then, I must be!' said Kenny.
Kenneth – unlike me, of course – took great pleasure in talking about his various ailments whenever he was on the chat show circuit, such as the occasion he was in his theatre dressing room, washing his rear end in a pot, when Noel Coward sprang through the door to congratulate him on his performance. Kenneth apologized for his appearance, and said that following his last operation, the doctor had advised that he should bathe rather than use paper after visiting the loo.
'My dear boy,' replied the Master, 'you have no need to explain. I had that very operation and know just how painful piles can be.'
'Piles! Oh nooooooo,' protested Williams, 'I don't have them! I have pipilles.'
Without missing a beat, Coward replied, 'Pipilles, dear boy, is an island in the South Seas.'
I can't write about Pinewood and not mention the biggest film never made there, can I? _Cleopatra_ with Elizabeth Taylor in the title role, Richard Burton as Mark Antony and Rex Harrison as Julius Caesar (though that part was actually cast with Peter Finch when production commenced in 1960). So many problems plagued the film, not least the runaway budget which, at the time, made it the most expensive film that had ever come out of the studio. In fact the trouble started when the proposed leading lady, Joan Collins, was rejected by director Rouben Mamoulian in favour of Elizabeth Taylor who demanded – and got – the previously unheard-of fee of $1million for her participation.
Following that huge investment, lavish sets of previously unprecedented dimensions were constructed on the back lot, including the harbour in Alexandria (which held one million gallons of water – that'd take a few days with a bucket, I'll bet) and the Egyptian desert, but the construction coincided with a plasterers' strike and, in desperation, the studio took out advertising on prime-time TV to fill the vacancies. Before even a foot of film had been exposed, the expenditure had exceeded £1 million.
Then there were the 5,000 extras to accommodate with extra trains, buses and shuttles and mobile toilets that were brought in from Epsom racecourse. But the one thing the financiers didn't bank on when they opened their wallets was the British weather. Torrential rain fell and shooting was abandoned. Worse still, Elizabeth Taylor fell dangerously ill and had to undergo an emergency tracheotomy, and Joan Collins was put on standby to replace her. But with the rain still falling incessantly, and news of Liz Taylor's recuperation likely to be a long one, the decision was made to relocate to a warmer climate, which would help aid their star's recovery, and so everyone shipped out to Rome to remount the production. Well, except for Finchie – he'd had enough. Rouben Mamoulian also left the production at that point and was replaced by Joseph Mankiewicz.
Spotting an opportunity to use some of the leftover sets and costumes, producer Peter Rogers made _Carry On Cleo_ and at one point fell foul of 20th Century Fox by emulating their posters. It wasn't the first time that had happened either, as when Peter made _Carry On Spying_ the Bond producers raised their concerns about his poster looking similar to _From Russia With Love_.
While we on the Bond films had the luxury of six-month schedules and generous budgets, the _Carry Ons_ were made for a few hundred thousand pounds and shot in about five weeks, with only occasional location work outside the studio – usually in Maidenhead. When _Carry On Up the Khyber_ was made they used Snowdonia in Wales to double for the Afghan mountains (again no expense lavished) and at the premiere, one of the invited dignitaries sidled up to the producer and said he'd served in the Army and, 'I remember so many of those locations in Afghanistan and India. What marvellous memories.' Peter Rogers didn't have the heart to tell him the truth. But that is the magic of movies.
Donald Sinden told me a great story about working at the studios, a story that typifies the ethos at the studio at that time. Donald was a familiar face at Pinewood as a Rank contract star and later on in the 1970s I worked with him on a film called _That Lucky Touch._ He used to love chatting about all the people he'd worked with and is a brilliant raconteur. Apparently he was in the bar at Pinewood one day, and bumped into the producer Joe Janni, who started telling Donald about a script that involved three months' location work cruising around the Greek islands. On hearing that, Donald said, 'Count me in!' and didn't even wait to read the script.
A few weeks before shooting, Janni called to say that, unfortunately, the budget wouldn't stretch to the Greek islands. It was to be the Channel Islands instead. It still sounded good though. However, a week or so before shooting, Donald went for a costume fitting and the wardrobe man said, 'Shame about the Channel Islands, isn't it?'
Donald didn't know what he meant... until the wardrobe man explained that the budget wouldn't stretch to the Channel Islands... and the location was now Tilbury docks near London! They shot out to sea on one side, turned the ship around and shot the other way, and spent three months in those wretched docks... the magic of the movies indeed!
Another great friend in the early years was Kenneth More. Kenny and I met in 1962 when he was filming _We Joined the Navy_ at ABPC Elstree and I was filming my first series of _The Saint_ on the next stage. We struck up an immediate friendship and saw one another all the time thereafter. Kenny had been a huge star over at Pinewood for Rank, starring in films such as _Genevieve_ , _Reach for the Sky_ and _Sink the Bismarck!_ and in 1960 Rank's Managing Director, John Davis, agreed to release him to appear in a big-budget film at Shepperton Studios called _The Guns of Navarone_. However, shortly before production commenced, Kenny made the mistake of heckling and swearing at Davis at a BAFTA dinner, and in one fell swoop lost both the role (which went to David Niven) and his contract with Rank.
My dear friend Kenny More was best man on my wedding day, 11 April 1969, seen here with his wife Angela Douglas.
At the time we met, Kenny was filing for divorce from his wife, Billie, as he'd fallen in love, and wanted to be with, Angela Douglas. I was, meanwhile, in the throes of seeking a divorce from Dot Squires. It's fair to say we both wondered if we'd ever receive a _decree absolute_ as the legal wranglings went on for years.
Kenny and I made a pact that we would both be each other's best man when we were finally able to remarry – he was the first, in March 1968, whereas I had to wait a little longer. In fact, I remember one time we were in Mario & Franco's restaurant in Soho and we had a discussion about what colour our respective wives-to-be wedding dresses should be. I suggested grey for my intended... they say hell hath no fury like a woman scorned, and did I feel hell's full fury that day!
Kenny was a shrewd man when it came to the business as, unlike some actors who think they are invincible, he knew his limitations and also what type of roles suited him. I know Sir Peter Hall suggested that he play Claudius to Albert Finney's Hamlet at the Royal National Theatre, but Kenny declined, saying, 'One part of me would have liked to, but the other part said that there were so many great Shakespearian actors who could have done it better. I stick to the roles I can play better than them.'
Sadly Kenny was forced to retire in 1980, when it was announced that he was suffering from Parkinson's disease. I regret not seeing as much of Kenny as I should have in his final years, but being mainly based in Hollywood and he in London it wasn't terribly easy.
Albert Finney was another of the great British actors of that time. I remember meeting Albie when he and his theatre-producing partner Michael Medwin were in Gstaad, though of course I had known him from his many great films. His breakthrough hit was probably _Saturday Night and Sunday Morning_ , which was produced by Harry Saltzman, and afterwards Harry placed Albie under some sort of contract, the terms of which meant that when not employing Albie, Harry still stood to gain from renting him out.
When a call came from legendary producer Sam Spiegel, wanting to see Albie for _Lawrence of Arabia_ , which David Lean was to direct, Harry sent him off post-haste to Shepperton. On his return Harry asked how it went.
'He blew smoke in my face,' said Albie.
'What are you talking about?' asked Harry.
'He blew fucking smoke in my face!' said the aggrieved star.
Apparently, Spiegel sat behind his desk at the studio smoking a big cigar throughout the interview and, perhaps not unreasonably, Albie resented it. A few days later, the phone rang and the message came through that Spiegel wanted to shoot some footage of Albie on set the following Monday, so he was obviously odds-on for the role.
'OK, you'll be there at 10 a.m. on Monday,' Harry said.
Nonplussed, Albie looked straight at Harry: 'No! He fucking blew smoke in my face!'
Later that evening Harry called Lew Wasserman, the doyen of agents and dealmakers, in California to tell him the situation.
'Get me Finney on the phone,' Lew barked back at Harry.
Harry had to then telephone around Albie's various girlfriends to find out which one he was shacked up with that week.
'Lew Wasserman wants to speak to you from Palm Springs about _Lawrence of Arabia_ ,' Harry said, when he finally tracked him down.
Albie dutifully phoned Wasserman back. 'Spiegel blew smoke in my fucking face!'
'It doesn't matter what he did!' Wasserman argued. 'Get your ass down to Shepperton at ten o'clock on Monday morning!'
'But I told you, he blew smoke in my face!'
'Look. Do you know who the hell I am?' asked Wasserman.
'Yes,' said Albie. 'You're my fucking agent – and he _still_ blew smoke in my fucking face!'
On another occasion Albie was appearing with Charles Laughton in a production of _The Party_ and every time Albie started his big speech, Laughton would very visibly start scratching his nose or arse, and generally making distracting moves. Albie suffered it a few nights and then told Harry Saltzman to, 'Tell Mr Laughton I'll kick him into the orchestra pit if he fucking does that again.'
'Oh will he?' chuckled Laughton.
'He WILL!' Harry replied matter-of-factly.
That's what I love about Albie, he is completely independent and speaks his mind without fear of upsetting anyone. Needless to say, he didn't go to Shepperton or get the part of Lawrence, though it never hampered his career. Nowadays he doesn't have an agent, he prefers to negotiate through his lawyer and you won't find him at film premieres or awards ceremonies as he hates all that – he'll turn up, give a great performance and then go home. That's what Albie does. He turned down both a CBE and Knighthood, saying it 'perpetuated snobbery'. (I guess that makes me a terrible snob?)
Back at the studio, you'd often find lots of actors and directors would zip away after a thirty-minute lunch to view rushes, which was the previous day's film back from the labs. It wouldn't be out of the ordinary to see a few extras dressed as centurions, or large chickens, cutting through the restaurant to the bar. Nobody flinched. It was, after all, a place of work.
Lunchtimes at the studio saw other regulars too, one being Christopher Reeve who'd walk in for his meal in full Superman costume. He was so polite and would always stop at the tables he passed to say hello to fellow diners. Many a waitress swooned after him.
The bar was quite a 'club' too. Often you'd find Peter Finch holding court at lunchtimes and evenings with tales of the outback and working in Hollywood. He and Diane Cilento once naughtily inserted a cigarette into the mouth of a rather expensive Laughing Cavalier-type painting on the wall, much to the annoyance of Peter Rogers, who'd just paid to have it restored. Finchie was very much the practical joker of Pinewood, hiding in cupboards to surprise passers-by, removing gargoyles from the entrance and taking them home, and jackarooing around the bar with Diane at lunchtime, rounding up the crew.
He was a wonderful character. British born, though raised in Australia, he probably received greatest acclaim for his last film, _Network_ – for which he posthumously received the Academy Award. I believe the only other person to win an Oscar after his death was another Australian, Heath Ledger – as my friend Michael Caine might say, not a lot of people know that.
Finchie very nearly didn't make it to his career in Hollywood, or Pinewood for that matter, as, when aged just nineteen, he almost died when he was in Melbourne for a play called _So This is Hollywood._ The cast all went out for a picnic one day and afterwards he and co-star Robert Capron took off for a walk to explore the Pound Bend Tunnel on the Yarra River at Warrandyte, Victoria. Finchie told me they saw a fox terrier puppy fall into the river and Capron dived in and tried to save it. Tragically the current proved too strong and despite Finchie's best efforts to save him, Capron drowned. Ironically, the dog survived and Finchie was later awarded a certificate of merit by the Humane Society, though I don't think he ever got over losing his friend; it weighed on his mind for the rest of his life.
In his obituary notices Finchie was invariably described as a 'hell-raiser' and to a great extent his drinking, womanizing and larger-than-life antics overshadowed his prolific acting career. In a poignant interview shortly before his untimely death in 1977, aged just sixty, he said:
'I'd like to have been more adventurous in my career. But it's a fascinating and not ignoble profession. No one lives more lives than the actor. Movie-making is like geometry and I hated maths. But this kind of jigsaw I relish. When I played Lord Nelson I worked the poop deck in his uniform. I got extraordinary shivers. Sometimes I felt like I was staring at my own coffin. I touched that character. There lies the madness. You can't fake it.'
My next visit to Pinewood after _The Persuaders!_ was in 1973 as Jimmy Bond. Cubby Broccoli had had any number of offers to take the series overseas, but no, he said, 'Pinewood is my home'. It wasn't just sentimentality, it was good business sense as the crews always delivered the very best and Cubby loved the environment.
Along with the good fortune and success, I've also seen Pinewood at its lowest ebb. When we went in to shoot _Octopussy_ there was nothing, and I mean nothing, else in the studio – no other films were being made. The whole industry was in the doldrums. Word had it that had we not returned it would have closed down. Shepperton shared a similar fate, with changes of ownership and asset-strippers bringing the studio to near collapse and closure. Soon afterwards, Pinewood was forced to go 'four-walled' – becoming a rental facility, rather than a fully crewed studio.
In between Bond films I earned a few bob playing Father Christmas. Listening to the little ones and their expectations from Santa was a treat I couldn't miss.
The studio quickly diversified into commercials and more TV work. The plan paid off and meant that scores of big-budget blockbuster movies all had a home in the British countryside, though at the turn of the century its future seemed unsure when in 2000 the Rank Organisation announced plans to pull out of all its film interests, including Pinewood; but a new hero was at hand to continue the Pinewood story. Enter Michael Grade.
I knew Michael of old, his uncle was Lew Grade of the Incorporated Television Company (ITC), and Michael was every bit as passionate about film and TV as Lew was, but he's also a very astute businessman. He spotted the potential and value of Pinewood and pulled together a financial consortium to buy the famous studio. He then made an offer to Ridley and Tony Scott, the owners of Shepperton, to merge the two studios.
The place is now much bigger than it was in 1947, with twenty stages as opposed to just five, and more on-site companies and services than you can shake a stick at, but the atmosphere remains the same. It is, I have decided, pure magic.
Tod Slaughter – looks a friendly chap, doesn't he?
CHAPTER 3
Stage-struck
'YOU CAN'T BE A SERIOUS ACTOR UNTIL YOU'VE TROD THE boards first!' was the adage of the old Shakespearian actors who toured until they dropped. They were quite inspiring to a young, hungry actor, though. While I spent much of the free time of my youth at the Odeon cinema in Streatham, south London, I also enjoyed visiting music halls and what you'd now call fringe theatres. One actor I seemed to follow around was a deliciously macabre chap named Tod Slaughter.
He was unintentionally hysterical and so over the top, playing maniacal villains with rolling eyes, a mad cackling laugh and over-exaggerated mannerisms. He was typecast to the point of being known as 'Mister Murder' and usually played in Victorian melodramas like _Sweeney Todd_ , _Burke and Hare_ and _Jack the Ripper_. In the latter, at the end of the play he was 'killed' on stage and I remember one night when the curtains came down his feet were left sticking out beneath the curtain. His size tens were slowly withdrawn, only for him to reappear in front of the tabs seconds later for his obligatory, and highly theatrical, curtain speech:
'Ladies and gentlemen, our story has been taken from the annals of Scotland Yard and is true in every aspect except the Ripper was never caught. But you saw him killed before your very eyes tonight and can now leave this theatre safe in the knowledge that you ladies don't have to wonder, "Is he there? Is he there?" when you walk into the darkness.'
These monologues were often better than the play and Slaughter was certainly the last of the great barnstormers, ceaselessly touring the provinces in his hoary old melodramas right up until the day he died.
Talking of ladies leaving the theatre safely, my former wife, Dorothy Squires – who was not a particularly tall person out of heels, and so liked to wear quite high ones – exited the stage door at the Brixton Empress one night when some lout on a bicycle came up behind her and 'goosed' her, leaving his thumb in her rear end so all she could do was totter along, screaming for help as he pushed her along. Ever after, she always looked left and right for cyclists.
Another film star of my youth who I also saw on stage was Lancastrian comic and music hall entertainer Frank Randle, who actually only made ten movies and usually without his false teeth in. He often fell foul of the censors, particularly in Blackpool where they banned him from performing some of his material on stage, and it's fair to say Randle did not take kindly to criticism or hecklers and would throw his dentures at them as a mark of protest. His many run-ins with the police led to a significant charge sheet being lodged with the Lancashire Constabulary!
With my wife, Dorothy Squires. Dot made her name on radio shows such as _Variety Bandbox_ , and then went on to success in the US.
He once performed at my local, the Hackney Empire, and when the resident comic came on, Frank appeared on stage and started talking over him, stepping on all his punch-lines. Someone called from the audience, 'Why don't you shut up and let the other comic have a word?'
'You bastards!' shouted Randle. 'You fucking ungrateful bastards!' Next thing I knew, the iron safety curtain came down and he was dragged off stage.
Randle had his caravan parked around the back of the theatre, and immediately let the tyres down so the management couldn't evict him from the premises. He then stood in its doorway, posing for the press with his shirt undone to the waist, showing off his torso.
A friend of mine was part of Randle's travelling theatre company and for the only time in the history of show business the pantomime they were performing, in Birmingham, went on strike.
'Ungrateful bastards!' Randall shouted at the cast. 'Who employs you?'
'You do, Frank, but you haven't paid us,' they reasoned.
'Well, I fucking employ you! What more do you want?'
The dressing rooms were all interconnected and the company discovered that by placing a glass on the radiator pipes they could listen in to Frank talking in the dressing room next door. One day they heard him talking to a police inspector. 'Aye, that second lead of mine, he's an iron hoof I tell you – he's taking it up the bum. And that tall blond London chap, he's been shoplifting and bragging about it.'
They realized that Frank was saying the most terrible things about the actors in order to get them arrested, so that he could get out of paying them.
Jimmy Wheeler was another music hall comedian I followed and got to know. He was a huge man who would come on stage with a violin, and his catchphrase was 'Aye, aye, that's ya lot!'.
A friend of mine was working with him in Australia and apparently one day Jimmy got into a taxi and banged the door hard shut. When they reached their destination the driver said, 'I'll thank you not to bang the door.'
Without missing a beat, Jimmy replied, 'The only doors I want to bang is Diana fucking Dors!'
He was once on the same variety bill as Dorothy Squires in Great Yarmouth. Dot was to close the show, while Wheeler was set to go on last in the first half. However, as the interval began drawing nearer, they couldn't find Wheeler. Dot, who was in the dressing room dressed in only her bra and knickers, suddenly heard her entrance music playing.
'Quick, Dot! We need you on!' came the call.
Dot threw on her dress, dashed out on stage and started singing, while the manager of the ABC was dispatched to the local pubs to look for Wheeler. Eventually they found him, half cut, propping up a bar:
'Jimmy! You're on!' snapped the manager.
'How am I fucking doing then?' he replied ... and no apology was forthcoming.
On another occasion I remember him chuntering on about young comedians, and his wife, who had finally got fed up of dear Jimmy, said, 'Why don't you give it a rest, Jimmy? You're only jealous!'
'And why don't you go and lie down for four-and-a-half fucking years?' quipped the great man.
Another theatrical great who used to like to bend his elbow at a bar was Wilfrid Lawson. He was serving as a Special Constable at Bow Street Police Station during the war – where my dad was also stationed. Part of their daily routine was to go on point duty, controlling the traffic long before traffic lights came into being.
One afternoon, Dad told me that he had to go out and help retrieve Special Constable Lawson from The Strand where, far from directing traffic, he was crawling across the roads on his hands and knees – in full uniform. It was not quite the image the Metropolitan Police wanted to portray.
On another occasion, Lawson was rehearsing a play on stage with Nicol Williamson, but failed to report back after lunch. Williamson went into a violent tirade on stage about what a drunk Lawson was, how unreliable he was and so on. When he paused for breath, Lawson's inimitable voice chirped up from the back of the theatre: 'I thought that speech had been cut!'
I often hear ageing actors today bemoaning the lack of good repertory theatre as a training ground for up-and-coming thespians. I must admit I enjoyed my time in the suburbs, as a young aspiring actor, as not only did the fast-changing programme instil a terrific ability to learn on your feet, it also brought with it a sense of stage discipline, invaluable performing experience and the chance to tackle a fascinating variety of plays. Money was never very generous though and our staple diet often revolved around baked beans on toast and fry-ups.
Sir Donald Wolfit has often been described as the last of the great actor-managers. With his repertory group of players he would travel the provinces bringing the great stories to the masses – well, maybe not quite _masses_ but a grateful few at least. Wolfit was an exuberant, larger-than-life personality who was perhaps most famously the inspiration for the character of 'Sir' in the film _The Dresser_ , as played by Albert Finney. I'm told by those who knew Wolfit that he was a temperamental and difficult man to deal with and was, like Frank Randle, enraged by any form of criticism. He was also tyrannical with the companies he led. One young actor was put in his place quite firmly when he asked what the next production would be and what his part in it might be: 'I'll ask you if I think you can play some speaking roles by then,' snapped Wolfit.
Wolfit was undoubtedly a fine actor, but one with indifferent feelings about film work – despite making thirty appearances on screen – considering film a 'poor relation' to theatre, though his talent and ambition in theatre was undoubtedly handicapped by the more modest venues he ended up appearing in. The great – if eccentric – English actress Hermione Gingold summed it up best by saying, 'Olivier is a tour-de-force, and Wolfit is forced to tour.'
At the end of each show Wolfit would perform a curtain call speech, much like Tod Slaughter did. One week, he came out front of the tabs and in his 'chewing the scenery' theatrical drawl said: 'Next week, ladies and gentlemen, we shall be bringing you _Macbeth_. I will be playing the title role and my wife ...'
'Your wife is a whore!' came a shout from the gallery.
'Nevertheless,' continued Wolfit, unfazed, 'she will be playing Lady Macbeth ...'
There were many stories and rumours about Wolfit, including him haranguing the people at a cinema queue to leave it and attend the theatre – in another part of town – where his company was playing. Some even suggested that, to earn a few bob on the side, he sold insurance on easy payment terms to actors. I'd like to believe that one!
It's not quite another Wolfit story, but one I'd like to think was from an actor-manager of his ilk (if only of lesser standing) for whom a young Welsh actor named Brian Ellis auditioned. When he offered Brian a role in the company he cautioned him that the pay wasn't very good, but, in a negotiating ploy to seal the deal, he proudly added, 'There is an edible pudding in act two every night.'
On his deathbed, so legend has it, Wolfit was still keen to offer words of wisdom to members of his rep company. One of his young actors asked, 'Sir Donald, after a life so filled with success and fame, dying must be hard ...'
'Dying is easy ... Comedy is hard,' replied the great man.
For some unknown reason, though I think it might well have been jealousy, Wolfit hated Sir John Gielgud and referred to him as 'the Enemy'. It obviously niggled at Gielgud, as when he was invited to speak at Wolfit's memorial programme, produced by the BBC, he responded, 'I couldn't. You see we always regarded him as something of a joke.'
I made a film with Gielgud called _Gold_ , though as is typical in this business we never actually met on set as neither of us appeared in a scene together. Gielgud had something of a reputation for putting his foot quite firmly in his mouth, and never really thought about what he was saying until after he'd actually said it. For example, when James Villiers (who played Bill Tanner in _For Your Eyes Only)_ assumed Laurence Olivier's role as Victor in _Private Lives_ , Olivier later asked Gielgud what his performance was like.
'Oh, Larry, I've never seen the part of Victor so perfectly realized. Oh, sorry, I didn't mean that!'
A great friend (and employer) of Gielgud's was theatre manager and producer Binkie Beaumont, who co-founded the highly successful theatrical production company H.M. Tennent and had control of half the theatres in London's West End. Binkie was a larger-than-life character. Hugely powerful in the London theatre world, his social circle was very wide and impressive. While I, as a lowly understudy for the firm, was never invited to mix in such company, I did hear a very amusing story about Binkie and Gielgud.
'John is coming over,' Binkie warned his weekend guests at his country home. 'And he is bringing a rather unsuitable young man with him, so be nice.'
Gielgud duly arrived with a young James Dean-type in tow, who, he said, 'wanted to be an actor' and, leaving him with Binkie in the hall, Gielgud went straight through to the kitchen. After a moment or two of awkward silence, Binkie asked, 'Would you like me to show you around, Hector?'
The boy smiled sheepishly, and nodded.
Five or six minutes later, Binkie returned to the kitchen, furious with Gielgud.
'Johnny! That boy is impossible! I took him to the garden and said, "This is the croquet lawn, Hector" and he just grunted. I took him around to the pool, and asked, "Do you swim, Hector?" and he just stood there, staring blankly at me and then stammered over a few incoherent words. And you say he wants to be an actor? He can't even speak the Queen's English!'
'For God's sake, Binkie,' said Gielgud. 'His name is Sebastian! Hector is what I call his cock!'
Gielgud's continuing gaffs really were the stuff of legend, and I remember as he visited the ailing Laurence Olivier towards the end of his life, he reportedly exclaimed, 'Larry! You're dead! I mean ... you're dying! I mean ... my poor darling Larry, you don't look at all well!'
He once upset Richard Burton too, after going to see him backstage following a production of _Hamlet_ , 'We'll have dinner when you're better ... I mean when you're ready.'
Despite delivering many fine performances in 'classical' parts on stage and screen, it was for his role as the butler in the film _Arthur_ that he was given an Academy Award. He very nearly didn't accept the role, however. 'I turned it down a couple of times ... I thought the script was rather smutty, rather common. They just wanted a posh-looking Englishman saying rather racy things. Every time I said "no" the price went up and finally, when I accepted, they said "how clever of you", which wasn't the case at all.'
Gielgud continued working well into his nineties, though at his own pace and mindful of his advancing years. To keep the insurance companies happy, producers usually scheduled his scenes all fairly tightly together. I remember him saying, 'In my last big parts, I kept thinking, "Suppose I die in the middle? What is it going to cost everyone?"'
When he was asked to join MPs Glenda Jackson and Gyles Brandreth at the House of Commons for his ninetieth birthday celebrations, he replied, 'Yes, I would be delighted to join them. You see, all my real friends are dead.'
Though perhaps some words of his I should heed, 'When you're my age, you never risk being ill, because everyone then says, "Oh, he's done for".'
Ah! the roar of the greasepaint and the smell of the crowds! Danny La Rue was unquestionably the most famous Dame – or if you'd prefer, drag act – in all Theatreland. In fact, he was one of Britain's highest-earning entertainers of the 1960s.
In the days of my exposure to the variety world, a married couple who were close friends of Dorothy Squires produced and toured a couple of drag shows and Dot used to sell her old ball gowns to members of the troupe – one of whom was Danny.
Jack, who was the husband in the husband-and-wife-producing team, said to me one day, 'You'd better believe in fairies.'
'Why?' I asked.
'They're real!' Jack exclaimed. 'One night I fired one of my leading ladies, and the next night, when I came out of the stage door, he pointed his unfurled umbrella at me and said, "The witch's curse on you, Jack Lewis!".'
Jack said that at the time he laughed it off and returned to his digs, only to open the newspaper the next morning to see headlines containing allegations about Lord Montagu and boy scouts. That incident, for which Baron Montagu of Beaulieu was jailed for twelve months, killed drag shows for months afterwards. You see, audiences – probably not helped by the press stories – incorrectly associated transvestites and homosexuals with 'doing things with boys'. Jack Lewis lost huge business – and never doubted the curse again.
Danny, meanwhile, with his dazzling coiffures, extravagant costumes, immaculate make-up, false eyelashes and high heels, went on to open a club in London's Hanover Square, where he performed his wonderful impressions of Elizabeth Taylor, Zsa Zsa Gabor, Judy Garland, Marlene Dietrich and many others. He also gave a break to a great many singers and comedians who were desperate for a leg-up in show business.
The club became extremely popular for its satirical revues long before _That Was The Week That Was,_ and attracted celebrities and royalty – as well as coach parties from all over the country – in huge numbers.
Danny made no bones about 'dressing up in a frock' being just a job, not a lifestyle choice. He enjoyed his job greatly but when off stage, liked his privacy. When Princess Margaret knocked on his dressing-room door after a show one night, a stark-naked Danny threw it open and shouted, 'Piss off!'
'I was mortified,' he said. 'I thought she was Peter Sellers messing about!'
Liberace was another tremendous showman. I met Liberace, or Lee as he liked to be called by his friends, in Hollywood in the very late 1950s – though I can't quite recall how I came to be invited to the party at his house. But you know me – anything for a free meal. To say the decor was kitsch would be the understatement of all understatements. I remember the ceilings being painted with frescos in the style of Rome's Sistine Chapel and there being ornate marble pillars, diamond-encrusted chandeliers, countless pianos, and many huge paintings and photos of Liberace decorated the house. Everything was done to excess, but that's how Lee liked it.
With Tommy Steele, Dot and Liberace at a party in LA. The hair is real – well, mine, at least.
Lee loved being around actors and he told me he'd always wanted to be a movie star. He did make a couple of films and I think even popped up as a villain in the _Batman_ TV series at one point, but it never quite worked out for him in that direction – though he did have his own hugely popular TV shows and guest-starred in many other variety and chat-style programmes.
I found him to be extremely polite, fascinating to talk with and very well read, and while I knew he was an accomplished pianist, it was only when I saw him perform in Vegas that I realized his genius was in taking popular well-known tunes, adding a huge dollop of bling and serving it up with masses of charm. There are many talented pianists and musicians in the world, but there are few who could match his showmanship.
It was relatively well known in Hollywood that Lee was gay, though the image he painted for his fans was of a man who hadn't yet met the right woman. I still remember the huge controversy in the UK when fifty years ago Lee sued the _Daily Mirror_ newspaper columnist William Connor (who wrote under the byline Cassandra) for implying that he was homosexual. Cassandra wrote that Liberace was:
'...The summit of sex – the pinnacle of masculine, feminine, and neuter. Everything that he, she, and it can ever want ... a deadly, winking, sniggering, snuggling, chromium-plated, scent-impregnated, luminous, quivering, giggling, fruit-flavoured, mincing, ice-covered heap of mother-love.'
It was the suggestion of him being a mincing mother's boy that led to a six-day hearing, during which Lee categorically denied being homosexual. The jury found for him and awarded a then-record £8,000 in damages (which I'm told would equate to about £500,000 in today's money).
His legion of female fans never doubted his heterosexuality, even when his chauffeur and companion Scott Thorson sued him for palimony in 1982.
When Lee died in 1987, his publicists, agent and doctor gave the cause of death as being heart failure, and everyone believed them. Only later, when an autopsy was called for after the coroner expressed his doubts about the death certificate, was it finally acknowledged he had actually died from an AIDs-related illness. Even in death, Liberace was keen that his image should not be tarnished.
Invariably, comedians are asked to 'tell us a joke' and magicians to 'show us a trick' and more often than not I'm asked to 'tell us about your practical jokes on set'. Of course, as there were so many over so many years, it's impossible to think of one specifically, and it's not something you necessarily plan or think about beforehand – it's usually a spontaneous reaction to a certain moment in time. The times I've squirted crews with hosepipes, dropped my trousers in a tense dramatic scene, started giggling to put a co-star off, or re-written dialogue, are all well documented elsewhere. But I much prefer talking about things that go wrong, as they're always far more memorable – so long as I'm not always the butt of the cock-up, that is.
The law of averages suggests that when you're in a long-running play, or on a lengthy film shoot, then, despite your best efforts, something, somewhere will mess up. In a film, of course, we have the luxury of 'take two' and a clever editor, but it's not quite the same in theatre.
As a young aspiring actor, I went to see John Gielgud in _Hamlet_ at the Haymarket theatre in London and, as the curtain went up, we discovered the stagehands were still putting the planks in place around the graves, the first gravedigger was shuffling on to take his place and the stage manager was walking across the back of the set looking up at the lights. I laughed – but I was probably the only one who did.
I have rather a name as a practical joker... but thankfully as someone who can take a joke as well, as seen here when I was presented with a rubber octopus during a dining scene in _Octopussy_.
When I was in repertory theatre at The Intimate in Palmers Green, I once pushed hard on what I thought was a sticky door only to realize it was in fact a pull-door, and took down the entire set. On another occasion I was in a play called _While Parents Sleep_ by Anthony Kimmins (the West End production of which starred Jack Hawkins) and in one scene I'm in a drawing room, having a bit of a fumble with a girl, when suddenly the lights come on and we both leap up to stand behind the settee as Lady Cattering enters. She, in turn, had some dialogue about forgiveness and I then had to bow my head down to her.
In rehearsal I said, 'Wouldn't it be funny if I bowed and banged my head on the settee?' The ensuing silence gave me my answer – no, it would not be!
Come the first night, I bowed down, and quite inadvertently hit my head on the said settee, ending up on my knees having almost passed out. The audience thought it was hilarious, of course.
Actors have all encountered stubborn doors, sets that wobble and stage revolves that just won't revolve, but there are also the technical bloopers such as phones not ringing on cue, gunshots sounding seconds after the actor has already fallen to the floor and doorbells that never ring, leaving the poor terrified soul awaiting his or her cue into having to ad lib. 'Oh! Is that Aunt Agatha at the door? I'm sure I heard her car pull up just now.'
And what about the props that don't behave as they ought to? Such as the time in a provincial tour of _The Importance of Being Earnest_ when one character was supposed to pour tea for another, but the handle dropped off the teapot. Or maybe a trap door opens when it shouldn't, as happened in the Broadway production of _Fiddler on the Roof_ , in which Nancy Opel broke her elbow.
Depending on the actor, they may prefer to ignore any mishap and continue, or perhaps admit something has gone wrong and let the audiences in on the gag, which can often add to the audience's pleasure as they feel they've then been part of something that doesn't ordinarily occur.
For example, in her 2013 record-breaking run of _The Audience_ at the Gielgud Theatre in London's glittering West End, Dame Helen Mirren made more headlines for her performance offstage – namely outside the stage door. Some friends of mine were in the audience on the fateful day. With the first half of the production beleaguered by lighting problems, the interval couldn't come quick enough for the cast. When the interval curtain went down, Dame Helen took the opportunity to apologize to the audience for the technical glitches, which, she told them, were being ironed out, but she also added that she had been incredibly mindful of the noise outside too, which no doubt spoiled the first act for patrons.
She went on to explain that the drumming parade, moving through Soho to promote a gay music festival, had decided to stop outside the stage door for a full eight minutes. While supporting the cause, she didn't feel it fair that they made so much noise during a performance and added, 'I told the band to "shut the f*** up" as people had paid "a lot of f***ing money to watch the show" and that they were "f***ing ruining it".'
One onlooker was later heard to say, 'It was strange to see this little woman in tiara and pearls shouting like that. It's not the behaviour you'd expect from the Queen.'
Dame Helen received a round of applause, and went on to offer the drumming band free tickets to see the show.
Michael Simkins was starring in a touring production of _Dial M for Murder_ and with the murderer revealed in the final scene the curtain was supposed to drop – end of play. However, on one particular evening the stagehand pulled the wrong lever and instead of dropping the tabs, activated the sprinkler system. Thinking on his feet, Michael's co-star, who had just left the stage, popped his head back through to help save the scene and said, 'And you ought to get that leaky roof fixed too!'
Then, of course, there was _Charlie and the Chocolate Factory_ at the Theatre Royal Drury Lane in June 2013, and a night the great glass elevator stopped working, leaving Douglas Hodge's Willy Wonka and the child actor playing Charlie stranded above the audience. Within ten minutes they were rescued and the performance continued, with an apology to all from the manager. Mind you, that wasn't quite as serious as in a Californian production of _Aladdin_ I read about, in which a magical flying carpet tipped off the actors and left them hanging by their safety harnesses.
Wardrobe malfunctions also play their part, as in the case of Maureen Lipman in _Candida_ in which she had opened the play and her skirt fell off, leaving her standing in only a slip while her co-star leapt to her defence, continuing to deliver their dialogue all the while.
It's no wonder that actors corpse with laughter, and when one is set off you can guarantee the rest will follow – all trying extremely hard to keep straight faces.
Perhaps the ultimate stage cock-up was at the Wood Green Empire in 1918 when Chinese magician Chung Ling Soo (or William Ellsworth Robinson, to give him his birth name) was performing his popular bullet-catching trick. A gunpowder flash, used in a second chamber of his revolver, which produced the illusion of the gun being fired, actually ignited the bullet in the main chamber, causing the magician to be shot in the chest. Having never before spoken English on stage while adopting his Chinese persona, Chung Ling Soo was heard to say, 'Something has happened. Lower the curtain.' ... I'm afraid it was curtains for him and no _encore_ for the audience.
We actors aren't infallible. Alas, we are only human, as I'm reminded when I read through the obituary columns. It's really quite worrying to see names I know in there, especially of people who are younger than me. Occasionally an actor, who, let's face it, spends much of the working day on stage or set, is reported as having croaked it during a performance and the writer will add, as though hugely consoling, 'he'd have liked to know he died on stage doing what he loved best'. Erm, that's not really very comforting, I assure you!
No one made a career out of things going wrong quite like Tommy Cooper – it was his whole act – though, of course, to be such a bad magician meant he was actually remarkably good at magic. But when he collapsed on stage, live on British TV, the audience laughed hysterically, thinking it was all part of the show.
Similarly, when Eric Morecambe suffered a heart attack after a show in Yorkshire in the late 1960s it wasn't perhaps treated as seriously as it would have been if he hadn't been a comedian. Having returned to his hotel he suffered terrible pains in his upper arm and, recognizing the danger signs, decided to drive himself to hospital, but as the pain spread to his chest, he became unable to continue the journey. Thankfully he was rescued by a passer-by who took him to hospital where, with Eric lying on a trolley waiting to go into the operating theatre, a nurse appeared with a pen and piece of paper and said, 'Before you go, would you mind giving me your autograph?'
Thankfully, Eric didn't 'go' anywhere that night, and dined out on the story for years afterwards. Sadly, he died after appearing in a Q&A session at a theatre in Tewkesbury, collapsing in the wings after a standing ovation.
Sid James also passed away at a theatre, but he did it on stage at the Sunderland Empire in front of a packed auditorium. Theatre manager Roy Todds phoned _The Mating Game_ 's producer, Bill Robertson, to tell him the shocking news, 'Sid James has just died in Sunderland!'
'Don't worry,' replied Bill. 'Everybody dies in Sunderland.'
Frank Sinatra, Montgomery Clift, Deborah Kerr and Burt Lancaster, leaving for location filming of _From Here to Eternity_ in the early 1950s. Burt Lancaster was a wonderful actor but boy did he have a terrible temper at times!
CHAPTER 4
On-set Tales
I'VE BEEN FORTUNATE TO WORK WITH SOME WONDERFUL actors – and I've also heard some wonderful stories about some actors I've not worked with!
Although I had a smattering of colloquial Italian, I never really understood the call sheets on my 1962 epic _The Rape of the Sabines_. The film was also known as _Romulus and the Sabines_ , and I had the pleasure of playing the role of our hero, Romulus. Though based at studios in Rome, we went off on various random locations too, and I got into the routine of just turning up at the studio and being ready to go wherever we were needed. One day we headed north, at least I think it was north because you could never really hear anything the driver was saying because of incessant chatter coming from the crew on board the bus.
At the first location stop I had a scene with the lovely Italian actress Scilla Gabel, before we then moved up high into a mountain range, where I remember there was a beautiful limpid blue pool, fed by a little stream, which in turn flowed down from the top of the mountain. It was very picturesque, and easy to see why the location manager had suggested it.
All the extras – women who had been kidnapped by the Romans, under my character's command – began bathing quite merrily and swimming around the pool. One in particular, a former Miss Austria, created quite a bit of interest among the crew in her wet top – they gazed at her with what can best be described as primitive lust and longing. A couple of minutes later the assistant director shouted, 'OK! Now we'll shoot the South American version.'
_The Rape of the Sabines_ was memorable, but perhaps for all the wrong reasons... Romulus is seen here with screen parents Rosanna Schiaffino as Venus and Jean Marais as Mars.
I didn't remember ever reading anything about filming other versions of scenes in my contract, so asked what was happening.
'Don't worry,' I was told. 'You're not in it. Just go and take a seat.'
So, that's what I did.
Suddenly all the ladies started taking off their tops – thus creating even greater interest among the crew, so much so that some of them actually stopped chatting. But what we didn't know was that, high up in the mountain, there was some sort of reservoir, from which, when a switch was thrown, a mini, ice-cold, Niagara rushed down into our limpid pool, making it a raging torrent. The women started screaming and the gallant crew gathered by the pool side, to help pull them out – with the exception of Miss Austria, whose bust was acting as a rather splendid buoyancy aid, carrying her fast across to the far side of the pool.
With them all being good Catholics, I can't quite imagine which South American country would have ordered the slightly more risqué version of our film but I do know that it wasn't only this production that had 'alternative' scenes filmed. My (later) producing partner Bob Baker made a number of movies in the UK long before he brought _The Saint_ to TV screens. One of them, _Jack the Ripper_ , was shot at Shepperton Studios on a 'closed set' – and with good reason. The scene involved a theatre's backstage dressing room, and all the young actresses were in there busy putting on their make-up, dressed only in petticoats and bras, while preparing for curtain up. That was considered lurid enough for the UK censors of the 1950s, but when the scene was in the bag the assistant director, matter-of-factly, called, 'OK, cut! Clothes off for the continental version now!'
Angie Dickinson was wonderful to work with on _The Sins of Rachel Cade_. The film was set in the sweat and heat of the Belgian Congo... but filmed entirely on the sweaty Warner back lot, with footage from _The Nun's Story_ for the location scenes.
Their petticoats and bras were swiftly dispensed with and the same scene was shot again – only with more in it, if you follow me.
One of these such scenes was filmed on a Friday afternoon very near to the 4 p.m. wrap time, but all the electricians and stage hands willingly offered to stay on to help oversee the 'continental' version being filmed – and no overtime was requested!
One of the more pleasurable bonuses of being an actor is to not only meet but to work with other actors whom you have admired in the past, and Ray Milland was one of those. Ray was born Reginald Alfred John Truscott-Jones, but took his stage name from a riverside street called Milland Road in Neath, South Wales, where he once lived.
I worked with Ray on _Gold_ in 1974 and he, rather like Niven, was a great teller of amusing tales, one being of when he was working on a film on location in Africa and a male member of the cast turned up two weeks ahead of schedule. The director asked said actor why he had come so early and he told a story of being at a dinner party in Beverly Hills a couple of nights earlier, and seated next to a rather attractive young lady from the East Coast who was complaining of jetlag, having arrived from New York the day before. The actor suggested that he had some sleeping suppositories that worked wonders for him, and that he would be happy to let her have a couple. As she was staying at the Beverly Hills Hotel he said he'd happily give her a lift back, and stop off at his place on the way to pick up the pills.
Upon arriving home, he thought it would be only polite to ask her in for a drink, and as the night was still young she obviously thought it was a good idea as well and a few minutes later they were drinking Jack Daniels in his living room. He popped into his bedroom for the suppositories.
'They will take about half an hour to take effect,' he explained.
'Good,' she said. 'Maybe you would put one of them in for me now, and then be so kind as to drop me at the hotel?'
The lights in his living room were rather low, so he asked her to bend over close to a lamp, enabling him to see what he was doing. She raised her skirt and bent down and as she lowered her underwear he took up a position behind. Just as he was about to put the suppository in the required position, the door opened and in walked his wife. She was supposed to be in Palm Springs. With there being no believable explanation about him performing an act of kindness to a jetlagged lady, he decided to leave for the location early. Very early.
'Who was the actor?' I asked Ray.
'Can't you guess?' he smiled.
With Ray Milland in _Gold_ in 1974. This one was filmed in South Africa, but only after a technician's dispute was averted. We almost had to film it in Wales!
I performed an act of kindness for Ray myself – though not with a suppository, I hasten to add. Ray was suffering with a prostate problem during our filming together and found it very difficult to urinate. He suffered particularly great pain during a late-night shoot and I sympathized greatly because over the years I had suffered terribly with renal colic which, for the uninitiated, is way up at the top of the hit parade for being the most painful experience a man can undergo – they do say that having a child can 'maybe' be as painful. As a precaution against having an attack while away filming, my urologist had prescribed powerful painkillers for me to carry and I duly offered one to Ray. I'm sure that medics would recoil in horror at the thought of one hypochondriac actor giving prescription drugs to another, but needs must at times, and he found almost immediate relief and was able to finish the night's filming.
Towards the end of his career Ray took on a number of, well, less-than-brilliant roles. When asked in an interview why he had appeared in so many bad films, he smiled and said, 'For the money, old chap, for the money!'
In 1976 I made another foray into Italian cinema, signing up for a film called _The Street People ..._ no, _The Executioners ..._ no, _Opium Road ..._ ah yes, _The Sicilian Cross_. It kept changing title but on paper it sounded quite promising, with a script by, among others, Ernest Tidyman, who'd written _The French Connection_. At that time I had a holiday home in Italy, and an Italian wife, so the idea of working there (and in San Francisco too) was rather appealing.
It was essentially the tale of how a Mafia boss is wrongly suspected of smuggling a heroin shipment into San Francisco from Sicily, inside a cross donated to a church there, and so dispatches his nephew, a hotshot lawyer (that's me, being typecast again!), to identify the real culprit. Along the way, I enlist the aid of my best friend, grand prix driver Stacy Keach, who has a penchant for destroying anything in his way. Well, that's the essence of the story, but the rest of it – and the finished film – I don't really understand... and neither did cinema audiences to be honest.
In an attempt to explain away my casting, lines such as: 'The smartest thing I ever did was to get you out of Sicily and into that English law school,' and 'Being half-English, half-Sicilian was a good deal for both of us', were written in.
Stacy was a delight to work with, and huge fun, but apart from he and me, all the cast and crew were from Italy and they all spoke in very loud Italian, leading to great confusion about our cues. It was all dubbed (badly) afterwards into English and I do wonder if it might have made more sense to American audiences if it had been left in Italian, or perhaps Double Dutch?
Our first day of shooting in San Francisco, down by the docks, was memorable in so much as we had terrible trouble closing off roads, and in fact any traffic control at all proved impossible. The director, Maurizio Lucidi, who was very, _very_ Italian, made a call to the San Francisco Police Chief, who just so happened to be of Italian descent. The next morning we had police outriders escorting us to the location, with roads closed off all around and traffic officers all calling out, ' _Ciao! Buon giorno_ , Roger!' as we drove by. We enjoyed a very smooth shoot in the city after that.
Stacy Keach and I were a little bewildered by the fast-talking Italians involved in the making of _The Sicilian Cross_. I don't think either of us (or indeed the audience) really understood the plot!
While shooting in Rome I met up with Liza Minnelli, who was in town making a film with Ingrid Bergman called _A Matter of Time_ , which her father, Vincente Minnelli, was directing. Liza had recently been in Mexico filming _Lucky Lady_ with Gene Hackman and Burt Reynolds, and it appeared a few retakes had now been called for. Rather than wait for Liza to return to the US after completing her current project, the studio flew Gene, Burt and the director over to Rome for a few days. We all met for dinner one evening in a restaurant on the Via Veneto, and I'll never forget seeing the lone figure of Burt Lancaster sitting at the far end of the long dinner table that we'd been shown to. He didn't speak to, nor acknowledge, anyone.
Towards the end of dinner, the 'curse of Rome' (otherwise known as the paparazzi) bundled their way through the door and started snapping. I'm not sure why, but all the actors immediately – almost instinctively – moved away from each other; not that there was anything going on, it just seemed to be a natural reflex to not be photographed with anyone else, thereby denying them a story.
At this point, Burt Lancaster now stirred into life. He placed his knuckles on the table and slowly raised himself to his full height. He slapped his left hand across his right forearm while pushing his clenched fist forward and upwards – the gesture is quite unmistakable in Italy. The paps did not need a second warning – they left, quickly, and we gazed with great envy at Burt.
Talking of Burt Lancaster, he did have a formidable reputation around the studios – apart from being a fine actor, he had one of the worst tempers in Hollywood. He had a well-known hatred of the press, and was invariably brusque and rude to them, once even resorting to physically throwing a hapless pap out of his hotel suite. But then he was well known for being 'physical' on set, too.
In front of the cameras Burt was a force to be reckoned with and, though prized as an actor, he was often to be found in the middle of a major row or the cause of some disruption. When filming one of his most famous films, _The Birdman of Alcatraz_ , he and director John Frankenheimer were almost constantly at loggerheads. Burt was always known as an actor who immersed himself in his roles and he reputedly became obsessed with the character of Robert Stroud, the real Birdman, researching his life and reading everything that was available on him. Apart from having the starring role, Burt was also co-producer on the film through his production company Hecht-Hill-Lancaster.
When shooting began, John Frankenheimer found Burt almost impossible to deal with – being the star and the money-man made the director's job even trickier – and they argued about camera set-ups, dialogue, the lot. At one point, in an argument over where the camera should be placed for one particular scene, Burt physically picked Frankenheimer up and carried him across the set, depositing him where he thought the camera should go. Despite that treatment, Frankenheimer always admired the actor and they worked together again.
Michael Winner told me a great story about when he worked with the tempestuous Burt on a Western called _Lawman_. This was the early 1970s, at a time when Burt was at the height of his fame. In fact, such was his fame that Michael referred to his star as 'Mr Lancaster' throughout the whole shoot. But fame had not 'tempered' Burt's temper: one time they were filming a scene where Burt had to shoot his horse, drawing his Colt .45 to do the deed. In a later shot for the same sequence Burt used a rifle.
'Cut!' came the shout from the director's chair. 'Sorry, Mr Lancaster, but earlier you shot the horse with a Colt .45.'
Well, according to Michael, Burt went crazy, storming over to the director and shouting, 'What the hell do you know? I shot it with a Winchester! Why the hell would I use a Colt?!' With which he picked dear Michael up, carried him over to a nearby cliff and hung him over the edge, saying, 'So, _now_ what did I shoot the fucking horse with?'
And Michael had to agree, 'You shot the horse with a Winchester, Mr Lancaster. Of course you did.'
Back on the set of _The Sicilian Cross_ , another memorable day (for all the wrong reasons) came when we moved to the south of Naples to shoot a sequence in a fish market. As I walked through the market, one of the fish vendors called out in Italian 'I have a fish for you', which was all part of the characters' secret codes. We first shot from behind the fish seller, over his shoulder on to me, with the crowd moving by behind. The cameraman, who seemed to be getting very agitated, kept cutting and asking us to go again, until, on the fourth take, he leapt up, scrambled over the ice and lunged at an old man behind me, hitting out at him and seemingly trying to beat him up.
'Whoah, whoah!' I cried. 'What are you _doing_?'
'Every time we turn the camera over, that old man comes into frame and starts jumping up and down, pulling faces behind you!' shouted the cameraman, as they started scuffling again. I couldn't stand it any more and asked that we move on to another shot. It was chaos, sheer chaos.
In _The Wild Geese_ I played Lieutenant Shawn Fynn and, along with my fellow officers, portrayed by Richard Burton, Richard Harris and Hardy Kruger, I had my own platoon to command and consequently the schedule offered me a few days off here and there, while the others were filming with their respective platoons. In my group was an actor called Percy Herbert who'd worked on a couple of episodes of _The Saint_ and so I'd got to know him fairly well. One day, on our mutual day off, he came over and gave me some pages to read, saying it was his life story. It was absolutely fascinating.
During the Second World War, after just a few weeks' training, Percy was put on a ship and was told he was going to North Africa. However, midway, the orders were changed and the ship made its way to Singapore instead. No sooner had they arrived in the harbour than the Japanese air force struck from the air and their ship was sunk.
Meanwhile, the Japanese army advanced in over land, and as British guns were pointed out to sea, the rear action meant the Japanese very quickly won and occupied Singapore. Though wounded, Percy reached the shore with his pal and they were put into a British hospital where, after a few days, they heard gunfire, followed by screaming. Suddenly the doors to their twelve-bed ward were kicked open and two Japanese privates, machine guns at the ready, entered. Before they could open fire, a Japanese officer entered and with his cane knocked the soldiers' arms – and guns – down.
All the hospital patients were taken prisoner and Percy eventually ended up in Changi Prison as a prisoner of war. He, along with many other men from the jail, was then taken to work on the railroad that became the basis for the film _The Bridge on the River Kwai._
In prison, Percy and his friend found that they were able to infiltrate the food stores in their camp and in their desperate need for sugar they took some, some of which they buried. When the theft was noticed, both Percy and his friend found themselves prisoners of the British who, for disciplinary purposes, ran a prison within the Japanese prison. For most of the time Percy found himself in British solitary confinement. There they were screamed at, shouted at and everything was 'on the double', so poor Percy's feet never touched the ground. He had a terrible time and, ironically, all at the hands of the British.
It was back to South Africa, and the (then) Rhodesian border, for _The Wild Geese_ – along with Richard Harris, Richard Burton and Hardy Kruger. And were we wild? No, of course not.
Throughout his internment Percy never received any mail from home and when the Australians finally liberated the camp he was handed a bundle of Red Cross envelopes. They were all from his fiancée, so before opening any he carefully put them in order of posted date. The first started, 'My Darling Percy', the second, 'My Dear Percy', but the third simply said, 'Dear Percy' and went on to say she had found a new love ...
On the troop ship home to the UK, Percy and his mate were so terrified of waking up in the night believing they were still prisoners of war and jumping over the side, that they struck a deal whereby each night one would tie the other into his bunk. Such was the horrific toll of their imprisonment.
However, eventually they arrived at Liverpool docks and were given a pass to travel to London. Percy went straight to his mother's house and, after all these terrible years as a prisoner of war, walked down the basement steps, knocked on the door. His mother peered out.
'Blige me, Perce!' she said, adding, with typical British restraint, 'I'll put the kettle on!'
After a few days readjusting at home, Percy wanted to visit his old girlfriend's brothers, with whom he'd been great friends, and between them they arranged a meeting at Wood Green Underground Station. As Percy stood at the top of the escalator waiting, into his frame of vision came a female sergeant in the Auxilliary Territorial Service (ATS, the women's branch of the British Army in the Second World War) – his ex-fiancée. She told him that she had realized she'd made a huge mistake and that she had asked her brothers to set up this meeting between her and Percy. Happily, it wasn't long after that that they married. They went on to take up amateur dramatics together as a hobby and Percy later turned professional, which is how he came into my life.
When director David Lean was casting for his film _The Bridge on the River Kwai_ , Percy was taken up to meet him at Film House in Wardour Street. Lean looked him up and down, and grunted, 'not the type' before waving him out.
Percy, who had, not surprisingly, developed a nasty tic that came out when he was anxious or upset, a hangover from his mistreatment during the war, dived straight across the desk at Lean, face twitching like mad, 'Not the fucking type? What do you mean "not the fucking type"? I was fucking there!'
'OK! OK! You're in the film!' said Lean.
When they arrived on location, David Lean got to hear Percy's whole story and asked him how the prisoners might have shown contempt to their captors. 'How would you do it so as you knew but the Japanese didn't?' asked Lean.
Percy said they'd whistle, and demonstrated with the tune 'Colonel Bogey', and that's how it came to be used in the film.
Just as Percy was an advisor on that film, we had our own advisor on _Wild Geese_. Colonel Mike Hoare – a well-known mercenary, also known as Mad Mike – used to give us pep talks, similar to those he gave to his men before a mission. He'd led mercenary groups in the Congo and was a formidable man to whom, it was said, Richard Burton's character of Faulkner paid more than an accidental resemblance. He and some of the ex-mercenary actors in the cast used to terrify me: one in particular would be talking to you and then, all of a sudden, you'd feel his bayonet at your throat and he'd say, 'That's how it's done, my son.'
In the film, Jack Watson played RSM Sandy Young. Jack was the son of music hall comedian Nosmo King, who took his name from sitting in a railway carriage – a 'no smoking' carriage. Over the years I made a few films with Jack, and he was a fine physical specimen of a man but took himself rather seriously, despite appearing in variety shows with his father and working on many comedy films in his early career.
Jack was always extremely nice, but could be very pompous at times. After tennis one Sunday, my make-up artist Paul Engelen, my hairdresser Mike Jones and I were sitting around the swimming pool relaxing and Jack stood on tiptoes at the edge of the pool, stretched his arms out and said, 'There's nothing like having your own pool, is there, Roger?' which rather annoyed the boys in the crew, to say the least.
He then did a perfect pike into the pool but his body stopped three feet into the water as he'd dived into the shallow end! His legs remained rigid, sticking out like the Eiffel Tower and not one person went over to see if he was OK, we just sat there laughing.
I'm ashamed to say that Jack's serious take on life was like a red rag to a bull with me. One morning on _The Sea Wolves_ I had a word with our make-up artist Paul Engelen and when Jack slipped into the chair Paul proceeded to paint his face black.
'What's this for?' asked Jack.
'Oh, the director wants everyone made up for the night shooting and wants to run a camera test to see how everyone looks,' replied Paul.
Jack nodded knowingly, and sat bolt upright awaiting his transformation. An hour later, he reported on set, every bit the image of Al Jolson but minus the white lips. Andy McLaglen (our director) walked across with a look of total bemusement on his face.
'What the fuck are you looking like that for, Jack?' he asked.
I made good my exit, telling myself I really ought not to instigate these practical jokes.
Back on set with _The Wild Geese_ , Richard Burton had some shoulder problems at the time and the script called for a long shot of his platoon in the bush, with Burton's character carrying President Limbani (played by Winston Ntshona). Derek Cracknell, our wonderful First Assistant Director, said to Burton that they could use his double for this particular shot. Jack Watson, who was also in the sequence, overheard and, realizing his principal was being doubled, wanted to know where _his_ stand-in was.
Derek, sensing that there could be trouble brewing with Jack's somewhat misplaced feeling of self-importance, said, 'Jack, look at you and then look at all these herberts here. You're the perfect physical specimen, aren't you?'
Jack was forced to nod in agreement.
'Well, then. Which one of these herberts could _possibly_ double you?'
Jack did the scene himself.
Being a cowardly actor, having to play heroes means I'm in no small way indebted to the stunt boys. Nosher Powell was the senior member of the Powell dynasty of stuntmen and in between film stunt work, he used to look after security at the Jack of Clubs, a club underneath the famous Isow's Restaurant on Brewer Street in Soho, which was run by Jack Isow's son.
One night Nosher said to him, 'This group that just came in, I think they're going to be trouble and we should get them out.'
'No, they'll be fine,' he assured Nosh.
Three or four hours later, a £500 bar tab had been run up and the group started to get up to leave. Remember, this was a few years ago and £500 was worth a lot more then. Nosher stood at the top of the stairs, 'Look, boys, you owe a heck of a lot of money and you're not leaving until you pay.'
'Oh yeah?' they said. 'Who's going to stop us, you and whose fucking army?'
'Just me and my dog.'
With that, Nosh pulled round his huge German shepherd dog, which was snarling and salivating at the end of a leash.
'Oh, and what's your dog gonna do, Nosh?'
'Well, when I say "attack", he attacks,' replied Nosher.
'Well, let's see then,' they goaded.
'Attack!' Nosher commanded. With that, there was a terrible sound of snarling and the crunch of bones, and blood squirted everywhere as dog tore trousers, ripping into the flesh beneath ... and poor old Nosh was flat on his back trying to push the damn thing off!
The gang, in hysterics, threw the £500 on top of Nosher, thanking him for the best laugh they'd had in years.
Speaking of stuntmen, one rather famous one whom I won't name ... let's call him Roy ... was late reporting on set at Elstree Studios one day for a sequence in an episode of _The Saint_. It turned out that he was on an adjacent stage, upstairs in a dressing room. In fact he was in the wardrobe peeping out as one of his fellow stuntmen gave the horizontal performance of _his_ life with a young aspiring starlet. All the time this was going on, our Assistant Director was calling out, 'Roy! You're wanted on set! Roy!'
After reaching the inevitable conclusion, our intrepid hero waited a few moments before he made good his exit from the wardrobe and dashed down the stairs, five at a time, towards the stage where the call of 'Roy! Roy! We're waiting!' was becoming more and more audible around the entire studio. However, he missed his footing on the last stair, sprained his ankle and was unable to work for days. Voyeurism is not without its penalties!
I used to love spending time with the stunt boys, over dinner or a game of cards between set-ups. I always enjoyed hearing all their stories. One in particular that tickled me was of a card game organized when filming _El Cid._ Apparently one of the local boys had a terrible habit of slamming his hand of cards down with great force whenever he had a win. It rather annoyed the other boys, so the next day they decided to rig the table with charges and, when a winning hand was slammed down, the whole table went up in smoke.
There was another occasion I witnessed when I was on set with one of the elder statesmen of stuntmen. During breaks, he used to enjoy sitting in a chair and puffing on his old pipe and on this particular day he nodded off mid-puff. His kind and caring colleagues gently removed the pipe from his mouth, filled it with gunpowder and carefully placed it back. When he awoke from his slumber, he relit his pipe and there was the most almighty bang. Our friend's only reaction was to blow the cloud of smoke away, revealing his blackened face, and say, 'That tobacco's a bit fucking strong!'
My wonderful long-time stunt double and co-ordinator Martin Grace was quite probably the bravest man I ever knew. We'd worked together on about a dozen films and his easy-going Irish charm made him a firm favourite with everyone on the crew, though curiously nobody really ever got to know much about Martin himself. That's the way he liked it, I guess.
In October 2008 we met up at Pinewood for a reunion dinner centred around _The Spy Who Loved Me_ and picked up the conversation just as though the intervening fifteen years hadn't occurred. There in the magnificent Pinewood Ballroom we met up with so many other Bond alumni, including director John Glen, writer Christopher Wood, stars Richard Kiel, Caroline Munro, Valerie Leon, Shane Rimmer and a host of wonderful friends from behind the camera. As I bade goodnight to Martin, I never for one moment thought it would be the last time I saw my super-fit friend.
Two years later, at his home in Spain, Martin was involved in a cycling accident that, on the face of it, seemed rather trivial compared to some of the great stunts he'd dared in my name, and indeed breaking almost every bone in his body on a train stunt in _Octopussy_ , but there were (unknown) complicating factors and aged just sixty-seven Martin died in hospital.
My fearless stunt double and dear friend Martin Grace stood in for me in many of my outings as Jimmy Bond.
I subsequently discovered – as did many of Martin's mates – that he had a daughter from a short-lived marriage. I know she was incredibly proud of her father too. In death Martin was, as in life, an enigma but one to whom I owe such a great debt – as he made this coward look like a hero.
When tragic fatalities have happened while filming movies, canny producers have sometimes turned a disadvantage into quite the opposite in terms of publicity. For example, when Oliver Reed was filming _Gladiator_ in Malta and, true to character, enjoying a few drinks in a local pub on his day off, he reportedly consumed liberal amounts of beer, suffered a massive heart attack and died. With most of his scenes in the can, the producers used a CGI mask on a double to complete the outstanding sequences and billed it as Reed's last film – and it was actually one of his best performances. A similar thing happened on _The Misfits_ when Clark Gable died before shooting had ended, and though CGI wasn't around then, a body double was used.
Other actors who died mid-shoot and were doubled include John Candy, Bruce Lee, Heath Ledger and Donald Pleasence. A sense of morbid curiosity may have helped to fill seats, but ultimately I don't think any of their final films fared tremendously well – you just can't fake a star's power (or even double it) and that was certainly the case when, aged just forty-four, Tyrone Power dropped dead on the set of _Solomon and Sheba_ in Madrid.
Tyrone's wife, Deborah, had asked the director King Vidor to ease the schedule but after a prolonged sword-fight scene with George Sanders, wearing heavy robes and working with real Roman swords that weighed fifteen pounds, on an elevated staircase landing, Sanders wasn't pleased with some of the shots and asked that the scene be shot over. Finally, after the eighth take, Tyrone said he could stand no more and threw down his sword.
'If you can't find anything there you can use, just use the close-ups of me,' he said angrily. 'I've had it!'
A short time later, after complaining of being tired and feeling pain in his arm, Tyrone was rushed to hospital, where he died within the hour. Tyrone Power had filmed seventy-five per cent of his scenes and, in order that the film could be completed, was replaced by Yul Brynner – who had to wear a wig.
Perhaps the most exploitative of all stories, and one which did curiously benefit the project, was when director Ed Wood filmed just a couple of minutes of silent footage of Bela Lugosi wearing a cape for a planned vampire project that never came to fruition. After Lugosi died, Wood decided to use the footage in _Plan 9 From Outer Space_ and edited it into his movie multiple times. A body double, Tom Mason – who looked absolutely nothing like Lugosi – was brought in to finish the film. Totally nonsensical, the film is regarded as one of the worst films ever made, though ironically it's now a cult classic.
Good guy or rascal? You decide. Robert 'Bobbie' Newton in one of his most famous roles as Long John Silver in _Treasure Island_.
CHAPTER 5
The Good Guys (and a Few Rascals)
WHEN I LOOK BACK OVER MY LIFE I CAN'T QUITE BELIEVE I've counted some of my childhood acting heroes as being friends, co-stars and drinking buddies. I suppose I first met Gregory Peck sometime in the early 1970s, at the home of David Niven on Cap Ferrat in the South of France. They had been friends for years and Greg (along with his lovely wife, Veronique) became one of my dearest friends.
Greg had been a huge star from the 1940s and of course won the Academy Award for his role in _To Kill a Mockingbird_ in 1962. As is often the way in this business, by the mid-70s leading roles in big films were not coming his way, but then he was cast in a fairly modest British-made horror film called _The Omen_. My dear old pal, publicist Jerry Pam, handled PR for the film, though ironically the studio told Jerry not to focus on Greg as they didn't think he was 'box office' enough; so instead Jerry centred his campaign around the sign of the beast – '666'. It was massively successful, the film became box-office dynamite, and Greg's star shone brightly once again, leading to terrific roles in _The Boys from Brazil_ and then _The Sea Wolves_ with yours truly.
Gregory Peck and David Niven were two very dear friends to me.
While making _The Omen_ , Greg rented fellow actor Michael York's house in Belgravia and one evening I went for dinner there. As I said, it was the mid-70s and at the height of the IRA attacks on London, and when I left, around midnight, I found Greg outside on his back, having crawled underneath my car, thoughtfully checking if there was a bomb.
That kindly heroic deed certainly fitted in with roles he played. Mind you, he wasn't averse to playing against type every now and again, as confirmed to me at a dinner one evening hosted by another pal, Johnny Mills. Fellow guest Laurence Olivier said, 'Amazing man, Greg. Doesn't worry about his image by playing a Nazi ...' They'd just worked together in _Boys from Brazil_ in which Greg played the evil Nazi, Dr Josef Mengele.
Anyhow, in _Sea Wolves_ Greg was to play British officer, Colonel Lewis Pugh, and worked with a dialogue coach, the same one from _Boys from Brazil_ I believe, to perfect his accent. He was a very meticulous actor and extremely well prepared, and though I later read somewhere that Greg felt insecure about his British accent in this film, well, all I can say is I was not aware of it. It was a super film directed by Andrew McLaglen, produced by Euan Lloyd and co-starring David Niven, Trevor Howard and, initially, Diana Rigg; but, alas, it didn't work out for one reason or another, and happily Barbara Kellerman took on her proposed role of Mrs Cromwell.
Greg, always handsome and debonair, was one of the kindest people I've ever known.
Of course, during the shoot, we all socialized frequently in Goa and New Delhi, and very pleasant it was, too. In fact, we all had bungalows in the compound of Fort Aguada and would dine at one or the other each evening. Greg was joined by Veronique, and they were the most compatible couple ever, totally adoring of one another. It wasn't an easy location, mind, as the heat was quite often unbearable. Our big relief after a day's shooting was to immerse ourselves in the Indian Ocean.
We had New Year's Day off and director Andy McLaglen and I ventured to the ocean early, I guess around 9 a.m. Having lain in the sun for a few minutes, Andy got up and stretched out his 6' 7" frame. He turned to me and said, 'Rog, I think it's time we hit the drink.' (Meaning went for a swim.)
At that very moment, Trevor Howard appeared as if from nowhere and, overhearing, said, 'Good idea! Do you think we can get a waiter down here?'
In actual fact, contrary to popular opinion, the Trevor I knew wasn't really the 'hell-raiser' the newspapers described him as. A great cricket lover as well as being entirely devoted to his wife, Helen, Trevor undoubtedly loved a drink but he was in fact rather a quiet drinker on the whole, preferring to have a longer session in the corner of a pub, with a few friends. Things could get a little noisy when the 'Howard Roar' went up, though. He always said, 'I don't raise hell, amigo, I just like to enjoy myself.' And he really did.
However, having said all that, Trevor loved life to the full and was always open to new experiences, not wanting to miss out on anything. Which is why he was led, along with show business reporter Bill Hall, to Pamplona to run with the bulls. This was in 1972, and Trevor was no spring chicken at the time – he was fifty-six years old. Perhaps he could be forgiven for having a few stiff drinks on this occasion.
With the inimitable Trevor Howard pointing out the location of the nearest bar, on a beach in Goa while filming _The Sea Wolves_.
One of my favourite photographs, featuring three of my favourite people.
The story goes that he'd been attending a film festival some miles away from Pamplona and seen that the world-famous bull run was taking place. After a few drinks with Bill Hall one night, Trevor suddenly announced that he wanted to do the run, and asked Hall to join him. Several hours later the two found themselves in Pamplona's main square – still the worse for wear after several Bloody Marys – along with 2,000 other runners, waiting for the beasts to be let loose. Having seen this event several times on the TV, I can't imagine what they were thinking of, but come the time, they set off running, given a head start before the bulls were released. After a few minutes, out of breath, Trevor slowed to a walk, with the crowds lining the streets and Bill Hall urging him on to keep running.
Suddenly the bulls appeared round the corner, thundering down the street at full pelt. Trevor, deciding enough was enough, tried to get over the barrier at the side of the road but got his leg stuck and simply had to cling on for dear life as the bulls charged past, missing them both but mowing down others in their wake. Luckily, both Trevor and Bill escaped unscathed and made their way to the nearest bar.
Good old Trevor!
John Mills, that rather straight-laced, fine English hero of many a war film and Knight Commander of the Order of the British Empire, had a rather unique party trick, which extended to most film sets he worked on – not least in the cosy tents during the making of _Scott of the Antarctic_ – where he would drop his trousers, bend over and let off the most furious fart you can imagine but – as if that wasn't enough to impress – he would have a naked flame on standby and would ignite said anal wind to the great merriment of all around.
With Johnny Mills and Lady Mary at Johnny's eightieth birthday party at the St James's Club on Sunset Boulevard, along with Dudley Moore and his wife Brogan Lane.
In the 1960s I was asked to become chairman of the Stars Organisation for Spastics (SOS, now SCOPE) but when I started playing Bond it became apparent, in 1977, that I would have to leave the UK if I wasn't to pay ninety-eight per cent tax on my salary: an actor's life in the spotlight is short, so we need to look after our pennies, and that's why I decamped to Switzerland with its lovely snow-capped tax benefits. I therefore realized I would have to cut back on a lot of my UK commitments, SOS being one.
The committee asked if I had any ideas who could step in to replace me and I suggested Johnny Mills. He joked as the 'new boy' coming in he was actually a lot older than the outgoing chairman, but that was nothing new as I was older when I took over Bond from Sean Connery. Cheek!
Johnny and I were near neighbours in the UK in the 1970s, when he lived, with his wife Mary, in a house previously owned by the film mogul Alexander Korda and his one-time wife Merle Oberon. (I actually appeared in an episode of a TV drama series with Miss Oberon called _Assignment Foreign Legion_ , which was not filmed on location in the desert, oh no, but at Beaconsfield Studios in rural Buckinghamshire. Mind you, I never actually met Miss Oberon as she filmed her bits elsewhere. Bit of a pointless digression really, but I do like to name drop when I can.)
Anyway, back to the story. One day in 1976 I suggested John and Mary join my then wife and me for dinner at a rather upmarket dining pub in Denham – you know, the sort of establishment where you pay a small fortune for steak and chips. I think it's fair to say Johnny probably hadn't eaten at this establishment since his recent ennoblement by the Queen.
The headwaiter welcomed us warmly, 'Ah, Mr and Mrs Mills ... and Mr and Mrs Moore ... let me show you to a nice table.' He then proffered some menus. 'Would you care for a drink first, Mrs Mills? ... And you, Mr Mills?'
'Tonight, Mr Mills, I would recommend the T-bone ... Oh, and Mrs Mills, the Dover Sole is absolutely beautiful ...' The friendly waiter probably said 'Mr Mills' and 'Mrs Mills' a couple of dozen times over the course of the dinner, just as he did 'Mr Moore' and 'Mrs Moore'. Until Mary snapped, that is.
'It is SIR John and LADY Mills,' she hissed through clenched teeth.
Johnny's head dropped down towards his dessert bowl and he said, quietly, 'Well, I have waited long enough for it!'
Some years later, at Johnny's eightieth birthday party in LA's St James's Club, Mary beckoned me over and asked, 'Who is that man sitting over there, I know the face but I just can't place him?' I told her it was Omar Sharif, to which she looked at me rather vaguely. Anyhow, I was invited to say a few words and, after congratulating my host on his four score years, told the assembled company that Johnny owed most of his career to being the only actor in England who could stand full height in a submarine. Cue much laughter – except from the direction of Mary who looked at me decidedly more vaguely than ever!
I know when to make good my escape.
Johnny died in April 2005, aged ninety-seven, and I attended his funeral along with many of his closest friends and family: Lord Attenborough, Stephen Fry, Leslie and Evie Bricusse, Anita Harris, Dame Helen Mirren, Dame Judi Dench, Jack Hawkins' widow Doreen and even Cherie Blair. Dickie Attenborough, moved to tears, spoke for us all when he said, 'We shall miss him desperately. But we shall have him with us always in the deep love and unmatched joy that he has bequeathed to all of us.' It never stopped raining that day, and I believe they were actually tears from heaven.
The cast of _The Sea Wolves_. This photograph includes so many old friends – it makes me smile every time I see it.
Back on the set of _Sea Wolves_ , Greg Peck and I often chatted over a drink at the end of the day, and one evening I realized we'd both worked with John Huston – as an actor in my _Sherlock Holmes in New York_ and as a director helming _Moby Dick_ in which Greg had starred.
'Didn't like him,' said Greg, taking me aback a little, as I found Huston to be the consummate professional and a joy to work with.
'As a director,' continued Greg, 'all he cared about was getting the shot. In the scene where I was tied to the model of the whale in the tank at Elstree Studios and the waves were crashing all around me, Huston gave various instructions to lower the model into the water, and each time he held it there for longer than I could reasonably hold my breath. I was furious – he nearly drowned me.'
On another evening we were chatting about _Roman Holiday,_ that great romantic movie Greg had made with Audrey Hepburn in 1953. In the final scene where Greg's character has to say goodbye to Audrey's Princess Ann, knowing it's the last time he'll ever see her, Greg said he felt it was going to be a hugely emotional scene and one that he was not going to hold back on. As the tears of sadness ran down his cheeks, the director William Wyler leaned across and said, 'No, Greg, don't get upset. Get angry! Angry! Angry!'
And that's how Greg was forced to play the scene – angry that he wouldn't be seeing his love again. He said it was one of the defining moments of his career and he realized just how important listening to a director like Wyler could be.
Oh, before I forget, I must mention a scene in _Sea Wolves_ that I shared with the lovely Barbara Kellerman. It's the part where my character discovers that she is in fact a Nazi spy – though not before taking a bullet in my arm. The script called for me to change jackets, wrap a bandage around my bloodied and wounded arm, with blood running down into the palm of my hand, and go to the ball, which was being staged as a grand diversion for the attack on the German ships in the harbour. I went off to make-up and, as I was sitting in a chair waiting for my call, the Indian unit nurse came in, took one look, and said 'Oh my goodness! You have been injured!' and proceeded to attend to my fake wound. I don't think she'd ever worked on a film before.
I wish I'd been able to make more films with Greg and seen him more often, but sadly our geography placed us on different continents. I remember we did go fishing together once, off Cap Ferrat. We chartered a boat and forged our way out beyond the headland looking for signs of any water-propelled creatures. Having not caught so much as a herring, the captain called on the radio to other fishing boats in the area and they reported nothing in sight either. We then saw the reason: about sixty feet from us an enormous whale poked its nose up out of the water, obviously happy after its huge breakfast of fish.
Other places we used to hang out together were Crescendo and Ciro's nightclubs on Sunset Strip, where I saw Don Rickles perform a few times. Don took great enjoyment from insulting his audience – in fact he was known as an 'insult comic' – particularly the more prominent famous people in the audience. One night, I was there at one table, Gary Cooper was at another and opposite him was a Mafia boss. Rickles began giving Cooper hell, then he turned his attention to me and then to the Mafia guy, saying the most terrible and goading things. But he always capped it off by saying, 'I'm only joking, sir, I'm only joking, sir.' And then, 'I'm now going to walk among you and squeeze venom all over you!'
He once said of Frank Sinatra, 'When you enter a room, you have to kiss his ring. I don't mind, but he has it in his back pocket.'
One time a group of us were at Frank's weekend house, relaxing by the pool, and Don started on Gregory Peck. It was the usual wisecracking routine, 'Who picks your clothes, Greg – Stevie Wonder?' and so on. For the most part, Greg took it in good humour but later that night Don went a step too far with a wisecrack and Greg leapt to his feet.
'Shall we step outside and settle this like gentlemen?' he challenged.
'I'm only joking, sir! I'm only joking,' Don guffawed.
The next day, Greg's wife Veronique had her white fluffy dog draped around her shoulders, like a stole, and we were all sitting in cast iron chairs having lunch on the terrace. Again, Don started on Greg.
'When actors get old like you, Greg, they get mouths like flounders and can pull their lower lip over their forehead,' he said. As he spoke, Don moved his chair, and the grating of iron against the tiled floor startled the dog, which leapt off Veronique's lap and right under Rickles' chair, getting snagged between the floor and chair leg as it did so. It let out such a cry. Greg placed his cutlery down, signalled to his wife and they both stood up, picked up the dog, and went home. He simply couldn't take any more.
Other times, Greg and I played tennis or poker, or sat around telling jokes over a Jack Daniels or two. Whenever Kristina gave me a birthday party at Le Dome in West Hollywood, on what is called The Strip, Greg and Veronique were always the first to arrive and we picked up our conversation as though the intervening months had never occurred. I remember on one occasion at the end of the 1990s, Greg said he'd wound back on film work and was greatly enjoying taking his 'pony and trap' around theatres, whereby he'd sit on a stage and tell stories from his life and career. I always remembered the phrase 'pony and trap' and when I was invited to take part in a small tour in 2012 to help promote my last book _Bond on Bond_ (copies still available in all good book shops) I thought of Greg, and readily accepted. I took my pony and trap out again in 2013 – who'd have thought this boy from Stockwell would not only become an author, but a raconteur with his own stage show? I'll keep going until they find me out!
Our mutual friend, David Niven, was a happy constant in much of my life, and I shared many dinners with him over the years and played audience to his wonderful stories. Of course, we all knew most of them were exaggerated, borrowed or downright lies – but he told them so well. I remember one such story about how Errol Flynn and he (along with Niv's new girlfriend) went out boating one day from LA.
Miles offshore in the Pacific Ocean they started water-skiing (according to Niv he had introduced the sport to America!) and Flynn decided to cut Niv free and sailed off so that he could get to know the girl for himself. Niv then told us how, with only two skis for buoyancy, he swam miles back towards the shore – pursued by sharks if you please – until he was rescued by Ronald Colman.
Well, another one of those 'is it quite true?' stories came when Niv shared a house in Malibu with notorious bad boy Errol Flynn, which they called 'Cirrhosis by the Sea'. One of their regular visitors was the actor John Barrymore, who used to sit in his favourite chair and smoke his pipe, looking out on to the ocean. By all accounts, after John Barrymore died he was taken to the Utter McKinley Funeral Directors store on Sunset Strip, which had a clock with a long swinging arm in the window ominously counting down time.
One evening, Niv and some friends broke in and retrieved the body, took it home, sat it in Barrymore's favourite chair ... and awaited Errol's return home!
Flynn's book, _My Wicked, Wicked Ways_ , was going to be made as a film by Roy Huggins, who was a producer at Warner Bros during my contract years there, and Roy wanted me to play Flynn. Alas, Errol had what you might best say was a 'difficult' relationship with Jack Warner, the famed studio head, stemming from the time when one of Jack's brothers suffered a heart attack after a big argument with Flynn. One day shortly afterwards, Jack walked into the dining room of the studio and noticed a new English writer who had joined the staff who had an uncanny resemblance to Flynn.
'Get that son of a bitch out of here!' Jack exploded. 'He looks like Flynn!'
'But it's not Flynn,' someone said.
'I don't care! He _looks_ like Flynn. Get him out!'
I probably didn't help matters myself when I was called in for a meeting with old Jack Warner to discuss the role.
'I understand Roy wants you to play Errol Flynn?' said the great man.
'Yes, it's a part I'd love,' I replied.
'I also understand he calls me a thief in his book?'
'Well, aren't you?' I asked.
'Son of a bitch – I won't make the movie!'
Meeting over.
I only met Errol Flynn once – when I was understudying David Tomlinson and Geoffrey Toone in _The Little Hut_. David was going out with a girl who he knew had been in a relationship with Flynn some years earlier, so on hearing Flynn was coming to town, David thought he'd have a little fun with his love and faked a telegram:
'Darling. Am coming to London. Let's pick up where we left off. Love Errol.'
It didn't take long for the young lady to twig it was Tomlinson who had sent it, and she told Flynn about the gag when he landed in town.
'I'll fix the son of a bitch,' said Flynn.
Flynn was a huge, tall, very imposing man and he arrived at the Lyric Theatre Shaftesbury Avenue, just in time to stand in the wings with his sleeves rolled up for when David glanced in his direction ... and Flynn was glaring at him wildly. Terrified, David couldn't get his lines out for the next five minutes!
Charmer though he undoubtedly was, Flynn also loved to fight – and I mean fists flying, down and dirty, slugging it out. In fact, he loved fighting to such an extent that he was often to be found sparring with a professional fighter, just to keep himself in shape.
Another legendary slugger was the director John Huston, and the story goes that one night, fed up and bored at yet another Hollywood party, Huston and Flynn decided to retire to the garden and knock seven bells out of each other. They were on good terms, there was no issue between them, they just wanted a fight. So for a goodly part of the rest of the evening, the other guests were treated to the sounds of the pair of them, toe to toe, knocking each other around the garden – at the end of which they both wound up in hospital for repairs!
Another great actor with a fondness for the bottle was Robert Newton, who was a wonderful Bill Sikes in _Oliver Twist_ as well as an unforgettable Long John Silver in _Treasure Island_. Legend has it he once walked the length of the corridors at Denham Studios with his honourable member for Wapping hanging out while under the influence.
Newton had an unfortunate way about him when sloshed, and while making a film with Herbert Wilcox and Anna Neagle he upset them in a way only Newton could. Later that day he got a call from his agent saying that unless he turned up the next morning sober, on time and apologized, he'd be fired.
At 8.30 a.m., Wilcox and Neagle appeared on set and Newton was nowhere to be seen, but suddenly they heard him coming down from the gantry, rather unsteadily, on a ladder.
'I am told that I have to apologize to you both for my unseemly behaviour yesterday, well I'd love to but am afraid I cannot.'
With that he walked off the set and out of the picture.
I remember one time when I was in LA with my then wife, Dot Squires, Trevor Howard phoned from the Beverly Hills Hotel and said he was with Newton.
'Bobby, come here,' said Trevor. 'I want you to speak to Dorothy Squires, a great artist!'
Bobby came on the phone and slurred, 'Madam, I admire all your paintings.'
There's one more story about Bobby Newton that always makes me smile. He was appearing in a play in London's West End and towards the end of the run one Saturday night the curtains didn't rise. The audience was getting more than a little restless when suddenly there was a commotion behind the curtain and a pair of shoes appeared at the base. The audience went quiet, the curtain parted and Bob's face appeared through it.
'Ladies and Gentlemen!' cried the star. 'The reason this curtain hasn't risen is because the stage manager has the fucking impertinence to suggest that I am pissed!'
One actor I've always admired – in fact I've always been a little envious of – is Peter O'Toole, as I would have dearly loved to have played Lawrence of Arabia. I wouldn't have been anywhere near as good as him mind, and I did get to know him later.
Michael Caine once told me a story of when he was cast to understudy O'Toole in the play _The Long and the Shortand the Tall_ at the Royal Court Theatre, in London. One Saturday night after the show O'Toole invited him to a restaurant. Eating a plate of egg and chips was the last thing Michael says he remembered, before he woke up in broad daylight in a strange flat.
'What time is it?' he asked O'Toole, holding his aching head in his hands.
'Never mind what time it is! What fucking _day_ is it?' came the reply from a similarly slumped O'Toole.
It turned out to be five o'clock on Monday afternoon – and the curtain was set to go up at eight. They rushed to the theatre, where the stage manager told them the restaurant owner had been in and had banned them from his establishment – for life. Michael was about to ask what they'd done when O'Toole whispered, 'Never ask what you did. It's better not to know ...'
O'Toole was legendary for his boozing and that probably wasn't helped when in his first film, _Kidnapped_ , he starred alongside Australian actor Peter Finch – an even mightier drinker, if that were possible.
Shooting in Ireland, they were refused a drink because it was after closing time, so the stars decided to buy the pub and wrote out a cheque there and then. After an all-night drinking session, and having sobered up a little, they later rushed back to the pub and were mightily relieved the landlord hadn't cashed the cheque.
By all accounts O'Toole and Finchie remained friends with the publican and when he died his wife invited them to his funeral. Both men sobbed loudly at the graveside, and an overcome Finchie eventually had to turn away ... only for his face to change from one of sadness to one of confusion as he realized they were at the wrong funeral – their friend was being buried 100 yards away!
I always admired Peter O'Toole; legendary boozer he may have been but he was also a legend in front of the camera and a good dancer by the looks of things.
Richard Harris had a fairly well-earned reputation as a hell-raiser – he certainly lived life to the maximum, that's for sure. I first became aware of his 'personality' when I was making _The Saint_ at Elstree. Harris was over at Pinewood shooting a film called _The Heroes of Telemark_ with Kirk Douglas.
The director, Anthony Mann, had originally been signed to direct _Spartacus_ some years earlier but he and Kirk Douglas didn't get on, and so Mann was replaced by Stanley Kubrick. Mann had also previously worked with Richard Harris and they didn't get along too well either, so bringing them all together wasn't perhaps the best of ideas.
Kirk Douglas and Richard Harris were very jealous of each other and were constantly arguing about who the 'star' was. I remember my friend, publicist John Willis, telling me that their demands became increasingly ridiculous – to the point of seriously affecting the production. For instance, Harris rolled up at the studio one day with a tape measure, measured Douglas's trailer and then announced he was going home – apparently it was a few inches bigger than his – leaving the cast and crew with nothing to do until a longer trailer was found and brought to the studio. On another occasion, Kirk Douglas fired his chauffeur after an argument and Richard Harris immediately turned round and hired him.
Doris Spriggs – who went on to become my personal assistant for twenty-nine years – worked on _Heroes_ in the production department and told me that they were nearly run out of town when they were filming on location in Norway at a lovely old Norwegian church. The church had survived the worst ravages of World War II, only to be burned to the ground by the English film crew when an arc light overheated after being left on overnight. Compounding matters further, all the extras marched through the town dressed in full Nazi uniform thinking nothing of the effect it might have on the elderly inhabitants, who were convinced the Reich had risen once more.
Things got worse between the two stars when they moved on location to Rome, and one evening attended a film premiere there. Earlier in the day, the British papers ran a story about all the childish behaviour and petty rivalry between them, and Harris was understandably furious. When he saw John Willis in the foyer of the cinema, he pushed everyone else out of the way and demanded to know who leaked the story. John said nothing so Harris threatened to hit him, and would have done had they not been pulled apart.
But then, ironically, on the last day of shooting back at Pinewood, both Douglas and Harris were in the corridor walking towards each other – a bit like the scene from _High Noon_ – and John Willis found himself right in between them. 'I couldn't believe my eyes,' John said to me. 'As they met, they shook each other's hands like they were old friends and walked off to their dressing rooms!'
In the late 1960s, Richard Harris divorced his wife Elizabeth, and afterwards Elizabeth sent him a bird in a silver cage, with the message, 'Here's one bird that will never get away.' In turn, he sent her an antique cowbell saying: 'Wherever you go, I'll now be able to hear you.'
I always try to help out an old friend where I can, and such was the occasion when Rex Harrison called me one day to ask a favour. Rex was starring in a play in Oxford in 1969 and couldn't therefore attend the London premiere of _Staircase_ – a film he had made with Richard Burton – so he asked if I would escort his then wife, Rachel Roberts. I agreed and duly arrived at the Connaught Hotel, where she was staying, and called up to her room to say I was ready with the car.
'Come up to the room, darling,' said Rachel.
Conscious of time, I went upstairs and found the door open, and there inside stood Rachel in her petticoat – one breast hanging below her brassiere and the other above – with another lady in the room, sipping champagne.
'Come in and have a drink, darling!'
'No, no, Rachel, I'll wait in the bar,' I said, making a hasty escape from what I thought might turn into a sticky situation. After what seemed like an age, she came down, I leapt up and bounded across to the door.
'Time for another drink?' she asked.
'We really ought to go, Rachel.'
'Nonsense! There's always time for another drink, darling,' she said, dragging me back to the bar.
Eventually we got away from the hotel and drove to the premiere. When we pulled into Haymarket, where the premiere was being held, a whole hoard of photographers converged around the car as I opened the door for Rachel to step out.
Rex Harrison and Rachel Roberts – there were always fireworks when these two were involved.
'Here, get a picture of my boyfriend!' she yelled. 'I'll make you famous, Roger!'
I rather hurriedly pushed my inebriated friend into the theatre, just as she screamed, in her strong Welsh lilt, 'Are they 'ere?' She wasn't referring to the Royal party, which, in this case, was Princess Margaret, but rather to the Burtons – Richard and Elizabeth were due to attend.
'No,' I said, as I hushed her from screaming further and pushed her through to our seats in the dress circle. After a few minutes, Richard and Elizabeth arrived and the Grenadier Guards started playing music. At almost the same moment, Princess Margaret arrived at her seat and, before I could restrain her, Rachel shouted, 'I've got to say hello to my Richard!' and, with that, clambered over everybody, including Her Royal Highness, much to Burton's great embarrassment.
She returned to her seat, grumbling under her breath and humming along to 'We'll Gather Lilacs In The Spring', kicked her shoes off, put her feet up on the balustrade in front, and continued chuntering about how 'Richard cut me dead'. I explained he was in company with HRH and we really ought to settle down for the film.
No sooner had the film started than I heard a gentle snoring coming from next to me and thanked goodness she'd fallen asleep. However, minutes later, there was a sudden and very loud cry, and Rachel sat bolt upright.
'The bastards! They cut the close-up of my lovely Rex!' she exclaimed.
Never had ninety minutes seemed so much an eternity as that evening, and afterwards we all went to the Savoy for the party. As any dutiful star/host would, Richard (with Elizabeth) stood at the entrance to receive all the guests and, as I went through, Burton whispered, 'Good luck, boyo,' in my ear.
When the Toastmaster called for everyone to 'Be upstanding' as HRH entered, Rachel said, 'I'm not bloody getting up!' I dragged her to her feet, saying, 'You will!' through my clenched teeth, and really was beginning to regret ever accepting Rex's invitation.
Meanwhile, the dancing started and a waiter came to our table to say Rex was on the phone for Rachel. He'd just come off after his play, in which he was starring with Elizabeth Harris (ex of Richard Harris) and asked how Rachel was behaving. I was suitably diplomatic and gentlemanly – and then he asked to speak to Burton.
At this time, Burton was dancing with HRH, and so I quietly stood behind them and coughed politely to attract his attention.
'Oh, sorry boyo, you want to dance with 'er? Be my guest!' he said as he pushed Princess Margaret towards me.
'No, no ... I mean yes, I would, Ma'am, but no ... Richard,' I said, 'Rex is on the phone and wants to speak to you.'
'Right-oh boy, look after things here, will you?'
I smiled politely ...
Rachel was a lovely lady and a wonderful actress, but a terrible handful. Rex was her second husband and she was Rex's fourth wife. Rachel always had a great aversion to sailing, but every year Rex chartered a boat in the Mediterranean and took Rachel's best friend with him for the trip – Elizabeth Harris ... Having discovered Rex's affair with Elizabeth, Rachel divorced him in 1971. She was devastated after the divorce, moved to the States and tried to carry on, but it was reported that her alcoholism and depression increased, eventually leading to her suicide in 1980, aged just fifty-three. It was terribly sad.
I continued to see Rex from time to time and he was always friendly towards me, apart from one day when I was staying with Leslie Bricusse in France. Rex had arrived (uninvited) two days earlier, after being besieged by the press at his house on Cap Ferrat following Rachel's death. Leslie explained I'd been promised the guest room and the screenwriter Jack Davies, who lived next door, suggested Rex went to stay with him.
We all joined up for dinner at a restaurant one evening and I guess there were about six of us. Throughout the evening I felt rather guilty about Rex having to move and I decided that the least I could do was to get the bill for dinner, at which Harry Belafonte and his wife and James Baldwin had joined us, at my invitation. When we had finished, I proffered my American Express card and Rex came up from behind me, tore the card from my hand and threw it on the floor.
'I don't want your damned plastic!' he shouted.
'But I want to get dinner,' I reasoned.
'No! I don't mind paying for YOUR friends,' he snapped.
The next morning I was having breakfast in a pagoda in the garden at Leslie's home, when Rex came across the lawn, sat down and looked at me, 'I'll get breakfast this morning ...' I said, with a smile.
Curiously, Rex and David Niven just didn't get on and Rex was always rather pissed off with Niv, saying, 'He's been on the Cap long before I arrived, yet he's never invited me for dinner or a drink.'
Rex could be a rather mean-spirited man, to put it mildly and, unfortunately, a lot of people in the business had been treated badly or spoken to nastily by him at some point or other, which was underlined when I joined Kirk and Anne Douglas, along with Greg and Veronique Peck, to see a play in LA one night. Claudette Colbert was co-starring in it with Rex. Afterwards, we went round to see Claudette and took her to dinner at Chasen's Restaurant. But we didn't ask Rex, I always felt a bit sniffy about that, but I'm afraid the Douglas's had no time for him. I never needed to ask why.
Rex's next wife was Elizabeth (Liz) Harris – she was the fifth of his six wives. I, of course, knew Liz through my sometime co-star Richard Harris, and she lived with Rex in a big house off Belgrave Square in London. She once told Richard (who told me) of their lifestyle. Rex would dress immaculately before taking the lift down from his bedroom to the first-floor dining room for breakfast. In fact, he'd take a good thirty minutes to dress, and would put on his cape and newly polished boots even if he was just popping to the corner to post a letter. After breakfast, he would call the butler in to discuss the wine list for lunch, then go upstairs to change, only to re-emerge a few hours later in his tweeds for his meal and to sample the wine. The shout would invariably go up, 'How dare you serve me corked wine!'
Rex always sent the wine back in restaurants – and is the only person I ever knew who did the same at home too.
The butler, in what must have been a well-worn and quite frustrating routine, would have to ensure there were buckets of ice available at 11.45 a.m. and 5.45 p.m. in three different rooms in case Rex wanted a drink in any one of them. He insisted Liz dress for dinner every night too, even if they were eating in alone, and was of the firm belief that children should be seen and most definitely not heard, which made life a little tricky for Liz and her three sons. It might not surprise you to hear the marriage was short-lived.
Rex wasn't regarded very warmly by those who knew him (or even knew of him) but I will say the one very decent thing he did do was look after my lovely friend Kay Kendall when she became ill. Kay and Rex had become an item in the mid-50s, when he was still married to Lilli Palmer, and when he discovered from her doctor that she was suffering from terminal myeloid leukaemia, he arranged a divorce from Lilli in order to marry Katie (as we all knew her) and care for her, on the understanding he'd remarry Lilli after Katie's death. In the event, Lilli was also having an affair with Carlos Thompson and married her lover, so she and Rex never got back together.
Rex kept the illness from Katie, who believed she was suffering from an iron deficiency, and cared for her until she died aged just thirty-two. He often said one of his greatest pleasures was to 'simply sit and admire Kay'.
Quite how Katie put up with him I'll never know, but when Rex was starring in _My Fair Lady_ on Broadway she used to have to stand at the side of the stage for every performance when he sang 'I've grown accustomed to your face' as he point blankly refused to sing it to his co-star Julie Andrews, whom he hated with a passion. He in fact suggested the song should be dropped, but the producers wouldn't hear of it and so Rex said the only compromise would be if he could sing it to Kay.
Ironically, when he won the Oscar for the film version in 1964, he smiled widely as he dedicated it to his two fair ladies – Julie Andrews and Audrey Hepburn.
I got on very well with Peter Sellers and I knew three of his wives quite well, too. He was a solitary character though, always preferring to hide behind a mask, and consequently you never _really_ got to know the real Sellers. This was, after all, the man who said, 'To see me as a person on screen would be one of the dullest experiences you could ever wish to experience'.
Don't ask me why Peter Sellers was attempting to paint my toenails at Cubby Broccoli's house – I simply can't remember.
Although a star of comedy films, Peter very desperately wanted to be a romantic lead, though knew he wasn't classically good-looking. Sadly, he humiliated his first wife, Anne, when he told her about a great affair he was having with Sophia Loren, which was actually all in his head as there never was any romance with Sophia whatsoever. After divorcing Anne he met Britt Ekland at the Dorchester, as she was in London for a PR junket having signed a contract with 20th Century Fox. They married two weeks later. The marriage only lasted four years, as Britt couldn't live with Peter and his violent mood swings any longer. A couple of years later he married Miranda Quarry and, though I didn't go to the wedding, I was there for the honeymoon.
They were staying on the Cap Ferrat in the South of France, and I was staying at the same hotel while filming _The Persuaders!._
One day Leslie Bricusse was bringing Johnny Gold around the Cap to the bay of Villefranche in his Riva and I was on Sellers' yacht. Sellers and I spoke with one of the customs patrol boats and, having supplied them with a few hundred cigarettes and a couple of bottles of Scotch, we suggested they pull Leslie over on the pretence of him coming into the bay too fast. From a safe distance aboard Sellers' yacht we cried with hysterics as their boat was indeed pulled over and we could see Leslie's face turning red with embarrassment as the officials produced this cargo of illicit contraband (supplied by us) from down below. Protesting his innocence, Johnny started waving a large white envelope around, which was addressed to Sellers from his London tailor.
'We're here to see Peter Sellers!' he shouted, evidently hoping that this information would be enough to secure their release.
Finally, when we couldn't bear watching them any longer, we waved to the customs men to let the errant 'pirates' off. When they arrived on shore, Johnny gave the envelope to Sellers, who opened it only to reveal a big bag of some white powdered substance, together with a note saying it was a 'gift for the honeymoon'.
That's the closest I've ever come to being arrested, let me tell you.
That evening, back at the hotel, Sellers called us down to his room, and he was – shall we say – rather 'far gone' on the contents of his envelope, telling us his bed was a flying carpet and he was going to fly around the harbour – and asking if we would like to go with him ...
Talking of things potent, one of Peter's good friends was Graham Stark. I worked with Graham on _The Sea Wolves_ but he is most probably more fondly and famously remembered for his many appearances in the _Pink Panther_ films with Peter, they were old mates and loved working together. In _The Pink Panther Strikes Again_ Graham played an old Austrian innkeeper, and had the most wonderful – and often quoted – scene where Clouseau walked in to book a room and looked down at a little dog in the reception area.
'Does your dog bite?' asks Clouseau.
'No,' replies the old innkeeper, at which point Clouseau lowers his hand to stroke the 'nice doggie' and it attacks him.
'I thought you said your dog does not bite?!' exclaims Clouseau.
'That is not my dog,' replies the innkeeper.
Anyhow, filming the scene, director Blake Edwards announced, 'Graham, Peter and I think that you'd look good if you smoked a Meerschaum pipe when we do this scene.'
Graham had never smoked in his life, but happily agreed to go along with the request. The only problem being that they didn't load it with tobacco, but hash.
Graham dutifully puffed away on the pipe but every time he opened his mouth to speak, only gibberish came out. Edwards, Sellers and the entire crew couldn't stop laughing. Graham, meanwhile, thought it was the best day of his comedy life as he'd never had this amazing comedic effect on anyone before. Poor Graham.
In later years, like me, Sellers lived in Gstaad and he had the most wonderful chalet there. After divorcing Miranda in 1974, he married a young actress named Lynne Frederick – she was twenty-three, he was fifty-two. Many of his friends felt that marrying the much-younger Frederick was a mistake and regarded her as an opportunist who married Sellers for his money. Just before his untimely death in 1979, Peter had made arrangements to see his lawyers to change his will and exclude Frederick, whom he was on the verge of divorcing. The night before he was due to sign the papers he suffered a massive heart attack and died, leaving Frederick, his widow, to inherit almost his entire estate, which was estimated at £4.5 million, plus all future royalties from his films. Meanwhile, he left his children £800 each in a calculated and deliberate move to make them find their own way in life. It is thought that the feeling of rejection ultimately led to his son Michael's early death. Very sadly, Michael died at fifty-two, exactly twenty-six years after his father's death.
Of course, Frederick continued to profit from the estate and even sued Blake Edwards and United Artists, the producers of _Trail of the Pink Panther_ , which was made after Sellers' death and used out-takes of the late actor. She was awarded $1.475 million in damages for 'insulting the memory' of her late husband.
After a very brief marriage to David Frost, she married a surgeon named Barry Unger, by whom she had a daughter, Cassie. Aged just thirty-nine, Frederick died in 1994, and her mother Iris inherited the estate until Cassie came of age. Which is how it came to pass that a person whom Sellers never knew now controls his estate and owns all of his belongings, while his own natural children remain disinherited.
As for my own experiences of Lynne Frederick? It was around 1977 or 1978 when Peter called me at my home in Tuscany, saying he was coming into the port nearby with his yacht and had Dr Christiaan Barnard (his heart surgeon, who had performed the first human heart transplant) and Lynne Frederick on board, and asking, 'Could we meet?'
I drove down to the port and found Peter leaning against a rail on the deck of his boat while Lynne was busy massaging his member, which in turn was popping out of his swimming shorts to say hello.
'Uh-oh!' I thought. 'She's trouble!' And I think I was right.
Gstaad and _The Pink Panther_ feature heavily in a story Victor Spinetti told me. Victor was the most wonderful raconteur and larger-than-life character. I first met him when he guest-starred in _The Saint_ , though of course he more famously went on to appear in several of the Beatles' films – as he would often tell anyone who happened to be in earshot.
A year or two before I moved to the Swiss ski resort, Victor decamped there to film _The Return of the Pink Panther_ , which my old friend Lord Lew Grade funded. Peter Sellers had been lured back to play Clouseau a decade after his last outing, due, no doubt, in part to him needing to re-establish his box office appeal following a few not very successful films. Victor had a few very funny scenes as a hotel worker, which left Peter in hysterics on set as it happens. A good thing, surely? No, I'm afraid not, as in the rushes screening Victor got more laughs from the crew than Peter did, and the editor was ordered to move in with his scissors.
Quite oblivious to this, Victor later accepted the invitation from Lew Grade to attend the star-studded premiere in Gstaad and to take part in the various press junkets. It was only when he arrived in Gstaad that he was told that most of his screen time had been left on the cutting-room floor. Meanwhile, all the posters in the town proclaimed welcome to the stars 'Peter Sellers, Christopher Plummer, Catherine Schell and Victor Spinetti' in huge four-foot-high lettering. Victor realized he could hardly back out and return to the airport, so he agreed to do whatever they wanted.
Victor Spinetti was a fantastic raconteur. His fund of stories was legendary and he told them with great humour and warmth.
The one line Victor had left in the film was when Clouseau asked Victor's hotel manager character, 'Do you have a rheum?' and Victor responded with, 'A _rheum_?' That was it.
Dreading that the press might ask him how he prepared for his role in the film, Victor called his old mate Richard Burton, who was then also resident in Gstaad, with his sometime wife Elizabeth Taylor, and confided in them that he was feeling rather uneasy, particularly about having to attend the big post-film party afterwards, where he was sure he would feel a bit of a sham celebrating his role. Richard, having not been invited, decided that he and Liz would support their old friend by turning up and – of course – the photographers went mad. Peter Sellers' face dropped and Liz Taylor proceeded to wind him up even further by saying things like, 'Why did you choose _this_ place for the party?' and 'What's that _awful_ music the band's playing?' (it was 'The Pink Panther Theme') and so on. Then, director Blake Edwards introduced his wife, Julie Andrews, to sing, at which Burton leaned over to Victor and said, 'Anything you can do to follow, old love? The world's press are here and might discover you all over again.'
Victor suggested he could do his monologue of 'When Alec Guinness was fucked by the Turks', to which Burton enthusiastically agreed, and as soon as Julie Andrews completed her song, Richard stood to his feet and said, 'Ladies and gentlemen, and now my great friend Victor Spinetti ...!'
Sellers leapt up and shouted 'NO!'
Burton smiled, 'Of course, Peter, I'm sorry. Come Victor. Come Elizabeth,' and they swept out, only stopping momentarily for the blinding series of camera flashes.
'We'll get you in the bloody papers yet, Victor!' said Burton.
The next morning, Burton arrived at Victor's hotel to take him back home for lunch with Elizabeth. On the way back they picked up the European newspapers and found their three faces plastered across most of the front pages.
'Success!' cried Richard. Only on closer inspection they discovered the captions all read, 'At the premiere of _The Return of the Pink Panther_ Richard Burton, Elizabeth Taylor and Mel Ferrer...'
That's show business, folks!
Dean Martin and Jerry Lewis in the early days. One of the greatest-ever comedy pairings.
CHAPTER 6
_The Rat Pack_
FUNNILY ENOUGH, I KNEW SOME OF THE _ORIGINAL_ RAT Pack from the 1950s... David Niven was one, as were Frank Sinatra, Humphrey Bogart, Lauren Bacall, Swifty Lazar, Cary Grant and Rex Harrison among others. You see, it was a moveable feast then, 'visiting members' were allowed and the name was actually coined by the lovely Bacall when she gazed upon the very motley crew of revellers returning from a show in Las Vegas.
Niv told me the story that Frank Sinatra had invited a few friends to join him at the Desert Inn in Vegas, where Noel Coward was opening in a show. There were about fourteen of them, so they took the overnight train, in a private coach, of course, for the overnight trip. It was champagne and caviar all night. Next day, they partied in the day and then watched Coward's triumphant first night, before partying some more. This went on for four days and four nights, and I remember Niv remarking that towards the end of those four nights, Frank was the only member of this merry band who seemed able to cope with it. It was at this point that Lauren Bacall made her famous remark, 'You lot look like a goddam Rat Pack!'
Humphrey Bogart was always hailed as the head of the Rat Pack, but Frank Sinatra was the President, and after Bogie's death he assumed the mantle. The Rat Pack continued to have a 'fluid' membership until the sixties, when it was used – by the press – to refer to Frank, Dean Martin, Sammy Davis Jr, Peter Lawford and Joey Bishop. I came to know Frank, Dean and Sammy in later years.
I first met Frank in the 1950s at a club called Moulin Rouge in Hollywood. I was then under contract to Warner Bros. and was invited, as a very minor celebrity, to attend a Thalian Society charity fundraiser that was themed around 'Cowboys and Indians'. At the dinner, my first memory of Frank is of him having a rather public confrontation with the most famous cowboy of all, John Wayne. The unpleasantness stemmed from the fact that Jack Warner, the head of Warner Bros., bid $1,000 to have a Warner Bros. contract artist, Gordon MacRae, sing at the event... only for John Wayne to bid $2,000 for him _not_ to sing.
Frank felt The Duke had insulted his friend and made a point of it. Seeing the tall-framed Duke face up to Frank – who was a lot shorter, thinner and, with his gaunt cheeks, certainly appeared less formidable – was something I'll never forget.
But Frank stood up for his friends. He was fiercely loyal. That was just one of the qualities that endeared him to me.
There had been history between Frank and The Duke going back some years, but eventually they patched up their differences – enough that when an All-Star Tribute to the great Wayne was produced in the mid-70s, Frank agreed to host the show. It's said that even The Duke was surprised to see Frank hosting, but the evening went well and the stars – including Maureen O'Hara, Jimmy Stewart, Lee Marvin, Angie Dickinson, Sammy, Bob Hope and the like – all shone bright. Towards the end of the show, after all the songs and tributes, John Wayne stole the show when he rose to address his guests, 'Tonight you've made an old man and an actor very happy. You are happy, aren't you, Frank?'
With two founder members of the _original_ Rat Pack, talent agent Swifty Lazar and David Niven.
The night Frank and I met, we only really exchanged pleasantries. Frank greeted me warmly and said, 'Hi, I'm Frank Sinatra,' as if I – or anyone else in that room come to that – didn't know who he was. His humility was another of his qualities that so impressed me. I thought this was a man I would like to meet again.
Francis Albert 'Frank' Sinatra's entry into this world had not been an easy one. For as long as I knew him, Frank continually said that he would like to meet and kill the doctor who delivered him. He had used forceps, which had badly damaged Frank's ear and scarred his face. Alarmed that the just-born infant was not breathing, and with the doctor seemingly not knowing what to do, his grandmother hurriedly grabbed Frank and placed his head under a cold water tap, startling the baby into taking his first gasps of New York air.
With dear Frank Sinatra and yet another princess, this time Princess Margaret at a swanky function in London.
In later life, Frank suffered from terrible hearing and ear problems. He maintained it was due to that doctor's negligence, which only fuelled his desire to 'bump into him' even more. As well as being loyal to friends, another of Frank's qualities was that he always bore a deserved grudge.
I didn't meet him again until a decade later, in the mid-60s, at a London restaurant with his then wife Mia Farrow, who was in town making a film called _Guns at Batasi_ (in which she replaced Britt Ekland whose new husband Peter Sellers had whisked her to LA for the weekend and refused to allow her back to Pinewood to continue filming) and Frank accompanied her.
'We just love watching _The Saint_ ,' said Mia, taking me totally unawares.
'We watch it in bed, in our hotel room. It's the best thing on TV,' Frank added.
We had dinner and chatted until the very late hours about movies, TV, London, family and just about everything else. Frank was very grounded and it was clear his success hadn't spoilt him. He also made time to smile and say hello to our fellow diners as they passed our table, which undoubtedly made their night – being in his company certainly made mine.
I can't help think about an old story here – probably apocryphal – of a young man who approached Frank in a restaurant, and said, 'Mr Sinatra, I'm so sorry to bother you, but I have my new girlfriend joining me on a dinner-date here this evening, and I wondered if you'd mind saying, "Hi, Al! How you doing?" when she gets here? It'll really impress her.'
Not wanting to deny the young chap a chance of getting his leg over, Frank said sure, no problem. A couple of minutes later the young lady duly arrived, Al leapt up to greet her and, in walking her to the table, they passed Frank.
'Hey, Al! How you doing, my old friend?' said Frank.
The young man turned around and snapped, 'Oh, fuck off, Frank! Can't you see I'm with my girlfriend?'
I was a huge fan of Frank's, both his music and films, and revelled in his stories over the dinner table. He told me of the making of his movies and the people he'd worked with and hoped to work with in the future. Frank began his musical career in the swing era, with Harry James and Tommy Dorsey – Dorsey in fact called _him_ 'kid' as a term of endearment. Frank told me that he realized a large part of their performing success was down to their ability to 'phrase', that is, to 'say play' their instruments and play without appearing to breathe and interrupting the rhythm. Frank took up underwater swimming to help increase his lung capacity and develop his breathing technique. From there, he practised his phrasing method, which you'll see in evidence on screen whenever he sings: he developed a slight twitch in the corner of his mouth through which he breathed in, while expelling air through the rest of his mouth in song.
As well as captivating all audiences with his singing, Frank also had a great screen presence. His ability to glide across it with seeming ease was one I envied greatly. Frank's acting came naturally to him. He was an all-round natural performer, but he only ever did one good take.
George Schlatter, who directed him in several advertisements and TV shows, told me that Frank psyched himself up so much for take one that if you didn't get it on that first shot, then you never would. While he rehearsed dance routines for the big musical numbers, Frank literally walked on to the set word-perfect and ready to go. He never rehearsed dialogue scenes and loathed re-takes.
'Bang-bang-bang, went the scene,' said George. 'And Frank would leave to await the next set-up.' He didn't suffer fools and expected everyone else to be as fully prepared as he was.
I often read about Frank's alleged personal and professional links with Mafia figures such as Sam Giancana, Lucky Luciano and Rocco Fischetti, so I asked him about it.
'Kid, most of the venues I play are, in one way or another, controlled by the Mob – they run Vegas for a start,' he said. 'I turn up at the places and am greeted by all these suited guys who want me to pose for a photo with them. I've no idea who they are, yet in the photos they're standing with their arm around me and appear to be long-lost friends. How many photos have you posed for with people you don't know?'
Mind you, I don't think Frank ever did anything to publicly dispel the rumours of his underworld connections. He rather liked it!
Frank had three children: Nancy Jr, Frank Jr and Tina by his first wife, Nancy Barbato. He was married three more times, to the actresses Ava Gardner, then Mia Farrow and finally to Barbara Marx, to whom he was married at the time of his death. He, like any loving father, supported his children in every way he could. I remember on Nancy Jr's thirtieth birthday, she opened her present from her father to discover it was a clear $1million wrapped up. Frank was very generous like that.
When his daughter Tina made the dramatized biography of Frank for TV, apart from mis-casting a rather – dare I say – plain-looking actress to play Ava Gardner, she showed her father having a string of affairs – and usually interrupted his bedroom scenes with a phone call to say his wife was in labour! It was a very unflattering biopic to say the least. I broached the subject with his wife, Barbara, and asked what Frank thought of it. She said that she had sat Frank down in the TV room with a large drink and ran the tape for him to watch, alone. When he emerged from the room he said, 'I don't think they quite got me, did they?'
My former wife and I socialized a lot with Frank and Barbara, and spent virtually every Thanksgiving and Easter at their lovely home in Palm Springs, along with Gregory and Veronique Peck, Don Rickles, George and Jolene Schlatter, Cary and Barbara Grant and sometimes the legendary agent Swifty Lazar. They were great days. We would chat, run movies, eat, drink and go swimming in the pool. It was a rare recipe for total relaxation – which Frank relished.
On Easter Sunday Frank would clear us out of the kitchen, and cook his favourite dish of spicy meatballs and pasta. It was the one day on which no one else was allowed to cook. Our only involvement was to help select the accompanying wine.
'Come on down the wine room,' he'd say, 'and choose a bottle.' Frank had one of the best-stocked (and biggest) wine coolers in the world, and was an expert sommelier. After an hour or so of exploring we chose many more than one bottle! Alas, I'm not a great authority on wine, but I can assure you that we drank some pretty good vintages, not to mention expensive ones. But that was Frank – a very generous and attentive host.
One of our other favourite pastimes, aside from drinking copious amounts of Jack Daniels into the night, was gambling. We played the tables in Vegas on many occasions – rarely ever winning much, I might add – and as Frank was very much a night person, he loved it when people stayed up into the early hours with him. It wasn't so much the gambling as the company he loved. He was a very social person and came to life in the late evenings. I couldn't always keep up with him, especially if I needed a clear head the next morning for work.
Frank had a view on hangovers, as he did most things, 'I feel sorry for people who don't drink. When they wake up in the morning, that's as good as they're going to feel all day.'
Despite increasing success, in 1971, at a concert in Hollywood to raise money for the Motion Picture and TV Relief Fund, and at the age of fifty-five, Frank announced that he was retiring. Perhaps he thought he was at the pinnacle of his career, and this was the time to bow out? His self-enforced retirement didn't last long, though – his public wouldn't allow it. Two years later, Frank returned with a television special and an album, both entitled _Ol' Blue Eyes is Back_. He certainly was.
As far as I was concerned, Frank had never changed from the first day I met him. He remained humble, kind, generous and warm. He only ever had praise for me when he saw one of my movies, and when I was cast as James Bond he called to say how delighted he was for me. He never once criticized or offered advice on any of my performances, such as they were, he merely expressed his satisfaction.
Despite various health problems, Frank toured extensively and remained the highest attraction on the worldwide concert circuit during the first half of the 1990s. Alas, at times his memory failed him as did his hearing. I asked how he could still pick up his musical cues. He said he 'felt' it through his feet, via the reverberations on stage. Nothing was going to get in the way of Frank performing.
Having fought bravely against cancer, Frank sadly suffered a heart attack and died at the Cedars-Sinai Medical Center at 10.50 p.m. on 14 May 1998, with wife Barbara and daughter Nancy by his side. His final words were 'I'm losing'.
The following night the lights on the Las Vegas Strip were dimmed. A light had gone out in all our lives. The tributes were manyfold and came from far and wide: Presidents, Prime Ministers, Royalty and, of course, his many friends and fans.
Frank's legacy to the world is a vast one: his many films and TV performances, his many concerts and his many recordings will live on forever, as will my memories of this amazing man who I was so fortunate to be able to call a friend.
Dean Martin was a former boxer turned comedian, actor, singer and hugely charismatic man. I first encountered him way back in 1952 when, with my then wife, Dorothy Squires, I was in New York and we trotted to Times Square to see 'Dean Martin and Jerry Lewis' at the Paramount Theatre. I'd heard a lot about Dino and Jerry from the cast of the play _Mr Roberts_ with whom I'd worked in London – and I was not disappointed. It struck me that, like Morecambe and Wise in the UK, Dino and Jerry were both very talented individuals in their own right, but together they were dynamite.
_The Cannonball Run_ , in which I played someone who thought he was Roger Moore.
A few years later, when I was a contract artist at Warner Bros., I bumped into Dean on the studio lot – where he and Jerry Lewis were making their caper films – several times and exchanged pleasantries.
Later in the decade, of course, he became a founding member of the Rat Pack with Frank Sinatra, Joey Bishop, Sammy Davis Jr and Peter Lawford, and went on to even greater movie success with them in _Ocean's 11._
With the advent of rock and roll, Dean's career as a crooner was eclipsed by his film work. He was riding high as a movie star and between 1966 and 1969, Dean starred in perhaps his most famous role in four Matt Helm films. They were, in essence, a US version of the Bond movies, though perhaps lighter in style. A planned fifth film, _The Ravagers_ , was cancelled after the murder of Dean's co-star and friend Sharon Tate. He said he was too distraught to consider ever playing the character again. In fact, Dean had always been fiercely loyal to his co-stars, as demonstrated when 20th Century Fox fired Marilyn Monroe in _Something's Got to Give_ (1962) and then attempted to replace her with Lee Remick. Dean reminded the studio that he had contractual approval of his leading lady and point blank refused to continue without Monroe. She was re-hired, but sadly died of a drug overdose before shooting could resume. Nine hours of unseen footage remained in the vaults at Fox for decades.
I had to wait until the 1980s before I had the chance to work with Dean in _The Cannonball Run_ and he became a very close friend thereafter. I'll never forget when we were shooting in Atlanta, where the cars were all lined up before the race started, and I noticed there were two very pretty showgirls engaged in conversation with Dean. When I sidled over, I smiled and asked, 'Have you just met the new Mrs Martin?' Talk about the eternal flirt!
Contrary to his image as a drinker, Dean told me he'd actually endured a lot of eye surgery in recent years and had been prescribed heavy painkillers, which meant he couldn't drink. In fact he became hooked on the pills for a time. His trademark cigarette and glass in hand was known the world over, but what wasn't known was that the glass was filled with apple juice.
When he toured with Frank and Sammy on the 'Rat Pack Reunion', I occasionally visited and took in a show. But I noticed Frank (who was first on) would always sing one of Dean's numbers, and then Sammy came on and told one of Dean's gags. So when Dino eventually appeared on stage and the music piped up, Frank would shout, 'We've already sung that!' So Dean would start telling a story...
'I've told that one already!' Sammy would call over. This went on for a few shows and really got under Dean's skin, until one night, as soon as the curtain fell, he took Frank's private plane back alone to LA and refused to ever work with Sinatra again. They only made it up a short time before Dean's death.
I know the greatest tragedy to befall Dean, and one I don't think he ever really recovered from, was when his son, Dean Paul Martin, died in a plane crash in 1987. So much so that, as cancer took hold of my friend in the mid-90s, he consoled his loved ones by saying we shouldn't worry as he'd soon be reunited with his son.
His tombstone carries the title of his most well-known song: 'Everybody Loves Somebody Sometime'.
Sammy Davis Jr was often billed as the 'greatest living entertainer in the world' and no wonder, as son of vaudeville star Sammy Davis Sr, he could do it all: sing, dance, act, perform stand-up comedy, play instruments and do just about anything else you'd care to throw at him.
I was working in television at Warner Bros. when I first met Sammy, and he was a real movie buff who loved nothing better than being around a film studio – whether he was working or not. When I was filming the TV series _Maverick_ , he became quite a professional pistol drawer and a past master at gun twirling. I socialized quite a lot with Sammy and realized his fascination with guns hadn't diminished when, in the 1980s, another old pal of mine, lyricist and playwright Leslie Bricusse, threw a dinner party at his home in LA and Sammy arrived, opened his jacket and removed his revolver – handing it to the maid as you would a hat.
Sammy was a hugely funny man and a deeply religious one too. His spirituality stemmed from a near death experience in a car accident in 1954, in San Bernardino, California, as he was making a return trip from Las Vegas to Los Angeles. It was as a result of this accident that he lost his left eye. At the time, it was feared he might lose his other eye, and his friend the actor Jeff Chandler offered one of his own eyes if it would save Sammy from blindness. Thankfully it wasn't needed, but that always struck me as a truly selfless offer. After wearing a patch for a while, Sammy was later fitted with a glass eye, which he wore for the rest of his life.
Though never one to miss an opportunity of sending himself up, Sammy once joked after overhearing someone complaining about discrimination, 'You got it easy! I'm a short, ugly, one-eyed, black Jew. What do you think it's like for me?'
Years later, when he was hosting the Oscars, Sammy remarked, 'Tonight, the Academy honours both my peoples with _Fiddler on the Roof_ and _Shaft_.' Audiences around the world loved Sammy's self-deprecating style and wit.
In 1957, Sammy famously became involved with the beautiful Kim Novak, who was then one of Columbia Studios' prized young contract stars. Fearful of any negative press this relationship might attract – and its effect on the studio – boss Harry Cohn called mobster Johnny Roselli and asked him to persuade Sammy to end the affair. Roselli did this by kidnapping Sammy for a few hours and eventually persuading him it was the only way of not having his one good eye ending up on the table. Sammy hastily arranged a marriage to black dancer Loray White for the following year in an attempt to put any controversy to bed, but the union only lasted fifteen months.
I was having dinner with Cubby and Dana Broccoli at the White Elephant in London in the early 1970s and Sammy was at a table across the room. Through a series of gestures he asked if we wanted to go and see a movie with him. Cubby gestured back asking what it was, and Sammy smiled, put a finger in his mouth and then pointed at the collar line of his neck. None the wiser, we agreed and he then took us to the private cinema at the Mayfair hotel and _Deep Throat_ started.
The whole cast on _The Cannonball Run_ – including Dean Martin, Sammy Davis Jr, Burt Reynolds and Dom DeLuise – was incredible and the director, Hal Needham, kept several cameras running at all times, just to make sure he didn't miss any of the comedy ad libs.
I guess we watched ten or fifteen minutes before Dana suggested we should leave. I think her decision was prompted by the leading lady, having oral sex performed on her, delivering the line: 'Do you mind eating while I smoke?'
Cubby and I grumbled we wouldn't have minded seeing the rest of the film, but our wives did not permit!
A little while later in Hollywood, Sammy invited us to dinner at his home and he'd had a giant screen installed in his living room to run movies. After dinner, we all settled down and waited for his projector to crank up for the promised 'great movie', and _Behind the Green Door_ – the follow-up to _Deep Throat_ – started. That was Sammy's sense of humour!
Sammy had been friends with Cubby for years and, in fact, he was offered a part in _Diamonds Are Forever_ , which, of course, was filmed in Vegas. Before the sinister assassins Mr Wint and Mr Kidd murder the smuggler Shady Tree, a scene was shot in which Sammy is seen playing roulette at the Whyte House club, and having an exchange with Bert Saxby about contract differences he has with Willard Whyte. The conversation stops when Sammy sees and recognizes James Bond, saying, 'They ain't never gonna get a cake big enough to put him on top of.' Sadly, the scene ended up on the cutting-room floor as it was felt it wasn't needed – and when watching the final running time any 'unnecessary' scenes are the first to go, I'm afraid.
Sammy and Frank Sinatra once (reluctantly) agreed to perform at the MGM Grand hotel in Vegas while sharing the stage with Leo the Lion between them. They were assured it would be safe, as the very old lion would be handled, with a choke chain, by its trainer. In the middle of the number, the beast looked at Sammy and licked its lips, then leaned back on its haunches, as if it was going to leap. Sammy made the sign of the cross. Luckily nothing happened, but after the show Sammy and Frank were having drinks in the lobby and Frank said, 'Boy, I thought that cat was going to come after you there. Hey, Smokey, didn't you turn Jewish? Why did you make the sign of the cross?'
'Well, babe,' Sammy replied, 'I didn't think I would have time to make the Star of David!'
You would often find Sammy in Vegas playing a residency at one of the hotels. Alas, he was a big gambler and, like many entertainers on the strip, he worked there to pay back his losses.
I last saw Sammy a month before he died; he was lying in bed and music legend Quincy Jones was sitting in an armchair next to him. Comatose and full of morphine as he lay dying, Sammy had always been a tiny, thin man but there really wasn't much of him at all that day. The throat cancer that had so tragically struck him had spread. Though when he was told that a laryngectomy would offer him the best chance of survival, Sammy replied he would rather keep his voice and face the illness than have a part of his throat removed.
After his death, Frank Sinatra paid off many of Sammy's outstanding debts.
With Bryan Forbes and his wife, Nanette Newman, in St Mark's Square, Venice.
CHAPTER 7
The Creative Geniuses
IT'S TRUE TO SAY THAT BEHIND EVERY GOOD ACTOR STANDS A terrific writer and a highly talented director. Bryan Forbes, or Brownie as I always called him, and I first met during our National Service sixty-odd years ago, when we were both stationed with the Combined Services Entertainment Unit in Hamburg. We became great friends, and I'm happy to say that that extended into a very happy working relationship when Bryan was head of ABPC films and, in 1970, greenlit what I have always believed is my best film, _The Man Who Haunted Himself_. He later directed me in _Sunday Lovers_ and, when Cannon Films approached me to make a film for them in 1985 and I suggested the Sidney Sheldon book _The Naked Face,_ they asked if I had a director in mind – without hesitation I told them to call Bryan.
Bryan's stories of his adventures in the film business were wonderful. For example, there was the time in the late 1950s when Cubby Broccoli came to England with Alan Ladd Jr to make _The Black Knight_. It was not a promising script, but a cast including Peter Cushing, Patricia Medina and some of the best technicians alive was assembled.
Bryan Forbes green-lit _The Man Who Haunted Himself_ (ABOVE), which I consider to be my best film (seen here with Hildegarde Neil); and directed _Sunday Lovers_ (seen here with Lynn Redgrave and Priscilla Barnes). The tagline on the poster for _Sunday Lovers_ was 'NEVER has turned to ALWAYS on a Sunday'. Fun to make? Yes, I think that's a fair assumption...
Bryan was drafted in to work on the script as, early on in his writing career, he was a contract script doctor, called in to perform emergency surgery on terminal cases. Bryan described _The Black Knight_ as 'the brainchild of half a dozen parents'.
One Saturday afternoon the producers rang him to say that they had reached an impasse. 'We've run out of pages,' they told him. 'Could you come up with something by Monday morning?'
Bryan was young and hungry, and with the misguided confidence that often goes with these two factors, he agreed and was shown footage of what had already been shot and what needed to be bridged.
The one big problem, he discovered, was that Sue Ladd – Alan's wife – had script approval. Sue, who had been an actress herself and was Ladd's agent, was quite a force to be reckoned with, and every word uttered by Ladd had to first be approved by her. Bryan came up with a few pages in which Ladd dodged arrows, vaulted from the castle battlements into a cart of hay, sliced a few of the villains in two with his sword, seized a horse and galloped across the rising drawbridge just in time.
What was Mrs Ladd's verdict?
'Alan Ladd does not steal horses.'
She went on to explain that if he did, they would lose the Boy Scouts Association, the Daughters of the American Revolution and probably half his fan club. Everyone was dumbfounded. However, Cubby's partner, Irving Allen said, 'Sue, he's not _stealing_ a horse, he's _borrowing_ one.'
She was not convinced. So Bryan came up with a solution. After Ladd had done all his vaulting and slicing he strode towards a sentry and uttered the immortal words: 'Is this the horse I ordered?' He then jumps onto it and gallops off. Sue agreed it! And that's what they shot.
You quite honestly could not make these stories up! It's almost as bad as dear Tony Curtis saying: 'Yonder lies da castle of my foddah,' in (reportedly) _The Black Shield of Falworth._
As much fun as that sounds, dear reader, it is a somewhat apocryphal story perpetuated by Debbie Reynolds. Tony actually delivered the line: 'Yonder in the valley of the sun is my father's castle' in a film called _Son of Ali Baba_. Debbie, in a TV interview, misquoted it (and the film) and somehow the story stuck in the minds of the public.
Years later, Tony was at Hugh Hefner's house at a party and Hef greeted him with, 'Yonder likes da castle of my foddah'.
'I never said that,' Tony replied coolly.
'Then don't tell anybody,' Hef said. 'It makes a great movie story!'
Bryan was never averse to sharing his movie stories with a wider audience, and many years ago he volunteered as a prison visitor to do just that. One time, he gave a lecture of some pith and moment (or so he believed) about the film industry and then asked for questions. A hand went up immediately: 'Is Lana Turner a good fuck?'
Bryan saw then that his erudition had fallen on stony ground!
Tom Mankiewicz, who had written a couple of my Bond scripts, was the son of Joseph L. Mankiewicz, who wrote and directed _Cleopatra_ , and consequently he had many a tale to regale us with on set. One of my favourites was how, in 1961, young Tom landed himself the job of production assistant on _The Comancheros_ , a Western starring John Wayne, Stuart Whitman and Lee Marvin. It was directed by the renowned Hungarian director Michael Curtiz, though John Wayne also took a co-director credit.
They were on location in Moab, Utah and were setting up a scene where 200 head of cattle were being driven towards a canyon. Seeing the twenty-foot sheer drop ahead of them, Mankiewicz asked Curtiz what would happen to the cattle.
'They'll go over the edge. Those that die we'll sell for carpets,' he was told.
'You can't kill the poor animals!' exclaimed Tom, amazed at the director's total lack of regard for life.
'You're fired!' snapped Curtiz. 'Get off my set!'
Tom headed back to his motel and had started packing when the phone rang. It was 'Duke' Wayne.
'What are you doing, kid?' he asked.
'I'm packing, Mr Wayne. I've been fired.'
'Were you the kid who saved my cattle?'
'Your cattle, sir?'
'Yes, my cattle!' said Wayne. 'Start unpacking again and get back to the set – you're now my assistant.'
The next day the production team were setting up a sequence in which the Indians charge over the hill towards the heroes. Back then, there were no such luxuries as walkie-talkies or mobile phones and as the cameras were set way back from the hill in order to capture the full panoramic widescreen shot, the director decided he needed someone up there to signal the Indians when to charge.
'Can anyone ride a pony?' he asked.
Being gung-ho, Tom stepped forward. 'Yes, sir, I can.'
The next thing he knew, Tom was in the make-up chair being blacked-up and, wearing a loincloth and with a full headdress of feathers, headed up to the brow of the distant hill on his mount. There he waited, watching for a white handkerchief being waved by the assistant – his signal to lead the charge downhill. All went swimmingly and Tom came galloping down towards the camera and, now fully into his character, as he drew near to the camera, threw himself off and at one of our cowboy heroes.
After the director called 'Cut!', Wayne sidled up to Tom.
'Was that you on that pony, kid?'
'Yes, sir!' said Tom, proudly.
'Kid, you looked like a monkey fucking a football!' the Duke offered as he walked away smiling.
Tom wisely gave up stunt work and moved into screenwriting and directing thereafter.
I suppose the one downside to reaching eighty-six years of age is that I'm now losing a lot of my mates. Sadly, in January 2013, Michael Winner joined the cutting-room staff upstairs. (When we were filming _The Persuaders!_ , whenever we heard the news of an actor or director dying, Tony Curtis would say, 'That's another one gone to the great cutting-room in the sky!') Despite being friends forever, I only made one film with Michael – called _Bullseye!_ – in which Michael Caine starred with me (or did I star with him?). The film wasn't... how can I put this... it wasn't the highlight of either of our careers (or Michael Winner's come to that). In fact, one day Michael Caine leaned across to me and said, ''Ere, Rog, this film is going to be our bleeding _Ishtar_ ,' likening the experience to the film that was at that point the biggest flop in the history of Columbia Pictures.
Say what you like about _Bullseye!_ but working with Michael Caine and Michael Winner was always a joy.
Mr Winner was certainly one of the more animated directors I worked with and one who liked to be heard on set. In fact, that reminds me of a story I heard about when he was shooting his first film at Pinewood in the early 1960s, called _Play It Cool_.
Michael decided he'd like to use a megaphone on set, and called out all of his instructions through it. The poor cameraman, who was sitting about three feet away from the director, even had his orders barked through the loudhailer. It got to a point where, during a short break, the cameraman in question excused himself from set and disappeared down the corridor. Ten minutes later, and ready to continue, Mr Winner was furious to see the camera devoid of an operator, and screamed, 'Where is he?'
I've known Michael Caine for over fifty years now. Not a lot of people know that.
'Gone to the toilet,' replied some helpful spark.
With that, dear Michael marched down the corridor, flung open the lavatory door and saw that the middle cubicle was occupied. 'Are you in there?' he shouted.
'Yes,' came the reply. 'But, Michael, please go away. I can only deal with one shit at a time!'
As well as directing, Michael often produced, wrote, cast and edited his films, though usually under the pseudonym Arnold Crust. Many of the credits for photographs that later appeared in his newspaper column were also credited to Arnold Crust or Arnold Crust Jr. Being modest, Michael didn't want to take all the credit himself.
On set, we noticed Winner would develop a small red blush on each cheek which, as it grew larger, gave us prior warning of him about to explode. It was a nuance we used to our advantage as whenever Michael Caine and I saw his cheeks reddening we'd say, 'Here it comes!' and warn any new actor on set, just as our beloved director gave 'what for' to some poor unsuspecting person. Never before had I known a director who would fire people instantly on set, though once the shouting was over, Winner would turn to us and say, 'Oh, how lovely! We've got rid of him. Right, let's continue, dears!'
I have to say, his bark was far worse than his bite, and he really was a very kind and caring man. When a rather famous production secretary confided in him that she was about to undergo treatment for cancer and would be unable to start his next film, he immediately arranged for her to go private, and in the best hospital too.
Sadly, Michael had been ill, and in and out of hospital himself, ever since consuming a bad oyster in 2007. He told me, on numerous occasions, of how he underwent scores of operations, dying several times on the table, but each time he bounced back and defied the doctors' pessimistic prognosis.
In the summer of 2012 he was told he had only two years left to live, and even joked about it in his newspaper column. Such was his spirit that, even from his hospital bed and throughout his long illness, he continued to write his _Sunday Times_ restaurant reviews – only he had the restaurants send the food in to him.
Michael had written his restaurant reviews 'Winner's Dinners' in the _Sunday Times_ since 1994, and he used to mail me photocopies of the column when I was abroad. Aside from being a critic, he was also a great host at any meal. Our last memorable dinner together was after his final 'Winner's Dinners Awards', which was held at the Belvedere Restaurant (chosen because of its close proximity to his house in Kensington) and afterwards Michael and his long-term partner Geraldine invited us back to their home for a superb dinner. At the table were Knights of the Realm Michael Parkinson, David Frost, Tim Rice and me, together with Lord Lloyd Webber and our respective wives. It was a hysterically funny dinner, with Michael being the host par excellence, despite his occasional good-humoured screams to the kitchen for the next course.
Although famous for having many girlfriends, there was one in particular who remained very dear to Michael. He'd known Geraldine since 1957, and they got together again in the early 2000s. Geraldine was at Michael's side every day he spent in hospital and helped nurse him back to health. Such was his love for Geraldine that after returning home he proposed to her and threw a big engagement party at the Ritz where he warned everyone, 'It's taken me this long to get engaged, so don't expect a wedding anytime soon.'
Over three years later, Mr Winner was enjoying all the benefits of wedded life but hadn't actually made the commitment of marrying Geraldine. One evening at Scott's restaurant in Mayfair, Kristina and I joined Michael and Geraldine, the Caines and the Bricusses and turned the conversation around to marriage. We told him he'd been engaged long enough and his reasons for remaining a bachelor were no longer convincing. We all urged him to make the commitment. The wedding followed in 2011 and I know it made Michael the happiest man on earth.
Aside from films, his other great passion was the Police Memorial Trust, a charity dedicated to raising plinths in memory of policemen and women killed in the line of duty. In 2005 he presided as the Queen unveiled the National Police Memorial in The Mall, designed by Norman Foster. Ever the film director, Michael called Her Majesty 'dear' when she unveiled the memorial.
Michael was offered an OBE for his work on behalf of the police, but turned it down remarking, 'an OBE is what you get if you clean the toilets well at King's Cross Station'.
My last conversation with him was towards the end of 2012, when I called to wish him a happy birthday. He'd just returned from another stay in intensive care and sounded terribly weak. Sadly, his liver was in failing health and a short time afterwards, Michael called 'Cut' on his final scene. As per the Jewish faith, his funeral took place within a couple of days of his passing and, sadly, I wasn't able to be there.
Fittingly, his memorial on 23 June 2013 was held at his beloved National Police Memorial and I was delighted to attend with Kristina and speak, along with Michael Caine, Michael Parkinson, Leslie Bricusse and a whole roster of police nobility, all keen to express their gratitude to Michael for establishing permanent memorials to their fallen colleagues.
Celebrating Michael Winner's seventieth birthday. He chartered a private plane and flew us all to Venice. ( _back row, l to r_ ) Leslie Bricusse, Terry O'Neill, Johnny Gold, me, Michael Caine, and ( _front row_ ) Andrew Lloyd Webber and Michael Winner.
He was a great director, a talented writer and tireless charity worker, but more than that he was a real character who enjoyed nothing more than taking the rise out of himself with his newspaper columns, letters pages and in person.
One story I wish I'd told Michael related to my assistant Gareth, who does a rather good impersonation of the great man. On occasion Gareth would call up our travel agent, driver and airport VIP service lady (who had all dealt with Michael Winner over the years, and had the scars to prove it), along with other unsuspecting innocents and would bark down the telephone at them with the most bizarre requests.
That was all well and good until a driver we used to use called Gareth in a fit of hysterics. It was fast approaching Christmas, and thus a busy time for car companies, so when Eddie took a call from 'Michael Winner', who was ranting on about wanting four cars that evening and how he dare not be late and so on, Eddie said, 'Fuck off, Gareth. I haven't got time for this today!' and hung up.
A few seconds later the phone rang again and the voice said, 'Eddie Wilcox, this is Michael Winner! Did you just hang up on me?'
'Oh! Hello, erm, Michael,' said a sheepish Eddie, 'I'm sorry, I think we had a crossed line there!'
'OK, dear,' said the great man. 'Now, about cars for my staff Christmas outing ...'
He'd have laughed so much had he known about Gareth's mischief. I really don't think we'll ever see his like again.
Another great pal who is mentioned many times within these pages is composer and lyricist Leslie Bricusse, and that's because Leslie's always been around throughout my life. It was when I moved to Stanmore during my days on _The Saint_ that I first met Leslie, at a restaurant called Maxim's. We hit it off immediately and a friendship developed. Mind you, soon after that I worked with Leslie's wife, Yvonne Romain (Evie), on an episode of _The Saint_ entitled 'The King of the Beggars', in which we both dressed down to play down-and-outs – so I know I was destined to get to know the lovely couple, one way or another.
As well as being a talented lyricist and composer with hit musicals _Stop The World – I Want To Get Off_ , _Pickwick_ and _Doctor Dolittle_ to his credit, Leslie is also an accomplished playwright and I've collaborated with him on two projects, _Sunday Lovers_ and _Bullseye!_ , one of which I'm sure he'd probably rather forget – I'll leave you to guess which.
We would often meet up for dinner with Jackie Collins and Oscar Lerman and Johnny and Jan Gold, along with Leslie and Evie at their house in Beverly Hills. Even when they weren't at home, I quite often stayed at the house when I had to go to Hollywood.
Evie and Leslie also had homes in Acapulco and Portofino at various times, and I stayed in those, too. In fact, I distinctly remember being at the Italian house when Neil Armstrong took the first step on the moon, as Leslie woke me up to watch it on TV.
It was on that trip that he told me he was building a home in St Paul in the South of France, and, then living by the water in Tuscany myself, I couldn't understand why someone would want to build up in the hills, miles from anywhere and a half-hour drive from the ocean. It was only me when I visited that house – and I'm actually very grateful to Leslie for the free holidays I've had – that I fell in love with the area and knew _exactly_ why he wanted to build there. On one occasion when I had to be out of England for a while, for tax reasons, Leslie was having some work done at the house in St Paul and asked me to move in and supervise the builders. A fair exchange, I reckoned.
With Yvonne – Evie – Romain in an episode of _The Saint_ called 'The King of the Beggars' back in the 1960s. Yes, that's me on the left, raising an eyebrow.
It was at that time I discovered, via local architect Robert Dallas, a plot of land on which foundations for a house had been laid was lying abandoned, with the developer having run out of money. I put in a bid, and that is how I came to own my pad in St Paul.
At St Paul, we have the best restaurant you could want right on our doorstep – the Colombe d'Or. Kristina and I still take our holidays there and in the five decades I've been visiting I don't think it's changed much at all. Whenever I come into the tree-lined courtyard, I think back to seeing regular visitors Simone Signoret and Yves Montand sitting in the corner, in their favourite spot. They loved it so much that they were married there, too. I had lunch with Yves in the mid-80s, a while after Simone passed away, and he explained that he'd met a lady who had fallen pregnant.
'I think I've made a mistake, Roger,' he said, 'and the child will grow up without knowing his father.' Yves died just three years later, aged seventy, on the set of _IP5: The Island of Pachyderms_. It was the very last day of shooting and, after his very last shot, he literally dropped dead of a heart attack.
Despite no longer living in St Paul, my connection with the village continues as my son Christian was married there, and since my last book there have been additions to the Moore ranks as Christian and his wife, Lara, are now the proud parents of Tristan, who was born in December 2009, and Maximillian who was born in November 2011. Both boys were baptised in the same church in which their parents were married.
It was also through Leslie Bricusse that I met Danny Kaye. Of course, I knew of him, and had seen many of his shows, but through getting to know him I came to appreciate how magical Danny was, not only as an entertainer but as a humanitarian as well. He had a natural ability to make children smile, quite often children who had little to smile about. Danny became the first UNICEF Goodwill Ambassador and I'm so grateful to be able to help continue, in my own modest way, the work he started.
After Danny's death, his daughter Dena invited me and the Bricusses to join her for one last meal in the kitchen Danny loved (and the room where we'd enjoyed so many marvellous meals – he was a terrific chef) before the house was sold. The only thing missing was Danny rushing in to another room afterwards to watch a VHS show of himself conducting an orchestra – he loved to conduct, and loved sharing the experience with friends, although I don't think he'd ever read music, he just felt the rhythm.
Director Roy Baker, with whom I'd happily worked on many episodes of _The Saint_ , had himself earlier worked with Alfred Hitchcock as an assistant director at Gainsborough Studios. Roy maintained he learned a tremendous amount from Hitch, chief amongst which was that 'time spent on preparation was seldom ever wasted'. However, Hitch didn't have a great love of actors, once famously describing them as 'cattle'. When Roy was assisting him, one hapless actress appeared on set for her first day of shooting, and after the first take Hitch looked up and said, 'Cut! I wind her up, I put her down but she don't go! Bring me _Spotlight_!' (the actors' directory). He could be a very cruel man at times.
Hitchcock reserved a particular dislike for those thespians who employed the 'method' technique. When directing Ingrid Bergman, the star turned and said, 'Hitch, what is my motivation for this scene?'
In front of the entire crew Hitchcock snapped back, 'It's only a bloody film, Ingrid!'
It reminded me of a story I heard about Laurence Olivier and Dustin Hoffman working together on _The Marathon Man_ – and you'll never feel the same about visiting the dentist again. One of the scenes between the duo took place just as Dustin's character arrived the worse for having been out running. Being a keen advocate of 'the method' Dustin actually went on a sprint around the studio grounds before reporting on set, and was genuinely out of breath, ready for the call of 'Action'. A rather concerned Olivier leaned across to him and said, 'Have you ever thought about just acting out of breath, dear boy?'
One of Roy Baker's more unusual films for Rank starred Dirk Bogarde and John Mills and featured veiled intimations of homosexuality in New Mexico – and it always really intrigued me as to why any of them made it.
After the success of two 1950s films for Rank – _The One that Got Away_ and _A Night to Remember_ – Roy signed a three-picture contract with the company, and Pinewood's head of production, Earl St John, handed him a book called _The Singer not the Song_.
'I thought it was awful, and I told him so,' Roy later told me. 'I came to the table with three other projects which were, one by one, rejected. Earl said that he really wanted me to do _The Singer not the Song_ with Dirk Bogarde.'
Bogarde was then Rank's highest-paid artist and, nearing the end of his seven-year contract, was keen to get away from the pretty boy heroes he'd largely played for the studio; though he'd made it clear he didn't really want to do _this_ film.
Catapulted into this strange scenario, Roy dutifully flew out to LA to see Dirk Bogarde and discuss the picture but he soon discovered that for some reason Bogarde was angry about Johnny Mills being cast to star opposite him. Mills, on the other hand, had a different take on the project and said when he was first offered the film, he was to co-star with Marlon Brando. Naturally, he was very excited, but for some reason Brando pulled out and that's when Dirk came in. All this caused a great deal of tension on the set but Roy soldiered on through it and made it look as good as possible. However, as good as it looked, they finished up with a picture that was panned by everyone.
The last film notwithstanding, Roy had enjoyed great success in the UK and later went to Hollywood, where he famously directed Marilyn Monroe in _Don't Bother to Knock._ The experience, however, was not a particularly happy one, as he later discovered that Joseph Schenck, one of the production chiefs at 20th Century Fox, had decided to green-light the film as an indulgence to Marilyn, despite her being terribly miscast in the role (Roy suggested Jane Wyman should have played the part by the way). The studio head, Darryl Zanuck, thought it was all a waste of time but as Monroe was under contract they were obliged to find her a film, and with nothing else on offer he let it go ahead, on the proviso that it was made as cheaply as possible.
Marilyn, meanwhile, demonstrating her great insecurity, was adamant she'd only work if her dialogue coach, Natasha Lytess, was on set at all times. Zanuck point-blank refused, and wrote to Marilyn, saying she had 'built up a Svengali and if you are going to progress with your career and become as important talent-wise as you have publicity-wise then you must destroy this Svengali before it destroys you. When I cast you for the role I cast you as an individual.'
Nevertheless, on day one Miss Lytess appeared on set and sat down next to the camera.
Roy had terrible problems with Marilyn's timekeeping as she'd often arrive two or three hours after everyone else was ready and on set. Such was her insecurity, that despite countless rehearsals she never quite knew when to pick up a cue or how to move across set; every take would be different. She'd seek constant reassurance from her Svengali and thought more about how she should _act_ rather than just naturally being the character.
Eventually, with schedules overrunning, Miss Lytess was barred from the set and Roy said that when he told Marilyn he had expected her to react badly, but she quietly accepted the situation. In fact, she later admitted to being so confused that she'd let anyone be her friend and advise her – though it went to extremes when she let her dialogue coach pass judgement on everything she said and did in her professional life and in private too.
Roy offered her every reassurance Marilyn needed on set and finished the movie on schedule, quite possibly drawing out one of Marilyn's best, if underrated, performances.
I never met Miss Monroe myself, but my hairdresser on _Ivanhoe_ – a lovely, if somewhat camp, chap named Gordon Bond – used to tell me stories of working with her, usually when I was in the make-up chair, where he'd make a habit of pressing his manhood against my shoulder while telling me stories about his Spanish boyfriend and saying things like, 'My matador is coming home for the weekend'.
Anyhow, Gordon had worked on _The Prince and The Showgirl_ at Pinewood with Laurence Olivier and Marilyn Monroe, and said he had terrible problems (as did everyone) with Marilyn's terrible timekeeping. But alas – poor Gordon! – having spent an hour or two preparing her hair for the shoot, Arthur Miller would come in to the dressing room, squeeze Marilyn's shoulder and take her into the room next door... from which they'd emerge thirty minutes later and Gordon would have to do her hair all over again.
One day on a location shoot, Gordon hitched a ride in Marilyn's car. The scene involved the Marines and Queen's Guards, who'd been drafted in, and as the Prince (as played by Olivier) arrived, they were to give him the full military welcome.
Marilyn got out of her car and, as Gordon stepped out behind her, half the guards, who'd dutifully ignored her, called out, 'Oh! Hello, Gordon!'
While working at Fox, Roy Baker said one of the production executives was a chap named Sol Wurtzel. Actually, the whole Wurtzel family were employed at the studio in one capacity or another, be it in the carpenter's shop, the transport department – and everywhere else. One young starlet who was signed to the ranks was advised that her career might progress if she 'stepped out' with Mr Wurtzel. Keen to advance, she willingly did so but after a few months realized she was getting nowhere and only then discovered she'd been going out with the wrong Mr Wurtzel – her guy worked in the paint shop. Only in Hollywood!
I couldn't write about Hollywood and not mention my dear friend Albert Romolo Broccoli, also known as Cubby, who was one of its biggest and most successful film producers. He was born in April 1909 in a tenement block in Astoria, Queens, New York, but almost didn't survive, as his was a breech birth and he had trouble breathing. His grandmother Marietta, who had emigrated from Calabria to America in 1897, resorted to a traditional remedy – the insertion of the head of a black chicken into the child's mouth. The treatment worked and the boy started to breathe again. Thank goodness for all of us!
Cubby didn't enjoy a financially rich childhood, far from it. But in 1934, with the encouragement of his cousin Pat de Cicco, who had left his family farming business to become a Hollywood agent, Cubby travelled to LA to break into show business.
At first, life was tough and Cubby sold hair products and Christmas trees to get by. However, his cousin was a successful agent and was well placed to introduce him to people such as the actor Cary Grant, with whom Cubby became great friends, director Howard Hughes, who employed him as an assistant on the film _The Outlaw_ , and Charles Feldman, who gave him a job at his talent agency. Cubby then made the transition from agent to film producer, working with partner Irving Allen.
Having established himself in the LA film industry, Cubby first travelled to Britain in 1948, where he cut an incongruous figure in our austerity shrouded post-World War II capital.
Dear Cubby Broccoli was a great friend to me – and fearless too (although the tiger is stuffed, I think).
'Where's the King's Arms?' he once asked a local.
'Around the Queen's arse!' he was told – a sardonic reply that delighted him. He still laughed about that years later.
Cubby booked into the Savoy on that first trip, and went into the Grill to have breakfast. A waiter came to the table and asked, 'What would you like, sir?'
'I'll have bacon and eggs and a pot of coffee,' came the reply.
The waiter had to explain that, due to rationing, that wouldn't be possible. However, a couple of days later, the same waiter turned up at Cubby's breakfast table saying he had a surprise. Underneath a big silver lid, he unveiled two boiled eggs. Cubby asked the waiter how he had got them. 'I brought them from home,' he said.
Cubby was so touched that he immediately became a champion of all things British, and every movie he made thereafter was in Britain. He loved the lifestyle: the horse racing, the gambling clubs (where I first met him) and the social scene.
His first film as a producer, and one he made in London, was _The Red Beret._ He and his producing partner, Irving Allen, decided Alan Ladd would be the perfect star and Cubby was despatched to meet with the actor. Cubby told me he had thought it was odd that Irving Allen didn't want to accompany him to the meeting, though he later discovered why. Alan Ladd, who started out in the business as a camera grip technician in the late 1930s, had worked on a series that Irving Allen directed and had mentioned his aspirations to become an actor.
'Why do you want to be an out-of-work actor?' Allen asked him. 'Stay as a grip, you'll make more money.'
Well, by the early 1950s Ladd had indeed become an actor – one of Hollywood's most popular – and was earning $100,000 a year under contract. By 1952, however, he felt he was worth more and asked for a raise. It was not forthcoming, so he refused to renew his contract.
Cubby told me that at the time he didn't even have enough money to pay his rent that month but the prospect of _The Red Beret_ being green-lit if he could secure Hollywood's biggest actor was too great an opportunity to miss, so he took a gamble and told Ladd's agent, Lew Wasserman, that they could pay $200,000 for this one film, plus ten per cent of the profits.
Wasserman turned Cubby down, saying he was just starting out and was 'an amateur'.
Realizing Ladd's wife, Sue, actually called the shots in that camp, and having heard that she fancied a trip to Europe, Cubby called her to explain his $200,000 offer had been rejected.
'Stay by the phone,' Sue snapped.
A few minutes later she called back, inviting Cubby to meet her later that day. Also at the meeting was a rather pale-faced Lew Wasserman, who'd obviously felt the wrath of Sue Ladd. She told him to go ahead and make the deal, but Lew told her that he felt it was crazy.
'You haven't got Alan any more than this offer!' she barked at him.
The deal was done, and in fact became the first of three pictures the producers made with Ladd, thus launching Cubby's career as a mainstream film producer. That first film, incidentally, was directed by Terence Young, and written by Richard Maibaum – both of whom, of course, went on to collaborate on the early Bond films.
At the opening of the second 007 stage at Pinewood with ( _l to r_ ) Fiona Fullerton, Cubby Broccoli, Tanya Roberts, Christopher Walken and Alison Doody.
I feel I should also mention the important role a young female writer named Johanna Harwood made to the early days of the Bond series too, because her involvement has often been overlooked and her pivotal role clouded by the vagaries of film history and the egos of those within. Johanna is actually a neighbour of mine in Monaco, though she's long since hung up her typewriter ribbon.
Having started in film continuity in her native Ireland, Johanna became 'assistant continuity' on _The Red Beret_ , but her involvement in Bond came not via Cubby, but Harry Saltzman. With a shortage of film work, Johanna had taken employment with a theatrical agency in London, but it was soon taken over by Harry Saltzman and she became his secretary and script reader. Her talent for writing developed and she authored many articles and stories, including a James Bond magazine spoof in 1959, and so when Harry had an idea for a film one day he asked Johanna to write an outline for him. Johanna was subsequently handed books Harry had read and felt could be film subjects.
'I had to write film outlines or first draft scripts,' she told me, 'so that Harry could tout them around financiers in the hope of raising funds. When he optioned Ian Fleming's books, and was trying to create interest in making a film, I adapted _Dr No_ for him. It was a first draft script.'
When Harry and Cubby came together to found EON Productions, they initially intended to film _Thunderball_ but legal issues surrounding the book prompted Harry to pull out Harwood's _Dr No_ script.
'Terence Young was a terrible misogynist,' Johanna told me, 'and so the idea of working with a female scriptwriter didn't appeal one bit. He brought in Wolf Mankowitz, who wanted to make the villain a monkey – which appalled Cubby – and then Richard Maibaum and Berkely Mather (whose novel _The Pass Beyond Kashmir_ had been optioned by the duo that year) were brought in to add a man's touch. However, it was largely my script they ended up filming, as it was closest to Fleming's book.'
During filming of _Dr No_ , Johanna was dispatched to Paris to start work on adapting another of Fleming's books, _FromRussia With Love_. Consequently she only ever visited the set of _Dr No_ once for a meeting with the producers. She later heard Terence Young dismiss her as 'my script girl', suggesting she only contributed one or two ideas in the screenplay.
Though she later wrote the screenplay for _Call Me Bwana_ , which EON produced in 1962, she moved to Paris to work for Reader's Digest, having grown somewhat dispirited by the (then) male-dominated industry.
_Bwana_ , incidentally, was all set in Africa but the crew didn't get quite that far. Gerrard's Cross Golf Club, a mile or so from Pinewood Studios, doubled for the location, with the addition of plastic palm trees and three imported giraffes, an elephant and a zebra. At night they used to let the animals roam around the course – simply closing the gate at the end of the day's shoot. Filmed in the freezing winter, back at the studio they would knock the snow off the potted palms and pretend it was Africa. The fact you can see the actors' breath in the film didn't matter – much like in my days shooting the glamorous locations of _The Saint_ on the Elstree backlot.
Aside from _Bwana_ and _Chitty Chitty Bang Bang_ in 1967, Cubby concentrated primarily on producing Bond films for the rest of his life. When once asked if he felt frustrated at having confined himself to 007, he said that he had a tiger by the tail and that he couldn't let it go!
It's incredible to think that we are all now looking forward to the twenty-fourth Bond film – twenty-four films and _still_ the world's biggest film franchise! I can't wait, personally. I think Daniel Craig is a tremendous Bond and I know that Cubby would have been delighted to see him in the role – he's the perfect 007 and looks as though he could actually kill... whereas I just hugged or bored them to death...
Producer Elliott Kastner – with whom I'd worked on _North Sea Hijack_ in the late 1970s – used every trick in the book to make sure he got the film he wanted. Nothing wrong with that – it's what all good producers do.
CHAPTER 8
The Producers
WHILE THE WRITERS AND DIRECTORS COME UP WITH interesting ways to spend the money, it's the producer's job to raise it. Apart from being perhaps the most prolific of all British film producers, Harry Alan Towers was one of the more colourful characters of the British film business, racking up over 100 big-screen producing credits, plus another fifty-odd as a writer. Granted, many of them were cheaply made action, horror and soft-porn movies, but Harry had no interest in winning plaudits or awards; he just wanted to entertain and make a quick profit.
Though I never worked with him, I knew many cast and crew who did and who, in turn, regaled me with their tales.
One such was from Fred Turner, who was the managing director of Rank Film Distributors in the 1980s and 1990s. When Fred took his annual holiday one year, Harry – or El Sombrero as he was known by his crews due to his initials HAT – rolled up to see Fred's deputy and said, 'George, I've got this most wonderful script by a very talented writer named Peter Welbeck and I think it's right up your street.'
Harry didn't bank on the Deputy MD knowing that Peter Welbeck was in fact his pseudonym and the script had been knocking around Wardour Street for months. That was Harry, always looking for the main chance.
Peter Manley, who was production manager on some of my _Saints_ and made a few films with El Sombrero, told me about one time he was with Harry on a film in Marrakesh. Apparently every weekend HAT would go off to London with the negative in his suitcase and the following Monday he would arrive back with a case full of unexposed film and cash to pay the crew – all highly illegal back then, of course.
One Monday Harry didn't come back – and by Wednesday there was still no sign of him and the crew were about to mutiny and stop shooting. Peter persuaded them to carry on, as he knew they would be in a better bargaining position if they were still working.
Sure enough, the following week Harry arrived and asked how it was all going, but he hadn't brought any cash with him. It wasn't untypical of Harry to have cash-flow problems. He asked for the film that had been shot in his absence but – thinking on his feet – Peter told him it was in the hotel safe and couldn't be released until Harry had paid the crew. Harry fumed and jumped up and down a bit but then shot off somewhere, only to return a day later with the cash.
Harry was a chancer, a charmer and a shrewd businessman who would put together the most obscure and complicated co-production deals involving countries you'd never heard of, and he found film-friendly tax shelters in the furthest-flung corners of the world. He churned his movies out at a rate of knots – mainly because he had to make the next film to pay off his debts and crews on the last.
Sir Larry Olivier and producer Harry Alan Towers – HAT or El Sombrero to his friends – another producer who knew a few tricks.
He would never _not_ pay, he was just very tardy.
John Llewellyn Moxey, who directed some of my _Saints_ , made one picture for Towers but 'stopped shooting until my cheque cleared,' he said.
Director Michael Tuchner checked in to a hotel somewhere in deepest Europe one day while on a recce for a (non-HAT) production. The manager greeted him warmly and said, 'I understand you are in the film business?'
'Yes, that's right,' said Michael.
'Do you know Mr Harry Alan Towers?'
'Well, I know _of_ him,' Tuchner replied.
'Then please would you take this bill to London and ask him to settle it?' asked the distraught manager.
HAT would often shoot two films back to back or in the same locale to benefit from the economies of scale. One such case was when _The Call of the Wild_ was on location a few miles from his big-screen version of _Treasure Island_ with Orson Welles, supposedly one of his oldest friends. Peter Manley, who was associate producer on _The Call of the Wild_ , suggested to HAT that it would be great to have Orson Welles appear in his film in a day role, and might he consider allowing Welles to come over from the other side of the valley? Realizing that Orson would ask for more money, Towers declined, saying, 'I wouldn't wish him on you, Peter!'
Incidentally, of the former film, its star, Charlton Heston said, 'The worst film I ever made was _The Call of the Wild_. How can you possibly screw up that story? You may well ask. The root of our troubles was the producer, a sort of rogue Brit who flickered shadowlike in and out of the country to avoid his various creditors. What we finally ended up with was a joint British/American/Norwegian/German/French/Italian/Spanish co-production. There are many good actors in all these countries whose English is perfectly competent. Our producer did not hire them.' Ouch.
For many years HAT could not enter the US without the threat of being arrested, after jumping bail there in 1961. By all accounts he would arrive in New York with some lovely ladies and, to help seal various finance deals for his films, he would leave the young ladies with the executives for the afternoon... and very obliging they were too, I hear. The vice ring was soon uncovered and Towers was arrested.
HAT made a living from adapting public domain stories and was not averse to a little thinly disguised plagiarism. He would often arrive on location for one film, having written a screenplay for the next film on the plane over. Yet, despite his sausage factory approach, Towers managed to attract big-name actors such as Orson Welles, Jack Palance, Michael Caine and Christopher Lee – sometimes more than once. In fact, my _Golden Gun_ adversary starred in five of HAT's _Fu Manchu_ films, based on the characters created by Sax Rohmer. The first, _The Face of Fu Manchu_ was rather good but then ...
' _Brides of Fu Manchu_ was tosh,' Christopher said. 'An extravagant publicity stunt almost sank the picture. At the instigation of producer Harry Alan Towers, who took an enthusiastic part, I toured European countries choosing from each the winner of a national beauty competition, whose prize was a part in the film. They titted about the set, draped themselves about pillars in Fu Manchu's great stone den, and between takes some draped themselves about members of the unit. But they could not show themselves off to best advantage because they were not members of Equity and therefore they had not a line to speak between the whole dozen.'
That didn't deter HAT though, who went on to film _The Castle of Fu Manchu_ – a Spanish/Italian/West German co-production shot in Turkey – beginning with Fu freezing the Atlantic and wrecking an ocean liner via spliced-in clips from _A Night to Remember_ , tinted a spectral blue in an attempt to disguise the fact that they were shot in black and white and the rest of the film is in colour!
On another occasion HAT was in Rio shooting a film with his favoured director, Jess Franco, who had a reputation for shooting very quickly. They were over a week ahead of schedule, with their next shots being of the finale of the Rio Carnival. Faced with putting his cast and crew on paid hiatus to await the annual festivities, HAT decided to write a script called _99 Women_ over the weekend and, with his three female leads, director and crew, shot its action and location sequences in five days, before resuming work on the other film. By the time he returned to London he had one complete film and another third of _99 Women_ in the can.
There is an anecdote, possibly apocryphal, about how Towers tried to persuade Herbert Lom, another of his favoured leading actors, to join the cast of a new Harry Palmer spy movie, _Bullet to Beijing_ , starring Michael Caine.
'It will be shot in Russia in an exciting location where few film crews have ever gone,' said Harry, without naming the place.
When Lom insisted on knowing where it was, Towers replied, after some hesitation, 'Um... Chernobyl.'
HAT had done a deal with Len Deighton to bring his famous spy creation back to the screens, although I believe he had to do a separate deal with Harry Saltzman, as in the first book the spy didn't have a name – that was a creation of the film that Harry Saltzman had produced. So to use the name 'Harry Palmer' didn't have a clear title of use in a new film and necessitated a bit of negotiation.
HAT had found some money in the newly opened-up Russia, tied it up with a Canadian/UK co-production and secured Michael Caine for the lead. Filming in the former Soviet Union meant buying certain friendships and protection. Towers said, 'Russia welcomed us with open palms.'
On another occasion, cameraman Ronnie Maasz told me about joining one of HAT's productions in Salzburg. Rather than pay to shoot in a studio, the prudent producer opted to adapt hotel bedrooms, houses, offices and any other setting he could acquire cheaply, with sets dressed as they went along. The actors were mainly German (as that's where his finance came from) and Ronnie discovered they had actually started the film in Prague, but for 'financial reasons' everything was transported across the border overnight and re-set in Austria.
The first shot was to be of an exotic white sports car, but a rusty old Czech Tatra turned up. HAT declared it only needed a quick re-spray – and he knew just where to get it done. An hour later, the car returned to the set sporting a new paint job that was a curiously flat shade of white. Soon after, it started to rain and the colour washed off the car – he'd had it painted on the cheap and the local painter had used emulsion! Not to be outdone, HAT approached a local shopper and asked if he could borrow his white sports car, to which, amazingly, the local agreed. Unfortunately, though, Harry failed to think ahead to the next day's shooting requirements. So, on day two another white car was found – an entirely different model – and HAT ordered shooting to continue, reasoning, 'No one will notice – they'll be too caught up in the plot!' Needless to say, everyone noticed!
With Tony Curtis and Lord Lew Grade, a formidable producer and a great friend.
One time, filming in Rome, I met for lunch with an old friend of mine. Vincenzo Labella was a lecturer in Vatican History and Art in Florence and Rome, and later became a very successful writer and producer, after starting out as a technical advisor on _Francis of Assisi_. Over lunch, Vincenzo told me his next project as a producer was going to be on the television series _Jesus of Nazareth_ for Lew Grade. In fact, it transpired, he was going to the airport that very afternoon to pick Lew up, as they had some meetings to attend about the new series.
It was an opportunity I couldn't resist. I asked Vincenzo if I could act as their chauffeur. I said I would wear a jacket and cap... all I asked was that Vincenzo went along with anything I said.
'OK,' he said, eyeing me suspiciously.
At the airport, Lew got into the back of the car, puffing on his trademark cigar, and, without looking around, in my best Italian I asked, ' _Dove vorresti che guidi?_ '
'What? What did he say?' barked Lew.
'Oh,' said Vincenzo, 'he just asked where are we going.'
'What sort of driver is he if he doesn't know where we're going?' Lew snapped.
'It's OK, he's merely confirming ...' said Vincenzo.
I started driving and waited for them to become engrossed in conversation before I interrupted them. ' _Mi scusi, signore, ma la prego di chiedere al signore si è alla guida con se posso avere uno dei suoi magnifici sigari?_ '
'What? What did he say now?' asked an irritated Lew.
'He wants to know if he can have one of your fine cigars?' said Vincenzo.
'What the bloody hell is going on here?' cried Lew.
I turned around and smiled.
'Stop the car!' he shouted. He got out, went to the boot, pulled out a huge box of Monte Cristos and gave them to me, laughing his head off. Lew's trademark was his wonderful, large Havana cigar. Comedian Dave Allen always said of him, 'Never wear a brown suit when you're around Lew as he'll pick you up and try to light you.'
Lew was a great friend to me, and although he was of Jewish-Russian extraction, I think he'd have converted to any religion if it meant getting a deal. Kathy, Lady Grade, is Catholic and when she was invited to an audience with the Pope in Rome in the 1970s, Lew naturally travelled with her.
'Roger,' he later told me, 'Pope held Kathy's hand and blessed her, blessed her family and blessed the work of her beloved husband. Then Pope called me over, and do you know what Pope said, Roger? He said "Mr Grade, I want to bless you for all of the great work you produce and in particular I want to thank you for making _Jesus of Nazareth_ ".'
'Oh really, Lew?'
'Yes, and do you know what else, Roger? Well, the following Sunday when Pope came out on his balcony to bless the crowds in St Peter's Square ...'
'Yes, Lew?' I asked.
'Well, Roger. That Sunday he held up his hands to bless the crowds and said, "Bless you my children, and be sure you watch _Jesus of Nazareth_ on TV tonight!"'
Lew initially sold the series to NBC in America, and such was its popularity that NBC asked Lew to fly over to negotiate a deal for repeat screening rights. Just before he left he received a call from Proctor & Gamble Entertainment saying that they wanted to buy the show and all future rerun rights with it. It was hot property.
On the plane going over to LA, Lew said he had a vision, 'And that vision was of the number 25'. He knew it wasn't $25 and it couldn't be $25,000, so when he went in to his first meeting with the P&G executives he told them the price was $25 million.
They told him he was crazy.
He went to see NBC next, as planned, and they asked how much he wanted. '$25 million,' he beamed.
They only wanted five re-runs and after an hour of negotiating said their top price was $15.5 million – Lew wasn't interested.
His next LA meeting was at CBS, where he was pitching a new mini-series with Sophia Loren and Omar Sharif, but he felt the executives just weren't biting and couldn't figure out why, so he asked them point blank if he'd done anything to upset them.
'What about offering us repeats of _Jesus of Nazareth_?' they asked.
'NBC have first option to buy it and I want $25 million,' said Lew.
'OK, when do you want to know?' they asked.
'I'm leaving for the airport at 4 p.m.'
'You'll hear from us by then.'
Lew, meanwhile, called back NBC and said they had twenty-eight minutes to reach a decision on buying the show, and the price was $25 million, take it or leave it. It was chutzpah on a grand scale. Ten minutes later, the chairman, Herbert Schlosser, called him. 'Lew, we've been friends for fifteen years. How can you do this to me? I need time.'
'You've got twenty minutes left,' Lew said, 'before I offer it to CBS.'
Sure enough, NBC came up with the $25 million.
When asked later how he arrived at the visionary figure, Lew shook his head and said, 'I don't know, but I couldn't believe they paid it!'
While _Jesus of Nazareth_ was a production triumph for Lew's company, ITC, the jewel in his weekly network television schedule was undoubtedly the variety show _Sunday Night at the London Palladium_ , and one Sunday Lew made a star out of a puppet act called 'Topo Gigio'. A few months later they were short of a 'top of the bill' act and on the Friday prior to the show word came back to Lew that he was not to worry as they'd secured Tito Gobbi (the famous Italian baritone).
'Aww, no!' cried Lew. 'Not that bloody mouse act again.'
Towards the end of his life, Lew was approached by the chiefs at Polygram Films, who hit the jackpot in 1994 with _Four Weddings and a Funeral,_ and who had then recently acquired the ITC library of programmes and films that Lew had green-lit, financed and/or produced – including _The Saint, The Persuaders!, Randall & Hopkirk Deceased, The Baron, The Champions, Raise the Titanic_, _Escape to Athena_ and so on. It was a vast library and had been wrestled from Lew in the 1980s after he lost control of his empire due to some unfortunate business dealings.
Anyhow, the proposition was for Lew to be appointed non-executive chairman of Polygram – a figurehead position more than anything else. They offered him the annual salary of £100,000.
'That's not enough,' said Lew abruptly. 'Make it £150,000.'
The two chiefs, Stewart Till and Michael Kuhn, were a little taken aback as they were effectively offering him a job with little work involved for a not inconsiderable sum of money.
'But, Lew, we can't go any higher! You only have to attend a couple of meetings a year,' they reasoned.
'Well, you boys think about it,' said Lew.
Deflated and somewhat stumped, Till and Kuhn said they'd get back to Lew after their impending trip to Germany, where they hoped to conclude a deal with the huge Kirsch Media TV empire to license some of the ITC shows.
'I'll come with you!' said Lew. 'If I'm going to be chairman I need to earn my £150,000.'
Despite their protests that they had already more or less sewn up the deal and didn't need any help, Lew was insistent.
Arriving at Heathrow airport to board their flight a couple of days later, they saw Lew emerge from his Rolls-Royce and enter the departure hall, trademark cigar firmly in his mouth. The 'no smoking' policy was something completely alien to Lew and consequently something he ignored. His young executive friends wondered what they might be getting themselves into with this larger-than-life character who lived life very much on his terms and who seemed adamant that lending his name to their already successful company was worth £150,000.
On arriving at Kirsch HQ, Lew was immediately whisked up to see Leo Kirsch, the head of the company, and launched straight into Yiddish with his old friend. They swiftly disappeared, arm-in-arm, into an office, while Kuhn and Till were left outside worrying that their long-gestating deal was about to be blown. A few minutes later, Lew emerged.
'OK, boys. The deal is done,' he told them. 'Let's go for lunch.'
Kuhn and Till, by now rather furious, told Lew he had no right to interfere with their negotiations and that he didn't even know how much they'd bargained on getting, so had undoubtedly sold them short.
'How much did you want?' asked Lew.
'We were edging towards £200,000,' said 'the boys'.
'I just got you £300,000, boys,' Lew smiled. 'Now you see why I'm worth that extra fifty?'
Needless to say, Lew became their chairman at his proposed fee.
There was much talk of remaking some of the classic TV shows, and I know Lew was very keen to revisit his past successes and took to the chat show sofas to talk about it all. Sadly, Lew passed away before many of his plans could come to fruition. Soon after, Polygram hit financial problems and the company's Dutch parent company announced it was withdrawing from film production, selling its catalogue to ITV.
Lew's brother, Lord Bernard Delfont – or Bernie to his friends – was also very active in film finance and production and was, in fact, the person who invited my old friend Bryan Forbes to head up ABPC's production activity in the late 1960s. Later on, in 1978, Bernie, as chairman of the now renamed EMI Films, had backed a little British film called _The Life of Brian_ , which was the brainchild of the Monty Python team.
Bernie was set on investing a great deal of his company's money in the movie but at the last minute got cold feet about the religious subject matter – and not least about the image of 'Brian' on a crucifix singing, 'Always look on the bright side of life'. Bernie pulled out just days before shooting was due to start.
Faced with finding another backer at short notice, the team were all set to throw in the towel when Eric Idle suddenly remembered a chap he recently met at a party – former Beatle George Harrison. Not having time to beat about the bush, Idle came straight to the point and asked Harrison if he would be interested in bailing out the film for $4 million. Harrison read the script the following day, loved it immediately, and agreed to come on board. Idle later described this moment as 'the most expensive movie ticket ever purchased'. Harrison formed Handmade Films with his business manager Denis O'Brien and production commenced.
The film was an enormous success and although Handmade was formed originally to produce only one film, they soon found themselves becoming involved with another when the gangster movie _The Long Good Friday_ came their way. The film had been completed but its production company, Black Lion Films, which was owned by Lew Grade, was becoming nervous about its prospects due to the high level of violence and a key subplot involving the IRA. Handmade made an offer to buy the rights for £700,000 and released the film, which proved to be another huge hit.
Harrison once said, 'As a musician I've been the person who's said of the people with the money, "What do they know?" and now I'm that person. But I know that unless you give an artist as much freedom as possible, there's no point in using that artist.'
The company continued producing movies into the early 1990s when it was sold, though, alas, they never gave me a job!
Another important producer in my life, and indeed in the movie business as a whole, was Harry Saltzman. The often brash, frequently loud and always extravagant producer was actually Canadian, born in Quebec, and not American as many people incorrectly assumed.
I know Harry didn't have a particularly happy childhood and ran away from home aged fifteen to join the circus. Such was his entrepreneurial spirit, two years later he was running his own circus troupe.
During the 1940s he joined the army, serving in World War II, where he was posted to Paris and was later recruited to the OSS – the Office of Strategic Services, an intelligence division. Such was the sensitivity of his work that when in 2003 his daughter Hilary wanted to move to Quebec from LA, she had to prove her father was a Canadian citizen, so contacted the Department of State to retrieve Harry's records. Those relating to his military service were said to need the permission of the Secretary of State himself before they could be released, and even then they were edited heavily – some sixty years after the fact.
After the war, Harry stayed on in Paris where he met Jacqueline, a Romanian who had escaped from the troubles of her homeland, and they married soon after. In Paris he found work as a casting agent but despite modest success he never really made his mark in the profession, nor a great deal of money. He next became involved in a TV series about the French Foreign Legion, which proved more profitable and, after joining British producer Betty Box to work on _The Iron Petticoat_ , Harry realized that there were more rewarding opportunities in the film business and, together with Tony Richardson and John Osborne, he formed a production company with the hopes of being able to find finance for more features. They called the collaboration Woodfall Productions and Harry finally found his real vocation in life.
_Look Back in Anger_ cast rising Welsh actor Richard Burton as the lead, and Harry sought out various other stage plays he thought fitting for the screen. _Saturday Night and Sunday Morning_ followed in 1960, starring a young Albert Finney. When the James Cagney film _The Gallant Hours_ opened in Leicester Square, it did terrible business and was pulled after just three days. The manager, keen to find something else to screen immediately, was offered _Saturday Night_ and the rest, as they say, is history ... The film scooped three BAFTA awards, including Best British Film and made a fortune!
Harry Saltzman ( _left_ ) and Cubby Broccoli ( _right_ ) with the man who created James Bond, Ian Fleming.
_The Entertainer_ , starring Laurence Olivier, was another award-winning production, but soon after that the Woodfall team parted ways to pursue their own projects. Harry was looking for another investment and turned to a successful series of novels by Ian Fleming. He reportedly paid $50,000 for the rights in 1961, a huge sum for the times, and his pricey offer earned him just a six-month option on the James Bond character. But in teaming with Cubby Broccoli on the eve of expiry of his option, a deal was done with United Artists and Jim Bond hit the big screen.
Jess Conrad told me a great story about when he came across Harry. Jess, who, aside from being a big singing star had also made a foray into films, went up for roles in a number of TV commercials but never seemed to get them. It was, he reasoned to his friend and fellow struggling thespian Gareth Hunt, who was experiencing similar problems at the time, because neither of them had blue eyes – all the Paul Newman lookalikes were getting the jobs they went up for. So they decided the best thing to do would be to invest in a pair of blue contact lenses between them. At the time, contact lenses were very expensive and they could only afford one each. They agreed to take it in turns to wear them.
When the call went out to find a new James Bond in 1968, Jess wangled himself an audition and called Gareth Hunt, 'Can I borrow the lens tomorrow?' When Gareth asked why, Jess wanted to throw him off the scent of this juicy role. 'Oh, it's nothing, just a commercial for something or other,' he lied.
So, Jess got the lenses. You must remember that at this time contact lens technology was pretty young, and the only problem with the lenses was that although they _looked_ good, you couldn't actually _see_ much through them.
Jess duly reported at the production company EON's South Audley Street office and, running a little late, told the receptionist that he had a meeting with Harry Saltzman. He was told to go up to the office so he dashed up the stairs and when he reached the top he realized he hadn't put the lenses in so, a little out of breath and somewhat nervous, he started trying to put the lenses in, only to drop one on the floor. On his hands and knees Jess started feeling through the deep carpet pile for the lens. Eventually he found it and stuck it straight in, knocked on the door and entered.
'The name's Conrad, Jess Conrad,' he stated confidently.
'I'm over here,' replied Harry Saltzman, wondering why this actor was talking to a hat stand.
'Oh, yes,' Jess said, swivelling around.
Unfortunately, the lens had picked up some fluff and grit from the carpet, and as if it wasn't bad enough that he couldn't see much with the lenses in, now tears were running down his cheek thanks to the grit in his eye – not _quite_ the persona for a fearless 007.
The interview lasted a few minutes and Jess left, still wearing the lenses, only to miss his footing on the top stair, and fall all the way down. As he clattered to a halt at the bottom, Harry appeared at the top of the stairwell, laughing riotously, and shouted, 'You'll never be James Bond – but I'd love to sign you up as a stunt man!'
Such was the success of the Bond films that Harry and Cubby began to make more money than they knew what to do with. Guy Hamilton was in Cubby's office one day after _Goldfinger_ opened and Harry called through on the speakerphone: 'Cubby, I know what we should do with our money,' he said. 'We'll buy gold!'
I'll always be grateful to Harry and Cubby for giving me the chance to play Jimmy Bond.
'But where would we keep it all?' Cubby asked, worriedly.
I had been great friends with Harry from the early 1960s, though that all changed when I started working for him. You see, Harry had this belief that if he paid you, then he owned you. I know he had a contract with Albert Finney, for example, where Albie had to seek his permission to work elsewhere.
When we started _Live and Let Die_ his partnership with Cubby was beginning to crack around the edges, and Harry effectively ran the production on the film by agreeing Cubby would do similar on the next – it was a way of minimizing the time they had to spend working together.
Harry loved making movies and made many other non-Bond films, whereas Cubby was content to concentrate on the franchise.
Once I signed as 007, Harry became very possessive and felt he owned me. He demonstrated it, for example, by not allowing my friend David Hedison to stay at the same hotel as me because he was jealous of our friendship. He argued about my long-time hairdresser, Mike Jones, joining the production ... and things like that.
I know director Guy Hamilton felt caught in the middle of the two producers and told UA that he could happily make a film with Harry, and could happily make a film with Cubby, but he wasn't keen on making a film with Cubby and Harry together.
However, for all his faults, Harry was still an amazing showman and he loved making movies.
Another legendary filmmaker and philanthropist, Sir Run Run Shaw. His use of ginseng root tea was thought to have helped keep him going to the ripe age of 106.
Greg Peck once told me about a meeting he had with the producer Run Run Shaw in Hong Kong. Shaw was a tremendously successful film mogul and philanthropist and founded Shaw Brothers Studios, which became one of the best-known film production companies in Hong Kong. Greg had gone over to meet with the great man and see his studio, where they shot all the Kung Fu and karate movies that were so popular all over the Far East. Anyhow, they were sitting in Run Run Shaw's office and Shaw brought out a beautiful, highly polished mahogany box, lined with red velvet, and lying inside was a large, knobbly ginseng root. Run Run Shaw told Greg that this particular piece of ginseng was from the mountains of mainland China, where mountaineers would hunt down the roots and dig them out with small spoons in order to preserve the root intact.
I should add that at this stage, Run Run Shaw was in his early seventies, incredibly rich and very charming.
'Gregory, I take three capsules of this every morning, and three more at night before I go to bed. I have a man grind this root up for me into a powder and I wash it down with a tot of Scotch,' declared the producer. 'And I can still do everything – I mean _everything_ – that I could when I was thirty,' he added, leaving Greg in no doubt as to what he meant.
Seeing how well ginseng worked at keeping Run Run Shaw on top form, dear Greg decided that it might just work for his old friend Niv, who had been ailing rather markedly the last time they had met. He duly sent some along to David Niven, but heard shortly after that Niv's doctors, already worried about him, were monitoring all his intake and wouldn't allow him to take it.
There's got to be something in it, though. I remembered that story when I read about Run Run Shaw's death in January 2014 – at the tender age of 106.
Of course, films featuring would-be heroic actors such as I only come about because of the ingenuity, negotiating and charming skills of producers. One who had all that in bucket loads was Elliott Kastner, who was most definitely one of the largest larger-than-life characters in the film business. He had a damn good eye for a commercial story, but then again he did start his career as a literary agent, so he ought to have known a good book when he read one.
I only made one film with Elliott, _North Sea Hijack_ , but knew him socially and from around Pinewood, where he kept an office for most of his working life. When he was in residence you'd know because there was no mistaking his voice booming down the corridor when he was on the phone, having rather heated conversations with financiers and executives, which often ended with Elliott lovingly telling them where to 'shove it'.
I think it's fair to say that Elliott was in litigation for most of his life with one person or another, but he also made more films than anyone else I know.
He was always on the lookout for well-heeled people in his never-ending pursuit of what he'd call the 'war chest', and the story goes that whenever Elliott arrived in LA he would stay at a well-known and rather luxurious hotel. On arrival, he would slip a very handsome gratuity to the reservations manager and ask if there were any residents from Texas in the large suites. He'd then go to their rooms, knock on the door and introduce himself. Over the years, Elliott had worked out that they were pretty likely to be wealthy oil executives and he'd always have a script to tell them about and an exciting opportunity for them to be in movies.
A couple of years before we started shooting _North SeaHijack_, Elliott had cast Marlon Brando and Jack Nicholson in a movie called _The Missouri Breaks_.
Brando was a great believer in 'the method' school of acting, so launched himself right into his characters, always trying to capture the character's psychological motivation and emotions. In this case he did so by catching grasshoppers in his downtime and eating a live frog. If that wasn't bizarre enough, the Western saw Brando speaking in an over-the-top Irish accent, and wearing a dress too.
When he was trying to pull the finance for the film together, Elliott heard that Jack Nicholson had just moved into a house near Marlon Brando's, and so he courted them both to star. He knew if both men agreed then the finance was assured. Brando was at the peak of his powers and Elliott happily yielded to demands both large and small in order to secure him, including hiring Arthur Penn to direct. But despite satisfying their every demand, neither Brando nor Nicholson would quite commit. So Elliott did the only thing he could – he resorted to subterfuge, telling both of them that the other _had_ agreed a deal. Not long after that they both happily signed on the dotted line! That was Elliott.
Postscript
A FEW YEARS AGO, WITH MY BRITISH PASSPORT COMING UP FOR renewal, I thought the easiest and quickest way of picking up a replacement was to book a one-day appointment at the Passport Office in London, where the plan was that I would fill in a form, bring a couple of photos and return after lunch to pick up my new document.
Upon handing my duly completed renewal paperwork to the man behind the counter, he tutted and said my signature was 'outside the box' and told me I'd need to go and fill in another complete form. About ten minutes later I returned, and ensured my signature was now well and truly within the appointed box.
'Are these photos recent ones?' he asked as I handed them over.
'Yes, I had them taken a week ago,' I replied. 'In Switzerland.'
'What?'
'A week ago ...' I repeated.
'You said Switzerland?' he asked, as he dropped them back on the desk. 'I'm afraid we can't accept these, as they are not on approved UK photographic paper.'
I was somewhat taken aback, but my interrogator was not someone I felt I could be in any way glib with, so, looking forlorn, I asked what I could do.
'Down that corridor,' he pointed. 'Turn left and follow it to the end and you'll see a photo machine there. It costs £5 and will print you four. When you have them, come back.'
After what seemed like a three-mile hike, I managed to obtain the photos, though obviously by then I wasn't in a terribly good mood as I look most perturbed in them. I dropped them back to my friend. He read through the form again, looked at the photos... 'I'm sorry, but this is a different person.'
'Pardon?' I asked.
'On your current passport you are named as _Mr_ Roger Moore, but now you want it to be in the name of _Sir_ Roger Moore – it's different.'
'Yes, I've been knighted in the meantime.'
'Ah,' he replied. 'Do you have any proof of that?'
'Proof?! What would you like?' I seethed through clenched teeth. 'A letter from the Queen?'
With that, he finished off the paperwork and matter-of-factly told me to report back to the office around the corner in three hours.
Following lunch and a much-needed glass of wine, I reported to what I can only really describe as a hatch in a wall, where I rang the bell and waited for said hatch to open.
'Name?' the little man said.
'Roger Moore.'
'Date of birth?'
'14 October 1927.'
'Got any ID on you?'
It's at times like this I'm incredibly tempted to say 'Do you know who I am?' (Or as in the case of when I was in New York for an interview promoting my first book, and the door security man [who had my name on his list] said, 'I can't let you in without photo ID', I slapped my book down in front of him and said, 'There! That's me and that's my name,' and walked through past him.) On this occasion, though, I think I produced either my driving licence or credit card.
He studied my ID very carefully and held it up against my passport. 'Ah, yes, that's fine.'
I took my passport and just as I was about to turn and exit, the little man called out, 'Excuse me!'
I returned to his hatch, and he smiled widely, 'I've always been a big fan, Sir Roger. Any chance of an autograph?'
I tell you this small anecdote to highlight the fact that whatever one's fame, whatever one has 'accomplished', whoever one has met and mingled with, wherever one has travelled, there are always times in life that one is brought back to earth with a bump. It's true what they say: 'You can take the boy out of south London, but you can't take south London out of the boy. This particular south London boy has always been very lucky. I've worked with the best and travelled the world, made friends with the great and the good and continue to live life to the full – but it's incidents like the one above that (eventually) make me smile and remember my roots.
Kristina and I visit some of the Kosovar refugees in FYR Macedonia in 1999.
On our first visit to Kazakhstan, Kristina and I met children with disabilities and were continually inspired.
In Closing ...
ACTING INTERESTS (AND ATTEMPTS) ASIDE, I AM, OF COURSE, still primarily kept busy as a UNICEF Goodwill Ambassador and I'm always delighted when my passion for UNICEF rubs off on those around me. One of my more recent fundraising projects is _Giving Tales_ , which came about when my son Christian and his business partner Klaus Lovgreen discussed the idea of producing an interactive, animated version of Hans Christian Andersen fairy tales. _Giving Tales_ is an app available on iPhones, iPads etc (although other tablet devices are available, of course...), and the plan is for a celebrity to read a Hans Christian Andersen story and, as an ever-continuing project, every year the catalogue is set to grow, generating an on-going royalty for UNICEF. I thought it sounded like a terrific project.
I called on a few friends to see if they might lend their tonsils, and the first to jump on board was the lovely Ewan McGregor. Soon afterwards, Stephen Fry, Joanna Lumley, Joan Collins and Michael Caine all joined me in recording tales. I'll keep you posted!
At UNICEF HQ with Namibian youth representative Livey Van Wyk and fellow Goodwill Ambassador Whoopi Goldberg, unveiling the winning poster of an international advertising competition to promote the Global Campaign on Children and AIDS.
Kristina and I still travel for UNICEF too, and not long ago we were in Germany for the committee's sixtieth anniversary, with various activities, interviews and recordings to tackle. I'm also now deploying my Twitter account to spread UNICEF news, appeals and share success stories – social media is certainly becoming all-important.
10 August 2004 in Beijing, China, I greet a girl, as other children and members of the press look on, during the launch of a UNICEF-supported summer camp for children orphaned by AIDS.
But thinking back to my early days as an Ambassador, having been recruited by Audrey Hepburn, I often wondered what drove Audrey's total and tireless devotion to the charity. I only discovered the answer at her funeral, when her son Sean read a poem that Audrey had herself read to her family from her sickbed only a few weeks earlier, on Christmas Eve 1992; it was written by American humorist and writer Sam Levenson for his granddaughter.
I found the words so very poignant that I wrapped up each of my recent theatre appearances with it, and indeed with an announcement that UNICEF collection buckets would be in the foyer on the way out – I've no shame in asking for help to save children's lives, and everyone gave so very generously.
_For attractive lips, speak words of kindness._
_For lovely eyes, seek out the good in people._
_For a slim figure, share your food with the hungry._
_For beautiful hair, let a child run his or her fingers through it once a day._
_For poise, walk with the knowledge that you never walk alone._
_People, even more than things, have to be restored, renewed, revived, reclaimed and redeemed; never throw out anyone._
_Remember, if you ever need a helping hand, you'll find one at the end of each of your arms._
_As you grow older, you will discover that you have two hands, one for helping yourself, the other for helping others._
Acknowledgements
_I would hereby like to acknowledge the invaluable help of:_
Gareth Owen for again sprinkling a little literacy on my work; Michael and Lesley O'Mara; Louise Dixon, my editor; Ron Callow, Claire Cater, Judy Palmer and Katy Parker; Lesley Pollinger and all at Pollinger Ltd; Iris Harwood; the National Motor Museum, Beaulieu; the BBC Written Archive, Caversham, Berkshire; _Cinema Retro_ magazine www.cinemaretro.com; Andy Boyle; Doris Spriggs; EON Productions Ltd; Alan Davidson of www.sirrogermoore.com
_And for supplying many of the photographs:_
Jaz Wiseman, UNICEF and EON Productions; Jenny Hanley; Geraldine Winner.
Picture Credits
Anthony Osmond-Evans
© United Archives/TopFoto
© Alpha Press
© Mirrorpix
REX/George Harris/Associated Newspapers
© Mirrorpix
REX/ITV
REX/ITV
REX/David Magnus
© Mirrorpix
Bert Hardy/Picture Post/Hulton Archive/Getty Images
REX
REX/Moviestore Collection
Alllied Artists/Getty Images
REX/ITV
© Alpha Press
Ron Galella Ltd/WireImage/Getty Images
Terry O'Neill/Getty Images
REX/Moviestore Collection
20th Century Fox/Golden Harvest/The Kobal Collection
NBC via Getty Images
Jack Garofalo/Paris Match via Getty Images
© UNICEF / Mark Thomas
© UNICEF
© UNICEF/Susan Markisz
© UNICEF/Wang Yongji
James Bond images are © All Rights Reserved Danjaq LLC and United Artists Corporation
Index
_A Matter of Time_ ref1
_A Night to Remember_ ref1, ref2
_A View to a Kill_ ref1
ABC TV ref1
ABPC Studios ref1, ref2, ref3, ref4, ref5, ref6, ref7
Abrams, J. J. ref1, ref2
Ackland-Snow, Terry ref1
ACTT ref1
_Aladdin_ ref1
Albert of Monaco, Prince ref1
Alexandra Palace 'Ally Pally', London ref1, ref2
_Alias_ ref1
Allen, Dave ref1
Allen, Irving ref1, ref2, ref3
Allen, Nat ref1
Anderson, Jean ref1
Andrews, Julie ref1, ref2
_Anna Karenina_ ref1
_Anniversary, The_ ref1
_Arabian Adventure_ ref1
Armstrong, Neil ref1
Army Film Unit, British ref1
_Arthur_ ref1
_Assignment Foreign Legion_ ref1
Assistant Stage Manager (ASM) roles, RM's ref1, ref2
Attenborough, Lord ref1
_Audience, The_ ref1
Automobile Association (AA) training films ref1
Bacall, Lauren ref1
_Bad Sister_ ref1
Baker, Bob ref1, ref2, ref3
Baker, Roy Ward ref1, ref2, ref3, ref4
_Bal Creole_ ref1
Baldwin, James ref1
Bankhead, Tallulah ref1
Barker, Lex ref1
Barnard, Dr Christiaan ref1
Barrymore, John ref1
Barth, Louis ref1
_Batman_ ref1
BBC (British Broadcasting Corporation) ref1, ref2
Beaconsfield Studios ref1
Beaumont, Binkie ref1
_Behind the Green Door_ ref1
Belafonte, Harry ref1
Bergman, Ingrid ref1, ref2
Berman, Monty ref1
Bertil of Sweden, Prince ref1
_Birdman of Alcatraz, The_ ref1
Bishop, Joey ref1, ref2
_Black Knight, The_ ref1, ref2
_Black Lion Films_ ref1
_Black Narcissus_ ref1
Black Park Studios ref1
_Black Shield of Falworth, The_ ref1
Blackman, Honor ref1
Blair, Cherie ref1
_Blue Lagoon, The_ ref1
Bogarde, Dirk ref1, ref2, ref3
Bogart, Humphrey ref1, ref2, ref3
Bond, Gordon ref1
_Bond on Bond_ (RM's book) ref1
_Boston Strangler, The_ ref1
Box, Betty ref1, ref2
Box, Sydney ref1
_Boys from Brazil, The_ ref1, ref2
Brando, Marlon ref1, ref2
Bray Studios ref1
Bricusse, Leslie ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9
_Brides of Fu Manchu, The_ ref1
_Bridge on the River Kwai, The_ ref1
British Lion Studios ref1, ref2
Broccoli, Albert R. 'Cubby' ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13
Broccoli, Dana ref1
Bryan, Dora ref1
Bryner, Yul ref1
_Bullet to Beijing_ ref1
_Bullseye_ ref1, ref2
Burton, Richard ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10
Caan, James ref1
Cagney, James ref1
Caine, Michael ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10
_Call Me Bwana_ ref1
_Call of the Wild, The_ ref1
_Candida_ ref1
Candy, John ref1
Cannes Film Festival ref1
_Cannonball Run, The_ ref1
Capron, Robert ref1
_Carry on Cleo_ ref1
_Carry On_ film series ref1, ref2, ref3
_Carry on Spying_ ref1
_Carry on up the Khyber_ ref1
_Castle of Fu Manchu, The_ ref1
CBS ref1
Chandler, Jeff ref1
Changi Prison, Singapore ref1, ref2
_Charlie and the Chocolate Factory_ ref1
_Chitty Chitty Bang Bang_ ref1
Christian Anderson, Hans ref1
Christian, Linda ref1
Chung Ling Soo ref1
Cilento, Diane ref1
Clark, Petula ref1
Clayton, Harold ref1
_Cleopatra_ ref1, ref2
Cobb, Lee J. ref1
Cohn, Harry ref1
Colbert, Claudette ref1
Collins, Elsa ref1
Collins, Jackie ref1, ref2
Collins, Joan ref1, ref2, ref3, ref4, ref5
Collins, Joe ref1
Colman, Ronald ref1
Columbia Studios ref1, ref2, ref3
_Comancheros, The_ ref1
Combined Services Entertainment Unit, Hamburg ref1, ref2
Connery, Sean ref1
Connor, Kevin ref1
Connor, William 'Cassandra' ref1
Conrad, Jess ref1, ref2, ref3
Cooper, Gary ref1
Cooper, Tommy ref1
_Coronation Street_ ref1
Cossins, James ref1
Cote d'Azur, France ref1
Coward, Noel ref1, ref2, ref3
Cowell, Elizabeth ref1
Cracknell, Derek ref1
Craig, Daniel ref1
Craig, Ivan ref1
Crawford, Joan ref1, ref2
_Crossplot_ ref1
Crown Film Units ref1
Crust, Arnold (Michael Winner pseudonym) ref1
Curtis, Leslie ref1, ref2
Curtis, Tony ref1, ref2, ref3, ref4, ref5
Curtiz, Michael ref1
Cushing, Peter ref1
_Daily Mirror_ ref1
Dallas, Robert ref1
_Dangerous_ ref1
Dark, John ref1
_Dark of the Moon, The_ ref1
Davies, Betty Ann ref1
Davies, Jack ref1
Davis, Bette ref1, ref2, ref3
Davis, John ref1, ref2, ref3
Davis Jr., Sammy ref1, ref2, ref3
Davis Sr., Sammy ref1
de Cicco, Pat ref1
_Death on the Nile_ ref1
_Deep Throat_ ref1
Deighton, Len ref1
Delfont, Lord 'Bernie' ref1
Dench, Dame Judi ref1
Denham Studios ref1, ref2
_Dial M for Murder_ ref1
_Diamonds are Forever_ ref1
_Diane_ ref1
Dickens, Monica ref1
Disney ref1
Disney, Walt ref1
_Doctor Dolittle_ ref1
_Doctor_ film series ref1
_Don't Bother to Knock_ ref1
Dors, Diana ref1, ref2, ref3
Dorsey, Tommy ref1
Douglas, Angela ref1
Douglas, Anne ref1
Douglas, Kirk ref1, ref2
_Dr No_ ref1, ref2
_Drawing Room Detective, The_ ref1, ref2
_Dresser, The_ ref1
Duke, Doris ref1
Eady Levy ref1
Ealing Studios ref1
Eden, Mark ref1
_Educating Archie_ ref1
Edwards, Blake ref1, ref2, ref3
Ekland, Britt ref1, ref2
Elizabeth II, Queen ref1
Ellis, Brian ref1
Elstree Studios ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8
EMI Films ref1
Engelen, Paul ref1
_Entertainer, The_ ref1
EON Productions ref1, ref2, ref3
Equity ref1, ref2
_Face of Fu Manchu, The_ ref1
_Fallen Idol, The_ ref1, ref2
Farrow, Mia ref1, ref2
Feldman, Charles ref1
_Fiddler on the Roof_ ref1
Finch, Peter ref1, ref2, ref3, ref4
Finney, Albert ref1, ref2, ref3, ref4
Fleming, Ian ref1, ref2, ref3
Flynn, Errol ref1
Foenander, Allan ref1
_For Your Eyes Only_ ref1
Forbes, Bryan ref1, ref2, ref3, ref4
Ford, Glenn ref1
Foster, Norman ref1
_Four Weddings and a Funeral_ ref1
_Francis of Assisi_ ref1
Franco, Jess ref1
Frankenheimer, John ref1
Frederick, Lynne ref1, ref2
Friedman, Dore ref1
_From Here to Eternity_ ref1
_From Russia with Love_ ref1, ref2
Frost, David ref1, ref2
Fry, Stephen ref1, ref2
_Fu Manchu_ film series ref1
Gabel, Scilla ref1
Gable, Clark ref1, ref2
Gabor, Eva ref1
Gabor, Zsa Zsa ref1
Gainsborough Studios ref1
_Gallant Hours, The_ ref1
Gardner, Ava ref1, ref2, ref3, ref4
Gayson, Eunice ref1
_Genevieve_ ref1
Gibson, Percy ref1
Gielgud, Sir John ref1, ref2
Gielgud Theatre, London ref1
Gilliat, Sidney ref1
Gingold, Hermione ref1
_Giving Tales_ fundraising project ref1
_Gladiator_ ref1
Glen, Archie ref1
Gobbi, Tito ref1
_Gold_ ref1, ref2
Gold, Jan ref1
Gold, Johnny ref1, ref2
_Goldfinger_ ref1, ref2
_Gone with the Wind_ ref1
Goodman, Johnny ref1, ref2, ref3
Gordon, Dorothy ref1
_Governess, The_ ref1
Grace, Martin ref1, ref2
Grace of Monaco, Princess
_see_ Kelly, Grace
Grade, Lady ref1
Grade, Lew ref1, ref2, ref3, ref4, ref5, ref6
Grade, Michael ref1
Granger, Stewart ref1, ref2
Grant, Cary ref1, ref2, ref3, ref4
Gray, Willoughby ref1
_Great Expectations_ ref1
Great Gingalee, The ref1
_Green Fire_ ref1
Guilaroff, Sydney ref1
_Guns at Batasi_ ref1
_Guns of Navarone, The_ ref1
Gustav, Carl ref1
H. M.Tennent ref1
Hackman, Gene ref1
Hackney Empire, London ref1
Hall, Bill ref1, ref2
Hall, Sir Peter ref1
Hamilton, Guy ref1, ref2, ref3
Hamilton, Patrick ref1
_Hamlet_ ref1, ref2
Hammer Films ref1, ref2, ref3
Hancock, Sheila ref1, ref2
Handmade Films ref1
Hanks, Tom ref1
Hanley, Jenny ref1, ref2
Harben, Joan ref1
Harbud, Gordon ref1
Harris, Anita ref1
Harris, Elizabeth ref1, ref2, ref3
Harris, Richard ref1, ref2, ref3
Harrison, George ref1
Harrison, Rex ref1, ref2, ref3, ref4
Harrison, Stephen ref1
Harwood, Johanna ref1
Hawkins, Jack ref1, ref2
Hedison, David ref1
Hefner, Hugh ref1
Hepburn, Audrey ref1, ref2, ref3
Hepburn Ferrer, Sean ref1
Hepburn, Katharine ref1
Herbert, Percy ref1, ref2
_Heroes of Telemark, The_ ref1
Herren, Kip ref1
Heston, Charlton ref1
Hird, Dame Thora ref1
Hitchcock, Alfred ref1
Hjartso, Ingvar ref1
Hoare, Colonel Mike ref1
Hodge, Douglas ref1
Hoffman, Dustin ref1
Holden, William ref1
Holder, Boscoe ref1
Holder, Geoffrey ref1
Hollywood Studios ref1
Hopper, Hedda ref1
_House on the Square, The_ ref1
Howard, Trevor ref1, ref2, ref3
Hudson, Rock ref1
Huggins, Roy ref1
Hughes, Howard ref1, ref2
Hunniford, Gloria ref1
Hunt, Gareth ref1
Huston, John ref1, ref2
Hutton, Barbara ref1
Idle, Eric ref1
_Importance of Being Earnest, The_ ref1
Intimate, The, Palmers Green ref1
_IP5:The Island of Pachyderms_ ref1
_Iron Petticoat, The_ ref1
_Ishtar_ ref1
ITC (Incorporated Television Company) ref1, ref2, ref3
ITV ref1
_Ivanhoe_ ref1
_Jack the Ripper_ (film) ref1
_Jack the Ripper_ (stage production) ref1
Jackson, Glenda ref1
James Bond films ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16
James, Harry ref1
James, Sid ref1, ref2
Janni, Joe ref1
Jefferies, Lionel ref1
_Jesus of Nazareth_ ref1, ref2
Jones, Mike ref1, ref2
Jones, Quincy ref1
Kastner, Elliott ref1
Kaye, Danny ref1
Kaye, Dena ref1
Keach, Stacy ref1
Kellerman, Barbara ref1, ref2
Kelly, Grace ref1, ref2
Kendall, Kay 'Katie' ref1
_Kidnapped_ ref1
_Killers, The_ ref1
Kimmins, Anthony ref1
King, Nosmo ref1
Kirsch, Leo ref1
Kirsch Media TV ref1
Knopf, Edwin ref1
Korda, Alexander ref1, ref2, ref3
Kruger, Hardy ref1
Kubrick, Stanley ref1, ref2
Kuhn, Michael ref1
La Rue, Danny ref1
Labella, Vincenzo ref1
Ladd, Alan ref1, ref2, ref3
Ladd, Sue ref1, ref2
_Ladykillers, The_ ref1
Lancaster, Burt ref1, ref2
_Land of the Pharaohs_ ref1
Las Vegas, California ref1, ref2, ref3, ref4, ref5
_Last Time I Saw Paris, The_ ref1
Laughton, Charles ref1
Launder, Frank ref1
Lawford, Peter ref1, ref2
_Lawman_ ref1
_Lawrence of Arabia_ ref1
Lawson, Wilfred ref1
Lazar, Swifty ref1, ref2
Lean, David ref1, ref2, ref3
Ledger, Heath ref1, ref2
Lee, Benny ref1
Lee, Bruce ref1
Lee, Christopher ref1, ref2, ref3, ref4
Lee, Jenny ref1
Leigh, Vivien ref1
Lerman, Oscar ref1
Levenson, Sam ref1
Lewis, Jack ref1
Lewis, Jerry ref1
Liberace ref1
_Life of Brian, The_ ref1
Lilian of Sweden, Princess ref1
Lipman, Maureen ref1
_Little Hut, The_ ref1
_Live and Let Let Die_ ref1, ref2
Lloyd, Euan ref1
Lloyd Webber, Lord ref1
Loew Jr., Arthur ref1
Lom, Herbert ref1
_Long and the Short and the Tall, The_ ref1, ref2
_Long Good Friday, The_ ref1
_Look Back in Anger_ ref1
Loren, Sophia ref1, ref2
Lovgreen, Klaus ref1
Lucasfilms ref1
Lucidi, Marizio ref1
_Lucky Dip_ ref1
_Lucky Lady_ ref1
Lugosi, Bela ref1
Lumley, Joanna ref1
Lytess, Natasha ref1
Maasz, Ronnie ref1
MacRae, Gordon ref1
Maibaum, Richard ref1, ref2
Mamoulian, Rouben ref1, ref2
_Man Who Haunted Himself, The_ ref1
_Man with the Golden Gun, The_ ref1, ref2
Mankiewicz, Joseph ref1, ref2
Mankiewicz, Tom ref1
Mankowitz, Wolf ref1
Manley, Peter ref1, ref2
Mann, Anthony ref1
_Marathon Man, The_ ref1
Margaret, Princess ref1, ref2, ref3
Marsden, Betty ref1
Martin, Dean ref1, ref2
Martin, Dean Paul ref1
Marvin, Lee ref1
Marx, Barbara ref1, ref2, ref3
Mason, Tom ref1
Mather, Berkely ref1
_Mating Game, The_ ref1
_Maverick_ ref1
Mayer, Louis B. ref1
McGregor, Ewan ref1
McLaglen, Andy ref1, ref2
Medina, Patricia ref1
Medwin, Michael ref1
MGM ref1, ref2, ref3, ref4, ref5, ref6, ref7
MGM Grand Hotel, Las Vegas ref1
Milland, Ray ref1, ref2
Miller, Arthur ref1
Mills, John ref1, ref2, ref3, ref4
Mills, Mary ref1
Minnelli, Liza ref1
Minnelli, Vincente ref1
Mirren, Dame Helen ref1, ref2
_Misfits, The_ ref1
_Missouri Breaks, The_ ref1
Mitchell, Leslie ref1
Mitchum, Robert ref1
_Moby Dick_ ref1
Monaco ref1, ref2
Monroe, Marilyn ref1, ref2
Montagu of Beaulieu, Baron ref1
Montand, Yves ref1
Monty Python team ref1
Moon, George ref1
Moore, Christian (RM's son) ref1, ref2
Moore, Evelyn ref1
Moore, George (RM's father) ref1
Moore, Lara (RM's daughter-in-law) ref1
Moore, Tristan and Maximillian (RM's grandsons) ref1
More, Billie ref1
More, Kenneth ref1, ref2
Morecambe, Eric ref1
Morton, Clive ref1
Moxey, John Llewellyn ref1
Murton, Peter ref1
_My Fair Lady_ ref1
_My Forbidden Past_ ref1
_My Wicked Wicked Ways_ ref1
_Naked Face, The_ ref1
_Nanny, The_ ref1
National Police Memorial, The Mall ref1, ref2
NBC ref1
Neagle, Anna ref1
Neame, Ronald ref1
Nesbit, E. ref1
_Network_ ref1
Newton, Robert ref1
Nicholson, Jack ref1
_99 Women_ ref1
Niven, David ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8
_North Sea Hijack_ ref1
Novak, Kim ref1
Ntshona, Winston ref1
Oberon, Merle ref1
O'Brien, Denis ref1
_Ocean'sref1_ ref2
_Octopussy_ ref1, ref2
_Ol' Blue Eyes Is Back_ ref1
_Oliver Twist_ ref1, ref2
Olivier, Laurence ref1, ref2, ref3, ref4, ref5, ref6
_Omen, The_ ref1
_On Her Majesty's Secret Service_ ref1
_One That Got Away, The_ ref1
Opel, Nancy ref1
Osborne, John ref1
O'Toole, Peter ref1, ref2
_Outlaw, The_ ref1
Owen, Gareth ref1
Palance, Jack ref1
Palmer, Lilli ref1
Pam, Jerry ref1
Pamplona, Spain ref1, ref2 paparazzi ref3
Paramount Pictures ref1, ref2
Parkinson, Michael ref1, ref2
Parsons, Louella ref1, ref2
_Party, The_ ref1
_Pass Beyond Kashmir, The_ ref1
passport renewal, British ref1
Pathé ref1
Peck, Gregory ref1, ref2, ref3, ref4, ref5, ref6, ref7
Peck, Veronique ref1, ref2, ref3, ref4, ref5
Penn, Arthur ref1
_Persuaders!, The_ ref1, ref2, ref3, ref4, ref5, ref6
Phillips, Leslie ref1
_Pickwick_ ref1
Pinewood Company of Youth ref1
Pinewood Studios ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18
_Pink Panther Strikes Again, The_ ref1
_Pink Panther, The_ ref1
_Plan 9 from Outer Space_ ref1
_Play it Cool_ ref1, ref2
Pleasence, Donald ref1
Police Memorial Trust ref1
Polygram Films ref1
_Potter's Wheel_ interlude film ref1
Powell, Michael ref1
Powell, Nosher ref1
Power, Tyrone ref1, ref2
Pressburger, Emeric ref1, ref2, ref3
_Prince and the Showgirl, The_ ref1
_Private Lives_ ref1
Proctor & Gamble Entertainment ref1
Purdom, Edmund ref1
Quarry, Miranda ref1, ref2
_Radio Times_ ref1
Rae, Lynette ref1
_Railway Children, The_ ref1
Rakoff, Alvin ref1
Randle, Frank ref1, ref2
Rank, Arthur J. ref1, ref2
'Rank Charm School' ref1
Rank Organisation ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10
_see also_ Pinewood Studios
_Rape of the Sabines, The_ ref1
Rat Pack ref1
_Reach for the Sky_ ref1
_Red Beret, The_ ref1, ref2
_Red Shoes, The_ ref1
Reed, Carol ref1, ref2
Reed, Maxwell ref1
Reed, Oliver ref1
Reeve, Christopher ref1
Reich Chancellery, Berlin ref1
Remick, Lee ref1
_Return of the Pink Panther, The_ ref1, ref2
Reynolds, Burt ref1
Reynolds, Debbie ref1
Rice, Tim ref1
Richardson, Tony ref1
Rickles, Don ref1, ref2
Rigg, Diana ref1
Ritchie, June ref1
Roberts, Rachel ref1, ref2
Robertson, Bill ref1
Robinson, William Ellsworth ref1
Rogers, Peter ref1, ref2, ref3, ref4, ref5
Romain, Yvonne 'Evie' ref1, ref2
_Roman Holiday_ ref1
_Romulus and the Sabines_ ref1
Rooney, Mickey ref1
Roselli, Johnny ref1
Rossiter, Leonard ref1
Rubirosa, Porfirio ref1
_Saint, The_ ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9, ref10, ref11, ref12, ref13
Saltzman, Harry ref1, ref2, ref3, ref4, ref5
Saltzman, Jacqueline ref1
Samms (Samuelson), Emma ref1
Sanders, George ref1, ref2
_Saturday Night and Sunday Morning_ ref1, ref2
Savoy Hotel, London ref1, ref2
_Scars of Dracula_ ref1, ref2
Schenck, Joseph ref1
Schlatter, George ref1, ref2
Schlosser, Herbert ref1
Scott, George C. ref1
_Scott of the Antarctic_ ref1
Scott, Ridley ref1
Scott, Tony ref1
screen tests, RM's ref1, ref2, ref3
_Sea Wolves, The_ ref1, ref2, ref3, ref4, ref5, ref6
Second World War ref1, ref2, ref3, ref4
Sellers, Anne ref1
Sellers, Michael ref1
Sellers, Peter ref1, ref2, ref3
Shanks, Bill ref1
Sharif, Omar ref1, ref2
Shaw, Artie ref1
Shaw Brothers Studios ref1
Shaw, Sir Run Run ref1
Shearer, Norma ref1
Sheldon, Sidney ref1
Shepperton Studios ref1, ref2, ref3, ref4, ref5, ref6
Sheridan, Dinah ref1
_Sherlock Holmes in New York_ ref1
_Sicilian Cross, The_ ref1, ref2
Signoret, Simone ref1
Simkins, Michael ref1
Simmons, Jean ref1
Sinatra (née Barbato), Nancy ref1, ref2, ref3
Sinatra, Frank ref1, ref2, ref3, ref4, ref5, ref6
Sinatra Jr., Frank ref1
Sinatra Jr., Nancy ref1, ref2
Sinatra, Tina ref1, ref2, ref3
Sinden, Donald ref1, ref2
Singapore ref1, ref2
_Singer Not the Song, The_ ref1
_Sink the Bismarck!_ ref1
Slaughter, Tod ref1
_So this is Hollywood_ ref1
_Solomon and Sheba_ ref1
_Some Like it Hot_ ref1
_Something's Got to Give_ ref1
_Son of Ali Baba_ ref1
_Spartacus_ ref1, ref2
Spiegel, Sam ref1
Spinetti, Victor ref1, ref2
Spriggs, Doris ref1
_Spy Who Loved Me, The_ ref1
Squires, Dorothy 'Dot' ref1, ref2, ref3, ref4, ref5, ref6, ref7
St John, Earl ref1
St Paul, France ref1, ref2, ref3
_Staircase_ premiere ref1
_Star Wars_ series ref1, ref2
Stark, Graham ref1
Stars Organisation for Spastics (SOS/SCOPE) ref1
Steel, Anthony ref1
_Stop the World – I Want to Get Off_ ref1
Stroud, Robert ref1
Studio Lisa ref1 stuntmen ref1, ref2
_Sunday Lovers_ ref1, ref2
_Sunday Night at the London Palladium_ ref1
_Sunday Times_ ref1
_Sweet Smell of Success, The_ ref1
Tate, Sharon ref1
Taylor, Elizabeth ref1, ref2, ref3, ref4, ref5
Taylor, Robert ref1
_That Lucky Touch_ ref1, ref2
Theatre Royal, Drury Lane ref1
_Third Man, The_ ref1
Tholstrup, Kristina (RM's wife) ref1, ref2, ref3, ref4, ref5, ref6
Thomas, Gerald ref1
Thomas, Ralph ref1
Thompson, Carlos ref1
Thorburn, Lew ref1
Thorson, Scott ref1
Tidyman, Ernest ref1
Till, Stewart ref1
_To Kill a Mockingbird_ ref1
Todds, Roy ref1
Tomlinson, David ref1
Tone, Franchot ref1
Toone, Geoffrey ref1
Towers, Harry Alan 'HAT' ref1
_Trail of the Pink Panther_ ref1
_Trapeze_ ref1
_Treasure Island_ ref1, ref2
Tuchner, Michael ref1
Turner, Frank ref1
Turner, Lana ref1, ref2, ref3
20th Century Fox ref1, ref2, ref3, ref4, ref5, ref6, ref7
_Twice Upon a Time_ ref1
Unger, Barry ref1
Unger, Cassie ref1
UNICEF ref1, ref2, ref3
United Artists ref1, ref2, ref3
Ustinov, Peter ref1
Vidor, King ref1
Villiers, James ref1
Wagner, Robert ref1
Wakefield, Duggie ref1
Wardour, Sir Archibald ref1
Warner Bros. ref1, ref2, ref3, ref4
Warner, Jack ref1, ref2, ref3
Wasserman, Lew ref1, ref2
Watson, Jack ref1
Wayne, John ref1, ref2, ref3
_We Joined the Navy_ ref1
Welbeck, Peter ref1
Welles, Orson ref1, ref2
_What Ever Happened to Baby Jane?_ ref1
Wheeler, Jimmy ref1
_While Parents Sleep_ ref1
White, Loray ref1
Whitman, Stuart ref1
Wilcox, Eddie ref1
Wilcox, Herbert ref1
_Wild Geese, The_ ref1, ref2, ref3
Wilder, Billy ref1
Williams, Kenneth ref1, ref2
Williamson, Nicol ref1
Willis, John ref1, ref2
_Wind Cannot Read, The_ ref1
Windsor, Barbara ref1
Winner, Geraldine ref1
Winner, Michael ref1, ref2, ref3
Winters, Shelley ref1
Wisdom, Norman ref1
Wolfit, Sir Donald ref1
Wood, Ed ref1
_Wooden Horse, The_ ref1
Woodfall Productions ref1, ref2
Wretman, Ewa ref1
Wretman, Tore ref1
Wurtzel, Sol ref1
Wyler, William ref1
Wyman, Jane ref1
York, Michael ref1
Young, Terence ref1, ref2, ref3
Zanuck, Darryl Francis ref1, ref2, ref3
Zukor, Adolph ref1
List of Illustrations
1. With the inimitable Elizabeth Taylor in my first movie in Hollywood, _The Last Time I Saw Paris_. Believe me, I was looking into the glass! _(REX/Moviestore Collection)_
2. Two shots from _Diane_ with Lana Turner. She taught me how to kiss with passion, but without the pressure. _(MGM/The Kobal Collection)_
3. With lovely Angie Dickinson in _The Sins of Rachel Cade_ , 1961. Sadly I wasn't allowed to sin (on screen at any rate!). _(REX/SNAP)_
4. The poster for an 'intriguing' film... _The Sicilian Cross_ , _The Executors_ , _Opium Road_ , _Street People_ , it went by several titles and the poster was most definitely better than the film – though Stacy Keach was a delight.
5. I recruited a few young starlets to help me find my missing halo on _The Saint_.
6. In1967 German magazine _Bravo_ voted me the best television actor – naturally! – and their editor presented me with this lovely award.
7. Me 'persuading' dogs not to use this tree.
8. We had attracted a big crowd for this set-up and had to stop Tony falling into the river by having someone clutch him from behind.
9. The Aston Martin DBS from _The Persuaders!_ sold for £545,000 at auction in May 2014... wish I'd kept it now...
10. A shotgun wedding! With Lee Marvin and Barbara Parkins in _Shout at the Devil_ in 1976, and that's Sir Ian Holm peering over the bride's shoulder. The black eyes were make-up – honestly!
11. Not _Hart to Hart_ , but with Stefanie Powers in _Escape to Athena_ , going in for a close up. My other co-stars included David Niven and Telly Savalas. The tagline on the poster was: 'The patriot, the professor, the comic and the stripper were fighting for what they believed in... GETTING RICH!' I couldn't have put it better myself!
12. They say an actor's life is a solitary one. I say, 'nah, nah, nah-nah-nah' to that!
13. Me looking well and truly goosed in a shot from _The Wild Geese_. It's not all wine, women and song, you know.
14. At the Oscars in 1989 Michael Caine, Sean Connery and I presented the Best Actor in a Supporting Role award to Kevin Kline for his part in _A Fish Called Wanda_ – and then tried to steal it off him.
15. On a holiday in Venice with my old friend Bryan Forbes.
16. Tony Curtis with his wife Jill, Kristina and myself met up at the _Empire_ Awards in London in 2006, where Tony picked up a much-deserved Lifetime Achievement Award.
17. Kristina always threw the best birthday parties for me, this one at Le Dome on Hollywood's famous Strip. ( _l to r_ ) Frank Sinatra, Joan Collins, Gregory Peck and my darling, at one of them.
18. With Frank and Barbara Sinatra and Chief Abbot William B. Williams (the American disc jockey who first called Frank 'The Chairman of the Board') at The Friars Club Man of the Year award in 1986. Believe it or not, I was that man!
19. Dean Martin was always thought of as a hard-drinking rascal but in reality his glass was usually filled with apple juice.
20. HSH Prince Albert II of Monaco presents me as a Monaco Goodwill Ambassador on behalf of the Ambassadors Club of Monaco in 2012, at the Hotel de Paris.
21. ( _l to r_ ) Kevin Costner, HSH Prince Albert II of Monaco, Kristina, myself and Sir Richard Branson at the Monaco Golf Club – we arrived by helicopter and left in Richard's hot air balloon!
22. Two of my favourite ladies: daughter Deborah and Kristina at the Cannes Film Festival in 2007 for _The Ladykillers_.
23. ( _l to r_ ), my daughter-in-law Loulou and her husband Geoffrey (that makes him my son!), Deborah, Kristina and me with the lovely Geraldine and Michael Winner in Gstaad.
24. My most recent film was _A Princess for Christmas_ , during which three huge Bulgarian technicians fell on my leg – hence the attractive footwear in this cast shot.
25. With ( _l to r_ ) Tore Wretman, HRH Princess Lilian (showing her sense of fun with that apron), Kristina and HM Queen Sylvia of Sweden at a lunch party.
26. ( _l to r_ ) Lady-in-waiting Lena Luttichaü, HM Queen Margrethe II of Denmark and Pastor Peter Parkov, who married Kristina and I in Copenhagen. HRH The Prince Consort is not in the picture – because he took it!
27. My eldest son Geoffrey with his wife Loulou and daughters Ambra and Mia, with Kristina and me.
28. Kristina's daughter Christina with her horse Lucky.
29. Kristina's son, Hans-Christian, his wife Henrietta and their two children Kathrine and Adrian.
30. Kristina with her grandson Lucas.
31. My youngest son Christian with his wife Lara, daughter Heidi and sons Tristan and Maximillian.
32. In 2011 the Royal Albert Hall staged a festive treat in aid of UNICEF – and here I am, with my daughter Deborah, introducing Captain Beaky and his Band, and a number called 'The Grasshopper'. ( _Christine Goodwin/courtesy of RAH, 2011_ )
33. In 2012 I was awarded an Honorary Doctorate of Arts from the University of Hertfordshire. Their Chancellor, the Marquess of Salisbury, kindly presented it to me.
34. Kristina and me on a recent UNICEF trip to Aachen, Germany, and the Opening Ceremony of the World Equestrian Festival, looking rather regal in our carriage ( _Henry Herrmann/UNICEF_ )
PLATES
1. With the inimitable Elizabeth Taylor in my first movie in Hollywood, _The Last Time I Saw Paris_. Believe me, I was looking into the glass! _(REX/Moviestore Collection)_
2. Two shots from _Diane_ with Lana Turner. She taught me how to kiss with passion, but without the pressure. _(MGM/The Kobal Collection)_
3. With lovely Angie Dickinson in _The Sins of Rachel Cade_ , 1961. Sadly I wasn't allowed to sin (on screen at any rate!). _(REX/SNAP)_
4. The poster for an 'intriguing' film... _The Sicilian Cross_ , _The Executors_ , _Opium Road_ , _Street People_ , it went by several titles and the poster was most definitely better than the film – though Stacy Keach was a delight.
5. I recruited a few young starlets to help me find my missing halo on _The Saint_.
6. In1967 German magazine _Bravo_ voted me the best television actor – naturally! – and their editor presented me with this lovely award.
7. Me 'persuading' dogs not to use this tree.
8. We had attracted a big crowd for this set-up and had to stop Tony falling into the river by having someone clutch him from behind.
9. The Aston Martin DBS from _The Persuaders!_ sold for £545,000 at auction in May 2014... wish I'd kept it now...
10. A shotgun wedding! With Lee Marvin and Barbara Parkins in _Shout at the Devil_ in 1976, and that's Sir Ian Holm peering over the bride's shoulder. The black eyes were make-up – honestly!
11. Not _Hart to Hart_ , but with Stefanie Powers in _Escape to Athena_ , going in for a close up. My other co-stars included David Niven and Telly Savalas. The tagline on the poster was: 'The patriot, the professor, the comic and the stripper were fighting for what they believed in... GETTING RICH!' I couldn't have put it better myself!
12. They say an actor's life is a solitary one. I say, 'nah, nah, nah-nah-nah' to that!
13. Me looking well and truly goosed in a shot from _The Wild Geese_. It's not all wine, women and song, you know.
14. At the Oscars in 1989 Michael Caine, Sean Connery and I presented the Best Actor in a Supporting Role award to Kevin Kline for his part in _A Fish Called Wanda_ – and then tried to steal it off him.
15. On a holiday in Venice with my old friend Bryan Forbes.
16. Tony Curtis with his wife Jill, Kristina and myself met up at the _Empire_ Awards in London in 2006, where Tony picked up a much-deserved Lifetime Achievement Award.
17. Kristina always threw the best birthday parties for me, this one at Le Dome on Hollywood's famous Strip. ( _l to r_ ) Frank Sinatra, Joan Collins, Gregory Peck and my darling, at one of them.
18. With Frank and Barbara Sinatra and Chief Abbot William B. Williams (the American disc jockey who first called Frank 'The Chairman of the Board') at The Friars Club Man of the Year award in 1986. Believe it or not, I was that man!
19. Dean Martin was always thought of as a hard-drinking rascal but in reality his glass was usually filled with apple juice.
20. HSH Prince Albert II of Monaco presents me as a Monaco Goodwill Ambassador on behalf of the Ambassadors Club of Monaco in 2012, at the Hotel de Paris.
21. ( _l to r_ ) Kevin Costner, HSH Prince Albert II of Monaco, Kristina, myself and Sir Richard Branson at the Monaco Golf Club – we arrived by helicopter and left in Richard's hot air balloon!
22. Two of my favourite ladies: daughter Deborah and Kristina at the Cannes Film Festival in 2007 for _The Ladykillers_.
23. ( _l to r_ ), my daughter-in-law Loulou and her husband Geoffrey (that makes him my son!), Deborah, Kristina and me with the lovely Geraldine and Michael Winner in Gstaad.
24. My most recent film was _A Princess for Christmas_ , during which three huge Bulgarian technicians fell on my leg – hence the attractive footwear in this cast shot.
25. With ( _l to r_ ) Tore Wretman, HRH Princess Lilian (showing her sense of fun with that apron), Kristina and HM Queen Sylvia of Sweden at a lunch party.
26. ( _l to r_ ) Lady-in-waiting Lena Luttichaü, HM Queen Margrethe II of Denmark and Pastor Peter Parkov, who married Kristina and I in Copenhagen. HRH The Prince Consort is not in the picture – because he took it!
27. My eldest son Geoffrey with his wife Loulou and daughters Ambra and Mia, with Kristina and me.
28. Kristina's daughter Christina with her horse Lucky.
29. Kristina's son, Hans-Christian, his wife Henrietta and their two children Kathrine and Adrian.
30. Kristina with her grandson Lucas.
31. My youngest son Christian with his wife Lara, daughter Heidi and sons Tristan and Maximillian.
32. In 2011 the Royal Albert Hall staged a festive treat in aid of UNICEF – and here I am, with my daughter Deborah, introducing Captain Beaky and his Band, and a number called 'The Grasshopper'. ( _Christine Goodwin/courtesy of RAH, 2011_ )
33. In 2012 I was awarded an Honorary Doctorate of Arts from the University of Hertfordshire. Their Chancellor, the Marquess of Salisbury, kindly presented it to me.
34. Kristina and me on a recent UNICEF trip to Aachen, Germany, and the Opening Ceremony of the World Equestrian Festival, looking rather regal in our carriage ( _Henry Herrmann/UNICEF_ )
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 4,472
|
End of June, our Indian dealer participated at the Logmat trade show in Chennai, India. It was the 4th edition of the Logmat trade show, the leading Business to Business exhibition for Logistics, Storage, Material Handling, Warehousing, Transportation, Supply chain and packaging.
The show was visited by leading Indian manufacturing companies, corporate houses, buying agencies, public sector units & government departments to source their requirements. An excellent opportunity for our dealer to present the DYNACO High Performance Doors to the growing Indian market. We congratulate the enthusiastic team for their excellent results.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 958
|
Statistics and Sourcess2equalpay2017-10-07T13:58:40-04:00
Statistics & Sources
If gender wage equality existed right now, the number of Florida women living in poverty would be reduced by more than 50%
SPECIFIC LANGUAGE: If working women in Florida aged 18 and older were paid the same as comparable men—men who are of the same age, have the same level of education, work the same number of hours, and have the same urban/rural status—the poverty rate among all working women would fall by 57.3 percent, from 8.2 percent to 3.5 percent.
SOURCE: (page 13) https://womensfundmiami.org/2015site/wp-content/uploads/2015/08/The-Status-of-Women-in-Florida-by-County-Poverty-Opportunity_fullreport.pdf
Working women in Florida earn 19.8 percent less than men
SOURCE: (Gender Pay Gap by State interactive infographic) http://www.payscale.com/data-packages/gender-pay-gap
One in five (20.5% or more than 250,000) women in Miami-Dade County live in poverty – many are single working moms struggling to feed their families
With gender wage equity, the poverty rate among working single mothers in Florida would drop from more than one in five single mothers in poverty (23.1 percent) to fewer than one in 10 (9.5 percent)
SPECIFIC LANGUAGE: Florida's working single mothers would see an even more dramatic reduction in poverty if they earned the same as comparable men; the poverty rate among working single mothers in Florida would drop from more than one in five single mothers in poverty (23.1 percent) to fewer than one in 10 (9.5 percent).
Florida will not achieve wage parity for at least 22 years, until 2038
SOURCE: https://iwpr.org/publications/projected-year-wage-gap-will-close-state
For 13 years, Florida has earned a D+ grade on the status of poverty and opportunity for women
SPECIFIC LANGUAGE: Since the 2004 publication of The Status of Women in the States, Florida's grade of D+ on the Poverty & Opportunity Composite Index has remained unchanged.
Even men working in the 20 most common occupations for women earn more than women working in those same occupations.
SOURCE: http://now.org/resource/women-deserve-equal-pay-factsheet/
SOURCE: https://iwpr.org/issue/employment-education-economic-change/pay-equity-discrimination/
Women earn less than men do, even when they make the same family and career decisions.
Dollar for dollar, women in the US earn 22 percent less than their male counterparts, with Latinas and African American women experiencing the sharpest pay disparities compared to white men
SPECIFIC LANGUAGE: Yet, on average, women continue to earn considerably less than men. In 2015, female full-time, year-round workers made only 80 cents for every dollar earned by men, a gender wage gap of 20 percent
SOURCE: https://iwpr.org/issue/employment-education-economic-change/pay-equity-discrimination
In nearly 2/3 of American households, women are the co-, primary or sole bread winners making their income critical to family stability
SPECIFIC LANGUAGE: In 2015—the year for which the most recent data are available—42 percent of mothers were sole or primary breadwinners, bringing in at least half of family earnings. Nearly another one-quarter of mothers—22.4 percent—were co-breadwinners, bringing home from 25 percent to 49 percent of earnings for their families.
SOURCE: https://www.americanprogress.org/
If working women were paid the same as men, 2.2 million children in Florida (56.0 percent) would benefit from their mother's increased earnings.
SOURCE: https://iwpr.org/publications/the-economic-impact-of-equal-pay-by-state/
The pay disparity not only affects women's spending power, it penalizes their retirement security by creating gaps in Social Security and pensions.
It has taken 44 years for the wage gap to close just 18 cents — a rate of less than half a penny a year. This narrowing of the gap has slowed substantially since the turn of the century.
SOURCE: https://www.washingtonpost.com/
Over fifteen years, a typical woman loses $499,101 due to pay inequity.
SOURCE: https://iwpr.org/the-wage-gap-and-occupational-segregation/
Our goal is to make sure every woman in Miami-Dade County is paid fairly — women doing the same jobs as men should be paid equally.
We're looking for leading organizations to join us in a compact to bring pay equity to every workplace.
Signing the Compact gives employers the opportunity to be part of a diverse group of companies – large and small, for-profit and nonprofit – leading the way toward gender pay equity.
Join Us – Sign the Compact
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,536
|
Simple Rating in your Fat Free Framework Project
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,092
|
The world meets Apple's newest iPhone
Apple Watch video presentation showed the hall of Kyiv main railway hub with two escalators
The dreams about new iPhone are now a real thing.
Viewers of today's presentation of Apple products were pleasantly surprised: during the presentation the building of Kyiv railway station was shown. When watching video Apple Watch presentation, you will recognize the lobby of the Kyiv station with two escalators, and you can even hear the announcement in Ukrainian. Next, the action takes place on a subway platform, similar to "Zoloti Vorota" station.
Recall, have just Apple Inc. showcased its latest iPhones - the iPhone 8, iPhone 8 Plus and iPhone X.
Related: Apple releases new iPhone today
Related: Putin's birthday: Caviar released collection of "presidential" iPhone 6s
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,694
|
Q: how to give order_by in find_in_set here is my code in that tool_id contains values stored as an array and want to get it back the result as stored in table and when I give like this am getting white screen.
public function view_tools()
{
$this->db->select('tu.*, GROUP_CONCAT(t.name) as tool_ids', FALSE);
$this->db->from('tbl_tools t');
$this->db->join('tbl_tool_use tu', 'ORDER BY FIND_IN_SET(t.id, t.tool_id) desc', 'inner', FALSE);
$this->db->group_by('tu.id');
$this->db->order_by('tu.id','desc');
$query = $this->db->get();
return $query->result();
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,493
|
Visit Springdale
The Town of Springdale has adopted a color palette to guide the selection of color for new structures and signage. In general, the Town allows a wide range of hue, but prohibits bright and saturated colors and very light colors.
Munsell Book of Color
The specific standards for color regulation are based on the Munsell Book of Color. The Munsell system classifies color based on three attributes: hue, chroma (saturation), and value (light reflective value). A copy of the Munsell Book of Color is available for review at the Town Office. Town staff can assist you in finding appropriate color ranges.
Although the Munsell system has existed for many years, it is new to many developers. This information is intended to demonstrate how the Town uses the system to regulate color.
Chroma & Value Ranges
The Munsell Book of Color is composed of 40 color pages. Each page represents one "hue." Individual colors are displayed on each hue page in a grid pattern with "chroma" on the horizontal axis and "value" on the vertical axis. Any hue in the Munsell system is acceptable, within the chroma and value ranges listed as follows:
Color ranges for all new building or structure construction, reconstruction, or expansion, and alterations or modifications of existing buildings and structures:
In Munsell hues BG (Blue-Green), B (Blue), PB (Purple-Blue), P (Purple), and RP (Red-Purple) acceptable color ranges are:
Values of three through seven if the proposed Chroma is one or two
Values of three through six if the proposed Chroma is four
In Munsell hues R (Red), YR (Yellow-Red), Y (Yellow), GY (Green-Yellow), and G (Green) acceptable color ranges are:
Values of three through five if the proposed Chroma is six
Values of three through four if the proposed Chroma is eight
Color ranges for all new signage:
In all Munsell hues acceptable color ranges for all sign elements and sign areas, except copy, are:
Values of three through seven if the proposed Chroma is one, two, or four
Values of three through six if the proposed Chroma is six
Values of three through five if the proposed Chroma is eight
Acceptable colors for copy are:
Any color identified in section 10-17-4 (A)(2)(a)
Colors with Values of three through nine and Chromas of one or two in all Munsell hues
In addition to falling within these ranges, proposed colors must compliment the natural surroundings (see Town Code section 10-17-4 (B)).
Construction Design Standards & Details (PDF)
Fee Schedule (PDF)
Flood Hazard Regulations
Landscaping & Approved Plant List
Zoning Design Standards Manual
Springdale Town Offices
118 Lion Boulevard
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,455
|
Trump called because "I have been very clear" about Mueller and the lawyers he's hired amounting to a "rigged game", Gingrich said.
Newsmax Media CEO Christopher Ruddy stood by his claims after the White House downplayed his comments and suggested he did not know what he was talking about.
Firing Mueller would be a politically explosive move that would raise new questions about Trump, whose dismissal of James Comey as Federal Bureau of Investigation director generated accusations of obstruction of justice and led to Mueller's appointment.
"I think it is a consideration the President has had, because (Robert) Mueller is illegitimate as special counsel", Ruddy told CNN's Chris Cuomo on "New Day" Tuesday.
Deputy Attorney General Rod Rosenstein Rosenstein says the attorney general would be the only one who could fire Mueller, and since Attorney General Jeff Sessions has recused himself from the investigation, Rosenstein is acting in that capacity. Should Trump decide to remove the special counsel, he could pursue two alternatives: order the Department of Justice to fire Mueller, or fire Mueller himself.
"These are people who are going to be after Trump", Gingrich said, adding that Mueller "apparently couldn't find a single pro-Trump attorney to hire, and I just think that's a rigged game, and I think that it's a mistake to pretend this is going to be some neutral investigation".
"I think we should let Bob Mueller do his work and get to the bottom of it and get to the bottom of it quickly so that he can be vindicated", Ryan said on "The Hugh Hewitt Show".
Speaking earlier, House Speaker Paul Ryan said politicians should allow the special counsel to do his work. Doubt, however, has been cast on that explanation because of cryptic testimony from former FBI Director James Comey last week.
Schiff told MSNBC's "Morning Joe" that if Trump fired Mueller, Congress might name its own independent counsel to investigate the case.
They say Trump did not collude with Russian Federation and see the investigation as a politically motivated sham that handicaps Trump's ability to execute his agenda, according to one person who advises the White House on how to handle the probe. "I never said I had a conversation", Ruddy said on CNN.
Sessions: But I'm not going to discuss any hypotheticals of what might be a factual situation in the future that I'm not aware of today because I know nothing about the investigation and have fully recused myself. Still, he added, "I can't imagine that that issue is going to arise". Ruddy did not immediately respond to questions seeking clarification.
The speaker listed the policy issues that congressional Republicans were focused on like a bill on the House floor this week to improve accountability for veterans' health care, telling reporters, "We are focused on solving people's problems". Don't waste time trying to undermine one of your few allies.
Soon after Ruddy's remarks became public Monday night, White House press secretary Sean Spicer said: "Mr".
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,210
|
{"url":"https:\/\/www.physicsforums.com\/threads\/charging-up-capacitors.339233\/","text":"Homework Help: Charging up capacitors\n\n1. Sep 21, 2009\n\nalexburns1991\n\n1. The problem statement, all variables and given\/known data\n\nA 50 $$\\mu$$F capacitor is being charged from a 6V battery via a 100k$$\\Omega$$ resistor . what is the initial charging current? after a period of time, the charging current is 30$$\\mu$$A. what are the pd's across the resistor and the capacitor at that moment? how much charge has been stored on the capacitor up to that time?\n\n2. Relevant equations\n\nQ = CV\n\nInitial Charging Current = emf\/resistance\n\nQ = EC (1 - e-t\/RC)\n\n3. The attempt at a solution\n\nthe initial charging current wasnt a problem, using V=IR.\nI0 = 60$$\\mu$$A\n\nfrom there on, i struggled to get the pd's but i realised that one half time had gone seeing as the current had halved so the pd must be 3V on each.\n\ni tried using Q = CV to find the charge on the capacitor at that point, but it gave me 1.5 x 10-4C which is not the answer given. i also used the third equation given to get the charge, given the time- but the e term was negative, giving a negative amount of charge. the answer given in the book is 6$$\\mu$$C, but i can't see how to get this.\n\nLast edited: Sep 22, 2009\n2. Sep 22, 2009\n\nRedbelly98\n\nStaff Emeritus\nDouble-check your math. 6V, 50k\u03a9, ___\u03bcA?\n\n3. Sep 22, 2009\n\nalexburns1991\n\nsorry i copied the question wrong, it was meant to be 100kilo ohms.\ni still dont see how you can get the answer given though.\n\n4. Sep 22, 2009\n\nturin\n\nWhat are \"pd's\"?\n\nWhat is a \"half time\"? Is this a \"half-life\"? If you meant \"time constant\" (i.e. RC), then that would be incorrect. The quantity (say current) reduces to one-half in a bit less than one time constant of decay. (Remember, the current is the time derivative of the charge, which is an exponential function of time.) Hmm... I guess your result for 3V on the cap and res should be correct anyway.\n\n5. Sep 22, 2009\n\nwillem2\n\nbecause the current has halved, the potential difference across the resistor must also have\nhalved because of ohms low. So it is indeed 3V. The rest of the 6V must be across the capacitor.\nif your statement of the problem is correct, 1.5*10^-4 C is the right answer.\n\nthe e term can't go negative BTW. if t>0 and R>0 and C>0 then -t\/RC < 0 so $e^{-t\/RC} < 1$ and $1- e^{-t\/RC} > 0$\n\nShare this great discussion with others via Reddit, Google+, Twitter, or Facebook","date":"2018-07-22 18:42:13","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\": 0.6418939828872681, \"perplexity\": 1202.9933849046702}, \"config\": {\"markdown_headings\": false, \"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-2018-30\/segments\/1531676593438.33\/warc\/CC-MAIN-20180722174538-20180722194538-00287.warc.gz\"}"}
| null | null |
Sandy Williams, née en aux États-Unis, est une auteure américaine de romans de fantasy urbaine.
Biographie
Œuvres
Série Sidhe
Série An Anomaly Novel
Liens externes
Sandy Williams chez Milady
Naissance en 1981
Naissance aux États-Unis
Romancière américaine du XXIe siècle
Écrivaine américaine de fantasy
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,194
|
package com.zx.sms.codec.smpp;
import io.netty.buffer.ByteBuf;
import java.io.Serializable;
import org.apache.commons.codec.binary.Hex;
import com.zx.sms.common.util.ByteBufUtil;
import com.zx.sms.common.util.PduUtil;
/**
* Simple representation of an Address in SMPP.
*
* @author joelauer (twitter: @jjlauer or <a href="http://twitter.com/jjlauer" target=window>http://twitter.com/jjlauer</a>)
*/
public class Address implements Serializable{
private byte ton;
private byte npi;
private String address;
public Address() {
this((byte)0, (byte)0, (String)null);
}
public Address(byte ton, byte npi, String address) {
this.ton = ton;
this.npi = npi;
this.address = address;
}
public byte getTon() {
return this.ton;
}
public void setTon(byte value) {
this.ton = value;
}
public byte getNpi() {
return this.npi;
}
public void setNpi(byte value) {
this.npi = value;
}
public String getAddress() {
return this.address;
}
public void setAddress(String value) {
this.address = value;
}
public int calculateByteSize() {
return 2 + PduUtil.calculateByteSizeOfNullTerminatedString(this.address);
}
public void read(ByteBuf buffer) throws UnrecoverablePduException, RecoverablePduException {
this.ton = buffer.readByte();
this.npi = buffer.readByte();
this.address = ByteBufUtil.readNullTerminatedString(buffer);
}
public void write(ByteBuf buffer) throws UnrecoverablePduException, RecoverablePduException {
buffer.writeByte(this.ton);
buffer.writeByte(this.npi);
ByteBufUtil.writeNullTerminatedString(buffer, this.address);
}
@Override
public String toString() {
StringBuilder buffer = new StringBuilder(40);
buffer.append("0x");
buffer.append(Hex.encodeHexString(new byte[]{this.ton}));
buffer.append(" 0x");
buffer.append(Hex.encodeHexString(new byte[]{this.npi}));
buffer.append(" [");
buffer.append(this.address);
buffer.append("]");
return buffer.toString();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,957
|
\section{Introduction}
\label{sec:intro}
Semi-Lagrangian methods offer an efficient and widely used approach to model
advection dominated problems.
Initially introduced in atmospheric and weather models
\cite{robert_stable_1981, robert_semi-lagrangian_1982}, these methods are now
widely used in all fields of fluid mechanics
\cite{staniforth_semi-lagrangian_1991, oliveira_comparison_1995,durran_numerical_1999}.
They have found a wide range of application in computational fluid
dynamics.
These methods have triggered a wide variety of schemes, including
spline interpolation methods \cite{knorr_representation_1980,
shoucri_numerical_1983,zerroukat_application_2007}, finite element
WENO algorithms \cite{liu_weighted_1994,qiu_conservative_2011,
huang_eulerianlagrangian_2012} or CIP methods \cite{
nakamura_exactly_2001,xiao_completely_2001}.
Considerable development has also been achieved in
application to hyperbolic problems (e.g. compressible
hydrodynamics \cite{lentine_unconditionally_2011}, Vlasov equation
\cite{sonnendrucker_semi-lagrangian_1999}) and fall out of the scope of this paper.
Semi-Lagrangian methods involve an advected field $\Phi$,
following the characteristics backward in time.
The procedure requires the estimation of field values that do not lie on the computational grid.
Semi-Lagrangian methods therefore rely on an interpolation of
$\Phi (t-\Delta t, \xvec - \U \Delta t ) $, which in general is not a known quantity on the discrete grid.
Because of their local nature, low order semi-Lagrangian methods perform remarkably well on
massively parallel computers \cite{liu_real-time_2004, wu_improved_2004}.
Limitations occur with high-order interpolation methods. As the width
of the stencil increases, the locality of the scheme is reduced and the resulting
schemes require larger communications stencils.
When the interpolation strategy is simple, multi-linear in the case of the
\CIR{} scheme \cite{courant_solution_1952},
the scheme is local and the computational cost is small.
If the interpolation stencil is not localized near the computational point, but near
the point where the interpolated value must be reconstructed, one can
show that the method is then unconditionally stable, in
the case of a uniform and steady velocity field
\cite{leonard_stability_2002}. Such
schemes are however prone to large inter-process communations, and are not
unconditionally stable for general flows.
Dupont \textit{et al.} \cite{dupont2003, dupont2007, kim2007}
introduced two new corrective algorithms: ``Forward Error Correction'' (here denoted \FEC )
and ``Backward Error Correction'' (here denoted \BEC). These algorithms
take advantage of the reversibility of the advection equation to
improve the order of most semi-Lagrangian schemes by using multiple
calls of an initial advection scheme. The resulting schemes yield an enhanced
accuracy. In that sense, they are built with a similar spirit
to the predictor-corrector method \cite{butcher_numerical_2008}
or the MacCormack scheme \cite{maccormack_effect_2003}.
Here we introduce a new scheme following this methodology, and
thus extend this approach to third order accuracy.
\section{Multi-stage approaches}
A possible strategy to increase the order of Semi-Lagrangian schemes is to use
higher order interpolation formula e.g. \cite{celledoni_high_2015}. This has the drawback of
relying on a wider stencil, which requires larger communication patterns
on a distributed memory computer. Another significant issue with wider
stencils is the handling of boundary conditions.
Equation \eqref{eq:intro:adv} models the advection of a passive scalar $\Phi$
by a velocity field $\U$,
\begin{align}
\Dt \Phi \equiv
\left[ \partial_t + (\U \cdot \grad ) \right] \Phi =0 \, .
\label{eq:intro:adv}
\end{align}
The Lagrangian derivative in \eqref{eq:intro:adv} is usually defined
as the limit, following the characteristic, of
\begin{align}
D_t \Phi = \lim _{\Delta t \to 0} \dfrac{ \Phi (t, \xvec) -
\Phi (t-\Delta t, \xvec - \U \Delta t ) }{\Delta t} \,.
\end{align}
Semi-Lagrangian methods rely on this expression to discretize the advective operator $D_t \Phi $
instead of expanding the sum in a temporal term $\partial_t \Phi $
and an advective term $( \U \cdot \grad) \Phi $, as in \eqref{eq:intro:adv}.
The semi-Lagrangian discretisation of \eqref{eq:intro:adv} therefore introduces
an interpolation operator
\(
L _{\U} \left[ \Phi ^n \right] = \widetilde \Phi ^n (\xvec - \U \Delta t) \, ,
\)
where $\widetilde \Phi$ denotes the interpolated value away from the grid
points.
A strategy introduced by Dupont \textit{et al.} \cite{dupont2003}
to increase the order of a semi-Lagrangian scheme, without requiring the use of high-order
interpolation formula, is based on two consecutive calls to the interpolation operator,
the second call involving the reversed flow.
This method is known as the ``Forward Error Correction'' \cite{dupont2003}.
The advantages of this procedure over the above high order schemes rely both on the
accurate implementation of boundary conditions and on the limited communication stencil.
The Forward Error Correction scheme is constructed as
\begin{align}
\bar{\Phi} &\equiv L_{-\U}\left[ L_{\U}\left[ \Phi ^n\right] \right] \, , \\
FEC \left[\Phi ^n\right] & \equiv
L_{\U}\left[\Phi ^{n} \right]+ \left(\Phi ^n - \bar{\Phi}\right)/2
\, .
\label{eq:intro:FEC}
\end{align}
The \FEC{} corrective algorithm has further been improved in \cite{dupont2007, kim2007}
using three calls to the interpolation operator for each time-step.
The resulting algorithm is known as the ``Backwards Error Correction''
(\BEC) algorithm. It is constructed using
\begin{align}
BEC \left[ \Phi ^n\right] \equiv
L_{\U} \left[ \Phi^{n} + (\Phi ^n - \bar{\Phi} )/2 \right ]
\,.
\label{eq:intro:BEC}
\end{align}
Both the \FEC{} and the \BEC{} algorithms suppress the leading
order error term when the interpolation operator is irreversible.
Both the \FEC{} and the \BEC{} schemes are
free of numerical diffusion. However, they introduce numerical
dispersive effects related to their truncation errors.
This truncation error can be advantageously used to construct a scheme free of numerical
dispersion and characterized by a fourth order derivative
truncation error. This is achieved for the same computational
cost as the \BEC{} scheme.
A new ``Combined Error Correction'' (\CEC) algorithm is introduced,
using a linear combination of the \FEC{} and \BEC{} algorithms,
\begin{align}
CEC \left [ \Phi \right ]
\equiv \cA \, FEC \left [ \Phi \right ]
+ \cB \, BEC \left [ \Phi \right ] \, .
\label{eq:intro:CEC}
\end{align}
When the \CIR{} scheme is used as the interpolation operator,
the scheme generated by the \FEC{} algorithm is similar, in
the Eulerian framework, to the one introduced in \cite{fromm_method_1968}.
In this case, the values of the coefficients \cA{} and \cB{}
in \eqref{eq:intro:CEC} can be explicitly determined and the stability of the resulting schemes assessed.
In one dimension, their expression is
\begin{align}
3 \, \cA = 2 - {\Delta x}/{( \vert u \vert \Delta t)}
\quad \text{and} \quad
\cB =1 - \cA
\, ,
\label{eq:intro:CECcoef}
\end{align}
where $\Delta t$ denotes the time-step and $\Delta x$ the grid-step.
In one dimension of space, the \CIR{} scheme is strictly equivalent to the
Eulerian upwind scheme. It is well known \cite{courant_uber_1928,
hirsch_numerical_2007, leveque_numerical_1992} that this scheme
is stable for Courant-Friedrichs-Lewy (CFL) numbers smaller
than unity and introduces diffusive errors.
The spurious diffusive effects are directly related to the truncation error of the scheme.
The generalization to $d$-dimension must be carried
out with care. As described later, the fields can be advected one
dimension at a time using a splitting technique similar to \cite{fromm_method_1968}.
In two or three dimensions, the interpolation can be done by applying the \CEC{}
scheme on each direction separately.
\section{One-dimensional algorithms}
\label{sec:algo1D}
\label{sec:1Dalgo}
In the semi-Lagrangian formalism, the advection equation can be discretized
using the \CIR{} scheme \cite{courant_solution_1952}. In one dimension,
the \CIR{} scheme has the same stencil as the Upwind scheme
\cite{durran_numerical_1999,
butcher_numerical_2008, hirsch_numerical_2007}
\begin{align}
\Phi^{CIR}_{i}=&
( 1 - U_i) \Phi^n[i] + U_i
\Phi^n[i- s_i] \,,
\end{align}
where $\Phi^{n}[i]=\Phi^{n}_{i}$ denotes the value of the
passive scalar at time $n \,\Delta t$ and position $i \,\Delta x$,
$s_i = \sgn ( u_i )$ the sign of the velocity and
$U_i = \vert u_i \vert {\Delta t} / {\Delta x} $
the reduced velocity with $u_i$ the velocity.
A Von Neumann stability analysis shows that the scheme is
strictly stable for $U\leq1$.
For a constant velocity, the modified equation takes the form
\begin{align}
\sme{ \Big[ \partial_t \Phi + u \partial_x \Phi \Big] }{\CIR}
= \sme{D}{\CIR} \, \partial^2_x \Phi + ...
\quad \text{with} \quad
\sme{D}{\CIR} = \left( 1- U \right)\frac{ \vert u \vert \Delta x }{2}
\, .
\label{eq:CIRdiff}
\end{align}
The \FEC{} scheme \eqref{eq:intro:FEC} is a multi-stage
version of the \CIR{} scheme.
The developed expression for the \FEC{} scheme
requires the first nearest neighbors
for the velocity and the second nearest neighbors for the passive scalar (see Appendix~A).
For a constant velocity, the expression of \FEC{} is
\begin{align}
\FEC[\Phi]_{i}
= - \tfrac{1}{2} U (1- U)\Phi^n[i+ 1]
+ (1- U^2 )\Phi^n[i]
+ \tfrac{1}{2} U (1 + U) \Phi^n[i- 1]
\, .
\label{eq:cstUHALO}
\end{align}
The stability analysis of \eqref{eq:cstUHALO} provides
the following expression for the amplification factor
\begin{align}
\xiF = 1 - U^2 + U^2\cos (k \Delta x ) - i U \sin( k \Delta x ) \,.
\end{align}
The \FEC{} scheme is stable for $U\leq 1$. The modified equation associated to this scheme is
\begin{align}
\sme{\Big[ \partial_t\Phi + u \partial_x\Phi \Big]}{\FEC} =
- ( 1- U^2 ) \frac{u \Delta x^2}{3!} \partial^{3}_x \Phi
- 3(1- U^2)
\frac{u^2 \Delta x^2 \Delta t }{4!} \partial^{4}_x \Phi
+ ...
\label{eq:final_FEC}
\end{align}
The \BEC{} scheme, presented in \eqref{eq:intro:BEC},
is a modified version of the \CIR{} scheme using
$\bar{\Phi}^{n}$ to correct the field before the advection step.
The developed expression of the \BEC{} scheme
requires the second nearest neighbors for the velocity and third nearest neighbors for the passive scalar (see Appeendix~A).
To avoid using this long development, the simplified case of a constant velocity
will be studied.
\begin{align}
\label{eq:cstUBEC}
\BEC [\Phi]_i = &
- \frac{U }{2}(1- U)^2 \Phi^n_{i+ 1}
+
\frac{( 1 - U)}{2}
\left ( 3 - (1 - U)^2 - 2 U^2
\right ) \Phi^n_{i}
\\ &\nonumber
+ \frac{U}{2} \left ( 3 - 2 (1 - U)^2 - U^2
\right ) \Phi^n_{i- 1}
-
\frac{U^2}{2} ( 1 - U)\Phi^n_{i- 2}
\, .
\end{align}
The stability analysis
on \eqref{eq:cstUBEC} leads to the following amplification factor
\begin{align}
\xiB & = 1 - 2 i U \sin (\tfrac{1}{2}k \Delta x ) \Big[
{\rm e}^{-\frac{1}{2} i k \Delta x}
U ( 1
+
2 [ 1 - U ] \sin ^2 (\tfrac{1}{2}k \Delta x) )
+
\cos(\tfrac{1}{2}k \Delta x ) ( 1 - U )
\Big] \, .
\end{align}
It can be shown analytically that the \BEC{} scheme is stable for $U\leq1$.
In fact, the \BEC{}
scheme is still stable for a CFL number smaller than $1.5$.
The truncation error analysis leads to
\begin{align}
\label{eq:final_bec}
\sme{\Big[ \partial_t \Phi + u \partial_x\Phi \Big]}{\BEC} =
&
- (1- U)(1 - 2U )
\tfrac{u \Delta x^2}{3! }
\partial^{3}_x \Phi
\\ &\nonumber
- 9 (1- U)^2
\tfrac{u^2 \Delta x^2 \Delta t}{4! } \partial^{4}_x \Phi
+ ...
\end{align}
Simulations with Heaviside, triangle and cosine distributions advected by a constant velocity were carried out for a CFL number $U>1$. For $U\lesssim 1.5$, the \BEC{} scheme gives finite results consistent with the stable results collected for $U<1$. The other schemes (\CIR , \FEC{} and \CEC) diverge for $U>1$ and the \BEC{} scheme diverges for $U \gtrsim1.5$. This extension of stability of the \BEC{} scheme can be understood in the following way: for $U>1$, the interpolation is performed with points that are not the nearest value to the reconstructed point. The contribution of the second nearest neighbors in the \BEC{} formula results in an enhanced stability of the scheme.
The \FEC{} and \BEC{} schemes both have modified equations with a
third order derivative truncation error. The \CEC{} scheme,
presented in \eqref{eq:intro:CEC} and \eqref{eq:intro:CECcoef}
is a linear combination of these two schemes. The weights
are computed to cancel the leading order of truncation error (see Appendix~A)
and generate a higher order scheme.
Using the linearity of the stability analysis, the amplification factor is
\begin{align}
\xiC = 1
&- \tfrac{2}{3} \sin(\tfrac{1}{2}k \Delta x) \Big[
U \Big( 3 + 2 [ 1 - U^2 ] \sin ^2 (\tfrac{1}{2}k \Delta x) \Big)
\sin (\tfrac{1}{2}k \Delta x )
\\ & \nonumber
+ \Big( 3 + 2 U [ 1 - U^2 ] \sin ^2 (\tfrac{1}{2}k \Delta x) \Big)
i \cos(\tfrac{1}{2}k \Delta x )
\Big] \,.
\end{align}
The \CEC{} scheme is stable for $U\leq 1$.
To leading order, the modified equation of the \CEC{} scheme is
\begin{align}
\sme{\Big[ \partial_t \Phi
+ u \partial_x \Phi \Big]}{\CEC}
\!\! =
- (1 + U) ( 1 - U )(2 - U)
\tfrac{\vert u\vert (\Delta x)^3}{4! }
\partial^4_x\Phi
+ ...
\end{align}
The essential properties of the different schemes
are reported in Tab.~\ref{tab:1dsumup}. The computational cost is evaluated
using the number of composed interpolation operators.
The complexity of the interpolation operator varies
with the interpolation method used. In the case of the \CIR{} scheme,
the complexity is $\mathcal{O}(N)$ where $N$ is the total number grid of points.
\begin{table}[H]
\centering
\small
\begin{tabular}{ c | c | c | c | c }
\hline \hline
Scheme & Formula
& Error & Stability & Nb of calls\\
\hline
&&&& \\
\specialcell{
\CIR} &
$\CIR
\big[\Phi \big]
\!=\! L_{+} \big[\Phi \big]$ &
$\left( 1- U \right) \frac{ \vert u\vert \Delta x }{ 2} \partial^2_x \Phi $ &
$U<1$ &
$1$ \\
&&&& \\
\FEC &
$\FEC
\big[\Phi \big] \!\!=\!\!
L_{+}\big[ \Phi \big] + \tfrac{1}{2} (\Phi-\bar{\Phi})$ &
\specialcell{$
- ( 1- U^2 ) \frac{u \Delta x^2}{3!} \partial^{3}_x \Phi$ \\[2mm]
$- 3(1- U^2) \frac{u^2 \Delta x^2 \Delta t }{4!} \partial^{4}_x \Phi $ }&
$U<1$ &
$2$ \\
&&&& \\
\BEC &
$\BEC
\big[\Phi \big] \!\!=\!\!
L_{+} \big[\Phi + \tfrac{1}{2} (\Phi-\bar{\Phi}) \big]$ &
\specialcell{$- (1- U)(1 - 2U ) \tfrac{u \Delta x^2}{3! } \partial^{3}_x \Phi $ \\[2mm]
$- 9 (1- U)^2 \tfrac{u^2 \Delta x^2 \Delta t}{4! } \partial^{4}_x \Phi $} &
$U\lesssim 1.5$ &
$3$ \\
&&&& \\
\CEC &
\specialcell{
$\CEC
\! \big[\Phi \big] \!\! = \!\!
L_{+} \!\Big[ \!
\Phi +\tfrac{1+U \!\! }{6U \!\! }
(\Phi-\bar{\Phi} \!) \!\Big] $ \\[2mm]
$+ \tfrac{1-2U}{6U} (\Phi-\bar{\Phi})$} &
$ - (1 + U) ( 1 - U )(2 - U)
\tfrac{\vert u \vert (\Delta x)^3}{4! }
\partial^4_x\Phi $ &
$U < 1$ &
$3$ \\[7mm]
\hline
\end{tabular}
\caption{Comparative table of one dimension schemes.}
\label{tab:1dsumup}
\end{table}
\section{Results for one-dimensional problems}
\label{sec:res1D}
To assess the efficiency of the schemes introduced previously, simulations
with a constant velocity were performed. A one-dimensional periodic domain is considered, and the solution is advected for 10 or 100 cycles. Fig.~\ref{fig:1Dhea}, \ref{fig:1Dtri} and \ref{fig:1Dcos}
show the advection of three density profiles with different regularities.
Because of the Fourier properties of sine functions, the first harmonic was
studied thoroughly to check that it matches the properties of the modified equation.
The first set of tests was performed using an Heaviside profile
$
\Phi(x,t=0) = \sgn \Big[ \sin \left (
2\pi x / l \right ) \Big] \, .
\label{eq:hea}
$
This is a demanding test, as this profile
is discontinuous at two cross-over positions ($0$ and $0.5$).
As time elapses, the high frequencies get damped
and the profile is nearly reduced to its first harmonic.
In fig.~\ref{fig:1Dhea},
the \CEC{} scheme is closer to the
analytical solution than the other schemes by three criteria: (i)
the ``flatness'' of its profile at the beginning of the simulation, (ii)
the distance from the analytic cross-over position at all time and
(iii) the phase drift of the profile at long time. These criteria may seem independent but they
are all linked to the Fourier properties of the modified equation.
\begin{figure}[H]
\centering
\subfigure[$10$ periods]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/HS_10b.eps}
}
\subfigure[$100$ periods]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/HS_100b.eps}
}
\caption{One dimension advection of a Heaviside with a resolution of $N=30$ at $CFL=0.75$.}
\label{fig:1Dhea}
\end{figure}
The second set of tests was performed using a triangular profile,
$
\Phi(x,t=0) = \vert x / l - 0.5 \vert \, .
\label{eq:tri}
$
This profile
is non differentiable at two cross-over position ($0$ and $0.5$).
In fig.~\ref{fig:1Dtri}, the observations reported in
the previous paragraph still hold for the triangular profile. As expected,
the \CEC{} scheme is closer to the analytic results
in the case of a continuous but non-derivable profile.
\begin{figure}[H]
\centering
\subfigure[$10$ periods]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/tri_10b.eps}
}
\subfigure[$100$ periods]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/tri_100b.eps}
}
\caption{One dimension advection of a triangle with a resolution of $N=30$ at $CFL=0.75$.}
\label{fig:1Dtri}
\end{figure}
The last tests were performed using the first harmonic cosine profile,
$
\Phi(x,t=0) = - \cos \left ( 2\pi {x}/{l} \right ) \,.
\label{eq:cos}
$
The properties of
the profile will be studied in more details in
fig.~\ref{fig:1DresplotDR} and \ref{fig:1DresplotPD}.
In fig.~\ref{fig:1Dcos}, the \CIR{} scheme is
so diffusive that a ``corrected \CIR{}'' (green diamond
line)\footnote{The corrected \CIR{} values are
equal to those of \CIR{} multiplied by ${\rm exp}({ \sme{D}{\CIR} \, k^2 t })$
where $\sme{D}{\CIR}$ is defined in \eqref{eq:CIRdiff}.} is plotted.
Even though the \CIR{} scheme is near zero in fig.~\ref{fig:1Dcos},
the norm of its difference to the analytic profile is smaller than the \FEC{} scheme
which drifted to such an extent that it is nearly opposite to the
reference profile.
As noted above, provided the interpolation strategy involves
non-neighboring points, semi-Lagragian methods can use $CFL$ number
larger than one. Using a non-local interpolation stencil, we can
reproduce the advection test of fig.~\ref{fig:1Dcos} using a $CFL$
number of
$3.75$, see fig.~\ref{fig:1Dcfl}.
The time-step being larger in this last case, fewer time-steps are needed
for the same integration time (here respectively $10$ and $100$ periods),
the effects of numerical dispersion and diffusion
are thus weakened compared to fig.~\ref{fig:1Dcos}
This is achieved with a simple modification of relations
\eqref{eq:intro:CECcoef} to compute the weights $\cA{}$ and $\cB{}$,
in the form
\begin{align}
3 \, \cA = 2 - \frac{1}{\left( \vert u \vert \Delta
t/\Delta x \right) \, \% 1}
\quad \text{and} \quad
\cB =1 - \cA
\, ,
\label{eq:intro:CECcoefSL}
\end{align}
(where $\%1$ denotes the remainder of the
division by unity),
the accuracy of the \CEC{} scheme is preserved for large CFL numbers.
\begin{figure}[H]
\centering
\subfigure[$10$ periods, i.e. $1200$ time-steps]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/cos_10b.eps}
}
\subfigure[$100$ periods, i.e. $12000$ time-steps]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/cos_100b.eps}
}
\caption{One dimension advection of the cosine function with a resolution of $N=30$ at $CFL=0.75$.}
\label{fig:1Dcos}
\end{figure}
\begin{figure}[H]
\centering
\subfigure[$10$ periods, i.e. $80$ time-steps ]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/CFL_375e-2_cycles_10.eps}
}
\subfigure[$100$ periods, i.e. $800$ time-steps ]{
\includegraphics[scale=0.36,trim= 20 10 45 40, clip=true]
{FIG/CFL_375e-2_cycles_100.eps}
}
\caption{One dimension advection of the cosine function with a resolution of $N=30$
at $CFL=3.75$.}
\label{fig:1Dcfl}
\end{figure}
\section{Multi-dimensional problems}
\label{sec:res2D}
The extension of the above procedures to multi-dimensional problems requires some care.
For instance in two dimensions, the \CIR{} scheme is
\begin{align}
\label{eq:CIR2D}
\CIR[\Phi]_{i,j}
= &
( 1 - U_{i,j}^x )(1 - U_{i,j}^y)\Phi^n_{i,j}
+
(1 - U_{i,j}^x )U_{i,j}^y \Phi^n_{i,j - s_{i,j}^y}
\, ,\\ \nonumber
&+
U_{i,j}^x( 1 - U_{i,j}^y )\Phi^n_{i- s_{i,j}^x , j}
+
U_{i,j}^x U_{i,j}^y\Phi^n_{ i - s_{i,j}^x , j - s_{i,j}^y }
\, .
\end{align}
The semi-Lagrangian \CIR{} scheme uses one more value ($\Phi[ i - s_{i,j}^x ][ j - s_{i,j}^y ] $) than the Eulerian Upwind scheme.
However, the \CIR{} scheme is very similar to the
directionally split Upwind scheme
\begin{align}
\Phi^{\star}_{i,j}=
&
( 1 - U_{i,j}^x) \Phi^n_{i,j} + U_{i,j}^x
\Phi^n[i- s_{i,j}^x][j]
\, ,
\label{eq:split:1} \\
\Phi^{\star\star}_{i,j} =
&
( 1 - U_{i,j}^y) \Phi^{\star}_{i,j}
+ U_{i,j}^y
\Phi^{\star}[i][j- s_{i,j}^y] \, .
\label{eq:split:2}
\end{align}
In the general case in multi-dimension, there is no expression for the $\cA$ and $\cB$ coefficients of the \CEC{} scheme.
It can be extended to any dimension if the scheme is directionnally split as done in \eqref{eq:split:1} and \eqref{eq:split:2}. However, if a simple
splitting method is used, the approximation is reduced to first
order. Special splitting methods, such as Strang splitting
\cite{strang_construction_1968}, are required to increase the order of the total scheme.
\label{sec:2dres:rotdif}
\begin{figure}[H]
\centering
\subfigure[Initial patch distribution]{
\includegraphics[height=\figcma, trim= 20 10 60 10, clip=true]
{FIG/initial_cbCROP.eps}
\label{fig:inipatch} }
\subfigure[Velocity profile $u(0,y)$ or $v(x,0)$]{
\includegraphics[height=\figcma, trim= 20 10 10 10, clip=true]
{FIG/rd_vel.eps}
\label{fig:velgraph} }
\subfigure[Pure Lagrangian advection]{
\includegraphics[height=\figcma]
{FIG/mass_cbCROP.eps}
\label{fig:endpatch} }
\caption{Initial condition, velocity profile and final distribution for the two-dimensional advection test case.}
\end{figure}
To illustrate applications of our strategy to higher dimensions, let us consider
an advection problem in two dimensions of space.
A squared patch is considered for the initial distribution of the passive scalar: one inside the square
and zero outside, as presented in fig.~\ref{fig:inipatch}.
The order of the schemes for regularly varying
velocities should be the same as the one for constant velocities.
Quantitative results being difficult, only qualitative observations will be made.
The following velocity field was used to test the schemes
\begin{align}
u(x,y) = \quad&
\frac{y}{l} \left(1- \frac{y}{l} \right) \left(\frac{1}{2}- \frac{y}{l}\right)
\left[\cos \Big( 2\pi \frac{y}{l} (1-\frac{y}{l}) \Big)+1 \right]/(2 \pi^2)
\, ,
\\
v(x,y) = - &
\frac{x}{l} \left(1-\frac{x}{l}\right) \left(\frac{1}{2}-\frac{x}{l}\right)
\left[\cos\Big( 2\pi \frac{x}{l}(1-\frac{x}{l}) \Big) + 1 \right]/(2 \pi^2)
\, ,
\end{align}
where $l$ is the length of the box in both directions.
In fig.~\ref{fig:velgraph}, the velocity cancels out on
the edges of the box and is divergence free.
With the profiles used, the patch
is not transported through the walls of the box even though the simulation has
periodic boundary conditions. The patch never intersects itself which
makes it easier to track. To compare the results,
a fully Lagrangian method was used as a reference.
The time-step of this method was twenty times smaller to have more accurate results.
The solution is represented in fig.~\ref{fig:endpatch}.
\begin{figure}[H]
\centering
\subfigure[\CIR{} advection]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/upwCOM.eps} }
\subfigure[\FEC{} advection]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/fecCOM.eps} }
\subfigure[\BEC{} advection]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/becCOM.eps} }
\subfigure[\CEC{} advection]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/cecCOM.eps} }
\caption{Two-dimensional patch advection using the different schemes.}
\label{fig:dis1}
\end{figure}
In fig.~\ref{fig:dis1} and \ref{fig:dis2}, the analysis of the gap between a scheme and
the reference solution should not only be guided by the intensity of the
difference but also by the area impacted.
The \CIR{} scheme clearly introduces the largest computational error.
\begin{figure}[H]
\centering
\subfigure[\CIR{} error.]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/upw_diffCOM.eps}
\label{fig:disCIR}}
\subfigure[\FEC{} error.]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/fec_diffCOM.eps}
\label{fig:disFEC}}
\subfigure[\BEC{} error.]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/bec_diffCOM.eps}
\label{fig:disBEC}}
\subfigure[\CEC{} error.]{
\includegraphics[width=\figcm, height=\figcm,
trim=80 50 90 80, clip=true, keepaspectratio=false]
{FIG/cec_diffCOM.eps}
\label{fig:disCEC} }
\caption{Error, as measured by the difference of the numerical
solutions to the reference solution obtained with pure Lagrangian advection.}
\label{fig:dis2}
\end{figure}
The perturbation of the distribution can also give an intuition of the leading error term
in the modified equation. The quick oscillations at the tail of the patch in fig.~\ref{fig:disFEC}
and \ref{fig:disBEC} can be related to the dispersive residuals of
the \FEC{} and \BEC{} schemes.
In fig.~\ref{fig:disCEC}, the \CEC{} solution is the closest to the reference solution obtained by the pure Lagrangian method.
The error is of small amplitude and only impacts the edges of the patch.
\section{Application to thermal convection}
\label{sec:therConv}
\begin{figure}[H]
\centering
\subfigure[\CIR{} scheme]{
\includegraphics[scale=0.5, trim=70 30 160 30, clip=true]
{FIG/upw.eps}
\label{sf:CIR50}
}
\subfigure[\FEC{} scheme]{
\includegraphics[scale=0.5, trim=70 30 80 30, clip=true]
{FIG/fec.eps}
\label{sf:FEC50}
}
\\
\subfigure[\BEC{} scheme]{
\includegraphics[scale=0.5, trim=70 30 160 30, clip=true]
{FIG/bec.eps}
\label{sf:bec50}
}
\subfigure[\CEC{} scheme]{
\includegraphics[scale=0.5, trim=70 30 80 30, clip=true]
{FIG/cec.eps}
\label{sf:CEC50}
}
\caption{Rayleigh-B\'enard evolution of a localized thermal
perturbation. The numerical resolution $N=50^2$ is intentionally
modest, in order to highlight numerical errors. }
\label{f:50}
\end{figure}
In this section, the comparison between the different
advection schemes is extended to a physically more relevant case:
thermal convection in a layer of fluid heated from below. This
canonical example is also known as the
Rayleigh-B\'enard setup. The schemes will not only be used on passive
scalars that do not influence the velocity, but on the velocity itself and
the temperature, which, in the Rayleigh-B\'enard instability,
modifies the velocity actively.
The system of equations describing the evolution of the velocity
\U ~and the temperature $T$ of the fluid
is solved on a two-dimensional
Cartesian domain of aspect ratio $\chi = L_z/L_x = 0.5$, bounded by solid and impermeable
walls.
The bottom and top plates are maintained at fixed temperatures
$T_0$ and $T_0-\Delta T$, respectively, whereas the vertical walls are
assumed to be insulating (no heat flux through the
vertical boundaries).
Gravity is assumed to be uniform and vertical~$\gG=-g\ez~.$
To retain the essential physics with a minimum
complexity, the Boussinesq approximation is used to describe the
fluid within the cell and assume that variations of all physical
properties other than density can be ignored. Variations in density
are also neglected ``except in so far as they modify the action of
gravity'' \cite{rayleigh1916}. The density~$\rho$ is assumed to be
constant everywhere in the governing equations except in the buoyancy force
where it is assumed to vary linearly with temperature,
$
\rho\left(T\right) = \rhozero \left( 1 - \alpha \left( T -
\Tzero \right)\right) \,,
$
where $\alpha$ is the thermal expansion coefficient of the
fluid.
The system admits the stationary diffusive solution:
$\U^\star=0$, $T^\star = T_0 - z \, \Delta T / L_z$, and $\grad
P^\star = -g \rho\left(T^\star\right) \ez$. Subtracting the
stationary solution, choosing $L_z$,
$L_z^2/\kappa$, and $\Delta T$ as units of length, time, and
temperature, respectively, and using the temperature perturbation
$\theta = T-T^\star$, the system can be written \cite{chandrasekhar1961} as
\begin{align}
\partial_t \U+ \left(\U\cdot \grad \right)
\U
&= -\grad \Pi + \Ra \Prandtl \theta \, \ez
+ \Prandtl \laplacien \U \,, \label{e:nsnd}\\
\partial_t \theta
+ \left(\U\cdot \grad \right) \theta
&= w + \laplacien \theta \,, \label{e:Tnd}\\
\diV \U &= 0 \label{e:mnd}
\,,
\end{align}
with $w\equiv\U\cdot \ez$ the vertical velocity.
The non-dimensional control parameters are
the Rayleigh number, defined by $\Ra = \alpha g \Delta T
L_z^3/(\kappa \nu) $ and which measures the convective driving, and
the Prandtl number, defined as the ratio of viscous to thermal
diffusion, $\Prandtl = \nu/\kappa $, with $\nu$ the kinematic viscosity,
$\kappa$ the thermal diffusivity.
Equations~\eqref{e:nsnd} and \eqref{e:Tnd} are discretized on a
uniform grid using finite volume formula of order two in space and
order one in time, with all the terms being treated explicitly. To
enforce the solenoidal constraint \eqref{e:mnd}, the pressure-correction
scheme \cite{chorin1968,guermond2006} is used. This
splitting method is composed of two steps. In the first step, a
preliminary velocity field $\U^{\star}$ is computed by neglecting the
pressure term in Navier-Stokes equation. Since this preliminary
velocity field is generally not divergence-free, it is then corrected
in a second step by a projection on the space of solenoidal vector
fields. Given the temperature and velocity distributions at time-step
$n$, the velocity $\U^{n+1}$ is computed by solving
\begin{eqnarray}
\U^{(1)} &=& L\left [ \U ^{n}, \U ^{n} \right ]
\,, \\
\U^{(2)}&=& \U^{(1)}
+ \Delta t \left( \Ra \Prandtl \theta^n \ez
+ \left[\laplacien \U\right]^n \right)
\,, \\
\laplacien \phi^n &=& \diV\U^{(2)}
\label{e:poisson}
\,, \\
\U^{n+1} &=& \U^{(2)} - \grad \phi^n \,. \label{e:proj}
\end{eqnarray}
In \eqref{e:poisson}, the algorithm
requires to solve at each time-step a Poisson equation for the pressure. The
necessary impermeability conditions for the field $\phi$ are found by multiplying
\eqref{e:proj} by the normal vector $\N$. Together with
the velocity boundary condition, they lead to
$
\N\cdot \grad \phi^n = 0
\,.
$
The boundary conditions for the velocity field are no-slip,
i.e. $\U = 0$, while the temperature satisfies $\theta(z=0)
=\theta(z=1) = 0$ on the horizontal boundaries, and $\partial_x \theta
=0 $ on the vertical boundaries.
Boundary conditions are imposed on the intermediate
velocity field~$\U^\star$ by introducing ghost points outside of
the domain. In consequence, the tangential component of the actual
velocity field $\U$ will not exactly satisfy the boundary
conditions (the error being controlled by the time-step).
In order to develop the instability (the Rayleigh number being
sufficiently large and the Prandtl number set to unity), the simulations
were always started with $\U=0$ and with a
small temperature perturbation. This temperature perturbation consisted
of a hot spot ($\theta = 0.1$) next to
a cold spot ($\theta = -0.1$). This perturbation, localized close to the lower left
corner, generates a rising and a sinking plume.
The different simulations were compared when the rising plume
has reached the top boundary (after roughly a thousand iterations).
A very low resolution, $N=50^2$, was deliberately chosen in order to highlight
the numerical errors associated to the different schemes.
Snapshots of the total temperature $T = T^\star + \theta$ associated with the thermal plume
are compared on figure~\ref{f:50}.
In fig.~\ref{sf:FEC50} and \ref{sf:bec50}, strong
ripples appear in the wake of the plumes. They are not physically relevant
and are characteristics of dispersive schemes. The
comparison of the plumes in fig.~\ref{sf:CIR50} and
fig.~\ref{sf:CEC50} clearly highlights that the \CEC{} scheme is less
diffusive than the \CIR{} scheme for practical physical applications.
The \CEC{} scheme offers an improved scheme, with significantly reduced
diffusive effects, and free of the strong dispersion characterizing the
\FEC{} and \BEC{} schemes.
\section{Conclusion}
Using the simplest semi-Lagrangian \CIR{} scheme introduced by
Courant-Isaacson-Rees, it has been demonstrated that a simple
multi-stage approach can increase the order of the scheme from first
to third order. The resulting scheme is, at leading order, non-dispersive. This
procedure was shown to yield significant improvement on a thermal
convection problem. It can easily be used to increase the order of
existing codes on parallel computers, as the communication stencil is
unaltered by the multi-stage approach. The communications among
parallel processes are then restricted to the strict miminum (one
layer of cell at each domain boundary).
The \CEC{} algorithm, introduced here, only requires a modest increase in the computational cost
and can easily be implement in existing codes.
Moreover, its implementation is not limited to
regular Cartesian finite differences schemes. It can be
generalized to other geometries and scheme types by
following two simple steps:
(i)~deriving the modified advection equation for the \FEC{} and \BEC{} schemes and
(ii)~combining both schemes to cancel out their leading order error.
\bibliographystyle{wileyj}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,811
|
Q: How to select rows in a postgres table based on a date and store it in a new table? I have a postgres table which has some data.Each row has a date associated with it.I want to extract rows for the dates which has the month as April.Here is a csv version of my postgres table data
,date,location,device,provider,cpu,mem,load,drops,id,latency,gw_latency,upload,download,sap_drops,sap_latency,alert_id
0,2018-02-10 11:52:59.342269+00:00,BEM,10.11.100.1,COD,6.0,23.0,11.75,0.0,,,,,,,,
1,2018-02-10 11:53:04.006971+00:00,VER,10.11.100.1,KOD,6.0,23.0,4.58,0.0,,,,,,,,
2,2018-03-25 20:28:36.186015+00:00,RET,10.11.100.1,POL,7.0,26.0,9.83,0.0,,86.328,5.0,4.33,15.33,0.0,23.0,
3,2018-03-25 20:28:59.155453+00:00,ASR,10.12.100.1,VOL,5.0,14.0,2.67,0.0,,52.406,12.0,2.17,3.17,0.0,28.0,
4,2018-04-01 13:16:44.472119+00:00,RED,10.19.0.1,SEW,6.0,14.0,2.77,0.0,,52.766,2.0,3.25,2.29,0.0,1.0,0.0
5,2018-04-01 13:16:48.478708+00:00,RED,10.19.0.1,POL,6.0,14.0,4.065,0.0,,52.766,1.0,6.63,1.5,0.0,1.0,0.0
6,2018-04-06 21:00:44.769702+00:00,GOK,10.61.100.1,FDE,4.0,22.0,3.08,0.0,,54.406,8.0,3.33,2.83,0.0,19.0,0.0
7,2018-04-06 21:01:07.211395+00:00,WER,10.4.100.1,FDE,3.0,3.0,9.28,0.0,,0.346,2.0,10.54,8.02,0.0,33.0,0.0
8,2018-04-13 11:18:08.411550+00:00,DER,10.19.0.1,CVE,14.0,14.0,7.88,0.0,,50.545,2.0,6.17,9.59,0.0,1.0,0.0
9,2018-04-13 11:18:12.420974+00:00,RTR,10.19.0.1,BOL,14.0,14.0,1.345,0.0,,50.545,1.0,2.26,0.43,0.0,1.0,0.0
So I want only the rows which has a month of april data such that I will have a table which looks something like this
4,2018-04-01 13:16:44.472119+00:00,RED,10.19.0.1,SEW,6.0,14.0,2.77,0.0,,52.766,2.0,3.25,2.29,0.0,1.0,0.0
5,2018-04-01 13:16:48.478708+00:00,RED,10.19.0.1,POL,6.0,14.0,4.065,0.0,,52.766,1.0,6.63,1.5,0.0,1.0,0.0
6,2018-04-06 21:00:44.769702+00:00,GOK,10.61.100.1,FDE,4.0,22.0,3.08,0.0,,54.406,8.0,3.33,2.83,0.0,19.0,0.0
7,2018-04-06 21:01:07.211395+00:00,WER,10.4.100.1,FDE,3.0,3.0,9.28,0.0,,0.346,2.0,10.54,8.02,0.0,33.0,0.0
8,2018-04-13 11:18:08.411550+00:00,DER,10.19.0.1,CVE,14.0,14.0,7.88,0.0,,50.545,2.0,6.17,9.59,0.0,1.0,0.0
9,2018-04-13 11:18:12.420974+00:00,RTR,10.19.0.1,BOL,14.0,14.0,1.345,0.0,,50.545,1.0,2.26,0.43,0.0,1.0,0.0
Now If I try to extract a particular date with the below query
select * from metrics_data where date = 2018-04-13;
I get the error message
No operator matches the given name and argument type(s). You might need to add explicit type casts.
How do I get the rows for the month of April and store it in a new table say april_data?
Below is the structure of my existing table
Column | Type | Modifiers | Storage | Stats target | Description
-------------+--------------------------+-----------+----------+--------------+-------------
date | timestamp with time zone | | plain | |
location | character varying(255) | | extended | |
device | character varying(255) | | extended | |
provider | character varying(255) | | extended | |
cpu | double precision | | plain | |
mem | double precision | | plain | |
load | double precision | | plain | |
drops | double precision | | plain | |
id | integer | | plain | |
latency | double precision | | plain | |
gw_latency | double precision | | plain | |
upload | double precision | | plain | |
download | double precision | | plain | |
sap_drops | double precision | | plain | |
sap_latency | double precision | | plain | |
alert_id | double precision | | plain | |
A: The type of column date in your table is timestamp with time zone which format will be YYYY:MM:DD HH24:MI:SS.MS. In the query you make an operation timestamp with time zone = date, so it will throw an error.
So, if you want to fix it, you should fix one side to the type of other.
In your case I suggest as below:
*
*Match exact 1 day.
select * from metrics_data where date(date) = '2018-04-13';
*Match within 1 month.
select * from metrics_data where date BETWEEN '2018-04-01 00:00:00' AND '2018-04-30 23:59:59.999';
OR
select * from metrics_data where date(date) BETWEEN '2018-04-01' AND '2018-04-30';
OR
select * from metrics_data where to_char(date,'YYYY-MM') = '2018-04';
*Match only April.
select * from metrics_data where to_char(date,'MM') = '04';
OR
select * from metrics_data where extract(month from date) = 4;
Hopefully this answer will help you.
A: You need single quotes around the string literal.
PostgreSQL will automatically cast it to the correct data type (timestamp with time zone).
You could use the extract function to select only the dates from April:
SELECT * FROM yourtable WHERE extract (month FROM yourtable.date) = 4;
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 318
|
Q: Need to return Distinct fields from ElasticSearch without the full search result documents using C# and Nest I have Logs stored in ElasticSearch and a Windows Application using C# and Nest which is executing searches against the ElasticSearch. The mapping in ElasticSearch is shown below:
"mappings": {
"qns": {
"properties": {
"@timestamp": {
"format": "strict_date_optional_time||epoch_millis",
"type": "date"
},
"Error_Description": {
"index": "not_analyzed",
"type": "string"
},
"Thread_Id": {
"index": "not_analyzed",
"type": "string"
},
"Error_Description_Analyzed": {
"type": "string"
},
"Error_Source": {
"index": "not_analyzed",
"type": "string"
},
"record": {
"type": "string"
},
"@version": {
"type": "string"
},
"Log_Level": {
"type": "string"
},
"Record": {
"type": "string"
},
"id": {
"type": "long"
},
"Error_Source_Analyzed": {
"type": "string"
},
"Timestamp": {
"format": "strict_date_optional_time||epoch_millis",
"type": "date"
}
}
}
}
The corresponding C# class is as follows:
[ElasticsearchType(IdProperty = "Id", Name = "qns")]
public class QNS
{
[Number(NumberType.Long, Name = "id")]
public long Id { get; set; }
[Date(Name = "Timestamp")]
public DateTime Timestamp { get; set; }
[String(Name = "Error_Description", Index = FieldIndexOption.NotAnalyzed)]
public string ErrorDescriptionKeyword { get; set; }
[String(Name = "Error_Description_Analyzed")]
public string ErrorDescriptionAnalyzed { get; set; }
[String(Name = "Error_Source", Index = FieldIndexOption.NotAnalyzed)]
public string ErrorSourceKeyword { get; set; }
[String(Name = "Error_Source_Analyzed")]
public string ErrorSourceAnalyzed { get; set; }
[String(Name = "Thread_Id", Index = FieldIndexOption.NotAnalyzed)]
public string ThreadId { get; set; }
[String(Name = "Log_Level")]
public string LogLevel { get; set; }
[String(Index = FieldIndexOption.NotAnalyzed)]
public string Record { get; set; }
}
I need a way to search for distinct error records that falls within a datetime range and matches a certain range of patterns. While I am able to get the result, but I am also getting all the documents that satisfy the search while I only need the distinct error strings. For the Distinct query I am using FluentNest(https://github.com/hoonzis/fluentnest).
The code for retrieving the results is as follows:
private List<string> FindDistinctErrorsByPatternAndTimeRangeInternal(DateTime fromDateTime, DateTime toDateTime, List<pattern> patterns, string indexName, string type)
{
var documents = new List<QNS>();
var fromTime = fromDateTime.ToString(Constants.IndexSearch.ES_DATETIME_FORMAT);
var toTime = toDateTime.ToString(Constants.IndexSearch.ES_DATETIME_FORMAT);
var patternQueries = new List<QueryContainer>();
foreach (var p in patterns)
{
var pType = PatternType.unknown;
if (Enum.TryParse<PatternType>(p.Pattern_Type.ToLowerInvariant(), out pType))
{
switch (pType)
{
case PatternType.word:
patternQueries.Add(Query<QNS>.Regexp(r =>
r.Field(f =>
f.ErrorDescriptionAnalyzed)
.Value(p.Pattern_Description)
)
);
break;
case PatternType.phrase:
patternQueries.Add(Query<QNS>.MatchPhrase(m =>
m.Field(f =>
f.ErrorDescriptionAnalyzed)
.Query(p.Pattern_Description)
)
);
break;
case PatternType.unknown:
default:
break;
}
}
}
var datetimeQuery = Query<QNS>.QueryString(q =>
q.DefaultField(f =>
f.Timestamp).Query($"[{fromTime} TO {toTime}]")
);
var searchResults = client.Search<QNS>(s => s.Index(indexName)
.Type(type)
.Query(q =>
q.Filtered(f =>
f.Filter(fq =>
fq.Bool(b =>
b.MinimumShouldMatch(1).Should(patternQueries.ToArray())
)
)
.Query(qd =>
qd.Bool(b =>
b.Must(datetimeQuery)
)
)
)
)
.Sort(sort => sort.Ascending(SortSpecialField.DocumentIndexOrder))
.Aggregations(agg => agg.DistinctBy(q => q.ErrorDescriptionKeyword)));
var results = searchResults.Aggs.AsContainer<QNS>().GetDistinct(d => d.ErrorDescriptionKeyword);
return results.ToList();
}
I need to modify this code to only return the distinct error strings and not the entire result set. The number of hits from the query is around 3500 and only 2 distinct error strings are present. So it does not make sense to get all those records back as I am not going to use it. Can someone help me get to the right aggregation query using the date range and pattern regex/phrase match to only return the distinct error records using Nest or Nest/FluentNest.
A: I think you are looking for the terms aggregation.
But you whole query is a bit strange. Do you have some legacy requirements.
First thing you have two fields ErrorDescriptionAnalyzed and ErrorDescriptionKeyword are you making a different field just to have one Analyzed and one not? Why don't you use multi-fields
Second Filtered method has obsolete for some time.
Here is a quick sample that I hope will help
ElasticClient db = new ElasticClient(uri);
db.DeleteIndex(indexName);
var mappings = new CreateIndexDescriptor(indexName).Mappings(ms => ms.Map<A>(map => map.
AutoMap().
Properties(props =>
props.String(p =>
p.Name(a => a.Text).
Fields(fields =>
fields.String(pr => pr.Name("raw").NotAnalyzed()))))));
db.CreateIndex(mappings);
foreach (var item in Enumerable.Range(0, 10).Select(i => new A
{
Price1 = random.NextDouble() * 1000,
Date = i % 3 == 0 ? new DateTime(1900, 1, 1) : DateTime.Now,
Text = i % 2 == 0 ? "ABC" : "EFG"
}))
{
db.Index(item, inx => inx.Index(indexName));
}
var toDate = DateTime.Now + TimeSpan.FromDays(1);
var fromDate = DateTime.Now - TimeSpan.FromDays(30);
var data = db.Search<A>(s =>
s.Index(indexName)
.Query(q=>
q.DateRange(r => r.Field(f => f.Date).GreaterThan(fromDate).LessThanOrEquals(toDate))
&&
(
//term query is for finding words by default all words are lowercase but you can set a different analyzer
q.Term(t => t.Field(f => f.Text).Value("ABC".ToLower()))
||
//Raw field is not analysed so no need to lower case you can use you query here if you want
q.Term(t => t.Field("text.raw").Value("EFG"))
)
).Aggregations(aggr => aggr.Terms("distinct", aterm => aterm.Field("text.raw"))));
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,968
|
Revisiting Wes Craven's New Nightmare
Wes Craven's Meta Masterpiece Is Waiting To Be Rediscovered
It's such a shame this movie doesn't get more respect. Or that it's considered a classic. Having not seen it in quite some time, I was both pleasantly surprised and honestly floored by how brooding, intense and all around engrossing this was, and I think a lot of that has to do with the fact that not a lot of people talk about it. And I'm not really sure why, because this was pretty fucking great.
It's safe to say that the series had taken a hard detour into the absurd. There are moments in Part 3: The Dream Warriors (undoubtedly the most popular in the franchise) where Freddy is beginning to turn into the wise-cracking trickster we would eventually be accustomed to, but it's also easy to forget that in the first 2 films he was serious, scary and mean. Part 4: The Dream Master (my personal favorite, even though it's basically just a long slick looking music video of random sequences that don't make any sense) is where he fully embraces the silly side of his character, before everything goes overboard and into shit with Part 5: The Dream Child, a film I have never been able to finish.
So with this film, Wes Craven wanted to re-establish the franchise and the character of Freddy by taking him back to his horror roots and away from the silliness and campiness that had run rampant in the series for years. And honestly, he fucking nailed it here. Everything about this film works, even in it's surreal meta film within a film world dichotomy. It's dark, bleak, mean-spirited and so immersive that you barely even realize that Freddy Kreuger doesn't even make his first appearance until a full hour and 20 minutes in! Because you're fully invested, and you're submerged into this story.
Craven is a very skilled filmmaker. And that's something that took me years to fully appreciate. When I was younger, of course I loved films like Shocker, Swamp Thing and A Nightmare on Elm Street like everyone else, but as I got older, and really looked at the specific way he shoots his films, I was wickedly impressed more and more. I mean, look at what he was able to do with his followup to this, Eddie Murphy's Vampire in Brooklyn. Marketed as a comedy (and I will agree it is actually funny in some parts), I was shocked to discover upon watching it for the first time that it's actually more of a gothic romance horror film than anything else, and it looks absolutely gorgeous. I never would have thought based on the trailers trying to sell it as an Eddie Murphy comedy. I guess the same thing can be said about Murphy's Metro (1997), a film marketed as a comedy but in reality is a pretty badass cop/action/thriller. But with Vampire in Brooklyn, Craven was able to really make something special out of something that could have turned into a dud. And I know it wasn't necessarily a hit, but I do know that it's gained a nice cult status since it's initial release. My point is that it's far better than it should have been, and that's all due to Wes Craven, not Eddie Murphy.
It's a shame that New Nightmare didn't spawn more films and re-ignite the franchise. He tried his best to steer it in the right direction, and honestly, he nailed it here. Everything works so effectively. The constant sense of dread is palpable, and with the exception of a few dodgy CGI effects, the design of everything that happens in the final act is just incredible. In regards to Freddy's new look, I love it. While a bit different than the makeup that defined his look from Parts 1 through 4, it's leaps and bounds better than his new makeup look from Part 5.
While this last film (not counting Freddy vs Jason that came almost 10 years later) didn't kickstart a new era of Freddy films, it is a glorious sendoff to one of the most popular horror icons, and horror franchises in film history, delivered by the man who created it exactly 10 years earlier.
Please consider a donation so that I may continue doing this for you. In these tough times, every little bit helps, and further allows me to continue to spend the time needed to write these articles for you. I know it's a bit unorthodox, but we are in strange times and I have not been resistant to the turmoil that COVID has caused financially. Even if it's just $1, $5 or whatever. It would be greatly appreciated and I thank you. You can make a donation to my Venmo: @robotgeek
Revisiting 1997's Spawn: The Movie
The Best Movie I Never Saw: Sidekicks (1992)
Chuck Norris' Karate Kommados & Mr. T's Saturday M...
The Cult Corner: Arena (1989)
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,045
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.